Ann. Henri Poincar´e 6 (2005) 1 – 30 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/010001-30 DOI 10.1007/s00023-005-0197-9
Annales Henri Poincar´ e
Quantum Inequalities in Quantum Mechanics Simon P. Eveson, Christopher J. Fewster and Rainer Verch
Abstract. We study a phenomenon occurring in various areas of quantum physics, in which an observable density (such as an energy density) which is classically pointwise non-negative may assume arbitrarily negative expectation values after quantization, even though the spatially integrated density remains non-negative. Two prominent examples which have previously been studied are the energy density (in quantum field theory) and the probability flux of rightwards-moving particles (in quantum mechanics). However, in the quantum field context, it has been shown that the magnitude and space-time extension of negative energy densities are not arbitrary, but restricted by relations which have come to be known as ‘quantum inequalities’. In the present work, we explore the extent to which such quantum inequalities hold for typical quantum mechanical systems. We derive quantum inequalities of two types. The first are ‘kinematical’ quantum inequalities where spatially averaged densities are shown to be bounded below. Specifically, we obtain such kinematical quantum inequalities for the current density in one spatial dimension (imposing constraints on the backflow phenomenon) and for the densities arising in Weyl–Wigner quantization. The latter quantum inequalities are direct consequences of sharp G˚ arding inequalities. The second type are ‘dynamical’ quantum inequalities where one obtains bounds from below on temporally averaged densities. We derive such quantum inequalities in the case of the energy density in general quantum mechanical systems having suitable decay properties on the negative spectral axis of the total energy. Furthermore, we obtain explicit numerical values for the quantum inequalities on the one-dimensional current density, using various spatial averaging weight functions. We also improve the numerical value of the related ‘backflow constant’ previously investigated by Bracken and Melloy. In many cases our numerical results are controlled by rigorous error estimates.
1 Introduction The uncertainty principle lies at the root of many of the counterintuitive features of quantum theory. Consider, for example, a quantum mechanical particle moving in one dimension, whose state is a superposition of right-moving plane waves. Although the expectation value of (any power of) its momentum is positive, nonetheless it is possible for the probability flux at, say, the origin to become negative. Thus the probability of finding the particle in the right-hand half-line can decrease! We will return to this phenomenon, which has come to be known as backflow [3, 7, 30], in Sect. 2.1. Another, related, phenomenon occurs in quantum field theory. Even if one starts with a classical field theory in which energy densities
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(as measured by all observers) are everywhere positive,1 one finds that the renormalized energy density of the quantized field can assume negative values [11] and (in all models known to date) can even be made arbitrarily negative at a given spacetime point by a suitable choice of state. For example, the energy density between Casimir plates is computed to be negative; a fact indirectly supported by experiment ([9]; see the recent review [5] for an exhaustive list of up to date references). Various authors have suggested employing such effects to sustain exotic spacetime geometries containing wormholes [31] or ‘warp drive’ bubbles [2]. Such suggestions are, however, severely constrained [23, 33] by the existence of bounds, known as quantum inequalities (QIs) or quantum weak energy inequalities (QWEIs) [13, 14, 16, 18, 19, 20, 21, 22, 24, 34] which impose limitations on the magnitude and duration of negative energy densities. To give an example, let ρ(t)ψ be the energy density of the free scalar field2 measured along an inertial worldline in Minkowski space. Then, for any real-valued smooth compactly supported g, the averaged energy density obeys [14, 17] ∞ dt g(t)2 ρ(t)ψ ≥ − du Q(u)| g(u)|2 (1.1) 0
for all physically reasonable (Hadamard) states ψ, where Q is a known function of polynomial growth. The purpose of this paper is to apply techniques developed in the field theoretic setting to quantum mechanical problems. In so doing we wish to draw attention to a circle of ideas – including sharp G˚ arding inequalities, dynamical stability and the QWEIs – which eventually ought to be seen in the wider context of quantization theory. We begin with a discussion of ‘kinematical QIs’ in Sect. 2, taking the probability flux as our main example. We develop bounds on spatially averaged fluxes which share some technical similarity with the QWEI proved by two of us for the Dirac field [19] (see also [15]). An important aspect of our treatment is the numerical analysis of these bounds. To some extent this is motivated by the recent observation of Marecki [29] that QIs may have observational consequences in quantum optics; it is therefore important to know how sharp analytically tractable bounds are. The techniques used here may also be of independent interest, and we give a detailed account in Sect. 5. Before that, in Sect. 3 we establish a very general form of kinematical QIs arising in the Weyl–Wigner approach to quantizing classical systems (see, e.g., [28] as a general reference on that approach). More precisely, we consider the (quantized) configuration space density Rn x → ρF (x)ψ for normalized wave-functions ψ ∈ S (Rn ) which are associated with classical observables F , i.e., functions on phase space Rn × Rn , by n d p ρF (x)ψ = F (x, p)Wψ (x, p) , (1.2) 2π 1 In
general relativity, one would say that the field obeyed the weak energy condition (WEC). statements can be made for the Maxwell, Proca and Dirac fields [32, 16, 15].
2 Similar
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where Wψ denotes the Wigner function of ψ. Even if F is everywhere non-negative, the density ρF (x)ψ may assume negative values owing to the indefinite sign of the Wigner function; in fact, we show that, under very general conditions on F , this quantity is unbounded above and below for arbitrary given x upon varying ψ. Conversely, if F belongs to a certain class of symbols (in the sense of microlocal analysis [26, 36]) which are of second order (or lower) in the momentum variables, and if F is everywhere non-negative, then we establish a kinematical quantum inequality of the form (1.3) dn x χ(x)ρF (x)ψ ≥ −C with a suitable constant C depending on the non-negative weight-function χ, but not on the (normalized) wave-function ψ. This is a straightforward consequence of the sharp G˚ arding inequality [12, 27, 36]. The general result will be illustrated by a direct derivation of a kinematical QI for the energy density. In Sect. 4 we focus attention on ‘dynamical’ QIs which bound temporal averages of the energy density in general quantum mechanical systems. (In fact our kinematical flux inequality may be regained as the special case, in which the evolution is the group of translations on the line.) These are conceptually much closer to the QIs which have been obtained in quantum field theory; in fact, the method we use to establish these dynamical QIs makes contact with the techniques employed in [19]. Some features of the general result will be illustrated by taking the harmonic oscillator as a concrete example. We summarize our main results in the conclusion, Sect. 6.
2 A motivating example 2.1
Probability backflow
We begin with a simple example: the motion of a quantum mechanical particle in one dimension. Some time ago, Allcock [3] pointed out the existence of rightwards-moving states (in which the velocity is positive with unit probability) but for which the probability of locating the particle in the right-hand half-line is instantaneously decreasing – a phenomenon known as probability backflow. This phenomenon was subsequently studied in much greater detail by Bracken and Melloy [7] (see also [30]). To illustrate the idea, let us suppose the normalized state ψ (as well as being square-integrable itself) has a continuous, square integrable first derivative. Then the corresponding probability flux at position x is given by Re ψ(x)(pψ(x)) , (2.1) m where the momentum operator is, as usual, p = −id/dx and the particle has mass m. Now the spatial integral of the flux is pψ Re ψ | pψ jψ (x) dx = = (2.2) m m jψ (x) =
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and therefore yields the expected velocity. If ψ is a normalized right-moving wavepacket, it may be written by means of the Fourier transform as a superposition of right-moving plane waves dk ikx e ψ(k) ψ(x) = (2.3) 2π with ψ(k) = 0 for k < 0, so pψ =
0
∞
dk 2 k|ψ(k)| >0 2π
(2.4)
and we see that the spatially integrated flux is positive. However this does not imply that the flux itself is everywhere non-negative. Indeed, suppose that √ (2.5) ψ k0 (k) = N χ[0,k0 ] (k) k 3 − k0 , √ where χΩ denotes the characteristic function of Ω and N = (k03 (2 − 3)/(2π))−1/2 is a normalization constant. One may calculate 1 k02 1 k 2 jψk0 (0) = −√ (2.6) ∼ −0.006 0 , 4πm 2 m 3 which is not merely negative, but can clearly be made as negative as we wish by tuning k0 . Because the probability flux is negative at the origin, the probability of locating the particle in the left-half line is instantaneously increasing, thereby providing an example of the backflow phenomenon mentioned above. Backflow provides a nice illustration of the inadequacy of the phase velocity alone to predict the motion of a wavepacket. The three plots in Figure 1 indicate the time evolution of the position probability density in time under the free Hamiltonian; although the packet moves to the right, the two main peaks are reshaped in such a way that net probability has passed from the right-hand half line to the left. The wavepacket is given by Eq. (2.5) at time t = 0 with k0 = 5, m = 1/2 and = 1. Although these plots were obtained using the free Hamiltonian, we expect qualitatively similar behavior for evolutions generated by a wide class of Schr¨ odinger operators, on sufficiently small timescales, because the flux describes the flow of probability density for all such evolutions and a negative flux at one instant must persist for an interval of time by continuity. Nonetheless the free evolution is of special significance because an initially right-moving state will be purely right-moving at all times: backflow is not a scattering effect.
2.2
A quantum inequality for the flux
As we will see in Sect. 3 the backflow effect may be traced to the uncertainty principle. From this point of view, it is natural to seek bounds on its magnitude
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Figure 1: Evolution of a wavepacket under the free dynamics, illustrating the backflow phenomenon. From left to right, the plots show the position probability density at times t = −0.1, t = 0 and t = 0.1. and extent. Bracken and Melloy [7] studied the probability Pψ (t) of finding the particle in the left-hand half-line at time t, given that its state at t = 0 is ψ. They showed that (2.7) sup(Pψ (t) − Pψ (0)) = λ , ψ
for all t > 0, where the supremum is taken over all right-moving states ψ (i.e., ψ ∈ L2 (R) with ψ supported in R+ ) and the dimensionless constant λ is the largest positive eigenvalue of the equation 1 ∞ sin u2 − v 2 ϕ(v) dv = λϕ(u) (2.8) − π 0 u−v (for ϕ ∈ L2 (R+ )). It is striking that λ is not only independent of t, but also of the particle mass and Planck’s constant: backflow is an example of a purely quantum effect with no dependence on ! Although no analytical solution of Eq. (2.8) is known, Bracken and Melloy presented numerical evidence that λ ∼ 0.04. Using the numerical methods described in Sect. 5, we have recalculated this quantity to a much higher accuracy, although we have been unable to obtain consonant analytical error estimates. It turns out to be convenient to change variables to x = u2 ; we then consider the truncation of the resulting integral kernel to [0, X]. The maximum eigenvalue λ(X) was then calculated for values of X ranging from 6000 to 24000, using X/2 quadrature nodes. This choice was based on calculations using a variety of densities for
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λ(X) 0.0382 0.03815 0.0381 0.03805 0.038 0
10000
5000
15000
20000
X √ Figure 2: The least squares fit of λ(X) to a + b/ X.
values of X around 2000 for which X/2 nodes provide accuracy to 5 significant figures. By contrast, the largest calculation conducted in [7] corresponds to X = 625, which reflects the increase in available computing power over the past decade. √ The resulting data may be fitted to a remarkable degree by the form λ(X) = a + b/ X (as already noted by Bracken and Melloy for their data). Using a least squares fit to this, we obtain the estimate λ = 0.03845182014 with a maximum percentage residual error under 4 × 10−4 %. Assuming the residual errors would be comparable for larger X, this suggests that λ = 0.038452 to this level of precision. Our data points and the best-fit curve are shown in Fig. 2. One may interpret the Bracken–Melloy bound (2.7) as a demonstration of the transitory nature of backflow: large negative fluxes for right-moving states must be short-lived. Here, we present an apparently new bound, which demonstrates that such fluxes are also of small spatial extent, and whose proof is related to the quantum weak energy inequalities derived by two of us for the Dirac quantum field [19] (see also [15]). We consider spatially smeared quantities of the form jψ (f ) =
jψ (x)f (x) dx ,
(2.9)
which may be regarded as the instantaneous probability flux measured by a spatially extended detector. For any smooth, compactly supported, complex-valued function g, we will show that
jψ (x)|g(x)| dx ≥ − 8πm 2
dx |g (x)|2
(2.10)
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for all normalized states ψ belonging to the class R of right-moving states defined by = 0 for k < 0 and ψ continuous and square-integrable} . R = {ψ ∈ L2 (R) : ψ(k) (2.11) In fact, the conditions on both g and ψ may be weakened slightly.3 Before giving the proof, let us make three observations. 1. First, we note that there is no upper bound on the smeared flux. To see this, choose any normalized ψ ∈ R and let ψλ (x) = eiλx ψ(x). We have ψλ ∈ R for λ ≥ 0; moreover, jψλ (x) = jψ (x) + so
λ |ψ(x)|2 m
(2.12)
jψλ (x)f (x) dx → +∞ as λ → +∞.
2. Second, the scaling behavior of the above bound may be investigated by replacing g by gλ (x) = λ−1/2 g(x/λ), whereupon the right-hand side of inequality (2.10) scales by a factor of λ−2 . The limit λ → 0 corresponds to the unboundedness below of the probability flux at a point, while the limit λ → ∞ is consistent with the fact that pψ ≥ 0 for ψ ∈ R (because the bound vanishes more rapidly than λ−1 ). Roughly speaking, our bound asserts that the magnitude of negative flux times the square of its spatial extent satisfies a state-independent upper bound on R. Thus the extent of backflow is limited both in space and in time. Note also that the bound (2.10) vanishes in both the classical limit → 0 and the limit of large mass. This differs from Bracken and Melloy’s inequality (2.7) in which the dimensionless constant λ is independent of and m. We remark that – again in contrast to [7] – our result is kinematical rather than dynamical: no specific Hamiltonian is invoked. Here, ‘kinematic’ refers to the kinematics of the Schr¨odinger representation, i.e., the (unique) regular representation of the Heisenberg commutation relations. 3. Finally, on integration by parts, Eq. (2.10) can be reformulated as the assertion that for each normalized ψ ∈ R, the Schr¨ odinger operator 2 d2 + 4πjψ (x) (2.13) 2m dx2 is positive on the space of smooth compactly supported functions g, in the sense that g(x)(Hψ g)(x) ≥ 0 (2.14) Hψ = −
3 In particular, continuity of ψ may be weakened to ψ ∈ AC(R) ∩ L2 (R) with ψ ∈ L2 (R) at the expense of augmenting some statements with the qualification ‘almost everywhere’; by an approximation argument it is easy to see that (2.10) holds for all g belonging to the Sobolev space W 1,2 (R) [1].
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for all such g. Although the physical significance of this reformulation is not clear, it can provide useful necessary conditions for a given function j(x) to be the flux of a right-moving state. Only if the corresponding Schr¨ odinger operator has no bound states can this be the case. This can be sharpened slightly: as an illustration, suppose jψ (x) is the flux of a state in R with jψ (x) ≤ −M on some open interval I of length a. Then positivity of Hψ on C0∞ (I) implies that the Friedrichs extension HM of the operator −
2 d2 − 4πM 2m dx2
on C0∞ (I) ⊂ L2 (I)
(2.15)
is also positive. Since the Friedrichs extension of this operator corresponds to the imposition of Dirichlet boundary conditions at the boundary of I, HM has spectrum En =
2 n2 π 2 − 4πM 2ma2
(n = 1, 2, 3, . . .)
(2.16)
so we deduce (from E1 ≥ 0) that M ≤ π/(8ma2 ). This provides a more quantitative version of the connection between the magnitude and spatial extent of negative fluxes. Similar ideas have been employed in the context of quantum weak energy inequalities [18] to cast light on the ‘quantum interest conjecture’ of Ford and Roman [25]. We now establish the quantum inequality (2.10). It is sufficient to prove this for the case in which g ∈ C0∞ (R) is real-valued. Setting f (x) = g(x)2 and writing Mf for the multiplication operator (Mf ψ)(x) = f (x)ψ(x), we have 1 jψ (x)f (x) dx = Re ψ | Mf pψ m 1 = Re (ψ | Mg pMg ψ + ψ | Mg [Mg , p]ψ) m 1 ψ | Mg pMg ψ = m dk 2 k Mg ψ(k) , (2.17) = m 2π where we have used the fact that Re ψ | Mg [Mg , p]ψ = Re iψ | Mgg ψ = 0. We therefore may obtain a bound by estimating the portion of this integral arising from k < 0: 2 0 dk 2 ∞ dk k Mg ψ(k) = − k Mg ψ(−k) . (2.18) jψ (x)f (x) dx ≥ m −∞ 2π m 0 2π By the convolution theorem M g ψ(k) =
0
∞
dk ψ(k ) g(k − k ) , 2π
(2.19)
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where the restriction to k ∈ R+ is permissible for ψ ∈ R. Now a straightforward application of Cauchy-Schwarz gives 2 ∞ dk | g (k + k )|2 , ψ(−k) (2.20) M g ≤ 2π 0 where we have also used | g(−k)|2 = | g(k)|2 (since g is real) and ψ = 1. Substituting in (2.18), we now calculate ∞ dk ∞ dk k| g(k + k )|2 jψ (x)f (x) dx ≥ − m 0 2π 0 2π u ∞ du 2 = − | g (u)| dk k m 0 (2π)2 0 ∞ du 2 u | g (u)|2 = − m 0 8π 2 ∞ du 2 = − u | g (u)|2 m −∞ 16π 2 dx |g (x)|2 , = − (2.21) 8πm where we have changed variables from (k, k ) to (u, k) with u = k + k , used evenness of | g(u)| and Parseval’s theorem. This completes the proof of the quantum inequality (2.10). The later stages of this argument may be rephrased as follows. The inequality (2.18) asserts that 2 jψ (x)f (x) dx ≥ − T ψ (2.22) m where the operator T acts on L2 (R+ , dk/(2π)) by dk √ (T ϕ)(k) = k g (−k − k )ϕ(k ) 2π
(2.23)
and is easily seen to be Hilbert–Schmidt. Varying over normalized ψ, the right-hand side of Eq. (2.22) is bounded below by − T 2, where T denotes the operator norm of T . This leads to the bounds (2.24) jψ (x)f (x) dx ≥ − T 2 ≥ − T 2H.S. , m m where the last inequality holds because the Hilbert–Schmidt norm T H.S. dominates the operator norm. The calculation in (2.21) in fact precisely computes this final bound. To summarize, we have seen that, even for a right-moving state ψ ∈ R, the flux jψ need not be pointwise non-negative; moreover by tuning the state, one may
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arrange the normalized flux jψ (x) to be as negative as one likes at a given fixed x. However, weighted spatial averages of the flux are bounded below in terms of the weight function alone. This condition may be reformulated as asserting the positivity of the Hamiltonian for a particle moving in a potential given (up to constants) by the probability flux of any state in R.
2.3
Numerical results and sharper bounds
We illustrate our bound by reference to four weight functions: a Gaussian, a squared Lorentzian and two compactly supported weights which we call the truncated cosine and the smoothed truncated cosine. (Neither of the compactly supported weights are C ∞ , but they have sufficient smoothness for the above argument to hold; see footnote 3 above.) Our weight functions are summarized in Table 1, along with the corresponding bound arising from Eq. (2.10). In each case, fλ has unit integral, the parameter λ controls the sampling width and gλ (x) = fλ (x). For later reference we have also given the Fourier transforms of fλ and gλ . We wish to compare the above bound with two sharper (but less analytically tractable) bounds: the bound arising from the first inequality in (2.24) and a direct numerical estimate of the infimum of the integrated flux. In the first case, we are required to find the operator norm of T . Our numerical approach proceeds by first truncating the kernel to an interval [0, K] – we are able to estimate the error incurred here by using bounds obtained from the Hilbert–Schmidt norm – and applying a numerical quadrature scheme due originally to Fredholm (see, e.g., Sect. 4.1 of [4] or Chapter 4 of [10]) to the truncated kernel. This leads to a matrix whose eigenvalues approximate those of the truncated kernel and hence the original operator, and which can be computed using standard numerical packages. Full details, including a discussion of error estimates, are given in Sect. 5. This leads to quantum inequalities C , (2.25) jψ (x)fλ (x) dx ≥ − mλ2 where the constant C depends on the particular weight function used and is given: Kernel Gaussian Squared Lorentzian Trunc. cosine Smooth trunc. cosine
C 0.01958128485 (16π)−1 0.08463957004 0.125047838
Accuracy 10 S.F. Exact 2 S.F.? At least 3 S.F.
Improvement 1.6% 0 16% 4.5%
Note that we were only able to obtain fairly weak error bounds in the truncated cosine case. In each case, the improvement on the analytical bound is relatively small. We may interpret these results as showing that T = R+ S where R has rank 1 and the Hilbert–Schmidt norm of S is small relative to that of T . This is most apparent in the squared Lorentzian case, in which T is itself exactly rank 1 and
−/(16πmλ2 )
−/(16πmλ2 ) −0.01989436788
QI bound
[≈ /(mλ2 )×] −0.09817477044
−π/(32mλ2 )
4π 4 sin(λk) λk(k 2 λ2 − 4π 2 )(k 2 λ2 − π 2 )
π 2 sin(λk) λk(π 2 − k 2 λ2 ) √ 4π λ cos(λk) π 2 − 4k 2 λ2
K 2000 2200 2400 2600 2800 3000
Trunc. cosine µ(K) −0.029012801924 −0.029012804495 −0.029012806174 −0.029012807318 −0.029012808114 −0.029012808686
Smoothed trunc. cosine K µ(K) 140 −0.036095566956 160 −0.036095567038 180 −0.036095567056 200 −0.036095567060 220 −0.036095567061 360
Table 2: Numerical estimation of the minimum eigenvalue µ(K) of the truncation of J to [0, K] for various kernels.
Gaussian Squared Lorentzian K µ(K) K µ(K) 10 −0.0048295212087 30 −0.002980544308 20 −0.0048295668511 40 30 −0.0048295668517 50 40 60 50 70 60 80
−0.1308996939
−π/(24mλ2 )
2π 2 sin(λk) √ k 3λ(π 2 − k 2 λ2 )
4ϑ(λ − |x|)/(3λ)cos(xπ/(2λ))4
Smoothed truncated cosine
ϑ(λ − |x|)/λ cos(xπ/(2λ))
Truncated cosine
Table 1: Compendium of sampling functions considered.
−0.01989436788
√ 2λπe−λ|k|
(1 + λ|k|)e−λ|k|
√ 2 2λπ 1/4 e−(uλ) /2
/4
gλ
2
e−(λu)
2λ3 π −1 /(x2 + λ2 )2
√ 2 (λ π)−1 e−(x/λ)
fλ
fλ
Squared Lorentzian
Gaussian
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no improvement is obtained by using the operator norm. It would be interesting to understand the origin of this apparently general phenomenon. Our second numerical calculation aims to compute the infimum of the spectrum of the unbounded integral operator ∞ dk (k + k )fλ (k − k )ϕ(k ) . (Jϕ)(k) = (2.26) 2 0 2π We proceed by truncating the kernel to the interval [0, K], computing the minimum eigenvalue µ(K) using sufficiently many quadrature points to obtain machine precision. We then increase K until convergence of µ(K) is obtained, again to machine precision. Our results are given in Table 2, in which we give µ(K) in units of /(mλ2 ). Blank entries indicate that the computed value was identical to the last printed number in that column. The density of quadrature points used (per unit K) was 5 for the Gaussian, 1 for the truncated cosine, and 5 for the smoothed truncated cosine, although higher densities were also used as a numerical check (40, 2, and 10 respectively). The results for the squared Lorentzian were rather slower to converge as the density increased (perhaps because the kernel fails to be everywhere smooth) and were computed using a density of 60. For K < 80, a density of 70 was used as a check. To summarize, we have seen that a) the limitations of our flux QI do not lie in the estimation of an operator norm by a Hilbert–Schmidt norm, but rather in the earlier stages of the derivation (probably the estimate (2.18)); b) the overall scope for improvement on our flux QI is roughly a factor of between 3 and 7 (in our examples), and it is clear that the sharp bound is not simply a multiple of our bound (2.10) (in contrast to the situation for two-dimensional massless quantum fields [21, 14]).
3 The Wigner function and kinematical quantum inequalities It is worth emphasizing that phenomena similar to those presented above arise naturally in the context of Weyl quantization, in which the phase space aspect of quantum mechanics is brought to the fore. In our discussion we will consider the phase space to be Rn × Rn (see, e.g., [28] for Weyl quantization on manifolds). We recall that the central object in this approach is the Wigner function Wψ defined on phase space by n 2 (3.1) dn y e2ipy/ ψ(x + y)ψ(x − y) , Wψ (x, p) = where ψ ∈ L2 (Rn ) is the corresponding normalized quantum mechanical state vector. The classical analogue of Wψ (x, p) would be a probability distribution on phase space; as is well known, however, Wψ is not itself a probability distribution because it is not guaranteed to be everywhere non-negative. This has important
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consequences for observables obtained via Weyl quantization, which proceeds as follows. Given an observable on the classical phase space, i.e., a smooth function4 F : Rn × Rn → R, Weyl quantization defines an operator Fw whose expectation values are given (for normalized ψ) by n n d xd p F (x, p)Wψ (x, p) . (3.2) Fw ψ = (2π)n The action of this operator may be written in the form n n d yd p F ([x + y]/2, p)ei(x−y)·p/ψ(y) . (Fw ψ)(x) = (2π)n
(3.3)
Let us note that this procedure also yields a natural definition for the quantum mechanical density associated with a classical observable. Namely, setting dn p F (x, p)Wψ (x, p) , (3.4) ρF (x)ψ = (2π)n it is clear that the spatial integral of ρF (x)ψ yields the expectation value Fw ψ for all F and ψ (modulo domain questions5 ). Now, because the Wigner function need not be everywhere positive, we see that the Weyl quantization of a non-negative classical observable may assume negative expectation values. This situation is exacerbated for the densities defined above (see statement (II) below). However, we will show that kinematical quantum inequalities may be derived, under certain conditions. Indeed, these bounds are obtained as applications of the so-called sharp G˚ arding inequalities in the theory of pseudodifferential operators [36, 12, 27]. It is interesting to note that Fefferman and Phong, to whom the most general sharp G˚ arding results are due, were guided by intuition arising from quantum mechanics: in particular, the uncertainty principle. We begin by specifying more precisely the class of classical observables. For m ) is defined to be m ∈ N, the symbol class S m (often denoted more precisely as S1,0 the set of smooth functions F : Rn × Rn → C such that, for each compact K ⊂ Rn and n-dimensional multi-indices α, β, there exists a constant CK,α,β such that |(Dxα Dpβ F )(x, p)| ≤ CK,α,β (1 + |p|)m−|β|
(3.5)
m for all (x, p) ∈ K × Rn (see, e.g., [26, 36] for multi-index notation). By Shom , we m denote the set of F ∈ S admitting a (unique) decomposition F = Fpr + Fsub such that the principal symbol Fpr belongs to S m , is homogeneous of degree m in momentum, i.e., Fpr (x, λp) = λm Fpr (x, p) for all (x, p) ∈ Rn and λ ∈ R+ , and 4 Precise
growth conditions will be specified below. example, this will certainly hold for F of polynomially bounded growth and ψ belonging to the Schwartz class. 5 For
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is non-zero except at vanishing momentum, while the sub-principal symbol Fsub belongs to S m−1 . For F ∈ S m , the Weyl quantization Fw is a continuous linear map from ∞ C0 (Rn ) to C ∞ (Rn ), so Eq. (3.2) holds for all normalized ψ ∈ C0∞ (Rn ). The density ρF (x)ψ is in fact defined (and indeed smooth in x) for all ψ belonging to the Schwartz class S (Rn ); however, it is only guaranteed to be integrable for ψ ∈ C0∞ (Rn ). Our first result now reads as follows. Theorem 1 m (I) Suppose F ∈ Shom is real, for some m ≥ 1. Then, for each x, the density ρF (x)ψ is unbounded from both above and below as ψ varies in C0∞ (Rn ) with ||ψ||L2 = 1. (II) Suppose F ∈ S 2 is non-negative, F (x, p) ≥ 0 for all (x, p) and let χ ∈ C0∞ (Rn ) be non-negative. Then there exists a constant C ≥ 0, depending on F and χ, such that (3.6) dn x χ(x)ρF (x)ψ ≥ −C
for all ψ ∈ S (Rn ) with ||ψ||L2 = 1. Proof. To establish (I) we may assume without loss of generality that x = 0, for which we have n n n 2 d pd y ρF (0)ψ = F (0, p)e2ipy/ ψ(y)ψ(−y) (3.7) (2π)n for normalized ψ ∈ C0∞ (Rn ). Setting ψλ (x) = λ−n/2 ψ(x/λ), (λ > 0) and making the obvious change of variables, n n n 2 d pd y ρF (0)ψλ = F (0, p/λ)e2ipy/ ψ(y)ψ(−y) (3.8) λ (2π)n so, bearing in mind that |F (0, p/λ) − Fpr (0, p/λ)| ≤ C(1 + |p/λ|)m−1
(3.9)
m and Eq. (3.5), we obtain by definition of Shom
λm+n ρF (0)ψλ −→
n n n 2 d pd y Fpr (0, p)e2ipy/ ψ(y)ψ(−y) (2π)n
(3.10)
as λ → 0+ . It now remains to show that the right-hand side of this expression attains values of both signs as ψ varies in C0∞ (Rn ). To this end, assume (without loss) that Fpr (0, p) depends non-trivially on the first coordinate, p1 , of p. Integrating by parts, the right-hand side of (3.10) in the form Py (ψ(y)ψ(−y))|y=0 , where
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Py is a homogeneous linear partial differential operator (in y) of order m with (possibly complex) constant coefficients cα . We now consider ψ of the form ψ(y) = f (y1 )ei(y2 +···+yn ) χ(y)
(3.11)
where χ ∈ C0∞ (Rn ) is equal to unity in a neighborhood of the origin and f ∈ C0∞ (R). For such ψ we have Py (ψ(y)ψ(−y))|y=0 = Qy1 (f (y1 )f (−y1 ))|y1 =0
(3.12)
for some ordinary differential operator Qy1 =
q
ck (−i)k
k=0
dk dy1k
(3.13)
of order 1 ≤ q ≤ m with constant real coefficients. (That Qy1 is of order at least one is a consequence of our assumption that Fpr (0, p) depends non-trivially on p1 ; reality of the ck holds because the right-hand side of Eq. (3.10) is manifestly real for all ψ ∈ C0∞ (Rn ).) We now choose f so that f (0) = 1 and f (k) (0) = 0 for 1 ≤ k ≤ q − 1. Then by Leibniz’ formula, Py (ψ(y)ψ(−y))|y=0 = cq (−i)q f (q) (0) + (−1)q f (q) (0) + c0 . (3.14) It is now obvious that f may be chosen so that the right-hand side of this expression adopts values of both signs, completing the argument. Statement (II) is straightforward: because χ ∈ C0∞ (Rn ) and F ∈ S 2 , the symbol χF obeys uniform bounds |(Dxα Dpβ χF )(x, p)| ≤ Cα,β (1 + |p|)2−|β|
(3.15)
for all (x, p) ∈ Rn × Rn , so the sharp G˚ arding inequality (Corollary 18.6.11 in [27] with δ = 0, ρ = 1; see also Eq. (18.1.1) therein) entails the existence of a constant C such that (3.16) dn x χ(x)ρF (x)ψ = (χF )w ψ ≥ −C for all normalized ψ ∈ S (Rn ). This is the required kinematical quantum inequality. We now give two examples to illustrate the above ideas. Example 1: Consider a classical Hamiltonian H(x, p) =
p2 + V (x) 2m
(3.17)
on Rn ×Rn with V ∈ C ∞ (Rn ). The Hamiltonian density obtained from Eq. (3.4) is ρH (x)ψ =
2 |∇ψ(x)|2 − Re ψ(x)(ψ)(x) + V (x)|ψ(x)|2 , 4m
(3.18)
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where = ∇2 is the Laplacian. Clearly ρH (x)ψ may be made arbitrarily negative as ψ varies in C0∞ (Rn ) by arranging that ∇ψ(x) = 0, ψ(x)ψ(x) > 0 and then – as in (I) above – scaling ψ about x, introducing ψλ (y) = λ−n/2 ψ(x + (y − x)/λ)
(3.19)
for which ρH (x)ψλ = −λ−(n+2)
2 Re ψ(x)(ψ)(x) + λ−n V (x)|ψ(x)|2 . 4m
(3.20)
As in the proof of (I), the subprincipal symbol drops out in the limit λ → 0+ , so ρH (x)ψλ → −∞ Since H ∈ S 2 we already know that a kinematical quantum inequality exists. However it is instructive to give a direct argument for this, which also yields an explicit bound. To this end, we note that, for any non-negative χ ∈ C0∞ (Rn ) and normalized ψ ∈ S (Rn ),
dn x χ(x)ρH (x)ψ =
1 1 pi ψ | Mχ pi ψ + ψ | (Mχ p2 + p2 Mχ )ψ + ψ | MχV ψ 4m 8m
(3.21)
where pi = −i∇i and (Mf ψ)(x) = f (x)ψ(x) is the operator of multiplication by f . Now [Mχ , p] = iM∇i χ , so Mχ p2 + p2 Mχ = 2pi Mχ pi + i[M∇i χ , pi ] = 2pi Mχ pi − 2 Mχ and hence
dn x χ(x)ρH (x)ψ =
1 pi ψ | Mχ pi ψ + ψ | ML ψ 2m
(3.22)
(3.23)
where
2 (χ)(x) + V (x)χ(x) . (3.24) 8m Since the first term in (3.23) is non-negative, we obtain the quantum inequality 2 n d x χ(x)ρH (x)ψ ≥ infn − (χ)(x) + V (x)χ(x) , (3.25) x∈R 8m L(x) = −
for all non-negative χ ∈ C0∞ (Rn ) and ψ ∈ S (Rn ). We note as a curiosity the appearance of a Schr¨ odinger operator applied to the weight χ (rather than the state ψ). In the case of a non-negative potential, we may obtain a QI (slightly weaker than that given above) in the form 1 inf (Hχ)(x) . (3.26) dn x χ(x)ρH (x)ψ ≥ 4 x∈Rn
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Example 2: We now show that (II) allows us to deduce the existence of a kinematical flux QI on rightwards moving states. Let f be a non-negative smooth compactly supported function. The averaged probability flux jψ (f ) is easily seen to be the expectation in state ψ of the Weyl quantization j(f ) of f (x)p/m, which is a (first order) element of the symbol class S 2 (but is of course negative for p < 0). Now let η(p) be smooth, vanishing for p < 0 and equal to p for p greater than some p0 , and set F (x, p) = f (x)η(p)/m. Then the quantization Fw differs from j(f ) on R ∩ S (R) only by a bounded operator. Accordingly (II) entails that jψ (f ) is bounded below for normalized ψ ∈ R ∩ S (R). Of course, this argument does not determine the magnitude of the bound, in contrast to the direct approach of Sect. 2.2. We should like to remark that in [6] there appears a result which is complementary to ours; in that reference, the authors consider the one-dimensional case n = 1 and show that there is a ψ-independent bound below (and above) on the integral of the Wigner function over elliptic sub-regions of the phase-plane which is much sharper than that implied by the a priori uniform bounds on the Wigner function. This is again an effect of averaging, this time over a region of finite extension in both x- and p-space. It would be interesting to see if this result can be generalized to higher dimensions through a generalization of (II) to a more general class of symbols; however, it is not at all clear that this can be accomplished as it apparently goes beyond the scope of sharp G˚ arding inequalities.
4 Dynamical quantum inequalities In this section we turn to a different type of QI, which is closer to those studied in quantum field theory. The focus here is on time-averages of the energy density at a fixed spatial point: we will refer to the QI bounds obtained as dynamical quantum inequalities. To keep the discussion fairly general, we assume that configuration space M is a topological space carrying a measure ν, so that the state space is H = L2 (M, dν). The dynamics is assumed to be generated by a self-adjoint Hamiltonian H which is defined on a dense domain in H . Each normalized state ψ belonging to the domain of H then determines both a position probability density ρ(t, x)ψ and a Hamiltonian density h(t, x)ψ by ρ(t, x)ψ
=
|ψt (x)|2
(4.1)
h(t, x)ψ
=
Re ψt (x)(Hψt )(x)
(4.2)
where ψt = e−iHt/ ψ. This definition of the energy density differs from that employed in Sect. 3; note that we are not assuming in this section that H is the quantization of a classical observable, so the above would appear to be the most natural definition. In particular, both the quantities defined are integrable with respect to the measure dν(x) for each t ∈ R, with integrals equal to unity and Hψ respectively. However, we will be interested mainly in time averages of these quantities at some fixed point x ∈ M . In so doing, we immediately encounter the
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problem that it does not generally make sense to speak of the value of an L1 function at a point.6 To avoid this, we introduce the spaces Hk = D((1 + H 2 )k/2 ) and assume that, for some k > 0, each element in Hk should be (almost everywhere equal to) a continuous function and that for each x ∈ M there is a vector ηx ∈ H such that ψ(x) = ηx | pk (H)ψ ∀ψ ∈ Hk . (4.3) Here, ψ(x) means the value at x of the continuous function to which ψ is almost everywhere equal, and we have written pk (E) = (1 + E 2 )k/2 . Therefore, the functional ψ → ηx | pk (H)ψ on Hk coincides with the δ-distribution concentrated at x, so that formally [as it is not an element of H ] pk (H)∗ ηx is the δ-distribution. In practice, these assumptions are fairly mild: in particular, for the case in which H is minus the Laplacian on some manifold they are simply a transcription of the content of Sobolev’s lemma. We remark that the necessary regularity in quantum field theoretic quantum inequalities is obtained by restricting to the class of Hadamard states, which would correspond to H∞ = k∈N Hk in the present context. It now makes sense to define the position and Hamiltonian densities as ρ(t, x)ψ
=
|ηx | pk (H)ψt |2
(4.4)
h(t, x)ψ
=
Re pk (H)ψt | ηx ηx | pk (H)Hψt
(4.5)
for normalized states ψ ∈ Hk+1 . Furthermore, one may easily check (using CauchySchwarz) that these quantities are bounded in t, so the time-averaged quantities ρx (f )ψ and hx (f )ψ given by ρx (f )ψ =
dt f (t)ρ(t, x)ψ
(4.6)
and the analogous equation for hx (f )ψ are well-defined for any smooth compactly supported function f . From now on, we denote the spectral measure of H by dPE . (In the case where H may be diagonalized by a basis of orthogonal eigenvectors φn with simple eigenvalues En ,
ψ | φn φn | ϕf (En ) dψ | PE ϕf (E) = n
=
dE
δ(E − En )ψ | φn φn | ϕf (E);
(4.7)
n
more generally, the projection-valued measure allows for the case of varying – even infinite – multiplicities and for both continuous and discrete spectrum.) While 6 Elements of the space L1 (M, dµ) are really equivalence classes of functions agreeing almost everywhere.
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there is some ambiguity in choosing k and ηx such that Eq. (4.3) holds, the measure on R defined by ηx | dPE ηx pk (E)2 (4.8) µx (∆) = ∆
for bounded Borel sets ∆ has an independent meaning. In fact, µx (∆) is simply the diagonal P∆ (x, x) of the integral kernel of the spectral projection P∆ of H on ∆,7 given by
µx (∆) = |φn (x)|2 , (4.9) n:En ∈∆
if H has purely discrete spectrum. Below, it will occasionally be useful to consider the corresponding measure arising from self-adjoint operators other than H; in (H ) these cases, we will write µx to denote the operator H involved. Finally, since 2 0 ≤ µx (∆) ≤ ηx supE∈∆ pk (E)2 , we see that µx is polynomially bounded. After these preliminaries, we come to the statement of our dynamical quantum inequalities. Theorem 2 (i) Let g be any real-valued, compactly supported function on R and set f = g 2 . Then given real numbers a < b, the inequalities bρx (f )ψ +
du Q+ (u)| g (u)|2 ≥ hx (f )ψ 2π ≥ aρx (f )ψ −
du Q− (u)| g (u)|2 2π
(4.10)
hold for all normalized ψ ∈ P[a,b] H , where Q− (u)
=
Q+ (u)
=
[a,b]
dµx (E){u + a − E}+
[a,b]
dµx (E){u − b + E}+
(4.11)
are non-negative, monotone increasing and polynomially bounded in u, and we have used the notation {λ}+ = max{0, λ}. (Similarly, we will write {λ}− = min{0, λ}.) Moreover, the first (resp., second) inequality in (4.10) also holds for all ψ ∈ P(−∞,b] Hk+1 (resp., P[a,∞) Hk+1 ) provided the integration range in (4.11) is replaced by (−∞, b] (resp., [a, ∞)). 7 Since,
for any ψ ∈ H , we have P∆ ψ ∈ D(pk (H)), it follows that (P∆ ψ)(x) = dν(y) P∆ (x, y)ψ(y) where P∆ (x, y) = (pk (H)P∆ ηx )(y) is continuous in y. This last quantity may easily be expressed as pk (H)P∆ ηx | pk (H)P∆ ηy , so in particular, P∆ (x, x) = pk (H)P∆ ηx 2 = µx (∆).
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(ii) Suppose R− dµx (E)(1 + |E|) < ∞ and let g be as in (i). Then, for any fixed c ∈ R, the inequality du S(H − c1; u)| g (u)|2 (4.12) hx (f )ψ ≥ cρx (f )ψ − 2π holds for all normalized ψ ∈ Hk+1 , where S(H; u) = dµ(H) x (E) {u − E}+
(4.13)
is non-negative, monotone increasing and polynomially bounded in u. (There
is of course a dual statement, for the case R+ dµx (E)(1 + |E|) < ∞.) Before proceeding to the proof of these statements, we illustrate them by drawing some consequences. The interpretation of (i) is that a state with energy between a and b has an averaged energy density between a and b, suitably weighted by the averaged position probability density, modulo a certain latitude bounded by quantum inequalities. Replacing g by gλ (t) = λ−1/2 g(t/λ), we may consider the two regimes λ → 0+ , representing tightly peaked averages, and λ → ∞, which represents widely spread averages. In the former case, we have µx ([a, b]) ∞ du du Q± (u)| u| g(u)|2 gλ (u)|2 ∼ (4.14) 2π λ 2π 0 provided µx ([a, b]) > 0 (failing which the left-hand side vanishes identically). Thus the latitude afforded by the quantum inequality bound grows as the sampling becomes more tightly peaked. As λ → ∞, the QI latitude tends to zero and one may show that b ≥ lim sup λ→∞
hx (gλ2 )ψ hx (gλ2 )ψ ≥ lim inf 2 2 ) ≥ a λ→∞ ρx (gλ ρx (gλ )ψ ψ
(4.15)
for all ψ ∈ P[a,b] H , provided ρ(t, x)ψ is non-zero for some t. This ergodic result shows that the spatial and temporal averages of energy densities obey related constraints. As a second illustration, consider (ii) in the case where H has a discrete spectrum {En } with corresponding orthonormal eigenfunctions {φn }, and satisfying the integrability condition on µx (for example, H might be semibounded). Then, in the case c = 0,
du 2 | g(u)|2 |φn (x)|2 (u − En ) hx (g )ψ ≥ − 2π En ≤u
αn |φn (x)|2 (4.16) = − n
where αn =
∞
En
du | g(u)|2 (u − En ) .
(4.17)
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These formulae may be used to compare the relative ease of obtaining negative energy densities at different spatial locations. For example, the eigenfunctions φn of the harmonic oscillator H = p2 /(2m) + 12 mω 2 x2 on L2 (R) obey the following bounds (cf. the Appendix to Sect. V.3 in [35]): For any j ∈ N0 there exists cj > 0 and rj ∈ N0 such that sup |(1 + xj )φn (x)| ≤ cj (1 + n)rj
(4.18)
x∈R
for all n ∈ N0 . Thus, for all normalized ψ ∈ S (R) (in fact, for all ψ in a considerably larger domain) we have hx (g 2 )ψ ≥ −
∞
cj αn (1 + n)rj . 1 + |x|j n=0
(4.19)
In this case, it is clear that – for a fixed sampling function g – the αn form a rapidly decaying sequence and so the sum converges for any j. Thus we have shown that the state-independent bound on energy density is itself a rapidly decaying function of x. It is therefore generally easier to maintain negative energy densities near the classical equilibrium point rather than far away. Finally, consider (i) for the case H = −id/dx on H = L2 (R) and a particle of mass m. In this instance, the spaces Hk coincide with the Sobolev spaces W k,2 (R) and Sobolev’s lemma permits us to take k > 1/2. Then the dynamical evolution amounts to spatial translation and the averaged Hamiltonian density is related to the spatially averaged probability flux by hx (f )ψ = m jψ (f˜x )
(4.20)
where f˜x (t) = f (x − t). Moreover, the measure µx is easily seen to be given by µx (∆) = |∆|/(2π), where | · | denotes the usual Lebesgue measure. Then the second inequality in (i) may easily be checked to reproduce the flux inequality (2.10) for all ψ ∈ P[0,∞) W k+1,2 (R). Proof of Theorem 2. The assertions (i) and (ii) are based upon two facts, which will be proved below. First, for any c ∈ R and normalized ψ ∈ Hk+1 , one may show that 2 d hx (f )ψ − cρx (f )ψ = ( − c) ψ | pk (H) g (−1 [1 − H])ηx . 2π
(4.21)
Second, if ∆ is a Borel set then 2 ψ | pk (H) g (−1 [1 − H])ηx ≤ for all normalized ψ ∈ P∆ H .
∆
2
dµx (E) | g([ − E]/)| .
(4.22)
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Putting these together, we obtain d { − c}− hx (f )ψ − cρx (f )ψ ≥ dµx (E) | g ([ − E]/)|2 2π ∆ du 2 | g (u)| = dµx (E) {u + E − c}− 2π ∆ du | g(u)|2 =− dµx (E) {u + c − E}+ , 2π ∆
(4.23)
for all normalized ψ ∈ P∆ H , where we have made the change of variables u → −u and exploited the fact that | g (u)|2 is even (because g is real-valued). The interchange of variables employed in the first step is justified provided the inner integral in the last line of (4.23) is polynomially bounded in u. To obtain the second inequality in (i), we set c = a and ∆ = [a, b] and observe that the inner integral in Eq. (4.23) reduces to Q− (u) and is polynomially bounded because µx is. The inequality clearly remains true for ψ ∈ P[a,∞) Hk+1 with ∆ = [a, ∞). To obtain (ii), we set ∆ = R and observe that the integrability condition R− dµx (E)(1 + |E|) < ∞ and polynomial boundedness of µx guarantee that S(H − c1, u) exists and is polynomially bounded. Inequality (4.12) follows
(H−c1)
(H) (E) F (E). from the above on observing that dµx (E) F (E − c) = dµx To obtain the first inequality in (i) and the dual statement to (ii), one argues in an analogous fashion from the calculation d 2 { − c}+ dµx (E) | g([ − E]/)| hx (f )ψ − cρx (f )ψ ≤ 2π ∆ du 2 | g(u)| = dµx (E) {u + E − c}+ , (4.24) 2π ∆ which holds for all normalized ψ ∈ P∆ H . It remains to prove the two facts presented as Eqs. (4.21) and (4.22) above. First, observe that for any normalized ψ ∈ Hk , ρx (f )ψ may be expressed as ρx (f )ψ = dt f (t) dψ | PE ηx dηx | PE ψei(E−E )t/ pk (E)pk (E ) (4.25) by the functional calculus. Performing the t integral first (which is legitimate since f is smooth and compactly supported) we obtain ρx (f )ψ = dψ | PE ηx dηx | PE ψf([E − E]/)pk (E)pk (E ) . (4.26) Since f = g 2 , the convolution theorem may be used to write d g([E − ]/) f ([E − E]/) = g([E − ]/) 2π
(4.27)
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using the fact that g(λ) = g(−λ) since g is real-valued. Substituting in (4.26), and again rearranging the order of integration, we obtain ρx (f )ψ
=
d 2π
dψ | PE ηx
dηx | PE ψ
g([E − ]/) g([E − ]/)pk (E)pk (E )
2 d dψ | PE ηx pk (E) = g ([E − ]/) 2π 2 d ψ | pk (H) = g (−1 [1 − H])ηx 2π
(4.28)
To treat hx (f )ψ for normalized ψ ∈ Hk+1 , we write hx (f )ψ =
1 2
dt f (t) dψ | PE ηx dηx | PE ψ ei(E−E
)t/
(E + E )pk (E)pk (E ) , (4.29)
by functional calculus and use the identity (E + E ) f ([E − E]/) = 2
d g ([E − ]/) g([E − ]/) 2π
(4.30)
in place of the convolution theorem. (See [19] and [15] for proofs of this identity.) By a derivation analogous to that used for ρx (f )ψ we then obtain hx (f )ψ =
2 d ψ | pk (H) g (−1 [1 − H])ηx 2π
(4.31)
and Eq. (4.21) follows from this equation and (4.28). The second assertion, Eq. (4.22), is proved by noting that 2 ψ | pk (H) g (−1 [1 − H])ηx ≤ P∆ pk (H) g (−1 [1 − H])ηx 2
(4.32)
using ψ = P∆ ψ and the Cauchy–Schwarz inequality (with ψ = 1). The righthand side may be written ∆
dηx | PE ηx pk (E)2 | g ([ − E]/)|2 =
∆
dµx (E)| g ([ − E]/)|2
(4.33)
which completes the derivation of Eq. (4.22) and hence the proof of Theorem 2.
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5 Numerical details In this section we provide more details on the numerical methods employed in Sect. 2 and discuss rigorous error estimates on the numerical errors. The basic numerical method is easily explained (see, e.g., Sect. 4.1 of [4] or Chapter 4 of [10]). Suppose T is an integral operator on L2 (R+ , dk) with kernel G, i.e., ∞ ) . (T ψ)(k) = dk G(k, k )ψ(k (5.1) 0
To handle this numerically, we first truncate the kernel to [0, K] × [0, K] for some K > 0, which amounts to studying a compression TK of T . Provided that the required properties of T are, for sufficiently large K, well-approximated by the corresponding properties of TK restricted to L2 (0, K), we proceed to approximate this restricted operator by a matrix. To do this, we suppose that (ξj )N j=0 and N (wj )j=0 are the nodes and weights for a suitable quadrature method on [0, K], and define the (N + 1)-square matrix A = (Ajk )N j,k=0 with (j, k) entry Ajk = 1/2
1/2
wj wk Gλ,K (ξj , ξk ). The relevant computations are performed on A and, if N and K are sufficiently large, this will provide a numerical approximation to the required quantity. This technique was applied to the Bracken–Melloy kernel as described in Sect. 2.2. In that case, we were unable to derive useful error estimates. However, the operator norm calculations of Sect. 2.3 are more controlled, as we now describe. The problem is to estimate the squared operator norms of the family of integral operators Tλ (λ > 0) defined in the above fashion8 with kernel 1 √ k gλ (−k − k ) . (5.2) Gλ (k, k ) = 2π It is straightforward to verify that Tλ 2 = λ−2 T1 2 , so we hereafter study only the operator T = T1 with kernel G = G1 and denote by TK its compression onto L2 (0, K). These compressions converge to T in the Hilbert–Schmidt norm, and therefore in operator norm, as K → ∞. That the corresponding N × N matrix approximations have operator norms converging to TK as N → ∞ is a consequence of the convergence of the quadrature formula to the integral for continuous functions. Thus our technique can legitimately be applied to this problem and it remains to control the errors inherent in the scheme for finite N and K. In general we have analytical control of the truncation errors (parametrised by K): in Hilbert–Schmidt norm, this is given by the integral of |G|2 over the region [0, ∞)×[0, ∞)\[0, K]×[0, K] which, by routine manipulations involving symmetry and a change of variables can be written as 2K ∞ 1 1 2 u(u − K)| g (−u)| du + u2 | g (−u)|2 du . (5.3) 4π 2 K 8π 2 2K 8 In Sect. 2.3 the operators were defined on L2 (R+ , dk/(2π)). Here, we absorb the factor of (2π)−1 into the kernel, which leaves the spectral data and operator norms unchanged.
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Although we are not able to control the discretization errors, we are able to observe apparent convergence to machine precision in most cases. We now consider the four functions used in Sect. 2.3. Starting with √ sampling 2 the Gaussian kernel g(k) = 2π 1/4 e−k /2 , the Hilbert–Schmidt norm of T can be found √by substituting K = 0 in (5.3) and evaluating the integral to give T H.S. = 1/(4 π) which is of course an upper bound for the operator norm. For more precise results, we turn to the quadrature method described above. For this kernel, the integrals in (5.3) can be evaluated explicitly to give a relative error in the Hilbert-Schmidt norm of
T − TK H.S. = (1 + erf(2K) − 2 erf(K))1/2 . (5.4)
T H.S. It can be numerically verified that the relative error falls below ε = 0.5×10−10 (for ten-digit precision) at approximately K = 6.756 (this calculation requires about 25-digit precision). The computations were performed in Maple 8 using c-panel repeated Clenshaw-Curtis quadrature (see Section 2.4.4 of [10]) on the interval [0, 6.9] and Maple’s NAG-based SingularValues routine. Using 33, 65 or 129 samples with c = 1, 2 gives in each case the same results for the first largest two singular values: σ1 = 0.1399331442, σ2 = 0.0175697912 to 10 figures. Notice that the second singular value is very much smaller than the first, which means that the matrices, and hence T , can be well approximated by operators of rank 1. This is consistent with the operator norm,√computed here to be .1399331442, being close to the Hilbert-Schmidt norm, 1/(4 π) = .1410473959 . . . . This similarity finally justifies our use of truncation constants based on the relative error in the Hilbert-Schmidt norm: the calculated value of the operator norm is certainly no larger than the true value, since it is the norm of a compression of T , so we have
T − TK
T − TK H.S. T H.S. ≤
T
T H.S.
T .1410473959 = .4912379017 × 10−10 (5.5) ≤ .4873572016 × 10−10 .1399331442 which is still less than the target figure of 0.5 × 10−10 . The next kernel of interest is the squared Lorentzian; however, in this case T is a rank-1 operator so the Hilbert–Schmidt and operator norms coincide and there is no need for numerical investigation. This leaves the two compactly supported kernels. In the truncated cosine case, the same techniques as above lead to an error estimate of the order of 2% relative error in the Hilbert–Schmidt norm for K = 1100. As this is a rather weak estimate, we suppress the details; the numerical estimate of the squared operator norm (for K = 1100, N = 1024) is given in Sect. 2.3. Our last example is the smoothed truncated cosine, defined by √ 2 3π 2 sin(k) . (5.6) g(k) = 3 (π 2 − k 2 )k
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The relatively slow decay of this function makes the precision obtained in the Gaussian example impractical, but we can obtain results to at least four significant figures. In fact the numerical results appear to be much more precise than would be suggested by this error estimate. Maple is able explicitly to evaluate the integrals in (5.3) to give a rather complicated formula involving the Si and Ci special functions, and from this to give the asymptotic formula 5π/(16K 3) + o(1/K 4 ) for the relative error in the Hilbert-Schmidt norm. Using only the leading term, we can predict that truncation at about K = 732.3 should give a Hilbert-Schmidt norm relative error less than 0.5 × 10−4 ; numerical investigation of the exact formula near this point confirms this value. Proceeding in the same way as for the Gaussian kernel but this time using the faster numerical engine in Matlab 6 to calculate the singular values, we obtain σ1 = 0.3536210388 and σ2 = 0.0733902259 using 513 samples. The same values are obtained if the number of samples is increased to 1025 or two panels are used. Once again, the fact that the second singular value is considerably smaller than the first can be used after the fact to justify the use of relative errors in the Hilbert-Schmidt norm (rather than in the operator norm) in choosing the truncation constant. Although the error analysis only allows us to be confident of the first four figures, it seems likely that this figure for the operator norm is considerably more accurate than that. Doubling the truncation constant and using 2049 points, again with 1 and 2 panels, gives exactly the same results to ten figures as the two 1025point methods above. The last set of calculations reported in Sect. 2.3 concern the unbounded operator J of Eq. (2.26). Here we have not succeeded in obtaining usable estimates of the errors introduced by truncation to [0, K]. However, it is nonetheless true that inf σ(JK ) → inf σ(J) as a consequence of the following arguments. Proposition 3 Suppose k is an absolutely bounded kernel9 on L2 (0, ∞), and let w be a measurable function on (0, ∞) (with respect to the Lebesgue measure). Let D = {f ∈ L2 (0, ∞) : wf ∈ L2 (0, ∞)} .
(5.7)
Suppose (w(x) + w(y))k(x, y) is a Hermitian function of x and y, and define an operator with domain D by ∞ (T f )(x) = (w(x) + w(y))k(x, y)f (y)dy (5.8) 0
and assume that T is bounded below. For K > 0 define the truncated operator K (TK f )(x) = (w(x) + w(y))k(x, y)f (y)dy . (5.9) 0
Then lim inf σ(TK ) = inf σ(T ) .
K→∞ 9 That
is, |k| is the integral kernel of a bounded operator.
(5.10)
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Proof. TK is a compression of T so inf σ(TK ) ≥ inf σ(T ) for all K. For any ε > 0 we can choose f ∈ D with f = 1 and such that f | T f < inf σ(T ) + ε/2. Now ∞ ∞ (w(x) + w(y))k(x, y)f (y)dy f (x)dx (5.11) f | T f = 0
0
and the integrand here is in L2 ((0, ∞) × (0, ∞)) by Lemma 4 (see below). It now follows from Lebesgue’s dominated convergence theorem and Fubini’s Theorem that, provided the above repeated integral can be interpreted as an integral on the measure space (0, ∞) × (0, ∞), K
f | T f = lim
K→∞
K
(w(x) + w(y))k(x, y)f (y)dy 0
f (x)dx .
(5.12)
0
If we let fK (x) = f (x)χ(0,K) (x) then this tells us that fK | TK fK → f | T f ; we also have fK → f = 1 as K → ∞, so fK | TK fK / fK 2 → f | T f as K → ∞. In particular, for sufficiently large K we have ε fK | TK fK < f | T f + < inf σ(T ) + ε
fK 2 2
(5.13)
which implies that inf σ(TK ) < inf σ(T ) + ε. In combination with the earlier in2 equality inf σ(TK ) ≥ inf σ(T ), this establishes the result. It remains to justify the treatment of the repeated integral as an integral on a product measure space. Lemma 4 In the notation of the above theorem, for any f ∈ D, the repeated integral in Eq. (5.11) is absolutely convergent, and so can be interpreted as the integral of a function in L2 ((0, ∞) × (0, ∞)). Proof. We calculate ∞ ∞ |(w(x) + w(y))k(x, y)f (x)f (y)|dy dx 0 0 ∞ ∞ |k(x, y)| |f (y)|dy dx ≤ |f (x)w(x)| 0 0 ∞ ∞ + |k(x, y)| |w(y)f (y)|dy dx . (5.14) |f (x)| 0
0
2
Since both f and wf are L functions and k is an absolutely bounded kernel, both terms on the right-hand side are finite by the Cauchy-Schwarz inequality. The final conclusion now follows from Tonelli’s Theorem.
6 Conclusion The main focus of this paper has been to draw attention to links between the failure of pointwise energy conditions in quantum field theory and a range of similar situations in quantum mechanics. In addition we have seen that there are links at the
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technical level between the QIs developed in quantum field theory and those obtained here in the quantum mechanical setting. We have also made contact with the ideas and methods of Weyl–Wigner quantization and sharp G˚ arding inequalities. In conclusion, we briefly summarise the new results we have obtained along the way. First, we have seen that the backflow phenomenon is limited in space (as well as in time [7]) as shown by our flux QI (2.10). In particular, the magnitude of the negative flux times the square of its spatial extent is bounded above for all right-moving states in R. We have also provided an improved numerical estimate of Bracken and Melloy’s backflow constant, and also given numerical evidence to support the conjecture that our flux QI is generally within an order of magnitude of the optimal bound (i.e., the infimum of the spectrum of J, given by Eq. (2.26)). Second, we have shown that similar phenomena occur for densities of observables obtained via Weyl quantization. This is a consequence of the indefinite sign of the Wigner function, and therefore an expression of the uncertainty principle. Moreover, for observables which are second order (or less) in momentum, we have seen that sharp G˚ arding inequalities entail the existence of kinematic quantum inequalities. We have also obtained explicit bounds in the case of Schr¨ odinger operators with smooth potentials. Finally, for general quantum mechanical systems describing dynamics on a topological measure space, we have shown that the time-averaged energy density obeys dynamical quantum inequalities (evolution being generated by the spatial integral of the energy density). For the 1-dimensional harmonic oscillator, we saw that the QI bound (for a given sampling function) is a Schwartz-class function: it becomes rapidly much harder to create sustained negative energy densities away from the classical equilibrium point. Moreover, we have seen that a bound on the spectral behaviour of the Hamiltonian on the negative spectral axis, expressed by the integrability condition on µx in (ii) of Thm. 2, already leads to dynamical QIs. This integrability condition can be viewed as a condition on the global dynamical stability of a quantum system, much in the sense of quantum systems in thermal equilibrium, where the spectral weight of the generator of the time-evolution (the Liouvillian) is exponentially suppressed on the negative half-axis (cf. Prop. 5.3.14 in [8]). This indicates again the link between (thermo)dynamical stability and dynamical QIs which was established in [20] and which originally motivated the introduction of QIs in [22]. All these findings corroborate the intimate connection between QIs and the fundamental principles of quantum mechanics: the uncertainty principle and dynamical stability. Acknowledgments. CJF thanks I˜ nigo Egusquiza for bringing ref. [7] to his attention. The work of CJF was assisted by EPSRC Grant GR/R25019/01 to the University of York; RV also thanks the EPSRC for support received under this grant during a visit to York. Numerical calculations were partly conducted using the White Rose Grid node hosted at the University of York. Some of this work was conducted at the Erwin Schr¨ odinger Institute, Vienna, during the programme
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on Quantum Field Theory in Curved Spacetime which took place in July–August 2002; CJF and RV thank the organizers of this programme and the institute for its hospitality. Particular inspiration was drawn from the Hotel Sacher.
References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York 1975. [2] M. Alcubierre, Class. Quantum Grav. 11, L73 (1994). [3] G.R. Allcock, Ann. Phys. 53, 311 (1969). [4] K.E. Atkinson, The numerical solution of integral equations of the second kind, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge 1997. [5] M. Bordag, U. Mohideen, V.M. Mostepanenko, Phys. Rep. 353, 1 (2001). [6] A.J. Bracken, H.-D. Doebner, and J.G. Wood, Phys. Rev. Lett. 83, 3558 (1999). [7] A.J. Bracken and G.F. Melloy, J. Phys. A: Math. Gen. 27, 2197 (1994). [8] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol 2, 2nd Ed., Springer-Verlag, Berlin 1997. [9] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wet. 51, 793 (1948). [10] L.M. Delves and J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge 1985. [11] H. Epstein, V. Glaser, and A. Jaffe, Nuovo Cimento 36, 1016 (1965). [12] C. Fefferman and D.H. Phong, Comm. Pure Appl. Math. 34, 285 (1981). [13] C.J. Fewster, Class. Quantum Grav. 17, 1897 (2000). [14] C.J. Fewster and S.P. Eveson, Phys. Rev. D 58, 084010 (1998). [15] C.J. Fewster and B. Mistry, Phys. Rev. D 68, 105010 (2003). [16] C.J. Fewster and M.J. Pfenning, J. Math. Phys. 44, 4480 (2003). [17] C.J. Fewster and E. Teo, Phys. Rev D 59, 104016 (1999). [18] C.J. Fewster and E. Teo, Phys. Rev D 61, 084012 (2000). [19] C.J. Fewster and R. Verch, Commun. Math. Phys. 225, 331 (2002). [20] C. J. Fewster and R. Verch, Commun. Math. Phys. 240, 329 (2003). ´ E. ´ Flanagan, Phys. Rev. D 56, 4922 (1997). [21] E. [22] L.H. Ford, Proc. Roy. Soc. Lond. A364, 227 (1978). [23] L.H. Ford and T.A. Roman, Phys. Rev. D 53, 5496 (1996). [24] L.H. Ford and T.A. Roman, Phys. Rev. D 55, 2082 (1997). [25] L.H. Ford and T.A. Roman, Phys. Rev. D 60, 104018 (1999).
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[26] L. H¨ormander, The Analysis of Linear Partial Differential Operators I, Springer Verlag, Berlin 1983. [27] L. H¨ormander, The Analysis of Linear Partial Differential Operators III, Springer Verlag, Berlin 1994. [28] N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer-Verlag, Berlin 1998. [29] P. Marecki, Phys. Rev. A 66, 053801 (2002). [30] G.F. Melloy and A.J. Bracken, Found. Phys. 28, 505 (1998). [31] M.S. Morris, K.S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988). [32] M.J. Pfenning, Phys. Rev. D 65, 024009 (2002). [33] M.J. Pfenning and L.H. Ford, Class. Quantum Grav. 14, 1743 (1997). [34] M.J. Pfenning and L.H. Ford, Phys. Rev. D 57, 3489 (1998). [35] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, New York 1975. [36] M.E. Taylor, Pseudodifferential Operators, Princeton University Press, 1981. Simon P. Eveson and Christopher J. Fewster Department of Mathematics University of York Heslington York YO10 5DD United Kingdom email:
[email protected] email:
[email protected] Rainer Verch Max-Planck-Institut for Mathematics in the Sciences Inselstr. 22 D-04103 Leipzig Germany email:
[email protected] Communicated by Yosi Avron submitted 27/01/04, accepted 05/05/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 31 – 84 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/010031-54 DOI 10.1007/s00023-005-0198-8
Annales Henri Poincar´ e
Integrated Density of States for the Periodic Schr¨ odinger Operator in Dimension Two Alexander V. Sobolev
1 Introduction The objective of the present paper is to study the high energy asymptotics of the density of states D(λ) for the Schr¨ odinger operator L2 (Rd ), d ≥ 1 with a periodic potential V : H = −∆ + V. (1.1) Here V is a real-valued function periodic with respect to a d-dimensional lattice Γ ⊂ Rd . Below we denote by O ⊂ Rd a standard fundamental domain of the lattice Γ, and by O† the fundamental domain of the dual lattice Γ† . For this operator, as well as for any other elliptic self-adjoint differential operator, the density of states is defined by the formula (R)
N (λ; HD ) . R→∞ Rd
(1.2)
D(λ) = lim (R)
Here HD is the restriction of H to the cube [0, R]d with the Dirichlet boundary (L) conditions, and N (λ; · ) is the counting function of the discrete spectrum of HD . The above limit exists for periodic and almost periodic potentials, see [17], [22]. To be precise, the quantity D(λ) is called the integrated density of states, but for the sake of brevity we call it simply the density of states. Calculation of the density of states D0 (λ) for the unperturbed operator H0 = −∆ is an elementary exercise: one easily proves (see, e.g., Proposition 2.4 below) that D0 (λ) =
d 1 wd λ 2 , λ ≥ 0, d (2π)
d
wd =
π2 , Γ(1 + d/2)
(1.3)
where wd is the volume of the unit ball in Rd . For d = 1 it was shown in [20] (see [19] for the almost periodic case) that the density of states admits a complete asymptotic expansion in the powers of λ−1 , as λ → ∞. On the basis of this result it is natural to conjecture that for general d ≥ 2 the asymptotics of D(λ) exists and has the form N bj λ−j + o(λ−N ) , λ → ∞, ∀N. D(λ) = D0 (λ) 1 + j=1
(1.4)
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For d = 1 the coefficients bj satisfy simple recursive relations, see [20]. Results for the multidimensional Schr¨ odinger operator are much less advanced: relation (1.4) has been proved only with N = 1 so far. The first formula was found in [21] (see also [22] for an elementary proof) for almost periodic potentials V : D(λ) = D0 (λ) + O(λ 2 −1 ), λ → ∞. d
A two-term asymptotics was first established in [6] for C∞ -smooth periodic potentials V : 3 d D(λ) = D0 (λ) 1 + b1 λ−1 + O(λ− 2 + ) , ∀ > 0, b1 = − V (x)dx. (1.5) 2|O| O The proof in [6] relied based on the powerful methods of microlocal analysis. In [8], via an advanced version of perturbation theory for periodic operators, this asymptotics was generalized to the case of the polyharmonic operator (−∆)l + V , l > 1/2, with an improvement of the remainder estimate: 1 D(λl ) = D0 (λl ) 1 + b1,l λ−l + O(λ 2 −2l ln λ) , b1,l = b1 l−1 .
(1.6)
A more precise result was obtained in [8] for the Schr¨odinger operator −∆ + V for d = 3: ˆ + O(λ−δ ), (1.7) D(λ) = D0 (λ) 1 + b1 λ−1 + D ˆ =D ˆ V . Recently it was observed with some small δ > 0 and a constant coefficient D ˆ = 0, and hence the above formula is consistent with the conjecture in [11] that D (1.4). Note that the density of states was also studied for the magnetic operator (−i∇ − a)2 + V with a periodic magnetic vector-potential a, see [14] and [8]. In [14] it was shown that D(λ) = D0 (λ) + O(λ(d−2)/2+ ) with an arbitrary > 0. Justification of the hypothesis (1.4) for large N would be a very hard problem. However, if one assumes that (1.4) holds, then the coefficients bj can be calculated relatively easily. They are obtained in [11] by a standard argument from the asymptotics of the heat kernel of the Schr¨ odinger operator. These coefficients are integrals of certain standard polynomials depending on V and derivatives of V . These polynomials are known both in mathematical and physical literature as heat kernel invariants. For recent work on the structure of these polynomials see, e.g., [16], [7]. In particular, this approach gives d(d − 2) V 2 (x)dx, b2 = 8|O| O which implies that b2 = 0 for d = 2. Moreover, for all even d the coefficients bj satisfy bj = 0 if j ≥ d/2 + 1. Our ultimate goal is to justify the conjecture (1.4) with N = 2 for all dimensions d ≥ 2. In this paper this is done for d = 2, while the general case d ≥ 3 will
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be addressed in a subsequent publication. The main result of the present paper is the formula 6 (1.8) D(λ) = D0 (λ) 1 + b1 λ−1 + O(λ− 5 + ), ∀ > 0, for the density of states of the Schr¨odinger operator with d = 2, see Theorem 2.3. From the technical point of view the paper is a continuation of [25] where the density of states was studied in the case d = 1 for elliptic operators of a more general form than the Schr¨ odinger operator. As in [25], our approach is a variant of the “near-similarity” method, which is usually applied in dimension one (see [18], [1] and [12], [13]). The central idea is to reduce the operator H with the help of a suitable similarity transformation, to an operator with constant coefficients. In the present paper the required similarity is implemented by a unitary operator eiΨ , where Ψ is a bounded self-adjoint PDO with the symbol ψ(x, ξ). In contrast to the one-dimensional case considered in [25], for d ≥ 2 a complete reduction to constant coefficients is not achievable, since instead of a smooth symbol ψ(x, ξ) (which is the case for d = 1) a straightforward application of the method produces a symbol ψ with singularities on a set Λ which is a union of hyper-planes {ξ ∈ Rd : θ(ξ + θ/2) = 0} where θ ∈ Γ† , Γ† being the dual lattice. To avoid the singularity, one studies the neighbourhood of Λ separately from the region away from Λ. Outside the set Λ the symbol ψ is found as in the case d = 1, from a series of commutator equations which emerge as a result of the requirement that the new operator should have constant coefficients. Then the density of states for the new operator is found by an elementary calculation. Near the “singular” set Λ the operator H is reduced to a “one-dimensional” effective operator of the Schr¨odinger type with a pseudo-differential perturbation. For this operator the density of states is found using the results of [25]. Although the set Λ emerges in a natural way in the context of the PDO calculus, there is also a perturbation-theoretic interpretation. Recall that the eigenvalues of the unperturbed Floquet Hamiltonian H0 (k) are given by λ(ω) (k) = (ω + k)2 , where ω ∈ Γ† are points of the dual lattice and k ∈ O† is the quasi-momentum. The analysis of these eigenvalues under the perturbation V is dramatically different for d = 1 and d ≥ 2. If d = 1, then the standard perturbation theory yields a complete asymptotic expansion of the eigenvalues. On the contrary, for d ≥ 2 the unperturbed eigenvalues split in two groups behaving differently under the perturbation V , which can be described with the help of the set Λ. The eigenvalues / Λ can be more or less completely described by the perturbation λ(ω) (k) with ω ∈ theory, see [4], [9]. The eigenvalues with ω ∈ Λ move by a quantity of order V under the perturbation V . This effect is due to the small divisors arising when the eigenvalues λ(ω) (k) get close together. In the relevant literature these exceptional eigenvalues are sometimes called resonant, unstable or singular, see [5], [9]. It was shown in [5], [9] that their behavior can be described by means of some effective one-dimensional Schr¨ odinger operators. The resonant set presents a major obstacle when studying spectral properties of the periodic Schr¨ odinger operator, and in particular, the asymptotics of the
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density of states D(λ). The precision of the asymptotics eventually depends on how well one knows the behavior of the resonant eigenvalues. For instance, estimating their number from above leads to the remainder estimate in (1.6). The more precise result (1.7) requires more thorough study of the set Λ, see [8]. In the present paper the asymptotics (1.8) is also derived via a detailed analysis of the set Λ. Although the study of the density of states is an object of independent interest in its own right, it can be also used to investigate other spectral properties of the Schr¨ odinger operator. One such problem is to justify the Bethe-Sommerfeld conjecture, that is to prove that the number of gaps in the spectrum of H is finite. The conjecture is known to be true for all dimensions d ≥ 2 under the condition that the lattice Γ is rational, see [23]. For general lattices it was justified so far only for dimensions d = 2, 3, 4, see [2], [24], [9], [6], [15] and references therein. This result is derived not directly from (1.5) or (1.6), but from the asymptotics of the same type for the so-called generalized density of states, see, e.g., [6] or [15] for definition. The restriction d ≤ 4 is then dictated by the remainder estimate in this asymptotics. An improved remainder estimate would lead to the inclusion of bigger d’s. Moreover, the justification of the asymptotics with more terms would allow one to increase the dimension even further. The paper is organized as follows. Section 2 contains the precise definitions of objects studied in the paper, and the statement of the main result (see Theorem 2.3). In Section 3 necessary information on the calculus of periodic PDO’s is collected, including their transformations under linear maps. Section 4 describes partitions of PDO’s which are central for their reduction to constant coefficients. Section 5 is devoted to the study of the density of states for the model operator, which is based on its decomposition in the invariant subspaces. Their structure is explicitly described in terms of the resonant set Λ. On each of the invariant subspaces the model operator reduces to a one-dimensional Schr¨ odinger-type operator, which makes possible the application of the asymptotics established in [25]. At the next step, in Section 6, the Schr¨ odinger operator (1.1) is reduced to the model operator with the help of the unitary operator having the form eiΨ with a suitable PDO Ψ. We loosely call this operator a gauge transformation. A further analysis of the model operator leads to the conclusion that its density of states is determined by its constant coefficients part Ao , see Section 7. The proof of the Main Theorem is completed in Section 8 together with the asymptotic formula for the density of states of the operator Ao . The calculation of an integral featuring in Section 8, is done in the Appendix. We emphasize that although the main result concerns the case d = 2, we do the calculations for arbitrary dimension d ≥ 2 whenever possible, indicating the points where the argument requires d = 2.
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2 Main result 2.1
Classes of PDO’s
Before we define the pseudo-differential operators (PDO’s) we introduce first the relevant classes of symbols. Let Γ ∈ Rd be a lattice. Denote by O its fundamental domain. For example, for O one can choose a parallelepiped spanned by a basis of Γ. The dual lattice and its fundamental domain are denoted by Γ† and O† respectively. Sometimes we reflect the dependence on the lattice and write OΓ and O†Γ . In particular in the case Γ = (2πZ)d one has Γ† = Zd and it is natural to take O = [0, 2π)d , O† = [0, 1)d . For any measurable set C ⊂ Rd we denote by |C| or vol(C) its Lebesgue measure (volume). The volume of the fundamental domain does not depend on its choice, it is called the determinant of the lattice Γ and denoted d(Γ) = |O|. By e1 , e2 , . . . , ed we denote the standard orthonormal basis in Rd . For any u ∈ L2 (O) and f ∈ L2 (Rd ) define the Fourier coefficients and Fourier transform respectively: 1 u ˆ(θ) = e−iθ,x u(x)dx, θ ∈ Γ† , d(Γ) O (Ff )(ξ) =
1 (2π)
d 2
Rd
e−iξ,x f (x)dx, ξ ∈ Rd .
Let us now define the periodic symbols and PDO’s associated with them. Let b = b(x, ξ), x, ξ ∈ Rd , be a Γ-periodic complex-valued function, i.e., b(x + γ, ξ) = b(x, ξ), ∀γ ∈ Γ. Let w : Rd → R be a locally bounded function such that w(ξ) ≥ 1, ∀ξ ∈ Rd and w(ξ + η) ≤ Cw(ξ) η κ , ∀ξ, η ∈ Rd ,
(2.1)
for some κ ≥ 0. We say that the symbol b belongs to the class Sα = Sα (w) = Sα (w, Γ), α ∈ R, if for any l ≥ 0 and any non-negative s ∈ Z the condition b
(α) l,s
= max max sup θ p w(ξ)−α+|s| |Dsξˆb(θ, ξ)| < ∞, p≤l |s|≤s ξ,θ
(2.2)
is fulfilled. If necessary, we reflect the dependence of this norm on the weight w and (α) write b l,s;w . Here we have used the standard notation t = 1 + |t|2 , ∀t ∈ Rd . Also, for any s ∈ Zd we denote |s| = s1 + s2 + · · · + sd . We mainly use two types of classes Sα : either with the weight w(ξ) = ξ , which satisfies (2.1) for κ = 1, or with a weight w(ξ) = L where a constant L is chosen in a convenient way. Note that Sα is an increasing function of α, i.e., Sα ⊂ Sβ for α < β. For later
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reference write the following convenient bounds that follow from Definition (2.2) and property (2.1): |Dsξ ˆb(θ, ξ)| ≤ b |Dsξ ˆb(θ, ξ
+ η) −
Dsξˆb(θ, ξ)|
≤C b
(α) −l α−s , l,s θ w(ξ)
(α) −l α−s−1
η κ|α−s−1| |η|, l,s+1 θ w(ξ)
(2.3) s = |s|, (2.4)
with a constant C depending only on α, s. We introduce a separate notation for (α) the set Pα = Pα (w, Γ) ⊃ Sα of symbols b such that b l,0 < ∞ for all l ≥ 0. Periodic functions V ∈ C∞ (Rd ) can be also viewed as symbols from P0 . For such (0) (0) functions V l,s = V l,0 for any s. Now we define the PDO Op(b) in the usual way: 1 b(x, ξ)eiξ,x (Fu)(ξ)dξ, Op(b)u(x) = d (2π) 2 the integral being taken over Rd . Under the condition b ∈ Pα the integral in the r.h.s. is clearly finite for any u from the class B(Rd ) of functions such that their Fourier transforms decay faster than any power of ξ, that is B(Rd ) = {u : sup ξ l |(Fu)(ξ)| < ∞, ∀l > 0}. ξ
(2.5)
In particular, Op(b) is well defined for u in the Schwarz class S(Rd ). Moreover, the condition b ∈ P0 guarantees the boundedness of Op(b) in L2 (Rd ), see Proposition 3.1. Unless otherwise stated, from now on S(Rd ) is taken as a natural domain for all PDO’s at hand, although sometimes we need to consider Op(b) on functions from the bigger class B(Rd ) as well. Observe that the operator Op(b) is symmetric if its symbol satisfies the condition ˆb(θ, ξ) = ˆb(−θ, ξ + θ).
(2.6)
We shall call such symbols symmetric. Note that Sα (L) = Sβ (L) for any α, β ∈ R and b
(α) l,s
= Lβ−α b
(β) l,s .
(2.7)
In fact, the introduction of different notation for the same class is done here to reflect the dependence on the parameter L. Throughout the entire paper we adopt the following convention. An estimate (or an assertion) is said to be uniform in a symbol b ∈ Sα (resp. Pα ) if the constants in the estimate (or assertion) at hand depend only on the constants Cl,s (α) (resp. Cl,0 ) in the bounds b l,s ≤ Cl,s . As was indicated in the introduction, our ultimate goal is to study the density of states of the Schr¨ odinger operator H = H0 + V in H = L2 (Rd ) with H0 = −∆
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and a smooth real-valued periodic potential V . However, some general definitions are more natural to give for more general operators. For these purposes it is not even necessary to assume that H0 = −∆, but it would be sufficient to suppose that H0 = Op(h0 ) with h0 (ξ) = |Fξ|m , m > 0, where F is a non-degenerate d × dmatrix. Also, the perturbation is allowed to be an arbitrary PDO of order lower than H0 . Another reason of considering more general operators is methodological: a number of intermediate results requires the use of such PDO’s. Precisely, we consider the operator H = Op(h), h(x, ξ) = h0 (ξ) + b(x, ξ), (2.8) h0 (ξ) = |Fξ|m , b ∈ Pα ( ξ ), α < m, with a symmetric symbol b. The operator Op(b) is H0 -bounded with an arbitrarily small relative bound. Thus H is self-adjoint on the domain D(H) = D(H0 ) = Hm (Rd ). Sometimes we call symbols (PDO’s) of this type elliptic symbols (PDO’s) of order m. In this paper we do not need to consider more general elliptic symbols. Due to the Γ-periodicity of the symbol b, the operator H commutes with the shifts along the lattice vectors, i.e., HTγ = Tγ H, γ ∈ Γ. with (Tγ u)(x) = u(x + γ). This allows us to use the Floquet decomposition.
2.2
Floquet decomposition
We identify the space H = L2 (Rd ) with the direct integral Hdk, H = L2 (O). G= O†
This identification is implemented by the Gelfand transform 1 e−ik,x e−ik,γ u(x + γ), k ∈ O† , (U u)(x, k) = d(Γ† ) γ∈Γ
(2.9)
which is initially defined on u ∈ S(Rd ) and extends by continuity to a unitary mapping from H onto G. In terms of the Fourier transform the Gelfand transform is defined as follows: (U u)(θ, k) = (Fu)(θ + k), θ ∈ Γ† . The unitary operator U reduces Tγ to the diagonal form: (U Tγ U −1 f )( · , k) = eik,γ f ( · , k), ∀γ ∈ Γ. Let us consider a self-adjoint operator A in H which commutes with Tγ for all γ ∈ Γ, i.e., ATγ = Tγ A. We call such operators Γ-periodic or simply periodic.
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Then A is partially diagonalized by U (see [17]), that is, there exists a measurable family of self-adjoint operators (fibres) A(k) acting in H, such that ∗ U AU = A(k)dk. (2.10) O†
It is easy to show that any periodic symmetric operator T , which is A-bounded with relative bound < 1, can be also decomposed into a measurable set of fibers T (k) in the sense that (U T f )( · , k) = T (k)(U f )( · , k), a.e. k ∈ O† , for all f ∈ D(A). Moreover, the fibers T (k) are A(k)-bounded with the bound , and if T is symmetric, then the operator A(k) + T (k) is self-adjoint on D(A(k)). Suppose that the operator A (and hence A(k)) is bounded from below and that the spectrum of each A(k) is discrete. Denote by λj A(k) , j = 1, 2, . . . , the eigenvalues of A(k) labelled in the ascending order. Using the min-max principle one easily sees that each λj (A( · )) is a measurable function of k. Suppose also that the counting function N λ, A(k) = #{j : λj A(k) ≤ λ}, λ ∈ R, is bounded as a function of k ∈ O† . Then we define the integrated density of states by the formula 1 N λ, A(k) dk. (2.11) D(λ) = D(λ; A) = d (2π) O† This definition makes sense for the operator A + T as well, since N λ, A( · ) + T ( · ) ∈ L∞ (O† ). Sometimes we need to reflect the dependence of the counting function and density of states on the lattice. In this case we use the notation NΓ λ, A(k) , DΓ (λ; A). Let us indicate some elementary general properties of the density of states, following directly from Definition (2.11). Proposition 2.1 Let A, A1 be self-adjoint Γ-periodic operators as defined above. (i) The density of states is monotone in A, that is, if A ≤ A1 , then D(λ; A) ≥ D(λ; A1 ). (ii) The density of states is a unitary invariant. Precisely, for any unitary Γperiodic operator W one has D(λ; W ∗ AW ) = D(λ; A). † Proof. The inequality in (i) follows from the inequality A(k) ≤ A1 (k), a.a.∗ k ∈ O . The unitary invariance follows from the identity N (λ, A(k)) = N λ, W (k)A(k) W (k) , where W (k) are the fibres of the operator W in the decomposition (2.10).
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d If A = Op(a) with a real-valued symbol a ∈ L∞ loc (R ) depending only on ξ, d d then A(k) is a self-adjoint PDO on the torus TΓ = R /Γ defined as follows:
† 1 A(k)u(x) = eiγ ,x a(γ † + k)ˆ u(γ † ). d(Γ) γ † ∈Γ† If a(ξ) → ∞ as |ξ| → ∞, then the spectrum of each A(k) is purely discrete with eigenvalues given by λ(m) (k) = a(m + k), m ∈ Γ† . Consequently, the number of eigenvalues below each λ ∈ R is essentially bounded from above uniformly in k ∈ O† . If T is a periodic symmetric operator which is A-bounded with a bound < 1, then the spectrum of A(k) + T (k) is also purely discrete and the counting function is also bounded uniformly in k. The above applies to the elliptic operator H defined in (2.8), and thus the quantity D(λ; H) is well defined. In fact, if necessary, one can obtain more information on the operators H0 (k) and H(k). Applying the Gelfand transform (2.9) to Op(b), one finds that, similarly to A considered above, the operator H(k) is a PDO on the torus TdΓ of the form † 1 H(k)u(x) = eiγ ,x h(x, γ † + k)ˆ u(γ † ). d(Γ) γ † ∈Γ† Moreover, if the symbol b(x, ξ) is smooth in ξ, then the family H(k) is smooth in k. Note however that for our purposes we need neither this explicit formula, nor the smoothness property. Recall that for periodic elliptic differential operators A there is another, equivalent, definition of the density of states given by (1.2). In the case of a pseudo-differential operator the formula (1.2) is not applicable, and we use (2.11) as definition. It is important however to convince oneself that the formula (2.11) preserves the invariance of D(λ) with respect to the choice of the lattice in the following sense: if the operator H happens to be periodic with respect to the latformula for tices Γ and Λ, then DΓ (λ) = DΛ (λ). To this end we write another D(λ) in terms of the spectral projection E(λ) = χ H; (−∞, λ] for H. Here and everywhere below we denote by χ( · ; C) the characteristic function of a measurable set C ⊂ Rm . Denote T (λ, C) = tr E(λ)χ(x; C) . It is easy to show that this trace is finite for any bounded set C ⊂ Rd . Theorem 2.2 Let the operator H be as defined in (2.8). Let D(λ; H) be its density of states defined by the formula (2.11). Then D(λ; H) = lim
R→∞
where KR = [0, R]d .
1 T (λ, KR ), Rd
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Proof. Let us make a preliminary calculation. Let B be a finite subset of Γ, and let C = ∪γ∈B (γ + O) be the set consisting of O and its translations by the vectors γ ∈ B. Using the invariance of the trace under unitary transformations, rewrite using the Gelfand transform (2.9): T (λ, C) = tr U E(λ)U ∗ U χ( · , C)U ∗ . A straightforward calculation shows that the operator U χU ∗ has the form ∗ M (x; k − k )f (x, k )dk , (U χU f )(x, k) = O†
M (x; t) =
1 −it,x+γ e . d(Γ† ) γ∈B
To find the trace T (λ, C) we use the formula tr K = (Kφn , φn ), n
which gives tr K for any trace-class operator K in a Hilbert space as the sum over an arbitrary orthonormal basis {φn }. Let us take the following orthonormal basis in G: ψγ,ω (x, k) = fγ (k)gω (x), 1 eix,ω , γ ∈ Γ, ω ∈ Γ† . fγ (k) = eik,γ , gω (x) = † d(Γ) d(Γ ) 1
Denote (U E(λ)U ∗ )(k) = E(λ, k). Then (E(λ, k)M ( · ; k − k )gω , gω )fγ (k)fγ (k )dkdk T (λ, C) = =
γ,ω
O†
ω
O†
O†
(E(λ, k)M ( · ; 0)gω , gω )dk.
Note that M (x; 0) =
1 1 1 {γ ∈ B} = |C| = |C|, d(Γ† ) d(Γ† ) d(Γ) (2π)d
where |C| denotes the Lebesgue measure of C. Consequently 1 1 |C| tr E(λ, k)dk = |C| N λ, H(k) dk = |C|D(λ). T (λ, C) = d d (2π) (2π) O† O† (2.12) Now let C> (resp. C< ) be the maximal (resp. minimal) set consisting of the domain O and its translations by the vectors γ ∈ Γ, such that C< ⊂ KR ⊂ C> . Then, clearly, T (λ, C< ) ≤ T (λ, KR ) ≤ T (λ, C> ), and both volumes |C< | and |C> | are Rd + o(Rd ), R → ∞. Applying (2.12) to C< and C> we obtain the required formula.
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Main result
We are now in a position to state the main result of the paper. Let V be a Γ-periodic function, and let H = −∆ + V . In the main Theorem below we assume that d = 2. The multidimensional case will be considered in a subsequent publication. Theorem 2.3 Let d = 2. Assume that V ∈ C∞ (Rd ) is Γ-periodic and that Vˆ (0) = 0. Then there is a number λ0 = λ0 (V ) > 0 such that D(λ; H) =
6 1 λ + Oδ (λ− 5 +δ ), ∀δ > 0, 4π
for all λ ≥ λ0 . The constant λ0 and the remainder estimate are uniform in V in (0) the sense that they depend only on the constants in the bounds V l,0 ≤ Cl . The condition Vˆ (0) = 0 does not restrict generality, since Vˆ (0) can be always incorporated into the spectral parameter λ. This simple argument allows one to deduce the formula (1.8) announced in the introduction, from Theorem 2.3.
2.4
“Partial” density of states
Now we define the density of states for PDO’s on their invariant subspaces. To this end we need to introduce a class of projection operators. Let C ⊂ Rd be a measurable set. Denote by χ(ξ) =χ(ξ; C) the characteristic function of C. Then the operator P(C) = χ(D; C) = Op χ( · ; C) is a projection in H on the subspace H(C) = P(C)H, and the operator P(k) = P(k; C) is a PDO on the torus with the symbol χ(γ † + k). Suppose that H(C) is an invariant subspace of the operator H defined in (2.8), that is PD(H) ⊂ D(H) and HPD(H) ⊂ H(C). Then the subspace H(k) = P(k)H, k ∈ O† , is invariant for H(k). Since the spectrum of H(k) is purely discrete, then so is the spectrum of the restriction of H(k) to the subspace H(k). The counting function of this restriction is denoted by N λ, H(k); C), and the density of states by D(λ; H; C). The same notation can be naturally introduced for any self-adjoint PDO A such that A(k) H(k) has a discrete spectrum. For instance, this is the case for any symbol a ∈ S0 ( ξ , Γ), such that for some bounded C the subspace H(C) is invariant for A. Note that the condition that H(C) is invariant for a bounded C automatically implies that the symbol h is a trigonometric polynomial in x, and that the operator H(k) H(k) is finite-dimensional. If C consists of two disjoint components, i.e., C = C1 ∪ C2 with C1 ∩ C2 = ∅, then obviously, D(λ; H; C) = D(λ; H; C1 ) + D(λ; H; C2 ). Sometimes we indicate which lattice of periodicity is used to compute the partial density of states and write DΓ (λ; H; C). On the other hand, similarly to the “full” density of states (see Theorem 2.2), one can easily show that D(λ; H; C) does not depend on the choice of the lattice of periodicity for the symbol h(x, ξ). The following three statements provide three important examples involving the density of states, in which the answer can be computed either completely or
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partially in terms of the symbol. The first statement provides an explicit formula for D(λ) in the case of a symbol with constant coefficients. Before proceeding we d introduce a convenient notation. Let a ∈ L∞ loc (R ) be a real-valued function. Denote E(λ; a) = {ξ ∈ Rd : a(ξ) ≤ λ}, ∀λ ∈ R.
(2.13)
d Proposition 2.4 Let A = Op(a) with a real-valued symbol a ∈ L∞ loc (R ) such that d a(ξ) → ∞ as |ξ| → ∞. Let C ⊂ R . Then
D(λ; A; C) =
1 vol C ∩ E(λ; a) . d (2π)
Proof. Denote
ϑ(t) =
0, t < 0; 1, t ≥ 0.
Observe that the eigenvalues of A(k)P(k) equal a(µ + k), µ + k ∈ C, µ ∈ Γ† , so that ϑ λ − a(µ + k) dk = dξ, (2π)d D(λ) = µ+k∈C∩Γ†
O†
a(ξ)≤λ, ξ∈C
as required. The next lemma deals with the integral of the density of states.
Lemma 2.5 Let C ⊂ Rd be a bounded set. Suppose that h(x, ξ) is a symbol of the form (2.8) such that the subspace H(C) is invariant for H. If λ ≥ P(C)H, then (2π)d
λ
−∞
D(µ; H; C)dµ = λ|C| −
1 d(Γ)
C
OΓ
h(x, ξ)dxdξ.
Proof. For any self-adjoint operator A in a finite-dimensional Hilbert space E one can write λ N (µ; A)dµ = λ dim E − tr A, ∀λ ≥ A. −∞
Therefore for H(k) = P(D + k, C)H, we have
λ
−∞
N µ; H(k); C dµ = λ#{γ † ∈ Γ† : γ † + k ∈ C} − tr(H(k) H(k)),
if λ ≥ P(C)B. Integrating the first term in k, we obtain λ|C|. To find the second term write the matrix elements of the operator H(k) on the subspace H(k): † † 1 e−ix,ω −γ h(x, γ † + k)dx, ω † , γ † ∈ C − k, Hω † ,γ † = d(Γ) OΓ
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so that tr H(k) H(k) =
Hγ † ,γ † =
γ † :γ † +k∈C
γ † :γ † +k∈C
1 d(Γ)
OΓ
h(x, γ † + k)dx,
After integrating it in k, we obtain the expected integral, thereby completing the proof. In the next example we consider a PDO which admits a partial separation of variables. To illustrate we use the second order elliptic periodic operators, although the argument below can be easily extended to more general operators. Let n, l be two natural numbers such that n + l = d. Let x = (y, t) and ξ = (η, ω) with y, η ∈ Rn and t, ω ∈ Rl . Let b = b(y, η) be a symmetric elliptic symbol of second order on Rn × Rn and periodic in y w.r.t. the lattice Λ ⊂ Rn . Let a(ω) = |Rω|2 , with a non-degenerate matrix R. Then the operator H = I ⊗ Op(a) + Op(b) ⊗ I is self-adjoint on H2 (Rd ). Lemma 2.6 Let the operator H be as above and let C ⊂ Rl , D ⊂ Rn be subsets of Rl and Rn . Then 1 D(λ, H; D × C) = D(λ − a(k), B; D)dk. (2π)l C Proof. The symbol of H is periodic w.r.t. the lattice Λ × (2πZ)l with the fundamental domain M = O × [0, 2π)l . The Floquet representative of the operator H is the operator H(K) = I ⊗ A(k) + B(µ) ⊗ I, K = (µ, k), k ∈ [0, 1)l , µ ∈ O† . Since the symbol a does not depend on x, after separating variables we have N (λ, H(µ, k); D × C) = N (λ, a(m + k) + B(µ); D), ∀µ ∈ O† , m∈Zl , m+k∈C
and thus D(λ, H; D × C) =
1 (2π)l+n
C
which leads to the required formula.
O†
N (λ, a(k) + B(µ); D)dµdk,
3 Properties of periodic PDO’s In this section we collect various properties of periodic PDO’s to be used in what follows.
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3.1
Ann. Henri Poincar´e
Some basic results on the calculus of periodic PDO’s
We begin by listing without proofs the results established in [25]. Recall that S(Rd ) is taken as a natural domain of Op(b). Unless otherwise stated, all the symbols are supposed to belong to the class Sα = Sα (w, Γ) with an arbitrary function w satisfying (2.1) and a lattice Γ. The function w and the lattice Γ are usually omitted from the notation. Proposition 3.1 Assume that b ∈ P0 . Then B = Op(b) is bounded in H and (0) B ≤ Cl b l,0 , ∀l > d, with a constant C = Cl independent of b. Remark 3.2 The above proposition automatically implies that PDO’s with symbols (α) b ∈ Sα (L, Γ) are bounded for any α ∈ R and Op(b) ≤ Cl Lα b l,0 . Since Op(b)u ∈ S(Rd ) for any b ∈ Sα and u ∈ S(Rd ), the product Op(b)Op(g), b ∈ Sα ,g ∈ Sβ , is well defined on S(Rd ). A straightforward calculation leads to the following formula for the symbol b ◦ g of the product Op(b) Op(g): (b ◦ g)(x, ξ) =
1 ˆ b(θ, ξ + φ)ˆ g (φ, ξ)ei(θ+φ)x , d(Γ) θ,φ
and hence 1 ˆb(θ, ξ + φ)ˆ (b ◦ g)(χ, ξ) = g (φ, ξ), χ ∈ Γ† , ξ ∈ Rd . d(Γ) θ+φ=χ
(3.1)
Here and below θ, φ ∈ Γ† . Proposition 3.3 Let b ∈ Sα , g ∈ Sβ . Then b ◦ g ∈ Sα+β and b◦g
(α+β) p,s
≤ Cl,s b
(α) l,s
g
(β) l,s ,
for some l = l(p). We are also interested in the estimates for symbols of commutators. For PDO’s A, Ψl , l = 1, 2, . . . , N , denote ad(A; Ψ1 , Ψ2 , . . . , ΨN ) = i ad(A; Ψ1 , Ψ2 , . . . , ΨN −1 ), ΨN , ad(A; Ψ) = i[A, Ψ], adN (A; Ψ) = ad(A; Ψ, Ψ, . . . , Ψ), ad0 (A; Ψ) = A. For the sake of convenience we use the notation ad(a;ψ1 ,ψ2 ,...,ψN ) and adN (a,ψ) for the symbols of multiple commutators. It follows from (3.1) that the Fourier coefficients of the symbol ad(b, g) are given by g)(χ, ξ) = i ˆb(θ, ξ + φ)ˆ g (φ, ξ) − ˆb(θ, ξ)ˆ g (φ, ξ + θ) , ad(b, d(Γ) θ+φ=χ χ ∈ Γ† , ξ ∈ Rd . (3.2)
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Proposition 3.4 Let b ∈ Sα and gj ∈ Sβj , j = 1, 2, . . . , N . Then ad(b; g1 , . . . , gN ) ∈ N Sγ with γ = α + j=1 (βj − 1), and ad(b; g1 , . . . , gN )
(γ) l,s
≤ Cl,s b
(α) p,s+N
N
gj
(βj ) p,s+N −j+1 ,
(3.3)
j=1
with some p = p(l, N, s, α, βj ), and a constant Cl,s independent of b, gj .
3.2
Asymptotics in the case d = 1
An important part is played by the result obtained in [25] for a PDO acting in L2 (R). We state this result in the form convenient for our purposes. Let b ∈ S0 ( ξ , Λ) and A = Op(a0 ) + Op(b) with a0 (ξ) = gξ 2 with a constant g > 0. We assume that Λ is a one-dimensional lattice with period τ > 0, so that d(Λ) = τ . Note that here and everywhere below, in the case d = 1 we use the notation x and ξ instead of the boldface letters x, ξ. Proposition 3.5 Let d = 1. Suppose that b ∈ S0 ( ξ , Λ) is a τ -periodic symmetric symbol. Then the density of states D(λ; A) is given by the formula 1 |λ| A(λ, g) + O( λ −3/2 ), ∀λ ∈ R, 2πD(λ; A) = 2 − g
λ g τ 1 b(x, |λ|g −1 ) + b(x, − |λ|g −1 ) dx. A(λ, g) = 2τ 0 This formula is uniform in the symbol b and in the parameters g and τ satisfying the bounds c ≤ g ≤ C, c ≤ τ ≤ C. Remark that the above formula has an asymptotic meaning only for large λ, and for bounded λ we can only claim that the density D(λ; A) is bounded from above uniformly in b. However it is useful to express these two facts in one formula which is valid for all λ ∈ R. In the study of the density of states for the multidimensional case we shall encounter integrals involving densities for lower-dimensional operators. In particular, there is a need to calculate integrals of the density of states for the one-dimensional case, of the type µ p D(t; A)(λ − t) 2 dt, µ < λ, p ∈ R. (3.4) Dp (µ, λ; A) = −∞
In the next lemma we apply Lemma 2.5 and Proposition 3.5 to compare the above integral for the operator A = A0 + Op(b) with that for the “unperturbed” operator Ao = A0 + Op(bo ), where bo = bo (ξ) is the mean value of b(x, ξ): 1 τ bo (ξ) = b(x, ξ)dx. (3.5) τ 0
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Lemma 3.6 Let = [−Lg −1/2 , Lg −1/2 ], L > 0. Suppose that b ∈ S0 ( ξ , τ Z) is a τ -periodic symmetric symbol such that Op(bo ) ≤ L2 , Op(b − bo ) ≤ L2 .
(3.6)
Suppose also that the subspace H() is reducing for the operator Op(b) and that P(R \ ) Op(b) = Op(bo ). Then for λ/2 ≥ µ ≥ 3L2 and p ∈ R one has |Dp (µ, λ; A) − Dp (µ, λ; Ao )| ≤ Cλ
p−2 2
L.
(3.7)
The constant C = C(p) in (3.7) is finite for all p ∈ R. In particular, D0 (µ, λ; A) = D0 (µ, λ; Ao ). The constant C(p) is independent of µ, λ and uniform in the symbol b and in the parameters g and τ satisfying the conditions c ≤ τ ≤ C, c ≤ g ≤ C. Proof. Compare the densities for A and Ao : Dp (µ, λ; A) = Dp (µ, λ; Ao ) µ p p + D(t; A) − D(t; Ao ) (λ − t) 2 − λ 2 dt −∞ µ p + λ2 D(t; A) − D(t; Ao ) dt.
(3.8)
−∞
Let us consider the second integral first. Since P(R \ ) Op(b) = Op(bo ), we have D(t; A) = D(t; A; ) + D(t; Ao ; R \ ), and hence D(t; A) − D(t; Ao ) = D(t; A; ) − D(t; Ao ; ). To find the integrals of the terms in the r.h.s. use Lemma 2.5. In view of (3.6), AP() ≤ L2 + Op(b) ≤ 3L2 . By the conditions of the lemma this does not exceed µ, and therefore, by Lemma 2.5 and (3.5), τ µ 1 −1/2 2π D(t; A; )dt = 2µLg − (gξ 2 + b(x, ξ))dxdξ τ 0 −∞ µ D(t; Ao ; )dt. = 2π −∞
Hence the second integral in (3.8) vanishes. To handle the first integral in (3.8) we note that in view of Proposition 3.5 |D(t; A) − D(t; Ao )| ≤ C|t|−3/2 ,
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for all t = 0 with a constant C uniform in the symbol b. Notice also that in view of the estimates (3.6) and the property that P(R \ ) Op(b) = Op(bo ), we have D(t; A) = D(t; Ao ) for all t ≥ 3L2 , and thus the limits of integration can be replaced by −3L2 and 3L2 . Now, using the straightforward estimate (λ − t) p2 − λ p2 ≤ Cλ p2 −1 |t|, |t| ≤ µ ≤ λ/2, with a constant C = C(p), we conclude that 3L2 p p o 2 − λ 2 dt (D(t; A) − D(t; A )) (λ − t) −3L2 3L2 p−2 ≤ Cλ 2 |t|−1/2 dt −3L2
˜ ≤ Cλ
p−2 2
L.
The last inequality and the formula (3.8) lead to (3.7).
3.3
Linear change of variables
It is an elementary exercise to describe how the symbols of PDO’s and their densities of state transform under a linear change of variables. Since in what follows we heavily use various changes of variables, below we state these elementary formulae in the form of Lemmas. Let a ∈ Pα (w, Γ) with a function w satisfying (2.1). Let M : Rd → Rd be a non-degenerate linear map. Note first of all that the lattice Γ transforms into an˜ = MΓ. Also, Γ ˜† = (MT )−1 Γ† . It is straightforward other lattice which we denote Γ to see that the sets O˜Γ = MOΓ , O†˜Γ = (MT )−1 O†Γ ˜ and Γ ˜ † respectively, and that are fundamental sets of the lattices Γ ˜ = det M d(Γ). d(Γ) Define the unitary operator W = WM : H → H for any u ∈ H as follows: √ (W u)(t) = det M u(Mt).
(3.9)
Lemma 3.7 Let M be a non-degenerate linear map from Rd to Rd , and let a ∈ Pα (w, Γ). Then the symbol b(x, ξ) of the operator B = W ∗ Op(a)W is given by b(x, ξ) = a(M−1 x, MT ξ).
(3.10)
˜ ˜ = MΓ, the Fourier transform ˆb(θ, ξ) of the symbol This symbol is Γ-periodic with Γ b is given by √ ˆb(θ, ξ) = det M a ˜† , ˆ(MT θ, MT ξ), θ ∈ Γ (3.11) T ˜ ˜ ˜ Γ), w(ξ) ˜ = w(M ξ). Moreover, if a ∈ Sα (w, Γ), then b ∈ Sα (w, ˜ Γ). and b ∈ Pα (w,
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Proof. The formula (3.10) for b follows by a direct elementary calculation. By ˜ To prove (3.11) inspection the symbol b is periodic with respect to the lattice Γ. write, using (3.10): ˆb(θ, ξ) = 1 ˜ d(Γ) det M = ˜ d(Γ)
O˜Γ
OΓ
a(M−1 x, MT ξ)e−iθ,x dx a(x, MT ξ)e−iM
T
θ,x
√ det M a ˆ(MT θ, MT ξ).
dx =
This proves (3.11). ˜ if a ∈ Pα (w, Γ), or, b ∈ Sα (w, ˜ ˜ Γ) ˜ Γ) Now it is immediate to see that b ∈ Pα (w, if a ∈ Sα (w, Γ). Let us find out how the density of states changes under the change of variables. Lemma 3.8 Let M be a non-degenerate linear map from Rd to Rd , and let a be a Γ-periodic elliptic symbol such that H(C), C ⊂ Rd is an invariant subspace for A = Op(a). Then the subspace H((MT )−1 C) is invariant for B = W ∗ AW, W = WM and DΓ (λ; A; C) = det M D˜Γ (λ; W ∗ AW ; (MT )−1 C). In particular, if C = Rd , then DΓ (λ; A) = det M D˜Γ (λ; W ∗ AW ). Proof. If H(C) is invariant for A, then the invariance of H(C ), C = (MT )−1 C, for B follows from the formula P(C ) = W ∗ P(C)W . The representatives in the Floquet decomposition of A and B = W ∗ AW ˜ k ˜ ∈ O† acting in L2 (OΓ ) and L2 (O˜ ) are the operators A(k), k ∈ O†Γ and B(k), ˜ Γ Γ respectively, with the symbols a(x, γ † + k), γ † ∈ Γ† , and ˜ = a(M−1 x, γ † + MT k), ˜† . ˜ γ ˜ † + k) ˜ † = (MT )−1 γ † ∈ Γ b(x, γ ˜ = W ∗ A(MT k)W ˜ As in Lemma 3.7 one checks directly that B(k) , and therefore
(2π) D˜Γ (λ; B; C ) = d
O˜† Γ
˜= ˜ C dk N λ, B(k);
1 = det M
O˜† Γ
˜ C dk ˜ N λ, A(MT k);
(2π)d DΓ (λ; A; C). N λ, A(k); C dk = det M O†Γ
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A special orthogonal transformation
A special role in what follows will be played by an orthogonal change of variables associated with a vector ν ∈ Rd . From now on we use the notation n(ν) = ν|ν|−1 , 0 = ν ∈ Rd . Recall that ej , j = 1, 2, . . . , d denote the vectors of the standard orthonormal basis in Rd . Let M = M(ν) be an orthogonal transformation M : Rd → Rd such that e1 = Mn(ν). Clearly, M(tν) = M(ν) for any real t > 0. Let us find out how this transformation acts on certain domains in Rd . Let L ≥ 0, s ≥ 0 be some numbers. Now define the domains Λν = {ξ ∈ Rd : | ξ, ν | ≤ L|ν|} = {ξ ∈ Rd : | ξ, n(ν) | ≤ L}. 2
2
(3.12) 2
Ων (s) = {ξ ∈ R : |ξ| − | ξ, n(ν) | ≥ s }. d
(3.13)
The number L will be kept the same throughout the paper and thus it is not reflected in the notation. The geometrical meaning of the sets Λν and Ων is simple: in particular, Ων is the set of all vectors ξ such that the distance from ξ to the one-dimensional subspace spanned by ν is greater than s. Clearly, for any t = 0 one has Λtν = Λν and Ωtν = Ων . For the case d = 2 the definition of Ων can be simplified. Namely, let 0 1 J= . (3.14) −1 0 Then for any t ∈ R2 the vector t⊥ = Jt is orthogonal to t. Now Ων can be rewritten as follows: Ων (s) = {ξ ∈ R2 : | n⊥ (ν), ξ | ≥ s}.
(3.15)
The next lemma describes how the sets Λν and Ων (s) transform under M(ν): Lemma 3.9 Let M = M(ν) be the orthogonal transformation defined above. Then MΛν = Λe1 = {ξ1 ∈ R : |ξ1 | ≤ L} × Rd−1 , ˆ ≥ s}. MΩν (s) = Ωe (s) = R × {ξˆ ∈ Rd−1 : |ξ| 1
Proof. As MC = {ξ ∈ Rd : MT ξ ∈ C} for any set C ⊂ Rd , we have MΛν = {ξ ∈ Rd : | MT ξ, n(ν) | ≤ L} = {ξ ∈ Rd : | MT ξ, MT e1 | ≤ L} = Λe1 . Here we have used the property e1 = Mn(ν). Similarly for Ων (s).
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4 Further properties of periodic PDO’s. Partition of symbols In this section we describe two procedures of partitioning the symbols, that will play a crucial role in the study of spectral properties of PDO’s. Roughly speaking, the aim of these partitions is to transform the operators to a form when separation of variables becomes possible.
4.1
Partition I
The first partition is designed to split the symbol into components supported on different parts of the dual space. The general definitions below will be given for symbols b ∈ Pα (w, Γ) with an arbitrary weight w, but later we shall make more restrictive assumptions. Let Υ ∈ C∞ 0 (R) be a non-negative function such that
0 ≤ Υ ≤ 1, Υ(t) =
1, |t| ≤ 1/4; 0, |t| ≥ 1/2.
(4.1)
Assume also for convenience that Υ is even, i.e., Υ(t) = Υ(−t). For a number L ≥ 1 define
θ, ξ + θ/2 ζθ (ξ; L) = Υ , |θ|L (4.2) ϕθ (ξ; L) = 1 − ζθ (ξ; L). We point out that ϕθ (ξ; L) = ϕ−θ (ξ + θ; L), ζθ (ξ; L) = ζ−θ (ξ + θ; L),
(4.3)
since the function Υ is even. Note that |Ds ϕθ (ξ; L)| + |Ds ζθ (ξ; L)| ≤ Cs L−s , s = |s|.
(4.4)
This inequality shows that the functions ζθ and ϕθ , viewed as symbols, belong to the class S0 (L). Using ϕθ , ζθ , we shall introduce the following linear operations on PDO’s. Fix a positive parameter r and let Θr = Θr (Γ) = {θ ∈ Γ† : 0 < |θ| ≤ r}, Θ0r = Θr ∪ {0}, Ξr = Ξr (Γ) = {θ ∈ Γ† : |θ| > r} = Γ† \ Θ0r (Γ).
(4.5)
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Given a symbol b ∈ Sα (w), define four new symbols bo , b , b , b↑ as follows: 1 ˆb(θ, ξ)eiθx , b↑ (x, ξ) = d(Γ) θ∈Ξr 1 ˆb(θ, ξ)ϕθ (ξ; L)eiθx , b (x, ξ) = d(Γ) θ∈Θr 1 ˆb(θ, ξ)ζθ (ξ; L)eiθx , b (x, ξ) = d(Γ) θ∈Θr 1 ˆ b(0, ξ). bo (x, ξ) = bo (ξ) = d(Γ) By definition of Υ
(4.6) (4.7) (4.8) (4.9)
b = b↑ + bo + b + b .
The symbol b↑ contains only the Fourier coefficients ˆb(θ, ξ) with large θ’s, and later it will be shown not to contribute to the answer. The remaining symbols are trigonometric polynomials in x. The Fourier coefficient ˆb (θ, · ) is supported near the hyperplane θ, ξ + θ/2 = 0, whereas ˆb (θ, · ) lives away from this hyperplane. It is easy to see that for any symbol b ∈ Pα the introduced symbols also belong to the same class. The corresponding operators are denoted by B ↑ = Op(b↑ ), B = Op(b ), B = Op(b ), B o = Op(bo ). We also denote B ,↑ = B + B ↑ . Assume now that b ∈ Sα (w, Γ) with a constant weight w = L ∈ [1, L]. Then the operations introduced above preserve the properties of the symbol b, that is for any b ∈ Sα (L , Γ) the symbols b , b , b↑ , bo belong to the same class and for all l, s and p < l − d
(α) (α) (α) (α) b l,s + b l,s + bo l,s ≤ C b l,s , (4.10) (α) (α) b↑ p,s ≤ Crp−l+d b l,s . The constant C in the above estimates depends on l, s and on the lattice Γ. The first estimate immediately follows from (4.4). The second bound is a consequence of (2.3) and (4.5). In the case d = 1 the operation “” possesses one more useful property: Lemma 4.1 Let d = 1 and let b ∈ Sα (L, Γ), α ≤ 0. Then for r ≤ L the symbol b belongs to Sα ( ξ , Γ) and b
(α) l,s;ξ
≤ Cl,s b
with a constant Cl,s depending only on l, s.
(α) l,s;L
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Proof. The proof is similar to the above proof of (4.10) for the classes with a constant weight. One uses the fact that |ξ| ≤ L/2 + r/2 ≤ L on the support of ζθ , and thus |∂ξsˆb(θ, ξ)| ≤ b
−l α−s l,s;L θ L
≤ Cl,s b
−l α−s , l,s;L θ ξ
for all ξ ∈ supp ζθ and θ ∈ Θr . It also follows from (4.4) that |∂ξs ϕθ (ξ; L)| + |∂ξs ζθ (ξ; L)| ≤ Cs ξ −s . The above estimates lead to the proclaimed estimate for the norm b
(α) l,s;ξ .
To check that the introduced operations preserve symmetry, calculate using (4.3) and remembering that θ and −θ do ( or do not ) belong to Θr simultaneously: ˆb (−θ, ξ + θ) = ˆb(−θ, ξ + θ)ζ−θ (ξ + θ; L) = ˆb(θ, ξ)ζθ (ξ; L) = ˆb (θ, ξ). Thus, by (2.6) the operator B is symmetric if so is B. Similarly for B and B ↑ . Let us find out how these symbols transform under an orthogonal change of variables M. Below we use the notation for the transformed objects, introduced in the beginning of Subsect. 3.3. Note first a direct consequence of the Definition (4.2) ζθ (MT ξ; L) = ζω (ξ; L), ˜† = MΓ† . (4.11) ω = Mθ ∈ Γ T ϕθ (M ξ; L) = ϕω (ξ; L), Lemma 4.2 Let M : Rd → Rd be an orthogonal transformation. Let a ∈ Pα (w) with some α ∈ R. Denote by any of the symbols ↑, , o or . Then ∗ ∗ WM AWM = WM A WM . ∗ AWM . We consider only the case = . By Definition (4.8) Proof. Let B = WM the symbol of the operator B coincides with
1 ˆb(ω, ξ)ζω (ξ; L)eiω,x . ˜ d(Γ) ω∈Θr (˜Γ) ˜ = Θr (Γ) and by (3.11) the Fourier transform of the On the other hand, MT Θr (Γ) ∗ symbol of the operator WM A WM is given by ˜† . a ˆ(MT ω, MT ξ)ζMT ω (MT ξ; L), ω ∈ Γ In view of (3.11) and (4.11) this coincides with ˆb(ω, ξ)ζω (ξ; L), as required.
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Partition II
Here we describe a way to split symbols into components associated with the socalled primitive vectors ν ∈ Γ† . For a symbol b ∈ Sα (w, Γ) introduce a family of associated symbols constructed as follows. For a vector 0 = ν ∈ Γ† define the subset (4.12) Γ†ν = {nν, 0 = n ∈ Z}. Let us introduce the symbol 1 ˆb(θ, ξ)eiθx = 1 ˆb(nν, ξ)einνx . bν (x, ξ) = d(Γ) d(Γ) † 0 =n∈Z θ∈Γν
Clearly, (bν )ν = bν . Notice that this symbol is symmetric if so is the initial symbol (α) (α) b(x, ξ), see (2.6). Besides, bν ∈ Sα (w, Γ) and bν l,s ≤ b l,s . Let C ⊂ Rd be a set, containing along with each point ξ ∈ C the straight line {ξ + tν, t ∈ R}. Then assuming that all the PDO’s at hand are defined on the set B(Rd ) (see (2.5)), it is straightforward to see H(C) is a reducing subspace of Op(bν ), and in particular, that P(C) Op(bν ) = Op(bν )P(C) (see Subsect. 2.4 for definition of P(C)). For example, the set Ων = Ων (s) defined in (3.13) possesses this property and therefore P(Ων ) Op(bν ) = Op(bν )P(Ων ). Below we decompose any symbol in the sum of symbols of the form bν . To explain how it is done we need to recall the definition of a primitive vector. Definition 4.3 A non-zero vector ν ∈ Λ is said to be a primitive vector of the lattice Λ ⊂ Rd if (i) The first non-zero coordinate of ν = (ν1 , ν2 , . . . , νd ) is positive; (ii) There are no vector η ∈ Λ distinct from ν and no integer n > 0 such that ν = nη. It follows from this definition that for each non-zero vector χ ∈ Λ there exist a uniquely defined integer n and a primitive vector ν ∈ Λ such that χ = nν. Also, every two primitive vectors are linearly independent. The set of all primitive vectors of the lattice Λ is denoted by P Λ. Now we can decompose any symbol b ∈ Sα (w, Γ) into a sum over primitive vectors ν ∈ P Γ† :
b(x, ξ) = bo (ξ) +
bν (x, ξ),
(4.13)
ν∈P Γ†
see (4.9) for definition of bo . From this and Definition (2.2) it follows that max
bo
(α) l,s ,
bη
(α) l,s
≤ b
(α) l,s
≤ max
bo
(α) l,s ,
sup ν∈P Γ†
bν
(α) l,s ,
∀η ∈ Γ† . (4.14)
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Denote
Ann. Henri Poincar´e
r = P Γ† ∩ Θ r . Θ
(4.15)
Let us now combine two types of partition introduced above. Observe first that by definitions (4.6)–(4.8) we always have (b )ν = (bν ) , =↑, , , so that one can use the notation bν without risk of confusion. Moreover, since b (x, ξ) is a trigonometric polynomial, one can rewrite b (x, ξ) = b ν (x, ξ). r ν∈Θ
Due to the presence of a cut-off in the definition, the operators b ν have an additional reducing subspace. Let Λν be the set defined in (3.12). Then the following lemma holds: Lemma 4.4 Let the operator B with the symbol b ∈ Pα (w) be defined on B(Rd ) (see (2.5)). Suppose that r ≤ L. Then BP(Λν ) = Bν P(Λν ) = Bν for any ν ∈ Θr . For a symmetric symbol b this lemma implies that the subspace H(Λν ) is reducing for the operator Bν and that Bν P(R2 \ Λν ) = 0. Proof. It suffices to check that the support of ζθ , θ = nν ∈ Θr is contained in the domain Λν . By Definition (4.1), under the condition r ≤ L we have for each ξ ∈ supp ζθ and n = 0: 1 1 | θ, ξ | ≤ | θ, ξ + θ/2 | + |θ|2 ≤ L|θ| + r|θ| ≤ L|θ|. 2 2
It remains to recall (3.12).
Suppose now that the symbol b is symmetric. As was already mentioned, the subspace H(Ων ) is then reducing for Bν . Together with Lemma 4.4 this implies that for the set ˆ ν (s) = Λν ∩ Ων (s) (4.16) Λ ˆ ν ) is reducing for B if r ≤ L. The properties of the operators the subspace H(Λ ν Bν on these reducing subspaces will be a key ingredient in our study of the density of states. In this study the set Λ = ∪ν∈Θr Λν plays the role of the resonant set described in the introduction. ˆν. First of all we need to establish some geometric properties of the sets Λ This will be done for the case d = 2 only.
4.3
Some geometric estimates for d = 2
Recall the notation n(θ) = θ|θ|−1 , and t⊥ = Jt, see (3.14) for definition of J. Lemma 4.5 Let d = 2. Suppose that θ, η ∈ Θr are linearly independent. Then | n(θ), n⊥ (η) | ≥ d(Γ† )r−2 .
(4.17)
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Proof. Let γ 1 , γ 2 be two basis vectors of the lattice Γ† , so that θ = n1 γ 1 + n2 γ 2 and η = m1 γ 1 + m2 γ 2 with some integers n1 , n2 , m1 , m2 . Consequently, ⊥ ⊥
θ, (η)⊥ = n1 m2 γ 1 , γ ⊥ 2 + n2 m1 γ 1 , γ 2 = (n1 m2 − n2 m1 ) γ 1 , γ 2 .
Since θ and η are linearly independent, the integer factor n1 m2 − n2 m1 never vanishes, and thus | θ, η ⊥ | ≥ | Jγ 1 , γ 2 | = d(Γ† ). Since |θ| ≤ r and |η| ≤ r, we obtain (4.17).
Lemma 4.6 Let d = 2. Suppose that ν, µ ∈ Θr are linearly independent. If d(Γ† )ρr−2 L−1 ≥ 4 and ρL−1 1 ≥ 4, then one has ˆ ν (ρ − L1 ). (4.18) | n(µ), ξ | ≥ 2−1 d(Γ† )r−2 ρ, ∀ξ ∈ Λ ˆ ν (ρ − L1 ), Λµ (L) ≥ 4−1 d(Γ† )r−2 ρ. ˆ ν (ρ − L1 ) ∩ Λµ (L) = ∅ and dist Λ Moreover, Λ Proof. Use the short-hand notation nµ = n(µ), nν = n(ν). Decompose the vector nµ into a sum as follows ⊥ nµ = nµ , nν nν + nµ , n⊥ ν nν
= anν + a⊥ n⊥ ν. Then
nµ , ξ = a⊥ n⊥ ν , ξ + a nν , ξ . ˆν = Λ ˆ ν (ρ − L1 ) then | nν , ξ | ≤ L and by (3.15), | n⊥ , ξ | ≥ ρ − L1 , so If ξ ∈ Λ ν that | nµ , ξ | ≥ |a⊥ |(ρ − L1 ) − L. By Lemma 4.5 |a⊥ | ≥ κr−2 with κ = d(Γ† ), and hence | nµ , ξ | ≥ κr−2 ρ − (κr−2 L1 + L) = κr−2 ρ 1 − L1 ρ−1 − κ −1 Lr2 ρ−1 . Since ρr−2 L−1 ≥ 4κ −1 and ρL1 ≥ 4, the r.h.s. is greater than 2−1 κρr−2 , which guarantees (4.18). The r.h.s. of (4.18) is also greater than L, which leads to the ˆ ν and t ∈ Λµ , we have ˆ ν ∩ Λµ = ∅. Also, for any ξ ∈ Λ identity Λ |ξ − t| ≥ | ξ, nµ | − | t, nµ | ≥ 2−1 κρr−2 − L. Again, under the condition imposed on ρ, r, L we have the required lower bound ˆ ν and Λµ . for the distance between Λ
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4.4
Ann. Henri Poincar´e
Separation of variables
Observe that the condition d = 2 is crucial for the previous two lemmas. It guarˆ ν and Λµ do not intersect for linearly independent ν and µ. antees that the sets Λ This fact allows one to ”separate variables” when studying the model operator A = Ao + B , Ao = Op(ao ), ao (ξ) = |ξ|2 + bo (ξ).
(4.19)
Lemma 4.7 Let d = 2. Suppose that b ∈ P0 (w) is a symmetric symbol. Let λ0 ≥ 0 be a fixed number. Suppose that 1 ≤ r ≤ L, ρL−1 1 ≥ 4, d(Γ† )ρr−2 L−1 ≥ 4, 2ρL1 ≥ λ0 + 3L2 + L21 , and that B o + B ≤ 2L2 . Then ˆ + ˆν) D(λ; A) = D λ; Ao ; R2 \ Λ D(λ; Ao + Bν ; Λ
(4.20)
r ν∈Θ
ˆν. ˆν = Λ ˆ ν (ρ − L1 ) and Λ ˆ = ∪ν Λ for all λ ≥ ρ2 − λ0 . Here Λ ˆ ν and Proof. Let Cν = Λν \ Λ ˆν. ˆ = ∪ν Λ C = ∪ν Cν , Λ = ∪ν Λν , Λ By Lemma 4.4 the subspace H(Λ) is invariant for the operator A, and AP(R2 \Λ) = ˆ and H(C) are also Ao P(R2 \ Λ). By virtue of Lemma 4.6 the subspaces H(Λ) invariant for A and thus ˆ . N (λ, A(k)) = N λ, Ao (k); R2 \ Λ + N λ, A(k); C + N λ, A(k); Λ (4.21) Furthermore, the third term in the last formula equals ˆ ν ), N λ, Ao (k) + Bν (k); Λ r ν∈Θ
ˆ ν are disjoint for distinct ν’s. Hence it remains to verify that since Λ N λ, A(k); C = N (λ, Ao (k); C . For any ξ ∈ Cν one has: |ξ|2 = | ξ, n(ν) |2 + | ξ, n⊥ (ν) |2 ≤ L2 + (ρ − L1 )2 = ρ2 − 2ρL1 + L21 + L2 . Since ρL−1 ≥ 4 and 2ρL1 ≥ λ0 + 3L2 + L21 , the r.h.s. is bounded from above 1 2 by ρ − λ0 − 2L2 ≤ λ − 2L2 . Consequently, Ao P(C) ≤ λ − 2L2 + B o . By AP(C) virtue of the condition B o + B ≤ 2L2 we have ≤ λ, which implies by an elementary perturbation argument that N λ, A(k); C = N λ, Ao (k); C , as required.
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5 Asymptotics in the invariant subspaces In this section we study the density of states for the model operator (4.19) introduced in the previous section. Our strategy is dictated by Lemma 4.7: since ˆ ν do not intersect for distinct ν ∈ Θ ˜ r , the investigation of each term in the sets Λ the r.h.s. of (4.20) is done independently. Furthermore, the symbol of the operator Ao + Bν depends only on the projection x, n(ν) (see (4.8)), so that in each ˆ ν ) the problem reduces to a one-dimensional one. subspace H(Λ We begin with studying the operator Ao + Bν . Emphasize that the analysis of this operator is carried out without any restrictions on the dimension d ≥ 2. On the contrary, in the closing subsection, dealing with the operator (4.19), we need to assume that d = 2 in order to use Lemma 4.7.
5.1
A new class of symbols
In order to describe the reduction to a one-dimensional problem we need to introduce a new class of symbols that encode the “one-dimensionality”. Let z = z(x, η), x ∈ Rd , η ∈ R be a function and L ≥ 1 be a constant. We say that this function belongs to the class Tα (L, Γ), α ∈ R, if it is Γ-periodic, C∞ -smooth in η and its Fourier coefficients satisfy the condition |∂ηs zˆ(θ, η)| ≤ Cl,s θ −l Lα−s , for all integer l ≥ 0 and s ≥ 0. Similarly to the class Sα (L, Γ) introduce the norm z
(α) l,s
= max max sup sup θ p L−α+r |∂ηr zˆ(θ, η)|. r≤s p≤l
θ
η
We are interested in the PDO’s with symbols having the following Fourier coefficients:
ˆb(θ, ξ) = zˆ θ, ξ, n(θ) , θ = 0, (5.1) 0, θ = 0. Clearly, b ∈ Sα (L, Γ) and −1 z Cl,s
(α) l,s
≤ b
(α) l,s
≤ Cl,s z
(α) l,s ,
(5.2)
with some constant Cl,s . We call the symbol z symmetric if zˆ(θ, η) = zˆ(−θ, −η − |θ|).
(5.3)
It is straightforward to check that under this condition the symbol (5.1) is symmetric in the sense of Definition (2.6).
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Reduction to the case d = 1
Assuming, that the symbol b is defined by (5.1) we study the density of states for an operator with the symbol a(x, ξ) = ao (ξ) + bν (x, ξ), ao (ξ) = |ξ|2 + f ξ, n(ν) ,
(5.4)
with some 0 = ν ∈ Γ† , and some real-valued uniformly bounded function f . We are interested in the “partial” density of states D(λ; A; C) with C = Ων ˆ ν . The first step is to perform a change of variables which reduces Ων or C = Λ ˆ ν to Ωe1 and Λ ˆ e1 . Let M = M(ν) be the orthogonal map from Subsect. 3.4, and Λ and let W = WM be the unitary operator defined in (3.9). Then by Lemma 3.7 the symbol of A˜ = W ∗ AW is given by a ˜(x, ξ) = |ξ|2 + f MT ξ, n(ν) + bν (M−1 x, MT ξ). Remembering that Mn(ν) = e1 , we have MT ξ, n(ν) = ξ1 . Using the last relation it is easy to find that bν (M−1 x, MT ξ) = ˜b(x1 , ξ1 ) with the symbol ˜b given by ˜b(x, η) = 1 zˆ lν, l|l|−1 η ei|ν|lx , x ∈ R, η ∈ R. (5.5) d(Γ) l =0 Consequently,
a ˜(x, ξ) = o a ˜ (ξ) =
a ˜o (ξ) + ˜b(x1 , ξ1 ), ˆ 2 + ξ 2 + f ξ1 , ξˆ = (ξ2 , ξ3 , . . . , ξd ). |ξ| 1
(5.6)
Note that the symbol ˜b is periodic in x with the period τ = τ (ν) = 2π
[|ν|] + 1 , |ν|
(5.7)
so that the whole symbol a ˜ is periodic w.r.t. the lattice Λ = (τ Z) × (2πZ)d−1 . Note that τ (ν) is bounded from above and below uniformly in ν ∈ P Γ† . By Lemma 3.8 we have ˜ = MΓ. ˜ MC), Γ DΓ (λ; A; C) = D˜Γ (λ; A; Since the density of states does not depend on the lattice (see Subsection 2.4), we ˜ with Λ defined above, and thus can replace Γ ˜ MC), Λ = (τ Z) × (2πZ)d−1 . DΓ (λ; A; C) = DΛ (λ; A;
(5.8)
In the next three lemmas we show that up to a controllable error the densities of states for the operator A = Op(a) and Ao = Op(ao ) coincide. We begin with a reduction to a one-dimensional operator T with the symbol t(x, η) = η 2 + f (η) + ˜b x, η , x ∈ R, η ∈ R. (5.9)
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The following lemma compares the densities of states for the operators A = Ao +Bν and Ao +Bν with those for the operators T and T o +T respectively, see subsection 4.1 for definitions. Lemma 5.1 Suppose that z ∈ Tα (L, Γ) is a symmetric symbol. Let b be a symbol defined as in (5.1), and let f be a real-valued uniformly bounded function on R. Let ˜b and t be the symbols defined in (5.5) and (5.9) respectively. Then (i) The symbol ˜b belongs to Sα (L, τ Z) with τ specified in (5.7), it is symmetric, (α) (α) and ˜b l,s ≤ Cl,s z l,s with a constant Cl,s depending only on the lattice Γ; (ii) For all λ ∈ R and s ≥ 0 one has D λ; A; Ων (s) = σDd−3 (λ − s2 , λ, T ), ωd−2 σ= , Dλ; Ao + B ; Ω (s) = σD (λ − s2 , λ; T o + T ), 2(2π)d−1 ν d−3 ν (5.10) where (5.11) ωp = (p + 1) wp+1 , p ≥ 1, is the surface area of a unit sphere in Rp+1 , ω0 = 2, and the quantity Dq is defined in (3.4). Proof. (i) The τ -periodicity of the symbol ˜b with the specified τ has already been observed. The estimate for the norm ˜b follows by inspection. The symmetry of ˜b follows from (5.3). (ii) To prove (5.10) we use Lemma 3.9: ˆ ≥ s}. MΩν (s) = Ωe1 (s) = R × {ξˆ ∈ Rd−1 : |ξ| Thus we can now use (5.8) and Lemma 2.6 to conclude that ˜ Ωe1 (s) (2π)d−1 DΓ λ, A; Ων (s) = (2π)d−1 DΛ λ, A; ˆ 2 ; T )dξˆ = ωd−2 = D(λ − |ξ| D(λ − ξ 2 ; T )ξ d−2 dξ ˆ |ξ|≥s
=
ωd−2 2
ξ≥s λ−s2
−∞
D(µ; T )(λ − µ)
d−3 2
dµ.
By Lemma 4.2 an analogous formula holds for operators Ao + Bν and T o + T . Definition (3.4) leads to the proclaimed formula (5.10). Before calculating the asymptotics of the r.h.s. of (5.10) we need to study the density of states for the operator Ao = Op(ao ) with the symbol ao defined in (5.4). We shall need the notation E(λ; · ) introduced in (2.13).
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Lemma 5.2 Let f be a uniformly bounded function. Suppose that s ≥ 0 and λ ≥ 0 and 0 ≤ λ0 ≤ λ/2 are numbers such that s2 ≤ λ − λ0 − L2 − sup |f (η)|,
(5.12)
η
L2 + sup |f (η)| ≤ η
λ − λ0 . 2
(5.13)
Then for any λ , λ ∈ [λ − λ0 , λ + λ0 ] D(λ , Ao ; Λ ˆ ν (s)) − D(λ , Ao ; Λ ˆ ν (s)) ≤ Cλ d−3 2 |λ − λ |L,
(5.14)
with a constant C independent of the numbers λ, λ , λ , L, s, vector ν and symbol f . Proof. Let M = M(ν) be the orthogonal map from Subsection 3.4. By Definition (4.16) and Lemma 3.9 ˆ ≥ s}. ˆ e1 (s) = {|ξ1 | ≤ L} × {|ξ| ˆ ν (s) = Λ MΛ According to (5.8), formula (5.6) and Proposition 2.4, under the condition (5.12) for any λ ∈ [λ − λ0 , λ + λ0 ] we have the following formula: ˆ e1 (s) ∩ E(λ ; a ˆ ν (s) = (2π)d D λ , A˜o ; Λ ˆ e1 (s) = vol Λ (2π)d D λ , Ao ; Λ ˜o ) L d−1 = wd−1 λ − η 2 − f (η) 2 dη − 2L wd−1 sd−1 . −L
Here wp is the volume of the unit ball in Rp . Under the conditions (5.12) and (5.13) the above formula yields (5.14). The next lemma yields an important intermediate result – it provides an asymptotic formula for the density of states of the operator Ao + Bν : Lemma 5.3 Let f be a uniformly bounded function, and let b be defined by (5.1) with a symmetric symbol z ∈ T0 (L, Γ). Suppose that r ≤ L and that for some λ0 ∈ [0, λ/2] Bν ≤ L2 , supη |f (η)| ≤ L2 , (5.15) λ + λ0 3L2 ≤ λ − λ0 − s2 , s2 ≥ . 2 Then for all λ , λ ∈ [λ − λ0 , λ + λ0 ] one has o D λ , A + Bν ; Λ ˆ ν (s) −
1 ˆ ν (s) ∩ E(λ ; ao ) ≤ Cλ d−5 2 L, vol Λ d (2π)
(5.16)
and ˆ ν (s) − D λ ; Ao + Bν ; Λ ˆ ν (s) | |D λ ; Ao + Bν ; Λ ≤ Cλ
d−3 2
L |λ − λ | + λ−1 . (5.17)
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The constants in bounds (5.16) and (5.17) depend on λ0 . They do not depend on λ, s, ν, L, f , and are uniform in the symbol z. Proof. We derive the required formulas from the relation (5.10) with the use of Lemma 3.6. Let us check that its conditions are satisfied. The symbol t = ˜b = ˜b (x, η) (see (5.9) for definition of t) has the form considered in Lemma 3.6 with g = 1. Moreover, since r < L, by Lemma 4.4 we have P() Op(˜b ) = Op(˜b ), where = [−L, L]. Furthermore, by Lemma 4.1 ˜b ∈ S0 ( η , τ Z), and in view of Lemma 5.1(i) we have ˜b (α) ≤ C ˜b (α) ≤ C˜ z (α) , l,s;L l,s l,s;η with some universal constants. And finally, the conditions (5.15) guarantee that Op(f ) ≤ L2 , T ≤ L2 and λ /2 ≥ λ − s2 ≥ 3L2 , for all λ ∈ [λ − λ0 , λ + λ0 ]. Now we can apply Lemma 3.6 with p = d − 3 to the r.h.s. of (5.10), which leads to Dd−3 (λ − s2 , λ ; T o + T ) − Dd−3 (λ − s2 , λ ; T o ) ≤ Cλ d−5 ˜ d−5 2 L ≤ Cλ 2 L, and hence
D(λ , Ao + B ; Ων (s)) − D(λ , Ao ; Ων (s)) ≤ C λ d−5 2 L, ν
(5.18)
ˆ ν (s) note for all λ ∈ [λ − λ0 , λ + λ0 ]. To establish a similar estimate for the set Λ that by Lemma 4.4 ˆ ν (s)) = D(λ ; Ao + B ; Ων (s) − D λ , Ao ; Ων (s) \ Λ ˆ ν (s) , D λ , Ao + Bν ; Λ ν and hence (5.18) yields ˆ ν (s)) − D(λ , Ao ; Λ ˆ ν (s)) ≤ C λ d−5 D(λ , Ao + Bν ; Λ 2 L,
(5.19)
for all λ ∈ [λ − λ0 , λ + λ0 ]. By virtue of Proposition 2.4 the second term in the l.h.s. coincides with 1 ˆ ν (s) ∩ E(λ ; ao ) , vol Ων (s) \ Λ (2π)d which implies (5.16). Note that the conditions (5.15) guarantee (5.12) and (5.13). Now to obtain (5.17) it suffices to use (5.19) for λ , λ , and (5.14).
5.3
Density of states for the model operator (4.19)
Our goal in this subsection is to establish a formula similar to (5.16) for the model operator (4.19). Now it is crucial to assume that d = 2. Let us first specify the symbols that we are working with. Let f (ν) = f (ν) (η), r be a collection of real-valued uniformly bounded functions. Also η ∈ R, ν ∈ Θ suppose that the quantity κν = sup |η|β |f (ν) (η)| |η|≥4L
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is finite for some β ≥ 0. This implies that |f (ν) (η)| ≤ κν |η|−β , ∀|η| ≥ 4L.
(5.20)
Instead of (5.4) assume that
ao (ξ) = |ξ|2 + f (ξ), f (ξ) =
f (ν) n(ν), ξ .
(5.21)
r ν∈Θ
All the subsequent results will be uniform in the function f in the sense that they depend only on the constant C in the bound supξ |f (ξ)| ≤ C. The perturbation symbol b is chosen in the same way as above, i.e., it is defined by the formula (5.1) for some symmetric z ∈ T0 (L, Γ). Our objective is to compare the density of states for the operator A = Ao + B with that of Ao . We are going to use the notation already exploited in the proof of Lemma 4.7: ˆ ˆ ˆ ˆ Λ = ∪ν∈Θ r Λν , Λν = Λν (ρ − L1 ), Λ = ∪ν∈Θ r Λν . For technical reasons we also need to include a bounded perturbation given by a self-adjoint PDO Q with a symbol q ∈ P0 (w) with an arbitrary weight w. Theorem 5.4 Let d = 2. Let the operator A be as described above with z ∈ T0 (L, Γ), and let q ∈ P0 (w) be a symmetric symbol for some weight w. Denote ˆ ν ), κ = δ = max Q ν P(Λ κν . r ν∈Θ
r ν∈Θ
For a fixed λ0 ≥ 0 denote λ1 = λ0 + δ + κ(4L)−β . Suppose that ρ2 ≥ 16λ1 and that 1 ≤ r ≤ L, ρ ≥ 4L1 , † −2 d(Γ )ρr ≥ 8L, 2ρL1 ≥ 2λ1 + 3L2 + L21 .
(5.22)
(5.23)
(5.24)
Then there exists a constant L0 = L0 (z, q, f ) such that under the condition L ≥ L0 , one has D(λ; Ao + B + Q ) −
(5.25)
1 vol E(λ; ao ) ≤ Cr2 ρ−1 L(κr2β ρ−β + δ) + Cr2 ρ−3 L, 2 (2π)
for all λ ∈ [ρ2 − λ0 , ρ2 + λ0 ]. The constants C, L0 are uniform in the symbols f, q and z, and C may depend on λ0 and λ1 .
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Proof. Since z ∈ T0 (L) and q ∈ P0 (w), the operators B, B , Q, Q are bounded by Proposition 3.1 and Lemma 5.1(i): Bν + B + B ≤ Cl z (0) l,0 , ∀ν ∈ Γ† , l > 2. Q + Q + Q ≤ C q (0) , l ν l,0 Choosing sufficiently large L0 one ensures that Op(f ) + B + Q ≤ 2L2 . Remembering also (5.23), (5.24), one guarantees that the conditions of Lemma 4.7 are fulfilled. Consequently, in view of (4.20) ˆ + ˆ ν ), (5.26) D(λ; Ao + B + Q ) = D(λ; Ao ; R2 \ Λ) D(λ; Ao + Bν + Q ν ; Λ r ν∈Θ
for all λ ≥ ρ2 − λ1 . By Proposition 2.4 the first term equals ˆ = D(λ; Ao ; R2 \ Λ)
1 ˆ ∩ E(λ; ao ) . vol R2 \ Λ 2 (2π)
(5.27)
r Let us consider each summand in the second term separately. Let us fix a ν ∈ Θ and define y o (ξ) = |ξ|2 + f (ν) n(ν), ξ , Y o = Op(y o ). Note that in view of (4.18) we have | n(µ), ξ | ≥ 2−1 d(Γ† )r−2 ρ ≥ 4L for all ˆ ν for any two ν, µ ∈ Θ r such that ν = µ, so that by (5.20) ξ∈Λ −β 2β −β (µ)
n(µ), ξ ≤ 2β d(Γ† ) max sup f κr ρ ≤ κ(4L)−β . r ξ∈Λ ˆν ν∈Θ
r ν =µ∈Θ
ˆ ν ) one has Consequently, with δν = Q ν P(Λ −β 2β −β κr ρ − δν ≤ Ao + Bν + Q ν Y o + Bν − 2β d(Γ† ) −β 2β −β ≤ Y o + Bν + 2β d(Γ† ) κr ρ + δν . (5.28) Here we assume that all the operators are considered on their invariant subˆ ν ). Choosing L0 sufficiently large we may assume that B ≤ L2 , space H(Λ ν (ν) supη |f (η)| ≤ L2 , so that the first half of the conditions (5.15) are satisfied. Using the bounds ρ2 ≥ 16λ1 and (5.24), under the condition λ ∈ [ρ2 − λ1 , ρ2 + λ1 ] (see (5.22) for definition of λ1 ) one proves that the second half of (5.15) is also satisfied with s = ρ − L1 and λ1 instead of λ0 . Therefore, by (5.16) D(λ; Y o + B ; Λ ˆν) − ν
1 ˆ ν ∩ E(λ; y o ) ≤ Cρ−3 L, vol Λ 2 (2π)
∀λ ∈ [ρ2 − λ1 , ρ2 + λ1 ].
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In view of monotonicity of the density of states (see Proposition 2.1) and the bounds (5.28), (5.17) we have ˆ ν ) − D(λ;Ao + Bν + Q ν ; Λ ˆ ν )| ≤ Cρ−3 L + C ρ−1 L(κr2β ρ−β + δ), |D(λ;Y o + Bν ; Λ ∀λ ∈ [ρ2 − λ0 ,ρ2 + λ0 ]. The last two estimates lead to the bound ˆν) − D(λ; Ao + Bν + Q ν ; Λ
1 ˆ ν ∩ E(λ; y o ) vol Λ 2 (2π)
≤ Cρ−3 L + C ρ−1 L(κr2β ρ−β + δ),
(5.29)
∀λ ∈ [ρ2 − λ0 , ρ2 + λ0 ]. Using this estimate for Q = B = 0, in combination with Proposition 2.4 one also concludes that vol Λ ˆ ν ∩ E(λ; ao ) − vol Λ ˆ ν ∩ E(λ; y o ) ≤ Cρ−3 L + C ρ−1 Lκr2β ρ−β . This shows that in the estimate (5.29) the set E(λ; y o ) can be replaced with ˜ r , taking into account E(λ; ao ). Adding together the formulae (5.29) for all ν ∈ Θ 2 ˜ that card Θr ≤ Cr , we obtain that ˆν) − D(λ; Ao + Bν + Q ν ; Λ ˜r ν∈Θ
1 ˆ ∩ E(λ; ao ) vol Λ 2 (2π) ≤ Cr2 ρ−3 L + C r2 ρ−1 L(κr2β ρ−β + δ).
It remains to combine the obtained formula with (5.27), using (5.26).
6 A “gauge transformation” In this and all the subsequent sections we use the notation Sα for the class Sα (L). For the classes Sα (w) with different weight w we use the full notation to avoid confusion.
6.1
Preparation
Our strategy will be to find a unitary operator which reduces an elliptic PDO H = H0 + Op(b) (see Definition (2.8)) with b ∈ Sα ( ξ ), α < m to another PDO, whose symbol, up to some controllable small errors, depends only on ξ. Very soon we shall focus on the operators of second order, but in this subsection the order is irrelevant and it is allowed to be any positive m > 0. The sought unitary operator will be constructed in the form U = eiΨ with a suitable bounded self-adjoint Γperiodic PDO Ψ. This is why we sometimes call it a “gauge transformation”. It is useful to consider eiΨ as an element of the group U (t) = exp{iΨt}, ∀t ∈ R.
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We assume that the operator ad(H0 , Ψ) is bounded, so that U (t)D(H0 ) = D(H0 ). This assumption will be justified later on. Let us express the operator At = U (−t)HU (t) via its (weak) derivative w.r.t. t: t U (−τ ) ad(H; Ψ)U (τ )dτ. At = H + 0
By induction it is easy to show that M 1 j (1) A1 =H + ad (H; Ψ) + RM+1 , (6.1) j! j=1 1 τM τ1 (1) dτ1 dτ2 . . . U (−τM+1 ) adM+1 (H; Ψ)U (τM+1 )dτM+1 . RM+1 = 0
0
0
The operator Ψ is sought in the form Ψ=
N
Ψk , Ψk = Op(ψk ), ψk ∈ Sk(α−m)+1 .
(6.2)
k=1
Substitute this formula in (6.1) and rewrite, regrouping the terms: A1 =H0 + B +
M M 1 j! j=1
ad(H; Ψk1 , Ψk2 , . . . , Ψkj )
l=j k1 +k2 +···+kj =l
(1)
(2)
+ RM+1 + RM+1 , (2)
RM+1 =
M 1 j! j=1
ad(H; Ψk1 , Ψk2 , . . . , Ψkj ).
(6.3)
k1 +k2 +···+kj ≥M+1
Rewrite: A1 = H0 + B +
M
ad(H0 ; Ψl ) +
ad(H0 ; Ψk1 , Ψk2 , . . . , Ψkj )
l=j k1 +k2 +···+kj =l
l=1 M M 1 + j! j=1
M M 1 j! j=2
(1)
(2)
ad(B; Ψk1 , Ψk2 , . . . , Ψkj ) + RM+1 + RM+1 .
l=j k1 +k2 +···+kj =l
Switch the summation signs: A1 = H0 + B +
M l=1
+
l−1 M+1 l=2 j=1
1 j!
l M 1 ad(H0 ; Ψl ) + j! j=2 l=2
k1 +k2 +···+kj =l−1
ad(H0 ; Ψk1 , Ψk2 , . . . , Ψkj )
k1 +k2 +···+kj =l (1)
(2)
ad(B; Ψk1 , Ψk2 , . . . , Ψkj ) + RM+1 + RM+1 .
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Introduce the notation B1 = B, Bl =
l−1 j=1
Tl =
1 j!
ad(B; Ψk1 , Ψk2 , . . . , Ψkj ), l ≥ 2,
(6.4)
k1 +k2 +···+kj =l−1
l 1 j! j=2
ad(H0 ; Ψk1 , Ψk2 , . . . , Ψkj ), l ≥ 2.
(6.5)
k1 +k2 +···+kj =l
We emphasize that the operators Bl and Tl depend only on Ψ1 , Ψ2 , . . . , Ψl−1 . One more rearrangement: A1 = H0 + B +
M
ad(H0 , Ψl ) +
l=1
RM+1 = BM+1 +
M
Bl +
l=2 (1) RM+1 +
M
Tl + RM+1 ,
l=2 (2) RM+1 .
(6.6)
Now we can specify our algorithm for finding Ψk ’s. The symbols ψk will be found from the following system of commutator equations: ad(H0 ; Ψ1 ) + B1 = 0, ad(H0 ; Ψl ) + and hence
Bl
+
Tl
= 0, l ≥ 2,
,↑ A1 = A0 + XM + RM+1 , M M XM = l=1 Bl + l=2 Tl , M o o A0 = H0 + M l=1 Bl + l=2 Tl .
(6.7) (6.8)
(6.9)
Below, in Lemma 6.3 we shall prove that all the symbols bl and tl belong to appropriate classes Sβ with some β, and thus by (4.10) the symbols b l , t l possess the same property. This means that Bl and Tl are bounded (see Proposition 3.1) and hence the commutators ad(H0 , Ψl ) are also bounded in view of (6.7), (6.8). This justifies the assumption that ad(H0 , Ψ) is bounded, made in the beginning of the formal calculations in this Section.
6.2
Commutator equations
Since our primary concern is the Schr¨ odinger operator, from now on we assume that m = 2 and F = I in Definition (2.8). Before proceeding to the study of the commutator equations (6.7), (6.8) note that the symbol τθ (ξ) = h0 (ξ + θ) − h0 (ξ) = 2 θ, ξ + θ/2
(6.10)
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|Dsξ τθ−1 | ≤ Cs |θ|−1 L−s−1 , θ = 0,
(6.11)
for all ξ in the support of the function ϕθ (see (4.2)). This estimate will come in handy in the next lemma. Lemma 6.1 Let A = Op(a) be a symmetric PDO such that a ∈ Sα . Then the PDO Ψ with the Fourier coefficients of the symbol ψ(x, ξ) given by ˆ (θ, ξ) ˆ ξ) = i a , ψ(θ, τθ (ξ)
(6.12)
ad(H0 ; Ψ) + Op(a ) = 0.
(6.13)
solves the equation Moreover, the operator Ψ is bounded and self-adjoint, ψ ∈ Sα−1 and ψ
(α−1) l,s
≤C a
(α) l−1,s .
The constant C is independent of the parameter L ≥ 1 and the symbol a. Proof. Let t be the symbol of ad(H0 ; Ψ). The Fourier transform tˆ(θ, ξ) is easy to find using (3.2): ˆ ξ) = iτθ (ξ)ψ(θ, ˆ ξ). tˆ(θ, ξ) = i h0 (ξ + θ) − h0 (ξ) ψ(θ, Therefore the equation (6.13) amounts to ˆ ξ) = −ˆ a (θ, ξ) = −ˆ a(θ, ξ)ϕθ (ξ; L). iτθ (ξ)ψ(θ, By definition of the function ϕθ , a solution ψˆ exists and is given by (6.12). This symbol satisfies the condition (2.6), so that Ψ is a symmetric operator. Note also that by (4.4) and (6.11) the symbol ψ belongs to Sα−1 and one easily shows that ψ
(α−1) l,s
≤C a
(α) l−1,s .
This estimate for s = 0 and Proposition 3.1 ensure the boundedness of Ψ.
Remark 6.2 Let the symbols a and ψ be as in Lemma 6.1 and consider the commutator Op(a) = ad(Op(g), Ψ) with some symmetric symbol g ∈ Sγ . By (3.2) i ˆ ξ) − gˆ(θ, ξ)ψ(φ, ˆ ˆ gˆ(φ, ξ + θ)ψ(θ, ξ + θ) a(χ, ξ) = d(Γ) θ+φ=χ gˆ(φ, ξ + θ)ˆ a (θ, ξ) gˆ(φ, ξ)ˆ a (θ, ξ + φ) 1 − = − . τθ (ξ) τθ (ξ + φ) d(Γ) θ+φ=χ
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Analogously, one can easily derive a formula for the commutator symbol aµ,η = ad(gµ , ψη ) with arbitrary µ, η ∈ Γ† , see Subsection 4.2 for definition of these symbols. It is the same formula as above, but with the summation restricted to appropriate subsets of the lattice Γ† : gˆ(φ, ξ + θ)ˆ a (θ, ξ) gˆ(φ, ξ)ˆ a (θ, ξ + φ) 1 ˆ − aµ,η (χ, ξ) = − , τθ (ξ) τθ (ξ + φ) d(Γ) θ+φ=χ θ∈Γ†η ,φ∈Γ†µ
(6.14) see (4.12) for definition of Γ†ν . Recalling that τ−θ (ξ + θ) = −τθ (ξ), and using the property (2.6) we obtain gˆ(−θ, ξ + θ)ˆ a g ˆ (θ, ξ)ˆ a (θ, ξ) (−θ, ξ + θ) 1 − tˆ(0, ξ) = − τθ (ξ) τ−θ (ξ + θ) d(Γ) θ 1 1 gˆ(θ, ξ)ˆ a (θ, ξ) + gˆ(θ, ξ)ˆ a (θ, ξ) . = − τ (ξ) d(Γ) θ θ Let us apply Lemma 6.1 to equations (6.7) and (6.8). Lemma 6.3 Let b ∈ Sα be a symmetric symbol. Then there exists a sequence of selfadjoint bounded PDO’s Ψl , l = 1, 2, . . . with the symbols ψl ∈ Sβl , βl = l(α−2)+1, such that (6.7) and (6.8) hold, and (α) l l) ψl (β (6.15) (i) r,s ≤ C b p,n ) , l ≥ 1; (ii) The symbols bl , tl of the corresponding operators Bl , Tl belong to Sγl with γl = l(α − 2) + 2 and bl
(γl ) r,s
+ tl
bo2 (ξ) + to2 (ξ) = −
(iii)
(γl ) r,s
≤ C( b
(α) l p,n ) ,
l ≥ 2;
1 |ˆb(θ, ξ)|2 1 − ζθ2 (ξ; L) . d(Γ) τθ (ξ)
(6.16) (6.17)
θ∈Θr
The constant C in (6.15) and (6.16) does not depend on b, but depends on l, r, s, α. The integer-valued parameters p, n in (6.15) and (6.16) depend on l, r, s, α. (α)
(iv) If b l,s ≤ Cl,s L2−α for all l and s, then for some positive integer n, p the following bounds hold: ˜ b RM+1 ≤ C( xM
(α) l,s
(α) M+1 (M+1)(α−2)+2 L , p,n )
≤ C˜ b
(α) p,n ,
p = p(M, α), n = n(M, α); (6.18)
p = p(l, s, α, M ), n = n(l, s, α, M ).
(6.19)
The constant C˜ depends only on the constants Cl,s and the parameters M, α.
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Proof. The existence of ψ1 ∈ Sβ1 with required properties follows from Lemma 6.1. Further proof is by induction. To make the calculations less cumbersome, throughout the proof we adopt the following notational convention. If two symbols, φ1 and φ2 satisfy the esti(β) (ω) mate φ1 l,s ≤ C φ2 p,n with some p = p(l, s) and n = n(l, s) we simply write φ1 (β) ≤ C φ2 (ω) . Suppose that ψk with k = 1, 2, . . . , K − 1 satisfy (6.15). In order to conclude that ψK also satisfies (6.15), first we need to check that bK and tK satisfy (6.16). Step I. Estimates for bl . To begin with we prove that all the symbols bl with l ≤ K, satisfy the estimate (6.16). We first obtain a bound for ad(b; ψk1 ψk2 , . . . , ψkj ) with k1 + k2 + · · ·+ kj = l − 1 . To this end we use (6.15) and Proposition 3.4 to conclude that ad(b; ψk1 , ψk2 , . . . , ψkj )
(γ)
≤C b
(α)
j
( b
(α) kn
)
= C( b
(α) l
)
(6.20)
n=1
with γ = α+
j
(βkj − 1) = α +
n=1
j
kj (α − 2) = (l − 1)(α − 2) + α − 2 + 2 = l(α − 2) + 2.
n=1
This implies that bl satisfies (6.16) for all l ≤ K. Step II. Estimates for tl . For the symbols tl the proof is by induction. First of all, note that ad(h0 ; ψ1 , ψ1 ) = − ad(b , ψ1 ), so that, by Proposition 3.4 ad(h0 ; ψ1 , ψ1 )
(2α−2)
≤ C( b
(α) 2
) ,
and thus t2 satisfies (6.16). Suppose that all tk with k ≤ l − 1 ≤ K − 1 satisfy (6.16). Then by Definition (6.8) and (4.10) all ad(h0 ; ψk ), k ≤ l − 1, satisfy the same bound. Remembering that the definition of tl involves only ψk with k ≤ l − 1, and applying Proposition 3.4, we obtain for k1 + k2 + · · · + kj = l, j ≥ 2: ad(h0 ; ψk1 , ψk2 , . . . , ψkj ) (γ) = ad ad(h0 ; ψk1 ); ψk2 , . . . , ψkj (γ) ≤ ( b (α) )l , (6.21) with j γ = k1 (α − 2) + 2 + (kn (α − 2) + 1 − 1) = l(α − 2) + 2. n=2
This leads to (6.16) for all tl , l ≤ K. Step III. To handle ΨK we use the solution Ψ of the equation (6.13) constructed in Lemma 6.1. Then from Definition (6.8) and steps I, II we immediately conclude that ψK ∈ Sγ with γ = β − 1, β = K(α − 2) + 2 and that ψK (β) ≤ C bK (β) + tK (β) ≤ C( b (α) )K , as required.
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Step IV. Proof of (iii). By (6.4) and by (6.5), (6.7) 1 B2 = ad(B; Ψ1 ), T2 = − ad(B ; Ψ1 ). 2 It follows from (6.12) that ˆb (θ, ξ) . ψˆ1 (θ, ξ) = i τθ (ξ)
(6.22)
Remark 6.2 and Definition (4.9) lead to the formulas bo2 (ξ) = −
2 2 |ˆb(θ, ξ)|2 1 |ˆb(θ, ξ)|2 ϕθ (ξ; L), to2 (ξ) = ϕθ (ξ; L) . d(Γ) τθ (ξ) d(Γ) τθ (ξ) θ∈Θr
θ∈Θr
Adding them up and recalling that ϕθ = 1 − ζθ , we get (6.17). Step V. Proof of (iv). The remainder RM+1 (see (6.6)) consists of three components. In view of (6.16), bM+1 ∈ S(M+1)(α−2)+2 , so that by Remark 3.2 the norm of BM+1 is bounded by ( b (α) )M+1 L(M+1)(α−2)+2 as required. M (1) (α) Consider now RM+1 defined in (6.1). Let ψ = ≤ l=1 ψl . Since b 2−α CL , according to (6.15), (2.7) we have ψl
(α−1)
≤ CL(l−1)(α−2)
b
(α) l
≤ C b
(α)
.
Similarly, by Definition (6.9) we have X (α) ≤ C b (α) in view of (6.16), which proves (6.19). It follows from (6.7) and (6.8) that ad(H0 , Ψ) + X = 0. Now, repeating the same argument as on Steps 1 and II, we conclude that adM+1 (H, Ψ) (γ) ≤ (α) M+1 b with γ = (M + 1)(α− 2)+ 2. By Remark 3.2 this leads to the required (1)
estimate for the norm RM+1 .
(2)
In the same way the norm of the error RM+1 defined in (6.3) can be shown to satisfy the same bound. This completes the proof of (6.18).
7 Density of states for operator A1 In this section we apply the transformation constructed in the previous section, to the Schr¨ odinger operator, that is to the operator (2.8) with m = 2 and F = I, b(x, ξ) = V (x), so that α = 0. For the proof of Theorem 2.3 we shall need the representation (6.9) with M = 2. We begin with deriving further consequences from Lemma 6.3. From now on we shall use the notation Vj instead of Bj , j = 1, 2, . . . . All the estimates below are uniform in V . The majority of the results below are obtained for d = 2, although for some intermediate results the condition d ≥ 2 will be sufficient.
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71
Operators V2 and T2
Let us investigate in more detail the operators V2 and T2 . Recall again that by (6.4), (6.5) and (6.7) 1 V2 = ad(V ; Ψ1 ), T2 = − ad(V ; Ψ1 ). 2 We start by studying the symbols 1 aµ,η = ad Vµ , (ψ1 )η , bµ,η = − ad Vµ , (ψ1 )η 2 with µ, η ∈ P Γ† , see Subsection 4.2 for definition of the symbols bν and of the set of the primitive lattice vectors P Γ† . By (4.14), (4.10) and Lemma 6.3 used with m = 2, α = 0, we have Vµ , Vµ ∈ S0 , (ψ1 )η ∈ S−1 , and Vµ
(0) l,s
+ Vµ
(0) l,s
(0) l,0 ,
≤ Cl,s V
(ψ1 )η
(−1) l,s
≤ Cl,s V
(0) p,0 ,
p = p(l, s).
Consequently by Proposition 3.4 aµ,η , bµ,η ∈ S−2 and aµ,η
(−2) l,s
+ bµ,η
(−2) l,s
≤ Cl,s ( V
(0) 2 p,0 ) ,
p = p(l, s),
(7.1)
uniformly in µ, η ∈ P Γ† . We shall need more detailed properties of these commutators. In particular, let us find bounds for symbols (aµ,η )ν and (bµ,η )ν , ν ∈ P Γ† . In the next lemma and further on we shall need the explicit formulas for these symbols, which follow from (6.14) and (6.22). For brevity we write only the formula for ˆ aµ,η : ϕθ (ξ; L) ϕθ (ξ + φ; L) 1 ˆ ˆ ˆ V (φ)V (θ) − aµ,η (χ, ξ) = − . (7.2) τθ (ξ) τθ (ξ + φ) d(Γ) φ+θ=χ φ∈Γ†µ , θ∈Γ†η ∩Θr
ˆν = In this formula χ ∈ Γ† and τθ is defined in (6.10). Recall the notation Λ ˆ Λν (ρ − L1 ) and (4.15). Lemma 7.1 Let d = 2. Let V be as above and ψ1 be as found in Lemma 6.3. Suppose that 1 ≤ r ≤ L, (7.3) ρ ≥ 4L1 , † −2 d(Γ )ρr ≥ 16L. r one has Then for any ν ∈ Θ ˆν) (aµ,η ) ν χ( · ; Λ r µ∈P Γ† ,η∈Θ µ =η
(0) l,0
ˆν) + (bµ,η ) ν χ( · ; Λ
(0) l,0
≤ Cl ( V
(0) 2 4 −2 , p,0 ) r ρ
(7.4)
r . (Here χ( · ; Λ ˆ ν ) denotes the multiplifor all l ≥ 0, p = p(l), uniformly in ν ∈ Θ ˆ cation by the function χ(ξ; Λν ).)
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Proof. Let us estimate each term in the sum (7.4) individually. For the sake of brevity we conduct the proof only for the case of the symbol aµ,η . For this we ˆ ν . Since we are interested in the use (7.2) with χ ∈ Γ†ν , assuming that ξ ∈ Λ operator a µ,η , we may assume that χ ∈ Θr (see Definition (4.8)), and hence we have φ ∈ Θ2r in (7.2). Let us estimate first the terms in the square brackets in (7.2). Since µ = η, the vectors ν and θ, φ in (7.2) are pairwise linearly independent. Consequently, in ˆ ν the bounds view of (4.18) and (7.3) we have for ξ ∈ Λ | n(θ), ξ | ≥ 2−1 d(Γ† )r−2 ρ ≥ 8L, | n(θ), ξ + φ | ≥ 2−1 d(Γ† )r−2 ρ − 2r ≥ 6L, |τθ (ξ)| = 2| θ, ξ + θ/2 | 1 ≥ |θ| d(Γ† )r−2 ρ − r ≥ |θ| d(Γ† )r−2 ρ, 2 |τθ (ξ + φ)| = 2| θ, ξ + φ + θ/2 | 1 ≥ |θ| d(Γ† )r−2 ρ − 5r ≥ |θ| d(Γ† )r−2 ρ. 2 By Definitions (4.1) and (4.2), in view of the above bounds we have
and
(7.5)
(7.6)
ˆν, ϕθ (ξ) = ϕθ (ξ + φ) = 1, ∀ξ ∈ Λ ˆ ν the symbol ˆaµ,η has the form: and hence for χ ∈ Θr ∩ Γ†ν , ξ ∈ Λ 1 1 1 ˆ Vˆ (φ)Vˆ (θ) aµ,η (χ, ξ) = − − . τθ (ξ) τθ (ξ + φ) d(Γ) φ+θ=χ φ∈Γ†µ , θ∈Γ†η ∩Θr
According to (7.5), (7.6), 1 2| θ, φ | 1 4 −2 τθ (ξ) − τθ (ξ + φ) = τθ (ξ)τθ (ξ + φ) ≤ C|φ|r ρ , and hence ˆν) aµ,η ν χ( · ; Λ
(0) l,0
≤ C max max χ l |ˆaµ,η (χ, ξ)| χ∈Θr ξ∈Λ ˆν
≤ Cl r4 ρ−2
φ∈Γ†µ
|φ|l+1 |Vˆ (φ)|
|θ|l |Vˆ (θ)|.
θ∈Γ†η
The r.h.s. is finite since V ∈ S0 . Summing these estimates over µ ∈ P Γ† and η ∈ P Γ† , we bound the r.h.s. by the product (0) 2 |φ|l+1 |Vˆ (φ)| |θ|l |Vˆ (θ)| ≤ Cp V p,0 , φ∈Γ†
θ∈Γ†
for any p > l + 3. Consequently, the estimate (7.4) is fulfilled. The symbol bµ,η can be treated in the same way. These calculations are omitted to avoid repetitions.
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Using (4.13) for V, V and ψ1 , we can now decompose the symbol V2 + t2 as follows:
f=
(aν,ν
V2 + t2 = V2o + to2 + f + g, + bν,ν )ν , g = (aµ,η + bµ,η )ν .
ν∈P Γ†
(7.7)
ν∈P Γ† µ,η∈P Γ† µ =η
Our objective is to show that the symbol f has the form (5.1) and the symbol g has a “small” norm. These properties are proved in the next lemma. Theorem 7.2 Let d = 2. The symbols f, g defined above, satisfy the following properties: (−2)
(i) f, g ∈ S−2 (L) and f l,s + g (ii) For some z ∈ T−2 (L) one has
(−2) l,s
≤ Cl,s ( V
(0) 2 p,0 ) ,
p = p(l, s);
ˆf(χ, ξ) = zˆ(χ, ξ, n(χ) ), χ = 0, z
(−2) l,s
≤ Cl,s ( V
(0) 2 p,0 ) ,
(7.8)
p = p(l, s).
(7.9)
(iii) Let the conditions (7.3) be fulfilled. Then ˆ ν ) ≤ Cp ( V Op(g ν )P(Λ
(0) 2 4 −2 , p,0 ) r ρ
ˆν = Λ ˆ ν (ρ − L1 ), Λ
(7.10)
r. with some integer p, uniformly in ν ∈ Θ The constants C in the above inequalities are independent of V and ρ, L, r. (−2)
(0)
Proof. Let us prove (i) first. By Lemma 6.3 we have V2 + t2 l,s ≤ Cl,s ( V p,0 )2 with some p = p(l, s). By (4.14) this guarantees the same estimate for V2o and to2 . Consequently, part (i) will be proved if we establish this estimate for f only. The required bound follows from (7.1) in view of (4.14). Proof of (ii). Use (7.2) with µ = η = ν, and a similar formula for bν,ν . Recalling that ϕθ (ξ; L) = 1 − Υ L−1 ( n(θ), ξ + 2−1 |θ|) , τθ (ξ) = 2 θ, ξ + θ/2 , by Definitions (4.2) and (6.10), we conclude that ˆfν = ˆaν,ν + ˆbν,ν has the form (5.1). This means that ˆf also has this property, that is ˆf satisfies (7.8) with some function z. Moreover, in view of (5.2) and part (i), the function z belongs to T−2 and satisfies the bound (7.9). One can write an explicit formula for the function z, but it is too cumbersome and is therefore omitted. Proof of (iii). The estimate (7.10) follows from Definition (7.7) by virtue of (7.4) and Proposition 3.1.
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A.V. Sobolev
7.2
Ann. Henri Poincar´e
Operator A1
Now we use the established results to study the operator A1 from Section 6 with M = 2. According to (6.9) A1 = Ao + V + (V2 + T2 ) + X2↑ + R3 , (7.11) Ao = H + V o + T o . 0 2 2 Recall that Vˆ (0) = 0, so that the term V1o drops out. The remainders X2↑ and R3 satisfy the bounds R3 ≤ CL−4 ,
X2↑ ≤ Cp r−p , ∀p > 0.
(7.12)
Indeed, the estimate for R3 follows from (6.18) used with α = 0 and M = 2. Furthermore, the estimate for X2↑ is a consequence of (6.19), (4.10) and Remark 3.2. In this section we establish a suitable asymptotic formula for the density of states of the operator Ao + V + V2 + T2 with the help of Theorem 5.4. Let us verify first that the symbol ao has the required form. Lemma 7.3 Let d ≥ 2. The symbol ao has the form (5.21) with f (ν) (η) = −
1 |Vˆ (lν)|2 ˜ r. 1 − Υ2 (η + l|ν|/2)L−1 , ν ∈ Θ d(Γ) 2l|ν| η + l|ν|/2 0 =l∈Z, lν∈Θr
(7.13) (ν) in (5.21) belongs to S−2 (L) uniformly in V . MoreThe function f = ν∈Θ r f over, under the condition 1 ≤ r ≤ L one has sup |η|2 |f (ν) (η)| ≤ κν ,
|η|≥4L
κ=
r ν∈Θ
κν ≤
1 V 2L2 . d(Γ)
(7.14)
Proof. We need to show that the symbol V2o + to2 has the form f (ξ) as specified in (5.21). To this end rewrite (6.17), replacing the sum over all θ ∈ Θr by the double sum . r θ∈Γ†ν ∩Θr ν∈Θ
Now, denoting θ = lν, 0 = l ∈ Z and η = ξ, n(ν) we can write for each θ ∈ Γ†ν that ζθ (ξ; L) = Υ (η + l|ν|/2)L−1 , τθ (ξ) = 2l|ν| η + l|ν|/2 . Now it is clear that f = V2o + to2 has the form (5.21) with f (ν) as in (7.13). Observe also that according to Lemma 6.3(ii), the function f belongs to S−2 uniformly
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in V . In order to establish (7.14) observe that for r ≤ L and |η| ≥ 4L the function Υ in the Definition (7.13) vanishes (see Definition (4.1)), and hence 1 1 1 (ν) 2 ˆ − |V (lν)| f (η) = − d(Γ)|ν| 2l η + l|ν|/2 2l η − l|ν|/2 0
1 1 . = |Vˆ (lν)|2 2 2 d(Γ) η − l2 |ν|2 /4 0
Remembering again that |η| ≥ 4L ≥ 4r ≥ 4|l||ν|, we conclude that for |η| ≥ 4L |f (ν) (η)| ≤ κν η −2 ,
κν =
1 ˆ |V (lν)|2 , d(Γ) l
and hence κ=
κν ≤
˜r ν∈Θ
1 ˆ 1 V 2L2 , |V (θ)|2 = d(Γ) d(Γ) θ =0
as required.
We need to specify the operators which will play the role of B and Q in Theorem 5.4. Let f and g be as defined in (7.7). We use Theorem 5.4 with B = Op(b), Q = Op(q) where b = V + f, q = g. (7.15) Let us establish a useful estimate for the operator Q stating the result in a form of a lemma for the reference convenience: Lemma 7.4 Let d = 2. Under conditions (7.3) one has ˆ ν ) ≤ Cr4 ρ−2 , Λ ˆν = Λ ˆ ν (ρ − L1 ), δ = max Q ν P(Λ r ν∈Θ
(7.16)
with a constant uniform in V . Proof. The sought result immediately follows from the Definition (7.15) and the bound (7.10). Theorem 7.5 Let d = 2 and let κ and δ be as defined in (7.14) and (7.16). Let λ ∈ [ρ2 − λ0 , ρ2 + λ0 ] with some λ0 ≥ 1, and let ρ2 ≥ 20λ0 . Assume that (7.3) and (5.24) are satisfied. Then there exists a constant L0 = L0 (V ) such that under the condition (5.25) one has D(λ; Ao + V + V2 + T2 ) −
1 vol E(λ; ao ) ≤ Cr6 ρ−3 L. 2 (2π)
The constants L0 , C are uniform in V and do not depend on λ, ρ, L, r.
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Proof. Note first of all that in view of (7.3) the conditions (5.23) are satisfied. Moreover, by the third estimate in (7.3), δ ≤ CL−2 and for sufficiently large L ≥ L0 the condition ρ2 ≥ 20λ0 will ensure that ρ2 ≥ 16λ1 with the number λ1 defined in (5.22). Let us check now that further conditions of Theorem 5.4 are fulfilled. Let B = Op(b) and Q = Op(q) with the symbols b, q defined in (7.15). From Theorem 7.2(i) we know that q ∈ P0 (L), and that the symbol b can be represented in the form (5.1) with some z ∈ T0 (L). Besides, according to Lemma 7.3 the symbol ao = h0 + V2o + to2 can be represented in the form (5.21), and the property (5.20) is satisfied with β = 2 and κν specified in (7.14). Remembering the bound (7.16) and applying Theorem 5.4, we obtain that D(λ;Ao +V +V2 +T2 )−
1 volE(λ;ao ) ≤ Cr2 ρ−1 L(r4 ρ−2 +r4 ρ−2 )+Cr2 ρ−3 L, (2π)2
which leads to the stated estimate.
8 Density of states for operator Ao . Proof of Theorem 2.3 8.1
Operator Ao
The last step of the proof of Theorem 2.3 is the asymptotics of the quantity vol E(λ; ao ). Recall that ao (ξ) = |ξ|2 + f (ξ), f = V2o + to2 . Now we do not need the formula (7.13) for f , but apply the initial formula (6.17): f (ξ) = −
1 |Vˆ (θ)|2 1 − ζθ2 (ξ; L) . d(Γ) τθ (ξ) θ∈Θr
Recall that τθ and ζθ are defined in (6.10) and (4.2) respectively. The calculations in this section are done for the case of a general dimension d ≥ 2. Our aim is to prove Theorem 8.1 Let ao be as defined above. Suppose that 2L ≤ ρ with some ∈ (0, 1). Then for any l > d/2 one has 2 vol E(ρ2 ; ao ) − wd ρd − d(d − 2) wd ρd−4 |V (x)| dx 8 d(Γ) O ≤ Cl ρd−4 ρ−1 + ρ4−3 ln ρ + r−2l+d + L−4 + ρL−5 . The constant Cl is uniform in V . To begin with, we need to find the formula describing the level surface ao (ξ) = ρ . Denote ξ = tω, t = |ξ|, |ω| = 1. 2
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Rewrite the equation ao (ξ) = ρ2 : t2 + f (tω) = ρ2 , and solve it for t. Since f ∈ S−2 (L) (see Lemma 7.3), one can write |∇ξ f (ξ)| ≤ CL−3 , so that the solution of this equation is t = t(ρ, ω) = ρ −
f (ρω) + O(ρ−3 L−4 ) + O(ρ−2 L−5 ). 2ρ
Here and further on all remainder estimates are uniform in V . Therefore t(ρ,ω) d 1 vol E(ρ2 ; ao ) = t(ρ, ω) dω τ d−1 dτ dω = d Sd−1 Sd−1 0 f 1 = ρd − dρd−2 dω + O(ρd−4 L−4 ) + O(ρd−3 L−5 ) d Sd−1 2 1 = wd ρd − ρd−2 f (ρω)dω + O(ρd−4 L−4 ) + O(ρd−3 L−5 ). 2 Sd−1 (8.1) We concentrate on the second term: 1 2
M (ρ) =
f (ρω)dω.
(8.2)
Sd−1
Let us rewrite the function f in a more manageable form: |Vˆ θ) |2 1 g n(θ), ξ + |θ|/2 , f (ξ) = − 2 d(Γ) 2|θ| n(θ), ξ + |θ| θ∈Θr
where
g(η) = 1 − Υ2 ηL−1 .
(8.3)
The value of f (ξ) will not change if we replace θ by −θ in the above sum. Remembering that the function Υ is even ( see (4.1)) and adding up both sums we conclude that |Vˆ θ) |2 1 z n(θ), ξ ; |θ|/2 , f (ξ) = − 4 d(Γ) |θ| θ∈Θr
z(η; η0 ) =
g η + η0 g η − η0 − . η + η0 η − η0
We begin the calculation of the integral M in (8.1) with finding the asymptotics of the integral z ρ ω, e1 ; η0 dω, Z(ρ; η0 ) = as ρ → ∞.
Sd−1
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Ann. Henri Poincar´e
Lemma 8.2 Let the functions g and z be as defined above, and let 0 ≤ η0 ≤ L, 2L ≤ ρ for some ∈ (0, 1). Then Z(ρ; η0 ) = 2d(d − 2) wd η0 ρ−2 + η0 O(ρ−3 ) + O(ρ4−5 ln ρ) + O(η03 ρ−4 ). Proof. The integral at hand can be rewritten as follows: π 1 d−2 Z(ρ; η0 ) = dφ z(ρ cos β; η0 ) sin βdβ = ωd−2 z(ρt, η0 )(1 − t2 )κ dt 0 −1 Sd−2 ρ d−3 −1 . z(t, η0 )(1 − t2 ρ−2 )κ dt, κ= =ωd−2 ρ 2 −ρ Here ωd−2 is defined in (5.11). Denote ρ g(t + η0 ) (1 − t2 ρ−2 )κ dt, S1 (ρ) = −ρ t + η0
S2 (ρ) =
ρ
−ρ
g(t − η0 ) (1 − t2 ρ−2 )κ dt, t − η0
so that Z = ρ−1 ωd−2 (S1 − S2 ). Step 1. Each of these two integrals is split into the sum of two integrals in the following way. Represent the integrals S1 , S2 in the form S1
=
S11
=
S12
=
S11 + S12 , S2 = S21 + S22 , ρ −η0 g(t + η0 ) (1 − t2 ρ−2 )κ dt, t + η 0 −ρ −η0 g(t + η0 ) (1 − t2 ρ−2 )κ dt, t + η0
(−ρ,ρ)\(−ρ −η0 ,ρ −η0 )
S21
=
S22
=
ρ +η0
−ρ +η0
g(t − η0 ) (1 − t2 ρ−2 )κ dt, t − η0 g(t − η0 ) (1 − t2 ρ−2 )κ dt, t − η0
(−ρ,ρ)\(−ρ +η0 ,ρ +η0 )
For the integrals S11 and S21 we use the decomposition (1 − t2 ρ−2 )κ = 1 − κt2 ρ−2 + O(ρ4(−1) ), |t| ≤ Cρ , Recalling Definitions (8.3), (4.1) one can write S11 (ρ) = S˜11 + O(ρ4(−1) ln ρ), with S˜11 (ρ) =
ρ −η0
−ρ −η0
S21 (ρ) = S˜21 + O(ρ4(−1) ln ρ),
ρ g(t + η0 ) g(t) 1 − κ(t − η0 )2 ρ−2 dt, (1 − κt2 ρ−2 )dt = t + η0 −ρ t ρ g(t) S˜21 (ρ) = 1 − κ(t + η0 )2 ρ−2 dt. −ρ t
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Consequently, S11 − S21 = S˜11 − S˜21 + O(ρ4(−1) ln ρ) ρ −2 = 4κη0 ρ g(t)dt + O(ρ4(−1) ln ρ) −ρ
= O(η0 ρ
−2
) + O(ρ4(−1) ln ρ).
Step 2. Let us now concentrate on the remaining integrals S12 , S22 . Introduce the following contours (paths): 12 = [−ρ, ρ] \ (−ρ − η0 , ρ − η0 ), 22 = [−ρ, ρ] \ (−ρ + η0 , ρ + η0 ), 11 = {z ∈ C : Im z ≥ 0, |z + η0 | = ρ }, 21 = {z ∈ C : Im z ≥ 0, |z − η0 | = ρ }, 2 = {z ∈ C : Im z ≥ 0, |z| = ρ}. The paths 12 , 22 consist of two segments each. Define also 1 = {z ∈ C : Im z ≥ 0, |z| = ρ }, so that 11 , 21 can be rewritten as 11 = {z ∈ C : z + η0 ∈ 1 }, 21 = {z ∈ C : z − η0 ∈ 1 }.
(8.4)
By Definition (8.3) and because of the conditions 2L ≤ ρ , η0 ≤ L, we have g(t ± η0 ) = 1 for all |t| ≥ ρ , and hence 1 1 S12 = (1 − t2 ρ−2 )κ dt, S22 = (1 − t2 ρ−2 )κ dt.
12 t + η0
22 t − η0 In view of the analyticity of the integrands away from ±t0 and ±ρ, these integrals can be rewritten as follows: 1 1 2 −2 κ − (1 − z ρ ) dz, S22 = − (1 − z 2 ρ−2 )κ dz. S12 = z + η z − η 0 0
11
2
21
2 The direction of integration is counter-clockwise. Let us show first that the integrals over 11 and 21 give a lower order contribution. Using (8.4), as in Step 1 write the difference of these integrals in the form: dz + O(ρ4(−1) ln ρ) = η0 O(ρ−2 ) + O(ρ4(−1) ln ρ). 4κη0 ρ−2
1
Consider now the difference of the integrals over the large semi-circle 2 : 1 1 1 2 −2 κ − ) dz − (1 − z 2 ρ−2 )κ dz = 2η0 2 (1 − z ρ 2 z − η0
2 z + η0
2 z − η0 1 = 2η0 (1 − z 2 ρ−2 )κ dz + O(η03 ρ−3 ). 2
2 z
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Ann. Henri Poincar´e
After rewriting the integral over 2 with substitution z = ρeiφ , φ ∈ [0, π], and collecting all the pieces together we obtain: S21 − S22 = 2η0 ρ−1 J + η0 O(ρ−2 ) + O(ρ4(−1) ln ρ) + O(η03 ρ−3 ), π J =i e−iφ (1 − e2iφ )κ dφ. 0
Adding to this the estimate for S11 − S21 obtained on Step I, recalling that Z = ωd−2 ρ−1 (S1 − S2 ), and using the formula (9.1) derived in the Appendix, we arrive at the expected formula. Let us now calculate M (ρ), see (8.2): Lemma 8.3 Let r ≤ 2L ≤ ρ for some ∈ (0, 1). Then for any l > d/2 M (ρ) + d(d − 2) wd V 22 ≤ Cl ρ−2 r−2l+d + ρ−3 + ρ4−5 ln ρ L (O) 8ρ2 d(Γ) with a constant Cl uniform in V . Proof. It follows from definition of Z(ρ; η0 ) that M =−
|Vˆ (θ)|2 1 Z(ρ; |θ|/2). 8 d(Γ) |θ| θ∈Θr
According to the Lemma 8.2 and Parceval’s identity the leading term of the r.h.s. is given by −
d(d − 2) wd d(d − 2) wd ˆ V 2L2 (O) + O(ρ−2 ) |V (θ)|2 = − |Vˆ (θ)|2 , 2 2 8ρ d(Γ) 8ρ d(Γ) θ∈Θr
θ∈Ξr
see (4.5) for definition of Ξr . The error term does not exceed, up to a multiplicative constant independent of ρ and V , (ρ−3 + ρ4−5 ln ρ)V 2L2 (O) + ρ−4 ∆V 2L2 (O) . In view of (2.3), for any l > d/2 one has (0) |Vˆ (θ)|2 ≤ V l,0 |θ|−2l ≤ Cl V θ∈Ξr
(0) −2l+d . l,0 r
|θ|≥r
This leads to the proclaimed formula.
Proof of Theorem 8.1. The required asymptotics immediately follows from Lemma 8.3 and formula (8.1).
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Completion of the proof of Theorem 2.3
In contrast to the previous subsection, where we could allow any dimension d ≥ 2, now we restrict ourselves to d = 2 only. Recall that the unitary operator U = eiΨ constructed in Section 6 is Γperiodic. Consequently, by Proposition 2.1(ii), we have D(λ; H) = D(λ; A1 ), and hence it remains to establish the sought asymptotics for the operator A1 only. To this end we shall use the formula (7.11). By monotonicity of the density of states (see Proposition 2.1(i)) the formulae (7.12) and (7.11) give the estimates D(ρ2 − CL−4 − Cr−p ; Ao + B + B2 + T2 ) ≤ D(ρ2 ; A1 ) ≤ D(ρ2 + CL−4 + Cr−p ; Ao + B + B2 + T2 ). In order to apply Theorem 7.5, assume that L = L1 = ρ /2, r = ρβ with some ∈ (0, 1) and β ∈ (0, min{, (1 − )/2}), so that the conditions (7.3), (5.23), (5.25) are satisfied for all ρ ≥ ρ0 with a sufficiently large ρ0 = ρ0 (V ) which is uniform in V . According to Theorems 7.5 and 8.1 we have D(ρ2 ; A1 ) − w2 ρ2 ≤ Cl ρ−2 ρ−1 + ρ4−3 ln ρ 2 (2π) −2l+d +r + L−4 + ρL−5 + r6 ρ−1 L + C L−4 + Cp r−p , ∀l > d/2, ∀p > 0. Substitute L = ρ /2 and r = ρβ : Cρ−2 r6β ρ−1 + ρ4−3 ln ρ + ρ2−4 + Cp ρ−pβ , ∀p > 0. Optimizing in we get = 3/5. Choose an arbitrarily small β and a suitably large p.
9 Appendix Our aim is to find the value of the integral π d−3 J =i , e−iφ (1 − e2iφ )κ dφ, κ = 2 0 featuring in the proof of Lemma 8.2. Lemma 9.1 The integral J is given by
√ d(d − 2) Γ d+1 wd 2 . J = d(d − 2) = π ωd−2 d − 1 Γ d+2 2
(9.1)
Proof. Recall that (see (1.3), (5.11)) d
wd =
d−1
π2 (d − 1)π 2 d+2 , ωd−2 = (d − 1) wd−1 = , Γ 2 Γ d+1 2
so that we need to prove only that J coincides with the r.h.s. of (9.1).
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A.V. Sobolev
Define J(t) = i
0
π
Ann. Henri Poincar´e
e−iφ (1 − te2iφ )κ dφ, |t| < 1.
Expanding the integrand in the absolutely convergent series, we find J = lim J(t) = lim i t↑1
t↑1
∞ π
tn e(2n−1)iφ (−1)n
0 n=0
∞ 1 κ κ (−1)n dφ = −2 . n 2n − 1 n n=0
(9.2) For d = 2 we have κ = −1/2 and we can use the formula −1/2 1/2 = (1 − 2n), n n so that J =2
∞
(−1)n
n=0
1/2 = 2(1 − 1)1/2 = 0. n
For d ≥ 3 use [3], formula 1.4(2), which implies that the r.h.s. of (9.2) coincides with √ Γ d−1 Γ − 12 Γ κ + 1 2 − = 2 π d−2 Γ κ + 12 Γ 2 In view of the relations d+1 d−1 d−2 d+2 d−1 d(d − 2) Γ Γ Γ = , Γ = , 2 2 2 2 4 2 this leads to (9.1).
Acknowledgments. The author is grateful to L. Parnovski for discussions. The paper was completed during the author’s stay at the Mittag-Leffler Institute in September 2002.
References [1] M.S. Agranovich, Elliptic operators on closed manifolds, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 63, 5–129 (1994); Engl. transl. in: Partial differential equations. VI, Encycl. Math. Sci. 63, 1–130 (1994). [2] B.E.J. Dahlberg, E. Trubowitz, A remark on two-dimensional periodic potentials, Comment. Math. Helvetici 57, 130–134 (1982). [3] A. Erd´elyi, Higher transcendental functions, V. I, McGraw-Hill 1953.
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[4] J. Feldman, H. Kn¨ orrer, E. Trubowitz, The perturbatively stable spectrum of a periodic Schr¨ odinger operator, Invent. Math. 100, 259–300 (1990). [5]
, Perturbatively unstable eigenvalues of a periodic Schr¨ odinger operator, Comment. Math. Helvetici 66, 557–579 (1991).
[6] B. Helffer, A. Mohamed, Asymptotics of the density of states for the Schr¨ odinger operator with periodic electric potential, Duke Math. J. 92, 1–60 (1998). [7] M. Hitrik, I. Polterovich, Regularized traces and Taylor expansions for the heat semigroup, J. London Math. Soc.II. Ser. 68, No. 2, 402–418 (2003). [8] Yu. Karpeshina, On the density of states for the periodic Schr¨ odinger operator, Ark. Mat. 38, 111–137 (2000). [9]
, Perturbation theory for the Schr¨ odinger operator with a periodic potential, Lecture Notes in Math. vol 1663, Springer Berlin 1997.
[10] T. Kato, Perturbation theory for linear operators, Springer 1966. [11] E.Korotyaev, A. Pushnitski, On the high energy asymptotics of the integrated density of states, Bull. London Math. Soc. 35, 770–776 (2003). [12] A.S. Milevskij, Similarity transformations and spectral properties of hypoelliptic pseudodifferential operators on a circle, Funk. Anal. Prilozen. 23 no 3, 71–72 (1989); Engl. transl. in Funct. Anal. Appl. 23 no 3, 231–233 (1989). [13]
, A simplification of hypoelliptic pseudodifferential operators on the circle via Fourier integral operators, Moscow 1988, dep. in VINITI 05.09.88, No 6856-B88.
[14] A. Mohamed, Asymptotics of the density of states for the Schr¨odinger operator with periodic electromagnetic potential, J. Math. Phys. 38, 4023–4051 (1997). [15] L. Parnovski, A.V. Sobolev, Lattice points, perturbation theory and the periodic polyharmonic operator, Ann. H. Poincar´e 2, 573–581 (2001). [16] I. Polterovich, Heat invariants of Riemannian manifolds, Israel J. Math. 119, 239–252 (2000). [17] M. Reed, B. Simon, Methods of modern mathematical physics, IV, Academic Press, New York, 1975. [18] G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators on the circle, Trudy Moskov. Mat. Obshch. 36, 59–84 (1978) (Russian).
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[19] A.V. Savin, Asymptotic expansion of the density of states for one-dimensional Schr¨ odinger and Dirac operators with almost periodic and random potentials, (in Russian), Sb. Nauchn. Tr. IFTP, 1988, Moscow. [20] D. Shenk and M. Shubin, Asymptotic expansion of the state density and the spectral function of a Hill operator, Math. USSR Sbornik 56 no. 2, 473–490 (1987). [21] M.A. Shubin, Weyl’s theorem for the Schr¨ odinger operator with an almost periodic potential, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 31, no 2, 84–88 (1976)(Russian). Engl. transl.: Moscow Univ. Math. Bull. 31, 133–137 (1976). [22]
, The spectral theory and the index of elliptic operators with almost periodic coefficients, Russian Math. Surveys 34 no 2, 109–157 (1979).
[23] M. Skriganov, Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators, Proc. Steklov Math. Inst. Vol. 171, 1984. [24]
, The spectrum band structure of the three-dimensional Schr¨ odinger operator with periodic potential, Inv. Math. 80, 107–121 (1985).
[25] A.V. Sobolev, Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one, to appear in Revista Matematica Iberoamericana (2005). Alexander V. Sobolev Department of Mathematics University of Sussex Falmer Brighton BN1 9RH United Kingdom email:
[email protected] Communicated by Bernard Helffer submitted 30/04/04, accepted 26/07/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 85 – 102 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/010085-18 DOI 10.1007/s00023-005-0199-7
Annales Henri Poincar´ e
Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock Jean-Marie Barbaroux, Maria J. Esteban and Eric S´er´e
Abstract. We study the ground state solutions of the Dirac-Fock model in the case of weak electronic repulsion, using bifurcation theory. They are solutions of a minmax problem. Then we investigate a max-min problem coming from the electronpositron field theory of Bach-Barbaroux-Helffer-Siedentop. We show that given a radially symmetric nuclear charge, the ground state of Dirac-Fock solves this maxmin problem for certain numbers of electrons. But we also exhibit a situation in which the max-min level does not correspond to a solution of the Dirac-Fock equations together with its associated self-consistent projector.
1 Introduction The electrons in heavy atoms experience important relativistic effects. In computational chemistry, the Dirac-Fock (DF) model [1], or the more accurate multiconfiguration Dirac-Fock model [2], take these effects into account. These models are built on a multi-particle Hamiltonian which is in principle not physically meaningful, and whose essential spectrum is the whole real line. But they seem to function very well in practice, since approximate bound state solutions are found and numerical computations are done and yield results in quite good agreement with experimental data (see, e.g., [3]). Rigorous existence results for solutions of the DF equations can be found in [4] and [5]. An important open question is to find a satisfactory physical justification for the DF model. It is well known that the correct theory including quantum and relativistic effects is quantum electrodynamics (QED). However, this theory leads to divergence problems, that are only solved in perturbative situations. But the QED equations in heavy atoms are nonperturbative in nature, and attacking them directly seems a formidable task. Instead, one can try to derive approximate models from QED, that would be adapted to this case. The hope is to show that the Dirac-Fock model, or a refined version of it, is one of them. Several attempts have been made in this direction (see [6, 7, 8, 9] and the references therein). Mittleman [6], in particular, derived the DF equations with “self-consistent projector” from a variational procedure applied to a QED Hamiltonian in Fock space, followed by the standard Hartree-Fock approximation. More precisely, let H c be the free Dirac Hamiltonian, and Ω a perturbation. We denote Λ+ (Ω) = χ(0,∞) (H c + Ω). The electronic space is the range H+ (Ω) of this projector. If one computes the QED energy of Slater determinants of N wave functions in this electronic space, one obtains the DF en-
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ergy functional restricted to (H+ (Ω))N . Let ΨΩ be a minimizer of the DF energy in the projected space (H+ (Ω))N under normalization constraints. It satisfies the projected DF equations, with projector Λ+ (Ω). Let E(Ω) := E(ΨΩ ). Mittleman showed (by formal arguments) that the stationarity of E(Ω) with respect to Ω implies that Λ+ (Ω) coincides, on the occupied orbitals, with the self-consistent projector associated to the mean-field Hartree-Fock Hamiltonian created by ΨΩ . From this he infers ([6], page 1171) : “Hence, Ω is the Hartree-Fock potential when the Hartree-Fock approximation is made for the wave function”. Recently rigorous mathematical results have been obtained in a series of papers by Bach et al. and Barbaroux et al. [10, 11, 12] on a Hartree-Fock type model involving electrons and positrons. This model (that we will call EP) is related to the works of Chaix-Iracane [9] and Chaix-Iracane-Lions [13]. Note, however, that in [10, 11, 12] the vacuum polarization is neglected, contrary to the Chaix-Iracane approach. In [10], in the case of the vacuum, a max-min procedure in the spirit of Mittelman’s work is introduced. In [12], in the case of N -electron atoms, it is shown that critical pairs (γ, P + ) of the electron-positron Hartree-Fock energy EEP give solutions of the self-consistent DF equations. This result is an important step towards a rigorous justification of Mittleman’s ideas. All this suggests, in the case of N -electrons atoms, to maximize the minimum E(Ω) with respect to Ω. It is natural to expect that this max-min procedure gives solutions of the DF equations, the maximizing projector being the positive projector of the self-consistent Hartree-Fock Hamiltonian. We call this belief (expressed here in rather imprecise terms) “Conjecture M”. In [14] and [15], when analyzing the nonrelativistic limit of the DF equations, Esteban and S´er´e derived various equivalent variational problems having as solution an “electronic” ground state for the DF equations. Among them, one can find min-max and max-min principles. But these principles are nonlinear, and do not solve Conjecture M. In this paper we try to give a precise formulation of Conjecture M in the spirit of Mittleman’s ideas and to see if it holds true or not, in the limit case of small interactions between electrons. We prove that in this perturbative regime, given a radially symmetric nuclear potential, Conjecture M may hold or not depending on the number of electrons. The type of ions which are covered by our study are those in which the number of electrons is much smaller than the number of protons in the nucleus, with, additionally, c (the speed of light) very large. The paper is organized as follows : in Section 2 we introduce the notations and state our main results (Theorems 9 and 11). Sections 3 and 4 contain the detailed proofs.
2 Notations and main results In the whole paper we choose a system of units in which Planck’s constant, , and the mass of the electron are equal to 1 and Ze2 = 4π0 , where Z is the number
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Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock 87
of protons in the nucleus. In this system of units, the Dirac Hamiltonian can be written as H c = −ic α · ∇ + c2 β, (1) 11 0 where c > 0 is the speed of light , β = , α = (α1 , α2 , α3 ), α = 0 −11 0 σ and the σ ’s are the Pauli matrices. The operator H c acts on 4-spinors, σ 0 i.e., functions from R3 to C4 , and it is self-adjoint in L2 (R3 , C4 ), with domain H 1 (R3 , C4 ) and form-domain H 1/2 (R3 , C4 ). Its spectrum is the set (−∞, −c2 ] ∪ [c2 , +∞). In this paper, the charge density of the nucleuswill be a smooth, radial and compactly supported nonnegative function n, with n = 1, since in our system of units Ze2 = 4π0 . The corresponding Coulomb potential is V := −n ∗ (1/|x|). Then V : R3 → (−∞, 0) is a smooth negative radially symmetric potential such that −
1 ≤ V (x) < 0 (∀x) |x|
,
|x| V (x) −1 for
|x| large enough .
Note that the smoothness condition on V is only used in step 3 of the proof of Proposition 15. Actually we believe that this condition can be removed. It is well known that H c + V is essentially self-adjoint and for c > 1, the spectrum of this operator is as follows: σ(H c + V ) = (−∞, −c2 ] ∪ {λc1 , λc2 , . . . } ∪ [c2 , +∞), with 0 < λc1 < λc2 < · · · and lim λc = c2 . →+∞
Finally define the spectral subspaces Mci = Ker(H c + V − λci 11) and let Nic denote Mci ’s dimension. Since the potential is radial, it is well known that the eigenvalues λci are degenerate (see, e.g., [16]). For completeness, let us explain this in some detail. To any A ∈ SU (2) is associated a unique rotation RA ∈ SO(3) such that ∀x ∈ R3 , (RA x) · σ = A(x · σ)A−1 , where σ = (σ1 , σ2 , σ3 ). This map is a morphism of Lie groups. It is onto, and its kernel is {I, −I}. It leads to a natural unitary representation • of SU (2) in the Hilbert spaces of 2-spinors L2 (S 2 , C2 ) and L2 (R3 , C2 ), given by −1 (A • φ)(x) := A φ(RA x) . (2) Then, on the space of 4-spinors L2 (R3 , C4 ) = L2 (R3 , C2 ) ⊕ L2 (R3 , C2 ), one can define the following unitary representation (denoted again by •) −1 Aφ(RA x) (A • φ)(x) φ (x) := = A• . −1 (A • χ)(x) χ Aχ(RA x)
(3)
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The radial symmetry of V implies that H c + V commutes with •. The eigenspaces Mci are thus SU (2) invariant. Now, let Jˆ = (Jˆ1 , Jˆ2 , Jˆ3 ) be the total angular momentum operator associated to the representation •. The eigenvalues of Jˆ2 = Jˆ12 + Jˆ22 + Jˆ32 are the numbers (j 2 − 1/4) , where j takes all positive integer values. If φ is an eigenvector of Jˆ2 with eigenvalue (j 2 − 1/4) , then the SU (2) orbit of φ generates an SU (2) invariant complex subspace of dimension 2j ≥ 2. This implies the following fact, which will be used repeatedly in the present paper: Lemma 1. If φ ∈ L2 (R3 , C2 ) is not the zero function, then there is A ∈ SU (2) such that φ and A • φ are two linearly independent functions. Proof of the Lemma. Assume, by contradiction, that C φ is SU (2) invariant. Then φ is an eigenvector of J for = 1, 2, 3, hence it is eigenvector of Jˆ2 . But we have seen that in such a case, the SU (2) orbit of φ must contain at least two independent vectors: this is absurd. As a consequence of the Lemma, the spaces Mci have complex dimension at least 2. The degeneracy is higher in general: for each j ≥ 1 , H c + V has infinitely many eigenvalues of multiplicity at least 2j. Note that in the case of the Coulomb potential, the eigenvalues are even more degenerate (see, e.g., [16]). Now, on the Grassmannian manifold GN (H 1/2 ) := {W subspace of H 1/2 (R3 , C4 ); dimC (W ) = N } we define the Dirac-Fock energy Eκc as follows Eκc (W )
:=
Eκc (Ψ)
κ + 2
:=
N i=1
R3 ×R3
R3
((H c + V )ψi , ψi )dx
ρΨ (x)ρΨ (y) − |RΨ (x, y)|2 dxdy , |x − y|
(4)
where κ > 0 is a small constant, equal to e2 /4π0 in our system of units, {ψ1 , . . . ψN } is any orthonormal basis of W , Ψ denotes the N -uple (ψ1 , . . . , ψN ), ρΨ is a scalar and RΨ is a 4 × 4 complex matrix, given by ρΨ (x) =
N ψ (x), ψ (x) ,
RΨ (x, y) =
=1
N
ψ (x) ⊗ ψ∗ (y) .
(5)
=1
Saying that the basis {ψ1 , . . . , ψN } is orthonormal is equivalent to saying that GramL2 Ψ = 11N . Eκc (W )
(6) Eκc (Ψ).
or The energy can We will use interchangeably the notations be considered as a function of W only, because if u ∈ U (N ) is a unitary matrix,
with the notation (uΨ)k =
Eκc (uΨ) = Eκc (Ψ)
l
ukl ψl .
(7)
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Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock 89
Note that since V is radial, the DF functional is also invariant under the representation • defined above. Its set of critical points will thus be a union of SU (2) orbits. Finally let us introduce a set of projectors as follows: Definition 2. Let P be an orthogonal projector in L2 (R3 , C4 ), whose restriction to 1 1 H 2 (R3 , C4 ) is a bounded operator on H 2 (R3 , C4 ). Given ε > 0, P is said to be 1 c 3 4 2 ε-close to Λ+ c := χ(0,+∞) (H ) if and only if, for all ψ ∈ H (R , C ), 14 P − Λ+ −c2 ∆ + c4 c ψ
14 ≤ ε −c2 ∆ + c4 ψ
L2 (R3 ,C4 )
L2 (R3 ,C4 )
.
In [14] the following result is proved: Theorem 3 ([14]). Take V , N fixed. For c large and 0 , κ small enough, for all P ε0 -close to Λc+ , c(P ) :=
sup
inf
W + ∈GN (P H 1/2 )
W ∈GN (H 1/2 ) P (W )=W +
Eκc (W )
is independent of P and we denote it by Eκc . Moreover, Eκc is achieved by a solution Wκ =span{ψ1 , . . . , ψN } of the Dirac-Fock equations:
c Hκ,Wκ ψi = ci ψic , 0 < ci < 1, (DF) GramL2 Ψ = 1N with c Hκ,W ϕ := (H c + V + κ ρΨ ∗
1 )ϕ − κ |x|
R3
RΨ (x, y)ϕ(y) dy . |x − y|
(MF)
Remark. It is easy to verify that ε0 > 0 given, for c large and κ small enough, c χ(0,∞) (Hκ,W ) is ε0 -close to Λ+ c . κ Corollary 4 ([14]). Take V, N fixed. Choose c large and κ small enough. If we define the projector + c Pκ,W = χ(0,∞) (Hκ,W ) c with Hκ,W given by formula (MF), then
Eκc =
min
W ∈Gn (H 1/2 ) + P W =W κ,W
Eκc (W ) =
min
W ∈GN (H 1/2 ) W solution of (DF)
Eκc (W ) .
(8)
Another variational problem was introduced in the works of Bach et al. and Barbaroux et al. ([10, 11, 12]): define κ = {P + = χ[0,∞) (H c ) ; W ∈ GN (H 1/2 )}, P κ,W κ,W
(9)
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and N Sκ, W
c {γ ∈ S1 (L2 ) , γ = γ ∗ , Hκ, γ ∈ S1 , W
:=
P + γP − = 0 , −P − ≤ γ ≤ P + , tr γ = N }, κ,W
κ,W
κ,W
κ,W
with the notation P − := 1I − P+ , and S1 being the Banach space of trace-class κ,W
κ,W
N operators on L2 (R3 , C4 ). For all γ ∈ Sκ, , let W
ργ (x)ργ (y) κ |γ(x, y)|2 κ dx dy − dx dy. = tr ((H + V )γ) + 2 |x − y| 2 |x − y| 4 Here, ργ (x) := s=1 γs,s (x, x) = n wn |ψn (x)|2 , with wn the eigenvalues of γ and ψn the eigenspinors of γ, and γ(x, y) = n wn ψn (x) ⊗ ψn (y), i.e., γ(x, y) is the kernel of γ. κ , the infimum of Fκc on In [12] it has been proved that for every P + ∈ P Fκc (γ)
c
κ,W
the set S N is actually equal to the infimum defined in the smaller class of Slater κ,W determinants. More precisely, with the above notations, κ , one has Theorem 5 ([12]). For κ small enough and for all P + ∈ P κ,W
inf
γ∈S N
Fκc (γ)
κ,W
=
inf
W ∈GN (P + H 1/2 )
Eκc (W )
(10)
κ,W
Moreover, the infimum is achieved by a solution of the projected Dirac-Fock equations, namely N
ψi , .ψi γmin = i=1
with
P + ψi κ,W
= ψi (i = 1, . . . , N ), and for Wmin := span(ψ1 , . . . , ψN ) ,
c P + Hκ,W P + ψi = i ψi , 0 < i < 1, min κ,W κ,W GramL2 Ψ = 1N
(11)
Let us now define the following sup-inf: ecκ :=
sup
κ P + ∈P κ,W
inf
W ∈GN (P + H 1/2 )
Eκc (W ) .
κ,W
Then, Theorem 5 has the following consequence: Corollary 6. If κ is small enough, ecκ =
sup
κ P + ∈P κ,W
inf
γ∈S N
κ,W
Fκc (γ).
(12)
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Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock 91
From the above definitions, Theorem 3, Corollary 4 and the remark made after Theorem 3, we clearly see that for all κ small and c large, Eκc ≥ ecκ .
(13)
One can hope more: Conjecture M: The energy levels Eκc and ecκ coincide, and there is a solution Wκc of the DF equations such that Eκc (Wκc ) = ecκ =
inf
+ 1/2 ) V ∈GN (Pκ,W cH
Eκc (V ) .
κ
In other words, the max-min level ecκ is attained by a pair (W, P + ) such that κ,W
= W. W
This paper is devoted to discussing this conjecture, which, if it were true, would allow us to interpret the Dirac-Fock model as a variational approximation of QED. In order to study the different cases that can appear when studying the problems Eκc and ecκ for κ small, we begin by discussing the case κ = 0. Proposition 7. Conjecture M is true in the case κ = 0. Proof. The case κ = 0 is obvious. Indeed, all projectors P + coincide with the 0,W
projector χ[0,∞) (H c + V ). The level E0c , seen as the minimum of Corollary 2, is achieved by any N -dimensional space Wmin spanned by N orthogonal eigenvectors of H c +V whose eigenvalues are the N first positive eigenvalues of H c +V , counted with multiplicity. Then E0c is the sum of these N first positive eigenvalues. Clearly, (Wmin , χ[0,∞) (H c + V )) realizes ec0 . The interesting case is, of course, κ > 0 , when electronic interaction is taken into account. For κ > 0 and small two very different situations occur, depending on the number N of electrons. The first situation (perturbation from the linear closed shell atom) corresponds to I N= Nic , I ∈ Z+ (14) i=1
is treated in detail in Section 3. We recall that Nic is the dimension of the eigenspace Mci = Ker(H c +V −λci 11) already defined. Under assumption (14), for κ = 0, there is a unique solution, W0c , to the variational problems defining E0c and ec0 , W0c =
I i=1
Mci .
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The “shells” of energy λci , 1 ≤ i ≤ I , are “closed”: each one is occupied by the maximal number of electrons allowed by the Pauli exclusion principle. The subspace W0c is invariant under the representation • of SU (2). We are interested in solutions Wκc of the Dirac-Fock equations lying in a neighborhood Ω ⊂ GN (H 1/2 ) of W0c , for κ small. Using the implicit function theorem, we are going to show that for each κ small, Wκc exists, is unique, and is a smooth function of κ. Information about the properties enjoyed by Wκc is given by Proposition 8. Fix c large enough. Under assumption (14), for κ small enough, Eκc = Eκc (Wκc ) =
inf
+ 1/2 ) W ∈GN (Pκ,W cH
Eκc (W ),
(15)
κ
and Wκc is the unique solution of this minimization problem. This proposition will be proved in Section 3. Our first main result follows from it: Theorem 9. Under assumption (14), for c > 0 fixed and κ small enough, Eκc = ecκ and both variational problems are achieved by the same solution Wκc of the self κ is P + c . consistent Dirac-Fock equations. For ecκ , the optimal projector in P κ,Wκ Proof. The above proposition implies that for κ small, ecκ ≥
inf
+ 1/2 ) W ∈GN (Pκ,W cH
Eκc (W ) = Eκc (Wκc ) = Eκc .
(16)
κ
Therefore, ecκ = Eκc . Moreover, by Proposition 8, ecκ is achieved by a couple (Wκc , P ) + c such that P = Pκ,W c , Wκ being a solution of the Dirac-Fock equations. This ends κ the proof. The second situation (perturbation from the linear open shell case) occurs when I c Nic + k, I ∈ Z+ , 0 < k < NI+1 . (17) N= i=1
It is treated in detail in Section 4. When (17) holds and when κ = 0, there exists a manifold of solutions, S0 , whose elements are the spaces I
c Mci ⊕ WI+1,k ,
i=1 c c for all WI+1,k ∈ Gk (MI+1 ). These spaces are all the solutions of the variational c problems defining E0 and ec0 . The (I + 1)th “shell” of energy λcI+1 is “open”: it is
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c occupied by k electrons, while the Pauli exclusion principle would allow NI+1 −k more. Note that we use the expression “open shell” in the linear case κ = 0 only: indeed, adapting an idea of Bach et al. [17], one can easily see that for κ positive and small, the solutions to (DF) at the minimal level Eκc have no unfilled shells. For κ > 0 and small we look for solutions of the DF equations near S0 (see Section 4). We could simply quote the existence results of [15], and show the convergence of solutions of (DF) at level Eκc , towards points of S0 , as κ goes to 0. But we prefer to give another existence proof, using tools from bifurcation theory. This approach gives a more precise picture of the set of solutions to (DF) near the level Eκc (Theorem 12). In particular, we obtain in this way all the solutions of (DF) with smallest energy Eκc (Proposition 13). − We now choose one of these minimizers, and we call it Wκc . We have Pκ,W c κ c c (Wκ ) = 0 . Since V is radial, Wκ belongs to an SU (2) orbit of minimizers. We are interested in cases where this orbit is not reduced to a point. Then the mean-field c operator Hκ,W c should not commute with the action • of SU (2), and one expects κ the following property to hold:
(P): Given c large enough, if κ is small, then for any solution Wκc of (DF) at level Eκc , there is a matrix A ∈ SU (2) such that − c Pκ,W c (A • Wκ ) = 0 . κ
(18)
The next proposition shows that whenever (P) holds, Conjecture M does not. This result will imply that Conjecture M is indeed wrong. Proposition 10. If (P) is satisfied, then for c large enough and κ small, given any solution Wκc of the nonlinear Dirac-Fock equations such that Eκc (Wκc ) = Eκ , we have inf Eκc (W ). (19) Eκc = Eκc (Wκc ) > W ∈GN (H 1/2 ) − P c W =0 κ,Wκ
This proposition will be proved in Section 4. Moreover, we verify (see Proposition 15) that (P) holds when I ≥ 1 and k = 1, i.e., when in the linear case there is a single electron in the highest nonempty shell. Our second main result follows directly from Propositions 10 and 15. Theorem 11. Take N=
I
Nic + 1,
I ≥ 1.
i=1
For c large and κ > 0 small, there is no solution W∗ of the nonlinear Dirac-Fock equations with positive Lagrange multipliers, such that the couple + (W∗ , Pκ,W ) ∗
realizes the max-min ecκ . So Conjecture M is wrong.
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I Remark. Note that the fact that Conjecture M is wrong in the case N = Nic +1,
I ≥ 1, is related to nonuniqueness of the minimizer for the problem inf
W ∈GN (H 1/2 ) − P c W =0 κ,Wκ
i=1
Eκc (W ) .
When such a situation happens, it is well known that one has to be very careful when considering max-min (resp. min-max) problems, since even when solvable, they do not always deliver critical points of the considered functional. A very simple example for this fact is provided by the function f : R2 → R defined by f (x, y) := (1 − x2 )2 + xy. It is easy to verify that sup inf f (x, y) = 0 ,
y∈R x∈R
that the unique maximizer is y = 0 and that there are exactly two minimizers of x → f (x, 0), x± = ±1. But neither (−1, 0) nor (1, 0) are critical points of f .
3 Perturbation from the linear closed shells case Let us recall that we are in the case N=
I
Nic ,
I ∈ Z+ ,
i=1
Nic being the dimension of the eigenspace Mci = Ker(H c + V − λci 11). We want to apply the implicit function theorem in a neighborhood of W0c , for κ small. For this purpose, we need a local chart near W0c . Take an orthonormal basis (ψ1 , . . . , ψN ) of W0c , whose elements are eigenvectors of H c +V , the associated eigenvalues being µ1 ≤ · · · ≤ µN (i.e., λc1 , . . . , λcI counted with multiplicity). Let Z be the orthogonal space of W0c for the L2 scalar product, in H 1/2 (R3 , C4 ). Then Z is a Hilbert space for the H 1/2 scalar product. The map C : χ = (χ1 , . . . , χN ) → span(ψ1 + χ1 , . . . , ψN + χN ) , defined on a small neighborhood O of 0 in Z N , is the desired local chart. Denote Gχ the N × N matrix of scalar products (χl , χ )L2 . Then Eκc ◦ C(χ) = Eκc (I + Gχ )−1/2 (ψ + χ) . The differential of this functional defines a smooth map Fκ : O ⊂ Z N → (Z )N , where Z ⊂ H −1/2 is the topological dual of Z for the H 1/2 topology, identified with the orthogonal space of W0c for the duality product in H −1/2 × H 1/2 . Note that Fκ depends smoothly on the parameter κ. A subspace C(χ) is solution of
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(DF) if and only if Fκ (χ) = 0. To apply the implicit function theorem, we just have to check that the operator L := Dχ F0 (0) is an isomorphism from Z N to its dual (Z )N . This operator is simply the Hessian of the DF energy expressed in our local coordinates: (20) Lχ = (Hc + V − µ1 )χ1 , . . . , (Hc + V − µN )χN . Under assumption (14), the scalars µk , k = 1, . . . , N , are not eigenvalues of the restriction of H c + V to the L2 -orthogonal subspace of W0c . This implies that L is an isomorphism. As a consequence, there exists a neighborhood of W0c × {0} in GN (H 1/2 ) × R, Ω × (−κ0 , κ0 ) and a smooth function hc : (−κ0 , κ0 ) → Ω such that for κ ∈ (−κ0 , κ0 ), Wκc := hc (κ) is the unique solution of the Dirac-Fock equations in Ω. Moreover, for all κ ∈ (−κ0 , κ0 ), the following holds: u(Wκc ) = Wκc , ∀u ∈ SU (2) .
(21)
Indeed, the subset A of parameters κ such that (21) holds is obviously nonempty (it contains 0) and closed in (−κ0 , κ0 ). Now, for κ in a small neighborhood of A, the SU (2) orbit of Wκc stays in Ω. But this orbit consists of solutions of the Dirac-Fock equations, so, by uniqueness in Ω, it is reduced to a point. This shows that A is also open. A is thus the whole interval of parameters (−κ0 , κ0 ). Now we are in the position to prove Proposition 8. + c Proof of Proposition 8. Remember that for κ = 0, P0,W c coincides with χ(0,∞) (H + 0 c c V ). Now, W0 is clearly the unique minimizer of E0 on the Grassmannian sub+ 1/2 manifold G+ ). More precisely, in topological terms, for any 0 := GN (P0,W0c H c 1/2 neighborhood V of W0 in GN (H ), there is a constant δ = δ(V) > 0 such that 1/2 E0c (W ) ≥ E0c (W0c ) + δ , ∀ W ∈ G+ ) \ V) . 0 ∩ (GN (H
(22)
Moreover, looking at formula (20), one easily sees that the Hessian of E0c on G+ 0 is positive definite at W0c . We now take κ > 0 small, and we consider again the chart + 1/2 C constructed above. We define the submanifold G+ ). Then κ := GN (Pκ,Wκc H + + N + c the restriction Cκ of C to (Pκ,Wκc Z) is a local chart of Gκ near Wκ . For κ small enough, there is a neighborhood U of 0 in Z N such that the second derivative of + N . The functional Eκc ◦ Cκ+ is Eκc ◦ Cκ+ is positive definite on Uκ+ := U ∩ (Pκ,W c Z) κ thus strictly convex on Uκ+ . Now, for κ small, there is a unique χκ ∈ Uκ+ such that Cκ+ (χκ ) = Wκc . Then the derivative of Eκc ◦ Cκ+ vanishes at χκ . As a consequence Wκc = Cκ+ (χκ ) is the unique minimizer of Eκc on Vκ+ := Cκ+ (Uκ+ ). Now, we choose, as neighborhood of W0c in GN (H 1/2 ), the set V := C(U), and we consider the constant δ > 0 such that (22) is satisfied. Taking κ > 0 even smaller, we can impose min Eκc + δ/2 ≤ inf Eκc . + Vκ
+ G+ κ \Vκ
Hence, Wκc is the unique solution to the minimization problem (15).
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4 Bifurcation from the linear open shell case Recall that here we are in the case N=
I
c Nic + k, I ∈ Z+ , 0 < k < NI+1 .
i=1
For κ = 0, there exists a manifold of solutions, S0 , whose elements are the spaces I
c Mci ⊕ WI+1,k ,
i=1 c c for all WI+1,k ∈ G (MI+1 ). These spaces are all the solutions of the variational c problems defining E0 and ec0 . For κ > 0 and small we want to find solutions of the DF equations near S0 , by using tools from bifurcation theory. If λI+1 has only multiplicity 2, then (17) implies k = 1 and by Lemma 1 of §2, S0 is an SU (2) orbit. Then, as in Section 3, one can find, in a neighborhood of S0 , a unique SU (2) orbit Sκ of solutions of (DF). But there are also more degenerate cases in which λI+1 has a higher multiplicity, and S0 contains a continuum of SU (2) orbits. In such situations, κ = 0 is a bifurcation point, and one expects, according to bifurcation theory, that the manifold of solutions S0 will break up for κ = 0, and that there will only remain a finite number of SU (2) orbits of solutions. To find these orbits, one usually starts with a Lyapunov-Schmidt reduction: one builds a suitable manifold Sκ which is diffeomorphic to S0 (see, e.g., [18]). When S0 contains several SU (2) orbits, the points of Sκ are not necessarily solutions of (DF), but Sκ contains all the solutions sufficiently close to S0 . Moreover, all critical points of the restriction of Eκc to Sκ are solutions of (DF). The submanifold Sκ is constructed thanks to the implicit I+1 function theorem. More precisely, we consider the projector Π : L2 → i=1 Mci . To each point z ∈ S0 we associate the submanifold Fz := {w ∈ GN (H 1/2 ) : Πw = z}. For w a point of Fz , let ∆w := Tw Fz ⊂ Tw GN (H 1/2 ). Then the following holds:
Theorem 12. Under the above assumptions, there exist a neighborhood Ω of S0 in GN (H 1/2 ), a small constant κ0 > 0, and a smooth function h : S0 ×(−κ0 , κ0 ) → Ω such that (a) h(z, 0) = z ∀z ∈ S0 (b) Denoting Sκ := h(S0 , κ), Sκ is also the set of all points w in Ω such that
(Eκc ) (w), ξ = 0, (c) h(z, κ) ∈ Fz ,
∀(z, κ) ∈ S0 × (−κ0 , κ0 ).
∀ξ ∈ ∆w
(23)
I+1 Proof. We first fix a point z in S0 . Let N be the orthogonal space of i=1 Mci in H 1/2 for the L2 scalar product. As in Section 3, we can define a local chart
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Cz : O ⊂ (N )N → Fz near z, by the formula C(χ) = span(ψ + χ), where ψ = (ψ1 , . . . , ψN ) is an orthonormal basis of z consisting of eigenvectors of H c + V , with eigenvalues µ1 ≤ · · · ≤ µN (i.e., λc1 , . . . , λcI counted with multiplicity). The Hessian of E0c ◦Cz at χ = 0 is given once again by formula (20). It is an isomorphism between (N )N and its dual. So, arguing as in Section 3, we find, by the implicit function theorem, a small constant κz > 0, a neighborhood ωz of z in Fz and a ˜ z : (−κz , κz ) → Ω z such that: function h ˜ z (0) = z (i) h ˜ z (κ) is the unique point w in Ω z such that (ii) h
(Eκc ) (w), ξ = 0,
∀ξ ∈ ∆w
(24)
Since S0 is compact and Eκc (w) a smooth function of (w, κ), it is possible to choose z such that κ0 := inf z∈S0 κz > 0, with Ω := κz , Ω z∈S0 Ωz a neighborhood of S0 , ˜ and h(z, κ) := hz (κ) a smooth function on S0 × (−κ0 , κ0 ) with values in Ω. This function satisfies (a,b,c). From (b) any critical point of Eκc in Ω must lie on Sκ . From (c) it follows that Sκ is a submanifold diffeomorphic to S0 , and transverse to each fiber Fz in GN (H 1/2 ). If z ∈ S0 is a critical point of Eκc ◦ h(·, κ), then, taking w = h(z, κ), the derivative of Eκc at w vanishes on Tw Sκ . From (b), it also vanishes on the subspace ∆w which is transverse to Tw Sκ in Tz GN (H 1/2 ), hence (Eκc ) (w) = 0. This shows that the set of critical points of Eκc in Ω coincides with the set of critical points of the restriction of Eκc to Sκ . Arguing as in the proof of Proposition 8, one gets more: Proposition 13. For κ > 0 small, the solutions of (DF) of smallest energy Eκc are exactly the minimizers of Eκc on Sκ . We are now ready to prove Proposition 10. Proof of Proposition 10. Since κ is small, for any matrix A ∈ SU (2) the map + + 1/2 ) and Pκ,A•W c induces a diffeomorphism between the submanifolds GN (Pκ,W c H κ κ + 1/2 GN (Pκ, A•Wκc H ) . Now, we fix A ∈ SU (2) such that (18) holds. Then there exists a unique point W + ∈ GN (H 1/2 ) such that − + Pκ,W = 0, cW κ
By (18), we have
+ + Pκ, = A • Wκc A•Wκc W
(25)
W + = A • Wκc .
On the other hand, in [14] it was proved that Eκc (A • Wκc ) =
sup W ∈GN (H 1/2 ) + c P c W =A•Wκ κ,A•Wκ
Eκc (W )
(26)
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and A • Wκc is the unique solution of this maximization problem. Therefore, Eκc (A • Wκc ) > Eκc (W + ) . But
Eκc (W + ) ≥
inf
+ 1/2 ) W ∈GN (Pκ,W cH
Eκc (W ) ,
κ
hence, by invariance of Eκc
=
Eκc
under the action of SU (2),
Eκc (A
• Wκc ) >
inf
+ 1/2 ) W ∈GN (Pκ,W cH
Eκc (W ) ,
κ
and the proposition is proved.
Since there are no solutions of (DF) under level Eκc , and ecκ ≤ Eκc , Proposition 10 has the following consequence: Corollary 14. If (P) is satisfied, then for c large enough and κ small, there is no solution W∗ of the nonlinear Dirac-Fock equations with positive Lagrange multipliers, such that the couple + ) (W∗ , Pκ,W ∗ realizes the max-min ecκ . So Conjecture M is wrong when (P) holds. We now exhibit a case where (P) holds. Proposition 15. Assume that N =
I
Nic + 1, I ≥ 1. Then (P) is satisfied.
i=1
Proof. Step 0. Fix c large enough and take a sequence of positive parameters (κ )≥0 converging to 0. Let (Wc )≥0 be a sequence in GN (H 1/2 ), with Wc a minimizer of Eκc on Sκ . Let ψc ∈ Wc be an eigenvector of the mean-field Hamiltonian Hκc ,W c , normalized in L2 and corresponding to the highest occupied level. Extracting a subsequence if necessary, we may assume that ψc → ψ c ∈ McI+1 = Ker(H c + V − λcI+1 ). Moreover, from Theorem 12 we have Wc → W0c =
I
Mci ⊕ C ψ c .
i=1
Pκ− ,W c ψc
Step 1. Fix c ≥ 1 . Since = 0, we can write, by a classical result due to Kato, +∞ 1 (27) (Hκc ,Wc − iη)−1−(Hκc ,A•Wc − iη)−1 ψc dη Pκ− ,A•W c ψc = 2π −∞ 1 +∞ c (Hκ ,Wc − iη)−1 (Hκc ,A•Wc −Hκc ,Wc )(Hκc ,A•Wc − iη)−1 ψc dη = 2π −∞ κ +∞ c (H + V − iη)−1 (ΩA•W0c − ΩW0c )(H c + V − iη)−1 ψ c dη + o(κ ) , = 2π −∞
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c where by ΩW we denote the nonlinear part of Hκ,W : c = H c + V + κ ΩW . Hκ,W I c But note that since the space i=1 Mi is invariant under the action of SU (2), ΩA•W0c − ΩW0c = ΩA•ψc − Ωψc . c , So, we just have to prove that for c sufficiently large and for all ψ c ∈ MI+1 there exists A ∈ SU (2) such that +∞ (H c + V − iη)−1 (ΩA•ψc − Ωψc )(H c + V − iη)−1 ψ c dη = 0 . (28) −∞
Since (H c + V − iη)−1 ψ c =
ψc − iη
λcI+1
and Ωψc ψ c = 0 ,
c , there exists A ∈ SU (2) what we need to prove is that for all nonzero ψ c ∈ MI+1 c c such that L (ΩA•ψc ψ ) = 0, with +∞ dη c . (H c + V − iη)−1 c L := λI+1 − iη −∞
Step 2. We give an asymptotic expression for Lc when c → +∞: 1 1 +∞ 1 η −1 d(η/c2 ) 1 c c (H + V ) − i 2 = 2 Lc + O 2 , L = 2 λcI+1 c −∞ c2 c c c − i η2 2 c
(29)
c
where Lc , in the Fourier domain, is the operator of multiplication by the matrix +∞ ˆ Lc (p) = (−iu + β + (α · p)/c)−1 (−iu + 1)−1 du . (30) −∞
Here, we have used the standard fact that 1 λcI+1 = 1 + O . c2 c2 We have (−iu + β + (α · p)/c)−1 = with ω c (p) :=
1 + |p|2 /c2 ,
1 1 ˆ c (p) + ˆ c (p) Λ Λ −iu + ω c (p) + −iu − ω c (p) − c ˆ c (p) = ω (p) ± (β + (α · p)/c) . Λ ± 2ω c (p)
Hence, by the residues theorem, |p|2 2ˆ (α · p) . Lc (p) = β − 1 + +O π c c2
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Step 3. It is well known (see [16]) that ψ c can be written as 1 φ c ψ = −i(σ·∇)φ + O 2 , c 2c φ ∈ L2 (R3 , C2 ) being an eigenstate of ( −∆ 2 + V ), with eigenvalue µ = limc→+∞ (λcI+1 − c2 ). Since we have assumed that V is smooth, this asymptotic result holds for the topology of the Schwartz space S(R3 ). So, 1 0 2c2 c i c L (ΩA•ψc ψ ) = +O 2 , π c f (A, φ) c where
x · σ x · σ 2 ∗ f (A, φ) := |A • φ|2 ∗ φ −
A • φ, φ (A • φ) . C |x|3 |x|3
(31)
What remains to prove is: Step 4. For any eigenvector φ of the Schr¨ odinger operator − ∆ 2 + V , there exists an A ∈ SU (2) such that f (A, φ) ≡ 0 . Proof of Step 4. We consider the integral
(x · σ)φ, f (A, φ)C2 (r ω)dω . IA,φ (r) := S2
x Since φ has exponential fall-off at infinity, the electrostatic field |A • φ|2 ∗ |x| 3 x 1 2 when |x| is large. The takes the asymptotic form |x|3 + O |x|3 R3 |A • φ| x same phenomenon holds for the convolution product < A • φ, φ >C2 ∗ |x| 3 . As a consequence, for r large, r IA,φ (r) = |A • φ|2 |φ|2 (r ω) dω R3 S2 −
A • φ, φC2
φ, A • φC2 (r ω) dω R3 S2 1 2 +O |φ| (r ω) dω . r S2
Since • is unitary, the Cauchy-Schwarz inequality gives |φ|2 (r ω) dω = |A • φ|2 (r ω) dω ≥
A • φ, φC2 (r ω) dω . S2
S2
S2
By Lemma 1 of Section 1, we can choose A such that φ and A • φ are not colinear. Then 2 2 |A • φ| = |φ| >
A • φ, φC2 . R3
R3
R3
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Some Connections between Dirac-Fock and Electron-Positron Hartree-Fock 101
So there is a constant δ > 0 such that, for r large enough, 2 |φ| |φ|2 (r ω) dω . |r IA,φ (r)| ≥ δ R3
(32)
S2
Being an eigenvector of the Schr¨ odinger operator − ∆ 2 + V , the function φ cannot have compact support. So the lower estimate (32) implies that the function IA,φ (r) is not identically 0, hence f (A, φ) ≡ 0 . Step 4 is thus proved, and (P) is satisfied. Acknowledgments. The authors wish to thank the referee for useful comments on the first version of this paper. Financial support of the European Union through the IHP networks “Analysis and Quantum” (HPRN-CT-2002- 00277) and “HYKE” (HPRN-CT-200200282), is gratefully acknowledged.
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[10] V. Bach, J.M. Barbaroux, B. Helffer, H. Siedentop, Stability of matter for the Hartree-Fock functional of the relativistic electron-positron field, Doc. Math. 3, 353–364 (1998). [11] V. Bach, J.M. Barbaroux, B. Helffer, H. Siedentop, On the stability of the relativistic electron-positron field, Comm. Math. Phys. 201(2), 445–460 (1999). [12] J.-M. Barbaroux, W. Farkas, B. Helffer, H. Siedentop, On the Hartree-Fock equations of the electron-positron field, Preprint. [13] P. Chaix, D. Iracane, P.L. Lions, From quantum electrodynamics to mean-field theory: II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation, J. Phys. B 22 (23), 3815–3828 (December 1989). [14] M.J. Esteban, E. S´er´e, Nonrelativistic limit of the Dirac-Fock equations, A.H.P. 2, 941–961 (2001). [15] M.J. Esteban, E. S´er´e, A max-min principle for the ground state of the DiracFock functional, Contemp. Mathem. 307, 135–139 (2002). [16] B. Thaller, The Dirac Equation, Springer-Verlag, 1992. [17] V. Bach, E.H. Lieb, M. Loss, J.P. Solovej, There are no unfilled shells in unrestricted Hartree-Fock theory, Phys. Rev. Lett. 72(19), 2981–2983 (1994). [18] A. Ambrosetti, M. Badiale, Homoclinics: Poincar´e-Melnikov type results via a variational approach, Annales de l’IHP, Analyse non lin´eaire 15(2), 233–252 (1998). Jean-Marie Barbaroux CPT-CNRS Luminy Case 907 F-13288 Marseille Cedex 9 France email:
[email protected] Maria J. Esteban and Eric S´er´e Ceremade (UMR CNRS no. 7534) Universit´e Paris IX-Dauphine Place de Lattre de Tassigny F-75775 Paris Cedex 16 France email:
[email protected] email:
[email protected] Communicated by Bernard Helffer submitted 19/03/04, accepted 30/07/04
Ann. Henri Poincar´e 6 (2005) 103 – 124 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/010103-22 DOI 10.1007/s00023-005-0200-5
Annales Henri Poincar´ e
Localization for a Family of One-dimensional Quasiperiodic Operators of Magnetic Origin S. Jitomirskaya, D.A. Koslover and M.S. Schulteis
Abstract. We show strong dynamical localization for a family of one-dimensional quasiperiodic Jacobi operators of magnetic origin, throughout the regime of positive Lyapunov exponents.
1 Introduction The study of electrons subjected to a perpendicular magnetic field and twodimensional periodic potentials can be reduced via an appropriate gauge choice to the study of spectral properties of discrete one-dimensional quasiperiodic Jacobi matrices. The simplest and best studied case is the almost Mathieu operator Hθ,λ,ω acting on 2 (Z) by (using a somewhat nonstandard rescaling) (Hθ,λ,ω ψ)(n) = λ(ψ(n + 1) + ψ(n − 1)) + 2 cos 2π(θ + nω).
(1)
This is obtained from the model through a Landau gauge with, in general, anisotropic nearest neighbor couplings (e.g., [15]). For a recent review of the spectral theory of operator (1) see [22]. Two other models that include the almost Mathieu operator for a certain choice of parameters have been proposed in the physics literature [26, 3, 11]. The first model is the case of a square lattice with anisotropic nearest neighbor coupling and isotropic next nearest neighbor coupling. This is also the model that applies if one exposes an ultracold cloud of atoms to a bichromatic standing light wave [11]. The second model has anisotropic coupling to nearest neighbors and next nearest neighbors on a triangular lattice. This paper considers a slightly more general model that includes both cases above. Namely, we study one-dimensional operators corresponding to the case of anisotropic coupling of both nearest neighbors and second nearest neighbors on a two-dimensional lattice. The corresponding Hamiltonian Hθ,λ,ω acting on 2 (Z) is given by (Hθ,λ,ω ψ)(n) = c(n, θ, ω)ψ(n + 1) + c(n − 1, θ, ω)ψ(n − 1) + v(n, θ, ω)ψ(n) (2) where v(n, θ, ω) = v(θ + nω)
c(n, θ, ω) = c(θ + nω)
(3)
The authors were supported in part by BSF grant 2002068 and NSF grant DMS-0300974.
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and v(θ) c(θ)
= v(θ, λ) = 2 cos 2π(θ) = c(θ, λ) = λ2 + λ3 e
{2πi(θ+ 12 ω)}
(4) + λ1 e
{−2πi(θ+ 12 ω)}
,
the normalized hopping terms λ = (λ1 , λ2 , λ3 ) and θ ∈ T = R/(2πZ). The hopping terms (see Figure 1) are proportional to the probability an electron will hop to a neighboring site. ω is the magnetic flux. We will identify c(n, θ, λ) with c(n) and v(n, θ, λ) with v(n) when it can be done without confusion. The almost Mathieu family of operators corresponds to the choice λ1 = λ3 = 0. λ1 r
rλ r1 * 3 @ I r - rλ2 λ2 r @ @@ Rr r r λ3 λ1 1 Figure 1: Hopping terms. In the case of the almost Mathieu operator, it has been shown that if λ < 1, Hθ,λ,ω has only pure point spectrum with exponentially decaying eigenfunctions. And further, it exhibits strong dynamical localization in that region [13]. If λ = 1, it has purely singular-continuous spectrum and if λ > 1, the spectrum is purely absolutely continuous. (See [16] for a history of results.) For our general family, Hθ,λ,ω , it has been shown that if λ1 + λ3 < 1 and λ2 < 1, then the operator has positive Lyapunov exponents, thus no absolutely continuous spectrum. If λ1 + λ3 < λ2 and 1 < λ2 , or if λ1 or λ3 = 0, and λ1 + λ3 > max(λ2 , 1), there is no pure point spectrum [23] and our preliminary results indicate absolutely continuous spectrum [21]. If λ1 = λ3 and λ1 + λ3 > max(λ2 , 1) there is no absolutely continuous spectrum and we expect to show singular continuous spectrum [21]. (See Figure 2.) In this paper, we will show that if λ1 + λ3 < 1 and λ2 < 1, then under certain arithmetic conditions on (ω, θ), Hθ,λ,ω has only pure point spectrum with exponentially decaying eigenvalues. This is exactly the region of positive Lyapunov exponents, and pure point spectrum is not expected elsewhere. Also, localization does not hold under certain complementary conditions in frequency [9] or in phase [18]. Results on the other regions will follow in a separate paper. As originally shown in [10], spectral localization may not have any dynamical consequences. Dynamical localization, defined as a non-spread of initially localized wave-packets, requires additional arguments. For ergodic families, such as (2)–(4), a stronger statement, so-called strong dynamical localization (see Section 3), is also desirable. This has been achieved for random potentials throughout the regime of multiscale analysis by [14]. Strong dynamical localization also follows naturally
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1
+
105
no a.c. spectrum
3
1 s.c. spectrum
p.p. spectrum
s.c. spectrum 1
no p.p. spectrum
2
a.c. spectrum
Figure 2: In the unlabeled region, we expect an interesting dependence of the spectrum on the parameters λ1 and λ3 . In certain cases, it is expected that there is no a.c. spectrum. In other cases, we expect a.c. spectrum. No p.p. spectrum is expected throughout this region. whenever the Aizenman-Mochanov method applies [1]. In the quasiperiodic (and skew-shift) case, it was shown in [6] that dynamical localization follows from the approach of [5]. Strong dynamical localization in the quasiperiodic setting has only been established for the almost Mathieu operator [13]. In this paper, we prove strong dynamical localization throughout the regime of positive Lyapunov exponents. The rest of the paper is organized as follows. In Section 2, we formulate and prove our main result on Anderson localization. We follow the general scheme of [16] highlighting important difference as well as providing more detail. In particular, our proof is designed to work equally well for phases θ = ω/2 or (ω −1)/2 where ∈ Z. That was not the case in [16]. Those cases have recently been shown to be important for the proof of Cantor spectrum [25]. In Section 3, we formulate and prove strong dynamical localization, following the strategy of [13] with a modification as outlined in [7].
2 Anderson localization We say that ω is Diophantine if there exists b(ω) and 1 < r(ω) < ∞ such that | sin πjω| >
b(ω) |j|r(ω)
for all j ∈ Z. We define resonant phases as Θ(ω, α) = {θ : ∃j,
|j| < 3k α ,
−1
| sin 2π(θ + (k/2)ω)| < exp −k (2r(ω))
holds for infinitely many k’s} .
(5)
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The Lebesgue measures of the complement of {ω : ω is Diophantine} and the set of resonant phases are zero. Theorem 1. Suppose ω is Diophantine, θ ∈ ΘC (ω, α), λ2 < 1, and λ1 +λ3 < 1, then Hθ,λ,ω has only pure point spectrum with exponentially decaying eigenfunctions. Remarks 1. As discussed in the introduction, localization does not hold or is not expected to hold in all other regions of (λ1 , λ2 , λ3 ). 2. For all proofs in this paper, we will assume without loss of generality that 0 ≤ λ2 < 1, 0 ≤ λ1 + λ3 < 1 and at least one of λ1 , λ2 , λ3 > 0 . As usual, a formal solution ψE of the eigenvalue equation Hθ,λ,ω ψE = EψE is called a generalized eigenfunction if |ψE (n)| ≤ C(1 + |n|) for C = C(ψE ) < ∞. We will prove exponential decay of all generalized eigenfunctions, which is sufficient by [4]. The n-step transfer matrix for the eigenvalue equation is 1 1 E − v(i) −c(i − 1) M (θ, n, E) = c(i) 0 c(i) i=n
where
ψ(n + 1) ψ(n)
= M (θ, n, E)
ψ(1) ψ(0)
.
M (θ, n, E) is defined when |c(n, ω, θ, λ)| is strictly greater than zero. Denote Pk (θ, E) = det[(E − Hθ,λ,ω )|[1,k] ], for k ∈ N. Define P−1 (θ, E) = 0 and P0 (θ, E) = 1. Then the n-step transfer matrix can be written as 1 −c(0)Pn−1 (θ + ω, E) Pn (θ, E) M (θ, n, E) = c(1) . . . c(n) c(n)Pn−1 (θ, E) −c(n)c(0)Pn−2 (θ + ω, E) 1 g(θ, n, E). (6) := c(1) . . . c(n) This can be shown, for example, by expanding the determinant of (E −Hθ,λ,ω )|[1,k] in its final row and then using induction in n. 1 Let γn (M (E)) = ln M (θ, n, E). By Kingman’s subadditive ergodic n theorem 1 γ(M (E)) := lim ln M (θ, n, E) n−→∞ n exists for almost every θ ∈ T and fixed E and further 1 1 1 1 γ(M (E)) = lim ln M (θ, n, E)dθ = inf ln M (θ, n, E)dθ. n−→∞ n 0 n n 0 γ(M (E)) is called the Lyapunov exponent. It has been shown that γ(M (E)) is strictly greater than zero in the region we are considering (e.g., [23]).
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Note that 0 < γ(M (E))
= :=
1 n−→∞ n
1
lim
0
1 n−→∞ n
ln g(θ, n, E)dθ − lim
0
1
ln |c(1) . . . c(n)|dθ
γ(g(E)) − C(λ),
(7)
where both limits exist by the subadditive ergodic theorem. Further, 1 1 ln g(θ, n, E)dθ. γ(g(E)) = inf n n 0 and γ(g(E)) = lim
n−→∞
(8)
1 ln g(θ, n, E) n
(9)
for any fixed E and almost every θ . Finally, 1 n 1 1 ln |c(k)|dθ = ln |c(0)|dθ. C(λ) = lim n−→∞ n 0 0 k=1
Applying the Poisson-Jensen formula and using (4), ln λ3 if λ3 ≥ λ1 ≥ 0 and λ1 + λ3 ≥ λ2 ≥ 0 ln λ 1
if λ1 ≥ λ3 ≥ 0 and λ1 + λ3 ≥ λ2
2λ λ C(λ) =
1 3 ln
if λ1 + λ3 ≤ λ2 , λ1 , λ3 = 0
−λ2 + λ22 − 4λ1 λ3
ln λ2 if λ1 + λ3 ≤ λ2 , λ1 or λ3 = 0 . In the region we are considering, C(λ) < 0. For all n ∈ N, by (7) and (8) 1 1 ln g(θ, n, E)dθ . γ(M (E)) + C(λ) ≤ n 0 So applying (6), en(γ(M(E))+C(λ)) ≤ max θ
Pn (θ, E) −c(0)Pn−1 (θ + ω, E) c(n)Pn−1 (θ, E) −c(n)c(0)Pn−2 (θ + ω, E)
.
Thus, 1 √ en(γ(M(E))+C(λ)) ≤ (10) 4 2 max max [|Pn (θ, E)|, |Pn−1 (θ, E)|, |Pn−1 (θ + ω, E)|, |Pn−2 (θ + ω, E)|] θ
where we used an obvious upper bound for c.
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1 √ exp[nγ(M (E)) + (n + 2)C(λ)] ≤ 8 2 |Pn (θ, E)|}. From the derivation above, at least one of every {n, n + 1, n + 2} belongs to K. We will use the notation H[x1 , x2 ](θ, λ, ω) for Hθ,λ,ω restricted to the interval [x1 , x2 ] with zero boundary conditions at x1 − 1 and x2 + 1. G[x1 , x2 ](E, θ, x, y) will be the corresponding Green’s function (H[x1 , x2 ](θ, λ, ω) − E)−1 (x, y). Let K = {n ∈ N : ∃θ ∈ [0, 1] :
Definition 1. Let m > 0, E ∈ R, be fixed. A point y ∈ Z will be called (m, k, E, θ)regular if there exists an interval [x1 , x2 ], x2 = x1 + k − 1 containing y such that |G[x1 , x2 ](y, xi )| < exp
−mk 9
and dist(y, xi ) ≥ 19 k for i = 1, 2. Otherwise, y will be called (m, k, E, θ)-singular. The value of any formal solution ψ of the equation Hψ = Eψ, energies E ∈ / σ(H[x1 , x2 ](θ, λ, ω)), at a point x ∈ [x1 , x2 ] ⊂ Z can be reconstructed from the boundary values via ψ(x) = −c(x1 − 1)ψ(x1 − 1)G[x1 , x2 ](x, x1 ) − c(x2 )ψ(x2 + 1)G[x1 , x2 ](x, x2 ). (11) This implies that if ψE is a generalized eigenfunction, then every point y ∈ Z with ψE (y) = 0 is (m, k, E, θ)-singular for k sufficiently large, k > k1 (E, m, θ, y) and m > 0. Define −1 j Θk (y1 , ω, α) = {θ : ∃j, |j| < 3k α , | sin 2π(θ + y1 + ω)| < exp −k (r(ω)) } 2
(12)
and Θ(y1 , ω, α) = =
lim sup Θk (y1 , ω, α) k
−1
{θ : ∃j, |j| < 3k α , | sin 2π(θ + y1 + (k/2)ω)| < exp −k (2r(ω)) for infinitely many k’s}, (13)
Note: Θ(0, ω, α) = Θ(ω, α), as defined in (5). By a Borel-Contelli argument, Θ(y1 , ω, α) has measure zero. Our key technical statement is the following: Lemma 1. Let λ and ω be as in Theorem 1. Suppose θ ∈ Θk (y, ω, α)C . Then if
∈ (0, 12 γ(M (E))) and 1 < α < 2, ∃k2 (θ, ω, y, , α, E) : ∀k > k2 (θ, ω, y, , α, E), if x and y are both (γ(M (E)) − , k, E, θ)-singular and |x − y| > 34 k, then |x − y| > (k − 2)α .
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Remarks 1. In Lemma 1, the y dependence of k2 comes entirely through k5 (y, θ, ω). (See the discussion after (20).) 2. The E-dependence of k2 comes entirely through k3 ( , E, ω). (See Lemma 2.) Proof. We start the proof with a series of statements given in Lemmas 2-7. In Lemmas 2-4, we find a uniform upper bound on |Pk (θ, E)|. Applying Lemma 2, we show in Lemma 6 that if y is (γ(M (E)) − , k, E, θ)-singular and x is within a certain range of distance from y, then |Pk (θ + (x− 1)ω)| is unusually small. Finally, we show that if two points y1 and y2 are relatively close together and they are both (γ(M (E)) − , k, E, θ)-singular for some θ, then |Pk (θ)| is uniformly small. Taking this in combination with the lower bound found in (10), for large enough k, we obtain a contradiction. Thus singular points can occur in small clusters, but the clusters must be far apart. First, we wish to bound n1 ln g(θ, n, E) from above uniformly in θ. This bound can actually be made uniform in E as well and that is important for our proof of strong dynamical localization. We will prove an E-independent bound in Section 3. We need to show that 1 for all θ ∈ T. (14) lim sup ln g(θ, k, E) ≤ γ(g(E)) k−→∞ k This result was obtained in [8] for transfer matrices associated with almost periodic Schr¨ odinger operators (for all θ, but a.e. θ is sufficient for our purposes). Examining the proof of [8] shows that it applies to any quasiperiodic cocycle which is analytic in E. Thus (14) holds. Next, in order to bound |Pn (θ, E)|, we will prove Lemma 2. For every E ∈ R, ω irrational, and > 0, there exists k3 ( , E, ω) such that for all n > k3 ( , E, ω), |Pn (θ, E)| ≤ en(γ(g(E))+) = en(γ(M(E))+C(λ)+) for all θ∈T Proof. From (14) for every > 0 and k > K(θ, , E, ω) 1 ln g(θ, k, E) ≤ γ(g(E)) +
k
for a.e. θ ∈ T
and, by (6), |Pk (θ, E)| ≤ g(θ, k, E) ≤ ek(γ(g(E))+)
for a.e. θ .
(15)
To show that K can be chosen uniformly in θ, we will apply the following lemma
k j Lemma 3. ([16]) Let f (z) = j=0 cj z be an arbitrary kth degree polynomial. k Suppose |f (z0 )| ≥ a for some a > 0, z0 ∈ [−1, 1]. Then for any 0 < b < a and sufficiently large k (k > k0 ( ab )) |{θ ∈ (0, 1) : |f (cos 2πθ)| < bk }| ≤ c(a, b) < 1 .
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To apply Lemma 3, we must show that Pk (θ, E) can be written as a polynomial in cos θ . Lemma 4. 1) Pk (θ, E) is an even function around θ = −(k + 1)/2. 2) It can be written as a kth degree polynomial in cos 2π(θ +
k+1 2 ω).
Proof. Define AS to be the operation which first permutes the rows of A so that they are in reverse order and then similarly permutes the columns of A. This operation requires an even number of permutations, so det AS = det A.
k+1
ω, E = det E − H−θ− k+1 ω,λ,ω
Pk −θ − 2 2 [1,k] S
= det E − H−θ− k+1 ω,λ,ω
2 [1,k]
= det E − Hθ− k+1 ω,λ,ω
2 [1,k] k+1 ω, E = Pk θ − 2 which completes part 1. From part 1, Pk (θ, E) =
∞
aj exp(2πiθ )
j=−∞
where θ = θ + k+1 2 ω and aj = a−j ∈ R. It remains to show that aj = 0 for |j| > k. From (3) and (4), 2n − k − 1 k+1 ω + ω v(n) = 2 cos 2π θ+ 2 2 = b1 (n) exp 2πiθ + b2 (n) exp −2πiθ and c(n) = λ2 + b3 (n) exp 2πiθ + b4 (n) exp −2πiθ where b1 (n), b2 (n) ∈ R and b3 (n), b4 (n) ∈ C. Taking the determinant of
E − Hθ− k+1 ω,λ,ω
, we can write [1,k]
2
Pk (θ, E) =
k
b(n) exp 2πinθ
n=−k
where b(n) ∈ R.
Let k+1 k+1 ω := Q cos 2π θ + ω . (16) am cos 2π θ + Pk (θ, E) = 2 2 m=0 k
m
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From now on, for sets, we will use | · | for Lebesgue measure. Let An = {θ ∈ (0, 1) : ∀k > n, |Q(cos 2πθ)| ≤ exp (γ(g(E)) + 2 )k}. By (15) and (16), |AC n| goes to zero as n goes to infinity. Let N be such that for all k ≥ N , |AC | n < 1 (1 − c(exp[γ(g(E)) +
], exp[γ(g(E)) + ])) 2 2 Assume there exists a θ, such that |Q(cos 2πθ)| > exp k(γ(g(E))+ ) for some k > N . Then by Lemma 3
θ ∈ (0, 1) : |Q(cos 2πθ)| < exp k γ(g(E)) +
2
≤ c exp [γ(g(E)) + ] , exp γ(g(E)) + <1. 2
> 0. |AC | > 1 − c exp [γ(g(E)) +
] , exp γ(g(E)) + n 2 This contradiction proves the lemma.
Thus
Remark. As far as more general models are concerned, the proof above uses Lemma 3, and therefore only holds for polynomial v and c. An alternate proof, that can be applied to any continuous v and under very mild restrictions on c, is given in Section 3. However, it requires an exclusion of an additional countable set of frequencies. In order to obtain bounds for the Green’s function, we will need to bound the product c(1) . . . c(n) as given in (6). We will use the following lemma. Lemma 5. ([16]) If f (x) is an analytic function, z ∈ [min f, max f ] and p(f ) the maximum number of times f assumes any value on [0, 1), then for all > 0 there exists a N ( ) such that for n > N ( ) and, if desired, any ∈ [0, . . . , n − 1] n−1
ln |z − f (θ + jω)| ≤ n
j=0 j= n−1
0
ln |z − f (θ + jω)| ≥ n
j=0 j=
+ p(f )D(ω)n
1
1
0
1−r(ω)−1
ln |z − f (θ)|dθ +
(17)
ln |z − f (θ)|dθ −
ln n
min
j=0...n−1 j=
(18)
|z − f (θ + jω)|
.
By (17), for all > 0, there exist k4 ( ) such that if b − a > k4 ( ) ln
b−1 j=a
|c(j)| ≤ (b − a)
0
1
1 ln |c(a)|dθ +
18
= (b − a)(C(λ) +
1
) . 18
(19)
The next lemma shows that an (m, k, E, θ)-singular point “produces” many phases such that, for fixed E, |Pk (θ, E)| is “abnormally” small.
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Lemma 6. If y ∈ Z is (γ(M (E)) − , k, E, θ)-singular, < 12 γ(M (E)) and k > 1 9 max [k3 ( 18
, E, ω), k4 ( ), 8], then for any x such that y−[ 78 k] ≤ x ≤ y−[ 87 k]+[ 34 k], |Pk (θ + (x − 1)ω)| ≤ exp{kγ(M (E)) + (k − 1)C(λ) −
1 k } . 18
Proof. Let x1 = x and x2 = x + k − 1. If y ∈ [x1 , x2 ] is (γ(M (E)) − , k, E, θ)singular, one of the following is true: 1. G[x1 , x2 ] does not exist or E ∈ σ(H[x1 , x2 ]). Thus Pk (θ + (x1 − 1)ω) = 0 and the lemma holds. 2. y − x1 or x2 − y < 19 k. However, k/9 < k − [7k/8] − 1 < x2 − y and k/9 < [7k/8] − [3k/4] < y − x1 since k > 72. 1 3. |G[x1 , x2 ](y, x1 )| ≥ exp − (γ(M (E)) − ) k 9 1 or |G[x1 , x2 ](y, x2 )| ≥ exp − (γ(M (E)) − )k . 9 By a straightforward computation using Cramer’s rule, if x1 < y < x2 , then
Px2 −y (θ + yω) y−1
|G[x1 , x2 ](y, x1 )| =
|c(j)| and Pk (θ + (x1 − 1)ω) j=x 1
2 −1
Py−x1 (θ + (x1 − 1)ω) x
|c(j)| . |G[x1 , x2 ](y, x2 )| =
Pk (θ + (x1 − 1)ω) j=y Thus, |Px2 −y (θ + yω)| exp
1 (γ(M (E)) − )k 9
|Py−x1 (θ + (x1 − 1)ω)| exp
y−1
|c(j)| ≥ Pk (θ + (x1 − 1)ω) or
j=x1
1 (γ(M (E)) − )k 9
x 2 −1
|c(j)| ≥ Pk (θ + (x1 − 1)ω) .
j=y
Since k > 9k3 ( /18, E, ω), we have y − x1 and x2 − y > k3 ( /18, E, ω), so by Lemma 2, 1 |Px2 −y (θ, E)| ≤ exp (x2 − y) γ(M (E)) + C(λ) +
18 8 1 ≤ exp kγ(M (E)) + (x2 − y)(C(λ) + ) 9 18 and similarly, |Py−x1 (θ, E)| ≤ exp
8 1 kγ(M (E)) + (y − x1 )(C(λ) + ) 9 18
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and since y − x1 and x2 − y > 9k4 , we have by (19) y−1 1 |c(j)| ≤ exp(y − x1 ) C(λ) +
18 j=x 1
x 2 −1 j=y
1 |c(j)| ≤ exp(x2 − y) C(λ) + . 18
7 Next, let y1 and y2 be (γ(M (E)), k, E, θ)-singular, y2 > y1 , ηi = yi − 8 k for i = 1, 2, d = y2 − y1 ≥ 34 k and θ + (η1 + 34 k − 13 k + k−1 0, . .. , [ 13 k] 2 + j)ω for j = θj = (20) 3 k−1 θ + (η2 + 4 k − 2 + j)ω for j = 13 k + 1, . . . , k Combining terms, we get the desired result.
where [x] is the greatest integer less than or equal to x. Since y2 − y1 ≥ 34 k, phases θj , for j = 1, . . . , k, are all distinct. Note that 3/4 here can be improved to any number bigger than 2/3. Additionally, note that cos 2πθm = cos 2πθn for 0 ≤ m, n ≤ k, m = n and k large enough. Indeed, cos 2πθm = cos 2πθn if and only if θm ± θn = 0 or 1. From our definition of θj , θm + θn is of the form 2θ + ω and θm − θn is of the form p ω, where , p ∈ Z. We can eliminate the second 1−ω case by ω ∈ / Q. Thus we should take care of the cases θ = ω 2 and θ = 2 . If ω 1−ω θ ∈ / { 2 , 2 } for ∈ Z, set k5 (y1 , θ, ω) = 1. Otherwise, let k5 (y1 , θ, ω) be the 1 7 −θ 1−2θ smallest k such that η1 + 34 k − 13 k + k+3 2 ≥ y1 + 24 k − 8 > max[ ω , 2ω ]. Then for k > k5 (y1 , θ, ω), we have, by a straightforward computation, that cos θj , for j = 0, . . . , k are distinct. Our choice of θj guarantees that Lemma 6 can be applied. Thus recalling (16)
k + 1
1
ω) < exp kγ(M (E)) + (k − 1)C(λ) − k
|Q (cos 2π (θj ))| = Pk (θj − 2 18 (21) for j = 0, . . . , k. Now we write Qk (z) in the Lagrange interpolation form using points cos 2πθ0 , . . . , cos 2πθk :
k (z − cos 2πθ ) =j
. (22) Qk (cos(2πθj ) |Qk (z)| =
j=0 =j (cos 2πθj − cos 2πθ )
Recalling (12), Lemma 7. Suppose d < k α , for some α ∈ (1, 2), θ ∈ Θk (y1 , ω, α)C , and ω Diophantine. Then for any > 0 there exists k6 ( , ω, α) such that for k > k6 ( , ω, α), for any z ∈ [−1, 1], | =j (z − cos 2πθ )| ≤ exp k . | =j (cos 2πθj − cos 2πθ )|
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Proof. Applying Lemma 5 for k > N ( 3 ) |I1 | : = ln | (z − cos 2πθ )| =j
≤ = and |I2 | :
= ln |
1
(k + 1) ln |z − cos 2πθ|dθ + 3 0
(k + 1) − ln 2 + 3
(cos 2πθj − cos 2πθ )| =
=j
ln | cos 2πθj − cos 2πθ |
=j
ln | cos 2πθj − cos 2πθ|dθ − 3 0 −1 +2D(ω)(k + 1)1−r(ω) ln(k + 1) min | cos 2πθj − cos 2πθ | =0...n =j
= (k + 1) − ln 2 − 3 −1 +2D(ω)(k + 1)1−r(ω) ln(k + 1) min | cos 2πθj − cos 2πθ | . =0...n ≥ (k + 1)
1
=j
Here the integrals were computed using the Poisson-Jensen formula. We want to bound from below min | cos 2πθj − cos 2πθ | = 2 min | sin π(θj + θ )|| sin π(θj − θ )| . We have
2([ 3 k]−[ 78 k]−[ 13 k])+k−1+j+ θ + y1 + 4 ω 2 2(d+[ 34 k]−[ 78 k])−k+1+j+ θ + y1 + ω 2 θj + θ = 2 2([ 34 k]−[ 78 k])−[ 13 k]+d+j+ ω θ + y1 + 2
j, ∈ {0, . . . , [ 13 k]}
j, ∈ {[ 13 k] + 1, . . . , k} j ∈ {0, . . . , [ 13 k]}, ∈ {[ 13 k] + 1, . . . , k} .
Since θ ∈ Θ(y1 , ω, α)C , we have for all j, ∈ {0, . . . , k} (2r(ω))−1 3 | sin π(θj + θ )| > exp − 2d + k + 1 . 4 Also, θj − θ =
(j − )ω
(−d − [ 13 k] + k − 1 + j − )ω
j, ∈ {0, . . . , [ 13 k]} or {[ 31 k] + 1, . . . , k} j ∈ {0, . . . , [ 13 k]}, ∈ {[ 31 k] + 1, . . . , k}
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Since ω is Diophantine | sin π(θj − θ )| > b(ω)
1 2 k− 3 3
−r(ω) .
So, min | cos 2πθj − cos 2πθ | ≥ 2b(ω)
=0...n =j
Combining terms, |I1 | |I2 |
≤
=
2 1 k− 3 3
−r(ω)
(2r(ω))−1 3 exp − 2d + k + 1 . 4
−1 2 (k + 1) + 2D(ω)(k + 1)1−r(ω) ln(k + 1) 3 (2r(ω))−1 −r(ω) 2 3 1 × 2d + k + 1 k− − ln 2b(ω) 4 3 3 2 (k + 1) + o(k) ≤ exp k
exp 3
exp
for k > k6 ( , ω, α).
We are now ready to finish the proofs of Lemma 1 and Theorem 1. Proof of Lemma 1. Let θ ∈ Θk (y, ω, α)C . Set kˆ = max[9k3 ( /18, E, ω), 9k4( ), k5 (y1 , θ, ω), k6 ( /36, ω, α, θ), 72]. [kγ(M(E))+(k+2)C(λ)] ˆ ≥ e √ Pick k ∈ K such that k > kˆ and θˆ such that |Pk (θ)| . Let 8 2 α zˆ = cos 2π(θˆ + k+1 2 ω). Assume d < k . Then, by (21) and Lemma 7
e[kγ(M(E))+(k+2)C(λ)] √ z )| ≤ (k + 1)ekγ(M(E))+(k−1)C(λ)−k/36 . ≤ |Qk (ˆ 8 2 There exists a k7 ( ), such that for k > k7 ( ) this statement is contradictory. So ˆ k7 ] and k ∈ K, if x and y are both (γ(M (E)) − , k, E, θ)-singular for k > max[k, and |x − y| > 34 k, then |x − y| > k α . We would like to eliminate the condition ˆ k4 ] + 2 and |x − y| > 3 k > 3 (k − 1) > 3 (k − 2). One k ∈ K. Thus let k > max[k, 4 4 4 ˆ k7 ] + 2 of k − 2, k − 1, k ∈ K. So at worst, |x − y| > (k − 2)α . Letting k2 = max[k, completes the proof. By a Borel-Cantelli argument, we obtain as an immediate corollary Lemma 1a. Let λ and ω be as in Theorem 1. Suppose θ ∈ Θ(y, ω, α)C . Then if
∈ (0, 12 γ(M (E))) and 1 < α < 2, ∃k2 (θ, ω, y, , α, E) : ∀k > k2 (θ, ω, y, , α, E), if x and y are both (γ(M (E)) − , k, E, θ)-singular and |x − y| > 34 k, then |x − y| > (k − 2)α .
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Proof of Theorem 1. Let E(θ) be a generalized eigenvalue of Hθ,λ,ω with ψ(E, x) the corresponding eigenfunction. Assume without loss of generality that ψ(E, 0) = 0. Let 0 < < 12 γ(M (E)), |x| > max[k1 (E,γ(M (E)) − ,θ,0), k2 (θ,ω,0, ,1.5,E),2] and 1.5 . By Lemma k = |x|. Thus 0 is (γ(M (E)) − , k, E, θ)-singular and 3k 4 < |x| < k 1a, x must be (γ(M (E)) − , k)-regular. So there exists an interval [x1 , x2 ] where |G[x1 , x2 ](x, x1 )| |G[x1 , x2 ](x, x2 )| with x − x1 ≥
1 k 9
1 < exp − (γ(M (E)) − )k 9 1 < exp − (γ(M (E)) − )k 9 and
x2 − x ≥
and
1 k. 9
Applying (11) |ψ(x)| ≤ |c(x1 − 1)||ψ(x1 − 1)||G[x1 , x2 ](x, x1 )|
+ |c(x2 )||ψ(x2 + 1)||G[x1 , x2 ](x, x2 )|
≤ C |x1 |e−(γ(M(E))−)k/9 + |x2 |e−(γ(M(E))−)k/9
≤ Ce−{|x|(γ(M(E))−)/10}
for large x.
3 Strong dynamical localization We will use the following Definition 2. An ergodic family H(θ), θ ∈ Θ acting on a Hilbert space H is strongly dynamically localized if, for any q > 0 and initial state ψ ∈ H, ψ = 1, that decays faster than any polynomial, there exists a constant C(q, ψ) < ∞ such that sup exp{−iHt}ψ, |X|q exp{−iHt}ψ dP ≤ C(q, ψ) (23) Θ
t
where X is the position operator and P is the probability measure. Remark. Dynamical localization (as opposed to strong dynamical localization) is defined by the same inequality without the integration and holds for almost every θ ∈ Θ. Theorem 2. Assume ω is Diophantine. 1) If λ2 < 1, λ1 + λ3 < 1, λ1 = λ3 , then the family (Hθ,λ,ω )θ∈T is strongly dynamically localized. 2) If λ1 = λ3 < 1/2, λ2 < 1, the same statement is true under the additional condition that −λ2 1 mod 1 S, (24) ωk = cos−1 π 2λ1 for all k ∈ Z\{0}.
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For the proof of strong dynamical localization, we will need to make some of our estimates uniform in energy. For this purpose, we will first establish certain more general continuity results. Given a function c(θ) ∈ C(T) with at most countably many zeros, we will define the countable set Ac ⊂ T by the following rule. Let c0 , . . . , ck , . . . be the zeros of c(θ). We set Ac = {ω : ω = ci − cj (mod 1) for some , i, j ∈ N}. Lemma 8. Let v(θ) and c(θ) in (2) and (3) be in C(T), c(θ) have at most countably many zeros and ω ∈ / Ac . Then ln g(θ, n, E) ∈ C(T). / Ac leads to an additional condition only Remark. For Hθ,λ,ω , the assumption ω ∈ when λ1 = λ3 . The condition is precisely the one given in (24), as can be obtained by a simple computation. Proof. Since ln g(θ, n, E) is a positive continuous function, the proof reduces to showing that g(θ, n, E) = 0 for all θ. For fixed θ and ω, the sequence c(n, θ, ω) can have two or more zeros only if ω ∈ Ac . We will show that g(θ, n, E) = 0 if c(n, θ, ω) has one or no zeros. For fixed θ and ω, we will use c(n), v(n) for c(n, θ, ω), v(n, θ, ω). We will use the following lemma. Lemma 9. 1) If Pn (θ + kω, E) = Pn−1 (θ + kω, E) = 0, then ∃ k + 1 ≤ j < n + k such that c(j)=0 and P (θ + kω, E) = 0 for ≥ j − k. 2) If Pn (θ, E) = Pn−1 (θ + ω, E) = 0, then ∃ 1 ≤ k < n such that c(k)=0 and Pn− (θ + ω, E) = 0 for ≤ k. Proof. Let Pn (θ + kω, E) = Pn−1 (θ + kω, E) = 0. Expanding the determinant Pn (θ + kω, E) in the final row, we get Pn (θ + kω, E) = Pn−1 (θ + kω, E)(E − v(k)) − |c(n + k − 1)|2 Pn−2 (θ + kω, E) . Thus either P (θ + kω, E) = 0 for all ∈ N, or |c(j)| = 0 for some k + 1 < j < n + k and P (θ + kω, E) = 0 for ≥ j − k. If P (θ + kω, E) = 0 for all ∈ N, then P1 (θ + kω, E) = E − v(k + 1) = 0 and P2 (θ, E) = |c(k + 1)|2 = 0. This completes part 1. Now assume Pn (θ, E) = Pn−1 (θ + ω, E) = 0. Expanding Pn (θ, E) in the first row, the argument follows exactly as part 1. We now complete the proof of Lemma 8. Assume g(θ, n) = 0 for some n and θ. Then by (6), Pn (θ, E) = 0 and either Pn−1 (θ, E) = 0 or c(n) = 0. Case 1. If Pn−1 (θ, E) = 0, then by part one of Lemma 9, c(j) = 0 for some 1 ≤ j < n, and Pk (θ, E) = 0 for k ≥ j. By (6) and the fact that there is at most one zero of c(), Pn−1 (θ + ω, E) = Pn−2 (θ + ω, E) = 0. By Lemma 9, part 1, c(˜j) = 0 for some 2 ≤ ˜j < n and P (θ + ω, E) = 0 for ≥ ˜j − 1. Since there is only one zero, 1 < j = ˜j < n − 1. Note particularly that c(1) = 0. We now know
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Pj (θ, E) = Pj−1 (θ + ω, E) = 0. Thus by Lemma 9, part 2, c(p) = 0 for some p < j which is a contradiction. Case 2. If c(n) = 0, then by (6), Pn−1 (θ + ω, E) = 0. As above this implies c(j) = 0 for some j < n, a contradiction. We will also need another continuity statement. Theorem 3. ([17]) Let v(θ) in (2) and (3) be real analytic and c(θ) in (2) and (3) be analytic such that T |c(θ)| dθ < ∞. Let ω be Diophantine. Then γ(M (E)) = limn→∞ n1 γ(g(E)) − C(λ) is continuous in E. Lemma 8 and Theorem 3 will be an important ingredients in the uniformity results that follow. Lemma 10. Let v, c and ω be as in Lemma 8. For any > 0 and for any interval I there exists L( , ω, I) such that for any k ≥ L( , ω, I), E ∈ I and θ ∈ T, one has 1 ln g(k, θ, E) < γ(g(E)) + . k Remark. In the case of Hθ,λ,ω , we could use the same technique as in Lemma 2 to bound k1 ln g(k, θ, E) uniformly in θ. That proof, however, does not apply to more general c(θ) and v(θ). Proof. We will start with the following theorem. Theorem 4. ([12]) Let {fn } be a continuous subadditive ergodic process on a uniquely ergodic system (X, µ, T ), i.e., fn ∈ C(X) and fn+m (x) ≤ fn (x)+fm (T n x) for all x ∈ X. Then for every x ∈ X and uniformly on X: lim sup n−→∞
1 fn (x) ≤ γ(f ). n
Let T θ = θ + ω. Then (T, µ, T ) is a uniquely ergodic system and ln g(θ, n) is a subadditive process. Thus by Lemma 9 and Theorem 4 , for every θ ∈ T and uniformly on T, lim sup n−→∞
1 ln g(θ, n) ≤ γ(g(E)) . n
(25)
Remark. Clearly, (25) also implies that for all > 0 there exists k( , E, ω) such that for all n ∈ N , n > k( , E, ω) 1 1 ln |Pn (θ, E)| ≤ ln g(θ, n) ≤ γ(g(E)) + . n n Next we show a uniform bound on ln g(θ, n) in E. First note that by subadditivity, the subsequence {supx 21n ln g(θ, 2n )} is monotone decreasing. Let I be
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a compact set containing σ(H). Recalling Theorem 3 and Lemma 8, we can apply Dini’s theorem on I to the subsequence above. Thus given > 0, there exists k0 such that for all k > k0 sup x
1
ln g(θ, 2n ) < γ(g(E)) + . 2n 2
Now let R > 2 be written in binary expansion R = 2k + · · · + 2k0 + M where M < 2k0 . Then, by subadditivity k0
sup x
1 ln g(θ, R) R
For large enough R,
2k 1 1 2k0 sup k ln g(θ, 2k ) + · · · + sup ln g(θ, 2k0 ) R x 2 R x 2k0 1 M ln g(θ, M ) + sup R x M 2k + · · · + 2k0
C < γ(g(E)) + + . R 2 R
≤
C R
<
2
and we get the desired result.
We will be applying Lemma 10 on an interval which contains the spectrum of Hθ,λ,ω . Since Hθ,λ,ω is self-adjoint, the spectral radius is H. By a straightforward computation, ∞ 12 2 |c(n − 1)ψ(n − 1) + v(n)ψ(n) + c(n)ψ(n + 1)| H = sup ψ=1
n=−∞
≤ 2(1 + λ1 + λ2 + λ3 ) . Set I(H) := [−2(1+λ1 +λ2 +λ3 ), 2(1+λ1 +λ2 +λ3 )]. By the above, σ(H) ⊂ I(H). We will apply Lemma 10 on this interval. Lemma 10 replaces Lemma 2 and eliminates the dependence of k3 on E. Lemma 6, therefore, becomes Lemma − , k, E, θ)-singular, < 12 γ(M (E)) and k > 6 .1 If y ∈ Z is (γ(M (E))
, ω, I(H)), k4 ( ), 8 , then for any x such that y − [ 87 k] ≤ x ≤ y − [ 87 k]+ 9 max L( 18 [ 34 k], 1 |Pk (θ + (x − 1)ω)| ≤ exp kγ(M (E)) + (k − 1)C(λ) − k . 18
While it is possible to use sets Θk (y, ω, α) as defined in (12) for our proof, we will modify them in order to somewhat simplify the argument. Define for any s > α and k ≥ 1
1 j α
!
Θk (y, ω, α) := θ ∈ T : ∃|j| ≤ 3k , sin 2π θ + y + ω ≤ s . 2 k ! k (y, ω, α)C ⊂ Θk (y, ω, α)C , so Lemma 7 holds on this set. Θ
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One easily shows
!
Θk (y, ω, α) ≤
C k s−α
Ann. Henri Poincar´e
.
In our proof of Lemma 1, k2 depended on the choice of singular point y1 . (See remarks after Lemma 1.) In the proof of strong dynamical localization, we wish to eliminate this dependence. We omit the measure zero set 1 − 2θ −2θ or ∈Z . Φ= θ: ω ω This removes the need for constraint k5 (y1 , θ, ω) and the dependence on y1 for our definition of k2 . The bound k2 in Lemma 1 becomes k2 ( , α, I(H)) = max[9L( /18, ω, I(H)), 9k4 ( /18), k6 ( /36, ω, α), k7 ( )] + 2. By Theorem 3, γ(M (E)) is continuous. So on I(H), it has a minimum, γ0 > 0. It follows from definition that for 0 < < 12 γ(M (E)), (γ(M (E)) − , k, E, θ)regularity implies ( 12 γ0 , k, E, θ)-regularity. So we can reformulate Lemma 1 as # "k (y, ω, α)C ΦC , 1 < Lemma 1b. Let λ and ω be as in Theorem 2. Suppose θ ∈ Θ ! ! α < 2, and s > α. Then ∃L(ω, α, s, I(H)) : ∀E ∈ I(H) and ∀k > L(ω, α, s, I(H)), if 34 k < |x − y| ≤ (k − 2)α , then either x or y is ( 12 γ0 ,k)-regular. The rest of the proof closely follows the argument of [13]. By Theorem 1, Hθ,λ,ω has pure point spectrum with exponentially decaying # eigenfunctions for all θ ∈ Θ(y)C ΦC#, a set of full measure. Let ϕ[θ, E] be the orthonormal eigenfunctions on Θ(y)C ΦC with corresponding eigenvalues E[θ]. Define ϕ[θ, E] ϕ[θ, ! E] = (26) Bϕ[θ, E]2 where B is the operator of multiplication by b(x) := (1 + |x|)−δ and δ > 1/2. Notice that uniformly in θ and in energy E[θ] for all x ∈ Z, |ϕ[θ, ! E](x)| ≤ (1 + |x|)δ .
(27)
Clearly |ϕ[θ, ! E](x)| are bounded functions of x. However, what we need here is a uniform bound in (E, θ) as given in (27). ! Lemma 11. Pick k > L(ω, α, s, I(H)) and y ∈ Z.
"k (y,ω,α)C # ΦC , 1) There exists a constant K1 (λ, δ, α, γ0 ) < ∞ such that ∀θ ∈ Θ E = E[θ] and all x ∈ Z such that 34 k < |x − y| < (k − 2)α , we have γ0 k . |ϕ[θ, E](x)||ϕ[θ, E](y)| ≤ K1 (λ, δ, α, γ0 )Bϕ[θ, E]2 (1 + |y|)2δ exp − 20 # 2) There exists a constant K2 (λ, δ, α, γ0 ) < ∞ such that ∀θ ∈ Θk (y)C ΦC γ0 k sup | exp{−iH[θ]t}δx, δy | ≤ K2 (λ, δ, α, γ0 )(1 + |y|)2δ exp − . 20 t
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Proof. By Lemma 1b, either x or y is ( γ20 , E, θ, k)-regular. Let u = x or y be the regular point and v the other one. Let [x1 , x2 ] be the interval such that |G[x1 , x2 ](y, xi )| < exp −γ0 k/18) where i = 1, 2. Note that x2 = x1 + k − 1. Then ϕ[θ, E](u)
=
−c(x1 − 1)ϕ[θ, E](x1 − 1)G[x1 , x2 ](u, x1 ) −c(x2 )ϕ[θ, E](x2 + 1)G[x1 , x2 ](u, x2 ) .
We have using (4), (27) and the definition of regularity |ϕ[θ, ! E](u)| ≤
≤ ≤
|c(x1 − 1)| |ϕ[θ, E](x1 − 1)||G[x1 , x2 ](u, x1 )| Bϕ[θ, E]2 |c(x2 )| + |ϕ[θ, E](x2 + 1)||G[x1 , x2 ](u, x2 )| Bϕ[θ, E]2 $ % γ0 k (λ1 + λ2 + λ3 ) (1 + |x1 − 1|)δ + (1 + |x1 + k|)δ exp − 18 γ0 k δ δ . C(λ, δ)(1 + k) (1 + |u|) exp − 18 −
Therefore, using (27) and the fact that |x − y| < (k − 2)α , we have |ϕ[θ, ! E](x)ϕ[θ, ! E](y)| ≤ ≤
γ0 k C(λ, δ)(1 + k)δ (1 + |x|)δ (1 + |y|)δ exp − 18 k γ 0 K1 (λ, δ, α, γ0 )(1 + |y|)2δ exp − . 20
Substituting (26), we obtain statement 1 of the Lemma. Next, in order to bound the integrand of (23), we find sup | exp{−iH[θ]t}δx, δy | ≤ t
|ϕE [θ, E](x)ϕE [θ, E](y)|
E∈{E[θ]}
≤ K1 (λ, δ, α, γ0 )(1 + |y|)2δ e−
γ0 k 20
Bϕ[θ, E]2
E∈{E[θ]}
≤ K1 (λ, δ, α, γ0 )(1 + |y|)2δ e
γ0 k − 20
2δ −
= K2 (λ, δ, α, γ0 )(1 + |y|) e
γ0 k 20
b2 .
Proof of Theorem 2. Pick α ∈ (1, 2), q > 0 and s > α(q + 1). Define a sequence ! L1 = L(ω, α, s, I(H)), Lj+1 = ( 43 Lj − 2)α . Then if Lj = 34 k, Lj+1 = (k − 2)α . # ! Lj (y, ω, α)C ΦC . By Lemma 11, for all j ≥ 1 and x and y such that Let = Θ
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Lj < |x − y| ≤ Lj+1
& '
sup e−iH[θ] δx , δy dθ
Ann. Henri Poincar´e
& '
sup e−iH[θ] δx , δy dθ t∈R
& '
+ sup e−iH[θ] δx , δy dθ
=
T t∈R
C t∈R
≤ ≤ ≤
! Lj (y)| K2 (λ, δ, α, γ0 )(1 + |y|)2δ e−γ0 Lj /15 + |Θ C K2 (λ, δ, α, γ0 )(1 + |y|)2δ e−γ0 Lj /15 + s−α Lj 1 γ0 |x − y| α 2δ K2 (λ, δ, α, γ0 )(1 + |y|) exp − 20 1
+C1 {|x − y| α }−s+α since |x − y| ≤ Lj+1 = ( 43 Lj − 2)α ≤ ( 43 Lj )α . Thus there exists a constant K3 (λ, δ, α, γ0 , s) such that for all x, y ∈ Z
& '
γ 1 s
0 sup e−iH[θ] δx , δy dθ ≤ K3 (1 + |y|)2δ exp − |x − y| α + |x − y|− α +1 . 20 T t∈R (28) The rest of the argument follows exactly as in [13]. Suppose ψ ∈ 2 decays faster than any polynomial and ψ = 1, then ( ) q/2 −iHθ t −iHθ t q −iHθ t sup |X| e ψ dθ = sup e ψ, |X| e ψ, δx δx dθ T t∈R
T t∈R
=
sup
T t∈R
≤
x
≤
y
|x|q
x
|x|q | e−iHθ t ψ, δx |2 dθ
x
sup | e−iHθ t ψ, δx |dθ
T t∈R
|ψ(y)|
x
|x|
q
* +
sup e−iHθ t δy , δx dθ .
T t∈R
Applying equation 28, sup |X|q/2 exp{−iHθ t}ψ dθ T t∈R γ 1 s 0 |ψ(y)| |x|q (1 + |y|)2δ exp − |x − y| α + |x − y|− α +1 . ≤ K3 20 y x Since s > α(q + 1) the sum in x converges and since ψ decays faster than any polynomial, the sum in y converges.
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References [1] M. Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys. 6, 1163–1182 (1994). [2] J. Avron and B. Simon, Almost periodic Schrodinger operators II. The integrated density of states, Duke Math. J. 50, 369–385 (1983). [3] J. Bellissard, C. Kreft, R. Seiler, Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods. J. Phys. A 24 , 2329–2353 (1991). [4] Ju.M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence, RI (1968). [5] J. Bourgain and M. Goldstein, On nonperturbative localization with quasiperiodic potential, Ann. of Math. 152, 825–879 (2000). [6] J. Bourgain and S. Jitomirskaya, Anderson localization for the band model. Geometric aspects of functional analysis, Lecture Notes in Math. 1745, 67–79, Springer, Berlin (2002). [7] J. Bourgain and S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potentials, J. Stat. Phys. 108, 1203–1218 (2002). [8] W. Craig and B. Simon, Subharmonicity of the Lyaponov index, Duke Math J. 50, 551–560 (1983). [9] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry, SpringerVerlag, New York, (1987). [10] R. Del Rio, S. Jitomirskaya, Y. Last and B. Simon, Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69, 153–200 (1996). [11] K. Drese and M. Holthaus, Phase diagram for a modified Harper model, Phys. Rev. B 55, 693–696 (1997). [12] A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems, Ann. Inst. Henri Poincar´e 33, 797–815 (1997). [13] F. Germinet and S. Jitomirskaya, Strong dynamical localization for the almost Mathieu model, Rev. Math. Phys. 13, 755–765 (2001). [14] F. Germinet and A. Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222, 415–448.
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[15] D. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239–2249 (1976). [16] S. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. Math. 150, 1159–1175 (1999). [17] S. Jitomirskaya, D.A. Koslover and M.S. Schulteis, Continuity of the Lyapunov exponent for quasiperiodic Jacobi matrices. Preprint, 2004. [18] S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum III. Almost periodic Schr¨ odinger operators, Comm. Math. Phys. 165, 201–205 (1994). [19] Y. Katznelson, An Introduction to Harmonic Analysis, Dover Publications Inc., New York (1976). [20] P. Koosis, The Logaritmic Integral II, Cambridge University Press, New York (1992). [21] D.A. Koslover, Jacobi Operators with Singular Continuous Spectrum, submitted to Lett. Math. Phys. [22] Y. Last, Spectral theory of Sturm-Liouville operators on infinite intervals: a review of recent developments. Preprint, 2004. [23] V.A. Mandelshtam and S.Ya. Zhitomirskaya, 1D-Quasiperiodic operators. Latent symmetries, Comm. Math. Phys. 139, 589–604 (1991). [24] K. Petersen, Ergodic Theory, Cambridge University Press, New York (1997). [25] J. Puig, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys. 224, 297–309 (2004). [26] D.J. Thouless, Bandwidths for a quasiperiodic tight-binding model, Phy. Rev. B 28, 4272–4276 (1983). S. Jitomirskaya(1), D.A. Koslover(1) and M.S. Schulteis(1,2) (1) Department of Mathematics University of California, Irvine Irvine, CA 92697, USA email:
[email protected] email:
[email protected] (2)
Deparment of Mathematics Concordia University, Irvine Irvine, CA 92612, USA email:
[email protected] Communicated by Jean Bellissard submitted 7/07/04, accepted 22/07/04
Ann. Henri Poincar´e 6 (2005) 125 – 154 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/010125-30 DOI 10.1007/s00023-005-0201-4
Annales Henri Poincar´ e
The Aharonov-Bohm Solenoids in a Constant Magnetic Field Takuya Mine Abstract. We study the spectral properties of a two-dimensional magnetic Schr¨ odinger operator HN = ( 1i ∇ + aN )2 . The magnetic field is given by rot aN = B+ N j=1 2παj δ(z − zj ), where B > 0 is a constant, 1 ≤ N ≤ ∞, 0 < αj < 1 (j = 1, . . . , N ) and the points {zj }N j=1 are uniformly separated. We give an upper bound for the number of eigenvalues of HN between two Landau levels or below the lowest Landau level, when N is finite. We prove the spectral localization of HN near the spectrum of the single solenoid operator, when {zj }N j=1 are far from each are the same, and the boundary conditions at zj are other, all the values {αj }N j=1 uniform. We determine the deficiency indices of the minimal operator and give a characterization of self-adjoint extensions of the minimal operator.
1 Introduction 2 Let N = 1, 2, 3, . . . or N = ∞. Let {zj }N j=1 be points in R satisfying
R := inf |zj − zk | > 0 j=k
(the notation X := Y means X is defined to be Y ). Put SN := ∪N j=1 {zj }. Define a differential operator LN on R2 \ SN by 1 LN := ( ∇ + aN )2 , i
√ where i = −1 and ∇ = (∂x , ∂y ). We assume that aN = (aN,x , aN,y ) ∈ C ∞ (R2 \ SN ; R2 ) ∩ L1loc (R2 ; R2 ) and rot aN (z) := (∂x aN,y − ∂y aN,x)(z) = B +
N
2παj δ(z − zj )
(1)
j=1
in D (R2 ) (in the distribution sense), where z = (x, y) ∈ R2 , B, αj are constants satisfying B > 0 and 0 < αj < 1 for every j = 1, . . . , N. We find a proof of the existence of the vector potential aN satisfying above conditions in the paper of Arai [Ar] (see also [Me-Ou-Ro]). Define a linear operator
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LN on L2 (R2 ) by LN u = LN u, u ∈ D(LN ) = C0∞ (R2 \ SN ), where D(L) is the operator domain of a linear operator L and C0∞ (U ) is the space of the compactly supported smooth functions in an open set U . Then, the operator LN is symmetric and positive. Moreover, we will prove that the deficiency indices of LN are (2N, 2N ) (see Proposition 5.7). Thus the operator LN has self-adjoint extensions parametrized by (2N × 2N )-unitary matrices (see [Re-Si, Theorem X.2]). We denote by HN any self-adjoint extension of LN . In particular, we denote by AB the Friedrichs extension of LN , which is called the standard Aharonov-Bohm HN Hamiltonian. The Hamiltonian HN describes the motion of a non-relativistic charged particle moving in the Euclidean plane in the presence of a homogeneous magnetic field B plus a magnetic field concentrated on infinitesimally thin solenoids placed at the points zj with flux 2παj , provided that the mass m = 1/2, the Planck constant (divided by 2π) = 1 and the charge of an electron e = 1. When B = 0, the quantum mechanical system corresponding to HN is known to be a model which explains the Aharonov-Bohm effect ([Ah-Bo]), and is extensively studied from the view point of the scattering theory (e.g., [Rui], [Nam], [It-Ta1], [It-Ta2]), the spectral theory (e.g., [St1], [St2], [St3], [Ta], [Me-Ou-Ro]), and the theory of self-adjoint extensions of symmetric operators (e.g., [Ad-Te], [Da-St]). However, there seem to be few results in the case B > 0. Nambu [Nam] has AB and obtained an integral studied the standard Aharonov-Bohm Hamiltonian HN representation of eigenfunctions corresponding to the Landau level (2n − 1)B, for n = 1, 2, . . .. Particularly in the single solenoid case N = 1, Nambu has obtained all eigenvalues of H1AB and an integral representation of corresponding eigenfunctions. ˇ Exner, St’ov´ ıˇcek and Vytˇras [Ex-St-Vy] have given a complete characterization of the self-adjoint extensions H1 of L1 , written the eigenequation in terms of special functions, and solved it numerically. Their results about σ(H1AB ) (the spectrum of H1AB ) are summarized as follows: σ(H1AB ) =
∞
{(2n − 1)B} ∪ {(2n + 2α − 1)B}
n=1
and mult{(2n − 1)B ; H1AB } = ∞ mult{(2n + 2α − 1)B ; H1AB } = n
for n = 1, 2, . . . , for n = 1, 2, . . . ,
(2)
where mult{λ ; H} := dim Ker(λ − H). In this paper, we consider the following problems: (I) to investigate the spectrum of HN between the gaps of Landau levels, when B > 0 and N ≥ 2, (II) to
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give a complete characterization of self-adjoint extensions HN of LN , when B > 0 and N ≥ 2. These problems seem not to be studied yet. For the former problem, we obtain the following two results. First, we consider the finite solenoids case. In the sequel, PI (H) denotes the spectral projection of a self-adjoint operator H corresponding to an interval I. Theorem 1.1. Let 1 ≤ N < ∞. Then, the following holds: (i) For any self-adjoint extension HN of LN , we have mult{(2n − 1)B ; HN } = ∞
for n = 1, 2, 3, . . . .
AB (ii) For the standard Aharonov-Bohm Hamiltonian HN , we have AB ) = dim Ran P(−∞,B) (HN AB ) dim Ran P((2n−1)B,(2n+1)B) (HN
≤
0, nN
for n = 1, 2, 3, . . . .
(3)
(iii) For any self-adjoint extension HN of LN , we have dim Ran P(−∞,B) (HN ) ≤ 2N, dim Ran P((2n−1)B,(2n+1)B) (HN ) ≤ (n + 1)N
for n = 1, 2, 3, . . . .
Next, we consider the case where solenoids are far from each other and the physical situation around all solenoids are the same. To describe this situation rigorously, we prepare some definitions. Definition 1.1. Let w ∈ R2 . Let U be a simply connected open set, and V = U +w = {z + w; z ∈ U }. Let S be at most countable subset of U with no accumulation points in U and T = S + w. Let a ∈ C ∞ (U \ S; R2 ) ∩ L1loc (U ; R2 ) and b ∈ C ∞ (V \ T ; R2 ) ∩ L1loc (V ; R2 ) such that rot a(z) = rot b(z + w) holds in D (U ). By the Poincar´e Lemma, we can prove that there exists an operator t−w given by t−w v(z) = Φ(z)v(z + w), Φ(z) ∈ C ∞ (U \ S), |Φ(z)| = 1 and satisfying p(a)t−w v = t−w p(b)v, L(a)t−w v = t−w L(b)v, where 1 ∇ + a, i L(a) = p(a)2 ,
p(a) =
1 ∇ + b, i L(b) = p(b)2 .
p(b) =
We call the operator t−w the magnetic translation operator from V to U intertwining L(b) with L(a). We denote the inverse operator of t−w by tw , that is, tw u(z) = Φ(z − w)u(z − w).
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Remark. The magnetic translation operators are introduced in the classical paper by Zak [Za] and discussed in detail by many authors; see, e.g., Arai [Ar] or Ge˘ıler [Ge]. In the sequel, χ and χj are functions satisfying 0 (|z| > R2 ) χ ∈ C0∞ (R2 ), 0 ≤ χ ≤ 1 on R2 , χ(z) = (4) 1 (|z| < R3 ), χj (z) := χ(z − zj ) for j = 1, . . . , N. Definition 1.2. Let HN be a self-adjoint extension of LN . We say the operator HN has uniform boundary conditions if the following two conditions hold: (i) There exists a constant α with 0 < α < 1 such that αj = α for every j = 1, . . . , N . (ii) There exists a self-adjoint extension H1 of L1 independent of j such that D(HN ) = u ∈ D(LN ∗ ); t−zj (χj u) ∈ D(H1 ) for every j = 1, . . . , N , (5) where t−zj be the magnetic translation operator from {|z − zj | < αj R 2 } intertwining LN with L1 (see (6)).
R 2}
to {|z| <
Theorem 1.2. Let 2 ≤ N < ∞ or N = ∞. Let HN be a self-adjoint extension of LN which has uniform boundary conditions and H1 be the single solenoid operator appeared in (5). Let I = [c, d] be a closed interval satisfying I ∩ {(2n − 1)B; n = 1, 2, . . .} = ∅, c, d ∈ / σ(H1 ) and σ(H1 ) ∩ I = {λ1 , λ2 , . . . , λk } = ∅ (σ(H1 ) ∩ I is a finite set, by Theorem 1.1). Then, there exist constants u > 0 and R0 > 0 dependent on B, α, I, H1 (independent of N, R) satisfying the following assertions: (i) If R ≥ R0 , we have σ(HN ) ∩ I ⊂
k
[λl − δ, λl + δ],
l=1 2
where δ = e−uR . (ii) If R ≥ R0 , we have dim Ran PI (HN ) = N dim Ran PI (H1 ). We shall try to give a physical interpretation of our results. In classical mechanics, an electron in a homogeneous magnetic field makes a cyclotron motion. In quantum mechanics, however, the energy of an electron is quantized by the wave property of an electron, and takes a value in Landau levels. If some solenoids are contained in the circle of the cyclotron motion, the Aharonov-Bohm effect causes the phase shift of the electron wave by e/ times the magnetic flux through solenoids in the circle. Thus the energy of the electron
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is obliged to change into a value in a Landau gap, in order to correct the phase shift. For this reason, the number of eigenstates of the single solenoid operator H1AB in the nth Landau gap is considered to be roughly estimated by the possible number of electrons with energy in the nth Landau level, in the circle of the Larmor radius centered at the position of the solenoid. This number can be calculated as follows. Under our normalization of physical constants, the cyclotron radius r of an electron with nth Landau energy (2n − 1)B equals (2n − 1)/B. It is known that the density of states (the number of eigenstates per unit area) for each Landau level is B/2π (see, e.g., [Nak, Proposition 15]). Thus, the number of possible eigenstates in the circle is 1 B =n− . πr2 × 2π 2 The difference between this estimate and the rigorous result (2) is only 12 . AB in the nth Landau gap is roughly Similarly, the number of eigenstates of HN estimated by the number of eigenstates with nth Landau energy in the union set, with respect to j = 1, . . . , N , of the disks of Larmor radius centered at zj . Each disk contains n eigenstates with nth Landau energy. Thus, we conclude that the AB in the nth Landau gap is bounded by nN . Our number of eigenvalues of HN Theorem 1.1 has justified this conclusion. Moreover, if solenoids are far from each other compared with the cyclotron radius, then the interaction between two disks can be ignored. Thus we conclude AB in the nth Landau gap equals to nN . Our that the number of eigenvalues of HN Theorem 1.2 combined with (2) has justified this conclusion. The proof of Theorem 1.1 (section 3.2) depends heavily on a perturbation argument of the canonical commutation relation (CCR) of the annihilation operator and the creation operator (section 3.1; Iwatsuka [Iw] has also used a similar argument). Since our magnetic fields have δ-like singularities, CCR formally holds with a δ-like perturbation. We interpret the δ-like perturbation as a difference of the boundary conditions at zj of two self-adjoint extensions HN appeared in CCR. Determining the operator domain of these operators explicitly (section 5.4) and using a known result by Deift (Lemma 3.2), we obtain Theorem 1.1. Theorem 1.2 is analogous to the result of Cornean and Nenciu ([Co-Ne, Theorem III.1, Corollary III.1]). The proof of Theorem 1.2 (section 4.2) is also analogous to that of their results. However, we need an additional assumption (Definition 1.2) to obtain the result, because of the non-essential self-adjointness of the minimal operator LN . In order to prove Theorem 1.1 and Theorem 1.2, we need a detailed information about the self-adjoint extension of LN . For this reason, we shall give a complete characterization of self-adjoint extensions of LN (Section 5.3). Although there are many results about the self-adjoint extension of Schr¨ odinger operators with singular potentials (e.g., [Al-Ge-Ho-Ho], [Ad-Te], [Da-St], [Ex-St-Vy]), this problem is considered to be rather difficult when N ≥ 2, because of the difficulty
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of determining the deficiency subspaces Ker(LN ∗ ∓ i) explicitly. The main strategy is due to the locality of self-adjoint extensions, which has been pointed out by Bulla and Gesztesy ([Bu-Ge]) in the δ-potential case. Combining their idea with the gauge transformation technique and using the result in the single solenoid case ([Ex-St-Vy]), we obtain our result (Theorem 5.11).
2 Preliminaries We shall prepare some notations used in later sections. We denote the Landau operator (the Schr¨ odinger operator with a constant magnetic field) by L0 , that is, 2 1 B B ∇ + a0 , a0 = − y, x . L0 = i 2 2 It is known that the operator L0 |C0∞ (R2 ) is essentially self-adjoint (see [Ik-Ka], [Le-Si]). We denote by H0 the unique self-adjoint extension of L0 |C0∞ (R2 ) . When we consider the single solenoid operator L1 , we always assume z1 = 0 and take the radial gauge, that is, 2 1 α α B B ∇ + a y − x + = , a (z) = − y, x . (6) L1 = Lα 1 1 1 i 2 |z|2 2 |z|2 The upper suffix α is sometimes used to indicate the value α1 = α explicitly α (similarly, we sometimes denote the self-adjoint extension of Lα 1 by H1 ). ∗ We regard the operator domain D(LN ) as a Hilbert space equipped with the graph inner product and norm (u, v)N = (LN u, LN v) + (u, v), u 2N = (u, u)N , u, v ∈ D(LN ∗ ),
where (u, v) = R2 u¯vdxdy. Notice that the functions LN u and LN v belong to L2 (R2 ) (see (i) of Proposition 5.4 below). We regard D(HN ) and D(LN ) as closed subspaces of D(LN ∗ ). For a Hilbert space H and a closed subspace H of H, we denote H/H the quotient Hilbert space equipped with the norm
[u] = P u H, u ∈ H, where [u] is the equivalence class of u and P is the orthogonal projection onto (H )⊥ .
3 Finite solenoids case 3.1
Perturbation of the canonical commutation relation
Define differential operators ΠN,x , ΠN,y , AN , A†N by ΠN,x := 1i ∂x + aN,x , ΠN,y := 1i ∂y + aN,y , AN := iΠN,x + ΠN,y , A†N := −iΠN,x + ΠN,y .
(7)
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Define linear operators AN , A†N on L2 (R2 ) by AN u := AN u, A†N u := A†N u,
D(AN ) := C0∞ (R2 \ SN ), D(A†N ) := C0∞ (R2 \ SN ).
By the assumption (1), we have the following lemma. Lemma 3.1. The following operator relations hold: (i) A†N AN = LN − B, AN A†N = LN + B. ∗ (ii) A†N ⊂ AN ∗ , AN ⊂ A†N . We review some properties of a pair of operators XX ∗ and X ∗ X. Lemma 3.2. Let X be a densely defined closed linear operator on a Hilbert space H. Then, we have the following: (i) The operators X ∗ X and XX ∗ are self-adjoint. (ii) The operator (XX ∗ )|(Ker XX ∗ )⊥ is unitarily equivalent to the operator (X ∗ X)|(Ker X ∗ X)⊥ .
Proof. (i) See [Re-Si, Theorem X.25]. (ii) See [De, Theorem 3]. Applying Lemma 3.2 to our operators, we have the following. Proposition 3.3. ∗
AB AB (i) We have A†N A†N = HN + B, AN ∗ AN = HN − B.
∗
− − (ii) There exists a self-adjoint extension HN of LN such that A†N A†N = HN − B. AB (iii) The inequality HN ≥ B holds in the form sense. ∗
Proof. (i) By Lemma 3.1, we have A†N A†N ⊃ AN A†N = LN + B. By Lemma 3.2, ∗
we have A†N A†N is self-adjoint. Thus, there is a self-adjoint extension X of LN ∗
AB such that A†N A†N = X + B. To show X = HN , it is sufficient to show that these 2 ∞ operators have a common form core C0 (R \ SN ). This fact follows from the proof of [Re-Si, Theorem X.25] and the definition of the Friedrichs extension. The proof of the second equality is similar.
(ii) The proof is similar to the first part of the proof of (i). AB = AN ∗ AN +B. (iii) This assertion immediately follows from the equality HN
The following lemma is necessary for our proof of Theorem 1.1. − − AB )/(D(HN ) ∩ D(HN )) = N . Lemma 3.4. dim D(HN The proof of Lemma 3.4 will be given in section 5.4. We quote several facts about the spectrum of self-adjoint extensions of a symmetric operator.
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Lemma 3.5. Let L be a symmetric operator on a Hilbert space H. Suppose that the deficiency indices of L are (n, n) and n < ∞. Let A and B be two self-adjoint extensions of L. Then, the following holds: (i) σess (A) = σess (B). (ii) For any open interval I = (c1 , c2 ) such that c1 < c2 and dim Ran PI (A) < ∞, we have dim Ran PI (B) < ∞ and | dim Ran PI (A) − dim Ran PI (B)| ≤ d, where d = dim D(A)/ (D(A) ∩ D(B)) . Proof. (i) See [We, Theorem 8.17]. (ii) This assertion is an immediate corollary of [We, Exercise 8.8].
3.2
Proof of Theorem 1.1
Proof. By (ii) of Lemma 3.2, we have
† ∗ † † † ∗ AN AN AN AN ∗
(Ker A†N A†N )⊥
∗
(Ker A†N A†N )⊥
,
where the notation X Y means that two operators X and Y are unitarily equivalent. By (i) and (ii) of Proposition 3.3, we have − AB + B)|(Ker(HNAB +B))⊥ (HN − B)|(Ker(H − −B))⊥ . (HN N
(8)
Let HN be any self-adjoint extension of LN . By (i) of Lemma 3.5, we have there exists a closed subset S ⊂ R such that − AB ) = σess (HN ) S = σess (HN ) = σess (HN
for any self-adjoint extension HN of LN . In particular we have S ⊂ [B, +∞), by (iii) of Proposition 3.3. Thus {−B} ∈ / S. By (8), we have S \ {B} = S + 2B = {x + 2B; x ∈ S}.
(9)
It is easy to show that the set satisfying (9) is {(2n − 1)B; n = 1, 2, . . .} or the empty set. We show S is not empty. To see this, it is sufficient to construct a Weyl sequence for the spectrum B, that is, an orthonormal sequence {un }∞ n=1 such that (HN − B)un → 0 as n → ∞. Take countable disjoint disks {B(n; wn )}∞ n=1 (B(n; wn ) = {z; |z −wn | < n}) contained in R2 \SN . Let χ be a function satisfying (4). Put z 2 B vn (z) = twn χ( )e− 4 |z| , n
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where twn is the magnetic translation operator from {|z| < n} to {|z − wn | < n} intertwining L0 with LN . It is easy to check that {vn / vn }∞ n=1 is a Weyl sequence by using the equality B
2
B
2
B
LN twn e− 4 |z| = twn L0 e− 4 |z| = Btwn e− 4 |z|
2
in |z − wn | < n
and the Leibniz rule. Thus we have σess (HN ) = {(2n − 1)B; n = 1, 2, . . .}
(10)
for any self-adjoint extension HN of LN . Next, we shall prove (ii). The assertion AB )=0 dim Ran P(−∞,B) (HN
(11)
follows from (iii) of Proposition 3.3. By (8), we have − AB dim Ran P((2n−1)B,(2n+1)B) (HN ) = dim Ran P((2n+1)B,(2n+3)B) (HN )
(12)
for every n = 0, 1, 2, . . .. By Lemma 3.4 and (ii) of Lemma 3.5, we have − AB ) ≤ dim Ran P((2n−1)B,(2n+1)B) (HN ) + N (13) dim Ran P((2n−1)B,(2n+1)B) (HN
for every n = 0, 1, 2, . . .. By (11), (12), (13) and an elementary induction argument, we have (3). Thus (ii) of Theorem 1.1 holds. Since the deficiency indices of LN are (2N, 2N ), we have − ) ≤ 2N (14) dim D(H)/D(H) ∩ D(HN for any self-adjoint extension HN of LN . Thus (iii) of Theorem 1.1 follows from (3), (12), (14) and (ii) of Lemma 3.5. To show (i) of Theorem 1.1, consider the following equality:
=
dim Ran P{(2n−1)B} (HN ) dim Ran P((2n−1)B−,(2n−1)B+) (HN ) − dim Ran P((2n−1)B−,(2n−1)B) (HN ) − dim Ran P((2n−1)B,(2n−1)B+) (HN ).
The first term of the last expression is infinity, by (10). The second term and the third are finite for small > 0, by (iii) of Theorem 1.1. Therefore we have (i) of Theorem 1.1.
4 Large separation case 4.1
Self-adjoint extensions with uniform boundary conditions
We summarize fundamental facts about a self-adjoint extension HN with uniform boundary conditions. The proofs will be given in Section 5.5.
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Lemma 4.1. Let N = 1, 2, . . . or N = ∞. Let 0 < α < 1 and suppose αj = α for every j = 1, 2, . . . , N . Let H1 be any self-adjoint extension of Lα 1 . Then, there exists a unique self-adjoint extension HN of LN satisfying (5). Lemma 4.2. Let HN be a self-adjoint extension of LN which has uniform boundary conditions. Let η ∈ C0∞ ({|z| < R2 }) with η = 1 on some open neighborhood of 0. Let 1 ≤ j ≤ N and put ηj (z) = η(z − zj ). Let t−zj be the magnetic translation operator from {|z − zj | < R2 } to {|z| < R2 } intertwining LN with L1 , and let tzj be the inverse operator. Then, the following holds: (i) For any u ∈ D(HN ), we have ηj u ∈ D(HN ), t−zj (ηj u) ∈ D(H1 ) and HN (ηj u) = tzj H1 t−zj (ηj u).
(15)
(ii) For any v ∈ D(H1 ), we have ηv ∈ D(H1 ), tzj (ηv) ∈ D(HN ) and t−zj HN tzj (ηv) = H1 (ηv). Lemma 4.3. Let HN be a self-adjoint extension of LN which has uniform boundary conditions. Let U be a simply connected open set in R2 \ SN , m ∈ R2 and V = U − m = {z − m; z ∈ U }. Let η ∈ C0∞ (V ) and put ηm = η(z − m). Let t−m be the magnetic translation from U to V intertwining LN with L0 and tm be the inverse operator. Then, the following holds: (i) For any u ∈ D(HN ), we have ηm u ∈ D(HN ), t−m (ηm u) ∈ D(H0 ) and HN (ηm u) = tm H0 t−m (ηm u).
(16)
(ii) For any v ∈ D(H0 ), we have ηv ∈ D(H0 ), tm (ηv) ∈ D(HN ) and t−m HN tm (ηv) = H0 (ηv).
4.2
Proof of Theorem 1.2
Most part of our proof is similar to that of [Co-Ne, Theorem III.1], so we omit the detail of the proof of some lemmas. The main difference between our proof and theirs is that the approximating argument they used in [Co-Ne, Corollary III.1] is not applicable in our case, because the space C0∞ (R2 ) is not a common operator core of HN for N = 1, 2, . . .. Thus we take an approximate eigenfunction ψ instead of an eigenfunction ψ in (17) and prove the statement for H∞ directly. Proof. We shall introduce the notation used in [Co-Ne]. For p = (px , py ) ∈ Z 2 and δ > 0, put K(p, δ) := {z = (x, y) ∈ R2 ; |x − and let m(p) :=
R 10 p.
Since
p∈Z2
R R δ δ px | ≤ , |y − py | ≤ } 10 2 10 2
K(p, δ) = R2
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R R for δ ≥ 10 , we can take pj = (pj,x , pj,y ) ∈ Z 2 such that zj ∈ K(pj , 10 ), for each j = 1, . . . , N . Put K ((pj,x + β, pj,y + γ) , δ) , for δ > 0, Kj (δ) := β,γ∈{−1,0,1} N
FN :=
j=1
Kj (
R ) 10
and let mj := m(pj ). Let Γ0 :=
{m(p) ∈ R2 ; p ∈ Z 2 , K(p,
Γ1 := Γ :=
{mj ; j = 1, . . . , N }, Γ0 ∪ Γ 1 .
R ) ⊂ FN }, 10
For m = m(p) ∈ Γ0 , let t−m be the magnetic translation operator from K(p, R8 ) to K(0, R8 ) intertwining LN with L0 , and tm be the inverse operator. For j = 1, . . . , N , let t−zj be the magnetic translation operator from {|z − zj | < R2 } to {|z| < R2 } α intertwining LN with L1 j , and tzj be the inverse operator. Proof of (i) . Let I = [c, d] be an interval on R satisfying I ∩ {(2n − 1)B; n = 1, 2, . . .} = ∅ and c, d ∈ / σ(H1 ). Take E ∈ σ(HN ) ∩ I. For any > 0, there is some ψ ∈ D(HN ) such that
ψ = 1, ξ < , ξ := (HN − E)ψ .
(17)
Notice that σ(HN ) ∩ I is a finite set when N is finite, by (iii) of Theorem 1.1. In this case, we can take ψ as an eigenfunction corresponding to the eigenvalue E and ξ as 0. We shall prepare three lemmas. Lemma 4.4. Let η ∈ C ∞ (R2 ). Suppose supp |∇η| is a compact set in R2 \ SN and sup |∇η|2 + sup |∇∂x η|2 + sup |∇∂y η|2 ≤ C for some constant C. Then, ηψ ∈ D(HN ) and there exists a constant C5 > 0 dependent only on E, C such that dxdy(|ψ (z)|2 + |ξ (z)|2 ), (18)
[HN , η]ψ 2 ≤ C5 supp |∇η|
where the bracket denotes the commutator, that is, [X, Y ] = XY − Y X. Proof. In the similar way to the proof of [Co-Ne, Lemma II.2]. The fact ηψ ∈ D(HN ) follows from Lemma 4.2.
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Lemma 4.5. Let p ∈ Γ0 . Then, there exist constants C6 > 0 and c > 0 dependent only on B, E such that dxdy|ψ (z)|2 R K(p, 10 )
≤
C6
e
−cR2
2
R K(p, R 8 )\K(p, 9 )
dxdy|ψ (z)| +
K(p, R 8 )
2
dxdy|ξ (z)|
,
if R ≥ 1. Proof. Similar to the proof of [Co-Ne, Lemma III.3], but we use tm (H0 − E)−1 t−m ([HN , ηm ]ψ + ηm ξ ) = ηm ψ instead of (3.28) in [Co-Ne]. The above equality is justified by using Lemma 4.3. Lemma 4.6. There exists a constant C7 > 0 dependent only on B, E such that 2 dxdy|ψ (z)|2 ≤ C7 e−cR , if R ≥ 1, (19) c FN
c
2
where = e− 2 R , c is a constant given in Lemma 4.5. Proof. Similar to the proof of [Co-Ne, Lemma III.4], but we use Lemma 4.4 and Lemma 4.5. Take η0 ∈ C0∞ (R2 ) such that 0 ≤ η0 ≤ 1 and η0 (z) =
1 (z ∈ K0 ( R9 )) 0 (z ∈ / K0 ( R8 )),
sup |∇η0 |2 + sup |∇∂x η0 |2 + sup |∇∂y η0 |2 ≤
C , R2
(20)
where C is a positive constant independent of R and
K0 (δ) =
K((β, γ); δ).
β,γ∈{−1,0,1}
Put ηpj (z) = η0 (z − pj ) for j = 1, . . . , N . By Lemma 4.2, we have ηpj ψ ∈ D(HN ), t−zj ηpj ψ ∈ D(H1 ) and (H1 − E)t−zj ηpj ψ
= t−zj (HN − E)(ηpj ψ ) = t−zj ([HN , ηpj ]ψ + ηpj ξ ).
(21)
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2
Let c be the constant given in Lemma 4.5 and put = e− 2 R . Then we have N
(H1 − E)t−zj ηpj ψ 2
j=1
≤
2
N
[HN , ηpj ]ψ 2 + ηpj ξ 2 j=1
≤
2(C5 + 1) C
2
R Kj ( R 8 )\Kj ( 9 )
j=1
≤
N
c FN
dxdy|ψ (z)| +
Kj ( R 8 )
2
dxdy|ψ (z)| +
2
2
≤ Ce−cR ,
2
dxdy|ξ |
if R ≥ 1,
(22)
where C is a constant dependent only on B, E. We used (21) in the first inequality, (18) in the second, (17) in the third and (19) in the last. Moreover, we have N
(H1 − E)t−zj ηpj ψ 2
j=1
≥
dist{E, σ(H1 )}2
N
ηpj ψ 2
j=1
≥
dist{E, σ(H1 )}2
≥
dist{E, σ(H1 )}2 (1 − C7 e−cR ),
FN
dxdy|ψ |2 2
if R ≥ 1,
(23)
2
by (19). Take a large number R0 > 1 such that C7 e−cR0 < 12 . Then we have by (22) and (23) √ c 2 dist{E, σ(H1 )} ≤ 2Ce− 2 R , if R ≥ R0 . Thus (i) of Theorem 1.2 holds. Proof of (ii). Let I be an interval satisfying the assumption of Theorem 1.2. By (i) of Theorem 1.2, there exists a large number R0 > 1 and a small number δ0 > 0 such that I ∩ σ(HN ) ⊂ (E − δ0 , E + δ0 ), if R ≥ R0 , E∈I∩σ(H1 )
(E − δ0 , E + δ0 ) ∩ σ(H1 ) = {E},
for every E ∈ I ∩ σ(H1 ).
Thus it is sufficient to show that there exists R1 > R0 such that dim Ran P(E−δ0 ,E+δ0 ) (HN ) = N k, where k = dim Ker(H1 − E).
if R ≥ R1 ,
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First we show that there exists R1 > R0 such that dim Ran P(E−δ0 ,E+δ0 ) (HN ) ≥ N k
(24)
for each eigenvalue E of H1 contained in I, if R > R1 (when N = ∞, this is all to prove). Let Vj = {tzj η0 v ; v ∈ Ker(H1 − E)}, where η0 is the function satisfying (20). We can prove dim Vj = k if R is sufficiently large, with the help of the following inequality: 2 |v|2 dxdy ≤ C7 e−cR , if R ≥ 1 (25) R c K(0, 10 )
N
for v ∈ Ker(H1 − E), which follows from Lemma 4.6. Put V = ⊕ Vj . Then we j=1
have dim V = N k.
(26)
By (ii) of Lemma 4.2, we have V ⊂ D(HN ). Suppose that, for any positive number R3 , there exists a sequence {zj }N j=1 such that R = inf j=j |zj − zj | ≥ R3 and dim Ran P(E−δ0 ,E+δ0 ) (HN ) ≤ N k − 1 (when N = ∞, we replace the latter assumption by ‘dim Ran P(E−δ0 ,E+δ0 ) (H∞ ) < N ∞’). By (26), there exists v = j=1 tzj η0 vj ∈ V , vj ∈ Ker(H1 − E), such that ⊥
v = 1 and v ∈ Ran P(E−δ0 ,E+δ0 ) (HN ) . Then we have
(HN − E)v ≥ δ0 .
(27)
By (ii) of Lemma 4.2, we have (HN − E)v =
N
tzj [H1 , η0 ]vj .
j=1
By Lemma 4.4 and Lemma 4.6, we have 2
tzj [H1 , η0 ]vj 2 ≤ C9 e−cR vj 2 ,
if R ≥ 1,
where C9 and c is a constant dependent only on B, E. Moreover, we have by (25)
vj 2 ≤ 2 η0 vj 2 for sufficiently large R. Thus we have there exists R3 > 0 such that 2
(HN − E)v 2 ≤ Ce−cR ,
if R ≥ R3 ,
where C is a constant dependent only on B, E. This contradicts (27). Thus (24) holds.
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Next, we assume N is finite and prove there exists a constant R2 > R0 dependent only on B, E, δ0 and α such that dim Ran P(E−δ0 ,E+δ0 ) (HN ) ≤ N k,
if R ≥ R2 .
(28)
We prepare three lemmas. Lemma 4.7. Let P be an orthogonal projection and P be a finite rank operator on a Hilbert space. Suppose that
P − P < 1. Then, dim Ran P ≤ dim Ran P .
Proof. See [Co-Ne, Proposition III.1]. Lemma 4.8. There exist smooth functions {ηm }m∈Γ , satisfying 0 ≤ ηm ≤ 1 for m ∈ Γ,
2 ηm = 1 on R2 ,
m∈Γ
R supp ηm ⊂ K(m, ) 9 R supp ηmj ⊂ Kj ( ) 9 C sup(|∇ηm |2 + |∇∂x ηm |2 + |∇∂x ηm |2 ) ≤ 2 R
for m ∈ Γ0 , for mj ∈ Γ1 , for m ∈ Γ, if R ≥ 1,
where C is a constant independent of m, R.
Proof. See e.g. [Cy-Fr-Ki-Si]. (m)
Lemma 4.9. For m ∈ Γ, put η0 AN (w) =
mj ∈Γ1
(mj )
tzj η0
(z) = ηm (z + m). Define an operator AN (w) by
(H1 − w)−1 t−zj ηmj +
(m)
tm η0
(H0 − w)−1 t−m ηm
m∈Γ0
for w ∈ C, |w − E| = δ0 . Then, the sums converge in the strong operator topology of the bounded operators from L2 (R2 ) to D(HN ). Moreover, there exists a constant C10 > 0 dependent on B, E, δ0 , and α such that
AN (w) B(L2 (R2 );D(HN )) ≤ C10 for every w ∈ C with |w − E| = δ0 , if R ≥ 1. Here, B(X, Y ) denotes the space of the bounded operators from X to Y . Proof. Similar to the proof of [Co-Ne, Lemma III.5]. The fact AN (w)u ∈ D(HN ) follows from Lemma 4.2 and Lemma 4.3.
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We have (HN − w)AN (w)
=
(mj )
(HN − w)tzj η0
Ann. Henri Poincar´e
(H1 − w)−1 t−zj ηmj
mj ∈Γ1
+
(m)
(HN − w)tm η0
(H0 − w)−1 t−m ηm
m∈Γ0
=
(mj )
(tzj [H1 , η0
2 ](H1 − w)−1 t−zj ηmj + ηm ) j
mj ∈Γ1
+
(m)
(tm [H0 , η0
2 ](H0 − w)−1 t−m ηm + ηm )
m∈Γ0
=
1 + TN (w),
where TN (w)
=
(mj )
tzj [H1 , η0
](H1 − w)−1 t−zj ηmj
mj ∈Γ1
+
(m)
tm [H0 , η0
](H0 − w)−1 t−m ηm .
m∈Γ0
We used the intertwining property of t−m and the equality (m )
m∈Γ
(29)
2 ηm = 1. Notice
that the operator [H1 , η0 j ](H1 − w)−1 is well defined by Lemma 4.2, and the (m) operator [H0 , η0 ](H0 − w)−1 is well defined by Lemma 4.3. Lemma 4.10. The two sums in the right-hand side of (29) converge in the strong operator topology of the bounded operators from L2 (R2 ) to L2 (R2 ). Moreover, there is a constant C11 > 0 dependent only on B, E, δ0 and α such that
TN (w) B(L2 (R2 );L2 (R2 )) ≤
C11 R
for every w ∈ C with |w − E| = δ0 , if R ≥ 1.
Proof. Similar to the proof of (3.71) in [Co-Ne].
C11 R
1 2.
Take a large number R > R0 such that < By Lemma 4.10, we have
TN (w) ≤ 12 and thus 1 + TN (w) is invertible for w ∈ C with |w − E| = δ0 , if R ≥ R . Then HN − w is also invertible and its inverse is given by (HN − w)−1 = AN (w) − AN (w)TN (w)(1 + TN (w))−1 . Integrating (30) on {w; |w − E| = δ0 }, we have (m ) tzj η0 j P(E−δ0 ,E+δ0 ) (H1 )t−zj ηmj + RN , P(E−δ0 ,E+δ0 ) (HN ) = mj ∈Γ1
where RN =
1 2πi
|w−E|=δ0
AN (w)TN (w)(1 + TN (w))−1 .
(30)
(31)
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By Lemma 4.9 and Lemma 4.10, there exists a constant R2 > R dependent only on B, E, δ0 and α such that
RN < 1, if R ≥ R2 .
(32)
Since the first term of the right-hand side of (31) is a linear operator of at most rank N k, we have (28) by Lemma 4.7 and (32). Thus the proof of Theorem 1.2 is completed.
5 Self-adjoint extensions 5.1
Single solenoid operator
Consider the single solenoid case N = 1. In the sequel, we often identify a vector z = (x, y) ∈ R2 with a complex number z = x + iy ∈ C. Under the gauge (6), the operators defined in (7) is explicitly written as the following: 1 ∂x − i 1 = ∂y + i
Π1,x = Π1,y
B y− 2 B x+ 2
α y, |z|2 α x, |z|2 α B z¯ + , 2 z α B = −2∂z¯ + z + , 2 z¯
A1 = iΠ1,x + Π1,y = 2∂z + A†1 = −iΠ1,x + Π1,y
(33)
where
1 1 (∂x − i∂y ), ∂z¯ = (∂x + i∂y ). 2 2 α α α Define four functions φα −1 , ψ1 , φ0 , ψ0 by ∂z =
B
2
2
B
α −1 − 4 |z| e , ψ1α (z) := |z|−α z¯1 e− 4 |z| , φα −1 (z) := |z| z 2 2 B B α − 4 |z| φα , ψ0α (z) := |z|−α e− 4 |z| . 0 (z) := |z| e
(34)
In the polar coordinate z = reiθ , we have B
2
B
2
α+m imθ − 4 r φα e e , ψnα (z) = r−α+n e−inθ e− 4 r . m (z) = r
(35)
Thus the four functions belong to L2 (R2 ). Lemma 5.1. The following holds: α α α L1 φα −1 = (2α − 1)Bφ−1 , L1 ψ1 = Bψ1 , α α α L1 φ0 = (2α + 1)Bφ0 , L1 ψ0 = Bψ0α .
(36)
Proof. By a simple calculation using (33), (34) and the equality L1 = A†1 A1 + B.
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ˇ We quote a part of the results by Exner, St’ov´ ıˇcek and Vytˇras ([Ex-St-Vy]) for later use. ˇ Proposition 5.2 (Exner-St’ov´ ıˇcek-Vytˇras). (i) We have
2 (R2 \ {0}); L1 u ∈ L2 (R2 ) . D(L1 ∗ ) = u ∈ L2 (R2 ) ∩ Hloc
(ii) The deficiency indices of L1 are (2, 2). (iii) The four linear functionals 2π α 1−α 1 Φ−1 (u) = lim r u(reiθ )eiθ dθ, r→+0 2π 0 2π 1 α −1+α iθ iθ α −1+α u(re )e dθ − Φ−1 (u)r , Ψ1 (u) = lim r r→+0 2π 0 2π α 1 (u) = lim r u(reiθ )dθ, Ψα 0 r→+0 2π 0 2π 1 −α iθ α −α (u) = lim r u(re )dθ − Ψ (u)r Φα 0 0 r→+0 2π 0 are well defined and finite for u ∈ D(L1 ∗ ). (iv) Every u ∈ D(L1 ∗ ) is uniquely decomposed as α α α α α α α u = Φα −1 (u)φ−1 + Ψ1 (u)ψ1 + Φ0 (u)φ0 + Ψ0 (u)ψ0 + ξ,
(37)
where ξ ∈ D(L1 ). In particular, the element u ∈ D(L1 ∗ ) belongs to D(L1 ) if and only if α α α Φα −1 (u) = Ψ1 (u) = Φ0 (u) = Ψ0 (u) = 0. Remark. The decomposition (37) is a paraphrase of [Ex-St-Vy, page 2158, line 8], since the four functions have the asymptotics α−1 −iθ α α −α e , ψ1α ∼ r1−α e−iθ , φα φα −1 ∼ r 0 ∼ r , ψ0 ∼ r
as r → 0. Lemma 5.3. (i) The following equalities hold:
where Γ(z) =
2
φα −1 1
=
π (2α − 1) B + 1
ψ1α 21
=
π(B 2 + 1)
2
2 B
π (2α + 1) B + 1
ψ0α 21
=
π(B 2 + 1)
0
2 B
2 B
α Γ(α),
Γ(2 − α),
=
2−α
2
φα 0 1
∞
2
2
2
1−α
2 B
1+α
Γ(1 − α),
e−t tz−1 dt is the Gamma function.
Γ(1 + α),
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∗ (ii) For u, v ∈ D((Lα 1 ) ), define α [u, v]1 = (Lα 1 u, v) − (u, L1 v).
Then, the following equalities hold: α α α [φα −1 , ψ1 ]1 = 4π(α − 1), [φ0 , ψ0 ]1 = 4πα, α α α α α α α [φα −1 , φ0 ]1 = [φ−1 , ψ0 ]1 = [ψ1 , φ0 ]1 = [ψ1 , ψ0 ]1 = 0, α α α [φα l , φl ]1 = [ψn , ψn ]1 = 0
for l = −1, 0, n = 0, 1.
α α α α ∗ (iii) The operators Φα −1 , Ψ1 , Φ0 , Ψ0 are bounded linear functionals on D((L1 ) ). Moreover, we have
1 1
ψ α 1 , Ψα
φα 1 , 1 ≤ 4π(1 − α) 1 4π(1 − α) −1 1 1
ψ0α 1 ,
φα 1 .
Φα
Ψα 0 ≤ 0 ≤ 4πα 4πα 0
Φα −1 ≤
Proof. (i), (ii) (35) and (36). (iii) Notice that
One can prove these equalities by a short calculation using
[u, v]1 = (L1 ∗ u, v) − (u, L1 v) = 0
for any u ∈ D(L1 ∗ ) and v ∈ D(L1 ). From this equality, (ii) of this lemma and (37), we have 1 1 [ψ1α , u]1 , Ψα [φα , u]1 , 1 (u) = 4π(1 − α) 4π(α − 1) −1 1 1 [ψ α , u]1 , [φα , u]1 . Φα Ψα 0 (u) = 0 (u) = −4πα 0 4πα 0
Φα −1 (u) =
(38)
Moreover, we have by the Schwarz inequality |[u, v]1 | ≤ L1 u v + u L1 v ≤ u 1 v 1 for u, v ∈ D(L1 ∗ ). By (38) and (39), the conclusion holds.
5.2
Deficiency indices
In the sequel, we denote U (r) =
N
{z ∈ R2 ; |z − zj | < r}.
j=1
Proposition 5.4. Let N = 1, 2, . . . , or N = ∞. Then the following holds: (i) We have 2 D(LN ∗ ) = {u ∈ L2 (R2 ) ∩ Hloc (R2 \ SN ); LN u ∈ L2 (R2 )},
where the derivative LN u is interpreted in the distribution sense.
(39)
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(ii) Let u ∈ D(LN ∗ ). Suppose that there exists a constant R1 with 0 < R1 < R such that supp u ⊂ R2 \ U (R1 ). Then, u ∈ D(LN ). Proof. (i) This assertion follows from the definition of the adjoint operator and the elliptic inner regularity (see [Ag]). (ii) Take u satisfying the assumption. Since the vector potential aN is smooth in R2 \ U (R1 ), we can approximate u with respect to the graph norm of LN ∗ by a sequence of functions in C0∞ (R2 \ U (R1 )), because of the essential self-adjointness of Schr¨ odinger operators with smooth vector potentials (see [Ik-Ka], [Le-Si]). Since C0∞ (R2 \ U (R1 )) ⊂ D(LN ), we have u ∈ D(LN ). The following lemma shows the continuity of the cut-off map with respect to the graph norm. Lemma 5.5. Let 1 ≤ N < ∞ or N = ∞. Let η ∈ C ∞ (R2 ) and suppose supp |∇η| is a compact set in R2 \ SN . Then, for any u ∈ D(LN ∗ ), we have ηu ∈ D(LN ∗ ) and
[LN , η]u 2 ≤ C0
supp |∇η|
(|LN u|2 + |u|2 )dxdy,
(40)
where C0 = 10 sup(|∇η|2 + |∇(∂x η)|2 + |∇(∂y η)|2 ). Moreover, we have (|LN u|2 + |u|2 )dxdy,
ηu 2N ≤ C1
(41)
supp η
where C1 = 2(sup |η|2 + C0 ). Proof. The proof of (40) is the same as the proof of (2.31) in [Co-Ne], and (41) follows immediately from (40). The fact ηu ∈ D(LN ∗ ) follows from (i) of Proposition 5.4 and the Leibniz rule. Let χ, χj be functions satisfying (4). Define a linear operator T from αj ∗ αj D(LN ∗ )/D(LN ) to ⊕N j=1 D((L1 ) )/D(L1 ) by N
T ([u]) = ⊕ [Tj u]
(42)
j=1
for u ∈ D(LN ∗ ), where α
Tj : D(LN ∗ ) u → t−zj (χj u) ∈ D((L1 j )∗ ) and t−zj is the magnetic translation operator from {|z − zj | < α intertwining LN with L1 j .
(43) R 2}
to {|z| <
R 2}
Lemma 5.6. The operator T defined above is well defined, bijective and bicontinuous.
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Proof. First we show the well-definedness. Let u ∈ D(LN ∗ ). By Lemma 5.5, the intertwining property and unitarity of t−zj , and (i) of Proposition 5.4, we have Tj u ∈ D(LN ∗ ) and N
[Tj u] 2 ≤
j=1
≤ C1
N j=1
N j=1
{|z−zj |< R 2 }
Tj u 21 =
N
χj u 2N
j=1
(|LN u|2 + |u|2 )dxdy ≤ C1 u 2N < ∞,
(44)
where C1 is a constant dependent only on χ. Thus the right-hand side of (42) α converges even when N = ∞. In addition, we can show that Tj D(LN ) ⊂ D(L1 j ) 2 2 ∞ ∞ by the fact Tj C0 (R \ SN ) ⊂ C0 (R \ {0}) and an approximating argument using (44). Thus T is well defined. Moreover, if we take the representative u from (D(LN ))⊥ (with respect to the graph inner product), the inequality (44) implies the continuity of T . We can prove the bijectivity of T by constructing its inverse, that is, N
α
α
T −1 : ⊕ D((L1 j )∗ )/D(L1 j ) ([u1 ], . . . , [uN ]) → j=1 N tzj χuj ∈ D(LN ∗ )/D(LN ). j=1
N α ∗ Here we choose representatives uj ∈ D(L1 j ) (j = 1, . . . , N ) satisfying j=1 uj 21 ⊥ α < ∞; this condition is satisfied if we take uj ∈ D(L1 j ) . The well-definedness and continuity of T −1 also follows from Lemma 5.5, the intertwining property and unitarity of tzj , and Proposition 5.4. The equalities T T −1 = Id and T −1 T = Id follow from (ii) of Proposition 5.4. We shall determine the deficiency indices of LN . Proposition 5.7. Let 1 ≤ N < ∞ or N = ∞. Then, the deficiency indices of LN are (2N, 2N ). Proof. Since the operator LN is symmetric and positive, the deficiency indices m± = dim Ker(LN ∗ ∓ i) are equal (see [Re-Si, Corollary of Theorem X.1]). Since D(LN ∗ ) = D(LN ) ⊕ Ker(LN ∗ − i) ⊕ Ker(LN ∗ + i) (see [Re-Si, (b) of Lemma in page 138]), it is sufficient to show that dim D(LN ∗ )/D(LN ) = 4N, which follows from Lemma 5.6 and (ii) of Proposition 5.2.
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Characterization of self-adjoint extensions
We shall introduce some notation used in the theory of self-adjoint extensions of a symmetric operator (see [Re-Si]). Let X be a densely defined, closed, symmetric operator on a separable Hilbert space H. Define a sesqui-linear form [·, ·]X on D(X ∗ ) by [u, v]X := (X ∗ u, v) − (u, X ∗ v), u, v ∈ D(X ∗ ). Define a sesqui-linear form ·, ·X on D(X ∗ )/D(X) by [u], [v]X := [u, v]X , u, v ∈ D(X ∗ ). This form is well defined, since [u, v]X = 0 if u ∈ D(X) or v ∈ D(X). For a closed subspace V of D(X ∗ )/D(X), define a closed subspace V [⊥] by V [⊥] = {[u] ∈ D(X ∗ )/D(X) ; [u], [v]X = 0 for any [v] ∈ V }. Let P denotes the quotient map from D(X ∗ ) onto D(X ∗ )/D(X). The following proposition is fundamental (a similar statement is seen in [ReSi, Lemma on p.138]). Proposition 5.8. The correspondence Y = X ∗ |P −1 V ↔ V = P D(Y ) is a one-to-one correspondence between the closed operators Y satisfying X ⊂ Y ⊂ X ∗ and the closed subspaces V of D(X ∗ )/D(X). Moreover, the adjoint operator Y ∗ corresponds the closed subspace V [⊥] , and the self-adjoint extensions Y correspond the closed subspaces V satisfying V [⊥] = V . In the sequel, we denote [·, ·]N = [·, ·]LN ∗ and ·, ·N = ·, ·LN ∗ . Let us find a simple expression of the form ·, ·N . For simplicity, we assume the following. Assumption 1. There exist constants α− , α+ such that 0 < α− ≤ αj ≤ α+ < 1 for every j = 1, . . . , N. Remark. This assumption is automatically satisfied when N is finite. α α Define a linear operator Ξj from D((L1 j )∗ )/D(L1 j ) to C 4 by α
α
α
α
j Ξj [u] := (Φ−1 (u), Ψ1 j (u), Φ0 j (u), Ψ0 j (u)),
α
α
[u] ∈ D((L1 j )∗ )/D(L1 j ),
for j = 1, . . . , N . This operator is well defined by (iv) of Proposition 5.2. Define a linear operator Ξ from D(LN ∗ )/D(LN ) to C 4N (when N = ∞, C 4N = l2 (N )) by Ξ[u] := (Ξ1 [T1 u], . . . , ΞN [TN u])
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where Tj is the operator defined by (43). We call Ξ the boundary value operator. Define αj α (j) (j) ), ψ1 := tzj (χψ1 j ), φ−1 := tzj (χφ−1 αj α (j) (j) φ0 := tzj (χφ0 ), ψ0 := tzj (χψ0 j ) for j = 1, . . . , N , where χ is a function satisfying (4) and tzj is the magnetic α translation operator from {|z| < R2 } to {|z − zj | < R2 } intertwining L1 j with LN . Proposition 5.9. Let 1 ≤ N < ∞ or N = ∞. Suppose that Assumption 1 holds. Then, the boundary value operator Ξ is well defined, bijective, bicontinuous, and its inverse Ξ−1 is given by N (j) (j) (j) (j) , c4j−3 φ−1 + c4j−2 ψ1 + c4j−1 φ0 + c4j ψ0 Ξ−1 c = j=1
for c = (c1 , . . . , c4N ) (when N = ∞, the sum in the bracket converges in D(LN ∗ )). Moreover, [u], [v]N = (Ξ[u], JΞ[v])C4N for u, v ∈ D(LN ∗ ), (45) where J is a bounded operator on C 4N defined by the 4N × 4N matrix 0 α−1 0 Jα1 O . . . O 1−α O Jα2 . . . O 0 0 J = . . . . . . . . . . . . . . . . . . . , Jα = 4π 0 0 0 0 0 −α O O . . . JαN
0 0 . α 0
Proof. By Lemma 5.3 and Lemma 5.5, we see that there exists a constant C dependent only on α− , α+ , B, R, χ (independent of j) such that
Ξj [u] C4 ≤ C [u] ,
α
α
[u] ∈ D((L1 j )∗ )/D(L1 j ).
The well-definedness and continuity of Ξ follows from this inequality and Lemma 5.6. By (i) of Lemma 5.3, Lemma 5.5 and the intertwining property of tzj , we have there exists a constant C dependent only on α− , α+ , B, R, χ such that
N
(j)
(j)
(j)
(j)
c4j−3 φ−1 + c4j−2 ψ1 + c4j−1 φ0 + c4j ψ0
N ≤ C c C4N .
j=1
Thus Ξ−1 is also well defined and continuous. The equality Ξ−1 Ξ = Id follows from (iv) of Proposition 5.2 and Lemma 5.6, and ΞΞ−1 = Id follows by definition. Thus, we see that the set of functions (j)
(j)
(j)
(j)
{[φ−1 ], [ψ1 ], [φ0 ], [ψ0 ]}N j=1
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forms a basis of D(LN ∗ )/D(LN ) (when N = ∞, the finite linear combinations of the above functions are dense in D(LN ∗ )/D(LN )). We can prove that (45) holds for any element [u], [v] in the above basis. For example, (j)
α
(j)
α
α
α
j j [φ−1 ], [ψ1 ]N = [χφ−1 , χψ1 j ]1 = [φ−1 , ψ1 j ]1 = 4π(αj − 1),
where we used the intertwining property of tzj in the first equality, the fact (1 − αj α χ)φ−1 ∈ D(L1 ) and (1 − χ)ψ1 j ∈ D(L1 ) in the second and (ii) of Lemma 5.3 in the last. Corollary 5.10. Under the assumption of Proposition 5.9, we have N
(j)
(j)
(j)
(j)
D(LN ∗ ) = D(LN ) ⊕alg ⊕ L.h.{φ−1 , ψ1 , φ0 , ψ0 }, j=1
where ⊕alg denotes the algebraic direct sum and L.h. the linear hull. We shall give a characterization of self-adjoint extensions of LN . Theorem 5.11. Let N = 1, 2, . . . or N = ∞. Suppose that Assumption 1 holds. Let M be a bounded operator on C 4N satisfying Ran M : closed, Ker M ∗ J = Ran M,
(46)
M where J is a bounded operator given in Proposition 5.9. Define an operator HN by M D(HN ) = M HN u =
{u ∈ D(LN ) ; Ξ[u] ∈ Ran M }, M LN u, u ∈ D(HN ).
M Then, HN is a self-adjoint extension of LN . Moreover, for any self-adjoint extension HN of LN , there exists a bounded operator M on C4N satisfying (46) and M HN = HN .
Remark. When N = 1, Theorem 5.11 is a paraphrase of the proposition in [ExSt-Vy, page 2159]. M [⊥] M ) = P D(HN ) is translated Proof. By Proposition 5.9, the condition P D(HN into Ker M ∗ J = Ran M through the isomorphism Ξ, where P denotes the quotient map from D(LN ∗ ) to D(LN ∗ )/D(LN ). Thus the first assertion follows from Proposition 5.8. The second assertion holds if we take M as the orthogonal pro jection onto ΞP D(HN ).
By the above proof, we have the following corollary. M is Corollary 5.12. Suppose that Assumption 1 holds. Then, the map M → HN a one-to-one correspondence between the set of the orthogonal projections M on C 4N satisfying Ker M J = Ran M
and the set of the self-adjoint extensions HN of LN .
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When the operator M has a simple form, the condition (46) is simplified as follows. Corollary 5.13. (i) Let N < ∞. Then, the condition (46) is equivalent to rankM = 2N, M ∗ JM = O, where rankM = dim Ran M . (ii) Let N = 1, 2, . . . or N = ∞. Suppose that Assumption 1 holds. Suppose that the operator M is the (finite or infinite) direct sum of 4 × 4 orthogonal projection matrices, that is, M1 O . . . O O M2 . . . O 2 ∗ M = . . . . . . . . . . . . . . . . . . . , Mj : 4 × 4, Mj = Mj , Mj = Mj . O O . . . MN Then, the condition (46) is equivalent to rankMj = 2, Mj Jαj Mj = O
5.4
for every j = 1, . . . , N.
Proof of Lemma 3.4
Lemma 5.14. Let 1 ≤ N < ∞ or N = ∞. Then, we have D(AN ) = D(A†N ) ⊂ {u ∈ L2 (R2 ) ; AN u ∈ L2 (R2 ), A†N u ∈ L2 (R2 )},
(47)
where the derivatives AN u and A†N u are interpreted in the distribution sense. Proof. Notice that the graph norm of D(AN ) and that of D(A†N ) are equivalent, since
AN u 2 = (A†N AN u, u) = ((AN A†N − 2B)u, u) = A†N u 2 − 2B u 2, for u ∈ C0∞ (R2 \ SN ). Thus we have D(AN ) = D(A†N ). By (ii) of Lemma 3.1, we have ∗ (48) D(AN ) ⊂ D(AN ∗ ) ∩ D(A†N ). The right-hand side of (48) coincides with the last term of (47), by the definition of the adjoint operator. Lemma 3.4 is an immediate corollary of the following two lemmas. AB Lemma 5.15. Let 1 ≤ N < ∞ or N = ∞. Let HN be the Friedrichs extension of LN . Suppose that Assumption 1 holds. Then, we have N
(j)
(j)
AB D(HN ) = D(LN ) ⊕alg ⊕ L.h.{φ0 , ψ1 }. j=1
(49)
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Proof. Let D be the right-hand side of (49). Then, the operator LN ∗ |D equals M AB
HN N , where AB MN
O ... O M AB O M AB . . . O = ......................... , O O . . . M AB M AB
By (ii) of Corollary 5.13, we have HN N that
AB HN
⊂
M AB
0 0 = 0 0
0 1 0 0
0 0 1 0
0 0 . 0 0
is self-adjoint. Thus, it suffices to show
M AB HN N .
AB In the proof of Proposition 3.3, we obtain Q(HN ) = D(A†N ). By (47), we
have AB ) ⊂ D(HN
⊂
D(LN ∗ ) ∩ D(A†N ) {u ∈ D(LN ∗ ) ; AN u ∈ L2 (R2 ), A†N u ∈ L2 (R2 )}.
(50)
By a short calculation using the intertwining property of tzj , we can prove (j)
(j)
(j)
(j)
(j)
(j)
(j)
(j)
AN φ−1 ∈ / L2 , AN ψ1 ∈ L2 , AN φ0 ∈ L2 , AN ψ0 ∈ L2 and
/ L2 A†N φ−1 ∈ L2 , A†N ψ1 ∈ L2 , A†N φ0 ∈ L2 , A†N ψ0 ∈
(51) (52)
for every j = 1, . . . , N . By Corollary 5.10, (51) and (52), we have that the last term of (50) equals D. − be the operator defined in Lemma 5.16. Let 1 ≤ N < ∞ or N = ∞. Let HN Proposition 3.3. Suppose that Assumption 1 holds. Then, we have N
(j)
(j)
− ) = D(LN ) ⊕alg ⊕ L.h.{ψ0 , ψ1 }. D(HN j=1
(53)
Proof. As well as the first part of the proof of Lemma 5.15, we conclude that it − ) is included in the right-hand side of (53). By the suffices to prove that D(HN − definition of HN and (47), we have − D(HN ) =
= ⊂
∗
D(A†N A†N ) ∗
{u ∈ D(A†N ) ; AN u ∈ D(A†N )}
{u ∈ D(LN ∗ ) ; A2N u ∈ L2 (R2 )}.
(54)
By a short calculation using the intertwining property of tzj , we can prove (j)
(j)
(j)
(j)
A2N φ−1 ∈ / L2 , A2N ψ1 ∈ L2 , A2N φ0 ∈ / L2 , A2N ψ0 ∈ L2
(55)
for every j = 1, . . . , N . By Corollary 5.10 and (55), we have that the last term of (54) equals the right-hand side of (53).
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Proof of lemmas in Section 4.1
Proof of Lemma 4.1. By Corollary 5.12, there exists a 4 × 4 orthogonal projection matrix M1 satisfying rankM1 = 2, M1 Jα M1 = O, ∗ D(H1 ) = {u ∈ D((Lα 1 ) ); Ξ1 [u] ∈ Ran M1 },
(56)
where Ξ1 is the map Ξ associated to the operator Lα 1 . Put M1 O . . . O O M1 . . . O MN = ................... . O O . . . M1 MN is self-adjoint by (ii) of Corollary 5.13 and Theorem 5.11. Then we have HN Moreover, MN ) D(HN
= =
{u ∈ D(LN ∗ ) ; Ξ[u] ∈ Ran MN } {u ∈ D(LN ∗ ) ; Ξ1 [Tj u] ∈ Ran M1 for every j = 1, . . . , N }
=
{u ∈ D(LN ∗ ) ; t−zj (χj u) ∈ D(H1 ) for every j = 1, . . . , N }.
MN satisfies (5). The uniqueness follows by construction. Thus HN = HN
(57)
∗
Proof of Lemma 4.2. Take u ∈ D(HN ). We have ηj u ∈ D(LN ) by Lemma 5.5. By (57), we have Ξ1 [Tj (ηj u)] = δj j Ξ1 [Tj u] ∈ Ran M1 for j = 1, . . . , N . Thus we have ηj u ∈ D(HN ) by (57), and we have t−zj ηj u ∈ D(H1 ) by (56). The equality (15) follows from the intertwining property of t−zj . Thus the assertion (i) holds. We can prove the assertion (ii) similarly. Proof of Lemma 4.3. Take u ∈ D(HN ). Then ηm u ∈ D(LN ) by (ii) of Proposition 5.4. Since HN ⊃ LN , we have ηm u ∈ D(HN ). Moreover, L0 (t−m ηm u) = t−m LN ηm u ∈ L2 (R2 ). This implies t−m ηm u ∈ D(H0 ) and (16) holds, since H0 coincides with the maximal operator, that is, 2 D(H0 ) = {u ∈ L2 (R2 ) ∩ Hloc (R2 ) ; L0 u ∈ L2 (R2 )}
(see [Ik-Ka], [Le-Si]). Thus the assertion (i) holds. Using Proposition 5.4, we can prove the assertion (ii) similarly.
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Acknowledgment. The author would like to thank Akira Iwatsuka and all other members of Operator Seminar in RIMS, for many helpful comments. In particular, the author would like to thank Shin-ichi Shirai for giving the author an opportunity to begin the present work. The author would also like to thank Hideo Tamura, Hiroshi Ito and Yuji Nomura for helpful comments at the seminar in Ehime University, Pavel Exner and Michael Melgaard for teaching the author some references concerning the present subject.
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Takuya Mine Department of Mathematics Faculty of Science Kyoto University Sakyo-ku, Kyoto 606-8502 Japan email:
[email protected] Communicated by Rafael D. Benguria submitted 28/05/04, accepted 23/07/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 155 – 194 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/010155-40 DOI 10.1007/s00023-005-0202-3
Annales Henri Poincar´ e
KIDs are Non-Generic Robert Beig, Piotr T. Chru´sciel and Richard Schoen Abstract. We prove that the space-time developments of generic solutions of the vacuum constraint Einstein equations do not possess any global or local Killing vectors, when Cauchy data are prescribed on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that nonexistence of global symmetries implies, generically, non-existence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors.
1 Introduction Let P be the linearization of the general relativistic constraints map, as defined by (8.1) below. Recall that a Killing Initial Data (KID) is a couple (N, Y ), defined on a spacelike hypersurface, where N is a function and Y is a vector field, such that P ∗ (Y, N ) = 0, see (5.3)–(5.4) below. In vacuum space-times, with or without cosmological constant, KIDs are in one-to-one correspondence with Killing vectors in the associated space-time [13, 19]. A local Killing vector field is a solution X of the Killing equations defined on an open subset of a pseudo-Riemannian1 manifold M ; local conformal Killing vector fields and local KIDs are defined in an analogous way. When attempting to glue general relativistic initial data sets [12] one is faced with the need of proving the following: Conjecture 1.1 Generic general relativistic vacuum initial data sets have no local KIDs. The object of this paper is to establish such a fact under some supplementary conditions. For U ⊂ M let K (U ) denote the set of KIDs on U . We show, first, that non-existence of global KIDs implies, generically, non-existence of local ones: Theorem 1.2 Let Λ ∈ R, and consider the collection of vacuum initial data sets with cosmological constant Λ on an n-dimensional manifold M with a C k,α topology, k ≥ k0 (n), for some k0 (n) (k0 (3) = 6)2 , α ∈ (0, 1). Let (K0 , g0 ) in this 1
In our terminology a Riemannian metric is also pseudo-Riemannian. function k0 (n) is obtained, in dimension n = 3, by chasing the differentiability thresholds throughout the proof. We do not have an explicit estimate for k0 (n) if n > 3, or for the function 0 (n) appearing in Theorem 8.7 below, because some steps of the proof in those dimensions proceed via non-constructive arguments, see Section 7. 2 The
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collection be such that K (M ) = {0} .
(1.1)
1. Let p ∈ M and consider the set Vp = {vacuum initial data such that K (U ) = {0} for any neighborhood U of p} . Then Vp is open and dense in a neighborhood of (K0 , g0 ). 2. Define further: V = {vacuum initial data such that K (U ) = {0} for any open subset U of M } . Then V is of second category in a neighborhood of (K0 , g0 ). Identical results hold in the class of initial data with fixed constant trg K, as well as in the class of time symmetric initial data K ≡ 0. (Recall that a set is of second category if it contains a countable intersection of open dense sets; in complete metric or Fr´echet spaces such sets are dense.) The C k,α topology in Theorem 1.2, as well as in the remaining results below unless explicitly stated otherwise, can be understood as follows: one chooses some smooth complete Riemannian metric h on M , which is then used to calculate norms of tensors and their h-covariant derivatives. Other choices are possible, and this is discussed in more detail in Appendix A. One expects that for generic initial data the no-global-KIDs condition (1.1) of Theorem 1.2 will be satisfied. Attempts to prove that require analytical tools which impose restrictions on the geometry. We concentrate therefore on three cases which seem to us to be the most important ones from the point of view of applications: compact manifolds without boundary, or asymptotically flat initial data sets, or conformally compactifiable initial data sets. Our next main result, when used in conjunction with Theorem 1.2, establishes Conjecture 1.1 in those cases: Theorem 1.3 Consider the following collections of vacuum initial data sets: 1. Λ = 0 with an asymptotically flat region, or 2. trg K = Λ = 0 with an asymptotically flat region, or 3. K = Λ = 0 with an asymptotically flat region, or 4. with a conformally compactifiable region in which trg K is constant, or
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5. the trace of K is constant and the underlying manifold M is compact, with (trg K)2 ≥
2n Λ, (n − 1)
(1.2)
or 6. K = 0, M is compact, and the curvature scalar R satisfies R = 2Λ ≤ 0, with a C k,α × C k,α (weighted in the non-compact region) topology, with k ≥ k0 (n) for some k0 (n) (k ≥ 6 if n = dim M = 3). For each such collection the subset of vacuum initial data sets without global KIDs is open and dense. The weights in the asymptotic region should be chosen so that the metrics approach the Euclidean one as r−β , for some β ∈ (0, n − 2]. In the conformally compactifiable regions a topology as in [11, Theorem 6.7] with 0 ≤ t < (n + 1)/2 should be used. It would be of interest to have a version of points 4 and 5 without the CMC condition. Since the collection of initial data sets which have no global KIDs is open (see Proposition 4.2 below), for compact manifolds the proof of Theorem 1.3 also provides a large open collection of initial data sets which are close to CMC data and which have no global KIDs. However, the general case remains open. We think that the removal of the CMC condition in point 5 is the most notable problem left open by our paper. Somewhat surprisingly, the above results require a considerable amount of non-trivial work. We first show that generic metrics have no local conformal Killing vectors, or local Killing vectors3 . This is done by reducing the problem to a finite system of linear algebraic equations for the candidate vector, as well as a few of its derivatives, at a given point. While the argument is conceptually straightforward, there is some messy algebra involved when one wishes to show that those algebraic equations lead to the desired conclusion for at least one metric. This result is then used in the proof of Theorem 1.3. A similar argument is used for local KIDs, with an appropriately messier algebra. That would have settled the problem, if not for the fact that we want initial data satisfying the constraint equations. In order to take care of that we first use Taylor expansions to construct approximate solutions of the constraint equations near a point p. The gluing techniques of Corvino-Schoen [14] type, as extended in [10, 11], are then used to go from an approximate solution to a real one, establishing Theorem 1.2. Some comments on the organization of this paper are in order. The heart of our analysis lies in Section 8, where we show how to perturb solutions of the vacuum constraint equations to solutions without KIDs, preserving the constraint 3 The
only related result known to us in the literature is in [15], where it is shown that on a compact boundaryless manifold the set of Riemannian metrics without nontrivial isometries is open and dense. The argument given there does not seem to be useful to get rid of local Killing vector fields, and makes essential use of the fact that M is compact without boundary. Moreover it is not clear how to adapt it to account for conformal Killing vectors, or for KIDs.
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equations. This requires several preliminary results, such as a) perturbing initial data to get rid of KIDs, without necessarily satisfying any constraint equations, and b) perturbing metrics to get rid of conformal Killing vectors. The argument needed for a) is presented in Section 5 in dimension three, and in Section 7 in all dimensions. The advantage of the argument in Section 5 is that it gives an explicit differentiability threshold for the construction, in the physically important case n = 3, while the one in Section 7 leads to some uncontrollable, dimension dependent threshold. In Section 6 we show how to get rid of KIDs in the timesymmetric case, while remaining in the time-symmetric class. In Section 2 we construct functions that control existence, or lack thereof, of conformal Killing vector fields in dimension three. This result is the key for getting rid of KIDs on CMC initial data sets; it also sets the stage for the structure of the argument for KID-removal. As before, the higher-dimensional proof is carried out in Section 7, with some non-explicit differentiability threshold. In Section 3 we construct the corresponding functions for controlling Killing vectors. Here we obtain explicit differentiability thresholds in all dimensions. Perturbations removing conformal Killing vectors do of course remove Killing vectors as well, but the differentiability thresholds we obtain in the Killing case are explicit in all dimensions, and smaller than the corresponding conformal Killing threshold in dimension three. In Section 4 we show how the local perturbation arguments of the previous sections can be translated into category-type statements. This leads immediately to the question of topologies appropriate in our context, this is briefly discussed in Appendix A. Appendix B presents a monodromy-type argument for analytic overdetermined PDE systems, needed in the proofs of Section 7. All the results just described join forces in Section 9, where Theorems 1.2 and 1.3 are established.
2 Metrics without conformal Killing vectors near a point, n=3 We start with some preliminaries. Unless explicitly specified otherwise we assume in this section that dimension equals three. Recall that the Schouten tensor Lij is given by 1 Lij = Rij − gij R , (2.1) 4 where gij is a pseudo-Riemannian metric and Rij and R are respectively its Ricci and scalar curvature. Furthermore we define the Cotton tensor Bijk Bijk = Li[j;k] .
(2.2)
The tensor Bijk has the following algebraic properties Bijk = Bi[jk] ,
B iik = 0 ,
B[ijk] = 0 ,
(2.3)
which makes five degrees of freedom per space point. It also satisfies Bi[jk;l] = 0 .
(2.4)
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Equivalently, we can take Hij = kli Bjkl .
(2.5)
The tensor Hij is symmetric, tracefree and divergence-free. Suppose a metric has a conformal Killing vector X, Di Xj + Dj Xi =
2 ϕgij , 3
(2.6)
where ϕ here is the divergence of the vector field X. Then it has to be the case that (2.7) LX Bijk = 0 . The reason is that the map Cotton sending a metric to its Cotton tensor satisfies Φ∗ Cotton[g] = Cotton[Φ∗ g] for any map Φ of M into itself. One now applies this relation to the case where Φ is a one- parameter family of diffeomorphisms generated by a conformal Killing vector X. Taking the derivative with respect to the parameter and using that the map Cotton is invariant under conformal rescaling of the metric one obtains (2.7). An equivalent form of (2.7) is the relation 1 LX Hij = − ϕHij , 3
(2.8)
Taking cyclic permutations of the equation obtained by differentiating (2.6) one has 2 (2.9) Di Fjk = −Rjki l Xl + ϕ[j gk]i , 3 where we have defined Fij = D[i Xj] and ϕi = Di ϕ. The Lie derivative of the tensor Rij − Rgij /2(n − 1) (here, for future reference, we work in general dimension n) equals R (n − 2) LX Rij − − gij = − Di Dj ϕ . (2.10) 2(n − 1) n In dimension n = 3, to which we return now, this reads Di ϕj = −3LX Lij .
(2.11)
The identities (2.9)–(2.11), together with the relation Di Xj = Fij + ϕgij /3 ,
(2.12)
imply that a conformal Killing vector, for which the quantities (X , Fij , ϕ, ϕi ) are all zero at the point p, has to vanish in a neighborhood of p. Using (2.9), (2.8) takes the form (2.13) X k Dk Hij + 2F (i k Hj)k + ϕHij = 0 .
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Next we take a derivative of (2.13) with the result that 4 F l k Dk Hij + ϕDl Hij +X k Dl Dk Hij +2(Dl F (i k )Hj)k +2F(i k D|l| Hj)k +ϕl Hij = 0 . 3 (2.14) We are ready now to pass to the proof of the main result of this section: Theorem 2.1 Let (M, g) be a smooth three-dimensional pseudo-Riemannian1 manifold. 1. There exists a non-trivial homogeneous polynomial Q(·, ·, ·) : R6 × R3×6 × R3×3×6 → R such that if
Q(H, DH, D2 H)(p) = 0
(the R6 arises here because H is symmetric), then there exists a neighborhood Op of p on which there are no local conformal Killing vectors. 2. Let Ω be a neighborhood of p ∈ M . For any k ≥ 5 and > 0 there exists a metric g ∈ C ∞ (M ) such that g − g C k (Ω) ¯ < , with g − g supported in Ω, and such that Q(H , DH , D2 H )(p) does not vanish. Remark 2.2 A corresponding result in higher dimensions is proved in Theorem 7.4. Remark 2.3 Recall that a polynomial in the curvature tensor and its derivatives is called invariant if it is independent of the frame used to evaluate its numerical value. Below we arbitrarily choose some orthonormal basis of Tp M to define Q, and it is unlikely that the polynomial Q defined in our proof will be an invariant polynomial if the signature of the metric is Lorentzian; moreover, it is not clear how to modify Q to make it invariant while preserving the claimed properties. Note that one can view Q as a function on the frame bundle. In the Riemannian case ˜ be the integral of Q over those fibers with respect to the Haar measure, we let Q ˜ then Q is a non-trivial invariant polynomial with the properties as above. We note that the polynomial constructed below provides a convenient tool to capture the fact that a certain geometrically defined matrix has rank larger than ten; the latter assertion provides an equivalent invariant statement, regardless of signature. Proof. Before passing to the proof, some auxiliary results will be useful., Let the ˚ij := (Dk Rij )(p). In the superscript “˚” denote “value at the point p”, e.g., Dk R calculations that follow we will assume that the metric has Riemannian signature. The remaining cases require trivial modifications, which we leave to the reader. We start with a Lemma:
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Lemma 2.4 Consider a metric such that ˚ij = 0, Dk R ˚ij = 0 . ˚ gij = δij , R
(2.15)
Furthermore let the second derivatives of the curvature be such that ˚ij = Axk y(i zj) + Byk x(i zj) + Czk x(i yj) , Dk H
(2.16)
where (x, y, z) form an orthonormal basis of Tp M and the three real numbers A, B, C are all non-zero. Then the set of algebraic equations for w := (Xi , Fij := D[i Xj] , ϕ := Dk Xk , ϕi := Di ϕ)(p) obtained from the equations [LX Hij + 13 ϕHij ](p) = 0 , m Hj)m + 13 Dk (ϕHij )](p) = 0 , [LX Dk Hij + 2Ck(i
(2.17) (2.18)
m m Dm Hij + 2(Dl Ck(i )Hj)m + [LX Dl Dk Hij + Ckl m m +2Cl(i D|k| Hj)m + 2Ck(i D|l| Hj)m + 13 Dl Dk (ϕHij )](p) = 0 , i with Cjk defined as i = Cjk
1 i 2ϕ(j δk) − gjk ϕi 3
(2.19)
(2.20)
implies w = 0. Remark 2.5 Equations (2.17)–(2.19) are necessarily satisfied by every conformal Killing vector field X: (2.17) is equivalent to (2.13), while Equations (2.18)–(2.19) are equivalent to the first and second covariant derivatives of (2.13). ˚ij in (2.16) satisfies the necessary algebraic Remark 2.6 It can be seen that Dk H requirements to arise from a metric (i.e., being symmetric in (ij) and trace-free on all index pairs, compare (2.3)–(2.5)); this follows in any case from Proposition 2.7 below. ˚ = 0. Proof. It immediately follows from Equations (2.15), (2.16) and (2.13) that X Let a, b and c be defined as the following components of F in the basis (x, y, z): ˚l k = a(xl y k − yl xk ) + b(zl xk − xl z k ) + c(yl z k − zl y k ). (2.21) F ˚ = 0 we find that Evaluating (2.14) at p, and using X 0 = [b(A + C)zl − a(A + B)yl ]y(i zj) + [a(A + B)xl − c(B + C)zl ]x(i zj) + [c(B + C)yl − b(A + C)xl ]x(i yj) + (aCzl − bByl )xi xj + (cAxl − aCzl )yi yj 4 ˚(Axl y(j zi) + Byl x(j zi) + Czl x(j yi) ). + (bByl − cAxl )zi zj + ϕ (2.22) 3
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It follows by inspection that a, b, c and ϕ ˚ have all to be zero. Differentiating (2.14) we find that ˚lk )Dk H ˚ij + 4 ϕ ˚ij ˚m Dl H 0 = (Dm F 3 ˚(i|k )Dm| H ˚j) k + 2(Dm F ˚(i|k )Dl| H ˚ j) k + ϕ ˚ij , + 2(Dl F ˚l Dm H (2.23) where we have used the vanishing of X and Di X j at p. Next observe that, by Equations (2.15) and (2.9), there holds ˚jk = Di F
2 ϕ ˚[j gk]i . 3
(2.24)
We now insert (2.24) into (2.23) to find that 0=
8 ˚ij − 1 gml ϕ ˚ij + 2 ϕ ˚j)l ϕ ˚(l Dm) H ˚(i D|m| H ˚k Dk H 3 3 3 2 ˚j)m − 2 ϕ ˚j k − 2 ϕ ˚i k . ˚(i D|l| H ˚k gi(l Dm) H ˚k gj(l Dm) H + ϕ 3 3 3
(2.25)
Direct algebra using (2.16) shows that ϕ ˚i vanishes, which is what had to be established. Let us show now that Proposition 2.7 A metric satisfying (2.15)–(2.16) exists. Proof. We start with two elementary lemmata: Lemma 2.8 Suppose we are given, on a star-shaped domain Ω in (Rn , δij ), a tensor field Bijk satisfying Bijk B[ijk] Bi[jk,l]
= Bi[jk] ,
(2.26)
= 0, = 0.
(2.27) (2.28)
Then there exists a tensor field Lij = L(ij) such that Bijk = Li[j,k] .
(2.29)
If B is a homogeneous polynomial of order p, then L can be chosen to be a homogeneous polynomial of order p + 1. Proof. By Equations (2.26)–(2.28), there exists a tensor field Mij , not necessarily symmetric in i and j, satisfying (2.29) with Lij replaced by Mij . From (2.27) it follows that there exists a covector field Λi with M[ij] = Λ[i,j] . Set Lij = Mij − Λi,j , then Lij = L(ij) and satisfies (2.29) thus proving Lemma 2.8. The fact that solutions can be chosen as polynomials follows from the explicit formula for the primitive of a form used in the proof of the Poincar´e Lemma. We will also need the following variation of a result of Pirani [21]:
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Lemma 2.9 Let Ω be as in Lemma 2.8 and on it a tensor field Rijkl having the symmetries of the Riemann tensor and obeying the differential identity Rij[kl,m] = 0 .
(2.30)
Then there exists hij = h(ij) such that Rijlm = 2∂[i hj][l,m] .
(2.31)
If moreover Rijkl is a homogeneous polynomial in the manifestly flat coordinates ξ i of order q, then hij can be chosen as a homogeneous polynomial of order q + 2. Proof. This is proved by inspection of the proof in Pirani [21, pp. 279–280], using the fact that the proof there consists of the repeated use of the Poincar´e Lemma. Returning to the proof of Proposition 2.7, let ξ be coordinates on Ω and define 1 Bijk = mjk (∂n Him )ξ n , (2.32) 2 where the constants ∂n Him are given by the right-hand-side of (2.16). The field Bijk defined by (2.32) obviously satisfies (2.26), while (2.27)-(2.28) hold because (2.16) is trace-free in all indices. Now let Lij be the homogenous quadratic polynomial guaranteed to exist by Lemma 2.8. As (2.16) is symmetric in i and j, the field Bijk satisfies the second equation in (2.3). This implies ∂ j Lij = ∂i L,
(2.33)
where L = δ ij Lij . Consider the field Sijkl defined by Sijkl = 2δk[i Lj]l − 2δl[i Lj]k ,
(2.34)
it is a homogeneous quadratic polynomial in ξ which clearly has the symmetries of a Riemann tensor. Equation (2.33) implies that (2.30) holds, hence all the assumptions of Lemma 2.9 are fulfilled. Let hij be the fourth order homogeneous polynomial guaranteed to exist by Lemma 2.9, set gij = δij + hij .
(2.35)
Since h vanishes to order three, both the Riemann tensor and its derivatives vanish at p, which justifies (2.15). Further, the Riemann tensor Rijkl of g coincides with Sijkl up to terms which give zero contribution at p in all the calculations relevant here, so that it is not difficult to show that gij satisfies (2.16), which proves Proposition 2.7.
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We can now pass to the Proof of Theorem 2.1. Consider the linear map L which to w = (Xi , Fij := D[i Xj] , ϕ := Dk Xk , ϕi := Di ϕ)(p) ∈ R10 assigns R10 w → Lw
:=
1 1 m LX Hij + ϕHij , LX Dk Hij + 2Ck(i Hj)m + Dk (ϕHij ) , 3 3 m m LX Dl Dk Hij + Ckl Dm Hij + 2(Dl Ck(i )Hj)m 1 m m +2Cl(i D|k| Hj)m + 2Ck(i D|l| Hj)m + Dl Dk (ϕHij ) (p) 3 ∈ R6 ⊗ R3×6 ⊗ R3×3×6 .
Here the Lie derivative is calculated using the usual formula for the Lie derivative of a tensor, and then the values of X and its derivatives as determined by w are inserted. Further, the second derivatives of ϕ are eliminated using (2.11). It follows from Lemma 2.4 and Proposition 2.7 that the set of metrics for which L is injective is not empty. Standard linear algebra implies that there exists a 10 × 10 matrix, say A, constructed by listing ten appropriately chosen rows of L, which has non-vanishing determinant when H arises from the metric of Proposition 2.7. Let Q be the sum of squares of determinants of all ten-by-ten submatrices of L, then Q ≥ (det A)2 and therefore Q is not identically vanishing by construction. Clearly L is injective whenever Q is non-zero, which proves point 1. To prove point 2, let g be an arbitrary metric, if Q(p), evaluated for the metric g, does not vanish, then the result is true with g = g. Otherwise, define J5
:=
{the set of fifth jets of g in normal coordinates at p as g varies in the set of all Riemannian metrics} .
(2.36)
This a linear space, an explicit parameterisation of which can be found in [23]. Let ei , i = 1, . . . , N , be any basis of J5 , thus every j ∈ J5 can be written as j = i e i , for some numbers i ∈ R. By definition of J5 , for every (i ) ∈ RN there exists some Riemannian metric for which j = i ei . Clearly the map g → (i ) is continuous ¯ ≥ 5, topology on the set of metrics, and a small variation of i can be in a C (Ω), realized by a small variation of g. In a frame such that gij (p) = δij , the map that assigns to the fifth jets of g, at p, the values of the tensors H, DH, and DD2 H at p, is a polynomial on J5 . We want to show that a small variation of g will make Q non-zero. Now, Q is a polynomial in the i ’s. Let i0 be the values of the i ’s corresponding to the metric g, and suppose that we have ∀ i 1 , . . . , in
∂ i1 +···+iN Q (i ) = 0 . ∂ i1 1 . . . ∂ iN N 0
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Then the polynomial Q would identically vanish, contradicting its construction. Hence there exists at least one of the above partial derivatives which does not vanish, and therefore an appropriate, no matter how small, variation of g will lead to a non-vanishing value of Q at p. As the argument depends only upon the jets of g at p, the variation can be made supported in a ball containing p with radius as small as desired.
3 Metrics without Killing vectors near a point Results on non-existence of Killing vectors follow of course immediately from those on non-existence of conformal Killing vectors, as established above. However, for Killing vectors in dimension three the differentiability threshold of Theorem 2.1 can be lowered to three. Further, for Killing vectors a simple proof can be given in all dimensions: Theorem 3.1 Let (M, g) be a n-dimensional pseudo-Riemannian manifold. 1. There exists a non-trivial homogeneous invariant polynomial Pn [g] := (DR, . . . , D2n+1 R) of degree n, where R is the Ricci scalar, such that if Pn (DR, . . . , D2n+1 R)(p) = 0 at a point p ∈ M , then there exists a neighborhood Op of p such that there are no non-trivial Killing vectors on any open subset of Op . In dimension n = 3 there exists such a polynomial Pˆ3 which depends upon Ric and D Ric. 2. Let Ω be a neighborhood of p ∈ M . For any k ≥ 2n + 1 and > 0 there exists a metric g such that g − g C k (Ω) (3.1) ¯ < , with g − g supported in Ω, and such that Pn (DR , . . . , D2n+1 R )(p) does not vanish. In dimension three we can arrange for the non-vanishing of Pˆ3 (Ric , D Ric )(p) using a perturbation supported in Ω and satisfying (3.1) for each arbitrarily chosen k ≥ 3. Remark 3.2 The differentiability required above in dimension n is certainly not optimal, but it allows the simple proof below. Remark 3.3 The polynomial Pn obtained here is completely useless from the point of view of Killing vectors in vacuum space-times, where the Ricci scalar vanishes. In this context it is of interest to have a statement as above with a polynomial depending only upon the Weyl tensor, and we prove existence of such polynomials in Theorem 7.4 below. Further, in Section 8 we will construct small perturbations of initial data which preserve the vacuum constraints.
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Proof. If X is a Killing vector we have LX (∆k R)=0 for all k, where ∆k denotes the kth power of the Laplace operator ∆. At p this gives the linear system of equations Aij X i (p) = 0 ,
Aij = Di (∆j R)(p) ,
j = 0, . . . , n − 1 .
Let Pn = det(Aij ). If Pn (p) does not vanish, then X(q) = 0 for all q in the neighborhood of p defined as {q : Pn (q) = 0}, hence X ≡ 0. It is not too difficult to check, using Taylor expansions of the metric (point 2 of Proposition 5.4 below is useful here), that there exist metrics for which Pn = 0, and the result follows by a repetition of the arguments of the proof of Theorem 2.1. In dimension 3 the number of the derivatives of the metric needed can be improved as follows: Let Gij = Rij − g kl Rkl gij /2, in the notation of Section 2 we assume that ˚ij = λ1 xi xj + λ2 yi yj + λ3 zi zj , G (3.2) with (λ1 − λ2 )(λ2 − λ3 )(λ3 − λ1 ) = 0. We set, as in (2.21), ˚ij = 2(ax[i yj] + bz[i xj] + cy[i zj] ) , F
(3.3)
so that ˚(i k G ˚j)k = b(λ1 − λ3 )x(i zj) + c(λ3 − λ2 )y(i zj) + a(λ2 − λ1 )x(i yj) , F
(3.4)
which has zero components on the diagonal. Finally we assume that ˚ij;k = µ1 (x(i δj)k − 2xk δij ) + µ2 (y(i δj)k − 2yk δij ) + µ3 (z(i δj)k − 2zk δij ) , (3.5) G where µ1 µ2 µ3 = 0. We now set ˚ = αxi + βyi + γzi . X
(3.6)
˚1 + G ˚2 + G ˚3 , we find Writing (3.5) in the form G ij;k ij;k ij;k ˚k ˚1 X G ij;k ˚2ij;k X ˚k G
=
µ1 α(−2yi yj − 2zi zj − xi xj ) + off diagonal terms ,
(3.7)
=
µ2 β(−2xi xj − 2zi zj − yi yj ) + off diagonal terms ,
(3.8)
˚3ij;k X ˚k G
=
µ3 γ(−2xi xj − 2yi yj − zi zj ) + off diagonal terms .
(3.9)
We first consider the relation LX Gij = 0 with i = j. Then (3.4) gives no contribution, while from (3.7) we obtain a linear homogenous system for (α, β, γ) with coefficient matrix ∆ given by µ1 2µ2 2µ3 ∆ = 2µ1 µ2 2µ3 . 2µ1 2µ2 µ3
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There holds det(∆) = 5µ1 µ2 µ3 = 0. Thus, the equation LX Gij = 0, satisfied by any Killing vector, leads to α = β = γ = 0. The off-diagonal components of LX Gij = 0 imply now, by (3.4), that a = b = c = 0. Since (3.2) is symmetric, and (3.5) satisfies the linearized Bianchi identities, the results in [23] show that there exists a metric gij = δij + hij , with hij = O(ξ 2 ), satisfying (3.2) and (3.5). The proof is completed by the same argument as already given for general n.
4 Generic non-existence of local Killing, or conformal Killing, vector fields In this section we only consider three-dimensional manifolds, the reader will easily formulate an equivalent statement and proof for local Killing vector fields in any dimension using Theorem 3.1, or for local conformal Killing vector fields using Theorem 7.4 below. Theorem 4.1 Let M be a three-dimensional manifold. Then 1. The set of pseudo-Riemannian metrics on M which have no local Killing vector fields is of second category in the C 3 topology. 2. The set of pseudo-Riemannian metrics on M which have no local conformal Killing vector fields is of second category in the C 5 topology. Proof. We start with the following: Proposition 4.2 Let Ω be a domain in M . Then: 1. The set of metrics on Ω which have no Killing vectors on Ω is open in a ¯ topology, k ≥ 2. C k (Ω) 2. The set of metrics on Ω which have no conformal Killing vectors on Ω is ¯ topology, k ≥ 3. open in a C k (Ω) 3. The set of initial data (g, K) on Ω which have no non-trivial KIDs on Ω is ¯ ⊕ C k (Ω) ¯ topology, k ≥ 1. open in a C k+1 (Ω) Remark 4.3 The openness established here holds for any metrisable topology Tk such that convergence in Tk implies uniform convergence in C k norm on compact sets, with k ≥ 2 for Killing vectors, etc; see also Appendix A. Proof. We will show that existence of Killing vectors, or conformal Killing vectors, or KIDs, is a closed property. We start with the slightly simpler case of conditionally compact Ω: Lemma 4.4 Proposition 4.2 holds if Ω has compact closure.
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Proof. 1. Let γi be a sequence of metrics with non-zero Killing vectors X(i). Rescaling X(i) we can assume that sup γi (X(i), X(i)) = 1 .
(4.1)
p∈Ω
We note that Killing vectors extend by continuity to Ω, we shall use the same symbol to denote that extension. Let pi ∈ Ω be such that the sup is attained, passing to a subsequence if necessary there exists p∗ in Ω such that pi → p∗ . Now, Killing vectors satisfy the system of equations Di Dj Xk = R ijk X ,
(4.2)
which shows that second covariant derivatives of all the X(i)’s are uniformly bounded on Ω. Interpolation [16, Appendix] shows that the sequence X(i) is uniformly bounded in C 2 . The existence of a subsequence converging in C 1 to a non-trivial Killing vector field follows from the Arzela-Ascoli theorem. 2. The argument is essentially identical, with the following modifications: we replace the normalization (4.1) by sup (|X(i)|γi + |DX(i)|γi ) = 1 .
(4.3)
p∈Ω
Equation (4.2) is replaced by the set of equations (2.9)–(2.12). Those equations easily imply boundedness of the sequence X(i) in C 3 , leading to a converging subsequence in C 2 . 3. Let (γi , Ki ) be a sequence of metrics with non-zero KIDs (Y (i), N (i)). We use the normalization sup (|Y (i)|γi + |DY (i)|γi + |N (i)| + |DN (i)|γi ) = 1 .
(4.4)
p∈Ω
From (5.4) and (5.5) one obtains a uniform C 2 bound on (Y (i), N (i)), and one concludes as before. Returning to the proof of point 1 of Proposition 4.2, let Ωj be an increasing sequence of conditionally compact domains such that Ω = ∪Ωj . By Lemma 4.4 we have K (Ωj ) = {0} for all j. The restriction map induces an injection ii,j : K (Ωi ) → K (Ωj ), i ≥ j, so that 1 ≤ dim ii,1 (K (Ωi )) for all i, with ii+1,1 (K (Ωi+1 )) ⊂ ii,1 (K (Ωi )) ⊂ K (Ω1 ). It follows that F := ∩i ii,1 (K (Ωi )) = {0}, and every element of F extends to a globally defined Killing vector field on Ω. Proof of Theorem 4.1. Let pi , i ∈ N be a dense collection of points and let B(pi , 1/j), j ≥ Ni , be a collection of coordinate balls with compact closure. Let Vi,j be the set of metrics such that K (B(pi , 1/j)) = {0}. By Proposition 4.2 the set Vi,j is open, and it is dense by Theorem 3.1. Then any metric in ∩i,j Vi,j has no local Killing vectors. The argument for conformal Killing vector fields is identical, based on Theorem 2.1.
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5 Three-dimensional initial data sets without KIDs near a point We now pass to the construction of initial-data sets without KIDS. C (Kij , gij ) := (Ji , ρ) be the constraints map,
Let
ρ := R + K 2 − Kij K ij − 2Λ ,
(5.1)
Ji := −2Dj (Kij − Kgij ) ,
(5.2)
where Λ ∈ R is the cosmological constant. In this section, and only is this section, the symbol K denotes the trace of Kij ; K stands for the full extrinsic curvature tensor elsewhere in this paper. Let P denote the linearization of C , and let P ∗ be the formal adjoint of P . By definition, a KID (N, X i ) is a solution of the set of equations P ∗ (N, X) = 0; explicitly, in dimension n (cf., e.g., [10]), D(i Xj) = −N Kij , Di Dj N = N (Rij +KKij −2Kil Kj l )−LX Kij +
1 (n − 1)
(5.3)
Jl X l − (ρ + 2Λ) N 2
gij .
(5.4) One checks that any KID (N, X i ) for which X i , Fij = D[i Xj] , N and Ni := Di N all vanish at p has to be zero in a neighborhood of p. This is proved in the usual way from (5.4) together with D Dj Xi = −Rji k Xk − D (N Kij ) − Dj (N Ki ) + Di (N Kj ) .
(5.5)
(Equation (5.5) is obtained by considering cyclic permutations of first derivatives of (5.3).) Since LX Ric(g) = Ric (LX g), the usual formula for Ric leads to LX Rij
= ∆(N Kij ) + Di Dj (N K) − 2D(i Dl (N Kj)l ) l
−2N Rlijm K lm − 2N R(i Kj)l = ∆(N Kij ) + Di Dj (N K) − 2Dl D(i (N Kj)l ) .
(5.6)
From now on we assume that n = 3. By taking the curl of (5.4) one also finds m Rlij k Dk N = −2LX D[l Ki]j − 2Cj[l Ki]m
Jm X m ρ m − + Λ N gi]j , +2D[l N (Ri]j + Ki]j K − 2Ki]m Kj ) + 4 2
(5.7)
where i = −g in [Dj (N Kkn ) + Dk (N Kjn ) − Dn (N Kjk )] . Cjk
(5.8)
We choose some α, β, λi , ai ∈ R and we consider initial data with the following properties at p: ˚jl = 0 , ˚ij = β δij , K ˚ij = α δij , Di K R (5.9) 3 3
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˚jk = λx xi y(j zk) + (cyclic), Di R
(5.10)
where (cyclic) means cyclic permutations of (x, y, z), and λx λy λz = 0. (This ansatz is general enough to lead to the required result, and simple enough so that the calculations are manageable. We will show shortly that such initial data exist.) We also assume that ˚lm = ax xi xj y(l zm) + (cyclic) , Di Dj K with λx − λy =
a2y a2x − λx λy
(5.11)
(5.12)
and λx + λy = 0 , λx + λz = 0 , λy + λz = 0 .
(5.13)
For further reference we note that, in local coordinates ξ such that p corresponds to ξ = 0, (5.9)–(5.11) imply R + K 2 − Kij K ij = β +
2α2 + O(ξ 2 ) , 3
Dj (Kij − Kgij ) = O(ξ 2 ) .
(5.14)
In particular, if β = 2Λ − 2α2 /3 then ρ = R + K 2 − Kij K ij − 2Λ = O(ξ 2 ) ,
Ji = −2Dj (Kij − Kgij ) = O(ξ 2 ) . (5.15)
Inserting (5.9) into (5.3) and (5.4) we find that ˚j) D(i X
=
˚ = Di Dj N = ˚ = ∆N
α˚ δij , − N 3 ˚ N 3ρ β + α2 − − 3Λ δij 2 3 ˚ Nβ δij , − 6 ˚ Nβ − . 2
(5.16)
(5.17) (5.18)
Evaluating (5.6) at p, it follows that ˚m Dm R ˚ij = N ˚∆K ˚ij , X
(5.19)
˚D[l R ˚i]j = X ˚i]j . ˚m Dm D[l K N
(5.20)
and, from (5.7), that ˚i = αx xi + αy y i + αz z i , that From (5.19) we find, using the expansion X ˚ax , αy λy = N ˚ay , αz λz = N ˚az , αx λx = N
(5.21)
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and from (5.20) ˚(λx −λy ) = αx ax −αy ay , N ˚(λy −λz ) = αy ay −αz az , N ˚(λz −λx ) = αz az −αx ax . N (5.22) Combining (5.22) with (5.21) and using (5.12), it follows that ˚ = 0 = αx = αy = αz . N
(5.23)
Using (5.23) in the first derivative of (5.4) and in (5.5), we infer that ˚j) = − α δij Dl N ˚ , Dl Di Dj N ˚ = − β δij Dl N ˚. Dl D(i X 3 6
(5.24)
We now take a derivative of (5.6) to obtain (recall that Fij is the anti-symmetric part of Di Xj ) ˚ij + 2F ˚j)m = ˚km Dm R ˚(i| m Dk| R F ˚)∆K ˚ij + 2(Dl N ˚)Dk Dl K ˚ij − 2(Dl N ˚)Dk D(i K ˚j) l . (5.25) (Dk N Somewhat surprisingly, all terms involving α and β have dropped out. We have to ˚ij as compute the different terms entering (5.25). Writing F ˚ij = Ax y[i zj] + (cyclic) , F
(5.26)
we obtain
˚ij = 1 λx (Ay zk − Az yk )y(i zj) + (cyclic) , ˚km Dm R F 2 1 ˚j m = (λx xk yj + λy yk xj )(Ax yi − Ay xi ) + (cyclic) . ˚im Dk R F 4 ˚ = ux xi + uy yi + uz zi , we have that Also, decomposing Di N ˚)∆K ˚ij = (ux xk + uy yk + uz zk )(ax y(i zj) + (cyclic)) . (Dk N
(5.27) (5.28)
(5.29)
and ˚)(Dk Dl K ˚ij −Dk D(i K ˚j) l ) = ux (2ax xk −ay yk −az zk )y(i zj) +(cyclic) . (5.30) 2(Dl N We now insert Equations (5.27)–(5.30) into (5.25). Contracting the resulting equation first with xk y i z j and cyclic permutations thereof, one sees that ux , uy , uz have to vanish. Contracting, then, with terms of the form xk xi y j , xk xi z j , y k y i xj , etc., we see that Ax , Ay , Az are also zero, due to (5.13). Thus (N, X i ) is zero near p. We have thus proved: Lemma 5.1 Consider an initial data set (gij , Kij ) satisfying Eqs. (5.9)–(5.11) together with the conditions on the coefficients spelled out above. For any α, β, Λ ∈ R the algebraic equations for r = (Xi , Fij , N, Di N ) obtained from (5.3)–(5.4) by taking derivatives up to order two imply the vanishing of r(p).
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We also have the following KID-analogue of Proposition 2.7: Proposition 5.2 1. A pair (gij , Kij ) satisfying (5.9)–(5.11) exists. 2. Further, one can choose gij = δij + hij and Kij so that, in local coordinates ξ, the tensor fields gij and Kij satisfy the vacuum constraints up to terms which are of O(|ξ|2 ). Proof. By Lemma 2.9 we can find hij of order O(|ξ|2 ), so that (5.9)–(5.10) are satisfied. For Kij we choose Kij =
α 1 ˚ α δij + (K ∂l ∂m˚ hij )ξ l ξ m , ijlm + 3 2 3
(5.31)
where the second term on the right-hand side of (5.31) is given by the right-hand side of (5.11). One checks that (5.11) is valid. Point 2 follows from (5.15). We are ready now to prove: Theorem 5.3 Let α, β ∈ R, p ∈ M , and consider the collection of all threedimensional data sets (M, Kij , gij ) with (Kij , gij ) ∈ C k × C k+1 , k ≥ 3, with the trace K(p) of Kij (p) equal to α, and with R(p) = β. 1. There exists a non-trivial homogeneous invariant polynomial Q[Kij , gij ] := Q(Rij , DRij , D2 Rij , Kij , DKij , D2 Kij , D3 Kij ) such that if Q[Kij , gij ](p) = 0 at a point p ∈ M , then there exists a neighborhood Op of p for which there exist no non-trivial KIDS on any open subset of Op . 2. Let Ω be a domain in M with p ∈ Ω. a) There exists a variation (δKij , δgij ) ∈ (C ∞ × C ∞ )(Ω), compactly supported in Ω, such that Q[Kij + δKij , gij + δgij ](p) = 0 for all small enough. b) The variation can be chosen so that it preserves the value of R(p) and of K(p). One can further arrange for the trace of Kij + δKij to be equal to K throughout Ω when K is a constant. 3. If (Kij , gij ) is vacuum (with perhaps non-zero cosmological constant) with (Kij , gij ) ∈ C k++1 × C k++2 , ≥ 0, then for any p ∈ Ω the variation of point 2 can be chosen to satisfy the linearized constraint equations up to error terms which are o(r ) in a C k (B(p0 , r)) norm, for small r. Proof. The proof of points 1 and 2 follows closely that of points 1 and 2 of Theorem 2.1. Given any constant α, the set J5 of (2.36) is replaced by the {the set of fourth jets of gij and of third jets of Kij at p that one obtains as gij varies in the set of all Riemannian metrics in normal coordinates near p and as Kij varies in the set of all symmetric tensors with trace equal to a prescribed constant α} .
(5.32)
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The intermediate elements of the proof are provided by Lemma 5.1 and the first part of Proposition 5.2. The variations of gij and of the trace-free part of Kij can be chosen to be polynomials multiplied by a smooth cut-off function, and are therefore smooth. One can then adjust the trace part of Kij to achieve K = α. We further note that the non-vanishing of some derivative of Q follows immediately from the fact that Q(p) is a polynomial, when viewed as a function depending upon the jets of gij and Kij in normal coordinates at p. Further details are left to the reader. In order to prove point 3, for r > 0 it is useful to introduce the following set: W+k
= { jets at p of order ( + k + 1, + k + 2) of (Kij , gij ) such that ρ = o(|ξ|+k ) , J = o(|ξ|+k ) in B(0, r)} .
Here ξ are supposed to be geodesic coordinates near p in the metric gij . Equivalently, if (Kij , gij ) ∈ C k++1 × C k++2 has jets in W+k , then we have Dα ρ(p) = 0 , Dα J(p) = 0 ,
0 ≤ |α| ≤ + k ,
(5.33)
where the α = (i1 . . . ij )’s are multi-indices, with |(i1 . . . ij )| = i1 +. . .+ij . Elements of W+k can be uniquely parameterized as follows: Taylor expanding gij and Pij := Kij − Kgij in geodesic coordinates around p, one can write gij = δij + hijα ξ α + O(|ξ|+k+3 ) , with hi(j1 ...jp ) = 0 , (5.34) 2≤|α|≤+k+2
˚ij + Pij = P
Pijα ξ α + O(|ξ|+k+2 )
(5.35)
1≤|α|≤+k+1
(see, e.g., [23] for a justification of the last condition in (5.34)). Then (5.33) can be solved by induction as follows: (5.33) with |α| = 0 gives ˚2 ˚ 2 i,j (hijij − hjjii ) = i,j Pij − ( i Pii ) + 2Λ , i Pijj = 0 . ˚ij ∈ Rm0 := R6 the first equation defines an affine subspace For any given P isomorphic to Rn2 for some n2 , in the vector space of second Taylor coefficients hijkl . The second equation defines a linear subspace isomorphic to Rm1 in the space of Pijk ’s, for some m1 . To understand (5.33) with |α| ≥ 1 we will need the following: Proposition 5.4 Let k ∈ N, and suppose that dim M = n ≥ 2. 1. For every Ji = Jij1 ...jk ξ j1 . . . ξ jk and p = pj1 ...jk+1 ξ j1 . . . ξ jk+1 there exists Pij = Pijj1 ...jk+1 ξ j1 . . . ξ jk+1 , symmetric in i and j, such that ∂j Pij = Ji , Pii = p . i
i
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2. For every f = fj1 ...jk ξ j1 . . . ξ jk there exists hij = hijj1 ...jk+2 ξ j1 . . . ξ jk+2 , symmetric in i and j, with hi(jj1 ...jk+2 ) = 0, such that (∂j ∂i hij − −∂i ∂i hjj ) = f . i,j
Proof. Consider a system of linear PDEs Pu = I ,
(5.36)
with constant coefficients, of order p, which can be written in the Cauchy-Kowalevska form with respect to a coordinate z. We claim that if I is a polynomial of order l, then there exists a solution of (5.36) which is a polynomial of order l + p. In order to see that, we note that (5.36) determines, at z = 0, the z-derivatives of u of order greater than or equal to p as polynomials in the remaining variables. So choosing zero Cauchy data on {z = 0} one obtains a polynomial solution in z with polynomial coefficients, hence a polynomial. If P is homogeneous of order p, and if I is in addition homogenous of order l, then the above solution is a homogeneous polynomial of order l + p. In order to prove point 1, we make the ansatz
1 p−2 ∂ W δij , Pij = ∂i Wj + ∂j Wi + n
which leads to a homogeneous second order elliptic system for W , and the above argument applies. In order to prove point 2, we first make the ansatz hij = n1 hll δij , solve the resulting Poisson equation in the class of homogeneous polynomials as described above, and introduce a metric gij = δij + hij . In geodesic coordinates y i the metric gij will have an expansion with some new coefficients satisfying the symmetry condition in (5.34) [23]. One has y i = ξ i + O(|ξ|k+3 ), which implies that the polynomial obtained from the y–Taylor coefficients of gij of order k + 2 provides the desired hij . Proposition 5.4 shows that (5.33) can be used to inductively determine higher order Taylor coefficients hijα and Pijβ in terms of lower order ones, as well as in terms of some free P –coefficients in Rm|β| , for some m|β| ∈ N, and some free h-coefficients in Rn|α| , for some n|α| ∈ N. It follows in particular that W+k is diffeomorphic to RN+k , for some N+k ∈ N. For solutions (Kij , gij ) of the constraint equations, the polynomial Q[Kij , gij ](p) can be expressed as a polynomial of (Kij , gij )-jets at p of order (2, 3), call this ˜ Since the W+k ’s are included in each other in the obvious way, polynomial Q. ˜ can actually be viewed as a function defined on W+k which depends only on Q those coefficients which parameterise W1 . The pair (Kij , gij ) constructed in Propo˜ is non-trivial on W1 . It then follows, sition 5.2 has jets in W1 , which shows that Q
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˜ as in the proof of Theorem 2.1, that any jets in W1 can be -perturbed so that Q(p) does not vanish on the perturbed jet, with the jets of the perturbation belonging ˜ with respect to its arguments to W+k ; by analyticity some of the derivatives of Q will not vanish at p. It should be clear from (5.33) that the perturbed solution satisfies the properties described in the statement of point 3 of Theorem 5.3.
6 Riemannian metrics without static KIDs near a point An interesting class of initial data is provided by the time-symmetric ones, K ≡ 0. In this case the KID equations (5.3)–(5.4) decouple, with X in (5.3) being simply a Killing vector field of g. It remains to analyse the equation for N , Di Dj N = N Rij + ∆N gij .
(6.1)
A solution of (6.1) will be called a static KID, and the set of static KIDs on a set Ω will be denoted by N (Ω). (The origin of the adjective “static” will be clarified shortly.) Since time-symmetric initial data are non-generic amongst all initial data, the results of the previous section do not say anything about non-existence of static KIDs, and separate treatment is required. Taking the trace of (6.1) one obtains, in dimension n ∆N = −
1 NR , n−1
(6.2)
so that (6.1) can be rewritten as Di Dj N = N (Rij −
1 gij R) . n−1
(6.3)
Calculating Dj of (6.3) and commuting derivatives one is led to (recall that the Einstein tensor is divergence-free) N Di R = 0 .
(6.4)
Since the zero-set of solutions of (6.1) has no interior except if N ≡ 0, we conclude that existence of non-trivial static KIDs implies that R is constant. It follows that a non-trivial solution of (6.3) does indeed correspond to initial data for a static solution of the vacuum Einstein equations with a cosmological constant. Further, one immediately obtains that generic C 2 metrics have no static KIDs: it suffices to vary the metric so that the scalar curvature is not constant. From now on we assume dim M = 3. In order to prepare the proof, that generic metrics with fixed constant value of scalar curvature have no static KIDs, we consider a metric g with Ricci tensor at p equal to ˚ij = Axi xj + Byi yj + Czi zj , R
(6.5)
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where we assume that (A − B)(A − C)(B − C) = 0, and we further suppose that ˚j]l = αx[i yj] zl + βz[i xj] yl + γy[i zj] xl − 1 (α + β + γ)ijl , D[i R 6 with (α, β, γ) = 0. We also impose the condition that ˚= 0 . Di R
(6.6)
(6.7)
Taking a curl of (6.1) we infer that (2Rj[l − Rgj[l )Di] N + gj[l Ri]k Dk N = N D[l Ri]j . The left-hand side of (6.8), with ˚ = axi + byi + czi , Di N
(6.8)
(6.9)
takes the form xj x[l [yi] b(A − C) + zi] c(A − B)] +yj y[l [xi] a(B − C) + zi] c(B − A)] +zj z[l [xi] a(C − B) + yi] b(C − A)] .
(6.10)
˚ vanSince no terms with this index structure occur in (6.6) we obtain that Di N ishes, and using (6.8) allows us to finally conclude that ˚ = Di N ˚=0. N (6.11) The arguments of proof of Proposition 2.7 apply and provide existence of a metric gij = δij + hij satisfying (6.5) and (6.6). Clearly A + B + C can be chosen so ˚ has any prescribed value. Now, we can multiply gij by 1 + α, where α is that R ˚ is zero. By conformal a homogeneous third order polynomial chosen so that Di R ˚ invariance this does not change the value of Bijk , hence of (6.6) (compare (2.1)– ˚ either. A repetition of the remaining (2.2)), and does not change the value of R arguments of Section 5, with Kij there set to zero, gives: Theorem 6.1 Let (M, g) be a Riemannian manifold with g ∈ C k , k ≥ 2. A necessary condition for a non-trivial N (Ω) is that the scalar curvature of g be constant on Ω. Further, in dimension three, and for k ≥ 3, the following hold: 1. There exists a non-trivial homogeneous invariant polynomial Q[g] := Q(Ric, D Ric)
such that if
Q(Ric, D Ric)(p) = 0
at a point p ∈ M , for a metric for which the gradient of the scalar curvature vanishes at p, then there exists a neighborhood Op of p for which there exist no non-trivial static KIDS on any open subset of Op . 2. Let Ω be a domain in M , and let p ∈ Ω. There exists a variation δg ∈ C ∞ (Ω), compactly supported in Ω, such that Q[g + δg](p) = 0 for all small enough. If g ∈ C k++2 , ≥ 0, has constant scalar curvature, then the variation above can be chosen to have the same scalar curvature up to error terms which are o(r ) in a C k (B(p0 , r)) norm, for small r.
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7 Results in general dimensions, with non-explicit orders of differentiability The results obtained so far did require rather unpleasant, tedious, and lengthy calculations, and we will present here an argument which avoids those. The draw-back is that one does not obtain an explicit statement on the number of derivatives involved. However, non-genericity of KIDs is obtained in higher dimensions. Further, the proof below generalizes immediately, e.g., to the Einstein-Maxwell equivalent of the KID equations, the details are left to the reader. The starting point of the analysis in this section is the following result (recall that n = dim M ): Lemma 7.1 1. For any n ≥ 2 and for any signature there exists a real analytic compact pseudo-Riemannian manifold (M, g) without local Killing vectors. 2. For any n ≥ 3 there exists a real analytic compact simply connected Riemannian manifold (M, g) without local conformal Killing vectors. 3. For any n ≥ 3, Λ ∈ R, τ ∈ R there exists a real analytic vacuum initial data set (M, g, K), with cosmological constant Λ, with trg K = τ , and without local KIDs. 4. For any n ≥ 3 and Λ ∈ R, there exists a real analytic Riemannian or Lorentzian manifold (M , g), with dim M = n + 1, satisfying the vacuum Einstein equations with cosmological constant Λ and without local Killing vectors. Remark 7.2 The main point of the Lemma is to construct one single example in each category listed. However, our argument makes it clear that there are actually lots of examples. For instance, the proof below shows that in point 1 for any analytic manifold M which is simply connected, compact, with dim M ≥ 2 one can find a g with the required properties. Remark 7.3 If τ2 ≥
2n Λ, (n − 1)
(7.1)
then one can find an M as in point 3 which is compact (without boundary). The proof in the strict inequality case is given below. If the inequality in (1.2) is an equality, in the proof below one should instead choose (M, γ0 ) to be any real analytic compact Riemannian manifold of positive Yamabe class. The monotone iteration scheme for solving the Lichnerowicz equation can then be handled by an argument in [17]. In that last reference only dimension three is considered, but the proof applies in any dimension. Proof. 1: Let M be any simply connected, compact, analytic Riemannian manifold with dim M ≥ 2, let p ∈ M and let g0 be any smooth Riemannian metric on M
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such that the polynomial Pn = Pn [g] of Theorem 3.1 does not vanish at p. We need analytic approximations of g, for example for 0 ≤ t < we can let gt be the family of metrics obtained by evolving g0 using the Ricci flow, then the metrics gt are indeed real analytic for t > 0. By continuity, reducing if necessary, we will have Pn [gt ](p) = 0, hence gt will have no Killing vectors in a neighborhood of p. Now, a theorem4 of Nomizu [20] shows that on a simply connected analytic manifold every locally defined Killing vector extends to a globally defined one. This implies that for 0 < t < the metrics gt have no Killing vectors on any open subset of M . 2: For n = 3 this follows from Theorem 2.1. For any n ≥ 3 one can argue as follows: Let M be any compact real analytic manifold of dimension not less than three. By [18] there exists on M a metric g with strictly negative Ricci curvature. It is well known that such metrics do not have non-trivial conformal Killing vectors, we recall the proof for completeness: from (2.6) with 2/3 replaced by 2/n it follows that 1 Di Dj Xk = −Rjki l Xl + (ϕi gjk + ϕj gik − −ϕk gij ) , (7.2) n hence 2 ∆Xk = −Rk l Xl − (1 − )ϕk . (7.3) n Multiplying by X k and integrating over M one finds (recall that ϕ = divX) 2 |DX|2 − Ric(X, X) + (1 − )ϕ2 = 0 , n M so that X ≡ 0 if Ric < 0. Approximating g by real-analytic metrics gt , gt → g as t → 0, one will have no conformal Killing vectors for gt when t is small enough by Proposition 4.2. It then follows from Theorem B.1, Appendix B, that the gt ’s will have no local conformal Killing vector field either. 3 and 4: We start by noting that in dimension n = 3, an example of initial data as in point 3 can be obtained using vacuum Robinson-Trautman space-times with cosmological constant Λ (cf., e.g., [5]). Because of the parabolic character of the Robinson-Trautman equation, those metrics are always analytic away from the initial data surface. Further, if the initial metric h0 on S 2 used in the RobinsonTrautman equation has no continuous global symmetries, then it follows from Proposition 4.2 that the evolved metrics ht will not have any continuous global symmetries either, at least for t small enough. It is clear that the resulting fourdimensional metric 4 g will then have no globally defined Killing vectors except the zero one. The non-existence of local Killing vectors follows then from Nomizu’s theorem [20]. Finally, the initial data set of point 3 can be obtained as that induced by 4 g on any hypersurface with trg K = τ in M ; such hypersurfaces can be obtained by solving a Dirichlet problem for the CMC equation on the boundary of a sufficiently small spacelike three-ball [4]. 4 We note that in [20] a Riemannian metric is assumed, but the proofs given there apply to any signature.
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In any case, whatever n ≥ 3 one can proceed as follows: consider, first, τ such that the inequality in (7.1) is strict, let (M, γ0 ) be any real analytic compact Riemannian manifold of negative Yamabe class. Let L0 be any non-zero, γ0 -transverse and traceless tensor on M ; such tensors exist by [6]. For t ∈ [0, ) let Lt be a family of analytic symmetric γ0 -trace free tensors converging to L0 . For example, Lt can be obtained from L0 by heat flow using any analytic metric on M , and removing the γ0 trace. Using the conformal5 method [17] with seed fields (γ0 , Lt ) one obtains a family of real analytic vacuum CMC initial data sets (gt , Kt ) with cosmological constant Λ. Since γ0 has no global conformal Killing vectors, gt will have no global Killing vectors. Now, trgt Kt = τ is a constant, which implies (see Remark 9.2 below) that any global KIDs for (gt , Kt ) are of the form (N = 0, Y ), where Y is a Killing vector of gt , therefore none of the (gt , Kt )’s has global KIDs. In the Lorentzian case we let (M , n+1 gt ) be the maximal globally hyperbolic vacuum development of (M, gt , Kt ), then M is diffeomorphic to R × M (hence simply connected), and n+1 g t is analytic by [1]. In the Riemannian case we let M be any simply connected and connected neighborhood Ut of M × {0} in M × (−1, 1), chosen so that there exists a vacuum metric n+1 gt on Ut with Cauchy data (gt , Kt ) on M × {0}, obtained from the Cauchy-Kowalewska theorem. Suppose that there exists an open nonempty subset Ωt ⊂ M such that Kt (Ωt ) = {0}, where Kt denotes the set of KIDs with respect to (gt , Kt ), then a standard argument [13]6 , using the Cauchy-Kowalewska theorem, shows that there exists a non-trivial Killing vector X in a neighborhood of Ωt in M . By Nomizu’s theorem [20] X extends to a globally defined Killing vector on M , hence (M, gt , Kt ) has a globally defined KID, a contradiction. Thus there are no local KIDs on (M, gt , Kt ), and (M , n+1 gt ) is a vacuum metric without local Killing vectors. This proves point 4, as well as Remark 7.3 in the case of a strict inequality there. To prove point 3 for the remaining values of τ one spans [4], within the Lorentzian solution M just constructed, a CMC hypersurface of prescribed τ = trg K on the boundary of a small spacelike ball. The data induced on the resulting CMC hypersurface provide the desired initial data set. We continue with the question of generic non-existence of KIDs, it should be clear that an identical argument applies to conformal Killing vectors (compare [7, Equation (1.15)]), or to Killing vectors. Let (g, K) be any vacuum analytic initial data on a simply connected manifold M which have no global KIDs. As explained above, it follows from a theorem of Nomizu [20], that such an initial data set will not have any local KIDs. Let r(p) ∈ RM be as in Lemma 5.1, for some appropriate M , and for α ∈ Nn , let us write Dα r = Pα r
(7.4)
5 Since the inequality in (1.2) is strict, a small and a large constant provide barriers for the monotone iteration scheme. 6 Compare the proof of [9, Theorem 2.1.1]; the Cauchy-Kowalewska theorem should be invoked for solvability of Eq. (2.1.5) there, or for uniqueness of solutions of Eq. (2.1.7) there.
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for the linear system of equations obtained by calculating the (k + 1)-st derivatives of r by differentiating (5.3)–(5.4) |α| times, and replacing the lower order derivatives that arise in the process by their values already calculated from the previous equations. Let us write Lk r = 0 for the system of equations that arise from first-order integrability conditions of the system (7.4) with |α| ≤ k. Choose some orthonormal frame, then Lk can be identified with a Nk ×M matrix, for some Nk , with entries built out of the extrinsic curvature tensor K, of the Riemann tensor, and of their derivatives. Let Qk denote the sum of squares of determinants of all M × M sub-matrices of Lk . Then the equation Lk r = 0 admits a non-trivial solution if and only if Qk = 0. Suppose that there exists r0 such that Lk r0 = 0 for all k. One can then use (7.4) to calculate all the jets of r with initial value r0 at p so that the Killing equations are satisfied to infinite order at p by a formal solution determined by those jets. To show convergence of the resulting Taylor series one can proceed as follows: let xi ∈ [−, ]n be local analytic coordinates around p = 0, we can solve the linear equation ∂r = P1 r ∂x1 along the path [−, ] x1 → (x1 , 0, . . . , 0), with initial data r0 at the origin, obtaining an analytic solution there. We can use the function so obtained as initial data for the equation ∂r = P2 r ∂x2 to obtain an analytic solution on [−, ]2 × {0} × · · · × {0}. An inductive repetition of this procedure provides an analytic solution on [−, ]n of the equation ∂r = Pn r , ∂xn such that the equation ∂r = Pk r holds on [−, ]k × {0} × · · · × {0}. ∂xk n − k factors
By choice of r0 the analytic functions Lk r have all derivatives vanishing at the origin, hence they vanish on [−, ]n . Standard arguments imply that the function r so obtained provides an analytic solution of the KID equations in a neighborhood of p. This gives a contradiction with the fact that (g, K) has no local KIDs near p. Therefore there exists k such that Qk is non-zero for the initial data set under consideration. This Qk provides the non-trivial polynomial needed in Theorem 5.3. When the metric involved is Riemannian we can integrate Qk , viewed as a function on the frame bundle, over the rotation group to obtain an invariant polynomial.
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We have therefore proved: Theorem 7.4 Theorem 6.1 remains valid in any dimension, with an invariant polynomial that depends upon some dimension-dependent number k of derivatives of g. Similarly Theorem 5.3 remains valid in any dimension, for some polynomial that depends upon k + 1 derivatives of g and k derivatives of K, for some dimensiondependent number k. In dimension n ≥ 4 Theorem 3.1 remains valid with a polynomial which depends upon some dimension-dependent number k of derivatives of the Weyl tensor. Finally, Theorem 2.1 remains valid in any dimension n ≥ 3 with a polynomial that depends upon some dimension-dependent number of derivatives of the Riemann tensor. Proof. The only statement which, at this stage, might require justification is the extension of Theorem 3.1: this result follows from point 4 of Lemma 7.1, as the polynomial obtained in that case by the proof above depends only upon the Weyl tensor.
8 From approximate linearized solutions to small vacuum perturbations The perturbation results of the previous sections can be used to prove nongenericity of KIDs when no restrictions on ρ and J are imposed. They also apply if, e.g., a strict dominant energy condition ρ > |J| is imposed, for then a sufficiently small perturbation of the data will preserve that inequality. However, some more work is needed when vacuum initial data are considered, and this is the issue addressed in this section. Let Ω ⊂ M be open and connected, and let K (Ω) denote the set of KIDs defined on Ω; each K (Ω) is a finite-dimensional, possibly trivial, vector space. If Ω ⊂ Ω we have the natural map iΩ : K (Ω) → K (Ω ) , with iΩ (x) being defined as the restriction to Ω of the KID x ∈ K (Ω). A local KID vanishing on an open subset vanishes throughout the relevant connected component of its domain of definition, which shows that iΩ is injective. We denote by B(p, r) the open geodesic ball of radius r, and for a < b we set Γp (a, b) := B(p, b) \ B(p, a). We will need the following result: Proposition 8.1 For every p ∈ M and R r > r1 > 0 there exists 0 < r2 < r1 such that iΓp (r2 ,r) : K (B(p, r)) → K (Γp (r2 , r)) is bijective.
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The proof rests on the following lemma: Lemma 8.2 For every p ∈ M and r1 > 0 there exists σ ∈ (0, 1) such that iσ : K (B(p, r1 )) → K (Γp (σr1 , r1 )) is bijective. Here iσ denotes iΓp (σr1 ,r1 ) . Proof. As already pointed out, injectivity always holds. Suppose that surjectivity fails, then for every σ ∈ (0, 1) there exists a KID xσ ∈ K (Γp (σr1 , r1 )) such that xσ ∈ iσ (K (B(p, r1 ))). Choose any scalar product h on K (Γp (r1 /2, r1 )). For σ < 1/2 without loss of generality we can assume that the restriction xˆσ of xσ to Γp (r1 /2, r1 ) is h-orthogonal to the image of i1/2 , and that h(ˆ xσ , x ˆσ ) = 1. Since K (Γp (r1 /2, r1 )) is finite-dimensional there exists a sequence σi → 0 such ˆ0 , with h(ˆ x0 , x ˆ0 ) = 1. Further x ˆ0 is h-orthogonal to that xˆσi converges to some x i1/2 (K (B(p, r1 ))). It should be clear from (5.4)–(5.5) that for i such that σ > σi , the sequence of KIDs on Γp (σr1 , r1 ) obtained by restricting xσi to Γp (σr1 , r1 ) converges, and defines a non-trivial KID which restricts to x ˆ0 on Γp (r1 /2, r1 ), with the limit being independent of σ in the obvious sense. This shows that there exists a ˆ0 is the restriction of x0 to Γp (r1 /2, r1 ). KID x0 defined on B(p, r1 )\{p1 } such that x But (5.4)–(5.5) further shows that x0 can be extended to a KID defined on B(p, r1 ), still denoted by x0 . It follows that x ˆ0 = i1/2 (x0 ), which contradicts orthogonality of x ˆ0 with the image of i1/2 . Proof of Proposition 8.1: Let r2 = σr1 , with σ given by Lemma 8.2. Every KID on Γp (r2 , r) induces, by restriction, a KID on Γp (r2 , r1 ), therefore dim K (Γp (r2 , r)) ≤ dim K (Γp (r2 , r1 )). By Lemma 8.2 we have dim K (Γp (r2 , r1 )) = dim K (B(p, r)). Again by restriction we have dim K (B(p, r)) ≤ dim K (Γp (r2 , r)), whence the result. Corollary 8.3 Suppose that K (B(p, r)) = {0}. Then for any > 0 there exists > r1 > 0 such that K (Γp (r1 , r)) = {0}. 2 Recall that the constraints map has been defined by the formula: J 2(−∇j Kij + ∇i trK) . (K, g) := ρ R(g) − |K|2 + (trK)2 − 2Λ
(8.1)
The following is one of the key steps of the proof: Theorem 8.4 For ∈ N, ≥ 2, α ∈ (0, 1), p ∈ M , r, η > 0, let the symbol P denote the linearization of the constraints operator (8.1) at (K, g) ∈ C +2,α × C +2,α (B(p, r)), and let
xη = (δKη , δgη ) ∈ C +2,α × C +2,α (B(p, r))
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be an “approximate solution” of the linearized constraint equations defined on B(p, r), in the sense that: P xη (C +1,α ×C ,α )(B(p,r)) ≤ η . 1. There exists a constant C such that if iΓp (σr,r) is surjective for some σ ∈ (0, 1/2], then there exists a solution x ∈ C +2,α × C +2,α (B(p, r)) of the linearized constraint equations supported in B(p, r) such that x − xη (C +2,α ×C +2,α )(B(p,σr)) ≤ Cη .
(8.2)
x is smooth if (K, g) and xη are. 2. For ≥ 4, for any (K0 , g0 ) in C +2,α × C +2,α (B(p, r)), and for any r0 such that B(p, r0 ) has smooth boundary, the constant C can be chosen independently of σ ∈ (0,1/2], (K, g), and r satisfying 0 < r ≤ r0 , for all (K, g) sufficiently close in C +2,α × C +2,α (B(p, r0 )) to (K0 , g0 ). Remark 8.5 The restriction σ ≤ 1/2 is arbitrary, the argument applies with any 0 < σ ≤ σ0 ∈ (0, 1), with a constant in (8.2) depending perhaps upon σ0 . Proof. We use the definitions and notation of [11]. In particular if Ω is a domain with smooth boundary, then Hs2 ∩ Csk,α . Λsk,α = ˚ Roughly speaking, functions in that space behave as o(xs ) near the boundary {x = 0}, with derivatives of order j, 0 ≤ j ≤ k, being allowed to behave as o(xs−j ). In particular if s > k + α then functions in the space above are in C k,α (Ω). We will need the following result [11, Proposition 6.5]: Proposition 8.6 Suppose that (K0 , g0 ) ∈ C k+2,α × C k+2,α (M ), k ≥ 2, α ∈ (0, 1), and let Ω ⊂ M be a domain with smooth boundary and compact closure. For all s = (n + 1)/2, (n + 3)/2, the image of the linearization P , at (K0 , g0 ), of −s+2 the constraints map, when defined on Λ−s+1 k+2,α × Λk+2,α (Ω), is
−s (J, ρ) ∈ Λ−s k+1,α × Λk,α such that (J, ρ), (Y, N )(L2 ⊕L2 )(Ω,dµg0 ) = 0 for all (Y, N ) ∈ H1s−n × H2s−n satisfying P ∗ (Y, N ) = 0 . −s+2 Further P −1 (0) ⊂ Λ−s+1 k+2,α × Λk+2,α splits.
The proof of point 1 of Theorem 8.4 will proceed in two steps: Step 1: We set M := B(p, r), k = , (g0 , K0 ) = (g, K), and we use Proposition 8.6 with s = s1 for some s1 < −1. Now, for such s the space K0 ⊂ K(B(p, r)) above is the space of KIDs on B(p, r) which vanish at S(p, r) := ∂B(p, r) together with
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their first derivatives; but Equations (5.4)–(5.5) imply that there are no such nontrivial KIDs. It follows that P is surjective, with the splitting property being −s+2 equivalent to the fact that there exists a closed subspace X ⊂ Λ−s+1 k+2,α × Λk+2,α such that the restriction of P to X is an isomorphism. This shows that there exists −s+2 x ˆη ∈ Λ−s+1 k+2,α × Λk+2,α satisfying ˆ xη Λ−s+1 ×Λ−s+2 ≤ Cη , k+2,α
k+2,α
and P (ˆ xη ) = −P (xη ) ⇔ P (xη + xˆη ) = 0. Step 2: Now, because s = s1 < −1, the correction term xˆη could be blowing-up near S(p, r), while we want a solution which vanishes there to rather high order. To correct that, let ϕ be any smooth non-negative function which is identically one on B(p, 5r/8), and vanishes on Γp (3r/4, r), set yη = ϕ(xη + x ˆη ) . 8.6 Then P (yη ) is supported in Γp (5r/8, 3r/4) ⊂ Γ(σr, r). We now use Proposition ˜η ∈ C +2,α × C +2,α (Γp (σr, r)), once again, with some s = s2 > + 3, to find x which extends by zero both through S(p, σr) and through S(p, r) in a C +2,α × C +2,α manner, such that P (˜ xη + yη ) = 0 ⇐⇒ P (˜ xη ) = −P yη =: zη . This will be possible if and only if zη is orthogonal in L2 (Γp (σr, r)) to K0 (Γp (σr, r)), where now K0 (Γp (σr, r)) coincides with the space of all KIDs on Γp (σr, r). Let, thus, w = (Y, N ) ∈ K0 (Γp (σr, r)), by hypothesis there exists a KID w ˆ defined on ˆ We then have B(p, r) such that w is the restriction to Γp (σr, r) of w. w, P yη = w, ˆ P yη = Pˆ ∗ w, ˆ yη = 0 . Γp (σr,r)
B(p,r)
B(p,r)
Here the first and the second equalities are justified because P yη is supported in Γp (σr, 3r/4), while the last one follows because, by definition of a KID, P ∗ w ˆ = 0. This provides the desired x ˜η . Setting xη = ϕ(x + xˆη ) + x ˜η , point 1 is proved. To prove point 2, we first note that the value of σ does not affect the constant C, as that constant arises from step 1 of the proof of point 1: the perturbation x ˜η from step 2, which could depend upon σ, is supported away from B(p, σr). The result is proved now by the usual contradiction argument: Consider the map ⊥
⊥
g −s −s −s πK⊥g L,x,xs−n/2 : K0 g ∩ (Λ−s k+3,α × Λk+4,α ) −→ K0 ∩ (Λk+1,α × Λk,α ) , 0
(8.3)
with L,x,xs−n/2 being a regularized version, as in [11], of the map Lx,xs−n/2 of [10, Section 5]. Equation (8.2) will fail to hold only if there exists a sequence of radii ⊥ rn and data (Kn , gn ) on B(p, rn ) near (K0 , g0 )|B(p,rn ) , with KIDs (Yn , Nn ) ∈ K0 g such that (Yn , Nn )Λ−s ×Λ−s = 1 and L,x,xs−n/2 (Yn .Nn )|Λ−s ×Λ−s ≤ 1/n . 3,α
4,α
1,α
0,α
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Consider an extracted sequence, still denoted by rn , converging to r∞ . If r∞ > 0, then (K0 , g0 )|B(p,r∞ ) would admit a KID vanishing, together with its first derivatives, at S(p, r∞ ), a contradiction. On the other hand suppose that r∞ = 0, introduce geodesic coordinates for the metrics (Kn , gn ) centred at p; this might lead to a loss of two derivatives of the metric, so we increase the threshold on from ˜ n , g˜n ) on B(p, 1) obtained by scaling up the two to four. Consider the sequence (K ˜ ball B(p, rn ) to B(p, 1). Then (Kn , g˜n ) converges to (0, δ), where δ is the Euclidean metric on B(p, 1). As before one obtains a contradiction because there are no KIDs vanishing, together with their first derivatives, on S(p, 1) for (K, g) = (0, δ). Smooth solutions can be obtained proceeding as above, but working instead with exponentially-weighted rather than power-weighted spaces. The main result of this section is the following (see footnote 2): Theorem 8.7 Let M be a compact manifold with boundary, suppose that ≥ 0 (n), α ∈ (0, 1) for some 0 (n) ( 0 (3) = 6), and let (M, K, g) be a C ,α × C ,α vacuum initial data set such that K (M ) = {0} . For any p ∈ M \ ∂M and for any > 0 there exists r > 0 and an -small, in a C ,α × C ,α topology, vacuum perturbation (K , g ) of (K, g) such that K (U ) = {0} for all U such that U ∩ B(p, r) = ∅ . Further, (K , g ) can be chosen to coincide with (K, g) in a neighborhood of ∂M . Proof. For definiteness in the proof we will assume n = 3, for n > 3 in the argument below Theorem 5.3 should be replaced by its higher-dimensional generalization provided by Theorem 7.4. If the polynomial Q of point 1 of Theorem 5.3 does not vanish at p, we let r > 0 be small enough so that Q has no zeros on B(p, r). Otherwise, let δx := (δK, δg) be as in point 3 of Theorem 5.3 with = 1 and k = 3. Let > 0, as Q is a polynomial we have Q[x + δx](p) = j (Q(j) [δx])(p) + O(j+1 ) , for some j ≥ 1 such that (Q(j) [δx])(p) = 0. By Proposition 8.1 for any r > 0 there exists σr ∈ (0, 1) such that the conditions of Theorem 8.4 are satisfied. We then have P δx(C 3,α ×C 2,α )(B(p,r)) ≤ P δx(C 4 ×C 3 )(B(p,r)) ≤ C1 r , and Theorem 8.4 provides a solution δ˜ x of the linearized constraint equations supported in B(p, r) such that δx − δ˜ xC 4,α (B(p,σr r)) ≤ CC1 r . Choosing r small enough so that CC1 r ≤ one obtains Q[x + δ˜ x](p) = j (Q(j) [δx + O()])(p) + O(j+1 )) = j (Q(j) [δx])(p) + O(j+1 ) .
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Since δ˜ x satisfies the linearized constraint equations and since K (M ) = {0}, it follows from [10, Theorem 5.6] together with the regularization technique from [11] that for small enough we can find δˆ x(), with δˆ x()C 4 (B(p,r)) ≤ C2 2 , such that x + δ˜ x + δˆ x() satisfies the vacuum constraint equations. Choosing small enough so that C2 ≤ 1/2 we then obtain Q[x + δ˜ x + δˆ x()](p)
x + O(1/2 )])(p) + O(j+1 ) = j (Q(j) [δ˜ = j (Q(j) [δx])(p) + O(j+1/2 ) = 0
for small enough. Replacing r by a smaller number if necessary so that Q[x + δ˜ x + δˆ x()] has no zeros on B(p, r), the resulting data set has no KIDs in any subset of B(p, r) by point 1 of Theorem 5.3. The construction described so far leads to a perturbed initial data set which agrees with the starting one at ∂M to arbitrarily high order. Consider, next, a collar neighborhood Ns = {p ∈ M : d(p, ∂M ) < s} of ∂M . Arguments as in Lemma 8.2 show that K (Ms := M \ Ns ) = {0} for s small enough. Applying the result already established to Ms one obtains a perturbation which vanishes on Ns . An identical proof, based on the results in Section 6, gives: Theorem 8.8 Let (M, g) be a n-dimensional C ,α compact Riemannian manifold with boundary, suppose that ≥ 0 (n), α ∈ (0, 1) for some 0 (n), and suppose that g has constant scalar curvature s. Assume that there are only trivial static KIDs, N (M ) = {0} . For any p ∈ M \ ∂M and for any > 0 there exists r > 0 and an -small, in a C ,α topology, perturbation g of g with scalar curvature s such that N (U ) = {0} for all U such that U ∩ B(p, r) = ∅ . Further, g can be chosen to coincide with g in a neighborhood of ∂M .
9 Proofs of Theorems 1.2 and 1.3 Proof of Theorem 1.2: Let Q be the polynomial of Theorem 5.3, set Vˆp = {vacuum initial data such that Q[K, g](p) = 0} , then Vˆp is open and contained in Vp . To show density, let Mi ⊂ M be a sequence of relatively compact domains with smooth boundary such that M = ∪i Mi . The argument of the proof of Lemma 8.2 shows that K (Mi ) = {0} for i large enough. Point 1 follows then from Theorem 8.7 with M there equal to M i . The timesymmetric case is obtained similarly by Theorem 8.8. Point 2 is established by repeating the argument of the proof of Theorem 4.1.
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Proof of Theorem 1.3: Openness follows from Proposition 4.2, it remains to establish density. We start by showing that for spatially compact CMC initial data KIDs are “purely spacelike”. Somewhat more generally, one has: Proposition 9.1 Consider a vacuum initial data set (M, g, K) with constant τ := trg K = 0, suppose that (1.2) holds, and assume that (M, g) is geodesically complete (perhaps as a manifold with boundary). Let (N, Y ) be a KID on M satisfying lim
sup |N (q)| = 0 ,
r→∞ q∈S (r) p
(9.1)
for some p ∈ M , where Sp (r) is the boundary of the geodesic ball of radius r centred at p. If N ≡ 0, then K is pure trace, M is compact and (M, g) is Einstein. Remark 9.2 A KID satisfying (9.1) will be called asymptotically tangential ; a KID with N ≡ 0 will be called tangential. In the compact boundaryless case we have Sp (r) = ∅ for r large enough, so all KIDs are asymptotically tangential. Proof. We note that if M has a boundary, then Sp (r) ∩ ∂M := ∂Bp (r) ∩ ∂M = ∅ for r large, so that (9.1) implies that N vanishes on ∂M . The KID equations imply 2 2Λ ˜ 2 + (trg K) − 2Λ ∆N = |K|2 − N = |K| N, (n − 1) n (n − 1) ˜ is the trace-free part of K. Equation (1.2) and the maximum principle where K ˜ ≡ 0, with (1.2) being an equality, and N = const, or N ≡ 0. show that either K In the former case the KID equations further imply Ricci flatness of g. The case N ≡ 0 is compatible with (9.1) only if Sp (r) = ∅ for r sufficiently large, which is equivalent to compactness of M . We note the following straightforward consequence of Theorem 2.1 and Proposition 4.2: Proposition 9.3 Consider the collection of Riemannian metrics on a three-dimensional manifold with a C k (weighted, with arbitrary weights, in the non-compact case) topology, k ≥ 5. The set of such Riemannian metrics which have no globally defined conformal Killing vectors is open and dense. Proof. Choose any relatively compact Ω ⊂ M , then by Theorem 2.1 and Proposition 4.2 there exists an open and dense set of metrics which have no conformal Killing vector fields on Ω, then those metrics do not have globally defined conformal Killing vector fields either. k,α defined in Appendix A. The weights ϕ and ψ Our next result uses spaces Cϕ,ψ in our next result have to be chosen in a way compatible with the conformal method in the asymptotically flat regions [8], similarly in the asymptotically hyperbolic regions [2], while ϕ = ψ = 1 in the compact case. The differentiabilities here are different, as compared to Theorem 1.3, because under the CMC restriction the conformal method can be used:
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Corollary 9.4 There exists k1 (n), with k1 (3) = 5, such that for k ≥ k1 (n) and α ∈ k,α k−1,α × Cϕ,ψ -open and dense collection (0, 1) the following holds: There exists a Cϕ,ψ of vacuum CMC initial data sets (M, g, K) which are either 1. asymptotically flat with compact interior (then Λ = 0), or 2. asymptotically hyperbolic as in [2], or 3. defined on a compact M , with Λ satisfying (1.2), and which do not have any asymptotically tangential KIDs. k,α k−1,α × Cϕ,ψ such that g Proof. Let U be the set of vacuum initial data (g, K) ∈ Cϕ,ψ is Einstein. This class of initial data obviously forms a closed set with no interior. k,α k−1,α × Cϕ,ψ initial Let V1 be the complement of U within the set of all vacuum Cϕ,ψ data, then V1 is open and dense. Choose any p ∈ M , and let V2 be the set of initial data in V1 such that K is not pure trace, and such that the polynomial Q[H, DH, D2 H] of Theorem 2.1 does not vanish at p, then V2 is open. Consider any (g, K) which is not in V2 and which is not in U , by Proposition 9.3 for any > 0 there exists a metric g () which is in V2 (and therefore has no conformal 5 Killing vectors) such that g − g ()Cϕ,ψ ≤ . Using K as the seed solution for the extrinsic curvature, the conformal method [2, 8, 17] allows one to solve for a nearby solution (M, g(), K()) of the vacuum constraint equations. Let (N, Y ) be a KID for g(). By Proposition 9.1 the KID (N, Y ) is tangential, N ≡ 0, which implies that Y is a Killing vector field of g(). Now g() is a conformal deformation of g (), therefore Y is a conformal Killing vector field of g (), hence Y = 0. It follows that V2 provides the desired open and dense set.
On compact boundaryless manifolds all KIDs are asymptotically tangential, and Theorem 1.3 is established in this case. Consider, next, the asymptotically flat case, with an r−β weighted topology, β ∈ (0, n − 2). Recall that we want to prove density of metrics without KIDs. For such β the result can be established as follows: consider the set of solutions of the constraint equations on R3 \ B(0, R), which approach (g, K) at S(0, R) exponentially fast as in [11, Theorem 6.6], and which are r−β -asymptotically flat. A straightforward generalization of [11, Corollary 6.3] applies to this space of initial data and shows that this collection forms a manifold. It follows that each linearized solution of the constraint equations constructed as at the beginning of the proof of Theorem 8.7 is tangent to a curve of solutions, which coincide with (g, K) away from the asymptotic region R3 \ B(0, R). This establishes point 1 of Theorem 1.3. We note that the condition trg K = 0 is not necessarily preserved by the perturbation just constructed. However, it follows from the implicit function theorem, or from the results of Bartnik [3], that the deformed initial data set on R3 \ B(0, R) can be deformed in the associated space-time to obtain a data set with vanishing mean extrinsic curvature, proving point 2. Point 3 is established as above using [11] in the K ≡ 0 setting. In the conformally compactifiable case the argument is identical, based on [11, Theorem 6.7].
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In the asymptotically flat case with β = n − 2 some more work is needed. For simplicity we consider only smooth initial data, but the construction works also in the finite differentiability case. The idea is to obtain solutions up to kernel using the techniques of [10, 14], and to show that one can correct for the kernel by changing the metric in the asymptotic region, the argument proceeds as follows. Let Γ(R, 2R) be a coordinate annulus, with inner radius R and outer radius 2R, contained in the asymptotically flat region, let x = (K, g). Let δx = (δK, δg) be a solution of the linearized constraint equations supported in Γ(5R/4, 7R/4), constructed as at the beginning of the proof of Theorem 8.7, so that x = x + δx has no KIDs on Γ(R, 2R) for all positive small enough. By construction x fails to solve the constraint equations by O(2 ). We use the terminology of [10, Sections 8.1 and 8.2]. Let Q0 = (m0 , p 0 , c0 , J0 ) denote the Poincar´e charges of x0 = x, and for Q in a neighborhood of Q0 let yQ = (KQ , gQ ) be a reference family of metrics obtained on Rn \B(R) as follows: by scaling, boosting, and space-translating (K, g) one is led to a family of initial data sets with mass m, ADM-momentum p, and centre of mass c covering a neighborhood of (m0 , p0 , c0 ). Choosing R large enough, a construction in [22] can be used to deform each of the solutions obtained so far to initial data sets with arbitrary angular momentum in a neighborhood of J0 .7 One can now glue x with yQ using the techniques described in detail in [10, 14] obtaining, for + |Q − Q0 | small enough, on Γ(R, 2R) a “solution up-to-kernel” z,Q = (K,Q , g,Q ) which smoothly extends across the inner sphere B(0, R) to x, which smoothly extends across the exterior sphere B(0, 2R) to yQ , and which differs from x by terms which are quadratic in and in Q − Q0 . Making and |Q − Q0 | smaller if necessary, the arguments presented in Sections 8.1 and 8.2 of [10] show that one can find Q() so that z,Q() solves the constraints, providing the desired solution without global KIDs.
A
Topologies
In this paper we prove both density and openness results, and there does not seem to be a topology which captures both features in an optimal way. The aim of this appendix is to discuss those issues in some detail. As already pointed out in the introduction, a possible topology for which our results hold is the following: one chooses some smooth complete Riemannian metric h on M , which is then used to calculate norms of tensors and their hcovariant derivatives; we shall denote this topology by T k (h). If M is compact, the resulting topology is h-independent, and all our results in the compact case hold with such topologies, for appropriate k’s. However, when M is not compact, there exist choices of h which will lead to different topologies; nevertheless, for each such 7 The point of the current construction is to obtain a “reference family”, as defined in [10], near the initial data we started with. An alternative way is to first deform the initial data to data which are exactly Kerr outside of a compact set with large radius, and use the Kerr family as the reference family.
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choice Theorems 1.2 holds. Further, all the results, except the perturbations that remove global KIDs in an asymptotically flat or asymptotically hyperbolic region, k,α remain true if, e.g., weighted Cφ,ϕ topologies defined with respect to h are used, as defined in [10], with norm k uC k,α(h) = supx∈M i=0 ϕφi ∇(i) u(x)h φ,ϕ , (i) (i) u(y) h + sup0 =dh (x,y)≤φ(x)/2 ϕ(x)φi+α (x) ∇ u(x)−∇ dα (x,y) h
k,α with any weight functions φ and ϕ; we shall denote such topologies by Tφ,ϕ (h). Finally, all openness and density results established in this paper, including statements involving the field equations, will hold with any choice of h and weight functions except for the following restriction: if (M, g, K) contains an asymptotically flat region, and one wishes to construct a perturbation that gets rid of a globally defined KID while preserving the field equations, then h should be chosen to be, e.g., the Euclidean metric in the asymptotically flat region, with the weights φ = r, ϕ = r−β , for some β ∈ (0, n − 2]. Similarly, in the context of Corollary 9.4 and of point 4 of Theorem 1.3, the weights in the asymptotically hyperbolic region should be chosen in a way compatible with the asymptotic conditions in the conformally compactifiable region as in [2]. While the above topologies seem satisfactory for most purposes, the optimal topology for perturbations that get rid, e.g., of Killing vectors, at a given point p, is that of convergence in the space of kth jets of the metric at p, with k ≥ k0 (n), for some k0 (n) as described above, on the space of metrics which coincide with the starting metric g away from a compact neighborhood of p. However, this space is unnecessarily small for our openness results, which do not hold in such a weak topology in any case; see also Remark 4.3.
B “Local extends to global” in the simply connected analytic setting In this appendix we wish to generalise Nomizu’s theorem [20] concerning Killing vectors to conformal Killing vectors and to KIDs. It should be clear that our argument applies to a large class of similar overdetermined systems with analytic coefficients, such as, e.g., those considered in [7]. In particular the proof given here applies to Killing vector fields in arbitrary signature, and seems to be somewhat simpler than the original one. Theorem B.1 Let (M, g) be a simply connected analytic pseudo-Riemannian manifold. 1. Every locally defined conformal Killing vector extends to a globally defined one. 2. If, moreover, K is also analytic then every locally defined KID extends to a globally defined one.
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Proof. We give the proof for KIDs, the argument for conformal Killing vector fields is identical. Let r, Pα and Lk be as in Section 7. We note the following: Lemma B.2 Consider a KID x defined on an open set Ω, let γ : [0, 1] → M be a differentiable path such that γ : [0, 1) → Ω, with γ(1) ∈ γ([0, 1)). Then there exists a neighborhood U of γ([0, 1]) and a KID xˆ defined on U such that x = xˆ on γ([0, 1)). Proof. Equation (7.4) shows that each covariant derivative Dα r of r satisfies along γ the linear equation D (Dα r ◦ γ) = γ˙ µ ((Dµ Dα r) ◦ γ) = (Pµα r) ◦ γ , ds with the multi-index µα in Pµα defined in the obvious way. It follows that each Fα (s) := (Dα r ◦ γ) (s) extends by continuity to some values, denoted by Fα (1), such that Fα (1) = Pα (γ(1))F (1) , where F (1) = lims→1 r(γ(s)). By continuity the integrability conditions Lk = 0 are satisfied by F (1), and therefore, by the argument given after (7.4), there exists > 0 and a solution of the KID equations defined on B(γ(1), ) for some > 0. We can cover γ([0, 1]) by a finite number of open balls Bi := B(γ(si ), ri ), i = 1, . . . , N , such that s1 = 0, sN = 1, rN ≤ , with the balls pairwise disjoint except for the neighboring ones: Bi ∩ Bj = ∅ if |i − j| > 1. It should be clear that the solution just constructed on B(γ(1), ) coincides with that which exists already on the overlap with B(γ(sN −1 ), rN −1 ). The desired neighborhood is obtained by setting U = ∪i B(γ(si ), ri ). Returning to the proof of Theorem B.1, let q be any point in Ω, let p ∈ M , and let γ : [0, 1] → M be any piecewise differentiable path without self-intersections with γ(0) = q, γ(1) = p. Let I ⊂ [0, 1] be the set of numbers s such that there exists a neighborhood Us of γ|[0,s] and a KID xs defined on Us such that xs = x near p. Then I is open by definition, it is closed by Lemma B.2, therefore I = [0, 1]. We have thus shown: Lemma B.3 For any piecewise differentiable path γ : [0, 1] → M without selfintersections, with γ(0) ∈ Ω, there exist a neighborhood U of γ and a KID xγ defined on U , coinciding with x on U ∩ Ω. Any γ as in Lemma B.3 allows us therefore to extend x to a neighborhood of p. It remains to show that this extension is γ–independent. Let thus γ and
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γˆ be two differentiable paths from q to p without self-intersections, since M is simply connected there exist a homotopy of differentiable paths γt : [0, 1] → M , t ∈ [0, 1], with γt (1) = p, γt (0) = q, γ0 = γ and γ1 = γˆ . If any γt self-intersects at s1 and s2 , with s1 < s2 , we replace it by a new path, still denoted by γt , obtained by staying at γt (s1 ) for s ∈ [s1 , s2 ]; this procedure is repeated until all self-intersections of γt have been eliminated. Let r(t) denote the value of r at p obtained from Lemma B.3 by following γt , then r is a continuous function of t. The set of t’s for which r(t) = r(0) is closed by continuity of r, it is open by Lemma B.3, hence r(0) = r(1), which establishes Theorem B.1. Acknowledgments. We are grateful to L. Andersson, A. Cap, E. Delay, A. Fischer, J. Isenberg, J. Lewandowski, M. McCallum and D. Pollack for comments, discussions and suggestions.
References [1] S. Alinhac and G. M´etivier, Propagation de l’analyticit´e des solutions de syst`emes hyperboliques non-lin´eaires, Invent. Math. 75, 189–204 (1984). [2] L. Andersson and P.T. Chru´sciel, On asymptotic behavior of solutions of the constraint equations in general relativity with “hyperboloidal boundary conditions”, Dissert. Math. 355, 1–100 (1996). [3] R. Bartnik, The existence of maximal hypersurfaces in asymptotically flat space-times, Comm. Math. Phys. 94, 155–175 (1984). [4]
, Regularity of variational maximal surfaces, Acta Math. 161, 145–181 (1988).
[5] J. Biˇc´ak and J. Podolsk´ y, The global structure of Robinson-Trautman radiative space-times with cosmological constant, Phys. Rev. D 55, 1985–1993 (1996), gr-qc/9901018. [6] J.-P. Bourguignon, D.G. Ebin, and J.E. Marsden, Sur le noyau des op´erateurs pseudo-diff´erentiels `a symbole surjectif et non injectif, C. R. Acad. Sci. Paris S´er. A-B 282, Aii, A867–A870 (1976). ˇ [7] T. Branson, A. Cap, M. Eastwood, and R. Gover, Prolongations of geometric overdetermined systems, (2004), math.DG/0402100v2. ´ Murchadha, The boost problem in general rela[8] D. Christodoulou and N. O tivity, Comm. Math. Phys. 80, 271–300 (1980). [9] P.T. Chru´sciel, On uniqueness in the large of solutions of Einstein equations (“Strong Cosmic Censorship”), Australian National University Press, Canberra, 1991.
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[10] P.T. Chru´sciel and E. Delay, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, M´em. Soc. Math. de France 94, 1–103 (2003), gr-qc/0301073v2. [11]
, Manifold structures for sets of solutions of the general relativistic constraint equations, Jour. Geom Phys. (2004), in press, gr-qc/0309001v2.
[12] P.T. Chru´sciel, J. Isenberg, and D. Pollack, Initial data engineering, (2004), gr-qc/0403066. [13] B. Coll, On the evolution equations for Killing fields, Jour. Math. Phys. 18, 1918–1922 (1977). [14] J. Corvino and R. Schoen, On the asymptotics for the vacuum Einstein constraint equations, gr-qc/0301071, 2003. [15] D.G. Ebin, The manifold of Riemannian metrics, Global Analysis, Proc. Sympos. Pure Math., vol. 15, 1970, pp. 11–40. [16] L. H¨ormander, The boundary problems of physical geodesy, Arch. Rat. Mech. Analysis 62, 1–52 (1976). [17] J. Isenberg, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Class. Quantum Grav. 12, 2249–2274 (1995). [18] J. Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140, 655– 683 (1994). [19] V. Moncrief, Space-time symmetries and linearization stability of the Einstein equations. II, Jour. Math. Phys. 17, 1893–1902 (1976). [20] K. Nomizu, On local and global existence of Killing vector fields, Ann. Math. 72, 105–120 (1960). [21] F.A.E. Pirani, Introduction to gravitational radiation theory, Lectures on general relativity, Brandeis, vol. 1, Prentice Hall, Englewood Cliffs, New Jersey, 1965. [22] R. Schoen, in preparation, (2003). [23] T.Y. Thomas, The differential invariants of generalized spaces, Cambridge University Press, 1934.
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Robert Beig∗ Institut f¨ ur Theoretische Physik Universit¨ at Wien Boltzmanngasse 5 A-1090 Vienna Austria email:
[email protected] Piotr T. Chru´sciel∗ D´epartement de Math´ematiques Facult´e des Sciences Parc de Grandmont F-37200 Tours France email:
[email protected] Richard Schoen∗ Department of Mathematics Stanford University Palo Alto USA email:
[email protected] Communicated by Sergiu Klainerman submitted 22/02/04, accepted 28/07/04
To access this journal online: http://www.birkhauser.ch
∗
ESI visiting scientist
Ann. Henri Poincar´e
Ann. Henri Poincar´e 6 (2005) 195 – 215 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020195-21 DOI 10.1007/s00023-005-0203-2
Annales Henri Poincar´ e
A Product Formula Related to Quantum Zeno Dynamics Pavel Exner and Takashi Ichinose Abstract. We prove a product formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P on a separable Hilbert space H, with the convergence in L2loc (R; H). It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P . The convergence in H is demonstrated in the case of a finite-dimensional P . The main result is illustrated in the example where the projection corresponds to a domain in Rd and the unitary group is the free Schr¨ odinger evolution.
1 Introduction The fact that the decay of an unstable system can be slowed down, or even fully stopped in the ideal case, by frequently repeated measurements checking whether the system is still undecayed was noticed first by Beskow and Nilsson [BN]. It was only decade later, however, when Misra and Sudarshan [MS] caught the imagination of the community by linking the effect to the well-known Zeno aporia about a flying arrow. While at first the subject was rather academical, in recent years the possibility of observing Zeno-type effects experimentally has become real and at present there are scores of physical papers discussing this topic. On the mathematical side, the first discussion of the continuous observation appeared in [Fr]. Two important questions, however, namely the existence of Zeno dynamics and the form of its effective Hamiltonian have been left open both in this paper and later in [MS]. The second problem is particularly important when the subspace into which the state of the system is repeatedly reduced has dimension larger than one. A partial answer was given in [Ex, Sec. 2.4] where it was shown that the results of Chernoff [Ch1, Ch2] allow to determine the generator of the Zeno time evolution naturally through the appropriate quadratic form. Our interest to the problem was rekindled by a recent paper by Facchi et al. [FPS] who studied the important special case when the presence of a particle in a domain of Ω ⊂ Rd is repeatedly ascertained. Using the method of stationary phase the authors showed that the Zeno dynamics describes in this case the free particle confined to Ω, with the hard-wall (Dirichlet) condition at the boundary of the domain. The result cannot be regarded as fully rigorous, because detailed properties of the convergence are not worked out, but the idea is sound without any doubt. In the present paper we combine the results of [Ch1, Ch2] with that of Kato [Ka2] to address this question in a general setting. We show that if the natural
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effective Hamiltonian mentioned above is densely defined – which is a nontrivial assumption – then the Zeno dynamics exists and the said operator is its generator in a topology which includes an averaging over the time variable – cf. Theorem 2.1 for exact statement (a part of the present result given in Corollary 2.3 was announced in [EI]). Our conclusion cannot be thus regarded as fully satisfactory from the mathematical point of view, because the natural topology to be used here is given by the norm of the Hilbert space, and in this respect an important part of the problem remains open. We demonstrate, however, the strong convergence in H for the particular case when the projections involved are finite-dimensional – cf. Theorem 2.4. On the other hand, from the physical point of view the result given in Theorem 2.1 is quite plausible taking into account that any real measurement is burdened with errors – see Remark 2.5 below. We will formulate the theorems together with their corollaries in the next section. Theorem 2.1 will be then proven in Sections 3 and 4, Theorem 2.4 in Section 5. As an example we discuss in the concluding section reduction of a free dynamics to a domain in Rd by permanent observation. We will establish that the Zeno generator mentioned above is in this case the Dirichlet Laplacian, obtaining thus in a different way the result of the paper [FPS].
2 The main result Throughout the paper H will be a nonnegative self-adjoint operator in a separable Hilbert space H, and P will be an orthogonal projection. The nonnegativity assumption is made for convenience; our main result extends easily to any selfadjoint operator H bounded from below as well as one bounded from above, i.e., to each semi-bounded self-adjoint operator in H. Consider the quadratic form u → H 1/2 P u2 with form domain D[H 1/2 P ]. Note that H 1/2 P involved here is a closed operator and HP has the same property. Let HP := (H 1/2 P )∗ (H 1/2 P ) be the self-adjoint operator associated with this quadratic form. In general, HP may not be densely defined in which case it is a self-adjoint operator in a closed subspace of H. More specifically, it is obviously defined and acts nontrivially in a closed subspace Ran P , the closure of the form domain D[H 1/2 P ], while in the orthogonal complement (Ran P )⊥ it acts as zero. The quadratic form u → H 1/2 P u2 defined on D[H 1/2 P ] is a closed extension of the form u → P u, HP u defined on D[HP ], but the former is not in general the closure of the latter. Indeed, if H is unbounded, D[H] is a proper subspace of D[H 1/2 ]. Take u0 ∈ D[H 1/2 ]\D[H] such that the vector H 1/2 u0 is nonzero, and set P to be the orthogonal projection onto the one-dimensional subspace spanned by u0 . Taking into account that D[HP ] = {u ∈ H; P u ∈ D[H]} which u0 = P u0 does not belong to, we find HP u = 0 for u ∈ D[HP ], while H 1/2 P u0 = H 1/2 u0 = 0 by assumption. To describe our results, we denote by L2loc ([0, ∞); H) = L2loc ([0, ∞)) ⊗ H the Fr´echet space of the H-valued strongly measurable functions v(·) on [0, ∞)
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such that v(·) is locally square integrable there, with the topology induced by T 1/2 the semi-norms v → 0 v(t)2 dt for a countable set {T }∞ =1 of increasing positive numbers accumulating at infinity, lim→∞ T = ∞. In a similar way one defines the Fr´echet space L2loc (R; H) = L2loc (R) ⊗ H. Our main result can be stated as follows: Theorem 2.1 Let H be a nonnegative self-adjoint operator on a separable Hilbert space H and P an orthogonal projection. Let t → P (t) be a strongly continuous function whose values are orthogonal projections in H, defined in some neighborhood of zero, with P (0) =: P . Moreover, suppose that D[H 1/2 P (t)] ⊃ D[H 1/2 P ] and limt→0 H 1/2 P (t)v = H 1/2 P v holds for v ∈ D[H 1/2 P ]. If the operator HP specified above is densely defined in the whole Hilbert space H, then for every f ∈ H and ε = ±1 it holds that [P (1/n) exp(−iεtH/n)P (1/n)]nf −→ exp(−iεtHP ) P f ,
(2.1)
[P (1/n) exp(−iεtH/n)]n f −→ exp(−iεtHP ) P f , [ exp(−iεtH/n) P (1/n)]n f −→ exp(−iεtHP ) P f ,
(2.2) (2.3)
in the topology of L2loc (R; H) as n → ∞. Note that HP differs in general from the operator P HP , which may not be selfadjoint in H, nor even closed, because P H is not necessarily closed, though HP is. HP is a self-adjoint extension of P HP under the requirement of the theorem that HP is densely defined in H, which means nothing else but that the domain D[H 1/2 P ] of the quadratic form in question is dense in H. Note also that for ε = 1, the theorem concerns a nonnegative self-adjoint operator εH = H, while for ε = −1, we get product formulae for the non-positive self-adjoint operator εH = −H. Moreover, the result is preserved when H is replaced with a shifted operator H + cI, i.e., for any semi-bounded self-adjoint operator in a separable Hilbert space. An important particular case, most often met in the applications, concerns the situation when the projection-valued function is constant. Corollary 2.2 Let H be a self-adjoint operator bounded from below in a separable Hilbert space H and P an orthogonal projection. If the operator HP specified above is densely defined, then for every f ∈ H and ε = ±1 we have in the topology of L2loc (R; H) the limiting relation [P exp(−iεtH/n)P ]n f −→ exp(−iεtHP ) P f
(2.4)
for n → ∞ as well as its nonsymmetric counterparts obtained by setting P (1/n) = P in (2.2) and (2.3). From the viewpoint of quantum Zeno effect described in the introduction the optimal result would be a strong convergence on H for a fixed value of the time variable, moreover uniformly on each compact interval in t. Our Theorem 2.1 implies the following weaker result on pointwise convergence.
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Corollary 2.3 Under the same hypotheses as in Theorem 2.1, there exist a set M ⊂ R of Lebesgue measure zero and a strictly increasing sequence {n } of positive integers along which we have
[P (1/n ) exp(−iεtH/n )P (1/n )]n f −→ exp(−iεtHP ) P f , n
[P (1/n ) exp(−iεtH/n )] f −→ exp(−iεtHP ) P f ,
n
[ exp(−iεtH/n ) P (1/n )] f −→ exp(−iεtHP ) P f ,
(2.5) (2.6) (2.7)
for every f ∈ H, strongly in H for all t ∈ R \ M . As we have indicated above, one need not resort to subsequences in the particular case when the projections involved are finite-dimensional. Theorem 2.4 In addition to the hypotheses of Theorem 2.1, assume that the orthogonal projection P as well as P (t) is of finite dimension. Then (i) the formulae (2.1)–(2.3) hold in the norm of H as n → ∞, uniformly on each compact interval of the variable t in R \ {0}, (ii) it also holds for ε = ±1 that as n → ∞, [P (t/n) exp(−iεtH/n)P (t/n)]n −→ exp(−iεtHP ) P , [P (t/n) exp(−iεtH/n)]n −→ exp(−iεtHP ) P , [ exp(−iεtH/n) P (t/n)]n −→ exp(−iεtHP ) P , strongly on H, uniformly on each compact interval in the variable t ∈ R. Before proving Theorems 2.1 and 2.4 and Corollary 2.3 let us comment briefly on several other aspects of the result. Remark 2.5 While the necessity to pick a subsequence makes the pointwise convergence result weaker than desired, let us notice that from the physical point of view the convergence in L2loc (R; H) can be regarded as satisfactory. The point is that any actual measurement, in particular that of time, is burdened with errors. Suppose thus we perform the Zeno experiment on numerous copies of the system. The time value in the results will be characterized by a probability distribution φ : R+ → R+ , which is typically a bounded, compactly supported function – in a precisely posed experiment it is sharply peaked, of course. Corollary 2.2 then gives 2 (2.8) φ(t) [P exp(−iεtH/n)P ]n f − exp(−iεtHP ) P f dt → 0 as n → ∞, in other words, the Zeno dynamics limit is valid after averaging over experimental errors, however small they are.
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Remark 2.6 While the proof of strong convergence in H in Theorem 2.1 and Corollaries 2.2 and 2.3 remains elusive without the finite-dimension assumption, such a claim can be easily established in the orthogonal complement of the subspace P H. Indeed, taking f ∈ QH, where Q := I − P , we have (P (1/n)e−iεtH/n P (1/n))n f = (P (1/n)e−iεtH/n P (1/n))n P (1/n)Qf , (e−iεtH/n P (1/n))n f = (e−iεtH/n P (1/n))n P (1/n)Qf , which converge to zero, uniformly on each compact t-interval in R, as n → ∞, s because P (τ ) → P as τ → 0. This gives the result for (2.5) and (2.7), while for (2.6) one has to employ in addition the relation (3.11) below. Remark 2.7 The fact that the product formulae require HP to be densely defined is nontrivial. Recall the example of [Ex, Rem. 2.4.9] in which H is the multiplication operator, (Hψ)(x) = xψ(x) on L2 (R+ ), and P is the one-dimensional projection onto the subspace spanned by the vector ψ0 : ψ0 (x) = [(π/2)(1+x2 )]−1/2 . In this case obviously HP is the zero operator on the domain D[HP ] = {ψ0 }⊥ . On the other hand, P e−itH P acts on Ran P as multiplication by the function v(t) := e−t −
i −t 2i e Ei(t) − et Ei(−t) = 1 + t ln t + O(t), π π
where Ei (−t) and E i (t) are exponential integrals [AS]; due to the rapid oscillations of the imaginary part as t ↓ 0 a pointwise limit of v(t/n)n for n → ∞ does not exist. Notice also that different limits may be obtained in this example along suitably chosen subsequences {n }. Remark 2.8 In their recent study of Trotter-type formulae involving projections Matolcsi and Shvidkoy [MaS] presented two examples in which expressions of the type [exp(−iH/n)P ]n do not converge strongly. This result does not answer the question, however, whether the product expressions considered here converge in the strong topology of H or not, because our assumptions are not satisfied there. In the first example of [MaS] the analogue of the operator HP is not densely defined, in the second one H is not semi-bounded.
3 Proof of Theorem 2.1 We present the argument for ε = 1, the case ε = −1 can be treated similarly. We first prove (2.1) in (a), and next (2.2), (2.3) in (b). (a) Let us begin with the symmetric product case and prove the formula (2.1) with ε = 1. We will check the convergence in (2.1) on an arbitrary compact t-interval in the closed right half-line [0, ∞). The proof for t-intervals in the closed left half-line (−∞, 0] is analogous, and in addition, it can be included in the case ε = −1 with the convergence in (2.1) on compact t-intervals of the closed right half-line [0, ∞).
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Put Q(t) := I − P (t) and Q := Q(0) = I − P (0) = I − P , where I is the identity operator on H. Since H is nonnegative by assumption, there ∞ exists a spectral measure E(dλ) on the nonnegative real line such that H = 0− λ E(dλ). For ζ ∈ C with Re ζ ≥ 0 and τ > 0, we put F (ζ, τ ) = P (τ ) e−ζτ H P (τ ) ,
(3.1)
which is a contraction, and S(ζ, τ ) = τ −1 [I − F (ζ, τ )] = τ −1 [I − P (τ ) e−ζτ H P (τ )],
(3.2)
which exists as a bounded operator on H with Re f, S(ζ, τ )f ≥ 0 for every f ∈ H. For definiteness we use here and in the following the physicist convention about the inner product supposing that it is antilinear in the first argument. For a non-zero ζ ∈ C with Re ζ ≥ 0, we put also H(ζ) := ζ −1 [I − e−ζH ] .
(3.3)
Each element v(·) in L2loc ([0, ∞); H) is an equivalence class such that any two representatives of it are equal a.e. on [0, ∞). However, at some places we will not avoid an abuse of notation using for a particular representative of such an element the same symbol v(·). At the same time, in the following the convergence of a family of vectors v(·, τ ) to v(·) in the topology of the space L2loc ([0, ∞); H) = L2loc ([0, ∞)) ⊗ H as τ tends to zero will be often written as v(t, τ ) −→ v(t); this will be the case when writing v(·, τ ) −→ v(·) would require to introduce a separate symbol for this v(t, τ ) the meaning of which is clear from the context. The key ingredient of the proof is the following lemma. Lemma 3.1 (I + S(it, τ ))−1 converges to (I + itHP )−1 P as τ → 0 strongly in L2loc ([0, ∞); H), in other words, for all f ∈ H and every finite T > 0 we have 0
T
(I + S(it, τ ))−1 f − (I + itHP )−1 P f 2 dt → 0 ,
τ → 0.
(3.4)
We postpone the proof of Lemma 3.1 to the next section. For the moment we will accept its claim and use it to show that it implies the symmetric case (2.1) of the product formula in Theorem 2.1. To this end, let {mn } be a strictly increasing sequence of positive integers, i.e., a subsequence of the sequence of all positive integers. We have only to show that there exists a subsequence {n } in any such sequence {mn } along which (2.1) holds. Then by a standard argument we can conclude that (2.1) actually holds along the sequence of all positive integers n. For if this were not the case, there would exist a subsequence {n } of strictly increasing positive integers along which
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(2.1) does not converge. However, we see that there is a subsequence {n } of {n } along which the convergence takes place to the same limit, which is a contradiction. Fix {mn } and f ∈ H. Lemma 3.1 holds, in particular, along the sequence {τn } with τn := 1/mn , and since L2 convergence implies pointwise convergence a.e. along a subsequence, there exist a subset Mf of Lebesgue measure zero of the variable t in [0, ∞) and a subsequence {τf,n } of {τn }, both dependent on f , such that (I + S(it, τf,n ))−1 f −→ (I + itHP )−1 P f holds strongly in H for t ∈ [0, ∞) \ Mf . Since H is separable by assumption, we can choose a countable dense subset D = {f }∞ =1 in H. Then we infer that for f1 ∈ D there exist a set M1 := Mf1 of Lebesgue measure zero and a subsequence {τ1,n } of {τn } along which (I + S(it, τ1,n ))−1 f converges to (I + itHP )−1 P f for every t ∈ / M1 . Next, for f2 ∈ D there exist a set M2 := Mf2 of Lebesgue measure zero and a subsequence {τ2,n } of {τ1,n } along which (I + S(it, τ2,n ))−1 f converges / M2 . Proceeding in this way, we associate in the to (I + itHP )−1 P f for every t ∈ th step with f ∈ D a set M := Mf of Lebesgue measure zero and a subsequence {τ,n } of {τ−1,n } along which (I + S(it, τ,n ))−1 f converges to (I + itHP )−1 P f for every t ∈ / M . Now we put τn := τn,n and n := 1/τn , so that {n } is a subsequence of the strictly increasing sequence {mn } of positive integers from which we have started. −1 f} Then it follows that for every t ∈ [0, ∞)\∪∞ =1 M , the sequence {(I +S(it, τn )) −1 converges to (I + itHP ) P f strongly in H as τn → 0 for every f ∈ D, and therefore also in H, because both (I +S(it, τ,n ))−1 and (I +itHP )−1 P are bounded operators on H with the norms not exceeding one. We denote M := ∪∞ =1 M , which is, of course, again a set of Lebesgue measure zero. In this way we have found a subsequence {τn } of {τn = 1/mn } and an exceptional subset M of [0, ∞) such that (3.5) (I + S(it, τn ))−1 f = (I + S(it, 1/n ))−1 f −→ (I + itHP )−1 P f strongly in H as τn → 0 or n → ∞ for every f ∈ H and for each fixed t ∈ / M ; it is important that M is independent of f . Lemma 3.2 For the sequence {n } specified above and every f ∈ H we have
[P (1/n ) exp(−itH/n )P (1/n )]n f −→ e−itHP P f
(3.6)
as n → ∞ strongly in H provided t ∈ / M. Notice that this claim is in fact the “symmetric” part of Corollary 2.3. Proof of Lemma 3.2. We use arguments analogous to those employed in derivation of Chernoff’s theorem – see [Ch2, Theorem 1.1], [Ch1] and [Ka1, Thm IX.3.6]. We divide the proof into two steps referring to f belonging to P H and to its orthogonal complement.
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Suppose first that f ∈ P H. For t ∈ / M and τ fixed, the operator S(it, τ ) generates a strongly continuous semigroup { e−θS(it,τ ) : θ ≥ 0 } on H, and the resolvent convergence (3.5) implies the convergence of the corresponding semigroups [Ka1, Thm IX.2.16], so we have
e−θS(it,1/n ) f −→ e−iθtHP f s
/ M , uniformly on each compact interval of the variable in H as n → ∞ for t ∈ θ ≥ 0. In particular, choosing θ = 1 we get for each t ∈ [0, ∞) \ M
e−S(it,1/n ) f −→ e−itHP f , s
n → ∞ .
(3.7)
The same equivalence implies for any λ ≥ 0 and t ∈ [0, ∞) \ M that (I + λS(it, 1/n ))−1 f −→ (I + iλtHP )−1 P f , s
in particular, using the diagonal trick we obtain
−1 1 s I + √ S(it, 1/n ) f −→ P f n
as n → ∞,
(3.8)
for every t ∈ [0, ∞) \ M . Next we use [Ch1, Lemma 2] which gives for any g ∈ H the inequality √ F (it, 1/n )n g − e−n (I−F (it,1/n )) g ≤ n (I − F (it, 1/n))g . Choosing g = I +
−1
√1 S(it, 1/n ) n
f we infer that
−1
1 f F (it, 1/n )n − e−S(it,1/n ) 1 + √ S(it, 1/n) n
−1 1 ≤ I + √ S(it, 1/n ) f − f , n where the right-hand side tends to zero as n → ∞ by (3.8). Using (3.8) once again we get (3.9) F (it, 1/n )n f − e−S(it,1/n ) f −→ 0 . The sought relation (3.6) immediately follows from (3.7) and (3.9), since by (3.1) we have F (it, 1/n)n = [P (1/n ) exp(−itH/n )P (1/n )]n . The case f ∈ QH is easier being independent of the arguments preceding Lemma 3.2. We have, along the sequence of all positive integers n, [P (1/n) exp(−itH/n)P (1/n)]n f → 0
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strongly in H and for each t ∈ [0, ∞), since P (1/n)f = P (1/n)Qf converges by assumption to P Qf = 0 as n → ∞, while exp(−itHP )P f = 0.
This yields the sought result because {[P (1/n ) exp(−itH/n )P (1/n )]n } is a bounded sequence for any t ≥ 0 and by Lebesgue dominated-convergence theorem it tends to the expected limit in L2loc ([0, ∞); H). Using the standard “subsequence” trick mentioned above we have thus shown that Lemma 3.1 implies the symmetric product formula (2.1) of Theorem 2.1. (b) Let us turn to the non-symmetric product-formula cases, i.e., to prove that (2.1) implies (2.2) and (2.3). Proof of (2.2). We employ the standard notation, [U, S] = U S − SU , for the commutator of bounded operators U and S. First we observe the following fact. s
Lemma 3.3 It holds that [ e−itτ H, P (τ ) ] −→ 0 as τ → 0, uniformly on each compact t-interval in R. Proof: By (3.3) with ζ = itτ we have [ e−itτ H, P (τ ) ] = i P (τ )tτ H(itτ ) − tτ H(itτ )P (τ ) , and hence for any v ∈ H we can estimate −itτ H [ e , P (τ ) ]v ≤ tτ H(itτ )v + tτ H(itτ )P (τ )v . We rewrite (3.3) with ζ = itτ as iH(itτ ) =
I − cos tτ H sin tτ H +i =: B(tτ ) + iA(tτ ) , tτ tτ
(3.10)
where B(tτ ) and A(tτ ) are obviously bounded self-adjoint operators on H, and B(tτ ) is in addition nonnegative. The definition makes sense if t = 0 but we need not exclude this case because what we really need is the operator tτ H(itτ ). For any w ∈ H we get tτ H(itτ )w2 = [tτ B(tτ ) + itτ A(tτ )]w2 = [(I − cos tτ H) + i sin tτ H]w2 = w, [(I − cos tτ H)2 + sin2 (tτ H)]w = 4 sin(tτ H/2)w2 → 0 , uniformly on compact t-intervals in R. In this way we have proved the claim, noting s that P (τ ) −→ P holds uniformly on each compact t-interval in R as τ → 0. Now we employ the following identity, n n P (1/n) e−itH/n v − P (1/n) e−itH/n P (1/n) v n−1 −itH/n [e , P (1/n) ]v , = − P (1/n)e−itH/n P (1/n)
(3.11)
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the right-hand side of which converges by Lemma 3.3 to zero uniformly on each n−1 compact t-interval for any v ∈ H, because P (1/n)e−itH/n P (1/n) is a contraction on H, and hence also in L2loc ([0, ∞); H). This yields the formula (2.2). Proof of (2.3). In view of the already proven formula (2.1) we have for every f ∈ H and T > 0 the following chain of relations T P f
2
≥ lim sup
T
(e−itH/n P (1/n))n f 2 dt
0
T
= lim sup 0
T
+ ≥ lim sup
T
= 0
e
0
0 T
P (1/n)(e−itH/n P (1/n))n f 2 dt Q(1/n)(e−itH/n P (1/n))n f 2 dt
(P (1/n)e−itH/n P (1/n))n f 2 dt
−itHP
P f 2 dt = T P f 2, s
with the lim sup taken along n → ∞, because I = P (1/n)+Q(1/n) and P (τ ) −→ P T as τ → 0. It follows that 0 Q(1/n)(e−itH/n P (1/n))n f 2 dt −→ 0 as n → ∞. Thus for any v(·) ∈ L2loc ([0, ∞); H) and every T > 0 we have, again by (2.1),
T
0
T
= 0
v(t), (e−itH/n P (1/n))n f dt P (1/n)v(t), (P (1/n)e−itH/n P (1/n))n f dt
T
+ −→
0
0 T
Q(1/n)v(t), Q(1/n)(e−itH/n P (1/n))n f dt
v(t), e−itHP P f dt
as n → ∞. It means that {(e−itH/n P (1/n))n f } converges to e−itHP P f weakly in L2loc ([0, ∞); H) together with all the seminorms, and therefore the convergence is strong in L2loc ([0, ∞); H). This yields the formula (2.3). It remains to prove Lemma 3.1 on which the above arguments were based.
4 Proof of Lemma 3.1 To demonstrate (3.4), we shall use the Vitali theorem – see, e.g., [HP] – for holomorphic functions and employ arguments analogous to those used in Kato’s paper
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[Ka2] for the self-adjoint Trotter product formula with the form sum of a pair of nonnegative self-adjoint operators. We do it in three steps. I. In the first step we will show the following lemma. Lemma 4.1 For a fixed ζ = t > 0, (I + S(t, τ ))−1 −→ (I + tHP )−1 P s
as
τ → 0.
(4.1)
Proof: The argument will be analogous to that in [Ka2], and indeed, validity of the result in the particular case when our projection-valued function is constant is remarked in [Ka2, Eq. (5.2), p. 194]. For ζ = tτ > 0 we have from (3.3) H(tτ ) = (tτ )−1 [I − e−tτ H ], which is a bounded, nonnegative and self-adjoint operator on H. It allows us to rewrite S(t, τ ) = τ −1 [I − P (τ )(I − tτ H(tτ ))P (τ )] = τ −1 Q(τ ) + tP (τ )H(tτ )P (τ ) , which is in this case also a bounded and nonnegative self-adjoint operator. To prove (4.1) take any f ∈ H and put uˆ(t, τ ) := (I + S(t, τ ))−1 f , so that f = (I + S(t, τ ))ˆ u(t, τ ) = [I + τ −1 Q(τ ) + tP (τ )H(tτ )P (τ )]ˆ u (t, τ ) .
(4.2)
Then we have u(t, τ )2 + tH(tτ )1/2 P (τ )ˆ u(t, τ )2 . ˆ u(t, τ ), f = ˆ u(t, τ )2 + τ −1 Q(τ )ˆ
(4.3)
Thus the families {ˆ u(t, τ )}, {τ −1/2 Q(τ )ˆ u(t, τ )} and {t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ )} are all bounded by f for all t > 0, uniformly as τ → 0, and therefore they are weakly compact in H. It follows that for each fixed t > 0 there exists a sequence {τn (t)} with τn (t) → 0 as n → ∞, in general dependent on t, along which these vectors converge weakly in H, w
u ˆ(t, τ ) −→ uˆ(t) ,
τ −1/2 Q(τ )ˆ u(t, τ ) −→ g0 (t) , w
t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ ) −→ h(t) , w
(4.4)
for some vectors u ˆ(t), g0 (t) and h(t) in H. Note that the sequence {τn (t)}∞ n=1 can be chosen the same for all three families. s From this result we see first that Q(τ )ˆ u(t, τ ) −→ 0 uniformly in t > 0 as τ → 0, so that we have Qˆ u(t) = 0 or u ˆ(t) = P u ˆ(t) ∈ P H. For every v ∈ D[H 1/2 ] we have, with the limit taken along {τn (t)}, u(t, τ ) v, h(t) = lim v, t1/2 H(tτ )1/2 P (τ )ˆ = t1/2 lim H(tτ )1/2 v, P (τ )ˆ u(t, τ ) = t1/2 H 1/2 v, P u ˆ(t) , s
because H(tτ )1/2 v −→ H 1/2 v as τ → 0. Hence u ˆ(t) = P u ˆ(t) belongs to D[H 1/2 ] 1/2 1/2 1/2 and h(t) = t H P u ˆ(t) because D[H ] is dense by assumption. Furthermore,
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multiplying (4.2) by τ 1/2 and taking the weak limit along the sequence {τn (t)} we get g0 (t) = 0. Similarly, multiplying (4.2) by P (τ ) we have for every v ∈ D[H 1/2 P ] v, P (τ )f = v, P (τ )ˆ u(t, τ ) + t1/2 H(tτ )1/2 P (τ )v, t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ ). Then taking the limit along the sequence {τn (t)} we get v, P f = v, P uˆ(t) + t1/2 H 1/2 P v, h(t), because by spectral theorem H(tτ )1/2 (P (τ )−P )v = H(tτ )1/2 (I + H)−1/2 (I + H)1/2 (P (τ )−P )v ≤ (I + H)1/2 (P (τ )−P )v , s
which tends to zero since P (τ ) → P as τ → 0, D[H 1/2 P (τ )] ⊃ D[H 1/2 P ] and H 1/2 P (τ )v → H 1/2 P v for v ∈ D[H 1/2 P ] by assumption1 . Hence H 1/2 P uˆ(t) ∈ D[H 1/2 P ] and Pf
= =
Pu ˆ(t) + t1/2 (H 1/2 P )∗ h(t) = uˆ(t) + t(H 1/2 P )∗ (H 1/2 P )ˆ u(t) ˆ(t) , (4.5) u ˆ(t) + tHP u
because D[H 1/2 P ] is supposed to be dense. Applying once again the standard argument mentioned after Lemma 3.1 to all the three families we conclude that the weak convergence in (4.4) takes place independently of a sequence {τn (t)} chosen. On the other hand, we infer from (4.3) that ˆ u(t), f
≥ lim inf ˆ u(t, τ )2 + lim inf τ −1 Q(τ )ˆ u(t, τ )2 + lim inf t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ )2 ≥ ˆ u(t)2 + g0 (t)2 + h(t)2 = ˆ u(t)2 + t1/2 H 1/2 P u ˆ(t)2 1/2
= ˆ u(t)2 + tHP u ˆ(t)2 with lim inf taken along τ → 0. Since by (4.5) the left-hand side of the above inequality is equal to 1/2
u(t), HP u ˆ(t) = ˆ u(t)2 + tHP u ˆ(t)2 , ˆ u(t), f = ˆ u(t), P f = ˆ u(t)2 + tˆ we see that the norms of these vectors converge to the norms of their limit vectors. It allows us to conclude that the H-valued families in question, {ˆ u(t, τ )}, {τ −1/2 Q(τ )ˆ u(t, τ )} and {t1/2 H(tτ )1/2 P (τ )ˆ u(t, τ )} converge to u ˆ(t), 0 and 1 This part of the proof shows that the hypotheses of Theorem 2.1 can be slightly weakened, because we need in fact only that s − limτ →0 H(tτ )1/2 P (τ )v = H 1/2 P v holds for any v ∈ D[H 1/2 P ].
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t1/2 H 1/2 P uˆ(t) strongly in H, respectively, as τ → 0. In particular, we have shown s u(t) and u ˆ(t, τ ) −→ u ˆ(t) = (I + tHP )−1 P f , or (4.1). This that P f = (I + tHP )ˆ proves Lemma 4.1. II. Next, for a fixed τ > 0, the function ζ → F (ζ, τ ) is holomorphic in the open right half-plane Re ζ > 0 and uniformly bounded in norm by one. This makes it possible to mimic the argument of Feldman [Fe], which is reproduced in Chernoff’s book [Ch2, p. 90], see also [Fr], to conclude by means of the Vitali theorem (see, e.g., [HP, Thm 3.14.1]) that for Re ζ > 0 (I + S(ζ, τ ))−1 −→ (I + ζHP )−1 P s
as τ → 0
(4.6)
holds uniformly on compact subsets of Re ζ > 0. At the boundary Re ζ = 0, or ζ = it with t real, (I + S(ζ, τ ))−1 still converges as τ → 0 but in a weaker sense only. Using the argument of [Fe] based on the Poisson kernel, we can check that for each pair of f, g ∈ H and all φ ∈ L1 (R) the following relation is valid, φ(t)g, (I + S(it, τ ))−1 f dt = φ(t)g, (I + itHP )−1 P f dt . (4.7) s − lim τ →0
R
R
This says that for each pair of f, g ∈ H the family {g, (I + S(it, τ ))−1 f } of functions of t in L∞ (R) converges to g, (I + itHP )−1 P f as τ → 0 weakly∗ , or equivalently, in the weak topology defined by the dual pairing between L∞ (R) and L1 (R) – see, e.g., [K¨ o]. III. Now we shall show the family of the bounded operators {(I + S(it, τ ))−1 } is weakly convergent in L2loc ([0, ∞); H), and in fact, strongly convergent there too. To do so, we will employ an argument analogous to that used in the proof of Lemma 4.1 on the Hilbert space H, however, this time on the Fr´echet space L2loc ([0, ∞); H). Using the decomposition (3.10) with t = 0, we find (cf. [Ich]) S(it, τ ) = τ −1 [I − P (τ )(I − itτ H(itτ ))P (τ )] = τ −1 Q(τ ) + tP (τ )(B(tτ ) + iA(tτ ))P (τ ) . To prove (3.4), take any f ∈ H and put u(t, τ ) := (I + S(it, τ ))−1 f. Note that this u(t, τ ) represents an element in L2loc ([0, ∞); H) as well as its unique representative in (0, ∞), because u(t, τ ) is strongly continuous at this interval as a function of t. Then f = (I + S(it, τ ))u(t, τ ) = [I + τ
−1
(4.8)
Q(τ ) + tP (τ )(B(tτ ) + iA(tτ ))P (τ )]u(t, τ ) ,
so we have u(t, τ ), f = u(t, τ ), (I + S(it, τ ))u(t, τ ) = u(t, τ )2 + τ −1 Q(τ )u(t, τ )2 + tB(tτ )1/2 P (τ )u(t, τ )2 +itP (τ )u(t, τ ), A(tτ )P (τ )u(t, τ ) .
(4.9)
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Observing the real part of (4.9) we see that for τ small enough, each of the H-valued families {u(t, τ )}, {τ −1/2 Q(τ )u(t, τ )} and {t1/2 B(tτ )1/2 P (τ )u(t, τ )} is bounded by f for all t > 0. Moreover, they are strongly continuous in t for fixed τ > 0, and locally bounded as H-valued functions of t in L2loc ([0, ∞); H), uniformly as τ → 0. s Hence we infer first of all that Q(τ )u(t, τ ) −→ 0, uniformly in t ∈ (0, ∞), as 2 τ → 0. Next, since Lloc ([0, ∞); H) is reflexive [GV, Chap. 1, Sec. 3.1, pp. 57-62], any bounded set in it is weakly compact [K¨ o, Sec. 23.5, pp. 302-304]. Consequently, with τ → 0 as n → ∞ along which the above families there is a sequence {τn }∞ n n=1 are weakly convergent in L2loc ([0, ∞); H): w
u(t, τ ) −→ u(t) ,
τ −1/2 Q(τ )u(t, τ ) −→ f0 (t) , w
t1/2 B(t, τ )1/2 P (τ )u(t, τ ) −→ z(t) , w
(4.10)
with some vectors u(·), f0 (·) and z(·) ∈ L2loc ([0, ∞); H). Note that as before the sequence {τn }∞ n=1 can be chosen the same for all three families. Lemma 4.2 These above mentioned vectors have the following properties, u(t) = P u(t) ∈ P H for a.e. t ,
z(·) = 0 ,
f0 (·) = 0 .
Proof. For B(tτ ) and A(tτ ) in (3.10), the spectral theorem gives ∞ 1 − cos tτ λ 1/2 2 E(dλ)H 1/2 v2 → 0 , v ∈ D[H 1/2 ] ; (B(tτ ) v = tτ λ 0− ∞ 1 − cos tτ λ 2 E(dλ)Hv2 → 0 , v ∈ D[H] (4.11) (B(tτ )v2 = tτ λ 0− as τ → 0 by the Lebesgue dominated-convergence theorem. s s Since Q(τ )u(t, τ ) −→ 0 uniformly in t ∈ (0, ∞) and Q(τ ) −→ Q as τ → 0, we have Qu(t) = 0, or in other words u(t) = P u(t) ∈ P H for a.e. t. Moreover, by (4.11) we infer that ∞ ∞ φ(t)v, z(t) dt = lim φ(t)v, t1/2 B(tτ )1/2 P (τ )u(t, τ ) dt 0 0 ∞ 1/2 ¯ B(tτ )1/2 v, P (τ )u(t, τ ) dt φ(t)t = lim 0 ∞ ¯ φ(t)0, P u(t) dt = 0 = 0
holds for every φ ∈ C0∞ ([0, ∞)) and v ∈ D[H 1/2 ], hence z(t) = 0 a.e. because D[H 1/2 ] is dense in H, so that z(·) is the zero element of L2loc ([0, ∞); H). Finally, the relation f0 (·) = 0 follows from (4.8) which implies τ 1/2 Q(τ )f = τ 1/2 (1 + τ −1 )Q(τ )u(t, τ ), yielding the result; this concludes the proof.
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Our next aim is to show that the weak limits in (4.10) do not depend upon a sequence chosen. The u(·, τn ) = (I + S(it, τn ))−1 f converge to u = u(·) weakly in L2loc ([0, ∞); H) as n → ∞. It obviously implies that for all φ ∈ C0∞ ([0, ∞)) and for every g ∈ H we have ∞ ∞ −1 φ(t)g, (I + S(it, τ )) f dt −→ φ(t)g, u(t) dt , 0
0
again along the sequence {τn }. It follows from (4.7) that u(t) = (I + itHP )−1 P f,
for a.e. t in [0, ∞),
(4.12)
¯ because the set of all such φ(·)g is total in L2loc ([0, ∞); H). This shows that for every −1 f ∈ H, (I + S(it, τn )) f converges to (I + itHP )−1 P f weakly in L2loc ([0, ∞); H) as n → ∞. Together with the fact that z(·) = 0, f0 (·) = 0 in view of Lemma 4.2, this yields the desired property, namely that the weak limits of (4.10) are independent of the particular subsequence {τn } chosen. The standard argument sketched below Lemma 3.1 shows that (4.10) holds as τ → 0 without any restriction on subsequences. s
Finally, we are going to check the strong convergence u(·, τ ) −→ u(·) in L2loc ([0, ∞); H) as τ → 0. In fact, we will prove two other limiting relations at the same time. Lemma 4.3 In the topology of L2loc ([0, ∞); H), the family {u(·, τ )} converges to the vector u = u(·) as τ → 0, and moreover, τ −1/2 Q(τ )u(t, τ ) −→ f0 (t) = 0 , t1/2 B(t, τ )1/2 P (τ )u(t, τ ) −→ z(t) = 0 . Proof. In the above reasoning we have already checked the weak convergence in (4.10) as τ → 0. Integrating the real part of (4.9) in t over the interval [0, T ] for any fixed T > 0 and taking lim inf as τ → 0, we get by Lemma 4.2 T T u(t), f dt ≥ lim inf u(t, τ )2 dt Re 0
0
T
τ −1 Q(τ )u(t, τ )2 dt
+ lim inf 0
T
+ lim inf 0
≥ ≥
T 0 T 0
u(t)2 dt + u(t)2 dt .
t1/2 B(tτ )1/2 P (τ )u(t, τ )2 dt
T 0
f0 (t)2 dt +
0
T
z(t)2 dt
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On the other hand, the left-hand side of the above inequality is by (4.12) equal to Re 0
T
u(t), f dt = Re
0
T
u(t), (I + itHP )u(t) dt =
0
T
u(t)2 dt.
Hence we conclude that all the Fr´echet-space semi-norms of the vectors u(t, τ ), τ −1/2 Q(τ )u(t, τ ) and t1/2 B(t, τ )1/2 P (τ )u(t, τ ) converge to the semi-norms of the weak-limit vectors u(t), 0 and 0, respectively, as τ → 0. Thus the convergence is strong with respect to each semi-norm, and since their family induces the topology in L2loc ([0, ∞); H) the lemma is proved. This completes the proof of Lemma 3.1, and by that the verification of our main result, Theorem 2.1.
5 The finite-dimensional case In this section, we will prove Theorem 2.4 in which we assume that P and P (t) are finite-dimensional orthogonal projections. Since the closed operator H 1/2 P is supposed to be densely defined, the domain D[H 1/2 P ] of H 1/2 P becomes the whole space H, for the restriction H 1/2 P |P H of the operator H 1/2 P to the finitedimensional subspace P H is densely defined, so its domain must coincide with P H, and it acts as zero on QH. The same is valid for H 1/2 P (t) when P (t) is of a finite dimension. As a result, H 1/2 P and HP = (H 1/2 P )∗ (H 1/2 P ) as well as H 1/2 P (t) are bounded operators on H by the closed-graph theorem. By the assumptions common with Theorem 2.1, for each fixed f ∈ D[H 1/2 P ] = H the family {H 1/2 P (t)f } converges to H 1/2 P f as t → 0, and hence is uniformly bounded with respect to t near to zero, say, for −1 ≤ t ≤ 1. Then by the uniform boundedness principle we can conclude that sup|t|≤1 H 1/2 P (t) < ∞. To prove the assertions (i) and (ii) of Theorem 2.4 simultaneously, take a fixed a ∈ R and consider instead of F (ζ, τ ), S(ζ, τ ) defined by (3.1) and (3.2), respectively, the following operators Fa (ζ, τ ) := P (aτ ) exp(−ζτ H)P (aτ ),
Sa (ζ, τ ) := τ −1 [I − Fa (ζ, τ )].
In fact, we shall employ Fa (it, τ ), Sa (it, τ ) instead of F (it, τ ), S(it, τ ) in the proof of Lemma 3.2 and Lemma 3.1. Similarly u(t, τ ) used above will be replaced by ua (t, τ ) = (I + Sa (it, τ ))−1 f corresponding to a given f ∈ H. Lemma 5.1 For any t, t ≥ 0 and 0 < τ ≤ 1 we have ua (t, τ ) − ua (t , τ ) ≤ C(a)|t − t | f with a positive C(a) independent of t, t , which is uniformly bounded as a function of a on each compact interval of R.
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Proof. By the resolvent equation we have ua (t, τ ) − ua (t , τ ) = (I + Sa (it, τ ))−1 f − (I + Sa (it , τ ))−1 f
= (I + Sa (it, τ ))−1 P (aτ )τ −1 [e−itτ H − e−it τ H ]P (aτ )(I + Sa (it , τ ))−1 f 1 t d −isτ H = (I + Sa (it, τ ))−1 P (aτ ) ds P (aτ )(I + Sa (it , τ ))−1 f e τ t ds t He−isτ H ds P (aτ )(I + Sa (it , τ ))−1 f = −i(I + Sa (it, τ ))−1 P (aτ ) t
−1
1/2
= −i(I + Sa (it, τ )) (H P (aτ ))∗ t × e−isτ H ds (H 1/2 P (aτ ))(I + Sa (it , τ ))−1 f . t
At the beginning of this section we have argued that the operators H 1/2 P (aτ ) are uniformly bounded on H for 0 < τ ≤ 1. It follows that ua (t, τ ) − ua (t , τ ) ≤ C(a)|t − t | f with C(a) := sup|aτ |≤1 H 1/2 P (aτ )2 . By the argument preceding the lemma the function C(·) is uniformly bounded on each compact a-interval in R; this yields the claim. Proof of Theorem 2.4. It follows from the lemma that the vector family {ua (t, τ )}, continuous in H, is uniformly bounded and equicontinuous. Hence we may infer by the Ascoli–Arzel`a theorem that the sequence {τn } used in part III of the proof of Lemma 3.1 can chosen to have an additional property, namely that the sequence {ua (t, τn )} converges strongly to u(t) also pointwise, uniformly on [0, ∞). Then the limit u(t) becomes strongly continuous in t ≥ 0, and coincides with (I + itHP )−1 f for all t ≥ 0. Thus we have instead of Lemma 3.1 the following claim: (I + Sa (it, τ ))−1 −→ (I + itHP )−1 P
(5.1)
as τ → 0, strongly on H and uniformly on each compact interval of the variable t in [0, ∞). Next we will modify the reasoning of Sec. 3 based on [Ch2, Theorem 1.1] with the aim to show the symmetric product case, s
[P (at/n) exp(−itH/n)P (at/n)]n −→ exp(−itHP )P,
n → ∞.
(5.2)
Let f ∈ H. The resolvent convergence (5.1) with t = 1 implies the convergence of the corresponding semigroups, so we have e−θSa (i,τ ) f −→ e−iθHP f s
(5.3)
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in H as τ → 0, uniformly on each compact interval of the variable θ ≥ 0. Using this equivalence once more we get for any λ ≥ 0 the relation (I + λSa (i, τ ))−1 f −→ (I + iλHP )−1 P f , s
τ → 0.
In particular, taking τ = θ/n and using the diagonal trick, we infer that θ s (I + √ Sa (i, θ/n))−1 f −→ P f , n
n → ∞,
(5.4)
holds uniformly on each compact θ-interval in [0, ∞). Then the mentioned lemma from [Ch1] yields √ Fa (i, θ/n)n g − e−n(I−Fa (i,θ/n)) g ≤ n (I − Fa (i, θ/n))g . Choosing again g = I +
−1
√θ Sa (i, θ/n) n
f we find that
−1
θ n −θSa (i,θ/n) f 1 + √ Sa (i, θ/n) Fa (i, θ/n) − e n
−1 θ ≤ I + √ Sa (i, θ/n) f − f , n where the right-hand side tends to zero as n → ∞ by (5.4). Using the last named convergence once more we get (5.5) Fa (i, θ/n)n f − e−θSa (i,θ/n) f −→ 0 uniformly on each compact θ-interval in [0, ∞). Choosing now θ = t we see that the validity of (5.2) on P H follows immediately from (5.3) and (5.5). Consequently, on the subspace P H the assertion (i) is obtained by taking a = 1/t for any t belonging to a compact interval in R \ {0} and (ii) by choosing simply a = 1. The case f ∈ QH can be treated as in the proof of Lemma 3.2; together this yields the relation (5.2) on H, i.e., the symmetric product case. The non-symmetric product cases can also be checked with the help of Lemma 3.3 – cf. part (b) of the proof of Theorem 2.1 in Section 3. This concludes the proof of Theorem 2.4.
6 An example As we have said, our investigation was motivated by the result by Facchi et al. [FPS] mentioned in the introduction. Let us thus look how the result looks in this case. To see this, consider an open domain Ω ⊂ Rd with a smooth boundary, and denote by P the orthogonal projection on L2 (Rd ) defined as the multiplication operator by the indicator function χΩ of the set Ω. Consider further the free
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quantum Hamiltonian H := −∆, i.e., the Laplacian in Rd which is a nonnegative self-adjoint operator in L2 (Rd ), and the Dirichlet Laplacian −∆Ω in L2 (Ω) defined in the usual way [RS, Sec. XIII.15] as the Friedrichs extension of the appropriate quadratic form. We consider the Zeno dynamics in the subspace L2 (Ω) corresponding to a permanent reduction of the wavefunction to the region Ω, which may be identified with the volume of a detector. In the sense of the L2loc (R; L2 (Rd )) topology, which is physically plausible as explained in Remark 2.5, we then claim that the generator of the dynamics in L2 (Ω) is just the appropriate Dirichlet Laplacian, (P e−it(−∆/n) P )n → e−it(−∆Ω ) P
(6.1)
as n → ∞, or in other words: Proposition 6.1 The self-adjoint operator −∆P = ((−∆)1/2 P )∗ ((−∆)1/2 P )
(6.2)
is densely defined in L2 (Rd ) and its restriction to the subspace L2 (Ω) is nothing but the Dirichlet Laplacian −∆Ω of the region Ω, with the domain D[−∆Ω ] = W01 (Ω) ∩ W 2 (Ω). Proof. Let u ∈ D[−∆P ], so that u and −∆P u belong to L2 (Rd ). We have −∆P u, ϕ = u, −∆ϕ = −∆u, ϕ, for any ϕ ∈ C0∞ (Ω) because ϕ has a compact support in Ω. Thus −∆P u = −∆u holds in Ω in the sense of distributions, which means that ∆u|Ω ∈ L2 (Ω). On the other hand, since (−∆)1/2 P u ∈ L2 (Rd ), we have χΩ u ∈ W 1 (Rd ). Since we have ∇(χΩ u) = ∇((χΩ )2 u) = (∇χΩ )χΩ u(x) + χΩ ∇(χΩ u), in order to belong to L2 (Rd ) the function ∇(χΩ u) must not contain the δ-type singular term, which requires u(·) = 0 on the boundary of Ω. This combined with the fact that u|Ω , ∆u|Ω ∈ L2 (Ω) – see, e.g., [LM, Thm 5.4] – implies that u|Ω belongs to W 2 (Ω) and W01 (Ω). Thus we have shown that u|Ω ∈ D[−∆Ω ] and (−∆P u)|Ω = −∆Ω (u|Ω ) or −∆Ω ⊃ −∆P |L2 (Ω) , but both operators are self-adjoint, so they coincide. In this sense therefore our result given in Theorem 2.1 provides one possible abstract version of the result by Facchi et al. [FPS].
Acknowledgments P.E. and T.I. are respectively grateful for the hospitality extended to them at Kanazawa University and at the Nuclear Physics Institute, AS CR, where parts of this work were done. We thank the referee who spotted a logical gap in the
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first version of the paper. The research has been partially supported by ASCR and Czech Ministry of Education under the contracts K1010104 and ME482, and by the Grant-in-Aid for Scientific Research (B) No. 13440044 and No. 16340038, Japan Society for the Promotion of Science.
References [AS]
M.S. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York 1965.
[BN]
J. Beskow and J. Nilsson, The concept of wave function and the irreducible representations of the Poincar´e group, II. Unstable systems and the exponential decay law, Arkiv Fys. 34, 561–569 (1967).
[Ch1] P.R. Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal. 2, 238–242 (1968). [Ch2] P.R. Chernoff, Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators, Mem. Amer. Math. Soc. 140; Providence, R.I. 1974. [Ex]
P. Exner, Open Quantum Systems and Feynman Integrals, D. Reidel Publ. Co., Dordrecht 1985.
[EI]
P. Exner and T. Ichinose, Product formula for quantum Zeno dynamics, to appear in Proceedings of the XIV International Congress of Mathematical Physics (M ∩ Φ), Lisbon, July 28–Aug 2, 2003.
[FPS] P. Facchi, S. Pascazio, A. Scardicchio, and L.S. Schulman, Zeno dynamics yields ordinary constraints, Phys. Rev. A 65, 012108 (2002). [Fe]
J. Feldman, On the Schr¨ odinger and heat equations for nonnegative potentials, Trans. Amer. Math. Soc. 108, 251–264 (1963).
[Fr]
C. Friedman, Semigroup product formulas, compressions, and continual observations in quantum mechanics, Indiana Math. J. 21, 1001–1011 (1971/72).
[GV] O.M. Gel’fand and N.Y. Vilenkin, Generalized Functions, IV. Applications of Harmonic Analysis, Academic Press, New York 1965. [HP]
E. Hille and R.S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloquium Publ. No. 31, rev. ed., Providence, R. I. 1957.
[Ich]
T. Ichinose, A product formula and its application to the Schr¨ odinger equation, Publ. RIMS Kyoto Univ. 16, 585–600 (1980).
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[Ka1] T. Kato, Perturbation Theory for Linear Operators, Springer, BerlinHeidelberg-New York 1966. [Ka2] T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in Topics in Functional Analysis (I. Gohberg and M. Kac, eds.), Academic Press, New York 1978; pp.185–195. [K¨ o]
G. K¨ othe, Topological Vector Spaces I, Springer, Berlin-Heidelberg-New York 1969.
[LM] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Springer, Berlin-Heidelberg-New York 1972. [MaS] M. Matolcsi and R. Shvidkoy, Trotter’s product formula for projections, Arch. der Math. 81, 309–317 (2003). [MS]
B. Misra and E.C.G. Sudarshan, The Zeno’s paradox in quantum theory, J. Math. Phys. 18, 756–763 (1977).
[RS]
M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, New York 1978.
Pavel Exner Department of Theoretical Physics Nuclear Physics Institute Academy of Sciences ˇ z 25068 Reˇ Czech Republic and Doppler Institute Czech Technical University Bˇrehov´ a7 11519 Prague Czech Republic email:
[email protected] Takashi Ichinose Department of Mathematics Faculty of Science Kanazawa University Kanazawa 920-1192 Japan email:
[email protected] Communicated by Gian Michele Graf submitted 21/06/04, accepted 12/10/04
Ann. Henri Poincar´e 6 (2005) 217 – 246 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020217-30 DOI 10.1007/s00023-005-0204-1
Annales Henri Poincar´ e
Precise Coupling Terms in Adiabatic Quantum Evolution Volker Betz and Stefan Teufel Abstract. It is known that for multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. For a family of two-state systems with real-symmetric Hamiltonian we construct such a superadiabatic representation and explicitly determine the asymptotic behavior of the exponentially small coupling term. First order perturbation theory in the superadiabatic representation then allows us to describe the time-development of exponentially small adiabatic transitions. The latter result rigorously confirms the predictions of Sir Michael Berry for our family of Hamiltonians and slightly generalizes a recent mathematical result of George Hagedorn and Alain Joye.
1 Introduction and main result The decoupling of slow and fast degrees of freedom in the adiabatic limit is at the basis of many important approximations in physics, as, e.g., the BornOppenheimer approximation in molecular dynamics and the Peierls substitution in solid state physics. We refer to [BMKNZ, Te] for recent reviews. Generically the decoupling is not exact and a coupling which is exponentially small in the adiabatic parameter remains. However, this small coupling has important physical consequences, as it makes possible, e.g., non-radiative decay to the ground state in molecules. Since Kato’s proof from 1950 [Ka] the adiabatic limit of quantum mechanics was considered also as a mathematical problem, with increased activity during the last 20 years. Some of the landmarks are [Ne1 , ASY, JoPf1 , Ne2 , HaJo]. We consider a two-state time-dependent quantum system described by the Schr¨ odinger equation iε∂t − H(t) ψ(t) = 0 (1) in the adiabatic limit ε → 0. For the moment we take the Hamiltonian H(t) to be the real-symmetric 2 × 2-matrix cos θ(t) sin θ(t) H(t) = ρ(t) . (2) sin θ(t) −cos θ(t) The eigenvalues of H(t) are ±ρ(t) and we assume that the gap between them does not close, i.e., that 2ρ(t) ≥ g > 0 for all t ∈ R. As to be explained, even for this simple but prototypic problem there are open mathematical questions. In order to explain the concern of our work, namely the time-development of the exponentially small adiabatic transitions, let us briefly
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recall some important facts about (1). Let U0 (t) be the orthogonal matrix that diagonalizes H(t), i.e., cos(θ(t)/2) sin(θ(t)/2) U0 (t) = . (3) sin(θ(t)/2) − cos(θ(t)/2) Then the Schr¨ odinger equation in the adiabatic representation becomes U0 (t) iε∂t − H(t) U0∗ (t) U0 (t)ψ(t) =: iε∂t − Hεa (t) ψ a (t) = 0
with Hεa (t) =
ρ(t)
iε 2 θ (t)
− iε2 θ (t)
−ρ(t)
and
ψ a (t) = U0 (t)ψ(t) .
Here and henceforth, primes denote time derivatives. First-order perturbation theory in the adiabatic representation (cf. proof of Corollary 1) and integration by parts yields the adiabatic theorem [BoFo, Ka]: The off-diagonal elements of the unitary propagator K a (t, s) in the adiabatic basis, i.e., the solution of iε∂t Kεa (t, s) = Hεa (t)Kεa (t, s) ,
Kεa (s, s) = id ,
vanish in the limit ε → 0. More precisely, let 1 0 0 0 P+ = , P− = , 0 0 0 1
(4)
which project onto the adiabatic subspaces in the adiabatic representation. Then P− Kεa (t, s) P+ = O(ε) .
(5)
Therefore the transitions between the adiabatic subspaces are O(ε). This bound is optimal in the sense that in regions where θ(t) is not constant the leading order term in the asymptotic expansion of P− Kεa (t, s) P+ in powers of ε is proportional to ε. However, if limt→±∞ θ (t) = 0 then in the scattering limit the transitions between the adiabatic subspaces are much smaller: if the derivatives of θ ∈ C ∞ (R) decay sufficiently fast, then for any n ∈ N A(ε) := lim P− Kεa (t, −t) P+ = O(εn ) . t→∞
(6)
If θ is analytic in a suitable neighborhood of the real axis, then transition amplitudes are even exponentially small, A(ε) = O(e−c/ε ) for some constant c depending on the width of the strip of analyticity, see [JoPf1 , Ma]. It is well understood, see [Le, Ga, Ne1 ], how to reconcile the apparent contrariety between the smallness of the final amplitudes in (6) and the optimality of (5): the adiabatic basis is not the optimal basis for monitoring the transition
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process. For any n ∈ N there exist unitary transformations Uεn (t) such that the Hamiltonian in this nth superadiabatic representation takes the form n ρε (t) cnε (t) n Hε (t) = with ρnε (t) = ρ(t) + O(ε2 ) and |cnε (t)| = O(εn+1 ) . cnε (t) −ρnε (t) (7) In the nth superadiabatic basis the off-diagonal components of the propagator and hence also the transitions are of order O(εn ), i.e., there are constants Cn such that P− Kεn (t, s) P+ ≤ Cn εn .
(8)
In the scattering regime, where θ(t) becomes constant, the superadiabatic bases agree with the adiabatic basis, i.e., limt→±∞ Uεn (t) = U0 (t), and therefore the bound in (8) basically yields (6). Typically limn→∞ Cn εn = ∞ for all ε > 0, i.e., choosing n larger while keeping ε fixed does not necessarily decrease the bound in (8). However, one can choose nε = n(ε) in such a way that Cnε εnε is minimal. If θ is analytic, one obtains the improved estimate P− Kεnε (t, s) P+ = O(e−c/ε ) in the optimal superadiabatic basis nε , see [Ne2 , JoPf2 ]. More interesting than bounds on A(ε) is its actual value. Since A(ε) is asymptotically smaller than any power of ε, this question is beyond standard perturbation theory. For the case of analytic coupling θ, asymptotic formulas of the type tc
A(ε) = C e− ε (1 + O(ε))
(9)
have been established, see, e.g., [JKP, Jo], where the constants C and tc depend on the type and location of the complex singularities of θ (t)/ρ(t). These results are obtained by solving (1) along a certain Stokes line in the complex plane except near the singular points, where a comparison equation is solved. As a consequence, the method gives no information at all about the way in which the exponentially small final transition amplitude A(ε) is built up in real time. This question of adiabatic transition histories is the concern of our paper. Berry [Be] and, in a refined way Berry and Lim [BeLi, LiBe], gave an answer on a non-rigorous level and explicitly left a mathematically rigorous treatment as an interesting open problem. Only very recently Hagedorn and Joye [HaJo] succeeded and confirmed Berry’s results rigorously for a specific Hamiltonian. Although our work has been strongly motivated by the findings of Berry, our approach is slightly different. Let us first state our main result before we discuss its relation to the earlier ones. Without loss of generality we assume that ρ(t) ≡ 12 . It was observed in [Be] that this can always be achieved by transforming (1) to the natural time scale t
τ (t) = 2 0
(s) ds .
(10)
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However, we can only treat a rather special class of Hamiltonians, since we must assume that in the natural time scale the coupling has the form 1 1 γtc θ (t) = iγ − (11) = 2 t + itc t − itc t + t2c with γ ∈ R and tc > 0. In other words we assume t 1 cos θ(t) sin θ(t) H(t) = . (12) with θ(t) = 2 γ arctan sin θ(t) − cos θ(t) 2 tc We shall comment below on the meaning of this special choice and remark here that the Hamiltonian in [HaJo] is (12) with γ = 12 . Our main result is the construction of an optimal superadiabatic basis in which the coupling term in the Hamiltonian is exponentially small and can be computed explicitly at leading order. This optimal basis is given as the nth ε superadiabatic basis where 0 ≤ σε < 2 is such that nε =
tc − 1 + σε ε
is an even integer.
(13)
Theorem 1. Let H(t) be as in (12) and nε as in (13), and let ε0 > 0 be sufficiently small. Then for every ε ∈ (0, ε0 ] one can construct a family of unitary matrices Uεnε (t) ∈ C2×2 , depending smoothly on t ∈ R, such that ε nε Uε (t) − U0 (t) = O (14) 1 + t2 and
n ρε ε (t) cnε ε (t) . Uεnε (t) iε∂t − H(t) Uεnε ∗ (t) = iε∂t − cnε ε (t) −ρnε ε (t)
=: Hεnε (t)
Here ρnε ε (t) = and for every α < cnε ε (t) with
= 2i
1 +O 2
ε2 1 + t2
(15)
,
3 2
πγ tc t2 2ε e− ε e− 2εtc cos sin πtc 2
t t3 σε t − + ε 3εt2c tc
t t2 α ε exp − εc 1 + 4t2c φα (ε, t) = 1 tc ln 2 1 + exp − ε 2 1 + t2
+ O (φα (ε, t)) , (16)
if |t| < tc , (17) if |t| ≥ tc .
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√ Remark 1. The explicit term in cnε ε is of order O(e−tc /ε ) only for times |t| = O( ε). For larger times all terms in cnε ε are exponentially small compared to the leading c /ε exponential e−t√ . As a consequence, Taylor expansion of the cosine in cnε ε around t/ε for |t| = O( ε) shows that it can be replaced by cos(t/ε) at the cost of lowering α to α < 1: for every α < 1 πγ tc t2 t 2ε nε e− ε e− 2εtc cos sin cε (t) = 2i + O (φα (ε, t)) . πtc 2 ε Remark 2. The slow time decay of the error in (17) for large times is due to the fact that nε is optimal for t near 0, but not for large t. Remark 3. Taking nε defined in (13) odd instead of even would yield slightly different off-diagonal elements in the effective Hamiltonian Hεnε (t). However, the resulting unitary propagator, cf. Corollary 1, would be the same at leading order. See the end of Section 5 for a discussion of this somewhat surprising fact. Let us shortly explain the idea of the proof of Theorem 1 and at the same time the structure of our paper. First we construct the nth order superadiabatic basis as in (7) in two steps: in Section 2 we construct the projectors on the superadiabatic basis vectors and in Section 3 we construct the unitary basis transformation Uεn (t). We cannot use existing results here, e.g., [Ga, Ne2 ], since we need to keep careful track of the exact form of the off-diagonal terms cnε (t) of the superadiabatic Hamiltonian, and since we aim at a scalar recurrence relation instead of a matrix recurrence relation for the cnε (t)’s. The main mathematical challenge is the asymptotic analysis of the resulting recurrence relation, which is done in Section 4. This is also the only part where we have to assume the special form (11) for θ . Theorem 1 then follows by choosing the order n of the superadiabatic basis as in (13), a choice which minimizes cnε (t) near t = 0. The details of this optimal truncation procedure and the proper proof of Theorem 1 are given in Section 5. Finally in Section 6 we use first order perturbation theory in the optimal superadiabatic basis in order to obtain the following Corollary, in which we abbreviate ∆(t, s) := arctan(t) − arctan(s) . x 2 Also recall that erf: R → (−1, 1) with erf(x) = √2π 0 e−x dx switches smoothly and monotonically from erf(−∞) = −1 to erf(∞) = 1. Corollary 1. The unitary propagator in the optimal superadiabatic basis + kε (t, s) kε (t, s) , Kεnε (t, s) = k ε (t, s) kε− (t, s) i.e., the solution of iε∂t Kεnε (t, s) = Hεnε (t)Kεnε (t, s) ,
Kεnε (s, s) = id ,
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satisfies kε± (t, s) = e∓
i(t−s) 2ε
and kε (t, s) =
sin
πγ
tc
e− ε e−
i(t+s) 2ε
+ O(ε∆(t, s))
2 √ tc +O εe− ε ∆(t, s) .
Ann. Henri Poincar´e
erf
t √ 2εtc
(18)
− erf
√
s 2εtc
(19)
Outside the transition region, more precisely for |t| > εβ and |s| > εβ for some tc β < 12 , (19) holds with the error term replaced by O(εα e− ε ∆(t, s)) for every α < 1. Corollary 1 immediately implies the existence of solutions to (1) of the form it √ e− 2ε 1 + O(ε(arctan(t) + π2 )) tc ∗ πγ − tc it εe− ε . (20) ψ(t) = Uε (t) +O t sin 2 e ε e 2ε erf √2εtc + 1 They start at large negative times in the positive energy adiabatic subspace and smoothly and monotonically develop the √ exponentially small component in the negative energy adiabatic subspace in a ε-neighborhood of t = 0. Berry and Lim [Be, BeLi] argue that this behavior is universal: whenever θ has the form θ (t) =
±iγ + O(|t ± itc |α ) t ± itc
for some α > −1
near its singularities ±itc closest to the real axis,√then (20) should hold. For the Landau-Zener Hamiltonian (i.e., (2) with ρ = t2 + δ 2 and θ = arccot(t/δ)), which describes the generic situation, one finds after the transformation (10) that γ = 13 and α = − 31 . Hagedorn and Joye [HaJo] proved (20) for the Hamiltonian (12) with γ = 12 . In the approach of Berry and, slightly modified, of Hagedorn and Joye, the optimal superadiabatic basis vectors are obtained through optimal truncation of an asymptotic expansion of the true solution of (1) in powers of ε. In contrast, in our approach the optimal superadiabatic basis is constructed by approximately diagonalizing the Hamiltonian. The main advantage of “transforming the Hamiltonian” over “expanding the solutions” is that the former approach can be applied, at least heuristically, to more general adiabatic problems, cf. [Te], as for example the Born-Oppenheimer approximation. While we cannot control the asymptotics for the Born-Oppenheimer model rigorously yet, the heuristic application of the idea yields new physical insight into adiabatic transition histories and new expressions for the exponentially small off-diagonal elements of the S-matrix for simple Born-Oppenheimer type models, cf. [BeTe]. Therefore we see the rigorous results obtained in this paper also as a first attempt to justify the application of analogous ideas to more complicated but also more relevant systems. Furthermore, the concept of an adiabatically renormalized Hamiltonian was
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used to derive a criterion for selecting possible transition sequences in multi-level problems [WiMo]. For the specific problem (1) the knowledge of two linearly independent solutions is of course equivalent to the knowledge of the propagator and the effective Hamiltonian in the optimal superadiabatic basis. Therefore we shortly explain which aspects of our result constitute an improvement compared to [HaJo]: Most importantly, our proof does not rely on the a priori knowledge of the scattering amplitude A(ε). Indeed, our result yields for the first time a proof of (9) based on superadiabatic evolution, as expressed in Corollary 2. Moreover, we allow for a slightly larger class of Hamiltonians and obtain more detailed error estimates, which, in particular, give rise to close to optimal error bounds in the expansion of the S-matrix, cf. Corollary 2. Finally, we also get explicitly the next order correction in (19) resp. (20), cf. Section 6. It should be noted, however, that the improved error estimates and the next order corrections could have been obtained also based on the proof in [HaJo]. We finally turn to the scattering limit. Let Kε0 (t, s) denote the propagator in the original basis and define the scattering matrix in the adiabatic basis by 1 iH0 t iH0 t 0 a 0 ∗ 2 ε ε Sε := lim e U0 (t) Kε (t, −t) U0 (−t) e , where H0 = . 0 − 12 t→∞ Since, according to (14), for large negative and positive times the optimal superadiabatic basis agrees with the adiabatic basis, Sεa can be computed with help of the optimal superadiabatic propagator from Corollary 1. Corollary 2. For β < 1 we have πγ − tc ε (1 + O(εβ )) e 1 + O(ε) 2 sin 2 − tc . Sεa = e ε (1 + O(εβ )) 1 + O(ε) 2 sin πγ 2 Proof. According to (14) we have Sεa = lim e t→∞
iH0 t ε
Uεnε (t) Kε0 (t, −t) Uεnε ∗ (−t) e
iH0 t ε
= lim e t→∞
iH0 t ε
Kεnε (t, −t) e
iH0 t ε
.
Now the claim follows from inserting (18) and (19) with the improved error estimate outside of the transition region. From Corollary 2 we conclude that the transition amplitude is given by πγ tc A(ε) = P− Sεa P+ = 2 sin e− ε 1 + O(εβ ) , for any β < 1 , 2 which agrees with the results of [Jo], as explained in [BeLi]. We conclude the introduction with two recommendations for further reading: The numerical results of Berry and Lim [LiBe] beautifully illustrate the idea of optimal superadiabatic bases and universal adiabatic transition histories. The
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introduction of the paper of Hagedorn and Joye [HaJo] gives a slightly different viewpoint on the problem and, in particular, a short discussion on how exponential asymptotics for the Schr¨odinger equation (1) fit into the broader field of exponential asymptotics for ordinary differential equations. Acknowledgments. We are grateful to Alain Joye and George Hagedorn for many helpful discussions.
2 Superadiabatic projections For the present and the following section we assume that H(t) has the form (12), but with some arbitrary θ ∈ C ∞ (R). The first aim is to construct time-dependent matrices π (n) ∈ R2×2 with (π (n) )2 − π (n) = O(εn+1 ), iε∂t − H, π (n) = O(εn+1 ).
(21) (22)
Here, [A, B] = AB−BA denotes the commutator two operators A and B. Likewise, we will later use [A, B]+ = AB + BA to denote the anti-commutator of A and B. Equation (21) says that π (n) is a projection up to errors of order εn+1 , while (22) implies that π (n) (t) is approximately equivariant, i.e., Kε0 (t, s) π (n) (s) = π (n) (t) Kε0 (t, s) + O(εn ) . Recall the Kε0 (t, s) is the unitary propagator for (1). Hence π (n) (t) is an almost projector onto an almost equivariant subspace. We construct π (n) inductively starting from the Ansatz π (n) =
n
πk εk .
(23)
k=0
By (12), H has two eigenvalues ±1/2. Let π0 be the projection onto the eigenspace corresponding to +1/2, and π (0) = π0 according to (23). It is easily checked that (21) and (22) are fulfilled for n = 0. In order to construct πn for n > 0, let us write Gn (t) for the term of order εn+1 in (21), i.e., (π (n) )2 − π (n) = εn+1 Gn+1 + O(εn+2 ) . Obviously, Gn+1 =
n
πj πn+1−j .
(24)
(25)
j=1
Proposition 1. Assume that π (n) given by (23) fulfills (21) and (22). Then a unique matrix πn+1 exists such that π (n+1) defined as in (23) fulfills (21) and (22). πn+1 is given by πn+1 = Gn+1 − π0 Gn+1 − Gn+1 π0 − i [πn , π0 ] . (26)
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Furthermore πn+1 is off-diagonal with respect to π0 , i.e., π0 = (1 − π0 )πn+1 (1 − π0 ) = 0, π0 πn+1
(27)
and Gn+1 is diagonal with respect to π0 , i.e., π0 Gn+1 (1 − π0 ) = (1 − π0 )Gn+1 π0 = 0.
(28)
Remark 4. The fact that the superadiabatic projections are unique answers the question raised in [Be] to which extent the superadiabatic basis constructed there is uniquely determined. Remark 5. Our construction can be seen as a special case of the construction in [EmWe], see also [Sj]. It was applied in the same context in [PST, Te]. The role and the importance of the superadiabatic subspaces as opposed to the superadiabatic evolution have been emphasized by Nenciu [Ne2 ]. He constructs the superadiabatic projections for much more general time-dependent Hamiltonians. However, Nenciu’s construction is less suitable for the explicit computations we need to perform. Proof. Let π (n+1) be given by (23) and suppose π (n) fulfills (21) and (22). Let π ˜n+1 be an arbitrary matrix, and define π ˜ (n+1) = π (n) + εn+1 π ˜n+1 . Then ˜ (n+1) , π ˜ (n+1) = (π (n) )2 − π (n) + εn+1 π ˜n+1 − π ˜n+1 . (˜ π (n+1) )2 − π +
Using (24), we see that terms of order εn+1 vanish if and only if Gn+1 = π ˜n+1 − [π0 , π ˜n+1 ]+ = (1 − π0 )˜ πn+1 (1 − π0 ) − π0 π ˜n+1 π0 .
(29)
Multiplying (29) with (1 − π0 ) and with π0 on both sides and subtracting the results, we find that π ˜n+1 must fulfill πn+1 (1 − π0 ) + π0 π ˜n+1 π0 = Gn+1 − [Gn+1 , π0 ]+ . (1 − π0 )˜
(30)
Similarly iε∂t − H, π ˜ (n+1) = iε∂t − H, π (n) + εn+1 [iε∂t − H, π ˜n+1 ] . Again terms of order εn+1 vanish if and only if ˜n+1 ] . iπn = [H, π
(31)
Since π0 is the projector onto the eigenspace of H, we have π0 H = Hπ0 = Eπ0 , where E = 1/2 is the positive eigenvalue of H, and similarly (1 − π0 )H = H(1 − π0 ) = −E(1 − π0 ). When we multiply (31) first with with π0 from the left and with
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1 − π0 from the right, then the other way round, and finally subtract the second result from the first, we get 2E(π0 π ˜n+1 (1 − π0 ) + (1 − π0 )˜ πn+1 π0 ) = −i [πn , π0 ] .
(32)
Now we divide (32) by 2E and add (30) to find π ˜n+1 = Gn+1 − [Gn+1 , π0 ]+ −
i [π , π0 ] . 2E n
(33)
˜ (n+1) should fulfill Thus π ˜n+1 is uniquely determined by the requirement that π (21) and (22). On the other hand, H, Gn+1 − [Gn+1 , π0 ]+ = 0 and π0 [πn , π0 ] π0 = (1 − π0 ) [πn , π0 ] (1 − π0 ) = 0 , and thus πn+1 given by the right-hand side of (33) indeed fulfills (30) and (31). This shows existence. (28) and (27) now follow directly from (29) and (31). The calculation of π (n) via the matrix recurrence relation (26) and (25) is now possible in principle, but extremely cumbersome. In order to make more explicit calculations possible, we introduce a special basis of R2×2 . Recall that U0 (t) as defined in (3) is the unitary transformation into the basis consisting of the eigenvectors of H, i.e., the adiabatic basis, and let V0 (t) = θ2(t) U0 (t). With P = P+ as in (4) we then have U02 = V02 = id and P U0 V0 P = P V0 U0 P = 0, and π0 = U0 P U0 . Moreover, since G1 = 0 by (25), (26) implies i π1 = − θ (V0 P U0 − U0 P V0 ). 2
(34)
Motivated by this, we put X = V0 P U0 − U0 P V0 , Z = V0 P U0 + U0 P V0 ,
Y = V0 P V0 − U0 P U0 , W = V0 P V0 + U0 P U0 .
It is immediate that this is a basis of R2×2 for all t, and in fact −1 0 −1 1 0 X= , W = , Y = −2H, Z = Y . 1 0 0 1 θ Our reason for representing X through Z via U0 and V0 is that the following important relations now follow without effort: X = 0, Y = −θ Z, Z = θ Y, [X, Y ]+ = [X, Z]+ = [Y, Z]+ = 0, −X 2 = Y 2 = Z 2 = W,
(35) (36)
[X, π0 ] = Z, [Y, π0 ] = 0, W − [W, π0 ]+ = Y.
(37) (38)
[Z, π0 ] = X,
These relations show that this basis behaves extremely well under the operations involved in the recursion (26). This enables us to obtain
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Proposition 2. For all n ∈ N, πn is of the form πn = xn X + yn Y + zn Z,
(39)
where the functions xn , yn and zn satisfy the differential equations xn yn
zn
= =
izn+1 , −θ zn ,
(40) (41)
=
ixn+1 + θ yn .
(42)
Moreover, i x1 (t) = − θ (t), 2
y1 (t) = z1 (t) = 0.
(43)
Remark 6. Hence, for all even n, xn = 0, while for all odd n, yn = zn = 0. Proof. (43) was already noticed in (34), or alternatively follows from π0 = (W − Y )/2, (35) and (37). Now suppose πn is given by (39). By (36) and (38), Gn+1 − [Gn+1 , π0 ]+ is proportional to Y with a prefactor given through (25), and by (26), (35) and (37), πn+1 =
n (−xj xn+1−j + yj yn+1−j + zj zn+1−j )Y + i(θ yn − zn )X − ixn Z. (44) j=1
Comparing with (39) shows (40) and (42). To show (41), we use (27). This gives 0 = π0 πn π0 = (yn + θ zn )π0 Y π0 + (zn − θ yn )π0 Zπ0 + xn π0 Xπ0 . Since π0 Zπ0 = π0 Xπ0 = 0 and π0 Y π0 = −π0 , the claim follows.
Remark 7. From (40) through (42) we may derive recursions for calculating xn or zn , e.g., d θ (t)zn (t) dt + C . (45) zn+2 (t) = − z (t) + θ (t) dt n The constant of integration C must (and in some cases can) be determined by comparison with (44). In the case where θ is given by (11), this strategy will lead to fairly explicit expressions of the coefficient functions xn , yn and zn , cf. Proposition 5; from these we will extract the asymptotic behavior of xn , yn and zn , cf. Theorem 3. Using (40)–(42), we can give very simple expressions for the quantities appearing in (21) and (22). As for (22), we use (35) and the differential equations to find iε∂t − H, π (n) = iεn+1 πn = iεn+1 (xn X + (yn + θ zn )Y + (zn − θ yn )Z) =
−εn+1 (zn+1 X + xn+1 Z).
(46)
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Now we turn to (π (n) )2 − π (n) , the term by which π (n) fails to be a projector. Let us write n (π (n) )2 − π (n) = εn+k Gn+1,k . (47) k=1
With our earlier convention, Gn+1,1 = Gn+1 . Explicitly, (23) and (47) give Gn+1,k = [πk , πn ]+ + [πk+1 , πn−1 ]+ + . . . =
n−k
πj+k πn−j .
(48)
j=0
Proposition 3. For each n ∈ N, there exist functions gn+1,k , k ≤ n with ((π
) −π
(n) 2
(n)
)(t) =
n
ε
n+k
gn+1,k (t) W.
(49)
k=1
For each k ≤ n, gn+1,k = 2i(xk zn+1 − zk xn+1 ).
Proof. By (36), each Gn+1,k is proportional to W . Using (39) additionally, we find [πk , πm ]+ = 2(−xk xm + yk ym + zk zm )W , and thus (48) yields gn+1,k =
n−k
−xj+k xn−j + yj+k yn−j + zj+k zn−j .
j=0
Thus by using Proposition 2, gn+1,k
=
n−k
i(zj+k+1 xn−j + xj+k zn−j+1 ) − (θ zj+k yn−j + θ yj+k zn−j )
j=0
+θyj+k zn−j + θ yj+k zn−j − i(xj+k+1 zn−j + zj+k xn−j+1 ) =
i
n−k
((zj+k+1 xn−j − zj+k xn−j+1 ) + (xj+k zn−j+1 − xj+k+1 zn−j )
j=0
=
2i(xk zn+1 − zk xn+1 ).
The last equality follows because the sum is a telescopic sum.
Since W = id is independent of t, Proposition 3 gives the derivative of the correction (π (n) )2 − π (n) to a projector. As above, this gives an easy way for estimating the correction itself provided we have some clue how to choose the constant of integration.
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3 Construction of the unitary We now proceed to construct the unitary transformation Uεn into the nth superadiabatic basis. By (23) and (26), π (n) is self-adjoint. Thus it has two orthonormal eigenvectors vn and wn . Let cos(θ/2) sin(θ/2) v0 = , w0 = sin(θ/2) − cos(θ/2) be the eigenvectors of π0 , and write vn = αv0 + βw0 ,
wn = αw0 − βv0
(α, β ∈ C).
(50)
We make this representation unique by requiring 0 ≤ α ∈ R. Let Uεn be the unitary operator taking (vn , wn ) to the standard basis (e1 , e2 ) of R2 , i.e., Uεn = e1 vn∗ + e2 wn∗ ,
(51)
where all vectors are column vectors. Note that the definition (3) of U0 is consistent with (51) for n = 0. Uεn diagonalizes π (n) , thus λ1 0 Uεn π (n) Uεn ∗ = D ≡ , (52) 0 λ2 where λ1,2 are the eigenvalues of π (n) . Although α, β and λ1,2 depend on n, ε and t, we suppress this from the notation. Lemma 1. U0 Uεn ∗ =
α −β β α
,
U0 Uεn ∗ =
and
α + β β − α
α−β α + β
.
Proof. The calculations are straightforward and we only show the second equality. First note that v0 = −w0 and w0 = v0 . Thus
Uεn ∗ = ((α + β)v0 + (β − α)w0 )e∗1 + ((α − β )v0 + (α + β)w0 )e∗2 , and using the orthogonality of v0 and w0 yields the claim,
U0 Uεn ∗ = e1 (α + β)e∗1 + e1 (α − β )e∗2 + e2 (β − α)e∗1 + e2 (α + β)e∗2 . It will turn out that β, α α, and β are small quantities, λ1 , λ2 , and λ2 are even much smaller, while α2 and λ1 are large, i.e., of order 1+O(ε). This motivates the form in which we present the following result.
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Proposition 4. Suppose λ1 = λ2 . Then for each n ∈ N, 2 Uεn (iε∂t −H)Uεn ∗
with R=
1 2
= iε∂t −
α2 εn+1 λ1 −λ2 (−xn+1
εIm(β(2α + β )) + |β|2 εn+1 β 2 λ1 −λ2 (xn+1
− zn+1 )
α εn+1 λ1 −λ2 (xn+1 − 12
− zn+1 ) n+1
Ann. Henri Poincar´e
− zn+1 )
2
− ελ1 −λβ2 (xn+1 + zn+1 )
−εIm(β(2α + β )) − |β|2
+R,
.
Proof. Let us write Uεn (iε∂t − H)Uεn ∗ = (Mi,j ), i, j ∈ {1, 2}. M1,1 and M2,2 are calculated in a straightforward manner, using Lemma 1 together with the fact 1/2 0 U0 HU0∗ = : 0 −1/2 Uεn (iε∂t − H)Uεn ∗
= =
iε∂t + iεUεn U0∗ U0 Uεn ∗ − Uεn U0∗ U0 HU0∗ U0 Uεn ∗ α β α + β α − β iε∂t + iε −β α β − α α + β 1 1 0 α −β α β − . 0 −1 −β α β α 2
Carrying out the matrix multiplication yields 1 M1,1 = −M2,2 = iε∂t + iε((α(α + β) + β(β − α)) − (α2 − |β|2 ). 2
(53)
We now use α2 + |β|2 = 1 to obtain 0 = 2αα + β β + β β = 2Re(αα + ββ ) and α2 − |β|2 = 1 − 2|β|2 . Plugging these into (53) gives the diagonal coefficients of M . Although we could get expressions for the off-diagonal coefficients by the same method, these would not be useful later on. Instead we use (52), i.e., Uεn ∗ D = π (n) Uεn ∗ together with (46) and obtain Uεn (iε∂t − H)Uεn ∗ D = DUεn (iε∂t − H)Uεn ∗ − εn+1 Uεn (zn+1 X + xn+1 Z)Uεn ∗ . (54) By multiplying (54) with ej e∗j from the left and by ek e∗k from the right (j, k ∈ {1, 2}) and rearranging, we obtain (λk − λj ) ej e∗j Uεn (iε∂t − H) Uεn ∗ ek e∗k
(55)
= −εn+1 ej e∗j Uεn (zn+1 X + xn+1 Z) Uεn ∗ ek e∗k − iδk,j ε λj ej e∗j .
0 1 0 −1 ∗ From the equalities = , U0 ZU0 = and Lemma 1 −1 0 −1 0 we obtain 2 2 α(β − β) α + β Uεn XUεn ∗ = , −(α2 + β 2 ) −α(β − β) U0 XU0∗
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Uεn ZUεn ∗
=
−α(β + β) −(α2 − β 2 )
2
−(α2 − β ) α(β + β)
231
.
The expressions for M1,2 and M2,1 follow by taking k = j in (55).
We now use our results from the previous section to express α, β and λ1,2 in terms of xk , yk and zk , k ≤ n. Let us define n (56) ξ ≡ ξ(n, ε, t) = k=1 εk xk (t), n k (57) η ≡ η(n, ε, t) = k=1 ε yk (t), n k ζ ≡ ζ(n, ε, t) = k=1 ε zk (t). (58) Moreover, let g ≡ g(n, ε, t) =
n
k=1 ε
n+k
gn+1,k (t)
(59)
be the quantity appearing in (49). Lemma 2. The eigenvalues of π (n) solve the quadratic equation λ21,2 − λ1,2 − g = 0. Proof. By (52) and Proposition 3 we obtain 2 g λ1 − λ1 0 n (n) 2 (n) n∗ n n∗ = U ((π ) − π ) U = U gW U = ε ε ε ε 0 λ22 − λ2 0
0 g
.
Lemma 3. α2 (λ1 − λ2 ) = 1 − η − λ2 , Proof. We have
and
αβ(λ1 − λ2 ) = −ξ − ζ.
π (n) = λ1 vn vn∗ + λ2 wn wn∗ .
(60)
Plugging in (50), we obtain π (n) v0
=
λ1 αvn − λ2 βwn = (λ1 α2 + λ2 |β|2 )v0 + (λ1 − λ2 )αβw0
=
(α2 (λ1 − λ2 ) + λ2 )v0 + (λ1 − λ2 )αβw0 .
In the last step, we used |β|2 + α2 = 1. On the other hand, from (23) and (26) we have n π (n) = π0 + εk (xk X + yk Y + zk Z) , (61) k=1
and since Xv0 = Zv0 = −w0 , π0 v0 = v0 and Y v0 = −v0 , we find π (n) v0 = (1 − η)v0 − (ξ + ζ)w0 . Comparing coefficients finishes the proof.
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Theorem 2. Let ε0 > 0 be sufficiently small. For ε ∈ (0, ε0 ] assume there is a bounded function q on R such that ξ(t), η(t), ζ(t) and their derivatives ξ (t), η (t), ζ (t) are all bounded in norm by εq(t). Then Uεn (iε∂t − H)Uεn ∗ = iε∂t −
1 2
+ O(ε2 q)
εn+1 (−xn+1 − zn+1 ) (1 + O(εq))
εn+1 (xn+1 − zn+1 ) (1 + O(εq)) − 21 + O(ε2 q)
(62) .
Proof. From (61) and our assumptions it follows that π (n) − π0 = O(εq). Thus λ1 = 1 + O(εq) and λ2 = O(εq), and from Lemma 2 we infer g = O(ε) and λ1 = 12 1 + 1 + 4g , λ2 = 12 1 − 1 + 4g . Since λ1 − λ2 = 0, Lemma 3 yields √ 1 + 1 + 4g − 2η √ , α2 = 2 1 + 4g
−ξ − ζ β= √ . 1 + 4gα
Hence α2 = 1 + O(εq), and β, β and αα = (α2 ) /2 are all O(εq). Plugging these into the matrix R in Proposition 4 shows the claim.
4 Solving the recursion: a pair of simple poles In order to make further progress, we need to understand the asymptotic behavior of the off-diagonal elements of the effective Hamiltonian in the nth superadiabatic basis for large n. According to (62) this amounts to the asymptotics of xn and zn as given by the recursion from Proposition 2. It is clear that the function θ alone determines the behavior of this recursion. We will study here the special case θ (t) =
iγ iγ γtc − = 2 . t + itc t − itc t + t2c
(63)
The reason lies in the intuition that the poles of θ closest to the real axis determine the superadiabatic transitions, and that these transitions are of universal form whenever these poles are of order one, see [Be, BeLi] for details. As in [HaJo], we have to restrict to the special case that θ has no contribution besides these poles in order to solve the recursion. We now have two parameters left in θ . The distance tc of the poles from the real axis determines the exponential decay rate in the offdiagonal elements of the Hamiltonian and the strength of the residue γ determines the pre-factor in front of the exponential. As is done in [HaJo], we could get rid of the parameter tc by rescaling time, but we choose not to do so because tc plays a nontrivial role in optimal truncation and the error bounds obtained therein, and keeping this parameter will make things more transparent.
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We use (45) in order to determine the asymptotics of zn . From Proposition 2 together with (44) it is clear that yn must be integrable on R. This fixes the constant of integration in (45), and we arrive at the linear two-step recursion zn+2 (t) = −
d dt
zn (t) + θ (t)
t −∞
θ (s)zn (s) ds .
(64)
The fact that the recursion is linear will make its analysis simpler than the one of the nonlinear recursion in [HaJo]. We rewrite θ as θ (t) =
γ (f + f ) tc
with
f (t) =
itc . t + itc
For zn , we will make an Ansatz as a sum of powers of f and f . The reason for the success of this approach is the fact that this representation is stable under differentiation and integration, and also under multiplication with θ through the partial fraction expansion. More explicitly, the following identities hold for the mth power f m of f : Lemma 4. For each m ≥ 1, θ Im(f m ) θ Re(f m ) Im(f m ) Re(f m ) Proof. We have f + f =
m−1 γ −k 2 Im(f m+1−k ), tc k=0 m−1 γ −k m+1−k −m = 2 Re(f )+2 θ , tc k=0 m = − Re(f m+1 ), tc m = Im(f m+1 ). tc
=
2t2c t2 +t2c
f kf =
θf
(66) (67) (68)
= 2f f , and thus
1 k−1 1 f (f + f ) = (f k + f k−1 f ) 2 2
and n−j
(65)
γ = tc
n−j
2
−k n+1−(j+k)
f
+2
−n+j
f
.
(69)
k=0
Taking the complex conjugate of (69) and adding it to resp. subtracting it from (69), we arrive at (65) and (66). To prove (67) and (68), it suffices to use that (f k ) = kf k+1 /(itc ) along with the complex conjugate equation.
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Proposition 5. For each even n ∈ N and j = 0, . . . , n − 1, let the numbers aj recursively defined through a0
(2)
=
(n+2)
=
aj
(2)
a1 = 0 , j k 1 (n) n+1−j (n) 2 a (n − j) aj − γ (n + 1) n n − k m=0 m
1,
be
(70) (j < n) , (71)
k=0
a(n+2) n
=
(n+2)
(n+2)
an−1 ,
an+1 = 0 .
Then zn yn
n−1 (n − 1)! −j (n) 2 aj Im(f n−j ) (n even) , tnc j=0 j n−1 1 (n) 2 (n − 1)! −j = γ 2 ak Re(f n−j ) tnc n − j j=0
= −γ
(72)
(n even) ,
(73)
k=0
xn
n−1 n (n+1) (n − 1)! −j aj = iγ 2 Re(f n−j ) tnc n − j j=0
Proof. We proceed by induction. We have x1 = iθ /2 = and (68), i γ z2 = x1 = − 2 Im(f 2 ). tc tc
(n odd) ,
iγ tc Re(f ),
(74)
and thus by (40)
This proves (72) for n = 2. Now suppose that (72) holds for some even n ∈ N. Then by (40) and (68), (74) holds for n − 1. To prove (73) for the given n, we want to use (41). (65) and the induction hypothesis on zn yield
θ zn
−γ
=
2 (n
−γ 2
=
n−1 − 1)!
tn+1 c
j=0 n−1
(n) aj
(n − 1)! tn+1 c m=0
n−j−1
2−(k+j) Im(f n+1−(j+k) )
k=0
m (n) 2−m aj Im(f n+1−m ).
(75)
j=0
Since (75) only contains second or higher order powers of f , it is easy to integrate using (68). Let us write m 1 (n) = a . (76) b(n) m n − m j=0 j Then by (68) we obtain yn = −
t
−∞
θ (s)zn (s) ds = r2
n−1 (n − 1)! −m (n) 2 bm Re(f n−m ), tn−1 c m=0
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proving (73) for n. It remains to prove (72) for n + 2. We want to use (64), and therefore we employ (66) and our above calculations in order to get θ (t)
t
−∞
θ (s)zn (s) ds
n−j+1 n−1 (n − 1)! = −γ 3 n+1 bj 2−(k+j) Re(f n+1−(k+j) ) + 2−n θ tc j=0 k=0 j n−1 n−1 (n − 1)! = −γ 3 n+1 2−j bk Re(f n+1−j ) + 2−n+1 bk Re(f ) . tc j=0 k=0 k=0 By (67), zn = γ
n−1 (n − 1)! −j (n) 2 aj (n − j)Re(f n+1−j ). tn+1 c j=0
Now we sum the last two expressions, differentiate again and obtain j n−1 (n − 1)! −j (n) 2 zn+2 = −γ n+2 2 (n + 1 − j) (n − j)aj − γ bk Im(f n+2−j ) tc j=0 k=0 n−1 −2γ 2 2−n bk Im(f 2 ) . k=0
Comparing coefficients, this proves (72) for n + 2.
(n)
We now investigate the behavior of the coefficients aj (n)
Proposition 6. Let aj (n)
(a) a0
=
as n → ∞.
be defined as in Proposition 5.
2 sin(γπ/2) . 1 + O nγ 2 γπ/2
(b) There exists C1 > 0 such that for all n ∈ N (n)
|a1 | ≤ C1
ln n . n−1
(c) For each p > 1 there exists C2 > 0 such that for all n ∈ N sup p−j |aj | ≤ (n)
j≥2
C2 . n−1
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(2)
Proof. (a) By (70), a0 = 1, and (n+2)
a0
(n)
= a0
γ2 1− 2 . n
Comparing with the product representation of the sine function ([AbSt], 4.3.89) sin(πx) = πx
∞ x2 1− 2 , n n=1
we arrive at (a). (n) (b) Put αn = (n − 1)a1 . Then by (71), 1 1 γ2 (n) 2 + αn+2 = αn 1 − − γ a0 . (n − 1)2 n n−1 thus for n − 1 > γ, we have |αn+2 | ≤ |αn | + γ
2
1 1 + n n−1
(m)
max |a0 |, m∈N
which shows (b). (n) (n) (n) (c) Put cj = (n − 1)p−j aj , and c(n) = maxj≥2 |cj |. We will show that the (n) sequence c is bounded. We have j k 1 −j+m (n) n+1−j (n+2) (n) cj (n − j)cj − γ 2 = p cm n(n − 1) n − k m=2 k=2 j j 1 1 (n) (n) −j 2 + a1 . (77) −(n − 1)p γ a0 n−k n−k k=0
Now
j k=2
k=1
k 1 −j+m (n) 1 p2 p cm ≤ c(n) , n − k m=2 n − j (p − 1)2
and p−j
j k=0
(j + 1)p−j 1 1 ≤ ≤ . n−k n−j (n − j) ln p
We plug these results into (77) and obtain (n + 1 − j)(n − j) (n + 1 − j) γ 2 p2 1 (n+2) |cj | ≤ c(n) + n(n − 1) (n − j)(p − 1)2 n(n − 1) 1 (n + 1 − j) p2 γ 2 (n) (n) (|a | + |a1 |). + n (n − j)(p − 1)2 ln p 0
(78)
(79)
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(n)
By (a) and (b), a0 and a1 are bounded. Taking the supremum over j ≥ 2 above, we see that there exist constants B1 and B2 with n−2 B1 B2 (n+2) (n) + , ≤c c + n n(n − 1) n hence c(n+2) − c(n) ≤
1 n
B1 −2 + c(n) + B2 . n−1
Now let n − 1 > B1 . Then for c(n) > B2 , the above inequality shows c(n+2) < c(n) , while for c(n) ≤ B2 , c(n+2) ≤ c(n) + B2 /n ≤ B2 (1 + 1/n). Thus c(n) is a bounded sequence. Remark 8. We will make no use of the fact that the logarithmic correction to the (n) 1/n-decay of the higher coefficients occurs only in the coefficient a1 . We chose to include this in the statement of the preceding theorem anyway, because this gives some insight into the nature of the recursion and is not hard to prove. (n)
Remark 9. Numerical calculations of the first few thousand aj suggest that (c) above continues to be true if we choose p = 1, but this seems to be much harder to prove. However, the estimate above is more than good enough for us. Remark 10. The constants appearing in the proof of Proposition 6 (b) and (c) are not optimal, and could be improved by more careful arguments. This is unimportant for our purposes, and for the sake of brevity and readability we chose to use the simple estimates given. (n)
Corollary 3. Let bj such that
be given by (76). Then for each p > 1, there exists C3 > 0 sup p−j bj
(n)
j≥0
≤
C3 . n−1
Proof. For j ≤ n − 1, we have n − 1 ≤ j(n − j), and thus Proposition 6 (c) gives p2 p−j C2 (n) (n) (n) p−j bj (pj−1 − 1) ≤ (a0 + a1 ) + n−j n−1p−1 ≤ p−j
(n)
(n)
p C2 1 C3 j(a0 + a1 ) + ≤ . n−1 p−1n−1 n−1 (n)
Having good control over the coefficients aj , we can now derive relatively sharp estimates on the functions xn , yn and zn . Let us fix α < 1 and define ! " n−2 n 2 1 1 t t c c max , √ . Rnα (t) = (n − 1)α t + itc t + itc 2
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Obviously, for t ≤ tc the first function in the maximum above dominates, for t > tc the second one does. For families of functions gn (t), Gn (t) we write gn (t) = O(Gn (t)) if there exists C > 0 such that |gn (t)| ≤ C|Gn (t)| for all n ∈ N and all t ∈ R. Theorem 3. For n > 1 and α < 1, we have −n t (n − 1)! 2 sin(γπ/2) α Re 1−i + O(Rn (t)) , xn (t) = i tnc π tc
(80)
(n − 1)! O(Rnα (t)), (81) tnc −n t (n − 1)! 2 sin(γπ/2) Im 1 − i − + O(Rnα (t)) , (82) tnc π tc
yn (t) = zn (t) =
Proof. With the definition of f and Proposition 6 (a) we get −n tc t 2 sin(γπ/2) 1 (n) n Im 1−i +O a0 Im(f ) = 2 πγ tc n t + itc
n
when n is even, and a similar formula for a0 Re(f n ) when n is odd. This covers the j = 0 terms in (74) and (72). For the remaining terms, let ! (n) aj if n is even, (n) cj = (n) naj /(n − j) if n is odd. (n+1)
Now n/(n − j) ≤ j for j < n, and thus by Proposition 6 (b) and (c) for each p > 1 we can find C > 0 such that C (n) cj ≤ jpj (n − 1)α for all j ≥ 1. (For j ≥ 2, we may even choose α = 1, but we will not exploit this.) For |t| ≤ tc , we have |tc /(t + itc )|−j ≤ 2j/2 , so we get n−j j n−1 n−1 n c(n) C tc p itc j √ ≤ j . 2j t + itc (n − 1)α j=2 t + itc 2 j=2
√ If we choose p < 2, the sum on the right-hand side is bounded uniformly in n. Combining this with our above calculations, (80) and (82) are proved for |t| < tc . √ For |t| > tc , we have |tc /(t + itc )| ≤ 1/ 2, and thus n−2 j=2
(n)
cj 2j
itc t + itc
n−j
C tc ≤ α (n − 1) t + itc
p j 1 n−2−j √ j . 2 2 j=2
2 n−2
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√ If we√choose again p < 2, the sum on the right-hand side is bounded by ˜ C(1/ 2)n−2 uniformly in n. For the term with j = n − 1, this does not work (n) since then n − 2 − j < 0. But for n even, this term vanishes since then cn−1 = 0, and for n odd, it equals n−2 (n) (n+1) 2 cn−1 nan−1 itc C˜ tc 1 t2c √ Re ≤ . = 2n t + itc 2n t2 + t2c n−1 t + itc 2 This proves (80) and (82) for |t| ≥ tc . The proof of (81) is similar and uses Corollary 3.
5 Optimal truncation By the results of the previous section πk grows like (k − 1)!/tkc . Hence, the sum n (n) π = k=0 εk πk does not converge to an exactly equivariant projection π (∞) as n → ∞. This is the reason why we see exponentially small transitions. The basis in which these transitions develop smoothly is the optimal superadiabatic basis: since we cannot go all the way to infinity with n, we fix ε and choose n = n(ε) such that the off-diagonal elements in (2) become minimal. Using Stirling’s formula and (80) resp. (82), it is easy to see that the place to truncate is at n(ε) = tc /ε. This n(ε) is in general not a natural number, but we will find that a change of n which is of order one does not change the results. Before we go into more details, we need a preliminary result. Lemma 5. Uniformly in x ∈ [0, 1] and for k > 0, we have (1 + x)−k = e−kx + e−kx/2 O k1 . Proof. We start with the equality
(1 + x)−k − e−kx = e−kx ek(x−ln(1+x)) − 1 .
(83)
At first consider x > 1/k. There we use the inequality (x − ln(1 + x)) ≤ x/3, valid for 0 ≤ x ≤ 1, in (83) and obtain |(1 + x)−k − e−kx | ≤ e−kx ekx/3 − 1 = e−kx/2 e−kx/6 − e−kx/2 . For x > 1/k, theterm in the last bracket above is O(1/k), and we are done in this case. For x ≤ 1/k, we use (x − ln(1 + x)) ≤ x2 /2 and rearrange (83) to get 2 ekx/2 ((1 + x)−k − e−kx ) ≤ e−kx/2 ekx /2 − 1 =: f (x, k). To find out where f (x, k) is maximal, we calculate 2 d k f (x, k) = e−kx/2 1 + ekx /2 (2x − 1) . dx 2
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The derivative is zero exactly at the solutions of the equation ln(1 − 2x)/x2 = −k/2.
(84)
+ R(x), where R(x) is a power series in x, convergent Now ln(1 − 2x)/x2 = −2/x for x < 1/2. Thus for x < 1/k and k sufficiently large, there exists exactly one solution x∗ (k) of (84), and x∗ (k) < C/k uniformly in k for some C > 0. Since d 2 ∗ dx f (x, k) > 0 for x < 1/k , f (x, k) has a maximum at x (k). Thus f (x, k) ≤ f (x∗ (k), k) ≤ e−C/2k − 1 = O(1/k) for x <
1/k, and the claim is proved.
Lemma 5 immediately yields 1+
a −k = e−a 1 + O k1 k
(85)
uniformly on compact intervals of a by taking x = a/k. We now turn to the proof of Theorem 1, which we deduce from Theorems 2 and 3. As stated already in (13), we will use nε =
tc − 1 + σε , ε
(86)
where σε ∈ [0, 2[ is such that nε is an even integer. The advantage of this convention about σε is that now the off-diagonal components in (62) are always given by εnε +1 xnε +1 since zn+1 = 0 for even n. Of course we could as well consider the asymptotic behavior of εn+1 zn+1 for odd n and one would expect to end up with the same result. However, it is obvious from (80) and (82) that xn+1 is purely imaginary and zn+1 is real at leading order. Thus the large n asymptotics of the off-diagonal component of the effective Hamiltonian do depend on whether we consider even or odd superadiabatic bases. On the other hand, the asymptotics of the propagator must be independent of the exact choice of basis. We will discuss this point after giving the proof of Theorem 1 based on the above convention. Proof of Theorem 1. We want to apply Theorem 2 and thus have to check that ξ, η and ζ defined in (56)–(58) together with their derivatives are O(εθ ). From Proposition 5 together with Proposition 6 we infer that there exists C > 0 such that |xk (t)| ≤ Cθ (t)(k − 1)!/tkc for each k. The same is true for yn and zn . Using the differential equations (40)–(42), we find that there is C > 0 with |xn (t)| ≤ C θ (t)n!/tn+1 . This means that c
|ξ (t)| ≤ εC θ (t)
n k=1
εk t−k−1 k! εk−1 , c
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with similar expressions for the other quantities. Now taking ε = tc /(nε − σε ), we find nε
k t−k c ε (k + 1)! =
k=0
nε (k + 1)! 2 3! = 1 + + + . . . . (nε − σε )k nε − σε (nε − σε )2 k=0
Each of the nε + 1 terms in the sum above is bounded by const/(nε − σε ) except the first which is 1. This shows ε , |ξ (t)| ≤ εθ C 1 + nεn−σ ε and Theorem 2 gives (15) with cnε ε (t) = εnε +1 xnε +1 (t)(1 + O(εθ (t)). Recall that znε +1 (t) = 0 due to our convention. It remains to determine the leading order asymptotics of εnε +1 xnε +1 . For convenience of the reader let us rewrite (80) as εnε +1 xnε +1 (t) = i
εnε +1 nε ! tnc ε +1
# 2 sin(γπ/2) π
Re
1 − i ttc
−(nε +1)
$ + O Rnβε +1 (t) . (87)
Lemma 6. With (86), we have εnε +1 nε ! = tnc ε +1
2πε − tc e ε (1 + O(ε)). tc
Proof. Stirling’s formula for (n + 1)! implies n! =
1 n+1
n+1 e
n+1
√
√ 1 . n + 1 2π 1 + O n+1
Together with (85) this yields ε
nε +1
−(nε +1) 2π σε 1 + O nε1+1 nε ! = 1− nε + 1 nε + 1 2π 1 + O nε1+1 = tnc ε +1 e−(nε +1) eσε nε + 1 tc 2πε (1 + O(ε)). = tnc ε +1 e− ε tc + εσε tnc ε +1 e−(nε +1)
Finally,
2πε = tc + εσε
2πε tc
−1/2 2πε εσε = (1 + O(ε)) . 1+ tc tc
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Lemma 6 takes care of the first factor in (87). Turning to the terms inside the square brackets in (87), let us first note that for |t| ≥ tc , both terms are O(2−(nε −1)/2 /(1 + t2 )) = O(exp(−tc ln 2/(2ε))/(1 + t2 )) , proving the theorem in this case. For |t| < tc , we investigate the modulus and the phase separately. Let 0 < β < 1. From Lemma 5 it follows that 1 + i ttc
nε +1
= 1+
t2 t2c
(tc /ε+σε )/2
= 1+
t2 t2c
σε /2 t2 t2 e− 2tc ε + O εe− 4tc ε .
For |t| ≥ εβ/2 , exp(−t2 /(2tc ε)) = O(ε exp(−t2 /(4tc ε))). Thus neither the prefactor involving σε above nor the phase play any role in this region. For |t| < εβ/2 , (1 + t2 /t2c )σε /2 = 1 + O(σε εβ ) and therefore 1 + i ttc
nε +1
t2 t2 = e− 2tc ε + O εβ e− 4tc ε .
The same reasoning applies to Rnβε +1 and gives t2 t2 Rnβε +1 (t) ≤ εβ e− 2tc ε + O εβ e− 4tc ε . Turning to the phase in the region |t| < εβ/2 , we find ei(nε +1) arctan(t/tc ) = exp i tεc + σε (t/tc ) − 13 (t/tc )3 + O((t/tc )5 ) σε t 5 3 t3 = exp i εt − 3εt + O(t (t/t ) /ε) + O(σ (t/t ) ) 2 + t c c ε c c c σε t t t3 1 + O(ε5β/2−1 ) + O(ε3β/2 ) . = exp i ε − 3εt2 + tc c
Now we just have to collect all the pieces and add the complex conjugate.
Let us now see what of the above would have changed for nε odd. Then xnε +1 = 0, and (82) together with Lemma 5 and 6 yields cnε ε (t) = =
−εnε +1 znε +1 (t) (1 + O(εθ )) (88) 3 2 tc t t t 2ε πγ σε t − e− ε e− 2εtc sin sin + 2 + O (φα (ε, t)) . πtc 2 ε 3εt2c tc
At first, this looks like an important difference, since now the off-diagonal elements in the transformed Hamiltonian are purely real-valued in leading order, while in the other case they were purely imaginary. However, in the computation of the propagator, another factor of exp(±it/ε) from the dynamical phase appears, cf. (89). At leading order only the resonant term of the Hamiltonian survives, which is the same for odd and even nε .
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6 First-order perturbation in the optimal superadiabatic basis In this section we prove Corollary 1. Since we use standard first-order perturbation theory, we stay sketchy in some parts. After splitting Hεnε (t), see (15), as 1 0 nε 2 + Vε (t) =: H0 + Vε (t) , Hε (t) = 0 − 12 Dyson expansion in the interaction picture (cf. [ReSi], Thm. X.69) yields itH iτ H isH0 i t iτ H0 nε − ε0 − ε0 ε Kε (t, s) = e e Vε (τ ) e dτ e ε id − ε s tc O(ε2 ) O(εe− ε ) ∆(t, s) . + tc O(ε2 ) O(εe− ε ) Thus we only need to evaluate the integral −
i ε
t
e
iτ H0 ε
s
Vε (τ ) e−
iτ H0 ε
dτ
=
iτ i t 0 e ε cnε ε (τ ) dτ iτ ε s e− ε cnε ε (τ ) 0 O(ε) 0 + ∆(t, s) . 0 O(ε)
−
Inserting (16) and using (17) gives −
i ε
t
iτ
e ε cnε ε (τ )dτ s iτ iτ 3 iστ t 3 πγ iτ tc τ2 iτ 2 − + − − iτ + iστ e− ε = sin e ε e− 2εtc e ε 3εt2c tc + e ε 3εt2c tc dτ επtc 2 s α − tεc ∆(t, s) = (∗) , (89) +O ε e ±(
iτ 3
− iστ )
3
iτ iστ for each α < 1. Now we replace the exponentials e 3εt2c tc by 1 ± ( 3εt 2 − t ). c c Using |eiϕ − 1 − iϕ| ≤ ϕ2 , we conclude that the resulting error is bounded by a constant times 6 ∞ tc tc τ2 1 τ τ4 2 + τ e− 2εtc + dτ = O(εe− ε ) . ε − 2 e− ε 2 ε ε −∞
Hence we obtain % (∗) =
2 επtc
sin
πγ 2
tc
e− ε
s α − tεc +O ε e ∆(t, s)
t
τ2
e− 2εtc
1+
iτ 3 3εt2c
−
iστ tc
+e
2iτ ε
1−
iτ 3 3εt2c
+
iστ tc
dτ (90)
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with α < 1, where the first summand in the integrand gives rise to the explicit term in (19). The remaining terms can be integrated explicitly as well, most conveniently √ tc using Maple or Mathematica. They are all of order O( εe− ε ∆(t, s)) uniformly tc in t and s resp. of order O(εα e− ε ∆(t, s)) for |t| and |s| larger than εβ for some β < 12 . To illustrate the reasoning note that
t s
s2 τ2 t2 e− 2εtc τ dτ = εtc e− 2εtc − e− 2εtc . 2β−1
This is uniformly of order O(ε), but of order O(e−ε ) for |t| and |s| larger than εβ . Finally we emphasize that we could get the next order corrections to (19) explicitly by evaluating (90).
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Volker Betz Institute for Biomathematics and Biometry GSF Forschungszentrum Postfach 1129 D-85758 Oberschleißheim Germany email:
[email protected] Stefan Teufel Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom email:
[email protected] Communicated by Yosi Avron submitted 29/06/04, accepted 14/08/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e
Ann. Henri Poincar´e 6 (2005) 247 – 267 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020247-21 DOI 10.1007/s00023-005-0205-0
Annales Henri Poincar´ e
Existence of the D0–D4 Bound State: a Detailed Proof ∗ Laszlo Erd¨os, David Hasler and Jan Philip Solovej
Abstract. We consider the supersymmetric quantum mechanical system which is obtained by dimensionally reducing d=6, N =1 supersymmetric gauge theory with gauge group U(1) and a single charged hypermultiplet. Using the deformation method and ideas introduced by Porrati and Rozenberg [1], we present a detailed proof of the existence of a normalizable ground state for this system.
1 Introduction The particular system, which we will consider, belongs to a class of supersymmetric quantum mechanical models. These models appear in the study of quantized membranes [2], D-brane bound states [3], and M-theory [4]. Especially the question of existence respectively absence of normalizable ground states, i.e., zero energy states, is of physical importance. The Hamiltonian of these models is of the form H = −∆ + V + HF . The scalar potential V is polynomial in the bosonic degrees of freedom and admits zero energy valleys extending to infinity while HF is quadratic in the fermionic degrees of freedom and linear in the bosonic degrees of freedom. Moreover, the Hilbert space carries a unitary representation of a gauge group. The physical Hilbert space consists of gauge invariant states. Due to supersymmetric cancellations, the zero energy valleys render the Hamiltonian to have continuous spectrum, which covers the positive real axis. Therefore, the Hamiltonian is non-Fredholm and the question about existence of ground states is subtle. The Witten index IW , i.e., the number of bosonic ground states minus the number of fermionic ground states, can be calculated by means of IW = lim lim Tr((−1)F χR e−βH ) , R→∞ β→∞
where χR denotes the characteristic function of the ball of radius R centered around the origin in configuration space, c.p. [5]. Since there is no gap in the ∗ Work partially supported by NSF grant DMS-0200235, by EU grant HPRN-CT-2002-00277, by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation, and by grants from the Danish research council.
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spectrum one has to deal with a delicate analysis of boundary contributions. As a different approach, Porrati and Rozenberg proposed in [1] a deformation method to detect the existence of normalizable ground states for systems with at least two real supercharges. One deforms the supercharges of the system with a real potential w, D → Dw := e−w Dew ,
† D † → Dw := ew D† e−w ,
† † + Dw Dw such that the spectrum of the deformed Hamiltonian Hw := Dw Dw becomes discrete. This might allow one to show the existence of a ground state Ψw for the deformed problem. Provided that e±w Ψw is normalizable, then, the original problem admits a ground state as well. Using this method the number of ground states for numerous models could be determined, [6]. In this paper, we consider the quantum mechanical system which is obtained by dimensionally reducing N = 1 supersymmetric gauge theory, with gauge group U(1) and with a single charged hypermultiplet from six dimensions. The system appears in the problem of counting H-monopole ground states in the toroidally compactified heterotic string [7]. Moreover, the same system describes the low energy dynamics of a D0-brane in the presence of a D4-brane [8, 9]. String duality arguments predict the existence of exactly one bound state at threshold for this system, c.p. [10]. The existence of such a state provides a check of the correctness of these duality hypotheses. In [5], an analysis was sketched of how to obtain the value one for the Witten index for this system. Combined with vanishing Theorems, [11], such a result implies that the model has a unique ground state. Independently of the work in [5], it was argued in [1] how a deformation method may be used to establish existence of a ground state. In this paper we use this deformation method and follow the main ideas of [1] to present a rigorous proof of the existence of a ground state. In particular, we make the argument in [1] mathematically precise in two important aspects. First we prove the existence of a ground state for the deformed problem: we have to do semiclassical analysis on the space of gauge invariant functions and we have to deal with the fact that HF is unbounded. In a second part we prove a decay estimate for the ground state of the deformed problem. In particular, we show that it decays sufficiently fast implying that the original problem also has a ground state. To obtain this decay property, we use an Agmon [12] estimate and combine it with a symmetry argument. We think this is a clear and direct way to obtain the necessary decay. Alternatively one could also determine the asymptotic form of the ground state by analyzing the effective dynamics along a potential valley. Such an analysis was indicated in [1]. Similarly one could use a supercharge analysis related to the one in [13] (which was used to determine the asymptotic form of the bound state of two D0-branes). Similar considerations have to be taken into account when using the deformation method to study the number of zero energy states for other supersymmetric models of the same type. Moreover, there are results about the structure of the D0-D4 bound state [14].
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The paper is organized as follows. In Section 2, we describe the model. In Section 3, we introduce the deformation method and give an outline of the proof, which is then presented in Section 4.
2 The model The model is obtained by dimensionally reducing N = 1, U(1) supersymmetric gauge theory with a single charged hypermultiplet, from 5 + 1 dimensions to 0 + 1 dimension [8, 14]. The bosonic degrees of freedom are given by q = (qj )j=1,...,4 ∈ R4 ,
and x = (xµ )µ=1,...,5 ∈ R5 ,
and their configuration space is X = R4 × R5 . Let pj , j = 1, . . . , 4, and pµ , µ = 1, . . . , 5, be the associated canonical momenta obeying, [qj , pk ] = iδjk ,
[xµ , pν ] = iδ µν .
The fermionic degrees of freedom are described by the real Clifford generators λa , a = 1, . . . , 8
and ψa , a = 1, . . . , 8 ,
i.e., λ†a = λa , ψa† = ψa , and {λa , λb } = δab ,
{ψa , ψb } = δab ,
{λa , ψb } = 0 .
(Here and below { · , · } stands for the anticommutator.) By F we denote the irreducible representation space of this Clifford algebra. The dimension of F is 28 . We introduce as a preliminary Hilbert space H0 = L2 (X; F ) = L2 (X) ⊗ F . As given in Appendix A, we choose an explicit real irreducible representation µ )a,b=1,...,8 , γ µ = (γab
µ = 1, . . . , 5 ,
of the gamma matrices in 5 dimensions, i.e., {γ µ , γ ν } = 2δ µν . Furthermore we consider the real 8 × 8 matrices si = (siab )a,b=1,...,8 ,
i = 1, . . . , 4 ,
as they are defined in Appendix A. We note that s1 = 1I8×8 and (sl )T = −sl for l = 2, 3, 4 and that each si commutes with the γ-matrices. We define Dab =
1 R 2 R (q s q )ab , 2
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with qR
=
s1 q1 + s2 q2 + s3 q3 + s4 q4 ,
qR
=
s1 q1 − s2 q2 − s3 q3 − s4 q4 .
We will use the convention of summing over repeated indices. The supercharges are given by Qa = (sj ψ)a pj + (γ µ λ)a pµ + Dab λb + (γ µ sj s2 ψ)a xµ qj ,
a = 1, . . . , 8 .
Note, for any 8 × 8 matrix A we set (Aψ)a = Aab ψb , ψAψ = ψa Aab ψb , and likewise for expressions containing λa . The Hilbert space H0 carries a unitary representation of U(1), called the gauge transformation, defined by the generator i J = W12 + W34 − ψs2 ψ , 2 where Wij = qi pj − qj pi . We set |x| := (xµ xµ )1/2 and |q| := (qi qi )1/2 . The full model satisfies µ µ x J, {Qa , Qb } = δab H + 2γab
(1)
with H
1 = pµ pµ + pi pi + |x|2 |q|2 + |q|4 − ixµ ψγ µ s2 ψ + i2qj λsj s2 ψ 4 = −∆ + V + HF ,
where we have defined 1 V = |x|2 |q|2 + |q|4 , and HF = −ixµ ψγ µ s2 ψ + i2qj λsj s2 ψ . 4 The Hilbert space of the model H is the U(1)-invariant subspace of H0 , i.e., H = {Ψ ∈ H0 | JΨ = 0 } . Note that the supercharges Qa are U(1) invariant and that on H the superalgebra (1) closes, i.e., {Qa , Qb }|H = δab H|H . The Hilbert space H0 carries a natural representation of Spin(5) defined by the infinitesimal generators i µν (λa λb + ψa ψb ) , µ, ν = 1, . . . , 5 , T µν = xµ pν − xν pµ − γab 4 with γ µν = 12 [γ µ , γ ν ]. Under this representation the supercharges Qa transform as spinors and the Hamiltonian H is invariant. The action of Spin(5) commutes with the gauge transformation, and thus leaves the Hilbert space H invariant.
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We introduce the fermionic number operator (−1)F := 28 λ1 λ2 . . . λ8 ψ1 ψ2 . . . ψ8 , which anti-commutes with Qa and commutes with H, and decompose the Hilbert space by means of (−1)F as H± := {Ψ ∈ H | (−1)F Ψ = ±Ψ } , i.e., into bosonic (+) and fermionic (–) sectors. We note that the operators Qa and H are essentially selfadjoint on C0∞ (X; F ). Furthermore their restriction to H is essentially selfadjoint on the space of U(1)-invariant functions in C0∞ (X; F ).
3 Result and outline of the proof The main Theorem is the following: Theorem 1. There exists a state Ψ ∈ H with HΨ = 0. To prove this theorem, we use the deformation method introduced in [1]. We consider the “complex” supercharges D and D† , 1 D = √ (Q1 + iQ2 ) , 2
1 D† = √ (Q1 − iQ2 ) . 2
2
On H, D2 = D† = 0 and
H = {D, D† } .
(2)
We define the U(1)-invariant function wk on X, by wk = k · x1 ,
for k ≥ 0 .
We introduce the deformed supercharges Dk = e−wk Dewk ,
Dk† = ewk D† e−wk .
We have i Dk = D − k √ ((γ 1 λ)1 + i(γ 1 λ)2 ) , 2
i Dk† = D† + k √ ((γ 1 λ)1 − i(γ 1 λ)2 ) . 2
As a little calculation shows, we have on H Hk = {Dk , Dk† } ,
with
Hk := H + k 2 + k(q32 + q42 ) − k(q12 + q22 ) .
We point out that the deformed Hamiltonian is Spin(5) invariant, despite that the function wk = k · x1 is not. The claim of Theorem 1 is an immediate consequence of the following three propositions.
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Proposition 2. If for some k there exists a state Ψk ∈ H with Hk Ψk = 0 such that e±wk Ψk ∈ H, then HΨ = 0 for some state Ψ ∈ H. Remark. Proposition 2 holds for more general supersymmetric quantum mechanical systems and deformations, c.p. [1]. The proof of Proposition 2, which is presented in Subsection 4.1, makes use of the Hodge decomposition and a cohomology argument. Proposition 3. For k large enough, there exists a unique state Ψ ∈ H with Hk Ψ = 0. Remark. Proposition 3 implies that Hk admits a zero energy ground state for all k > 0. This follows from the stability of the Fredholm index of the continuous family of Fredholm operators, (0, ∞) k → Ak := 2−1/2 (Dk + Dk† )|H− : H− → H+ , where the topology is given by the graph norm with respect to A0 , see for example [15]. However, we will not use this fact to prove Theorem 1. To prove Proposition 3, which is done in Subsection 4.2, we first observe that the set of points, in which the scalar potential of the deformed Hamiltonian, i.e., Vk = V + k 2 − k(q12 + q22 ) + k(q32 + q42 ) , vanishes, is a circle in configuration space X (see, e.g., (5)). Its radius is proportional to k 1/2 . The circle is an orbit of the U (1) action on X. In the direction orthogonal to the circle the Hessian of Vk is non degenerate. Note that up to gauge transformations the scalar potential vanishes exactly in one point. Moreover, at infinity the potential Vk is bounded below by k 2 . Using semiclassical analysis of eigenvalues, as given for example in [16], together with a gauge fixing procedure, we show that there exists only one low lying eigenvalue for k → ∞. In particular, we have to consider the fact that HF is unbounded from below. By supersymmetry this low lying eigenvalue must equal zero for large k. Proposition 4. For k > 0, a state Ψ ∈ H with Hk Ψ = 0 satisfies e±wk Ψ ∈ H. For the proof of Proposition 4, which is given in Subsection 4.3, we need to show that Ψ decays sufficiently fast as |x| → ∞. We write the Hamiltonian as a sum of a free Laplacian in the x-variables and an x-dependent operator, which describes the dynamics in the transverse direction. We show that the latter is bounded below by k 2 − c|x|−2 for some constant c and |x| large. Using an Agmon estimate we then conclude that |x|−1 ek|x| Ψ is square integrable at infinity. As will be shown, this together with the fact that Ψ is invariant under Spin(5) yields e±wk Ψ ∈ H.
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Remark. To be precise, the operators D, D† , Dk , Dk† and Hk are defined in H0 and H as the closure on C0∞ (X; F ) and C0∞ (X; F )∩H, respectively. The domain of D is the set of all Ψ in H0 and H such that DΨ (defined in the sense of distributions) is again in H0 and H, respectively (and analogous for the domains of D† , Dk , Dk† , H, and Hk ). Indeed, D† (resp. Dk† ) is the adjoint of D (resp. Dk ).
4 Proofs 4.1
Proof of Proposition 2
We shall first show the Hodge decomposition H = ker H ⊕ RanD ⊕ RanD† .
(3)
To show the orthogonality, we note that (DΨ, D† Φ) = (D2 Ψ, Φ) = 0 , with Ψ ∈ D(D) and Φ ∈ D(D† ), and Ψ ∈ ker H iff DΨ = 0 and D† Ψ = 0, by (2). To show the completeness, we note that for each Ψ ∈ (ker H)⊥ , Ψ
= = = =
lim P(a,∞) (H)Ψ a↓0
1 1 lim (DD† + D† D) P(a,∞) (H)Ψ a↓0 2 H 1 1 † 1 † 1 D(D P(a,∞) (H)Ψ) + D (D P(a,∞) (H)Ψ) lim a↓0 2 H 2 H 1 1 1 1 lim D(D† P(a,∞) (H)Ψ) + lim D† (D P(a,∞) (H)Ψ) . a↓0 2 a↓0 2 H H
By PΩ (H) we denoted the projection valued measure of H, and the last equality follows since the two terms belong to different orthogonal subspaces. Hence we have shown (3). The equation Hk Ψk = 0 implies Dk Ψk = 0 and Dk† Ψk = 0, and further wk De Ψk = 0 and D† e−wk Ψk = 0. Assume ker H = {0}. Then ewk Ψk ∈ ker D = RanD†
⊥
= RanD
by the Hodge decomposition. It follows that ewk Ψk = limn→∞ DΦn for some Φn , but then (Ψk , Ψk ) = (ewk Ψk , e−wk Ψk ) = =
lim (DΦn , e−wk Ψk )
n→∞
lim (Φn , D† e−wk Ψk ) = 0.
n→∞
This is a contradiction, and hence ker H = {0}.
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Proof of Proposition 3
We shall first rescale the operators Hk , Dk and Dk† . For Ψ ∈ H and t > 0, we define the unitary operator (U (t)Ψ)(ξ) = t9/2 Ψ(t ξ) , where ξ = (q, x). Furthermore, we define Kt
:=
Ft
:=
Ft†
:=
t2/3 U (t1/3 )Ht2/3 U ∗ (t1/3 ) t1/3 U (t1/3 )Dt2/3 U ∗ (t1/3 )
t1/3 U (t1/3 )Dt†2/3 U ∗ (t1/3 ) .
It follows that on H {Ft , Ft† } = Kt and
,
Ft2 = 0 ,
2
Ft† = 0 ,
Kt = −∆ + t2 V1 + tHF ,
where
1 V1 = |x|2 |q|2 + |q|4 + 1 + (q32 + q42 ) − (q12 + q22 ) . 4 Proposition 3 follows from Lemma 5. Let En (t) denote the nth eigenvalue of Kt counting multiplicity. Then lim E1 (t)/t = 0 and lim inf E2 (t)/t ≥ r > 0 .
t→∞
(4)
t→∞
By supersymmetry, each non-zero eigenvalue of Kt must be two fold degenerate, i.e., occur as the eigenvalue of a pair consisting of a bosonic and a fermionic eigenvector (see Theorem 6.3., [16]). In view of (4), for large t, two fold degeneracy of E1 (t) is not possible. Hence E1 (t) = 0. Moreover, this eigenvalue is nondegenerate. Proof of Lemma 5. Writing the deformed potential V1 as 2 1 1 1 V1 = |x|2 |q|2 + (q12 + q22 ) − 1 + (q32 + q42 ) 1 + (q32 + q42 ) + (q12 + q22 ) (5) 2
4
2
we see that the set of points Γ in which the potential V1 vanishes is given by Γ
:= {(q, x) ∈ X |V1 (q, x) = 0 } = {(q, x) ∈ X | q12 + q22 = 2, q3 = 0, q4 = 0, x = 0 } .
√ The set Γ is a circle in the (q1 , q2 )-plane about the origin with radius 2. The Hessian of V1 at points lying in Γ is 2 0 0 (2qr qs ) ∂ V1 = , α, β = 1, . . . , 9, (HessV1 )αβ |Γ = 0 41I2×2 0 ∂ξ α ∂ξ β Γ 0 0 41I5×5
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with (ξ 1 , . . . , ξ 9 ) := (q1 , . . . , q4 , x1 , . . . , x5 ) and r, s = 1, 2. At a point p ∈ Γ, the tangent to Γ is the only degenerate direction of the Hessian. To show that there exists only one low lying eigenvalue, we will fix the U(1) gauge. For ω ∈ L1 (X) with (W12 +W34 )ω = 0, we may integrate out the coordinate q1 as follows. We introduce the coordinates Φ : [0, 2π] × [0, ∞) × R2 −→ R4 α q1 cos α − sin α 0 ρ q2 sin α cos α v3 −→ q3 = cos α − sin α 0 v4 q4 sin α cos α
0 ρ v3 v4
with α = arctan(q2 /q1 ) and ρ = (q12 + q22 )1/2 . The metric determinant is √ det DΦT DΦ = | det DΦ| = ρ, and R4 ×R5
dq1 . . . dq4 d5 xω(q, x) = 2π
(0,∞)×R2 ×R5
dρdv3 dv4 d5 xρω((0, ρ, v3 , v4 ), x) .
The integration on the right-hand side is reduced to the gauge fixed configuration
:= {0} × (0, ∞) × R2 × R5 ⊂ X. We introduce the Hilbert space space X
:= L2 (X;
F) H
and we denote its canonical scalar product by w.r.t. the Lebesgue measure of X, · , · GF . We define the isometry H Ψ
−→ H
:= 2πρ Ψ| . −→ Ψ X
(6)
we may recover Ψ through By M = − 2i ψs2 ψ we denote the spin part of J. From Ψ Ψ(q, x) = √
1
ρ, q3 cos α − q4 sin α, q4 cos α + q3 sin α, x) . e−iαM Ψ(0, 2πρ
(7)
Under the isometry (6), the corresponding transformation for the operators A ∈
∈ L(H),
is characterized by L(H), i.e., A → A
Ψ
= AΨ . A For f ∈ C0∞ (X), one has
∂ i f = − W12 f , ∂q1 q2
X X
(8)
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only after the derivatives are performed. where the function f is restricted to X Applying this result to the function ∂f /∂q1 , using the commutation relation [W12 , ∂/∂q1 ] = i∂/∂q2 and again (8), one finds 1 ∂ ∂ ∂ 1 2 f = − 2 W12 f . ∂q1 ∂q1 q2 ∂q2 q2
X X We set L := J − W12 . Then for Ψ ∈ H, W12 Ψ = −LΨ .
= v3 (−i∂/∂v4) − v4 (−i∂/∂v3 ) − i ψs2 ψ. For Ψ ∈ H ∩ C ∞ (X; F ), a Note that L 0 2 straightforward calculation yields 1 2 1 ∂ ∂ 1 ∂ + 2L Ψ Ψ + 2Ψ − Ψ =− ∂q1 ∂q1 ρ ∂ρ 2ρ ρ and
As a result
∂ ∂ ∂ ∂ 1 ∂ 3 1 Ψ. − Ψ+ Ψ− Ψ =− ∂q2 ∂q2 ∂ρ ∂ρ ρ ∂ρ 4 ρ2
2 − 1 Ψ
, (−∆Ψ) = −∆X + ρ−2 L 4
(9)
We will use eq. (9) only for functions in where ∆X is the formal Laplacian on X. ∞ C0 (X; F ). We use the following partition of unity. We define √ j1,t (ξ) = χr (t2/5 ((q12 + q22 )1/2 − 2)) · χa (t2/5 (q3 , q4 , x)) , ξ = (q, x) , where for α = r, a, we have chosen rotation invariant functions χα ∈ C0∞ (Rnα ) with nr = 1, na = 7, 0 ≤ χα ≤ 1, χα (x) = 1 if |x| ≤ 1 and 0 if |x| ≥ 2. Let R ≥ 1 be fixed as t → ∞. We choose j2 ∈ C ∞ (X) with j2 (ξ) = j2 (|ξ|), 0 ≤ j2 ≤ 1, j2 (ξ) = 1 for |ξ| ≥ 2R and j2 (ξ) = 0 for |ξ| < R. Furthermore we set 2 j0,t := (1 − j1,t − j22 )1/2 .
→ X := {0} × R8 and the For technical matters we consider the embedding X 2 9
coordinates √ (0, η , . . . , η ) ∈ X. By η0 we denote the intersection of X with Γ, i.e., η0 = (0, 2, 0, . . . , 0). We define 9 1 V10 (η) = (HessV1 )αβ (η0 )(η α − η0α )(η β − η0β ) . 2 α,β=2
F) and introduce the following operator on L2 (X; Gt = −∆X + t2 V10 + tHF (η0 ) ,
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√ where HF (η0 ) = −i2 2(λ1 ψ1 + ...λ8 ψ8 ) : F → F denotes the evaluation of HF at For χ ∈ L2 (X; F ), we define η0 and −∆X the eight-dimensional Laplacian on X. the unitary transformation (T (t)χ)(η) = t2 χ(t1/2 (η − η0 )) . Then 1 1 T (t)∗ Gt T (t) = −∆X + (HessV1 )αβ (η0 )η α η β + HF (η0 ) . t 2 The eigenvalue problem for this operator can be solved easily. It has purely discrete spectrum and its ground state Φ0 has zero energy and is non degenerate: the sum and has ground of the first two√terms is a harmonic oscillator, which acts on L2 (X) state √ energy 8 2, and HF (η0 ) acts on F and has a unique ground state with energy −8 2, see Appendix B (i). Define
F ) → L2 (X; F) .
t := j1,t T (t)Φ0 ∈ C0∞ (X; Ψ We recall that the corresponding U (1)-invariant wave function in Ψt ∈ H is obtained using (7). Calculating the energy of this state, we find
t
t, K
tΨ Ψt , Kt Ψt = Ψ GF
2 − 1 ) + t2 V 1 + tH
F )Ψ
t = Ψt , (−∆X + ρ−2 (L GF 4 1 −2 2
= Ψt , Gt Ψt GF + Ψt , ρ (L − )Ψt GF (10) 4
t , t(H
t , t2 (V 1 − V10 )Ψ
t
F − H
F (η 0 ))Ψ
t + Ψ . + Ψ GF GF For the first term in (10), we find for t → ∞,
t
t , Gt Ψ Ψ = T (t)Φ0 , j1,t Gt j1,t T (t)Φ0 GF GF 1 2 1 2 = T (t)Φ0 , ( + |∇X j1,t Gt + Gt j1,t j1,t |2 )T (t)Φ0 GF 2
= O(t
4/5
2
),
by ∇ , and we used that Gt T (t)Φ0 = 0 where we denoted the gradient on X X 2 4/5 and ∇X j1,t ∞ = O(t ). By rotation invariance of Φ0 in the v3 , v4 variables, the second term in (10) is an order one term. The estimate 2 2 (V1 − V10 )| ≤ const · t2 |η − η0 |3 ≤ const · t2 · t−6/5 |t2 j1,t j1,t
(11)
t , t2 (V 1 − V 0 )Ψ
t GF = O(t4/5 ). And a similar estimate, yields Ψ 1 2 2
F )| ≤ const · t |t j1,t (HF (η0 ) − H j1,t |η − η0 | ≤ const · t · t−2/5 ,
gives
t , t(H
F − H
F (η0 ))Ψ
t Ψ = O(t3/5 ) , GF
(12)
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as t → ∞. Collecting terms, we find Ψt , Kt Ψt = O(t4/5 ) , which implies that limt→∞ E1 (t)/t = 0. This shows the first part of (4). To prove the second part, i.e., lim inf E2 (t)/t ≥ r > 0 ,
(13)
t→∞
it suffices to show that there exists an r > 0 such that Kt ≥ (t · r + o(t))1I + Rt ,
(14)
where Rt is a symmetric, rank one operator. To see this, suppose (14) holds. Let ω1,t and ω2,t be the eigenvectors to the eigenvalues E1 (t) and E2 (t) of Kt , respectively. There exists a ωt ∈ Span{ω1,t , ω2,t } in the kernel of Rt . Hence E2 (t)ωt 2 ≥ ωt , Kt ωt ≥ (t · r + o(t))ωt 2 which implies (13). To show (14), we use the IMS localization formula 1
=
Kt
ja,t Kt ja,t + j2 Kt j2 −
a=0
1
|∇ja,t |2 − |∇j2 |2 .
(15)
a=0
Now, supp(j0,t ) ⊂ {ξ ∈ X|dist (ξ, Γ) ≥ t−2/5 }. We have ∇ja,t 2∞ = O(t4/5 ) for a = 0, 1, and ∇j2 2∞ = O(1). We estimate j0,t Kt j0,t
≥
t2 j0,t V1 j0,t + tj0,t HF j0,t
≥
2 (t2 t−4/5 cV − tcF )j0,t ,
≥
t·
2 rj0,t
,
for some cV > 0, cF > 0
for some r > 0 ,
(16)
F) and for t large. By fixing the gauge, we have on L2 (X;
t
j1,t K j1,t
0
1 − V 1 ) = j1,t Gt j1,t t2 (V j1,t j1,t + 1
F − H
F (η0 ))
2 ) + j1,t t(H j1,t + j1,t ρ−2 (− + L j1,t 4
≥ j1,t Gt j1,t + O(t4/5 ) ≥ j1,t t r 1 − |T (t)Φ0 GF · GF T (t)Φ0 | j1,t + O(t4/5 ) ,
2 , the estimates (11, 12), for some r > 0, where we have used the positivity of L and the gap in the spectrum of Gt . On H, this yields 2 − t · r|Ψt · Ψt | + O(t4/5 ) . j1,t Kt j1,t ≥ t · rj1,t
(17)
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To estimate the term j2 Kt j2 , we recall the explicit form of Kt : 1 Kt = pi pi + pµ pµ + t2 |x|2 |q|2 + |q|4 + 1 − (q12 + q22 ) + (q32 + q42 ) 4
+t(−ixµ ψγ µ s2 ψ + 2iqj λsj s2 ψ) . We recall the notation ξ = (q, x). Define a function θ ∈ C ∞ (X) with√θ(ξ) = θ(|q|), 0 ≤ θ ≤ 1, θ(|q|) = 1 if |q| > 4 and θ(|q|) = 0 if |q| < 3. Define θ¯ := 1 − θ2 . Then ¯ 2 )j2 . ¯ t θj ¯ 2 − j2 (|∇θ|2 + |∇θ| j2 Kt j2 = j2 θKt θj2 + j2 θK ¯ 2 = The localization error gives order 1 contributions, i.e., ∇θ2∞ = O(1), ∇θ ∞ O(1). First we consider the case where |q| is large and estimate (see Appendix B (i) for the terms containing fermions) pµ pµ ≥ 0 ,
−ixµ ψγ µ s2 ψ ≥ −4|x| ,
2iqj λsj s2 ψ ≥ −8|q| ,
pi pi + t2 |x|2 |q|2 + t(−ixµ ψγ µ s2 ψ) ≥ pi pi + t2 |x|2 |q|2 − 4t|x| ≥ 0 , where the last inequality follows from the ground state energy of the harmonic oscillator. This yields j2 θKt θj2
1 4
≥
j2 θ(t2 ( |q|4 − |q|2 + 1) − 8t|q|)θj2
≥
t2 · cj22 θ2 ,
for some c > 0 and t sufficiently large. For points ξ = (q, x) ∈ supp j2 , if |q| < 4, then |x| is large for sufficiently large R. We have ¯ t θj ¯2 j2 θK
≥ ≥ ≥
¯ i pi + t2 |x|2 (1 − |x|−2 )|q|2 − 4t|x| + t2 − 8t|q|)j2 θ¯ j2 θ(p ¯ j2 θ(t(4|x|(1 − |x|−2 )1/2 − 4|x|) + t2 − 32t)j2 θ¯ t2 · cj 2 θ¯2 , 2
for some c > 0 and t sufficiently large. Hence there exists an r > 0 such that for large t, j2 Kt j2 ≥ t · rj22 . (18) Now, inserting eqns. (16–18) into (15) yields (14) and therefore (13).
4.3
Proof of Proposition 4.
We decompose the Hilbert space H0 as a constant fiber direct integral [17], with fiber F := L2 (R4 ; F ), H0 =
⊕
R5
F dx ,
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the isomorphism being Ψ → (x → Ψx := Ψ(·, x)). The Hamiltonian has a direct integral decomposition, Hk = p µ p µ +
⊕
R5
Hk,x dx ,
where the fibers Hk,x , acting on F , are given by 1 Hk,x = Hx0 + |q|4 + 2iqj λsj s2 ψ + k 2 − k(q12 + q22 ) + k(q32 + q42 ) , 4 with
Hx0 := pi pi + |x|2 |q|2 − ixµ ψγ µ s2 ψ .
The scalar product, the norm and operator norm in F will be denoted by (·, ·)F and · F , respectively. Let Px be the projection onto the eigenspace of Hx0 corresponding to its lowest eigenvalue, which is, in fact, zero. We set Px⊥ := 1 − Px , and we define the projection P =
⊕
R5
Px dx ,
(19)
and its complement P ⊥ = 1 − P . As is shown in Appendix B (ii), for x = 0, RanPx = { Ξx · ξ | ξ ∈ F with (uψ)ξ = 0, ∀ u : −iγ µ xµ s2 u = |x|u} , where
1 Ξx (q) := (|x|π)−1 exp(− |x||q|2 ) . 2
Lemma 6. There exists an R > 0 and a constant c > 0 depending on k, such that for |x| > R Hk,x ≥ k 2 − c|x|−2 . Proof. Since all elements in RanPx are spherically symmetric in q it immediately follows that (20) Px Hk,x Px ≥ k 2 Px . We estimate, c.p. Appendix B (i), 2iqj λsj s2 ψ ≥ −8|q| ,
and − k(q12 + q22 ) ≥ −|x|−1 k(|x|−1 pi pi + |x||q|2 ) .
Hence Hk,x ≥ |x|(1 − |x|−1 )(|x|−1 pi pi + |x||q|2 ) + |x|−1 pi pi + |x||q|2 −ixµ ψγ µ s2 ψ − 8|q| − |x|−1 k(|x|−1 pi pi + |x||q|2 ) + k 2 ≥ |x|(1 − |x|−1 − k|x|−2 )(|x|−1 pi pi + |x||q|2 ) − ixµ ψγ µ s2 ψ − 16|x|−1 + k 2 ,
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where we used |x||q|2 − 8|q| ≥ −16|x|−1 in the last inequality. The range of Px⊥ is given by the closure of the set of linear combinations of states which are a product of an eigenstate of pi pi + |x|2 |q|2 and an eigenstate of −ixµ ψγ µ s2 ψ, excluding states which are a product of two ground states. Thus Px⊥ Hk,x Px⊥ ≥ (c0 |x| + k 2 )Px⊥ ,
(21)
for some c0 > 0 and large |x|. Using that |q|α Px F ≤ cα |x|−α/2 from the Gaussian decay of states in RanPx , ⊥ P Hk,x Px = Px⊥ |q|4 /4 − k(q12 + q22 ) + k(q32 + q42 ) + 2iqj λsj s2 ψ Px F x F ≤
c|x|−1/2
for some c > 0 and large |x|. By the selfadjointness of Hk,x also Px Hk,x P ⊥ ≤ c|x|−1/2 . x F
(22)
Let ux ∈ D(Hk,x ) ⊂ F . Then from (20), (21) and (22) (ux , Hk,x ux )F
2
ux 2F
Px ux F Px⊥ ux F
≥ k + Ak,x 2 ≥ k + inf ξ=1 (ξ, Ak,x ξ) ux 2F ,
with
Ak,x :=
0 −c|x|−1/2
−c|x|−1/2 c0 |x|
Px ux F Px⊥ ux F
.
We have inf ξ=1 (ξ, Ak,x ξ) ≥ −c|x|−2 , for some c > 0 and large |x|. Hence the Lemma follows.
Let R ≥ 1 be as in Lemma 6, and let η : R5 → R be a smooth function with η(x) = η(|x|), 0 ≤ η ≤ 1, ∇η∞ ≤ 1, η(x) = 0 for |x| ≤ R and η(x) = 1 for |x| ≥ 3R. The deformed supercharge 1 Q1,k = Q1 + k(γ 1 λ)2 = √ (Dk + Dk† ) 2
(23)
satisfies on H, 2(Q1,k )2 = {Dk , Dk† } = Hk . Hence for Ψ ∈ H, Hk Ψ = 0 iff Q1,k Ψ = 0. Lemma 7. Let Ψ ∈ H with Hk Ψ = 0. Then for any > 0, |x|−1/2− ek|x| ηΨ ∈ H.
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Proof. It is sufficient to show the claim for arbitrarily small . To prove the lemma, we use an Agmon estimate [12]. Let h : R5 → [0, ∞) be a smooth function such that the set K = {x ∈ R5 | k 2 − c|x|−2 − |∇h(x)|2 < 0 } is compact. Then, as we will show, η 2 Ψx 2F (k 2 − c|x|−2 − |∇h(x)|2 )e2h dx ≤ M0 Ψ2 .
(24)
R5
where M0 :=
sup R≤|x|≤3R
(1 + 2|∇h(x)|)e2h(x) < ∞ .
Define hα := h(1 + αh)−1 . Then, by Lemma 6, (ηehα Ψx , Hk,x ηehα Ψx )F dx (ηehα Ψ, Hk ηehα Ψ) ≥ 5 R ≥ η 2 e2hα (k 2 − c|x|−2 )Ψx 2F dx .
(25)
R5
We estimate hα ηe Ψ, Hk ηehα Ψ = 2 Q1,k ηehα Ψ, Q1,k ηehα Ψ = 2 [Q1,k , ηehα ]Ψ, [Q1,k , ηehα ]Ψ ≤ |∇(ηehα )|2 Ψ, Ψ ≤ (|∇η|2 + 2(∇η)(∇hα )η + |∇hα |2 η 2 )e2hα Ψ, Ψ . Inserting this into inequality (25), we obtain Iα := η 2 e2hα (k 2 − c|x|−2 − |∇hα |2 )Ψx 2F dx R5 ≤ (|∇η|2 + 2|∇η||∇hα |η)e2hα Ψ, Ψ ≤
M0 Ψ2 .
Using Fatou’s Lemma on the set K c and dominated convergence on K yields + η 2 Ψx 2F (k 2 − c|x|−2 − |∇h|2 )e2h dx ≤ lim inf Iα ≤ M0 Ψ2 K
α
Kc
and hence (24). We choose h such that on the support of η, h(x) = k|x| − log |x|. Then k 2 − c|x|−2 − |∇h(x)|2 = 2k|x|−1 − (c + 2 )|x|−2 ≥ k|x|−1 , for large |x|. Hence, by (24) dxη 2 Ψx 2F e2k|x| k|x|−2−1 < ∞ , R5
which proves the Lemma.
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Proof of Proposition 4. We recall that the deformed Hamiltonian commutes with the action of Spin(5). Let k be sufficiently large such that Ψ is the unique zero energy state of Hk . Thus Ψ belongs to a one-dimensional representation of Spin(5), and therefore it is Spin(5) invariant. Let R(S) denote the image of S under the canonical projection Spin(5) → SO(5). By R(S) we denote the spin part of the µν (λa λb + ψa ψb ). Then Spin(5) action, i.e., the representation generated by − 4i γab Ψ(q, R(S)x) = R(S)Ψ(q, x) ,
∀ S ∈ Spin(5) .
This implies that for x, x ∈ R5 with |x| = |x |, |Ψx |F = |Ψx |F . We set ω = x/|x|. Let dΩ denote the surface measure of the unit sphere. Then π 1 e−2k|x|±2kx dΩ(ω) = vol(S 3 ) e−2k|x|(1∓cos θ) sin3 θdθ S4
≤ vol(S 3 )
0
1
−1 −2
≤ const |x|
e−2k|x|(1−cos θ) 2(1 − cos θ) d cos θ .
For the ground state Ψ ∈ H of Hk , we have 1 1 e±2kx |Ψ(q, x)|2 dqdx = e±2kx |Ψx |2F dx 1 1 = (1 − η)e±2kx |Ψx |2F dx + ηe±2kx |Ψx |2F dx 1 ≤ const + ηe±2kx e−2k|x| e2k|x| |Ψx |2F dx ≤ const + const η|x|−2 e2k|x| |Ψx |2F dx <∞ , where in the last step we have used Lemma 7.
Appendix A In this appendix we mainly follow [14]. We consider the quaternions with generators 1, I, J, K satisfying the relations I 2 = −1 , J 2 = −1 , K 2 = −1 , IJK = −1 . A quaternion can be expanded as q = q1 1 + q2 I + q3 J + q4 K .
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The conjugate is given by q = q1 1 − q2 I − q3 J − q4 K . We note that qq = qq = sentation, with respect to 1, I, J, K, respectively. We 0 −1 1 0 IR = 0 0 0 0
|q|2 . By 1R , I R , J R , K R we denote the matrix reprethe basis (1, I, J, K), of the right multiplication with have 0 0 0 0 −1 0 0 0 J R = 0 0 0 −1 1 0 0 0 1 0 −1 0 0 1 0 0
KR
0 0 = 0 1
0 0 −1 0
0 −1 1 0 . 0 0 0 0
Note that (AB)R = B R AR with A, B ∈ {1, I, J, K}. We define the matrices R R 1 I 0 0 2 = s1 = , s , 0 1R 0 IR
3
s =
JR 0
0 JR
4
, s =
KR 0
0 KR
.
We remark that (sl )T = −sl for l = 2, 3, 4. We choose the gamma matrices as 1I4×4 0 1I4×4 0 1 2 γ = , , γ = 0 −1I4×4 1I4×4 0 γ3 = with IL =
0 −K L
−iσ 2 0
KL 0
0 −iσ 2
, γ4 =
, JL =
0 −I L
IL 0
0 σ3
−σ 3 0
, γ5 =
, KL =
0 −J L
0 σ1
JL 0
−σ 1 0
,
,
where σ i , i = 1, 2, 3, are the Pauli matrices and the superscript L indicates that the matrix corresponds to left multiplication. Using that left multiplication commutes with right multiplication one sees that [γ µ , sj ] = 0.
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Appendix B (i) Consider a real antisymmetric 16 × 16 matrix S and the Clifford generators denoted as (ϑ1 , . . . , ϑ8 , ϑ9 , . . . , ϑ16 ) = (ψ1 , . . . , ψ8 , λ1 , . . . , λ8 ). We will show that the map 16 ϑa Sab ϑb : F → F i a,b=1
has a ground state ξ ∈ F, which is determined by the condition that 16
v a ϑa ξ = 0
(26)
a=1
for all eigenvectors v of iS with strictly positive eigenvalue. The ground state √ energy is − 21 tr S t S. If S is invertible the ground state is unique. The matrix iS is hermitian. Let v be an eigenvector of iS with eigenvalue λ, then v is an eigenvector with eigenvalue −λ. Hence we have the spectral decomposition 8 iS = λj (P j+ − P j− ) , λj ≥ 0 , j=1 t
where P j± are orthogonal projectors with (P j± ) = P j∓ . This yields 16
ϑa iSab ϑb
=
8 j=1
a,b=1
=
8 j=1
16
λj
a,b=1
2λj
16 a,b=1
j+ j− ϑa Pab ϑb − ϑa Pab ϑb
j+ ϑa Pab ϑb −
8
λj .
j=1
8 Therefore, the ground state ξ satisfies (26) and has energy − j=1 λj = − 12 tr √ S t S. If S is invertible then there are exactly 8 linearly independent eigenvectors with strictly positive eigenvalue. By the irreducibility of F , the condition (26) then determines the ground state uniquely. (ii) Now, let us consider the special case −iψxµ γ µ s2 ψ. The vector ξ ∈ F is a ground state of −iψxµ γ µ s2 ψ if and only if (uψ)ξ = 0 for all u satisfying −iγ µ xµ s2 u = |x|u. We define Wx := { ξ ∈ F | ξ is a ground state of − iψxµ γ µ s2 ψ } .
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The operators λa leave this space invariant and act irreducibly on it. Thus dim Wx = 24 . The ground state of the harmonic oscillator pi pi + |x|2 |q|2 is 1 2
Ξx (q) = (|x|π)−1 exp(− |x|q 2 ) . By Px we denote the projection onto the ground state of Hx0 = pi pi + |x|2 |q|2 − iψxµ γ µ s2 ψ . The harmonic oscillator part commutes with the fermionic part. The ground state energy of Hx0 is zero and RanPx = { Ξx · ξ | ξ ∈ F with (uψ)ξ = 0, ∀ u : −iγ µ xµ s2 u = |x|u} .
Acknowledgment D.H. and L.E. want to thank the Mathematics Department of the University of Copenhagen, at which this work was started. J.P.S. wants to thank the Institute for Advanced Study, where part of this work was done. Moreover, D.H. wants to thank G.M. Graf, J. Hoppe, and J. Fr¨ ohlich for discussions.
References [1] M. Porrati and A. Rozenberg, Bound States at Threshold in Supersymmetric Quantum Mechanics, Nucl. Phys. B515, 184–202 (1998), hep-th/9708119. [2] B. de Wit, J. Hoppe, and H. Nicolai, On the quantum mechanics of supermembranes, Nucl. Phys. B305, 545–581, (1988). [3] E. Witten, Bound States of Strings and p–Branes, Nucl. Phys. B460, 335–350 (1996), hep-th/9510135. [4] T. Banks, W. Fischler, S.H. Shenker, and L. Susskind, M Theory as a Matrix Model: a Conjecture, Phys. Rev. D55, 5112–5128 (1997), hep-th/9610043. [5] S. Sethi and M. Stern, A Comment on the Spectrum of H–Monopoles, Phys. Lett. B398, 47–51 (1997), hep-th/9607145 . [6] V.G. Kac and A.V. Smilga, Normalized Vacuum States in N = 4 Supersymmetric Yang–Mills Quantum Mechanics with any Gauge Group, Nucl. Phys. B571 , 515–554 (2000), hep-th/9908096 . [7] E. Witten, Small Instantons in String Theory, Nucl. Phys. B460, 541 (1996), hep-th/9511030. [8] M. Berkooz and M.R. Douglas, Five-branes in m(atrix) theory, Phys. Lett. B395, 196–202 (1997), hep-th/9610236.
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[9] M.R. Douglas, D. Kabat, P. Pouliot, and S.H. Shenker, D-branes and short distances in string theory, Nucl. Phys. B485, 85–127 (1997), hep-th/9608024. [10] J. Polchinski, String Theory, Volume II. Cambridge University Press, 1998. [11] S. Sethi and M. Stern, Invariance Theorems for Supersymmetric Yang-Mills Theories, Adv. Theor. Math. Phys. 4, 487–501 (2000), hep-th/0001189. [12] S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schr¨ odinger Operators, Princeton University Press, 1982. [13] J. Fr¨ ohlich, G.M. Graf, D. Hasler, J. Hoppe, and S.-T. Yau, Asymptotic form of zero energy wave functions in supersymmetric matrix models, Nucl. Phys. B567, 213–248 (2000), hep-th/9904182. [14] S. Sethi and M. Stern, The Structure of the D0-D4 Bound State, Nucl. Phys. B578, 163–198 (2000), hep-th/0002131. [15] P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, CRC Press, 2nd edition, 1994. [16] H.L Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schr¨ odinger Operators, Springer–Verlag, 1986. [17] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV Analysis of Operators, Academic Press, New York, 1978. Laszlo Erd¨os1 Mathematisches Institut University Munich Theresienstr. 39 D-80333 Munich, Germany email:
[email protected] Jan Philip Solovej Department of Mathematics University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen, Denmark email:
[email protected] Communicated by Yosi Avron submitted 12/08/04, accepted 22/08/04 1
On leave from School of Mathematics, Georgia Tech.
David Hasler Department of Mathematics University of British Columbia V6T 1Z2 Vancouver, BC, Canada email:
[email protected]
Ann. Henri Poincar´e 6 (2005) 269 – 281 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020269-13 DOI 10.1007/s00023-005-0206-z
Annales Henri Poincar´ e
A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary Carmen Cortazar, Manuel Elgueta and Julio D. Rossi∗
Abstract. Let J : R → R be a nonnegative, smooth compactly supported function such that R J(r)dr = 1. We consider the nonlocal diffusion problem x−y J ut (x, t) = dy − u(x, t) in R × [0, ∞) u(y, t) R with a nonnegative initial condition. Under suitable hypotheses we prove existence, uniqueness, as well as the validity of a comparison principle for solutions of this problem. Moreover we show that if u(·, 0) is bounded and compactly supported, then u(·, t) is compactly supported for all positive times t. This implies the existence of a free boundary, analog to the corresponding one for the porous media equation, for this model.
1 Introduction Let J : R → R be a nonnegative, smooth function with R J(r)dr = 1. Assume also that J is supported in [−1, 1], is strictly increasing in [−1, 0] and strictly decreasing in [0, 1]. Equations of the form J(x − y)u(y, t)dy − u(x, t), (1.1) ut (x, t) = J ∗ u − u(x, t) = R
and variations of it, have been recently widely used to model diffusion processes, see [2], [4], [5], [6], [8]. As stated in [5] if u(x, t) is thought of as a density at the point x at time t and J(x − y) is thought of as the probability distribution of jumping from location y to location x, then (J ∗ u)(x, t) is the rate at which individuals are arriving to position x from all other places and −u(x, t) = − R J(y − x)u(x, t)dy is the rate at which they are leaving location x to travel to all other sites. This consideration, in the absence of external sources, leads immediately to the fact that the density u satisfies equation (1.1). Equation (1.1), so-called nonlocal diffusion equation, shares many properties with the classical heat equation ut = ∆u ∗ Supported by Universidad de Buenos Aires under grant TX048, by ANPCyT PICT No. 0300000-00137, by CONICET (Argentina) and by FONDECYT (Chile) project number 1030798 and Cooperacion Internacional 7040093.
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such as: bounded stationary solutions are constant, a maximum principle holds for both of them and, even if J is compactly supported, perturbations propagate with infinite speed. By this we understand that if u is a nonnegative nontrivial solution, then u(x, t) > 0 for all x ∈ R and all t > 0 no matter whether the nontrivial initial condition u(x, 0) vanishes in some region. Another classical equation that has been used to model diffusion is the wellknown porous medium equation, ut = ∆um with m > 1. This equation also shares several properties with the heat equation but there is a fundamental difference, in this case if the initial data u(·, 0) is compactly supported, then u(·, t) has compact support for all t > 0. In such a case, if the support of the initial condition is a finite interval, one can define the right and left free boundaries of the solution by s+ (t) = sup{x / u(x, t) > 0} and s− (t) = inf{x / u(x, t) > 0} respectively. Properties and the behavior of the free boundary for the porous medium equation have been largely studied over the past years. See for example [1], [7] and the corresponding bibliography. It is worth mentioning that this phenomena also arises in the context of the Stefan problem, see [3] and the references therein. The purpose of this note is to present a simple nonlocal model for diffusion whose solutions, with compactly supported bounded initial data, develop a free boundary. To do this we propose a model where the diffusion at a point depends on the density. The simplest situation we can think of is when the probability distribution of jumping from location y to location x is given by x−y 1 J u(y, t) u(y, t) when u(y, t) > 0 and 0 otherwise. In this case the rate at which individuals are arriving to position x from all other places is x−y J dy u(y, t) R and the rate at which they are leaving location x to travel to all other sites is y−x −u(x, t) = − J dy. u(x, t) R
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As before this consideration, in the absence of external sources, leads immediately to the fact that the density u has to satisfy x−y ut (x, t) = J dy − u(x, t). u(y, t) R As for the initial data, although we are mostly interested in functions u(·, 0) ∈ L1 (R) ∩ L∞ (R) it is more convenient, for technical reasons that will become clear later, to consider a slightly more general set of initial conditions. So in this paper we will deal with the problem x−y ut (x, t) = J dy − u(x, t) in R × [0, ∞). u(y, t) R (1.2) u(x, 0) = c + w0 (x) on R, where c ≥ 0, w0 ∈ L1 (R) and w0 ≥ 0. Most of the results contained in this note can be obtained in several dimensions without many changes in the elementary arguments but, we have chosen to treat the one-dimensional case for the sake of simplicity of the exposition. We will address in this paper the questions of existence, uniqueness, comparison principles and some basic facts about the free boundary for solutions of problem (1.2). Several further questions, such as the decay rate of solutions, the speed at which the free boundary moves, the existence of the so-called waiting times for the free boundary and many others, are left open. Also one can consider equations involving a source term and to study, for example, the blow-up phenomena. We hope such questions can be answered by us or by someone else in the near future.
2 Existence and uniqueness The existence and uniqueness result will be a consequence of Banach’s fixed point theorem and it is convenient to give some preliminaries before giving its proof. Fix t0 > 0 and consider the Banach space C([0, t0 ]; L1 ) with the norm |w| = max w(·, t)L1 . 0≤t≤t0
Let
Xt0 = w ∈ C([0, t0 ]; L1 ) / w ≥ 0
which is a closed subset of C([0, t0 ]; L1 ). We will obtain the solution in the form u(x, t) = w(x, t) + c where w is a fixed point of the operator Tw0 : Xt0 → Xt0 defined by t x−y e−(t−s) J Tw0 (w)(x, t) = dy ds w(y, s) + c 0 R +e−t w0 (x) − c(1 − e−t ).
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The following lemma is the main ingredient of our proof. Lemma 2.1 Let z0 , w0 be nonnegative functions such that w0 , z0 ∈ L1 (R) and w, z ∈ Xt0 , then |||Tw0 (w) − Tz0 (z)||| ≤ (1 − e−t0 )|||w − z||| + ||w0 − z0 ||L1 (R) . Proof. We have |Tw0 (w)(x, t) − Tz0 (z)(x, t)| dx R t x−y x−y e−(t−s) ≤ J −J dy dx ds w(y, s) + c z(y, s) + c 0 R R −t +e |w0 − z0 |(y) dy. R
Now set
A+ (s) = {y / w(y, s) ≥ z(y, s)}
and
A− (s) = {y / w(y, s) < z(y, s)}.
We have now x−y x−y J − J dy dx w(y, s) + c z(y, s) + c R
R
≤
A+ (s)
R
+ A− (s)
R
J J
x−y w(y, s) + c x−y z(y, s) + c
−J
−J
x−y z(y, s) + c
x−y w(y, s) + c
dy dx dy dx.
Since the integrands are nonnegative we can apply Fubini’s theorem to get x−y x−y J −J dy dx w(y, s) + c z(y, s) + c R A+ (s) = A+ (s)
(w(y, s) − z(y, s))dy
and similarly for the integral over A− (s). Therefore we obtain x−y x−y J − J dy dx w(y, s) + c z(y, s) + c R R |w(y, s) − z(y, s)| dy. ≤ R
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Hence we get |Tw0 (w) − Tz0 (z)| ≤ (1 − e−t0 )|w − z| + ||w0 − z0 ||L1 (R)
as desired. We can state now the main result of this section.
Theorem 2.1 For every nonnegative w0 ∈ L1 and every constant c ≥ 0, there exists a unique solution u, such that (u − c) ∈ C([0, ∞); L1 ), of 1.2. Moreover, the solution verifies u(x, t) ≥ c and preserves the total mass above c, that is (u(y, t) − c) dy = w0 (y) dy for all t ≥ 0. (2.1) R
R
Proof. We check first that Tw0 maps Xt0 into Xt0 . Since w ≥ 0 we have x−y x−y J ≥J w(y, s) + c c and hence Tw0 (w)(x, t) ≥
t
0
e−(t−s)
R
J
x−y c
dy ds (2.2)
+e−t w0 (x) − c(1 − e−t ) = e−t w0 (x) ≥ 0. Taking z0 ≡ 0, z ≡ 0 in Lemma 2.1 we get that Tw0 (w) ∈ C([0, t0 ]; L1 ). Now taking z0 ≡ w0 in Lemma 2.1 we get that Tw0 is a strict contraction in Xt0 and the existence and uniqueness part of the theorem follows from Banach’s fixed point theorem. We finally prove that if u = w + c is the solution, then the integral in x of w is preserved. Since t x−y 0= e−(t−s) J dy ds − c(1 − e−t ), c 0 R we can write w(x, t) =
0
t
e
−(t−s)
(J R
x−y w(y, s) + c
−J
x−y ) dy ds + e−t w0 (x). c
The integrand in the above formula is nonnegative so we can integrate in x and apply Fubini’s theorem to obtain t −(t−s) −t w(x, t)dx = e w(y, s) dy ds + e w0 (x)dx (2.3) R
0
R
R
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C. Cortazar, M. Elgueta and J.D. Rossi
from where it follows that
d dt
Ann. Henri Poincar´e
R
w(x, t)dx = 0
and the theorem is proved.
We will need in what follows the following lemma which is a direct corollary of the proof of Theorem 2.1 and is a first version of the comparison principle of Section 3 below. Lemma 2.2 With the above notation if 0 ≤ w(x, 0) ≤ M for all x ∈ R, then w(x, t) ≤ M for all (x, t) ∈ R × [0, ∞). Proof. Under the given hypotheses one has that if w(x, t) ≤ M , then Tw0 (w)(x, t) =
t
e
0
−(t−s)
≤
J
x−y w(y, s) + c
R t
0
e
−(t−s)
R
J
x−y M +c
dy ds + e−t w0 (x) − c(1 − e−t )
dy ds + e−t M − c(1 − e−t ) = M.
The lemma follows by the uniqueness of the fixed point for Tw0 .
Lemma 2.1, Theorem 2.1, Lemma 2.2 and their proofs have several immediate consequences that we state as a series of remarks for the sake of future references. Remark 2.1 Solutions of 1.2 depend continuously on the initial condition in the following sense. If u and v are solutions of 1.2, then max u(·, t) − v(·, t)L1 (R) ≤ et0 ||u(·, 0) − v(·, 0)||L1 (R)
0≤t≤t0
for all t0 ≥ 0. Remark 2.2 The function u is a solution of 1.2 if and only if t x−y −(t−s) u(x, t) = e J dy ds + e−t u(x, 0). u(y, s) 0 R Remark 2.3 From the previous remark and Lemma 2.2 we get that if c > 0 and u(·, 0) ∈ C k (R) with 0 ≤ k ≤ ∞, then u(·, t) ∈ C k (R) for all t ≥ 0. Moreover if u(·, 0) is a compactly supported C 1 function, then there exists a constant K depending on c, J and w0 such that ∂u |ut (x, t)| , (x, t) ≤ K. ∂x
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Remark 2.4 A consequence of Remark 2.3 and of (2.1) is that if c > 0 and w0 is a compactly supported C 1 function, then lim u(x, t) = c uniformly on compact intervals [0, T ].
|x|→∞
Remark 2.5 It follows from inequality (2.2) that w(x, t) ≥ e−t w(x, 0). In particular, in the case that u(·, 0) ∈ L1 (R), the support of u(·, t) does not shrink as time increases. By this we understand that if u(x0 , t0 ) > 0, then u(x0 , t) > 0 for all t ≥ t0 .
3 Comparison Principle Comparison principles like the one below have proven to be a very useful tool in studying diffusion problems. Theorem 3.1 Let u and v be continuous solutions of 1.2. If u(x, 0) ≤ v(x, 0) for all x ∈ R, then u(x, t) ≤ v(x, t) for all (x, t) ∈ R × [0, ∞).
(3.1)
Proof. We assume first that u(x, 0) = c + w(x, 0)
and
v(x, 0) = d + z(x, 0)
with 0 < c < d and u(x, 0) < v(x, 0). Moreover we assume for a moment that w(x, 0) and z(x, 0) are compactly supported C 1 functions. In this case there exists δ > 0 such that u(x, 0) + δ < v(x, 0). Assume, for a contradiction that the conclusion does not hold. In view of Remark 2.4 we have that there exists a time t0 > 0 and a point x0 ∈ R such that u(x0 , t0 ) = v(x0 , t0 ) and u(x, t) ≤ v(x, t) for all (x, t) ∈ R × [0, t0 ]. Let us consider the set B = {x ∈ R / u(x, t0 ) = v(x, t0 )}. Clearly B is nonempty and closed. Let x1 ∈ B. We have then x1 − y x1 − y 0 ≤ (u − v)t (x1 , t0 ) = J −J dy ≤ 0 u(y, t0 ) v(y, t0 ) R which implies u(y, t0 ) = v(y, t0 ) for all y ∈ (x1 − c, x1 + c).
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Hence B is open. It follows that B = R which is the desired contradiction since (u(·, t0 ) − c) ∈ L1 (R). We now get rid of the extra hypothesis that w(x, 0) and z(x, 0) are compactly supported C 1 functions. In order to do this let wn (x, 0) and zn (x, 0) be sequences of compactly supported C 1 functions such that wn (x, 0) → w(x, 0) and zn (x, 0) → z(x, 0) in L1 (R) as n → ∞ and, moreover, un (x, 0) = c + wn (x, 0) < vn (x, 0) = d + zn (x, 0). Let un and vn be the solutions with initial data un (x, 0) and vn (x, 0) respectively. By the previous argument one has un ≤ vn an the result follows by letting n → ∞ in view of Remark 2.1. In order to prove the theorem in the general case pick strictly decreasing sequences an and bn such that 0 < an < bn and bn → 0 as n → ∞. Let un and vn be the solutions with initial conditions un (x, 0) = u(x, 0) + an and vn (x, 0) = v(x, 0) + bn respectively. According to the previous argument one has un ≤ vn . Moreover un+1 ≤ un and vn+1 ≤ vn . By Remark 2.2, after an application of the monotone convergence theorem, it follows that un (x, t) → u(x, t) and vn (x, t) → v(x, t) as n → ∞ and the theorem is proved. An immediate consequence of the comparison principle and Remark 2.4 is the following corollary that extends Remark 2.4 to the case c = 0. Corollary 3.1 If c = 0 and w0 is a compactly supported C 1 function, then lim u(x, t) = 0 uniformly on compact intervals [0, T ].
|x|→∞
4 The free boundary In this section we will prove that solutions of (1.2), with compactly supported continuous initial data, do have a free boundary in the sense that s+ (t) = sup{x / u(x, t) > 0} < +∞ and s− (t) = inf{x / u(x, t) > 0} > −∞ for all t ≥ 0. It follows from Remark 2.5 that s+ and s− are nondecreasing and nonincreasing functions respectively. Moreover we will also prove in this section that the supports of u(·, t) eventually fill at least half a ray of the space, in particular either lim s+ (t) = ∞ or lim s− (t) = −∞. In the case that J is even, that is t→∞ t→∞ the case of an isotropic media, the supports eventually cover the whole of R. The following theorem implies the existence of free boundaries. Theorem 4.1 If u(·, 0) is compactly supported and bounded then u(·, t) is also compactly supported for all t ≥ 0.
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Proof. Due to the scaling invariance of the equation, namely if u(x, t) is a solution then for any λ > 0 the function vλ (x, t) = λu( xλ , t) is also a solution, we can restrict ourselves to initial data supported in [−1, 1] and such that sup u(x, 0) ≤ 1. x∈R
We note first that
ut (x, t) ≤
R
J
x−y u(y, t)
dy.
(4.1)
Therefore, since 0 ≤ u ≤ 1, we get by (4.1) that u(x, t) ≤
1 1 for all t ≤ and all x such that |x| ≥ 1. 2 2
Now if |x| ≥ 2 and t ≤ 12 we have that |x − y| ≤ u(y, t) implies that |y| ≥ 1 and hence u(y, t) ≤ 12 . Therefore, again by (4.1), we have u(x, t) ≤
1 1 for all t ≤ and all x such that |x| ≥ 2. 4 2
We look now at the case |x| ≥ 2 + 12 and t ≤ 12 . In this case |x − y| ≤ u(y, t) implies that |y| ≥ 2 and hence u(y, t) ≤ 14 . Again by (4.1), we have 1 1 1 for all t ≤ and all x such that |x| ≥ 2 + . 8 2 2 Repeating this procedure we obtain by induction that for any integer n ≥ 1 one has u(x, t) ≤
u(x, t) ≤
1 2n+2
n
for all t ≤
1 1 and all x such that |x| ≥ 2 + . 2 2k k=1
It follows that the support of u(·, t) is contained in the interval [−3, 3] for all t ≤ 12 as we wanted to prove. In order to prove our next result we need a preliminary lemma. Lemma 4.1 If u(x, 0) is continuous and not constant, then the function M (t) = max u(x, t) x∈R
is strictly decreasing. Proof. It is clear, by comparison with a constant, that M (t) decreases as t increases. Moreover by Remark 2.5 one has M (t) > c for all t ≥ 0. Fix t0 ≥ 0 and let t1 > t0 . Let us consider the set C = {x / u(x, t1 ) = M (t0 )}.
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The set C is clearly closed. Since u(x, t) ≤ M (t0 ) for all t ≥ t0 we have that at any point x0 ∈ C one must have x0 − y 0 ≤ ut (x0 , t1 ) = J dy − u(x0 , t1 ) ≤ 0. u(y, t1 ) R This implies that u(x, t1 ) = M (t0 ) for all x in a neighborhood of x0 and hence C is open. Consequently either C = R or C is empty. It is clear that C = R, so C = ∅ and the lemma is proved. We are now in a position to prove that at least one of the free boundaries go to infinity. Theorem 4.2 Let u be the solution of problem 1.2 with c = 0 and w0 = 0. Then either or lim s− (t) = −∞ lim s+ (t) = ∞ t→∞
t→∞
and the supports of u(·, t) eventually cover an infinite half-ray of R. If J is an even function the supports eventually cover the whole of R. Proof. By comparison, and the invariance under translations of the equation, it is enough to prove the theorem under the assumptions that w0 ∈ C 1 , its support is the interval [−A, A] and it is symmetric with respect to the origin. We claim first that the support of u(·, t) is not uniformly bounded. Assume for a contradiction that there exists L > 0 such that u(x, t) = 0 for all x such that |x| ≥ L and all t ≥ 0. Since R u(x, t)dx = R u(x, 0)dx > 0 there exists C > 0 such that lim M (t) = C. t→∞
Let v(x, 0) be a smooth function supported in [−L − 1, L + 1] such that 0 ≤ v(x, 0) ≤ C and v(x, 0) ≡ C if x ∈ [−L, L]. Let us denote by v(x, t) the solution of (1.2) with this initial condition. By Lemma 4.1 we have that max v(x, 1) < C. x∈R
Now for any integer n > 0 let vn (x, 0) be a smooth compactly function supported in [−L − 2, L + 2] such that 0 ≤ v(x, 0) ≤ C + n1 and vn (x, 0) ≡ C + n1 if x ∈ [−L, L]. Assume further that vn+1 (x, 0) ≤ vn (x, 0) and denote by vn (x, t) the solution of (1.2) with initial condition vn (x, 0). By comparison it follows that vn+1 (x, t) ≤ vn (x, t).
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Using Remark 2.2 and the monotone convergence theorem one has vn (x, 1) → v(x, 1) in [−L − 2, L + 2] as t → ∞. Moreover, being the limit continuous the convergence is uniform by Dini’s theorem. Consequently there exists n0 such that max vn0 (x, 1) < C. x∈R
On the other hand there exists t0 such that u(x, t0 ) ≤ vn0 (x, 0). This implies, by comparison, that max u(x, t0 + 1) < C x∈R
a contradiction that proves the claim. We are ready now to prove the statement of the theorem. We claim that if there exists x0 ≥ A such that u(x0 , t) = 0 for all t ≥ 0, then u(x, t) = 0 for all (x, t) ∈ [x0 , ∞) × [0, ∞). Indeed, let d > 0 and we will prove that u(x, t) ≤ d for all x ≥ x0 and all t ≥ 0.
(4.2)
Since u(x0 , t) ≡ 0 one has u(x, t) ≤ |x − x0 | for all x ∈ R and all t ≥ 0. Moreover u(x, 0) = 0 for all x ≥ x0 . So if (4.2) does not hold, using Corollary 3.1, there exists a point x1 ∈ R with x1 ≥ x0 + d and a time t1 > 0 such that u(x1 , t1 ) = d and u(x, t) ≤ d for all (x, t) ∈ R × [0, t1 ]. As in the proof of Theorem 3.1 we consider the set B = {x ≥ x0 + d / u(x, t1 ) = d} which is clearly closed. Also at a point x2 ∈ B one has x2 − y x2 − y J 0 ≤ (d − u)t (x2 , t1 ) = −J dy ≤ 0 d u(y, t0 ) R which implies u(y, t0 ) = d for all y ∈ (x2 − d, x2 + d).
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It follows that B is open and hence B = [x0 , ∞) which is a contradiction that proves (4.2). Since d > 0 was chosen arbitrarily the claim follows. An analog of the above claim holds for points −x1 < −A such that u(−x1 , t) = 0 for all t ≥ 0. Such a points x0 and x1 can not exist simultaneously because this contradicts the fact that the supports of u(·, t) are not uniformly bounded. This, plus the fact that if J and u(·, 0) are even functions then u(·, t) is even for all t ≥ 0, proves the theorem. Finally we give an example of a nonsymmetric function J such that the supports of solutions u(·, t), with compactly supported bounded initial data, do not eventually cover the whole of R. We will show that for a special choice of J the function 0 if x ≤ 0 u(x) = x+ = x if x ≥ 0 satisfies
0= R
J
x−y u(y)
dy − u(x).
(4.3)
It is immediate that if x ≤ 0, then x−y J dy = 0 u(y) R and hence (4.3) is satisfied. As for the case x > 0 we have that
R
J
=
x−y u(y)
∞ x 2
J
=x
|x−y| y+
≤ 1 implies 0 < x ≤ 2y and hence
dy − u(x) x − 1 dy − x y
dr J(r) −1 . (1 + r)2 −1 1
Now we choose J such that, in addition to the hypotheses already made, satisfies 1 dr J(r) =1 (1 + r)2 −1 and (4.3) also holds. The desired example follows now by a comparison argument, like the one of the proof of Theorem 3.1, using the function x+ , or a translation of it, as a barrier.
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References [1] D.G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems, A. Fasano and M. Primicerio eds. Lecture Notes in Math. 1224, Springer Verlag, (1986). [2] P. Bates, P- Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions. Arch. Rat. Mech. Anal. 138, 105–136 (1997). [3] J.R. Cannon, Mario Primicerio, A Stefan problem involving the appearance of a phase, SIAM J. Math. Anal. 4, 141–148 (1973). [4] X. Chen, Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations, Adv. Differential Equations 2, 125–160 (1997). [5] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, 153–191, Springer, Berlin, 2003. [6] C. Lederman and N. Wolanski, A free boundary problem from nonlocal combustion, preprint. [7] J.L. Vazquez, An introduction to the mathematical theory of the porous medium equation, in “Shape optimization and free boundaries” (M.C. Delfour ed.), Dordrecht, Boston and Leiden, 347–389, 1992. [8] X. Wang, Metaestability and stability of patterns in a convolution model for phase transitions, preprint. Carmen Cortazar, Manuel Elgueta and Julio D. Rossi Departamento de Matem´atica Universidad Cat´ olica de Chile Casilla 306, Correo 22 Santiago Chile email:
[email protected] email:
[email protected] email:
[email protected] Communicated by Rafael D. Benguria submitted 29/01/04, accepted 09/09/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 283 – 308 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020283-26 DOI 10.1007/s00023-005-0207-y
Annales Henri Poincar´ e
The Darwin Approximation of the Relativistic Vlasov-Maxwell System Sebastian Bauer and Markus Kunze Abstract. We study the relativistic Vlasov-Maxwell system which describes large systems of particles interacting by means of their collectively generated forces. If the speed of light c is considered as a parameter then it is known that in the Newtonian limit c → ∞ the Vlasov-Poisson system is obtained. In this paper we determine the next order approximate system, which in the case of individual particles usually is called the Darwin approximation.
1 Introduction and main results The relativistic Vlasov-Maxwell system ∂t f + vˆ · ∇x f + (E + c−1 vˆ × B) · ∇v f c∇ × B c∇ × E = −∂t B, ∇ · E = 4πρ, ∇·B ρ := f dv, j
= 0, = =
∂t E + 4πj, 0, := vˆf dv,
(RVMc)
describes the time evolution of a single-species system of particles (with mass and charge normalized to unity) which interact by means of their collectively generated forces. The distribution of the large number of particles in configuration space is modelled through the non-negative density function f (x, v, t), depending on position x ∈ R3 , momentum v ∈ R3 , and time t ∈ R, where vˆ = (1 + c−2 v 2 )−1/2 v ∈ R3
(1.1)
is the relativistic velocity associated to v. The Lorentz force E + c−1 vˆ × B realizes the coupling of the Maxwell fields E(x, t) ∈ R3 and B(x, t) ∈ R3 to the Vlasov equation, and conversely the density function f enters the field equations via the scalar charge density ρ(x, t) and the current density j(x, t) ∈ R3 , which act as source terms for the Maxwell equations. It is supposed that collisions in the system are sufficiently rareso that they can be neglected. The parameter c denotes the speed of light, and always means R3 . At time t = 0, the initial data f (x, v, 0) = f ◦ (x, v),
E(x, 0) = E ◦ (x),
and B(x, 0) = B ◦ (x)
are prescribed. In this work we treat the speed of light as a parameter and study the behavior of the system as c → ∞. Conditions will be established under which
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the solutions of (RVMc) converge to a solution of an effective system. We recall that in [21] it has been shown that as c → ∞ the solutions of (RVMc) approach a solution of the Vlasov-Poisson system with the rate O(c−1 ); see [1, 5] for similar results and [15] for the case of two spatial dimensions. The respective Newtonian limits of other related systems are derived in [20, 4]. It is the goal of this paper to replace the Vlasov-Poisson system by another effective equation to achieve higher order convergence and a more precise approximation. This will lead to an effective system whose solution stays as close as O(c−3 ) to a solution of the full VlasovMaxwell system, if the initial data are matched appropriately. In the context of individual particles, this post-Newtonian order of approximation is usually called the Darwin order, see [23, 13] and the references therein. Let us also mention that weak convergence properties of other kinds of Darwin approximations for the Vlasov-Maxwell system were studied in [6, 2]. In the present paper we mainly view the Darwin approximation as a rigorous intermediate step towards the next order, where in analogy to the case of individual particles [14] radiation effects are expected to play a role for the first time. Since at the radiation order the corresponding dynamics of the Vlasov-Maxwell system most likely will have to be restricted to a center manifold-like domain in the infinite dimensional space of densities (to avoid “run-away”-type solutions [23, 14]), it is clear that several new mathematical difficulties will have to be surmounted in this next step. Then the ultimate goal would be to determine the effective equation for the Vlasov-Maxwell system on the center manifold, which should finally lead to a slightly dissipative Vlasov-like equation, free of “run-away” solutions; see [11, 12] for a model of this equation and more motivation. Compared to systems of coupled individual particles, for the Vlasov-Maxwell system one immediately encounters the problem that so far in general only the existence of local solutions is known. These solutions are global under additional conditions, for instance if a suitable a priori bound on the velocities is available; see the pioneering work [8], and also [10, 3], where this result is reproved by different methods. This means that from the onset we will have to restrict ourselves to solutions of (RVMc) which are defined on some time interval [0, T ] that may be very small. On the other hand, in [21] it has been shown that such a time interval can be found which is uniform in c ≥ 1, so it seems reasonable to accept this restriction. In order to find the desired higher-order effective system, we formally expand all quantities arising in (RVMc) in powers of c−1 : f E
= =
f0 + c−1 f1 + c−2 f2 + · · · , E0 + c−1 E1 + c−2 E2 + · · · ,
B ρ
= =
B0 + c−1 B1 + c−2 B2 + · · · , ρ0 + c−1 ρ1 + c−2 ρ2 + · · · ,
j0 + c−1 j1 + c−2 j2 + · · · , where ρk = fk dv and jk = vfk dv for k = 0, 1, 2, . . .. Moreover, vˆ = v − (c−2 /2)v 2 v + · · · by (1.1), where v 2 = |v|2 . The expansions can be substituted into j
=
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(RVMc), and comparing coefficients at every order gives a sequence of equations for these coefficients. At zeroth order we obtain ∇ × E0 = 0,
∇ · E0 = 4πρ0 ,
∇ × B0 = 0,
∇ · B0 = 0.
(1.2)
If we set B0 = 0, then the Vlasov-Poisson system ∂t f0 + v · ∇x f0 + E0 · ∇v f0 = 0, E0 (x, t) = − |z|−2 z¯ ρ0 (x + z, t) dz, ρ0 = f0 dv, f0 (x, v, 0) = f ◦ (x, v),
(VP)
is found, with z¯ = |z|−1 z. Next we consider the equations at first order in c−1 . Here ∇×E1 = −∂t B0 = 0,
∇·E1 = 4πρ1 ,
∇×B1 = ∂t E0 +4πj0 ,
∇·B1 = 0, (1.3)
needs to be satisfied for the fields; also see [12]. Using (1.2), we get ∆B1 = −4π∇× j0 and therefore define (1.4) B1 (x, t) = |x − y|−1 ∇ × j0 (y, t) dy = |z|−2 z¯ × j0 (x + z, t) dz. Regarding the density f1 , we obtain the linear Vlasov equation ∂t f1 + v · ∇x f1 + E1 · ∇v f0 + E0 · ∇v f1 = 0. Hence if we suppose that f1 (x, v, 0) = 0, then we can set f1 = 0 and E1 = 0 consistently. The field equations at the order c−2 are ∇ × E2 = −∂t B1 ,
∇ · E2 = 4πρ2 ,
∇ × B2 = ∂t E1 + 4πj1 = 0,
∇ · B2 = 0.
Therefore we can define B2 = 0. Calculating the equation for the density f2 and taking into account (1.3), we arrive at the following inhomogeneous linearized Vlasov-Poisson system, for which
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we choose homogeneous initial data: 1 2 ∂t f2 + v · ∇x f2 − 2 v v · ∇x f0 + E0 · ∇v f2 + (E2 + v × B1 ) · ∇v f0 = 0, ∆E2 = ∂t2 E0 + 4π(∇ρ2 + ∂t j0 ), f2 (x, v, 0) = 0. (LVP) At this point we need to discuss the solvability of the Poisson equation for E2 . Restricting our attention to initial data f ◦ for (RVMc) with compact support, it will turn out below that both ρ2 and j0 have compact support and thus lead to unproblematic sources. The first term ∂t2 E0 has to be examined more closely. Since ρ0 (·, t) has compact support for all t, see (2.1) below, we can calculate the iterated Poisson integrals dy 1 ∂ 2 E0 (y, t) ∆−1 (∂t2 E0 )(x, t) = − 4π |x − y| t 1 dw = |z|−2 z¯ ∂t2 ρ0 (x + w + z, t) dz 4π |w| 1 2 = dy ∂t ρ0 (y, t) du |y − x − u|−1 |u|−3 u 4π 1 1 dy (y − x) ∂t2 ρ0 (y, t) = = z¯ ∂t2 ρ0 (x + z, t) dz (1.5) 2 |y − x| 2 1 z · v)¯ z ) ∂t f0 (x + z, v, t) dz dv, = |z|−1 (v − (¯ 2 where we used (VP), eq. (5.27) from the appendix, and ∂t ρ0 + ∇ · j0 = 0 in conjunction with an integration by parts, the continuity equation itself being a direct consequence of (VP). In view of (1.5) and (LVP) we thus define 1 E2 (x, t) = z¯ ∂t2 ρ0 (x + z, t) dz − |z|−1 ∂t j0 (x + z, t) dz 2 − |z|−2 z¯ ρ2 (x + z, t) dz. (1.6) By (1.6), (VP), and a further integration by parts, we obtain the alternative expression 1 z · v)2 − v 2 ) f0 (x + z, v, t) dz dv E2 (x, t) = |z|−2 z¯ (3(¯ 2 1 |z|−1 (1 + z¯ ⊗ z¯) (E0 ρ0 )(x + z, t) dz − |z|−2 z¯ ρ2 (x + z, t) dz. (1.7) − 2 The first aim of this paper is to show that fD
:=
f0 + c−2 f2 ,
ED
:=
E0 + c−2 E2 ,
B
D
:=
c
−1
B1 ,
(1.8)
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yields a higher-order pointwise approximation of (RVMc) than the Vlasov-Poisson system; we call (1.8) the Darwin approximation. It is clear that for achieving this improved approximation property also the initial data of (RVMc) have to be matched appropriately by the data for the Darwin system. For a prescribed initial density f ◦ , we are able to calculate (f0 , E0 ), B1 , and (f2 , E2 ) according to what has been outlined above. We then consider (RVMc) with initial data f (x, v, 0) = f ◦ (x, v), (IC) E(x, 0) = E ◦ (x) := E0 (x, 0) + c−2 E2 (x, 0), B(x, 0) = B ◦ (x) := c−1 B1 (x, 0). Before we formulate our main theorem let us recall that solutions of (RVMc) with initial data (IC) exist at least on some time interval [0, T ] which is independent of c ≥ 1; see [21, Thm. 1], and cf. Proposition 2.2 below for a more precise statement. This time interval [0, T ] is fixed throughout the paper. Theorem 1.1 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. From f ◦ calculate (f0 , E0 ), B1 , and (f2 , E2 ), and then define initial data for (RVMc) by (IC). Let (f, E, B) denote the solution of (RVMc) with initial data (IC) and let (f D , E D , B D ) be defined as in (1.8). Then there exists a constant M > 0, and also for every R > 0 there is MR > 0, such that |f (x, v, t) − f D (x, v, t)| D
|E(x, t) − E (x, t)| |B(x, t) − B D (x, t)|
≤ M c−3 −3
≤ MR c ≤ M c−3
(x ∈ R3 ), (|x| ≤ R), (x ∈ R3 ),
(1.9)
for all v ∈ R3 , t ∈ [0, T ], and c ≥ 1. The constants M and MR are independent of c ≥ 1, but do depend on the initial data. Note that if (RVMc) is compared to the Vlasov-Poisson system (VP) only, one obtains the estimate |f (x, v, t) − f0 (x, v, t)| + |E(x, t) − E0 (x, t)| + |B(x, t)| ≤ M c−1 ; see [21, Thm. 2B]. Approximate models have the big advantage that, since by now the VlasovPoisson system is well understood, the existence of (f0 , E0 ), and here also of B1 and (f2 , E2 ), does no longer pose serious problems; note that in (LVP) the equation for f2 is linear. Therefore one can hope to get more information on (RVMc) by studying the approximate equations. As a drawback of the above hierarchy, one has to deal with two densities f0 , f2 and two electric fields E0 , E2 to define f D and E D . Therefore it is natural to look for a model which can be written down using only one density and one field. It turns out that the appropriate (Hamiltonian) system is ∂t f + (1 − 12 c−2 v 2 )v · ∇x f + (E + c−1 v × B) · ∇v f = 0, c∇ × E = −∂t B, ∇ · E = 4πρ, (DVMc) c ∆B = −4π∇ × j, ρ = f dv, j = (1 − 12 c−2 v 2 )v f dv,
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which we call the Darwin-Vlasov-Maxwell system. We note that (f D , E D , B D ) solves (DVMc) up to an error of the order c−3 . Theorem 1.2 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. Then there exist c∗ ≥ 1 and T ∗ > 0 such that the following holds for c ≥ c∗ . (a) If there is a local solution of (DVMc), then the initial data E ◦ and B ◦ of (DVMc) at t = 0 are uniquely determined by the initial density f ◦ . (b) The system (DVMc) has a unique C 2 -solution (f ∗ , E ∗ , B ∗ ) on [0, T ∗] attaining that initial data (f ◦ , E ◦ , B ◦ ) at t = 0. This solution conserves the energy 1 2 1 −2 4 ∗ 1 v − c v f dx dv + |∇φ∗ |2 + |∇ ∧ A∗ |2 dx, H= 2 8 8π where the potentials φ∗ and A∗ are chosen in such a way that B ∗ = ∇ ∧ A∗ , ∇ · A∗ = 0, and −∇φ∗ = E ∗ + c−1 ∂t A∗ . (c) Let (f, E, B) denote the solution of (RVMc) with initial data (f ◦ , E ◦ , B ◦ ). Then there exists a constant M > 0, and also for every R > 0 there is MR > 0, such that |f (x, v, t) − f ∗ (x, v, t)|
≤
M c−3
(x ∈ R3 ),
|E(x, t) − E ∗ (x, t)|
≤
MR c−3
(|x| ≤ R),
∗
|B(x, t) − B (x, t)|
≤
Mc
−3
(x ∈ R3 ),
for all v ∈ R3 , t ∈ [0, min{T, T ∗}], and c ≥ c∗ . Instead of performing the limit c → ∞ in (RVMc) it is possible to reformulate Theorem 1.1 in terms of a suitable dimensionless parameter. Taking this viewpoint means that we consider (RVMc) at a fixed c (say c = 1) by rescaling a prescribed nonnegative initial density f ◦ , for which we suppose that f ◦ ∈ C ∞ (R3 × R3 ) has compact support. To be more precise, let v¯ = vˆf ◦ (x, v) dx dv, where vˆ is taken for c = ε−1/2 ; cf. (1.1). Then v¯ is viewed as an average velocity of the system. Now we introduce f ε,◦ (x, v) = ε3/2 f ◦ (εx, ε−1/2 v) and consider f ε,◦ for c = 1. It follows that √ √ ε ε,◦ v¯ = vˆf (x, v) dx dv = ε wf ˆ ◦ (y, w) dy dw = ε v¯,
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i.e., the system with initial distribution function f ε,◦ has small velocities compared to the system associated to f ◦ . Starting from f ◦ , we next determine (f0 , E0 ), B1 , and (f2 , E2 ), and then the initial data for (RVMc) via (IC) with c = ε−1/2 , as in Theorem 1.1. Next we note that (f, E, B) is a solution of (RVMc) with c = ε−1/2 if and only if f ε (x, v, t)
=
E ε (x, t) = B ε (x, t) =
ε3/2 f (εx, ε−1/2 v, ε3/2 t), ε2 E(εx, ε3/2 t), ε2 B(εx, ε3/2 t),
is a solution of (RVMc) with c = 1. We further introduce f0ε (x, v, t)
E0ε (x, t) B1ε (x, t)
f2ε (x, v, t) E2ε (x, t) ρε0 (x, t) j0ε (x, t) ρε2 (x, t)
= ε3/2 f0 (εx, ε−1/2 v, ε3/2 t), = ε2 E0 (εx, ε3/2 t), = ε5/2 B1 (εx, ε3/2 t), = ε5/2 f2 (εx, ε−1/2 v, ε3/2 t), = ε3 E2 (εx, ε3/2 t), = f0ε (x, v, t) dv = ε3 ρ0 (εx, ε3/2 t), = vf0ε (x, v, t) dv = ε7/2 j0 (εx, ε3/2 t), = f2ε (x, v, t) dv = ε4 ρ2 (εx, ε3/2 t).
Straightforward calculations then confirm the following statements: (a) (f0 , E0 ) is a solution to (VP) with initial data f ◦ if and only if (f0ε , E0ε ) is a solution to (VP) with initial data f ε,◦ , (b) B1 solves ∆B1 = −4π∇ × j0 if and only if B1ε solves ∆B1ε = −4π∇ × j0ε , (c) (f2 , E2 ) is a solution to (LVP) if and only if (f2ε , E2ε ) is a solution to 1 2 ε ε ε ε ε ε ε ε ∂t f2 + v · ∇x f2 − 2 v v · ∇x f0 + E0 · ∇v f2 + (E2 + v × B1 ) · ∇v f0 = 0, ∆E2ε = ∂t2 E0ε + 4π(∇ρε2 + ∂t j0ε ), ε f2 (x, v, 0) = 0. Therefore Theorem 1.1 may be reformulated in a way which parallels [13, Thm. 2.2], where the case of individual particles is considered √ which are far apart (of order O(ε−1 )) and have small velocities (of order O( ε)) initially. Note that in this result the Lorentz force is determined up to an error of order O(ε7/2 ), and the dynamics of the full and the effective system can be compared over long times of order O(ε−3/2 ); see [13, p. 448].
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Theorem 1.3 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. From f ◦ calculate (f0 , E0 ), B1 , and (f2 , E2 ), and then define initial data for (RVMc) by (IC) with c = ε−1/2 . Let (f, E, B) denote the solution of (RVMc) on [0, T ] for c = ε−1/2 with initial data (IC). Moreover, let f ε,◦ , f ε , E ε , B ε , f0ε , E0ε , B1ε , f2ε , E2ε , ρε0 , j0ε , and ρε2 be defined as above. Then (f ε , E ε , B ε ) is a solution of (RVMc) on [0, ε−3/2 T ] for c = 1 with initial data (f ε , E ε , B ε )(x, v, 0) = (f ε,◦ (x, v), E0ε (x, 0) + E2ε (x, 0), B1ε (x, 0)). In addition, there exists a constant M > 0, and also for every R > 0 there is MR > 0, such that |f ε (x, v, t) − f0ε (x, v, t) − f2ε (x, v, t)| |E ε (x, t) − E0ε (x, t) − E2ε (x, t)| |B ε (x, t) − B1ε (x, t)|
≤ M ε3 (x ∈ R3 ), 7/2 ≤ MR ε (|x| ≤ ε−1 R), ≤ M ε7/2
(x ∈ R3 ),
for all v ∈ R3 , t ∈ [0, ε−3/2 T ], and ε ≤ 1. The constants are independent of ε. By definition of the rescaled fields, these fields are slowly varying in their space and time variables, which means that we are considering an adiabatic limit. It is clear that also Theorem 1.2 could be restated in an analogous ε-dependent version. The paper is organized as follows. Some facts concerning (VP), (LVP), and (RVMc) are collected in Section 2. The proof of Theorem 1.1 is elaborated in Section 3, whereas Section 4 contains the proof of Theorem 1.2. For the proofs we will mostly rely on suitable representation formulas for the fields (refined versions of those used in [8, 21]), which are derived in the appendix, Section 5. Notation: B(0, R) denotes the closed ball in R3 with center at x = 0 or v = 0 and radius R > 0. The usual L∞ -norm of a function ϕ = ϕ(x) over x ∈ R3 is written as ϕ x , and if ϕ = ϕ(x, v), we modify this to ϕ x,v . For m ∈ N the W m,∞ -norms are denoted by ϕ m,x , etc. If T > 0 is fixed, then we write g(x, v, t, c) = Ocpt (c−m ), if for all R > 0 there is a constant M = MR > 0 such that |g(x, v, t, c)| ≤ M c−m
(1.10)
for |x| ≤ R, v ∈ R3 , t ∈ [0, T ], and c ≥ 1. Similarly, we write g(x, v, t, c) = O(c−m ), if there is a constant M > 0 such that (1.10) holds for all x, v ∈ R3 , t ∈ [0, T ], and c ≥ 1. In general, generic constants are denoted by M .
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2 Some properties of (VP), (LVP), and (RVMc) There is a vast literature on (VP), see, e.g., [7, Sect. 4] or [19] and the references therein. For our purposes we collect a few well-known facts about classical solutions of (VP). Proposition 2.1 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. Then there exists a unique global C 1 -solution (f0 , E0 ) of (VP), and there are nondecreasing continuous functions PVP , KVP : [0, ∞[→ R such that
f0 (t) x,v ≤ f ◦ x,v , supp f0 (·, ·, t) ⊂ B(0, PVP (t)) × B(0, PVP (t)),
f0 (t) 1,x,v + E0 (t) 1,x ≤ KVP (t),
(2.1)
for t ∈ [0, ∞[. This result was first established by Pfaffelmoser [18], and simplified versions of the proof were obtained by Schaeffer [22] and Horst [9]; a proof along different lines is due to Lions and Perthame [17]. For our approximation scheme we also need bounds on higher derivatives of the solution. This point was elaborated in [16], where it was shown that if f ◦ ∈ C k (R3 × R3 ), then (f0 , E0 ) possess continuous partial derivatives w.r.t. x and v up to order k. The existence of continuous time-derivatives then follows from the Vlasov equation. Thus (f0 , E0 ) are C ∞ , if f ◦ is C ∞ , and by a redefinition of KVP we can assume that
f0 (t) 3,x,v ≤ KVP (t),
t ∈ [0, ∞[.
(2.2)
The existence of a unique C 1 -solution (f2 , E2 ) of (LVP) follows by a contraction argument, but we omit the details. Furthermore it can be shown that there are nondecreasing continuous functions PLVP , KLVP : [0, ∞[→ R such that supp f2 (·, ·, t) ⊂ B(0, PLVP (t)) × B(0, PLVP (t)),
f2 (t) 1,x,v + E2 (t) 1,x ≤ KLVP (t),
(2.3) (2.4)
for t ∈ [0, ∞[. Concerning solutions of (RVMc), we have from [21, Thm. 1] the following Proposition 2.2 Assume that f ◦ ∈ C ∞ (R3 × R3 ) is nonnegative and has compact support. If E ◦ and B ◦ are defined by (IC), then there exits T > 0 (independent of c) such that for all c ≥ 1 the system (RVMc) with initial data (IC) has a unique C 1 solution (f, E, B) on the time interval [0, T ]. In addition, there are nondecreasing continuous functions (independent of c) PVM , KVM : [0, T ] → R such that f (x, v, t) = 0 if |v| ≥ PVM (t), |E(x, t)| + |B(x, t)| ≤ KVM (t), for all x ∈ R3 , t ∈ [0, T ], and c ≥ 1.
(2.5) (2.6)
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In fact E ◦ and B ◦ do not depend on c in [21, Thm. 1], but an inspection of the proof shows that the assertions remain valid for initial fields defined by (IC).
3 Proof of Theorem 1.1 In Section 5.1.1 below we will show that the approximate electric field E D from (1.8) admits the following representation: D D D E D = Eext + Eint + Ebd + Ocpt (c−3 ),
(3.1)
with D Eext (x, t) = − |z|−2 z¯ (ρ0 + c−2 ρ2 )(x + z, t) dz |z|>ct 1 |z|−1 ∂t j0 (x + z, t) dz + c−2 z¯ ∂t2 ρ0 (x + z, t) dz, (3.2) −c−2 2 |z|>ct |z|>ct D −2 −2 Eint (x, t) = − |z| z¯ (ρ0 + c ρ2 )(x + z, tˆ(z)) dz |z|≤ct −c−1 z · v)¯ z ) f0 (x + z, v, tˆ(z)) dv dz |z|−2 (v − 2(¯ |z|≤ct −2 z · v)v + v 2 z¯ − 3¯ z (¯ z · v)2 ) f0 (x + z, v, tˆ(z)) dv dz |z|−2 (2(¯ +c |z|≤ct −2 +c |z|−1 (¯ z ⊗ z¯ − 1)E0 ρ0 (x + z, ˆt(z)) dz, (3.3) |z|≤ct D (x, t) = c−1 (ct)−1 (¯ z · v)¯ z f ◦ (x + z, v) dv ds(z) Ebd |z|=ct −2 −1 +c (ct) ((¯ z · v)v − (¯ z · v)2 z¯) f ◦ (x + z, v) dv ds(z), |z|=ct
where the subscripts ‘ext’, ‘int’, and ‘bd’ refer to the exterior, interior, and boundary integration in z. We also recall that z¯ = |z|−1 z and tˆ(z) = t − c−1 |z|. On the other hand, according to Section 5.1.2 below we have E = Eext + Eint + Ebd + O(c−3 ),
(3.4)
with
1 |z|−2 z¯ ρ0 + t∂t ρ0 + t2 ∂t2 ρ0 (x + z, 0) dz 2 |z|>ct 1 −2 + c z¯ ∂t2 ρ0 (x + z, 0) dz − c−2 |z|−1 ∂t j0 (x + z, 0) dz, (3.5) 2 |z|>ct |z|>ct
Eext (x, t) = −
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|z|−2 z¯ρ(x + z, ˆ t(z)) dz |z|−2 (2(¯ z · v)¯ z − v) f (x + z, v, ˆt(z)) dv dz +c−1 |z|≤ct +c−2 |z|−2 (v 2 z¯ + 2(¯ z · v)v − 3(¯ z · v)2 z¯) f (x + z, v, ˆt(z)) dv dz |z|≤ct |z|−1 (¯ z ⊗ z¯ − 1)(Ef )(x + z, v, ˆt(z)) dv dz, (3.6) +c−2
Eint (x, t) = −
|z|≤ct
|z|≤ct
Ebd (x, t) =
D Ebd (x, t).
In order to verify (1.9), we start by comparing the exterior fields. Let x ∈ B(0, R) with R > 0 be fixed. Then we obtain from (3.5) and (3.2), due to |¯ z | = 1, and taking into account ρ2 (x, 0) =
f2 (x, v, 0) dv = 0
by (LVP), as well as (2.1), (2.2), (2.3), and (2.4), D |Eext (x, t) − Eext (x, t)| ≤ |z|−2 ρ0 (x + z, t) − ρ0 (x + z, 0) − t∂t ρ0 (x + z, 0) |z|>ct
−
|z|−1 |v| |∂t f0 (x + z, v, 0) − ∂t f0 (x + z, v, t)| dv dz |z|>ct 1 −2 + c |∂t2 f0 (x + z, v, t) − ∂t2 f0 (x + z, v, 0)| dv dz 2 |z|>ct
t −2 2 3 ≤M |z| (t − s) PVP (s) KVP (s)1B(0,PVP (s)) (x + z) ds dz + c−2
1 2 2 t ∂t ρ0 (x + z, 0) dzρ2 (x + z, 0)| dz 2
|z|>ct
+ M c−2
0
|z|>ct
+ M c−2
|z|>ct
−2
+ Mc 3 ≤ Mt
|z|−1
|z|>ct
|z|>ct
+ Mt c
|z|−2
−2
t 0 t 0
t
PLVP (s)3 KLVP (s)1B(0,PLVP (s)) (x + z) ds dz PVP (s)4 KVP (s)1B(0,PVP (s)) (x + z) ds dz 3
PVP (s) KVP (s)1B(0,PVP (s)) (x + z) ds dz
|z|−2 1B(0,R+M0 ) (z) dz
|z|>ct
≤ MR c−3 ;
0
|z|−1 (|z|−1 + 1 + |z|)1B(0,R+M0 ) (z) dz (3.7)
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note that here we have used M0 = max PVP (s) + KVP (s) + PLVP (s) + KLVP (s) < ∞, s∈[0,T ]
and for instance t3 |z|−2 1B(0,R+M0 ) (z) dz ≤ (ct)−3 t3 |z|>ct
|z|≤R+M0
To bound |Eint (x, t) −
D Eint (x, t)|,
(3.8)
|z| dz ≤ MR c−3 .
we first recall from [21, Thm. 2B] that
|E(x, t) − E0 (x, t)| = O(c−1 ).
(3.9)
Actually the initial conditions in [21] are different, but we only added terms of order c−2 , so that an inspection of the proof in [21] leads to (3.9). Next we define H(t) = sup {|f (x, v, s) − f D (x, v, s)| : x ∈ R3 , v ∈ R3 , s ∈ [0, t]},
as well as M1 = max
s∈[0,T ]
PVM (s) + PVP (s) + PLVP (s) < ∞.
Then f (x, v, s) = f0 (x, v, s) = f2 (x, v, s) = 0 for x ∈ R3 , |v| ≥ M1 , and s ∈ [0, T ]. Also if R0 > 0 is chosen such that f ◦ (x, v) = 0 for |x| ≥ R0 , introducing the constant M2 = R0 + T M1 + max PVP (s) + PLVP (s) < ∞ s∈[0,T ]
it follows that f (x, v, s) = f0 (x, v, s) = f2 (x, v, s) = 0 for |x| ≥ M2 , v ∈ R3 , and s ∈ [0, T ]. Let x ∈ B(0, R) with R > 0 be fixed. From (3.6), (3.3), (3.8), (3.9), (IC), and 0 ≤ tˆ(z) ≤ t for |z| ≤ ct we obtain D (x, t)| |Eint (x, t) − Eint −2 −2 ˆ ≤ |z| (f − f0 − c f2 )(x + z, v, t(z)) dv dz |z|≤ct −1 −2 −2 ˆ z · v)¯ z − v) (f − f0 − c f2 )(x + z, v, t(z)) dv dz |z| (2(¯ +c |z|≤ct z · v)¯ z − v) f2 (x + z, v, ˆt(z)) dv dz +c−3 |z|−2 (2(¯ |z|≤ct −2 −2 +c |z| (v 2 z¯ + 2(¯ z · v)v − 3(¯ z · v)2 z¯) |z|≤ct −2 ˆ (f − f0 − c f2 )(x + z, v, t(z)) dv dz −4 −2 2 2 ˆ +c |z| (v z¯ + 2(¯ z · v)v − 3(¯ z · v) z¯) f2 (x + z, v, t(z)) dv dz |z|≤ct +c−2 |z|−1 (1 − z¯ ⊗ z¯)([E − E0 ]f )(x + z, v, ˆt(z)) dv dz |z|≤ct
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≤
M (M13 + M14 )H(t) |z|−2 1B(0,M2 ) (x + z) dz |z|≤ct |z|−2 1B(0,M2 ) (x + z) dz +M M14 M0 c−3 |z|≤ct +M M15 H(t) c−2 |z|−2 1B(0,M2 ) (x + z) dz |z|≤ct |z|−2 1B(0,M2 ) (x + z) dz +M M15 M0 c−4 |z|≤ct +M M13 f ◦ x,v c−3 |z|−1 1B(0,M2 ) (x + z) dz
≤
MR (c−3 + H(t)),
|z|≤ct
since for instance |z|≤ct
|z|−2 1B(0,M2 ) (x + z) dz ≤
(3.10) |z|≤R+M2
|z|−2 dz ≤ MR .
D Recalling that the Ebd (x, t) = Ebd (x, t), we can summarize (3.4), (3.1), (3.7), and (3.10) as |E(x, t) − E D (x, t)| ≤ MR (c−3 + H(t)), (3.11)
for |x| ≤ R and t ∈ [0, T ]. Formulas (5.11), (5.13), (5.23), (5.17), (5.18), and (5.19), and an analogous (actually more simple) calculation also leads to |B(x, t) − B D (x, t)| ≤ M (c−3 + H(t)),
(3.12)
for x ∈ R3 and t ∈ [0, T ]. It remains to estimate h = f − f D . Using (RVMc), (1.8), (VP), and (LVP), it is found that ∂t h + vˆ · ∇x h + (E + c−1 vˆ × B) · ∇v h = −∂t f D − vˆ · ∇x f D − (E + c−1 vˆ × B) · ∇v f D 1 = v − c−2 v 2 v − vˆ · ∇x f0 + c−2 (v − vˆ) · ∇x f2 2 +(E D − E) · ∇v f0 + c−2 (E D − E) · ∇v f2 − c−4 E2 · ∇v f2 +c−2 ((v − vˆ) × B1 ) · ∇v f0 + c−1 (ˆ v × (B D − B)) · ∇v f0 − c−3 (ˆ v × B) · ∇v f2 . v | = (1 + c−2 v 2 )−1/2 |v| ≤ |v| ≤ M1 uniformly in c, and If |v| ≤ M1 , then also |ˆ hence 1 v − 1 − c−2 v 2 v ≤ M c−4 . ˆ 2 Next we note the straightforward estimate |B1 (x, t)| ≤ M for |x| ≤ M2 and t ∈ [0, T ], with B1 from (1.4). In view of the bounds (2.1), (2.4), and (2.6), thus by (3.11) and (3.12), |∂t h(x, v, t) + vˆ · ∇x h(x, v, t) + (E(x, t) + c−1 vˆ × B(x, t)) · ∇v h(x, v, t)| ≤ M (c−3 + H(t))
(3.13)
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for |x| ≤ M2 , |v| ≤ M1 , and t ∈ [0, T ]. But in {(x, v, t) : |x| > M2 } ∪ {(x, v, t) : |v| > M1 } we have h = f − f D = 0 by the above definition of M1 > 0 and M2 > 0. Accordingly, (3.13) is satisfied for all x ∈ R3 , v ∈ R3 , and t ∈ [0, T ]. Since h(x, v, 0) = 0, the argument from [21, p. 416] yields t H(t) ≤ M (c−3 + H(s)) ds, 0 −3
and therefore H(t) ≤ M c for t ∈ [0, T ]. Then due to (3.11) and (3.12), |E(x, t)− E D (x, t)| ≤ MR c−3 for |x| ≤ R and t ∈ [0, T ], as well as |B(x, t) − B D (x, t)| ≤ M c−3 for x ∈ R3 and t ∈ [0, T ]. This completes the proof of Theorem 1.1.
4 Proof of Theorem 1.2 In this section we will be sketchy and omit many details, since the proof is more or less a repetition of what has been said before. First let us assume that there is a C 2 -solution (f ∗ , E ∗ , B ∗ ) of (DVMc), existing on a time interval [0, T ∗ ] for some T ∗ > 0, such that supp f ∗ (·, ·, t) ⊂ R3 × R3 is compact for all t ∈ [0, T ∗ ]. Then ∗ −1 (4.1) |z|−2 z¯ × j ∗ (x + z, t) dz, B (x, t) = c ∆E ∗ (x, t)
= 4π∇ρ∗ (x, t) + c−1 ∂t ∇ × B ∗ (x, t). (4.2) 1 −2 2 ∗ ◦ ∗ ∗ Since f (x, v, 0) = f (x, v) and j (x, 0) = (1 − 2 c v )vf (x, v, 0) dv = (1 − 1 −2 2 v )vf ◦ (x, v) dv, it follows that B ∗ (x, 0) is determined by f ◦ . In order to 2 c compute the Poisson integral for E ∗ , we calculate by means of the transformation y = w − z, dy = dw, and using (5.27) below, c−1 ∆−1 (∂t ∇ × B ∗ )(x, t) 1 dy ∇ × ∂t B ∗ (y, t) = − 4πc |x − y|
1 dy −2 ∗ = − ∇y × |z| z¯ × ∂t j (y + z, t) dz 4πc2 |x − y| 1 = − z · ∇)∂t j ∗ (w, t) dw dz |z|−2 |x − w + z|−1 z¯ ∇ · (∂t j ∗ )(w, t) − (¯ 2 4πc 1 dw ∗ ∗ [x − w] ∇ · (∂ = j )(w, t) − ([x − w] · ∇)∂ j (w, t) t t 2c2 |x − w| dz 1 (1 + z¯ ⊗ z¯)∂t j ∗ (x + z, t). = − 2 2c |z| If we invoke the Vlasov equation for f ∗ and integrate by parts, this can be rewritten as c−1 ∆−1 (∂t ∇ × B ∗ )(x, t) 1 dz 1 1 − c−2 v 2 v ∂t f ∗ (x + z, v, t) dv (1 + z¯ ⊗ z¯) = − 2 2c |z| 2
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dz 1 1 −2 2 ∗ 2 2 c v f (x + z, v, t) dv z ¯ (3(¯ z · v) − v ) 1 − 2c2 |z|2 2 1 dz − 2 (1 + z¯ ⊗ z¯) ((E ∗ + c−1 v × B ∗ )f ∗ )(x + z, v, t) dv 2c |z| 1 dz + 4 (1 + z¯ ⊗ z¯) v 2 v ∂t f ∗ (x + z, v, t) dv. 4c |z|
Therefore the solution E ∗ of (4.2) has the representation E ∗ (x, t) = 4π∆−1 (∇ρ∗ )(x, t) + c−1 ∆−1 (∂t ∇ × B ∗ )(x, t) = − |z|−2 z¯ ρ∗ (x + z, t) dz 1 1 z · v)2 − v 2 ) 1 − c−2 v 2 f ∗ (x + z, v, t) dv dz |z|−2 z¯ (3(¯ + 2 2c 2 1 − 2 |z|−1 (1 + z¯ ⊗ z¯) ((E ∗ + c−1 v × B ∗ )f ∗ )(x + z, v, t) dv dz 2c 1 + 4 z · v)¯ z ) ∂t f ∗ (x + z, v, t) dv dz. (4.3) |z|−1 v 2 (v + (¯ 4c Comparison with (VP) and (1.7) reveals the analogy to E D at the relevant orders of c−1 . In particular, if we evaluate this relation at t = 0, the Banach fixed point theorem applied in Cb (R3 ) shows that for c ≥ c∗ sufficiently large the function E ∗ (x, 0) is uniquely determined by f ◦ (x, v) = f ∗ (x, v, 0). Thus f ◦ alone already fixes E ◦ and B ◦ . Concerning the local and uniform (in c) existence of a solution to (DVMc) and the conservation of energy, one can use (4.1) and (4.3) to follow the usual method by setting up an iteration scheme for which convergence can be verified on a small time interval; cf. [7, Sect. 5.8]. Finally, by similar arguments as used in the proof of Theorem 1.1 it can be shown that solutions of (DVMc) approximate solutions of (RVMc) up to an error of order c−3 .
5 Appendix 5.1
Representation Formulas
5.1.1 Representation of the approximation fields E D and B D Here we will derive the representation formula (3.1) for the approximate field E D from (1.8). Since the calculations for the electric and the magnetic field are quite similar, we will only analyze in detail the electric field and simply state the result for its magnetic counterpart. From (1.8) we recall E D = E0 + c−2 E2 , where E0 (x, t) = − |z|−2 z¯ ρ0 (x + z, t) dz, (5.1)
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E2 (x, t)
=
1 2
−
z¯ ∂t2 ρ0 (x + z, t) dz −
Ann. Henri Poincar´e
|z|−1 ∂t j0 (x + z, t) dz
|z|−2 z¯ ρ2 (x + z, t) dz,
(5.2)
cf. (VP) and (1.6). We split the domain of integration in {|z| > ct} and {|z| ≤ ct}, and to handle the interior part {|z| ≤ ct} we expand the densities w.r.t. t about the retarded time tˆ(z) := t − c−1 |z|. To begin with, we have −2 |z| z¯ ρ0 (x + z, t) dz = − |z|−2 z¯ ρ0 (x + z, ˆt(z)) dz − |z|≤ct |z|≤ct −1 −1 −2 1 ˆ −c |z| z¯ ∂t ρ0 (x + z, t(z)) dz − c z¯ ∂ 2 ρ0 (x + z, ˆt(z)) dz 2 |z|≤ct t |z|≤ct t 1 − |z|−2 z¯ (t − s)2 ∂t3 ρ0 (x + z, s) ds dz. (5.3) 2 |z|≤ct t(z) Using (2.1) and (2.2), the last term is Ocpt (c−3 ); note that |x| ≤ R for some R > 0 together with the support properties of f0 imply that we only have to integrate in z over a set which is uniformly bounded in c ≥ 1. Since ∂t ρ0 + ∇ · j0 = 0 by (VP), we also find −1 −1 −1 ˆ |z| z¯ ∂t ρ0 (x + z, t(z)) dz = c |z|−1 z¯ ∇x · j0 (x + z, ˆt(z)) dz −c |z|≤ct |z|≤ct −1 = c |z|−1 z¯ v · ∇x f0 (x + z, v, tˆ(z)) dv dz |z|≤ct −1 = c |z|−1 z¯ v · ∇z [f0 (x + z, v, tˆ(z))] + c−1 z¯ ∂t f0 (x + z, v, ˆt(z)) dv dz |z|≤ct
= I + II,
(5.4)
with I
=
|z|−1 z¯ v · ∇z [f0 (x + z, v, tˆ(z))] dv dz
−1 ∇z · |z|−1 z¯i v i=1,2,3 f0 (x + z, v, tˆ(z)) dv dz −c |z|≤ct +c−1 (ct)−1 z¯(¯ z · v)f ◦ (x + z, v) dv ds(z) |z|=ct −c−1 z · v)¯ z )f0 (x + z, v, tˆ(z)) dv dz |z|−2 (v − 2(¯ |z|≤ct z¯(¯ z · v)f ◦ (x + z, v) dv ds(z); +c−1 (ct)−1 c
−1
|z|≤ct
=
=
|z|=ct
(5.5)
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observe that tˆ(z) = 0 for |z| = ct was used for the boundary term. Similarly, by (VP), −2 z · v) ∂t f0 (x + z, v, tˆ(z)) dv dz |z|−1 z¯(¯ II = c |z|≤ct = −c−2 |z|−1 z¯(¯ z · v)(v · ∇x f0 + E0 · ∇v f0 )(x + z, v, tˆ(z)) dv dz |z|≤ct z · v)v)i=1,2,3 f0 (x + z, v, ˆt(z)) dv dz ∇z · (|z|−1 z¯i (¯ = c−2 |z|≤ct −c−2 (ct)−1 z¯(¯ z · v)2 f ◦ (x + z, v) dv ds(z) |z|=ct −c−3 z · v)2 ∂t f0 (x + z, v, tˆ(z)) dv dz |z|−1 z¯(¯ |z|≤ct −2 |z|−1 z¯ z¯ · E0 f0 (x + z, v, tˆ(z)) dv dz +c |z|≤ct −2 = c z · v)v + v 2 z¯ − 3¯ z (¯ z · v)2 ) f0 (x + z, v, tˆ(z)) dv dz |z|−2 ((¯ |z|≤ct −2 +c |z|−1 z¯ z¯ · E0 ρ0 (x + z, ˆt(z)) dz |z|≤ct −2 −1 (5.6) z¯(¯ z · v)2 f ◦ (x + z, v) dv ds(z) + Ocpt (c−3 ). −c (ct) |z|=ct
Next, due to (2.1) and (2.2) we also have 1 1 z¯ ∂t2 ρ0 (x + z, tˆ(z)) dz = −c−2 z¯ ∂ 2 ρ0 (x + z, t) dz + Ocpt (c−3 ). −c−2 2 |z|≤ct 2 |z|≤ct t (5.7) Thus so far by (5.1) and (5.3)–(5.7), −2 |z| z¯ ρ0 (x + z, t) dz − |z|−2 z¯ ρ0 (x + z, t) dz E0 (x, t) = − |z|>ct |z|≤ct = − |z|−2 z¯ ρ0 (x + z, t) dz |z|>ct −2 −2 1 ˆ |z| z¯ ρ0 (x + z, t(z)) dz − c z¯ ∂ 2 ρ0 (x + z, t) dz − 2 |z|≤ct t |z|≤ct −c−1 z · v)¯ z )f0 (x + z, v, ˆt(z)) dv dz |z|−2 (v − 2(¯ |z|≤ct +c−2 z · v)v + v 2 z¯ − 3¯ z(¯ z · v)2 ) f0 (x + z, v, ˆt(z)) dv dz |z|−2 ((¯ |z|≤ct |z|−1 z¯ z¯ · E0 ρ0 (x + z, ˆt(z)) dz +c−2 |z|≤ct
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+c−1 (ct)−1 −c−2 (ct)−1
|z|=ct
|z|=ct
Ann. Henri Poincar´e
(¯ z · v)¯ z f ◦ (x + z, v) dv ds(z) (¯ z · v)2 z¯ f ◦ (x + z, v) dv ds(z) + Ocpt (c−3 ).
(5.8)
D −2 Now we turn to E2 , cf. (5.2). Since E2 enters E with the factor c , we first note that c−2 12 |z|≤ct z¯ ∂t2 ρ0 (x + z, t) dz cancels a term on the right-hand side of (5.8). In addition, by analogous arguments, |z|−1 ∂t j0 (x + z, t) dz −c−2 |z|≤ct
t −2 −1 ˆ = −c |z| ∂t j0 (x + z, t(z)) dz − c |z| ∂t2 j0 (x + z, s) ds dz tˆ(z) |z|≤ct |z|≤ct = −c−2 |z|−1 v ∂t f0 (x + z, v, ˆ t(z)) dv dz + Ocpt (c−3 ) |z|≤ct |z|−1 v (v · ∇x f0 + E0 · ∇v f0 )(x + z, v, tˆ(z)) dv dz + Ocpt (c−3 ) = c−2 |z|≤ct = c−2 |z|−1 v v · ∇z [f0 (x + z, v, tˆ(z))] + c−1 z¯ ∂t f0 (x + z, v, ˆt(z)) dv dz |z|≤ct +c−2 |z|−1 v E0 · ∇v f0 (x + z, v, ˆt(z)) dv dz + Ocpt (c−3 ) |z|≤ct z · v)vf0 (x + z, v, tˆ(z)) dv dz |z|−2 (¯ = c−2 |z|≤ct +c−2 (ct)−1 (¯ z · v)vf ◦ (x + z, v) dv ds(z) |z|=ct −2 −c |z|−1 (E0 ρ0 )(x + z, tˆ(z)) dz + Ocpt (c−3 ). (5.9) −2
−1
|z|≤ct
Finally, −c−2
|z|≤ct
|z|−2 z¯ ρ2 (x+z, t) dz = −c−2
|z|≤ct
|z|−2 z¯ ρ2 (x+z, ˆt(z)) dz+Ocpt (c−3 ).
(5.10) Therefore if we write 1 c−2 E2 (x, t) = c−2 z¯ ∂t2 ρ0 (x + z, t) dz − c−2 |z|−1 ∂t j0 (x + z, t) dz 2 |z|>ct |z|>ct 1 −c−2 |z|−2 z¯ ρ2 (x + z, t) dz + c−2 z¯ ∂ 2 ρ0 (x + z, t) dz 2 |z|≤ct t |z|>ct |z|−1 ∂t j0 (x + z, t) dz − c−2 |z|−2 z¯ ρ2 (x + z, t) dz, −c−2 |z|≤ct
|z|≤ct
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use (5.9) and (5.10), and thereafter add the result to (5.8), it turns out that E D = E0 + c−2 E2 can be decomposed as claimed in (3.1). Similar calculations for B D (x, t) = c−1 B1 (x, t) using (1.4) yield B D (x, t) = c−1 |z|−2 z¯ × j0 (x + z, t) dz |z|>ct −1 |z|−2 z¯ × j0 (x + z, tˆ(z)) dz +c |z|≤ct −2 −2c z · v)(¯ z × v) f0 (x + z, v, ˆt(z)) dv dz |z|−2 (¯ |z|≤ct −2 +c |z|−1 z¯ × E0 ρ0 (x + z, ˆt(z)) dz (5.11) |z|≤ct (¯ z · v)(¯ z × v) f ◦ (x + z, v) dv ds(z) + O(c−3 ). −c−2 (ct)−1 |z|=ct
5.1.2 Representation of the Maxwell fields E and B In this section we will verify the representation formula (3.4) for the full Maxwell field E, by expanding the respective expressions from [8, 21] to higher orders. Once again the computation for the corresponding magnetic field B is very similar and therefore omitted. Let (f, E, B) be a C 1 -solution of (RVMc) with initial data (f ◦ , E ◦ , B ◦ ). We recall the following representation from [21, (A13), (A14), (A3)]:
where ED (x, t) EDT (x, t) ET (x, t) ES (x, t)
E
=
ED + EDT + ET + ES ,
(5.12)
B
=
BD + BDT + BT + BS ,
(5.13)
t = ∂t E (x + ctω) dω + ∂t E(x + ctω, 0) dω, 4π |ω|=1 |ω|=1 = −(ct)−1 z , vˆ)f ◦ (x + z, v) dv ds(z), KDT (¯ |z|=ct = − |z|−2 KT (¯ z , vˆ)f (x + z, v, ˆt(z)) dv dz, |z|≤ct = −c−2 |z|−1 KS (¯ z , vˆ)(E + c−1 vˆ × B)f (x + z, v, tˆ(z)) dv dz, t 4π
◦
|z|≤ct
and
t t B ◦ (x + ctω) dω + ∂t B(x + ctω, 0) dω, 4π |ω|=1 4π |ω|=1 (ct)−1 z , vˆ)f ◦ (x + z, v) dv ds(z), LDT (¯
BD (x, t)
=
BDT (x, t)
=
∂t
|z|=ct
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BT (x, t)
=
BS (x, t)
=
c−1
|z|≤ct
c−2
|z|≤ct
|z|−2 |z|−1
Ann. Henri Poincar´e
z , vˆ)f (x + z, v, ˆt(z)) dv dz, LT (¯
z , vˆ)(E + c−1 vˆ × B)f (x + z, v, ˆt(z)) dv dz, LS (¯
with z¯ = |z|−1 z and tˆ(z) = t − c−1 |z|. The kernels are given by z , vˆ) = KDT (¯ KT (¯ z , vˆ) = z , vˆ) = KS (¯
(1 + c−1 z¯ · vˆ)−1 (¯ z − c−2 (¯ z · vˆ)ˆ v ), (1 + c−1 z¯ · vˆ)−2 (1 − c−2 vˆ2 )(¯ z + c−1 vˆ), (1 + c−1 z¯ · vˆ)−2 (1 + c−2 v 2 )−1/2 z · vˆ)¯ z − vˆ) ⊗ vˆ − (¯ z + c−1 vˆ) ⊗ z¯ ∈ R3×3 , (1 + c−1 z¯ · vˆ) + c−2 ((¯
and z , vˆ) = (1 + c−1 z¯ · vˆ)−1 (¯ z × c−1 vˆ), LDT (¯ LT (¯ z , vˆ) = (1 + c−1 z¯ · vˆ)−2 (1 − c−2 vˆ2 )(¯ z × vˆ), −1 −2 −2 2 −1/2 (1 + c−1 z¯ · vˆ)¯ LS (¯ z , vˆ) = (1 + c z¯ · vˆ) (1 + c v ) z × (. . .)
−c−2 (¯ z × vˆ) ⊗ (c¯ z + vˆ) ∈ R3×3 .
Next we expand these fields in powers of c−1 . According to (2.5) we can assume that the v-support of f (x, ·, t) is uniformly bounded in x ∈ R3 and t ∈ [0, T ], say f (x, v, t) = 0 for |v| ≥ P := maxt∈[0,T ] PVM (t). Thus we may suppose that |v| ≤ P in each of the v-integrals, and hence also |ˆ v | = (1 + c−2 v 2 )−1/2 |v| ≤ |v| ≤ P uniformly in c. It follows that 1 vˆ = 1 − c−2 v 2 v + O(c−4 ). 2 For instance, for the kernel KDT of EDT this yields z , vˆ) KDT (¯
= (1 + c−1 z¯ · vˆ)−1 (¯ z − c−2 (¯ z · vˆ)ˆ v) −1 −2 2 = 1 − c z¯ · v + c (¯ z · v) + O(c−3 ) z¯ − c−2 (¯ z · v)v + O(c−4 ) z · v)¯ z + c−2 (¯ z · v)2 z¯ − c−2 (¯ z · v)v + O(c−3 ). = z¯ − c−1 (¯
If we choose R0 > 0 such that f ◦ (x, v) = 0 for |x| ≥ R0 , then −1 −(ct) O(c−3 )1B(0,R0 ) (x + z) dv ds(z)
= ct
|z|=ct
|ω|=1
|v|≤P
1B(0,R0 ) (x + ctω) ds(ω) O(c−3 ) = O(c−3 )
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by [21, Lemma 1], uniformly in x ∈ R3 , t ∈ [0, T ], and c ≥ 1. Therefore we arrive at −1 EDT (x, t) = −(ct) z · v)¯ z + c−2 [(¯ z · v)2 z¯ − (¯ z · v)v] z¯ − c−1 (¯ |z|=ct ◦
f (x + z, v) dv ds(z) + O(c−3 ).
(5.14)
Concerning ET , we note that f (x, v, t) = 0 for |x| ≥ R0 + T P =: R1 . Since, by distinguishing the cases |x − y| ≥ 1 and |x − y| ≤ 1, |z|−2 1B(0,R1 ) (x + z) dz = |x − y|−2 1B(0,R1 ) (y) dy = O(1) |z|≤ct
|x−y|≤ct
uniformly in x ∈ R3 , t ∈ [0, T ], and c ≥ 1, similar computations as before show that −2 z¯ + c−1 [v − 2(¯ ET (x, t) = − |z| z · v)¯ z ] + c−2 [3(¯ z · v)2 z¯ |z|≤ct
z · v)v] f (x + z, v, ˆt(z)) dv dz + O(c−3 ). −v z¯ − 2(¯ 2
(5.15)
In the same manner, elementary calculations using also (2.6) can be carried out to get ES (x, t) = −c−2 |z|−1 (1 − z¯ ⊗ z¯)(Ef )(x + z, v, tˆ(z)) dv dz +O(c BDT (x, t)
|z|≤ct −3
= (ct)−1
),
|z|=ct
(5.16)
c−1 z¯ × v − c−2 (¯ z · v)¯ z × v f ◦ (x + z, v) dv ds(z)
BT (x, t)
(5.17) +O(c−3 ), = c−1 |z|−2 (¯ z × v − c−1 2v · z¯z¯ × v)f (x + z, v, ˆt(z)) dv dz
BS (x, t)
+O(c−3 ), (5.18) = c−2 |z|−1 z¯ × (Ef )(x + z, v, ˆt(z)) dv dz + O(c−3 ). (5.19)
|z|≤ct
|z|≤ct
Next we consider the data term
t t ◦ ED (x, t) = ∂t E (x + ctω) dω + ∂t E(x + ctω, 0) dω, 4π |ω|=1 4π |ω|=1 =: III + IV. Since f2 (x, v, 0) = 0 by (LVP), we have ρ2 (x, 0) = 0.
(5.20)
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Thus we get from (IC), (VP), and (1.6), E ◦ (x)
= =
E0 (x, 0) + c−2 E2 (x, 0)
1 − |z|−2 z¯ ρ0 (x + z, 0) dz + c−2 z¯ ∂t2 ρ0 (x + z, 0) dz 2 − |z|−1 ∂t j0 (x + z, 0) dz .
Using the formulas (5.25), (5.26), and (5.24) below, we calculate − |z|−2 z¯ ρ0 (x + ctω + z, 0) dz dω |ω|=1 = − ρ0 (y, 0) |y − x − ctω|−3 (y − x − ctω) dω dy |ω|=1 |z|−2 z¯ ρ0 (x + z, 0) dz, = −4π |z|>ct z¯ ∂t2 ρ0 (x + ctω + z, 0) dz dω |ω|=1 |y − x − ctω|−1 (y − x − ctω) dω dy = ∂t2 ρ0 (y, 0) |ω|=1 1 z¯ − (ct)2 |z|−2 z¯ ∂t2 ρ0 (x + z, 0) dz = 4π 3 |z|>ct 8π + z ∂ 2 ρ0 (x + z, 0) dz, 3ct |z|≤ct t |z|−1 ∂t j0 (x + ctω + z, 0) dz dω − |ω|=1 = − ∂t j0 (y, 0) |y − x − ctω|−1 dω dy |ω|=1 4π −1 = −4π |z| ∂t j0 (x + z, 0) dz − ∂t j0 (x + z, 0) dz. ct |z|≤ct |z|>ct Therefore we get
t III = ∂t E ◦ (x + ctω) dω 4π |ω|=1
t = ∂t − t |z|−2 z¯ ρ0 (x + z, 0) dz + 2 z¯ ∂t2 ρ0 (x + z, 0) dz 2c |z|>ct |z|>ct t3 1 −2 2 − |z| z¯ ∂t ρ0 (x + z, 0) dz + 3 z ∂ 2 ρ0 (x + z, 0) dz 6 |z|>ct 3c |z|≤ct t t 1 − 2 |z|−1 ∂t j0 (x + z, 0) dz − 3 ∂t j0 (x + z, 0) dz c |z|>ct c |z|≤ct
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=
−
|z|>ct
1 − t2 2
|z|−2 z¯ ρ0 (x + z, 0) dz +
|z|>ct
+(ct)−1
1 −2 c 2
|z|>ct
|z|−2 z¯ ∂t2 ρ0 (x + z, 0) dz − c−2
|z|=ct
305
z¯ ∂t2 ρ0 (x + z, 0) dz
|z|>ct
|z|−1 ∂t j0 (x + z, 0) dz
z¯ ρ0 (x + z, 0) ds(z),
(5.21)
note that several terms have cancelled here. Now we discuss the second part IV of the data term ED , cf. (5.20). To begin with, by (IC), (1.4), and (VP), B(x, 0)
due to j0 = obtain
◦
−1
−1
|z|−2 z¯ × j0 (x + z, 0) dz = B (x) = c B1 (x, 0) = c = c−1 z × v)f0 (x + z, v, 0) dv dz, |z|−2 (¯ vf0 dv. Therefore using (VP) for f0 and integration by parts, we
|z|−2 (¯ z × v)f0 (x + z, v, 0) dv dz ∇ × B(x, 0) = c−1 ∇ × = c−1 z × v) dv dz |z|−2 ∇x f0 × (¯ z − (¯ z · ∇x f0 )v dv dz |z|−2 (v · ∇x f0 )¯ = c−1 −1 = c |z|−2 z¯ (−∂t f0 − E0 · ∇v f ) dv dz −c−1 |z|−2 v z¯ · ∇z [f0 (. . .)] dv dz |z|−2 z¯ ∂t f0 (x + z, v, 0) dv dz + c−1 4π vf0 (x, v, 0) dv, = −c−1 by observing that in general for suitable functions g, invoking the divergence theorem, |z|−2 z¯ · ∇z g(x + z) dz = g(x + εω) dω → 4πg(x), ε → 0. − |z|>ε
|ω|=1
t) = c∇ × B(x, From Maxwell’s equations we have ∂t E(x, t) − 4πj(x, t), and thus in view of j(x, 0) = vˆf (x, v, 0) dv = vˆf ◦ (x, v) dv = vˆf0 (x, v, 0) dv, ∂t E(x, 0) = =
c∇ × B(x, 0) − 4πj(x, 0) −2 − |z| z¯ ∂t f0 (x + z, v, 0) dv dz + 4π (v − vˆ)f0 (x, v, 0) dv.
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Hence due to (5.25), v − vˆ = O(c−2 ), and by [21, Lemma 1], t ∂t E(x + ctω, 0) dω IV = 4π |ω|=1 t = − |z|−2 z¯ ∂t f0 (x + ctω + z, v, 0) dv dz dω 4π |ω|=1 (v − vˆ)f ◦ (x + ctω, v) dv dω + c−1 (ct) |ω|=1 = −t |z|−2 z¯ ∂t ρ0 (x + z, 0) dz + O(c−3 ).
(5.22)
|z|>ct
If we combine (5.12), (5.20), (5.21), (5.22), (5.14), (5.15), and (5.16), then we see that (3.4) is satisfied. A similar calculation yields −1 |z|−2 z¯ × j0 (x + z, 0) dz BD (x, t) = c |z|>ct +c−1 t |z|−2 z¯ × ∂t j0 (x + z, 0) dz |z|>ct z¯ × j0 (x + z, 0) ds(z), (5.23) −c−1 (ct)−1 |z|=ct
and an analogous decomposition of B into B = Bext + Bint + Bbd + O(c−3 ).
5.2
Some explicit integrals
We point out some formulas that have been used in the previous sections. For z ∈ R3 and r > 0 an elementary calculation yields 4πr−1 : r ≥ |z| |z − rω|−1 dω = . (5.24) 4π|z|−1 : r ≤ |z| |ω|=1 Differentiation w.r.t. z gives −3 |z − rω| (z − rω) dω = |ω|=1
Similarly,
|ω|=1
|z − rω| dω =
4πr + 4π|z| +
and thus by differentiation |z − rω|−1 (z − rω) dω = |ω|=1
0 : r > |z| . 4π|z|−2 z¯ : r < |z|
4π 2 −1 3 z r 4π 2 −1 3 r |z|
4π¯ z−
: r ≥ |z| : r ≤ |z|
8π 3r z 4π 2 −2 z¯ 3 r |z|
(5.25)
,
: r > |z| : r < |z|
.
(5.26)
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Finally, for z ∈ R3 \ {0} also
z |z − v|−1 |v|−3 v dv = 2π¯
307
(5.27)
can be computed. Acknowledgments. The authors are indebted to G. Rein, A. Rendall and H. Spohn for many discussions.
References [1] K. Asano, S. Ukai, On the Vlasov-Poisson limit of the Vlasov-Maxwell equation, in Patterns and Waves, Eds. Nishida T., Mimura M. & Fujii H., Stud. Math. Appl., Vol. 18, North-Holland Publishing Co., Amsterdam 1986, pp. 369–383. [2] S. Benachour, F. Filbet, Ph. Lauren¸cot, E. Sonnendr¨ ucker, Global existence for the Vlasov-Darwin system in R3 for small initial data, Math. Methods Appl. Sci. 26, 297–319 (2003). [3] F. Bouchut, F. Golse, Ch. Pallard, On classical solutions to the 3D relativistic Vlasov-Maxwell system: Glassey-Strauss’ theorem revisited, Arch. Rational Mech. Anal. 170, 1–15 (2003). [4] S. Calogero, H. Lee, The non-relativistic limit of the Nordstr¨ om-Vlasov system, ArXiv preprint math-ph/0309030. [5] P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equation for infinite light velocity, Math. Methods Appl. Sci. 8, 533–558 (1986). [6] P. Degond, P. Raviart, An analysis of the Darwin model of approximation to Maxwell’s equations, Forum Math. 4, 13–44 (1992). [7] R.T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia 1996 [8] R.T. Glassey, W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal. 92, 59–90 (1986). [9] E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci. 16, 75–85 (1993). [10] S. Klainerman, G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal. 1, 103–125 (2002). [11] M. Kunze, A.D. Rendall, The Vlasov-Poisson system with radiation damping, Ann. H. Poincar´e 2, 857–886 (2001).
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[12] M. Kunze, A.D. Rendall, Simplified models of electromagnetic and gravitational radiation damping, Classical Quantum Gravity 18, 3573–3587 (2001). [13] M. Kunze, H. Spohn, Slow motion of charges interacting through the Maxwell field, Comm. Math. Phys. 212, 437–467 (2000). [14] M. Kunze, H. Spohn, Post-Coulombian dynamics at order c−3 , J. Nonlinear Science 11, 321–396 (2001). [15] H. Lee, The classical limit of the relativistic Vlasov-Maxwell system in two space dimensions, Math. Methods Appl. Sci. 27, 249–287 (2004). at der L¨ osungen des Vlasov-Poisson-Systems par[16] A. Lindner, C k -Regularit¨ tieller Differentialgleichungen, Diplom Thesis, LMU M¨ unchen 1991. [17] P.-L. Lions, B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105, 415–430 (1991). [18] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95, 281– 303 (1992). [19] G. Rein, Selfgravitating systems in Newtonian theory – the Vlasov-Poisson system, in Proc. Minisemester on Math. Aspects of Theories of Gravitation 1996, Banach Center Publications 41, part I, 179–194 (1997). [20] A.D. Rendall, The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system, Comm. Math. Phys. 163, 89–112 (1994). [21] J. Schaeffer, The classical limit of the relativistic Vlasov-Maxwell system, Comm. Math. Phys. 104, 403–421 (1986). [22] J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations 16, 1313–1335 (1991). [23] H. Spohn, Dynamics of Charged Particles and their Radiation Field, Cambridge University Press, Cambridge 2004. Sebastian Bauer1 and Markus Kunze Universit¨ at Essen, FB 6 – Mathematik D-45117 Essen Germany email:
[email protected] email:
[email protected] Communicated by Rafael D. Benguria submitted 07/01/04, accepted 30/07/04 1 Partially
supported by DFG priority research program SPP 1095
Ann. Henri Poincar´e 6 (2005) 309 – 326 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020309-18 DOI 10.1007/s00023-005-0208-x
Annales Henri Poincar´ e
Absence of Continuous Spectral Types for Certain Non-Stationary Random Schr¨ odinger Operators Anne Boutet de Monvel, Peter Stollmann and G¨ unter Stolz In Memory of Robert M. Kauffman Abstract. We consider continuum random Schr¨ odinger operators of the type Hω = −∆ + V0 + Vω with a deterministic background potential V0 . We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of −∆ + V0 . The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (“random tube”) in arbitrary dimension.
1 Introduction In this article we are concerned with spectral properties of certain non-stationary random operators. More specifically, we consider Schr¨odinger operators of the form Hω = −∆ + V0 + Vω in L2 (Rd ). Here V0 is a deterministic background potential and Vω an Anderson-type random potential which is either sparse near infinity, or concentrated near a lower dimensional surface, or both. This type of models has attracted considerable interest as it allows to study a transition from pure point to continuous spectrum. Here, we are mainly concerned with the former phenomenon. We obtain our results by essentially “deterministic” techniques from [27, 22, 28], establishing conditions on Vω such that Hω has no absolutely continuous spectrum or no continuous spectrum outside the spectrum of −∆ + V0 . This gives us considerable flexibility in the choice of our model. In particular, we are able to avoid some of the typical technical restrictions that come with the usual multiscale analysis or fractional moments proofs of localization. E.g., we can allow for perturbations of changing sign and single site distributions without any continuity. On the other hand, we need decaying randomness in the sense that near infinity the random perturbation is not too effective. That excludes identically distributed random parameters in most cases. An important exception is our result on 1-D “surfaces” (rather tubes) in arbitrary dimensions, see Theorem 4.1 below. The paper is organized in the following way: In Section 2 we present the deterministic techniques we use, recalling the relevant notions and results from [27, 22, 28]; in fact we will need results that are a little stronger than what is explicitly stated in the above cited articles. The common flavor of these methods is that they provide comparison criteria for the absence of continuous and absolutely
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continuous spectra, respectively. These criteria are formulated in the following way: We consider Schr¨ odinger operators with two potentials that differ only on a set that is “small near infinity in a certain geometrical sense”. Then the spectrum of the first operator has no absolutely continuous component on the resolvent set of the second one. To exclude continuous spectrum one needs a bit more complicated assumptions involving randomization. In Sections 3 and 4 we state and prove our main new results, Theorems 3.1, 4.1 and 4.3. In Section 3 we are dealing with sparse random potentials. The framework we introduce is fairly general and includes as special cases the sparse random models considered in [7], e.g., random scatterers are distributed quite arbitrarily in space and the single site perturbations are assumed to be picked with probabilities that tend to zero near infinity. Then, throughout the resolvent set of the unperturbed operator there is no absolutely continuous spectrum (3.1(a)). Since we can treat quite general unperturbed operators, this includes cases with gaps in the spectrum of the unperturbed operator, a case that is completely new. In the proof we combine elementary combinatorial arguments, Lemma 3.2, with the methods discussed above. In the same fashion, under a bit more incisive conditions concerning the background and at least one random scatterer but with the same condition concerning the decay of probabilities near infinity, we can even deduce absence of continuous spectrum outside the spectrum of the unperturbed operator (3.1(b)). That is, all the new spectrum generated by the random perturbation is pure point. This is quite different from what one can obtain with the usual localization proofs, which require a large disorder condition, or apply to energies near the spectral boundaries of the perturbed operator only (with the exception of the one-dimensional case). In Section 4 we study surface-like structures. This means we consider potentials that are concentrated near a subset of lower dimension. Our strongest result, Theorem 4.1, concerns what we call quasi-1D surfaces. There is quite some literature on surface potentials. Most are dealing with the discrete case [4, 5, 8, 9, 11, 10, 13, 14] while in [3, 7] and the present paper continuum models are treated. Here again, our goal was to be able to exclude absolutely continuous spectrum on all of the unperturbed resolvent set and not just near band edges. Theorem 4.3 deals with absence of absolutely continuous and continuous spectrum, respectively, for m-dimensional surface potentials in Rd under an additional sparseness assumption. In the last section we conclude with a discussion of some possible extensions of our results and a comparison with other works, in particular the results in [10] and [7].
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2 Comparison criteria for absence of (absolutely) continuous spectrum In this section we present our methods of proof, essentially taken from [27, 22, 28]. These methods rely on comparison of the spectral properties of Schr¨ odinger operators H1 = −∆ + V1 and H2 = −∆ + V2 whose “difference” is “small” in the sense that the set {V1 = V2 } := {x ∈ Rd | V1 (x) = V2 (x)} is sufficiently sparse. To this end we introduce the following concept, following [27]: Definition. A sequence (Sn )n∈N of compact subsets of Rd with Lebesgue measure |Sn | = 0 (n ∈ N) is called a total decomposition if there exists a family (Ui )i∈I of disjoint, open, bounded sets such that Rd \ Sn = Ui . n∈N
i∈I
A typical example would be Sn = ∂B(0, n), where B(x, r) denotes the closed ball of radius r, centered at x. (Let us stress that the Sn ’s need not be pairwise disjoint.) The sparseness of {V1 = V2 } will be expressed by the existence of a total decomposition (Sn )n∈N with sufficient distance of Sn to {V1 = V2 } compared with the size of Sn . An appropriate notion of size is given by the generalized surface area of a set, a notion introduced in [22] in the following way; here S ⊂ Rd is compact: |{x ∈ Rd | r ≤ dist(x, S) ≤ r + 1}| . σ(S) := sup rd + 1 r≥0 It is easily seen that σ(S) ≤ C ((diam S)d + 1),
(2.1)
i.e., σ(S) is at worst a volume, while for sufficiently regular surfaces it is a surface area measure, for example σ(∂B(x, r)) ≤ C(rd−1 + 1). We cite the following result, essentially taken from [27]: Theorem 2.1 Assume that for each γ > 0 there exists a total decomposition (γ) (Sn )n∈N = (Sn )n∈N such that δn = δn(γ) := dist({V1 = V2 }, Sn ) → ∞ as n → ∞ and
σ(Sn )e−γδn < ∞.
n
Then σac (H1 ) ∩ (H2 ) = ∅.
(2.2) (2.3)
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The following figure is to help visualizing the geometry one is confronted with in the Theorem.
Sn
δn
Sn−1
Figure 1. {V1 = V2 } must not intersect the shaded region. Here, and in what follows, all potentials V are assumed to be locally uniformly in Lp , where p ≥ 2 if d ≤ 3 and p > d/2 if d > 3, i.e., V pp,unif := sup |V (y)|p dy < ∞. (2.4) x
B(x,1)
Theorem 2.1 is essentially Theorem 4.2 from [27]. We will need the slightly stronger version provided above in which the decomposition Sn may vary with γ. The proof provided in [27] goes through under this weaker assumption. This is roughly seen as follows: It suffices to show that σac (H1 ) ∩ J = ∅
(2.5)
for all compact subsets J of (H2 ). For fixed J the argument in [27] provides a γ > 0 (roughly the exponential decay rate in a Combes-Thomas type bound on the resolvent of H2 for energies in J) such that the validity of (2.2) and (2.3) for a suitable decomposition will imply (2.5). Also, in [27] all potentials are assumed to have locally integrable positive parts and negative parts in the Kato class. Our Lp -type assumptions are a special case. The second result we use is taken from [28] and excludes continuous spectrum. It is clear that a statement of the form of Theorem 2.1 above has to be false, since dense pure point spectrum is extremely unstable and can be destroyed by “tiny” perturbations [26]. The geometry is somewhat similar to what we had above but more restrictive. Namely, consider an increasing sequence (An )n∈N of bounded open sets with n An = Rd . Then Sn := ∂An is a total decomposition. For the arguments in [28] it is not necessary that |∂Sn | = 0, but this will be the case in all our applications.
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We assume that δn := min{dist(Sn , {V1 = V2 }), 12 dist(Sn , Sn−1 ∪ Sn+1 )} > 0. d+1
Theorem 2.2 Assume that V1 ∈ Lloc2 (Rd ), W ∈ L∞ with compact support, of fixed sign and such that |W | ≥ cχB(0,s) for suitable c > 0 and s > 0. Moreover, assume that for every γ > 0 there exist An = An (γ) as above such that δn = δn (γ) → ∞ and |An+1 \ An−1 | e−γδn < ∞. (2.6) n
Then for the family Hλ := H1 + λW , λ ∈ R there exists a measurable subset M0 ⊂ R such that |R \ M0 | = 0 and σc (Hλ ) ∩ (H2 ) = ∅ for all λ ∈ M0 . See [28] for the proof which extends to the case of W as specified above. Again, as with Theorem 2.1 above, the possible γ-dependence of the sets An is not explicitly stated in [28], but allowed for by the proof provided there. The requirement that the summability conditions (2.3), (2.6) have to hold for all γ > 0 (and suitable decompositions) comes from the fact that we want to exclude (absolutely) continuous spectrum up to the edges of σ(H2 ). It is possible to quantify and refine the results in a way which says that validity of (2.3), (2.6) for a fixed γ implies absence of (absolutely) continuous spectrum in regions above a certain (γ-dependent) distance from σ(H2 ).
3 Sparse random models In this section we will show how to use the methods from the preceding section to prove absence of continuous or absolutely continuous spectrum for sparse random potentials. As mentioned in the introduction, these models have been set up to study situations in which a transition from singular to absolutely continuous spectrum occurs. This has attracted some interest in the last decade as can be seen in the articles [15, 16, 19, 20, 21, 23, 24] dealing with discrete Schr¨ odinger operators and [7] for the continuum case. We will be concerned mainly with absence of a continuous spectral component away from the spectrum of the unperturbed operator. For this reason we state our results in a generality that does include cases in which no absolutely continuous spectrum survives. As model examples, let us mention two families of models that have been treated in [7].
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Specific models of sparse random potentials, as considered in [7], are Model I Vω (x) =
ξi (ω)f (x − i),
ω∈Ω
i∈Zd
where f is a compactly supported single site potential and the ξi are independent Bernoulli variables. Set pi := P(ξi = 1). If pi → 0 as |i| → ∞ the random potential will no longer be stationary. In fact, it will be sparse in the sense that almost surely large islands near ∞ will occur where Vω vanishes. For the second model f and the ξi , pi will have the same meaning and, additionally, the qi are i.i.d. nonnegative random variables. Model II Vω (x) =
qi (ω)ξi (ω)f (x − i).
i∈Zd
Again, Vω is sparse in the above sense. Of course, for pi ≡ 1 we would get the usual Anderson model. Hundertmark and Kirsch study in [7] the metal insulator transition for H(ω) = −∆ + Vω in L2 (Rd ) for the case that pi → 0 as |i| → ∞ but not too fast in order to make sure that σess (H(ω)) ∩ (−∞, 0) = ∅. Our Model In the following we consider: (A1 ) V0 : Rd → R which is locally uniformly Lp with p ≥ 2 if d ≤ 3 and p > d/2 if d > 3. (A2 ) Σ ⊂ Rd a set of sites that is uniformly discrete in the sense that inf{|j − i| | j, i ∈ Σ, j = i} =: rΣ > 0. (A3 ) For each i ∈ Σ a single site potential fi ∈ Lp such that, for finite constants ρ and M , supp fi ⊂ B(0, ρ) and fi p ≤ M. (A4 ) Vω (x) =
i∈Σ
ωi fi (x − i)
where ω = (ωi )i∈Σ ∈ (Ω, P) = (RΣ , i∈Σ µi ), i.e., the ωi are independent random variables with distribution µi , and supp µi ⊂ [0, 1] for all i ∈ Σ.
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For our results on absence of continuous spectrum, in order to apply Theorem 2.2, we will also require (d+1)/2
(A5 ) Let V0 , fi ∈ Lloc (Rd ) for all i ∈ Σ. There exists one k ∈ Σ with fk of definite sign, bounded, and such that |fk | ≥ cχB(0,s) for some c > 0 and s > 0. For further reference denote pi (ε) := µi ([ε, 1]) = P{ωi ≥ ε}.
(3.1)
mk := (µk )ac ([0, 1])
(3.2)
Also, denote by the total mass of the absolutely continuous component (µk )ac of µk . We will only use this for the fixed k ∈ Σ given in (A5 ). We consider the self-adjoint random Schr¨ odinger operator H(ω) = H0 + Vω in L2 (Rd )
(3.3)
where H0 = −∆ + V0 . Our assumptions guarantee that the local Lp -bounds (2.4) for V0 + Vω are uniform not only in x, but also in ω. Of course, our model contains Models I and II above as special cases and pi (ε) ≤ pi for any ε > 0 in these cases. We have the following result: Theorem 3.1 Let H(ω) be as above, satisfying (A1 ) to (A4 ), and assume that for all ε > 0, (3.4) pi (ε) = o(|i|−(d−1) ) as |i| → ∞. Then (a) σac (H(ω)) ∩ (H0 ) = ∅ almost surely. (b) Assume, moreover, that (A5 ) holds. Then, with k as in (A5 ), P{σc (H(ω)) ∩ (H0 ) = ∅} ≥ mk .
(3.5)
In particular, σc (H(ω)) ∩ (H0 ) = ∅ holds almost surely if µk is purely absolutely continuous, without any assumption on the distribution at the other sites. In order to apply the results from Section 2 we need to find sufficiently many and sufficiently large regions in which the random potential Vω is small and thus Hω close to H0 . We start by showing that these regions appear with probability one. Definition. Call a set U ε-free for ω if ωi ≤ ε for all i ∈ Σ ∩ U . Denote by Ar,R = B(0, R) \ B(0, r) the annulus with inner radius r and outer radius R.
(3.6)
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Lemma 3.2 Fix ε > 0 and a > 1. For n ∈ N let an := P Ar,r+n is not ε-free for all r ∈ [an , an+1 − n] . Then
n
(3.7)
an < ∞.
Proof. Choose η > 0 such that a(1 − η) > 1. Using uniform discreteness of Σ we get that for all n ∈ N and r ≥ 1, #(Ar,r+n ∩ Σ) ≤ Cnrd−1 ,
(3.8)
where C depends on d and rΣ . Here # A is the cardinality of a set A. With C from (3.8) choose δ ∈ (0, η/(Cad−1 )). By (3.4), pi (ε) ≤ δ|i|−(d−1) for i sufficiently large. Thus, for sufficiently large n and each r ∈ [an , an+1 − n],
P(Ar,r+n is ε-free) =
(1 − pi (ε))
i∈Ar,r+n ∩Σ
≥ (1 − δ|i|−(d−1) )#(Ar,r+n ∩Σ) ≥ (1 − δa−n(d−1) )Cna
(n+1)(d−1)
≥ (1 − Cδad−1 )n ≥ (1 − η)n .
(3.9)
Aan ,an+1 contains at least n1 (an+1 − an ) − 1 disjoint annuli Aj := Arj ,rj +n of width n. Thus, using independence and (3.9), an ≤ P(no Aj is ε-free) = P(Aj is not ε-free) j −1
≤ (1 − (1 − η)n )n ≤ e−(1−η)
n
(an+1 −an )−1
(n−1 an (a−1)−1)
.
As (1 − η)a > 1, the an are summable. By the Borel-Cantelli lemma we conclude P(Ωε,a ) = 1, where
Ωε,a := ω ∈ Σ : For each sufficiently large n the annulus Aan ,an+1 contains a sub-annulus Arn ,rn +n which is ε-free for ω . Therefore Ωε =
∈N
also has full measure.
Ωε,1+1/
(3.10)
(3.11)
(3.12)
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Based on this we can now complete the Proof of Theorem 3.1. Fix a compact K ⊂ (H0 ). Since (H0 ) can be exhausted by an increasing sequence of compact subsets, it suffices to prove that σac (H(ω)) ∩ K = ∅ almost surely.
(3.13)
It can be shown, using the general theory of uniformly local Lp potentials, e.g., [25], that there is an ε > 0 such that σ(H0 + V ) ∩ K = ∅,
(3.14)
for each V with V p,unif ≤ ε . Thus, by the properties of Σ and fi , there is an ε > 0 such that
δi fi (x − i) ∩ K = ∅ (3.15) σ H0 + i∈Σ
if |δi | ≤ ε for all i ∈ Σ. Fix this ε > 0 and let Ωε be the full measure set found above. For given ˜ i := min{ωi , ε}, i ∈ Σ, and ω ∈ Ωε let ω V2 (x) := ω ˜ i fi (x − i). i∈Σ
By (3.15) we have σ(H0 + V2 ) ∩ K = ∅. Thus, in order to apply Theorem 2.1 and (γ) conclude (3.13), it suffices to find for every γ > 0 a total decomposition (Sn ) of {Vω = V2 } which satisfies (2.2) and (2.3). For given γ > 0 choose an integer > 2(d − 1)/γ. This implies (d − 1) log a < γ/2, where a := 1 + 1/ . As ω ∈ Ωε,a , for each sufficiently large n the annulus Aan ,an+1 contains an ε-free annulus Arn ,rn +n . (γ) Choose Sn := ∂B(0, rn + n2 ). Then δn(γ) := dist({Vω = V2 }, Sn(γ) ) ≥
n −ρ 2 (γ)
since Arn ,rn +n is ε-free (recall that supp fk ⊂ B(0, ρ)). Thus δn → ∞. Also using (γ) that σ(Sn ) ≤ Can(d−1) , we conclude (γ) σ(Sn(γ) )e−γδn ≤ Ceγρ en((d−1) log a−γ/2) < ∞. n
n
This proves part (a) of Theorem 3.1. In order to apply Theorem 2.2 to prove part (b) we slightly modify the above construction, essentially replacing Σ by Σ \ {k}. Let Ω := RΣ\{k} with measure P = ⊗i∈Σ\{k} µi . As the property defining Ωε,a in (3.11) does not depend on the value of ωk , we get that also P (Ωε,a ) = P (Ωε ) = 1, where Ωε,a and Ωε are defined as in (3.11) and (3.12), but as subsets of Ω .
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For compact K ⊂ (H0 ) choose ε > 0 as in the proof For ω ∈ Ωε of part (a). let := min{ωi , ε} (i ∈ Σ \ {k}). Also let Vω (x) = i∈Σ\{k} ωi fi (x − i) and ˜ i fi (x − i). As before, σ(H0 + V2 ) ∩ K = ∅. V2 (x) = i∈Σ\{k} ω For γ > 0 choose > 2d/γ, a = 1 + 1/ . With rn from (3.11), let An = B(0, rn + n2 ) and Sn = ∂An . This yields ω ˜ i
|An+1 \ An−1 | ≤ cd a(n+2)d and
1 n δn = min dist(Sn , {Vω = V2 }), dist(Sn , Sn−1 ∪ Sn+1 ) ≥ − ρ. 2 2 The choice of a guarantees that n |An+1 \An−1 | e−γδn < ∞. By Theorem 2.2 this proves the existence of a measurable subset M0,ω ⊂ R with |R \ M0,ω | = 0 and such that σc (H(λ, ω )) ∩ K ⊂ σc (H(λ, ω )) ∩ (H0 + V2 ) = ∅
for all λ ∈ M0,ω , where H(λ, ω ) = H0 + λfk (x − k) + Vω (x). As µk (M0,ω ) ≥ (µk )ac (M0,ω ) = (µk )ac (R) = mk it follows by Fubini that P{ω ∈ Ω : σc (H(ω)) ∩ K = ∅} ≥ mk . Since this bound is independent of K and we can exhaust (H0 ) by an increasing sequence Kn we arrive at the assertion. This completes the proof of Theorem 3.1. Remarks. (1) While the “volume” term |An+1 \ An−1 | in (2.6) has to be considered larger than the “surface” term σ(Sn ) in (2.3), this did not make a significant difference in the above proof. The same total decomposition Sn can be used to prove absence of absolutely continuous spectrum and absence of continuous spectrum. The difference will become more significant for the quasi-1D surfaces considered in the next section. (2) Crucial for our method to apply is the almost sure appearance of a sequence of ε-free annular regions which must (i) grow in thickness and (ii) not be too far apart, as found in Lemma 3.2. In Theorem 3.1 this was enforced through the assumptions on the distribution of the coupling constants. In Section 4 it will follow from sparseness of the single site set Σ. (3) Note that our methods are sufficiently “soft” to allow for considerable flexibility of the model. The single site potentials fi may depend on the site, do not need to be sign definite, and may include Lp -type singularities. We can deal with quite arbitrary single site distributions. Only for the proof of absence of continuous spectrum we need one of the distributions to be absolutely continuous. These assumptions are weaker than what usually enters into the proof of localization properties through the multiscale analysis or fractional moment methods. (4) The assumption supp µj ⊂ [0, 1] is just a normalization. For our methods to apply, the random potentials have to obey uniform bounds, e.g., in the sense of · p,unif from (2.4).
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Let us finally state the following result for our model which easily follows from the “Almost surely free lunch Theorem” in [7]. For the case V0 = 0 it can be combined with Theorem 3.1 to provide examples with purely singular or pure point (while not discrete) spectrum below zero and an absolutely continuous spectral component above zero. Theorem 3.3 Let µk , fk , V0 be as above, V0 = 0 and assume that, additionally, the fk ∞ are uniformly bounded and that the second moments of the ηk obey 1 2 x2 dµk ≤ C|k|−β E(ηk ) = 0
for some β > 2. Then σac (H(ω)) ⊃ [0, ∞) P -a.s. Proof. The assumptions clearly make sure that 1
W (x) := E(Vω (x)2 ) 2 ≤ C(1 + |x|)−(1+ε) so that we can apply Theorem 2.4 from [7] to see that Cook’s criterion is applicable for P -a.e. ω ∈ Ω. For general V0 the corresponding result, namely that σac (−∆+V0 ) ⊂ σac (Hω ) almost surely, is probably false. It should be true for certain periodic potentials, see [2, 6, 29].
4 Quasi-1D surfaces In Section 3 sparseness of the potential Vω in (A4 ) resulted from an assumption on decaying randomness, e.g., (3.4). In the present section we will modify our methods and results for the case where sparseness of Vω arises directly through sparseness of the deterministic set Σ. By this we mean situations where Σ does not have positive d-dimensional density in Rd , i.e., #(Σ ∩ B(0, R)) = o(Rd ) as R → ∞. A special case would be an m-dimensional sublattice, e.g., Σ = Zm × {0} ⊂ Rm × Rd−m , 0 < m < d, in which case Vω would model a random surface potential. Our most interesting result holds for m = 1, where our methods cover the following more general situation: Definition. A uniformly discrete subset Σ of Rd is called quasi-one-dimensional (quasi-1D) if there exists C < ∞ such that #(Σ ∩ AR,R+1 ) ≤ C
(4.1)
for all R ≥ 0. Theorem 4.1 Let H(ω) = H0 + Vω satisfy (A1 ) to (A4 ). In addition, assume that Σ is quasi-1D and that sup pi (ε) < 1 (4.2) i∈Σ
for every ε > 0. Then σac (H(ω)) ∩ (H0 ) = ∅ almost surely.
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If Σ is quasi-1D, then by Theorem 4.1, no spatial decay in the randomness of the ηi is required to conclude absence of absolutely continuous spectrum in gaps of σ(H0 ). For example, (4.2) is satisfied for independent, identically distributed random variables ηi such that 0 ∈ supp µ for their common distribution µ. In particular, as every uniformly discrete Σ ⊂ R is quasi-1D, this strengthens Theorem 3.1(a) in the case d = 1, which would require pi (ε) = o(1) as k → ∞. Of course, in the case d = 1 our result is hardly new as (essentially) much stronger results are known for one-dimensional random potentials. More interesting is the case d > 1, where special cases of quasi-1D sets include discrete tubes of the form Σ = Z×S, with S a bounded subset of Zd−1 . Theorem 4.1 shows the absence of absolute continuity in the “surface spectrum” generated by the random (1D) surface potential V (ω). Also, within certain limitations, we can allow for curvature in the tubes Σ, thus covering rather general “random sausages”. Proof. We start with a modification of Lemma 3.2. Lemma 4.2 Fix ε > 0. Let δ = supi pi (ε) < 1, C as in (4.1) and a > the an , as defined in (3.7), are summable.
1 (1−δ)C
. Then
Proof. This follows with the same argument as in the proof of Lemma 3.2, using that now P(Ar,r+n is ε-free) ≥ (1 − δ)Cn . Thus the set Ωε,a , defined as in (3.11), has full P-measure. of Theorem 3.1 to find Fix K ⊂ (H0 ) compact and argue as in the proof ˜ i fi (x − i), ω ˜i = ε > 0 such that σ(H0 + V2 ) ∩ K = ∅, where V2 (x) = i∈Σ ω min{ωi , ε}. Choose a > 1 as in Lemma 4.2 and ω ∈ Ωε,a , i.e., Aan ,an+1 contains ε-free Arn ,rn +n for all sufficiently large n. As before, the spheres Sn = ∂B(0, rn + n2 ) give a total decomposition with dist({Vω = V2 }, Sn ) ≥ n2 − ρ. But, as Lemma 4.2 prevents us from choosing a arbitrarily close to 1, this will not yield convergence of (2.3) for all γ > 0. We will therefore refine our construction by splitting the Sn in two parts. One part is a union of spherical caps for which, due to points of Σ close to Arn ,rn +n , the distance n2 − ρ from {Vω = V2 } can’t be improved. The second part (the remaining “Swiss cheese”) has much bigger distance to {Vω = V2 } and, due to the sparseness of Σ, contains most of Sn . The details of this construction are as follows: Fix α > 1. Let Pn := (Arn −nα ,rn ∪ Arn +n,rn +n+nα ) ∩ Σ α
(4.3)
be the points of Σ in the n -neighborhood of Arn ,rn +n (but outside Arn ,rn +n ). For each j ∈ Pn define the spherical cap
n j Sn,j := Sn ∩ B rn + (4.4) , nα . 2 |j| Also let Sn := Sn \
j
Sn,j .
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j ∈ Pn
Sn
Arn ,rn +n
Sn,j := Sn ∩ B
rn +
n 2
j α |j| , n
Figure 2. The geometry in the proof of Lemma 4.2: the bold face line shows a part of Sn , the shaded region is Arn ,rn +n , the point in the small circle a j ∈ Pn and j the small circle the boundary of B((rn + n2 ) |j| , nα ). Since Sn ∪
j
Sn,j = Sn , we have that {Sn,j : n ∈ N, j ∈ Pn } ∪ {Sn : n ∈ N}
(4.5)
is a total decomposition of Rd . As above, since Arn ,rn +n is ε-free, δn,j := dist({Vω = V2 }, Sn,j ) ≥
n − ρ. 2
(4.6)
If x ∈ Sn and j ∈ Σ ∩ (Arn ,rn +n )c , then, by elementary geometric considerations, dist(x, j) ≥ nα for sufficiently large n. Using this and again that Arn ,rn +n is ε-free, we find (4.7) δn := dist({Vω = V2 }, Sn ) ≥ nα − ρ. From the simple volume bound (2.1) on the generalized surface area one gets σ(Sn,j ) ≤ Cndα ,
(4.8)
σ(Sn ) ≤ Cadn .
(4.9)
Checking (2.3) for the partition (4.5) amounts to proving that σ(Sn ) e−γδn < ∞
(4.10)
n
and
n j∈Pn
σ(Sn,j ) e−γδn,j < ∞
(4.11)
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for each γ > 0. (4.10) follows from (4.7) and (4.9) since α > 1. (4.11) follows from (4.6) and (4.8), noting that #Pn ≤ 2nα + 2 since Σ is quasi-1D. From Theorem 2.1 we conclude σac (H(ω)) ∩ K ⊂ σac (H(ω)) ∩ (H0 + V2 ) = ∅. Remark. It is possible to prove Theorem 4.1 under a slightly weaker assumption on the set Σ, namely that there exists C < ∞ such that #(Σ ∩ B(0, R)) ≤ CR
(4.12)
for all R ≥ 1. (4.12) is weaker than (4.1) in that it allows the number of points in Σ ∩ AR,R+1 to be unbounded with respect to R. (4.12) is also somewhat more natural as it doesn’t depend on the norm used to define B(0, R) nor on the choice of the center of the ball. A simple counting argument shows that, under the assumption (4.12), for each annulus of the form Aan ,an+1 most sub-annuli AR,R+n satisfy a bound #(Σ ∩ AR,R+n ) ≤ Cn. Here “most” means at least a non-vanishing fraction. One finds sufficiently many disjoint such annuli to construct ε-free regions as before. Moreover, by an additional counting argument, one argues that most of these annuli do not have more than C nα points of Σ in their nα -neighborhoods. Based on this one can construct a partition {Sn , Sn,j } as above and carry through the proof. We skip the somewhat tedious details of this generalization. We are not able to prove a result like Theorem 3.1(b), i.e., absence of continuous spectrum in (H0 ) with positive probability, under the assumptions of Theorem 4.1 (plus (A5 )). For the partition Sn = ∂An , An = B(0, rn + n2 ) the volumes |An+1 \ An−1 | grow too fast to get validity of (2.6) for all γ > 0. A trick like the introduction of {Sn , Sn,j } as above is not applicable here since in Theorem 2.2 the Sn need to arise as boundaries of a growing sequence An . However, if one replaces (4.2) by pi (ε) = o(1) as |i| → ∞ for all ε > 0, then Lemma 4.2 will hold for any a > 1, which allows for an application of Theorem 2.2 with a γ-dependent choice of the Sn , as in the proof of Theorem 3.1(b). Sparseness of the random potential is achieved here through a combination of sparseness of Σ and decaying randomness pi (ε) = o(1), as opposed to Theorem 3.1, where sparseness follows exclusively from stronger decay pi (ε) = o(|i|−(d−1) ). In fact, the correlation between the degree of sparseness of Σ and the rate of decay of pi (ε) can be made more specific. For this, call a uniformly discrete set Σ ⊂ Rd quasi-m-dimensional (1 ≤ m ≤ d, not necessarily integer) if for some C < ∞ and all R ≥ 0, (4.13) #(Σ ∩ AR,R+1 ) ≤ CRm−1 . Then the following result is found with the same methods as above: Theorem 4.3 Let H(ω) satisfy (A1 ) to (A4 ), Σ be quasi-m-dimensional and, for all ε > 0, (4.14) pi (ε) = o(|i|−(m−1) ) as |i| → ∞, then σac (H(ω)) ∩ (H0 ) = ∅ almost surely. If, moreover, (A5 ) holds, then P{σc (H(ω)) ∩ (H0 ) = ∅} ≥ mk .
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5 Concluding remarks Among the known results for discrete surface models, the one most closely related to Theorem 4.1 above is the result of Jakˇsi´c and Molchanov [10]. They consider the discrete Laplacian on Z × Z+ with random boundary condition ψ(n, −1) = Vω (n)ψ(n, 0), where the Vω (n) are i.i.d. random variables. They show that the spectrum outside [−4, 4], i.e., outside the spectrum of the two-dimensional discrete Laplacian, is almost surely pure point. This is stronger than our continuum analogue in the sense that we can only prove absence of absolute continuity outside the spectrum of the deterministic background operator H0 . The proof in [10] requires a technical tour de force. The two-dimensional problem can be reduced to a one-dimensional problem with long range interactions. Anderson localization for the latter has been proven in [12] with methods based on an approach developed in [17] (which is also behind Theorem 2.2 above). The onedimensional problem depends nonlinearly on the spectral parameter, a difficulty which is resolved by adapting some ideas from the Aizenman-Molchanov fractional moment method [1]. Our methods are comparatively soft. In particular, they work directly in the multi-dimensional PDE setting and do not require a reduction to d = 1. One-dimensionality of the random surface only enters through its probabilistic consequences (Lemma 4.2) for the frequency of the appearance of ε-free regions, which constitute the “potential barriers” required in Theorem 2.1. This makes our methods very flexible. In addition to the extension to continuum models, they allow for rather general quasi-1D surfaces (e.g., curved tubes, unions of tubes), work in arbitrary dimension d and allow for the presence of an additional deterministic background potential V0 . It is possible to adapt our methods to lattice operators and prove absence of absolutely continuous spectrum outside the spectrum of the discrete Laplacian for much more general geometries than the half-plane considered in [10]. Also, our methods can easily be adjusted to work for operators of the type (3.3) on L2 (Ω), Ω = Rd . For example, for H(ω) = −∆ + Vω in L2 ((0, a) × Rd−1 ) with Dirichlet boundary conditions and Vω given through (A2 ) to (A4 ) with i.i.d. coupling constants ωi , we would get that σac (H(ω)) ∩ (−∞, 0) = ∅ almost surely. Of course, for this physically one-dimensional operator (with no bulk space), one would expect the much stronger result that σc (H(ω)) = ∅. But the corresponding result for discrete strips, e.g., [18], does not seem to extend easily to the continuum. Finally, we mention that Hundertmark and Kirsch [7] announce some results on pure point spectrum for continuum models similar to the ones studied here. They will use suitable adaptations of multiscale analysis to show that the negative spectrum of −∆ + Vω is almost surely pure point. Here Vω is either of the type of Model II above or a random potential at the surface of a half space Schr¨ odinger operator. In situations where the multiscale analysis can be carried out, their results should be stronger than ours.
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Acknowledgment Our collaboration has been supported by the University Paris 7 Denis Diderot where part of this work was done, by the DFG in the priority program “Interacting stochastic systems of high complexity” and through the SFB 393, as well as through US-NSF grant no. DMS-0245210.
References [1] M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: An elementary derivation, Commun. Math. Phys. 157, 245–278 (1993). [2] M.Sh. Birman and D.R. Yafaev, The scattering matrix for a perturbation of a periodic Schr¨ odinger operator by decreasing potential, (Russian) Algebra i Analiz 6, no. 3, 17–39 (1994); translation in St. Petersburg Math. J. 6, no. 3, 453–474 (1995). [3] A. Boutet de Monvel and P. Stollmann, Dynamical localization for continuum random surface models, Arch. Math., 80, 87–97 (2003). [4] A. Boutet de Monvel and A. Surkova, Localisation des ´etats de surface pour une classe d’op´erateurs de Schr¨odinger discrets `a potentiels de surface quasip´eriodiques, Helv. Phys. Acta 71, no. 5, 459–490 (1998). [5] A. Chahrour and J. Sahbani, On the spectral and scattering theory of the Schr¨ odinger operator with surface potential, Rev. Math. Phys. 12, no. 4, 561–573 (2000). [6] C. G´erard and F. Nier, Scattering theory for the perturbations of periodic Schr¨ odinger operators, J. Math. Kyoto Univ. 38, no. 4, 595–634 (1998). [7] D. Hundertmark and W. Kirsch, Spectral theory of sparse potentials, in “Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999),” Amer. Math. Soc., Providence, RI, 2000, pp. 213–238. [8] V. Jakˇsi´c and Y. Last, Corrugated surfaces and a.c. spectrum, Rev. Math. Phys. 12, no. 11, 1465–1503 (2000). [9] V. Jakˇsi´c and Y. Last, Spectral structure of Anderson type hamiltonians, Invent. Math. 141, no. 3, 561–577 (2000). [10] V. Jakˇsi´c and S. Molchanov, On the surface spectrum in dimension two, Helv. Phys. Acta 71, no. 6, 629–657 (1998). [11] V. Jakˇsi´c and S. Molchanov, On the spectrum of the surface Maryland model, Lett. Math. Phys. 45, no. 3, 189–193 (1998).
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[12] V. Jakˇsi´c and S. Molchanov, Localization for one-dimensional long range random hamiltonians, Rev. Math. Phys. 11, 103–135 (1999). [13] V. Jakˇsi´c and S. Molchanov, Localization of surface spectra, Commun. Math. Phys. 208, no. 1, 153–172 (1999). [14] V. Jakˇsi´c, S. Molchanov and L. Pastur, On the propagation properties of surface waves, in “Wave propagation in complex media (Minneapolis, MN, 1994),” IMA Math. Appl., Vol. 96, Springer, New York, 1998, pp. 143–154. [15] W. Kirsch, Scattering theory for sparse random potentials, Random Oper. Stochastic Equations 10, no. 4, 329–334 (2002). [16] W. Kirsch, M. Krishna and J. Obermeit, Anderson model with decaying randomness: Mobility edge, Math. Z., 235, 421–433 (2000). [17] W. Kirsch, S. Molchanov and L. Pastur, One-dimensional Schr¨ odinger operators with high potential barriers, Operator Theory, Adv. Appl. 57, 163–170 (1992). [18] A. Klein, J. Lacroix and A. Speis, Localization for the Anderson model on a strip with singular potentials, J. Funct. Anal. 94, no. 1, 135–155 (1990). [19] M. Krishna, Anderson model with decaying randomness: Existence of extended states, Proc. Indian Acad. Sci. Math. Sci. 100, no. 3, 285–294 (1990). [20] M. Krishna, Absolutely continuous spectrum for sparse potentials, Proc. Indian Acad. Sci. Math. Sci. 103, no. 3, 333–339 (1993). [21] M. Krishna and K.B. Sinha, Spectra of Anderson type models with decaying randomness, Proc. Indian Acad. Sci. Math. Sci. 111, no. 2, 179–201 (2001). [22] I. McGillivray, P. Stollmann and G. Stolz, Absence of absolutely continuous spectra for multidimensional Schr¨ odinger operators with high barriers, Bull. London Math. Soc. 27, no. 2, 162–168 (1995). [23] S. Molchanov, Multiscattering on sparse bumps, In “Advances in differential equations and mathematical physics” (Atlanta, GA, 1997), Contemp. Math. 217, Amer. Math. Soc., Providence, RI, 1998, pp. 157–181. [24] S. Molchanov and B. Vainberg, Multiscattering by sparse scatterers, In “Mathematical and numerical aspects of wave propagation” (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, 2000, pp. 518–522. [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978. [26] B. Simon, Spectral analysis of rank one perturbations and applications, CRM Proc. Lecture Notes, 8, 109–149 (1995).
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[27] P. Stollmann and G. Stolz, Singular spectrum for multidimensional operators with potential barriers, J. Operator Theory 32, 91–109 (1994). [28] G. Stolz, Localization for Schr¨ odinger operators with effective barriers, J. Funct. Anal. 146, no. 2, 416–429 (1997). [29] D. Yafaev, Eigenfunctions of the continuous spectrum for the N -particle Schr¨ odinger operator, In “Spectral and scattering theory” (Sanda, 1992), 259–286, Lecture Notes in Pure and Appl. Math., 161, Dekker, New York, 1994. Anne Boutet de Monvel Institut de Math´ematiques de Jussieu Universit´e Paris 7 2, place Jussieu, case 7012 F-75251 Paris France email:
[email protected] Peter Stollmann Fakult¨ at f¨ ur Mathematik Technische Universit¨ at D-09107 Chemnitz Germany email:
[email protected] G¨ unter Stolz Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294 USA email:
[email protected] Communicated by Jens Marklof submitted 07/04/04, accepted 19/08/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 327 – 342 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020327-16 DOI 10.1007/s00023-005-0209-9
Annales Henri Poincar´ e
Spectrum of the Magnetic Schr¨ odinger Operator in a Waveguide with Combined Boundary Conditions Denis Borisov, Tomas Ekholm and Hynek Kovaˇr´ık Abstract. We consider the magnetic Schr¨ odinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann1 . We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.
1 Introduction The existence of bound states of the Laplace operator in the strip with Dirichlet boundary conditions and Neumann window was proven in [1] and independently also in [2]. The so called Neumann window is represented by the segment of the length 2l of the boundary, on which the Dirichlet condition is changed to Neumann. A discrete spectrum of the Laplace operator with Neumann window appears for any nonzero length of the Neumann segment. In particular, for small values of l the eigenvalue emerges from the continuous spectrum proportionally to l4 . The asymptotical estimate for small l was established in [3]. The asymptotics expansion of the emerging eigenvalue for small l was constructed formally in [4], while the rigorous results were obtained in [5]. On the other hand, the results on the discrete spectrum of a magnetic Schr¨ odinger operator in waveguide-type domains are scarce. A planar quantum waveguide with constant magnetic field and a potential well is studied in [6], where it was proved that if the potential well is purely attractive, then at least one bound state will appear for any value of the magnetic field. Stability of the bottom of the spectrum of a magnetic Schr¨odinger operator was also studied in [7, Sec. 9] In this work we consider the system, where the discrete spectrum in the absence of magnetic field appears due to the perturbation of the boundary of the domain rather than due to the additional potential well. We also assume that the magnetic field is localized in the sense to be specified below. This assumption rules out the case of a constant field. As it has been recently shown in [8] the presence of a suitable magnetic field can prevent the existence of bound states in the Dirichlet strip with a sufficiently small “bump”. Changing the boundary 1 For
the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)
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conditions to Neumann is however a stronger perturbation in the sense that the existence of a bound state in a waveguide with the bump added to a certain segment of the boundary implies the existence of a bound state in a waveguide with Neumann conditions on the same segment, see [1, Cor. 1.3]. Therefore we cannot mimic the arguments of [8] in the case of the waveguide with Neumann window and a different approach is needed. The main technical tool used in [8] is a modified version of the Hardy inequality for the magnetic Dirichlet quadratic form in the two-dimensional strip. In the present paper we establish a similar inequality in order to prove the absence of a discrete spectrum of the magnetic Schr¨ odinger operator in the straight strip with Neumann window. More exactly speaking, we give sufficient conditions on the magnetic field and the length of the window, under which the discrete spectrum is empty. The above mentioned version of Hardy inequality enables us to reduce the problem to the study of a one-dimensional Laplacian with a purely attractive potential well of a width 2l and a small but fixed positive potential, see Section 4.2 for the details. We then show that for l small enough such a system has no bound state. The main profit of our method is that it gives us an explicit estimate on the critical length of the window, depending on the magnetic field, which guarantees the absence of discrete spectrum. It is of course natural to ask whether a sufficiently large Neumann window will lead to the existence of eigenvalues also in the presence of the magnetic field. In the case of a smooth and compactly supported field we give an answer to this question using a minimax-like argument. The article is organized as follows. In Section 2 we define the mathematical objects that we work with and describe the problem. We also give the statements of the main results separately for the case of a compactly supported bounded magnetic field and for the Aharonov-Bohm field. In Section 3 we show that the essential spectrum of the Dirichlet Laplacian is not affected by the magnetic field, neither by the presence of a Neumann window. Sufficient conditions for the absence of the discrete spectrum are proved in Section 4. Finally, the question of presence of eigenvalues is discussed in Section 5.
2 Statement of the problem and the main results Let x = (x1 , x2 ) be Cartesian coordinates, Ω be the strip {x : 0 < x2 < π}, and γ be the interval {x : |x1 | < l, x2 = 0}. The rest of the boundary will be indicated by Γ, i.e., Γ = ∂Ω \ γ. We denote by B = B(x) a real-valued magnetic field and assume that A is a magnetic vector potential associated with B, i.e., A = A(x) = (a1 (x), a2 (x)) and B = curl A = ∂x1 a2 − ∂x2 a1 . In what follows we will consider two main cases of magnetic fields B. The first case is a smooth compactly supported field. Hereinafter by this we denote the field B belonging to C 1 (Ω) and vanishing in the neighborhood of infinity. The second one is the
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Aharonov-Bohm field originated by the potential with components a1 (x) = −
Φ · (x2 − p2 ) , (x1 − p1 )2 + (x2 − p2 )2
a2 (x) =
Φ · (x1 − p1 ) , (x1 − p1 )2 + (x2 − p2 )2
(2.1)
where Φ is a constant and 2πΦ is the flux through the point p = (p1 , p2 ) which is assumed to be inside the strip Ω. We denote by M0 the operator 2
2
(−i∂x1 + a1 ) + (−i∂x2 + a2 )
on the domain D(M0 ) consisting of all functions u ∈ C ∞ (Ω) vanishing in a neighborhood of Γ and in a neighborhood of infinity and satisfying the boundary condition (2.2) (−i∂x2 + a2 )u(x) = 0 on γ. We will call it magnetic Neumann boundary condition. In the case of AharonovBohm field, the functions u ∈ D(M0 ) are assumed to vanish in a neighborhood of the point p. Clearly, the operator M0 is non-negative and symmetric in L2 (Ω) and therefore it can be extended to a self-adjoint non-negative operator by the method of Friedrich. In what follows we will denote this extension by M . The main object of our interest is the spectrum of the operator M . In order to formulate the main results we need to introduce some auxiliary notations. By Ω(α, β) we will indicate the subset of Ω given by {x ∈ Ω : α < x1 < β} and Ω± will be the subsets {x ∈ Ω : x1 > l}, {x ∈ Ω : x1 < −l}, respectively. The symbol Br (q) denotes a ball of radius r centered at a point q in R2 . The flux of the field through the ball Br (q) is given by 1 Φq (r) = B(x) dx. 2π Br (q) Below we give the summary of the main results of the article. Theorem 2.1. The essential spectrum of the operator M coincides with [1, +∞). Theorem 2.2. Assume that the field B is smooth and compactly supported and (1) There exist two balls BR− (p− ) ⊂ Ω− , BR+ (p+ ) ⊂ Ω+ so that at least one of the fluxes Φp± (r) is not identically zero for r ∈ [0, R± ]; (2) The inequality 1 (2.3) (κ− + κ+ ) l≤ 12 holds true, where π κ± := min πc± , , (2.4) 4 ln 2 + π|p± 1 | c± are defined in Lemma 4.1. Then the operator M has empty discrete spectrum.
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Theorem 2.3. Assume that the field B is the Aharonov-Bohm one with the potential given by (2.1) and (1) The point p is (p1 , p2 ), where p1 < −l; (2) The inequality l< holds true, where
κ := min πc,
κ 6
π 4 ln 2 + π|p1 |
(2.5) ,
(2.6)
c is defined in Lemma 4.2. Then the operator M has empty discrete spectrum. The next theorem provides a condition, that guarantees the existence of discrete eigenvalues in the case of a smooth and compactly supported field. Theorem 2.4. Let the field B be smooth and compactly supported, λ = λ(l) be the lowest eigenvalue of the Laplacian −∆N ,D in the strip Ω subject to the Dirichlet condition on Γ and Neumann condition on γ. Assume that the inequality λ(l) + inf max |A(x)|2 < 1 A
(2.7)
Ω
holds, where infimum is taken over all potentials associated with the field B. Then the operator M has non-empty discrete spectrum. Remark 2.5. In the case of a smooth compactly supported field B we did not define the magnetic potential uniquely. In fact, this is not needed, since the spectrum of the operator M is invariant under the gauge transformation A → A + ∇ϕ, where ϕ is a real-valued function. We will employ this property in section 5 to show that under the hypothesis of this theorem the potential A can be chosen such that |A| is bounded and of compact support. This will imply that the quantity inf max |A(x)|2 A
Ω
in (2.7) is finite. Remark 2.6. The constants κ± and κ in Theorems 2.2 and 2.3 giving the estimates for window length depend on the magnetic field. The constants c± and c in (2.4) and (2.6) are determined by the rational part of the flux and the distance from the support of the field to the boundary (see (4.3) and (4.16)). The important role of the fractional part of the flux is the usual property of the system with magnetic field (see, for instance, [7, Sec. 10], [9, Sec. 6.4]); this is a case in our work too. The distance between the magnetic field and the window is taken into account by π in (2.4) and by the similar term in (2.6). the presence of the terms 4 ln 2+π|p ± | 1
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Throughout the article we will often make use of some notations and it is convenient to introduce them now. The spectrum of an operator T will be indicated by σ(T ) while the essential spectrum will be denoted by σess (T ). We will employ the symbol qT = qT [·, ·] for the sesquilinear form associated with a self-adjoint operator T and D(qT ) will be the domain of the quadratic form produced by the sesquilinear form qT . The Hilbert space we will work in is L2 (Ω); we preserve the notation (·, ·) and · for the inner product and norm in this space. In all other cases the notations of the inner product and norm in a Hilbert space H will be equipped by a subscript H.
3 Proof of Theorem 2.1 To prove the theorem we will need some auxiliary notations and statements. Let H be a Hilbert space and S be a positive definite operator in H whose domain is dense in H. By S1 we indicate the Friedrich’s extension of the operator S and by S2 another self-adjoint positive definite extension of S. By definition, D(qS2 ) is a Hilbert space endowed with the inner product and the norm originated by the quadratic form qS2 . Since S1 is the Friedrich’s extension of S it follows that D(qS1 ) is a subspace of D(qS2 ). Let Q be the orthogonal complement D(qS1 )⊥ in D(qS2 ) in the inner product qS2 [·, ·]. The proof of the theorem is based on the following lemma proven in [10, Lemma 3.1]. Lemma 3.1. If each bounded subset of Q (in the norm · D(qS2 ) ) is compact in H, then the operator T := S2−1 − S1−1 is compact in H. In our case L2 (Ω) plays the role of H and S := (−i∇ + A)2 + 1 with D(S) := C0∞ (Ω). The Friedrich extension S1 of S is in fact the extension of (−i∇ + A)2 + 1 subject to Dirichlet boundary condition. We know from [8] that σess (S1 ) = [2, +∞). We set S2 := M + 1; we naturally can treat M + 1 as an extension of S. If we prove that T := S2−1 − S1−1 is compact, then the essential spectra of the operators S1 and S2 will coincide by the Weyl theorem (see for instance [11, Ch. 9, Sec. 1]). We will prove the compactness of T by Lemma 3.1. First we will establish an auxiliary lemma. By ω we indicate some bounded subdomain of Ω with infinitely differentiable boundary such that dist (γ, Ω \ ω) > 0. In the case of Aharonov-Bohm field we also assume that the point p does not belong to ω. Lemma 3.2. For each function u ∈ Q the inequality u ≤ cuL2 (ω) , holds true, where the constant c is independent on u. Proof. In the proof of the lemma we follow the ideas of the proof of Lemma 3.3 in [10]. The domains D(qS1 ) and D(qS2 ) are completions of C0∞ (Ω) and D(M0 ),
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respectively, in the norm (−i∇ + A) · 2 + · 2 . In the case of compactly supported field we can choose the vector potential A being from C 1 (Ω) which will make this potential bounded on ω. In the case of Aharonov-Bohm field the potential is in C 1 (ω) as well since the point p does not belong to ω by assumption. Therefore, each element v of D(S2 ) belongs to H 1 (ω) due to the inequality: v2H 1 (ω) = (−i∇ + A)v − Av2L2 (ω) + v2L2 (ω) ≤ 2 (−i∇ + A)v2L2 (ω) + Av2L2 (ω) + v2L2 (ω) ≤ C (−i∇ + A)v2L2 (ω) + v2L2 (ω) = C(S2 v, v),
(3.1)
where the constant C is independent on v. We denote by χ = χ(x) an infinitely differentiable function taking values from [0, 1] and being equal to one in some neighborhood of γ, which is a subdomain of ω, and vanishing outside ω. Since S2 ≥ 1 it follows that S2−1 u ≤ u.
(3.2)
χ)S2−1 u
Let u ∈ Q. Clearly, (1 − ∈ D(qS1 ) ∩ D(S2 ), thus S2 (1 − χ)S2−1 u, u = (1 − χ)S2−1 u, u D(qS
2)
= 0.
Using this equality we deduce
u2 = (u, u) − S2 (1 − χ)S2−1 u, u = (S2 χS2−1 u, u).
(3.3)
Since
S2 χS2−1 u = χu − 2 ∇(S2−1 u), ∇χ R2 − (S2−1 u)∆χ − 2 i (A, ∇χ)R2 S2−1 u
due to (3.1)–(3.3) we have 2 u ≤ χ|u|2 dx + cuL2(ω) S2−1 uH 1 (ω) Ω
−1 ≤ CuL2(ω) u + (S2 u, u) ≤ CuL2 (ω) u, where C is independent on u. This proves the lemma. Let us finish the proof of the Theorem. Given a subset K of Q bounded in the norm · D(qS1 ) , we conclude that it is also bounded in H 1 (ω) due to (3.1). By the well known theorem on compact embedding of H 1 (ω) in L2 (ω) for each bounded domain with smooth boundary (see, for instance, [12, Ch. 1, Sec. 6]) we have that the set K is compact in L2 (ω). Applying now Lemma 3.2, we conclude that K is compact in L2 (Ω). Hence, the assumption of Lemma 3.1 is satisfied and the operator T introduced above is compact. The proof of Theorem 2.1 is complete.
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4 Absence of the discrete spectrum This section is devoted to the proof of Theorems 2.2 and 2.3. By Theorem 2.1 we know that the essential spectrum of the operator M is [1, +∞). Thus, the equivalent formulation of the absence of the discrete spectrum is the following inequality inf σ(M − 1) = inf (−i∇ + A)u2 − u2 ≥ 0. (4.1) u=1 u∈D(qM )
It will be enough to check the infimum for a · D(qM ) -dense subset of D(M ). Hence inf σ(M − 1) = inf (−i∇ + A)u2 − u2 ≥ 0 . (4.2) u=1 u∈D(M0 )
In order to prove this we will need some auxiliary statements which will be established in the next two subsections.
4.1
A Hardy inequality
Here we state a Hardy inequality for the quadratic form of the operator M , which will be one of the crucial tools in the proofs of Theorems 2.2 and 2.3. Let p = (p1 , p2 ) ∈ Ω be some point and the number R be such that BR (p) ⊂ Ω. Given a smooth compactly supported field B, we define the function µ(r) := dist (Φp (r), Z), where we recall that Φp (r) is the flux of the field B through the ball Br (p). We introduce the function 1 , if Φp (r) ≡ 0 as r ∈ [0, R], 16 + c (R)c 1 2 (p, R) c(p, R) = (4.3) 0, if Φp (r) ≡ 0 as r ∈ [0, R], where
64 + 4R2 , R4 2R2 c3 (p2 )c4 (R) + 4c4 (R) + 4R2 c2 (p, R) = , c3 (p2 ) cos2 (|p2 − π2 | + R)
c1 (R) =
c3 (p2 ) = π 2 min{p−2 , (π − p2 )−2 } − 1, 2 µ(r) c4 (p, R) = max , r [0,R] 2 2 c5 (R) = max 2µ0 + 4c5 c6 µ40 , c6 , r02 2R3 − 3R2 r0 + r03 c6 (R) = 4 max 2 , j0,1 6r0
(4.4)
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and µ0 and r0 are defined by µ0 :=
1 r0 = , max r−1 µ(r) µ(r0 ) [0,R]
j0,1 is a smallest positive root of the Bessel function J0 . It was shown in [8] that the function c(p, R) is well defined. Finally, let us define 1, if |x1 | > l, (4.5) g(x1 ) = 1 , if |x1 | ≤ l. 4 Lemma 4.1. Assume that the field B is smooth and compactly supported and the − condition (1) of Theorem 2.2 is satisfied for the points p− = (p− 1 , p2 ) and p+ = + + (p1 , p2 ), then |(−i∇ + A)u|2 − g(x1 )|u|2 dx, (4.6) ρ(x1 )|u|2 dx ≤ Ω
Ω
holds for all u ∈ D(M0 ), where c− − 2, 1 + (x1 − p1 ) 0, ρ(x1 ) = c + , 2 1 + (x1 − p+ 1)
if − ∞ < x1 < p− 1, + if p− 1 < x1 < p1 ,
(4.7)
if p+ 1 < x1 < +∞,
and the constants c± = c(p± , R± ) are given by (4.3). Proof. We start the proof from the estimate |u|2 |(−i∇ + A)u|2 − |u|2 dx, dx ≤ c− − 2 1 + (x1 − p1 ) Ω(−∞,p− Ω(−∞,p− 1 ) 1 )
(4.8)
which is valid for all u ∈ D(M0 ). The proof of this estimate follows from the calculations of [8, Sec. 6], where the similar inequality |u|2 |(−i∇ + A)u|2 − |u|2 dx, (4.9) dx ≤ c − 2 Ω 1 + (x1 − p1 ) Ω is proved for all u ∈ H01 (Ω) with some constant c. The approach employed in [8, Sec. 3] can be applied to prove the inequality (4.8). We will not reproduce all the details of this proof and just note that the only modification needed is to replace the function ϕ defined in [8, Eq. (3.28)] by R 1 if x1 < p− 1 − √ , 2 √ − 2(p1 − x1 ) R (4.10) ϕ(x) := − if p− 1 − √ < x1 < p1 , R 2 0 elsewhere,
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In the same way the inequality |u|2 2 2 c+ dx ≤ − |u| |(−i∇ + A)u| dx, (4.11) + 1 + (x1 − p1 )2 Ω(p+ Ω(p+ 1 ,+∞) 1 ,+∞) holds for all u ∈ D(M0 ), where c+ = c(p+ , R+ ). We will make use of the diamagnetic inequality (see [13]) |∇|u|(x)| ≤ |(−i∇ + A)u(x)|
(4.12)
which holds pointwise almost everywhere in Ω for each u ∈ D(M0 ). In addition the trivial inequality π π |∂x2 u|2 dx2 ≥ g|u|2 dx2 (4.13) 0
0
holds for each fixed x1 and all u ∈ D(M0 ). The diamagnetic inequality (4.12) and the last estimate lead us to the inequality |(−i∇ + A)u|2 dx ≥ |∇|u||2 dx ≥ g|u|2 dx, Ω(α,β)
Ω(α,β)
Ω(α,β)
which is valid for all α < β. Combining now this inequality with (4.8), (4.11) we arrive at the statement of the lemma. In the case of the Aharonov-Bohm field the similar statement is true. Lemma 4.2. Assume that the field is generated by Aharonov-Bohm potential given by (2.1) and that the condition (1) of Theorem 2.3 is satisfied for the point p = (p1 , p2 ). Then ρ(x1 )|u|2 dx ≤ |(−i∇ + A)u|2 − g(x1 )|u|2 dx, (4.14) Ω
Ω
holds for all u ∈ D(M0 ), where c , 1 + (x − p1 )2 1 ρ(x1 ) = 0,
−∞ < x1 < p1 ,
(4.15)
p1 < x1 < +∞,
the constant c = c(p, Φ) is given by R2 µ2 c3 (p2 ) cos2 (|p2 − π2 | + R) , c(p, Φ) = 2 2 8 2µ R c3 (p2 ) + (8µ2 + 8 + c3 (p2 ))(9R2 + 16π 2 )
(4.16)
µ := dist {Φ, Z}, c2 (p2 ) is the same as in (4.4). The proof of this lemma is the same as the one of Lemma 4.8. It is also based on similar calculations of [8, Sec. 7.1], where the inequality (4.9) was proven for Aharonov-Bohm field. Here one also needs to replace the function ϕ in [8, Eq. (3.28)] by the function ϕ defined in (4.10) with p− 1 = p1 .
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A one-dimensional model
In this section we will show that the inequality (4.2) holds true if the oned2 2 dimensional Schr¨ odinger operator − dx 2 + V in L (R) with certain potential V 1 is non-negative. We will consider the case of a compactly supported field and the Aharonov-Bohm field simultaneously. In view of Lemmas 4.1 and 4.2 we have 1 (−i∇ + A)u2 − u2 = (−i∇ + A)u2 − (g u, u) 2 1 1 + (−i∇ + A)u2 + ((g − 2) u, u) 2 2 1 1 ≥ (−i∇ + A)u2 + ((ρ + g − 2) u, u) , 2 2 where g is given by (4.5). Here ρ is determined by (4.7) in the case of a compactly supported field and by (4.15) in the case of the Aharonov-Bohm field. Thus, inf (−i∇ + A)u2 − u2 u=1 u∈D(M0 )
≥
1 2
inf
u=1 u∈D(M0 )
(−i∇ + A)u2 + ((ρ + g − 2) u, u) .
By the diamagnetic inequality (4.12) we have inf (−i∇ + A)u2 − u2 u=1 u∈D(M0 )
≥ =
1 2
inf
u=1 u∈D(M0 )
1 2
1 = 2
∇|u|2 + ((ρ + g − 2) u, u)
inf
u=1 u∈D(M0 )
∇u2 + ((ρ + g − 2) u, u)
inf
u=1 u∈D(M0 )
Ω
|∂x1 u|2 + |∂x2 u|2 dx
+ ((ρ + g − 2) u, u) . Using now (4.13) we arrive at inf (−i∇ + A)u2 − u2 u=1 u∈D(M0 )
≥
1 2
inf
u=1 u∈D(M0 )
∂x1 u2 + (ρ u, u) + 2((g − 1) u, u) .
(4.17)
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In order to establish the inequality (4.2) it is therefore enough to show that π |ux1 (x)|2 + ρ(x1 )|u(x)|2 + 2(g(x1 ) − 1)|u(x)|2 dx1 dx2 ≥ 0, 0
R
which is equivalent to the inequality 2 |v | + ρ|v|2 + 2(g − 1)|v|2 dx1 ≥ 0,
(4.18)
R
for all v ∈ C0∞ (R). In other words, to prove Theorems 2.2 and 2.3 it is sufficient to show that the one-dimensional Schr¨ odinger operator −
d2 + ρ + 2(g − 1) dx21
is non-negative in L2 (R). The proof of this fact is the main subject of the next section.
4.3
The proofs of Theorems 2.2 and 2.3
As it has been shown in the previous section to prove the absence of the eigenvalues it is sufficient to check the inequality (4.18). Due to the definition of g it can be rewritten as 3 l |v (t)|2 + ρ(t)|v(t)|2 dt ≥ |v(t)|2 dt. (4.19) 2 −l R Let us show that under the assumptions of Theorems 2.2, respectively 2.3 this inequality holds true. We will show it in detail for the case of compactly supported field only (i.e., for Theorem 2.2); the case of the Aharonov-Bohm field is similar. We introduce a function π − c− + arctan(t − p− 1 ) , t < p1 , 2 φ− (t) := πc (4.20) −, t ≥ p− . 1 2 − We remind that c− and p− 1 are given in Lemma 4.1. Clearly, φ− (t) = ρ(t) for t < p1 − and φ− (t) = 0 if t ≥ p1 . Keeping these properties in mind for each t ∈ (−l, l) we deduce the obvious equality
πc− v(t) = φ− (t)v(t) = 2
t −∞
(φ− (s)v(s)) ds
p− 1
=
t
ρ(s)v(s) ds + −∞
−∞
φ− (s)v (s) ds,
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where we also employ the fact that by the assumption of Theorem 2.2 we have p− 1 < −l. The equality obtained, definition of φ− and Cauchy-Schwarz inequality give rise to an estimate 2 2 p− t 1 π 2 c2− |v(t)|2 ≤ 2 ρ(s)v(s) ds + φ− (s)v (s) ds −∞ 4 −∞ − p− t t p1 1 2 2 2 (4.21) ρ(s) ds ρ(s)|v(s)| ds + φ (s) ds |v (s)| ds ≤2 ≤2
−∞
πc− 2
−∞
p− 1
−∞
2
t
ρ(s)|v(s)| ds + −∞
−∞
φ2− (s) ds
−
−∞
l
−∞
2
|v (s)| ds .
Since the function φ− (t) is constant for t > p− 1 it follows that
t
−∞
φ2− (s) ds
p− 1
= −∞
= c2−
− φ2− (s) ds + φ2− (p− 1 )(t − p1 ) 0
−∞
π
2 π 2 c2− + arctan(s) ds + (t − p− 1) 2 4
= c2− π ln 2 +
π 2 c2− (t − p− 1 ). 4
Substituting the last equality into (4.21) and using the expression for φ− (p− 1 ) (see (4.20)) we arrive at p− 1 2 2 ρ(s)|v(s)|2 ds |v(t)| ≤ 2 πc− −∞ (4.22)
l 4 ln 2 − 2 + + (t − p1 ) |v (s)| ds . π −∞ In the case c− = 0 the fraction c1− in this inequality is understood as +∞, so the inequality is valid for all possible values of c− . Integration (4.22) over (−l, l) and using the obvious equality 0 p− 1 2 ρ(s)|v(s)| ds = ρ(s)|v(s)|2 ds −∞
−∞
lead us to the estimate
l l 0 4 ln 2 2 − 2 2 2 − p1 |v(t)| dt ≤ 4l ρ(s)|v(s)| ds + |v (s)| ds πc− −∞ π −l −∞ l 0 4l ≤ ρ(s)|v(s)|2 ds + |v (s)|2 ds , 2 κ− −∞ −∞
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where κ− is given by (2.4). We can rewrite this inequality as l l 0 2 2 2 κ− |v(t)| dt ≤ 4l 2 ρ(s)|v(s)| ds + |v (s)| ds .
(4.23)
−l
−∞
−∞
This inequality is valid also in the case of c− = 0. In the same way one can easily prove similar inequality κ+
l
−l
|v(t)| dt ≤ 4l 2 2
+∞
2
+∞
ρ(s)|v(s)| ds +
0
−l
2
|v (s)| ds ,
(4.24)
where κ+ is given by (2.4). We sum the inequalities (4.23) and (4.24) to get l l 2 2 (κ− + κ+ ) |v(t)| dt ≤ 4l 2 ρ(s)|v(s)| ds + |v (s)|2 ds −l
R
+∞
+ −l
−∞
|v (s)|2 ds .
This implies that
l
−l
|v(t)|2 dt ≤
8l κ
ρ(s)|v(s)|2 ds +
R
R
|v (s)|2 ds ,
where κ = κ− + κ+ . An immediate consequence of the last inequality is that to satisfy (4.19) it is sufficient to set l≤
κ , 12
which coincides with the inequality (2.3). This completes the proof of Theorem 2.2. The proof of Theorem 2.3 is similar. One just needs to use the inequality (4.23) rewritten in a slightly different way: l 0 2 2 |v(t)| dt ≤ 4l ρ(s)|v(s)|2 ds πc− −∞ −l
l 4 ln 2 − 2 + − p1 |v (s)| ds π −∞ l 0 4l 2 2 ≤ ρ(s)|v(s)| ds + |v (s)| ds , κ −∞ −∞ with κ given by (2.6). This inequality will immediately imply the estimate (4.19) if the relation (2.5) is satisfied.
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5 Presence of eigenvalues In this section we will prove Theorem 2.4. We will use the formula (−i∇ + A)u2 − u2 . inf σ(M − 1) = inf u=1 u∈D(qM )
If we find a test function u ∈ D(qM ) such that (−i∇ + A)u2 − u2 < 0 this will prove the presence of the discrete spectrum due to Theorem 2.1. Clearly, D(qM ) is a subspace of H 1 (Ω) consisting of functions that vanish on Γ. The eigenfunction ψ of −∆N ,D associated with the lowest eigenvalue λ(l) belongs to D(qM ). We can choose this eigenfunction being real-valued and normalized in L2 (Ω). Choosing ψ as a test function we have (−i∇ + A)ψ2 = ∇ψ2 + Aψ2 = λ(l) + Aψ2 ≤ λ(l) + max |A|2 . Ω
(5.1)
Here we used the normalization condition for ψ and an obvious relation λ(l) = ∇ψ2 . The left-hand side of inequality (5.1) is invariant under the gauge transformation of the magnetic potential A. Bearing this fact in mind we take the infimum in (5.1) over all potentials associated with the field B what leads us to (−i∇ + A)ψ2 − ψ2 ≤ λ(l) + inf max |A|2 − 1. A
Ω
By the assumption the right-hand side of the last inequality is less than zero, hence the theorem is proved. In conclusion let us show that the second term on the left-hand side of (2.7) is finite. It is sufficient to show that it is finite for some A. Let A be some potential associated with B. Since B is smooth and compactly supported, the potential A can be chosen in C 1 (Ω). Therefore it is bounded on each bounded subset of Ω. The support of B is a compact set, so there exists number b > 0 such that B = 0 as x ∈ Ω \ Ω(−b, b), i.e., ∂x2 a2 − ∂x1 a1 = 0 as x ∈ Ω \ Ω(−b, b). Since both domains Ω(−∞, −b) and Ω(b, +∞) are simply connected, this immediately implies the existence of functions h− ∈ C 1 (Ω(−∞, −b)), h+ ∈ C 1 (Ω(b, +∞)) such that ∇h− = A as x ∈ Ω(−∞, −b), ∇h+ = A as x ∈ Ω(b, +∞). We introduce the function h− (x)ζ(x1 ), x1 < −b, −b ≤ x1 ≤ b, h(x) = 0, h+ (x)ζ(x1 ), x1 > b, where ζ(x1 ) is equal to one as |x1 | > 2b and vanishes as |x1 | ≤ b. By definition h ∈ := A−∇h leads us to a new vector potential A C 1 (Ω). The gauge transformation A is compactly supported associated with the same field B. Moreover the potential A ∈ C 1 (Ω), it follows that max |A| 2 is since ∇h = A if |x1 | is large enough. Since A finite.
Ω
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6 Acknowledgments D.B. has been supported by DAAD (A/03/01031) and partially supported by RFBR and the program “Leading scientific schools” (NSh-1446.2003.1). T.E. has been supported by ESF Program SPECT. The work has also been supported by the DAAD project 313-PPP-SE/05-lk. D.B. and T.E. thank the Stuttgart University, where this work has been done, for the hospitality extended to them. Authors would like to thank T. Weidl for suggesting them the study of the initial problem and for numerous stimulating discussions.
References [1] W. Bulla, F. Gesztesy, W. Renger and B. Simon, Weakly coupled bound states in quantum waveguides, Proc. Amer. Math. Soc. 125, no. 5, 1487–1495 (1997). ˇ [2] P. Exner, P. Seba, M. Tater, D. Vanˇek, Bound states and scattering in quantum waveguide coupled through a boundary window, J. Math. Phys. 37, no. 10, 4867–4887 (1996). [3] P. Exner and S. Vugalter, Asymptotic estimates for bound states in quantum waveguides coupled laterally through a narrow window, Ann. Inst. H. Poincar´e: Phys. th´eor. 65, no. 1, 109–123 (1996). [4] I.Yu. Popov, Asymptotics of bound states for laterally coupled waveguides, Rep. Math. Phys. 43, no. 3, 427–437 (1999). [5] R.R. Gadyl’shin, On regular and singular perturbations of acoustic and quantum waveguides, C.R. M´ecanique 332, no. 8, 647–652. [6] P. Duclos, P. Exner, B. Meller, Resonances from perturbed symmetry in open quantum dots. Rep. Math. Phys. 47, no. 2, 253–267 (2001). [7] T. Weidl, Remarks on virtual bound states for semi-bounded operators, Comm. in Part. Diff. Eq. 24, no. 1&2, 25–60 (1999). [8] T. Ekholm and H. Kovaˇr´ık, Stability of the magnetic Schr¨ odinger operator in a waveguide, to appear in Comm. in Part. Diff. Eq., Preprint: arXiv:math-ph/0404069. [9] H.L. Cycon, R.G. Froese, W. Kirsh, B. Simon, Schr¨ odinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer Study Edition. Springer-Verlag, Berlin-New York. 1987. [10] M.S. Birman, Perturbation of the continuous spectrum of a singular elliptic operator under a change of the boundary and the boundary condition, Vestnik Leningradskogo universiteta. 1, 22–55 (1962).
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[11] Michail S. Birman and Michail Z. Solomyak, Spectral theory of self-adjoint operators in Hilbert space, D. Reidel Publishing Company, 1987. [12] O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Applied Mathematical Sciences, v. 49. Springer-Verlag, New York, 1985. [13] D. Hundertmark and B. Simon, A diamagnetic inequality for semigroup differences, J. Reine Angew. Math. 571, 107–130 (2004).
Denis Borisov Department of Physics and Mathematics Bashkir State Pedagogical University October rev. st., 3a 450000 Ufa Russia email:
[email protected]
Tomas Ekholm Department of Mathematics Royal Institute of Technology Lindstedtsv¨ agen, 25 S-100 44 Stockholm Sweden email:
[email protected]
Hynek Kovaˇr´ık Faculty of Mathematics and Physics Stuttgart University Pfaffenwaldring, 57 D-70569 Stuttgart Germany email:
[email protected] Communicated by Vincent Rivasseau submitted 11/05/04, accepted 21/09/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 343 – 367 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020343-25 DOI 10.1007/s00023-005-0210-3
Annales Henri Poincar´ e
The Hopf Algebra of Rooted Trees in Epstein-Glaser Renormalization Christoph Bergbauer and Dirk Kreimer Abstract. We show how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization and coordinate space renormalization in general. In particular, we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions. Twisting the antipode with a renormalization map formally solves the Epstein-Glaser recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator B+ .
Introduction The Epstein-Glaser framework [5, 16] and its modern variants [15, 2, 14] provide a mathematically rigorous approach to perturbation theory and renormalization in coordinate space. Let M = R1,3 denote the Minkowski space. Epstein and Glaser constructed, for a scalar φk field theory say, a sequence of operator-valued distributions Tn on M n respectively, which replace the ill-defined time-ordered products in the standard approach to perturbation theory. The result is a perturbation theory which is a priori finite in each order – no removal of short-distance singularities is needed since all expressions are well defined from the very beginning. The appropriate notion of renormalization in the Epstein-Glaser framework is extension of distributions onto diagonals. Indeed, the objects of interest Tn are a priori determined outside the diagonals by causality. Finite renormalizations correspond to different ways of extending distributions onto diagonals. Moreover, in this approach the S-matrix is local by construction. On the other hand, the combinatorics of momentum space renormalization have been most efficiently described [4, 11] in terms of the Hopf algebra and associated Lie algebra of Feynman graphs. Renormalization and in particular the Bogoliubov recursion boil down to twisting the antipode S of that Hopf algebra by renormalization maps into some target ring of Laurent or formal power series. This is possible due to a coproduct which disentangles 1PI graphs into divergent 1PI subgraphs. There is a universal object behind all Hopf algebras of this kind: the Hopf algebra of rooted trees [9, 3] which encodes nested subdivergences in terms of a tree and their recursive removal in terms of its coproduct and the resulting antipode. We will show how the Hopf algebra of rooted trees works in the realm of Epstein-Glaser renormalization in almost complete analogy to other renormalization programs like BPHZ. In fact it is even easier to understand
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its role in Epstein-Glaser renormalization since no regularization is required and overlapping divergences do not exist in the coordinate space language. This paper is organized as follows: In the first section we give a short review of the Epstein-Glaser construction of time-ordered products, emphasizing the point of view of diagonals [2]. The second section recalls the powerful notion of a Hochschild 1-cocycle on a connected graded bialgebra, giving rise to two equivalent presentations of the Hopf algebra of rooted trees. A new convolution-like product is introduced which in cooperation with the antipode allows to recursively generate all terms needed for an Epstein-Glaser time-ordered product, as will be proved using explicit renormalized “Feynman rules” in the final theorem which we already state in a short version: Theorem (Main result) There is a map Φ : H•∗ → F V such that the n-th Epstein-Glaser time-ordered product Tn is given by Φ(SR id)(t) Tn = t∈Tn •∗
where H is a Hopf-algebra of rooted trees, F V something like the tensor algebra of distributions on M , Tn the set of all binary trees with n leaves, SR the twisted antipode of H•∗ and a modified “convolution product” in H•∗ .
1 Some background on Epstein-Glaser renormalization For simplicity we restrict ourselves to a massive neutral scalar field theory with interaction Lagrangian λ LI = φk , (1) k! on the flat Minkowski space-time M := R1,3 . Generalizations to Quantum Electrodynamics and globally hyperbolic space-times have been worked out in [16] and [2], respectively, which though does not affect the combinatorics we are primarily interested in.
1.1
Motivation
As a starting point for the Epstein-Glaser construction of time-ordered products [5] we consider the symbolic Dyson series for the S-matrix S = T ei
LI (x)dx
(2)
which is formally derived from the Schwinger differential equation of motion by transforming it into an iterated integral equation and applying the time-ordering operator T to each summand (i)n T (LI (x1 ) . . . LI (xn ))dx1 . . . dxn n! M n
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which has the benefit that we are integrating now over M n rather than over an n-simplex × R3n . Let A, B be operator-valued functions on M. The time-ordering operator T is usually defined by T (A(x1 )B(x2 )) := Θ(x01 − x02 )A(x1 )B(x2 ) + Θ(x02 − x01 )B(x2 )A(x1 )
(3)
where Θ denotes the Heaviside characteristic function of R≥0 . Analogously one defines T on more than two factors. Now S and LI are obviously supposed to be operator-valued distributions, for which (3) does not make sense since distributions can not just be multiplied by noncontinuous functions like Θ. It does make sense though outside the thick diagonal Dn = {x ∈ M n : xi = xj for some i = j} where products of Θ(x0i − x0j ) are continuous. In fact the mathematical origin for the appearance of short-distance singularities in perturbation theory is the ill-defined notion of time-ordering reviewed above. Epstein and Glaser proposed a way to construct well-defined time ordered products Tn , one for each power n of the coupling constant, that satisfy a set of suitable conditions explained below, the most prominent being that of locality or micro-causality. The power series S constructed by (2) using the Epstein-Glaser time-ordered product T is a priori finite in every order, and renormalization corresponds then to stepwise extension of distributions from M n − Dn to M n . In general, distributions can not be extended uniquely onto diagonals. The resulting degrees of freedom are in one-to-one correspondence with the degrees of freedom (finite renormalizations) in momentum space renormalization programs like BPHZ and dimensional regularization. The notion of locality, crucial to the following construction of time-ordered products, can be motivated as follows: Suppose x = (x1 , . . . , xn ) ∈ M n , ∅ I N := {1, . . . , n} and for each i ∈ I, the point xi is not in the past causal shadow of any of the xj for j ∈ N −I. We denote this situation xi xj ∀i ∈ I, j ∈ N −I. Then our time ordered product Tn is supposed to satisfy (in the sense of operator-valued distributions) Tn (x1 , . . . , xn ) = T|I| (xi )i∈I T|N −I|(xj )j∈N −I (4) because we think of the xi to happen after (or at least not before) the xj . If both xi xj and xj xi , ∀i, j, so if all pairs (xi , xj ) are spacelike, we have [T|I| (xi )i∈I , T|N −I| (xj )j∈N −I ] = 0.
1.2
Construction of time-ordered products
In this subsection we give a short review of the mathematical core of EpsteinGlaser renormalization in its modern variant [15, 2, 14] which emphasizes the point of view of nested diagonals. For the proofs, the reader is referred to [2].
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The Minkowski metric on M provides a relation on M as follows: x y − − iff x is not in the past causal shadow of y, that is x ∈ / y + V where V := {z ∈ M : (z)2 ≤ 0, z 0 ≤ 0} is the closed past lightcone. Now, for n ∈ N let N := {1, . . . , n} and ∅ I N. The set CI := {(x1 , . . . , xn ) ∈ M n : xi xj ∀i ∈ I, j ∈ N − I} is obviously a translation invariant open subset of M n . Lemma 1 (Geometric lemma)
CI = M n − ∆n
∅IN
where ∆n = {x ∈ M n : x1 = . . . = xn } is the “thin” diagonal in M n . The proof is an easy induction on n. The geometric lemma tells us that the causality condition (4) determines the time-ordered product Tn everywhere outside the thin diagonal ∆n , once the Tk for k < n are known on whole M k , respectively. It is important to understand that the geometric lemma does not really constitute a specific feature of the Minkowski space. Indeed, the lemma holds if one replaces by any relation such that x y or y x whenever y = x, and such that is “weakly transitive” in the sense that x y and ¬(z y) imply x z. Definition 2 A causal partition of unity {pI,N −I }∅IN is a smooth partition of unity subordinate to the cover {CI }∅IN of M n − ∆n . For simplicity, we will sometimes drop the curly brackets in the subscript, for example p1,2 denotes p{1},{2} . Let D(M ) = C0∞ (M ) denote the space of test functions on M with the usual topology. Let H denote the Hilbert space of the free field theory and D a suitable dense subspace. In principle an Epstein-Glaser time-ordered product is a collection (Tnr )n∈N (r = (r1 , . . . , rn ) an n-multiindex) of operator-valued distributions Tnr : (r ,...,rn ) D(M n ) → End(D), such that Tn 1 replaces the time ordering of the n Wick monomials : φr1 :, . . . , : φrn : . Definition 3 A collection (Tnr ) of operator-valued distributions Tnr : D(M n ) → End(D) is called an (Epstein-Glaser) time-ordered product if (i) T1k (f ) = : φk : (f ) where : φk : (f ) denotes the Wick monomial : φk : smeared with the test function f, (ii) T is symmetric Tnr (f1 ⊗ . . . ⊗ fn ) = Tnr (fπ(1) ⊗ . . . ⊗ fπ(n) )
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when π is a permutation of N := {1, . . . , n}. This allows for the notation T (N ) = Tnr (f1 ⊗ . . . ⊗ fn ) when the fi and ri are clear from the context, (iii) T splits causally: Let ∅ I N. Then T (N ) = T (I)T (N − I)
(5)
for all test functions with support in CI ⊂ M n , (iv) T is translation covariant U (a, 1)T (f1 , . . . , fn )U (a, 1)−1 = T (τa f1 , . . . , τa fn ) where U (·, 1) . . . U (·, 1)−1 is the representation of the translation part of the Poincar´e group in D, and τa f (x) = f (x − a) denotes translation by a. (v) The Wick expansion relates time-ordered products corresponding to different Wick-powers Tn(r1 ...rn ) (f1 ⊗ · · · ⊗ fn ) = Ω, Tn(r1 −i1 ,...,rn −in ) (f1 ⊗ · · · ⊗ fn )Ω i1 ,...,in
r1 rn × ··· : φi1 . . . φin : (f1 ⊗ · · · ⊗ fn ) (6) i1 in with Ω the vacuum state in D ⊂ H. Note that by the so-called Theorem 0 in [5] the summands in the right-hand side of (6) as products of translation invariant numerical distributions and Wick monomials are well-defined operator-valued distributions. Once a time-ordered product T = (Tnr ) is given, the S-matrix for the φk -theory is obtained as the formal power series ∞ in S(f ) = T (k,...,k) (f ⊗n ), (7) n!(k!)n n n=0 possibly taking the adiabatic limit f → λ later on, which is a highly nontrivial task we shall not be concerned about in the present work. The S-matrix (7) and the relative S-matrices constructed from T are local. If one imposes additional normalization conditions (Lorentz covariance, Hermiticity etc., see [2]) on T, the S-matrix becomes Lorentz covariant and unitary, etc. Moreover, the interacting field constructed from the relative S-matrices are Lorentz covariant, Hermitean and satisfy the interacting field equation. Theorem 4 Time-ordered products exist.
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A constructive proof is given in [2] and of course, but in a somewhat different notation, in the original paper [5]. The idea is as follows: Provided all (Tm ) for m < n are constructed, the Geometric Lemma 1 ensures that Tn is determined on M n − ∆n by causality (iii). We define TI = T (I)T (N − I) as a distribution on CI . One easily shows that T (I)T (N − I) = T (J)T (N − J) on the intersection CI ∩CJ . Therefore, we can patch the TI together using a causal partition of unity {pI,N −I } 0 T (N ) := pI,N −I T (I)T (N − I) (8) ∅IN
which is a well-defined distribution on M n − ∆n . As usual, 0 T (N ) is independent on the choice of the partition of unity. It remains to extend it to a distribution on M n . Using the Wick expansion (v) and translation invariance, this amounts to an extension problem of numerical distributions 0 tn from M n−1 − {0} to M n−1 . Having quantified the behavior of a numerical distribution at the origin by the Steinmann scaling degree (see [2] for details), a generalization of the degree of homogeneity, one can show that there is a unique extension tn of 0 tn to M n−1 preserving the scaling degree, provided the scaling degree sd(0 tn ) of 0 tn is smaller than the dimension 4(n − 1). Otherwise, if it is larger or equal but still finite, there is a finite dimensional space of extensions obtained as follows: Let f ∈ D(M n−1 ). The distribution
0 α ωα ∂ f (0) (9) tn : f → tn f − α
where the sum goes over all 4(n − 1)-multiindices α such that |α| ≤ sd(0 tn ) − 4(n − 1) and the ωα ∈ D(M n−1 ) such that ∂ β ωα (0) = δα,β , has then scaling degree sd(0 tn ) < 4(n − 1) and is hence uniquely extendible (preserving the scaling degree). There is an ambiguity due to the ωα however, and it is exactly this ambiguity which corresponds to the freedom of finite renormalizations. We call the linear operator id − w on test functions ωα ∂ α f (0) id − w : f → f − α
Taylor subtraction operator and, motivated by the fact that tn = (id − w∗ )0 tn holds on the level of numerical distributions, we write by abuse of notation the extension of 0 T (N ) to the diagonal by ∗ )0 T (N ) T (N ) = (id − W1...n
(10)
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although there is no linear operator W ∗ on the space of operator valued distributions doing this duty. Our abuse of notation is justified though because we are only concerned with the combinatorics with respect to n in the following, and the Wick expansion leaves n obviously unchanged. So we understand W ∗ as the symbolic “operator” which unpacks the operator valued distributions into Wick monomials and numerical distributions, Taylor subtracts the test function for those numerical distributions and produces then a “counterterm” such that (id − W ∗ ) maps a distribution on M n − ∆n to an extension on M n while the possible ambiguity (depending on the scaling degrees) is fixed by a choice of the ωα . The subscript in ∗ W1...n indicating to which coordinates it applies will be useful later on. This constructive proof of Theorem 4 actually proves more than the theorem demands: that in each extension step the scaling degree does not increase. If we make this an additional condition on time-ordered products, we can state Corollary 5 All time-ordered products are uniquely (up to the ωα , more precisely up to the finite set of constants 0 tn (ωα ) in every order n) characterized by equations (8) and (10). Feynman graphs enter the game when one applies Wick’s theorem. It might be instructive to have a look at the examples in [14]. We also note that the usual notions of renormalizable theories, critical dimension etc. can be traced back to the behavior of the scaling degrees as n and the space-time dimension vary. In particular, the scaling degree coincides with the usual power-counting techniques in momentum space.
2 The Hopf algebra of rooted trees in Epstein-Glaser renormalization The combinatorics of renormalization in coordinate space can be most easily described in terms of rooted trees. Given some space-time points, •
•
•
•
•
•
we consider them as leaves of a tree (to be constructed). Whenever some of these points come together on a diagonal in M n , we connect the corresponding vertices to a new vertex such that subdivergences (subdiagonals) correspond to subtrees, for example • @ • @ @ @ @• @• • A A A • A• • A• • A• So a tree represents the (partially ordered) nested or disjoint subdiagonals which are relevant to renormalization. It is now possible to construct a suitable coproduct
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on the free algebra generated by these trees such that the Bogoliubov recursion is essentially solved by the antipode of the resulting Hopf algebra on trees, as will be made precise in Subsection 2.2. This remarkable property and the fact that local counterterms result [12] are the consequence of the fact that a certain operator on the Hopf algebra is a Hochschild 1-cocycle.
2.1
Hochschild cohomology of bialgebras
All algebras are supposed to be over some field k of characteristic zero, associative and unital, analogously for coalgebras. The unit (and by abuse of notation also the unit map) will be denoted by I, the counit map by . All algebra homomorphisms ∞ are supposed to be unital. A bialgebra (A = i=0 Ai , m, I, ∆, ) is called graded connected if Ai Aj ⊂ Ai+j and ∆(Ai ) ⊂ j+k=i Aj ⊗ Ak , and if ∆(I) = I ⊗ I and ∞ A0 = kI, (I) = I and = 0 on i=1 Ai . We call ker the augmentation ideal of A and denote P the projection A → A onto the augmentation ideal, P = id − I . Let (A, m, I, ∆, ) be a bialgebra. We think of linear maps L : A → A⊗n as n-cochains and define a coboundary map b by bL := (id ⊗ L) ◦ ∆ +
n
(−1)i ∆i ◦ L + (−1)n+1 L ⊗ I
(11)
i=1
where ∆i denotes the coproduct applied to the i-th factor in A⊗n . It is easy to see (using essentially the coassociativity of ∆) that b2 = 0, which gives rise to a cohomology theory called Hochschild cohomology. It is also easy to see that, for A finite dimensional say, the cohomology theory (11) is the dual of the usual Hochschild homology of the dual algebra A∗ . In case n = 1, (11) reduces to, for L : A → A, bL = (id ⊗ L) ◦ ∆ − ∆ ◦ L + L ⊗ I.
(12)
It is known [3] that the category of objects (A, C) consisting of a commutative bialgebra A and a Hochschild 1-cocycle C on A with morphisms bialgebra morphisms commuting with the cocycles has an initial object (H, B+ ), with H the Hopf algebra of (non-planar) rooted trees and the operator B+ which grafts a product of rooted trees together to a new root as described in the next subsection. While the higher (n > 1) Hochschild cohomology of H vanishes [6], the closedness of B+ will turn out to be crucial for what follows. The next lemma will provide a convenient way to construct Hopf algebras out of free or free commutative algebras by choosing linear endomorphisms Ci and demanding that the Ci be Hochschild 1-cocycles. ∞ Lemma 6 Let A = n=0 An be a free or free commutative graded algebra (generated by a graded vector space) such that A0 = kI, and let (Ci )i∈I be a collection
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of injective linear endomorphisms of A such that Ci (A) ∩ Cj (A) = {0} for i = j and such that each free generator y in degree n is the image under some Ci of an x ∈ An−1 for n ≥ 1. Then there is a unique connected graded bialgebra structure (∆, ) on A such that the Ci are Hochschild closed with respect to ∆. In particular, A is a Hopf algebra (with this property) in a unique way. Proof. We will construct ∆ by induction on n. The Hochschild closedness of the Ci demands that ∆ ◦ Ci = (id ⊗ Ci ) ◦ ∆ + Ci ⊗ I.
(13)
∆(I) = I ⊗ I by convention, so ∆ is known on A0 . Now let y be a free generator in An+1 . By assumption there is a unique x ∈ An such that y = Ci x. Assume ∆ is known on x, then by (13) it is also known on y. Hence we can uniquely extend ∆ to an algebra homomorphism on An+1 . By induction, this uniquely defines ∆ as an algebra morphism on A. From (13) it also follows inductively that ∆ respects the grading in all orders: ∆(An ) ⊂
n
Ak ⊗ An−k .
k=0
For the coassociativity (∆ ⊗ id)∆ = (id ⊗ ∆)∆ we note that (∆ ⊗ id)∆Ci
=
(∆ ⊗ id)((id ⊗ Ci )∆ + Ci ⊗ I)
= =
(∆ ⊗ Ci )∆ + ∆Ci ⊗ I (∆ ⊗ Ci )∆ + (id ⊗ Ci ⊗ id)(∆ ⊗ I) + Ci ⊗ I ⊗ I
=
(id ⊗ id ⊗ Ci )(∆ ⊗ id)∆ + (id ⊗ Ci ⊗ id)(∆ ⊗ I) + Ci ⊗ I ⊗ I.
On the other hand, (id ⊗ ∆)∆Ci
=
(id ⊗ ∆)((id ⊗ Ci )∆ + Ci ⊗ I)
= =
(id ⊗ ∆Ci )∆ + Ci ⊗ I ⊗ I id ⊗ ((id ⊗ Ci )∆ + Ci ⊗ I)∆ + Ci ⊗ I ⊗ I
=
(id ⊗ id ⊗ Ci )(id ⊗ ∆)∆ + (id ⊗ Ci ⊗ id)(∆ ⊗ I) + Ci ⊗ I ⊗ I
which proves the coassociativity by induction on the grading. Now setting (I) = I and = 0 elsewhere finishes the proof. Note that any connected graded bialgebra is a Hopf algebra in a unique way.
2.2
The Hopf algebra of rooted trees, relation to previous work
In this section we collect well-known results [3, 4, 9, 12] on Hopf algebra methods in momentum space renormalization which will turn out to be applicable to EpsteinGlaser renormalization as well.
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A rooted tree is a connected contractible compact graph with a distinguished vertex, the root. A forest is a disjoint union of rooted trees. Isomorphisms of rooted trees or forests are isomorphisms of graphs preserving the distinguished vertex/vertices. Let t be a rooted tree with root o. The choice of o determines an orientation of the edges of t : we draw the root on top and let the rest of the tree “hang down.” Vertices of t having no outgoing edges are called leaves, the other vertices (and the root) are called internal vertices. The set of forests is graded, for instance by the number of vertices a forest has (the weight grading). Let H be the free commutative algebra generated by rooted trees with the weight grading. The commutative product in H will be visualized as the disjoint union of trees, such that monomials in H are scalar multiples of forests. We demand that the linear operator B+ on H, defined by B+ (I)
=
B+ (t1 . . . tn ) =
• • A @ • A@• t1 . . . tn
is a Hochschild 1-cocycle, which makes H a Hopf algebra by virtue of Lemma 6. It is easy to see that the resulting coproduct can be described as follows Pc (t) ⊗ Rc (t) (14) ∆(t) = I ⊗ t + t ⊗ I + adm.c
where the sum goes over all admissible cuts of the tree t. By a cut of t we mean a nonempty set of edges of t that are to be removed. The product of subtrees which “fall down” upon removal of those edges is called the pruned part and denoted Pc (t), the part which remains connected with the root Rc (t). Now a cut c(t) is admissible, if for each leaf l of t it contains at most one edge on the path from l to the root. For instance, • • • A A A • • • A • A ∆ = ⊗I+I⊗ • A + AA ⊗ + A A A A A A • • • A A A • • • • • • • • • • • • A • + •⊗ • + 2 • ⊗ • A• + AA • ⊗ • +2 • • ⊗ • + A • • • • A• • • • . + • • ⊗ AA + • • • ⊗ • • • H is obviously not cocommutative. Let V be a unital ring with multiplication mV . Given ring homomorphisms φ, ψ : H → V, one can define their convolution product φ ψ : H → V, x →
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mV (φ ⊗ ψ)∆x, which is a ring homomorphism again. In particular, the antipode S is the inverse of id : H → H with respect to this convolution product. Let Q be the linear endomorphism of H ⊗ H such that Q(I ⊗ I) = −I ⊗ I and Q = id ⊗ P otherwise. So (up to the sign) Q is a projection onto H ⊗ ker ⊕ kI ⊗ kI. The shorthand notation φ Q ψ := mV (φ ⊗ ψ)Q∆ will be useful. Now in any Hopf-algebra approach [9, 12, 3, 4] to perturbative quantum field theory, renormalization boils down to twisting the antipode which, (in any graded Hopf algebra) satisfies the recursive equation S = −m(S ⊗ id)Q∆ = −S Q id, by a homomorphism Φ : H → V, called “Feynman rules”, for example into a ring V of Laurent series (dimensional regularization) or formal power series (BPHZ), and a “renormalization scheme” R : V → V which delivers the counterterm. More explicitly, one considers Φ Φ Φ := −RmV (SR ⊗ Φ)Q∆ = −R(SR Q Φ). SR
(15)
While Φ means application of unrenormalized Feynman rules, the renormalized expression is then given by Φ SR Φ. (16) For details the reader is referred to [3]. In Epstein-Glaser renormalization, essentially the same happens, but in an easier way because no regularization is required. The target ring V is most suitably chosen to be something like the tensor algebra of distributions on M, Φ will then map a given “subdivergence situation” encoded in a rooted tree to the corresponding distribution in V. The meaning of Φ is much easier to understand however if we give a somewhat different presentation of the Hopf algebra and define a modified convolution product.
2.3
The cut product and the Bogoliubov recursion
We enlarge the Hopf algebra H to H•∗ by allowing for two types of vertices: • and ∗. This yields two Hochschild 1-cocycles B+• and B+∗ depending on which type the newly adjoined root has. B+• (I) = • • A@ B+• (t1 . . . tn ) = • A@• t1 . . . tn
B+∗ (I) = ∗
∗ A @ B+∗ (t1 . . . tn ) = • A@• t1 . . . tn
It is easy to see that the coproduct ∆ which we endow H•∗ with using B+• , B+∗ and Lemma 6 has the same form (14) as in H. Now let R be the algebra endomorphism of H•∗ which changes the type of the root to ∗, whatever it was
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before. ∗ • A@ @ A = • A@• . R • A@• t1 . . . tn t1 . . . tn
R(•) = ∗,
R(∗) = ∗,
Once again we remark that all our algebra endomorphisms are supposed to be unital, so we will not specify their values at I explicitly. Our aim is now to construct a new product called cut product of linear endomorphisms of H•∗ . The usual convolution product (φ, ψ) → φ ψ = m(φ ⊗ ψ)∆ in Endk (H) or Endk (H•∗ ) has the disadvantage that, applied several times with the projection P onto the augmentation ideal, it gets rid of the structure of trees. For example, for any tree t there is an n ∈ N such that P n (t) = (P . . . P )(t) = polynomial in • . Our new product (φ ψ)(t) is supposed to apply φ to Pc (t) and ψ to Rc (t) as well, but reassemble the tree afterwards rather than taking the disjoint union of pruned and root parts using m. For instance, (φ ψ)
• •
:= φ
• •
ψ(I) + φ(I)ψ
• •
+
ψ(•) φ(•)
which should be compared to (φ ψ)
• •
=φ
• •
ψ(I) + φ(I)ψ
• •
+ φ(•)ψ(•).
This is however only possible for a rather small class of φ and ψ which do not change the trees too much. For example, φ is supposed to map trees to trees while ψ is not allowed to kill the vertices where something has been cut. We leave it to the reader to find the most general notion of those maps, because the only ones we need here are B+ and id, P, R, where all this is possible in a rather trivial way. ˜ •∗ be the Hopf algebra of trees as in H•∗ with an additional decoration of Let H ˜ •∗ → H•∗ the vertices by subsets of N. There is an obvious forgetful projection π : H •∗ •∗ ˜ and an inclusion j : H → H decorating all vertices by the empty set. We lift ˜ •∗ → H ˜ •∗ by the any of the maps φ = B+ , id, P, R : H•∗ → H•∗ to a map φ˜ : H prescription that newly created vertices are to be decorated by the empty set while the decorations of the old vertices is to be preserved.
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˜ :H ˜ •∗ → H ˜ •∗ ⊗ H ˜ •∗ which does the same as ∆ in H•∗ We consider the map ∆ but decorates each root in Pc and each vertex in Rc that got separated by a cut by the same integer (by the smallest unused integer say), preserving the existing decoration. For example, ˜ ∆
• • • •1 •2 A = AA ⊗ I + I ⊗ AA + •1 ⊗ + •2 ⊗ + •1 •2 ⊗ •12 . A • • • • • • • •
Here we do not display the empty set and set brackets for simplicity. Note that ˜ is a coproduct. The decoration has the only purpose to we do not contend that ∆ provide “glueing” information. ˜ •∗ → H ˜ •∗ which reconstructs the preimage of ∆ ˜ ˜ •∗ ⊗ H We define a map m ˜ :H by inserting edges between vertices that have been decorated by the same integers ˜ −1 on the image of ∆ ˜ and and discards the used decoration afterwards. So m ˜ =∆ ˜ •∗ . For otherwise, if no decorations match, m ˜ is the free multiplication mH˜ •∗ of H instance, • •1 • m ˜ •1 •2 •3 ⊗ •2 •4 = AA •3 •4 • • • • m ˜ is obviously not an algebra homomorphism. Definition 7 Let φ ∈ {id, P, R} and ψ ∈ {id, P, R, B+ }. Then the linear endomorphism φ ψ of H•∗ , ˜ ∆j ˜ (φ ψ) = π m( ˜ φ˜ ⊗ ψ) is called the cut product of φ and ψ. It is easy to see that if φ and ψ are algebra endomorphisms, so is φ ψ. As a shorthand notation, we will be using ˜Q ˜ ∆j ˜ (φ Q ψ) := π m( ˜ φ˜ ⊗ ψ) ˜ is the obvious lift of Q to (H ˜ •∗ )⊗2 . In analogy to the approach presented in where Q the preceding subsection, we recursively define the twisted antipode by S˜R (I) = I and ˜ m( ˜∆ ˜ = −R ˜ m(− ˜ m(. ˜∆ ˜ ⊗id)Q ˜ ∆. ˜ S˜R := −R ˜ S˜R ⊗ id)Q ˜ R ˜ . . ⊗ id)Q (17) ˜R S
Let SR := π S˜R j. If one is willing to ignore the fact that jπ = id, one can view SR as defined by SR := −R(SR Q id)
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which might be a helpful motivation when compared to (15). Note that these are ˜∆ ˜ reduces the number of edges and SR (I) = I recursive definitions indeed since Q terminates the recursion. SR will turn out to be the counterterm map in the Epstein-Glaser framework. Remember that R is an idempotent algebra endomorphism, hence in particular a Rota-Baxter operator. Therefore SR and SR id are algebra endomorphism as well by a general inductive argument [10]. Lemma 8 (SR id)B+• = (id − R)B+• (SR id). Proof. We use the Hochschild closedness of B+• , ∆B+• = (id ⊗ B+• )∆ + B+• ⊗ I.
(18)
˜ •∗ )⊗2 in order to apply it to (SR id) : Now we want to lift this equation to (H ˜B ˜+ = C(id ⊗ B ˜+• )∆ ˜ +B ˜+• ⊗ I ∆
(19)
˜ •∗ ⊗ H ˜ •∗ → H ˜ •∗ ⊗ H ˜ •∗ which decorates vertices affected by a where C is a map H cut by the same integer. This is the only adjustment we have to make when going ˜ and j∆ differ only by decoration. This yields from (18) to (19) because ∆j (SR id)B+•
˜ +• = π m( ˜B ˜+• j π m( ˜ S˜R ⊗ id)∆jB ˜ S˜R ⊗ id)∆ ˜ ˜ ˜ ˜ π m( ˜ SR ⊗ id)(C(id ⊗ B+• )∆ + B+• ⊗ I)j ˜+• j ˜+• )∆j ˜ + π S˜R B π m( ˜ S˜R ⊗ id)C(id ⊗ B ˜+• )∆j ˜ − πR ˜ m( ˜∆ ˜B ˜+• j ˜ S˜R ⊗ id)Q π m( ˜ S˜R ⊗ id)C(id ⊗ B
= = = =
= =
˜+• )∆j ˜ − πR ˜ m( ˜+• )∆j ˜ π m( ˜ S˜R ⊗ id)C(id ⊗ B ˜ S˜R ⊗ id)C(id ⊗ B ˜+• )∆j ˜ (id − R)π m( ˜ S˜R ⊗ id)C(id ⊗ B ˜+• )∆j ˜ (id − R)π mC( ˜ S˜R ⊗ id)(id ⊗ B ˜ ˜ +• )(S˜R ⊗ id)∆j (id − R)π mC(id ˜ ⊗B
=
(id − R)B+ (SR id),
= =
where we have used (19), Q(id ⊗ B+• ) = id ⊗ B+• , Q(B+• ⊗ I) = 0 which are obvious, and (S˜R ⊗ id)C = C(S˜R ⊗ id) and mC(id ˜ ⊗ B+• ) = B+• m ˜ which follow from the definition of C. This finishes the proof. Example 9 We illustrate the action of the map ˜∆ ˜ ⊗ id)∆j ˜ SR id = −πRm(−R ˜ m(. ˜ . . ⊗ id)Q on the two trees
• •
• and AA . • •
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•
H•∗
•
˜ ∆j
? ˜ •∗ ⊗ H ˜ •∗ H
• •
⊗I+I⊗
•
+ •1 ⊗ •1
•
˜ Q∆⊗id
? •∗ ⊗3 ˜ (H )
• • + •1 ⊗ •1 ⊗ I − (I ⊗ I) ⊗ + I ⊗ •1 ⊗ •1 I⊗ • •
S˜R ⊗id⊗id
? ˜ •∗ )⊗3 (H
• • − ∗1 ⊗ •1 ⊗ I − (I ⊗ I) ⊗ + I ⊗ •1 ⊗ •1 I⊗ • •
−Rm⊗id ˜
? ˜ •∗ ⊗ H ˜ •∗ H
∗ ∗ • + − ∗1 ⊗ •1 − ⊗I+I⊗ ∗ • •
πm ˜
? H•∗
−
∗ •
+
∗ ∗
+
• •
−
• ∗
.
˜ •∗ Note that we do not need to go into higher than the third tensor power of H because SR (I) = I and hence SR (•) = −∗ terminate the recursion. Now the second, less trivial example:
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• A • A•
˜ ∆j
? ˜ •∗ ⊗ H ˜ •∗ H
• •1 •2 • A ⊗ I + I ⊗ A + •1 ⊗ + •2 ⊗ + •1 •2 ⊗•12 • • • A• • A•
˜ Q∆⊗id
? •1 •2 • • ˜ •∗ )⊗3 I ⊗ A + •1 ⊗ (H + • ⊗ + • • ⊗• ⊗ I − I ⊗ I ⊗ AA 2 1 2 12 A • • • • • • •1 •2 + I ⊗ •2 ⊗ + (I ⊗ •1 •2 + •1 ⊗ •2 + •2 ⊗•1 ) ⊗ •12 +I ⊗ •1 ⊗ • • S˜R ⊗id⊗2
? •1 •2 • • ˜ •∗ )⊗3 I ⊗ A − ∗1 ⊗ (H − ∗ ⊗ + ∗ ∗ ⊗• ⊗ I − I ⊗ I ⊗ AA 2 1 2 12 A • • • • • • •1 •2 +I ⊗ •1 ⊗ + I ⊗ •2 ⊗ + (I ⊗ •1 •2 − ∗1 ⊗ •2 − ∗2 ⊗•1 ) ⊗ •12 • • −Rm⊗id ˜
? ˜ •∗ ⊗ H ˜ •∗ H
• ∗ ∗ ∗ ∗ ⊗ I + I ⊗ AA − AA + AA + AA − AA • • ∗ ∗ • ∗ ∗ • • • •1 •2 − ∗1 ⊗ − ∗2 ⊗ + ∗1 ∗2 ⊗•12 • •
πm ˜
? H•∗
2.4
• • • ∗ ∗ ∗ − AA + 2 AA − AA + AA − 2 AA + AA . ∗ ∗ ∗ • • • ∗ ∗ ∗ • • •
An alternative presentation of the Hopf algebra
In this subsection we give a somewhat different presentation H of H which will turn out to be more instructive for Epstein-Glaser renormalization. The basic idea is as follows: We consider a tree t of the preceding subsections as a trunk and let two more branches, called “hair”, grow out of each leaf and one more branch out
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of each unary vertex of the trunk. This yields a tree t in the presentation H. • A • t= → t = • A A • ◦ A◦ A◦ While the trunk will correspond to an abstract nest of subdivergences, the leaves of the hairy tree actually represent (some unordered set of) space-time points to which that particular subdivergence situation applies. For the reader’s convenience, we visualize hair by ◦ and the trunk vertices by •. This is only to make it easier to distinguish between the bold trees in H and the hairy trees in H, so we are not talking about trees with “two types of vertices” here. Now in order to underline the power of the Hochschild 1-cocycle and to illustrate Lemma 6, we will prescribe the cocycle and see what the coproduct looks like then. Let H be the free commutative algebra generated by rooted trees the leaves of which descend exclusively from binary vertices. In other words each leaf must have one and only one sibling (which is not necessarily a leaf too). For example, the trees • @ • A • @ A , • A , @ @• A A • • ◦ A◦ A A A ◦ A◦ ◦ A A ◦ ◦ ◦ ◦ ◦ A◦ are in H while • A • • A , • •A• , A • •A• • • A• are not. The tree • consisting only of the root is not in H by convention, so the most “primitive” generator is • AA . ◦ ◦ Now we demand B + to act as follows: B + (I)
=
• B + ( AA ) ◦ ◦
=
• A ◦ A◦ • A • A A A ◦ A◦ ◦
B + (t)
=
• • A • A◦ , so t is grafted to a leaf of •AA◦ t
and for a forest, B + (t1 . . . tn )
=
• A@ A@• • t1 . . . tn
in general, for any tree t,
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Lemma 10 There is a unique Hopf algebra structure (∆, , S) on H such that B + is Hochschild closed. ∆ is given on trees t by
∆(t) = I ⊗ t + t ⊗ I +
P c (t) ⊗ Rc (t)
adm c
where the definition of admissible cuts and P c , Rc is as in the preceding subsections with the following modifications: (i) cuts containing external edges (hair) are not admissible here (ii) if a vertex v of Rc (t) has no more outgoing edges due to cut edges in c, that • vertex v is to be replaced by AA in Rc (t). ◦ ◦ If a vertex v of Rc (t) is left with only one outgoing edge due to cut edges in c, an additional branch is to be adjoined to v in Rc (t). The map β : H → H, given by removing the hair, i.e., all leaves and adjacent edges, is an isomorphism of Hopf algebras. β −1 in turn replaces vertices with fertility 0 or 1 by binary vertices. Sketch of proof. First of all we note that whole H − kI is the iterated image of B + and the multiplication. Moreover, H is graded as an algebra by the number of internal (non-hairy) vertices. The existence and uniqueness of (∆, , S) is then a consequence of Lemma 6. The remaining statements are easy to check using the map β, in particular β
• A = •, ◦ A◦
• A • . β • A = AA A • ◦ ◦ ◦
Therefore H is nothing but a somewhat different presentation of H. Using β, we can transfer all notions developed in the preceding subsections to H (which we denote by underlining everything). Note that in H•∗ only internal vertices can ˜ •∗ only internal vertices are decorated etc. From now on, we have type ∗, in H ˜ •∗ . work only in the presentation H, H•∗ , H
2.5
Feynman rules and counterterms. Main result
On the Hopf algebra level, a tree represents a certain subdivergence situation. Internal vertices of type • mean that the unrenormalized Feynman rules have been applied to the respective subdivergence, while ∗ denotes the corresponding counterterm. For example, • A ◦ A◦
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corresponds to the distribution 0 T2 : f1 ⊗ f2 → (p1,2 T1 ⊗ T1 )(f1 ⊗ f2 ) + (p2,1 (T1 ⊗ T1 )(f2 ⊗ f1 ), defined on M 2 − ∆2 . Again we do not display the Wick multiindex r for simplicity. The tree ∗ A ◦ A◦ ∗ 0 represents the counterterm −W12 T2 . We already know that their sum (id − ∗ 0 W12 ) T2 = T2 is the well-defined Epstein-Glaser time-ordered product on whole M 2 . In less trivial cases subtrees represent subdivergences, the root represents the overall divergence. For example
∗ @ • @ @ @ @• @ ∗ ∗ A A A ◦ A◦ ◦ A◦ ◦ A◦ yields ∗ ∗ ∗ W123456 p1234,56 p12,34 W12 p1,2 W34 p3,4 p5,6 T1⊗6 + suitable perm. of indices.
Epstein-Glaser renormalization is essentially a binary operation since in each step only products T (I)T (N − I) of two operator-valued distributions are considered. Indeed, it is impossible to extend a distribution from M n − Dn (for n > 2) onto the thin diagonal in (M n − Dn )∪∆n without extending it to the thicker diagonals, e.g., {xi = xj for some i, j} first. So we will be needing only binary trees here. Now let t be a binary tree in H. All of its internal vertices are of type •. We need a map which changes the types of internal vertices of t in all possible combinations and sums up the resulting trees in order to take care of the Bogoliubov recursion. This is essentially done by S R id, as we have proved in Lemma 8. In order to avoid overcounting, we will have to take care of the symmetry factors which show up whenever the coproduct is applied. For instance, in the second part ∗ of Example 9 we got 2 AA because two cuts, one on the “left”, the other on the ∗ • “right-hand side”, yield the same result. We will compensate that by eventually dividing by symmetry factors. Let T 1 = {I} and for n ≥ 2 let T n be the subset of H of binary trees with n leaves. Furthermore, let F V be the free commutative algebra generated by the graded vector space ∞ V := Dop (M n − Dn ) n=0 (M n − Dop n
Dn ) is the space of collections (Tnr ) of operator-valued distriwhere butions on M − Dn (again r is an n-multiindex referring to the Wick powers under consideration). By Dn we continue to denote the thick diagonal in M n .
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Thus elements of F V are formal free commutative products of operator-valued distributions on the configuration spaces M n − Dn carrying a Wick-multiindex. The free commutative product is supposed to model the analogue of the disjoint union of trees. We don’t actually need it to state the theorem, but it is instructive to keep it in mind. The reader might wish to review the notation for Epstein-Glaser time-ordered products in Subsection 1.2 at this point. Theorem 11 (Main result) Let Φ : H•∗ → F V be the homomorphism of free commutative algebras such that Φ(I) = T1 where T1k =: φk : and n for n ≥ 2, 1 ≤ i ≤ n − 1, ti ∈ T i , tj ∈ T n−i and f1 . . . fn ∈ D(M ) such that i=1 supp fi = ∅, Φ(B+• (ti tj )) is the collection of distributions defined by Φ(B+• (ti tj ))(f1 ⊗ . . . ⊗ fn ) = =
1 S(ti , tj )
pI,N −I Φ(ti )(⊗k∈I fk )Φ(tj )(⊗l∈N −I fl )+
I⊂N,|I|=i
+pN −I,I Φ(tj )(⊗l∈N −I fl )Φ(ti )(⊗k∈I fk ), ∗ = W1...n Φ(B+• (ti tj )).
Φ(B+∗ (ti tj ))
while Φ(t ) = 0 on non-binary trees t . The symmetry factor S(ti , tj ) := 2 if the root of ti has type • and tj = R(ti ), and S(ti , tj ) := 1 otherwise. Using these renormalized Feynman rules Φ, the n-th Epstein-Glaser time-ordered product is (the unique extension onto M n of ) Tn :=
Φ(S R id)(t).
(20)
t∈T n
Note that in an obvious abuse of notation we consider the counterterms as distributions on M n − Dn too. Recall that the extension onto M n is only unique up to the ωα as discussed in Corollary 5. We assume here that for each n, those ωα have been chosen once and forever according to some renormalization scheme. Proof. For n = 1 and n = 2 the statement is obviously true (take t1 = t2 = I). ∗ Now for t ∈ T n it is easy to see that (ΦR)(t) = (W1...n Φ)(t) (note that W ∗ is idempotent as well) and ΦB +• is the very sum of causal partitions times lower order time-ordered products that shows up in the equation ∗ ) Tn = (id − W1...n
∅IN
pI,N −I T (I)T (N − I)
(21)
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which defines the time-ordered product Tn by Corollary 5. Symbolically, the diagrams H•∗
- FV
Φ
H•∗
×
B +•
? H•∗
pI,N −I ...
- FV W∗
R
? H•∗
? - FV
Φ
Φ
Φ
? - FV
commute. This can be seen as follows: T n = B + (T n−1 ) ∪
n−2
B + (T i T n−i ) =
i=2
n−1
B + (T i T n−i )
i=1
where we are overcounting since H is commutative. Using the Hochschild closedness of B+ in the form of Lemma 8 and the fact that S R id is an algebra homomorphism, we get by induction on n, using the symmetry factor S (ti , tj ) := 2 if ti = tj and S (ti , tj ) := 1 otherwise: Tn
=
Φ(S R id)(t)
t∈T n
=
n−1 1 2 i=1
S (ti , tj )Φ(S R id)B + (ti tj )
ti ∈T i tj ∈T n−i
=
=
1 2
n−1
S (ti , tj )Φ(id − R)B +• (S R id)(ti tj )
i=1 ti ∈T i tj ∈T n−i n−1
1 ∗ (id − W1...n ) ΦB +• 2 i=1
ti ∈T i
= =
(S R id)(ti )
(S R id)(tj ) + C
tj ∈T n−i
n−1 1 ∗ (id − W1...n ) pI,N −I T (I)T (N − I) + pN −I,I T (N − I)T (I) 2 i=1 I⊂N,|I|=i ∗ (id − W1...n ) pI,N −I T (I)T (N − I) ∅IN
where C is eventually C = t (S R id)(t)(S R id)(t) (for each t such that ti = tj =: t has occurred in the sum above, thus in particular for all t ∈ T n/2 if n is even) which cancels the symmetry factor S(ti , tj ) in the statement of the theorem. This finishes the proof. While the preceding theorem just defines Φ inductively by pushing it forward along B + , which is a perfectly natural way of doing so, one might also work out a
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non-recursive formula for Φ as follows: Draw the tree, scan it from the top to the bottom and wherever you see an ∗, apply W ∗ . Then symmetrize in all possible ways. Since H is nothing but a different presentation of H, one could also have stated the theorem in terms of trees of H from the very beginning, which would have required a grading on H that is isomorphic to the grading of H by the number of external (hairy) vertices. We encourage the reader to check that one could obtain the same result in complete analogy to momentum space renormalization (BPHZ, dimensional regularisation, etc.) [3, 4, 9, 12] as reviewed in Subsection 2.2 by the following approach: Define the (unrenormalized) Feynman rules Φ : H → H •∗ → F V as in Theorem 11, but let now R : F V → F V be the idempotent algebra endomorphism T → W ∗ T. Note that R is a Rota-Baxter operator. Then replace the cut product by the usual convolution product again, and an adaptation of Theorem 11 yields (S Φ Tn = R Φ)(t) t∈T n
which should be compared to (16). The reason why we preferred the method of letting R act in the Hopf algebra H•∗ and using is that like this we achieved a complete decoupling of the combinatorics (which happen in H•∗ ) and the analysis (which happens in V ), making it easier to see how the essential work is being done on the Hopf algebra side while the renormalized Feynman rules Φ : H•∗ → F V is a rather trivial map translating abstract subdivergence situations into the appropriate operator valued distributions.
3 Conclusions and outlook We have seen how Hopf algebras of rooted trees take care of the combinatorics of Epstein-Glaser renormalization. It is the twisted antipode SR which provides a complete set of counterterms and formally solves the Bogoliubov recursion thanks to the Hochschild closedness of B+ . The statement of Lemma 8 also amounts to the fact that the counterterms are local. Indeed, once the subdivergences are taken care of, it suffices to subtract the superficial divergence, i.e., to extend a distribution onto the thin diagonal. Although we do not claim that the statement of Theorem 11 makes actual calculations easier, it closes the gap between the Epstein-Glaser approach and the Hopf algebra picture in momentum space. Starting from Theorem 11, one rather easily derives Feynman rules Φ for the vaccum expectation values of time-ordered products. One can also try to construct a coproduct on the vacuum expectation values of time-ordered products. Finally, we would like to mention another issue which seems to be intimately related to the above approach to coordinate space renormalization: constructing
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an analogy between extension of distributions from M n − Dn to M n and compactification of the configuration space M n − Dn of n points in M. Indeed, we can already see how this leads to rooted trees if we look at the Fulton-MacPherson compactification of configuration spaces [7, 1, 8] defined as follows: Let M be a smooth manifold. There is an obvious inclusion of the configuration space into a product of blowups, Bl(M |I| , ∆|I| ) (22) M n − Dn → M n × I⊂N,|I|≥2
where Bl(M i , ∆i ) is the (differential-geometric) blowup of M i along ∆i of M i , i.e., M i where the thin diagonal ∆i is replaced by the sphere bundle in the normal bundle over ∆i . For the details, the reader is referred to [1]. The Fulton-MacPherson compactification M [n] of M n − Dn is then the closure of M n − Dn upon this inclusion. Obviously M [n] has only a chance to be compact if M is compact. Now a closer look at what happens in the right-hand side of (22) when a sequence approaches the thin diagonal in M n leads to a nice description of M [n] in terms of nested screens [7, 1]. In particular, it can be shown that there is a stratification of the manifold with corners M [n], M (S) M [n] = S∈S
where S is the set of all nests of subsets of N = {1, . . . , n} with at least 2 elements. Now nested sets are perfectly described by the forests in H. Moreover, if we restrict our attention to M = Rk and replace M n − Dn by the moduli space F˙k (n) := (M n −Dn )/G(k) where G(k) is the subgroup (acting diagonally) of affine transformations of Rk generated by translations by elements of Rk and dilatations by elements of R+ , there is an operad structure behind the Fulton-MacPherson compactifications Fk (n) of the moduli spaces F˙k (n) [8, 13]. The compactifications M [n] of the configuration spaces still furnish a right module over the operad Fk . Operads arise in a natural way when rooted trees are grafted to each other: • @ • • • • @ A , A , A A , → @ A A A A @• • • • • • • • • ••• A A A • A• • A• • A• It seems tempting to explore possible relations between the operad µF M of FultonMacPherson compactified moduli spaces Fk (n), the operad µEG which arises when the trees in H we used for Epstein-Glaser renormalization are grafted to each other, and finally the operad of Feynman graph insertions µF G [13, 11]. The operad µF G is closely related to the pre-Lie structure of Feynman graphs which is dual in a certain sense to the coproduct in H. This might establish a true analogy between the Fulton-MacPherson compactification M [n] of M n −Dn and the renormalization of time-ordered products in the sense of Epstein-Glaser.
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Acknowledgments It is a pleasure to thank Henri Epstein and Ivan Todorov for valuable discussion and helpful comments during a series of talks given on the subject. The first named author would also like to thank the IHES for generous hospitality and the German Academic Exchange Service (DAAD) for financial support.
References [1] S. Axelrod and I.M. Singer, Chern-Simons perturbation theory 2, J. Diff. Geom. 39, 173–213 (1994); hep-th/9304087. [2] R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds, Commun. Math. Phys. 208, 623–661 (2000); math-ph/9903028. [3] A. Connes and D. Kreimer, Hopf algebras, renormalization and non-commutative geometry, Commun. Math. Phys. 199, 203–242 (1998); hep-th/9808042. [4] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210, 249–273 (2000); hep-th/9912092. [5] H. Epstein and V. Glaser, The role of locality in perturbation theory, Ann. Inst. Henri Poincar´e, Section A, Vol. XIX, n. 3, 211 (1973). [6] L. Foissy, Les alg`ebres de Hopf des arbres enracin´es d´ecor´es, Thesis, 2001, Department of Mathematics, University of Reims. [7] W. Fulton and R. MacPherson, A compactification of configuration spaces, Ann. of Math. 139, 183–225 (1994). [8] E. Getzler and J.D.S. Jones, Operands, homotopy algebra and iterated integrals for double loop spaces, Preprint hep-th/9403055. [9] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theory, Adv. Theor. Math. Phys. 2.2, 303–334 (1998); q-alg/9707029. [10] D. Kreimer, Chen’s Iterated Integral represents the Operator Product Expansion, Adv. Theor. Math. Phys. 3, 3 (2000); Adv. Theor. Math. Phys. 3, 627–670 (1999); hep-th/9901099. [11] D. Kreimer, Combinatorics of (perturbative) quantum field theory, Phys. Rept. 363, 387–424 (2002); hep-th/0010059. [12] D. Kreimer, Factorization in quantum field theory: an exercise in Hopf algebras and local singularities, Contributed to Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry, Les Houches, France, 9–21 Mar 2003; hep-th/0306020.
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[13] M. Markl, S. Shnider and J. Stasheff, Operads is Algebra, Topology and Physics, volume 96 of Mathematical surveys and monographs. Amer. Math. Soc., Providence, RI, 2002. [14] G. Pinter, The Action Principle in Epstein Glaser Renormalization and Renormalization of the S-Matrix in φ4 -Theory, Annalen Phys. 10, 333–363 (2001); hep-th/9911063. [15] G. Popineau and R. Stora, A Pedagogical Remark on the Main Theorem of Perturbative Renormalization Theory, Unpublished preprint. [16] G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, SpringerVerlag, 1995, 2nd edition.
Christoph Bergbauer Freie Universit¨ at Berlin II. Mathematisches Institut Arnimallee 3 D-14195 Berlin Germany email:
[email protected] and ´ Institut des Hautes Etudes Scientifiques F-91440 Bures-sur-Yvette France
Dirk Kreimer ´ Institut des Hautes Etudes Scientifiques 35, route de Chartres F-91440 Bures-sur-Yvette France email:
[email protected] and Boston University Department of Mathematics and Statistics Center for Mathematical Physics Boston, MA 02215 USA
Communicated by Vincent Rivasseau submitted 29/03/04, accepted 01/06/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 369 – 395 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020369-27 DOI 10.1007/s00023-005-0211-2
Annales Henri Poincar´ e
Integrable Renormalization II: The General Case Kurusch Ebrahimi-Fard, Li Guo and Dirk Kreimer Abstract. We extend the results we obtained in an earlier work [1]. The cocommutative case of ladders is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the double Rota-Baxter construction, respectively Atkinson’s theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.
1 Introduction The perturbative approach to quantum field theory (QFT) has been spectacularly successful in the past. It is based on a priori formal series expansions of Green functions in orders of a coupling constant, measuring the strength of the corresponding interaction. Terms in these series expansions are indexed by Feynman diagrams, a graphical shorthand for the corresponding Feynman integrals. Physically relevant quantum field theories when treated perturbatively develop short-distance singularities present in all superficially divergent contributions to the perturbative expansion. Renormalization theory [2] allows nevertheless for a consistent way to treat these divergent Feynman integrals in perturbative QFT. The intricate combinatorial, algebraic and analytic structure of renormalization theory within QFT is by now known for almost 70 years. Within the physics community the subject reached its final and satisfying form through the work of Bogoliubov, Parasiuk, Hepp, and Zimmermann. It was very recently that one of us in [3] discovered a unifying scheme in terms of Hopf algebras and its duals underlying the combinatorial and algebraic structure. This Hopf algebraic approach to renormalization theory, as well as the related Lie algebra structures, were exploited in subsequent work [4, 5, 6, 7, 8, 9, 10, 11]. The focus of an earlier work of us [1]1 and this article is on the algebraic Birkhoff decomposition discovered first in [4, 8, 9], and the related Lie algebra of rooted trees, respectively Feynman graphs [10, 11]. The Rota-Baxter algebra structure on the target space of (regularized) Hopf algebra characters showed to be of crucial importance with respect to the Birkhoff decomposition (in [4] this relation appeared under the name multiplicativity constraint). Using a classical r-matrix ansatz, coming simply from the Rota-Baxter map, we were able to derive in (I) the formulae for the factors φ± [8] for the decomposition of a Hopf algebra character φ in the case of the Hopf subalgebra of rooted ladder 1 For
the rest of this work we will cite paper [1] by (I).
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¯ trees. Bogoliubov’s R-map finds its natural formulation in terms of a character with values in the double Rota-Baxter algebra of the above target space Rota-Baxter φ algebra. The counterterm SR = φ− and the renormalized character φ+ simply lie ˜ = R − id, respectively, of the in the images of the group homomorphisms R, −R Bogoliubov character. In this work we would like to extend these results to the general case, i.e., the full Hopf algebra of arbitrary rooted trees. The main difference lies in the fact that in the rooted ladder tree case we worked with a cocommutative Hopf algebra, or dually, with the universal enveloping algebra of an Abelian Lie algebra. In the general case the Lie algebra of infinitesimal characters is non-Abelian, correspondingly the Hopf algebra is non-cocommutative, necessitating a more elaborate treatment due to contributions from the Baker-Campbell-Hausdorff (BCH) formula. The modifications coming from these BCH contributions have to be subtracted order by order in the grading of the Hopf algebra. Hence we will define in a recursive manner an infinitesimal character in the Lie algebra which allows for the Birkhoff decomposition of the Feynman rules regarded as an element of the character group of the Hopf algebra. The factors of the derived decomposition φ = φ− give the formulae for the renormalized character φ+ and the counterterm SR ¯ introduced in [3, 4, 8]. Bogoliubov’s R-map becomes a character with values in the double of the target space Rota-Baxter algebra. It should be underlined that the above ansatz in terms of an r-matrix solely depends on the algebraic structure of the Lie algebra of infinitesimal characters, i.e., the dual of the Hopf algebra of (decorated) rooted trees or Feynman graphs, and on the Rota-Baxter structure underlying the target space of characters. When specializing the Rota-Baxter algebra to be the algebra of Laurent series with pole part of finite order, this approach naturally reduces to the minimal subtraction scheme in dimensional regularization, or to the momentum scheme, which are both widely used in perturbative QFT and thoroughly explored in [3, 4, 5, 6], which extend to non-perturbative aspects still using the Hopf algebra [7]. The paper is organized as follows. In the following section, we introduce the notion of Rota-Baxter algebras and recall some related basic algebraic facts, like the relation to the notion of classical Yang-Baxter type identities, the double Rota-Baxter construction and Atkinson’s theorem. After that, we review the notion of a renormalization Hopf algebra by introducing the universal object [8] for such Hopf algebras, the Hopf algebra of rooted trees. Having this at our disposal, generalizations to the Hopf algebra of decorated rooted trees or Feynman graphs are a straightforward generalization which we will outline later on. Its dual, containing the Lie group of Hopf algebra characters, and the related Lie algebra of its generators, i.e., the infinitesimal characters, is introduced without repeating the details which are by now standard [8]. In section four, which contains the main part of this paper, the notion of a regularized character is introduced as a character with values in a Rota-Baxter algebra. Note that Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormaliza-
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tion falls into this class, even though it makes no use of a regulator, but of a Taylor operator on the integrand instead, which provides a Rota-Baxter map similarly. This is immediate upon recognizing that disjoint one-particle irreducible graphs allow for independent Taylor expansions in masses and momenta. This allows us to lift the Lie algebra of infinitesimal characters to a Rota-Baxter Lie algebra, giving the notion of a classical r-matrix on this Lie algebra. We then review briefly the results of (I), i.e., the Birkhoff factorization in the cocommutative case. Motivated by this result for the simple case of rooted ladder trees, we solve here the factorization problem for the non-cocommutative Hopf algebra of rooted trees by defining a BCH-modified infinitesimal character. Section four closes with some calculations using the notion of normal coordinates intended to make the construction of the modified character in terms of the BCH-corrections more explicit, and a remark on decorated, non-planar rooted trees and Feynman graphs.
2 Rota-Baxter algebras: from Baxter to Baxter The Rota-Baxter (RB) relation first appeared in 1960 in the work of the American mathematician Glen Baxter [12]. Later it was explored especially by the mathematicians F.A. Atkinson, G.-C. Rota and P. Cartier [13, 14, 15]. In particular, Rota underlined its importance in various fields of mathematics, especially within combinatorics [16]. But it was very recently that after a period of dormancy it showed to be of considerable interest in several so far somewhat disconnected areas like Loday type algebras [17, 18, 19, 20, 21], q-shuffle and q-analogs of special functions through the Jackson integral [22], differential algebras [23, 24], number theory [25], and the Hopf algebraic approach to renormalization theory in perturbative QFT [1, 4, 9]. In particular, in collaboration with Connes, the connection to Birkhoff decompositions based on Rota-Baxter maps was introduced in [9, 10]. It is the latter aspect on which we will focus in this work. In its Lie algebraic version the RB relation found one of its most important applications within the theory of integrable systems, where it was rediscovered in the 1980s under the name of (operator form of the) classical and modified classical Yang-Baxter2 equation [26, 27, 28]. There some of its main features, already mentioned in [13], and Atkinson’s theorem itself, were analyzed in greater detail. Especially the double Rota-Baxter construction introduced in the work of Semenov-Tian-Shansky [26, 27], and the related factorization theorems [29] will be of interest to us. In the following we will collect a few basic results on Rota-Baxter algebras, some of which we will need later, and some of which we state just to indicate interesting relations of these algebras to other areas of mathematics. Of course, the list is by no means complete, and a more exhaustive treatment needs to be done. 2 Referring
to C.N. Yang and the Australian physicist Rodney Baxter.
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Let us start with the definition of a Rota-Baxter algebra [16, 23, 24]. Suppose K is a field of characteristic 0. A K-algebra neither needs to be associative, nor commutative, nor unital unless stated otherwise. Definition 2.1 Let A be a K-algebra with a K-linear map R : A → A. We call A a Rota-Baxter (RB) K-algebra and R a Rota-Baxter map (of weight θ ∈ K) if the operator R holds the following Rota-Baxter relation of weight θ ∈ K 3 : R(x)R(y) + θR(xy) = R R(x)y + xR(y)), ∀x, y ∈ A. (1) Remark 2.2 1) For θ = 0 a simple scale transformation R → θ−1 R gives the so-called standard form: R(x)R(y) + R(xy) = R R(x)y + xR(y)). (2)
2) 3) 4)
5)
6)
For the rest of the paper we will always assume the Rota-Baxter map to be of weight θ = 1, i.e., to be in standard form. ˜ := id−R fulfills the same Rota-Baxter relation. If R fulfills relation (2) then R The images of R and id − R give subalgebras in A. The free associative, commutative, unital RB algebra is given by the mixable shuffle algebra [23] which is an extension of Hoffman’s quasi-shuffle algebra [30, 31]. The case θ = 0, R(x)R(y) = R R(x)y + xR(y)), naturally translates into the ordinary shuffle relation, and finds its most prominent example in the integration by parts rule for the Riemann integral. A relation of similar form is given by the associative Nijenhuis identity [32]: (3) N (x)N (y) + N 2 (xy) = N N (x)y + xN (y) . Given a RB algebra with an idempotent RB map R, the operator Nγ := ˜ γ ∈ K fulfills relation (3). See [20, 33, 34] for recent results with R − γ R, respect to this relation.
Example 2.3 1) The q-analog of the Riemann integral, or Jackson-integral [16, 22], on a well-chosen function algebra F is given by: x J[f ](x) := f (y)dq y 0 := (1 − q) f (xq n )xq n . (4) n≥0
It may be written in a more algebraic version, using the operator: Pq [f ] := Eqn [f ],
(5)
n>0 3 Some
λ = −θ.
authors denote this relation in the form R(x)R(y) = R R(x)y + xR(y) + λxy . So
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where Eq [f ](x) := f (qx), f ∈ F. Pq and id + Pq =: Pˆq are RB operators of weight −1, 1, respectively. Now let us define a multiplication operator Mf : F → F, f ∈ F, Mf [g](x) := [f g](x) = f (x)g(x) which fulfills the associative Nijenhuis relation (3). The Jackson integral is given in terms of the above operators as: J[f ](x) = (1 − q)Pˆq Mid [f ](x),
(6)
and fulfills the following mixed RB relation J[f ] J[g] + (1 − q)JMid [f g] = J J[f ] g + f J[g] .
(7)
In a forthcoming work two of us (K.E.-F., L.G.) will report some interesting implications of this fact with respect to some recent results on q-analog of multiplezeta-values [31]. 2) A rich class of Rota-Baxter maps is given by certain projectors. Within renormalization theory, dimensional regularization together with the minimal subtraction scheme play an important rˆ ole. Here the RB map RMS is of weight θ = 1 and defined on the algebra of Laurent series C , −1 ] [4] with finite pole part. For −1 ∞ k ] it gives: k=−m ck ∈ C , RMS
∞ k=−m
−1 ck k := ck k .
(8)
k=−m
Of equal importance is the projector which keeps the finite part, closely related to the momentum scheme. We now introduce the modified Rota-Baxter relation. Its Lie algebraic version already appeared in [26, 28]. Definition 2.4 Let A be a Rota-Baxter algebra, R its Rota-Baxter map. Define the operator B : A → A, B := id − 2R to be the modified Rota-Baxter map and call the corresponding relation fulfilled by B: B(x)B(y) = B B(x)y + xB(y) − xy, ∀x, y ∈ A (9) the modified Rota-Baxter relation. Remark 2.5 In the following proposition (2.6), we mention the notion of pre-Lie algebras. Let us state briefly its definition. A (left) pre-Lie K-algebra A is a Kvector space, together with a bilinear pre-Lie product : A × A → A, holding the (left) pre-Lie relation: a (b c) − (a b) c = b (a c) − (b a) c, ∀a, b, c ∈ A. The commutator [a, b] := a b − b a, ∀a, b ∈ A fulfills the Jacobi identity.
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Proposition 2.6 For the Rota-Baxter algebra A to be either an associative or preLie K-algebra, the (modified) Rota-Baxter relation naturally extends to the Lie algebra LA with commutator bracket [x, y] := xy − yx, ∀x, y ∈ A: [R(x), R(y)] + R([x, y]) = R [R(x), y] + [x, R(y)] [B(x), B(y)] = B [B(x), y] + [x, B(y)] − [x, y].
(10) (11)
The proof is a straightforward calculation. The relations (10) and (11) are well known as the (operator form of the) classical Yang-Baxter and modified YangBaxter equation, respectively. Remark 2.7 1) The same is true for the associative Nijenhuis relation (3). In its Lie algebraic version, identity (3) was investigated in [35, 36]. 2) Let A be an associative K-algebra. We regard A⊗A as an A-bimodule, x⊗y ∈ A⊗A and a(x⊗y)b = (ax⊗y)b = ax⊗yb. A solution r := i s(i) ⊗t(i) ∈ A⊗A of the extended associative classical Yang-Baxter relation: r13 r12 − r12 r23 + r23 r13 = θr13 , θ ∈ K gives a RB map β : A → A of weight θ, defined by β(x) := i s(i) x t(i) . The notation rij means, for instance, r13 := i s(i) ⊗1⊗t(i) . This example implies many more interesting results with respect to unital infinitesimal bialgebras, which will be presented elsewhere. The case θ = 0 was already treated in [21], implying a RB map of weight 0. Atkinson gave in [13] a very nice characterization of general RB K-algebras in terms of a so-called subdirect Birkhoff decomposition: Theorem 2.8 (Atkinson [13]): For a K-algebra A with a linear map R : A → A to be a Rota-Baxter K-algebra, it is necessary and sufficient that A has a subdirect Birkhoff decomposition. The proof of this theorem may be found in [13] and will not be given here. Essentially, the subdirect Birkhoff decomposition in this case means that the Cartesian ˜ product D := (R(A), −R(A)) ⊂ A × A is a subalgebra in A × A and that every ˜ element x ∈ A has a unique decomposition x = R(x) + R(x). This should be compared to the results in the Lie algebra case (10) to be found in [26, 27, 29, 37]. We come now to one of the main facts about RB algebras. In the following we assume every RB algebra A to be either an associative algebra or a pre-Lie or Lie algebra. The RB relation then implies furthermore a possibly infinite hierarchy of the same RB structure in each of the former cases. We call this the double RotaBaxter construction of the RB-hierarchy on the RB algebra A, given as follows.
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Proposition 2.9 Let A be a Rota-Baxter algebra with (modified) Rota-Baxter map R, set B = id − 2R. Equipped with the new product: a ∗R b :=
R(a)b + aR(b) − ab 1 = − B(a)b + aB(b) , 2 A is again a Rota-Baxter algebra of the same type, denoted by AR .
(12) (13)
The proof of this proposition is immediate by the definition of ∗R . Following the terminology in [26, 27], we call this new Rota-Baxter algebra AR the double RB algebra of A. It is also in [26] where already the notion of the double RB structure for associative algebras equipped with a modified Rota-Baxter operator was suggested. Remark 2.10 1) Let A be an associative RB algebra. The composition ab := R(a)b−bR(a)+ ab defines a pre-Lie structure on A. This aspect becomes more apparent in the context of Loday’s dendriform structures, for which associative RB algebras give a rich class of interesting examples, see [17, 18, 19, 20, 21] and references therein. 2) It is obvious that Proposition 2.9 implies a whole, possibly infinite, hierarchy (i) (0) (1) of double RB algebras AR (here, ∗ = ∗R and ∗R = ∗R ): (0)
(1)
(i)
(i)
AR := A, AR := (A, ∗R ), . . . , AR := (A, ∗R ), . . . (i)
a ∗R b :=
1 1 di e− 2 tB (a) e− 2 tB (b), a, b ∈ A. dti |t=0
(i)
Let us call AR the ith double RB algebra of A, or equivalently the double of (i−1) AR . The following diagram serves to visualize the so-called RB-hierarchy: ∗
(1)
(1) ∗
(2)
(2) ∗
(3)
(3)
R R R A −− → AR −− → AR −− → AR → · · ·
3) The RB-hierarchy becomes cyclic of period 2 at level i = 3, for R being an (k) (k+2) idempotent RB map, R2 = R, i.e., the kth double product ∗R = ∗R . (i)
4) The Rota-Baxter map R becomes an K-algebra homomorphism between AR (i−1) and AR , i ∈ N: (i) (i−1) R(b). (14) R(a ∗R b) = R(a) ∗R ˜ := id − R, we have 5) For the Rota-Baxter map R (i−1) ˜ ˜ ∗(i) b) = −R(a) ˜ R(a R(b). ∗R R
We therefore have the following diagram of K-algebra homomorphisms: ˜ R,R
(1)
˜ R,R
(2)
˜ R,R
(3)
A ←−− AR ←−− AR ←−− AR ← · · ·
(15)
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We introduce the following composition, using the shuffle product notion formally. For an associative K-algebra A and a, b ∈ K, we define a
A b := ab + ba. For fixed ai , bj ∈ A, 1 ≤ i ≤ m, 1 ≤ j ≤ n, define recursively (a1 a2 · · · am )
A (b1 b2 · · · bn ) =
a1 ((a2 · · · am )
A (b1 · · · bn )) +b1 ((a1 · · · am )
A (b2 · · · bn )),
Proposition 2.11 Let A be an associative Rota-Baxter algebra. For n ∈ N, x ∈ A we have ˜ 1) integer powers of R(x) and R(x) can be written explicitly as: (−R(x))n = (−1)n R x∗R n = ˜ n R(x)
= =
n n−1 (−R(x))n−k
A xk −R x +
(16)
k=1 (n−1) ∗R n
˜ (−1) R
˜ xn + R
x
n−1
(−R(x))n−k
A xk .
(17)
k=1
2) for A also being commutative the above formulae simplify to: n
=
˜ n R(x)
=
(−R(x))
n
n n−1 −R x + (−R(x))(n−k) xk , k k=1
n−1 n ˜ xn + R (−R(x))(n−k) xk . k
(18)
(19)
k=1
The proof of this proposition follows by induction on n.
3 The Hopf algebra of rooted trees Rooted trees naturally give a convenient way to denote the hierarchical structure of subdivergences appearing in a Feynman diagram [3], and the structure maps of their Hopf algebras describe the combinatorics of renormalization of local interactions, encapsulating Zimmermann’s forest formula. For a renormalizable theory, the hierarchy of subdivergences can always be resolved into decorated rooted trees (the parenthesized words of [3]) upon resolving overlapping divergences using maximal forests [38] corresponding to Hepp sectors. This amounts to a determination of the closed Hochschild one-cocycles of the Hopf algebra of renormalization for a given quantum field theory. This is always possible as the rooted trees Hopf algebra with its one-cocycle B+ is the universal object [8] of graded commutative Hopf algebras. Hence it suffices to study this universal object, while the details
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of a specific Hopf algebra of renormalization of a chosen quantum field theory only provide additional notational excesses, albeit cumbersome, see [5, 6, 7] for applications. The main ingredient of this universal commutative Hopf algebra of rooted trees is given by a well-suited non-cocommutative coproduct, defined in terms of admissible cuts on these rooted trees. The aforementioned forest formula is then given essentially by the recursively defined antipode of this Hopf algebra, coming for free from mathematical structure.
Figure 1. A rainbow diagram and corresponding rooted tree of weight 8. Having the Hopf algebra of rooted trees, organizing the algebraic and combinatorial aspects of renormalization, the description of the analytical structure in terms of the group of so-called (regularized) Hopf algebra characters takes place within the dual of this Hopf algebra, being an associative algebra with respect to convolution. Let us introduce the Hopf algebra of rooted trees [8, 39, 40], which we will denote as Hrt . The base field K is assumed once and for all to be of characteristic zero. By definition a rooted tree T is made out of vertices and nonintersecting oriented edges, such that all but one vertex have exactly one incoming line. We denote the set of vertices and edges of a rooted tree by V (T ), E(T ) respectively. The root is the only vertex with no incoming line. Each rooted tree is effectively a representative of an isomorphism class, and the set of all isomorphism classes will be denoted by Trt .
···
···
Definition 3.1 The commutative, unital, associative K-algebra of rooted trees Art is the polynomial algebra, generated by the symbols T , each representing an isomorphism class in Trt . The unit is the empty tree, denoted by 1, and the product of rooted trees is denoted by concatenation, i.e., mArt (T, T ) =: T T . We define a grading on the rooted tree algebra Art in terms of the number of vertices of a rooted tree, #(T ) := |V (T )|. This is extended to monomials, i.e., so-called
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n (n) forests of rooted trees, by #(T1 · · · Tn ) := i=1 #(Ti ), so that Art = n≥0 Art becomes a graded, connected, unital, commutative, associative K-algebra. Let us introduce now the notion of admissible cuts on a rooted tree. A cut cT of a rooted tree is a subset of the set of edges of T , cT ⊂ E(T ). It becomes an admissible cut, if and only if along a path from the root to any of the leaves of the tree T , one meets at most one element of cT . By removing the set cT , E(T ) − cT , each admissible cut cT produces a monomial of pruned trees, denoted by PcT . The rest, which is a rooted tree containing the original root, is denoted by RcT . We exclude the cases, where cT = ∅, such that RcT = T, PcT = ∅ and the full cut, such that RcT = ∅, PcT = T . We extend the rooted tree algebra Art to a bialgebra Hrt by defining the co-unit : Hrt → K: 0 T1 · · · Tn = 1 (T1 · · · Tn ) := (20) 1 else. The coproduct ∆ : Hrt → Hrt ⊗ Hrt is defined in terms of the set of all admissible cuts CT of a rooted tree T : ∆(T ) = T ⊗ 1 + 1 ⊗ T + PcT ⊗ RcT . (21) cT ∈CT
It is obvious, that this coproduct is non-cocommutative. We extend this by definition to an algebra morphism. Definition 3.2 The graded connected Hopf algebra Hrt := (Art , ∆, ) is defined as the algebra Art equipped with the above defined compatible coproduct ∆ : Hrt → Hrt ⊗ Hrt (21), and co-unit : Hrt → K (20). Remark 3.3 1) The coproduct can be written in a recursive way, using the B + operator, which is a Hochschild 1-cocycle [4, 8, 39]: ∆(B + (Ti1 · · · Tin )) = T ⊗ 1 + {id ⊗ B + }∆(Ti1 · · · Tin ).
(22)
B + : Hrt → Hrt is a linear operator, mapping a (forest, i.e., monomial of) rooted tree(s) to a rooted tree, by connecting the root(s) to a new adjoined root: B + (1) = , B + ( ) = , B + ( ) =
, B+(
)=
, B+(
)=
···
It therefore raises the degree by 1. Every rooted tree lies in the image of the B + operator. Its conceptual importance with respect to fundamental notions of physics was illuminated recently in [41]. 2) The Hopf algebra Hrt contains a commutative, cocommutative Hopf subl algebra Hrt , generated by the so-called rooted ladder trees, denoted by the
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symbol tn , n ∈ N and recursively defined in terms of the B + operator, t0 := 1, m tm = B + (1). The coproduct (21) therefore can be written as: ∆(tn ) = tn ⊗ 1 + 1 ⊗ tn +
n−1
ti ⊗ tn−i .
(23)
i=1
The bialgebra Hrt actually is a graded connected Hopf algebra, since due to its grading and connectedness, it comes naturally equipped with an antipode S : Hrt → Hrt , recursively defined by: S(T ) := −T − S(PcT )RcT . (24) cT ∈CT ∗ of the Hopf algebra of rooted trees, i.e., linear We come now to the dual Hrt ∗ , T ∈ maps from Hrt into K. It is convenient to denote f (T ) =: f, T ∈ K, f ∈ Hrt Hrt . Equipped with the convolution product:
f g(T )
:= (21)
=
mK (f ⊗ g)∆(T ), T ∈ Hrt . f (T ) + g(T ) + f (PcT )g(RcT )
(25)
cT ∈CT ∆
f ⊗g
m
K Hrt −→ Hrt ⊗ Hrt −−−→ K ⊗ K −−→ K
it becomes an associative K-algebra. Its unit is given by the co-unit . Remark 3.4 Higher powers of the convolution product are defined as follows: f1 f2 · · · fn := mK (f1 ⊗ f2 ⊗ · · · ⊗ fn )∆(n−1)
(26)
∆(0) := id, ∆(k) := (id ⊗ ∆(k−1) ) ◦ ∆. ∗ contains the set charK Hrt of Hopf algebra characters, i.e., multiplicative Hrt linear maps with values in the field K.
Definition 3.5 A linear map φ : Hrt → K is called a character if φ(T1 T2 ) = φ(T1 )φ(T2 ), Ti ∈ Hrt , i = 1, 2, i.e., φ(1) = 1K . We denote the set of characters by charK Hrt . Proposition 3.6 The set of characters charK Hrt forms a group with respect to the convolution product (25). The inverse of φ ∈ charK Hrt is given in terms of the antipode (24), φ−1 := φ ◦ S. Definition 3.7 A linear map Z : Hrt → K is called derivation, or infinitesimal character if Z(T1 T2 ) = Z(T1 )(T2 )+(T1 )Z(T2 ), Ti ∈ Hrt , i = 1, 2, i.e., Z(1) = 0. The set of infinitesimal characters is denoted by ∂ charK Hrt .
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Lemma 3.8 For any Z ∈ ∂ charK Hrt and T ∈ Hrt of degree #(T ) = n < ∞, we have for m > n, Z m (T ) = 0. Remark 3.9 The last result implies that the exponential exp∗ (Z)(T ) := k≥0 Z k k! (T ),
Z ∈ ∂ charK Hrt , is a finite sum, ending at k = #(T ).
The above facts culminate into the following important results [8, 40]. Given the explicit base of rooted trees generating Hrt , the set of derivations ∂ charK Hrt is generated by the dually defined infinitesimal characters, indexed by rooted trees: ZT (T ) = ZT , T := δT,T .
(27)
Proposition 3.10 The set ∂ charK Hrt defines a Lie algebra, denoted by LHrt , and equipped with the commutator: ZT ZT − ZT ZT n(T , T ; T ) − n(T , T ; T ) ZT ,
[ZT , ZT ] := =
(28)
T ∈Trt
where the n(T , T ; T ) ∈ N denote so-called section coefficients, which count the number of single simple cuts, |cT | = 1, such that PcT = T and RcT = T . The exponential map exp∗ : LHrt → charK Hrt defined in remark (3.9) is a bijection. Generated by the infinitesimal characters ZT (27), the Lie algebra LHrt carries naturally a grading in terms of the grading of the rooted trees in Hrt , deg(ZT ) := (n) #(T ), and LHrt = n>0 LHrt . The commutator (28) implies then:
(n) (m) (m+n) LHrt , LHrt ⊂ LHrt .
(29)
Let us calculate a few commutators, to get a better feeling for the structure of LHrt : [Z , Z ] = Z + 2Z − Z = 2Z (30) [Z , Z ] = Z + Z
[Z , Z ] =
+ 2Z
−Z =Z
1 [[Z , Z ], Z ] = Z 2
− 3Z
+ 2Z
−Z .
This Lie algebra received more attention recently [11, 42, 43], but needs further structural analysis, since it captures in an essential way the whole of renormalization and the structure of the equations of motion [7] in perturbative QFT. This remark is underlined by the results presented in the next section.
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4 Classical r-Matrix and Birkhoff decomposition For a renormalizable theory, the process of renormalization removes the shortdistance singularities order by order in the coupling constant. For this to work one has to choose a renormalization scheme which determines the remaining finite part. This choice is of analytic nature but also contains an important algebraic combinatorial aspect, which lies at the heart of the Birkhoff factorization, found in [4, 9]. It is the goal of this section to clarify how this algebraic step implies the Birkhoff decomposition in a completely algebraic manner. We derive the corresponding theorem for graded connected Hopf algebras quite generically. The main ingredient is a generalized notion of regularization in terms of a Rota-Baxter structure, which is supposed to underlie the target space of the characters of Hrt . Following the Hopf algebraic approach to renormalization in perturbative QFT, we henceforth introduce the notion of regularized (infinitesimal) characters, maps from Hrt into a commutative, associative, unital Rota-Baxter algebra A. The choice of the Rota-Baxter map is determined by the choice of the renormalization scheme, which can be a BPHZ scheme (Taylor subtractions of the integrand), the before-mentioned minimal subtraction and momentum schemes, and others, which all provide Rota-Baxter maps. Here is not the space to give a complete census of renormalization schemes in use in physics, but we simply assume Feynman rules and a Rota-Baxter map being given. Let us mention that sometimes we write R-matrix, instead of the standard notation r-matrix, to underline its operator form, and origin in the Rota-Baxter relation. ∗ to L(Hrt , A), consisting of K-linear maps from We therefore generalize Hrt Hrt into the Rota-Baxter algebra A, i.e., φ, T ∈ A, φ ∈ L(Hrt , A), T ∈ Hrt . Due to the double RB structure on the Rota-Baxter algebra (12) we naturally get L(Hrt , AR ). We then lift the Rota-Baxter map R : A → A to L(Hrt , A), which is possible since it is linear. Proposition 4.1 Define the linear map R : L(Hrt , A) → L(Hrt , A) by f → R(f ) := R ◦ f : Hrt → R(A). Then L(Hrt , A) becomes an associative, unital Rota-Baxter algebra. The Lie algebra of infinitesimal characters LHrt ⊂ L(Hrt , A) with bracket (28) becomes a Lie Rota-Baxter algebra, i.e., for Z , Z ∈ ∂ charA Hrt , we have the notion of a classical R-matrix respectively classical Yang-Baxter relation: (31) [R(Z ), R(Z )] = R [Z , R(Z )] + R [R(Z ), Z ] − R [Z , Z ] . Notice that we replaced K by A for the target space of the regularized infinitesimal characters. The proof of this proposition was given in (I). Using the double RB construction and Atkinson’s theorem of Section 2 we have the following Lemma 4.2 The Rota-Baxter algebra L(Hrt , A) equipped with the convolution product: f R g = f R(g) + R(f ) g − f g (32)
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gives a Rota-Baxter algebra structure on the set of linear functionals with values in the double RB algebra AR of A, denoted by L(Hrt , AR ). An analog for LHrt exists, denoted by LHrt R , equipped with the R-bracket: [Z , Z ]R
= [Z , R(Z )] + [R(Z ), Z ] − [Z , Z ] −1 = [Z , B(Z )] + [B(Z ), Z ] . 2
The R map becomes a (Lie) algebra morphism (LHrt R → LHrt ) L(Hrt , AR ) → L(Hrt , A). Remark 4.3 ˜ := id − R ˜ (see Re˜ := id − R, respectively R 1) The above is also true for R mark 2.10). + ˜ 2) We will denote the Lie subalgebras R(LHrt ) by L− Hrt and R(LHrt ) by LHrt . We now apply Atkinson’s theorem to the Lie algebra LHrt of infinitesimal characters, the generators of the group of Hopf algebra characters charA Hrt . Lemma 4.4 Every infinitesimal character Z ∈ LHrt has a unique subdirect Birkhoff ˜ decomposition Z = R(Z) + R(Z). Remark 4.5 1) In the case of an idempotent Rota-Baxter map R we have a direct decompo+ sition A = A− + A+ respectively LHrt = L− Hrt + LHrt . 2) Let Z ∈ LHrt be the infinitesimal character generating the character φ = exp (Z) ∈ charA Hrt . Using the result in Proposition 2.11, we then see that for elements in ker(), the augmentation ideal, we have ˜ ˜ expR (−Z) . = −R exp (−R(Z)) = R expR (−Z) , exp (R(Z))
4.1
Review of the Ladder case
For the Hopf subalgebra of rooted ladder trees, introduced in the last section, we found in (I) the following simple factorization for a regularized character l l φ = exp∗ (Z), Z ∈ ∂ charA Hrt due to the abelianess of LHrt charA Hrt l , and induced by Atkinson’s theorem, i.e., the lifted Rota-Baxter map R:
=
exp∗ (Z) ˜ exp∗ R(Z) + R(Z)
(34)
=
φ−1 −
(35)
φ =
φ+
(33)
where: φ− = exp (−R(Z))
˜ φ+ = exp (R(Z)),
(36)
and such that we arrive at the following formulae [8] for φ± , using Proposition 2.11:
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l Proposition 4.6 In the rooted ladder tree case, Hrt , we find for the factors (36) in the Birkhoff decomposition (35) the following explicit formulae: φ− (tn ) = R expR (−Z) (tn ) (37) n−1 = −R φ(tn ) + φ− (tk )φ(tn−k )
(38)
k=1
˜ expR (−Z) (tn ) φ+ (tn ) = −R ˜ φ(tn ) + = R
n−1
(39)
φ− (tk )φ(tn−k ) .
(40)
k=1
We emphasize that the map: b[φ](tn ) := =
expR (−Z)(tn ) n−1 −φ(tn ) − φ− (tk )φ(tn−k )
(41) (42)
k=1 l → AR , which we will call Bogoliubov character, is a Hopf algebra character Hrt i.e., into the double RB algebra AR of A. This gives a natural algebraic expression ¯ for Bogoliubov’s R-map. In the next section, where we treat the general case, we ˜ i.e., the Lie algebra will formally introduce b[φ]. The Rota-Baxter maps R and −R, ˜ R,−R
homomorphisms, become group homomorphisms charAR Hrt −−−−→ charA Hrt . We will generalize this to arbitrary rooted trees in the following section, using equation (17).
4.2
The general case
As stated above, in (I) we introduced a classical R-matrix coming from the RotaBaxter structure underlying the target space of regularized characters. We saw that in the case of the Hopf subalgebra of rooted ladder trees, the abelianess of the related Lie algebra implies a somehow simple Birkhoff factorization (35, 36) respectively the formulae for the factors φ± (38, 40). The general, i.e., noncocommutative case can be solved due to the graded, connectedness of the Hopf algebra of rooted trees. Suppose we start with an infinitesimal character Z ∈ ∂ charA Hrt generating the regularized Hopf algebra character φ = exp∗ (Z) ∈ charA Hrt . The above mentioned properties of the Hopf algebra allow for a recursive definition of an infinitesimal character χ = χ(Z) ∈ ∂ charA Hrt , defined in terms of Z, using the lower central series of Lie algebra commutators. Setting ∞ (k) χZ (43) χ(Z) = Z + k=1
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we proceed in the following manner. We first introduce the related series χ(u; Z) = Z +
∞
(k)
u k χZ ,
(44)
k=1 (0)
where we assume that 0 < u < 1 is a real parameter. We also set χZ = Z. We next introduce the Baker-Campbell-Hausdorff (BCH) series
1 1 A, [A, B] − B, [A, B] + · · · . BCH(A, B) := [A, B] + (45) 2 12 See [44] for more details on the BCH formula. Let us write equation (45) in the form ∞ ck K (k) (A, B), (46) BCH(A, B) = k=1 (k)
such that the K are the appropriate nested- or multicommutators of depth k ∈ N, i.e., K (1) = [A, B], K (2) = [A, [A, B]] − [B, [A, B]] and so on. Then, the (k) χZ are defined as the solution of the fix point equation χ(u; Z) = Z −
∞
˜ . ck uk K (k) R(χ(Z)), R(χ(Z))
(47)
k=1
Note that χ(u; Z)(T ) is a polynomial in u of degree m − 2 for any finite tree T of degree #(T ) = m say, i.e., with m vertices and therefore is well defined at u = 1. We hence set χ(Z) ≡ χ(1; Z). Furthermore, χ(Z) is an infinitesimal character as it is a finite linear combination of infinitesimal characters, and thus the above definition on trees implies its action on forests as a derivation in the sense of definition (3.7). (k) It is immediate that χZ vanishes for all k ≥ 1 when applied to cocommutative Hopf algebra elements. Let us work out the cases k = 1, 2 as examples: 1 1 ˜ = − [R(Z), Z] χ(1) = − [R(Z), R(Z)] 2 2 and for k = 2 we have: χ(2)
1 1 ˜ (1) )] ˜ = − [R(χ(1) ), R(Z)] − [R(Z), R(χ 2 2
1 ˜ R(Z), [R(Z), Z] − R(Z), [R(Z), Z] − 12 1 1 ˜ ˜ = + [R([R(Z), Z]), R(Z)] + [R(Z), R([R(Z), Z])] 4 4
1 ˜ − R(Z), [R(Z), Z] − R(Z), [R(Z), Z] 12
(48)
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385
1 R([R(Z), Z]), Z 4
1 + R(Z), [R(Z), Z] − [R(Z), Z], Z . 12
(49)
˜ has completely vanished. This nontrivial fact comes partly from R ˜ = where R (k) id − R, and in a moment we show that the χZ solve the simpler recursion: χ(u; Z) = Z +
∞
ck uk K (k) (−R(χ(Z)), Z)) .
(50)
k=1 (k)
Indeed, from the relation (47) and the recursive definition of the χZ we have the ∗ : following factorization for group like elements in Hrt Proposition 4.7 Using the infinitesimal character χ ∈ ∂ charA Hrt defined in (47), we have the following decomposition of a character φ ∈ charA Hrt given in terms of its generating infinitesimal character Z ∈ ∂ charA Hrt : ˜ exp∗ (Z) = exp∗ R(χ(Z)) exp∗ R(χ(Z)) . (51) ˜ is apparThis then implies the simpler recursion (50), in which the vanishing of R ent. Remark 4.8 The above formal derivation of the factorization of Hrt characters using the BCH formula in (47) and (50) to define the infinitesimal character χ(Z) (43) may be summarized in a more suggestive manner by the following two recursive formulae: ˜ χ(Z) = Z − BCH R(χ(Z)), R(χ(Z)) = Z + BCH − R(χ(Z)), Z . Let us define the factors, i.e., characters φ± ∈ charA Hrt : ∗ ˜ R(χ(Z)) , φ+ := exp∗ R(χ(Z)) φ−1 − := exp
(52)
and introduce the Bogoliubov character now in general via the following definition. Definition 4.9 Let exp∗ (Z) = φ ∈ charA Hrt . We define the following character b[φ] ∈ charAR Hrt with values in the double RB algebra of A, and call it Bogoliubov character: (53) b[φ] := exp∗R (−χ(Z)). Remembering the crucial property of exponentiated Rota-Baxter maps, coming from (12), respectively (14, 15): exp∗ − R(Z) = R exp∗R (−Z) (54) ˜ ˜ exp∗R (−Z) , = −R exp∗ R(Z) (55)
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for elements in the augmentation ideal ker(), we have: φ− := R(b[φ]),
˜ φ+ := −R(b[φ]).
(56)
We use the factorization (51) in proposition (4.7) to derive an explicit formula for the characters b[φ] respectively φ± . Let T ∈ ker(), using the coproduct (21), we get: ˜ −R(b[φ])(T ) = =
exp∗ (−R(χ(Z))) exp∗ (Z)(T )
(57)
∗
exp (Z)(T ) + R(b[φ])(T ) 1 n−1 n
+ R(−χ(Z))(n−j) Z j (T ), n! j=1 j
(58)
n≥0
where (57) again implies the simpler recursion equation (50). Remark 4.10 1) All expressions are well defined since they reduce to finite sums for an element T ∈ ker() of finite order #(T ) = m < ∞. 2) In the last expression in equation (58), the primitive part in ∆(T ) is mapped to zero, since only strictly positive powers of infinitesimal characters appear. Continuing the above calculation, we get the following: ˜ −R(b[φ])(T ) − R(b[φ])(T )
= −b[φ](T ) = exp∗ (Z)(T ) 1 n−1 n
+ R(−χ(Z))(n−j) Z j (T ), n! j=1 j n≥0
and therefore we find the well-known formula: R exp∗R (−χ(Z)) (T ) = +
−R exp∗ (Z)(T ) 1 n−1 n
R(−χ(Z))(n−j) Z j (T ) . n! j=1 j n≥0
Finally, we rederive the results of [3, 4, 8] which gave the counterterm and the renormalized contribution as the image of the Bogoliubov character under the ˜ now derived from the double RB construction group homomorphisms R and −R, for any algebraic Birkhoff decomposition based on a suitable R, i.e., Rota-Baxter type map:
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Theorem 4.11 For T ∈ Hrt , #(T ) = m we have the following formulae for the factors in (51): R(b[φ])(T ) = = =
φ− (T )
m n−1
1 n −R φ(T ) + R(−χ(Z))(n−j) Z j (T ) . n! j=1 j n≥0 φ− (PcT )φ(RcT ) . −R φ(T ) + cT ∈CT
and
˜ b[φ] (T ) = R = =
φ+ (T )
m n−1
1 n ˜ φ(T ) + R R(−χ(Z))(n−j) Z j (T ) . n! j=1 j n≥0 ˜ φ(T ) + φ− (PcT )φ(RcT ) . R cT ∈CT
This should be compared to the general equation (16) including the shuffle: 1 − exp∗R (−χ(Z))(T ) = exp∗ (Z)(T ) + n! n−1
n≥0
n R(−χ(Z))(n−j) Z j (T ). j
j=1 ∗
= exp (χ(Z))(T ) + 1 n−1 R(−χ(Z))(n−j)
χ(Z)j (T ). n! j=1
n≥0
It allows us to define the infinitesimal character χ = χ(Z) to order k > 0 in another (j) way recursively by using the χZ , j < k. We therefore get to order k: (k)
χZ
= −
k−1
(j)
χZ −
j=1
−
k+2 l=1
k+2 n≥0
+
1 n!
1 1 χ(Z) l + Z l l! l!
n−1
k+2
l=1
R(−χ(Z))(n−j)
χ(Z)j
j=1
k+2 n≥0
n−1
1 n R(−χ(Z))(n−j) Z j . n! j=1 j
After these formal arguments based on the general results for Rota-Baxter operators and the structure of the rooted tree Hopf algebra, we end this section and the paper with a remark on calculational aspects.
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When treating the rooted ladder case, we mentioned at the end of (I) the use of normal coordinates introduced by Chryssomalakos et al. in [43]. Given a regularized character φ ∈ charA Hrt , this provided a very easy way to define the coefficients for its generator Z ∈ ∂ charA Hrt : Z := α(n) Ztn , α(n) ∈ A (59) n>0
such that exp∗ (Z)(tn ) = φ(tn ) ∈ A. Remark 4.12 1) Here and also later, we omit for notational reasons the tensor sign between the α(n) and Ztn , i.e., α(n) Ztn ∈ A ⊗ ∂ charA Hrt , n > 0. 2) The α(n) were given in terms of Schur polynomials, i.e., α(n) := φ(P (t1 , · · · , tn )). And φ− was just exp∗ (−R(Z)). In the general case, i.e., for arbitrary rooted trees, the simple Schur polynomials get replaced by the following set of polynomial equations. Details may be found in [43]. Introducing the symbols xT , indexed by rooted trees, and defining the new coordinates, which are characterized by exp∗ (Z)(xT ) = φ(xT ), φ ∈ charA Hrt , where: T αx ZT ∈ ∂ charA Hrt , (60) Z= T ∈Trt
we arrive at the following infinite set of coupled equations, expressing the coordinates T in terms of the new xT : 1 mHrt (P⊗n+1 )∆(n) (xT ), T ∈ Trt . T = (61) (n + 1)! n≥0
Here, the map P denotes the projector into the augmentation ideal: 0, xT1 · · · xTn = 1 T1 Tn P(x · · · x ) := xT1 · · · xTn , else.
(62)
∆ denotes the coproduct reduced to single simple cuts |cT | = 1, and ∆(n) := this op(id ⊗ ∆(n−1) ) ◦ ∆ , such that ∆(0) := id, ∆(1) = ∆ . One should compare ˆ eration with the formal linear map exp∗ (Z)(T ) on Hrt , where Zˆ := T ∈Trt T ZT , T ZT (T ) = T δT,T , and T1 ZT1 T2 ZT2 := T1 T2 ZT1 ZT2 . The first four equations for the rooted trees , , , =
x
=
1 x + x x 2
=
1 x +x x + x x x , 6
=x
are:
1 +x x + x x x 3
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The final step, done in [43], is to invert the above equations, giving the xT ’s in terms of the original rooted trees. In the ladder case we just get the Schur polynomials. In general, we have for example:
x = −
+
1 3
,
x
=
−
+
1 6
.
T
Therefore, the coefficients αx := φ(xT ) ∈ A in (60). Let us briefly dwell on the generalization to decorated non-planar rooted trees, and to Feynman graphs, following [38, 45]. Every Feynman graph provides a number r of maximal forests. The integer r counts the number of terms pi , i = 1, . . . , r in the coproduct which are primitive on the rhs, and in the augmentation ideal on the lhs of the coproduct on graphs. If r > 1, we call the graph overlapping divergent. It is then mapped to a linear combination of r decorated rooted trees, where each of those trees has a root decorated by one of the pi . Iterating this procedure, one obtains a map from Feynman graphs to decorated rooted trees where the decorations are provided by subdivergence free skeleton contributions. Having resolved the overlapping sectors into trees, one then proceeds as before. We close this paper with a study of a simple example on decorated rooted trees using two decorations. The generalization to Feynman graphs including form factor decompositions for theories with spin is somewhat excessive on the notational side, but provides no difficulty for the practitioner of quantum field theory, making full use of the Hopf and Lie algebra of Feynman graphs with external structures. See [5, 38, 45] where examples can be found. We consider the example of vertices with a decoration D by 2 elements {a, b}. Let us denote them by a vertex and a vertex . For the Lie bracket (28) of these two vertices we get: [Z , Z ] = Z − Z .
(63)
Note that though we have here the analog of a simple nesting of one graph in another, this has already a non-vanishing commutator in the Lie algebra. This fact makes it necessary to include the BCH-corrections (50) already at this level. For the above example (63), we have to add the correction (48). Let us do the calculation of the counterterm φ− explicitly for the decorated rooted ladder tree , using the normal coordinates in (60). We have to use χ = Z + χ(1) (43), with χ(1) given in (48). The infinitesimal character Z generating the character φ is given to order 2 in terms of the normal coordinates xT as:
Z = φ(x )Z + φ(x )Z + φ(x )Z + φ(x )Z ,
(64)
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where: φ(x )
= φ( ),
φ(x )
1 = φ( ) − φ( )φ( ), 2
φ(x ) = φ( ) 1 φ(x ) = φ( ) − φ( )φ( ). 2
(65)
And therefore we have for the infinitesimal character χ(Z) to order k = 1, i.e., including the first correction (48):
χ
=
φ(x )Z + φ(x )Z + φ(x )Z + φ(x )Z −
1 [R(φ(x )Z ), φ(x )Z ] + [R(φ(x )Z ), φ(x )Z ] . 2
So that when the counterterm character φ− = exp∗ (−R(χ)) is applied to get: φ− ( ) = = =
we
exp∗ (−R(χ))( ) 1 −R(χ)( ) + R(χ) R(χ)( ) 2 1 −R φ( )Z ( ) − φ( )φ( )Z ( ) 2 −
1 R(φ( )Z ( )) φ( )Z ( ) 2 −φ( )Z ( ) R(φ( )Z ( ))
=
(66)
(67)
1 + R(φ( )Z ( ))R(φ( )Z ( )) 2 −R φ( ) + R(φ( )) φ( ) ,
(68) (69)
which is the correct result. In line (67) no higher order terms can appear. In the next line we used relations (65). From (68) to the last equality (69) we used the RB relation: 1 1 R φ( )Z ( ) R φ( )Z ( ) + R φ( )Z ( ) φ( )Z ( ) 2 2 1 R R φ( )Z ( ) φ( )Z ( ) = 2 +φ( )Z ( ) R φ( )Z ( ) .
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Remark 4.13 1) Note that now χ(u; Z)(T ), equation (44), will be a polynomial of degree at most m−1 when acting on decorated trees with m vertices, as the tree studied above with two vertices differently decorated is already non-cocommutative under the coproduct. 2) Using the standard example of the QFT Φ36dim , the result (69) should be , which has an compared to the counterterm for the Feynman graph additional factor two reflecting the fact that it is overlapping divergent, r = 2, and it resolves into two identical rooted trees [38].
5 Conclusion and outlook In this work we generalized the results of (I) to arbitrary rooted trees, i.e., we showed how to derive the Birkhoff factorization for characters of the Hopf algebra of rooted trees. Using the Rota-Baxter structure underlying the target space of the characters of a renormalization Hopf algebra, the notion of a classical r-matrix was introduced on the corresponding Lie algebra defined on rooted trees. A couple of simple results for Rota-Baxter algebras were collected which allowed for a straightforward derivation of the twisted antipode formula, defined in [3, 8] concerning the study of the Hopf algebraic approach to perturbative QFT. This gives a firm algebraic basis to any renormalization scheme using an algebraic Birkhoff decomposition together with a suitable double RB construction. We regard this work as a further step towards a more interesting connection to the realm of integrable systems. Sakakibara’s result [46] also points into this direction. This connection was already apparent in [10], in which effectively the grading operator Y served as a Hamiltonian providing the ”scaling evolution” of the coupling constant, and hence the renormalization group flow initiated by scaling transformations, and can and should be worked out for the corresponding flow of many other physical parameters of interest.
Acknowledgments The first author would like to thank the Ev. Studienwerk for financial support. ´ Also the I.H.E.S. and its warm hospitality is greatly acknowledged. We would like to thank Prof. Ivan Todorov, Prof. Olivier Babelon, and Igor Mencattini for valuable discussions, and helpful comments. D.K. is in parts supported by NSF grant DMS-0401262.
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References [1] K. Ebrahimi-Fard, L. Guo, D. Kreimer, Integrable renormalization I: the ladder case, J. Math. Phys. 45, No 10, 3758–3769 (2004). [2] J.C. Collins, Renormalization, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, (1985). [3] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2, 303 (1998). [4] D. Kreimer, Chen’s iterated integral represents the operator product expansion, Adv. Theor. Math. Phys. 3, no. 3, 627 (1999). [5] D.J. Broadhurst, D. Kreimer, Renormalization automated by Hopf algebra, J. Symb. Comput. 27, 581 (1999). [6] D.J. Broadhurst, D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation, Phys. Lett. B 475, 63 (2000). [7] D.J. Broadhurst, D. Kreimer, Exact solutions of Dyson-Schwinger equations for iterated one-loop integrals and propagator-coupling duality, Nucl. Phys. B 600, 403 (2001). [8] A. Connes, D. Kreimer, Hopf algebras, Renormalization and Noncommutative Geometry, Comm. in Math. Phys. 199, 203 (1998). [9] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. in Math. Phys. 210, 249 (2000). [10] A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216, 215 (2001). [11] I. Mencattini, D. Kreimer, Insertion and elimination Lie algebra: the ladder case, Lett. in Math. Phys. 67, 61–74 (2004). [12] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10, 731 (1960). [13] F.V. Atkinson, Some aspects of Baxter’s functional equation, J. Math. Anal. Appl. 7, 1 (1963). [14] G.-C. Rota, Baxter algebras and combinatorial identities. I, II., Bull. Amer. Math. Soc. 75, 325 (1969); ibid. 75, 330 (1969).
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[15] P. Cartier, On the structure of free Baxter algebras, Advances in Math. 9, 253 (1972). [16] G.-C. Rota, Baxter operators, an introduction, In: “Gian-Carlo Rota on Combinatorics, Introductory papers and commentaries”, Joseph P.S. Kung, Editor, Birkh¨ auser, Boston, (1995). [17] M. Aguiar, J.-L. Loday, Quadri-algebras, J. Pure Applied Algebra 191, 205– 221 (2004). [18] K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation, Letters in Mathematical Physics 61, no. 2, 139 (2002). [19] Ph. Leroux, Ennea-algebras, Nov 2003, preprint: arXiv:math.QA/0309213. [20] K. Ebrahimi-Fard, L. Guo, On product and Duality of Binary, Quadratic Regular Operads, preprint 2004, J. Pure Applied Algebra, in press. [21] M. Aguiar, Prepoisson algebras, Letters in Mathematical Physics 54, no. 4, 263 (2000). [22] G.-C. Rota, Ten mathematics problems I will never solve, Mitt. Dtsch. Math.Ver., no. 2, 45 (1998). [23] L. Guo, W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150, no. 1, 117 (2000). [24] L. Guo, Baxter algebras and differential algebras, in ”Differential algebra and related topics”, (Newark, NJ, 2000), World Sci. Publishing, River Edge, NJ, 281, (2002). [25] L. Guo, Baxter Algebras, Stirling Numbers, and Partitions, Feb. 2004, preprint, http://newark.rutgers.edu/ liguo/lgpapers.html, to appear in J. Algebra and Its Appl. [26] M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funct. Ana. Appl. 17, no.4., 254 (1983). [27] A.G. Reyman, M.A. Semenov-Tian-Shansky, Group theoretical methods in the theory of finite dimensional integrable systems, in: Encyclopedia of mathematical science, v.16: Dynamical Systems VII, Springer, 116, (1994). [28] A. Belavin, V. Drinfeld, Triangle Equations and Simple Lie-Algebras, Classic Reviews in Mathematics and Mathematical Physics, 1. Harwood Academic Publishers, Amsterdam, (1998), viii+91 pp. [29] M.A. Semenov-Tian-Shansky, Integrable Systems and Factorization Problems, Lectures given at the Faro International Summer School on Factorization and Integrable Systems (Sept. 2000), Birkh¨auser 2003, Sept. 2002, preprint: arXiv: nlin.SI/0209057.
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[30] M. Hoffman, Quasi-shuffle products, J. Algebraic Combin., 11, no. 1, 49 (2000). [31] K. Ebrahimi-Fard, L. Guo, Rota’s q-shuffle relation for the Jackson integral, work in progress. [32] J. Cari˜ nena, J. Grabowski, G. Marmo, Quantum bi-Hamiltonian systems, Internat. J. Modern Phys. A 15, no. 30, 4797 (2000). [33] K. Ebrahimi-Fard, On the associative Nijenhuis relation, The Electronic Journal of Combinatorics 11 (1), R38 (2004) . [34] Ph. Leroux, Construction of Nijenhuis operators and dendriform trialgebras, Nov. 2003, preprint: arXiv:math.QA/0311132. [35] I.Z. Golubchik, V.V. Sokolov, One more type of classical Yang-Baxter equation, Funct. Anal. Appl. 34, no. 4, 296 (2000). [36] Y. Kosmann-Schwarzbach, F. Magri, Poisson-Nijenhuis structures, Ann. Inst. H. Poincar´e Phys. Th´eor. 53, no. 1, 35 (1990). [37] O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, (2003). [38] D. Kreimer, On overlapping divergences, Commun. Math. Phys. 204, 669 (1999). [39] H. Figueroa, J.M. Gracia-Bondia, J.C. Varilly, Elements of Noncommutative Geometry, Birkh¨ auser, (2001). [40] D. Manchon, Hopf algebras, from basics to applications to renormalization, Comptes-rendus des Rencontres math´ematiques de Glanon 2001. [41] D. Kreimer, Factorization in quantum field theory: An exercise in Hopf algebras and local singularities, Proceedings From Number Theory to Physics and Geometry, Les Houches March 2003, in press, arXiv:hep-th/0306020. [42] A. Connes, D. Kreimer, Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs, Ann. Henri Poincar´e 3, no. 3, 411 (2002). [43] C. Chryssomalakos, H. Quevedo, M. Rosenbaum, J.D. Vergara, Normal coordinates and primitive elements in the Hopf algebra of renormalization, Comm. in Math. Phys. 225, no. 3, 465 (2002). [44] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, (1984).
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[45] D. Kreimer, Structures in Feynman graphs: Hopf algebras and symmetries, arXiv:hep-th/0202110, Proceedings of GRAPHS AND PATTERNS in Mathematics and Theoretical Physics, Department of Mathematics and Institute for Mathematical Sciences Stony Brook, New York June 14–21, 2001. To be published in “Proceedings of Symposia in Pure Mathematics”, AMS. [46] M. Sakakibara, On the Differential equations of the characters for the Renormalization group, Mod. Phys. Lett. A 19, 1453–1456 (2004). Kurusch Ebrahimi-Fard* and Dirk Kreimer ´ Institut des Hautes Etudes Scientifiques 35, Route de Chartres F-91440 Bures-sur-Yvette France and *Universit¨ at Bonn Physikalisches Institut Nussallee 12 D-53115 Bonn Germany email:
[email protected] Li Guo Rutgers University Department of Mathematics and Computer Science Newark, NJ 07102 USA email:
[email protected] Communicated by Vincent Rivasseau submitted 16/03/04, accepted 09/09/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 397 – 398 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/020397-2 DOI 10.1007/s00023-005-0212-1
Annales Henri Poincar´ e
Erratum to “Negative Discrete Spectrum of Perturbed Multivortex Aharonov-Bohm Hamiltonians” Ann. Henri Poincar´e, 5 (2004) 979–1012 M. Melgaard, E.-M. Ouhabaz and G. Rozenblum
Regrettably, there is an error in the proof of Proposition 4.1. To correct it the following changes are needed. On page 989 : ∂u ∂u The quantity ∂x in the last term of (4.2) has to be replaced by ∂x . 1 2 Line 10 (the last line of the first non-numbered formula for I2 ): the two signs − must be +; the lines 13 and 14 (the last two lines in the second non-numbered formula for I2 ): again the last two − must be +, which gives + instead of − in (4.3), thus (4.3) has to be replaced by
I2 =
Ωn
−
2A1 Im
Im
Ωn
∂u ∂u sign u¯ |v| + 2A2 Im sign u ¯ |v| ∂x1 ∂x2 (A21 + A22 )|u||v| ≥− Ωn
2 2 ∂u ∂u |v| |v| χ{u=0} − χ{u=0} . sign u ¯ sign u ¯ Im ∂x1 |u| ∂x2 |u| Ωn (4.3)
After this, (4.4) follows, and then, on page 990, the rest of the proof, after the first line must be replaced by:
Re hn [u, v] ≥ ln [|u|, |v|] for all u, v ∈
H01 (Ωn )
obeying u · v¯ ≥ 0. This proves (4.1).
We use the occasion to make a comment on the results of the paper. For the case γ ≥ 1/2, alternatively to Theorem 1.2 in the paper, the Lieb-Thirring type estimates can also be obtained by an approximation procedure based on the results of the paper by A. Laptev and T. Weidl ([18] in the reference list, Theorem 3.2) and of the paper by D. Hundertmark, A. Laptev, T. Weidl, New bounds on
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the Lieb-Thirring constants. Invent. Math. 140 (2000), no. 3, 693–704, Theorem 4.2. This reasoning would give better constants in the estimates, up to 7 times smaller than the ones obtained in our Theorem 1.2. M. Melgaard Department of Mathematics Uppsala University Polacksbacken S-751 06 Uppsala Sweden email:
[email protected] E.-M. Ouhabaz Laboratoire Bordelais d’Analyse et G´eom´etrie Universit´e de Bordeaux 1 351, Cours de la Lib´eration F-33405 Talence cedex France email:
[email protected] G. Rozenblum Department of Mathematics Chalmers University of Technology and University of Gothenburg Eklandagatan 86 S-412 96 Gothenburg Sweden email:
[email protected] Communicated by Bernard Helffer received 03/01/05
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Ann. Henri Poincar´e 6 (2005) 399 – 448 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/030399-50 DOI 10.1007/s00023-005-0213-0
Annales Henri Poincar´ e
Renormalization of the 2-Point Function of the Hubbard Model at Half-Filling St´ephane Afchain, Jacques Magnen and Vincent Rivasseau Abstract. We prove that the Hubbard model at finite temperature T and half-filling is analytic in its coupling constant λ for |λ| ≤ c/| log T |2 , where c is some numerical constant. We also bound the self-energy and prove that the Hubbard model at half-filling is not a Fermi liquid (in the mathematically precise sense of Salmhofer), modulo a simple lower bound on the first non-trivial self-energy graph, which will be published in a companion paper.
I Introduction In [1] we introduced the tools for a multiscale analysis of the two-dimensional Hubbard model at half-filling: momentum slices, sectors and their conservation rules. In this paper we achieve the proof that the correlation functions of the model at finite temperature T are analytic in the coupling constant λ for |λ| ≤ c/| log T |2 , by treating the renormalization of “bipeds” (two-particle subgraphs), that was missing in [1]. This proof requires a new tool which is a constructive two-particle irreducible analysis of the self-energy. This analysis according to the line form of Menger’s theorem ([2]) leads to the explicit construction of three line-disjoint paths for every self-energy contribution, in a way compatible with constructive bounds. On top of that analysis, another one which is scale-dependent is performed: after reduction of some maximal subsets provided by the scale analysis, two vertex-disjoint paths are selected in every self-energy contribution. This requires a second use of Menger’s theorem, now in the vertex form. This construction allows to improve the power counting for two-point subgraphs, exploiting the particle-hole symmetry of the theory at half-filling, and leads to our analyticity result. In the last section we write the upper bounds on the self-energy that follow from our analysis. These upper bounds strongly suggest that the second momentum derivative of the self energy is not uniformly bounded in the region |λ| ≤ c/| log T |2 . A rigorous proof of this last statement follows from a rigorous lower bound of the same type than these upper bounds, but for the smallest nontrivial self-energy graph, so as to rule out any “miraculous cancellation”. This lower bound, which we have now completed, is the tedious but rather straightforward study of a single finite-dimensional integral. Since it is not related to the main analysis in this paper, we postpone it to a separate publication [5].
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Taken all together these bounds prove that the model is not a Fermi liquid in the sense of Salmhofer’s criterion (see [3] and [4]). Indeed to be such a Fermi liquid the second derivative would have to be uniformly bounded in a larger region (of type |λ| ≤ c/| log T |) than the one for which we prove it is unbounded. The scaling properties of the self energy and its derivatives in fact mean that the model is not of Fermi but of Luttinger type, with logarithmic corrections if we compare to the standard one-dimensional Luttinger liquid. Let us state precisely the main result of this paper: Theorem The radius of convergence of the Hubbard model perturbative series at half-filling is at least c/ log2 T , where T is the temperature and c some numerical constant. As T and λ jointly tend to 0 in this domain, the self-energy of the model does not display the properties of a Fermi liquid in the sense of [3], but those of a Luttinger liquid (with logarithmic corrections). Let us also put our paper in perspective and relation with other programs of rigorous mathematical study of interacting Fermi systems. Recall that in dimension 1 there is neither superconductivity nor extended Fermi surface, and Fermion systems have been proved to exhibit Luttinger liquid behavior [6]. The initial goal of the studies in two or three dimensions was to understand the low temperature phase of these systems, and in particular to build a rigorous constructive BCS theory of superconductivity. The mechanism for the formation of Cooper pairs and the main technical tool to use (namely the corresponding 1/N expansion, where N is the number of sectors which proliferate near the Fermi surface at low temperatures) have been identified [8]. But the goal of building a completely rigorous BCS theory ab initio remains elusive because of the technicalities involved with the constructive control of continuous symmetry breaking. So the initial goal was replaced with a more modest one, still important in view of the controversies over the nature of two-dimensional “Fermi liquids” [7], namely the rigorous control of what occurs before pair formation. The last decade has seen excellent progress in this direction. As is well known, sufficiently high magnetic field or temperature are the two different ways to break the Cooper pairs and prevent superconductivity. Accordingly two approaches were devised for the construction of “Fermi liquids”. One is based on the use of non-parity invariant Fermi surfaces to prevent pair formation. These surfaces occur physically when generic magnetic fields are applied to two-dimensional Fermi systems. The other is based on Salmhofer’s criterion [3], in which temperature is the cutoff which prevents pair formation. In a large series of papers [9], the construction of two-dimensional Fermi liquids for a wide class of non-parity invariant Fermi surfaces has been completed in great detail by Feldman, Kn¨orrer and Trubowitz. These papers establish Fermi liquid behavior in the traditional sense of physics textbooks, namely as a jump of the density of states at the Fermi surface at zero temperature, but they do not apply to the simplest Fermi surfaces, such as circles or squares, which are parity invariant.
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Another program in recent years was to explore which models satisfy Salmhofer’s criterion. Of particular interest to us are the three most “canonical” models in more than one dimension namely: • the jellium model in two dimensions, with circular Fermi surface, nicknamed J2 , • the half-filled Hubbard model in two dimensions, with square Fermi surface, nicknamed H2 , • and the jellium model in three dimensions, with spherical Fermi surface, nicknamed J3 . The study of each model has been divided into two main steps of roughly equal difficulty, the control of convergent contributions and the renormalization of the two-point functions. In this sense, five of the six steps of our program are now completed. J2 is a Fermi liquid in the sense of Salmhofer [10]–[11], H2 is not, and is a Luttinger liquid with logarithmic corrections, according to [1], to the present paper, and to [5]. Results similar to [10]–[11] have been also obtained for more general convex curves not necessarily rotation invariant such as those of the Hubbard model at low filling, where the Fermi surface becomes more and more circular, including an improved treatment of the four-point functions leading to better constants [12]. Therefore as the filling factor of the Hubbard model is moved from halffilling to low filling, we conclude that there must be a crossover from Luttinger liquid behavior to Fermi liquid behavior. This solves the controversy [7] over the Luttinger or Fermi nature of two-dimensional many-Fermion systems above their critical temperature. The short answer is that it depends on the shape of the Fermi surface. Up to now only the convergent contributions of J3 , which is almost certainly a Fermi liquid, have been controlled [13]. The renormalization of the two-point functions for J3 , the last sixth of our program, remains still to be done. This last part is difficult since the cutoffs required in [13] do not conserve momentum. This means that the two-point functions that have to be renormalized in this formalism are not automatically one particle irreducible, as is the case both in [11] and in this paper. This complicates their analysis.
II Slices, sectors, propagator decay and momentum conservation We recall here some generalities that were explained in [1], in order to make this paper self-contained. Given a temperature T > 0, the Hubbard model lives on [−β, β[ × Z2 , where β = T1 . Indeed, the real interval [−β, β[ should be thought of as the circle of radius β. A generic element of [−β, β[ × Z2 will be denoted → → x = (x0 , − x ), where x0 ∈ [−β, β[ and − x = (n1 , n2 ) ∈ Z2 .
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→ Like in every Fermionic model, the propagator C(x0 , − x ) 1 is antiperiodic 1 in the variable x0 , with antiperiod T . Therefore, for the Fourier transform of → ˆ 0, − the propagator C(k k ), the relevant values for k0 are discrete and called the Matsubara frequencies: (2n + 1)π k0 = , n ∈ Z, (II.1) β → − whereas the vector k lives on the two-dimensional torus R2 /(2πZ)2 . At half-filling and finite temperature T , we have: Cˆa,b (k) = δa,b
1
→ , − ik0 − e( k )
(II.2)
→ − with e( k ) = cos k1 + cos k2 . a and b are spin indices (elements of the set {↑, ↓}), and may sometimes be dropped when they are not essential. Hence the expression of the real space propagator is: π π 1 dk dk2 eik.x Cˆa,b (k) . (II.3) Ca,b (x) = 1 (2π)2 β −π −π k0
The notation k0 really means the discrete sum over the integer n in (II.1). When T → 0+ (which means β → +∞), k0 becomes a continuous variable, the corresponding discrete sum becomes an integral, and the corresponding propagator C0 (x) becomes singular on the Fermi surface √ defined by k0 = 0 and e(k) = 0. This Fermi surface is a square of side size 2π (in the first Brillouin zone) joining the corners (±π, 0), (0, ±π). We call this square the Fermi square, its faces and corners are called the Fermi faces and corners. Considering the periodic boundary conditions, there are really four Fermi faces, but only two Fermi corners. In the following, to simplify notations, we will write:
1 dk1 dk2 d k ≡ β [−π,π]2 3
,
k0
1 d x ≡ 2
β
3
−β
dx0
.
(II.4)
x∈Z2
The interaction of the Hubbard model is simply SV = λ V
d3 x
2 ψ a (x)ψa (x) ,
(II.5)
a∈{↑,↓}
where V := [−β, β[×V and V is an auxiliary volume cutoff in two-dimensional space, that will be sent to infinity eventually. Remark that in (II.1) |k0 | ≥ π/β = 0 1 Indeed, the propagator should be seen as depending on two variables x, y ∈ [−β, β[ × Z2 , but by translational invariance, we have C(x, y) = C(0, y −x) and we shall write in the following simply C(x) instead of C(0, x).
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hence the denominator in C(k) can never be 0 at non-zero temperature. This is why the temperature provides a natural infrared cutoff. We use in this paper the same slices and sectors than in [1] and recall the main points for completeness. Introducing a fixed large number M > 1, we perform a slice analysis according to geometric scales of ratio M . Like in [1] since we have a finite temperature, this analysis should stop for a scale imax (T ) such that M −imax (T ) 1/T . We write simply imax for imax (T ). As in [1] we use the tilted orthogonal basis in momentum space (e+ , e− ), defined by e+ = (1/2)(π, π) and e− = (1/2)(−π, π). In the corresponding coordinates (k+ , k− ) the Fermi surface is given by k+ = ±1 or k− = ±1. This follows from the identity πk− πk+ cos . (II.6) cos k1 + cos k2 = 2 cos 2 2 We also use the convenient notations q± = k± − 1 if k± ≥ 0 ; q± = k± + 1 if k± < 0
(II.7)
so that 0 ≤ |q± | ≤ 1. Picking a Gevrey compact support function u(r) ∈ C0∞ (R) of order α < 1 which satisfies: u(r) = 0
for |r| > 2 ; u(r) = 1 for |r| < 1 ,
(II.8)
we consider the partition of unity:
imax (T )
1=
i=0
with
2 2 πk+ 2 πk− cos ui k0 + 4 cos , 2 2
u0 (r) = 1 − u(r) , ui (r) = u M 2(i−1) r − u M 2i r for i ≥ 1.
(II.9)
(II.10)
The sum over i a priori runs from 0 to +∞ to create a partition of unity, but in fact since k02 is at least of order M −2imax (T ) , the sum over i stops as imax (T ). This is similar to [1]. The i slice propagator Ci (k) = C(k)ui (k) is further sliced into the ± directions exactly as in [1]: Ci (k) = Cσ (k) , (II.11) σ=(i,s+ ,s− )
where Cσ (k) = Ci (k) vs+
πk+ πk− cos2 vs− cos2 2 2
(II.12)
using a second partition of unity 1=
i s=0
vs (r) ,
(II.13)
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= 1 − u(M 2 r) , = us+1 for 1 ≤ s ≤ i − 1 , = u(M 2i r) .
(II.14)
Like in [1] we need s+ + s− ≥ i − 2 for non-zero Cσ (k), and the depth l(σ) of a sector is defined as l = s+ + s− − i + 2, with 0 ≤ l ≤ i + 2. 2 . We have the scaled decay ([1], Lemma 1):
| Cσ (x, y)| ≤ c.M −i−l e−c [dσ (x,y)]
α
(II.15)
where c, c are some constants and dσ (x, y) = M −i |x0 − y0 | + M −s+ |x+ − y+ | + M −s− |x− − y− | .
(II.16)
Furthermore we recall the momentum conservation rules for the four sectors σj , j = 1, . . . , 4 meeting at any vertex ([1], Lemma 4): Proposition 1: Momentum conservation at a vertex. For M large enough, the two smallest indices among sj,+ , j = 1, . . . , 4 differ by at most one unit, or the smallest one, say s1,+ must coincide with i1 with i1 < ij , j = 1. Exactly the same statement holds independently for the minus direction. We say that the sectors which have smallest indices at a vertex in a direction “collapse” in that direction. We also introduce a new index for each sector, r(σ) = E(i(σ) + l(σ)/2) (where E means the integer part like in [1], section 4) and the corresponding slice propagator Cσ (k) . (II.17) Cr (k) = σ | r(σ)=r
We remark that this slice cutoff respects the symmetries of the theory. It is with respect to this slice index that our main multislice analysis will be performed. The propagator with infrared cutoff r is defined as C≤r (k) =
Cσ (k) .
(II.18)
σ | r(σ)≤r 2 This definition of sectors looks at first sight a bit complicated. It is designed for Proposition 1 to hold, in order to get an analyticity in |λ| ≤ c/ log2 T . With less detailed sectors one could prove analyticity in a radius, e.g., only in |λ| ≤ c/ log4 T . This would be well enough to prove that the Hubbard model is not a Fermi liquid in Salmhofer’s sense. The reader only interested in this result could therefore skip some of the technicalities below by putting everywhere the l index to zero, and skipping Proposition 1. But he should not skip Sections VI–VII because the two particle irreducible analysis and the ring construction are there to fix the correct power counting, not just the logarithmic power counting.
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III Renormalization of the two-point function Let us define S2,≤r (k0 , k) as the connected amputated two-point function with infrared cutoff r, and define also: 1 S2,≤r (k0 , k) + S2,≤r (−k0 , k) . (III.19) G2,≤r (k0 , k) = 2 Consider k such that e(k) = 0. If our cutoff respects the symmetries of the theory, which is the case here, the nesting or particle-hole symmetry forces G2 to vanish for such k. Using the variables q+ and q− defined in (II.7), this is expressed by Lemma III.1 The following equality holds: G2,≤r (k0 , q+ , q− )
q+ =0 or q− =0
=0.
(III.20)
Proof. Using the symmetries of the theory, it is easy to check that for any Feynman two-point function graph G, the Feynman amplitude IG satisfies: IG (k0 , k1 , k2 ) = IG (k0 , k2 , k1 ) ,
(III.21)
IG (k0 , k1 , k2 ) = IG (k0 , −k1 , k2 ) ,
(III.22)
IG (k0 , k1 , k2 ) = −IG (−k0 , k1 + π, k2 + π) .
(III.23)
The last symmetry, the particle-hole symmetry, is the only non-trivial one and it can be checked because it changes all the propagators in momentum space into their opposite with all the momentum conservation laws respected. Since there is an odd number of propagators in a two-point subgraph, (III.23) holds. Now we consider a point k in the first quadrant with 0 ≤ k1 ≤ π and 0 ≤ k2 ≤ π. On the Fermi curve whose equation in this quadrant is k2 = π − k1 , we apply the relation (III.23) and get 0 = IG (k0 ,k1 ,k2 ) + IG (−k0 ,k1 + π,k2 + π) = IG (k0 ,k1 ,k2 ) + IG (−k0 ,2π − k2 ,2π − k1 ). (III.24) By the symmetries (III.21), (III.22) and periodicity 2π we obtain that IG (k0 , k) + IG (−k0 , k) = 0. By symmetry this relation holds also for the other quadrants, hence on all the Fermi square. Summing over all Feynman graphs we obtain the vanishing of G2,≤r (k0 , q+ , q− ) on the Fermi surface whose equation is q+ = 0 or q− = 0. The function being constant on the straight lines of the Fermi square, obviously its partial derivatives to any order along these straight directions also vanish on the Fermi surface. Recall that in [1] analyticity of a simplified Hubbard model at half filling was established in a domain of the expected optimal form |λ| ≤ c/| log T |2 . Indeed
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and more precisely the result was established only for a model called “bipedfree” in which all two-point subgraphs appearing in the multislice expansion were suppressed. A straightforward extension of the bounds given in [1] is not enough to prove analyticity in the expected domain for the full model. Naive power counting in the style of [1] is indeed not sufficient to sum geometric series made of insertions of a two-point subgraph at a scale r and a propagator at scale s >> r. Consider, e.g., the simplest such sum, made of the chain of Figure 1, where the three internal lines of the biped have main scale r and the external one has main scale s >> r. The naive bound for the contribution of such a chain is M −r−l/2 per propagator at scale r, M −s−l /2 per propagator at scale s, and contains for each irreducible biped one integral over the position of one vertex evaluated through the decay of a propagator of scale s and one evaluated through the decay of a propagator of scale r. Let us neglect the auxiliary “depth indices” l and l which are not essential. The bound is therefore a geometric series with ratio M −3r M −s M 2r M 2s = M s−r .
(III.25)
Figure 1. A simple chain of bipeds. This bad factor M s−r appears always in the naive bounds for any similar twopoint function; it is exponential, not logarithmic in s − r, and certainly prevents a proof of analyticity, not only for |λ| ≤ c/| log T |2 , but for |λ| ≤ c/| log T |q for any integer q as well. As remarked in [1], this is however only a bound, and the true contribution is much smaller due to the particle-hole symmetry of the model at half-filling. To exploit this, and to treat the true model, we must “renormalize” the two-point functions of the theory instead of suppressing them. This is accomplished by a second order Taylor expansion of the two-point function with given cutoff in the style of [11]. In momentum space we change first k0 to the smallest possible values ±πT :
+
1 S2,≤r (k0 , q+ , q− ) − S2,≤r (πT, q+ , q− ) 2 S2,≤r (k0 , q+ , q− ) − S2,≤r (−πT, q+ , q− )
+
G2,≤r (πT, q+ , q− ) .
S2,≤r (k0 , q+ , q− ) =
(III.26)
Then we use (III.20) to write G2,≤r (πT, q+ , q− )
= G2,≤r (πT, q+ , q− ) − G2,≤r (πT, 0, q− ) − G2,≤r (πT, q+ , 0) + G2,≤r (πT, 0, 0) ,
(III.27)
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where the variables (q+ , q− ) are the usual k variables translated, so as to vanish on the Fermi surface. They depend on the patch of coordinates chosen. This patch can be determined by the sector of the external line to which S2 is hooked. For constructive purpose one cannot however work in momentum space and one should write an equivalent dual formula in direct space. In practice a two-point function S2 is integrated in a bigger function against a kernel always made of one external propagator C and a rest called R, which (in momentum space) may be in general a function of the set Pe of external momenta. So in the momentum representation we have to compute not S2 itself but integrals such as I=
dpdq S2 (p)C(q)R(p, q, Pe )
(III.28)
where from momentum conservation R(p, q, Pe ) = δ(p − q)R (p, q, Pe ). To get the corresponding direct space representation we have to pass to the Fourier transform. Using same letters for functions and their Fourier transforms we write (III.29) I = dydz S2 (x, y)C(y, z)R(z, x, Pe ) (this integral being in fact by translation invariance independent of x) where S2 (x, y) = dp S2 (p)eip(x−y) ; C(y, z) = dq C(q)eiq(y−z) ; dpdq R(p, q, Pe )eip(z−x) , (III.30) R(z, x, Pe ) = where the last integral is not really a double integral because of the δ function hidden in R. Any counterterm for I that is expressed in momentum space by an operator τ acting on S2 (p), such as putting S2 to a fixed momentum k, hence τ S2 (p) = S2 (k), can also be represented by a dual operator τ ∗ acting in direct space, but on the external propagator. This τ ∗ is not unique, but a convenient choice is to use x as the reference point for τ ∗ : τ I = dp dq S2 (k)C(q)R(p,q,Pe ) = dy dz S2 (x,y)[eik(x−y) C(x,z)]R(z,x,Pe ), (III.31) hence τ ∗ C(y, z) = eik(x−y) C(x, z).
(III.32)
The dual version of the more complicated expressions (III.26–III.27) is given by (we write the expressions in the patch where q+ = k+ − 1, q− = k− − 1) I = dpdq S2 (p)C(q)R(p, q, Pe ) = I1 + I2
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I1 =
I2 =
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dydz S2 (x, y) C(y, z) − cos πT (x0 − y0 ) C (x0 , y+ , y− ), z R(z, x, Pe )
dydz S2 (x, y) cos πT (x0 − y0 ) C (x0 , y+ , y− ), z R(z, x, Pe ) − dydz S2 (x, y) cos πT (x0 − y0 ) ei(x+ −y+ ) C (x0 , x+ , y− ), z R(z, x, Pe ) − dydz S2 (x, y) cos πT (x0 − y0 ) ei(x− −y− ) C (x0 , y+ , x− ), z R(z, x, Pe ) + dydz S2 (x, y) cos πT (x0 − y0 ) ei[(x+ −y+ )+(x− −y− )] C(x, z)R(z, x, Pe ) (III.33)
where the propagator C is now the natural extension of the propagator to the continuum. Each integral I1 and I2 will be bounded separately. We need to exploit the differences as integrals of derivatives. This means that in I1 we write: C(y, z) − cos πT (x0 − y0 ) C (x0 , y+ , y− ), z 1 d C (ty0 + (1 − t)x0 , y+ , y− ), z cos πT (1 − t)(x0 − y0 ) dt = dt 0 1 1 dt (y0 − x0 ) eiπT (1−t)(x0 −y0 ) (∂0 + iπT )C (ty0 + (1 − t)x0 , y+ , y− ), z = 2 0 + e−iπT (1−t)(x0 −y0 ) (∂0 − iπT )C (ty0 + (1 − t)x0 , y+ , y− ), z (III.34) and in I2 we write C (x0 , y+ , y− ), z − ei(x+ −y+ ) C (x0 , x+ , y− ), z − ei(x− −y− ) C (x0 , y+ , x− ), z + ei[(x+ −y+ )+(x− −y− )] C(x, z) = F (1, 1) − F (0, 1) − F (1, 0) + F (0, 0) (III.35) where
x0 , sy+ +(1−s)x+ , ty− +(1−t)y+ , z ei[(1−s)(x+ −y+ )+(1−t)(x− −y− )] . (III.36) Finally we can use 1 1 d2 F (s, t) . (III.37) F (1, 1) − F (0, 1) − F (1, 0) + F (0, 0) = dsdt dsdt 0 0
F (s, t) = C
to obtain: I2 = dydz S2 (x, y) cos πT (x0 − y0 ) R(z, x, Pe )ei[(1−s)(x+ −y+ )+(1−t)(x− −y− )] (y+ −x+ )(y− −x− )(∂+ +i)(∂− +i)C x0 , sy+ +(1−s)x+ , ty− +(1−t)y+ , z . (III.38)
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IV Multislice expansion We perform a multi-slice expansion, and get a Gallavotti-Nicol`o or clustering tree structure as in [1]. In that paper a tree formula was used to express a typical function for the model, namely the pressure, but the analysis applies to any thermodynamic function. Now we would like to focus on the self-energy. A good starting point for this is the connected amputated two-point Schwinger function. We fix here some conventions and notations that have not been introduced in [1]. We will call a “field” (between inverted commas) a five-tuple (x, a, σ, nature, order) where: x ∈ V , a ∈ {↑, ↓} , σ ∈ Sect(T ) , nature ∈ {+, −} , order ∈ {1, 2}. (IV.39) x is the spacetime position of the “field”, a its spin and σ its sector. nature is an element of the set whose elements are denoted + and −; this parameter is introduced in order to distinguish between the fields and the antifields (corresponding respectively to the Grassmann variables ψ and ψ). Thus in the following, it may happen that we use the term field (without inverted commas) to mean a “field” such that nature = + and of course an antifield will be a “field” such that nature = −. At last, the parameter order allows to distinguish between the two copies antifield involved in the expansion of the quartic action: and of each field a∈{↑, ↓} ψ a ψa = a, b ψ a ψa ψ b ψb , in such a way that order = 1 corresponds to the first (anti)field represented by the Grassmann variables ψ a and ψa , while order = 2 corresponds to the second ones, represented by ψ b and ψb . Given an integer n ≥ 1, an n-tuple (x1 , . . . , xn ) of elements of V , two n-tuples (a1 , . . . , an ) and (b1 , . . . , bn ) of elements of {↑, ↓} and four n-tuples of elements of Sect(T ), denoted (σ1j , . . . , σnj ), j ∈ {1, 2, 3, 4}, we define the family of the antifields: AF = (x1 , a1 , σ11 , −, 1), (x1 , b1 , σ12 , −, 2), . . . , (xn , an , σn1 , −, 1), (xn , bn , σn2 , −, 2) . (IV.40) We can imagine it as a 2n-tuple indexed by the set [n] × {1, 2} (where [n] denotes the set {1, . . . , n}), lexicographically ordered: (1, 1) ≺ (1, 2) ≺ (2, 1) ≺ (2, 2) ≺ · · · ≺ (n, 1) ≺ (n, 2) .
(IV.41)
In the same way we introduce the family of the fields: F = (x1 , a1 , σ13 , +, 1), (x1 , b1 , σ14 , +, 2), . . . ,
(xn , an , σn3 , +, 1), (xn , bn , σn4 , +, 2) . (IV.42)
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Observe that AF and F are defined as families and not as sets. Hence the cardinality of AF and F is 2n, whatever may be the values of the parameters {xv }, {av }, {bv } and {σvj }. Given f ∈ AF and g ∈ F, we will simply denote by C(f, g) the propagator: C(f, g) = δa(f ), a(g) δσ(f ), σ(g) C x(f ) − x(g) , (IV.43) where the notations a(f ), a(g), σ(f ), σ(g), x(f ), x(g) have an immediate obvious meaning. With all these notations, we can express the partition function of the model as: ∞ λn d3 x1 . . . d3 xn det C(f, g) . (IV.44) Z(V ) = n! V n (f,g)∈AF ×F j n=0 {av }, {bv } {σv }
AF for the Fermionic determinant (Cayley’s F notation). To write the unnormalized unamputated two-point Schwinger function: 2 3 ψ a (x)ψa (x) , S2 (Y, Z)σ0 = dµC (ψ, ψ) ψ ↑, σ0 (Y )ψ↑, σ0 (Z)exp λ d x Sometimes we shall write simply
V
a
(IV.45) we only need to add the source terms (Y, ↑, σ0 , −) to AF and (Z, ↑, σ0 , +) to F 3 . Since AF and A are indeed totally ordered families, we must specify in which position (y, ↑, σ0 , −) and (z, ↑, σ0 , +) are inserted. Clearly, they must be added in first position, that is, we have: AF = (y, ↑, σ0 , −), (x1 , a1 , σ11 , −, 1),...,(xn , an , σn1 , −, 1), (xn , bn , σn2 , −, 2) (IV.46) and
F = (z, ↑, σ0 , +), (x1 , a1 , σ11 , −, 1),...,(xn , an , σn1 , −, 1), (xn , bn , σn2 , −, 2) . (IV.47) Observe that, with a slight abuse of notation, we denote these two families again by AF and F . With this convention, the expression of the two-point function S2 (y, z)σ0 is exactly the same as the one of Z(V ): ∞ λn C(f, g) . d3 x1 . . . d3 xn det S2 (Y, Z)σ0 = n! V n (f,g)∈AF ×F j n=0 {av }, {bv } {σv }
(IV.48) The main tool to express the connected two-point function is a Taylor jungle formula [14], that is a forest formula which is ordered according to the main slice index, namely r, attached to the propagator, to expand the Fermionic determinant. 3 Note
that these two external “fields” have no order parameter.
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To extract the connected part of the two-point function, namely S2 (Y, Z)c, σ0 = Z −1 S2 (Y, Z)σ0 , we only need to factorize the contributions of the vacuum clusters of the jungle, and we get a tree formula: S2 (Y, Z)c,
σ0 =
∞ λn n=0
∈T
1
dw
0
n!
Vn
d3 x1 ...d3 xn
treesT {av }, {bv }, {σvj } oriented over V
C f ( ,Ω), g( ,Ω)
∈T
det
(f, g)∈AF left ×Fleft
field attributions Ω
C(f, g,{w }) . (IV.49)
The amputated connected two-point function S2 (y, z)c,a is given by a similar formula, in which we should delete the two external sources Y and Z and the two propagators which connect them to two particular external distinguished vertices4 . Let us rename the position of these vertices as y and z, and rename all remaining internal positions as x1 , . . . , xn . So, after integration over positions of these n internal vertices, this amputated function is a function of the positions y and z of the two particular special external vertices. We shall denote V the family of the vertices: V = (y, z, x1 , . . . , xn ). We recall that a tree over V = {y, z, x1 , . . . , xn } is a set of pairs of vertices {v, v } (called the links of the tree), such that the corresponding graph has no loop and connects all the elements of V. As |V| = n + 2; any tree over V has n + 1 links. Once a tree T over V is chosen, a field attribution Ω for T is a family of the form (ω , ω ) where ω is a map from the pair to {1, 2} and ω a one-to-one ∈T
map from to {+, −}. Hence Ω is simply the choice, for each “half-line” of the tree T of a precise “field” of the vertex to which this half-line hooks. We have taken into account the constraint that a field must contract with an antifield by the fact that the maps ω : → {+, −} are one-to-one. Given ∈ T and a field attribution Ω, we denote respectively by f ( , Ω) and g( , Ω) the antifield and the field attached to by Ω. AF left and Fleft are the families of the remaining “fields”: AF left = AF \{f ( , Ω), ∈ T } and Fleft = F \{g( , Ω), ∈ T } .
(IV.50)
At last we must precise the expression of the entries of the remaining Fermi. onic determinant that depends now on the interpolation parameters w ∈P2 (V)
We recall that (see [14] – [1] for details) the data w allows to define a vector 4 Indeed we can forget the graphs where these two external sources Y and Z connect to the same external vertex, the “generalized tadpoles”, since they are zero by the particle hole symmetry.
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X T {w } whose components are indexed by P2 (V), the set of the (unordered) pairs of vertices. By definition, for {v, v } ∈ P2 (V), X T {w } is the in {v, v }
fimum of the w parametersover the unique path in T from v to v . Then, the expression of C f, g, {w } is simply obtained by multiplying C(f, g) by the component of X T {w } corresponding to the vertices v(f ) and v(g) of f and g. Hence we have: C f, g, {w } = X T {w } C(f, g) . (IV.51) {v(f ), v(g)}
IV.1 The Gallavotti-Nicol` o tree In order to analyze further this sum, it is well known that the main tool is the “Gallavotti-Nicol` o” or clustering tree which represents the inclusion relations of the connected components of “higher scales” (smaller r indices) into those of “lower scales” (bigger r indices) [6]. This tree is also the key tool to identify the components that require some renormalization (here the two-point functions). But before doing this, we want to describe precisely the constraints on the sum over the sectors {σvj }. Indeed, this sum could be let free of constraints, but due to the expression of the propagator: C(f, g) = C (x(f ), a(f ), σ(f )); (x(g), a(g), σ(g)) = δa(f ), a(g) δσ(f ), σ(g) Cσ(f ) (x(f ), x(g)) , (IV.52) we see easily that the sectors and spin indices are conserved along each line of the tree T . Therefore, once T has been fixed, the sum over the σvj ’s can be understood as a sum over the families of sectors indexed by the lines of T , denoted σ ∈T , and the families of sectors indexed by the remaining “fields”, σf f ∈AF ∪F . left left Now let suppose we are given an oriented tree T over V, and an attribution us o tree is defined as of sectors, σ ∈T and σf f ∈AF ∪F . The Gallavotti-Nicol` left left follows: for each index r ∈ [0, rmax (T )], we define a partition Πr . Πr is the set of the connected components of the graph whose set of vertices is V and whose internal tree lines are the lines of T such that r ≤ r. The family r∈[0, rmax ] Πr is partially ordered by the inclusion relation and forms the nodes of the GallavottiNicol` o tree. To visualize better the situation, let us take the example of Figure 2 for an amputated two-point function with external vertices at y and z (the external amputated legs in slice 6 are represented as dotted lines in Figure 2). The total number of vertices is 8, hence there are 7 lines in the tree T represented as bold lines, and 16 internal fields in the determinant represented as thin half-lines. In the attribution of r indices chosen we see that there is a two-point subfunction to renormalize, the one in the dotted box, which is completed at scale 3 with external lines at scale 5.
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3 5
5
6
5
5
x1
5 5
x4 2 5
3
3 3
0
y
3
x3
3 3
3
1
x6
x5
3
413
5
5
z
5
6
4 x2
Figure 2. A contribution with eight vertices to the two-point function at scale 6. The corresponding Gallavotti-Nicol` o tree is pictured in Figure 3 (with determinant fields omitted for simplicity). This abstract tree should not be confused with T , whose lines are the bold lines of Figure 3. As in [1], we can now write an expression of S2 (y, z)c,a re-ordered in terms of these “clustering tree structures”, in which all nested sums have to be compatible: S2 (y, z)c,a =
∈T
1
dw 0
∞ λn+2 d3 x1 . . . d3 xn n! n V n=0 × {av }, {bv }
clustering tree treesT field attributions structures C over V Ω
C f ( , Ω), g( , Ω)
∈T
{σvj }
det
(f, g)∈AF left ×Fleft
C(f, g, {w }) . (IV.53)
In the Gallavotti-Nicol` o tree, of particular interest to us are the nodes such as the dotted box of Figure 2 between scales 3 and 5 which correspond to two-point functions. They are the ones that were artificially suppressed in the simplified model [1]. We need to renormalize them to solve the divergent power counting explained in Section III. But we can choose to renormalize only the two-point functions for which external lines have r index bigger than the maximum index of internal lines plus 2, so as to create a gap between internal and external supports5 . Such two-point functions are the dangerous nodes of the GN tree. The gap ensures that all such dangerous two-point functions, which are those that we need to renormalize, are automatically one-particle irreducible by momentum conservation6 . Hence they correspond to the so-called self-energy. 5 The
two-point functions for which the external r index is the maximum r index of internal lines plus 1 don’t really need renormalization, as is obvious from power counting (see (III.25)). 6 Indeed any one particle reducible two-point function would have its external momentum also flowing through any internal one-particle-reducibility line, which is a contradiction with the fact that the internal and external cutoffs have empty intersection.
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y
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x5
x6
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x6
y x1
x3
x5
z
x2
Figure 3. The Gallavotti-Nicol` o tree corresponding to Figure 2. We can re-order the expression of S2 (y, z)c,a in terms of these dangerous two-point subgraphs, in the spirit of [1]: S2 (y, z)c,
∈T
0
a
∞ λn+2 = d3 x1 . . . d3 xn n! V n n=0 {av }, {bv }
biped structures external fields clustering tree treesT field attributions B EB structures C over V Ω
1
dw
C f ( , Ω), g( , Ω)
∈T
det
(f, g)∈AF left ×Fleft
{σvj }
C(f, g, {w }) . (IV.54)
V
Main theorem on the self-energy
We have given in the last section an expression for the connected amputated 2point Schwinger function. Now we would like to consider the self-energy Σ(y, z). This quantity can be defined either through its Feynman graph expansion, or through a Legendre transform.
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In the first approach, which we use, Σ(y, z) is given by the same sum (IV.54) than S2 (y, z)c, a but restricted to the contributions which are 1-particle-irreducible in the channel y−z, that is, in which y and z cannot be disconnected by the deletion of a single line. This definition does not look very constructive because in principle we would have to expand out all the remaining determinant in (IV.54) to know which contributions are 1-PI or not. But in the next section we shall see that to extract this information a partial (still constructive) expansion of the determinant is enough. In this section we only formulate our main bound on this connected amputated and one particle irreducible (1-PI) 2-point function or self-energy Σ. Note that, for convenience, we shall simply write in the following “1-PI” to mean: “1particle-irreducibility in the channel y − z”. The sum of all contributions to the self-energy with infrared cutoff r and fixed external positions y and z will be called Σ2 (y, z)≤r . Consider the set Σr of triplets σ ¯ = (i(¯ σ ), s+ (¯ σ ), s− (¯ σ )) with 0 ≤ i ≤ r and 0 ≤ s± ≤ r, also called “generalized sectors”. We can obviously also define the scale distance dσ¯ (y, z) for such triplets as in (II.16), and the index r(¯ σ) = σ ) + s− (¯ σ ))/2 . Then with all the notations of the previous section, the (i(¯ σ ) + s+ (¯ following bound holds: Theorem V.1 There exists a constant K such that: α
|Σ2 (y, z)≤r | ≤ (λ| log T |)2 sup KM −3r(¯σ) e−cdσ¯ (y,z) ,
(V.55)
σ ¯ ∈Σr
α
|y+ − z+ |.|y− − z− |.|Σ2 (y, z)≤r | ≤ (λ| log T |)2 sup KM −2r(¯σ) e−cdσ¯ (y,z) , (V.56) σ ¯ ∈Σr
≤r
|y0 − z0 |.|Σ2 (y, z)
α
| ≤ (λ| log T |) sup KM −2r(¯σ) e−cdσ¯ (y,z) . 2
σ ¯ ∈Σr
(V.57)
For the second equation (V.56), a naive bound would have M −r instead of . So the crucial point is to gain a factor M −r in the bound (V.56). (V.55) M and (V.57) are easy. The next four sections are dedicated to the proof of this theorem. We call a self energy contribution “primitively divergent” if there is no smaller biped in it. The sum of all such “primitively divergent” contributions to the self-energy with infrared cutoff r and fixed external positions y and z is called Σ2,pr (y, z)≤r . We first prove in the next three sections that the bounds (V.56) and (V.57) hold for Σ2,pr (y, z)≤r , then by an inductive argument we extend the bound to the general unrestricted self-energy. The most naive bounds don’t work. Indeed we should optimize power counting and positions integrals separately in the 0 and ± directions in order to bound correctly the effect of the (y − z)± factors in (V.56). But the problem is how to do this constructively. One cannot simultaneously build the three spanning trees that would optimize spatial integrations with respect to the 0 and ± directions, as this −2r
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may typically develop too many loops out of the determinant. The road to solve this problem is to derive not only a 1-PI, but a 2-PI expansion inside each twopoint contribution to renormalize. This expansion can be controlled constructively; then one can optimize over the 0 and ± multiscale analysis, using only the tree T and the additional loops which the expansion has taken out of the determinant. In this way one obtains a better bound than the one obtained naively by simply exploiting a single tree formula as in [1]. This is the key to our problem of the renormalization of the 2-point function.
VI Multiarch expansion Consider the self-energy of the model. The previous tree expansion insured the connexity of the graphs but not their 1 or 2-particle-irreducibility. We are going now to expand out explicitly some additional lines from the determinant, in order to complete the tree T into a 2-PI graph. Nevertheless it is not trivial to ensure that this additional expansion does not generate “too many” terms, or in other words that it is “constructive”. In the following section, we explain in detail this expansion for an expression of the type F = ∈T Cσ() (f ( ), g( )) detleft,T .
VI.1 1-particle-irreducible arch expansion First, we fix some conventions. We consider the tree T connecting all the vertices y, z, x1 , . . . , xn . We distinguish in T the unique path connecting y and z through T , denoted by P (y, z, T ). Each vertex of this path is numbered by an integer starting with 0 for y and increasing towards z, which is the end of the path (with number p). The set of the remaining 2(n + 2) fields and antifields, denoted by Fleft,T = AF left,T ∪ Fleft,T , is divided into p + 1 disjoint subsets or “packets” F0 , . . . , Fp : by definition, an element f ∈ Fleft,T belongs to Fk if and only if k is the first vertex of P (y, z, T ) met by the unique path in T joining the vertex to which f is hooked to y. Figure 4 allows to visualize better the situation. When f belongs to the packet Fk we also say that the packet index of f is k. F2
F0
Fp 00 11 0 1 0 1 0 1 00 11 00 11 0 1 0 1 0 1 00 11 1 0 0 1 00 11 00 11 1 0 0 1 0 1 0 1 00 11 00 11 1 0 0 00 11 0 1 1 0 00 11 11 00 11 00 11 00 1 0 1 11 1 00 0 00 11 0 00 11 0 1 0 1 1 00 11 00 11 00 11 0 1 0 1 0 1 00 11 0 1 00 11 0 1 11 00 0 1 00 11 00 11 1 0 00 11 0 1 0 1 00 11 0 1 00 11 00 11 00 11 0 1 00 11 00 11 00 11 0 1 1 0 00 11 1 0 00 11 0 1 00 11 00 11 11 00 0 1 00 11 00 11 11 00 00 11 00 11 0 1 00 11 00 11 00 11 0 1 00 11 00 0 1 F1 11 0 1 11 00 0 1 00 11 00 11 00 11 00 11 00 11 00 11 0 1 1 0 0 1 00 11 00 11 00 11 00 11 00 11 00 11 0 1 0 1 00 11 00 11 0 1 11 00 00 11 0 1 0 1 00 11 00 11 00 11 11 00 00 11 00 00 11 0 1 0 1 00 11 00 0 1 00 0 1 00 0 1 0011 11 00 00 00 00 11 00 11 00 11 11 0 11 1 0 11 1 00 11 11 00 11 0 11 1 00 11 0 11 1 111 000 1111 0000
0 y
1
2
3
p−1
p z
Figure 4. The tree T and the “field packets” F0 , . . . , Fp .
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In this figure we have represented the external (amputated) propagators by dotted lines, the links of P (y, z, T ) by bold lines, the other links of T by thin lines and at last the remaining fields in the determinant by thinner half-lines. Once the ordered family of subsets of fields F0 , . . . , Fp has been defined, the arch expansion is carried out in the standard way of [11], Appendix B1. Let us recall this expansion here for self-completeness. Among all the possible contraction schemes implicitly contained in detleft, T , we select through a Taylor expansion step with an interpolating parameter s1 those which have a contraction between an element of F0 and ∪pk=1 Fk . Given such a contraction, we call k1 the index of the precise packet joined to F0 by this contraction. Thus we have added to T an explicit line 1 joining F0 to Fk1 . At this stage, the graph obtained is 1-particle-irreducible in the channel y − xk1 (see Figure 5).
1
F2
F0 1 0 0 1 0 1 0 1 0 00 1 0 1 11 00 11 00 11 00 11 00 11 00 11 11 00
0 1 0 1 0000 1111 0 1
0 y
0 1 0 1 1 0 0 1 0 1 0 1 01 1 01 1 0 1 0 0 0 1 0 1 0 1 01 1 0 1 0 1 0 11 0 1 000 1 01 1 0 1 0 0 1 0 0 1 1 0000 1111 0000 1111 1 0 0000 1111 0 1 0000 1111 0 1 0 1 0111 1 0000 1111 01 1 0 111 000 000 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 00001 1111 000 111 0 0 1 0 1 0 0 1 0 1 01 1 0 1 0 1 0 1 0 1 0 1 01 1 0 1 0 1 0 1 0 0 1 01 1 0 0 1 0 01 1
F1
1
2
3
Fp
Fk 1
0 1 1 0 00 11 0 1 011 1 0 1 0 1 0 1 0 1 0 01 1 000 1 11 00 00 11 0 1 0 1 00 11 0 1 11 00 00 00 11 0 1 011 1 0 1 0 1 00 11 11 00 0 1 0 1 0 1 11 0 0011 11 0011 11 111 000 0 1 11 00 0 00 11 00 0 1 0000 1 0 1 0 1 1 0 00 11 00 111 0 1 00 11 0 1 00 11 0011 11 00 11 0 1 00 11 00 11 00 00 11 0011 11 00 11 00 11 00 11 00 00 11 0011 11 00 11 00 11 00 11 00 00 11 00 11 00 11 0011 11 00 11 00 11 00 11 00 11 00 11 00 11 00 0011 11 00 00 11 00 11 0011 11 00
k1
00 11 00 11 00 11 11 00 00 11 00 11 00 11
p−1
0 0 01 1 01 1 0 1 0 1 1 0 0 1 0 0 01 1 01 1 0 1 0 1 00 11 00 0 1 0011 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 11111 0000000 11 00 11
p z
Figure 5. The tree T completed by a first line from the arch expansion.
Then we continue the procedure, testing whether there is a contraction be1 Fk and one of ∪pk=k1 +1 Fk . If there is not, the line from k1 tween an element of ∪kk=0 to k1 +1 of the path P (x, y, T ) is certainly a line of 1-particle-reducibility (i.e., its deletion would disconnect y and z), and therefore the corresponding contraction schemes do not contribute to the self-energy. But on the contrary, if there exists 1 Fk and ∪pk=k1 +1 Fk , we select it and we have the picture of a line 2 between ∪kk=0 Figure 6. The graph T ∪ { 1 , 2 } is clearly 1-particle-irreducible in the channel y − xk2 . Observe that 0 < k1 < k2 therefore, in at most p steps, we shall reach certainly the end vertex z and we shall have a 1-particle-irreducible graph (in the channel y − z). Any final set of m arches derived in this way is called an m-arch system.
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1
F2
F0 0 0 1 01 1 0 1 0 1
0 0 1 01 1 0 1 0 1 01 1 0 1 1 0 0 1 0 0 0 1 01 1 0 1 01 1 0 1 0 1 0 0 1 0 1 1 01 1 0 0 0 1 1 000 11 0 0 1 0000 1111 11 00 1 0 1 0 1 00 11 0000 1111 1 0 00 11 0000 1111 0 1 00 11 0000 1111 10 0 1 0 0111 1 00 11 0000 1111 1 0 1 111 000 000 00 11 0000 1111 000 111 00 11 0000 1111 11 00 000 111 0000 1111 000 111 0000 1111 000 111 00001 1111 000 111 0 0 1 0 0 1 11 0 0 1 0 1 0 1 0 1 0 1 01 1 0 1 0 1 0 1 0 0 1 01 1 0 0 1 0 01 1
F1
1 0 0 1 0 1
0 y
1
2
3
2
Fp
011 1 0 1 0 1 00 0 1 0 1 0 1 00 0 1 0 1 0 1 01 1 011 1 11 00 0 1 00 11 0 00 11 0 1 11 00 00 00 11 0 1 011 1 0 1 0 1 00 11 00 11 0 1 0 1 1 0 0 1 0 1 001 11 00 0 1 0 1 1 0 0011 11 0011 01 1 111 000 0 0 00 11 0 1 00 11 0011 11 00 11 0 1 00 11 00 11 00 00 11 0011 11 00 00 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 11 00 00 11 00 11 0011 11 00 11 00 00 11 00 11 0011 11 00 11 0011 11 00 00 11 00 11 00
k1
k2
00 11 11 00 00 11 00 11 00 11 00 11 00 11
p−1
0 1 0 1 0 1 0 1 1 0 0 1 0 0 01 1 01 1 0 1 1 0 00 11 00 0 1 0011 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 0011 11 00 11 00 11 00
p z
Figure 6. The tree T completed by two lines from the arch expansion. We obtain the 1-PI part of the determinant as: !m " m 1 dsr C(fr , gr )(s1 , . . . , sr−1 ) F1−PI = r=1 m−arch systems r=1 0 (f1 ,g1 ),...,(fm ,gm ) with m≤p
∂ m detleft, T {sr } . (VI.58) m r=1 ∂C(fr , gr )
The expansion respects positivity of the interpolated propagator at any stage, because all sr interpolations are always performed between a subset of packets and its complement, hence the final covariance as function of the sr parameters is a convex combination with positive coefficients of block-diagonal covariances. This ensures that the presence of the sr parameters does not alter Gram’s bound on the remaining determinant, which is the same than with all these parameters set to 1 ([10]–[11]). Furthermore it is constructive in the sense that it does not generate any factorial in the bounds for the sum over all derived arches. Here is a subtlety. Once the departure and arrival fields joined by the arches have been fixed (which costs at most 4n ), the arrival fields are determined because their packet indices are strictly growing. But the departure fields are not, and in principle this could create a constructive problem. For example, if the line 1 joins F0 to Fk1 , it is possible for the second one, 2 , to join F0 to Fk2 (see Figure 7). Remark that in this case a posteriori 1 is useless. With three arches, an arch system such as Figure 8 shows the same phenomenon, in the sense that a posteriori 2 is useless. This is not a great disadvantage, because in spite of this lack of minimality, the expansion can indeed be controlled in a constructive way. The reason is that
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2 1 F0
0 0 1 1 0 01 1 00 0 1 0 11 1 1 110 00 00 11 00 11 00 11 00 11 00 11 11 00
1 0 0 1 0 1
0 y
00 11 0 1 1 0 00 11 0 1 0 1 0 1 0 1 00 11 01 0 01 1 0 1 00 11 0 1 00 11 0 001 11 0 1 1 00 11 0 0 001 11 0 1 1 1111 0000 0000 0 1111 0 1 0000 1111 0 1 0000 1111 10 0 1 0 11 00 000 111 0111 1 0000 1111 1 0 1 11 00 000 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 00 11 0 1 00 11 0 1 11 0 00 11 0 0 1 0 1 0 1 0 1 10 0 1 1 0 1 0 00 1 1 0 0 1 1 0 1 0 1
F1
1
2
Fp 00 11 1 0 00 11 1 000 0 0 111 00 11 1 0 11 0 0011 11 00 0 1 00 11 00 0011 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00
3
0011 11 00 11 00 0 1 00 11 00 11 11 00 00 11 0 1 00 11 00 11 000 111 00 11 00 11 00 11 00 11 0 1 00 11 000 111 00 11 00 11 00 11 000 111 00 11 00 11 000 111 00 11 00 11 000 111 00 11 1111 0000 11 00 000 111 00 11 000 111 11 00 000 111 000 111 000 111 000 111 000 111 000 111 00 11 1 0 000 111 00 11 1 0 000 111 00 11
k2
k1
11 00 00 11 0 1 00 11 0 1 00 11 0 1 00 11
p−1
00 11 0 111 00 11 00 0 00 1 11 00 00 11 11 0 1 11 00 00 11 00 11 0 1 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00
p z
Figure 7. A pair of arches which is not minimal from the 1-PI point of view. 3
1
2 0 1 0 1 1 0 0 1 0 0 1 01 1 01 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 01 1 01 0 1 0 01 1 1 0 0 1 0 1 0 1 0 1 0000 1111 11 00 1 0 0 0000 1111 0000 1 1111 1111 0 0000 1111 0 1 0 000 111 000 1 0 0000 1111 01 1 0 1 000 111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0 1 0 1 0 1 0 0 1 0 1 01 1 0 1 0 1 0 1 0 1 0 1 00 1 1 0 1 0 1 1 0 1 00 1
1 0 1 0 0 1 0 1 1 0 1 0 0 1 11 00 00 11 00 11 00 11 00 11 00 11 11 00
0 1 0 1 0000 1111 0 1
0 y
1
2
3
0 1 0 1 0 1 0 1 0 1 00 11 0 1 0 1 1 0 0 1 00 0 0 1 00 11 11 0 011 1 11 00 0 1 00 11 0 1 00 11 0 1 11 00 00 00 11 0 1 011 1 1 0 0 1 001 11 00 11 0 1 0 1 0 1 1 0 00 11 00 11 000 11 00 0 0 1 00 11 00 0 1 0111 1 0 1 01 1 0011 11 0011 11 000 111 0 1 0 001 11 0 1 00 11 00 11 00 0 00 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 11 00 00 11 0011 11 00 11 00 11 00 11 00 00 11 00 11 0011 11 00 11 11 00 00 00 11 00 11 0011 11 00 11 00 11 0000 11 00 11
k2
k1
00 11 00 11 00 11 11 00 00 11 00 11 00 11
p−1
0 1 0 1 1 0 0 1 0 1 0 1 01 1 01 1 0 0 0 1 0 1 0 1 0 1 0011 11 00 11 0 1 00 11 00 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 11111 00000 0011 11 00
p z
Figure 8. Another example of a “non-minimal” system of three arches. the arches for which the departure fields indices do not grow are damped by small s interpolation parameters, so that the result is indeed bounded by K n [11]. More precisely the dependence in the sr parameters in front of each arch system is a q m monomial r=1 srr,m−arch , so that we have: m m m q C(fr , gr )(s1 , . . . , sr−1 ) = C(fr , gr ) srr,m−arch . (VI.59) r=1
r=1
r=1
The reader can check that the integer qr,m−arch ≥ 0 is the number of arches which fly entirely over the rth arch, making it useless.
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Lemma VI.1 There exists some numerical constant K such that (for n ≥ 1): m m p 1 q dsr srr,m−arch ≤ c.K n . (VI.60) 0 m−arch systems m=1 r=1 r=1 (f1 ,g1 ),...,(fm ,gm ) with m≤p
Proof. The proof is identical to [11], Lemma 9. We reproduce it here for completeness. Consider Fkr the arrival packet of the rth arch, which joins the field fr to the field gr ∈ Fkr . The set of possible departure packets to which fr must belong is Er = F0 ∪ F1 ∪ · · · ∪ Fkr−1
(VI.61)
We also define ei = |Ei − Ei−1 | as the number of fields and antifields in Ei and not in Ei−1 . The sum over all m-arch systems which we have to bound is p
m=1 0
gr ∈Fk r r=1,...,m
1
1
ds1 . . . 0
dsm 0
m
fr ∈Er r=1,...,m
r=1
q
srr,m−arch .
(VI.62)
We start observing that
m
fr ∈Er r=1,...,m
r=1
q
srr,m−arch ≤
m
ar (s1 , . . . , sr−1 )
(VI.63)
r=1
where ar is defined inductively by a1 = e 1
and ar (s1 , . . . , sr−1 ) = er + sr−1 ar−1 (s1 , . . . , sr−2 ).
To see this we remark that we have e1 choices to choose f1 . In the same way, we have e2 choices to choose f2 if it does not hook to F1 . If it does hook to F1 , we have e1 = a1 choices, but we also have a multiplicative factor s1 coming from q s11,m−arch . This iterates into (VI.63). Remark that (VI.63) is an overestimate, not an equality, as, once f1 is fixed we have only e1 − 1 choices for f2 if it falls in the F1 packet, and so on. We have also 1 m m q dsr ar (s1 , . . . , sr−1 ) ≤ e r=1 er . (VI.64) 0 r=1
r=1
Indeed this follows from the inductive use of 1 1 (as + b)ds ≤ eas+b ds ≤ (1/a)ea+b , for a > 0, b > 0. 0
0
(VI.65)
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Now, as er = |Er \Er−1 |, we have m
er ≤ 2(n + 2)
(VI.66)
r=1
since 2(n + 2) is the total number of remaining fields (after extraction of the tree) in the amputated two-point function considered. Finally it is easy to check that 1 ≤ Kn . (VI.67) m
gr ∈Fr r=1,...,m
0
Indeed
1=
gr ∈Fr r=1,...,m
m
|Fkr )| < 2(n + 2)
(VI.68)
r=1
and 0
{σvj }
m r=1
biped structures external fields clustering tree treesT field attributions B EB structures C over V Ω
m−arch systems
(f1 ,g1 ,...,(fm ,gm )) with m≤p
1
dw
0
∈T
C(fr , gr )(s1 , . . . , sr−1 )
m
r=1
0
1
dsr
Cσ() (f , g )
∈T
∂ m detleft,T {w }, {sr } . ∂C(fr , gr )
m r=1
(VI.69)
The result of the first expansion is however complicated and it is convenient to select from the arch system an optimized sub-system, called a minimal 1-PI arch system. This defines a map φ which to any m-arch system A associates a minimal 1-PI m ¯ arch-system M. To define this map we select as first arch of M the unique arch in A which starts in F0 and ends in Fq1 with q1 maximal. If q1 = p we are done. If q1 = p, we 1 select as second arch of M the unique one in A which starts in ∪qk=0 Fk and ends in Fq2 with q2 maximal, and so on. Starting from the tree T , we have now a minimal arch system of lines which completes it into a 1-PI graph. For simplicity, let us first describe these graphs
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when the arch system M has no “coinciding packet” (i.e., no Fk contains more than one arch extremity). We have: • the path P (y, z, T ), • the m ¯ arches (f1 , g1 ), . . . , (fm ¯ , gm ¯ ) of M completed by the unique path joining fr to xk r and the unique path joining gr to xkr through T , • the remaining links which form subtrees of T . These three kinds of links are illustrated on Figure 9, where the links of P (y, z, T ) are drawn in bold lines, those of the completed arches in “normal” lines and the remaining ones in dashed lines.
1 0 0 1 0 1 0 1 0 1 0 1
1 0 0 1 0 1 0 1
y
0 0 1 00 11 01 1 0 1 0 1 00 11 0 01 1 0 1 0 1 0 1 1 00 11 0 1 0 1 0 00 11 0 1 00 11 0 1 00 011 1 0 1 1111 0000 00 11 000011 1111 00 0000 1111 01 1 0 1 1 1 0 0000 1111 0 1 0 0000 1111 00 1 0 1 00 11 0000 1111 0000 1111 0000 1111 0000 1111 00 11 0 1 0000 1111 00 11 0 1 0 1 00 11 0 01 1 00 11 0 1
0 1 1 0 1 0 0 001 11 00 11 00 11 00 11 00 11 00 11 11 00
10 0 1 0 0 1 1
1 0 1 0
11 00 11 00
00 11 00 11 0 1 111 000 00 11 0 1 1 0 000 0 111 000 00 11 0111 1 01 1 000 111 0 1 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111
00 11 00 11 00 11 00 11 0 1 00 00 11 00 11 11 11 00 00 11 0 1 00 11 00 11 00 11 0 1 00 11 00 11 00 11 0 1 00 11 1 0 001 11 0 11 00 00 11 0 1 001 11 000 111 0 000 111 000 111 000 111 000 111 000 111 00 11 00 11 000 111 00 11 00 11 1 0 000 111 00 11 00 11 1 0 000 111 00 11 00 11
00 11 11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11
00 11 11 00 11 00 11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 000 111 00 11 000 111 000 111 000 111 000 111 000 111 000 111 000 111
z
Figure 9. The three kinds of links after a first arch expansion. When the packet indices are all different, the structure of the minimal 1PIarch-system is therefore the one represented on Figure 10:
y
z
Figure 10. The minimal 1-PI structure without the remaining links of T . which can also be represented as a kind of “fish”, whose borders are shown with corresponding arrows in Figure 11. Now let us examine the case where the minimal 1PI-arch-system has “coinciding packets”, i.e., where the end of some arch and the origin of the next one belong to the same Fk . We shall distinguish various cases, according to the way the two arches are branching to Fk . From the vertex xk , apart from the two links
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Renormalization of the 2-Point Function of the Hubbard Model
11 00 00 11
423
11 00 00 11 111 000
111 000
Figure 11. The “fish” structure. of P(y, z, T ), hook two half-lines or lines which are potentially the beginning of two subtrees of T . First, consider the case where xk has two half-lines. Then the branching of the arches is like on Figure 12 (case 1):
xk Figure 12. The branching of two arches when xk has two half-lines (case 1). If xk bears a half-line and a subtree of T , we must distinguish two sub-cases: both arches can hook to the subtree (cases 2), or only one of them can hook to the subtree whereas the other one hooks to the half-line (cases 3). These two situations are pictured on Figures 13 and 14.
xk
Figure 13. Two arches branching on the same subtree of T (case 2).
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xk
Figure 14. One arch branching on a subtree and the other one on the half-line of xk (case 3). At last, if xk is the root of two subtrees of T , we have two sub-cases: both arches can hook to the same subtree (case 4), or each of them can hook to distinct subtrees (case 5). These two sub-cases are represented on Figures 15 and 16.
xk
Figure 15. Two arches branching on the same subtree, the other subtree being not touched (case 4).
xk
Figure 16. Two arches branching on different subtrees (case 5). The inspection of these five cases reveals that the “fish structure” of Figure 11 iterates. Cases 1, 3 and 5 induce a pinch leading to a “new fish” separated from
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the previous one by a vertex of reducibility (1-VR). Cases 2 and 4 do not induce any pinch but simply enlarge the “fish”. In the end we obtain a sequence of “fishes” separated by vertices of reducibility, as in Figure 17.
y
z
Figure 17. The general 1-PI structure. This object is called a fish structure, and it is made of an upper and a lower path, together with middle bars and middle 1-VR vertices. Any vertex on the upper or lower path which is neither a middle 1-VR vertex nor on a middle bar is called a ladder vertex. In the next section we use the minimal 1PI-arch-system as a guide for a 2-PI expansion, just like the initial tree T was the guide for the 1-PI expansion.
VI.2 2-particle-irreducible arch expansion The self-energy Σ2 (y, z) is defined as the sum of the 1-PI contributions, but it has automatically a stronger property in our model: it is 2-PI and one-vertex irreducible (1-VI) in the yz channel. This is just a consequence of the fact that all vertices in our theory have coordination 4. To take advantage of this fact, we devise an additional arch expansion, which derives explicitly more lines out of the determinant. These additional lines, which insure 2-PI, are necessary for the proof of Theorem V.1. Nevertheless, we must be careful in performing this second arch expansion to respect again the positivity property so that Gram’s bound is not deteriorated, and also to check the analog of Lemma VI.1, that is the constructive character of the expansion. A naive approach could consist in keeping the definition of the previous “field packets” Fk (in which, of course, the fields used in the first expansion are deleted), but this would not select exactly the 2-PI contributions. For example, if the first arch of the first expansion is of the type of Figure 18, that is, if the starting field is not hooked directly to the vertex y, the second arch expansion could arise as in Figure 19, and the two cuts indicated on the picture would still disconnect y and z. In order to avoid this difficulty, we need to use the general structure of the graph obtained after the first arch expansion. to define the new “field packets” and there is a small additional difficulty, which is that these packets are not totally ordered but only partially ordered in a natural way. The new definition of the field packets is the following: a field packet contains either all the fields whose path to y first meets the fish structure in a given middle
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1
1 0 1 00 0 0 1 00 1 11 1 1 0 0 001 11 00 11 00 11 00 11 00 11 00 11 11 00
1 0 0 1 0 1
0 y
00 11 0 1 1 0 00 11 0 11 0 1 0 0 1 00 11 01 1 0 00 11 1 0 1 0 00 11 0 1 00 11 0 00 001 11 011 1 00 11 1 0 0 1 001 11 0 0 1 1111 0000 0 0000 1 1111 0000 1111 1111 0 0000 1111 0 1 0 1 00 11 000 1 0 0000 1111 01 1 0 00 11 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 00 11 0 1 00 11 0 1 11 0 00 11 0 0 1 0 1 0 1 0 1 10 0 1 1 0 1 0 00 1 1 0 0 1 1 0 1 0 1
1
2
3
00 11 0 1 00 11 0 1 00 11 00 1 0 0 111 000 00 11 0 1 0011 11 00 11 01 1 0 1 00 11 00 0011 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 00 11 00 0011 11 00 11 00 11 00 0011 11 00 11
0011 11 00 11 0 1 00 11 0000 11 11 00 00 11 0 1 00 11 00 11 000 111 00 11 00 11 00 11 00 11 0 1 00 11 000 111 00 11 00 11 00 11 000 111 00 11 11 00 000 111 00 11 111 000 00 11 0000 1111 000 111 00 11 0000 1111 00 11 000 111 00 11 000 111 00 11 000 111 000 111 000 111 000 111 000 111 000 111 00 11 00 11 1 0 000 111 00 11 00 11 1 0 000 111 00 11 00 11
00 11 11 00 0 1 00 11 0 1 00 11 0 1 00 11
p−1
k1
111 00 00 11 00 11 000 11 00 11 0 11 1 00 11 00 11 00 0 1 00 11 00 11 00 11 0 1 00 11 00 11 0011 11 00 11 00 11 00 0011 11 00 11 00 11 00 00 11 00 0011 11 00 11
p z
Figure 18. An arch not hooked directly to y. 1
2
0 1 0 1 1 0 0 1 1 0 0 1 0 00 1 1 0 11 00 11 00 11 00 11 00 11 11 00 11 00 0 1 0 1 0 1 0 1 0000 1111
0 y
11 00 0 01 1 0 1 0 1 00 11 0 1 0 1 0 1 0 00 11 01 01 11 00 01 1 0 1 00 11 0 1 00 11 0 1 11 00 11 0 1 000 1 001 11 0 1 0 00 11 0 0 1 0000 1111 1 0000 1111 1 0 0 1 0000 1111 0 1 0000 1111 10 0 1 0 11 00 000 111 0111 1 0000 1111 1 0 1 000 11 00 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 00 11 0 1 0 1 00 11 0 1 01 1 00 11 0 0 1 0 1 0 1 01 1 0 1 1 0 0 1 0 0 1 01 1 0 0 1 0 01 1
1
2 cuts
3
0 11 1 00 0 000 111 00 11 1 0 11 0 0011 11 00 11 0 1 000 111 00 11 1 0 0 1 00 11 0 1 0000 11 00 11 00 11 00 0011 11 00 11 00 11 00 11 00 11 00 11 0011 11 00 11 00 11 0000 11 00 11 00 11 00 11
k1
0011 11 00 11 00 11 00 0 1 00 11 00 11 11 00 0 1 00 11 00 11 00 11 000 111 00 11 11 00 00 11 00 11 0 1 00 11 000 111 11 00 00 11 00 11 00 11 000 111 00 11 00 11 000 111 00 11 1111 0000 00 11 111 000 000 111 00 11 00 11 1111 0000 000 111 00 11 11 000 111 00 000 111 000 111 000 111 000 111 000 111 00 11 00 11 000 111 00 11 00 11 0 1 000 111 00 11 00 11 0 1 000 111 00 11 00 11
11 00 0 1 00 11 0 1 00 11 0 1 00 11 0 1 00 11
p−1
00 11 0 001 11 00 11 0 11 1 00 11 11 00 00 00 11 0 1 00 11 00 11 0 1 11 00 00 11 00 11 00 0 1 0011 11 00 11 00 11 00 11 0011 11 00 11 00 11 00 00 11 00 11 0011 11 00 11 00 11 00 11111 00000
p z
Figure 19. The beginning of a wrong 2-PI arch expansion.
bar, or all the fields whose path to y meets the fish structure at a given ladder vertex. In the first case we say that we have a “bar packet”, in the last case we have a “ladder packet”. Finally we could also add packets for each middle reducibility vertex, also called bar packets; although they do not contain any field, it is convenient to introduce them for consistency of the partial ordering defined below. These packets are shown in Figure 20 as dotted ellipses: in this figure there are 6 “bar packets” and 9 “ladder packets”. Furthermore we have an ordering on these packets, but it is only a partial ordering, noted ≺. If we put arrows from y to z on the two outer paths in the fishes, packets A and B satisfy A ≺ B if and only if one can go from A to B by a path which does not run against any arrow.
Vol. 6, 2005
Renormalization of the 2-Point Function of the Hubbard Model
B
D
0110 1010 1010 1010 10
y A
427
z
C
Figure 20. The partial ordering in a multi-fish structure. For instance in Figure 20, we have A ≺ B ≺ D and A ≺ C ≺ D but there is no relation between B and C. To grasp this partial ordering better, we can label the bar packets as G0 , G1 , . . . , Gq and label the ladder packets between bar packets r and r + 1 as Gr,a , Gr,b , . . . on the upper path and Gr,a , Gr,b , . . . on the lower path. This is illustrated on Figure 21:
G0,a
111 000 000 G0 111 000 111 00 11 00 11 00 11
y
G1,a 00 11 11 00 000 111 00 11 00 11 000 00 111 11 00 11 000 111 G3 00 11 000 111 00 11 000 111 00 11 00 11 000 111 00 11 G2 G1 00 11 000 111 00 11 000 111 00 11 000 111 000 111 00 11 000 111 000 111 00 11 000 111 000 G2,a 00 111 11 000 111 000 111 000 000 111 111 G1,a
G3,a
111 000 000 111 000 111
11 00 00 11 00 11
G3,b 11 00 00 11 00 11
00 11
000 G3,a 111 11 000 00 111 00 11 G3,b G3,c
1 0 0 1 0 1 0 1 G5 000 111 0 1 000 111 0 1 G4 111 000 0 1 0 1 z 0 1 0 1 0 1
Figure 21. The numbering of the field packets for the 2-PI arch expansion. Once this is done, the 2-PI expansion is carried out in a similar way than before, but with a few modifications. We introduce successive interpolation parameters s1 , s2 , . . . . The first one tests the packet G0 with the complement, that is the set of all later packets in the partial ordering. Hence the first Taylor expansion step creates a first arch joining this packet G0 to a bar packet Gr or to a ladder packet Gr,i or Gr,i called the first arrival packet. Such an arch insures 2-PI only for the block of all packets which are smaller or equal than the arrival packet in the sense of the partial ordering ≺. So at second stage we have to launch the second arch from this 2-PI block to the set of all the remaining packets not in this block, and so on. At any given stage of the induction, the 2-PI block is a “fish-commencing section”, that is either the set of packets smaller or equal to a single given packet (of any type r, (r, i) or (r, i )), or the set of packets smaller or equal to one among
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two ladder packets (r, i) and (r, i ) with same index r, one on the lower and the other on the upper part of the fish. From this block the next arch is launched to the remaining packets. This defines uniquely inductively our expansion.
5
6
2 3
y
z
4
1
Figure 22. A possible arch system for the 2-PI arch expansion. Now the result is a system of arches which insures 2-PI from y to z. On Figure 22 we have shown a possible example of such a system. The arches are represented by bold lines 1 , 2 , . . . and the corresponding successive 2-PI blocks are shown by the successive larger and larger dotted surrounding contours. The final result is therefore given by the same kind of formula than VI.69. If we call the second arch system an m -arch system, we have: ∞ λn+2 d3 x1 . . . d3 xn Σ(y, z) = n! Λn biped structures n=0
{av }, {bv }
external fields clustering tree treesT field attributions EB structures C over V Ω
∈T
dw
0
m r =1
1
Cσ() (f , g )
1
dsr
0
m
∂
m r=1 ∂C(fr , gr )
m−arch systems
(f1 ,g1 ),...,(fm ,gm )
m 0
1
m −arch systems
(f ,g ),...,(f ,g ) 1 1 m m
dsr
m
C(f r , g r )(s1 , . . . , sr −1 )
r =1
detleft,T m r =1
C(fr , gr )(s1 , . . . , sr−1 )
r=1 m+m
{σvj }
r=1
∈T
B
∂C(fr , gr )
{w }, {sr }, {sr }
(VI.70)
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In such a formula the nested sums are other all compatible possibilities, in particular the m -arch system has to be one of the possible ones that can arise using the fish structure of the m-arch system as the guide for the second expansion. This formula displays explicit 2-PI. Using the fact that vertices have coordination four, it also displays explicit 1-VI. We have to check that it also respects positivity and remains constructive, i.e., satisfies an analog of Lemma VI.1. The expansion respects again positivity of the interpolated propagator at any stage, for the same reasons than the first one, namely all the sr interpolations are always performed between a subset of packets and its complement, so the final covariance as function of the sr parameters is a convex combination with positive coefficients of block-diagonal covariances. This ensures that the presence of these sr parameters again does not alter Gram’s bound on the remaining determinant. We need finally to check that the expansion is still constructive. Arches system such as those of Figure 22 obey some constraints. For two arches i and j with i < j, the arrival packets Ai and Aj cannot coincide and it is not possible to have Aj ≺ Ai , hence arrivals respect the partial ordering ≺. Furthermore let us say that the arch is of upper type if the arrival packet is a bar packet with index r or an upper ladder packet (r, i) and is of lower type if the arrival packet is a lower ladder packet with index (r, i ). Then the set of arrival points for upper type arches is strictly ordered under ≺, and so is the set of arrival points for lower type arches. Hence we can fix separately the set of arrival fields, the set of departure fields, and for each arch for r = 1, . . . , m whether it is an upper or lower arch. This choice costs at most 42(n+2) 2m ≤ 25(n+2) , since the total number of fields is at most 2(n + 2) (this is not an optimal bound!). Once this choice is fixed we know exactly the arrival points gr for each arch. Then the choice of the corresponding departure points is determined using the sr parameters exactly as in Lemma VI.1, where the Er are now the sets of strictly growing “commencing sections”, that is the successive regions surrounded by dotted contours in Figure 22, and the numbers ei = |Ei − Ei−1 | are now the total number of fields hooked to the region between two successive contours with labels i − 1 and i. Therefore Lemma VI.1 also holds for the m arch system.
VI.3 Three disjoint paths By the previous double arch expansion, we have explicitly displayed the 2-PI structures contributing to the self-energy. The advantage of this expansion is that we have now at our disposal more explicitly derived links, which can be used to bound # in a better way the integrations v dxv,0 dxv,+ dxv,− . Integrating the vertices positions in the standard way using the decay of the lines of a single tree connecting all the vertices is (apparently) not sufficient to obtain the requested bounds of theorem V.1. Thanks to the 2-PI structure extracted by the double arch expansion, we are going to forge a better scheme of integration.
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We need a theorem (in fact, two versions of the same theorem) known as Menger’s theorem. Roughly speaking, it states that in a p-particle-irreducible graph, there exists (at least) p + 1 line-disjoint paths joining two given vertices. A cautious statement of this result is the following one: Theorem (“edge version” of Menger’s theorem): Let G be a graph, u and v two distinct vertices of G. Suppose that u and v cannot be disconnected by the deletion of p lines (edges) of G, for p ∈ N. Then there exists p+1 line-disjoint paths joining u and v through G. Two (or more) paths P1 and P2 are said line-disjoint if P1 ∩P2 = ∅ (remember that a path is by definition a set of lines). We stress the fact that these paths whose existence is insured by the edge version of Menger’s theorem may go across some identical vertices; in other words, they are not necessarily vertex-disjoint, even if we take away the end vertices u and v. But there exists another version of Menger’s theorem: Theorem (“vertex version” of Menger’s theorem): Let G be a graph, u and v two distinct vertices of G. Suppose that u and v cannot be disconnected by the deletion of p vertices (p ∈ N). Then there exist p + 1 internally vertex-disjoint paths joining u and v. We say that two paths P1 and P2 are internally vertex-disjoint if P1 and P2 , once deprived from their end vertices, have no vertex in common. For more details about these two versions of Menger’s theorem, the reader may consult [2] or any textbook on graph theory. Although Menger’s theorems are very simple, their proof is quite subtle. They can be seen as corollaries of a famous powerful theorem of graph optimization, the so-called “max flow-min cut theorem” [2]. It is easy to give examples of 2-PI graphs for which the previous theorem naturally holds, but in which it is impossible to exhibit three vertex-disjoint paths, for instance the graph of Figure 23:
x
u
v
Figure 23. A 2-PI graph with no triplet of vertex-disjoint paths joining u and v. Note also that the theorem does not state that the set of the paths is unique in general. Unicity can be insured only in the very special case of graphs having the (rather trivial) structure of Figure 24. But if the graph G−{u, v} has vertices linked to more than two neighbors it is possible to find several sets of three line-disjoint paths connecting u and v. Finally,
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u
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v
Figure 24. A graph with a unique triplet of line-disjoint paths joining u and v. we remark that these paths cannot be determined naively and independently of the other ones. For example, in the graph of Figure 25 , if we choose the first two
u
x
y
v
Figure 25. A 2-PI graph for which Menger’s theorem is not trivial. paths as being {{u, v}} and {{u, x}, {x, y}, {y, v}}, we cannot find a third one. Thus the result of the theorem is quite subtle and not totally obvious. The set of lines we derived explicitly thanks to our initial tree expansion and our two successive arch expansions is by construction 2-PI in the channel y − z. Then a straightforward application of the edge version of Menger’s theorem insures that, if we call G the graph whose vertices are V and lines those of T plus the ones explicitly derived by the two arch expansions, there exist (at least) 3 line-disjoint paths P1 , P2 and P3 joining y to z. From now on for vertex integration purposes we use only the the lines in L = T ∪ P1 ∪ P2 ∪ P3 , hence forget any arch line not in P1 ∪ P2 ∪ P3 and the remaining fields in the determinant or remaining lines. Remark that the union T ∪ P1 ∪ P2 ∪ P3 is not necessarily disjoint, since some lines of the Pi ’s may belong to T .
VII Ring construction In this section the scales of the lines enter the picture. Out of the lines of L we shall extract a subset, called a ring, which is the union of two line-and-vertex-disjoint paths from y to z. This ring has to satisfy Lemma VII.1 below and its extraction depends therefore on the Gallavotti-Nicol` o tree structure associated to the scales assignments over all lines and fields. We consider therefore the forest F of those connected parts or nodes Γ of the Gallavotti-Nicol` o tree associated to the scale decomposition (including the initial bare vertices, with four legs). The full two-point contribution G that we analyze is itself such a node Γ0 = G. Recall that for any such Γ and any pair of its external
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legs, there exists a unique path in T ∩ Γ joining the vertices to which these two external lines are hooked (this path being eventually empty when both lines hook to the same vertex!). We call this path the “tree shortcut” for the pair. This is because T ∩ Γ is a tree of Γ. Since we are studying primitively divergent two-point subgraphs, any Γ except G itself has at least 4 external legs. (If Γ contains y or z, we count the corresponding external lines of G as external legs of Γ.) Lemma VII.1 There exists a ring R ⊂ L which is the union of two line-and-vertexdisjoint paths from y to z, with the additional property that for any Γ ∈ F, at least two external legs of Γ are not in the ring R. Proof. An element Γ is called a “cut” if removing it separates y from z, or in other words if every path in L from y to z touches Γ. It is called “contractible” if it is not a cut. We consider the set S of all maximal contractible elements in F (by our convention they can be ordinary bare vertices). Elements of S must be all disjoint by the forest character of F . We reduce each element of S to a point, that is we ignore the interior of any element of S, and keep all the elements of S plus all the lines and determinant fields attached to them connected as before. In this way we obtain a new graph G , which has generalized vertices with 4 legs or more, in particular it has one such vertex for each element of S. It must still have three line-disjoint paths P1 , P2 and P3 , made of those lines in P1 , P2 and P3 which were not internal to any contractible element of F . The graph G is therefore still 2-lines irreducible in the channel y → z. 1 0 0 1
1 0
1 0
1 0
1 0 0 1
11 00 11 00 1 0
11 00
1 0
11 00
The graph G
1 0 11 00 00 11
ordinary vertices contractible subgraphs cuts
11 00 00 11 00 11
11 00 11 00
11 00 00 11 00 11 00 11
1 0
111 000 000 111 000 111 000 111
1 0 1 0
11 00 00 11 00 11
1 0 1 0
reduced vertices cuts
The reduced graph G’
Figure 26. The process of contraction. G is also one-vertex irreducible in the channel y → z, since by definition for any vertex v of G distinct from y and z there is a path in L from y to z which avoids v, so the corresponding reduced path in G also avoids the vertex v.
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By the vertex-version of Menger’s theorem, there is therefore a ring R ⊂ G in this graph, namely a subset of lines which is the union of two vertex disjoint paths R1 and R2 from y to z. We consider now the graph G − R . It must connect y to z. Otherwise there would be a connected component C(y) of G − R containing y and not z, and removing the two last exits of R1 and R2 from that component would disconnect G , hence G would not be 2-PI in the channel y → z. Therefore there exists a path R3 from y to z in G entirely line-disjoint from the ring R . 11 00 00 11 00 11
1 0 0 1
ring lines
1 0 11 00 00 11 00 11
11 00 00 11
11 00 00 11 00 11 00 11 00 11
the third path
Figure 27. The ring in G . We complete the ring R of G into a true ring R of G by adding, for every non-bare vertex of G touched by the ring the shortcut between the entrance and exit in T . Clearly this defines R ⊂ L in a unique way. Let us check that R has the desired property. It is obvious for Γ’s which are contractible. Indeed • either they are maximal contractible, in which case they are touched by at most only one of the two vertex-disjoint paths R1 and R2 of the ring, and the number of their external legs in the ring R is at most two, the entrance and exit of that path. We are done, since Γ has at least four external legs, of which only two belong to R (the ones in R − R are internal to Γ or disjoint from Γ). • or Γ is not maximal contractible, hence strictly inside a reduced vertex of G . Then recall that the ring R is made of a corresponding tree shortcut in T . Again it can touch Γ only twice (it cannot enter and reexit; this is due to the key property of T , whose restriction to any Γ node in the GN tree is a spanning tree in Γ). We conclude in the same way. Therefore we have to consider Γ’s which are cuts. But such Γ appear as subgraphs Γ in G which must be still cuts of G . Therefore they must be touched by R3 . Following R3 , its first entrance into Γ (when y ∈ Γ) and its last exit out of Γ (when z ∈ Γ) give external legs of Γ which do not belong to R , hence two external legs of Γ which do not belong to R.
VII.1 Regrouping ring sectors We shall now regroup some sectors for the ring lines in a way compatible with the vertex momentum conservation rules and the propagator’s decay, keeping the tree
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T and the clustering tree structure C fixed. This is necessary to adjust correctly the logarithmic power counting in the next section, and perform the sums over the auxiliary l indices of the ring lines. The reader who wants solely to understand the true power counting can skip this subtlety. For any ring vertex we fix with a finite factor which pairs of legs collapse in the + and in the − direction in the sense of [1] and Proposition 1 above. Recall that an index collapses, e.g., in the plus direction when it is the smallest at the vertex, up to plus or minus 1. Now consider any ring line not in the tree T , joining vertices v and v , which does not collapse in any direction at any end. For such a line we want to sum up all e ring sectors s+ ( ) or s− ( ) into extended sectors σ± ( ). What are the constraints on the range of s+ ( ) or s− ( ) if we keep the tree T with all its indices and the structure C fixed? Let us introduce s±,v , s±,v as the value of the collapsing indices at v or v plus two. The condition of no collapse means that s± ( ) ≥ max{s±,v , s±,v } = s±,v,v .
(VII.71)
Moreover we have the constraint s+ ( ) + s− ( ) ≥ i − 2.
(VII.72)
where i is the main i index of line . The condition that the line is not in T means that its r index is larger or equal to the smallest r index at which v and v fall into a single Γ component of C. Let us call r( , T ) this smallest value. Since r = (s+ + s− + i)/2 the condition is (VII.73) s+ ( ) + s− ( ) ≥ 2r( , T ) − i and we can regroup (VII.72) and (VII.73) into the single condition s+ ( ) + s− ( ) ≥ ¯i( , T ) − 2
(VII.74)
with ¯i( , T ) = max{i − 2, 2r( , T ) − i}. So we regroup the following packets of sectors: a) If s+,v,v + s−,v,v ≥ ¯i( , T ), the condition (VII.71) imply (VII.74) and we can forget (VII.74). We regroup all sectors satisfying (VII.71) into an enlarged sector σ e ( ) = (i, se+,v,v , se−,v,v ) where an enlarged index se+ or se− means that the corresponding slice cutoff vs+ (r) or vs− (r) in (II.14) is replaced by the “sum-of slices” cutoff u(M 2s+ r) or u(M 2s− r)7 . The lines of R − T in this case form a set called Ra . b) If s+,v,v + s−,v,v < ¯i( , T ) = i − 2 ≤ 2r( , T ) − i, or c) if s+,v,v + s−,v,v < ¯i( , T ) = 2r( , T ) − i < i − 2; in both cases b and c, we pick one preferred direction, say the + direction and regroup the sectors into 7 Remark
that indices in (II.14) s+ = i or s− = i are already enlarged in this sense.
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– one sector extended in both directions σ e ( ) = (i, se+,v,v , se− = ¯i( , T ) − s+,v,v ) – a certain number (in fact ¯i( , T )− s+,v,v − s−,v,v ) of sectors extended only in direction +, σ e ( ) = (i, se+ , s− = ¯i( , T ) − se+ ), with s+,v,v < se+ ≤ ¯i( , T ) − s−,v,v . This resummation being performed at fixed value of the i index, it is easy to check that these enlarged sectors have exactly the same bounds and decay properties (II.15) than the ordinary ones. Once the regrouping has been performed we arrived therefore for the ring lines not in T to two types of such resumed sectors: • for the lines of Ra , the two (extended) indices are completely determined by the collapsing indices at the ends of the line. • in case b, this is no longer necessarily true, but the new extended sectors have an auxiliary index l which is 0, since the sum of their s+ and s− indices, extended or not, is exactly i − 2. The lines of R − T in this case form a set called Rb . • finally in case c the new extended sectors have not necessarily an auxiliary index l which is 0, but since s+ + s− = 2r( , T ) − i, they have an index r( ) = (s+ + s− + i)/2 = r( , T ). This means that these lines although not in T “could” have been in T . Let us make this slightly more precise. We call Rc the set of lines of the ring in this case, and Γ1 ( ), Γ2 ( ) the two connected components at level r( ) − 1 containing the ends of . They are connected at level r( ) = r( , T ) by a path P ( , T ), so to any we can associate a line t( ) in this path so that (T − t( )) ∪ { } is still a tree which is a subtree in every connected component Γ of C. ¯ has at most two elements for The map t from Rc to T is also such that t−1 ( ) any ¯ ∈ T . This is because if t( ) = t( ) for , ∈ Rc , then Γ1 ( ) = Γ1 ( ) and Γ2 ( ) = Γ2 ( ) (or the converse), and these components must be “cuts” in the sense of the previous subsection (otherwise the reduced ring R would not be vertex disjoint, see Figure 27). These cuts must even be consecutive in the natural order of the cuts from y to z. Therefore there cannot be a third line ” ∈ Rc with t( ) = t( ) = t( ”).
VII.2 Ring sector Let us now return to the bound on the primitively divergent self-energy contribumax tion with cutoff rmax , namely |Σ≤r 2,pr (y, z)|. There is a first scale rT at which y and z fall into a common connected component of the GN tree. It is the largest index on the initial path in T from y to z. Let us call rR the first r index at which the ring connects y and z. Obviously
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since T is optimized with respect to the r indices, we have rT ≤ rR . rR can be expressed as a minimax over the two disjoint paths PR,1 and PR,2 which compose the ring: rR = min rR,j ; rR,j = max r(k) (VII.75) j=1,2
k∈PR,j
Obviously we have rR ≤ rmax . In the same vein we should define a (generalized) sector σ ¯R associated to the ring R and the tree T . It is a triplet (iR,T , s+,R , s−,R ) depending on the sector attributions of the lines of the tree T and of the ring R we have just built. s+,R and s−,R are also minimax of the corresponding indices over the two disjoint paths PR,1 and PR,2 which compose the ring. More precisely s+,R = min s+,R,j ; s+,R,j = max s+ (k)
(VII.76)
s−,R = min s−,R,j ; s−,R,j max s− (k).
(VII.77)
j=1,2
k∈PR,j
j=1,2
k∈PR,j
The index iR,T is optimized both over T and R. More precisely we define, if P (y, z, T ) is the unique path from y to z in T : iT =
max
k∈P (y,z,T )
i(k) ;
(VII.78)
iR = min iR,j ; iR,j = max i(k) ;
(VII.79)
iR,T = min{iR , iT } .
(VII.80)
j=1,2
k∈PR,j
Using the relations 0 ≤ s± ≤ i for ordinary sectors, one has s±,R,j ≤ iR,j ≤ rR,j , hence 0 ≤ s±,R ≤ rmax . Furthermore iT ≤ rT ≤ rmax so that the three indices iR,T , s+,R and s−,R being all bounded by rmax are indeed those of a generalized sector σ ¯R,T of Σrmax , in the sense of Section V. We define the associated r index of this generalized sector as rR,T : rR,T ≡
iR,T + s+,R + s−,R . 2
(VII.81)
We also define the scaled distance for that ring sector σ ¯R,T = (iR,T , s+,R , s−,R ) as dR,T (y, z) = diR,T ,s+,R ,s−,R (y, z) .
(VII.82)
VIII Power counting Everything is now prepared for the bounds. We do not repeat all details but concentrate on what is new with respect to [1]. We introduce all the momentum constraints χj (σ) for all the vertices of the primitively divergent self energy contribution. After that we apply Gram’s
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bound on the remaining determinant. This replaces the remaining determinant by a product over its entries of the corresponding power counting factors (see [1]). We shall first perform the spatial integration over the positions of internal vertices, using the propagators decay and the fields and propagators prefactors. This is really power counting. Then we shall perform the sector sums, using the coupling constants, which is a kind of logarithmic power counting. The spatial integration are themselves divided in two steps. We write: (cλ)n max χj (σ)I1,n (y, z)I2,n (y, z, xj,± ) (VIII.83) |Σ≤r 2,pr (y, z)| ≤ n! n σ j T ,R...
max z− ||Σ≤r 2,pr (y, z)|
|y+ − z+ |.|y− − (cλ)n ≤ χj (σ)I1,n,± (y, z)I2,n (y, z, xj,± ) n! n σ j
(VIII.84)
T ,R...
max |y0 − z0 |.|Σ≤r 2,pr (y, z)| n
≤
n
(cλ) n!
T ,R... σ
χj (σ)I1,n,0 (y, z)I2,n (y, z, xj,± ) .
(VIII.85)
j
In I2,n (y, z, xj,± ) we keep the positions of y, z and the spatial positions of the ring vertices xj,± fixed and integrate other all the remaining positions. To pay for all these integrations we put in I2,n (y, z, xj,± ) a fraction (say 1/2) of the decay of every line in L, all the determinant fields prefactors and the line prefactors for the lines not in the ring. Hence: α dxv,0 d3 xv M −rf /2−lf /4 e−c.dσ(k) /2 I2,n (y, z, xj,± ) = v ∈R
v∈R, v =y,z
f ∈R
k∈L
(VIII.86) (recall that f here runs over either determinant fields or propagator lines, where a line counts for two fields, one at each end). Then in I1,n (y, z) we gather the remaining factors and integrations. We first prove a uniform bound on I2,n (y, z, xj,± ) independent of the fixed positions y, z, xj,± : Let 0 be the line with largest value r( 0 ) = rT of r index in the path P (y, z, T ) from y to z in T . Let R be those lines of R which are of type a in section VII.1, plus 0 if it belongs to T . Lemma VIII.1 The following bound holds: I2,n (y, z, xj,± ) ≤ K n M −iT M −lf /12 Γ ring−intersecting y and z disjoint, e (Γ)>2
f ∈R
M −e(Γ)/6
Γ ring−disjoint, e(Γ)>4
M −e (Γ)/4
Γ ring−intersecting containing y or z e (Γ)>1
M −e (Γ)/3
(VIII.87)
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where e (Γ) for a connected component Γ is the number of external fields of Γ not in the ring. The product f ∈R runs over all fields and lines except those of R . Proof. At fixed + and − positions for the ring vertices we integrate • all x0 positions with the T propagators decay. This results in the factor i() (the 0 line is missing because both y and z are hold fixed). ∈T , =0 M We have also other slightly different bound: for any line of Rc as defined at the end of subsection VII.1, we could replace the integral performed with the line by the one performed with line if t( ) = ; it has the same r index, but may result in a better bound if i( ) < i(t( )). Let us note ¯i( ) = ¯ min{i( ), i( ) | t( ) = }. We have therefore really a bound ∈T , =0 M i() for the integration of all x0 positions. • all x± positions for the vertices not in the ring with the T propagator decays. Remark that a vertex v not in the ring integrated with a tree line of scale (s+ , s− ) costs exactly M s+ +s− . These integrals are made exactly with the set of lines T1 of T with index r such that at scale r the two connected components Γ at their ends do not both touch the ring. In particular 0 ∈ T1 , and every line in R ∩ T is not in T1 . Therefore M −l()/2 ≤ M −l()/2 . (VIII.88) ∈T −T1 , =0
∈R∩T , =0
Joining all factors we obtain, defining ¯l = 2[r( ) − ¯i( )]: I2,n (y, z, xj,± ) ≤ K n M −rf /2−lf /4 M s+ +s− =
K
n
f ∈R
M
−rf /2−lf /4
f ∈R
= Kn
∈T1
M
s+ +s− +i
M −rf /2−lf /4
f ∈R
M −lf /12
∈T , =0
M 2r
∈T1
f ∈R
¯
M i()
M
¯i()
∈T −T1 , =0
∈T1
f ∈R
≤ Kn
M −rf /2
¯
M r −l /2
∈T −T1 , =0
∈T1
M 2r
M r , (VIII.89)
∈T −T1 , =0
where in the last line we separated the r and l dependence and used (VIII.88). Indeed we have all the necessary factors M −lf /4 for the fields not in ; for the fields in Rb we have l = 0 so we can freely add that factor; and for the fields ¯ in Rc there at most two lines with t( ) = t( ”) = , and we have M −l /2 ≤ ¯ M −l /6 M −l /6 M −l” /6 , which gives the factor M −l /12 for each ends of the lines in Rc . We divide then as usual every factor M 2r , M r or M −r/2 as a product over all scales and we collect everything scale by scale. Following the previous section, we should distinguish the connected components Γ which have empty intersection
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with the ring, also called ring-disjoint, and those which contain at least one vertex of the ring, called ring-intersecting. Among these one should also distinguish those who contain neither y nor z, called y and z disjoint, and those who contain y or z or both. There is then at each scale a factor M 2 to pay for each ring-disjoint component (corresponding to one particular vertex which plays the role of a center of mass for that component, which is integrated both on time and spatial position); a factor M to pay for each ring–intersecting, y-z disjoint component, for which only the time position of a ring vertex has to be integrated, and no factor to pay for the components containing y or z or both, since y and z are fixed. Therefore we get M −rf /2 M 2r M r = M 2−e(Γ)/2 f ∈R
×
∈T −T1 , =0
∈T1
M
1−e (Γ)/2
Γ ring−intersecting, y and z disjoint
Γ ring−disjoint
M −e (Γ)/2 .
(VIII.90)
Γ containing y or z
Remark that this is not an upper bound but an equality. Using the previous section we know that for every connected component Γ which is y ∪ z disjoint, e (Γ) ≥ 2; and since we consider a primitively divergent contribution, e(Γ) ≥ 4. Furthermore for every connected component containing y or z but not both (these have odd numbers of external legs, since we amputated the external legs at y and z), we have e (Γ) ≥ 1, so that we have a decay factor M −1 from the first scale r = 0 until at least the first scale r( 0 ) at which y and z become connected in T . Hence following the usual argument as in [1] we get M 2−e(Γ)/2 M 1−e (Γ)/2 M −e (Γ)/2 Γ ring−intersecting, y and z disjoint
Γ ring−disjoint
≤ M −rT
M −e(Γ)/6
M −e (Γ)/4
Γ ring−intersecting y and z disjoint, e (Γ)>2
Γ ring−disjoint, e(Γ)>4
Γ containing y or z
M −e (Γ)/3 .
(VIII.91)
Γ ring−intersecting containing y or z e (Γ)>1
Now using that rT ≥ iT completes the proof of the Lemma.
We treat now the power counting of the ring lines and the space integration of the ring vertices together in I1 . We have (calling x1 , . . . , xp the internal positions of the vertices of the ring not equal to y or z): Lemma VIII.2 For some constant K p α I1,n (y, z) = dxj,+ dxj,− M −(r+l/2)(k) e−c.dσ(k) /2 j=1
k∈R
≤ K p M −s+,R,1 −s−,R,1 −s+,R,2 −s−,R,2 e
k∈L −c.dα R,T
(y,z)/4
,
(VIII.92)
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I1,n,± (y, z) =
p
dxj,+ dxj,− |y − z|+ |y − z|−
j=1
≤K M p
I1,n,0 (y, z) =
Ann. Henri Poincar´e
M −(r+l/2)(k)
k∈R
−s+,R −s−,R −c.dα R,T (y,z)/4
e
dxj,+ dxj,− |y0 − z0 |
j=1
≤K M
α
e−c.dσ(k) /2
k∈L
, (VIII.93)
p p
M −(r+l/2)(k)
k∈R
−s+,R,1 −s−,R,1 −s+,R,2 −s−,R,2
α
e−c.dσ(k) /2
k∈L
M
iR,T
e
−c.dα R,T (y,z)/4
. (VIII.94)
Proof. We use simply the triangular inequality αand a fraction of the decay of the ring lines or of the T lines to get the decay e−c.dR,T (y,z) . We keep an other fraction of the ring lines decay to perform the integration over positions of the ring vertices. To check the prefactor we remark that we can separately optimize the + and − integration. The + integration consumes all the lines M −s+ prefactors except two, namely the smallest ones on the two paths of the ring, which are M −s+,R,1 and M −s+,R,2 . Finally in the second bound (VIII.93) the |y − z|+ factor consumes an M −s+,R factor, and we can keep the other in the bound, and bound it again by a factor M −s+,R (this seems not optimal, but works!). The same is true with the − integrations. In the third bound (VIII.94) we keep both factors M −s±,R,1 and M −s±,R,2 but have to pay M iR,T for the |y − z|0 factor. Remark that to get this factor and not simply M iR , it is important to keep in I1,n a fraction of the decay of all the lines of T ∪ R, not only of R. Let us now combine the factor M −iT in (VIII.87) with the other power counting factors M −s+,R,1 −s−,R,1 −s+,R,2 −s−,R,2 of (VIII.92), M −s+,R −s−,R of (VIII.93) and M −s+,R,1 −s−,R,1 −s+,R,2 −s−,R,2 +iT of (VIII.94). We have (VIII.95) iR,T ≤ iR ≤ max s+ (k) + max s− (k) + 2 k∈R
k∈R
(where we used i ≤ s+ + s− + 2), hence s+,R + s−,R + iR,T ≤ 2(max s+ (k) + max s− (k)) + 2 k∈R
k∈R
(VIII.96)
so that, since iR,T ≤ iT 3(s+,R + s−,R + iR,T ) ≤ 2(s+,R,1 + s−,R,1 + s+,R,2 + s−,R,2 + iT ).
(VIII.97)
Therefore in (VIII.92) we have M −s+,R,1 −s−,R,1 −s+,R,2 −s−,R,2 M −iT ≤ M −(3/2)(s+,R +s−,R +iR,T ) = M −3rR,T .
(VIII.98)
Similarly combining the factors in (VIII.93) with M −iT ≤ M −iR,T we have M −s+,R −s−,R M −iT ≤ M −2rR,T
(VIII.99)
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and finally we also have, using (VIII.95), M −(s+,R,1 +s−,R,1 +s+,R,2 +s−,R,2 ) = M −(s+,R +s−,R +maxk∈R s+ (k)+maxk∈R s− (k)) ≤ M −(s+,R +s−,R +iR,T −2) ,
(VIII.100)
so that we obtain from (VIII.94) M −s+,R,1 −s−,R,1 −s+,R,2 −s−,R,2 M +iR,T M −iT ≤ M −s+,R −s−,R −iR,T +2 = KM −2rR,T .
(VIII.101)
The power counting factors of the right-hand side of (VIII.98)–(VIII.99)–(VIII.101) are exactly those of bounds (V.55)–(V.56)–(V.57) of Theorem V.1. Let us finally give indications on the logarithmic power counting, namely how to pay the sums over all sectors and indices r, l, s± . This is similar to [1] except for two new difficulties: The first additional difficulty comes when fixing all the r indices. Not only the 4-legged ring-disjoint components (also called “quadrupeds”) for which e(Γ) = 4 hence 2 − e(Γ)/2 = 0 but also any connected component Γ touching the ring and y, z disjoint with e (Γ) = 2, or containing y or z with e (Γ) = 1 have now apparently marginal power counting. So fixing their largest internal r scale once their first external r scale is known costs also for them one factor log T . It is however easy to check as in [1], that each new such component gives rise to one additional coupling constant, so the total number of such log T factors to pay is still bounded by n and in fact n − 1 where n is the total number of vertices in the self-energy. Once all r scales are fixed, we have to sum over the auxiliary indices l and the s+ and s− indices, subject to the constraint s+ + s− = r + l/2. This is again done at a cost of one factor log T per vertex using the momentum conservation rule and the auxiliary decay factor f M −lf /12 as in [1], Lemma 5. In fact knowing all r and l indices and the sector of the external line of the self energy which is conserved, so present both at y and z, we have in fact only to pay (log T )n−1 to know all s+ and s− indices. However there is a second difficulty here, which is that there is no exponential M −l/12 decay factor for the lines in Ra , nor for the 0 line if it is in R. But to find the corresponding l indices leads to at most one single new logarithmic factor log T to pay, to fix the l index of the line 0 (this is not a problem because we have at least two unemployed such factors corresponding to the last coupling constant at y). Indeed for all lines in Ra . both sectors s+ and s− collapse at some end, hence are determined by the corresponding collapsed sectors. Their sum (plus the r index) determines the l index of the line so there is no new logarithm to pay to know the l index of these lines. This is essentially what subsection VII.1 was about.
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The result is at most a cost log T 2n−1 to fix all internal sectors of the selfenergy when the two external are known, like in [1]. This is why we get analyticity only in a domain λ ≤ c/ log2 T . Finally we have to sum over the tree T , the arch constructions and over n. This is standard or explained above. In this way the proof of Theorem V.1 is achieved for primitively divergent contributions. In the next section we use an induction to extend this proof to the general case.
IX Chains of bipeds We turn now to the renormalization of a non-primitively divergent two-point subgraph. The method is to perform all arch expansions for all two-point subgraphs first, then apply Gram’s inequality to all the remaining determinant, then apply inductively the bounds of the previous section from the smallest primitively divergent subgraphs towards the larger ones. In a non-primitive two-point subgraph, with external vertices at positions Y and Z the “higher” self energy insertions can appear either as chains decorating a propagator from a vertex y to a vertex z, like in Figure 28, or as chains decorating a determinant field hooked at a vertex y, like in Figure 29. Using the bounds of the previous section, in which Theorem V.1 was proved for primitive bipeds, we are now in a position to bound such maximal chains Chainr of primitively divergent 1PI bipeds B1 , . . . , Bq , such as the one of Figure 28 with fixed ends y = z0 and z = yq+1 or the one of Figure 29 with fixed end y = z0 . The two external vertices of each biped Bj are called yj and zj .
y
y=z0
z
1 1 111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111
11111 00000 00000 11111 00000 11111 00000 00000 11111 y211111 z2 00000 11111
y3
z
3 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111
11111 00000 00000 11111 00000 11111 00000 11111 00000 y411111 z4 00000 11111
z= y5
Figure 28. A chain of bipeds on a propagator.
y111111 z 000001 y=z0
11111 00000 00000 11111 00000 11111 00000 11111 00000 11111
111111 000000 000000 111111 000000 111111 000000 000000z 111111 y111111 000000 111111 2 2
y3
z
3 11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111
111111 000000 000000 111111 000000 111111 000000 111111 000000 y111111 z4 000000 111111 4
Figure 29. A chain of bipeds on a field. Remark that by momentum conservation the q + 1 ordinary lines in Figure 28, or the q ordinary lines and the determinant field at the end of the chain in Figure 29 all belong to the same sector σ of scale r, not yet summed (or its nearest
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neighbors). We have to compare the bound for such a decorated propagator to that of a regular bare propagator. We have first to evaluate the action of the τ ∗ operators in (III.38) on the external lines. We find (∂+ + i)(∂− + i)Cσ x0 , sy+ + (1 − s)x+ , ty− + (1 − t)y+ , z (IX.102) α α ≤ M −2r(σ)−l(σ) sup e−dσ (x,z) , e−dσ (y,z) (∂0 − iπT )C ty0 + (1 − t)x0 , y+ , y− , z (IX.103) α α ≤ M −2r(σ) sup e−dσ (x,z) , e−dσ (y,z) We should take into account that there are q + 1 ordinary lines in the chain, and only q of them bear τ ∗ operators, hence have “improved” power counting prefactor M −2r(σ) . The last one has the usual prefactor M −r(σ)−l(σ)/2 . Multiplying these bounds by the correct factors for the bipeds, namely those of the previous section (having spared as usual the necessary factors for summing over the trees), we obtain that the chain of Figure 28 is bounded by q
λq K q M −(2q+1)r(σ)−l(σ)/2 q α α −dα (y,y1 ) σ dyj dzj sup e−dσ (yj ,yj+1 ) , e−dσ (zj ,yj+1 ) e j=1
e
−c.dα Rj ,Tj (yj ,zj )
α M −2rRj ,Tj ≤ KM −r(σ)−l(σ)/2 e−dσ¯ (y,z) . (IX.104)
with σ ¯ = (max{i(σ), iRj ,,Tj }, max{s+ (σ), s+,Rj ,,Tj }, max{s− (σ), s−,Rj ,,Tj }). Here the attentive reader could worry that the decay of this chain is no longer in dσ but in dσ¯ , and that the associated value of r can be quite higher than r(σ). Could this fact perturb the later multiscale analysis? For instance if i(σ) = 110, s+ (σ) = s− (σ) = 55, hence r(σ) = 110, and if i(Rj ) = s+ (Rj ) = s− (Rj ) = 100, the decay is in σ ¯ = (110, 100, 100) which has an associated index r¯ = 155, quite higher than r(σ) = 110. Fortunately there is no need to worry, because we never use the bound (IX.104) in later analysis. We should never compare the bound (IX.104) of the dressed chain to the bound (II.16) of a bare line of the initial tree T , because it is not necessary. Indeed in the multiscale analysis a line of the tree is always used to bound the integral over one end (e.g., z = yq+1 ) with respect to the other one
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kept fixed (e.g., y = z0 ). So what we should compare is bounds for integrals over one end. For the dressed chain, this integral is bounded by
λq K q M −(2q+1)r(σ)−l(σ)/2
q
α
dyq+1 e−dσ (y,y1 )
q
dyj dzj
j=1
α α −c.dα Rj ,Tj (yj ,zj ) sup e−dσ (yj ,yj+1 ) , e−dσ (zj ,yj+1 ) e M −2rRj ,Tj ≤ KM +r(σ)−l(σ)/2 . (IX.105)
which is identical to the bound obtained by integrating over z at fixed y with a single bare line with sector [r(σ), l(σ)] (see (II.16), where the prefactor M −i−l is also M −r−l/2 , and the weight of integration over z is M i+s+ +s− = M 2r ). The dressed field of Figure 29 is bounded in the same way but with one propagator replaced by a field: q
λq K q M −(2q+1/2)r(σ)−l(σ)/4
q
α α dyj dzj sup e−dσ (zj−1 ,yj ) , e−dσ (zj−1 ,zj )
j=1
−c.dα Rj ,Tj (yj ,zj ) e M −2rRj ,Tj ≤ KM −r(σ)/2−l(σ)/4 . (IX.106)
which is again identical to the bound for a bare undecorated field with sector [r(σ), l(σ)] in a determinant. So when chains of primitively divergent bipeds are inserted into convergent graphs, they do not spoil at all the analysis and bounds of [1]. The last thing to check for the attentive reader is that when such new decorated lines, appear inside a new (i.e., non-primitive biped), the bound including the |y − z|+ and |y − z|− factors (or the easier |y − z|0 factor) of the previous sections do not deteriorate. We now explain why this is true. The arch expansions have been performed for all 1PI bipeds which appear in the GN tree and need to be renormalized, so we know the rings for all these bipeds, not only for the primitive ones. For a given biped, not necessarily primitive, when a dressed lines is a tree line not in the ring, it is always used in combination with an integral over one end, and the bound (IX.105) is identical to the bound with a bare line. Finally when a dressed line is part of the ring for a later biped one could worry that it could “bear the |y − z|+ or |y − z|− factor”. But again remark that in the ring analysis of Lemma VIII.2 we never bound separately the |y − z|+ or |y − z|− factors (there are cases where they would indeed cost about M 2r instead of M r . . . ) but always in combination with all the dxj,+ and dxj,− integrals on the intermediate vertices. So here again, remark that when dressed lines with quite big values of s+ and s− “bear the cost of the |y − z|+ or |y − z|− factors”, they are not used for the corresponding integrals over any of their ends, which will be performed with the help of other lines with better decay. This is why the proof of
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Lemma VIII.2 using (IX.105) remains absolutely unchanged when rings lines are dressed. Hence the lemma holds throughout the induction from smaller to bigger bipeds. We have finally to explain why constants such as K in bounds (IX.105) or (IX.106) do not deteriorate in the induction. This is because at each stage of this induction the new decorated lines or fields hook to new distinct vertices which provide new unspoiled small coupling constants. Therefore Theorem V.1 holds for the sum over all contributions to the selfenergy, not only over the primitively divergent ones. Plugging that theorem into the analysis for convergent graphs of [1] finally completes the proof of analyticity of any correlation function for |λ|(log T )2 ≤ const.
X
Self-energy bounds
In this section we summarize what has been achieved by the previous sections into two bounds on the self-energy, one with the first non-trivial graph G2 included and the other with that graph excluded8 . We apply the analysis above to Σr (k), the sum of all self-energy contributions with lines of index ≤ r and at least one line of index r. We must keep track of the maximal number of logarithmic factors due to quadrupeds in the clustering tree structure, which is n − 1 at order n. Each vertex costs a logarithm for sector summation (except one, thanks to the fixed external legs) and each quadruped (except the last one) costs another logarithm for power counting. Hence the total number of | log T | factors in Σn , the nth order contribution to the self energy is at most 2n − 2. But each time we add a vertex there is a new factor |λ| ≤ c| log T |−2 that compensates for the new logarithmic factors. Hence perturbation series is bounded by a geometric convergent series, and its resumed size is of same order than the leading first term. It is therefore straightforward to obtain the following bounds analogs to Theorem V.1, in which M −r is the cutoff: |Σr (k)| ≤ K|λ|2 r2 M −r , (X.107) ∂ r 2 2 (X.108) ∂kµ Σ (k) ≤ K|λ| r , ∂2 r ≤ K|λ|2 r2 M r . Σ (k) (X.109) ∂kµ ∂kν where K is some constant. The same quantities but with the particular graph G2 taken out give similar but slightly better bounds since the series start with contributions of order 3: |Σrn≥3 (k)| ≤ K|λ|3 r4 M −r ,
(X.110)
8 This first non-trivial graph is the elementary biped in the chain of Figure 1, since the tadpole vanishes.
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∂ r 3 4 ∂kµ Σn≥3 (k) ≤ K|λ| r , ∂2 r 3 4 +r ∂kµ ∂kν Σn≥3 (k) ≤ K|λ| r M .
(X.111) (X.112)
(These bounds are certainly not optimal; they do not take into account the three paths built in the previous section, which show that for instance if µ = + and ν = −, there is no linearly divergent factor M +r in (X.112)). One can now sum over r ≤ imax (T ) the self energy contributions Σr (k), r Σn≥3 (k) and their first momentum derivatives in the domain λ| log T |2 ≤ c for small c, and obtain the bounds |Σ(k)| ≤ K|λ|2 ≤ Kc2 | log T |−4 , ∂ ≤ K|λ|2 | log T |3 ≤ Kc2 | log T |−1 , Σ(k) ∂kµ ∂2 3 4 −1 3 −2 −1 . ∂kµ ∂kν Σn≥3 (k) ≤ K|λ| | log T | T ≤ Kc | log T | T
(X.113) (X.114) (X.115)
This proves that the self energy is uniformly C 1 in the domain of analyticity of the theory, namely |λ|.| log T |2 ≤ c. However the bounds for second derivatives grow with r, strongly suggesting that the self-energy is not uniformly of class C 2 in the domain |λ| ≤ c/| log T |2 , just like the Luttinger liquid in one dimension. More precisely we proved by a tedious analysis ([5]) the following lower bound for the amplitude of the single graph G2 ∂2 2 −1 , (X.116) ∂kµ ∂kν Σn=2 (k) = |IG2 (k)| ≥ K λ T in the special case of µ, ν in the (+, +) direction and incoming momentum (k0 = πT, k+ = 1, k− = 0). This completes the proof that the Hubbard model at half-filling is not a Fermi liquid in the sense of [3]. Indeed for |λ|| log T |4 ≤ c and c smaller than K /2K , the rest of the series, bounded in (X.115) by K|λ|3 | log T |4 T −1 , hence by Kc|λ|2 T −1 , is smaller than half the right-hand side of (X.116). When we add Σn=2 = IG2 2 and Σn≥3 the modulus of the full quantity ∂k+∂∂k+ Σ(πT, 1, 0) therefore diverges at least as (K /2)c2 | log T |−8 T −1 along the curve |λ|| log T |4 = c as T → 0, which means that Salmhofer’s criterion for Fermi liquids is violated. 2 In fact ∂k+∂∂k+ Σ(πT, 1, 0) diverges also along the curve |λ|| log T |2 ≤ c, because bound (X.112) is not optimal. A more careful analysis using the three paths built in the previous sections would spare two additional logarithms and prove ∂2 ≤ K|λ|3 | log T |2 T −1 ≤ Kc3 | log T |−4 |T −1 , (X.117) Σ (πT, 1, 0) ∂k+ ∂k+ n≥3 but this is left to the reader.
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Acknowledgments. We thank A. Abdesselam for useful discussions on Menger’s theorem, and our referee for his careful work, which lead to significant improvements of the first manuscript.
References [1] V. Rivasseau, The two-dimensional Hubbard Model at half-filling: I. Convergent Contributions, Journ. Stat. Phys. 106, 693–722 (2002). [2] J.A. Bondy and U.S.R. Murty, Graph theory with applications, North-Holland Editions, 1979. [3] M. Salmhofer, Continuous renormalization for Fermions and Fermi liquid theory, Commun. Math. Phys. 194, 249 (1998). [4] M. Salmhofer, Renormalization, an introduction, Springer Verlag, 1999. [5] S. Afchain, J. Magnen and V. Rivasseau, Lower bound on the first self energy graph of the Hubbard model, to appear. [6] G. Benfatto and G. Gallavotti, Renormalization Group, Physics Notes, Chapter 11 and references therein, Vol. 1 Princeton University Press, 1995. [7] P.W. Anderson, Luttinger liquid behavior of the normal metallic state of the 2D Hubbard model, Phys. Rev. Lett. 64, 1839–1841 (1990). [8] J. Feldman, J. Magnen, V. Rivasseau and E. Trubowitz, Two-dimensional Many Fermion Systems as Vector Models, Europhys. Letters 24, 521 (1993). [9] J. Feldman, H. Kn¨ orrer and E. Trubowitz, A two-dimensional Fermi Liquid, series of papers in Commun. Math. Phys. 247, 1–319 (2004), and Reviews in Math. Physics 15, 9, 949–1169 (2003). [10] M. Disertori and V. Rivasseau, Interacting Fermi liquid in two dimensions at finite temperature, Part I: Convergent Attributions, Comm. Math. Phys. 215, 251 (2000). [11] M. Disertori and V. Rivasseau, Interacting Fermi liquid in two dimensions at finite temperature, Part II: Renormalization, Comm. Math. Phys. 215, 291 (2000). [12] G. Benfatto, A. Giuliani and V. Mastropietro, Low temperature Analysis of Two-Dimensional Fermi Systems with Symmetric Fermi surface, Ann. Henri Poincar´e 4 137 (2003). [13] M. Disertori, J. Magnen and V. Rivasseau, Interacting Fermi liquid in three dimensions at finite temperature, part I: Convergent Contributions, Ann. Henri Poincar´e 2, 733–806 (2001).
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[14] A. Abdesselam and V. Rivasseau, Trees, Forests and Jungles: A Botanical Garden for Cluster Expansions, in “Constructive Physics”, LNP 446, Springer Verlag, 1995.
St´ephane Afchain and Jacques Magnen Centre de Physique Th´eorique CNRS, UMR 7644 ´ Ecole Polytechnique F-91128 Palaiseau cedex France email:
[email protected] email:
[email protected] Vincent Rivasseau Laboratoire de Physique Th´eorique CNRS, UMR 8627 Universit´e de Paris-Sud F-91405 Orsay France email:
[email protected] Communicated by Joel Feldman submitted 09/09/04, accepted 20/12/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 449 – 483 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/030449-35 DOI 10.1007/s00023-005-0214-z
Annales Henri Poincar´ e
The Hubbard Model at Half-Filling, Part III: the Lower Bound on the Self-Energy St´ephane Afchain, Jacques Magnen and Vincent Rivasseau
Abstract. We complete the proof that the two-dimensional Hubbard model at halffilling is not a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing a lower bound on a second derivative in momentum of the first nontrivial self-energy graph.
I Introduction This paper is the third of a series ([1, 2]) devoted to the rigorous mathematical study of the two-dimensional Hubbard model at half-filling above the transition temperature to the expected low temperature region, which becomes N´eel-ordered at zero temperature. The goal of this series was to prove that this model does not obey Salmhofer’s criterion for Fermi liquid behavior of interacting Fermion systems at equilibrium ([3, 4]). In this way, this model differs sharply from those with a Fermi surface close to the circle, which obey Salmhofer’s criterion ([5, 6, 7]). In the first paper [1] the convergent contributions of the model were bounded in the domain |λ| log2 T ≤ K. In the second one [2], renormalization of the selfenergy was performed to complete the proof of analyticity in the coupling constant of all the correlation functions in that domain. Salmhofer’s criterion requires beyond this analyticity that the self-energy (in momentum space) is uniformly bounded together with its first and second derivatives in a domain |λ|| log T | ≤ K. In this paper we prove that a certain second derivative of the self-energy at a particular value of the external momentum is not uniformly bounded in the domain |λ| log2 T ≤ K where we have established analyticity. This domain being smaller than the Salmhofer’s one, it completes the proof that the two-dimensional halffilled Hubbard model is not a Fermi liquid. In conclusion, when we move from low filling to half-filling, the Hubbard model must undergo a cross-over from Fermi to non-Fermi (in fact Luttinger) liquid behavior. This solves the controversy on the nature of two-dimensional Fermionic systems in their ordinary phase [8]. We refer to [1, 2, 4] for a more complete review and further references on mathematical study of interacting Fermions.
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II Recall of notations The two-dimensional Hubbard model is defined on the lattice Z2 . Fixing a temperature T > 0, the “imaginary time”, denoted x0 , belongs to the real interval − T1 , T1 . In the following, we shall denote β = T1 . Indeed this interval should be thought of as a circle of length 2β, that is R/2βZ. Consequently, the momentum space, which is the dual of R/2βZ × Z2 in the sense of the Fourier transform, is πT Z × [R/2πZ]2 . The torus [R/2πZ]2 will be represented by the square [−π, π[2 , with periodic boundary conditions. In Fourier variables, the expression of the propagator at half-filling reads: C(k0 , k1 , k2 ) =
1 ik0 − cos k1 − cos k2
(II.1)
if k0 = (2n + 1)πT for some n ∈ Z. If k0 = 2nπT , C(k0 , k1 , k2 ) = 0 because in the formalism of Fermionic theories at finite temperature, the propagator has an antiperiod β with respect to the x0 variable and therefore each Fourier coefficient of even order vanishes. With a slight abuse of language, we can say that C(k0 , k1 , k2 ) is only defined for k0 = (2n + 1)πT . This set of values is called the Matsubara frequencies. The expression of the propagator in real space is deduced by Fourier transform: eik.x 1 C(x0 , x1 , x2 ) = (II.2) dk dk dk 0 1 2 (2π)3 ik0 − cos k1 − cos k2 where we adopt the notations of [1], namely the integral dk0 really means the discrete sum over the Matsubara frequencies 2πT n∈Z ((2n + 1)πT ) (with k0 = (2n+1)πT ), whereas the integrals over k1 and k2 are “true” integrals, for (k1 , k2 ) ∈ [−π, π[2 . (We do not need any ultraviolet cutoff for the graph studied in this paper, since it is ultraviolet convergent.) For our analysis, it will be convenient to introduce another parametrization of the spaces [−π, π[2 and Z2 . The idea is to “rotate” the Fermi surface of Figure II by an angle of π4 . In the k0 = 0 plane, it is defined by cos k1 + cos k2 = 0, which is equivalent to k2 = π ± k1 or k2 = −π ± k1 . k1 = π2 (k+ + k− ) k1 ±k2 Introducing the variables k± = π ⇐⇒ , the domain k2 = π2 (k+ − k− ) of integration (k1 , k2 ) ∈ [−π, π[2 becomes the set: D = (k+ , k− ) ∈ [−2, 2]2 with −2 ≤ k+ ≤ 0 0 ≤ k+ ≤ 2 . (II.3) or −2 − k+ ≤ k− ≤ 2 + k+ −2 + k+ ≤ k− ≤ 2 − k+ As cos k1 + cos k2 = 2 cos π2 k+ cos π2 k− , the Fermi surface in the variables k± is simply defined by k+ = ±1 , k− = ±1. The new domain of integration, with the Fermi surface is represented on Figure 2.
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π
451
k2
π −π
k1
−π
Figure 1. The square [−π, π[2 and the Fermi surface. k− 2
1 -2
1
-1
2
k+
-1
-2
Figure 2. The domain of integration in (k+ , k− ) and the Fermi surface. In a dual way, we introduce new variables in real space, x+ and x− in such a way that k1 x1 + k2 x2 = k+ x+ + k− x− . We have: x+ = π2 (x1 + x2 ) (II.4) x− = π2 (x1 − x2 ) . Observe that the image of the lattice Z2 by this change of variable is not
π 2 2Z
but
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π
n , (m, n) ∈ Z2 , m ≡ n[2] . (II.5) 2 2 In other words, the integers m and n must have same πparity.
π 2 k1 k+ 2 2 As the Jacobian of the transformation = π is J = − π2 , π k2 − k − 2 2 we have: ei(k1 x1 +k2 x2 ) ei(k+ x+ +k− x− ) π2 dk1 dk2 = dk+ dk− . ik0 − cos k1 − cos k2 2 D ik0 − 2 cos π2 k+ cos π2 k− [−π,π]2 (II.6) But the domain D is not very convenient for practical computations, and therefore we would like the k+ k− integration domain to factorize. Since the complement set [−2, 2[2 \D is another fundamental domain for the torus R2 /2πZ2 , we have: ei(k+ x+ +k− x− ) ei(k+ x+ +k− x− ) dk+ dk− = dk+ dk− . π π ik0 − 2 cos 2 k+ cos 2 k− ik0 − 2 cos π2 k+ cos π2 k− D [−2,2[2 \D (II.7) Hence: ei(k+ x+ +k− x− ) ei(k+ x+ +k− x− ) 1 dk+ dk− = dk+ dk− . π π ik0 − 2 cos 2 k+ cos 2 k− 2 [−2,2]2 ik0 − 2 cos π2 k+ cos π2 k− D (II.8) Recapitulating, the expression of the propagator that we take as our starting point is: ei(k0 x0 +k+ x+ +k− x− ) C(x0 , x+ , x− ) = d3 k (II.9) ik0 − 2 cos π2 k+ cos π2 k− for x± satisfying the parity condition (II.5). In II.9 the notation d3 k means 1 dk+ dk− , (II.10) dk0 32π [−2,2]2 where we recall that dk0 means 2πT n∈Z η((2n+1)πT )), since k0 = (2n+1)πT . Now, let us consider, in Fourier space, the amplitude of the graph G represented on Figure 3, with an incoming momentum k = (k0 , k+ , k− ). This amplitude ¯ 2 e−ik.x (where is denoted AG (k) and written as AG (k0 , k+ , k− ) = d3 x C(x)C(x) arrows join antifields to fields).
the subset
S=
π
m,
k
k 0
x
Figure 3. The first non-trivial graph contributing to the self-energy.
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More precisely, we shall consider the second momentum derivative in the + direction of this quantity, which up to a global inessential minus sign is: 2 ¯ 2 e−ik.x . ∂+ AG (k) = d3 x x2+ C(x)C(x) (II.11) The quantity we are going to study is explicitly written: 2 AG (πT, 1, 0) ∂+
=
3
d x
x2+
d3 k1
eik1 .x ik1,0 − 2 cos π2 k1,+ cos π2 k1,−
d3 k2
eik2 .x −ik2,0 − 2 cos π2 k2,+ cos π2 k2,−
d3 k3
eik3 .x ei(πT x0 +x+ ) , (II.12) −ik3,0 − 2 cos π2 k3,+ cos π2 k3,−
where again d3 x includes the parity condition (II.5). We state now the main result of this paper:
Theorem II.1 There exists some strictly positive constant K such that, for T small enough: 2 ∂+ AG (πT, 1, 0) ≥ K . (II.13) T We recall that this result, joined to the analysis of [2], leads to the result that the self-energy of the model is not uniformly C 2 in the domain |λ| log2 T < K and therefore that the two-dimensional Hubbard model at half-filling is not a Fermi liquid.
III Plan of the proof Theorem (II.1) will be proven thanks to a sequence of lemmas. But before presenting these lemmas, let us give an overview of our strategy. We use the sector decomposition introduced in [1] to write: 2 ∂+ AG (πT, 1, 0) = d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ ) , (III.14) σ1 ,σ2 ,σ3
where a sector σ is a triplet (i, s+ , s− ) with 0 ≤ s± ≤ i and s+ + s− ≥ i. The main idea is that in the sum over sectors of equation (III.14), the leading contribution is given by a restricted sum corresponding to sectors close to the “vertical part” of the Fermi surface, defined by k+ = ±1. To express this more precisely, let Λ be an integer (whose value will be chosen later), which will play the role of a cut-off for the sectors. We want to prove that as soon as one sector is not
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close to k+ = ±1, then we have a small contribution. Let us denote Λ − {ij },{s+ j },{sj } the sum in which at least one sector is “far” from the vertical sides of the Fermi surface. Precisely, this means that at least one index s+ j is smaller than imax (T )−Λ, where, as in [1], M −imax (T ) ≈ T . This constrained sum can be written explicitly: Λ
inf(i1 ,imax (T )−Λ)
i1 ,s− 1 ,σ2 ,σ3
s+ 1 =0
=
− {ij },{s+ j },{sj }
+
inf(i2 ,imax (T )−Λ)
i1
− + i1 ,s− 1 ,i2 ,s2 ,σ3 s1 =imax (T )−Λ
s+ 2 =0
i1
i2
inf(i3 ,imax (T )−Λ)
− − i1 ,s− 1 ,i2 ,s2 ,i3 ,s3
s+ 1 =imax (T )−Λ
s+ 2 =imax (T )−Λ
s+ 3 =0
+
.
(III.15)
Defining: AΛ G (πT, 1, 0)
=
Λ
d3 x Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ ) , (III.16)
− {ij },{s+ j },{sj }
we write: 2 2 2 Λ AG (πT, 1, 0) = ∂+ AG,Λ (πT, 1, 0) + ∂+ AG (πT, 1, 0) ∂+
(III.17)
2 2 2 Λ where ∂+ AG,Λ (πT, 1, 0) = ∂+ AG (πT, 1, 0) − ∂+ AG (πT, 1, 0) is expressed as a sum over sectors that are all close to k+ = ±1, i.e., such that each s+ j index is greater than imax (T ) − Λ. 2 Each sector appearing in the sum expressing ∂+ AG,Λ (πT, 1, 0) will be divided into two disjoint subsectors, according to the sign of k+ . We recall that in [1], the sectors were defined as: + − π π π π ik0 − 2 cos k+ cos k− ≈ M −i , cos k+ ≈ M −s , cos k− ≈ M −s . 2 2 2 2 (III.18) We shall call σ r and σ l (“right” and “left”) the subdomains of σ corresponding to k+ > 0 and k+ < 0 respectively. The underlying motivation is that, if a momentum, say k1 , is close to the side k+ = 1, by momentum conservation at each vertex, the other ones are necessarily close to the other side k+ = −1. Let us state precisely this point: 2 AG,Λ (πT, 1, 0), there must be one sector of Lemma III.1 In the sum expressing ∂+ the right type, and two of the left type.
The proof is obvious by momentum conservation in the + direction.
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This Lemma implies that: 2 ∂+ AG,Λ (πT, 1, 0) =
+ >imax (T )−Λ j σ1 right
d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ )
{σj },ij ,s
+
d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ )
+ >imax (T )−Λ j σ2 right
{σj },ij ,s
+
d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ ) .
(III.19)
+ {σj },ij ,s >imax (T )−Λ j σ3 right
Among these three contributions, the last two ones are indeed equal, and we have: 2 AG,Λ (πT, 1, 0) ∂+ =
d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ )
+ {σj },ij ,s >imax (T )−Λ j σ1 right
+2
d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x)e−i(πT x0 +x+ ) .
(III.20)
+ {σj },ij ,s >imax (T )−Λ j σ2 right
In each sum, we replace the cos π2 k+ appearing in the propagators by their Taylor expansions in the neighborhood of +1 in a right sector, and in a neighborhood of −1 in a left sector. We have cos π2 k+ ≈ − π2 (k+ − 1) for k+ in the neighborhood of 1, in which case we put q+ = (k+ − 1) and cos π2 k+ ≈ π2 (k+ + 1) for k+ in the neighborhood of −1, in which case we put q+ = (k+ + 1). This 2 ˜ AG,Λ (πT, 1, 0): replacement gives an expression that we call ∂+ 2 ˜ ∂+ AG,Λ (πT, 1, 0)
=
3
d x
x2+
d3 k1
uΛ (q1,+ )eik1 .x ik1,0 + πq1,+ cos π2 k1,−
uΛ (q2,+ )eik2 .x uΛ (q3,+ )eik3 .x 3 k e−i(πT x0 +x+ ) d 3 −ik2,0 − πq2,+ cos π2 k2,− −ik3,0 − πq3,+ cos π2 k3,− uΛ (q1,+ )eik1 .x + 2 d3 x x2+ d3 k1 ik1,0 − πq1,+ cos π2 k1,− uΛ (q2,+ )eik2 .x uΛ (q3,+ )eik3 .x 3 k e−i(πT x0 +x+ ) , d3 k2 d 3 −ik2,0 + πq2,+ cos π2 k2,− −ik3,0 − πq3,+ cos π2 k3,− (III.21) d3 k2
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where the uΛ (q+ )’s is now the smooth scaled cutoff function u(M imax (T )−Λ q+ ) which expresses the former sector constraint s+ ≥ imax (T ) − Λ (u is a usual fixed Gevrey function which is 1 on [−1, 1] and 0 out of [−2, 2], see [1]). In (III.21) we can freely change each integral over dk+ which ran over [−2, 2] into an integral on dq+ which runs from [−∞, ∞]. We still denote d3 k the corresponding integrals. We write now for each propagator in (III.21), uΛ (q+ ) = 1 + u1 (q+ ) + uΛ 1 (q+ ) where u1 (q+ ) = u(q+ ) − 1 and uΛ (q ) = u (q ) − u(q ). In this way we generate + Λ + + 1 three terms: • one in which all three functions uΛ (q+ ) are replaced by 1. We call this term 2 ˜ AG (πT, 1, 0) ∂+ 1 • one in which there is at least one factor uΛ 1 (q+ ) and no factor u (q+ ). We 2 Λ call this term ∂+ AG,1 (πT, 1, 0).
• finally one in which there is at least one factor u1 (q+ ). We call this term 2 1 ∂+ AG (πT, 1, 0). At this stage, we recapitulate: 2 2 ˜ 2 Λ 2 1 ∂+ AG (πT, 1, 0) + ∂+ AG (πT, 1, 0) = ∂+ AG,1 (πT, 1, 0) + ∂+ AG (πT, 1, 0)
2 2 ˜ 2 Λ AG,Λ (πT, 1, 0) + ∂+ + ∂+ AG,Λ (πT, 1, 0) − ∂+ AG (πT, 1, 0) . (III.22) 2 This relation shows that the quantity under study, ∂+ AG (πT, 1, 0), is equal to the 2 ˜ approximation ∂+ AG (πT, 1, 0), up to the four error terms 2 Λ 2 Λ AG (πT, 1, 0), ∂+ AG,1 (πT, 1, 0), ∂+
2 2 ˜ 2 1 AG,Λ (πT, 1, 0) , ∂+ ∂+ AG,Λ (πT, 1, 0) − ∂+ AG (πT, 1, 0) .
(III.23)
Now we are going to prove a lower bound similar to the one of Theorem 2 ˜ AG (πT, 1, 0), and establish an upper bound on each II.1, but on the quantity ∂+ 2 ˜ of the four error terms. More precisely, if we have ∂+ AG (πT, 1, 0) > K T for some constant K > 0 and if the modulus of each error term is smaller than K << K, we shall conclude that:
K T
with
2 ∂+ AG (πT, 1, 0) > K − 4K , (III.24) T 2 ˜ which shall prove Theorem II.1. The result that ∂+ AG (πT, 1, 0) > K T is really the most difficult to establish, and its proof is the heart of this paper. But the control of the error terms is easier, and each one will correspond to a lemma. We shall begin by these lemmas in the next section, and then turn to the lower bound 2 ˜ AG (πT, 1, 0). on ∂+
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IV The control of the error terms First we state a result that is not necessary for proving Theorem II.1 but whose proof illustrates the way the sector decomposition allows us to establish quite easily upper bounds. Lemma IV.1 There exists some constant K1 > 0 such that: 2 ∂ AG (πT, 1, 0) ≤ K1 . + T
(IV.25)
Proof: We use the decay property of C(i,s+ ,s− ) (x) (see [1], Lemma 1):
α
C(i,s ,s ) (x) ≤ c.M −s+ −s− exp −c dσ (x) , + −
(IV.26)
where α ∈]0, 1[ is a fixed number, c is a constant and dσ (x) = M −i |x0 | + M −s+ |x+ | + M −s− |x− |. We have: 2 + − ∂ AG (k) ≤ c3 .M − 3j=1 sj − 3j=1 sj + 3
α + − M −ij |x0 | + M sj |x+ | + M −sj |x− | . d3 x x2+ exp −c − {ij },{s+ j },{sj }
j=1
(IV.27) Among the indices i1 , i2 and i3 , we keep the best one, i.e., the smallest one, to perform the integration over x0 . We proceed in an analogous way for the indices + + − − − (s+ 1 , s2 , s3 ) and (s1 , s2 , s3 ) respectively. Thus we have:
2 ∂+ AG (k) ≤ c3
M−
3
j=1
s+ j −
3
j=1
s− j
+
−
M inf{ij } M 3 inf{sj } M inf{sj
}
.
− {ij },{s+ j },{sj }
(IV.28) To carry out our discussion, we introduce several notations. If (a1 , a2 , a3 ) is a family of three (not necessarily distinct) real numbers, we denote as usual inf{aj } the smallest number among the aj ’s, but we define also
inf {aj } = inf {a1 , a2 , a3 }\{inf{a1 , a2 , a3 }} (IV.29) 2
and:
inf {aj } = inf {a1 , a2 , a3 }\{inf{a1 , a2 , a3 }, inf {a1 , a2 , a3 }} . 3
2
(IV.30)
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Remark that inf 3 {aj } is indeed sup{aj }. Finally in this paragraph we shall write 3 − simply aj instead of j=1 aj , and similarly for the s+ j ’s and the sj ’s. With these notations, it is very easy to check the following identity: inf{aj } =
3 1 1 1 inf {aj } − inf{aj } − inf {aj } − inf{aj } . aj − 3 j=1 3 2 3 3
We introduce the abbreviation: ∆{aj } = inf {aj } − inf{aj } + inf {aj } − inf{aj } , 2
3
(IV.31)
(IV.32)
so that we have:
1 1 (IV.33) aj − ∆{aj } . 3 3 We use this identity to replace inf{ij } and inf{s± j } in formula (IV.28), and we obtain: 2 ∂ AG (k) + + + − + − 1 1 ≤ c3 M − sj − sj M 3 ij − 3 ∆{ij } M sj −∆{sj } M inf{sj } . inf{aj } =
− {ij },{s+ j },{sj }
(IV.34) Since inf{s− j } ≤
1 3
2 ∂+ AG (k) ≤ c3
s− j , we can write: − + 2 1 1 M 3 ij − 3 ∆{ij } M −∆{sj } M − 3 sj .
(IV.35)
− {ij },{s+ j };{sj }
− Now, we use the constraints in the sum to write, for each j ∈ {ij },{s+ j };{sj } {1, 2, 3}: + (IV.36) s− j ≥ i j − sj − 2 . We deduce that: and
1 1 + 1 − sj ≥ ij − sj − 2 3 3 3 1
M−3
s− j
1
≤ M 2M − 3
ij + 13
s+ j
.
(IV.37) (IV.38)
Replacing in equation (IV.35), we get: + − 2 + 1 1 1 ∂+ AG (k) ≤ c3 M 2 M − 3 ∆{ij } M 3 sj −∆{sj } M − 3 sj , (IV.39) − {ij },{s+ j },{sj }
and using relation (IV.33), we have: − 2 + + 2 1 1 ∂+ AG (k) ≤ c3 M 2 M − 3 ∆{ij } M inf{sj }− 3 ∆{sj } M − 3 sj . (IV.40) − {ij },{s+ j },{sj }
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+ At last, let us denote κ the value of the index j such that s+ κ = inf{sj }. We write + inf{sj } = iκ − (iκ − s+ κ ). Finally we obtain: − 2 + + 2 1 1 ∂+ AG (k) ≤ c3 M 2 M iκ M − 3 ∆{ij } M −(iκ −sκ )− 3 ∆{sj } M − 3 sj . − {ij },{s+ j },{sj }
(IV.41) − − Clearly the sums over s− 1 , s2 and s3 can be bounded by K2 = +
2
M . (M 1/3 −1)3
The decay M − 3 ∆{sj } can be used to perform the sums over s+ j for j = κ, also at 1 a cost K2 . In the same way, we use the decay M − 3 ∆{ij } to sum over the values ij , j = κ also at cost K2 per sum. It remains to sum over s+ κ:
+
M −(iκ −sκ ) ≤
0≤s+ κ ≤iκ
M . M −1
(IV.42)
At last, we have: imax (T ) 2 M imax (T )+1 ∂+ AG (k) ≤ K M iκ = K M −1 i =0
(IV.43)
κ
and we have M
imax (T )
∼
1 T
(see [1]), which proves lemma IV.1.
We have then the following lemma, which is a slight refinement of lemma IV.1: Lemma IV.2 2 Λ 2 Λ K1 ∂+ AG (πT, 1, 0) , ∂+ AG,1 (πT, 1, 0) ≤ Λ M T
(IV.44)
where K1 is the constant of Lemma IV.1. 2 Λ Proof: It is similar to the proof of Lemma IV.1. The case of ∂+ AG,1 (πT, 1, 0) can 2 Λ be decomposed into sectors exactly in the same way than ∂+ AG (πT, 1, 0) because away from the singularity and in a bounded domain in k+ , the presence of πq+ instead of cos π2 k+ does not change anything to the bounds on the propagators in sectors. Each step is then similar to the proof of of lemma IV.1 until we arrive at the last sum which we decompose in two pieces. The first piece corresponds to the domain iκ ≤ imax (T ) − Λ and gives imax (T )−Λ
M iκ
=
M imax (T )−Λ+1 − 1 M −1
(IV.45)
≤
K M M imax (T ) . = , M −1 MΛ T.M Λ
(IV.46)
iκ =0
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and the second piece corresponds to the domain imax (T ) − Λ < iκ ≤ imax (T ). In this case one should improve on equation (IV.42), to get + M −(iκ −sκ ) ≤ K.M −(iκ −(imax (T )−Λ)) , (IV.47) 0≤s+ κ ≤imax (T )−Λ
so that this second piece is bounded by imax (T )
M iκ K.M −(iκ −(imax (T )−Λ)) ≤ K .M imax (T )−Λ ,
(IV.48)
iκ =imax (T )−Λ
hence the bound for this second piece is the same as for the first. This proves the lemma. The following lemma bounds the contributions with at least one large infrared cutoff u1 on one propagator: Lemma IV.3
2 1 ∂+ AG (πT, 1, 0) ≤ K2
(IV.49)
where K2 is some new constant. Proof: The main idea is that a propagator bearing cutoff u1 = 1 − u on q+ decays 2 1 AG is now harmless, and this on a length scale O(1) in x+ , so the factor x2+ in ∂+ prevents the divergence in 1/T of the bound. 2 1 We remark first that in the amplitude ∂+ AG we can change the sum over x+ into a sum over the non zero values of x+ , because of the x2+ integrand. Since a propagator bearing cutoff u1 = 1 − u on q+ is not absolutely integrable at large q+ , we first prepare all such propagators (there are between 1 and 3 of them) using integration by parts. For any such propagator we first split the q+ integration into the two regions ∞ −1 dq + and −∞ dq+ and treat only the first term, the other one being identical. 1 Similarly we can assume that we work on a ‘right’ propagator, so that q+ = k+ −1, the other case being identical. The corresponding object is then:
D(x) = eix+
dk0
ieix+ =− x+
dk0
−2 2
−2
2
dk−
dk−
1
∞
1 ∞
dq+ dq+
[1 − u(q+ )]ei(k0 x0 +k− x− +q+ x+ ) ik0 + πq+ cos π2 k−
[π cos π2 k− ][1 − u(q+ )]ei(k0 x0 +k− x− +q+ x+ ) [ik0 + πq+ cos π2 k− ]2 u (q+ )ei(k0 x0 +k− x− +q+ x+ ) + . (IV.50) ik0 + πq+ cos π2 k−
The last term, having a compact support u is similar to the ones of the previous lemma, and left to the reader. Let us treat the first term.
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We divide it with a partition of unity into new sectors i, s+ , s− according to the size of the denominator ik0 + πq+ cos π2 k− , which is M −i , the size of q+ which is now of order M +s+ , with s+ > 0, and of k− which is of order M −s− = M −i−s+ , with s− = i + s+ . The bounds are: s+
−i
−s−
α
x− ] ≤ K|x+ |−1 M +i M s+ M −2s− e−c[M x0 +M x+ +M 2K −i−s+ −c[M −i x0 +M s+ x+ +M −s− x− ]α M ≤ e , (IV.51) π −1 Hence taking since for non zero x+ , on the tilted lattice |x+ | is bounded by 2/π. into account that the “integral” dx+ is really a discrete sum on π2 Z: −s− −i x− ]α /2 dx+ x2+ |Di,s+ ,s− (x)| ≤ KM −i−3s+ e−[M x0 +M . (IV.52)
|Di,s+ ,s− (x)|
Finally we need to optimize the dx0 and dx− using the best of the three other propagators. This leads to a bound which obviously is uniform in T . For instance if the three propagators have large infrared cutoffs u1 = 1 − u, we get the bound KM − j ij − j s+,j −2 sup s+,j +inf{i}+inf{i+s+ } i1 ,i2 ,i3 s+,1 ,s+,2 ,s+,3
≤
KM −(1/3)
j
ij −(4/3)
j
s+,j
≤ K , (IV.53)
i1 ,i2 ,i3 s+,1 ,s+,2 ,s+,3
and the other cases, when one or two propagators are of ordinary type, are similar and left to the reader. Finally we state the lemma that allows us to control the replacement of cos π2 k− by its Taylor expansion: Lemma IV.4 There exists a constant K3 > 0 such that: 2 2 ˜ AG,Λ (πT, 1, 0) ≤ K3 . ∂+ AG,Λ (πT, 1, 0) − ∂+
(IV.54)
Proof: 2 2 ˜ ∂+ AG,Λ (πT, 1, 0) AG,Λ (πT, 1, 0) − ∂+ d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x) − C˜σ1r (x)C¯˜σ2 C¯˜σ3 (x) = + {σj }, ij ,s >imax (T )−Λ j σ1 right
+2
e−(πT x0 +x+ )
d3 x x2+ Cσ1 (x)C¯σ2 (x)C¯σ3 (x) − C˜σ1 (x)C¯˜σ2r C¯˜σ3r (x) ,
+ {σj }, ij ,s >imax (T )−Λ j σ2 right
(IV.55)
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where
d3 k
uσr (k)eik.x ik0 + π(k+ − 1) cos π2 k−
(IV.56)
d3 k
uσl (k)eik.x . ik0 − π(k+ + 1) cos π2 k−
(IV.57)
Observing that there exists a constant K4 such that: π cos k+ + (π/2)(k+ − 1) ≤ K4 (k+ − 1)2 2 π cos k+ − (π/2)(k+ + 1) ≤ K4 (k+ + 1)2 2
(IV.58)
C˜σr (x)
=
C˜σ (x)
=
uniformly in k+ , we have: α Cσr() (x) − C˜σr() (x) ≤ c .M −2s+ −s− e−c dσ (x) .
(IV.59)
(IV.60)
Using the relation Cσ1 C σ2 C σ3 − C˜σ1 C˜ σ2 C˜ σ3 = (Cσ1 − C˜σ1 )C σ2 C σ3 + C˜σ1 (C σ2 − C˜ σ2 )Cσ3 + C˜σ1 C˜ σ2 (C σ3 − C˜ σ3 ) , (IV.61) ˜ we gain M −s+ ≤ M −(imax −Λ) in the power to create differences of the type C − C, counting with respect to a single propagator.
V
Main lower bound
Now we can state our main lower bound: Theorem V.1 There exists a constant K5 > 0 such that: K 2 ˜ 5 . ∂+ AG (πT, 1, 0) ≥ T
(V.62)
This theorem with the lemmas of the previous section obviously imply Theorem II.1, hence the remaining of this paper is devoted to the proof of this Theorem V.1.
V.1 Integration over k1,+ , k2,+ and k3,+ We return to equation (III.21), in which all three cutoffs uΛ have been replaced 2 ˜ 2 ˜ 2 ˜ AG (πT, 1, 0) as ∂+ AG,1 + 2∂+ AG,2 and by 1. Let us write in equation (III.21) ∂+ 2 ˜ let us consider the first term ∂+ AG,1 .
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The first propagator (after a change of variable to call the dummy variable q+ again k+ ): eik1 .x eix+ . d3 k1 (V.63) ik1,0 + πk1,+ cos π2 k1,− For cos
π
2 k1,−
+∞
−∞
dk1,+
= 0 we have:
eik1,+ x+ = ik1,0 + πk1,+ cos π2 k1,− +∞ 1 eik1,+ x+ π
. (V.64) dk1,+ ik π cos 2 k1,− −∞ k1,+ + π cos( π1,0k1,− ) 2
The corresponding residue is exp
k1,0 x+ π cos( π 2 k1,− )
. If x+ > 0, then we move the
path of integration upwards. It is oriented in the positive direction, so we get:
k1,0 x+ k1,0 > 0 2iπ exp π χ(x+ > 0) χ − . π cos( π2 k1,− ) π cos 2 k1,−
(V.65)
If x+ < 0, then the path of integration is moved downwards, and we get a minus sign owing to the negative direction. Hence: eik1,+ x+ 2i k1,0 x+ = exp dk1,+ ik1,0 + πk1,+ cos π2 k1,− cos π2 k1,− π cos π2 k1,− −∞ k1,0 k1,0 > 0 − χ(x+ < 0) χ − <0 . χ(x+ > 0) χ − π cos π2 k1,− π cos π2 k1,− (V.66)
+∞
We treat analogously the integrations over k2,+ and k3,+ . The only difference with the previous case is that these propagators were near the left singularity k+ −1, so there are some sign changes in q2,+ and q3,+ ≈ −1. We obtain:
eik2,+ x+ −2i k2,0 x+ = exp + π π π cos( π2 k2,− ) −ik2,0 − πk2,+ cos 2 k2,− cos 2 k2,− −∞ k2,0 k2,0 < 0 − χ(x+ < 0) χ >0 . χ(x+ > 0) χ π cos π2 k2,− π cos π2 k2,− (V.67)
+∞
dk2,+
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2 ˜ AG,1 (πT, 1, 0) = −8i d3 x dk1,0 dk1,− dk2,0 dk2,− dk3,0 dk3,− ∂+
k1,0 k2,0 k3,0 + π cos π k + x+ exp π cos( π ( 2 2,− ) π cos( π2 k3,− ) 2 k1,− ) π π x2+ cos 2 k1,− cos 2 k2,− cos( π2 k3,− )
ei(k1,0 +k2,0 +k3,0 +πT )x0 ei(k1,− +k2,− +k3,− )x− k1,0 k2,0 <0 χ <0 χ(x+ > 0) χ π cos π2 k1,− π cos π2 k2,− k1,0 k3,0 < 0 − χ(x+ < 0) χ >0 χ π cos π2 k3,− π cos π2 k1,− k2,0 k3,0 >0 χ >0 χ . (V.68) π cos π2 k2,− π cos π2 k3,−
V.2 Integration over x0 and k3,0 The calculation is done integrating over x0 , which leads to a delta function in the integrand, denoted with a slight abuse of notation by δ(k1,0 + k2,0 + k3,0 + πT = 0). 1 In fact, compensates the T factor of dk3,0 : remember there is a prefactor T that that dk3,0 means precisely: 2πT k3,0 ∈πT +2πT Z . This yields: 2 ˜ ∂+ AG,1 (πT, 1, 0)
= −8i
x2+
e
dx+ dx−
dk1,0 dk1,− dk2,0 dk2,− dk3,0 dk3,−
k1,0 k k + π cos( 2,0 + π cos( 3,0 πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−
x+
cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )
ei(k1,− +k2,− +k3,− )x− δ(k1,0 + k2,0 + k3,0 + πT = 0)
k2,0 k3,0 k1,0 < 0) χ( < 0)χ( < 0) π cos( π2 k1,− ) π cos( π2 k2,− ) π cos( π2 k3,− ) k2,0 k3,0 k1,0 > 0) χ( > 0)χ( > 0) . − χ(x+ < 0) χ( π cos( π2 k1,− ) π cos( π2 k2,− ) π cos( π2 k3,− ) (V.69) χ(x+ > 0) χ(
At this stage, we can use the delta function to integrate, for instance, over k3,0 : 2 ˜ ∂+ AG,1 (πT, 1, 0) = −8i
x2+
e
dx+ dx−
dk1,0 dk1,− dk2,0 dk2,− dk3,−
k1,0 k k +k2,0 +πT + π cos( 2,0 − π1,0 πk ) π cos( π k1,− ) cos( π k3,− ) 2 2 2,− 2
cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )
x+
ei(k1,− +k2,− +k3,− )x−
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k2,0 k1,0 < 0 χ < 0 × χ(x+ > 0) χ π cos( π2 k1,− ) π cos( π2 k2,− )
k1,0 + k2,0 + πT k1,0 χ > 0 − χ(x > 0 < 0) χ + π cos( π2 k3,− ) π cos( π2 k1,− )
k2,0 k1,0 + k2,0 + πT χ > 0 χ < 0 . (V.70) π cos( π2 k2,− ) π cos( π2 k3,− )
V.3 Simplification This rather complicated expression can be slightly simplified. Indeed, if we perform the change of variables: x+ = −x+ k = −k1,0 (V.71) 1,0 k2,0 = −k2,0 the integral
dx+ dx−
e
dk1,0 dk1,− dk2,0 dk2,− dk3,− x2+
k1,0 k k +k2,0 +πT + π cos( 2,0 − π1,0 πk ) π cos( π k1,− ) cos( π k3,− ) 2 2 2,− 2
k1,0 < 0) χ χ(x > 0 + cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− ) π cos( π2 k1,− )
k1,0 + k2,0 + πT k2,0 > 0 < 0 χ (V.72) χ π cos( π2 k2,− ) π cos( π2 k3,− ) x+
becomes: dx+ dx− dk1,0 dk1,− dk2,0 dk2,− dk3,− x2 +
e
−πT k1,0 k k +k2,0 + π cos( 2,0 − π1,0 πk ) π cos( π k1,− ) cos( π k3,− ) 2 2 2,− 2
k1,0 < 0 cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− ) π cos( π2 k1,− )
k2,0 k1,0 + k2,0 − πT χ <0 χ > 0 . (V.73) π cos( π2 k2,− ) π cos( π2 k3,− ) x+
χ(x+ > 0) χ
2 ˜ Consequently the previous expression of ∂+ AG,1 (πT, 1, 0) can be factorized: 2 ˜ AG,1 (πT, 1, 0) = −8i ∂+
x2+
e
dx+ dx−
dk1,0 dk1,− dk2,0 dk2,− dk3,−
k1,0 k k1,0 +k2,0 + π cos( 2,0 − π cos( πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−
cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )
x+
ei(k1,− +k2,− +k3,− )x−
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k2,0 k1,0 < 0 χ < 0 × χ(x+ > 0) χ π cos( π2 k1,− ) π cos( π2 k2,− )
−T x+ k1,0 + k2,0 + πT π e cos( 2 (k3,− )) χ > 0 π cos( π2 k3,− )
T x+ k1,0 + k2,0 − πT cos( π (k3,− )) 2 −e >0 . (V.74) χ π cos( π2 k3,− )
VI Integration over x− and k3,− We now are going to perform the integration over x− , which will provide a conservation rule for the moments k1,0 , k2,0 and k3,0 , but onlymodulo 2. To understand that, remember that dx+ dx− means more precisely: (x ,x )∈( π Z)2 , where the +
−
2
prime in the sum means that one has to respect a parity condition between x+ and x− . By slight abuse of language, we say that x+ and x−have thesame parity when x+ + x− ∈ πZ. So (x ,x )∈( π Z)2 does not mean: x+ ∈ π Z x− ∈ π Z but + 2 2 − 2 + . Now, π π x+ ∈πZ x− ∈πZ x+ ∈ +πZ x− ∈ +πZ 2
2
ei(k1,− +k2,− +k3,− )x− = δ(k1,− + k2,− + k3,− = 0[2])
(VI.75)
x− ∈πZ
where by δ(k1,− + k2,− + k3,− = 0[2]), we denote: Then it is clear that
n∈Z
δ(k1,− + k2,− + k3,− = 2n).
π
ei(k1,− +k2,− +k3,− )x− = ei 2 (k1,− +k2,− +k3,− ) δ(k1,− + k2,− + k3,− = 0[2]) .
x− ∈ π 2 +πZ
(VI.76) π Indeed, the factor ei 2 (k1,− +k2,− +k3,− ) can take only two values: 1 if k1,− + k2,− + k3,− = 0[4], and −1 if k1,− +k2,− +k3,− = 2[4]. Hence it is convenient to distinguish these two cases and write: δ(k1,− + k2,− + k3,− = 0[2]) = δ(k1,− + k2,− + k3,− = 0[4]) + δ(k1,− + k2,− + k3,− = 2[4]) (VI.77) and π
ei 2 (k1,− +k2,− +k3,− ) δ(k1,− + k2,− + k3,− = 0[2]) = δ(k1,− + k2,− + k3,− = 0[4]) − δ(k1,− + k2,− + k3,− = 2[4]) . (VI.78)
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At this stage, we can gather the previous remarks in the following formula : 2 ˜ AG,1 (πT, 1, 0) = −8i ∂+
x2+
e
dk1,0 dk2,0 dk1,− dk2,− dk3,−
∗ x+ ∈ π 2N
k1,0 k k1,0 +k2,0 + π cos( 2,0 − π cos( πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−
x+
δ(k1,− + k2,− + k3,− = 0[4]) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )
k2,0 k1,0 < 0 χ < 0 χ π cos( π2 k1,− ) π cos( π2 k2,− )
−T x+ k1,0 + k2,0 + πT π e cos( 2 (k3,− )) χ > 0 π cos( π2 k3,− )
T x+ k1,0 + k2,0 − πT cos( π (k3,− )) 2 −e >0 χ π cos( π2 k3,− ) − 8i dk1,0 dk2,0 dk1,− dk2,− dk3,−
x2+
e
[χ(x+
∗ x+ ∈ π 2N k1,0 k k1,0 +k2,0 + π cos( 2,0 − π cos( πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−
x+
δ(k1,− + k2,− + k3,− = 2[4]) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )
k2,0 k1,0 <0 χ <0 even) − χ(x+ odd)]χ π cos( π2 k1,− ) π cos( π2 k2,− )
−T x+ k1,0 + k2,0 + πT π e cos( 2 (k3,− )) χ >0 π cos( π2 k3,− )
T x+ k1,0 + k2,0 − πT π −e cos( 2 (k3,− )) χ > 0 . (VI.79) π cos( π2 k3,− )
Then we can perform the integration over k3,− . Formally, we only need to replace cos( π2 k3,− ) by cos( π2 (k1,− +k2,− )) for the first piece and with − cos( π2 (k1,− +k2,− )) for the second piece. We obtain: 2 ˜ AG,1 (πT, 1, 0) ∂+
= −8i
dk1,0 dk2,0 dk1,− dk2,−
∗ x+ ∈ π 2N
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
k1,0 χ <0 cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) π cos( π2 k1,− )
−T x+ k2,0 k1,0 + k2,0 + πT cos( π (k1,− +k2,− )) 2 χ < 0 e > 0 χ π cos( π2 k2,− ) π cos( π2 (k1,− + k2,− ))
T x+ k1,0 + k2,0 − πT π − e cos( 2 (k1,− +k2,− )) χ > 0 π cos( π2 (k1,− + k2,− ))
x2+
e
x+
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e
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dk1,0 dk2,0 dk1,− dk2,−
∗ x+ ∈ π 2N k1,0 k k +k2,0 + π cos( 2,0 + π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− ))
k2,0 k1,0 < 0 χ < 0 even) − χ(x+ odd)]χ π cos( π2 k1,− ) π cos( π2 k2,− )
T x+ k1,0 + k2,0 + πT π e cos( 2 (k1,− +k2,− )) χ < 0 π cos( π2 (k1,− + k2,− ))
−T x+ k1,0 + k2,0 − πT π − e cos( 2 (k1,− +k2,− )) χ < 0 . (VI.80) π cos( π2 (k1,− + k2,− ))
2 ˜ Now it is clear that ∂+ AG,1 (πT, 1, 0) is a purely imaginary number. The first piece gives the leading behavior as T → 0. Indeed the second piece is much smaller, thanks to the compensation in [χ(x+ even) − χ(x+ odd)]. Indeed the sum
∗ x+ ∈ π 2N
x2+ [χ(x+ even) − χ(x+ odd)] . . .
(VI.81)
can be written as a sum of two terms of the type
dke−2A(k)n [(2n)2 − (2n + 1)2 e−A(k) ]B(k)
(VI.82)
n∈N∗
where A and B are independent of n and A(k) > 0. Then we can decompose the remaining integrals dk into two zones, according to whether A(k) ≥ T 1/3 or A(k) ≤ T 1/3 . In the first zone we do not need to exploit the subtraction, but we 1/3 have simply n∈N∗ n2 e−2T n ≤ c.T −2/3 << T −1 , and in the second zone, we use |(2n)2 − (2n + 1)2 e−A(k) | ≤ 4n + 1 + (2n + 1)2 A(k) ≤ 4n + 1 + (2n + 1)2 T 1/3 . The first term in 4n + 1 is then bounded with the same techniques than Lemma IV.1, + − + − but the factor M inf{ij }+3 inf{sj }+inf{sj } is replaced by M inf{ij }+2 inf{sj }+inf{sj } and the bound corresponding to equation (IV.40) gives now
1
2
+
1
M − 3 ∆{ij } M − 3 ∆{sj } M − 3
s− j
≤ 0(1) ,
(VI.83)
− {ij },{s+ j },{sj }
hence this piece does not diverge at all when T → 0. Finally the piece with the factor (2n+1)2 T 1/3 is similar to previous pieces, except for the new factor T 1/3 , so that it is bounded in the manner of Lemma IV.1 by a factor c.T −1 T 1/3 = c.T −2/3 .
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So we are left to study:
A1 (T ) = −8i
∗ x+ ∈ π 2N
e
dk1,0 dk2,0 dk1,− dk2,− x2+
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
k1,0 <0 π cos( π2 k1,− )
χ cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− ))
−T x+ k2,0 k1,0 + k2,0 + πT cos( π (k1,− +k2,− )) 2 χ < 0 e > 0 χ π cos( π2 k2,− ) π cos( π2 (k1,− + k2,− ))
T x+ k1,0 + k2,0 − πT cos( π (k1,− +k2,− )) −e 2 >0 . (VI.84) χ π cos( π2 (k1,− + k2,− ))
VII Leading contribution VII.1 Symmetry properties Henceforward, we shall denote the integrand by F (x+ , k1,0 , k2,0 , k1,− , k2,− ) so that: A1 (T ) = −8i
dk1,0 dk2,0 dk1,− dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) .
∗ x+ ∈ π 2N
(VII.85) The couple of variables of integration (k1,− , k2,− ) describes the square [−2, 2]2 . To pursue the calculation, we shall make a partition of [−2, 2]2 , according to the signs of cos( π2 k1,− ), cos( π2 k2,− ) and cos( π2 (k1,− + k2,− )). This partition is represented in Figure 4. The signs of the three cosines determine eight cases we can discuss separately. In fact, it is possible to restrict the domain of integration thanks to symmetries of the integrand involving the variables k1,− and k2,− together with the variables k1,0 and k2,0 , which describe independently the set πT + 2πT Z. It is evident, by the parity of the cosine function, that the integrand is invariant under the replacement k1,− → −k1,− and k2,− → −k2,− , which corresponds to the central symmetry with respect to the origin (0,0). Hence we have: A1 (T ) = −16i
∗ x+ ∈ π 2N
dk1,0 dk2,0
2
−2
dk1,−
0
2
dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ). (VII.86)
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k−
1 +2 0 0 1 00000000000000 11111111111111 0 1 00000000000000 11111111111111 000000000000001111 11111111111111 (−,−,+) 00000000000000 11111111111111 0(+,−,−) 0000 1 0000 1111 00000000000000 00000000000000 11111111111111 (−,−,+)11111111111111 0 1 0000 1111 00000000000000 11111111111111 00000000000000 11111111111111 0 1 00000000000000 11111111111111 0000 1111 00000000000000 11111111111111 (−,−,−) (+,−,+) 1 0 00000000000000 11111111111111 0000 000000000000001111 11111111111111 00000000000000 11111111111111 0 1 0000 1111 00000000 11111111 00000000000000 11111111111111 0 1 (+,+,−) (−,+,+) 00000000 11111111 00000000000000 11111111111111 0 1 00000000 00000000000000 11111111111111 011111111 1 00000000 (−,+,−) 00000000000000 11111111111111 011111111 1 00000000 11111111 00000000000000 11111111111111 0 1 00000000 11111111 00000000000000 11111111111111 11111111111111111 00000000000000000 (+,+,+) 111k + 000 00000000 11111111 00000000 11111111 00000000000000 11111111111111 0 1 00000000 11111111 −2 +2 00000000000000 11111111111111 0 1 00000000 11111111 00000000000000 11111111111111 00000000 0 1 (−,+,−)11111111 00000000000000 11111111111111 00000000 11111111 0 1 00000000000000 11111111111111 00000000 11111111 0 (+,+,−) 1 (−,+,+) 00000000000000 11111111111111 00000000 11111111 00000000000000 11111111111111 0 1 00000000000000 11111111111111 00000000000000 11111111111111 00000000 11111111 00000000000000 11111111111111 0000 1111 0 1 00000000000000 11111111111111 (+,−,+) (−,−,−) 00000000000000 11111111111111 0000 1111 0 1 00000000000000 11111111111111 00000000000000 (−,−,+) 11111111111111 0000 1111 00000000000000 11111111111111 0 1 00000000000000 11111111111111 (+,−,−) 00000000000000 11111111111111 0000 1111 0 1 00000000000000 11111111111111 (−,−,+) 00000000000000 11111111111111 0000 1111 0 00000000000000 11111111111111 00000000000000 11111111111111 0000 1 1111 0 1 0 1 −2 0 1 0 1 Figure 4. The domain of integration in (k+ , k− ). Symmetry properties of F (x+ , k1,0 , k2,0 , k1,− , k2,− ) can be exploited further. The above integral may be separated into two pieces: 0 dk1,− dk1,0 dk2,0 A1 (T ) = −16i −2
∗ x+ ∈ π 2N
+
0
2
dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− )
dk1,0 dk2,0
∗ x+ ∈ π 2N
2
0
dk1,−
0
2
dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) . (VII.87)
For the first integral, one can easily verify that the integrand F (x+ , k1,0 , k2,0 , k1,− , k2,− ) is invariant under the change of variables: k1,0 = k2,0 , k2,0 = k1,0 , k1,− = −k2,− , k2,− = −k1,− .
(VII.88)
We get:
dk1,0 dk2,0 2
0
−2
dk1,−
dk1,0 dk2,0
0
2
dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) =
0 −2
dk1,−
2
−k1,−
dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) .
(VII.89)
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We treat analogously the other integral; we set: k1,0 = k2,0 , k2,0 = k1,0 , k1,− = k2,− , k2,− = k1,− .
Hence: dk1,0 dk2,0 2
0
2
dk1,−
dk1,0 dk2,0
0
2
(VII.90)
2
0
dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) =
2
dk1,−
dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) .
(VII.91)
k1,−
Finally, we have established owing to symmetry properties that: A1 (T ) = −32i dk1,0 dk2,0 dk1,− dk2,− F (x+ , k1,0 , k2,0 , k1,− , k2,− ) , (VII.92) T
the domain of integration in (k1,− , k2,− ) being the triangle T delimited by the lines k2,− = 2, k2,− = k1,− and k2,− = −k1,− .
VII.2 Discussion of the various cases VII.2.1 The (+, +, +) case As we have said, it is now convenient to carry a discussion about the signs of cos( π2 k1,− ), cos( π2 k2,− ) and cos( π2 (k1,− + k2,− )), which allows us to perform explicitly the summation over k1,0 and k2,0 in each case. We first begin with the case: > 0 cos( π2 k1,− ) > 0 , cos( π2 k2,− ) (VII.93) cos( π2 (k1,− + k2,− )) > 0 that we will denote as (+, +, +). The corresponding contribution to A1 (T ) is: (+,+,+) A1 (T ) = −32i dk1,− dk2,− dk1,0 dk2,0 ∗ x+ ∈ π 2N
e x2+
T(+,+,+)
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
χ(k1,0 < 0) χ(k2,0 < 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > −πT ) T x+ π −e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > πT ) , (VII.94)
where T(+,+,+) denotes the subset of T where the signs of the cosines are (+, +, +) respectively. Since the conditions k1,0 < 0, k2,0 < 0 and k1,0 + k2,0 > ±πT are (+,+,+) =0. incompatible, A1
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VII.2.2 The (+, +, −) case Let us consider the case:
> cos( π2 k1,− ) > cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) <
0 0 0
,
(VII.95)
corresponding to the sign configuration (+, +, −). We have: (+,+,−) A1 (T ) = −32i dk1,− dk2,− dk1,0 dk2,0 ∗ x+ ∈ π 2N
e x2+
T(+,+,−)
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
χ(k1,0 < 0) χ(k2,0 < 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < −πT ) T x+ π − e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < πT ) . (VII.96)
The conditions χ(k1,0 + k2,0 < ±πT ) can obviously be omitted. We must compute the following expression: 2
(2πT )
e
+∞ +∞
e
n=0 p=0 −(2p+1) cos( π1k 2
−(2n+1)
2,− )
1 − cos( π k 1 +k cos( π k1,− ) 2,− ) 2 2 1,−
1 1,− +k2,− )
T x+
− cos( π k 2
T x+
−T x+ T x+ π π e cos( 2 (k1,− +k2,− )) − e cos( 2 (k1,− +k2,− )) ,
(VII.97) which gives: 2e
(2πT )
−
1 + cos( π1k − cos( π k 1 +k ) cos( π k1,− ) 2,− ) 2 2 2,− 2 1,−
1−e
−2 cos( π1k 2
1,− )
1 1,− +k2,− )
− cos( π k 2
T x+
T x+
2T x+ π 1 − e cos( 2 (k1,− +k2,− ))
−2 1−e
1 − cos( π k 1 +k cos( π k2,− ) 2,− ) 2 2 1,−
T x+
.
(VII.98)
This is clearly a positive real number, and therefore we conclude that (+,+,−)
iA1
(T ) ≤ 0 .
(VII.99)
Indeed, the minus sign of the prefactor −32i is compensated by the minus sign of the product cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k1,− + k2,− ).
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VII.2.3 The (+, −, +) case We now consider the (+, −, +) case. The corresponding contribution writes: (+,−,+) A1 dk1,0 dk2,0 (T ) = −32i dk1,− dk2,− ∗ x+ ∈ π 2N
x2+
e
T(+,−,+)
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
χ(k1,0 < 0)χ(k2,0 > 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > −πT ) T x+ cos( π (k1,− +k2,− )) 2 −e χ(k1,0 + k2,0 > πT ) . (VII.100)
Here like in all the other cases, we have to sum geometric sequences whose ratio is explicitly strictly smaller than 1. This facilitates the discussion of the signs of the , corresponding quantities, as we shall see. If we perform the summation over k1,0
we are lead to a geometric sequence whose ratio is e which leads to a factor −2 1−e
T x+ Tx − cos( π k ++k cos( π k1,− ) 2,− ) 2 2 1,−
−2
T x+ Tx − cos( π k ++k cos( π k1,− ) 2,− ) 2 2 1,−
,
!−1
whose sign is not uniform in (k1,− , k2,− ). Consequently we introduce the variable s = k1,0 + k2,0 and replace k2,0 by s − k1,0 . We must compute:
k1,0 s−k + π cos( π1,0 − π cos( π (ks +k k ) π cos( π k1,− ) 1,− 2,− )) 2 2 2,− 2
x+
χ(k1,0 < 0)χ(s > k1,0 ) dk1,0 ds e −T x+ T x+ π π e cos( 2 (k1,− +k2,− )) χ(s > −πT ) − e cos( 2 (k1,− +k2,− )) χ(s > πT ) . (VII.101) The variable s describes the set 2πT Z and the condition χ(s > k1,0 ) can be omitted. Thus the previous expression writes: (2πT )2
+∞
e
−(2n+1)
T x+ Tx − cos( π k+ ) cos( π k1,− ) 2 2 2,−
n=0
e
−T x+ cos( π (k1,− +k2,− )) 2
+∞ p=0
−e
−T x+
cos( π (k1,− +k2,− )) 2
+∞ p=0
e
e
−2p cos( π (k
1 − cos( π1k ) 1,− +k2,− )) 2 2,−
2
−2p cos( π (k 2
1 − cos( π1k ) 1,− +k2,− )) 2 2,−
T x+
T x+
(VII.102)
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which is equal to: 2
(2πT )
e
−
1 − cos( π1k + cos( π (k 1 +k ) cos( π k1,− ) 1,− 2,− )) 2 2 2,− 2
−2 1−e
1 − cos( π1k ) cos( π k1,− ) 2 2 2,−
−2 1−e
T x+
T x+ 2T x+ π
1 − e cos( 2 k2,− )
1 − cos( π1k ) cos( π (k1,− +k2,− )) 2 2 2,−
T x+
. (VII.103)
This quantity is positive, thus the conclusion follows: (+,−,+)
iA1
(T ) ≤ 0 .
(VII.104)
VII.2.4 The (+, −, −) case Let us examine now the (+, −, −) case. The contribution is: (+,−,−)
A1
(T ) = −32i
dk1,0 dk2,0
∗ x+ ∈ π 2N
e x2+
dk1,− dk2,−
T(+,−,−)
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
χ(k1,0 < 0)χ(k2,0 > 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < −πT ) T x+ π − e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < πT ) . (VII.105)
We set k1,0 = s − k2,0 and we compute:
ds dk2,0 e
s−k2,0 k + π cos( 2,0 − π cos( π (k s +k πk ) π cos( π k1,− ) 1,− 2,− )) 2 2 2,− 2
x+
χ(s < k2,0 )χ(k2,0 > 0)
−T x+ T x+ π π e cos( 2 (k1,− +k2,− )) χ(s < −πT ) − e cos( 2 (k1,− +k2,− )) χ(s < πT ) . (VII.106)
The condition χ(s < k2,0 ) may be omitted and we must evaluate: (2πT )2
+∞
e
(2n+1)
1 − cos( π1k ) cos( π k2,− ) 2 2 1,−
n=0
+∞ −T x+ −2p π e cos( 2 (k1,− +k2,− )) e p=1
T x+
1 − cos( π (k 1 +k cos( π k1,− ) 1,− 2,− )) 2 2
T x+
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−e
T x+ cos( π (k1,− +k2,− )) 2
+∞
e
−2p cos( π1k 2
1,− )
475 1 1,− +k2,− ))
− cos( π (k 2
T x+
. (VII.107)
p=0
We find:
(2πT )2
e
−
1 − cos( π1k − cos( π (k 1 +k ) cos( π k1,− ) 1,− 2,− )) 2 2 2,− 2
1−e
−2 cos( π1k 2
1,− )
2
T x+
− cos( π1k
T x+
2,− )
−2T x+ π e cos( 2 k1,− ) − 1 1−e
−2 cos( π1k
1,−
2
1 1,− +k2,− ))
− cos( π (k ) 2
. (VII.108)
T x+
This is a negative number, therefore (+,−,−)
iA1
(T ) ≤ 0 .
(VII.109)
VII.2.5 The (−, +, +) and (−, +, −) cases There is no discussion to carry out: in fact, for (k1,− , k2,− ) ∈ T , we have never cos( π2 k1,− ) < 0, cos( π2 k2,− ) > 0 and cos( π2 (k1,− + k2,− )) < 0 simultaneously. We also conclude in the same way for the (−, +, −) case. VII.2.6 The (−, −, +) case
(−,−,+)
A1
(T ) = −32i
∗ x+ ∈ π 2N
e x2+
dk1,0 dk2,0
dk1,− dk2,−
T(−,−,+)
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
χ(k1,0 > 0)χ(k2,0 > 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > −πT ) T x+ π −e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 > πT ) . (VII.110)
We remark that the conditions χ(k1,0 + k2,0 > ±πT ) are superfluous, and that there is no need to introduce the variable s.
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We have: 2
(2πT )
+∞
e
(2n+1)
1 − cos( π (k 1 +k cos( π k1,− ) 1,− 2,− )) 2 2
n=0 +∞
e
T x+
1 − cos( π (k 1 +k cos( π k2,− ) 1,− 2,− )) 2 2
(2p+1)
T x+
p=0
−T x+ T x+ cos( π (k1,− +k2,− )) cos( π (k1,− +k2,− )) 2 2 −e e =
(2πT )2
e
−
1 − cos( π1k − cos( π1k ) ) cos( π (k1,− +k2,− )) 2 2 1,− 2 2,−
π
T x+
−2T x+
e cos( 2 (k1,− +k2,− )) − 1 1−e 1−e
−2 cos( π (k
−2 cos( π (k
1 − cos( π1k ) 1,− +k2,− )) 2 1,−
2
1 − cos( π1k ) 1,− +k2,− )) 2 2,−
2
T x+
. (VII.111) T x+
This quantity is negative and we conclude that (−,−,+)
iA1
(T ) ≤ 0 .
(VII.112)
VII.2.7 The (−, −, −) case We finally discuss the last case: (−,−,−)
A1
(T ) = −32i
dk1,0 dk2,0
∗ x+ ∈ π 2N
e x2+
dk1,− dk2,−
T(−,−,−)
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
χ(k1,0 > 0)χ(k2,0 > 0) cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− )) −T x+ π e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < −πT ) T x+ π −e cos( 2 (k1,− +k2,− )) χ(k1,0 + k2,0 < πT ) . (VII.113)
But it is clear that the conditions k1,0 > 0, k2,0 > 0 and k1,0 + k2,0 < ±πT are incompatible (as in the (+, +, +) case), hence (−,−,−)
A1
(T ) = 0 .
(VII.114)
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Lemma VII.1 There exists a constant K > 0 such that: K (+,+,−) (+,−,+) (+,−,−) (−,−,+) . (T ) + A1 (T ) + A1 (T ) + A1 (T ) > A1 T
(VII.115)
Proof: As each one of the quantities are purely imaginary, with non-negative imag(+,+,−) inary part, it is sufficient to prove the inequality |A1 (T )| > KT1 for some constant K1 . We have: (+,+,−)
|A1
(T )| = 32(2πT )2 dk1,− dk2,−
∗ x+ ∈ π 2N
T (+,+,−)
− 2 e x+
1 cos π k1,− 2
1−e
1
cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,−
+ cos π1k − cos π (k 1 +k T x+ )
−2 cos
2,−
2
1 πk 2 1,−
1,−
2
− cos
2,−
1 π (k 1,− +k2,− ) 2
π
+ k2,− ))
T x+
2T x+
1 − e cos 2 (k1,− +k2,− ) 1−e As 1 − e
−2 cos
and 1 − e
− cos
1 πk 2 2,−
1 π (k 1,− +k2,− ) 2
− cos
1 π (k 1,− +k2,− ) 2
(T )| ≤ 32(2πT )2
∗ x+ ∈ π 2N
1 πk 2 2,−
− cos
1 π (k 1,− +k2,− ) 2
. (VII.116)
T x+
1 πk 2 1,−
−2 cos
(+,+,−)
|A1
−2 cos
T x+
T x+
≤1 ≤ 1, we get:
T (+,+,−)
dk1,− dk2,−
1 cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− ))
2T x+ 1 + cos π1k − cos π (k 1 +k T x+ π k1,− ) 2 − cos π 2,− 1,− 2,− 2 2 2 1 − e cos 2 (k1,− +k2,− ) . (VII.117) x+ e As we are seeking a lower bound, we can restrict the integration over the open (+,+,−) ⊂ T (+,+,−) ,where is a strictly domain T (+,+,−) to a compact T positive 1 constant (for example = 10 ), in which we have cos π2 k1,− ≥ , cos π2 k2,− ≥
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(+,+,−) and cos π2 (k1,− + k2,− ) ≥ . For (k1,− , k2,− ) ∈ T , we have:
0<
x2+ .e
−
1 cos π k1,− 2
+ cos
1 πk 2 2,−
− cos
1 π (k 1,− +k2,− ) 2
T x+
2T x+ π 1 − e cos 2 (k1,− +k2,− ) ≤ e−3T x+ . (VII.118)
By Lebesgue domination theorem, we can invert x+ and (+,+,−) dk1,− dk2,− T and write: ! 3 (+,+,−) 2 |A1 (T )| ≥ 32(2πT ) dk1,− dk2,− x2+ .e− T x+ 1 − e−2T x+ , (+,+,−)
T
∗ x+ ∈ π 2N
(VII.119) or: (+,+,−)
|A1
(T )| ≥ 32π 4 T 2
+∞
3π n2 e− 2 T n 1 − e−πT n .
(VII.120)
n=0
Now we use the formula: +∞
n2 e−an =
n=0
to write: (+,+,−)
|A1
e−a + e−2a (for a > 0) (1 − e−a )3
(VII.121)
− 3π 2 T
e (T )| ≥ 32π 4 T 2
≥ 32πT −1
+e
1−e
− 3π T
− 3π 2 T
3 −
e
−( 3π 2 +π )T
−( 3π +2π )T
+e
3 −( 3π 1 − e 2 +π)T
2 + O(T ) 2 + O(T ) − 3 (3/2 ) (3/2 + 1)3
(VII.122)
.
(VII.123)
Hence for T small enough, we obtain the desired result: (+,+,−)
|A1
(T )| ≥ KT −1 .
(VII.124)
for some explicit K and the lemma is proven.
VIII Study of the other configurations We now are going to treat the other configuration, corresponding to: k1,+ ≈ −1 k1,+ ≈ −1 and k2,+ ≈ 1 k2,+ ≈ −1 k3,+ ≈ −1 k3,+ ≈ 1
(VIII.125)
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479
2 which are equal and form the term called 2∂+ AG,2 (πT, 1, 0). Let us concentrate on the first case. We have to consider the propagator:
+∞
−∞
dk1,+
eik1,+ x+ = ik1,0 − πk1,+ cos( π2 k1,− ) +∞ eik1,+ x+ −1 dk 1,+ ik1,0 π cos( π2 k1,− ) −∞ k1,+ − π
. (VIII.126)
π cos( 2 k1,− )
The pole of the integrand is e
k1,0 x+ − π cos( πk ) 2 1,−
ik1,0 π cos( π 2 k1,− )
and the corresponding residue writes
. Therefore we have:
+∞
k1,0 x+ −2i eik1,+ x+ − π cos( πk ) 2 1,− = e dk1,+ π π ik1,0 − πk1,+ cos( 2 k1,− ) cos( 2 k1,− ) −∞
k1,0 k1,0 > 0 − χ(x < 0 . < 0)χ χ(x+ > 0)χ + π cos( π2 k1,− ) π cos( π2 k1,− ) (VIII.127)
Now, let us consider the integration over k2,+ . We have:
+∞
−∞
dk2,+
eik2,+ x+ = −ik2,0 + πk2,+ cos( π2 k2,− ) +∞ 1 eik2,+ x+ dk2,+ π ik2,0 π cos( 2 k2,− ) −∞ k2,+ − π
. (VIII.128)
π cos( 2 k2,− )
In fact, the only change with the previous case is a global change of sign. We can immediately write:
+∞
k2,0 x+ 2i eik2,+ x+ − π = e π cos( 2 k2,− ) π π −ik + πk cos( k ) cos( k ) 2,0 2,+ 2,− 2,− −∞ 2 2
k2,0 k2,0 > 0 − χ(x+ < 0)χ <0 . χ(x+ > 0)χ π cos( π2 k2,− ) π cos( π2 k2,− ) (VIII.129)
dk2,+
For the integration over k3,+ , we have cos( π2 k2,+ ) ≈
+∞
−∞
π 2 (k2,+
−1 eik3,+ x+ = dk3,+ π −ik3,0 − πk3,+ cos( 2 k3,− ) π cos( π2 k3,− )
+ 1) and we consider:
+∞
−∞
eik3,+ x+ k3,+ +
ik3,0 π cos( π 2 k3,+ )
.
(VIII.130)
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Ann. Henri Poincar´e
k3,0 x+ π
ik
In this case, the pole is − π cos( π3,0k3,− ) and the residue e π cos( 2 k3,− ) . Therefore the 2 above integral writes:
+∞
k3,0 x+ eik3,+ x+ −2i π cos( π k3,− ) 2 = e −ik3,0 − πk3,+ cos( π2 k3,− ) cos( π2 k3,− ) −∞
k3,0 k3,0 χ(x+ > 0)χ < 0)χ < 0 − χ(x > 0 . + π cos( π2 k3,− ) π cos( π2 k3,− ) (VIII.131)
dk3,+
Hence we obtain: 2 ˜ AG,2 (πT, 1, 0) = −8i ∂+
k
k
− π cos( 1,0 πk 2
e
dk1,0 dk1,− dk2,0 dk2,− dk3,0 dk3,− x2+
dx 1,− )
− π cos( 2,0 πk 2
2,− )
k
+ π cos( 3,0 πk 2
3,− )
x+
cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− ) ei(k1,0 +k2,0 +k3,0 +πT )x0 ei(k1,− +k2,− +k3,− )x−
k2,0 k1,0 >0 χ >0 χ(x+ > 0)χ π cos( π2 k1,− ) π cos( π2 k2,− )
k3,0 k1,0 χ < 0 − χ(x+ < 0) χ <0 π cos( π2 k3,− ) π cos( π2 k1,− )
k2,0 k3,0 χ < 0 χ > 0 . (VIII.132) π cos( π2 k2,− ) π cos( π2 k3,− )
Then we integrate over x0 and perform the sum over k3,0 : 2 ˜ AG,2 (πT, 1, 0) = −8i ∂+
x2+
e
k
− π cos( 1,0 πk 2
1,−
dx+ dx− k
− π cos( 2,0 πk ) 2
2,−
− )
dk1,0 dk1,− dk2,0 dk2,− dk3,− k1,0 +k2,0 +πT π cos( π k3,− ) 2
x+
ei(k1,− +k2,− +k3,− )x− cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )
k2,0 k1,0 >0 χ >0 χ(x+ > 0) χ π cos( π2 k1,− ) π cos( π2 k2,− )
k3,0 k1,0 χ < 0 − χ(x+ < 0) χ <0 π cos( π2 k3,− ) π cos( π2 k1,− )
k2,0 k3,0 χ < 0 χ > 0 . (VIII.133) π cos( π2 k2,− ) π cos( π2 k3,− )
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Thanks to the change of variables x+ = −x+ , k1,0 = −k1,0 , k2,0 = −k2,0 , we get: 2 ˜ ∂+ AG,2 (πT, 1, 0) = −8i
x2+
e
−
dx+ dx−
dk1,0 dk1,− dk2,0 dk2,− dk3,−
k1,0 k k1,0 +k2,0 + π cos( 2,0 + π cos( πk πk ) ) π cos( π k1,− ) 2 2 2,− 2 3,−
x+
ei(k1,− +k2,− +k3,− )x− cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 k3,− )
k2,0 k1,0 >0 χ >0 χ(x+ > 0) χ π cos( π2 k1,− ) π cos( π2 k2,− ) −T x+
k1,0 + k2,0 + πT π e cos( 2 k3,− ) χ > 0 π cos( π2 k3,− )
T x+ k1,0 + k2,0 − πT π −e cos( 2 k3,− ) χ > 0 . (VIII.134) π cos( π2 k3,− )
Then we perform the sum over x− as previously and integrate over k3,− . There is a small contribution with a compensating factor [χ(x+ even)−χ(x+ odd)] that can be bounded as in Section VI, and we have again to study the dominant contribution: A2 (T ) = −8i dk1,0 dk2,0 dk1,− dk2,− x2+
e
−
∗ x+ ∈ π 2N
k1,0 k k +k2,0 + π cos( 2,0 + π cos( π1,0 πk ) (k1,− +k2,− )) π cos( π k1,− ) 2 2 2,− 2
x+
k1,0 >0 π cos( π2 k1,− )
χ cos( π2 k1,− ) cos( π2 k2,− ) cos( π2 (k1,− + k2,− ))
−T x+ k2,0 k1,0 + k2,0 + πT cos( π (k1,− +k2,− )) 2 χ > 0 > 0 e χ π cos( π2 k2,− ) π cos( π2 (k1,− + k2,− ))
T x+ k1,0 + k2,0 − πT π − e cos( 2 (k1,− +k2,− )) χ > 0 . (VIII.135) π cos( π2 (k1,− + k2,− )) Fortunately, we do not have to carry again a discussion about the signs of the three cosines. In fact, we can remark that A1 (T ) = A2 (T ). To see that, let us perform the following change of variables in A1 (T ): $ k1,− = k1,− +2 , (VIII.136) k2,− = k2,− +2 to obtain: A2 (T ) = −8i
e
∗ x+ ∈ π 2N
dk1,0 dk2,0
dk1,− dk2,− x2+
T
k1,0 k k +k2,0 + π cos( 2,0 − π cos( π1,0 π k ) ) (k +k )) π cos( π k 1,− 2,− 2 1,− 2 2,− 2
x+
) cos( π k ) cos( π (k cos( π2 k1,− 1,− + k2,− )) 2 2,− 2
χ
k1,0 ) < 0 π cos( π2 k1,−
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S. Afchain, J. Magnen and V. Rivasseau
×χ
k1,0 + k2,0 + πT e χ )) > 0 π cos( π2 (k1,− + k2,− T x+ + k − πT k 1,0 2,0 )) +k cos( π (k − e 2 1,− 2,− χ , (VIII.137) )) > 0 π cos( π2 (k1,− + k2,−
k2,0 ) <0 π cos( π2 k2,−
Ann. Henri Poincar´e
−T x+ )) +k cos( π (k 2,− 1,− 2
where T is the triangle T translated by the vector (−2, −2). Using the invariance under central symmetry and translations by vectors of the form (4n+ , 4n− ), (n+ , n− ) ∈ Z2 , we conclude that T may be replaced by T . Hence we have proved that A1 (T ) = A2 (T ). This concludes the proof of Theorem V.1 hence of Theorem II.1. Acknowledgments. We thank our referee for its very attentive reading and comments.
References [1] V. Rivasseau, The two dimensional Hubbard Model at half-filling: I. Convergent Contributions, Journ. Stat. Phys. 106, 693–722 (2002). [2] S. Afchain, J. Magnen and V. Rivasseau, Renormalization of the 2-point function of the Hubbard model at half-filling, cond-mat/0409231. [3] M. Salmhofer, Continuous renormalization for Fermions and Fermi liquid theory, Commun. Math. Phys. 194, 249 (1998). [4] M. Salmhofer, Renormalization, an introduction, Springer Verlag, 1999. [5] M. Disertori and V. Rivasseau, Interacting Fermi liquid in two dimensions at finite temperature, Part I: Convergent Attributions, Commun. Math. Phys. 215, 251 (2000). [6] M. Disertori and V. Rivasseau, Interacting Fermi liquid in two dimensions at finite temperature, Part II: Renormalization, Commun. Math. Phys. 215, 291 (2000). [7] G. Benfatto, A. Giuliani and V. Mastropietro, Low temperature Analysis of Two-Dimensional Fermi Systems with Symmetric Fermi surface, Ann. Henri Poincar´e 4, 137 (2003). [8] P.W. Anderson, Luttinger liquid behavior of the normal metallic state of the 2D Hubbard model, Phys. Rev. Lett. 64, 1839–1841 (1990).
Vol. 6, 2005
The Hubbard Model at Half-Filling, Part III
St´ephane Afchain and Jacques Magnen Centre de Physique Th´eorique CNRS, UMR 7644 ´ Ecole Polytechnique F-91128 Palaiseau cedex France email:
[email protected] email:
[email protected] Vincent Rivasseau Laboratoire de Physique Th´eorique CNRS, UMR 8627 Universit´e de Paris-Sud F-91405 Orsay France email:
[email protected] Communicated by Joel Feldman submitted 20/12/04, accepted 10/02/05
To access this journal online: http://www.birkhauser.ch
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Ann. Henri Poincar´e 6 (2005) 485 – 552 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/030485-68 DOI 10.1007/s00023-005-0215-y
Annales Henri Poincar´ e
Phase Space Reduction of Star Products on Cotangent Bundles Niels Kowalzig, Nikolai Neumaier and Markus J. Pflaum
Abstract. In this paper we construct star products on Marsden-Weinstein reduced spaces in case both the original phase space and the reduced phase space are (symplectomorphic to) cotangent bundles. Under the assumption that the original cotangent bundle T ∗ Q carries a symplectic structure of form ωB0 = ω0 + π ∗ B0 with B0 a closed two-form on Q, is equipped by the cotangent lift of a proper and free Lie group action on Q and by an invariant star product that admits a G-equivariant quantum momentum map, we show that the reduced phase space inherits from T ∗ Q a star product. Moreover, we provide a concrete description of the resulting star product in terms of the initial star product on T ∗ Q and prove that our reduction scheme is independent of the characteristic class of the initial star product. Unlike other existing reduction schemes we are thus able to reduce not only strongly invariant star products. Furthermore in this article, we establish a relation between the characteristic class of the original star product and the characteristic class of the reduced star product and provide a classification up to G-equivalence of those star products on (T ∗ Q, ωB0 ), which are invariant with respect to a lifted Lie group action. Finally, we investigate the question under which circumstances ‘quantization commutes with reduction’ and show that in our examples non-trivial restrictions arise.
1 Introduction Already in the first and fundamental article on deformation quantization by Bayen et al. [3], the problem how to construct a star product on a reduced phase space out of a known star product on the initial phase space has been considered. In particular, the example of the cotangent bundle T ∗ S n−1 of the n − 1-sphere has been discussed, and for reduction the Weyl-Moyal product on T ∗ (Rn \ {0}) has been used. Even after general existence proofs for deformation quantizations on symplectic and Poisson manifolds have meanwhile appeared, it remains an interesting question which relations one can establish between star products on the original phase space and those on the reduced phase space. In physics terms, this corresponds to the question, whether extrinsic and intrinsic quantization are equivalent, which in some sense would mean that ‘quantization commutes with reduction’ (cf. [16] and [12] for a discussion of these topics in the framework of geometric quantization and the conventional Hilbert space approach to quantum mechanics). Moreover, the question of existence of star products on symplectic stratified spaces is still unsolved, and it appears to be very promising to attack
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this problem first for singular reduced phase spaces, which have been studied in detail by Sjamaar and Lerman [30]. Considering particular examples, there are various explicit constructions for phase spaces with additional structure, for instance CP n [7, 31] or more general complex Grassmann manifolds [29] for which a deformation quantization analogue of phase space reduction can be constructed. But these examples all seem to be well tailored to special situations and do not apply to arbitrary symplectic phase spaces. In [15], Fedosov introduced a reduction scheme for arbitrary symplectic manifolds (M, ω) with a compact, free and symplectic Lie group action and a regular value of the momentum map. Fedosov starts with a certain Fedosov star product obtained from a certain G-invariant torsion free symplectic connection. Adapting the original star product appropriately to the fibering structure of the principal G-bundle M0 → Mred = M0 /G, where M0 denotes the inverse image of 0 ∈ g∗ with respect to the classical momentum map, he showed that one can always achieve that the resulting star product on the reduced phase space (Mred , ωred ) is equivalent to a canonical Fedosov star product on it. Thus, Fedosov is able to prove a ‘reduction commutes with quantization’ theorem within his particular situation. Another very general approach to reduction in the framework of deformation quantization is the BRST-method as presented by Bordemann et al. in [8]. Using a quantum BRST complex, the authors of this work are able to produce quite an explicit formula for the reduced star product under the following three assumptions: 1) the symmetry group acts properly and freely, 2) 0 is a regular value of the momentum map, 3) the initial star product on the phase space with symmetry is strongly invariant (cf. Section 2.2 for a definition). By results obtained in [19] and [25], which reveal some obstructions in the characteristic class for a star product to be strongly invariant, the last of these assumptions imposes a restriction on the possible characteristic class of the original star product. Note that implicitly, the same restriction appears in the Fedosov reduction scheme. The scope of the present paper is to develop a reduction scheme for star products on cotangent bundles with respect to a symplectic form which is the sum of the canonical symplectic form and the pull-back of a closed two-form on the base manifold which can be interpreted as a magnetic field. Additionally, we assume that the reduced phase space is again a cotangent bundle or, more precisely, symplectomorphic to a cotangent bundle via a non-canonical diffeomorphism. It is known that this latter assumption holds true (cf. [16, 21, 24]) if the action is the cotangent lift of a proper and free Lie group action on the base manifold and if the momentum value for which the reduced space is considered is an invariant element of the dual of the Lie algebra. Our construction is adapted to the particular geometry of a cotangent bundle and we have to restrict our reduction scheme to a
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certain class of star products namely those for which the space of formal functions which are polynomials in the momenta form a -subalgebra. Fortunately, this class of star products is rich enough to obtain star products of arbitrary characteristic class, hence we actually provide a reduction scheme which does not depend on the characteristic class of the initial star product. Our paper is organized as follows: In Section 2 we recall from [9, 10, 11] the construction of various star products on cotangent bundles which have the common property that the formal functions polynomial in the momenta form a subalgebra. We also collect some notions of invariance with respect to Lie group actions in deformation quantization and recall the definition of the deformation quantization analogue of a G-equivariant classical momentum map. The resulting quantum momentum maps will play a fundamental role in our framework of phase space reduction. In Section 3 we then address the reduction of a certain class of star products on (T ∗ Q, ωB0 ) which includes the examples considered in Section 2. Under the assumptions imposed on the initial data we establish a relation between the Poisson algebra of functions on T ∗ (Q/G) polynomial in the momenta and the Poisson algebra of horizontal invariant functions on T ∗ Q which are polynomial in the momenta as well. Moreover, we succeed to construct a deformation of this classical correspondence which enables us to define an associative product on the polynomial functions on T ∗ (Q/G) which is induced by a star product on (T ∗ Q, ωB0 ). It turns out that this product can be uniquely extended to the whole space C ∞ (T ∗ (Q/G))[[ν]]. We thus obtain a star product with respect to some symplectic form on the reduced phase space which differs from the canonical symplectic structure by an additional magnetic field term and which depends on B0 , the chosen classical momentum map, the curvature of the chosen connection, and the momentum value used for the phase space reduction. Also in this section we investigate the behavior of our reduction scheme with respect to natural operations on star product algebras like isomorphisms, automorphisms, and derivations and we give conditions on which these transfer to the reduced star products. In Section 4 we relax our assumptions somewhat to arbitrary Lie group actions on Q and return to consider the examples of Section 2. We derive first necessary and sufficient conditions on the geometric data which guarantee the considered star products to be invariant with respect to the lifted Lie group action on the base manifold. Furthermore, we find additional conditions which guarantee that these products admit a G-equivariant quantum Hamiltonian and thus a Gequivariant quantum momentum map in the sense of Xu [32]. In these particular cases it turns out that the quantum momentum maps are polynomials in the momenta, a result which is important for our reduction scheme to work. In addition, if there is a G-invariant torsion free connection on Q, we can give a classification of star products on (T ∗ Q, ωB0 ) invariant with respect to a lifted group action up to G-equivalence.
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In Section 5 we succeed in computing the characteristic class of the reduced star products. This clarifies how the choices made in the course of our reduction scheme affect the equivalence class of the resulting reduced star product. Moreover, we compare the resulting star products to naturally given star products on the reduced cotangent space in Section 6 and derive conditions for ‘reduction commutes with quantization’ results to hold. We conclude Section 6 by comparing our construction to known results and examples considered in the literature [3, 8, 15].
2 Preliminaries and notation In this section we first recall some notation and several canonical constructions of star products on cotangent bundles (see [9, 10, 11] for further details). Then we collect various notions of invariance in deformation quantization with respect to Lie group actions and recall the definition of the quantum analogue of a Gequivariant classical momentum map (cf. [2, 4, 32]). In our framework, this notion will turn out to be fundamental for the formulation of phase space reduction.
2.1
Constructions of star products on cotangent bundles
Throughout this article, Q will always denote a smooth n-dimensional manifold. Recall that the cotangent bundle π : T ∗ Q → Q is equipped with the canonical symplectic form ω0 = −dθ0 , where θ0 denotes the canonical one-form. The zero section of T ∗ Q is denoted by i : Q → T ∗ Q by means of which we consider Q as embedded into T ∗ Q. Local coordinates on Q will be denoted by x1 , . . . , xn , the induced coordinates on T ∗ Q by q 1 , . . . , q n , p1 , . . . , pn . Given k one-forms β1 , . . . , βk ∈ Γ∞ (T ∗ Q) we define a fiberwise acting differential operator F (β1 ∨ · · · ∨ βk ) : C ∞ (T ∗ Q) → C ∞ (T ∗ Q) of order k by (F (β1 ∨ · · · ∨ βk ) f ) (ζx ) :=
∂k ∂t1 · · · ∂tk t1 =···=tk =0
f (ζx + t1 β1 (x) + · · · + tk βk (x)), ζx ∈ Tx∗ Q. (2.1)
Clearly, F extends to an injective algebra morphism from Γ∞ ( T ∗ Q) into the algebra of differential operators on C ∞ (T ∗ Q). k To every contravariant symmetric tensor field T ∈ Γ∞ ( T Q) one can assign 1 a smooth function P (T ) ∈ C ∞ (T ∗ Q) by (P (T ))(ζx ) := k! is (ζx ) . . . is (ζx )T (x). Note that P (T ) is a homogeneous polynomial of degree k in the momenta. Let us denote the space of these fiberwise polynomial functions by P k (Q). Obviously, P then between the Z-graded commutative algebras extends to an isomorphism k P (Q). Γ∞ ( T Q) and P(Q) = ∞ k=0 Fixing a torsion free connection ∇ on the base manifold Q one can assign to every formal function f ∈ C ∞ (T ∗ Q)[[ν]] a formal series with values in the
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differential operators on C ∞ (Q) by 0 (f ) χ := i∗ F (exp(−νD)χ) f =
∞ (−ν)l l=0
l!
i∗
∂lf ∂pj1 · · · ∂pjl
1 is (∂xj1 ) . . . is (∂xjl ) Dl χ. (2.2) l! Here, D denotes the operator of symmetric covariant derivation which in local coordinates is given by D = dxi ∨ ∇∂xi . It has been shown in [9] that the restriction of 0 to P(Q)[[ν]] is injective and that the image of P(Q)[[ν]] under 0 is closed with respect to composition of differential operators. Hence, one can define an associative product on P(Q)[[ν]] by F, F ∈ P(Q)[[ν]]. (2.3) F 0 F := 0 −1 ( 0 (F ) 0 (F )) , Moreover, it has been shown that 0 can be described by bidifferential operators. Hence, one can uniquely extend 0 to a product on C ∞ (T ∗ Q)[[ν]] yielding a star product on (T ∗ Q, ω0 ) which also will be denoted by 0 and which will be called the standard ordered star product (corresponding to ∇). By definition of 0 it is obvious that 0 defines a representation of (C ∞ (T ∗ Q)[[ν]], 0 ) on C ∞ (Q)[[ν]]. Now we want to define further star products depending on a so-called order parameter κ ∈ [0, 1]. To this end consider a smooth positive density υ on Q. Then υ and the connection ∇ define a one-form α ∈ Γ∞ (T ∗ Q) by ∇X υ = α(X)υ,
X ∈ Γ∞ (T Q).
(2.4)
It is immediate to check that this one-form satisfies dα = −tr (R) ,
(2.5)
where R denotes the curvature tensor of ∇ and tr (R) the trace of the curvature endomorphism. Let us denote by hor∇ the horizontal lift (with respect to ∇) of vector fields on Q to vector fields on T ∗ Q (cf. also Definition A.2). Using local coordinates we then define a differential operator ∆ on C ∞ (T ∗ Q) by ∆ := ∆0 + F (α) := F dxi Lhor∇ (∂xi ) + F (α) =
∂2 ∂2 ∂ + pl π ∗ Γljk + π ∗ (Γllk + αk ) . (2.6) k ∂q ∂pk ∂pj ∂pk ∂pk
Here, Γljk denote the Christoffel symbols of ∇ and αk the components of α in the chosen local chart. After some immediate computation it turns out that ∆ is independent of the choice of local coordinates. In [11] we also considered the formal series Nκ of differential operators given by Nκ := exp(−κν∆).
(2.7)
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Then this operator induces the κ-ordered star product f κ f := Nκ−1 ((Nκ f ) 0 (Nκ f ))
(2.8)
which is obtained from 0 by the equivalence transformation Nκ−1 . In the cases κ = 1 and κ = 1/2 the corresponding star products are called star product of antistandard ordered type and Weyl ordered star product. For a further discussion of these star products we again refer the reader to [9, 10, 11], where one can particularly find a motivation for the above definitions by means of the applicability of the GNS construction to 1/2 . For later use let us recall the following factorization property of Nκ from [11, Lemma 3.6]: exp(κνD) − id α . (2.9) Nκ = exp(−κν∆0 ) exp −F D At this point let us mention two further important formulas which explicitly determine the κ -left- and the κ -right-multiplication with a formal function pulled-back from Q (cf. [11, Prop. 3.2]): π ∗ χ κ f f κ π ∗ χ
= F (exp(κνD)χ) f, = F (exp((κ − 1)νD)χ) f,
(2.10) (2.11)
where χ ∈ C ∞ (Q)[[ν]] and f ∈ C ∞ (T ∗ Q)[[ν]]. Let us also note that by κ (f ) := 0 (Nκ f ) one obtains a representation of (C ∞ (T ∗ Q)[[ν]], κ ) on C ∞ (Q)[[ν]]. Due to the properties of 0 and Nκ the restriction of κ to P(Q)[[ν]] is injective as well. Up to now we have only considered so-called homogeneous star products on (T ∗ Q, ω0 ), i.e., star products for which H = Lξ0 + ν∂ν is a derivation. Here ξ0 ∈ Γ∞ (T (T ∗ Q)) denotes the canonical Liouville vector field defined by iξ0 ω0 = −θ0 . From this property and the fact that the bidifferential operator describing 1/2 at order 2 in the formal parameter is symmetric it is easy to deduce (cf. [11, Thm. 4.6]) that the characteristic class of the star products κ is given by c(κ ) = c(1/2 ) = [0]. Now we want to recall a construction which yields star products with charac2 (Q)[[ν]] is an arbitrary formal series of closed teristic class ν1 [π ∗ B], where B ∈ ZdR two-forms on Q. The thus obtained star products comprise deformation quantizations of the symplectic manifold (T ∗ Q, ωB0 = ω0 + π ∗ B0 ), where B0 – which is assumed to be real – denotes the term of zeroth order in the formal parameter of B. For A ∈ Γ∞ (T ∗ Q)[[ν]] with real A0 and all κ ∈ [0, 1] consider the operator Ak : C ∞ (T ∗ Q)[[ν]] → C ∞ (T ∗ Q)[[ν]] defined by exp(κνD) − exp((κ − 1)νD) ∗ A − A0 Aκ := t−A0 exp −F , (2.12) νD where t−A0 denotes the fiberwise translation by −A0 that is t−A0 (ζx ) = ζx −A0 (x). The important property of Ak is that it defines an automorphism of κ if and only
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if dA = 0 (cf. [11, Thm. 3.4]). To define star products reflecting the presence of a magnetic field B on Q consider a good open cover {Oj }j∈I of Q together with local formal potentials Aj ∈ Γ∞ (T ∗ Oj )[[ν]] of B. This means B|Oj = dAj , where in addition Aj0 is chosen to be real. Let us denote by Ajκ the operator determined by Eq. (2.12) with Aj used instead of A. Then one defines an associative product jκ on C ∞ (T ∗ Oj )[[ν]] by f jκ f := Ajκ ((Ajκ )−1 f ) κ ((Ajκ )−1 f ) , f, f ∈ C ∞ (T ∗ Oj )[[ν]]. (2.13) Now one makes the crucial observation that the operator (Akκ )−1 Ajκ corresponds via (2.12) to the closed formal one-form Aj |Oj ∩Ok − Ak |Oj ∩Ok . Hence, this operator is an automorphism of (C ∞ (T ∗ (Oj ∩ Ok ))[[ν]], κ ) and one can define a star ∞ ∗ product B κ on C (T Q)[[ν]] by setting j f, f ∈ C ∞ (T ∗ Q)[[ν]]. (2.14) f B κ f T ∗ Oj = f |T ∗ Oj κ f |T ∗ Oj , These star products do not depend on the particular choice of the covering nor on the choice of the potentials but only on B (and of course on κ ). By a straight∗ forward computation one checks that B κ is a star product on (T Q, ωB0 ) for all κ ∈ [0, 1]. Moreover, in [11, Thm. 4.6] it has been shown that the characteristic 1 ∗ B class of B κ is given by c(κ ) = ν [π B]. Thus there exists for every equivalence ∗ class of star products on (T Q, ωB0 ) a representative B κ.
2.2
Notions of invariance in deformation quantization
Let G be a Lie group and denote by g = Lie(G) its Lie algebra. In addition, assume ϕ : G × M → M to be a (left) action of G on a manifold M equipped with a symplectic form ω, and denote for every g ∈ G by ϕg : M → M the map m → ϕ(g, m). Then a star product on (M, ω) is called invariant with respect to ϕ : G × M → M or, for short, G-invariant in case no confusion can arise, if every ϕ∗g is an automorphism of the star product, i.e., if ϕ∗g (f f ) = ϕ∗g f ϕ∗g f
for all f, f ∈ C ∞ (M )[[ν]], g ∈ G.
(2.15)
In other words this means that r defined by r(g)f := ϕ∗g−1 f defines a left action on (C ∞ (M )[[ν]], ) by automorphisms. By anti-symmetrization of Eq. (2.15) with respect to f and f one checks that the action ϕ necessarily has to be symplectic. Given a G-invariant star product on a symplectic manifold (M, ω), differentiation of r yields an action ρ of the Lie algebra g = Lie(G) on C ∞ (M )[[ν]] by derivations of . Explicitly, ρ(ξ)f = −LξM f
for all f ∈ C ∞ (M )[[ν]], ξ ∈ g,
(2.16)
where ξM denotes the fundamental vector field associated to ξ. Clearly, in the G-invariant case, every fundamental vector field ξM is symplectic.
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Now let be a G-invariant star product and denote by C 1 (g, C ∞ (M )) the space of linear forms on g with values in C ∞ (M ). Then an element J = J0 + J+ ∈ C 1 (g, C ∞ (M ))[[ν]] with real-valued J0 ∈ C 1 (g, C ∞ (M )) and J+ ∈ νC 1 (g, C ∞ (M ))[[ν]] is called a quantum Hamiltonian for r, if ρ(ξ) = −LξM =
1 ad (J(ξ)) ν
for all ξ ∈ g.
(2.17)
In other words this means that the Lie derivative with respect to the generating vector fields is a quasi-inner (or essentially inner) derivation of . J is called a Gequivariant quantum Hamiltonian, if it is a quantum Hamiltonian and additionally satisfies (2.18) ϕ∗g J(ξ) = J(Ad(g −1 )ξ) for all ξ ∈ g, g ∈ G. A quantum Hamiltonian J is called a quantum momentum map if in addition 1 (J(ξ) J(η) − J(η) J(ξ)) = J([ξ, η]) ν
for all ξ, η ∈ g.
(2.19)
Clearly, differentiation of (2.18) with respect to g at e shows that a Gequivariant quantum Hamiltonian always defines a quantum momentum map. Note that the converse generally does not hold true. In the sequel we will refer to a Gequivariant quantum Hamiltonian as a G-equivariant quantum momentum map. The zeroth order parts of (2.17) and (2.18) mean that J0 is a G-equivariant classical momentum map for r, i.e., ξM is a Hamiltonian vector field with Hamiltonian function J0 (ξ) and the smooth mapping Jˇ0 : M → g∗ defined by Jˇ0 (m), ξ = J0 (ξ)(m) for all ξ ∈ g and all m ∈ M , where , denotes the natural pairing between g∗ and g, is Ad∗ -equivariant. Note that we follow the convention to denote by Ad∗ (g) = (Ad(g −1 ))∗ the coadjoint action of g. Now recall that a G-invariant star product is called strongly G-invariant, if J = J0 defines a quantum Hamiltonian, where J0 is a G-equivariant classical momentum map. Finally, let us briefly introduce the notions of isomorphisms and equivalence transformations in the G-invariant framework. A star product on (M, ω) is called G-isomorphic to the star product on (M, ω ), if one can find an isomorphism from (C ∞ (M )[[ν]], ) to (C ∞ (M )[[ν]], ) which commutes with r(g) for every g ∈ G. If in addition ω = ω and there exists an equivalence transformation from (C ∞ (M )[[ν]], ) to (C ∞ (M )[[ν]], ) which commutes with r(g) for every g ∈ G, one calls a star product G-equivalent to . From these definitions it is obvious how to define the notions of G-automorphisms and G-self equivalences. As an immediate consequence note that for a G-isomorphism T from (C ∞ (M )[[ν]], ) to (C ∞ (M )[[ν]], ) and J a G-equivariant quantum momentum map for the transformed map J (ξ) := T J(ξ) is a G-equivariant quantum momentum map for . But one has to observe that the notion of strong G-invariance is not preserved under G-isomorphisms, in general, that means that for a strongly G-invariant , a G-isomorphic need not be strongly G-invariant.
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3 Reduction of star products on cotangent bundles In this section, we present a general procedure for the phase space reduction of a certain class of star products on cotangent bundles. As we show in Section 4.1, this reduction method applies in particular to the star products κ and B κ constructed in the previous section. Since we will be concerned with different cotangent bundles to indicate T ∗ Q, T ∗ Q at the same instance, we adopt the convention to use objects related to the cotangent bundle π : T ∗ Q → Q.
3.1
Classical phase space reduction of (T ∗ Q, ω0 + π ∗ B0 ): Geometric and algebraic properties
Given a Lie group G we denote by {ei }1≤i≤dim (G) a basis of its Lie algebra g and by {ei }1≤i≤dim (G) the corresponding dual basis of g∗ . Assume that φ : G × Q → Q is a left action on the base manifold Q. This action gives rise to a G-action Φ : G × T ∗ Q → T ∗ Q on T ∗ Q, the cotangent lift of φ. Explicitly, it is defined by Φ(g, ζx ) := Φg (ζx ) := (T ∗ φg−1 )(ζx ) and satisfies π ◦ Φg = φg ◦ π. We then have Φ∗g θ0 = θ0 and consequently Φ∗g ω0 = ω0 . Let us denote the fundamental vector fields of φ by ξQ ∈ Γ∞ (T Q) and those of Φ by ξT ∗ Q ∈ Γ∞ (T (T ∗ Q)). Clearly, these vector fields are π-related, i.e., T π ◦ ξT ∗ Q = ξQ ◦ π for all ξ ∈ g. Recall that a Gequivariant classical momentum map for the canonical symplectic form ω0 is given by J0 (ξ) = θ0 (ξT ∗ Q ) = P (ξQ ), where P (ξQ ) ∈ P 1 (Q) denotes the function linear in the momenta corresponding to the vector field ξQ . In canonical coordinates this i reads J0 (ξ) = pi π ∗ ξQ . 2 (Q) the following For the symplectic form ωB0 = ω0 + π ∗ B0 with B0 ∈ ZdR well-known result gives first conditions which have to be satisfied in order to be able to construct the reduced phase space. Lemma 3.1 Let Φ act on (T ∗ Q, ωB0 ) as above. Then the following holds true: i) Φ is a symplectic action with respect to ωB0 if and only if B0 is G-invariant. ii) If B0 is G-invariant, then there is a G-equivariant classical momentum map for Φ if and only if there is a real-valued element j0 ∈ C 1 (g, C ∞ (Q)) such that φ∗g j0 (ξ) = j0 (Ad(g −1 )ξ)
for all g ∈ G, ξ ∈ g. (3.1) In this case J0 (ξ) = P (ξQ ) + π ∗ j0 (ξ) defines a G-equivariant classical momentum map which is unique up to elements of the space g∗ G of invariants. iii) If the relations (3.1) are satisfied, one has in particular dj0 (ξ) = iξQ B0
and
j0 ([ξ, η]) = B0 (ξQ , ηQ )
for all ξ, η ∈ g.
(3.2)
Proof. The proof of i) is obvious, so let us show ii). To this end check first that J0 defines a classical Hamiltonian for Φ if and only if d(J0 (ξ) − P (ξQ )) = π ∗ iξQ B0 for all ξ ∈ g. After application of i∗ one notes that this is equivalent to the
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existence of j0 such that the first condition in Eq. (3.1) is fulfilled. Now observe that the canonical momentum map J 0 for the case B0 = 0, which is given by J 0 (ξ) = P (ξQ ), is G-equivariant. Therefore, G-equivariance of J0 is equivalent to the G-equivariance of j0 . The statement about the ambiguity of J0 is a general fact which holds true for arbitrary Hamiltonian G-spaces. Assertion iii) is obtained by differentiating φ∗g j0 (ξ) = j0 (Ad(g −1 )ξ) with respect to g at e and using the first condition in Eq. (3.1). From now on we will assume that the above conditions for the existence of a G-equivariant classical momentum map are satisfied and that the action of G on Q is proper and free. This implies in particular that the orbit space of the G-action is smooth and even that p : Q → Q = Q/G is a left principal G-bundle. Now fix an element µ0 ∈ g∗ G and recall that it gives rise to the reduced phase space −1 Jˇ0 (µ0 ) G, where Jˇ0 : T ∗ Q → g∗ is defined by Jˇ0 (ζx ), ξ = J0 (ξ)(ζx ). Since µ0 is a regular value of Jˇ0 , classical Marsden–Weinstein reduction [23] applies and the reduced phase space naturally carries the structure of a symplectic manifold. The induced symplectic form on the reduced space will be denoted by ωµ0 . It is uniquely characterized by the relation πµ∗ 0 ωµ0 = i∗µ0 ωB0 ,
(3.3)
where iµ0 : Jˇ0−1 (µ0 ) → T ∗ Q denotes the inclusion and πµ0 the projection of Jˇ0−1 (µ0 ) onto the orbit space. In case B0 = 0, J0 (ξ) = P (ξQ ) it is well known that the reduced phase space is symplectomorphic to T ∗ Q equipped with a symplectic structure of the form ω 0 + π ∗ b0 , where b0 is a closed two-form on Q which vanishes for µ0 = 0. Note that the construction of an appropriate symplectomorphism is not canonical unless µ0 = 0 (cf. [16, 21, 24]). An analogous result holds in the case of non-vanishing B0 (cf. [27, Thm. 15]). For the convenience of the reader we briefly present it here. To this end we need some tools from the theory of (left) principal G-bundles. Let γ be a connection oneform on the principal bundle p : Q → Q. Note that γ transforms according to the rule φ∗g γ = Ad(g)γ since G acts from the left on Q. Moreover, let λ := dγ − 12 [γ, γ]∧ denote the corresponding curvature form; observe the minus sign in front of the bracket which is due to the fact that we work with a left principal G-bundle. Recall that λ is a g-valued horizontal two-form on Q. Finally, we associate to every smooth map ˇ : Q → g∗ a one-form Γˇ ∈ Γ∞ (T ∗ Q) by , γ. Γˇ := ˇ
(3.4)
Then φ∗g Γˇ = ΓAd∗ (g−1 )φ∗g ˇ for all g ∈ G. Thus, if ˇ is G-equivariant, then Γˇ is G-invariant as well. Now we are prepared to formulate our result. Theorem 3.2 (cf. [28, 1, 21, 16, 24, 27]) With the notations and assumptions from above, the reduced phase space Jˇ0−1 (µ0 )/G with symplectic form ωµ0 induced by
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ωB0 is symplectomorphic to (T ∗ Q, ωb0 ) = (T ∗ (Q/G), ω0 + π ∗ b0 ), where b0 is the uniquely determined closed two-form on Q satisfying p∗ b0 = B0 + dΓˇj0 −µ0 : ((T ∗ Q)µ0 , ωµ0 ) := (Jˇ0−1 (µ0 )/G, ωµ0 ) ∼ = (T ∗ Q, ωb0 ).
(3.5)
Proof. Let us consider the one-form Γˇj0 −µ0 , where the smooth mapping ˇj0 : Q → g∗ is defined by ˇj0 (x), ξ = j0 (ξ)(x). This one-form induces a fiber translation tΓˇj0 −µ0 given by tΓˇj0 −µ0 (ζx ) = ζx +Γˇj0 −µ0 (x). Due to the properties of the connection oneform tΓˇj0 −µ0 maps Jˇ0−1 (µ0 ) to (T ∗ Q)0 := {ζx ∈ T ∗ Q | ζx (ξQ (x)) = 0 for all ξ ∈ g}. Next observe that tΓˇj0 −µ0 commutes with every Φg by the G-invariance of µ0 and the G-equivariance of j0 . Therefore, tΓˇj0 −µ0 passes to the quotient and defines a diffeomorphism Ψµ0 from Jˇ0−1 (µ0 )/G to (T ∗ Q)0 /G ∼ = T ∗ (Q/G). In order to ∗ ∗ determine the symplectic form that is carried over to T Q = T ∗ (Q/G) via (Ψ−1 µ0 ) , ∗ ∗ one first has to compute t−Γˇj −µ ωB0 = ω0 + π (B0 + dΓˇj0 −µ0 ). Now an easy 0 0 computation using the relations between B0 and j0 yields that B0 + dΓˇj0 −µ0 = H(B0 ) + ˇj0 − µ0 , λ, (3.6)
dim (G)
dim (G) where H(B0 ) = B0 − i=1 Γei ∧iei Q B0 − 12 i,k=1 Γei ∧Γek iei Q iek Q B0 denotes the totally horizontal part of B0 . But from Eq. (3.6) it follows that B0 + dΓˇj0 −µ0 is horizontal. Due to the fact that it is G-invariant and that p is a surjective submersion, this implies the existence of a unique closed two-form b0 as stated in the theorem. But then Ψµ0 induces a symplectic form on T ∗ Q. Explicitly, this form is given by ωb0 = ω 0 + π ∗ b0 which can be derived from the defining equation i∗µ0 ωB0 = πµ∗ 0 ωµ0 and the commutative diagram tΓˇ
j −µ
◦iµ0
Jˇ0−1 (µ0 ) πµ 0
0 −−−0−−− −−→
Jˇ0−1 (µ0 )/G
0 −−−− →
(T ∗ Q)0 0 π
π◦i0
−−−−→ Q p
(3.7)
(T ∗ Q)0 /G ∼ = T ∗ Q −−−−→ Q,
Ψµ
π
where iµ0 , i0 are the inclusions into T ∗ Q and πµ0 , π 0 the projections onto the respective orbit spaces. Now consider the space C ∞ (T ∗ Q)G of G-invariant smooth functions on T ∗ Q and the space Iµ0 ,·
dim (G)
∞ ∗ G := f ∈ C (T Q) f = hi (J0 (ei ) − µ0 , ei ) with hi ∈ C ∞ (T ∗ Q) . i=1
Since Iµ0 ,· is a Poisson ideal in C ∞ (T ∗ Q)G , the pointwise product and the Poisson bracket { , }B0 corresponding to ωB0 induce the structure of a Poisson algebra on the quotient C ∞ (T ∗ Q)G /Iµ0 ,· by [f ]µ0 ,· ·red [f ]µ0 ,· := [f f ]µ0 ,· ,
{[f ]µ0 ,· , [f ]µ0 ,· }red := [{f, f }B0 ]µ0 ,· .
(3.8)
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The obtained Poisson algebra is known to be isomorphic to (C ∞ (T ∗ Q), { , }b0 ), where { , }b0 denotes the Poisson bracket corresponding to the symplectic form ωb0 . Unfortunately, there is no canonical construction of such an isomorphism, but restricting to functions polynomial in the momenta, we are able to find a natural isomorphism between the Poisson subalgebras P(Q)G /IµP0 ,· ⊆ C ∞ (T ∗ Q)G /Iµ0 ,· and P(Q) ⊆ C ∞ (T ∗ Q) which depends only on the choice of the connection γ. Hereby we have used the abbreviations P(Q)G := P(Q) ∩ C ∞ (T ∗ Q)G and IµP0 ,· := P(Q) ∩ Iµ0 ,· . Let us now provide the details. Due to the choice of the connection γ the tangent bundle of Q can be written as the direct sum of the horizontal bundle HQ and the (trivial) vertical bundle V Q. Clearly, this decomposition induces a decomposition of the symmetric powers of T Q. Hence the space of sections Γ∞ ( T Q) can be written as ∞ ∞ k k−r r ∞ k Γ ( T Q) = Γ ( HQ) ⊕ Γ∞ ( HQ ∨ V Q). ∞
k=1 r=1
k=0
l r V Q) two deObviously, one can assign to each section T ∈ Γ∞ ( HQ ∨ grees by calling Thorizontal of degree l and vertical of degree r. We k ∞ k k−r r will re∞ ∞ Γ ( HQ) and Γ ( HQ ∨ VQ) as fer to the spaces ∞ k=0 k=1 r=1 the space of totally horizontal sections and partially vertical sections in T Q, respectively. Moreover, we denote by H the projection onto the totally horizontal sections and by PV the projection onto the partially vertical sections. Since a set of basis sections of the vertical bundle, there exist {ei Q }1≤i≤dim (G) is for every T ∈ PV(Γ∞ ( T Q)) uniquely determined tensor fields Ri (T ) ∈ Γ∞ ( T Q)
dim (G) i ∞ R (T ) ∨ ei Q . Hence Ri : PV(Γ ( T Q)) → Γ∞ ( T Q) is such that T = i=1 ∞ Q) by setting Ri (T ) := 0 for a well-defined mapping that extends to all of Γ ( T T ∈ H(Γ∞ ( T Q)). Using the isomorphism P : Γ∞ ( T Q) → P(Q) we then get the following decomposition of P(Q) into the spaces of so-called totally horizontal and partially vertical polynomial functions: P(Q) = h(P(Q)) ⊕ pv(P(Q)). Hereby, we have used h = P◦H◦P−1 and pv = P◦PV◦P−1 . Under the isomorphism P the mapping Ri transforms to ri : P(Q) → P(Q) which explicitly is given by 1 i if F is vertical of degree r ≥ 1, i r F (Γe ) F r (F ) = (3.9) 0 if F is vertical of degree 0. Thus, every F ∈ P(Q) can be written as
dim (G)
F = h(F ) +
i=1
ri (F )P ei Q .
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Now consider again the G-invariant one-form Γˇj0 −µ0 on T ∗ Q defined in the proof of Theorem 3.2. Observe that by γ(Y ) = 0 for every Y ∈ Γ∞ (HQ) the equality t∗Γˇj −µ F = F is satisfied for every totally horizontal polynomial function F . Hence 0 0 every F ∈ P(Q) can be written in the form
dim (G)
F = h(t∗−Γˇj
0 −µ0
F) +
i=1
t∗Γˇj
0 −µ0
ri (t∗−Γˇj
0 −µ0
F )(J0 (ei ) − µ0 , ei ),
(3.10)
where we have used that t∗Γˇj −µ P (ξQ ) = P (ξQ ) + π ∗ j0 (ξ) − µ0 , ξ for ξ ∈ g. After 0 0 these rather technical preparations we can now prove the following result. Proposition 3.3 i) The space P(Q) decomposes into the direct sum dim (G) P(Q) = h(P(Q)) ⊕ F ∈ P(Q) F = H i (J0 (ei ) − µ0 , ei )
i=1
with H i ∈ P(Q)
(3.11)
and this decomposition is G-invariant. Moreover, the projections onto the respective subspaces are given by hˇj0 −µ0 := h ◦ t∗−Γˇj −µ and pvˇj0 −µ0 := 0 0 t∗Γˇj −µ ◦ pv ◦ t∗−Γˇj −µ . 0
0
0
0
ii) According to i) the space of G-invariant polynomial functions P(Q)G decomposes into the direct sum P(Q)G = h(P(Q)G ) ⊕ IµP0 ,· .
(3.12)
iii) The space h(P(Q)G ) of totally horizontal G-invariant polynomial functions becomes a Poisson algebra with the usual pointwise product of functions and the Poisson bracket { , }ˇj0 −µ0 defined by {F, F }ˇj0 −µ0 := hˇj0 −µ0 ({F, F }B0 ) = h(t∗−Γˇj
0 −µ0
{F, F }B0 ), F, F ∈ h(P(Q)G ).
(3.13)
iv) As a Poisson algebra, (h(P(Q)G ), { , }ˇj0 −µ0 ) is isomorphic to P(Q) with the Poisson bracket induced by ωb0 = ω 0 + π ∗ b0 , where b0 denotes the uniquely determined closed two-form on Q such that p∗ b0 = B0 + dΓˇj0 −µ0 . If h : Γ∞ ( T Q) → Γ∞ ( T Q) denotes the horizontal lift (which is obtained by extension from Γ∞ (T Q) to Γ∞ ( T Q) as homomorphism with respect to ∨, particularly χh = p∗ χ for χ ∈ C ∞ (Q)), an explicit Poisson algebra isomorphism is given by l : P(Q) P (t) → P th ∈ h(P(Q)G ), t ∈ Γ∞ ( T Q). (3.14)
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v) Finally, (h(P(Q)G ), { , }ˇj0 −µ0 ) is isomorphic to P(Q)G /IµP0 ,· with Poisson algebra structure defined by Eq. (3.8). An isomorphism is given by h(P(Q)G ) F → [F ]µ0 ,· ∈ P(Q)G /IµP0 ,· .
(3.15)
Proof. Statements i) and ii) are obvious from the above considerations. Since the kernel of hˇj0 −µ0 |P(Q)G is a Poisson ideal in P(Q)G , it is straightforward to verify that { , }ˇj0 −µ0 defines a Poisson bracket on h(P(Q)G ). For the proof of iv), first note that l : P(Q) → h(P(Q)G ) is an isomorphism of commutative algebras. Thus it remains to show that the Poisson bracket obtained on P(Q) by pull-back by l of the Poisson bracket { , }ˇj0 −µ0 on h(P(Q)G ) coincides with the Poisson bracket induced by ωb0 . To check this, it is enough to compute the induced bracket of P (t) and P (s) for t, s ∈ Γ∞ (T Q). By a straightforward computation one obtains {l(P (t)), l(P (s))}ˇj0 −µ0 = −P [t, s]h − π ∗ ((B0 + dΓˇj0 −µ0 )(th , sh )). Since T p th = t◦p and analogously for s, we get (B0 +dΓˇj0 −µ0 )(th , sh ) = p∗ (b0 (t, s)) by definition of b0 . Then, using l(π ∗ χ) = π ∗ p∗ χ for χ ∈ C ∞ (Q), we find l−1 ({l(P (t)), l(P (s))}ˇj0 −µ0 ) = −P ([t, s]) − π ∗ (b0 (t, s)). But this proves iv) since this last expression coincides with the Poisson bracket of P (t) and P (s) with respect to the symplectic form ωb0 = ω0 + π ∗ b0 . For the proof of v), one again has to check the compatibility of the Poisson brackets, but this is straightforward observing that the kernel of hˇj0 −µ0 restricted to P(Q)G is the ideal IµP0 ,· .
3.2
Reduction of a certain class of star products on cotangent bundles
In view of the reduction of the classical structures considered in the preceding section, it should be possible to analogously determine the reduction of certain star products on (T ∗ Q, ωB0 ) in order to obtain star products on (T ∗ Q, ωB0 ). To this end recall first our general assumptions that the G-action on Q is proper and free and that µ0 ∈ g∗ , which is a regular value of Jˇ0 , is chosen to be invariant with respect to the coadjoint action of G. For the quantum reduction we have to assume in addition that we are given a star product on (T ∗ Q, ωB0 ), a G-equivariant quantum momentum map J and a deformation µ of the classical momentum value µ0 such that the following properties hold true: • is G-invariant, i.e., invariant with respect to the lifted action Φ of G on T ∗ Q. • J = J0 + J+ ∈ C 1 (g, C ∞ (T ∗ Q))[[ν]] is a G-equivariant quantum momentum map for , where J0 denotes a G-equivariant classical momentum map of the form J0 (ξ) = P (ξQ ) + π ∗ j0 (ξ) as in Lemma 3.1. • P(Q)[[ν]] is a -subalgebra and J(ξ) ∈ P(Q)[[ν]] for all ξ ∈ g.
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• The quantum momentum value has the form µ = µ0 + µ+ with µ+ ∈ νg∗c [[ν]] and g∗c the complexification of g∗ . Moreover, µ is invariant with respect to the coadjoint action of G. The third of the above assumptions will enable us to compute the reduced star product by means of polynomial functions, only. Note at this point that later in Corollary 4.14 ii) we will prove that the assumption J(ξ) ∈ P(Q)[[ν]] for all ξ ∈ g is actually no additional assumption but a consequence of the fact that P(Q)[[ν]] is a -subalgebra. As for the algebraic part of the classical reduction consider C ∞ (T ∗ Q)G [[ν]] and the subspace Iµ,
dim (G)
∞ ∗ G := f ∈ C (T Q) [[ν]] f = hi (J(ei ) − µ, ei ) i=1
with hi ∈ C ∞ (T ∗ Q)[[ν]] .
Then Iµ, is a two-sided ideal of C ∞ (T ∗ Q)G [[ν]], since J is a quantum Hamiltonian, and the quotient space C ∞ (T ∗ Q)G [[ν]]/Iµ, becomes an associative algebra by [f ]µ, red [f ]µ, := [f f ]µ, .
(3.16)
In order to interpret the associative product red as a star product on the reduced phase space, one has to find a C[[ν]]-module isomorphism between C ∞ (T ∗ Q)G [[ν]]/Iµ, and (C ∞ (T ∗ Q)G /Iµ0 ,· )[[ν]]. Since the latter space is isomorphic to C ∞ (T ∗ Q)[[ν]] one then defines an associative product on C ∞ (T ∗ Q)[[ν]] by declaring the isomorphism in question to be a homomorphism of associative algebras. If the thus constructed product is a star product indeed, then we have obtained the reduced star product we are looking for. At this point one has to mention that there is no canonical construction of such an isomorphism for the space of all smooth functions, but like in the classical case one can find a natural isomorphism between the subspaces obtained by restriction to polynomial functions. Let us now provide the details for the construction of this latter isomorphism. To this end observe first that by the third assumption above the product red can be P P restricted to P(Q)G [[ν]]/Iµ, ⊆ C ∞ (T ∗ Q)G [[ν]]/Iµ, , where Iµ, := P(Q)[[ν]]∩Iµ, . We now claim that there is a naturally constructed C[[ν]]-module isomorphism beP and (P(Q)G /IµP0 ,· )[[ν]] where the latter space is isomorphic tween P(Q)G [[ν]]/Iµ, to P(Q)[[ν]] by Proposition 3.3 iv) and v). Clearly, the projections hˇj0 −µ0 , pvˇj0 −µ0 and the maps rˇij −µ := t∗Γˇj −µ ◦ ri ◦ 0 0 0 0 t∗−Γˇj −µ extend by C[[ν]]-linearity from P(Q) to P(Q)[[ν]]. Hence, by Proposition 0 0 3.3 i), every F ∈ P(Q)[[ν]] decomposes uniquely into a sum of the form
dim (G)
F = hˇj0 −µ0 (F ) + pvˇj0 −µ0 (F ) = hˇj0 −µ0 (F ) +
i=1
rˇij0 −µ0 (F )(J0 (ei ) − µ0 , ei ).
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After defining µ, : P(Q)[[ν]] → P(Q)[[ν]] by the equation dim (G) 1 i rˇj0 −µ0 (F )(J0 (ei ) − µ0 , ei ) − rˇij0 −µ0 (F ) (J(ei ) − µ, ei ) , ν i=1 (3.17) the above decomposition can be rewritten as
µ, F :=
dim (G)
F = hˇj0 −µ0 (F ) +
rˇij0 −µ0 (F ) (J(ei ) − µ, ei ) + νµ, F.
i=1
Now repeat this ad infinitum and decompose at every step the remaining term
dim (G) i rˇj0 −µ0 (F ) (J(ei ) − µ, ei ). This procedure not of the form hˇj0 −µ0 (F ) + i=1 finally yields F = hˇj0 −µ0
id F id − νµ,
dim (G)
+
rˇij0 −µ0
i=1
id F id − νµ,
(J(ei ) − µ, ei ). (3.18)
Like in the classical case we obtain: Lemma 3.4 i) The space P(Q)[[ν]] decomposes into the direct sum P(Q)[[ν]] = h(P(Q))[[ν]]⊕ dim (G)
F ∈ P(Q)[[ν]] F = H i (J(ei ) − µ, ei ) with H i ∈ P(Q)[[ν]] i=1
(3.19) and this decomposition is G-invariant. Moreover, the projections onto the respective subspaces are given by hˇj0 −µ0 ◦
id id − νµ,
dim (G)
and
i=1
rˇij0 −µ0
id F (J(ei )−µ, ei ), id − νµ, F ∈ P(Q)[[ν]].
ii) According to i) the space of formal series of G-invariant polynomial functions P(Q)G [[ν]] decomposes into the direct sum P . P(Q)G [[ν]] = h(P(Q)G )[[ν]] ⊕ Iµ,
(3.20)
Proof. The fact that every element of P(Q)[[ν]] can be decomposed as stated in i) is obvious from the above considerations. To see that the sum in (3.19) is direct,
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dim (G) i one just has to observe that the lowest order in ν of i=1 H (J(ei )−µ, ei ) is
dim (G) i
k i given by i=1 H0 (J0 (ei ) − µ0 , ei ) (where we have written H i = ∞ k=0 ν Hk ).
dim (G) i H (J(ei )−µ, ei ) is horizontal, the fact that the But then assuming that i=1 sum in Eq. (3.11) is direct implies that H0i = 0 for all 1 ≤ i ≤ dim (G). Repeating this argument order by order, we finally get H i = 0 for all 1 ≤ i ≤ dim (G) proving that the decomposition is direct. Using the invariance properties of µ, J and , it
dim (G) i H (J(ei ) − µ, ei ) is easy to check that Φ∗g maps elements of the form i=1 to elements of the same form. Thus, the above decomposition turns out to be G-invariant, since obviously Φ∗g maps totally horizontal polynomials to totally P . horizontal polynomials. ii) is a direct consequence of i) and the definition of Iµ, After these preparations we obtain one of the main results of this section. Theorem 3.5 i) The space h(P(Q)G )[[ν]] of formal series of totally horizontal G-invariant polynomial functions becomes an associative algebra by means of the product •J,µ defined by id J,µ F • F := hˇj0 −µ0 (F F ) , F, F ∈ h(P(Q)G )[[ν]]. (3.21) id − νµ, ii) The pull back of •J,µ to P(Q)[[ν]] via the isomorphism l : P(Q)[[ν]] → h(P(Q)G )[[ν]] defined in Eq. (3.14) gives rise to a star product J,µ on P(Q)[[ν]], where the underlying Poisson bracket is induced by the symplectic form ωb0 = ω0 + π ∗ b0 given in Proposition 3.3 iv). P with the associative iii) (h(P(Q)G )[[ν]], •J,µ ) is isomorphic to P(Q)G [[ν]]/Iµ, algebra structure defined in Eq. (3.16). An isomorphism is given by P . h(P(Q)G )[[ν]] F → [F ]µ, ∈ P(Q)G [[ν]]/Iµ,
(3.22)
iv) The star product J,µ on P(Q)[[ν]] can be described by bidifferential operators, hence can be uniquely extended to a star product on C ∞ (T ∗ Q)[[ν]], which also will be denoted by J,µ . id restricted Proof. Using the fact that the kernel of the projection hˇj0 −µ0 ◦ id−ν
µ, G G to P(Q) [[ν]] is a two-sided ideal in P(Q) [[ν]], it is straightforward to see that the composition •J,µ defined in (3.21) on h(P(Q)G )[[ν]] is associative. For the id proof of ii), first note that F •J,µ 1 = hˇj0 −µ0 ( id−ν
F ) = F = 1 •J,µ F for all µ, G ∗ F ∈ h(P(Q) )[[ν]] since µ, F = 0 and t−Γˇj −µ F = F for totally horizontal 0
0
polynomial functions, implying that f J,µ 1 = f = 1 J,µ f for f ∈ P(Q)[[ν]]. Now, an immediate computation yields F •J,µ F − F •J,µ F = νhˇj0 −µ0 ({F, F }B0 ) + O(ν 2 ),
for all F, F ∈ h(P(Q)G ),
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but according to statement iv) of Proposition 3.3 this implies that for f, f ∈ P(Q) the lowest order in f J,µ f − f J,µ f is given by the Poisson bracket corresponding to ωb0 . Assertion iii) is obvious from Lemma 3.4 ii) and the definition of the P . Since is a differential associative product on h(P(Q)G )[[ν]] resp. P(Q)G [[ν]]/Iµ, star product and µ, a differential operator, it is obvious that the product •J,µ on h(P(Q)G )[[ν]] can be described by bidifferential operators. But this also implies that the corresponding star product on P(Q)[[ν]] is given by bidifferential operators. Since bidifferential operators are completely determined by their values on polynomial functions, J,µ extends to a star product on C ∞ (T ∗ Q)[[ν]] in a unique way. Remark 3.6 Clearly, pulling back the star product J,µ on C ∞ (T ∗ Q)[[ν]] to C ∞ ((T ∗ Q)µ0 )[[ν]] via the symplectomorphism Ψµ0 constructed in the proof of The∗ J,µ orem 3.2 one obtains a star product J,µ on the symplectic manifold Ψµ0 := Ψµ0 −1 ˇ (J (µ0 )/G, ωµ0 ). All results we derive in the sequel about the star products J,µ 0
will then transfer almost literally to the star products J,µ Ψµ . 0
In the remaining section we prove several properties of the reduction scheme introduced above. This will partially clarify the dependence of J,µ on the chosen µ ∈ g∗ G + νg∗c G [[ν]]. Moreover, we thus explain the relation between automorphisms resp. derivations of and those of J,µ . In particular, we show that our reduction is natural with respect to G-isomorphisms of star products which preserve P(Q)[[ν]] and satisfy an appropriate compatibility relation for the momentum values. Proposition 3.7 J ,µ
denote the products on h(P(Q)G )[[ν]] obtained by reduci) Let •J,µ and • tion of (, J, µ) and ( , J , µ ). Moreover, let T be an isomorphism (equivalence transformation) from to which preserves P(Q)[[ν]] and which commutes with every Φ∗g . Define T : h(P(Q)G )[[ν]] → h(P(Q)G )[[ν]] by TF := hˇj0 −µ0
id TF id − νµ ,
,
F ∈ h(P(Q)G )[[ν]].
(3.23) J ,µ
Then T is an isomorphism (equivalence transformation) from •J,µ to • , if µ = µ+δµ, where δµ ∈ g∗ G +νg∗c G [[ν]] is given by δµ, ξ = J (ξ)−T J(ξ). The inverse is given by id −1 −1 T F , F ∈ h(P(Q)G )[[ν]]. (3.24) T F = hˇj0 −µ0 id − νµ, If J (ξ) = T J(ξ), then T is an isomorphism (equivalence transformation) J ,µ from •J,µ to • .
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ii) Let •J,µ and •J,µ denote the products on h(P(Q)G )[[ν]] obtained by reduction of (, J, µ) and (, J, µ ). Assume A to be an automorphism of (starting with id) which preserves P(Q)[[ν]] and commutes with every Φ∗g . Define A : h(P(Q)G )[[ν]] → h(P(Q)G )[[ν]] by id AF , F ∈ h(P(Q)G )[[ν]]. (3.25) AF := hˇj0 −µ0 id − νµ ,
Then A is an isomorphism (equivalence transformation) from •J,µ to •J,µ , if µ = µ + δµ, where δµ ∈ g∗ G + νg∗c G [[ν]] is given by δµ, ξ = J(ξ) − AJ(ξ). The inverse is given by id A−1 F , F ∈ h(P(Q)G )[[ν]]. (3.26) A−1 F = hˇj0 −µ0 id − νµ, If J(ξ) = AJ(ξ), then A is an automorphism of •J,µ (starting with id). iii) Let D denote a C[[ν]]-linear derivation of which preserves P(Q)[[ν]] and commutes with every Φ∗g . Define D : h(P(Q)G )[[ν]] → h(P(Q)G )[[ν]] by id DF := hˇj0 −µ0 DF , F ∈ h(P(Q)G )[[ν]]. (3.27) id − νµ, Then D is a C[[ν]]-linear derivation of •J,µ , if DJ(ξ) = 0 for all ξ ∈ g. iv) Every isomorphism resp. equivalence transformation, automorphism (starting with id) or derivation constructed according to i), ii) or iii) transfers to an isomorphism resp. equivalence transformation, automorphism (starting with id), or derivation on P(Q)[[ν]] with the reduced star product as algebra structure. Moreover, the induced map extends in a unique way to an isomorphism resp. equivalence transformation, automorphism (starting with id), or derivation on C ∞ (T ∗ Q)[[ν]] with the reduced star product as algebra structure. v) All the above constructions can be localized in the sense that starting from a local isomorphism resp. equivalence transformation, automorphism (starting with id) or derivation on C ∞ (T ∗ U )[[ν]] resp. P(U )[[ν]], where U ⊆ Q is an open subset of Q, one obtains a local isomorphism resp. equivalence transformation, automorphisms (starting with id) or derivation on P(U )[[ν]] resp. C ∞ (T ∗ U )[[ν]], where U = p(U ) ⊆ Q. Proof. For the proof of the first three statements it suffices to show i) since ii) is just a special case of i) and the proof of iii) is only a slight adaption of the argument which proves i). Since T commutes with Φ∗g it also commutes with LξT ∗ Q for every ξ ∈ g. By definition of a quantum Hamiltonian this implies that ν1 ad (J (ξ) − T J(ξ)) = 0. But this entails that for every such isomorphism T one has J (ξ) − T J(ξ) ∈ C[[ν]], hence δµ, ξ = J (ξ)−T J(ξ) defines an element δµ ∈ g∗ +νg∗c [[ν]]. Using the G-equivariance of the quantum momentum maps and the G-invariance
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of T it is evident that δµ is G-invariant. Moreover, it is straightforward to see P that the condition µ = µ + δµ implies that T maps elements of Iµ, to elements J ,µ
P
of I µ , . Using the definition of the products •J,µ and • on h(P(Q)G )[[ν]] an easy computation then shows that T as defined in Eq. (3.23) satisfies the stated properties and that its inverse is given by Eq. (3.24). Statement iv) is obvious by observing that the induced mappings on P(Q)[[ν]] can be described as formal series of differential operators possibly composed with the pull-back of a diffeomorphism of T ∗ Q preserving P(Q). This latter situation occurs in case T and A do not start with id. Statement v) is evident from the fact that we are only concerned with local operators. Note that, evidently, the above construction of mappings on the reduced star product algebras is unfortunately not rich enough to yield all these mappings but only describes under which circumstances such mappings on the original star product on C ∞ (T ∗ Q)[[ν]] descend to such mappings on them. Now we want to consider star products which possess certain additional properties and will show under which preconditions these properties transfer to the reduced star products. Let us first recall some notions of special star products (cf., e.g., [11, 19]): i) A star product s resp. as on T ∗ Q is said to be of standard ordered resp. anti-standard ordered type, if for all f ∈ C ∞ (T ∗ Q)[[ν]] and χ ∈ C ∞ (Q)[[ν]] π ∗ χ s f = π ∗ χ f
resp. f as π ∗ χ = f π ∗ χ.
(3.28)
ii) A star product is called of Vey type or a natural star product, if the bidifferential operator describing the star product at order r in the formal parameter is of order r in each argument. Lemma 3.8 i) If s resp. as is a star product of standard ordered resp. anti-standard ordered type on (T ∗ Q, ωB0 ), then for every quantum momentum map J and every possible choice of a momentum value µ the reduced star product sJ,µ resp. asJ,µ on (T ∗ Q, ωb0 ) is also of standard ordered resp. anti-standard ordered type. ii) If is a star product of Vey type on (T ∗ Q, ωB0 ), then for every quantum momentum map J and every possible choice of a momentum value µ the reduced star product J,µ on (T ∗ Q, ωb0 ) is of Vey type as well. Proof. For the proof of i) we only consider the case of a standard ordered star product since the proof for anti-standard ordered star products is completely analogous. First, observe that for χ ∈ C ∞ (Q) we have l(P (χ)) = l(π ∗ χ) = P (p∗ χ) = π ∗ p∗ χ.
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Thus, an easy computation yields ∗ ∗ h id J,µ π p χ s P t l(P (χ))•s l(P (t)) = hˇj0 −µ0 id − νµ,s id π ∗ p ∗ χ P th = hˇj0 −µ0 = l(P (χ) P (t)). id − νµ,s =P((χt)h )=l(P(χt))
This implies that π ∗ χsJ,µ P (t) = π ∗ χ P (t) for all t ∈ Γ∞ ( T Q), hence sJ,µ is of standard ordered type. The proof of ii) consists of a rather lengthy but straightforward argument counting the order of differentiation in every order of id . the formal parameter within the projection hˇj0 −µ0 ◦ id−ν
µ, Now recall (cf., e.g., [26, Def. 3]) that a star product is called Hermitian, if the operation of complex conjugation C, where we set Cν := −ν, is an anti-automorphism of . Analogously, is said to have the ν-parity property, if P := (−1)degν is an anti-automorphism of , where degν := ν∂ν . Finally, a star product which is Hermitian and has the ν-parity property is called star product of Weyl type. In contrast to the above lemma, where the properties of the original star product transfer to the reduced star product without any further conditions, the property of being Hermitian and the ν-parity are not stable with respect to reduction, in general, unless certain additional conditions on the quantum momentum map J and the momentum value µ are satisfied. Lemma 3.9 i) Let be a Hermitian star product on (T ∗ Q, ωB0 ). If the relation ν ν C J+ (ξ) − µ+ , ξ − tr (ad(ξ)) = J+ (ξ) − µ+ , ξ − tr (ad(ξ)) (3.29) 2 2 is satisfied for all ξ ∈ g, then the reduced star product J,µ is Hermitian as well. ii) Let be a star product on (T ∗ Q, ωB0 ) which has the ν-parity property. If the equation ν ν P J+ (ξ) − µ+ , ξ − tr (ad(ξ)) = J+ (ξ) − µ+ , ξ − tr (ad(ξ)) (3.30) 2 2 holds true for all ξ ∈ g, then the reduced star product J,µ also has the ν-parity property. iii) Let be a star product of Weyl type on (T ∗ Q, ωB0 ). If Eqs. (3.29) and (3.30) are satisfied, then the reduced star product J,µ is of Weyl type, too. Proof. To prove i) we observe first that CJ defines a G-equivariant quantum Hamiltonian since is assumed to be Hermitian. Suppose that Eq. (3.29) holds true.
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Then it is easy to verify that C(νµ, F ) = νµ, CF for all F ∈ P(Q)G [[ν]]. Since hˇj0 −µ0 and l obviously commute with C, this implies that J,µ is Hermitian. The proof of ii) is completely analogous to the one of i) replacing C by P and iii) follows by combination of i) and ii). Now we consider homogeneous star products on (T ∗ Q, ω0 ), i.e., star products for which H = Lξ0 + ν∂ν is a derivation. Observe that for every κ ∈ [0, 1] both the star product κ and the star product B κ , B = νB1 constructed in Section 2.1 are of this kind. Therefore, the following results directly apply to some of the examples we will discuss in more detail in Sections 4.1 and 6.1. Lemma 3.10 i) Let denote a homogeneous star product on (T ∗ Q, ω0 ) and recall that the Gequivariant quantum momentum map J has the form J(ξ) = P (ξQ )+˜ µ0 , ξ+ ˜0 ∈ g∗ G . Then the reduced star product J,µ on (T ∗ Q, ω0 ) turns J+ (ξ) with µ out to be homogeneous as well, if J and the momentum value µ satisfy HJ(ξ) − J(ξ) = ν∂ν µ, ξ − µ, ξ
for all ξ ∈ g.
(3.31)
ii) Under the preconditions of i), there exists another G-equivariant quantum momentum map J of particular form J (ξ) = P (ξQ ) + νJ1 (ξ) = P (ξQ ) + νπ ∗ j1 (ξ) with j1 ∈ C 1 (g, C ∞ (Q)) and another momentum value µ = νµ1 such that J ,µ coincides with J,µ . Proof. First observe that Eq. (3.31) implies µ ˜0 = µ0 , since Lξ0 P (ηQ ) = P (ηQ ). Therefore J,µ is a star product with respect to the canonical symplectic form on T ∗ Q. Using (3.31) it is easy to show that the mapping H, which evidently P commutes with every Φ∗g , preserves Iµ, . This implies that the mapping H : h(P(Q)G )[[ν]] → h(P(Q)G )[[ν]] , which is defined by HF := h
id HF id − νµ,
,
F ∈ h(P(Q)G )[[ν]],
is a derivation with respect to the product •J,µ . Since according to Eq. (3.9) the map ri , 1 ≤ i ≤ dim (G) is homogeneous of degree −1 with respect to ξ0 , an easy computation shows that H commutes with νµ, . Thus HF = ν∂ν F + h(Lξ0 F ) holds true. From this observation it is evident that the composition l−1 ◦ H ◦ l, which is a derivation of the star product J,µ , equals H = ν∂ν + Lξ , where ξ 0 0 denotes the canonical Liouville vector field on (T ∗ Q, ω 0 ). This proves i). For the proof of ii) one has to analyze Eq. (3.31). By J(ξ) ∈ P(Q)[[ν]] and since every eigenvalue of Lξ0 has to be a non-negative one observes that J has the
integer, ∞ µ0 , ξ + νπ ∗ j1 (ξ) + r=2 ν r µr , ξ. Using µ ˜0 = µ0 we then form J(ξ) = P (ξQ ) + ˜ obtain the relation J(ξ) − µ, ξ = P (ξQ ) + νπ ∗ j1 (ξ) − νµ1 , ξ. This implies that J,µ equals J ,µ which proves ii).
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Remark 3.11 Proposition ii) of the preceding lemma shows in particular that two reduced star products J,µ and J ,µ coincide, if J(ξ)−µ, ξ = J (ξ)−µ , ξ. This follows from the fact that only the difference of the quantum momentum map and the momentum value enter the construction of the reduced star products. In view of this observation it is equivalent to either fix a quantum momentum map and vary the momentum values or vary the quantum momentum map and choose the momentum value to be zero. In the general considerations of the following sections we will nevertheless treat J and µ as independent parameters of the construction, but in concrete examples it will turn out to be convenient to restrict the consideration to one fixed quantum momentum map and to vary the momentum values arbitrarily.
4 Invariant star products on T ∗ Q and quantum moment maps In this section we prepare the grounds for the phase space reduction of the star products κ and B κ defined in Section 2.1. To this end we will provide conditions on the data entering the construction of κ and B κ which guarantee that the obtained star products are G-invariant. Furthermore, we provide conditions which are necessary and sufficient for the existence of a G-equivariant quantum momentum map. Note that all results derived in this section hold for the lifted action of an arbitrary Lie group action on the base manifold Q and that the assumption made earlier for phase space reduction, namely that the action is proper and free, is not needed here. Only assuming that there exists a G-invariant torsion free connection on Q we will show in particular that every G-invariant star product on (T ∗ Q, ωB0 ) is G-equivalent to some star product B 0 . This will actually turn out to be one of the key results for the computation of the characteristic class of the reduced star products in Section 5. In the course of these investigations, we also obtain a complete classification up to G-equivalence of star products on (T ∗ Q, ωB0 ) which are invariant with respect to lifted group actions. This turns out to be a slight refinement of the classification results in [4] for our special geometric situation.
4.1
Invariance of κ and B κ and their quantum moment maps
In order to derive necessary and sufficient conditions for the G-invariance of the star products κ , we prove the following: Proposition 4.1 Let φ denote a diffeomorphism of Q, let Φ = T ∗ (φ−1 ) be the lift to the cotangent bundle and assume that the connection ∇ is invariant with respect to φ, i.e., that φ∗ ∇X Y = ∇φ∗ X φ∗ Y holds true for all X, Y ∈ Γ∞ (T Q). Then there exists for every κ ∈ [0, 1] a uniquely determined formal series of differential operators Sκ,φ on C ∞ (T ∗ Q) which starts with id and commutes with H = Lξ0 +ν∂ν such that φ∗ κ (f ) (φ−1 )∗ = κ (Sκ,φ Φ∗ f )
for all f ∈ C ∞ (T ∗ Q)[[ν]].
(4.1)
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In addition Sκ,φ , is explicitly given by exp(κνD) − id ∗ Sκ,φ = exp −F (φ α − α) . D
(4.2)
Moreover, S0,φ and S1,φ are automorphisms of 0 and 1 , respectively. Furthermore, Sκ,φ is an automorphism of κ for κ = 0, 1 if and only if D(φ∗ α − α) = 0. Proof. A straightforward computation using the of ∇ and that F φ-invariance satisfies Eq. (A.8) yields φ∗ κ (f ) (φ−1 )∗ = κ Nκ−1 Φ∗ Nκ f . Next observe that Φ∗ ∆0 f = ∆0 Φ∗ f by Eqs. (A.7). Using the factorization property of Nκ given in Eq. (2.9), this proves that the formal series of differential operators Sκ,φ given in (4.2) satisfies (4.1). The facts that Sκ,φ starts with id and that it commutes with H are obvious from its explicit form. For the proof of the uniqueness assume that S κ,φ is a second formal series of differential operators having the same properties as Sκ,φ . Restricting to the space of functions polynomial in the momenta we obtain κ (S κ,φ F ) = κ (Sκ,φ F ) for all F ∈ P(Q)[[ν]]. Now observe that the homogeneity of S κ,φ and Sκ,φ implies that they map elements F ∈ P(Q)[[ν]] to elements of P(Q)[[ν]]. Since the restriction of κ to P(Q)[[ν]] is injective one thus concludes S κ,φ F = Sκ,φ F . But this implies S κ,φ = Sκ,φ since differential operators on C ∞ (T ∗ Q) are completely determined by their values on P(Q), proving the uniqueness of Sκ,φ . In case κ = 0, S0,φ is an automorphism of 0 since it coincides with id. For κ = 1 and A = ν(φ∗ α − α) consider the operator A1 defined by Eq. (2.12). Then A1 coincides with S1,φ and is an automorphism of 1 if and only if d(φ∗ α − α) = 0. Now recall that dα = −tr (R) and that φ∗ R = R due to the φ-invariance of ∇. Then φ∗ α − α is obviously closed, hence S1,φ is an automorphism of 1 . Finally, let us consider the case κ = 0, 1. Assuming that D(φ∗ α − α) = 0, the map Sκ,φ = exp (−κνF (φ∗ α − α)) coincides with Aκ for A = κν(φ∗ α − α), hence it is an automorphism of κ due to the closedness of φ∗ α − α. Conversely, let us assume that Sκ,φ is an automorphism of κ . Then Sκ,φ (Aκ )−1 is again an automorphism of κ , where the latter is taken for A = κν(φ∗ α − α). Now a straightforward expansionof this automorphism yields
Sκ,φ (Aκ )−1 = exp κ(κ−1) ν 2 F (D(φ∗ α − α)) + O(ν 3 ) . But since every star prod2 uct automorphism starting with id is of the form exp(νD), where D is a derivation of the star product, and since the lowest order of a derivation is given by a symν 2 F (D(φ∗ α − α)) must be zero since plectic vector field, one concludes that κ(κ−1) 2 otherwise it can never define a symplectic vector field. Together with the injectivity of F and the precondition κ = 0, 1 we thus obtain D(φ∗ α − α) = 0 and the proposition is proved. Using the explicit formulas for the κ -left- and κ -right-multiplication with functions π ∗ χ, χ ∈ C ∞ (Q), we also get the following. Lemma 4.2 Let φ denote a diffeomorphism of Q and let κ be invariant with respect to Φ = T ∗ (φ−1 ), i.e., Φ∗ (f κ f ) = Φ∗ f κ Φ∗ f for all f, f ∈ C ∞ (T ∗ Q)[[ν]]. Then the connection ∇ is invariant with respect to φ.
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Proof. Let us consider the case κ = 0 first. Using Eq. (2.11) and property (A.8) for F, a comparison of the terms of second order in the formal parameter within the equation Φ∗ (f 0 π ∗ χ) = Φ∗ f 0 π ∗ φ∗ χ yields F ((φ∗ D)dχ ) = F (Ddχ ) for all χ ∈ C ∞ (Q). Since F is injective, we thus conclude that (φ∗ D)dχ = Ddχ . Evaluating this for local coordinate functions xl and using φ∗ D = D − dxi ∨ dxj ∨ is (Sφ (∂xi , ∂xj )), we thus obtain that the tensor field Sφ from Lemma A.1 has to vanish. Hence ∇ has to be invariant with respect to φ. For κ = 0 we consider ∗ ∗ ∗ ∗ ∗ the order terms 2 of Φ (π χ κ f) = π∞φ χ κ Φ f . Then Eq. (2.10) yields second 2 ∗ F κ (φ D)dχ = F κ Ddχ for all χ ∈ C (Q). Since κ = 0, we may conclude as above that this implies φ-invariance of ∇. Combining the results of Proposition 4.1 and Lemma 4.2 we get: Theorem 4.3 i) The star products 0 and 1 are G-invariant if and only if the connection ∇ is G-invariant. ii) For κ = 0, 1 the star product κ is G-invariant if and only if the connection ∇ is G-invariant and D(φ∗g α − α) = 0
for all g ∈ G.
(4.3)
φ∗g κ (f ) φ∗g−1 = κ Sκ,φg Φ∗g f ,
(4.4)
iii) In either case we have
where according to Eq. (4.2) the automorphism Sκ,φg of κ is given by exp(κνD) − id ∗ Sκ,φg = exp −F φg α − α . (4.5) D for More explicitly, this means S0,φg = id, Sκ,φg = exp −F κν φ∗g α − α exp(νD)−id ∗ κ = 0, 1, and S1,φg = exp −F φg α − α . D Proof. Let us assume that ∇ is G-invariant. Then apply (4.1) with φ = φg and use the representation property of κ to check κ Sκ,φg Φ∗g (f κ f ) = κ Sκ,φg Φ∗g f κ Sκ,φg Φ∗g f , f, f ∈ C ∞ (T ∗ Q)[[ν]]. Restricting to F, F ∈ P(Q)[[ν]], the injectivity of κ and the fact that Sκ,φg Φ∗g preserves P(Q)[[ν]] imply that Sκ,φg Φ∗g (F κ F ) = Sκ,φg Φ∗g F κ Sκ,φg Φ∗g F
for all F, F ∈ P(Q)[[ν]].
Since κ is described by bidifferential operators and since these are completely determined by their values on P(Q)[[ν]], this yields that Sκ,φg Φ∗g is an automorphism of κ . For κ = 0 resp. κ = 1 we already know by Proposition 4.1 and the
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G-invariance of ∇ that S0,φg resp. S1,φg is an automorphism of 0 resp. 1 , and so is Φ∗g . In case κ = 0, 1 the additional condition D(φ∗g α − α) = 0 is equivalent to Sκ,φg being an automorphism of κ , but this is equivalent to Φ∗g being an automorphism of κ . Conversely, let us assume that κ is G-invariant. Then Lemma 4.2 implies that ∇ is also G-invariant. But now we can use the above consideration to conclude that for κ = 0, 1 the additional equation D(φ∗g α − α) = 0 must hold. Together this proves claims i) and ii) of the theorem. Assertion iii) is obvious by the proof of i), ii), and Eq. (4.2). Finally, we consider the star products B κ . Before we can state the generalization of Theorem 4.3 we provide some rather trivial but nevertheless crucial results: Lemma 4.4 2 i) For every κ ∈ [0, 1] and formal series B ∈ ZdR (Q)[[ν]] of closed two-forms on B ∗ Q with real B0 , κ -left-multiplication by π χ, χ ∈ C ∞ (Q)[[ν]] coincides with ∗ κ -left-multiplication by π ∗ χ. Analogously, B κ -right-multiplication by π χ co∗ incides with κ -right-multiplication by π χ. ii) For every κ ∈ [0, 1] a mapping Ak of the form given in Eq. (2.12) is an automorphism of κ , if and only if it is an automorphism of B κ. B ∗ B j j −1 ∗ ∗ ∗ Proof. By definition of κ we have π χ κj f |T Oj = Aκ ((Ajκ )−1 π∗ χ|T O∗j ) j −1 κ ((Aκ ) f |T ∗ Oj ) . By the explicit form of Aκ it is obvious that (Aκ ) π χ = π χ holds true and that (Ajκ )−1 commutes with F (β) for every β ∈ Γ∞ ( T ∗ Q)[[ν]]. Using these observations together with the expression for π ∗ χ κ f given in Eq. ∗ B (2.10) one obtains π ∗ χ B κ f = π χ κ f . The proof for κ -right-multiplication is completely analogous. Assertion ii) is obvious from the fact that mappings as given in Eq. (2.12) commute with the local isomorphisms Ajκ from κ to B κ. After these preparations we can state one of the main results of this section: Theorem 4.5 B i) The star products B 0 and 1 are G-invariant, if and only if ∇ is G-invariant ∗ and φg B = B for all g ∈ G. ii) For κ = 0, 1 the star product B κ is G-invariant, if and only if ∇ and B are G-invariant and D(φ∗g α − α) = 0 for all g ∈ G. Proof. First assume that ∇ and B are G-invariant. Additionally, assume for κ = 0, 1 that D(φ∗g α − α) = 0 for all g ∈ G. Then the star products κ are all G-invariant by Theorem 4.3. But this implies that Ajκ Φ∗g (Ajκ )−1 = t∗−(Aj −φ∗ Aj ) g 0 0 exp(κνD) − exp((κ − 1)νD) j (A − φ∗g Aj ) − (Aj0 − φ∗g Aj0 ) exp −F Φ∗g νD
(4.6)
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∗ is a local automorphism of (C ∞ (T ∗ Oj )[[ν]], B κ ). Obviously, the equality φg B = B j ∗ j entails the relation d(A − φg A ) = 0. By Lemma 4.4 ii) this implies that
t∗−(Aj −φ∗ Aj ) g 0 0 exp(κνD) − exp((κ − 1)νD) j (A − φ∗g Aj ) − (Aj0 − φ∗g Aj0 ) exp −F νD ∗ is a local automorphism of (C ∞ (T ∗ Oj )[[ν]], B κ ). But then Φg is also an automorphism of B κ , proving one direction of i) and ii). For the converse statement assume that B κ is G-invariant. Then Lemma 4.4 i) and Lemma 4.2 imply that ∇ is Ginvariant. For the cases κ = 0 and κ = 1 this implies that 0 and 1 are invariant. Together with the above considerations this entails by Lemma 4.4 ii) that id − exp(−νD) j t∗−(Aj −φ∗ Aj ) exp −F (A − φ∗g Aj ) − (Aj0 − φ∗g Aj0 ) g 0 0 νD
and t∗−(Aj −φ∗ Aj ) g 0 0
exp(νD) − id j j ∗ j ∗ j (A − φg A ) − (A0 − φg A0 ) exp −F νD
define local automorphisms of (C ∞ (T ∗ Oj )[[ν]], 0 ) and (C ∞ (T ∗ Oj )[[ν]], 1 ), respectively. But this implies that Aj − φ∗g Aj is closed, hence B is G-invariant. For the case κ = 0, 1 we need a more detailed argument. By invariance of the connection the mapping Sκ,φg Φ∗g is an automorphism of κ , hence Ajκ Sκ,φg Φ∗g (Ajκ )−1 is a local B automorphism of B κ . Since by assumption κ is G-invariant, this yields that t∗−(Aj −φ∗ Aj ) 0
g
0
exp(κνD) − exp((κ − 1)νD) j (A − φ∗g Aj ) − (Aj0 − φ∗g Aj0 ) exp −F νD exp(κνD) − id ∗ × exp −F (φg α − α) D
is a local automorphism of B κ . Considering the order zero part in the formal parameter this implies that t−(Aj −φ∗ Aj ) has to be a local symplectomorphism 0
g
0
with respect to ωB0 . Thus Aj0 − φ∗g Aj0 is closed. Factorizing the local automorphism corresponding to Aj0 − φ∗g Aj0 we now obtain that exp(κνD) − exp((κ − 1)νD) j (A − Aj0 − φ∗g (Aj − Aj0 )) exp −F νD exp(κνD) − id ∗ + (φg α − α) D
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j ∗ j is a local automorphism of B κ . In order one of ν this means that −F A1 − φg A1 +κ(φ∗g α − α) defines a symplectic vector field with respect to ωB0 . Therefore Aj1 − φ∗g Aj1 + κ(φ∗g α − α) is closed. But since ∇ is invariant, φ∗g α − α is closed and so is Aj1 − φ∗g Aj1 . Factorizing again the local automorphism corresponding to the closed one-form ν(Aj1 − φ∗g Aj1 + κ(φ∗g α − α)), we end up with another local automorphism of B κ . To lowest exponent of this automorphism order in the formal parameter the ∗ D(φ α − α) . This term again has to define is given by F −Aj2 + φ∗g Aj2 + κ(κ−1) g 2
a symplectic vector field. For κ = 0, 1 this is only possible, if D(φ∗g α − α) = 0, hence Sκ,φg is an automorphism of κ and κ is G-invariant. This means that the mapping Ajκ Φ∗g (Ajκ )−1 Φ∗g−1 from Eq. (4.6) is a local automorphism of B κ . Now this local automorphism is an operator of form Aκ as given in Eq. (2.12) which in turn is an automorphism of κ (and hence of B κ by Lemma 4.4 ii)), if and only if Aj − φ∗g Aj is closed. But this implies finally that φ∗g B − B = d(φ∗g Aj − Aj ) = 0, i.e., that B is G-invariant.
After having investigated the invariance of the star products κ and B κ , we will next consider G-equivariant quantum momentum maps for these star products. More precisely, we now state one of the key results on phase space reduction of the star products κ . To this end we henceforth assume κ to be G-invariant. Moreover, we assume the connection ∇ to be G-invariant and that for κ = 0, 1 the relation D(φ∗g α − α) = 0 holds true for all g ∈ G. Proposition 4.6 For every κ ∈ [0, 1], the G-invariant star product κ on (T ∗ Q, ω0 ) is strongly G-invariant, i.e., the map J ∈ C 1 (g, C ∞ (T ∗ Q)) with J(ξ) := J0 (ξ) = P (ξQ ) is a G-equivariant quantum momentum map for the lifted Lie group action. In case J , where J0 is assumed to be real, is another G-equivariant quantum momentum map for the same action, then J is given by J (ξ) = J0 (ξ) + ˜ µ, ξ with µ ˜ ∈ g∗ G + νg∗c G [[ν]]. Proof. Put g = exp(tξ) in Eq. (4.4), differentiate the resulting equation with respect to t and evaluate at t = 0. Then one obtains exp(κνD) − id LξQ α f + LξT ∗ Q f . [LξQ , κ (f )] = κ −F D From the definition of κ one concludes immediately that κ (P (ξQ )) = −νLξQ − κν κ (π ∗ (div(ξQ ) + α(ξQ ))), where div(ξQ ) = tr (Y → ∇Y ξQ ) denotes the covariant divergence of the vector field ξQ . Using the representation property of κ this implies 1 κ − adκ (P (ξQ ) + κνπ ∗ (div(ξQ ) + α(ξQ )))f + ν exp(κνD) − id LξQ α f = κ LξT ∗ Q f . F D
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At this point we need a little technical result. Sublemma 4.7 If the star product κ is G-invariant, then the following equality holds true: exp(κνD) − id LξQ α f κadκ (π ∗ (div(ξQ ) + α(ξQ )))f = F D for all f ∈ C ∞ (T ∗ Q). (4.7) Proof. For κ = 0 there is nothing to show, since both sides coincide with 0. ∗ For κ = 1 we obtain from (2.10) and (2.11) that ad1 (π (div(ξQ ) + α(ξQ ))) = F exp(νD)−id d(div(ξQ ) + α(ξQ )) . Now observe that (LξQ ∇)Y Z = LξQ ∇Y Z − D ∇LξQ Y Z − ∇Y LξQ Z = 0 by invariance of ∇. A straightforward computation then shows −tr (R) (ξQ , Y ) = dα(ξQ , Y ) = d(div(ξQ ))(Y ), but from this equation and Cartan’s formula Eq. (4.7) is immediate in case κ = 1. For κ = 0, 1 the additional relation D(φ∗g α − α) = 0 obviously implies DLξQ α = 0, hence F exp(κνD)−id LξQ α = κνF LξQ α = κF exp(κνD)−exp((κ−1)νD) LξQ α . By the D D argument above LξQ α = d(div(ξQ ) + α(ξQ )), hence F exp(κνD)−id L α equals ξ Q D ∗ κadκ (π (div(ξQ ) + α(ξQ ))) due to the explicit formulas (2.10) and (2.11). 1 By the sublemma we now obtain κ − ν adκ (P (ξQ ))f = κ LξT ∗ Q f . Recall that κ restricted to P(Q)[[ν]] is injective and observe that LξT ∗ Q and adκ (P (ξQ )) preserve P(Q)[[ν]]. Then we obtain 1 − adκ (P (ξQ ))F = LξT ∗ Q F ν
for all F ∈ P(Q)[[ν]].
Since κ is described by bidifferential operators, this implies −LξT ∗ Q = ν1 adκ (P (ξQ )), hence J(ξ) = J0 (ξ) = P (ξQ ) is a quantum Hamiltonian which is known to be G-equivariant. The claim about the ambiguity of the quantum momentum map is a general result which holds over arbitrary symplectic manifolds (cf. [32, Prop. 6.3]). For the study of reduction of the invariant star products κ the G-equivariant quantum momentum map given by J(ξ) = P (ξQ ) appears to be the preferred one, since it coincides with the (canonical) G-equivariant classical momentum map. In the sequel we therefore will mostly work with this momentum map in order to compute the reduced products of κ . ∗ Now we consider the star products B κ on (T Q, ωB0 ) and will derive necessary and sufficient conditions for the existence of G-equivariant quantum momentum maps for G-invariant star products of form B κ . In view of Theorem 4.5 we assume additionally to the conditions which guarantee κ to be G-invariant that B is an 2 element of ZdR (Q)G [[ν]].
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∗ Proposition 4.8 Suppose that the star product B κ on (T Q, ωB0 ) is G-invariant. Then there is a G-equivariant quantum momentum map for the G-action under consideration, if and only if there is an element j ∈ C 1 (g, C ∞ (Q))[[ν]] with realvalued j0 such that
dj(ξ) = iξQ B
and
φ∗g j(ξ) = j(Ad(g −1 )ξ)
for all ξ ∈ g, g ∈ G.
(4.8)
In this case we particularly have j([ξ, η]) = B(ξQ , ηQ )
for all ξ, η ∈ g.
(4.9)
Moreover, the map J ∈ C 1 (g, C ∞ (T ∗ Q))[[ν]] given by J(ξ) := P (ξQ ) + π ∗ j(ξ) defines a G-equivariant quantum momentum map, which is unique up to elements of g∗ G + νg∗c G [[ν]]. Proof. Consider the following equation over T ∗ Oj : exp(κνD) − exp((κ − 1)νD) ∗ j φg A − φ∗g Aj0 Φ∗g Ajκ Φ∗g−1 = t∗−φ∗ Aj exp −F . g 0 νD By differentiation at the neutral element of G we obtain exp(κνD) − exp((κ − 1)νD) LξQ Aj . LξT ∗ Q = Ajκ LξT ∗ Q (Ajκ )−1 − F νD Now the invariance of B κ implies that κ is invariant as well, hence we conclude from Proposition 4.6 that LξT ∗ Q = − ν1 adκ (P (ξQ )). By a direct computation using that Ajκ is a local homomorphism from κ to B κ and that P (ξQ ) is a polynomial in (P (ξQ ) − the momenta of degree one this implies that Ajκ LξT ∗ Q (Ajκ )−1 = − ν1 adB κ π ∗ (Aj (ξQ ))). On the other hand LξQ Aj = iξQ B + d(Aj (ξQ )). By the explicit (π ∗ (Aj (ξQ ))) we thus obtain formula for adB κ LξT ∗ Q = −
1 ν
exp(κνD) − exp((κ − 1)νD) i (P (ξ )) + F B . adB Q ξQ κ D
From this equation and from the explicit form of the inner derivations adB (π ∗ χ), κ ∞ χ ∈ C (Q)[[ν]] it is clear that there is a quantum Hamiltonian for the considered action, if and only if there is an element j ∈ C 1 (g, C ∞ (Q))[[ν]] such that dj(ξ) = iξQ B for all ξ ∈ g. Observe that the condition necessary for the solvability of this equation is satisfied, since by invariance of B κ we have that diξQ B = LξQ B = 0. Evidently, J with J(ξ) = P (ξQ ) + π ∗ j(ξ) is additionally G-equivariant, if and only if the second condition in Eq. (4.8) is satisfied. Finally, differentiating this equation in g at e and using the equality iξQ B = dj(ξ), it is straightforward to check that Eq. (4.9) holds true. The claim about the ambiguity of J is well known, hence the proposition is proved.
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Remark 4.9 It is immediate to show that B κ admits a quantum Hamiltonian J which additionally satisfies ν1 adB (J(ξ))J(η) = J([ξ, η]), if and only if there is an κ element j ∈ C 1 (g, C ∞ (Q))[[ν]] such that dj(ξ) = iξQ B and B(ξQ , ηQ ) = j([ξ, η]). Moreover, these conditions determine j up to elements of g∗ + νg∗c [[ν]] which are invariant with respect to the coadjoint action of g, i.e., which vanish on [g, g]. ∗ Corollary 4.10 Suppose that the star product B κ on (T Q, ωB0 ) is G-invariant ∗ and that J0 with J0 (ξ) = P (ξQ ) + π j0 (ξ) is a G-equivariant classical momentum 2 G map. Then B κ is strongly G-invariant, if and only if B+ ∈ νZdR (Q) [[ν]], where B = B0 + B+ , is horizontal, i.e., if and only if
for all ξ ∈ g.
iξQ B+ = 0
4.2
(4.10)
General invariant star products on T ∗ Q: Relations to B 0 and classification
In this section we consider star products on (T ∗ Q, ωB0 ) which are G-invariant with respect to a lifted group action. Under the assumption that there is a Ginvariant torsion free connection ∇ on Q we will in particular construct for every such a G-equivalent star product of form B 0 . Incidently, our results show that there is a G-equivariant quantum momentum map for an arbitrary G-invariant star product in the above sense if and only if there is a G-equivariant quantum momentum map for a certain star product B 0 . But for these star products we have already derived criteria for the existence of G-equivariant quantum momentum maps. Thus we obtain necessary and sufficient conditions for the existence of Gequivariant quantum momentum maps for an arbitrary G-invariant star product. Finally, we use the G-equivalence between a G-invariant and and an appropriate ∗ star product B 0 to give a classification of star products on (T Q, ωB0 ) up to Gequivalence. Actually, our result is a slight refinement of the general classification results of [4] in the particular case of a cotangent bundle with lifted G-action. First we need a few results which allow for a comparison of two different star
(k) (k+1) l products B and B in case B (k) = kl=0 ν l Bl and B (k+1) = k+1 0 0 l=0 ν Bl . Note that the following lemma holds for arbitrary ordering parameters κ ∈ [0, 1] but since we only need it for κ = 0 we restrict the proof to this particular case. The proof for general κ is an immediate adaption.
k+1
k Lemma 4.11 For k ∈ N let B (k+1) = l=0 ν l Bl and B (k) = l=0 ν l Bl be series of B
(k+1)
closed two-forms on Q. Then the describing bidifferential operators Cr 0 B Cr 0
(k)
of the corresponding star products (k+1) B 0
Cr (k+1) B 0
Ck+2
(k+1) B 0
(f, f ) − Ck+2
(k+1) B 0
(k) B 0
= Cr
and
(k) B 0
on (T Q, ωB0 ) satisfy
for r = 0, . . . , k + 1
(k) B 0
and
(k) B 0
(f , f ) = Ck+2 (f, f ) − Ck+2 (f , f ) − (π
∗
and
∗
Bk+1 )(XfB0 , XfB0 ),
(4.11) (4.12)
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where XfB0 denotes the Hamiltonian vector field of f ∈ C ∞ (T ∗ Q) with respect to the symplectic form ωB0 . Proof. Let O be an element of a good open cover of Q. Over O consider a local potential A(k+1) = A(k) + ν k+1 Ak+1 of B (k+1) , where A(k) is a local potential of B (k) and Ak+1 is a local potential of Bk+1 . From the very definition of (k+1) the star products corresponding and B (k) one obtains that Sk+1 := to B id−exp(−νD) k+1 defines a local equivalence from (C ∞ (T ∗ O)[[ν]], exp −F ν Ak+1 νD (k)
(k+1)
B ) to (C ∞ (T ∗ O)[[ν]], B ). Expanding Sk+1 = id − ν k+1 F (Ak+1 ) + O(ν k+2 ) 0 0 (k) (k+1) and B one obtains Eqs. (4.11) and (4.12) and expanding the products B 0 0 by an immediate computation. The following results (which are essentially due to Lichnerowicz [22, Lemma 1 and 2] and Bertelson et al. [4, Prop. 2.1]) will turn out to be crucial for our further investigations. Lemma 4.12 i) Suppose that there is a G-invariant torsion free connection ∇ on Q. Then every G-invariant differential C ∞ (T ∗ Q)-Hochschild p-coboundary C (p ≥ 1) which vanishes on constants is the coboundary of a G-invariant differential p − 1-cochain c vanishing on constants. In case C(F1 , . . . , Fp ) ∈ P(Q) for all F1 , . . . , Fp ∈ P(Q) one can additionally achieve that c(F1 , . . . , Fp−1 ) ∈ P(Q) for all F1 , . . . , Fp−1 ∈ P(Q). ii) For every closed G-invariant p-form Ω (p ≥ 1) on T ∗ Q there exists a Ginvariant closed p-form β on Q and a G-invariant p − 1-form Ξ on T ∗ Q such that i∗ Ξ = 0 and Ω = dΞ + π ∗ β. (4.13) If Ω(XFB10 , . . . , XFBp0 ) ∈ P(Q) for all F1 , . . . , Fp ∈ P(Q), then Ξ can be chosen
0 ) ∈ P(Q) for all F1 , . . . , Fp−1 ∈ P(Q). such that Ξ(XFB10 , . . . , XFBp−1
Proof. For the proof of i) recall from [9, Def. 4] that every torsion free connection ∗ ∇ on Q defines a torsion free connection ∇T Q on T ∗ Q which is G-invariant if the original connection ∇ is invariant. Moreover, having chosen a torsion free connecC on C∞ (M ) can be tion ∇M on a manifold M it is well known thatevery p-cochain uniquely written as C(f1 , . . . , fp ) = C I1 ;...;Ip (DM )
|I1 |
|Ip |
f1 (∂yI1 ) . . . (DM ) I1 ;...;Ip
fp
are components of (∂yIp ), where I1 , . . . , Ip denote multi-indices and the C |I1 | |Ip | ∞ tensor fields in Γ ( TM ⊗ ··· ⊗ T M) with respect to local coordinates y 1 , . . . , y m of M . In case C is a coboundary one can explicitly build a p − 1cochain c such that δH c = C (δH denotes the Hochschild differential), where the tensor fields defining c are given as combinations of those of C. But this implies that in case C and the connection are invariant c is also invariant (cf. [4, Remark 2.1]) proving the first part of i). For the proof of the second part of i)
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one first has to observe that the covariant derivative with respect to the above ∗ connection preserves P(Q) since the Christoffel symbols of ∇T Q in a local bundle chart are polynomials in the momenta. Together with the assumption that C preserves P(Q), which implies that the components of the corresponding tensor fields are polynomials in the momenta, this entails that c also preserves P(Q). For the proof of ii) consider the closed p-form β := i∗ Ω on Q which is evidently G-invariant by G-invariance of Ω. Therefore the closed two-form Ω − π ∗ β is Ginvariant and i∗ (Ω − π ∗ β) = 0. Now, consider the homotopy H : R × T ∗ Q → T ∗ Q, (t, ζx ) → H(t, ζx ) := tζx . By means of this homotopy one can explicitly define a 1 p − 1-form Ξ by Ξ(X1 , . . . , Xp−1 ) := 0 (H ∗ (Ω − π ∗ β))(∂t , X1 , . . . , Xp−1 )dt. This Ξ satisfies dΞ = Ω − π ∗ β by the classical proof of Poincar´e’s lemma. Due to the compatibility of the above homotopy with the Φg the thus defined p − 1-form is G-invariant. Since H(t, i(x)) = i(x) we also have i∗ Ξ = 0. From the explicit shape 0 of Ξ it is also obvious that Ξ(XFB10 , . . . , XFBp−1 ) ∈ P(Q) for all F1 , . . . , Fp−1 ∈ P(Q) in case Ω(XFB10 , . . . , XFBp0 ) ∈ P(Q) for all F1 , . . . , Fp ∈ P(Q).
Using these technical preparations we can adapt the proof of [4, Prop. 4.1] to the present situation and obtain: Proposition 4.13 Suppose that there is a G-invariant torsion free connection ∇ on Q. Then we have: i) For every star product on (T ∗ Q, ωB0 ) which is invariant with respect
∞ to the lifted action of a G-action on Q there is a formal series ν1 B+ = l=1 ν l−1 Bl 2 ∈ ZdR (Q)G [[ν]] of G-invariant closed two-forms on Q and a G-equivalence transformation T from to the G-invariant star product B 0 , where B = B0 + B+ . ii) In case P(Q)[[ν]] is a -subalgebra one can find B+ and a G-equivalence transformation T from to B 0 as in i) such that T F ∈ P(Q)[[ν]] for all F ∈ P(Q)[[ν]]. Proof. Let be an arbitrary G-invariant star product on (T ∗ Q, ωB0 ) as above (0) and consider the G-invariant star product B , where B (0) = B0 . In order zero 0 (0) trivially coincide. Since of the formal parameter, the star products and B 0 B
(0)
both are star products with respect to ωB0 the anti-symmetric part of C1 0 − C1 vanishes. Therefore Lemma 4.12 i) implies that there is a G-invariant 1-cochain c1 B
(0)
with C1 0 − C1 = δH c1 . Now put T (0) := id − νc1 , which is clearly G-invariant. Define another G-invariant star product by (0) := T (0) , i.e., let T (0) (f f ) = B
(0)
(0)
(T (0) f ) (0) (T (0) f ). Then an easy computation shows that Cr 0 = Cr for r = 0, 1. By associativity of these two star products one obtains that the antiB
(0)
symmetric part of C2 0 via B
C2 0
(0)
B
(f, f ) − C2 0
(0)
− C2
(0)
defines a G-invariant closed two-form Ω1 on T ∗ Q
(0) (0) (f , f ) − C2 (f, f ) + C2 (f , f ) = Ω1 XfB0 , XfB0 .
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Again from Lemma 4.12 ii) we get that B1 := i∗ Ω1 is a G-invariant closed twoform on Q and that there is a G-invariant one-form Ξ1 on T ∗ Q such that Ω1 = (1) dΞ1 + π ∗ B1 . Then we consider the G-invariant star product B with B (1) = 0 (0) := T(0) (0) , where T(0) f := B (0) + νB1 and the G-invariant star product B
f + νΞ1 (XfB0 ). According to Eq. (4.11) we have Cr 0 Now it is straightforward to check that Cr anti-symmetric part of
B C2 0
(1)
(0) − C2
(0)
(1)
(1) B 0
= Cr
B
= Cr 0
(0)
for r = 0, 1.
for r = 0, 1 and that the (0)
vanishes due to the definition of (1) B 0
and Eq.
(0)
(4.12). But then Lemma 4.12 i) yields C2 − C2 = δH c2 with a G-invariant (1) (1)(0) 1-cochain c2 . Putting := T with T (1) := id − ν 2 c2 we then obtain B
(1)
(1)
Cr 0 = Cr for r = 0, 1, 2. Proceeding inductively we thus can find G-invariant operators T (l) = id−ν l+1 cl+1 for l = 0, . . . , k, G-invariant T(m) for m = 0, . . . , k−1 with T(m) f = f + ν m+1 Ξm+1 (XfB0 ) and G-invariant closed two-forms B1 , . . . , Bk (k)
k (k) B on Q such that Cr 0 = Cr for r = 0, . . . , k + 1. Hereby, B (k) = l=0 ν l Bl and (k) = T (k) with T (k) := T (k) T(k−1) T (k−1) . . . T(0) T (0) . For k → ∞ we thus (∞) obtain a well-defined G-invariant
∞ l formal series of differential operators T := T and a formal series B = l=0 ν Bl of G-invariant closed two-forms on Q such that the G-invariant star product T coincides with B 0 . Hence T is a G-equivalence , and B is given by B = B − B proving i). For the proof of ii) one from to B + + 0 0 B (0) just has to observe that P(Q)[[ν]] is a 0 -subalgebra and that by assumption P(Q)[[ν]] is a -subalgebra. Using Lemma 4.12 this implies that in every step of the above construction one can achieve that T (l) and T(m) map elements of P(Q)[[ν]] to elements of P(Q)[[ν]]. To verify this check by induction that P(Q)[[ν]] is both (l) (k) -subalgebra since P(Q)[[ν]] is a B -subalgebra a (k) -subalgebra as well as a 0 for all occurring B (l) . Note that even after having fixed a G-invariant torsion free connection on Q one cannot use the construction of ν1 B+ in the proof of the preceding proposition to define a map from the space of G-invariant star products on (T ∗ Q, ωB0 ) to the space of formal series of closed G-invariant two-forms on Q. This fact is caused by the freedom of choice in the equivalence transformations which in fact affects the explicit form of ν1 B+ . For instance, in the definition of T (0) we could have replaced c1 by c1 + LX , where X is a G-invariant vector field on T ∗ Q. An easy computation then shows that this gives rise to a modification of B1 by the additional term −di∗ (iX ωB0 ). Later on, we will show that the above construction nevertheless induces a bijection between the G-equivalence classes of G-invariant star products 2 2 G ∞ ∗ G and formal series of elements of the space HdR ,G (Q) = ZdR (Q) /d(Γ (T Q) ) of second degree cohomology classes of G-invariant de Rham cohomology. Moreover, this bijection will actually turn out to be independent of the chosen connection, hence is canonical (cf. [4, Thm. 4.1] for an analogous statement on general symplectic manifolds).
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As an immediate corollary Proposition 4.13 implies: Corollary 4.14 Under the assumptions of the proposition the following holds true: i) There is a G-equivariant quantum momentum map for a G-invariant star product on (T ∗ Q, ωB0 ), if and only if there is a G-equivariant quantum momentum map for the star product B 0 , where B = B0 + B+ denotes a formal series of closed G-invariant two-forms on Q as in Proposition 4.13 i). ii) If P(Q)[[ν]] is in addition a -subalgebra, then every G-equivariant quantum momentum map J for satisfies J(ξ) ∈ P(Q)[[ν]] for all ξ ∈ g. Proof. For the proof of i) consider a G-equivariant quantum momentum map J B with J B (ξ) := for and a G-equivalence T to the star product B 0 . Then J T J(ξ) clearly defines a G-equivariant quantum momentum map for B 0 . Vice versa, every G-equivariant quantum momentum map J B for B 0 defines a G-equivariant quantum momentum map J for via J(ξ) := T −1 J B (ξ). For the proof of ii) apply Proposition 4.13 to show that T can be chosen to preserve P(Q)[[ν]]. Given a G-equivariant quantum momentum map J for we then get one for B 0 by J B (ξ) := T J(ξ). But from Proposition 4.8 we have that J B is of form J B (ξ) = P (ξQ ) + π ∗ j(ξ) ∈ P(Q)[[ν]], where j ∈ C 1 (g, C ∞ (Q))[[ν]] satisfies the conditions in Eq. (4.8). This implies in particular that J(ξ) = T −1 J B (ξ) ∈ P(Q)[[ν]]. Since any other G-equivariant quantum momentum map J for differs from J by an element of g∗ G + νg∗c G [[ν]] we obtain J (ξ) ∈ P(Q)[[ν]] for every G-equivariant J which is a quantum momentum map for . In view of the second part of the above corollary it now becomes clear that one of the assumptions we made for our reduction scheme – namely that J(ξ) ∈ P(Q)[[ν]] – is in fact not an additional assumption but a consequence of the assumption that P(Q)[[ν]] is a -subalgebra. For the purposes of the following section, where we will compute the characteristic class of a reduced star product the results achieved so far would completely suffice. But with a little more effort we can give a classification of the G-invariant star products on cotangent bundles up to G-equivalence, a result which is of independent interest. To this end we show in a first step the following proposition. Its proof is rather technical but yields the key tools for the main results of the last part of this section. B
∗ Proposition 4.15 Let B 0 resp. 0 be G-invariant star products on (T Q, ωB0 ) which are obtained from G-invariant torsion free connections ∇ resp. ∇ and Ginvariant formal series of closed two-forms B resp. B on Q starting with B0 . 1 1 1 B Then B 0 and 0 are G-equivalent, if and only if ν B+ = ν (B − B0 ) and ν B+ = 1 ν (B − B0 ) are G-cohomologous, i.e., if and only if there is a G-invariant formal series of one-forms β on Q such that ν1 B+ = ν1 B+ + dβ. Proof. Let us first assume that B+ and B+ are G-cohomologous. Then we want to B
prove that B 0 and 0 are G-equivalent. To this end we need the following result about standard ordered star products obtained from torsion free connections.
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Sublemma 4.16 Let ∇ and ∇ denote two torsion free connections on Q and 0 resp. 0 the corresponding standard ordered representation of the star product 0 resp. 0 on (T ∗ Q, ω0 ). Then there is a uniquely determined formal series S of differential operators on C ∞ (T ∗ Q) such that SF ∈ P(Q)[[ν]] for all F ∈ P(Q)[[ν]] and for all f ∈ C ∞ (T ∗ Q)[[ν]]. (4.14) 0 (f ) = 0 (Sf ) This implies that S is an equivalence transformation from (C ∞ (T ∗ Q)[[ν]], 0 ) to (C ∞ (T ∗ Q)[[ν]], 0 ). Moreover, S satisfies Sπ ∗ χ = π ∗ χ for all χ ∈ C ∞ (Q)[[ν]]. Consequently, one has 1 1 Sad0 (π ∗ χ)S −1 = ad0 (π ∗ χ). ν ν
(4.15)
Proof. First recall that the operators of symmetric covariant derivation D and D satisfy D = D − dxi ∨ dxj ∨ is (S(∂xi , ∂xj )), where the symmetric tensor field S is given by ∇X Y = ∇X Y + S(X, Y ). Now it is easy to see that 0 (F ) lies in the image of 0 for all F ∈ P(Q)[[ν]]. By injectivity of the restriction of 0 to P(Q)[[ν]] one can define a map S : P(Q)[[ν]] → P(Q)[[ν]] by SF := 0 −1 ( 0 (F )). Using the explicit form of the standard ordered representations it is immediate to check that this map is given by a formal series of differential operators on P(Q) and that this series starts with id. By definition, S satisfies Eq. (4.14) on polynomial functions in the momenta and S(F 0 F ) = SF 0 SF for all F, F ∈ P(Q)[[ν]]. This implies that S, which clearly extends uniquely to a mapping on C ∞ (T ∗ Q)[[ν]], is an equivalence from 0 to 0 and satisfies Eq. (4.14). Uniqueness of S is again a direct consequence of 0 being injective when restricted to P(Q)[[ν]]. For the proof of the further properties of S observe first that 0 satisfies 0 (π ∗ χF ) ψ = χ 0 (F ) ψ for all χ, ψ ∈ C ∞ (Q)[[ν]] and that an analogous relation holds for 0 . Using the definition of S this yields that S(π ∗ χF ) = π ∗ χSF for all F ∈ P(Q)[[ν]], hence S commutes with left-multiplications by formal functions pulled-back from Q. In particular, we obtain by setting F = 1 that Sπ ∗ χ = π ∗ χ. From this relation Eq. (4.15) is immediate, since S is an equivalence from 0 to 0 . Using G-equivariance of 0 and 0 the preceding sublemma entails that 0 (f ) = 0 (Sf ) = 0 Φ∗g SΦ∗g−1 f for all f ∈ C ∞ (T ∗ Q)[[ν]]. Clearly, Φ∗g SΦ∗g−1 preserves P(Q)[[ν]]. But since S is the uniquely determined map which satisfies Eq. (4.14) and preserves P(Q)[[ν]] we conclude that S = Φ∗g SΦ∗g−1 , i.e., S is a G-equivalence from 0 to 0 . We claim that if ν1 B+ = ν1 B+ + dβ with B
β ∈ Γ∞ (T ∗ Q)G [[ν]] one can use S to build a G-equivalence from 0 to B 0 . To j j j this end consider A = Aj0 + A + with local potentials Aj0 of B0 and A + of B+ over some Oj which is assumed to be an element of a good open cover of Q. Then j Aj = Aj0 + A + + νβ is a local potential of B. Composition of S with the local j isomorphisms (A 0 )−1 and Aj0 defined in Eq. (2.12) gives rise to a local isomorj B j phism Tj := A0 S(A 0 )−1 from (C ∞ (T ∗ Oj )[[ν]], 0 ) to (C ∞ (T ∗ Oj )[[ν]], B 0 ). We
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now have to show that these local isomorphisms actually glue together to a globally j i defined G-equivalence. To verify this recall from [11, Thm. 3.4] that (A 0 )−1 A 0 f = ∞ ∗ ˜ ˜ f (1) for every f ∈ C (T (Oj ∩ Oi ))[[ν]], where f is given by the unique solud ˜ tion of the differential equation dt f (t) = ν1 ad0 (π ∗ aji )f˜(t) with f˜(0) = f . Here, j i aji ∈ C ∞ (Oj ∩ Oi )[[ν]] satisfies daji = A − A over Oj ∩ Oi . Since S is an equivalence from 0 to 0 and Sπ ∗ χ = π ∗ χ, this implies that S(A j0 )−1 A i0 S −1 f = f˜˜(1), ˜ ˜ ˜ d ˜ where f˜ solves dt f˜(t) = ν1 ad0 (π ∗ aji )f˜(t) with f˜(0) = f . But from the choice j of the local potentials Aj and A we obtain daji = Aj − Ai over Oj ∩ Oi . j i Therefore, S(A 0 )−1 A 0 S −1 is equal to (Aj0 )−1 Ai0 , i.e., over C ∞ (T ∗ (Oj ∩ Oi ))[[ν]] we have Tj Ti −1 = id. But this entails that we can define a global isomorphism B T from (C ∞ (T ∗ Q)[[ν]], 0 ) to (C ∞ (T ∗ Q)[[ν]], B 0 ) by (T f )|T ∗ Oj := Tj f |T ∗ Oj . j j From the fact that A and A coincide at zeroth order in ν it is obvious that this isomorphism is in fact an equivalence transformation. It remains to show that T is G-invariant. Let us fix g ∈ G. Using that S is G-invariant we obtain j Φ∗g Tj Φ∗g−1 = Φ∗g Aj0 Φ∗g−1 SΦ∗g (A 0 )−1 Φ∗g−1 over φg−1 (Oj ). Now consider an index i such that φg−1 (Oj ) ∩ Oi = ∅. Then we claim that Φ∗g Tj Φ∗g−1 Ti −1 = id over φg−1 (Oj ) ∩ Oi . To prove this we may assume without loss of generality that φg−1 (Oj ) ∩ Oi is contractible since in case it were not contractible we could cover it by open contractible subsets and use the following argument for each j i element of this covering. By G-invariance of B we have d(φ∗g A − A ) = 0. Hence j i there exist formal functions bji over φg−1 (Oj ) ∩ Oi such that φ∗g A − A = dbji . j −1
Then Φ∗g A 0 Φ∗g−1 A 0 turns out to be the local automorphism of 0 generated by ν1 ad0 (π ∗ bji ). Like in the argument which showed that T is well defined one j i concludes that SΦ∗g (A 0 )−1 Φ∗g−1 A 0 S −1 is the local automorphism of 0 generated i
by ν1 ad0 (π ∗ bji ). On the other hand Φ∗g (Aj0 )−1 Φ∗g−1 Ai0 coincides with this local automorphism, since by G-invariance of β and the choice of the local potentials j j i Aj and A the equation φ∗g Aj − Ai = φ∗g A − A = dbji is valid. But this implies −1 that Φ∗g Tj Φ∗g−1 Ti = id over φg−1 (Oj ) ∩ Oi . Hence T is G-invariant. So we have B
shown that B 0 and 0 are G-equivalent, if
1 ν B+
and
1 ν B+
are G-cohomologous. B
For the proof of the converse statement assume that B 0 and 0 are G-equivalent and B are not G-cohomologous. and that l ≥ 1 is the smallest index such that B l
∞ l
∞ As usual we have set hereby B = B0 + r=1 ν r Br and B = B0 + r=1 ν r Br .
∞ r r Now consider B := B0 + l−1 r=0 ν Br + r=l ν Br . Then B+ is G-cohomologous B
. Consequently, we know from above that 0 is G-equivalent to B to B+ 0 . But B this implies that B and are also G-equivalent. It is now immediate to de0 0 B
B
duce from Lemma 4.11 that the describing bidifferential operators Cr 0 and Cr 0 coincide for r = 0, . . . , l and that B
B
B
B
0 0 0 0 Cl+1 (f, f ) − Cl+1 (f , f ) − Cl+1 (f, f ) + Cl+1 (f , f ) = (π ∗ Bl − π ∗ Bl )(XfB0 , XfB0 ).
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B But then the G-equivalence of B implies that π ∗ Bl − π ∗ Bl is G-exact 0 and 0 (cf. [4, Thm. 2.1]), i.e., there is a G-invariant one-form Ξ on T ∗ Q such that π ∗ Bl − π ∗ Bl = dΞ. Thus Bl = di∗ Ξ + Bl , where i∗ Ξ is G-invariant. Hence Bl and Bl are G-cohomologous, which is a contradiction, proving the other direction of the statement of the proposition.
Using the construction in the proof of Proposition 4.13 and the preceding proposition we can state the following classification result: Theorem 4.17 i) To every star product on (T ∗ Q, ωB0 ) which is invariant with respect to the lifted action of a G-action on Q one can assign a formal series in the second G-invariant de Rham cohomology of Q by cG : → cG () :=
1 2 [B+ ]G ∈ HdR ,G (Q)[[ν]], ν
(4.16)
where ν1 B+ denotes a formal series of G-invariant closed two-forms on Q as in Proposition 4.13. ii) The map cG in Eq. (4.16) is independent of the chosen G-invariant torsion free connection on Q which was used to define ν1 B+ . Moreover, cG induces by []G → cG () a bijection between the set of G-equivalence classes of G2 invariant star products as in i) and HdR ,G (Q)[[ν]]. Proof. For the proof of i) we just have to verify that cG is well defined. To this end consider two G-equivalence transformations T , T and two formal series B, B of closed G-invariant two-forms on Q starting with B0 such that T = B 0 and B B T = B as in Proposition 4.13. Then and are G-equivalent, clearly. 0 0 0 Proposition 4.15 implies that ν1 B+ and ν1 B+ are G-cohomologous, therefore cG is well defined. For the proof of ii) consider G-equivalences T , T and two formal series B, B of closed G-invariant two-forms on Q starting with B0 such that T = B 0 B B and T = 0 , where B 0 and 0 are obtained from different connections ∇ B and ∇ . Then B 0 and 0 are G-equivalent. Proposition 4.15 implies again that 1 1 ν B+ and ν B+ are G-cohomologous which shows that cG is independent of the connection used to construct ν1 B+ . Furthermore, observe that for G-equivalent star products and one has cG () = cG ( ), since there exists a G-equivalence B from to B 0 , hence we obtain a G-equivalence from to 0 . This implies that cG induces a mapping from the set of G-equivalence classes of G-invariant star 2 products as in i) to HdR ,G (Q)[[ν]] as given above. To prove surjectivity of this map B 2 consider 0 , where B = B0 +νβ and β ∈ ZdR (Q)G [[ν]] is an arbitrary formal series of closed G-invariant two-forms on Q. By definition of cG and choosing id as Gequivalence we obtain cG (B 0 ) = [β]G . Since β is arbitrary this proves surjectivity. To prove injectivity let , be star products with cG () = cG ( ). By Proposition B 4.15 the corresponding star products B 0 and 0 are G-equivalent which implies that and are G-equivalent.
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5 The characteristic class of the reduced star products J,µ In this section we want to compute the characteristic class of the reduced star products J,µ in order to clarify how the equivalence classes of these products depend on the initially chosen parameters of the reduction scheme. To this end we proceed in two steps. Under the general assumption of a proper and free G∗ action on Q we first consider a star product B 0 on (T Q, ωB0 ) constructed from a G-invariant torsion free connection ∇ on Q and a formal series B of G-invariant closed two-forms on Q. Recall that due to the properness of the G-action such a G-invariant connection exists on Q by Palais’ Theorem and that the resulting B 0 is G-invariant by the results of the preceding section. Additionally, we assume that j ∈ C 1 (g, C ∞ (Q))[[ν]] satisfies the conditions given in Eq. (4.8) so that we can use J B with J B (ξ) = P (ξQ ) + π ∗ j(ξ) as G-equivariant quantum momentum map. Then it is possible to compute the characteristic class of the reduced star J B ,µ
explicitly in terms of B, j, µ and the connection on p : Q → Q. In a product B 0 second step we use the relation between a star product that satisfies all necessary assumptions for the applicability of our reduction procedure and a G-equivalent J B ,µ
J,µ star product B to the one of B . 0 to relate the characteristic class of 0 Let us now provide a few results needed in the sequel for the computation of characteristic class of star products. For more details we refer the reader to [18, 26] which treats the case of arbitrary symplectic manifolds and to [11], where the special case of cotangent bundles is considered. Recall that the characteristic class of a star product on (M, ω) is an element 2 of [ω] ν + HdR (M )[[ν]]. For its computation one first has to find local derivations of ∞ (C (Oj )[[ν]], ), so-called local ν-Euler derivations, where {Oj }j∈I is a good open cover of M . These local ν-Euler derivations are of form ∞ Ej = ν∂ν + Lξj + ν r Dj,r , (5.1) r=1
where ξj is a local conformally symplectic vector field (i.e., Lξj ω|Oj = ω|Oj ), and the Dj,r are locally defined differential operators on C ∞ (Oj ). With the help of these the characteristic class can be determined except for the part of order zero in the formal parameter. For the computation of that term one additionally needs an explicit expression for the anti-symmetric part C2− (f, f ) = 12 (C2 (f, f ) − C2 (f , f )) of the bidifferential operator describing the considered star product in the second order of the formal parameter. More explicitly, to determine the characteristic class from the Ej one considers Ei − Ej over Oi ∩ Oj , which is a quasi-inner derivation, i.e., there are local formal functions dij ∈ C ∞ (Oi ∩ Oj )[[ν]] such that Ei −Ej = ν1 ad (dij ). Now, whenever Oi ∩Oj ∩Ok = ∅ the sums dijk = djk −dik +dij ˇ lie in C[[ν]] and define a 2-cocycle whose Cech class [dijk ] ∈ H 2 (M, C)[[ν]] is in2 dependent of the choices made. The corresponding class d() ∈ HdR (M )[[ν]] is − called Deligne’s derivation-related class of . In addition, let C2 denote the image of C2− under the projection onto the second component of the decomposition
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2 ∞ 2 HChev, (M ), C ∞ (M )) = C ⊕ HdR (M ) which describes the second cohomolnc (C ogy group of the null-on-constants Chevalley cohomology of (C ∞ (M ), { , }), taken with respect to the adjoint representation. Together, d() and C2− define the characteristic class c() of by
c()0 = −2C2−
and
∂ν c() =
1 d(). ν2
(5.2)
2 The so-defined element of [ω] ν +HdR (M )[[ν]] classifies the equivalence classes of star products on a symplectic manifold (M, ω) in a functorial way (cf. [18, Thm. 6.4]). In the following {Oi }i∈I denotes a G-invariant good open cover of Q which projects via p to a good open cover {Oi }i∈I = {p(Oi )}i∈I of Q. Such a cover exists due to the fact that the action on Q is proper. Our first goal is to define local νJ B ,µ
Euler derivations of B on every Oi using certain local ν-Euler derivations of 0 B B 0 on Oi . Unfortunately, an arbitrary ν-Euler derivation of 0 cannot be projected J B ,µ
down to such a derivation of B in general, since such derivations usually nei0 ther preserve P(Q)[[ν]] nor are G-invariant and even then do not preserve the ideal of G-invariant formal functions generated by the G-equivariant quantum momentum map. Therefore, we have to find appropriately modified ν-Euler derivations of B 0 , where we let us lead by intuition rather than by a deductive procedure. Actually, the relation between b0 and B0 in the lowest order of the characteristic J B ,µ
classes of B and B 0 0 suggests that a similar relation might also hold in higher orders. In analogy to classical phase space reduction we therefore consider the following formal series of closed two-forms on Q: B + dΓˇj−µ ,
(5.3)
where Γ has been defined by Eq. (3.4) and has been extended by C[[ν]]-linearity. A straightforward argument which is completely analogous to the computation in the proof of Theorem 3.2 now shows by G-equivariance of j that B + dΓˇj−µ is a formal series of G-invariant closed horizontal two-forms on Q. Hence there exists a uniquely defined formal series b of closed two-forms on Q such that B + dΓˇj−µ = p∗ b.
(5.4)
Now we choose local potentials ai of b over Oi , i.e., dai = b|Oi . For Oi ∩ Oj we choose local formal functions fij with dfij |Oi ∩Oj = (ai − aj )|Oi ∩Oj . Furthermore, we consider the local formal one-forms Ai on Oi defined by Ai := p∗ ai − Γˇj−µ . Then Ai − Aj = dp∗ fij holds true on Oi ∩ Oj by construction. Using these Ai we now consider the local isomorphisms Ai0 : (C ∞ (T ∗ Oi )[[ν]], 0 ) → (C ∞ (T ∗ Oi )[[ν]], B 0 ) from Eq. (2.12) and claim the following:
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Proposition 5.1 With notations from above the following holds true: i) The mappings Ei : C ∞ (T ∗ Oi )[[ν]] → C ∞ (T ∗ Oi )[[ν]] defined by Ei := Ai0 H(Ai0 )−1
(5.5)
are G-invariant local ν-Euler derivations of B 0 which preserve P(Oi )[[ν]] and P(Oi )[[ν]] ∩ IB . ,µ 0 ii) The mappings Ei : h(P(Oi )G )[[ν]] → h(P(Oi )G )[[ν]] defined by id Ei F , F ∈ h(P(Oi )G )[[ν]] Ei F := hˇj0 −µ0 id − νµ,B 0 are local derivations of (h(P(Oi )G )[[ν]], •B 0 Ei := l
−1
J B ,µ
). The corresponding mappings
◦ Ei ◦ l are local ν-Euler derivations of (P(Oi )[[ν]], B 0
uniquely extend to such derivations of (C ∞ (T ∗ Oi )[[ν]], B 0 iii) On C ∞ (T ∗ (Oi ∩ Oj ))[[ν]] one has Ei − Ej =
c(B 0
J B ,µ
)=
J B ,µ
J B ,µ
J B ,µ
) which
).
1 ad J B ,µ ((ν∂ν − id)π ∗ fij ). ν B 0
This implies that the characteristic class c(B 0
(5.6)
(5.7)
) is given by
1 ∗ J B ,µ [π b] − [π ∗ b1 ] + c(B )0 . 0 ν
(5.8)
Proof. For the proof of i) first observe that by definition of Ai one has dAi = B|Oi . Therefore, the mappings Ai0 are in fact local isomorphisms from (C ∞ (T ∗ Oi )[[ν]], 0 ) to (C ∞ (T ∗ Oi )[[ν]], B 0 ). By their explicit form it is obvious that they preserve P(Oi )[[ν]]. Moreover, they also turn out to be G-invariant due to the G-invariance of the Ai . Since H = Lξ0 + ν∂ν is evidently G-invariant and preserves P(Oi )[[ν]] these properties hold for Ei , as well. In addition, Ei is a local ν-Euler derivation of B 0 , as H is a global ν-Euler derivation of 0 . It remains to show that Ei preserves . But this follows from a straightforward proof of Ei J B (ξ)|T ∗ Oi = P(Oi )[[ν]] ∩ IB 0 ,µ B J (ξ)|T ∗ Oi −µ, ξ+ν∂ν µ, ξ which uses the explicit shape of J B and Ai0 . Using i) J B ,µ
it is rather evident that Ei defines a local derivation of •B on h(P(Oi )G )[[ν]], 0 and we only have to show that Ei is of form provided in Eq. (5.1). To this end id−exp(−νD) i (ν∂ν − id)A which recall from [11, Lemma 4.4] that Ei = H + F νD
∞ r −1 directly implies that Ei = l ◦ Ei ◦ l has form ν∂ν + r=0 ν Di,r , where the Di,r are locally defined differential operators. Hence, these mappings uniquely extend to C ∞ (T ∗ Oi )[[ν]], since differential operators are completely determined by their values on polynomial functions in the momenta. We only have to show that Di,0 = Lξ i with a locally defined vector field ξ i ∈ Γ∞ (T (T ∗ Oi )) satisfying
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Lξ ωb0 = ωb0 . But this follows from an easy computation expanding the exponent i in the above given expression for Ei . For the proof of iii) this expression again together with Ai − Aj = dp∗ fij entails 1 id ∗ F ((id − exp(−νD))(ν∂ν − id)p fij ) F (Ei − Ej )F = hˇj0 −µ0 id − νµ,B ν 0 =
1 ad J B ,µ (π ∗ p∗ (ν∂ν − id)fij )F, ν •B 0
where the second equality follows from Eqs. (2.10) and (2.11) together with Lemma 4.4 i). Conjugation of this result by l−1 yields Eq. (5.7). From the very definition of Deligne’s derivation related class and the definition of the local functions fij we J B ,µ
thus obtain d(B ) = (ν∂ν − id)[π ∗ b]. By definition of the characteristic class 0 this implies Eq. (5.8). J B ,µ
J B ,µ
To determine the missing part c(B )0 of the characteristic class of B 0 0 one has to compute the anti-symmetric part of the bidifferential operator describJ B ,µ
ing B in the second order of the formal parameter. As this is a rather cum0 bersome but nevertheless important computation we only give here the important intermediate steps and omit details of the proof. Lemma 5.2
B
B
2 3 0 0 i) Writing f B 0 f = f f + νC1 (f, f ) + ν C2 (f, f ) + O(ν ) for f, f ∈ ∞ ∗ C (T Q) we have:
1 ({f, f }B0 − ∆0 (f f ) + (∆0 f )f + f (∆0 f )) 2 (5.9) B B 1 C2 0 (f, f ) − C2 0 (f , f ) = − (∆0 {f, f }B0 − {∆0 f, f }B0 − {f, ∆0 f }B0 ) 2 1 ∗ − (π (B1 − tr (R)))(XfB0 , XfB0 ), (5.10) 2 B
C1 0 (f, f ) =
where tr (R) denotes the trace of the curvature tensor of ∇ and ∆0 the differential operator defined in Eq. (2.6). ii) For s, t ∈ Γ∞ ( T Q) the anti-symmetric part of the bidifferential operator B
J B ,µ
C2 0
B
which describes the star product B 0
J B ,µ
B
J B ,µ
on P(Q)[[ν]] is given by
J B ,µ
(P (s) , P (t)) − C2 0 (P (t) , P (s)) B = l−1 hˇj0 −µ0 µ,B P sh , P th B + C2 0 P sh , P th 0 0 B −C2 0 P th , P sh
C2 0
(5.11)
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1 δ0 {P (s) , P (t)}b0 − {δ0 P (s) , P (t)}b0 − {P (s) , δ0 P (t)}b0 2 1 1 b0 b0 + π ∗ −b1 − r + τλ (XP(s) , XP(t) ). 2 2 =−
527
(5.12)
Hereby, r and τλ are the unique closed two-forms on Q determined by p∗ r = −tr (R) + dΓdˇ and p∗ τλ = tr (ad(λ)), where d ∈ C 1 (g, C ∞ (Q)) is defined by d(ξ) := div(ξQ ). Moreover, mapping δ0 : P(Q) → P(Q) is given by the δ0 (P (s)) := l−1 (hˇj0 −µ0 (∆0 P sh )). Proof. For the proof of i) first recall from [9, Thm. 10] that the star product 1/2 is of Weyl type. Hence the describing bidifferential operators at order one and two of the formal parameter satisfy C1 1/2 (f, f ) = 12 {f, f }0 and C2 1/2 (f, f ) − 1/2 C2 (f , f ) = 0. Using these two relations together with the equalities (N1/2 f ) 0 i (N1/2 f ) = N1/2 (f 1/2 f ) and Ai0 (f 0 f |T ∗ Oi ) = (Ai0 f |T ∗ Oi ) B 0 (A0 f |T ∗ Oi ) a i straightforward computation which involves expansion of N1/2 , A0 and the products 1/2 , 0 , and B 0 up to the second order in ν then shows Eqs. (5.9) and (5.10). The first equality stated in ii) is obtained by an immediate computation using J B ,µ
the definition of B . In contrast, the proof of the second equality turns out to 0 be more involved but requires nothing more than a consequent application of the definitions. Last, we provide the argument showing that r and τλ are well defined. For τλ this is well known by Chern-Weil theory, and [τλ ] is a characteristic class of the principal G-bundle p : Q → Q. To prove that r is well defined observe that −iξQ tr (R) = dd(ξ) and φ∗g d(ξ) = d(Ad(g −1 )ξ) for all ξ ∈ g and repeat the argument used for Theorem 3.2 showing that b0 is well defined. By definition of the zeroth order of the characteristic class and the above result one directly obtains c(B 0
J B ,µ
τλ r . )0 = π ∗ b1 + − 2 2
(5.13)
At this point one might expect that the characteristic class of the reduced star product depends on the chosen G-invariant connection ∇ and that the geometry J B ,µ
of the principal bundle enters the characteristic class c(B ) via [τλ ]. As we will 0 show in the next lemma none of these dependencies occurs. Lemma 5.3 i) Let ∇ and ∇ be two G-invariant torsion free connections on Q. Then the corresponding closed two-forms r and r constructed as in Lemma 5.2 ii) are cohomologous. ii) With notations from Lemma 5.2 ii) we have r − τλ = −tr R − dw.
(5.14)
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Hereby, R denotes the curvature of the torsion free connection ∇ on Q defined by T p ∇th uh = ∇t u◦p for t, u ∈ Γ∞ (T Q). Moreover, w ∈ Γ∞ (T ∗ Q) is defined
(G) by p∗ w = H(W ) = W − dim W (ei Q )Γei with W ∈ Γ∞ (T ∗ Q) given by i=1
dim (G) Γel (∇X el Q ). Consequently, the cohomology classes of r W (X) = l=1 and τλ coincide: [r − τλ ] = [0]. Proof. For the proof of i) write ∇X Y = ∇X Y + S(X, Y ) with a symmetric G2 invariant tensor field S ∈ Γ∞ ( T ∗ Q ⊗ T Q). Observe that the G-invariant oneform tr (S) defined by tr (S) (X) := dxi (S(X, ∂xi )) satisfies tr (R ) = tr (R) + d(tr (S)) and div (X) = div(X) + tr (S) (X). Defining σ ∈ C 1 (g, C ∞ (Q)) by σ(ξ) := tr (S) (ξQ ) it is easy to see that there exists a unique one-form s on Q which satisfies p∗ s = tr (S) − Γσˇ . From the above identities relating the traces of the curvature tensors and the covariant divergences it is evident that r = r − ds. To prove ii) first verify that ∇ actually defines a torsion free connection on Q and that w is well defined. Then the proof consists of an easy computation showing that (−tr (R) + dΓdˇ)(th , uh ) − tr ad(λ(th , uh )) = p∗ (−tr R (t, u) − dw(t, u)), This computation mainly relies on splitting the definition of tr (R) into horizontal and vertical part. But since the trace of the curvature tensor of a torsion free connection is an exact two-form (cf. [9, Lemma 16]) this implies [r] = [τλ ]. Collecting our results we have shown: Theorem 5.4 The characteristic class of the star product B 0 given by 1 J B ,µ ) = [π ∗ b], c(B 0 ν
J B ,µ
on (T ∗ Q, ωb0 ) is (5.15)
2 where J B (ξ) = P (ξQ ) + π ∗ j(ξ) and b ∈ ZdR (Q)[[ν]] is determined by p∗ b = B + dΓˇj−µ .
Now we consider an arbitrary star product satisfying the assumptions for our reduction procedure and present the main result of this section. Theorem 5.5 Let be a G-invariant star product on (T ∗ Q, ωB0 ) such that P(Q)[[ν]] is a -subalgebra. Let J denote a G-equivariant quantum momentum map for and let µ ∈ g∗ G + νg∗c G [[ν]]. i) Assume that T is a G-equivalence from to B 0 as in Proposition 4.13 ii), where B = B0 +B+ is an appropriate formal series of G-invariant closed twoJ B ,µ
with G-equivariant forms. Then the star product J,µ is equivalent to B 0 quantum momentum map J B given by J B (ξ) := T J(ξ). ii) With notations from i), the characteristic class of the star product J,µ is given by 1 J B ,µ c(J,µ ) = c(B ) = [π ∗ b], (5.16) 0 ν
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where b denotes the uniquely determined formal series of two-forms on Q such that p∗ b = B + dΓˇj−µ and where j ∈ C 1 (g, C ∞ (Q))[[ν]] is given by j(ξ) = i∗ T J(ξ). Proof. i) is a direct consequence of Proposition 3.7 i) and iv) since a G-equivalence T exists due to Proposition 4.13 and since J B is a G-equivariant quantum moJ B ,µ
J,µ mentum map for B ) = c(B ) 0 by Corollary 4.14. By i) the equality c( 0 follows immediately. By Theorem 5.4 we obtain the claim about the explicit form of the characteristic class of J,µ . We only have to verify that j is actually given by j(ξ) = i∗ T J(ξ), but this is obvious since J B satisfies J B (ξ) = P (ξQ ) + π ∗ j(ξ) by Proposition 4.8.
Corollary 5.6 Let γ and γ denote two connection one-forms on p : Q → Q and let J,µ resp. (J,µ ) be the corresponding reduced star product on (T ∗ Q, ωb0 ) resp. (T ∗ Q, ωb0 ) obtained by reduction of the star product on (T ∗ Q, ωB0 ). Then the characteristic classes of J,µ and (J,µ ) coincide and there is an isomorphism from (C ∞ (T ∗ Q)[[ν]], J,µ ) to (C ∞ (T ∗ Q)[[ν]], (J,µ ) ). Moreover, the corresponding star J,µ products J,µ )Ψµ on ((T ∗ Q)µ0 , ωµ0 ) (cf. Remark 3.6) are equivalent. Ψµ and ( 0
0
Proof. Let Γˇj −µ be the one-form defined by ˇj0 − µ0 , γ . Then one observes 0 0 first that the translation tΓˇj −µ −Γˇj −µ along the fibre which maps the zero level 0
0
0
0
set (T ∗ Q)0 = {ζx ∈ T ∗ Q | ζx (ξQ (x)) = 0 for all ξ ∈ g} to itself clearly passes to the quotient defining a diffeomorphism of T ∗ Q. Moreover, it is easy to see that this diffeomorphism consists of a translation tβ0 along the fibres on T ∗ Q, where β0 is the unique one-form on Q such that p∗ β0 = Γˇj0 −µ0 − Γˇj0 −µ0 . By definition of b0 and b0 , where b0 is defined as in Theorem 3.2 using Γˇj0 −µ0 instead of Γˇj0 −µ0 , we have b0 = b0 − dβ0 . Hence, this diffeomorphism is in fact a symplectomorphism from (T ∗ Q, ωb0 ) to (T ∗ Q, ωb0 ). Let β be the unique formal series of one-forms on Q such that p∗ β = Γˇj−µ − Γˇj−µ . Then we analogously find that b = b − dβ, therefore c((J,µ ) ) = ν1 [π ∗ b ] = ν1 [π ∗ b] = c(J,µ ). Now we consider the star product (J,µ ) := t∗β0 J,µ on (T ∗ Q, ωb0 ), which has characteristic class c((J,µ ) ) = t∗β0 c(J,µ ) = ν1 [t∗β0 π ∗ b] = ν1 [π ∗ b] = c((J,µ ) ). Therefore it is equivalent to (J,µ ) . But then the composition of an equivalence transformation T from (J,µ ) to (J,µ ) with t∗β0 yields an isomorphism from (C ∞ (T ∗ Q)[[ν]], J,µ ) to (C ∞ (T ∗ Q)[[ν]], (J,µ ) ). One finally concludes that the corresponding star products on ((T ∗ Q)µ0 , ωµ0 ) are equivalent, since the characteristic class is natural with respect to diffeomorphisms (cf. [18, Thm. 6.4]) and since tβ0 = Ψµ0 ◦ (Ψµ0 )−1 by the above result, where Ψµ0 is defined as in Theorem 3.2 using γ instead of γ.
Similarly, we also find the dependence of the characteristic class on different choices of the G-equivariant quantum momentum map and different choices of the momentum value:
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Corollary 5.7 i) Let J,µ and J ,µ denote the star products on (T ∗ Q, ωb0 ) and (T ∗ Q, ωb0 ) obtained from two possibly different G-equivariant quantum momentum maps J and J satisfying J − J = µ ˜ ∈ g∗ G + νg∗c G [[ν]]. Then the characteristic classes fulfill 1 ˜λ ], (5.17) c(J,µ ) − c(J ,µ ) = [π ∗ µ ν 2 where µ ˜λ ∈ ZdR (Q)[[ν]] is determined by p∗ µ ˜λ = ˜ µ, λ. J,µ J,µ and denote the star products on (T ∗ Q, ωb0 ) and (T ∗ Q, ωb0 ) ii) Let obtained from two possibly different momentum values µ and µ and let µ ˜= µ − µ ∈ g∗ G + νg∗c G [[ν]]. Then the characteristic classes satisfy 1 ˜λ ], c(J,µ ) − c(J,µ ) = − [π ∗ µ ν
(5.18)
2 (Q)[[ν]] is determined by p∗ µ ˜λ = ˜ µ, λ. where µ ˜λ ∈ ZdR
Proof. Both claims are direct consequences of Theorem 5.5. For the proof of i) one just has to observe that for a G-equivalence T from to some B 0 we have j(ξ) − j (ξ) = i∗ (T J(ξ) − T J (ξ)) = ˜ µ, ξ, whereas the second claim is obvious. Finally, we are able to recover a relation between the characteristic class of the original star product on (T ∗ Q, ωB0 ) and the characteristic class of the reduced ∗ star product J,µ Ψµ0 (cf. Remark 3.6) on ((T Q)µ0 , ωµ0 ) which already has been observed by M. Bordemann [5] (cf. also [6]) for arbitrary symplectic manifolds. Corollary 5.8 Denote by iµ0 the inclusion of Jˇ0−1 (µ0 ) into T ∗ Q and by πµ0 the projection from Jˇ0−1 (µ0 ) to (T ∗ Q)µ0 . Then the characteristic classes of and J,µ Ψµ0 satisfy (5.19) i∗µ0 c() = πµ∗ 0 c(J,µ Ψµ ). 0
Proof. By naturality of characteristic classes with respect to diffeomorphisms we ∗ J,µ ). The commutative diagram (3.7) and Theorem 5.5 obtain c(J,µ Ψµ0 ) = Ψµ0 c( then entail πµ∗ 0 c(J,µ Ψµ ) = 0
1 ∗ ∗ ∗ 1 [π Ψ π b] = [i∗µ0 t∗Γˇj −µ π ∗ (B + dΓˇj−µ )] = i∗µ0 c(), 0 0 ν µ0 µ0 ν
where the last equality follows from π◦tΓˇj0 −µ0 = π and the fact that c() = ν1 [π ∗ B]. To conclude this section we discuss some special cases of Theorem 5.5 which clarify the result and allow for some interesting observations. First consider the star product 0 and the G-equivariant quantum momentum map J0 (ξ) = P (ξQ ). Then c(0 J0 ,µ ) = − ν1 [π ∗ µλ ], where µλ is the unique
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formal series of closed two-forms on Q defined by p∗ µλ = µ, λ. By Chern-Weil theory one knows that [µλ ] is independent of the chosen connection and defines a formal series with values in the first characteristic classes of the principal Gbundle which only depends on µ. Now choose µ0 in order to fix the symplectic form on T ∗ Q to be ω0 − π ∗ µ0 λ . In general, one then cannot obtain star products on (T ∗ Q, ω 0 − π ∗ µ0 λ ) in every possible characteristic class by reduction of the star product 0 on the original phase space. The reason for this is that not every de 2 Rham cohomology class in HdR (Q) is equal to a characteristic class, which is a direct consequence of the fact that the Chern-Weil homomorphism is not surjective, in general. Moreover, by our result it is clear that in general ‘quantization does not commute with reduction’. Consider for example µ0 = 0. Then the star product 0 J0 ,µ+ is equivalent to an intrinsically defined star product 0 (starting from a torsion free connection ∇ on Q), if and only if [µ+ λ ] = [0]. Furthermore, we will see in the following section that there are even more conditions which have to be fulfilled in order to achieve that 0 J0 ,µ+ equals 0 . ∗ Starting with the star products B 0 on (T Q, ωB0 ) one can actually get representatives for every characteristic class of star products on (T ∗ Q, ωb0 ) by varying B+ and the corresponding mappings j+ . This follows from the observation that [b+ ], where b = b0 + b+ with fixed b0 , corresponds to the formal series of G-equivariant cohomology classes defined by the pair (B+ , ˇj+ − µ+ ) and the wellknown fact that the G-equivariant cohomology of Q (which is by definition the cohomology of the complex of basic differential forms on Q) is isomorphic to the de Rham cohomology of the quotient Q = Q/G (cf. [17]).
6 Applications and examples 6.1
Reduction of κ and B κ
In this section we apply the reduction scheme developed in Section 3 to several concrete examples of star products on (T ∗ Q, ωB0 ). As we want to identify the resulting reduced products on (T ∗ Q, ωb0 ) with naturally defined star products we will often make use of the fact that certain star products are determined by their representations. Hence we will construct representations of the reduced star products, which is also of independent interest because of some strong relations to the results of [12, 13, 16], where the quantization is formulated in terms of representations alone neglecting the algebra of observables. As a first step towards the concrete computation of reduced star products derived from κ and B κ , we are going to establish a relation between the reduced star products obtained from different momentum values µ ∈ g∗ G + νg∗c G [[ν]]. This will allow us to restrict our further considerations to the case µ = 0. In the following we assume that the connection used to define κ and B κ is G-invariant; since the group action on Q is assumed to be proper such a connection always exists. 2 (Q)G [[ν]]. In case κ = 0, 1 we finally assume Moreover, we assume that B ∈ ZdR ∗ that D(φg α − α) = 0. Note that one can even achieve φ∗g α = α using a different
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G-invariant volume density for the definition of α, namely the Riemannian volume corresponding to an invariant Riemannian metric on Q; if ∇ is the pertinent Levi Civita connection one even has α = 0. For reduction of κ we always use the canonical G-equivariant classical momentum map with J0 (ξ) = P (ξQ ) as quantum momentum map J 0 . By Remark 3.11 this causes no loss of generality. For reduction B of the products B κ we use a quantum momentum map of the form J (ξ) = ∗ ∗ −1 P (ξQ ) + π j(ξ), where dj(ξ) = iξQ B and φg j(ξ) = j(Ad(g )ξ). J B ,µ
0
In order to relate the reduced star products κ J ,µ and B for different κ momentum values with each another we are going to construct local isomorphisms between them. For µ ∈ g∗ G + νg∗c G [[ν]] consider the formal series of one-forms 2 (Q)[[ν]] Γµ = µ, γ ∈ Γ∞ (T ∗ Q)G [[ν]]. Clearly, there is a uniquely defined bµ ∈ ZdR such that p∗ bµ = −dΓµ = −µ, λ. Now let {Oi }i∈I denote a good open cover of Q such that {Oi }i∈I with Oi = p(Oi ) is a good open cover of Q. Over every Oi choose a local potential aiµ of bµ which means bµ |Oi = daiµ , and consider the formal series of locally defined one-forms Aiµ := Γµ + p∗ aiµ ∈ Γ∞ (T ∗ Oi )[[ν]]. Then the Aiµ turn out to be closed, hence the operator Aiµ,κ defined by Eq. (2.12) using Aiµ instead of A is a local automorphism of κ and also of B κ by Lemma i 4.4 ii). Furthermore, Aµ,κ is G-invariant due to the invariance of the connection and the invariance of Aiµ . By its form it is clear that Aiµ,κ preserves P(Oi )[[ν]]. Moreover, an easy computation shows that Aiµ,κ P (ξQ ) |T ∗ Oi = P (ξQ ) |T ∗ Oi − µ, ξ and Aiµ,κ (P (ξQ ) + π ∗ j(ξ))|T ∗ Oi = (P (ξQ ) + π ∗ j(ξ))|T ∗ Oi − µ, ξ for all ξ ∈ g. By these observations and Proposition 3.7 ii) we obtain: Lemma 6.1 With notations from above, the mapping Aiµ,κ : h(P(Oi )G )[[ν]] → h(P(Oi )G )[[ν]] defined by Aiµ,κ F
:= h−µ0
id Ai F id − νµ,κ µ,κ
,
F ∈ h(P(Oi )G )[[ν]], 0
(6.1) 0
yields a local isomorphism from (h(P(Oi )G )[[ν]], •κ J ,0 ) to (h(P(Oi )G )[[ν]], •κ J ,µ ) which fulfills for all F ∈ h(P(Oi )G )[[ν]]. (6.2) Aiµ,κ F = Aiµ,κ F, 0 B i ˇ Replacing κ by B κ , J by J , and −µ0 by j0 − µ0 in the definition of Aµ,κ , one
obtains a local isomorphism from (h(P(Oi )G )[[ν]], •B κ J B ,µ •B ) which κ −1 i l ◦ Aµ,κ ◦ l is
J B ,0
) to (h(P(Oi )G )[[ν]],
coincides with Aiµ,κ . Moreover, the corresponding operator Aiµ,κ := 0 0 a local isomorphism from (P(Oi )[[ν]], κ J ,0 ) to (P(Oi )[[ν]], κ J ,µ ) J B ,0
J B ,µ
) to (P(Oi )[[ν]], B ). and from (P(Oi )[[ν]], B κ κ This operator extends uniquely to a local isomorphism, also denoted by Aiµ,κ , from (C ∞ (T ∗ Oi )[[ν]], κ J
0
,0
)
to
(C ∞ (T ∗ Oi )[[ν]], κ J
0
,µ
)
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and from (C ∞ (T ∗ Oi )[[ν]], B κ Together with the product κ J
0
J B ,0
,0
)
to
resp. B κ
completely determine the product κ J
0
,µ
(C ∞ (T ∗ Oi )[[ν]], B κ B
J ,0
J B ,µ
).
the local isomorphisms {Aiµ,κ }i∈I
resp. B κ
J B ,µ
.
Proof. First we observe that translations along the fibre and the operators defined by Eq. (2.1) preserve h(P(Q)G )[[ν]]. But this implies that Aiµ,κ preserves h(P(Oi )G )[[ν]], therefore the contributions involving µ,κ vanish. Observing that h−µ0 F = F for all F ∈ h(P(Oi )G )[[ν]] this implies Aiµ,κ F = Aiµ,κ F for F ∈ h(P(Oi )G )[[ν]]. A similar argument shows that the operator analogous to Aiµ,κ 0 B ˇ which is obtained by replacing κ by B κ , J by J , and −µ0 by j0 − µ0 also i coincides with Aµ,κ . Last, it remains to show that the product κ J
0
,µ
resp. B κ
J B ,0 B κ
J 0 ,0
J B ,µ
is completely
{Aiµ,κ }i∈I .
determined by κ resp. and the local isomorphisms In order to check this one just has to observe that for Oi ∩ Ok = ∅ the composi0 tions (Akµ,κ, )−1 Aiµ,κ define automorphisms of (h(P(Oi ∩ Ok )G )[[ν]], •κ J ,0 ) resp. (h(P(Oi ∩Ok )G )[[ν]], •B κ f κ J
0
J B ,0
). Therefore, κ J
0
,µ
resp. B κ
,µ
f |T ∗ Oi = Aiµ,κ ((Aiµ,κ )−1 f |T ∗ Oi κ J
0
,0
J B ,µ
is globally defined by
(Aiµ,κ )−1 f |T ∗ Oi )
resp. f B κ
J B ,µ
f |T ∗ Oi = Aiµ,κ ((Aiµ,κ )−1 f |T ∗ Oi B κ
J B ,0
(Aiµ,κ )−1 f |T ∗ Oi ),
where f, f ∈ C ∞ (T ∗ Q)[[ν]].
0
Clearly, the choice of J and J J ,µ
µ J 0 ,µ−˜
B
in the above lemma causes no loss of gener-
J ,µ
˜ ˜ J B ,µ−µ
ality since κ = κ , if J (ξ) = J 0 (ξ) + ˜ µ, ξ, and B = B , if κ κ B ˜ ˜ , ξ (cf. Remark 3.11). Therefore, the above result allows us to J (ξ) = J (ξ) + µ compute the star products obtained from κ and B κ by our reduction scheme in case we have at least determined one reduced star product explicitly for a special choice of the quantum momentum map and a special choice of the momentum value. Now let us consider the standard ordered star product 0 and the reduced 0 product 0 J ,0 more closely. Our goal is to find out whether the reduced star 0 product 0 J ,0 is again a standard ordered star product corresponding to a certain torsion free connection ∇ on Q. Lemma 6.2 i) Let us assign to every f ∈ P(Q)[[ν]] a formal series ˜0 (f ) of differential operators on C ∞ (Q) by the following relation: p∗ ˜0 (f )χ = 0 (l(f )) p∗ χ,
χ ∈ C ∞ (Q).
(6.3)
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Then ˜0 defines a representation of (P(Q)[[ν]], 0 J C[[ν]]-linear extension.
0
,0
Ann. Henri Poincar´e
) on C ∞ (Q)[[ν]] by
ii) If ˜0 coincides with the standard ordered representation 0 with respect to 0 some torsion free connection on Q, then 0 J ,0 coincides with the standard ordered star product 0 corresponding to the connection ∇ determined by T p ∇sh th = ∇s t ◦ p,
s, t ∈ Γ∞ (T Q).
(6.4)
0
iii) If 0 J ,0 coincides with the standard ordered star product 0 on (T ∗ Q, ω0 ) corresponding to some torsion free connection on Q, then ˜0 coincides with the standard ordered representation 0 with respect to the connection ∇ determined by Eq. (6.4). Proof. By equivariance of 0 (see Eq. (4.1)) and by the fact that l(f ) ∈ h(P(Q)G )[[ν]] the right-hand side of Eq. (6.3) is a G-invariant element of C ∞ (Q)[[ν]], therefore ˜0 (f )χ ∈ C ∞ (Q)[[ν]] is well defined by this equation, indeed. By the form of 0 (l(f )) one concludes that ˜0 (f ) is a formal series of differential operators, hence it can be extended to C ∞ (Q)[[ν]] by C[[ν]]-linearity. 0 To prove that ˜0 is a representation of (P(Q)[[ν]], 0 J ,0 ) let us first note that P 0 (F ) p∗ χ = 0 for all F ∈ I0, , since 0 is a representation of 0 and since 0 0 ∗ ∗ 0 J (ξ) p χ = −νLξQ p χ = 0 for all ξ ∈ g. Using this observation and the defi0
nition of 0 J ,0 one immediately verifies that ˜0 is a representation. This proves i). Let us consider ii). In case ˜0 coincides with the standard ordered representa0 tion 0 with respect to some torsion free connection on Q the star product 0 J ,0 coincides with the corresponding standard ordered star product 0 , since the representation completely determines the star product (cf. Section 2.1). Moreover, considering ˜0 (P (s ∨ t))χ = 0 (P (s ∨ t))χ for s, t ∈ Γ∞ (T Q) one finds that the torsion free connection used to define 0 is uniquely determined and is given by ∇ as in Eq. (6.4). 0 For the proof of iii) assume that 0 J ,0 = 0 , where 0 is obtained from some torsion free connection on Q. Using the definition of ˜0 it is immediate to verify that ˜0 (f ) = 0 (f ) for all f ∈ P 0 (Q) ⊕ P 1 (Q); note that the torsion free connection used to define 0 isof no importance, hereby. Let us now asr sume that ˜0 (f ) = 0 (f ) for all f ∈ k=0 P k (Q) and consider P (x1 ∨ · · · ∨ xr+1 ) 0 ∞ with xj ∈ Γ (T Q). By Lemma 3.10 we know that 0 J ,0 is a homogeneous 0 0 star product. Therefore P (x1 ) 0 J ,0 . . . 0 J ,0 P (xr+1 ) = P (x1 ∨ · · · ∨ xr+1 ) +
r+1 l r+1−l (Q). Furthermore, we have l=1 ν fl = P (x1 ) 0 . . . 0 P (xr+1 ), where fl ∈ P J 0 ,0 J 0 ,0 ˜0 (P (x1 ) 0 . . . 0 P (xr+1 )) = 0 (P (x1 ) 0 . . . 0 P (xr+1 )) by the representation properties and the fact that ˜0 (P (xj )) = 0 (P (xj )). Using the above ex0 0 pression for P (x1 ) 0 J ,0 . . . 0 J ,0 P (xr+1 ), this equation implies that 0 (P (x1 ∨ · · · ∨ xr+1 )) = ˜0 (P (x1 ∨ · · · ∨ xr+1 )).
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By an induction argument we then conclude that ˜0 coincides with 0 on P(Q), hence ˜0 = 0 . Like for ii) it then follows that the connection used to define 0 is given by ∇ as in Eq. (6.4). Unfortunately, it is in general not true that ˜0 equals 0 by the following equality: 0 − 0 )(P (x1 ∨ x2 ∨ x3 ))χ p∗ (˜ =
dim (G) (−ν)3 Γei (∇xh xhσ(2) )(dp∗ χ)(∇xh ei Q ). σ(1) σ(3) 3 i=1 σ∈S3
In fact, the analysis of the condition ˜0 = 0 turns out to be rather involved, but at least we can give two conditions which guarantee that this equality holds true 0 and then 0 J ,0 = 0 . Lemma 6.3 If the connection ∇ satisfies either ∇X V ∈ Γ∞ (V Q)G
for all X ∈ Γ∞ (T Q)G , V ∈ Γ∞ (V Q)G ,
(6.5)
or ∇xh y h ∈ Γ∞ (HQ)G
for all x, y ∈ Γ∞ (T Q),
(6.6)
the representation ˜0 coincides with the standard ordered representation 0 with respect to the torsion free connection ∇ defined by Eq. (6.4). Proof. To show that ˜0 actually coincides with the standard ordered representation defined by means of ∇, if one of the conditions (6.5), (6.6) is satisfied, note first that it suffices to prove the equality of the representations on elements of P(Q) of form P (x1 ∨ · · · ∨ xk ), where x1 , . . . , xk ∈ Γ∞ (T Q). Using the definition of ∇ it is k straightforward to check that (∇k p∗ χ)(xh1 , . . . , xhk ) = p∗ ((∇ χ)(x1 , . . . , xk )) holds true for for the k-fold covariant derivative, if one of the assumptions (6.5), (6.6) is satisfied. But from this and the definition of 0 we immediately obtain 0 l(P (x1 ∨ · · · ∨ xk )) p∗ χ = 0 P xh1 ∨ · · · ∨ xhk p∗ χ = k
(−ν)k ∗ k p (is (x1 ) . . . is (xk )D χ). k! k
This implies that ˜0 (P (x1 ∨ · · · ∨ xk ))χ = (−ν) k! is (x1 ) . . . is (xk )D χ. The last expression now is the standard ordered representation of P (x1 ∨ · · · ∨ xk ) with respect to the connection ∇. This proves the lemma. Finally, let us note that if Eq. (6.5) is satisfied, we moreover have (∇k p∗ χ)(Y1 , . . . , Yk ) = 0 for Y1 , . . . , Yk ∈ Γ∞ (T Q)G in case at least one Yi is vertical.
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As a direct consequence of the lemma we obtain: Proposition 6.4 Let 0 be the standard ordered star product on (T ∗ Q, ω0 ) obtained from a G-invariant torsion free connection ∇ on Q which satisfies one of the 0 conditions (6.5), (6.6). Then the reduced star product 0 J ,0 on (T ∗ Q, ω 0 ) coincides with the standard ordered star product 0 corresponding to the connection ∇ on Q defined by Eq. (6.4). Proof. The claim follows immediately from Lemma 6.2 and Lemma 6.3.
In other words the preceding proposition just means that, using appropriate connections, standard ordered quantization commutes with reduction. Symbolically we have the following commutative diagram: Q0 (∇)
(C ∞ (T ∗ Q), { , }0 ) −−−−→ R(J 0 ,0,·)
(C ∞ (T ∗ Q)[[ν]], 0 ) R(J 0 ,0,0 )
Q0 (∇)
(C ∞ (T ∗ Q), { , }0 ) −−−−→ (C ∞ (T ∗ Q)[[ν]], 0 J
0
,0
(6.7)
= 0 ).
Hereby, Q0 (∇) resp. Q0 (∇) denotes the standard ordered quantization using the respective connection and R(J 0 , 0, ·) resp. R(J 0 , 0, 0 ) denotes classical resp. quantum reduction using the indicated momentum map, momentum value and associative product. Note that the condition expressed by Eq. (6.6) is rather restrictive, since it particularly implies, by using that ∇ is torsion free, that the horizontal distribution has to be integrable, hence the principal connection corresponding to γ has to be flat. In contrast, the next lemma shows that given a G-invariant torsion free connection ∇, we can always find another G-invariant torsion free connection ˆ which satisfies Eq. (6.5) and even induces the same connection ∇ on Q. ∇ ˆ by Lemma 6.5 Let ∇ denote a torsion free G-invariant connection on Q. Define ∇ ˆ H H ∇ ˆ HV ∇
:= ∇H H , := ∇H V − H(∇H V ),
ˆV H ∇ ˆVV ∇
:= ∇V H − H(∇H V ), := ∇V V − H(∇V V ),
(6.8)
ˆ is a torsion free Gwhere H, H ∈ Γ∞ (HQ) and V, V ∈ Γ∞ (V Q). Then ∇ invariant connection on Q such that the induced connection on Q coincides with ˆ X V ∈ Γ∞ (V Q) for all X ∈ Γ∞ (T Q), V ∈ the one induced by ∇ and such that ∇ ˆ ˆ satisfying these conditions as well is of form Γ∞ (V Q). Any other connection ∇ ˆ ˆ XY = ∇ ˆ X Y + S(X, Y ) with S ∈ Γ∞ ( 2 T ∗ Q ⊗ V Q)G . ∇ ˆ X Y is well defined for all X, Y ∈ Proof. Since T Q = HQ ⊕ V Q, the vector field ∇ Γ∞ (T Q) by Eq. (6.8). The claim now follows by a straightforward argument using ˆ the definition of ∇. Analogously, if the principal connection corresponding to γ is flat, there exists a G-invariant torsion free connection on Q satisfying condition (6.6).
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Lemma 6.6 Let ∇ denote a torsion free G-invariant connection on Q and assume ˇ by that the principal connection corresponding to γ is flat. Define ∇ ˇ H H := H(∇H H ), ∇
ˇ Y Y := ∇Y Y , ∇
(6.9)
where H, H ∈ Γ∞ (HQ) and at least one of the vector fields Y, Y ∈ Γ∞ (T Q) is ˇ is a torsion free G-invariant connection on Q such that the invertical. Then ∇ ˇ HH ∈ duced connection on Q coincides with the one induced by ∇ and such that ∇ ˇ ∞ ∞ Γ (HQ) for all H, H ∈ Γ (HQ). Any other connection ∇ satisfying these con ˇ ˇ XY = ∇ ˇ X Y + S(X, Y ) with S ∈ Γ∞ ( 2 T ∗ Q ⊗ T Q)G ditions as well is of form ∇ satisfying S(H, H ) = 0 for all H, H ∈ Γ∞ (HQ). Proof. Again the proof is straightforward; the only crucial point to observe is ˇ HH − ∇ ˇ H H = [H, H ]. that the flatness of the principal connection implies ∇ ˇ ˇ induces the same connection like ∇ ˇ and that it Moreover, the condition that ∇ ˇ ∞ ∞ ˇ satisfies ∇H H ∈ Γ (HQ) for all H, H ∈ Γ (HQ) entails that S(H, H ) has to be both vertical and horizontal. Thus S(H, H ) has to vanish for all H, H ∈ Γ∞ (HQ). 0
To conclude our study of the reduced star product 0 J ,0 let us mention that one can also use the relation between the standard ordered representation 0 and a symbolic calculus for pseudo-differential operators on C ∞ (Q) (see [10, Sect. 6] and [11, Sect. 10]) to obtain the above ‘reduction commutes with quantization’ result. For details about reduction of star products in this framework we refer the interested reader to the thesis [20]. Now we consider the reduction of the products κ with κ = 0. These investigations will turn out to be slightly more involved. First of all let us recall that the reduction scheme introduced in Section 3 works for all the star products κ under the general assumption that the connection ∇ is G-invariant and that D(φ∗g α − α) = 0 holds true additionally, if κ = 0, 1. We will now show that without 0 additional assumptions the reduced star products κ J ,0 for κ = 0 are in general 0 0 not even equivalent to 0 J ,0 . This destroys the expectation that κ J ,0 could coincide with some naturally defined star product κ (cf. Section 6.2 for a concrete example in case κ = 1/2). κ Lemma 6.7 For all κ ∈ [0, 1] the G-invariant star product κ is G-equivalent to B 0 with Bκ = κνtr (R). Hence, κ is G-equivalent to 0 , if and only if [Bκ ]G = [0]G . 0 Consequently, the characteristic class of κ J ,0 is given by
c(κ J
0
,0
) = −κ[π ∗ r] = −κ[π ∗ τλ ],
(6.10)
where we have used the notation of Lemma 5.3. Proof. Consider the operator defined by Eq. (2.12) using Aκ = −κνα instead of Bκ κ A and denote it by AB is an equivalence transformation from 0 0 . Clearly, A0
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Bκ κ to B 0 , where Bκ = −κνdα = κνtr (R). Therefore, A0 Nκ defines an equivalence Bκ transformation from κ to Eq. (2.9), a straightforward computation
∞0 . 1Using r r r−1 κ shows AB N = exp F (κ − κ)(−ν) D α exp(−κν∆ ). For κ = 0 κ 0 0 r=2 r! B1 0 and κ = 1 we have AB N = id and A N = exp(−ν∆ ), which are both G0 1 0 0 0 κ N is G-invariant as well due invariant operators. For κ = 0, 1 the operator AB κ 0 to the additional condition D(φ∗g α − α) = 0. This proves that κ is G-equivalent κ to B 0 . Thus, by Proposition 4.15 the star product κ is G-equivalent to 0 , if and only if [Bκ ]G = [0]G . The result for the characteristic class of κ J0 ,0 is an immediate consequence of Theorem 5.5 and the observation that the equations 0 ∗ κ jκ (ξ) = i∗ AB jκ with bκ = −κνr hold 0 Nκ J (ξ) = −κνdiv(ξQ ) and p bκ = Bκ + dΓˇ true.
Clearly, one could now change the momentum value to the one defined by 0 µ, ξ = −κνtr (ad(ξ)). Thus one could achieve c(κ J ,µ ) = [0], but the result would not be a naturally defined star product κ for κ = 0. Instead, we remain at the choice of 0 momentum value and merely adjust the parameters entering the construction of κ in order to obtain a star product in the desired equivalence class. Henceforth, we thus assume that the volume density υ is G-invariant. Consequently, the one-form α defined in Eq. (2.4) is also G-invariant. Then Nκ is a 0 G-equivalence from κ to 0 , implying that κ J ,0 is equivalent to 0 Jκ ,0 , where Jκ (ξ) = Nκ J 0 (ξ) = J 0 (ξ) − κνπ ∗ (div(ξQ ) + α(ξQ )). But by G-invariance of υ we have LξQ υ = 0, which entails div(ξQ ) + α(ξQ ) = 0 by definition of α. Hence
0 Jκ ,0 = 0 J
0
,0
. Analogously to Lemma 6.2 we now obtain:
Lemma 6.8 Assume that κ = 0 and that the volume density υ is G-invariant. Then assign to every f ∈ P(Q)[[ν]] a formal series ˜κ (f ) of differential operators on C ∞ (Q) by p∗ ˜κ (f )χ = κ (l(f )) p∗ χ
for all χ ∈ C ∞ (Q).
Then ˜κ gives rise to a representation of (P(Q)[[ν]], κ J C[[ν]]-linear extension.
0
,0
(6.11)
) on C ∞ (Q)[[ν]] by
Proof. The proof of the claim is along the lines of the proof of Lemma 6.2 i). The only additional relation which should be noted for the proof of the representation property is the equality κ J 0 (ξ) = 0 Nκ J 0 (ξ) = −νLξQ which holds by Ginvariance of υ. 0
In order to interpret certain star products κ J ,0 as naturally defined star products κ we need a further condition which guarantees that the volume density υ induces a volume density υ on Q. The function υ(e1 Q , . . . , edim (G) Q , xh1 , . . . , xhn−dim (G) ) turns out to be G-invariant, if the group G is unimodular. Unimodularity hereby means that | det(Ad(g))| = 1 for all g ∈ G, whence tr (ad(ξ)) = 0 for all ξ ∈ g. With the additional assumption of G to be unimodular we can define a
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volume density υ on Q by p∗ (υ(x1 , . . . , xn−dim (G) )) = υ(e1 Q , . . . , edim (G) Q , xh1 , . . . , xhn−dim (G) ).
(6.12)
Evidently, the so-defined volume density υ depends on the chosen basis {ei }1≤i≤dim (G) of g. But the choice of a different basis {ei }1≤i≤dim (G) yields a volume density υ = aυ with a ∈ R+ . Therefore, the one-forms α, α defined by ∇x υ = α(x)υ and ∇x υ = α (x)υ with x ∈ Γ∞ (T Q) coincide. Hence, the corresponding κ-ordered star products also coincide and are independent of the above choice of a basis of g. Lemma 6.9 Assume that κ = 0, that the volume density υ is G-invariant and that G is unimodular. Then ˜κ coincides with the κ-ordered representation κ induced by the connection ∇ defined by Eq. (6.4) and the volume density υ defined 0 by Eq. (6.12), if and only if κ J ,0 coincides with the κ-ordered star product κ corresponding to ∇ and the volume density υ. Proof. First we note that ˜κ (f ) = κ (f ) for all f ∈ P 0 (Q) ⊕ P 1 (Q), where κ is defined using the connection ∇ and the one-form α ∈ Γ∞ (T ∗ Q) determined by
dim (G) p∗ α = H(α + W ). Here, the one-form W is given by W (X) = i=1 Γei (∇X ei Q ) (cf. Lemma 5.3). In order to interpret α as the one-form defined by ∇ and some volume density υ we must necessarily have dα = −tr R . Using the results and notation of Lemma 5.3 it is easy to compute that dα = τλ − tr R , since r = d(α − w). Actually, this relation suggests to assume that the Lie group G is unimodular, since then τλ = 0. In this case, one finds ∇x υ = α(x)υ for all x ∈ Γ∞ (T Q). Furthermore, it turns out that ˜κ (f ) = κ (f ) also holds for all f ∈ P 2 (Q) without any further conditions on the connection ∇. With these observations, the proof of the claim is completely analogous to the one of Lemma 6.2 ii) and iii). Like in the case κ = 0, the operator ˜κ (f ) coincides with κ (f ) without any 2 further conditions on ∇, if f ∈ k=0 P k (Q), but for polynomials in the momenta of higher degree one does not have ˜κ (f ) = κ (f ), in general. Fortunately, the conditions imposed on ∇ in case κ = 0 turn out to be also guarantee that the 0 reduced star product κ J ,0 coincides with κ defined by ∇ and υ resp. α. Proposition 6.10 For κ = 0 let κ be the κ-ordered star product on (T ∗ Q, ω0 ) obtained from a G-invariant volume density υ and a G-invariant torsion free connection ∇ on Q which satisfies one of the conditions (6.5), (6.6). If G is unimodular, 0 then the reduced star product κ J ,0 on (T ∗ Q, ω 0 ) coincides with the κ-ordered star product κ which corresponds to the connection ∇ on Q defined by Eq. (6.4) and to the volume density υ determined by Eq. (6.12). Proof. By Lemma 6.9, we only have to prove that each of the conditions (6.5), (6.6) We implies ˜κ = κ . First, we consider the case, where condition (6.6) is satisfied. have to determine κ l(P (x1 ∨ · · · ∨ xk )) p∗ χ = 0 Nκ P xh1 ∨ · · · ∨ xhk p∗ χ for
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the proof of the claim. Using the explicit form of the operator ∆ it is easy to find (cf. [11, Eq. (1.3)]) that k h h h h h h h h ∆P x1 ∨ · · · ∨ xk = P x1 ∨ · · · ∨ xl−1 (div(xl ) + α(xl ))xl+1 ∨ · · · ∨ xk l=1
+P
k
l j ∇xhj xhl ∨ xh1 ∨ . ˆ. . . ˆ. . ∨xhk ,
l,j=1 j=l
j
where . ˆ. . denotes omission of the j th term. Using the definition of ∇ and α, it is straightforward to show that div(xhl ) + α(xhl ) = p∗ (div(xl ) + α(xl )). Furthermore, the assumption about the connection ∇ implies ∇xhj xhl = (∇xj xl )h . Putting these formulas together we get ∆(l(P (x1 ∨ · · · ∨ xk ))) = l(∆(P (x1 ∨ · · · ∨ xk ))), where the differential operator ∆ on C ∞ (T ∗ Q) is defined completely analogously to ∆ using the connection ∇ and the one-form α. Defining Nκ := exp(−κν∆) we obtain by induction that Nκ l(f ) = l(Nκ f ) for all f ∈ P(Q)[[ν]]. Using Lemma 6.3 we then get p∗ ˜κ (f )χ = 0 (Nκ l(f )) p∗ χ = 0 l(Nκ f ) p∗ χ = p∗ 0 (Nκ f )χ = p∗ κ (f )χ, which says that ˜κ coincides with κ . Now we consider the case, where condition −1 (6.5) is satisfied. As a first step we will to show )))) that h(∆F ) =h l(∆(l (h(F G for all F ∈ P(Q) . To this end consider F = P V1 ∨ · · · ∨ Vr ∨ x1 ∨ · · · ∨ xhk with V1 , . . . , Vr ∈ Γ∞ (V Q)G . After application of ∆ to F several types of terms appear according to the above formula. The terms involving div(Vi ) + α(Vi ) vanish, since div(Vi ) + α(Vi ) = 0 by unimodularity of G and the fact that div(ξQ ) + α(ξQ ) = 0. Moreover, as in the first part of the proof we have div(xhj ) + α(xhj ) = p∗ (div(xj ) + α(xj )). From the second sum in the above formula four types of terms arise, namely those involving ∇Vi Vj , ∇xhi Vj , ∇Vj xhi , and ∇xhi xhj . By the assumption on the connection it is evident that ∇Vi Vj , ∇xhi Vj , and ∇Vj xhi are all vertical, hence these contributions vanish after projection to the total horizontal part. Projecting to the horizontal part we may also replace ∇xhi xhj by (∇xi xj )h . Combining these results we get h(∆F ) = 0, if r ≥ 1, and h(∆F ) = l(∆(P (x1 ∨ · · · ∨ xk ))), −1 G if r = 0. But this implies h(∆F ) = l(∆(l (h(F ))))h for all Fh ∈ P(Q) , since these are sums of terms of form P V1 ∨ · · · ∨ Vr ∨ x1 ∨ · · · ∨ xk . By induction, k
this implies that h(∆k F ) = l(∆ (l−1 (h(F )))) for all k ∈ N. Finally, one has to observe that 0 (P (Y1 ∨ · · · ∨ Yr )) p∗ χ = 0 for Y1 , . . . , Yr ∈ Γ∞ (T Q)G in case at least one of the Yi is vertical (cf. proof of Lemma 6.3). But then one finds k ∗
∞ (−κν)k (−κν)k p∗ ˜κ (f )χ = ∞ 0 h(∆k l(f )) p∗ χ = k=0 k=0 k! 0 ∆ l(f ) p χ = k!
∞ (−κν)k k 0 l(∆ f ) p∗ χ = p∗ ( 0 (Nκ f )χ) = p∗ κ (f )χ. k=0 k! This shows that ˜κ = κ . Symbolically, the above proposition can be expressed by a commutative diagram analogous to the one for κ = 0. More precisely, for κ ∈ (0, 1] and the ap-
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propriate connections and volume densities the κ-ordered quantization commutes with reduction, if G is unimodular: Qκ (∇,υ)
(C ∞ (T ∗ Q), { , }0 ) −−−−−−→ R(J 0 ,0,·)
(C ∞ (T ∗ Q)[[ν]], κ ) R(J 0 ,0,κ )
Qκ (∇,υ)
(C ∞ (T ∗ Q), { , }0 ) −−−−−−→ (C ∞ (T ∗ Q)[[ν]], κ J
0
,0
(6.13)
= κ ).
Hereby, Qκ (∇, υ) resp. Qκ (∇, υ) denotes the κ-ordered quantization using the respective connection and the respective volume density, and R(J 0 , 0, ·) resp. R(J 0 , 0, κ ) denotes classical resp. quantum reduction using the indicated momentum map, momentum value and associative product. Remark 6.11 At this point let us mention that the result of Proposition 6.10 does not contradict [12, Thm. 4], where it was shown that in the special case of a certain G-invariant Riemannian connection ∇g the horizontal distribution has to be integrable in order to achieve ˜1/2 = 1/2 . In contrast, our result shows that the, in general, weaker condition (6.5) suffices for the equality ˜κ = κ to hold. The reason for this peculiarity is that in the special case, where the G-invariant Riemannian metric g has been chosen such that g(V, H) = 0 for all V ∈ Γ∞ (V Q) and all H ∈ Γ∞ (HQ), condition (6.5) implies that (6.6) has to be satisfied, too. To check this, one observes g(∇gxh V, y h ) = −g(V, ∇gxh y h ). But this term has to vanish for all V ∈ Γ∞ (V Q)G due to Eq. (6.5) and the orthogonality of vertical and horizontal vector fields. This implies that ∇gxh y h ∈ Γ∞ (HQ)G for all x, y ∈ Γ∞ (T Q), i.e., (6.6) holds true. After the investigation of the reduced star products obtained from κ , we finally consider the reduced star products B κ 0
J B ,0
obtained from B κ . Having idenJ B ,0
tified the products κ J ,0 we now want to relate these products to B by κ means of local isomorphisms. In fact, the main steps for the construction of these isomorphisms have already been achieved in Section 5, where we have consid2 (Q)[[ν]] by ered the case κ = 0. Like in Eq. (5.4) let us now define bB,ˇj ∈ ZdR ∗ p bB,ˇj = B + dΓˇj . Denote by {Oi }i∈I a good open cover of Q as in Section 5 and by {Oi }i∈I the corresponding good open cover of Q. Then we obtain formal local one-forms AiB,ˇj = p∗ aiB,ˇj − Γˇj on the Oi , where aiB,ˇj denotes a local potential of bB,ˇj on O i . These formal local one-forms induce local isomorphisms AiB,ˇj,κ : (C ∞ (T ∗ Oi )[[ν]], κ ) → (C ∞ (T ∗ Oi )[[ν]], B κ ) as defined by Eq. (2.12). With these preparations we now get: Lemma 6.12 With notations from above, the mapping AiB,ˇj,κ : h(P(Oi )G )[[ν]] → h(P(Oi )G )[[ν]] defined by id AiB,ˇj,κ F := hjˇ0 AiB,ˇj,κ F , F ∈ h(P(Oi )G )[[ν]], (6.14) id − ν0,B κ
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yields a local isomorphism from (h(P(Oi )G )[[ν]], •κ J
0
,0
)
to
(h(P(Oi )G )[[ν]], •B κ
J B ,0
)
which fulfills AiB,ˇj,κ F = AiB,ˇj,κ F
for all F ∈ h(P(Oi )G )[[ν]].
(6.15)
The induced mapping AiB,ˇj,κ := l−1 ◦ AiB,ˇj,κ ◦ l is a local isomorphism from (P(O i )[[ν]], κ J
0
,0
∞
) to (P(O i )[[ν]], B κ ∗
phism from (C (T Oi )[[ν]], κ noted by AiB,ˇj,κ as well.
J 0 ,0
J B ,0
). It extends uniquely to a local isomor-
) to (C ∞ (T ∗ O i )[[ν]], B κ
J B ,0
) which will be de-
Proof. The claim is evident by Proposition 3.7 ii), since AiB,ˇj,κ P (ξQ ) |T ∗ Oi = (P (ξQ ) + π ∗ j(ξ))|T ∗ Oi and since AiB,ˇj,κ preserves h(P(Oi )G )[[ν]]. In the following lemma we give sufficient conditions which allow for a concrete computation of the above local isomorphisms {AiB,ˇj,κ }i∈I . Lemma 6.13 2 i) Assume that ∇ satisfies condition (6.5) and that B ∈ ZdR (Q)G [[ν]] is horizontal so that we can choose j = 0; this means in particular that B κ is strongly 0 G-invariant. Then the local isomorphism AiB,0,κ : (C ∞ (T ∗ O i )[[ν]], κ J ,0 ) → (C ∞ (T ∗ Oi )[[ν]], B κ
J 0 ,0
(C ∞ (T ∗ Oi )[[ν]], B κ
J B ,0
) is given by exp(κνD) − exp((κ − 1)νD) i i ∗ i , AB,0,κ = t−(ai )0 exp −F aB,0 − (aB,0 )0 B,0 νD (6.16) where D denotes the operator of symmetric covariant derivation with respect to ∇. ii) Assume that the G-invariant torsion free connection ∇ satisfies condition 0 (6.6). Then the local isomorphism AiB,ˇj,κ : (C ∞ (T ∗ O i )[[ν]], κ J ,0 ) →
AiB,ˇj,κ
) is given by exp(κνD) − exp((κ − 1)νD) i = t∗−(ai )0 exp −F aB,ˇj − (aiB,ˇj )0 . B,ˇ j νD (6.17)
Proof. In order to determine AiB,ˇj,κ explicitly, it suffices to compute AiB,ˇj,κ
P (x1 ∨ · · · ∨ xk ) for x1 , . . . , xk ∈ Γ∞ (T Q) and arbitrary k ∈ N \ {0}, since for χ ∈ C ∞ (Q) the equation AiB,ˇj,κ π ∗ χ = π ∗ χ is evident. But from the definition of F this means that we have to evaluate terms of form (Dk−1 (p∗ aiB,ˇj −Γˇj ))(xh1 , . . . , xhk ).
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Let us first consider the term involving p∗ aiB,ˇj . To this end observe that p∗ aiB,ˇj is a formal series of sums consisting of terms of form p∗ (χdχ ) = (p∗ χ)(Dp∗ χ ) with χ, χ ∈ C ∞ (O i ). Hence it suffices to determine (Dk−1 ((p∗ χ)(Dp∗ χ )))(xh1 , . . . , xhk ). Now recall that D is a derivation and observe that the result is a sum of terms we have already computed in the proof of Lemma 6.3. Since D is a derivation as well, it is now straightforward to compute that (Dk−1 p∗ (χdχ ))(xh1 , . . . , xhk ) = k−1
(χdχ ))(x1 , . . . , xk )), if one of the conditions (6.5), (6.6) is satisfied. If p∗ ((D B is horizontal and we may choose j = 0, this already implies Eq. (6.16). Now we turn to consider the term (Dk−1 Γˇj )(xh1 , . . . , xhk ) in case j is arbitrary and ∇ satisfies condition (6.6). But in this case a straightforward induction argument shows that (Dk−1 Γˇj )(xh1 , . . . , xhk ) = 0, since Γˇj vanishes on horizontal vector fields. Therefore, AiB,ˇj,κ then is given by Eq. (6.17). Using the result of the preceding lemma we can now easily identify the star J B ,0
products B with naturally defined star products on (T ∗ Q, ωb0 ), if certain κ conditions are satisfied. Moreover, observing that the local isomorphisms Aiµ,κ relating the reduced star products obtained from different momentum values that were constructed in Lemma 6.1 are of the same shape as the operators AiB,ˇj,κ we can – slightly modifying the above proof – obtain the main result of this section. Theorem 6.14 Let κ and B κ be the star products obtained from a G-invariant 2 (Q)G [[ν]]. For κ = 0, we moreover torsion free connection ∇ on Q and B ∈ ZdR assume that the volume density υ used to define α is G-invariant and that the Lie group G is unimodular. i) If
∇ satisfies condition (6.5) and B is horizontal, then the star product
J 0 ,0 B κ
on (T ∗ Q, ω(bB )0 ) coincides with the naturally defined star product bκB 2 obtained from the connection ∇, the volume density υ, and bB ∈ ZdR (Q)[[ν]] ∗ J 0 ,0 defined by p bB = B. In particular, the star product κ on (T ∗ Q, ω0 ) coincides with κ . J B ,µ
ii) If ∇ satisfies condition (6.6), then the star product B on (T ∗ Q, ωb0 ) coκ b incides with the naturally defined star product κ obtained from the connection 2 ∇, the volume density υ, and b ∈ ZdR (Q)[[ν]] defined by p∗ b = B + dΓˇj−µ . In J 0 ,µ particular, the star product κ on (T ∗ Q, ω(bµ )0 ) coincides with bκµ , where 2 (Q)[[ν]] is defined by p∗ bµ = −dΓµ . bµ ∈ ZdR 0
Proof. The claim about κ J ,0 in i) is just a restatement of Proposition 6.4 and Proposition 6.10. Moreover, Lemma 6.13 shows that the local isomorphisms from 0
J 0 ,0
κ J ,0 to B are of the same form as the operators Aκ of Eq. (2.12) using the κ connection ∇ and local potentials of the formal series bB = bB,0 of closed twoJ 0 ,0
coincides with bκB forms on Q defined by p∗ bB = B. But this implies that B κ which is evidently a star product with respect to ω(bB )0 . For the proof of ii) we
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observe that the composition of the operators Aiµ,κ and AiB,ˇj,κ , which is a local isomorphism from κ J
0
,0
to B κ
J B ,µ
, is given by
exp(κνD) − exp((κ − 1)νD) i ∗ i i i t−(ai +ai )0 exp −F (aB,ˇj + aµ ) − (aB,ˇj + aµ )0 µ B,ˇ j νD and that d(aiB,ˇj + aiµ ) = bB,ˇj + bµ = b , J B ,µ
where p∗ b = B + dΓˇj−µ . But this implies that B coincides with bκ using that κ 0 0 0 κ J ,0 = κ . Finally, the local isomorphism Aiµ,κ from κ J ,0 to κ J ,µ is given 0 by t∗−(ai )0 exp −F exp(κνD)−exp((κ−1)νD) aiµ − (aiµ )0 . Hence κ J ,µ equals bκµ , νD µ where p∗ bµ = −dΓµ .
6.2
Comparison to existing results
In this section we establish some relations between our construction of reduced star products and known concepts of reduction in deformation quantization. Additionally, we consider the more specific example T ∗ S n−1 which has been discussed in the literature. Remarks on Fedosov’s method In order to relate our results to the investigations of Fedosov [15], we will assume in this paragraph that the group acting on Q is compact. Then we consider the usual Fedosov star product F on (T ∗ Q, ωB0 ) associated to a G-invariant torsion free ∗ symplectic connection ∇T Q on T ∗ Q with c(F ) = ν1 [π ∗ B0 ]. In order to achieve that P(Q)[[ν]] is a F -subalgebra, we restrict our considerations to connections whose Christoffel symbols in a bundle chart are polynomials in the momenta (cf. [9, Appx. A]). Under these general assumptions it is clear that F is G-invariant and even strongly G-invariant, i.e., we can use J0 (ξ) = P (ξQ ) + π ∗ j0 (ξ) as Gequivariant quantum momentum map for reduction. Moreover, Proposition 4.13 tells us that F is G-equivalent to some G-invariant star product B 0 with B = 2 (Q)G [[ν]]. But then the characteristic classes of B B0 + B+ , where B+ ∈ νZdR 0 and F coincide, therefore B+ has to be exact. Since the group acting on Q is compact, one can therefore find a G-invariant formal potential A+ for B+ . Hence B0 B 0 is G-equivalent to 0 by Proposition 4.15. Together with Theorem 5.5 this implies that the characteristic class of F J0 ,0 is given by c(F J0 ,0 ) =
1 ∗ 1 [π b0 ] + [π ∗ µ ˜λ ], ν ν
(6.18)
where p∗ b0 = B0 +dΓˇj0 and µ ˜ ∈ νg∗c [[ν]] is defined by ˜ µ, ξ := i∗ T J0 (ξ)−j0 (ξ) with 2 0 a G-equivalence T from F to B ˜λ ∈ νZdR (Q)[[ν]] is determined by 0 . Here again, µ
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p∗ µ ˜λ = ˜ µ, λ. Thus, the reduced star product F J0 ,0 is in general not equivalent to a canonical star product on the reduced phase space with characteristic class 1 ∗ ν [π b0 ]. In view of this fact – similar to the situation for the star products κ – the ‘reduction commutes with quantization’ theorem proved by Fedosov appears to be a consequence of the appropriate choice of the original star product on the large phase space rather than a general principle in deformation quantization. Finally, let us emphasize that the picture changes in case the Lie algebra g is semi-simple, since then there are no non-zero elements of g∗ vanishing on [g, g] ˜λ ] which implies that in this case µ ˜ = 0. Another example, where the term [π ∗ µ obviously vanishes, is the case of a trivial principal G-bundle p : Q → Q, where all the characteristic classes of the bundle are zero. The star product on T ∗ S n−1 `a la Bayen et al. In this paragraph we consider a very concrete example of reduction. The reduced phase space is the cotangent bundle of the n − 1-sphere, which is obtained from T ∗ (Rn \ {0}) by classical Marsden-Weinstein reduction. Using our reduction method for star products we will show that the reduced star product obtained from the Weyl-Moyal star product 1/2 on T ∗ (Rn \ {0}) coincides with the deformation quantization obtained by Bayen et al. in [3]. Let G be the group R+ of positive real numbers with the usual multiplication as composition, and consider the action of G on Rn \ {0} given by φg (x) = gx for g ∈ R+ , x ∈ Rn \ {0}. Then the quotient (Rn \ {0})/R+ is isomorphic to the sphere S n−1 ⊂ Rn \ {0}. Then, the corresponding projection p : Rn \ {0} → S n−1 is given 1 x, where |x| denotes the Euclidean length of x ∈ Rn \{0}. For ξ ∈ g = by p(x) = |x| R, the generating vector field is explicitly given by ξRn \ {0} (x) = ξxi ∂xi . Hence, the (canonical) G-equivariant classical momentum map reads J0 (ξ)(q, p) = ξq i pi . xi i It is straightforward to check that γ ∈ Γ∞ (T ∗ (Rn \ {0})) with γ(x) = |x| 2 dx is a + connection one-form for the principal R -bundle under consideration. Clearly, γ is closed, whence the connection is flat. In order to compute the horizontal lift of a to describe t by a smooth vector field t ∈ Γ∞ (T S n−1 ), it turns out%to be convenient & 1 mapping t˜ : S n−1 → Rn which satisfies t˜ |x| x x = 0 for all x ∈ Rn \ {0}, where ·|· denotes the Euclidean inner product on Rn . With this mapping, the horizontal 1 lift th ∈ Γ∞ (T (Rn \ {0})) is easily computed and given by th (x) = |x|t˜ |x| x . But from this the following relation is immediate: ' 1 q|p ( (l(P (t)))(q, p) = P th (q, p) = t˜ q |q|p − q = (Π∗ P (t))(q, p), (6.19) |q| |q| 1 where Π : T ∗ (Rn \{0}) → T ∗ S n−1 , (q, p) → |q| q, |q|p − q|p q denotes the canon|q|
ical projection onto T ∗ S n−1 . Note hereby, that T ∗ S n−1 is naturally embedded in 2 T ∗ (Rn \ {0}) as the submanifold defined by the constraints |q| 1 and q|p = 0. =n−1 ∞ Clearly, the formula in Eq. (6.19) also holds for all t ∈ Γ ( T S ), since l and
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Π∗ are homomorphisms with respect to pointwise multiplication. The inverse l−1 is just given by restriction of F ∈ h(P(Rn \ {0})G) to T ∗ S n−1 , i.e., l−1 = I ∗ , where I : T ∗ S n−1 → T ∗ (Rn \ {0}) denotes the embedding of T ∗ S n−1 into T ∗ (Rn \ {0}). After these preparations recall that the Weyl-Moyal star product 1/2 on T ∗ (Rn \ {0}) can be written as ν f 1/2 f = m ◦ exp ∂qi ⊗ ∂pi − ∂pi ⊗ ∂qi (f ⊗ f ), 2 f, f ∈ C ∞ (T ∗ (Rn \ {0}))[[ν]], (6.20) where m is defined by m(f ⊗ f ) := f f . Given µ ∈ R + νC[[ν]], we now want to compute the reduced star product 1/2 J0 ,µ . According to Theorem 3.5 and the above results for l and l−1 it is given by id ∗ ∗ J0 ,µ P (s) 1/2 P (t) = h−µ0 ((Π P (s)) 1/2 (Π P (t))) . ∗ n−1 id − νµ,1/2 T S (6.21) qi (∂ f )(q, p) turns out to be The differential operator F (γ) with (F (γ) f )(q, p) = |q| 2 pi a derivation of 1/2 , which is easily verified using the explicit formula for 1/2 and the fact that γ is closed. Hence (Π∗ P (s)) 1/2 (Π∗ P (t)) lies in h(P(Rn \ {0})G)[[ν]], and Eq. (6.21) simplifies to P (s)1/2 J0 ,µ P (t) = (Π∗ P (s)) 1/2 (Π∗ P (t))T ∗ S n−1 = I ∗ ((Π∗ P (s))1/2 (Π∗ P (t))), (6.22) since µ,1/2 F = 0 and h−µ0 F = F for every invariant totally horizontal polynomial function F . Thus, the resulting star product 1/2 J0 := 1/2 J0 ,µ does not depend on the momentum value µ and therefore not on the particular choice of the G-equivariant classical momentum map (cf. Remark 3.11). This means in particular that 1/2 J0 is a star product with respect to the canonical symplectic form on T ∗ S n−1 . Moreover, the above expression in Eq. (6.22) coincides with the star product constructed by Bayen et al. in [3]. There, using a slightly different approach, T ∗ S n−1 has been regarded as the quotient T ∗ (Rn \ {0})/G , where the two-dimensional non-Abelian group G = R+ R is the semi-direct product of (R+ , ·) and (R, +) over σ : R+ → Aut(R), σ(g)h = g −1 h, which acts on T ∗ (Rn \ {0}) by Φ(g,h) (q, p) = (gq, g −1 p + hq). Finally, let us mention that our construction of reduced star products can be applied to all star products κ , but for κ = 1/2 the concrete computation turns out to be much more involved. This is caused by the fact that h(P(Rn \ {0})G)[[ν]] is not a κ -subalgebra unless κ = 1/2. id In particular, additional contributions arising from id−ν
have to be taken µ,κ into account, if κ = 1/2, which makes a concrete computation of κ J0 ,µ more difficult. To conclude the discussion of this example note that the usual connection on Rn \ {0} neither satisfies condition (6.5) nor condition (6.6) for the above choice of γ. Moreover, the canonical volume density on Rn \ {0} from which 1/2 has been obtained, is not invariant with respect to the considered action. Therefore,
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it is not to be expected that 1/2 J0 coincides with an intrinsically constructed star product 1/2 on T ∗ S n−1 using the induced connection ∇ and an appropriate volume density on S n−1 . The counterexample of the BRST-method In this paragraph we do not want to make an attempt to apply the BRST-method in deformation quantization as developed in [8] to the example (T ∗ Q, ωB0 ) in full generality, since this would be far beyond the scope of the present paper but might be an interesting topic for a future project. Instead, we merely consider two concrete intimately related examples also considered in [8], where in the first one the BRST-method can be applied without any problems but turns out to fail in the second example. In the first case we will show in particular that our reduction procedure yields the same result like the BRST-method. Moreover, we want to point out that due to our slightly more restrictive definition of a quantum momentum map the peculiarity occurring in the counterexample of the BRSTmethod is avoided in our framework. Let us consider T ∗ (S 1 × S 1 ) ∼ = T ∗ S 1 × T ∗ S 1 with the canonical symplectic form. As symmetry group we choose S 1 which acts on the base of the second factor by group multiplication. Thus the reduced phase space is T ∗ S 1 . Using the canonical flat covariant derivative on S 1 we can equip each factor T ∗ S 1 with the κ-ordered star product κ yielding a well-defined star product κ1 ,κ2 on T ∗ (S 1 × S 1 ), where we may even use different ordering parameters κ1 , κ2 in each factor. Denoting by Π1 : T ∗ S 1 × T ∗ S 1 → T ∗ S 1 the projection onto the first factor, one clearly has (Π∗1 f ) κ1 ,κ2 (Π∗1 f ) = Π∗1 (f κ1 f ) for all f, f ∈ C ∞ (T ∗ S 1 )[[ν]] by definition of the star product κ1 ,κ2 . Using local coordinates (exp(ix1 ), exp(ix2 )) for S 1 × S 1 and the induced coordinates (exp(iq 1 ), exp(iq 2 ), p1 , p2 ) of T ∗ (S 1 × S 1 ), it is evident that ξS 1 × S 1 = −iξ∂x2 for ξ ∈ g = iR and that γ = idx2 defines a connection one-form for the trivial S 1 -bundle p : S 1 × S 1 → S 1 . Moreover, the canonical classical momentum map is given by J0 (ξ) = −iξp2 , which is easily verified to be a quantum momentum map for κ1 ,κ2 . Now one can compute the reduced star product κ1 ,κ2 J0 ,µ using that l(f ) = Π∗1 f for all f ∈ P(S 1 )[[ν]], which is rather obvious from the explicit choice of γ. Completely analogously to the argumentation in the preceding paragraph we then find f κ1 ,κ2 J0 ,µ f = f κ1 f ,
f, f ∈ P(S 1 )[[ν]].
(6.23)
Thus, the reduced star product just gives back the star product we started from in the first factor of T ∗ S 1 × T ∗ S 1 , which is in perfect agreement with the results of [8]. Following [8], we now consider the S 1 -invariant equivalence transformation T := exp(−νp1 ∂p2 ) and the S 1 -invariant star product κ1 ,κ2 := T κ1 ,κ2 . According to our results of Proposition 3.7, it is immediately clear that the correJ ,µ is equivalent to κ1 ,κ2 J0 ,µ , where J (ξ) = sponding reduced star product κ1 ,κ2
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T J0 (ξ) = J0 (ξ) + iνξp1 is used as S 1 -equivariant quantum momentum map. AcJ ,µ even coincides with κ1 ,κ2 J0 ,µ , since T −1 ◦ Π∗1 = T ◦ Π∗1 = Π∗1 , tually, κ1 ,κ2 J ,µ
hence (Π∗1 f ) κ1 ,κ2 (Π∗1 f ) = Π∗1 (f κ1 f ) which implies κ1 ,κ2 = κ1 . Now, the crucial point, which lets the BRST-method fail in this case, is that J0 is an allowed ‘quantum momentum map’ for BRST-quantization, since it satisfies 1 J0 (ξ) κ1 ,κ2 J0 (η) − J0 (η) κ1 ,κ2 J0 (ξ) = 0 = J0 ([ξ, η]) ν by dim (S 1 ) = 1. But in contrast to the properties of J (ξ) we have 1 − adκ1 ,κ2 (J0 (ξ)) = LξT ∗ (S1 ×S1 ) ν hence the ‘quantum momentum map’ J0 (ξ) does not generate the classical symmetry via the quasi-inner derivation with respect to κ1 ,κ2 , whereas it does so with respect to κ1 ,κ2 . Thus, our slightly more restrictive definition of a quantum momentum map – which of course imposes additional conditions to be satisfied a priori (cf. Section 4) – completely avoids the peculiarity appearing in the BRSTmethod when using a non-appropriate ‘quantum momentum map’.
A
Equivariance properties of certain differential operators
Throughout this appendix, φ will always denote a diffeomorphism of Q and Φ = T ∗ (φ−1 ) the diffeomorphism lifted to T ∗ Q. Moreover, we continue to use the notation as introduced in Section 2. Lemma A.1 Let ∇ be a torsion there is a uniquely de2 free connection on Q. Then ∗ ∇)X Y −∇X Y = Sφ (X, Y ) fined tensor field Sφ ∈ Γ∞ ( T ∗ Q ⊗ T Q) such that (φ for all X, Y ∈ Γ∞ (T Q). Furthermore, for all β ∈ Γ∞ ( T ∗ Q) one has φ∗ D(φ−1 )∗ β = (φ∗ D)β,
(A.1)
where φ∗ D denotes the operator of symmetric covariant derivation corresponding to the connection φ∗ ∇ which is explicitly given by φ∗ D = D − dxi ∨ dxj ∨ is (Sφ (∂xi , ∂xj )).
(A.2)
Proof. The claim about the existence and uniqueness of the tensor field Sφ and the fact that it is symmetric follows immediately from the observation that φ∗ ∇ is a torsion free connection. In order to prove Eq. (A.1), it is enough to prove it for β ∈ C ∞ (Q) and β ∈ Γ∞ (T ∗ Q), since φ∗ D(φ−1 )∗ and φ∗ D are derivations with respect to ∨. It is straightforward to verify the formula for these cases. Analogously, it suffices to show that (A.2) is satisfied on C ∞ (Q) and Γ∞ (T ∗ Q). An easy computation shows this as well.
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Now we briefly recall some well-known basic definitions concerning horizontal and vertical lifts of vector fields and one-forms on Q to vector fields on T ∗ Q. Moreover, we study their behavior with respect to pull-back by Φ. Definition A.2 (cf. [9, Def. 2]) Let ∇ be a connection on Q. Consider the connection mapping K : T (T ∗Q) → T ∗ Q defined by d ζ(t) := ∇c∂t ζ t=0 (A.3) K dt t=0 for a curve ζ in T ∗ Q, where ∇c denotes the connection pulled-back along the footpoint-curve c = π ◦ ζ. Then (T π × K) : T (T ∗Q) → T Q ⊕ T ∗ Q is a fibrewise isomorphism. The horizontal and vertical lifts with respect to ∇ then are well defined and unique by the following. The section hor∇ (X) ∈ Γ∞ (T (T ∗ Q)) is called the horizontal lift of X ∈ Γ∞ (T Q), if and only if T π hor∇ (X) = X ◦ π
and
K(hor∇ (X)) = 0,
(A.4)
and ver∇ (β) ∈ Γ∞ (T (T ∗ Q)) is the vertical lift of β ∈ Γ∞ (T ∗ Q), if and only if K(ver∇ (β)) = β ◦ π
and
T π ver∇ (β) = 0.
(A.5)
Working in a local bundle chart one finds that hor∇ (X) = (π ∗ X i )∂qi + pj π ∗ (X k Γjki )∂pi
and
ver∇ (β) = (π ∗ βi )∂pi ,
(A.6)
where the Γjki denote the Christoffel symbols of ∇ and X i resp. βk the components of X resp. β in some chart of Q. In particular, it turns out that the vertical lift does not depend on the connection, henceforth we will simply denote it by ver. Lemma A.3 Let X ∈ Γ∞ (T Q), β ∈ Γ∞ (T ∗ Q) and β ∈ Γ∞ ( T ∗ Q). Then one has the following equivariance properties of hor∇ , ver and F with respect to pull-back by Φ: (A.7) Φ∗ hor∇ (X) = horφ∗ ∇ (φ∗ X) and Φ∗ ver(β) = ver(φ∗ β). Moreover, one has for all f ∈ C ∞ (T ∗ Q) Φ∗ F (β ) f = F (φ∗ β ) Φ∗ f.
(A.8)
Proof. The proof of Eqs. (A.7) consists of a straightforward computation using the definitions of horizontal and vertical lifts. Observe that the second of these identities implies Φ∗ F (β) f = F (φ∗ β) Φ∗ f , since F (β) = Lver(β) . But since F is compatible with the ∨-product and since Φ∗ F (χ) f = Φ∗ ((π ∗ χ)f ) = (π ∗ φ∗ χ)Φ∗ f = F (φ∗ χ) Φ∗ f for all χ ∈ C ∞ (Q), this also implies (A.8). Acknowledgments. N. N. is most indebted to M. Bordemann for pointing out reference [6] and for many clarifying discussions. N. N. and M. P. gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft. M. P. would like to thank Tudor Ratiu for helpful references on reduction of cotangent bundles.
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References [1] R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd ed., AddisonWesley Publishing Company, Reading (1978). [2] D. Arnal, J.C. Cortet, P. Molin, G. Pinczon, Covariance and geometrical invariance in ∗ quantization, J. Math. Phys. 24, 276–283 (1983). [3] F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization, Ann. Phys. 111, Part I: 61–110, Part II: 111–151 (1978). [4] M. Bertelson, P. Bieliavsky, S. Gutt, Parametrizing equivalence classes of invariant star products, Lett. Math. Phys. 46, 339–345 (1998). [5] M. Bordemann, talk at the Miniworkshop Quantization of Poisson spaces with singularities, MFI Oberwolfach, January 2003. [6] M. Bordemann, (Bi)Modules, morphismes et r´eduction des star-produits: le cas symplectique, feuilletages et obstructions, Preprint, March 2004, math.QA/0403334 v1. [7] M. Bordemann, M. Brischle, C. Emmrich, S. Waldmann, Phase space reduction for star products: an explicit construction for CP n , Lett. Math. Phys. 36, 357–371 (1996). [8] M. Bordemann, H.-C. Herbig, S. Waldmann, BRST cohomology and phase space reduction in deformation quantization, Commun. Math. Phys. 210, 107– 144 (2000). [9] M. Bordemann, N. Neumaier, S. Waldmann, Homogeneous Fedosov star products on cotangent bundles I: Weyl and standard ordering with differential operator representation, Commun. Math. Phys. 198, 363–396 (1998). [10] M. Bordemann, N. Neumaier, S. Waldmann, Homogeneous Fedosov star products on cotangent bundles II: GNS representations, the WKB expansion, traces, and applications, J. Geom. Phys. 29, 199–234 (1999). [11] M. Bordemann, N. Neumaier, M.J. Pflaum, S. Waldmann, On representations of star product algebras over cotangent spaces on Hermitian line bundles, J. Funct. Anal. 199, 1–47 (2003). [12] C. Emmrich, Equivalence of extrinsic and intrinsic quantization for observables not preserving the vertical polarisation, Commun. Math. Phys. 151, 515–530 (1993). [13] C. Emmrich, Equivalence of Dirac and intrinsic quantization for non-free group actions, Commun. Math. Phys. 151, 531–542 (1993).
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[14] B.V. Fedosov, A Simple geometrical construction of deformation quantization, J. Diff. Geom. 40, 213–238 (1994). [15] B.V. Fedosov, Non-abelian reduction in deformation quantization, Lett. Math. Phys. 43, 137–154 (1998). [16] M.J. Gotay, Constraints, reduction and quantization, J. Math. Phys. 27, 2051–2066 (1986). [17] V. Guillemin, S. Sternberg, Supersymmetry and equivariant de Rham theory, Springer Verlag, Berlin (1999). [18] S. Gutt, J. Rawnsley, Equivalence of star products on a symplectic manifold; ˇ an introduction to Deligne’s Cech cohomology classes, J. Geom. Phys. 29, 347–392 (1999). [19] S. Gutt, J. Rawnsley, Natural star products on symplectic manifolds and quantum moment maps, Lett. Math. Phys. 66, 123–139 (2003). [20] N. Kowalzig, Zur Anwendung der Deformationsquantisierung. Diploma thesis, Institut f¨ ur Theoretische Physik der Technischen Universit¨ at, Berlin (2001), (available at: http://staff.science.uva.nl/~nkowalzi/) [21] M. Kummer, On the construction of the reduced phase space of a Hamiltonian system with symmetry, Indiana Univ. Math. J. 30, 281–291 (1981). [22] A. Lichnerowicz, Connexions symplectiques et -produits invariants, C. R. Acad. Sc. Paris 291,A, 413–417 (1980). [23] J.E. Marsden, A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121–130 (1974). [24] R. Montgomery, The structure of reduced cotangent phase spaces for non-free group actions, Preprint 143 of the U. C. Berkeley Center for Pure and Applied Math., 1983. (available at: http://count.ucsc.edu/~rmont/papers/list.html) [25] M.F. M¨ uller-Bahns, N. Neumaier, Some remarks on g-invariant Fedosov star products and quantum momentum mappings, J. Geom. Phys. 50, 257–272 (2004). [26] N. Neumaier, Local ν-Euler derivations and Deligne’s characteristic class of Fedosov star products and star products of special type, Commun. Math. Phys. 230, 271–288 (2002). [27] M. Perlmutter, Symplectic reduction by stages, Ph. D. thesis, Department of Mathematics of the University of California, Berkeley (1999). (available at: http://www.cds.caltech.edu/~perl/)
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[28] W.J. Satzer, Canonical reduction of mechanical systems invariant under Abelian group actions with an application to celestial mechanics, Indiana Univ. Math. J. 26, 951–976 (1977). [29] J. Schirmer, A star product for complex Grassmann manifolds, Preprint, September 1997, q-alg/9709021. [30] R. Sjamaar, E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. 134, 373–422 (1991). [31] S. Waldmann, A remark on non-equivalent star products via reduction for CP n , Lett. Math. Phys. 44, 331–338 (1998). [32] P. Xu, Fedosov ∗-products and quantum momentum maps, Commun. Math. Phys. 197, 167–197 (1998). Niels Kowalzig Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam Plantage Muidergracht 24 NL-1018 TV Amsterdam The Netherlands email:
[email protected] Nikolai Neumaier and Markus J. Pflaum Fachbereich Mathematik Universit¨ at Frankfurt Robert-Mayer-Straße 10 D-60054 Frankfurt a. M. Germany email:
[email protected] email: pfl
[email protected] Communicated by Klaus Fredenhagen submitted 08/03/04, accepted 10/09/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 553 – 606 c Birkh¨ auser Verlag, Basel, 2005 1424-0637/05/030553-54 DOI 10.1007/s00023-005-0216-x
Annales Henri Poincar´ e
Scattering of an Infraparticle : The One Particle Sector in Nelson’s Massless Model Alessandro Pizzo
Abstract. In the one-particle sector of Nelson’s massless model, we construct scattering states in the time-dependent approach. On the so-defined scattering subspaces, the convergence of the asymptotic Weyl operators related to the boson field as well as the asymptotic limit of the mean velocity of the infraparticle are established. The construction relies on some spectral results concerning the one-particle (improper) states of the system. Moreover, in the region of physical interest, we assume a positive bound from below for the second derivative of the ground state energy as a function of the total momentum, uniform in the limit of no infrared cut-off in the interaction term.
Introduction In this paper we aim at describing the scattering behavior of a non-relativistic quantum particle interacting (only) with a quantized relativistic massless scalar field, when an ultraviolet cut-off is imposed on the interaction and no infrared regularization is adopted. This model is also known as the one-particle sector of the translation invariant Nelson’s massless model [Ne.]. The interest in Nelson’s massless model is related to the infrared features of Q.E.D., in spite of the various approximations here introduced: The charge is not described by a field (no pair production), an ultraviolet cut-off is imposed, the “photons” are scalar particles and the “electron” is a spinless non-relativistic particle. In particular, the analysis of the counterpart of Compton scattering in the given scalar model meets with problems (infrared divergences) analogous to the Q.E.D. case in some substantial respects. The general features of the asymptotic states – as they arise from perturbative computations and from rigorous results in related solvable models (dipole approximation, see also [Bl.]) – suggest the following intuitive picture: A “free” massive particle, that we call the electron or the charged particle, always surrounded by a cloud of asymptotic soft bosons, that we call photons even though they are scalar particles. The momenta distribution in the photon cloud, in the limit of zero energy, turns out to be linked to the electron asymptotic velocity according to a “Bloch and Nordsieck” [B.N.] type factor. The infrared features of the model are at the origin of the following difficulties in the control of scattering:
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Because of the massless dispersion of the bosons, the construction of the so-called asymptotic L.S.Z. (Lehmann-Symanzik-Zimmermann) operators [L.S.Z.] associated with the boson field requires a careful application of the stationary phase methods for the decay estimates of the solutions of Klein Gordon equation (in this respect, see [R.S.]). In our context, due to the non-relativistic description of the charge, it implies a restriction of the physical Hilbert space to keep the asymptotic (mean) velocity of the charge smaller with respect to the speed of the light. Due to the arbitrarily large number of photons emitted in the scattering at arbitrary long time, the explanation and the exact meaning of the asymptotic decoupling between the bosons and the non-relativistic particle require a different characterization of the “free” dynamics (which is only asymptotic) of the massive particle. The structural issue, as far as spectral properties are concerned, consists in the disappearance of one-particle states from the joint spectrum of the operators Hamiltonian and total momentum, in other words the absence of a proper mass shell for the charged particle. In literature, particles sharing such feature are called infraparticles [Sc.]. Because of this missing ingredient, an asymptotic description based on concepts and techniques which stem from “Haag-Ruelle scattering theory” (see [Ha.]) is conceptually not adequate. Haag-Ruelle scattering theory provides a recipe to construct scattering states for quantum relativistic fields satisfying Wightman axioms and with mass gap. In our model, while the notion of relativistic locality can be easily replaced by a non-relativistic one (that is at fixed time), the absence of one-particle states is a genuine infrared feature which modifies the collision picture at a substantial level with the appearance of non-Fock representations for the asymptotic boson algebra. Moreover the rigorous definition of the asymptotic degrees of freedom that describe the infraparticle cannot be accomplished without some further information on the (improper) mass shell structure. The first systematic analysis of scattering in the translationally invariant Nelson’s model has been done by Fr¨ohlich in two papers [Fr.1] and [Fr.2], where the second one provides spectral results exploited in [Fr.1]. In this study, indications coming from solvable models are mastered, two different approaches to collision theory are developed and many useful technical tools are provided. Starting from the intuitive picture, a recipe is attempted for the vector in the Hilbert space, ψ out(in) , corresponding (in the Heisenberg picture) to an asymptotic electron of given wave function. According to the time-dependent approach to scattering, the vector ψ out(in) is singled out by the time convergence of a related approximating vector ψ (t). The content of the present paper is strongly connected with Fr¨ ohlich’s attempt to provide a definition for the generic vector ψ (t), whose limit in time has to be consistent with an asymptotic description. Therefore a brief review of that work is carried on in a subsequent paragraph. Then we can better justify new conceptual and technical steps sufficient for a consistent collision theory both for the infraparticle and the bosons.
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In the framework of general quantum field theory, analogous issues has been treated by Buchholz [Bu.1] [Bu.2], who established the asymptotic convergence of massless boson fields applying Huyghens’ principle. Also the problem of the asymptotic description of an infraparticle has been addressed by Buchholz, Pormann and Stein [B.P.S.] in the context of a more general definition of the particle content of a theory. The infraparticles are described starting from weights; they are positive linear forms over the algebra of some operators that, in broad terms, represent detectors. The (pure) weights turn out to carry the properties of single one-particle improper states that means one-particle states with sharp energy-momentum p. For Nelson’s massless model with a confining potential, Gerard and Derezinski [D.G.] have recently faced the problem to define wave operators for non-Fock coherent sectors. In this respect, it is worthwhile to point out that the main physical feature in the model discussed in our paper is never expected in the confined case, namely the coexistence of inequivalent non-Fock representations of the asymptotic boson field labelled by the asymptotic (mean) velocity of the electron. The asymptotic completeness in the confined and infrared regularized case has been discussed by Gerard in [Ge.]. The asymptotic convergence of the radiation field in non-relativistic Q.E.D. has been established in [F.G.S.] for small energy configurations of the system. The approach is slightly different with respect to the point of view developed for the massless field in the present paper.
1 Preliminaries 1.1
Definition of the model
The physical system consists of a non-relativistic spin-less quantum particle of mass m, linearly coupled to a quantized relativistic scalar boson field, which is massless and real. The non-relativistic particle is described by position and momentum variables with usual canonical commutation rules (c.c.r.) [xl , pj ] = iδl,j ( = 1) l, j = 1, 2, 3; the (scalar) boson field, which we will call also photon field, at time t = 0 is: d3 k † 1 a (k) e−ik·y + a (k) eik·y , (1.1) A (0, y) = √ 3 · 2 |k| 2π (having assumed c = = 1), where a† (k) , a (k) are standard creation and annihilation operator-valued tempered distributions obeying the c.c.r. a (k) , a† (q) = δ 3 (k − q) , [a (k) , a (q)] = a† (k) , a† (q) = 0. The spatial translations are implemented by the total momentum P := p + ka† (k) a (k) d3 k .
(1.2)
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The dynamics of the system is generated by the covariant Hamiltonian ([H, P] = 0) κ d3 k p2 ph +g a (k) eik·x + a† (k) e−ik·x √ (1.3) H := 1 + H 2m 0 2 |k| 2 where κ is an ultraviolet cut-off, g (g > 0) is the coupling constant and H ph is the free Hamiltonian of the photon field (1.4) H ph := |k| a† (k) a (k) d3 k . The Hilbert space of the system is H = L2 R3 ⊗ F where F is the Fock space to the creation and annihilation operator-valued distributions † with respect a (k) , a (k) : j
2 R3 . (1.5) S L F = ⊕∞ j j=0 An element of H is a sequence {ψ n } of functions on R3(n+1) with ψ < ∞, where ∞ 2 ψ n (x, k1 , . . . , kn )ψ n (x, k1 , . . . , kn ) d3 k1 . . . d3 kn d3 x ψ = n=0
and each ψ n (x, k1 , . . . , kn ) is symmetric in k1 , . . . , kn . The n = 0 component corresponds to the tensor product of the vacuum subspace {Cψ0 } of F with the non-relativistic particle space L2 R3 . Standard results about H and P : i) The operators
P=p⊗I +I ⊗
ka† (k) a (k) d3 k ,
where I is the identity operator, are essentially self-adjoint (e.s.a.) in D := h ⊗ ψn , n∈N
i.e., the set of the finite linear combinations of vectors h (x) ψ n (k1 , . . . , kn ), where h (x) ∈ S R3 (the space of Schwartz test functions), ψ n (k1 , . . . , kn ) ∈ S s R3n (symmetric Schwartz test functions) ψ 0 vacuum component. Since p and
† 3 and 3 ka (k) a (k) d k are e.s.a. in S R and n∈N ψ n respectively, the result easily follows for the P operators. The spectrum of each component of P is the real axis, the spectral measure is absolutely continuous with respect to the Lebesgue measure. ii) The interaction term in the Hamiltonian is an infinitesimal small perturbation (in the sense of Kato) with respect to H0 :=
p2 + H ph . 2m
(1.6)
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Hence H is bounded from below, it is e.s.a. in D and its self-adjointness domain (s.a.d.), D (H), coincides with D (H0 ) (s.a.d. of H0 ). iii) The groups eia·P and eiτ H , τ ∈ R, a ∈ R3 , commute. iv) The joint spectral decomposition of the Hilbert space with respect to the P
⊕ operators is H = HP d3 P where HP is a copy of F . Indeed to the improper eigenvectors of the P operators, ΦnP , where − 32
ΦnP (x, k1 , . . . , kn ) := (2π)
ei(P−k1 −···−kn )·x ϕnP (k1 , . . . , kn )
ϕnP (k1 , . . . , kn ) ∈ S s R3n ,
we can relate a natural scalar product: m m 3 3 (Φn , Φ ) = δ ϕn n,m P P P (k1 , . . . , kn )ϕP (k1 , . . . , km ) d k1 . . . d kn .
(1.7)
The vector space n∈N ΦnP is defined as the closure of the finite linear combinations of the wave functions ΦnP (x, k1 , . . . , kn ) in the norm which arises from the scalar product (1.7) trivially extended to n = 0. Starting from this space, we uniquely define the linear application IP :
ΦnP → F b
(1.8)
n∈N
by the prescription: IP (ΦnP (x, k1 , . . . , kn )) 1 := √ b† (k1 ) . . . b† (kn ) ϕnP (k1 , . . . , kn ) d3 k1 . . . d3 kn ψ0 , n!
(1.9)
where b (k) , b† (k), which formally correspond to a (k) eik·x , a† (k) e−ik·x , are annihilation and creation operator-valued tempered distributions in the Fock space Fb ∼ = F , and ψ0 is the related vacuum. The given norm for ΦnP is equal to IP (ΦnP )F (·F is the Fock norm). The application IP is onto and isometric.
v) Since [H, P] = 0, we have that H = HP d3 P , where HP : HP → HP is e.s.a. in Db := n∈N ΦnP . In terms of the variables P, b (k) , b† (k), the operator HP is written as follows: ph 2 κ d3 k P −P +g b (k) + b† (k) (1.10) + H ph . HP = 2m 2 |k| 0
being H ph ≡ |k| b† (k) b (k) d3 k and Pph ≡ k b† (k) b (k) d3 k when applied on the fiber spaces HP .
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Notations We collect standard notations and some conventions which are used throughout the paper: 3 1) C n R3 denotes thespace of functions on R with continuous derivatives n 3 up to degree n, C0 R denotes the subset of compact support functions contained in C n R3 . 2) The symbol |.| will denote the absolute value for C numbers as well as the Euclidean norm for vectors in Rn , n > 1. Scalar products of vectors in Rn , n > 1, are denoted by the multiplication sign “·”. Multiplication of real numbers is often denoted by the same symbol. 3) Given a function χ (u), supp χ is the support of the function. 4) For 3-dimensional integrals we use only one integration symbol and the explicit integration bounds are referred to the radial part of the integration variable. If necessary the notations are less compressed. 5) The notation s − lim means strong limit. 6) Given a self-adjoint operator A, D (A) is the corresponding domain. The notation h.c. means hermitian conjugate. 7) The operators, ∇E σ (P), Wσ (∇E σ (P)), are functions of the total momentum operator P. For brevity the dependence on P is some times differently σ ). indicated (e.g., EP 8) In the estimates that we produce throughout the paper, we generically call C all the multiplicative constants which are are time independent, uniform in the infrared cut-off and in the cell partition.
1.2
Fr¨ ohlich’s construction
The issues and the results we are going to discuss concern the model with an ultraviolet cut-off and are connected to the infrared difficulties which affect the formulation of scattering theory. Focusing our attention on such aspect, we recall that in Fr¨ ohlich’s paper [Fr.1] the following cases are investigated and compared: 1) The massive and the massless cases as far as the boson field is concerned; 2) Both the non-relativistic and the relativistic dispersion law for the charged particle kinetic energy in the Hamiltonian. The scattering problem is studied in a time-dependent approach, by adapting the “Haag-Ruelle” framework [Ha.] to the mixed character of the model. In fact quantum mechanical non-relativistic matter coexists with a quantum relativistic field. The adopted procedure is successful as far as one particle states for the charge are available. It is always the case in presence of massive bosons; in the massless case only if an infrared regularization, for instance a cut-off, is imposed on the interaction. Starting from the one-particle states and the asymptotic limit of the L.S.Z. smeared field, the asymptotic picture is simply given by a free electron
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with a renormalized dispersion law and free bosons in the Fock representation. We recall that in the massless case the control of the asymptotic convergence of the L.S.Z. smeared field requires, as additional condition, some constraints on the asymptotic velocity of the non-relativistic particle. We therefore select states such that the asymptotic (mean) velocity of the non-relativistic particle is strictly smaller with respect to the boson velocity (the speed of the light). Such a physical description fails in the true (no infrared regularization) Nelson’s massless model and two alternative scattering descriptions are therefore considered. The first one is an attempt to generalize Haag-Ruelle theory by a limiting construction starting from the model with an infrared cut-off. This approach is reconsidered and developed in this paper, where it is proved to be consistent. The second one assumes the existence of the asymptotic boson (free) algebra to define the time-space translation generators for the asymptotic charge as a difference: These are obtained by subtracting from the full generators the corresponding ones for the asymptotic bosons (for details see [Fr.1]). Similar concepts were later exploited in Q.E.D. (see [F.M.S.]) in the Wightman framework of quantum field theory. In that context a tentative recipe has been provided for the construction of the asymptotic charged fields. The first approach (the only one we are interested in) requires a careful analysis of the one-particle improper states or, equivalently, of the one-particle states corresponding to Hamiltonians with smaller and smaller infrared-cutoff σ in the interaction term. The underlying conjecture is that a sufficiently refined control on the one-particle states (which disappear from the Hilbert space H in the limit σ → 0) should predict the low energy behavior of the boson cloud appearing in the scattering states. This aspect is clearly crucial in order to define an approximating vector ψ (t) of ψ out . To motivate why, in our opinion, the way followed in [Fr.1] is the correct one to understand the scattering behavior, we review that analysis before filling some conceptual steps towards a modified definition of ψ (t) and the proof of its convergence in time. The spectral results behind the definition of ψ (t) in [Fr.1] are concerned with the ground states of the Hamiltonians HP . They are achieved through a nonconstructive method already used by Glimm and Jaffe [G.J.]. The main results (for precise estimates see [Fr.1]) are: The ground state energy E (P) = E (|P|) is absolutely continuous, therefore ∂E(P) ∂|P| exists almost everywhere, moreover ∂E(P) p2 + H ph for P : |P| < m if H0 = 2m ∂|P| < 1 ∂E(P) P ∈ R3 if H0 = p2 + m2 + H ph ; ∂|P| < 1 for any The absence, in the not (infrared) regularized case, of a ground state for HP in the Hilbert space HP ∼ = F and its existence and uniqueness in the P-dependent,
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|P| < m, coherent representation of b (k) , b† (k) with coherent factor c (k) singled out by the infrared behavior g c (k) →k→0 − √ 3 · ∇E (P) 2 |k| 2 1 − k
P ∈ {P : |P| < m} .
(1.11)
A crucial technical tool is involved in the above results. It is an almost explicit σ expression for the action of b (k) on the ground state ψP of the Hamiltonian HP,σ , i.e., with an infrared cut-off σ in the interaction term. The tool is 1 g σ σ = σ ≤ |k| ≤ κ (1.12) ψP b (k) ψP 2 |k| E σ (P) − |k| − HP−k,σ σ σ σ where E σ (P) is the eigenvalue of ψP , HP,σ ψP = E σ (P) ψP . The resolvent formula (1.12) clearly plays an important role also in the present paper because it contains a structural information about the logarithmic divergence, in the infrared limit σ → 0, of the boson number operator, N := b† (k) b (k) d3 k, evaluated on the ground state of HP,σ . Once the previous spectral information is known, the main issue consists in σ the following question: How to define a vector ψh,κ (t) ∈ H with the property 1 σ that limt→∞,σ→0 ψh,κ1 (t) represents, in Heisenberg picture, an asymptotic electron with wave function h in the asymptotic momentum of the charged particle and with the expected freely moving (soft) photon cloud surrounding it, where the boson frequency is up to the threshold κ1 . The wave function of the asymptotic (soft) photons which form the cloud is suggested by the spectral analysis of oneparticle states. More precisely it is linked to the coherent representations (1.11) singled out in the limit σ → 0 for different P. This interpretation of the limiting vectors requires an a posteriori justification from the action of the asymptotic observables (to be constructed) on them. σ (t), Fr¨ ohlich starts from the In order to construct the generic vector ψh,κ 1 wave function, in terms of the charged particle position operator and the bosons momenta variables, of a one-particle state corresponding to the model with (infrared) cut-off σ in the interaction term of the Hamiltonian (1.3). Let it be given by the sequence 2 3 3 σ(n) σ(n) 3 (x,k1 ,...,kn ) : n ∈ N (x,k1 ,...,kn ) d xd k1 ...d kn < ∞ ψ ψ n
where ψ
σ(n)
− 32
(x, k1 , . . . , kn ) = (2π)
eip·x ψpσ(n) (k1 , . . . , kn ) d3 p
(1.13)
σ In ψh,κ (t) the subscript h is referred to the wave function in P of the one1 particle state. The support of h is restricted to a neighborhood of P = 0 such that
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|∇E σ (P)| < 1. In the expression (1.13) the P-dependence is hidden in the (symσ(n) metric) function ψp (k1 , . . . , kn ) where p is the charged particle momentum. By the substitution p = P − k1 − · · · − kn , we have ψ σ(n) (x, k1 , . . . , kn ) = =
σ(n) −3 (2π) 2 eiP·x e−i(k1 +···+kn )·x ψp (k1 , . . . , kn ) d3 p
iP·x −iPph ·x σ(n) e h (P) ψP (k1 , . . . , kn ) d3 P e (1.14)
where − 32
(2π)
σ(n) σ(n) ψp=P−k1 −···−kn (k1 , . . . , kn ) = h (P) ψP (k1 , . . . , kn )
(1.15)
with the normalization 2 σ(n) ψP (k1 , . . . , kn ) d3 k1 . . . d3 kn = 1 . n
In order to properly control the action, on the one-particle state, of the Weyl operator“carrying” the boson cloud, one first defines the operator-valued distributions a (k) , a† (k) smeared out with functions f (k, P) , where P is the total σ(n) momentum, and then applied to the vector ψp (k1 , . . . , kn ). Exploiting the decomposition of H on the spectrum of P, the definition is: a (k) f (k, P) d3 kψ σ(n) (x, k1 , . . . , kn ) := ph σ(n) = a (k) f (k, P) eiP·x−iP ·x h (P) ψP (k1 , . . . , kn ) d3 P d3 k (1.16) − 32 √ −iPph ·x+iP·x = (2π) n e σ(n) × f (k, P) e−ik·x ψP−k−k2 ···−kn (k, k2 , . . . , kn ) d3 kd3 P .
A similar procedure is used for the action of the operator a† (k) f (k, P) d3 k. On the basis of the previous definitions, the final expression we are interested in can be handled after having expanded, in terms of the generator, the formal expression for the L.S.Z. Weyl operator “carrying” the boson cloud. In other words, σ (t) is defined starting from each it means that the approximating vector ψh,κ 1 projection on the n-particle subspace, namely: (n) −iHσ t σ ψh,k1 (t) (x, k1 , . . . , kn ) := (1.17) e (n) a(k)ei|k|t −h.c. −g κ1 d3 k 3 σ √ σ ph σ (P)) σ 3 2|k| 2 (1−k·∇E = e e−iE (P)t eiP·x e−iP ·x h (P) ψP d P (k1 , . . . , kn )
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=
eiP·x e−iP
ph
·x
−g
e
κ1 a(k)ei|k|t−ik·x −h.c. d3 k 3 σ √ σ (P)) 2|k| 2 (1−k·∇E
Ann. Henri Poincar´e
(n) e−iE
σ
(P)t
σ 3 h (P) ψP d P
(k1 , . . . , kn ) . σ (t) is obtained For a fixed infrared cut-off σ, the time limit of the vector ψh,κ 1 exploiting Hepp’s method [He.]. The proof basically relies on the estimates of σ the expectation values of polynomials in b (k) , b† (k) on the ground state ψP (generalized resolvent formulas, see [Fr.1]) and on an implicit propagation estimate for the electron, contained in the constraint |∇E σ (P)| < 1 which holds in a neighborhood of P = 0. The ultimate motivation for the previous construction is however the limit in time of ψh,κ1 (t) with no infrared cut-off σ. In the physical situation without infrared cut-off, it indeed represents a minimal (with respect to the photon cloud) description of an infraparticle in a scattering state. It means that a photon cloud of soft photons is unavoidable, i.e., κ1 can be arbitrarily small but not zero. The subspace generated by such vectors can be seen as a one-particle subspace, up to an observability threshold in the energy of the asymptotic photons.
1.3
Minimal asymptotic electron
Let us inquire about the features and the problems of the previous construction. As already pointed out, no problem arises in the norm control and in the convergence σ in time of ψh,κ (t) as long as σ = 0, because the series expansion of the Weyl 1 operator in terms of the generator can be controlled, basically, due to the regularity σ in a neighborhood of P = 0. The situation changes drastically properties in P of ψP for σ = 0. If we remove the cut-off σ in the expression (1.17), the definition of the vector at finite times becomes a delicate issue. The previous method fails because of divergences appearing in the series expansion of the Weyl operator −g
e
a(k)−a† (k) κ1 d3 k 3 σ √ σ (P)) 2|k| 2 (1−k·∇E
.
(1.18)
The expansion is technically forced because the P-fiber spaces HP are not preserved under the action of the operator (1.18). The definition at finite times of each n-component in (1.17) and summability in n is still well founded by assuming some regularity properties which can be eventually reconciled with the existence of the second derivative of the ground state energy E (P). However, even in these assumptions, the time asymptotic behavior is practically out of control. The difficulties coming from the expression without the cut-off σ are at the origin of an alternative and, for some aspects, conceptually different recipe for the approximating vector that we denote by ψh,κ1 (t). The new main ingredients that we introduce are: A convergence scheme based on a diagonal limiting procedure to better follow the slow asymptotic decoupling due to the interaction with infrared bosons.
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It means that the infrared cut-off in the approximating vectors is removed only asymptotically in time; A constructive characterization of one particle states provided in [Pi.] which enables us to heavily use strong regularity properties (for details see Section 3 in [Pi.]). The propagation estimate provided by the non-relativistic locality of the model, namely the decoupling mechanism used in Haag-Ruelle theory can be reproduced in terms of cluster properties at fixed time of the photon field and of the current density field associated to the electron. Going to technical details, our proposal for the generic vector ψh,κ1 (t) introduces: A time dependent cut-off σt , which is removed at a rate faster than 1t ; The transformation of the integral in d3 P to a Riemann sum by a time dependent cell-partition of the P-momentum space; A phase factor, already somehow present in the tentative construction by Fr¨ ohlich for the case σ = 0, which is here exploited, in applying Cook’s argument, iHt x −iHt , that is the electron mean velocity as a function of the variable x(t) t := e te (at time t) up to a correction of order t−1 . The two main differences with respect to Fr¨ohlich’s proposal, namely the diagonal limit and the cell partition of the P-space, represent the building blocks of a strategy controlling, simultaneously, the logarithmic divergences arising in the two limits σ → 0 and t → ∞. To implement our strategy, the use of different time scales is crucial. They are basically: The rate σt of the removal of the infrared cut-off, by which we approach the limit σ = 0 of no infrared cut-off cutting away the frozen degrees of freedom at the given time scale t; The slower rate of the partition governed by an exponent , determined by the (estimated) time scale of the decoupling. Let us anticipate the expected advantages of these constructive modifications in controlling the two quantities ψh,κ1 (t)2
,
ψh,κ1 (t2 ) − ψh,κ1 (t1 )2
(1.19)
that we will study in the paper. 1) By the transformation of the integral to a Riemann sum: We can replace the series expansion of the Weyl operators by a “cell-expansion” in the P-space which we can easily control exploiting the cluster property of the system. In this respect, we anticipate here that different values of P in the expansion correspond to different asymptotic velocities of the charged particle; For all finite times, we deal with an expression in terms of bounded operators in the Hilbert space, that we can actually handle without considering, in general, any particular wave function representation but simply abstract calculus. 2) By introducing a time-dependent cut-off σt : We can exploit the unitarity property of the Weyl operators as long as σt > 0. For each cell, it provides a priori
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estimates without resuming contributions which are logarithmically divergent in the infrared limit. Moreover the a priori estimates match easily with the power law decay of the vanishing quantities which neutralize the divergent terms; We can extend or simply push to the limit of no infrared cut-off some properties which hold for the model with a fixed infrared cut-off. The properties are: 2i) The propagation estimate x e−iHt →t→∞ f (∇E (P)) eiHt f t
f ∈ C0∞ R3
which morally holds on one particle improper states, as it can be deduced from Theorem 4.2. This extrapolated property is nothing but the limiting case for σ → 0 of the analogous convergence which can be easily proved in the case of a fixed σ-cutoff dynamics [T.S.]; 2ii) The fact that, for a fixed σ-cutoff dynamics, the one particle states are vacua for the annihilation part of the asymptotic boson field. It turns out to be extremely useful in treating the off-diagonal terms, with respect to the partition, of the quantities (1.19); 3) The phase factor is employed in Cook’s argument, in analogy with Dollard’s treatment of Coulomb scattering [Do.] (see also [K.F.]), though the present phase factor is only a technical tool in the following sense. In contrast to the Coulomb phase, it is in fact convergent for t → ∞. It is seemingly avoidable, nevertheless it is helpful in our framework because provides a useful subtraction in the application of Cook’s argument. The explicit construction of the generic asymptotic vector will clarify the motivations for the strategy invoked so far. In the new recipe for the vector ψh,κ1 (t), the key different points of view to be stressed are: The construction of the vector is analyzed in terms of a “regular” block given w (see the expression by the one-particle states of transformed Hamiltonians HP,σ (1.21) written later) and a “dressing” block which is different from the physical dressing photon cloud; The infrared cut-off removal is an a posteriori result and a byproduct of the asymptotic decoupling. Our construction should be simplified in order to treat generalizations, for instance more than one electron. Hopefully some constructive device is not necessary or can be made less cumbersome in a modified and improved recipe. However the present construction represents a starting point for simpler descriptions of the asymptotic decoupling and for a precise analysis of the involved time scales. The entire construction is self-contained assuming the results in [Pi.] together with the resolvent formula (1.12) (for details, [Fr.1]). The only crucial constructive hypothesis not proven yet (but physically reasonable) concerns a positive bound from below for the second derivative of the ground state energy E σ (P) uniform in σ > 0 and in the region of P we are interested in.
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1.3.1 Assumptions for the construction Spectral properties We recall some spectral results stated in [Pi.] which hold for P ∈ Σ m Σ := P : |P| < , 20 k when the coupling constant g and the ratio m are sufficiently small. The constraint on Σ reflects the mixed character of the model, which forces to restrict the physical region to the set {P : |P| < m}; the adopted more restrictive constraint is only due to technical reasons.
Given the Weyl operator σ Wσ (∇EP )=e
−g
κ
b(k)−b† (k) σ |k| 1−k·∇E σ P
(
)
3k 2|k|
√d
(1.20)
and the transformed Hamiltonian w σ σ := Wσ (∇EP ) HP,σ Wσ† (∇EP ), HP,σ
(1.21)
the corresponding non-degenerate ground eigenvector in HP 1 1 − 2πi w γ HP,σ −E dE ψ0 σ σ σ γ : E ∈ C , |E − E (1.22) φP := | = P 1 4 − 2πi γ H w 1 −E dE ψ0 P,σ
is regular as function of σ and P in the space F b , according to the following results: Theorem 3.2 [Pi.] For P ∈ Σ, the limit s − limσ→0 φσP =: φP exists. Moreover the convergence of φσP to the non-zero vector φP in HP ∼ = F b and the conver 1 −δ 1 σ → ∇EP are estimated with errors at most of order σκ 4 and σκ 4 gence ∇EP respectively, where δ > 0 is arbitrarily small. Lemma 3.3 [Pi.] The following H¨ older estimate holds: 1
|∇E σ (P) − ∇E σ (P + ∆P)| ≤ C · |∆P| 16 where the constant C is uniform in 0 < σ < κ , in P,!P + ∆P ∈ Σ and ∆P ∈ 8 1 where I := ∆P : |∆P| ≤ 1 3 , m 34 |∆P| 4 ≤ κ and C is a constant I, I m 3C I
sufficiently larger than 1. k Theorem 3.4 [Pi.] Under the constructive hypotheses, for m and g sufficiently σ σ small, the norm difference between φP and φP+∆P is H¨ older in |∆P| with co1 − δ, δ > 0 and arbitrarily small. The multiplicative constant, Cδ , is efficient 16 uniform in 0 ≤ σ < κ , in P, P + ∆P ∈ Σ and ∆P ∈ I, I a sufficiently small fixed ball around ∆P = 0.
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In our construction we will assume the results above (Theorem 3.2 [Pi.], 1 Lemma 3.3 [Pi.] and Theorem 3.4 [Pi.]) with coefficients 14 and 16 respectively, with no substantial difference for our procedure and the content of the final results. From the analysis in [Fr.1] and [Pi.] we get that for P ∈ Σ we have small electron “velocities” that means |∇E σ (P)| < 1 ∀σ. In later constructions we assume that the upper frequency, κ1 , in the boson cloud of ψh,κ1 (t) is small enough such that for P ∈ Σ |∇E σ (P + k)| < v max < 1
∀σ, ∀k : 0 < |k| ≤ κ1
(1.23)
for a given and fixed v max > 0. Remark 1.1 We will treat the convergence in H of a vector given as a direct integral on the fiber spaces HP . In order to avoid any confusion in dealing with vectors belonging to different fiber spaces, we will use explicitly the isomorphism IP in our notations differently from [Pi.]. Therefore, for instance, the property in Theorem 3.4 [Pi.] is rewritten as follows: 1 IP+∆P φσP+∆P − IP (φσP ) ≤ Cδ · |∆P| 16 −δ . F Spectral hypothesis We also assume the following not proven hypothesis, which allows the construction of a (time-dependent) cell partition with the desired properties: Hypothesis H0. For P ∈ Σ , there exists a positive constant following inequalities hold uniformly in σ > 0: ∂ 2 E σ (P) 2
∂ |P|
1 mr
≥
1 mr
such that the
|P| ∂E σ (P) ≥ . ∂ |P| mr
Assuming this hypothesis, the application Jσ : P → ∇E σ (P)
P ∈ Σ, σ >0
(1.24)
is a bijection and the determinant of the Jacobian satisfies the inequality det dJσ =
1 2
|P|
·
∂E σ (P) ∂ |P|
2
·
∂ 2 E σ (P) 2
∂ |P|
≥
1 ; m3r
σ concerning the calculation of the determinant, we recall that the function E (P) ∞ 3 R (see [Fr.1]). Under this asis invariant under rotations and belongs to C σ sumption, given OP ⊂ Σ and the corresponding region O∇EPσ in the ∇EP -space, −1 σ OP = Jσ O∇EP , the following relation holds between their volumes:
VOP ≤ m3r · VO∇Eσ . P
(1.25)
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Cell partition Let us consider a region contained in Σ, for convenience a cube of volume V = L3 . We now construct a time-dependent, t 1, cell-partition Γ(t) of the volume V , according to the following recipe: At time t 1, the linear dimension of each cell is 2Ln where n ∈ N, is such that 1 1 >0 (2n ) ≤ t < 2n+1 and the small exponent 1 is fixed only a posteriori. 3 This definition implies that the total number of cells at time t is N (t) = (2n ) , (t) where n = [log2 t ], [.] is the integer part. We call Γj the j th cell, centered in Pj , belonging to the partition, Γ(t) , at time t. 1.3.2 Definition of the vector ψh,κ1 (t) The generic vector ψh,κ1 (t), t 1, is constructed starting from a one-particle state for the Hamiltonian Hσt , of wave function h in P-variables. A P-dependent L.S.Z. Weyl operator, in properly evolved photon variables is applied, cell by cell, on the considered one-particle state. The smearing function in the generator of the Weyl operator has frequency support in the set σt ≤ |k| ≤ κ1 < κ where σt → 0 for |t| → +∞ and κ1 is an arbitrarily small positive number which satisfies the constraint (1.23). The behavior of the smearing function at k = 0 is labelled by the spectral values of the operator ∇E σ (P). 1) We start from the vector (t) σt 3 ψj,σt := h (P) ψP d P (1.26) (t)
Γj
where: – h (P) ∈ C01 R3 \ 0 has support inside the cube V ; – σt = t−β , where β 1 is fixed only a posteriori; σt σt := Wσ†t (∇EP ) φσPt is the unique ground state of HP,σt . – ψP Notice that " # 12 (t) 2 3 |h (P)| d P ψj,σt = (t) Γj
− 12
is of order (N (t))
. (t)
2) We consider for each ψj,σt a corresponding dressing “cloud” carried by the L.S.Z. Weyl operator eiHt e−iH
ph
t
Wσt (vj ) eiH
here Wσt (vj ) := e
−g
ph
t −iHσt t
e
κ1 a(k)−a† (k) σt |k|(1−k·v j)
3k 2|k|
√d
,
(1.27)
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A. Pizzo
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where vj ≡ ∇E σt Pj is the “velocity” at time t corresponding to the cen(t)
ter Pj of the cell Γj . In order not to overburden the notations, the time dependence of vj is not explicit. However it can be easily recovered from the (t) time which labels the corresponding cell Γj . This will be carefully taken into account in the study of the convergence of ψh,κ1 (t), precisely in Subsection The 3.1.1. c-number vj clearly commutes with the algebra generated by a (k) , a† (k) . 3) Finally we define: N (t)
ψh,k1 (t)
:= eiHt e−iH
ph
t
Wσt (vj ) eiH
ph
σ
σ t iγσt (vj ,∇EPt ,t) −iEPt t
e
e
(t)
ψj,σt
j=1 N (t)
=
eiHt
σt σt (t) Wσt (vj , t) eiγσt (vj ,∇EP ,t) e−iEP t ψj,σt
(1.28)
j=1
with the definitions Wσt (vj , t) := e−iH
ph
t
Wσt (vj ) eiH
σt eiγσt (vj ,∇EP ,t) := e
where
στS
=τ
−α
−i
t 1
g
2
ph
t
=e
−g
κ1 a(k)ei|k|t −a† (k)e−i|k|t σt |k|(1−k·v j)
σt S cos k·∇EP τ −|k|τ στ σt (1−k·v j)
(
(1.29)
) dΩd|k|
3k 2|k|
√d
dτ
(1.30)
with α, 0 < α < 1, fixed only a posteriori.
The definitions (1.29), (1.30) require some comments contained in the remarks below, which give us the opportunity to come back to the motivations of the recipe here presented. Remark 1.2 As far as the boson cloud and the coherent factor (1.11) −g
1 · ∇E (P) |k| 2 |k| 1 − k
are concerned, by the introduction of the c-number vj we implement in our formalism a crucial physical feature which is not exploited in all its consequences in Fr¨ ohlich’s formalism. More precisely, the operator that actually labels the coherent factor in the photon cloud is the asymptotic (mean) velocity of the electron, that, differently from ∇E (P), has to commute with the asymptotic boson algebra. The two operators would coincide on the one-particle states if the latter ones existed, as happens when a fixed infrared cut-off in the interaction is considered. We want to keep track of this concept in a limiting construction, involving the time dependent partition and the discretized velocities ∇E σt Pj . We will see that the chosen recipe for the dressing cloud is in the end a technically convenient way to approximate the operator asymptotic (mean) velocity of the electron
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inside the wave function of the photon cloud. The reason is that we can easily exploit the cluster property which implements the asymptotic orthogonality between off-diagonal terms in the partition of the vector. We stress that the operator asymptotic (mean) velocity of the electron is not constructed yet. We only use the values that it is expected to take on the minimal asymptotic electron states for k1 smaller and smaller, in the region of momenta P which is physically meaningful. Moreover the expression given for the smearing function in the Weyl operators (1.29) encodes somehow, already at finite times, the commutation property we expect at asymptotic times between the asymptotic boson algebra and the asymptotic (mean) velocity of the electron, up to an error which becomes smaller and smaller as time increases and the partition gets finer and finer. Remark 1.3 The introduction of the phase factor (1.30) is related to Cook’s argument. The “fast” cut-off στ is of order τ −β , where β is larger than 1. The integration bound στS is a “slow” infrared cut-off, στS = τ −α where α is a positive number less than 1. The infinitesimal upper bound στS for the integral in (1.30) σt with x(t) (Corollary A3) for asympenables us to replace the argument ∇EP t totic times. Therefore we get that the time derivative of the phase factor kills an infrared tail term arising from the application of Cook’s argument, which is not (absolutely) convergent as function of t. On the basis of partial estimates, α is eventually chosen sufficiently close to 1 and β large enough with respect to 1 in order to achieve the strong convergence of the vector ψh,κ1 (t).
1.4
Survey of results and plan of the paper
After having constructed the generic vector ψh,κ1 (t), we prove the existence of the strong limit out(in)
s − lim ψh,κ1 (t) =: ψh,κ1 t→±∞
.
The construction is explicitly performed in the case “out”, the case “in” is completely analogous. In our notations we use the superscript (in) to mean either that an analogous structure holds for the ingoing case or to denote both the two ones, for instance both the two asymptotic subspaces. However we do not claim anything about their relations. By analogy with the regularized case, we define the invariant (under spacetime translation) subspaces Hκ1 1out(in) :=
out(in)
ψh,κ1
(τ, a) :
h (P) ∈ C01 (Σ \ 0) , τ ∈ R , a ∈ R3
where the subscript κ1 denotes the upper frequency in the boson cloud and the out(in) vector ψh,κ1 (τ, a) corresponds to the evolution τ in time and to a displacement out(in)
a in space of the state associated to the vector ψh,κ1 . Because of the presence of the boson cloud, the electronic wave functions {h} cannot fully characterize the set
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A. Pizzo
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of states we are interested in. The next step consists in adding “hard” asymptotic bosons as result of the limits s − lim eiHt e−iH t→±∞
ph
t i(a(µ)+a† (µ)) iH ph t −iHt
e
e
e
out(in)
ψh,κ1
out(in)
(τ, a) =: ψh,µ
†
where a† (µ) := (a (µ))† = a (k) µ (k) d3 k , µ (y) = eiky µ (k) d3 k, µ (k) ∈ ∞ 3 C0 R \ 0 , and the dependence on κ1 , τ , a is omitted in the final expression. Finally, the proposed scattering subspaces are out(in) Hout(in) := : h (P) ∈ C01 (Σ \ 0) , µ ∈ C0∞ (R3 \ 0) . ψh,µ On these subspaces the C0∞ functions f of the variable eiHt xt e−iHt converge. This means that the limits x e−iHt s − lim eiHt f t→±∞ t out(in)
exist and generate the commutative algebra Avel . Analogously the canonical out(in) associated to a free massless boson field is generated by the Weyl algebra Aph strong time limits of the L.S.Z. Weyl operators acting on the space Hout(in) : W out(in) (ζ) := s − lim eiHt e−iH t→±∞
ph
t i(a(ζ)+a† (ζ)) iH ph t −iHt
e
e
e
(1.31)
with ζ (k) ∈ L2 R3 , 1 + |k|−1 d3 k . out(in)
out(in)
The two algebras Avel , Aph and therefore commute each other.
are related to decoupled degrees of freedom
Remark 1.4 A warning is necessary at this point. Previous definitions are arbitrary to some extent, due to the coherent factor in the definition of the minimal asymptotic electron states, which is arbitrary except in the infrared limit. Nevertheless, 1 out(in) and Hout(in) we want to point through the (artificial) distinction between Hκ1 out that: From a technical point of view, our construction of the scattering subspaces 1 out(in) just to focus on the infrared dressing; is based on some, not unique, Hκ1 From a physical point of view, whether the “hard” photon cloud described by the smearing functions {µ} is totally removable, the photon cloud linked to the vectors 1 out(in) is not completely removable. It means that all scattering states contain in Hκ1 asymptotic photons. The physical quantities must be independent of the choice of the “one1 out(in) , in particular of the choice of κ1 . Indeed, once Σ is particle” subspace Hκ1 1 out(in) , i.e., with a different upper frequency, fixed, if we considered a different Hκ 1
out(in)
κ 1 > κ1 , in the photon cloud of the generic vector ψh,κ 1 , it is not difficult out(in)
to check that ψh,κ 1
out(in)
can be expressed as a dressing of ψh,κ1
by means of
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out(in)
out(in)
asymptotic photons and, vice versa, that ψh,κ1 can be obtained from ψh,κ 1 subtracting asymptotic photons. Following the procedures developed in the next two sections, it is enough to properly choice the partition rates for combining estimates and to take into account the mechanism used in Subsection 3.1.1. This is important in order to identify limits obtained with different partition rates. In this analysis it is also necessary to assume the following result proved in [Fr.2] (formula(3.5)) inf {E σt (P − k) + |k| − E σt (P)} = ∆ (σt , P) > 0
|k|≥σt
(1.32)
which holds for any P ∈Σ and provides the inequality (5.35) in Theorem A5 when κ1 is too large to fulfill the constraint (1.23). out(in) out(in) Therefore the space Hout(in) and the algebras Avel , Aph are indepen1 out(in)
dent of the construction of the “one-particle” space Hκ1 on Σ.
but depend only
About the structure of the paper: In Section 2, we study the time behavior of the norm of the approximating vector ψh,κ1 (t); In Section 3 we prove the strong convergence of ψh,κ1 (t) for t → +∞; out(in)
Section 4 contains the construction of the scattering subspace Hout(in) , of Avel out(in) and Aph ; Section 5 contains the Appendix where we collect some results employed in Sections 2,3,4. Lemmas and theorems in the Appendix are denoted by the letter A (i.e., Lemma A1).
2 Control of the norm of the approximating vector The squared norm (ψh,κ1 (t) , ψh,κ1 (t)) , t 1, corresponds to: σt (t) Wσt (vl , t) eiγσt (vl ,∇EP ,t) e−iHσt t ψl,σt ,
N (t)
l,j=1 σt (t) Wσt (vj , t) eiγσt (vj ,∇EP ,t) e−iHσt t ψj,σt
.
The diagonal terms are easily under control because their sum is constant in time: N (t)
j=1
(t) (t) ψj,σt , ψj,σt
N (t)
=
j=1
2
(t)
Γj
3
|h (P)| d P =
|h (P)|2 d3 P .
(2.1)
The non-trivial step consists in proving that it is indeed the limit of the squared norm provided the partition rate is properly chosen. For this purpose it is enough
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A. Pizzo
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$N (t) to show that each off-diagonal term, in the sum l,j=1 , asymptotically vanishes with an order in t substantially not related to the dimension of the cell. In the $N (t) end, we obtain that the sum of the off-diagonal terms, l,j=1,l =j , vanishes for t → +∞, provided the exponent , which determines the growth rate of the total number of cells, N (t) ≤ t3 , is sufficiently small.
2.1
Control of the off-diagonal terms
The generic off-diagonal term is (l = j) σt σt (t) (t) Ml,j (t) = eiγσt (vl ,∇EP ,t) ψl,σt , eiHσt t Wσt ,l,j (t) e−iHσt t eiγσt (vj ,∇EP ,t) ψj,σt (2.2) where
i|k|t −a† (k)e−i|k|t ) √d3 k − κ1 a(k)e ηl,j (k |k| 2|k| Wσt ,l,j (t) := e σt (2.3) and
:= ηl,j k
· (vj − vl ) gk . · vj · 1 − k · vl 1−k
(2.4)
Now, let us consider Ml,j (t) as a two-variable function, by distinguishing the variable t, which parameterizes the partition Γ(t) and the infrared cut-off σt , from the variable, s, of the dynamical evolution. Then, for s ≥ t we define: σ %l,j (t, s) := eiγσt (vl ,∇EPt ,s) ψ (t) , M l,σt σt (t) eiHσt s Wσt ,l,j (s) e−iHσt s eiγσt (vj ,∇EP ,s) ψj,σt
(2.5)
where σt γσt (vl , ∇EP , s) := !
s 2 τ ·τ −α cos(q·∇EPσt −|q|) dΩd|q| − 1 g τ ·σt dτ (1− q·vl ) τ
=−
σt− α1 1
g2
τ ·τ −α τ ·σt
σ
cos(q·∇EPt −|q|) dΩd|q| (1− q·vl ) τ
(2.6) for s :
s−α ≥ σt
! dτ
for s :
s−α < σt .
%l,j (t, t) ≡ Ml,j (t) follows by definition. The property M Theorem 2.1 Under the constructive assumptions and for α(< 1) sufficiently close %l,j (t, s): to 1, the following properties hold for the off-diagonal terms M %l,j (t, s) = 0 %l,j (t, +∞) := lims→+∞ M I) M % % II) |Ml,j (t)| = M (t, t) − M (t, +∞) ≤ C · t−7 l,j l,j provided 4 < η where η is a positive exponent α-dependent.
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Proof. Analysis of I). For s ≥ t, let us consider σt (t) λ %l,j M (t, s) := eiγσt (vl ,∇EP ,s) ψl,σt , σt (t) eiHσt s Wσλt ,l,j (s) e−iHσt s eiγσt (vj ,∇EP ,s) ψj,σt
where Wσλt ,l,j (s) := e
−λ
κ1 a(k)ei|k|s −a† (k)e−i|k|s |k| σt
3k 2|k|
) √d ηl,j (k
(2.7)
(2.8)
λ being a real parameter. %λ (t, s) with respect to the real parameter λ, the From the derivative1 of M l,j following differential equation is determined: %λ (t, s) dM l,j %λ (t, s) + rλ (t, s) = −λCl,j,σt · M l,j σt dλ where
κ1
Cl,j,σt = σt
(2.9)
2 d3 k ηl,j k 3 2 |k|
(2.10)
rσλt (t, s) =
σt (t) −Wσλ†t ,l,j (s) eiγσt (vl ,∇EP ,s) e−iHσt s ψl,σt ,
,
+
κ1 σt
κ1 σt
) a(k)ei|k|s ηl,j (k |k|
) a(k)ei|k|s ηl,j (k |k|
σt (t) k eiγσt (vl ,∇EP ,s) e−iHσt s ψj,σt 2|k|
√d
3
σt 3 (t) k eiγσt (vl ,∇EP ,s) e−iHσt s ψl,σt 2|k|
√d
σt (t) Wσλt ,l,j (s) eiγσt (vl ,∇EP ,s) e−iHσt s ψj,σt
,
(2.11)
The solution of the differential equation (2.9) at λ = 1 is %l,j (t, s) = e− M
Cl,j,σ t 2
λ=0 %l,j M (t, s) +
0
1
rσλt (t, s) · e−
Cl,j,σ t 2
(1−λ2 ) dλ .
(2.12)
Now, notice the following facts: %λ=0 (t, s) = 0 ∀t, s, because the P-supports of ψ (t) and ψ (t) , l = j, are M l,j j,σt l,σt disjoint; 1 ψ (t) ∈ D (H ) implies that it belongs to the σ j,σ t t ∈ L2 R3 \ Bσt with Bσt := k ∈R3 : |k| ≤ σt ;
f well defined.
domains of the operators a (f ) and a† (f ), therefore the derivative with respect to λ is
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A. Pizzo
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Thanks to Theorem A5 (in Appendix), the vector κ1 a (k) ei|k|s ηl,j k d3 k −iHσt s iγσt (vj ,∇EPσt ,s) (t) ψj,σt e s − lim eiHσt s e s→+∞ |k| 2 |k| σt is well defined and can be written as κ1 aout (k) ηl,j k −1 σt d3 k iγσt vj ,∇EPσt ,σt α (t) ψj,σt , e |k| 2 |k| σt where aout σt (k) is the asymptotic annihilation operator-valued distribution corresponding to the dynamics governed by the Hamiltonian Hσt ; (t) Since the vector ψj,σt is a vacuum vector for aout σt (k) (see Theorem A5), we get (2.13) lim rσλt (t, s) = 0 . s→+∞
Starting from the solution (2.12) and exploiting the dominated convergence theorem, we have 1 Cl,j,σ 2 t %l,j (t, +∞) = lim M rσλt (t, s) · e− 2 (1−λ ) dλ = 0 . (2.14) s→+∞
0
Analysis of II). Let us consider: σt d iHσt s (t) e e−iHσt s Wσt (vl , s) eiγσt (vl ,∇EP ,s) e−iHσt s ψl,σt ds (2.15) σt σ dγσt (vl , ∇EP , s) iγσt (vl ,∇EPt ,s) −iHσ s (t) t ψ = iWσt (vl , s) ϕσt ,vl (x, s) + e e l,σt ds where ϕσt ,vl (x, s) := g 2
κ1
σt
cos (k · x − |k| s) dΩd |k| . · vl 1−k
(2.16) (t)
The formal derivative in (2.15) is operatorially well defined because ψj,σt ∈ D(Hσt ) ≡ D(H).2 2 More
precisely, the result follows because: The operators κ1 d3 k p2 a (k) eik·x + a† (k) e−ik·x Hσt , H ph , H0 = + H ph and g 2m 2 |k| σt
have a common e.s.d. D. The derivative ph d eiH s e−iHσt s ds is an operator which has a closure. Approximating the vectors in D (Hσt ) with vectors in D (in the norm H0 ψ + ψ) and applying the formal calculus we get convergent sequences.
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First we discuss some preliminary quantities useful to estimate the norm of the expression (2.15). i) From the definition (2.6) we have σt , s) dγσt (vl , ∇EP ds
s1−α cos(q·∇EPσt −|q|) dΩd|q| −g 2 σt ·s (1− q·vl ) s = 0
By analogy we define dγσt vl , xs , s ds
s1−α −g 2 σt ·s :=
cos(q· x s −|q|) dΩd|q| (1− q·vl ) s
0
for
(2.17)
1 −α
s < σt
1 −α
for
s ≥ σt
for
s < σt
for
s ≥ σt
.
(2.18)
1 −α 1 −α
.
ii) The function ϕσt ,vl (x, s) can be decomposed as + ϕσt ,vl (x, s) = ϕ− σt ,vl (x, s) + ϕσt ,vl (x, s)
(2.19)
where the two terms on the right-hand side of the equation (2.19) are defined as ϕ− σt ,vl
(x, s) :=
s−α σt
cos(k·x−|k|s) l ) dΩd |k| (1−k·v
g2
κ1
s−α
1 −α
for s < σt
1 −α
(2.20)
for s ≥ σt
0
ϕ+ σt ,vl (x, s) :=
g2
1 −α
cos(k·x−|k|s) l ) dΩd |k| (1−k·v
for s < σt
ϕσt ,vl (x, s)
for s ≥ σt
1 −α
(2.21)
iii) For implementing the propagation estimate concerning the position of the electron, taking into accountHypothesis H0 (Spectral hypothesis, Subsection 1.3.1) we can consider a C0∞ R3 \ 0 function χh with the following property: σt χh (∇EP )≡1
for
P ∈ supp h
(2.22)
uniformly in t. What we want now to check is that the norm of the expression (2.15) goes to zero for s → ∞ with an integrable rate substantially independent of the partition rate. For this purpose we exploit the decomposition (2.19) of ϕσt ,vl (x, s) and the σt function χh (∇EP ). We break the expression (2.15) in separate contributions and
576
A. Pizzo
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estimate the norm for each of them to control the norm of the original vector: d σ (t) iγσt (vl ,∇EPt ,s) −iHσt s iHσt s (2.23) e ψ e W (s) e σt l,σt ds x σt σt (t) σt ≤ ϕσt ,vl (x, s) χh (∇EP ) − χh e−iEP s eiγσt (vl ,∇EP ,s) ψl,σt s x σ σ + −iEPt s iγσt (vl ,∇EPt ,s) (t) + ϕσt ,vl (x, s) χh e e ψl,σt s " x # dγ x −iE σt s iγσt (vl ,∇EPσt ,s) (t) σt vl , s , s − + ϕσt ,vl (x, s) + ψl,σt χh e P e ds s dγ v , x , s x σt σt σt l s (t) σt + χh (∇EP e−iEP s eiγσt (vl ,∇EP ,s) ψl,σt ) − χh ds s " # σt dγσt vl , xs , s dγσt (vl , ∇EP , s) + + − ds ds σt σt (t) σt ×χh (∇EP ) e−iEP s eiγσt (vl ,∇EP ,s) ψl,σt Now we explain how in the expression above each term is controlled and why we 3 2 can fix η > 0 such that a leading order is s−1 · s−η · |ln σt | · t− 2 . From Lemma A4, Theorem A2 and Corollary A3 we deduce that: After the subtraction of the infrared tail ϕ− σt ,vl (x, s) and exploiting the electron dispersion, the decoupling is estimated from above by x (x, s) χ sup ϕ+ ≤ C · s−2 · sα ; h σt ,vl s x The remainder is controlled by combining the bounds ln (σt ) sup |ϕσt ,vl (x, s)| ≤ C · s x dγ v , x , s ln (σt ) σt l s sup ≤C · ds s x
(2.24)
with the propagation estimates x σt σt 3 (t) σt ) − χh e−iEP s eiγσt (vl ,∇EP ,s) ψl,σt ≤ C · s−υ · |ln σt | · t− 2 χh (∇EP s (2.25) " # x σt σt σt dγσt vl , s , s dγσt (vl , ∇EP , s) (t) + e−iEP s eiγσt (vl ,∇EP ,s) ψl,σt − ds ds 3
≤ C · s−1 · s−υ · |ln σt | · t− 2
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where υ > 0 for α sufficiently close to 1 and small enough (Theorem A2, Corollary A3). Moreover, assuming the following constraint 4 < η , we easily obtain |Ml,j (t)| ≤
(2.26)
d iγσt (vl ,∇EPσt ,s) (t) iHσt s e Wσ,l,j (s) ψl,σt , e ds t σt (t) ×e−iHσt s eiγσt (vj ,∇EP ,s) ψj,σt ds +∞ d σ iγσt (vl ,∇EPt ,s) −iHσt s (t) (t) iHσt s e W (v , s) e ψ e ≤ σt l l,σt · ψj,σt ds ds t +∞ d σ (t) iγσt (vj ,∇EPt ,s) −iHσt s (t) iHσt s + Wσt (vj , s) e ψj,σt e ds e · ψl,σt ds t +∞
≤ C · t−7
(2.27)
Hence the sum of the off-diagonal terms is bounded by C · t− .
3 Strong convergence of the approximating vector In order to prove the strong convergence of ψh,κ1 (t) for t → +∞, we study the norm of the vector ∆t2 ,t1 ψh,κ1 := ψh,κ1 (t2 ) − ψh,κ1 (t1 )
(3.1)
for arbitrary times t2 > t1 1. For a time difference, t2 − t1 , sufficiently large we have different partitions corresponding to t2 and t1 respectively and then N (t2 ) = N (t1 ). The t2 -partition $N (t ) $N (t1 ) $ sum, j=12 , is therefore generally written as l(j) , where the index l (j) j=1 counts the subcells, relative to the t2 -partition, which are contained in the j th (t2 ) cell of Γ(t1 ) , with 1 ≤ l (j) ≤ N N (t1 ) . In accordance to these notations, the vector ∆t2 ,t1 ψh,κ1 corresponds to N (t1 )
eiHt2
σt 2 (t2 ) Wσt2 vl(j) , t2 eiγσt2 (vl(j) ,∇E (P),t2 ) e−iHσt2 t2 ψl(j),σ t
2
j=1 l(j) N (t1 )
−e
iHt1
Wσt1 (vj , t1 ) eiγσt1 (vj ,∇E
σt 1
(P),t1 ) −iHσt t1
e
(t )
ψj,σ1 t .
(3.2)
with ρ > 0 .
(3.3)
1
1
j=1
Our final goal is to obtain the following estimate ∆t2 ,t1 ψh,κ1 = ψh,κ1 (t2 ) − ψh,κ1 (t1 ) ≤ C ·
2
|ln t2 | t2ρ 1
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A. Pizzo
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This estimate is sufficient to prove the strong Cauchy property of ψh,κ1 (t) by a telescopic argument (Theorem 3.1).
3.1
Outline of the proof
Due to the constructive recipe, the time variation, t2 → t1 , yields many modifications in the vector ψh,k1 (t). In addition to the time evolution, the partition Γ(t) and the infrared cut-off σt are time dependent. Before going into details, it is worthwhile to explain the general mechanisms which prevent the rising of not convergent terms and then imply the Cauchy property. The increase in the number of cells is a potential source of problems in the control of the difference (3.3). Concerning the norm of the piece of vector ∆t2 ,t1 ψh,κ1 corresponding to each cell in Γ(t2 ) , if we only used the bound coming from the restriction of the support in the P-variable, the estimate for the norm of the entire 1 vector would diverge like N (t2 ) 2 . On the other hand, the (non-relativistic) cluster property of the system implies that the components with different (electronic) “velocities” in the cell-partition are asymptotically orthogonal as vectors in the Hilbert space. In order to exploit this mechanism in Theorem 2.1, the rate of the partition is chosen slower than the decoupling rate (constraint (2.26)). Besides the increase of number of cells in time, we must handle two delicate aspects concerning the convergence in each single cell, namely: A correction to the asymptotic dynamics, by means of the phase factor, is required in the application of Cook’s argument; The regularity properties concerning the vector φσPt (Subsection 1.3.1) which come into game and must be exploited in order to check that the dressing cloud combined with the one particle state gives rise to a well-defined vector in the limit σt → 0. Variation of the partition As preliminary step in the analysis of ∆t2 ,t1 ψh,κ1 , we control the variation of the approximating vector when the cell partition changes from Γ(t2 ) to Γ(t1 ) , all the other variables remaining fixed at time t2 . This means that we perform the following replacements σt σt Wσt2 vl(j) , t2 vl(j) ≡ ∇EP 2 , vj ≡ ∇EP 1 → Wσt2 (vj , t2 )
e
iγσt
2
σt 2
vl(j) ,∇EP
,t2
(t2 ) ψl(j),σ t2
→ e →
iγσt
2
σt 2
vj ,∇EP
j
l(j)
,t2
(t )
ψj,σ1 t
2
so that the corresponding modification of the approximating vector is D0) N (t1 )
e
iHt2
Wσt2
iγ vl(j) , t2 e σt2
σt 2
vl(j) ,∇EP
,t2
(t )
2 e−iHσt2 t2 ψl(j),σ t
2
j=1 l(j) N (t1 )
→
eiHt2
j=1
Wσt2 (vj , t2 ) e
iγσt
2
σt 2
vj ,∇EP
,t2
(t )
e−iHσt2 t2 ψj,σ1 t . 2
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Once the partition is Γ(t1 ) , we can study the variation of the vector N (t1 )
(t )
1 ψh,κ (t) := 1
σt σt (t ) eiHt Wσt (vj , t) eiγσt (vj ,∇EP ,t) e−iEP t ψj,σ1 t
(3.4)
j=1
between t = t2 and t = t1 . The initial constructive hypotheses are not sufficient to perform the time derivative: d (t1 ) ψ (s) |s=t . ds h,κ1 In fact the (strong) continuity in σt = 0 proved in [Pi.] for the vector σt σt ) ψP φσPt = Wσt (∇EP
does not imply that φσPt is Lipschitz in σt in a neighborhood of σt = 0. Assuming Hypothesis H0 and exploiting only the H¨older property of φσPt in neighborhoods of σt = 0 and P = 0 (see Spectral properties, Subsection 1.3.1) we (t1 ) (t1 ) perform some intermediate steps from ψh,κ (t2 ) to ψh,κ (t1 ) corresponding to 1 1 finite differences and we study the norm of each contribution. In the next lines, we carefully single out the intermediate variation D1), involving Cook’s argument, and then the variations D2), D3.1), D3.2) and D3.3), related to the removal of infrared cut-off. Indeed the order in the subsequent modifications is important to get the desired estimate. Cook’s argument The backwards time evolution, at fixed cut-off σt2 , corresponds to the modification D1) N (t1 )
eiHt2 Wσt2 (vj , t2 ) e
iγσt
2
σt 2
vj ,∇EP
σt 2
e−iEP
,t2
t2
(t )
ψj,σ1 t
2
j=1 N (t1 )
→
eiHt1 Wσt2 (vj , t1 ) e
iγσt
2
σt 2
vj ,∇EP
σt 2
e−iEP
,t1
t1
(t )
ψj,σ1 t
(3.5)
2
j=1
and the study of the difference consists in a standard Cook’s argument with the subtraction of the infrared tail as in Theorem 2.1. Variation of the infrared cut-off Under the variation of the infrared cut-off, σt2 → σt1 , the vector (3.5) changes as follows N (t1 )
eiHt1 Wσt2 (vj , t1 ) e
iγσt
2
σt 2
vj ,∇EP
,t1
σt 2
e−iEP
t1
(t )
ψj,σ1 t
2
j=1 N (t1 )
→
j=1
eiHt1 Wσt1 (vj , t1 ) e
iγσt
1
σt 1
vj ,∇EP
,t1
σt 1
e−iEP
t1
(t )
ψj,σ1 t
1
.
(3.6)
580
A. Pizzo
Ann. Henri Poincar´e
For convenience we consider each cell-vector in the sum at the line (3.6) as the composition of two blocks σt (3.7) eiHt1 Wσt2 (vj , t1 ) Wσ†t2 ∇EP 2 σt σt 2 σt (t ) iγ v ,∇EP 2 ,t1 e−iEP t1 Wσt2 ∇EP 2 ψj,σ1 t (3.8) e σt2 j 2
b(k)−b† (k) κ σt σt 2 2 |k| 1−k·∇E P
3 √d k 2|k|
−g σ ( ) where we recall that Wσt2 ∇EPt2 = e . Let us call the operator in (3.7) dressing block and the vector in (3.8) regular block. The contribution due to the infrared cut-off variation, σt2 → σt1 , can therefore be split in: The variation of the regular block (3.8) σt σ σ (t ) v ,∇EP 2 ,t1 −iE t2 t1 iγ e P Wσt2 ∇EPt2 ψj,σ1 t D2) e σt2 j 2 σt σ σt1 (t1 ) iγσt vj ,∇EP 1 ,t1 −iE t1 t1 1 P → e e Wσt1 ∇EP ψj,σt 1
which is substantially related to the convergence for σ → 0 of the vectors φσP in the fiber spaces HP ; The variation of the dressing block (3.7) up to the term eiHt1 σ D3) Wσt2 (vj , t1 ) Wσ†t2 ∇EPt2 σ → Wσt1 (vj , t1 ) Wσ†t1 ∇EPt1 . Let us analyze the variation D3) in further details. It can be written as σt Wσt2 (vj , t1 ) Wσ†t2 (vj ) Wσt2 (vj ) Wσ†t2 ∇EP 2 σ → Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt1 (vj ) Wσ†t1 ∇EPt1 where Wσt2 (vj ) = e smaller ones: D3.1) →
−g
b(k)−b† (k) κ σt |k|(1−k·v j) 2
3k 2|k|
√d
, so that we split the step D3) in three
σ Wσt2 (vj , t1 ) Wσ†t2 (vj ) Wσt2 (vj ) Wσ†t2 ∇EPt2 σ Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt2 (vj ) Wσ†t2 ∇EPt2
in this step the logarithmic divergence arising from the variation σt2 → σt1 in the two Weyl operators on the very left is neutralized by the strong H¨ older property in P of the regular block (3.8), on which the full operator is applied; σ D3.2) Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt2 (vj ) Wσ†t2 ∇EPt2 σ → Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt2 (vj ) Wσ†t2 ∇EPt1 σt D3.3) Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt2 (vj ) Wσ†t2 ∇EP 1 σt → Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt1 (vj ) Wσ†t1 ∇EP 1
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the last two steps account respectively for the difference between the gradients ∇E σt2 (P),∇E σt1 (P) and for the shift σt2 → σt1 in the two Weyl operators on the very right. The analysis of each difference D0), D1), D2), D3.1), D3.2) and D3.3) is the content of the remaining part of the Section. The discussion is carried out in Subsections 3.1.1, 3.1.2, 3.1.3 and 3.1.4, where we describe the physical ingredients and the technical steps used to control them. It is rather detailed and complete with the use of some results proved in the Appendix. Though it is not always explicitly written, we assume that α(< 1) is sufficiently close to 1 and that the constraint (2.26) is satisfied. We here anticipate the result: Let us assume β > 1 and large enough (which means that the removal of the infrared cut-off σt = t−β is sufficiently fast in time). Then the bounds (3.23), (3.32), (3.38), (3.47), (3.49) and (3.54) – which are obtained respectively for the norms of the vectors corresponding to the variations D0), D1), D2), D3.1), D3.2) and D3.3) – are such that a leading order term is 2 ln (t2 ) tρ1 with ρ > 0. We can now state the main theorem of the paper. Theorem 3.1 For
β > 1 large enough and α sufficiently close to 1, the vector ψh,κ1 (t), with |h (P)|2 d3 P > 0, converges strongly for t → +∞ to a non-zero out vector ψh,κ , with an error of order t1ρ at most, where ρ > 0 is a proper small 1 coefficient. Proof. Starting from Theorem 2.1, the time scale related to the partition is tuned according to the constraint 4 < η. Therefore, for β > 1 large enough and α sufficiently close to 1, we can estimate 2 ln (t2 ) (3.9) ψh,κ1 (t2 ) − ψh,κ1 (t1 ) < C · tρ1 where ρ > 0 e C > 0 are independent of t1 and t2 (t2 ≥ t1 > t 1). Now let us consider the sequence t1 , t21 , . . . , tn1 , . . . and assume tn1 ≤ t2 < . Due to the norm properties, it follows that: tn+1 1 ψh,κ1 (t2 ) − ψh,κ1 (t1 ) ≤ ψh,κ1 t21 − ψh,κ1 (t1 ) + · · · + ψh,κ1 (t2 ) − ψh,κ1 (tn1 )
≤
" #2 " #2 n+1 2 C + ··· + · ln (t1 ) ρ · ln (t1 ) nρ ρ · t1 t 2 t2 1
(3.10)
(3.11)
1
For t1 sufficiently large, t1 ≥ t1 > t 1, the series inside the brackets in (3.11) is bounded by a constant less than 1.
582
A. Pizzo
Ann. Henri Poincar´e
We can conclude that ∀t1 , t2 , where t2 ≥ t1 ≥ t1 , ψh,κ1 (t2 ) − ψh,κ1 (t1 ) ≤
C . tρ1
(3.12)
Because of Theorem 2.1, the limiting vector is non-zero if
2
|h (P)| d3 P > 0.
3.1.1 Variation of the partition The squared norm of the difference D0) is N (t1 ) iγσt vl(j) ,∇EPσt2 ,t2 −iHσ t2 (t2 ) t2 2 Wσt2 vl(j) , t2 e e ψl(j),σt 2 j=1 l(j) N (t1 )
−
Wσt2 (vj , t2 ) e
iγσt
2
σt vj ,∇EP 2
,t2
j=1 l(j)
where we have used that σt2 3 (t1 ) ψj,σt = h (P) ψP d P = (t ) Γj 1
2
(t2 ) Γl(j)
l(j)
2 (t2 ) e−iHσt2 t2 ψl(j),σ t2
σt
h (P) ψP 2 d3 P =
(3.13)
(t )
2 ψl(j),σ . t 2
l(j)
For brevity, let us define %σt (vj , t2 ) : = W 2 %σt W 2
vl(j) , t2 : =
Wσt2 (vj , t2 ) e
iγσt
Wσt2 vl(j) , t2 e
2
σt 2
vj ,∇EP
iγσt
2
,t2
σt 2
vl(j) ,∇EP
(3.14)
,t2
so that the squared norm (3.13) can be written as follows $N (t1 ) $ %σt vl(j) , t2 − W %σt (vj , t2 ) e−iHσt2 t2 ψ (t2 ) W l(j),l(j) j,j=1 l(j),σt , 2 2 2
%σt (vj , t2 ) e−iHσt2 t2 ψ (t2 ) %σt vl (j ) , t2 − W W l(j ),σt2 2 2 (3.15) The sum of the terms where j = j and l (j) = l (j) vanishes for t2 → +∞ and its rate is surely bounded (from above) by a quantity of order t− 2 , as we can estimate by the same decoupling mechanism exploited in the norm control of ψh,k1 (t) (Theorem 2.1). Keeping aside this estimate, we can focus on the following sum over the diagonal terms with respect to the partition Γ(t2 ) : ," #N (t1 ) %† (vj , t2 ) W %σt vl(j) , t2 2−W σt2 2 (3.16) %σ† vl(j) , t2 W %σt (vj , t2 ) −W −iHσ t2 (t ) j=1 l(j)
t2
2
e
t2
2 ψl(j),σ
t2
where, for a given operator O and a vector ϕ, Oϕ denotes (ϕ, Oϕ).
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Now from the sum (3.16) we extract the leading contribution. Let us start analyzing . / σt σt −iγσt vj ,∇EP 2 ,t2 iγσt vl(j) ,∇EP 2 ,t2 2 2 Wσt2 ,j,l(j) (t2 ) e e
−iHσt t2 (t2 ) 2 ψl(j),σ t2
(3.17)
e
and for s ≥ t2 . / σt σt iγσt vl(j) ,∇EP 2 ,s −iHσt s iHσt s −iγσt2 vj ,∇EP 2 ,s 2 e 2 Wσt2 ,j,l(j) (s) e 2 e e
(t )
2 ψl(j),σ
t2
(3.18) with the trivial property that (3.18) coincides with (3.17) for s = t2 . The limit for s → +∞ of the expression (3.18) is: " σ σ −1 −1 e
−
Cj,l(j),σ t2 2
·
where Cj,l(j),σt2
e
t2
vj ,∇EP
iγσt
2
,σt2 α
iγσt (t2 ) ψl(j),σ ,e 2 t2
2
κ = σt1 ηj,l(j) k 2
d3 k 2|k|3
t2
vl(j) ,∇EP
:= and ηl(j),j k
,σt2 α
#
(t2 ) ψl(j),σ t2
(3.19) (vj −vl(j) ) gk· l(j) ) (1−k·vj )·(1−k·v
.
Then we rewrite the expectation value (3.17) as the limit for s → +∞ of the corresponding quantity (3.18) plus a remainder. The limit corresponds to the expression (3.19). The sum of the remainders over the cells in (3.16) amounts to at most. a quantity which is of order t−4 2 Hence the discussion is now restricted to the following sum , σt Cl(j),j,σ t2 N (t1 ) i∆γσt vl(j) −vj ,∇EP 2 ,t2 − 2 2 e 2−e . (3.20) σt Cj,l(j),σ t2 i∆γσt vj −vl(j) ,∇EP 2 ,t2 2 2 −e e− j=1 l(j) (t2 ) ψ l(j),σt 2
where
e
i∆γσt
2
σt 2
vj −vl(j) ,∇EP
,t2
:= e
−iγσt
2
σt 2
vj ,∇EP
−1
,σt2 α
e
iγσt
2
σt 2
vl(j) ,∇EP
−1
,σt2 α
. (3.21)
An estimate from above of the sum (3.20) is given by − 16 − (t2 ) 2 ψl(j),σt · C · t1 · |ln (σt2 )| ≤ C · t1 16 · |ln (σt2 )|
N (t1 )
j=1 l(j)
2
taking into account that Cl(j),j,σt2 :=
κ1
σt2
2 d3 k ηl(j),j k 3 2 |k|
(3.22)
584
A. Pizzo
Ann. Henri Poincar´e
i∆γσ vl(j) −vj ,∇E σt2 ,t2 t2 P e
and
are both bounded by
− 16
C · t1
· |ln (σt2 )|
because of the difference vj − vl(j) and the regularity properties of the gradient of the ground state energy (Subsection 1.3.1 and Lemma A1). In the end, collecting all the partial estimates, we can conclude that the norm of the difference D0) is surely bounded from above by a quantity of order − 16
t1
· |ln (σt2 )| .
(3.23)
3.1.2 Cook’s argument The difference corresponding to the variation D1) is N (t1 )
e
iHt2
Wσt2 (vj , t2 ) e
iγσt
2
σt 2
vj ,∇EP
σt 2
e−iEP
,t2
j=1
−e
iHt1
Wσt2 (vj , t1 ) e
iγσt
2
t2
(t )
ψj,σ1 t
2
σt 2
vj ,∇EP
,t1
e
σt 2
−iEP
t1
(t ) ψj,σ1 t 2
! .
We estimate the contribution for each cell by expressing the difference of the two related vectors as the following integral from t1 to t2 : ! t2 σt σ d iγ v ,∇EP 2 ,s −iE t2 s (t1 ) e P ψj,σt ds . (3.24) eiHs Wσt2 (vj , s) e σt2 j 2 t1 ds Moreover we use the inequality ! σt t2 d iγσt vj ,∇EP 2 ,s −iE σt2 s (t1 ) iHs 2 P W (v , s) e e ψ e ds σt2 j j,σt2 t1 ds ≤
t2 d t1 ds
eiHs Wσt2 (vj , s) e
iγσt
2
σt 2
vj ,∇EP
,s −iE σt2 s P
e
! ds . 2
(t )
ψj,σ1 t
(3.25) The derivative in the expression (3.24) can be split as follows: # σ dγσt2 vj , ∇EPt2 , s (vj , s) ϕσt2 ,vj (x, s) + ds "
ie
iHs
Wσt2
×e +ie
iHs
Wσt2
iγσt
iγ (vj , s) H − Hσt2 e σt2
2
σt 2
vj ,∇EP σt 2
vj ,∇EP
,t2
σ ,s −iE t2 s P
e
σt 2
e−iEP
s
(t )
ψj,σ1 t
2
(t )
ψj,σ1 t
2
(3.26) (3.27)
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where the term (3.26) is analogous to the expression (2.15) in Theorem 2.1 a part from the evolution operator on the very left. Term (3.26) As in Theorem 2.1, Section 2, we decompose ϕσt2 ,vj (x, s) as + ϕ− σt2 ,vj (x, s) + ϕσt2 ,vj (x, s)
where: ϕ− σt2 ,vj ϕ+ σt2 ,vj
(x, s) : = g
2
s−α
cos (k · x − |k| s) dΩd |k| · vj 1−k
σt2
(x, s) : = g
2
κ1
s−α
cos (k · x − |k| s) dΩd |k| . · vj 1−k
(3.28)
(3.29)
By the same procedure, exploiting the subtraction of the infrared tail ϕ− σt2 ,vj (x, s)
iγ
v ,∇E
σt 2
,s
P by means of the derivative of e σt2 j and assuming the constraint (2.26), we obtain the following estimate from above for the norm of the expression (3.26):
− 3 2
C · s−1 · s−4 · t1
· (ln σt2 )2 .
Term (3.27) As far as the norm is concerned, the vector (3.27) is equivalent to the vector iγ H − Hσt2 e σt2
=e
iγσt
2
σt 2
vj ,∇EP
,s
(t )
ψj,σ1 t
(3.30)
2
σt 2
vj ,∇EP
,s
g 0
σt2
d3 k (t1 ) ψj,σt b (k) + b† (k) 2 2 |k|
whose norm can be estimated starting from σt2 d3 k (t1 ) † ψj,σt b (k) + b (k) g 2 0 2 |k| σt2 d3 k (t1 ) † ≤ g b (k) ψj,σt 2 0 2 |k|
(3.31)
− 3 2
≤ C · σt2 · t1 (t )
because b (k) ψj,σ1 t = 0 for k ∈ {k : |k| ≤σt2 }. 2 In conclusion the norm of the vector corresponding to the difference D1) is bounded by a quantity of order − 5 2
t1
3
· (ln σt2 )2 + t2 · σt2 · t12 .
(3.32)
586
A. Pizzo
Ann. Henri Poincar´e
3.1.3 Variation of the infrared cut-off: regular block Let us study the difference N (t1 )
j=1
σt eiHt1 Wσt2 (vj , t1 ) Wσ†t2 ∇EP 2 σt 2 σ (t ) Wσt2 ∇EPt2 e−iEP t1 ψj,σ1 t − 2 × σt σt1 −iE σt1 t1 (t1 ) iγσt vj ,∇EP 1 ,t1 1 P Wσt1 ∇EP e ψj,σt −e
e
iγσt
2
σt 2
vj ,∇EP
,t1
1
for each single cell in Γ(t1 ) . The norm of the cell-vector is controlled as follows iγσ vj ,∇E σt2 ,t1 −iE σt2 t σ (t ) P e t2 e P 1 Wσt2 ∇EPt2 ψj,σ1 t 2 σt σ t σt1 (t1 ) iγσt vj ,∇EP 1 ,t1 −iEP 1 t1 1 −e e Wσt1 ∇EP ψj,σt 1
4 5 σt iγσ vj ,∇E σt2 ,t1 iγσt vj ,∇EP 1 ,t1 t2 P 1 −e ≤ e σt 2 σ (t ) × e−iEP t1 Wσt2 ∇EPt2 ψj,σ1 t 2
σt iγσ vj ,∇E σt1 ,t1 −iE σt2 t σt2 (t1 ) −iEP 1 t1 t1 P 1 P e + e W ∇E ψ − e σt2 j,σt2 P 6 7 σt2 (t1 ) σ iγσt1 vj ,∇EPt1 ,t1 −iEPσt1 t1 Wσt2 ∇EP ψj,σt2 − + e e σ (t ) −Wσt ∇EPt1 ψj,σ1 1
(3.33) (3.34) (3.35)
t1
and each term (3.33), (3.34), (3.35) is infinitesimal in the limit in which the infrared cut-off is removed. Let us explain in details. Term (3.33) We can easily estimate 4 5 σt σt iγσ vj ,∇E σt2 ,t1 2 σ (t ) iγσt vj ,∇EP 1 ,t1 P e t2 1 −e e−iEP t1 Wσt2 ∇EPt2 ψj,σ1 t
2
σt iγσ vj ,∇E σt2 ,t1 iγσt vj ,∇EP 1 ,t1 (t1 ) t2 P 1 ≤ sup e · −e ψ j,σt2 P∈Σ
and the sup is bounded in terms of (see Lemma A1): 1 γσt vj , ∇E σt2 , t1 − γσt vj , ∇E σt1 , t1 ≤ C ·(σt1 ) 4 ·t2(1−α) +C ·t1 ·σt1 (3.36) 1 P P 2 1 − 3 2 +1
so that for α sufficiently close to 1 we can surely provide a bound of order t1 1 (σt1 ) 4 for the expression (3.33);
·
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Term (3.34) σ , P ∈ Σ, we easily get Because of the regularity properties of the energy EP σt σt 1 1 −iEP 2 t1 − e−iEP t1 ≤ C · σt41 · t1 . (3.37) e 1
− 3 2 +1
which implies a bound from above of order σt41 · t1
for the norm (3.34).
Term (3.35) Exploiting the Spectral properties, Subsection 1.3.1, we can estimate: σt (t ) σt (t ) Wσt2 ∇EP 2 ψj,σ1 t − Wσt1 ∇EP 1 ψj,σ1 t 2
1
σ σ σ σ = Γ(t1 ) h (P) ( Wσt2 ∇EPt2 ψPt2 − Wσt1 ∇EPt1 ψPt1 ) d3 P j
≤
(t ) Γj 1 1
12 σ σ 2 2 |hP | IP φPt2 − IP φPt1 F d3 P − 3 2
≤ C · (σt1 ) 4 · t1
.
In conclusion the norm of the difference D2) is surely bounded by 3
C · t12
+1
1
· (σt1 ) 4 .
(3.38)
3.1.4 Variation of the infrared cut-off: dressing block Analysis of D3.1) (t )
We first define ϕj,σ1 t the regular block corresponding to the time t1 1
(t )
ϕj,σ1 t := e
iγσt
1
σt 1
vj ,∇EP
,t1
1
σt 1
e−iEP
t1
σ (t ) Wσt1 ∇EPt1 ψj,σ1 t . 1
(3.39)
Then the difference involved in the step D3.1) can be written as: " # N (t1 ) † W (v , t ) W (v ) − σ j 1 j σ (t ) t σ t2 2 eiHt1 Wσt2 (vj ) Wσ†t2 ∇EPt2 ϕj,σ1 t . (3.40) 1 −Wσt1 (vj , t1 ) Wσ†t1 (vj ) j=1
We can restrict the analysis to each single cell. By standard algebraic steps we have Wσt2 (vj , t1 ) Wσ†t2 (vj ) − Wσt1 (vj , t1 ) Wσ†t1 (vj )
σt1 a(k)(ei|k|t1 −eik·x )−h.c. d3 k √ −g σ t2 |k|(1−k·v 2|k| j) − I = Wσt1 (vj , t1 ) Wσ†t1 (vj ) Z e †
+ Wσt1 (vj , t1 ) Wσt1 (vj ) (Z − I)
(3.41) (3.42)
588
A. Pizzo
with Z := e Since the vector
ig2
σt sin(−|k|t +k·x) 3 1 1 σt 2 d k 2 2|k|3 (1−k·v j)
Ann. Henri Poincar´e
.
σ (t ) Wσt2 (vj ) Wσ†t2 ∇EPt2 ϕj,σ1 t
1
belongs to the domain of the operator σt1 a (k) ei|k|t1 − eik·x − h.c. d3 k −g · vj 2 |k| σt2 |k| 1 − k we can use the identity e
−g
(
)
i|k|t1 −eik·x −h.c. σt a(k) e 1 σt |k|(1−k·v 2 j)
= −g
1
e
−gλ
√d
3k
− I
2|k|
0
(
)
i|k|t1 −eik·x −h.c. σt a(k) e 1 σt |k|(1−k·v 2 j)
×
σt1
σt2
3k 2|k|
√d
dλ
a (k) ei|k|t1 − eik·x − h.c. d3 k . (3.43) · vj 2 |k| |k| 1 − k
Moreover we can estimate σt 3 1 a (k) ei|k|t1 − eik·x − h.c. d k σt2 (t1 ) † W ∇E ϕ (v ) W σt2 j σt2 j,σt1 P · vj 2 |k| σt2 |k| 1 − k
≤
σt 1 σt2 σt 1 + σt2
≤
σt 1 2 σt2 +
a(k)(ei|k|t1 −eik·x ) j) |k|(1−k·v
√d
3
k Wσt2 2|k|
a† (k)(e−i|k|t1 −e−ik·x ) j) |k|(1−k·v b(k)(ei|k|t1 −ik·x −1) j) |k|(1−k·v
√d
σ (t ) (vj ) Wσ†t2 ∇EPt2 ϕj,σ1 t 1
3
(vj ) Wσ†t2
k Wσt2 2|k|
3 √d k Wσt 2 2|k|
. /1
σt1 |ei|k|t1 −eik·x |2 3 2 d k σt2 2|k|3 (1−k·v j )2 Wσ
t2
(vj ) Wσ†t2
(vj )Wσ†t
2
σt2 (t1 ) ∇EP ϕj,σt 1
(3.44)
.
(3.45)
σt 2
∇EP
σt2 (t1 ) ∇EP ϕj,σt 1
(t ) t1
1 ϕj,σ
As checked in Lemma A6, the two expressions (3.44) and (3.45) are logarithmically divergent in t2 but vanishing with a power law in t1 due to the smoothness of the regular block in its P-dependence and because of the upper integration bound σt1 .
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589
Concerning the term (3.42) the corresponding norm 2 σt1 sin(−|k|t1 +k·x) 3 ig σt2 2|k|3 1−k·v d k σ (t ) ( j )2 e − I Wσt2 (vj ) Wσ†t2 ∇EPt2 ϕj,σ1 t
1
(3.46)
can be treated like the expression (3.45). In the end we obtain that the norm of the term D3.1) is surely bounded by a quantity of order: 1 16 t3 . (3.47) 1 · |ln σt2 | · (σt1 ) Analysis of D3.2) The difference to be analyzed is N (t1 )
σ (t ) eiHt1 Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt2 (vj ) Wσ†t2 ∇EPt2 ϕj,σ1 t
1
j=1 N (t1 )
−
σ (t ) eiHt1 Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt2 (vj ) Wσ†t2 ∇EPt1 ϕj,σ1 t . 1
j=1
For each single cell, by an argument similar to that one described in Theorem A2 for the expression (5.18), we have σt σt σt (t ) (3.48) Wσ†t2 ∇EP 2 − Wσ†t2 ∇EP 1 Wσt1 ∇EP 1 ψj,σ1 t 1
1 4
≤ C · (σt1 ) · |ln σt2 | ·
− 3 t1 2
so that the norm of the vector corresponding to the variation D3.2) is bounded by 3
1
C · t12 · (σt1 ) 4 · |ln σt2 | .
(3.49)
Analysis of D3.3) The difference involved in this step is N (t1 )
σt (t ) eiHt1 Wσt1 (vj , t1 ) Wσ†t1 (vj ) Wσt2 (vj ) Wσ†t2 ∇EP 1 ϕj,σ1 t
1
j=1 N (t1 )
−
eiHt1 Wσt1 (vj , t1 ) e
iγσt
1
σt 1
vj ,∇EP
,t1
σt 1
e−iEP
t1
(t )
ψj,σ1 t . 1
j=1
This variation can be written as N (t1 )
j=1
eiHt1 Wσt1 (vj , t1 ) Λe
iγσt
1
σt 1
vj ,∇EP
,t1
σt 1t 1
e−iEP
(t )
ψj,σ1 t
1
(3.50)
590
A. Pizzo
with
Ann. Henri Poincar´e
σ σ σ Λ := W |σtt12 (vj ) W † |σtt12 ∇EPt1 − I
and the definitions σ
W |σtt12 (vj ) := σ W † |σtt12
σ ∇EPt1
:=
Wσ†t1 (vj ) Wσt2 (vj ) = e Wσ†t2
−g
σt1 σt 2
b(k)−b† (k) |k|(1−k·v j)
g σ σ ∇EPt1 Wσt1 ∇EPt1 = e
σt1 σt 2
3k 2|k|
√d
b(k)−b† (k) σt 1 P
|k| 1−k·∇E
(
(3.51) )
3k 2|k|
√d
.
(3.52) The discussion of this contribution requires the study of the squared norm of the vector (3.50) and the control of the off-diagonal terms, with respect to the cellpartition Γ(t1 ) , in the corresponding scalar product. We first check that the sum of the off-diagonal terms vanishes for t1 → +∞ with a rate fast enough. Then we turn to consider the diagonal contribution. Off-diagonal terms Let us consider the generic l − j term and try to reply the same procedure as in Theorem 2.1, by the insertion of the real parameter λ in the dressing operator Wσt1 (vj , t1 ) and the subsequent derivative with respect to λ. The only obstacles in repeating the usual steps come from the lack of commutativity between
κ1 σt1
) a(k)ei|k|s ηl,j (k |k|
χh dγσt
1
x s
(vl , xs ,s) ds
3 k 2|k|
√d
σ σ and W † |σtt12 ∇EPt1 σt σt and W † |σt12 ∇EP 1 σt σt and W † |σt12 ∇EP 1
in the study of the limit s → ∞. However these problems can be easily circumvented by means of an “ 3 argument”, by exploiting the resolvent equation (1.12) in addition to the usual regularity properties. Therefore an analogous estimate is obtained: the sum of the absolute values of the off-diagonal terms is bounded by C · t− 1 · |ln σt2 |. Diagonal terms Considering that the norm (t ) σt σt σt W |σt12 (vj ) W |σt12 ∇EP 1 − I ψj,σ1 t1 can be estimated from above in terms of a quantity of order − 3 2
sup |vj − ∇E σt1 (P)| · |ln σt2 | · t1 (t1 )
P∈Γj
(3.53)
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with
591
− sup ∇E σt1 (P) − ∇E σt1 Pj ≤ C · t1 16 , (t1 )
P∈Γj
−
the sum of the diagonal terms amounts to a contribution of order t1 16 · |ln σt2 | at most. We can conclude that the norm of the vector corresponding to the difference D3.3) is bounded by − (3.54) C · t1 16 · |ln σt2 | .
4 Scattering subspaces and asymptotic observables out(in)
In this section, at first we consider the family of vectors ψh,κ1 (τ, a) corresponding to the evolution τ in time and to a displacement a in space of the state associated out(in) to the vector ψh,κ1 previously constructed. Then we construct the covariant, 1 out(in)
under space-time translations, subspace H as the norm closure of the finite κ1 out(in) linear combinations of vectors in the set ψh,κ1 (τ, a) : Hκ1 1out(in) :=
out(in)
ψh,κ1
(τ, a) :
h (P) ∈ C01 (Σ \ 0) , τ ∈ R , a ∈ R3
out(in) obtained from the strong Later, in Theorem 4.1, we define the vectors ψh,µ (time) limit of the L.S.Z. Weyl operators, with smearing (k) functions {µ : µ out(in) ∈ C0∞ R3 \ 0 , applied to the total set ψh,κ1 (τ, a) of the Hilbert space 1 out(in)
Hκ1 closure of the finite linear combinations of the vectors in . The norm out(in) is a reasonable candidate for the scattering subspace Hout(in) . the set ψh,µ The physical meaning of this definition stems from the characterization of the states belonging to Hout(in) in terms of quantum numbers associated with the asymptotic variables which are well defined on them: the asymptotic photon Weyl operators and the asymptotic electron mean velocity. In Theorem 4.2 the asymptotic convergence of the C0∞ functions of the variable eiHt xt e−iHt is established on the vectors of Hout(in) . These functions generate the commutative algebra out(in) out(in) Avel . In Theorem 4.4, we construct the canonical Weyl algebra Aph , generated by the strong the L.S.Z. limits of Weyl operators smeared with functions −1 ζ : ζ (k) ∈ L2 R3 , 1 + |k| and acting on the space Hout(in) . The ald3 k out(in)
gebra Aph
is associated with a free massless boson field and commutes with
out(in) Avel
the algebra as consequence of the asymptotic decoupling. The spectral restriction on the electron (mean) velocity (strictly less than 1) implies a restriction of Hout(in) , as subspace of H, that can be explained with the partial non-relativistic character of the model. However no issue regarding
592
A. Pizzo
Ann. Henri Poincar´e
completeness is addressed in our discussion, even under the restriction on the energy configurations of the system. out Definition of the vector ψh,κ (τ, a). 1 out Applying the operator e−ia·P e−iHτ to the generic vector ψh,κ , we obtain: 1 out e−ia·P e−iHτ ψh,κ 1 N (t)
= s − lim e−ia·P e−iHτ eiHt t→+∞
σt Wσt (vj , t) eiγσt (vj ,∇EP ,t)
j=1 σt
× e−iEP N (t+τ )
= s − lim eiHt t→+∞
σ
(t−τ ) −iEPt τ
e
(t)
ψj,σt
σt+τ
W a σt+τ (vj , t + τ ) eiγσt+τ (vj ,∇EP
,t+τ )
j=1 σt+τ
× e−iEP
t
(t+τ )
ψj,σt+τ (τ, a)
τ,a (t) = s − lim ψh,k 1
(4.1)
t→+∞
where −g
a(k)eik·a ei|k|(τ +t) −h.c.
κ1
√d
3k
σt+τ |k|(1−k·vj ) 2|k| W a σt+τ (vj , t + τ ) := e σt+τ σ (t+τ ) ψj,σt+τ (τ, a) := e−ia·P e−iEP τ h (P) ψPt+τ d3 P (t+τ )
Γj
τ,a (t) corresponds to the approximating vector ψh,κ1 (t) with translated and ψh,κ 1 wave function both in the electron and the photon variables. The final equality (4.1) can be easily derived exploiting the estimates involved in the construction of out . Therefore the definition ψh,κ 1 out out (τ, a) := e−ia·P e−iHτ ψh,κ ψh,κ 1 1
is consistent with the expected asymptotic interpretation.
(4.2)
The definition of the subspace of the minimal asymptotic electron states is out (τ, a) : Hκ1 1out(in) := h (P) ∈ C01 (R3 \ 0) , supp h ⊂ Σ, τ ∈ R, a ∈ R3 . ψh,κ 1 (4.3) Definition of the scattering spaces out(in)
In next theorem we construct the vector ψh,µ out(in)
ψh,κ1
(τ, a)
in
starting from a vector Hκ1 1out(in)
and a cloud of photons represented by an L.S.Z. Weyl operator with smearing function µ. Concerning notations we omit the dependence on κ1 , τ, a due to out(in) ψh,κ1 (τ, a).
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593
Theorem 4.1 The strong limit out s − lim ψh,µ (t) := ψh,µ
(4.4)
t→+∞
exists, where: † ψh,µ (t) := eiHt ei(a(µt )+a (µt )) e−iHt ψh,κ1 (t) ; µ (k) ∈ C0∞ R3 \ 0 , µ t (k) := e−i|k|t µ (k) ; † † a† (µ) := (a (µ)) = a (k) µ (k) d3 k ;
out ∈ Hk11out . ψh,k 1 σt ) introduced in Proof. Taking into account Theorem 3.1, the function χh (∇EP Theorem 2.1 and stationary phase method, the result follows from Cook’s argument once the constraint (2.26) is assumed.
The scattering subspaces out(in) : ψh,µ Hout(in) :=
h (P) ∈ C01 (Σ \ 0) , µ ∈ C0∞ (R3 \ 0)
(4.5)
1 out(in)
are invariant under space-time translations because the subspaces Hκ1 invariant by construction.
are
Asymptotic algebras Theorem 4.2 The C0∞ functions f of the variable eiHt xt e−iHt , that is the electron mean velocity (at time t) up to a correction of order t−1 , have strong limits in Hout for t → +∞, namely x out out s − lim eiHt f e−iHt ψh,µ =: ψh·f (4.6) ∇E ,µ t→+∞ t where f∇E (P) := limσ→0 f (∇E σ (P)). Proof. Exploiting Theorem 3.1 and using the fact that the operators x † f , eiHt ei(a(µt )+a (µt )) e−iHt t are uniformly bounded in t, we obtain x out s − lim eiHt f e−iHt ψh,µ t→+∞ t = s − limt→+∞ eiHt f
x i(a(µt )+a† (µt )) −iHt out e ψh,κ1 t e †
= s − limt→+∞ eiHt ei(a(µt )+a =
(µt ))
†
s − limt→+∞ eiHt ei(a(µt )+a
f
x t
out e−iHt ψh,κ 1
(µt )) −iHt
e
out ψh·f ∇E ,κ1
(4.7) (4.8)
594
A. Pizzo
Ann. Henri Poincar´e
where the last step, from (4.7) to (4.8), is proved by means of the same technique out used in Theorem A2 and the notation ψh·f is justified starting from the ∇E ,κ1 σ regularity properties of ∇E (P). The extension to allof Hout is straightforward because f xt is uniformly out bounded in t and the set ψh,µ is dense in Hout , by construction. out(in)
We call Avel the norm closure of the *algebra generated by the C0∞ functions of the asymptotic electron mean velocity defined in Hout(in) by the strong limits (4.6) (for the out and the in case respectively) in Theorem 4.2. Corollary 4.3 In the space Hout , the unitary operators −1 d3 k W out (ζ) : ζ (k) ∈ L2 R3 , 1 + |k| are well defined starting from the strong limit: †
W out (ζ) := s − lim eiHt ei(a(ζt )+a
(ζt )) −iHt
t→+∞
e
.
(4.9)
The following properties hold: −1 d3 k i) The operators W out (ζ) : ζ ∈ L2 R3 , 1 + |k| satisfy the Weyl commutation rules W out (ζ) W out (ζ ) = W out (ζ + ζ) e− ζ (k) ζ (k) d3 k ; where ρ (ζ, ζ ) = 2iIm
ρ(ζ,ζ ) 2
(4.10)
ii) The mapping R s → W out (sµ) defines a strongly continuous, one parametric group of unitary operators. Proof. The existence of out s − lim eiHt W (ζt ) e−iHt ψh,µ t→+∞
with
†
W (ζt ) := ei(a(ζt )+a
(4.11)
(ζt ))
out on a generic ψh,µ implies that the bounded operators W out (ζ) can be extended out from the dense set ψh,µ to all of Hout , by continuity. In order to prove the existence of the limit (4.11), let us consider for t2 > t1 the difference out out − eiHt2 W (ζt2 ) e−iHt2 ψh,µ eiHt1 W (ζt1 ) e−iHt1 ψh,µ
8n (k) ∈ C0∞ R3 \ 0 , n ∈ N such that and a sequence of functions µ ζ − µn L2 (R3 ,(1+|k|−1 )d3 k) →n→+∞ 0 .
(4.12)
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595
Then we exploit the following identity out out − eiHt2 W (ζt2 ) e−iHt2 ψh,µ eiHt1 W (ζt1 ) e−iHt1 ψh,µ out = eiHt1 W (ζt1 ) e−iHt1 ψh,µ − eiHt1 W (ζt1 ) e−iHt1 ψh,µ (t1 ) + eiHt1 W (ζt1 ) e−iHt1 ψh,µ (t1 ) − eiHt1 W µnt1 e−iHt1 ψh,µ (t1 ) + eiHt1 W µnt1 e−iHt1 ψh,µ (t1 ) − eiHt2 W µnt2 e−iHt2 ψh,µ (t2 ) + eiHt2 W µnt2 e−iHt2 ψh,µ (t2 ) − eiHt2 W (ζt2 ) e−iHt2 ψh,µ (t2 )
+e
iHt2
W (ζt2 ) e
−iHt2
ψh,µ (t2 ) − e
iHt2
W (ζt2 ) e
−iHt2
out ψh,µ
(4.13) (4.14) (4.15) (4.16) (4.17)
and observe that: Concerning (4.13) and (4.17), the corresponding norms are at most of order t−ρ 1 and t−ρ 2 respectively, for some positive ρ, because of Theorem 4.1; Concerning (4.14) (and equivalently (4.16)), we can estimate iHt e 1 W (ζt1 ) e−iHt1 ψh,µ (t1 ) − eiHt1 W µnt e−iHt1 ψh,µ (t1 ) 1 ≤ C · ζ − µn L2 (R3 ,(1+|k|−1 )d3 k) # 12 " ph 12 1 1 · Hσt + a 2 e−iHt1 ψh,µ (t1 ) · H 1 Hσt1 + a where a is a sufficiently large positive number. Both the positive constant C and the two norms " # 12 ph 12 1 1 , Hσt + a 2 e−iHt1 ψh,µ (t1 ) H 1 Hσt1 + a are bounded uniformly in t1 (in t2 for the analogous expression in the case (4.16)); In term (4.15), at fixed n, the norm is infinitesimal for t1 → +∞ because of Theorem 4.1. Hence the convergence at line (4.9) follows. Moreover the so-defined operators −1 d3 k W out (ζ) : ζ (k) ∈ L2 R3 , 1 + |k| are unitary in Hout . Concerning the properties i) and ii): i) The operator W out (ζ) W out (ζ ) : Hout → Hout is the time limit of the equal time product of the corresponding approximating operators (4.9). The approximating operators obey the Weyl rules by construction. Hence the property is satisfied in the limit. ii) This property follows basically by means of the same approximation arguments used to justify the limit (4.9).
596
A. Pizzo
Ann. Henri Poincar´e
out(in)
Theorem 4.4 For the asymptotic boson algebra Aph defined as the norm closure of the *algebra generated by the set of unitary operators (4.9) (in the out and in the in case respectively) acting on Hout(in) , the following properties hold: i) Starting from the τ -evolved generators eiHτ W out(in) (ζ) e−iHτ = W out(in) (ζ−τ )
(4.18)
where ζ−τ ( ζ−τ (k) := ei|k|τ ζ (k)) is the freely evolved test function ζ in the out(in) is uniquely defined: time −τ , the automorphism ατ of Aph ατ W out(in) (ζ) := W out(in) (ζ−τ ). (4.19) out(in)
Therefore Aph field;
is the Weyl algebra associated with the free massless scalar
out(in)
ii) The algebra Aph
out(in)
commutes with the algebra Avel
.
Proof. i) The τ -evolved generators eiHτ W out(in) (ζ) e−iHτ are well defined on Hout(in) because e−iHτ :Hout(in) → Hout(in) . By inserting the expression (4.9) for W out(in) (ζ), we easily get the equality eiHτ W out(in) (ζ) e−iHτ = W out(in) (ζ−τ ). The Weyl commutation rules are trivially conserved by ατ because 3 ρ ζ−τ , ζ−τ = 2iIm (4.20) ζ (k) ζ (k) d k = ρ (ζ, ζ ) . out(in)
Hence ατ can be uniquely extended to all the algebra Aph . ii) By an approximation argument, the considered property follows from the defiout(in) out(in) nition of the generators of Avel and Aph in Theorem 4.2 and Corollary 4.3 respectively.
5 Appendix In the following lemmas and theorems we assume the properties discussed in Subsection 1.3.1. As in the previous Sections, we use the convention to generically call C the constants which are time independent, uniform in the infrared cut-off and in the cell partition. The bounds are intended from above, unless otherwise indicated. We now provide some results about the phase factor “ eiγ ” which enters in the definition of the approximating vector ψh,k1 (t). Lemma A1 Under the assumptions for the construction (Subsection 1.3.1) and because of the definition (2.6), the following estimates hold: σt σt −1 −1 γσt2 vj , ∇EP 2 , (σt2 ) α − γσt2 vl(j) , ∇EP 2 , (σt2 ) α ≤ C · vj − vl(j) (5.1)
Vol. 6, 2005
Scattering of an Infraparticle in Nelson’s Massless Model σ
σ
where vj ≡ ∇EPt1 and vl(j) ≡ ∇EPt2 j
597
are related to the partitions Γ(t1 ) and Γ(t2 )
l(j)
respectively; For t2 > t1 1 1 γσt vj , ∇E σt2 , t1 − γσt vj , ∇E σt1 , t1 ≤ C ·(σt1 ) 4 ·t2(1−α) +C ·t1 ·σt1 ; (5.2) 1 P P 2 1 For q ∈ q : |q| < s(1−α) , 1 > α > 0 σt σt −α 2(1−α) , s) − γσt vj , ∇EP+ . (5.3) q , s ≤ C · s 16 · s γσt (vj , ∇EP s
Proof. The bounds can be obtained by standard computations taking into account the assumptions and the results in the paragraph Spectral properties, Subsection 1.3.1. Moreover the bounds are intended not to be optimal but sufficient for our purposes. The following theorem and the related corollary are concerned with the convergence “eiHt xt e−iHt →t→∞ ∇E (P)”. Theorem A2 Under the assumptions for the construction (Subsection 1.3.1), for 0 < α (< 1) sufficiently close to 1 and > 0 sufficiently small, the following propagation estimate holds true with υ > 0: σt σt σt x (t) h (q) e−iq·∇EP − e−iq· s d3 qe−iEP s eiγσt (vj ,∇EP ,s) ψj,σt χ 3
≤ C · s−υ · t− 2 · |ln σt | σt where χ h (−q) is the Fourier transformed of χh , s ≥ t 1 and vj ≡ ∇EP is
referred to the partition Γ(t) .
Proof. Let us start from the following Hilbert inequality: σ σ σ −iq·∇EPt −iq· x 3 −iEPt s iγσt (vj ,∇EPt ,s) (t) χ s d ψ (q) e − e qe e h j,σt # " σ σ σt σ i EPt −E t q s (t) iγ v ,∇E ,s −iq·∇EPt 3 P+ ( ) σ j P s ≤ χ ψj,σt −e d qe t h (q) e σ σ σt i EPt −E t q s x (t) P+ s e−iq· s − 1 d3 qeiγσt (vj ,∇EP ,s) ψj,σt + χ h (q) e
j
(5.4)
(5.5)
In order to estimate the integrals in (5.4) and (5.5), we separate “large” and “small” q: h (q) ∈ S R3 , For large q, that is q : |q| ≥ s(1−α) , we exploit that χ therefore 1 ∀n ∈ N ∃ Cn > 0 s.t. | χh (q)| < Cn · for |q| > 1; (5.6) n |q| σt older properties in P of ∇EP and For small q, that is q : |q| < s(1−α) , the H¨ σt of φP provide the desired result.
598
A. Pizzo
Ann. Henri Poincar´e
Term (5.4) The inequality # " σ σ σt σ i EPt −E t q s (t) iγ v ,∇E ,s −iq·∇EPt 3 P+ ( ) σ j P s h (q) e −e d qe t ψj,σt χ (1−α)
s +∞ |q| · | χh (q)| d3 q (t) (t) ≤ C · ψj,σ · | χh (q)| d3 q + C · ψj,σ · 0 α s 16 s(1−α) holds because of the Spectral properties, Subsection 1.3.1, which imply: σt σt σt − sEP+ = −q · ∇EP with |P − P | ≤ qs sEP q s
σt σt |∇EP − ∇EP ≤ |
1
C · |P − P | 16
Therefore the term (5.4) is surely bounded by a quantity of order 3
s− 16 · t− 2 . α
Term (5.5) Let us start from the trivial inequality σ σ σt i EPt (P)−E t q s x (t) P+ s h (q) e e−iq· s − 1 d3 qeiγσt (vj ,∇EP ,s) ψj,σt χ +∞ σ σ 3 iγσ (vj ,∇E σt ,s) (t) i EPt −E t q s x −iq· P+ t P s − 1 d qe s ≤ ψj,σt e χ h (q) e (5.7) s(1−α) (1−α) s σ σ σt i EPt −E t q s x (t) P+ s e−iq· s − 1 d3 qeiγσt (vj ,∇EP ,s) ψj,σt . (5.8) χ h (q) e + 0 The integral in (5.7) involves large q, therefore it is easily under control thanks to (5.6). For the second term (5.8) we add and subtract the same quantities to eventually obtain three expressions, (5.10), (5.11) and (5.12), which can be controlled due to: The convergence rate of the vector φσP for σ → 0; σt The regularity properties in P of h (P), eiγσt (vj ,∇EP ,s) and φσ as a vector in F b ; P
The vanishing (for s → ∞) volume O qs which is the difference between the cell (t)
Γj and the same cell under a displacement
q s.
In the derivation of (5.10), (5.11) and (5.12) we warn the reader about the following crucial facts: σt σt q := e−iq· xs ψ σt belong to the same fiber space i) Both the vectors ψP− q , ψ P P− s s HP− qs ;
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σt t ii) As vectors in Fock space, ψσP− q and ψ P coincide s
x σt σt IP− qs e−iq· s ψP ≡ IP (ψP )
(5.9)
where the isomorphism IP is defined by (1.9) in Subsection 1.1. iii) Inside the integral (t) Γj
e
σ i E t
P−
q s
σ −EPt s
σt 3 t hP eiγσt (vj ,∇EP ,s) ψσP− qd P s
the integration variable P is the spectral value of the corresponding (vectorial) operator. iv) In an expression like σt σt σt Wσt ∇EP− ψP− IP− qs Wσ†t ∇EP− q q q s
s
s
σt ∇EP− q is the vectorial operator in HP− q obtained by the multiplication of the s s identity operator with the gradient of the ground state energy evaluated in P − qs . The treatment of (5.8) proceeds as follows (1−α) σ σ s 3 iγσ (vj ,∇E σt ,s) (t) i EPt −E t q s x −iq· P+ t P s − 1 d qe s ψj,σt e χ h (q) e 0 (1−α) σ σ s σt i E t q −EPt s 3 3 t χ h (q) e P− s hP eiγσt (vj ,∇EP ,s) ψσP− = qd Pd q (t) s 0 Γj s(1−α) σ σ σt i EPt −E t q s σ iγ v ,∇E ,s P+ ) ψ t d3 P d3 q P s − χ h (q) e hP e σt ( j P (t) 0 Γj (1−α) σ σ s σt i E t q −EPt s 3 3 t χ h (q) e P− s hP eiγσt (vj ,∇EP ,s) ψσP− = qd Pd q (t) s 0 Γj
(5.10) −
χ h (q)
0
+
0
s(1−α)
(t) Γj
e
χ h (q)
(t) Γj
P−
q s
σ −EPt s
σt σt 3 3 hP eiγσt (vj ,∇EP ,s) ψP− qd Pd q s
s(1−α)
σ i E t
e
σ i E t
P−
q s
σ
−EPt s
σt σt 3 3 hP eiγσt (vj ,∇EP ,s) ψP− qd Pd q s
(5.11)
600
A. Pizzo
−
χ h (q)
0
χ h (q)
0
(t)
(t) Γj
e
(1−α)
s
0
e
Γj
s(1−α)
+
−
s(1−α)
χ h (q)
(t) Γj
e
σ i E t
P−
σ i E t
P−
q s
σ −EPt s
hP− qs e
σ iγσt vj ,∇E t
P−
σ
−EPt s
σ σ i EPt −E t
P+
q s
hP− qs e
q s
,s
σt 3 3 ψP− qd Pd q s
q s
Ann. Henri Poincar´e
iγσt vj ,∇E
σt P−
q ,s s
σt 3 3 ψP− qd Pd q s
(5.12)
σ s iγσt (vj ,∇EPt ,s) σt 3 3 hP e ψP d P d q
We now study the three differences (5.10), (5.11), (5.12) which are controlled respectively by 12 s(1−α) 2 2 σt σ 3 t | χh (q)| |hP | IP (ψP ) − IP− qs ψP− q d P d3 q (5.13) (t)
0
s(1−α)
| χh (q)|
0
s(1−α)
| χh (q)|
0
Γj
s
F
2 2 σ P 3 σt iγσt (vj ,∇EPt ,s) q [ hP e ] IP− qs ψP− q ∆ d P P− s (t)
Γj
2
Oq
2
σt |hP | IP (ψP )F d3 P
12
d3 q
F
s
12
(5.14) d3 q
(5.15)
s
where ∆P P− q s
[ hP e
σ
iγσt (vj ,∇EPt ,s)
] := hP e
σ
iγσt (vj ,∇EPt ,s) (t)
and O qs is the difference between the cell Γj ment qs .
− hP− qs e
σ iγσt vj ,∇E t
P−
q s
,s
. (5.16) and the same cell under a displace-
Difference (5.10) (t)
Using the fact that P ∈ Γj ⊂ Σ, we estimate: σt σt ) − IP− qs ψP− q IP (ψP s
F
σt σt σt = IP Wσ†t (∇EP ) Wσt (∇EP ) ψP σt σt σt Wσt ∇EP− ψP− −IP− qs Wσ†t ∇EP− q q q s
s
s
σt σt σt σt ψP− ≤ IP (Wσt (∇EP ) ψP ) − IP− qs Wσt ∇EP− q q s
s
F
(5.17)
F
σt σt σt σt Wσ†t (∇EP Wσt ∇EP− ψP− + IP− qs ) − Wσ†t ∇EP− q q q (5.18) s
s
s
F
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Scattering of an Infraparticle in Nelson’s Massless Model
Now notice that:
The norm (5.17) is bounded by a quantity of order
|q| s
1 16
601
as consequence
φσPt
of the H¨ older regularity in P of the vector (Spectral properties, Subsection 1.3.1); The norm (5.18) can be estimated starting from the norm of the vector κk · ∇E σt − ∇E σt q b (k) − b† (k) 3 P d P− s k σ φ t q (5.19) IP− qs g · ∇E σt q · ∇E σt 1 − k 2 |k| P− s σt |k| 1 − k P
P− s
therefore in terms of the following quantities:
12 2 σt σt κ g k · ∇EP − ∇EP− q s √ d3 k 3 σ σ t t 2 σt 2 |k| 1 − k · ∇EP 1 − k · ∇EP− q
(5.20)
s
which is bounded by 1 1 1 σt σt C · ∇EP− q − ∇E · |ln σt | 2 ≤ C · s−α 16 · |ln σt | 2 ; P s 1 κ 2 3 2 σt σt q IP− s b (k) Wσt ∇EP− q ψP− q d k s s F σ t κ κ gχ (k) IP− q Wσt ∇E σt q b (k) + √ σt = 3 P− s s · ∇E σt q 2 σt 2 |k| 1 − k P− s 1 2 3 2 σt (5.21) × ψP− q d k s
F
for which, using the resolvent equation (1.12) in Subsection 1.2 1 g σt b (k) ψP,σt = σt ≤ |k| ≤ κ , ψP σ 2 |k| EPt − |k| − HP−k,σt
(5.22)
1 3 it is easy to provide a bound by a quantity O |ln σt | 2 · t− 2 (uniform in s) which is enough for our purposes. The difference (5.10), through the term (5.13), can be estimated in terms of 1 1 3 3 C · s−α 16 · t− 2 + C · s−α 16 · t− 2 · |ln σt | so that it is surely bounded by a quantity of order 3
s− 16 · t− 2 · |ln σt | . α
(5.23)
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A. Pizzo
Ann. Henri Poincar´e
Differences (5.11) and (5.12)
They are easily under control because: h ∈ C01 R3 \ 0 and the estimate (5.3) in Lemma A2 holds; this implies that for (5.14) and then for (5.11) we can surely provide a bound with the quantity 3
C · s− 16 · s2(1−α) · t− 2 ; α
(5.24)
Starting from a difference between volumes, the expression (5.15) can be bounded by a quantity of order sup |q|≤s(1−α)
|q| s
12
· t− ≤ s− 2 · t− . α
(5.25)
Conclusion For α, 0 < α (< 1), sufficiently close to 1 and > 0 sufficiently small, there exists υ > 0 such that the sum of the terms (5.4) and (5.5) is bounded by 3
C · s−υ · |ln σt | · t− 2 .
Corollary A3 Under the same assumptions as in Theorem A2, for s 1 and such that s−α ≥ σt , the norm of the vector 6
s·s−α
σt ·s
"
# 7 σt cos q · xs − |q| cos (q · ∇EP − |q|) dΩd |q| − · vj ) · vj ) (1 − q (1 − q s σt σt (t) × e−iEP s eiγσt (vj ,∇EP ,s) ψj,σt
σt (with vj ≡ ∇EP ) is surely bounded by a quantity of order j
3
s−1 · s−υ · |ln σt | · t− 2
(5.26)
for some υ > 0. Proof. The proof proceeds along the same lines as for the terms (5.4) and (5.8) in Theorem A2. In the next lemma we provide some upper estimates for the absolute value of the function κ1 cos (k · x − |k| s) dΩd |k| ϕt−α ,vj (x, s) := g 2 · vj t−α 1−k σt . The proof only requires some where 1 > α > 0, t 1 and s > t, vj ≡ ∇EP j integrations by parts; therefore it is left to the reader.
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603
Lemma A4 The following two bounds hold for t 1 and s > t: Uniformly in x ∈ R3
ϕt−α ,v (x, s) ≤ C · ln t ; j s
In the region x ∈ R3 : (1 − ρ ) s < |x| < (1 − ρ) s
,
(5.27) 0 < ρ < ρ < 1
α ϕt−α ,v (x, s) ≤ Cρ,ρ · t j s2
(5.28)
where the positive constant Cρ,ρ depends on ρ, ρ . We now discuss some properties for the annihilation operator associated to the asymptotic boson field when the Hamiltonian is Hσt , σt > 0. Theorem A5 The limit s − lim eiHσt s s→+∞
κ1 σt
is well defined, with
a (k) ei|k|s ηl,j k d3 k −iHσt s (t) (t) e ψj,σt =: aout ηl,j ) ψj,σt σt (ˇ |k| 2 |k| ηl,j k χκσ1 (k) ηˇl,j (k) := |k| · 2 |k| t
(5.29)
where χκσt1 (k) is the characteristic function of the set {k : σt < |k| ≤ κ1 }. The out(in) (t) (ˇ ηl,j ) ψj,σt belongs to D (Hσt ). vector aσt (t)
Under the assumption that P + k ∈ Σ for P and k belonging respectively to Γj and {k : 0 < |k| ≤ κ1 }, the following identity holds: (t)
(ˇ ηl,j ) ψj,σt = 0 . aout(in) σt Proof. The existence of the limit is a simple application of the propagation estimate σt in Theorem A2 and of Cook’s argument by using the function χh (∇EP ) as in Theorem 2.1 and exploiting the estimates (5.27) and (5.28) in Lemma A4. out(in) (t) (ˇ ηl,j ) ψj,σt belongs to D (Hσt ). Indeed, for each s, the vector The vector aσt Hσt eiHσt s e−iH is well defined because
ph
s
a (ˇ ηl,j ) eiH (t)
ph
s −iHσt s
e
a (ˇ ηl,j ) ψj,σt ⊂ D (Hσt ) .
(t)
ψj,σt
(5.30)
(5.31)
The inclusion (5.31) is proved by an approximation argument, exploiting the fact 1 (t) that |k| 2 · ηˇl,j (k) ∈ L2 R3 , d3 k and ψj,σt ⊂ D Hσ2t . Now, since Hσt is a closed operator, it is enough to prove the convergence for s → +∞.
604
A. Pizzo
Ann. Henri Poincar´e
We rewrite the vector (5.30) as Hσt eiHσt s e−iH =
ph
s
eiHσt s e−iH
a (ˇ ηl,j ) eiH
ph
s
ph
s −iHσt s
a (ˇ ηl,j ) eiH
e
ph
(t)
ψj,σt
s −iHσt s
e
(5.32) (t)
E σt (P) ψj,σt
ph ph (t) +eiHσt s Hσt − H ph , e−iH s a (ˇ ηl,j ) eiH s e−iHσt s ψj,σt
ph ph (t) +eiHσt s H ph , e−iH s a (ˇ ηl,j ) eiH s e−iHσt s ψj,σt . Because of the first part of the theorem and due to the following equality which holds on D (Hσt ) κ1
η k l,j ph ph d3 k , H ph , e−iH s a (ˇ ηl,j ) eiH s = − a (k) ei|k|s 2 |k| σt each term on the right-hand side of the expression (5.32) has a well-defined limit for s → +∞. (t) (t) We denote as ψj,σt the projection of ψj,σt on the fiber space HP+k . P+k
Starting from the spectral decomposition of H with respect to the P operators and because of the equation (5.32), we deduce that κ1 ηl,j k (t) ψj,σt aout (k) d3 k (5.33) σt P+k |k| 2 |k| σt is a vector in HP and it belongs to the domain of HP,σt . Then the procedure consists in studying the mean value of the positive operator HP,σt − E σt (P) + ∆ on the given vector in HP : HP,σt − E σt (P) + ∆ κ1 σt
aout σt (k)
) ηl,j (k (t) √ d3 k ψj,σ t P+k |k| 2|k|
(5.34)
where ∆ is a properly small positive number. The condition (1.23) implies the inequality E σt (P + k) − |k| − E σt (P) < 0
(5.35)
for k ∈ {k : 0 < |k| ≤ κ1 }, so that we can conclude that the original vector (5.33) is zero. The next lemma provides the estimates of the expressions (3.44) and (3.45) involved in the control of the difference D3.1 ) in Subsection 3.1.4.
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605
Lemma A6 Under the assumptions for the construction (Subsection 1.3.1) and for α, 0 < α (< 1), sufficiently close to 1, the estimates below are valid: σt 3 1 b (k) ei|k|t1 −ik·x − 1 d k σt2 (t1 ) † g W ∇E ϕ (v ) W σt2 j σt2 j,σt1 P · vj 2 |k| σt2 |k| 1 − k ,
σt1
σt2
i|k|t 2 1 e − eik·x 3 2 d k · vj 2 |k|3 1 − k
≤ C · t1− · |ln σt2 | · (σt1 ) ; 1 1 2
Wσt (vj )Wσ†t 2
2
σt 2
∇EP 1
≤ C · t12
− 5 4
(t ) t1
1 ϕj,σ
1
1
· (|ln σt2 |) 2 · (σt1 ) 16
σ
where vj ≡ ∇EPt1 . j
Proof. The proof is only outlined because the estimates involve similar procedures as in Theorem A2 on the basis of the known spectral properties. The key ingredients to be exploited are: The pull-through formula for the action of b (k); The k-regularity of σ (t ) ei|k|t1 e−ik·x − 1 Wσt2 (vj ) Wσ†t2 ∇EPt2 ϕj,σ1 t
1
which is related to the Spectral properties, Subsection 1.3.1; The fact that the considered momenta k belong to the set {k : |k| ≤σt1 }.
(5.36)
Acknowledgments. I would like to thank: G. Dell’Antonio for many helpful discussions during my Ph.d. in S.I.S.S.A.-Trieste; J. Fr¨ ohlich for his interest in my work, his criticism and our present, fruitful collaboration; G. Morchio for all he taught me about the infrared problem in Q.E.D.
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Alessandro Pizzo Theoretische Physik ETH-Hoenggerberg CH-8093 Z¨ urich Switzerland email:
[email protected] Communicated by Yosi Avron submitted 13/04/04, accepted 24/09/04
Ann. Henri Poincar´e 6 (2005) 607 – 624 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/04607-18, Published online 28.07.2005 DOI 10.1007/s00023-005-0217-9
Annales Henri Poincar´ e
Geometric Modular Action and Spontaneous Symmetry Breaking Detlev Buchholz and Stephen J. Summers
Dedicated to the memory of Siegfried Schlieder Abstract. We study spontaneous symmetry breaking for field algebras on Minkowski space in the presence of a condition of geometric modular action (CGMA) proposed earlier as a selection criterion for vacuum states on general space-times. We show that any internal symmetry group must commute with the representation of the Poincar´ e group (whose existence is assured by the CGMA) and each translationinvariant vector is also Poincar´e invariant. The subspace of these vectors can be centrally decomposed into pure invariant states and the CGMA holds in the resulting sectors. As positivity of the energy is not assumed, similar results may be expected to hold for other space-times.
1 Introduction There are a number of physically relevant mechanisms which entail a degeneracy of the vacuum state in quantum field theory. Primary among these is the spontaneous symmetry breaking of an internal symmetry group. Initiated by Borchers and by Reeh and Schlieder, systematic study [1,5,6,19,29,30,35] in quantum field theories satisfying the Wightman axioms [36] or the standard axioms of algebraic quantum field theory [22, 23] has shown that the presence of multiple vacua determines much of the global structure of the theory. Common to these approaches is the assumption of the positivity of the energy, with its concomitant analyticity properties. In Minkowski space the spectrum condition is a natural and physically meaningful assumption, but in other space-times it is neither. It is therefore of interest to revisit both spontaneous symmetry breaking and the structural consequences of degenerate vacua with the standard axioms for Minkowski space theories replaced by a recently proposed condition of geometric modular action (CGMA) [10, 13]. This condition is designed to characterize those elements in the state space of a quantum system which admit an interpretation as a “vacuum”. It is expressed in terms of the modular conjugations associated to the state and given family of algebras indexed by suitable subregions (wedges) of the underlying space-time and, in principle, can be applied to theories on any spacetime manifold. For a motivation of this condition and applications to theories in Minkowski, de Sitter, anti-de Sitter and a class of Robertson–Walker space-times, we refer the interested reader to [11–15, 17]. In this paper we shall restrict our
608
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attention to four-dimensional Minkowski space, but the arguments are applicable to other space-times, yielding similar results. We shall consider an arbitrary group G as the internal symmetry group of a quantum field theory formulated in the algebraic context [23]. Hence, we shall assume there exists a net {R(W )}W ∈W of von Neumann algebras indexed by the set W of all wedges (Poincar´e transforms of the set {x = (x0 , x1 , x2 , x3 ) ∈ R4 | x1 > |x0 |}) in Minkowski space and acting upon a separable Hilbert space H, and that there exists a unitary representation V of the group G such that V (g)R(W )V (g)−1 = R(W )
,
g ∈ G,W ∈ W .
We shall assume that there is a vacuum vector Ω0 ∈ H invariant under V (G) but make no assumption about the invariance properties of the other vacua. Indeed, one of the situations we are interested in including in our analysis is the case where the various vacua are permuted among themselves by the action of V (G). After specifying the working assumptions of this paper in Section 2, we shall show that in the presence of the CGMA, the internal symmetries must commute with the representation of the Poincar´e group, whose existence is assured by the CGMA and which is constructed using the modular conjugations. In Section 3 we shall investigate the global structure of the observable algebras and prove that any translation-invariant vectors must also be Poincar´e-invariant, in contrast to what is known about vectors invariant under representations of the translation group which satisfy the spectrum condition but do not arise from modular objects [1,19]. We then prove that under the central decomposition of the global observable algebra all relevant structures are preserved. Finally, in Section 4 we show that the CGMA and the modular stability condition introduced in [13] manifest some remarkable rigidity properties.
2 Modular action and internal symmetries Although the arguments presented here apply more generally, for convenience we assume that the net {R(W )}W ∈W is locally generated in the sense defined in [16] with a generating family C of convex compact space-time regions O. Roughly speaking, this means that every algebra R(W ) is generated by the family of all algebras R(O) with O ∈ C and O ⊂ W . This subsumes such familiar examples as nets generated by algebras associated with the set of double cones. Note that nets affiliated with quantum field theories satisfying the Wightman axioms are locally generated in this sense [37]. For notational simplicity, we shall only consider bosonic theories here. We shall be assuming that the V (G)-invariant unit vector Ω0 ∈ H is cyclic and separating for R(W ), for every W ∈ W.1 Thus, the Tomita–Takesaki modular theory will be applicable, cf. [8, 25]. In the following JW , resp. ∆W , will 1 The fundamental insight that under physically motivated conditions the vacuum vector is cyclic and separating for the quantum fields localized in wedge regions is due to Reeh and Schlieder [34].
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denote the modular conjugation, resp. the modular operator, associated to the pair (R(W ), Ω0 ) by the modular theory. Also, we shall use J to represent the group generated by the set {JW | W ∈ W}. The following are included in the standing assumptions of this paper. (a) W → R(W ) is an order-preserving bijection. (b) Ω0 is cyclic and separating for R(W ), given any W ∈ W. (c) For all W0 , W ∈ W, JW0 R(W )JW0 = R(λW0 W ), where λW0 ∈ P+ is the reflection through the edge of the wedge W0 . In [13,16] the Condition of Geometric Modular Action (CGMA), formulated solely in terms of the vector Ω0 and the net {R(W )}W ∈W without any a priori assumptions about the specific form of the adjoint action of the modular conjugations on {R(W )}W ∈W or even the existence of an isometry group, was shown to entail conditions (a)–(c). It has also been shown in [3] that (c) must hold for any nets {R(W )}W ∈W locally associated with finite-component quantum fields satisfying the Wightman axioms. Note that condition (c) implies that the adjoint action of any modular conjugation JW leaves the set {R(W )}W ∈W invariant. As the surjectivity of the map in (a) is automatic and the order-preserving property is just the operationally motivated condition of isotony, only the significance of the injectivity assumption is not immediately clear. It is shown in the Appendix that if the injectivity condition is dropped, the remaining assumptions imply that the algebras R(W ) are all Abelian and independent of localization region W . Such a situation is of no interest in quantum field theory. Hence, there is no loss of physical generality to include in our standing assumptions the requirement that in no subrepresentation of the net {R(W )}W ∈W are the wedge algebras abelian. Condition (c) and modular theory imply R(W ) = JW R(W )JW = R(W ) for any wedge W ∈ W, where W ∈ W denotes the causal complement of W . Thus, the net fulfills wedge duality and hence a fortiori locality. An immediate consequence of this fact is the following result about the type of the global algebra generated by the wedge algebras. It is in perfect concord with the idea that the CGMA characterizes elementary states. Proposition 2.1 Let {R(W )}W ∈W , Ω0 be a net and vector satisfying the standing assumptions, and let R = W ∈W R(W ). Then R ⊂ R and R is of type I. Proof. Because of wedge duality, R = W ∈W R(W ) = W ∈W R(W ) ⊂ R. Hence R coincides with the center of R, proving that R is of type I. As shown in [13, 16], the standing assumptions also imply that there exists a strongly continuous (anti)unitary representation U of the proper Poincar´e group P+ on four-dimensional Minkowski space, which is constructed in a canonical manner from products of the modular conjugations JW ∈ J so that U (λW ) = JW ,
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for all W ∈ W. Indeed, one has J = U (P+ ), so that J is closed in the strong-*topology. One therefore has U (λ)Ω0 = Ω0 , for all λ ∈ P+ . The representation U acts covariantly upon the net: U (λ)R(W )U (λ)−1 = R(λW ) , for all λ ∈ P+ , W ∈ W. Since the representation of the Poincar´e group is constructed out of modular involutions, a number of results which are difficult or not possible to obtain in other settings follow easily in the presence of the CGMA. Indeed, since V (g)Ω0 = Ω0 and V (g)R(W )V (g)−1 = R(W ), for all g ∈ G and W ∈ W, a basic result of modular theory entails that V (g) commutes with all modular involutions JW , W ∈ W, cf. [8, Corollary 2.5.32]. The commutation of V (G) with U (P+ ) is therefore immediate. Theorem 2.2 If the standing assumptions are fulfilled, V (g)U (λ) = U (λ)V (g), for all g ∈ G and λ ∈ P+ . We note that the CGMA, and hence also the standing assumptions, can be satisfied by examples in which the spectrum condition is violated [13]. Landau and Wichmann showed that in the context of a local net in an irreducible vacuum representation (with spectrum condition) the internal symmetry group must commute with the representation of the translation group [28]. With the further assumptions that there is a mass gap in the theory and that for each particle in the theory there exists a field with non-zero matrix elements between the vacuum and the one-particle states, Landau proved that the internal symmetry group must commute with the representation of the Poincar´e group [27]. From another more technical set of assumptions, Bisognano and Wichmann [4] were able to derive the same conclusion. Common to all these earlier approaches is the assumption of the spectrum condition.
3 Invariance and decomposition Let Z denote the center of the algebra R = W ∈W R(W ) and Z(W ) denote the center of R(W ). Furthermore, let Zs represent the set of all self-adjoint elements of Z. We recall from Proposition 2.1 that Z = R and continue with some useful properties of these algebras. ↑ ) , Zs ⊂ U (P+ ) and Proposition 3.1 Under the standing assumptions, Z ⊂ U (P+ Z ⊂ Z(W ), for all W ∈ W. ↑ Remark. Since Z = R, it follows from this result that the unitaries U (P+ ) are elements of the global algebra R, i.e., the Poincar´e transformations are weakly inner. Again, this is in accord with the idea that the CGMA characterizes elementary states.
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Proof. As Z = R ⊂ R(W ) = R(W ) for any W ∈ W, one obtains Z ⊂ Z(W ) for all W ∈ W. But one knows from [2, Lemma 3] that JW AJW = A∗ , for all ↑ A ∈ Z(W ). Since every element of U (P+ ) is a product of an even number of modular conjugations and every element of U (P+ ) is the product of JW and an ↑ element of U (P+ ) [13], the remaining claims follow at once. Let E0 be the orthogonal projection onto the subspace of H consisting of U (R4 )-invariant vectors. Hence, we have Ω0 ∈ E0 H. It therefore follows from the preceding result that ZΩ0 ⊂ E0 H. We shall see that the converse also holds. But, first, we adapt classic arguments [1,5,19,35] to prepare some intermediate results. Let a ∈ R4 be a spacelike translation, and set an = na, for each n ∈ N. Let O ∈ C, A ∈ R(O) and A(an ) = U (an )AU (an )−1 . Since the sequence {A(an )} is uniformly bounded in norm and H is separable, there exists a subsequence {A(ank )} which ∈ C there exists is weakly convergent. By the standing assumption on C, for any O ⊂ WN and, for every n ≥ N , an N ∈ N and a wedge WN ∈ W such that O . Since R is generated O + na ⊂ WN , i.e., A(an ) ∈ R(WN ) = R(WN ) ⊂ R(O) O ∈ C, the weak limit of the corresponding subsequence by the algebras R(O), {A(ank )}, call it A∞ , is an element of R = Z. Moreover, [5, Lemma 4] implies A∞ Ω0 = w − lim A(ank )Ω0 = w − lim U (ank )AΩ0 = E0 AΩ0 . k→∞
k→∞
(3.1)
Let Y = {A∞ | A ∈ R(O), O ∈ C} ⊂ Z denote the set of all such weak limit points. Since Ω0 is cyclic for R it follows from relation (3.1) that YΩ0 is a dense subset of E0 H. Thus, since Y ⊂ Z and ZΩ0 ⊂ E0 H we arrive at the following statement. Proposition 3.2 Under the standing assumptions, one has E0 H = ZΩ0 = YΩ0 . The following result is an easy consequence of the preceding proposition and the inclusion Y ⊂ Z, established before. Corollary 3.3 Given the standing assumptions, one has the equality Y = Z. Proof. It was shown in Proposition 3.2 that YΩ0 = E0 H. Thus, the restriction of the abelian algebra Y to the subspace E0 H has Ω0 as a cyclic vector. It follows that Y is maximally abelian on E0 H. Since Y is contained in the abelian algebra Z, the restrictions of Y and Z to E0 H must coincide. The desired assertion then follows, because Ω0 is separating for Z. In a vacuum representation fulfilling the standard assumptions, including the spectrum condition, it is known [19] that Z = Z(W ), for all W ∈ W, but this need not be the case in the setting considered here. After these preparations, we proceed to the central decomposition of R. Since the center Z of R coincides with the commutant R , this amounts to a decomposition of the underlying Hilbert space into irreducible subsectors. Moreover, as the
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Poincar´e transformations are weakly inner, the representation U (P+ ) decomposes into a continuous unitary representation of P+ in each sector. In particular, the Lorentz group is not spontaneously broken by this decomposition, which is to be contrasted with the existence of examples of nets in vacuum representations (satisfying the spectrum condition but not the CGMA) in which the Lorentz group is spontaneously broken in the central decomposition of R [1, 19]. In [19] it was shown that modular covariance (see below for a definition) prevents spontaneous breaking of the Lorentz group; here it is the CGMA which assures the stability of each vacuum sector under the action of the Lorentz group. Note that the CGMA is known to hold more generally than modular covariance does [13]. The proof of our decomposition theorem rests upon the theory of direct integral decomposition of a von Neumann algebra presented in [18]. The algebra R is decomposed with respect to the abelian algebra Z to yield a standard Borel measure space (S, ν) and measurable families ζ → H(ζ) of Hilbert spaces and ζ → R(ζ) of von Neumann algebras such that H=
⊕
S
H(ζ) dν(ζ) , R =
⊕
S
R(ζ) dν(ζ) .
For ν-almost all ζ, R(ζ) is a factor [18, Thm. II.3.3]. But here we are concerned with the decomposition of a great deal more structure. Though it is clear from Proposition 3.1 and 3.2 that the algebras R(W ), W ∈ W, and the group U (P+ ) = J also decompose, it is necessary to find a set N ⊂ S with ν(N ) = 0 such that for every ζ ∈ S \ N all of the decomposed structures still have the original properties. However, this involves prima facie uncountably many conditions, which could lead to a zero-set catastrophe. The standard technique to handle this technical problem is to impose only countably many of these conditions, each of which would hold for all ζ except in a set of measure zero. Since ν is countably additive, all countably many conditions would hold except in a possibly larger set N of measure zero. One then employs a suitable limit argument to assure that the remaining conditions also hold for all ζ ∈ S \ N . Of course, a countable union of countable sets is countable, and it is only a matter of taste or convenience whether one imposes in the argument the countable union of conditions at once or each countable subset after the other. Since the decomposition of many of the structures we are concerned with here has already been carefully treated in the literature, we shall only indicate details which seem to involve new arguments. ↑ acts transiWe recall some facts from [13,16]. Making use of the fact that P+ ↑ tively on W, we identify W, as a topological space, with the quotient space P+ /P0 , ↑ where P0 ⊂ P+ is the invariance subgroup of any given wedge W0 ∈ W; note that ↑ the topology does not depend on the choice of W0 . As P+ /P0 is separable, so is W. In order to successfully decompose all the structures of interest in such a manner that the zero set catastrophe is avoided, we need to be able to choose a countable,
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↑ ⊂ W satisfying the dense subgroup P ⊂ P+ and a countable, dense subset W following conditions:
leave W stable and P acts transitively upon W. (i) The elements of P (ii) For any W1 , W2 ∈ W such that W1 ⊂ W2 , there exist two sequences such that {Wi,n } converges to Wi , i = 1, 2, and {W1,n }, {W2,n } ⊂ W W1,n ⊂ W2,n , for all n ∈ N. is chosen to be the The reader may verify that these conditions are fulfilled if P semi-direct product of rational translations with the image under the canonical projection homomorphism of the subgroup of the covering group SL(2,C) whose is chosen elements have entries with only rational real and imaginary parts, and W to be P W0 for some fixed wedge W0 . Theorem 3.4 Under the standing assumptions, the central decomposition of R leads to a unique2 integral decomposition of the given structures into irreducible, Poincar´e-covariant nets. Precisely, there exists a measure ν on the spectrum S of Z and measurable families of Hilbert spaces ζ → H(ζ), von Neumann algebras ζ → R(ζ) ⊂ B(H(ζ)), and strongly continuous (anti)unitary representations of the proper Poincar´e group ζ → U (P+ )(ζ) such that H=
⊕
S
H(ζ) dν(ζ) , R =
⊕
S
R(ζ) dν(ζ) , U (λ) =
⊕
U (λ)(ζ) dν(ζ) , S
for all λ ∈ P+ . Moreover, for each W ∈ W, there exists a measurable family of von Neumann algebras ζ → R(W )(ζ) ⊂ B(H(ζ)) such that R(W ) =
⊕
S
R(W )(ζ) dν(ζ) ,
(3.2)
and such that isotony is satisfied by {R(W )(ζ)}W ∈W ν-almost everywhere. For ν-almost all ζ, R(ζ) = B(H(ζ)), E0 (ζ)H(ζ) = (E0 H)(ζ) is one-dimensional, and U (λ)(ζ) R(W )(ζ) U (λ)(ζ)−1 = R(λW )(ζ) ,
(3.3)
for all λ ∈ P+ and W ∈ W. Proof. The decomposition of the Hilbert space and algebra R is explained in [18]. As already mentioned, the factorial components R(ζ) in the central decomposition of R indeed act irreducibly on the respective subspaces H(ζ) since Z = R ↑ by Proposition 2.1. The representation U (P+ ) of the identity component of the Poincar´e group and the subspace E0 H are decomposed in [19], and the attendant 2 The measure space (S, ν) is unique up to isomorphism, and given (S, ν) the measurable fields are unique up to unitary equivalence. See Section II.6.3 in [18] for details.
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assertions made above are proven there, using results in [24]. Although the net {R(W )}W ∈W was also decomposed in [19], there the argument was framed for locally generated nets for which C is the set of double cones; a concrete choice of a countable “dense” subcollection of double cone algebras was given there. To obtain the assertion in the generality made here, one must provide another argument. Instead, here one decomposes the elements of the countable set {R(W )}W ∈W to obtain for each W ∈ W a measurable family ζ → R(W )(ζ) such that (3.2) holds. By enlarging the zero set N , if necessary, the covariance (3.3) in ν-almost and λ ∈ P. all sectors holds for all W ∈ W Theorem II.3.1 in [18] guarantees that for a fixed pair of wedges such that W1 ⊂ W2 , the containment R(W1 )(ζ) ⊂ R(W2 )(ζ) holds ν-almost everywhere. with After a possible change of the set N , the same is true for all W1 , W2 ∈ W W1 ⊂ W2 . ↑ such that W = For an arbitrary W ∈ W, there exists an element λ0 ∈ P+ which converges to λ0 . λ0 W0 . By construction, there exists a sequence {λn } ⊂ P Define R(W )(ζ) = {w − lim U (λn )(ζ)A(ζ)U (λn )(ζ)−1 | A(ζ) ∈ R(W0 )(ζ)} . n→∞
↑ The strong continuity of U (P+ )(ζ) in these sectors entails that R(W )(ζ) is independent of the choice of such a sequence. Moreover, the same continuity im↑ , and the definition of R(W )(ζ) plies that (3.3) is valid for all W ∈ W, λ ∈ P+ is compatible with all elements of the construction. In particular R(W )(ζ) = ↑ U (λ0 )(ζ)R(W0 )(ζ)U (λ0 )(ζ)−1 . By the measurability of ζ → U (P+ )(ζ) and the covariance of the original net, it follows that the family ζ → R(W )(ζ) is measurable and that (3.2) holds for all W ∈ W. The isotony in ν-almost all sectors for wedge algebras indexed by the elements of W now follows easily from property (ii) above and the already-established isotony for wedge algebras indexed by W. ↓ ↑ Finally, as U (P+ ) = U (λW )U (P+ ), for fixed W ∈ W, the assertion concerning U (P+ ) follows, since the complex antilinearity of U (λW ) = JW , i.e., the fact that U (λW ) commutes with Zs but not with Z, poses no problems [24, Thm. III.2].
It is noteworthy that, in our general setting, the above central decomposition always results in irreducible sectors even though the spectrum condition need not hold. This is in contrast to the situation in the Wightman formalism where the extremal states resulting from a corresponding decomposition need not be pure states [6, 7] – cf. [20] for a discussion of this matter. We close this section with a comment about unbroken symmetries in the internal symmetry group G. The group G will be unitarily implemented in a given sector if and only if V (G) commutes with the corresponding projection in Z. On the other hand, if G is a separable topological group, the representation g → V (g) is strongly continuous, and there exists a subgroup H ⊂ G such that Z ⊂ V (H) ,
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then the above arguments entail that there exists a measurable family of strongly continuous unitary representations ζ → V (H)(ζ) such that ⊕ V (h)(ζ) dν(ζ) , V (h) = S
and
V (h)(ζ) R(W )(ζ) V (h)(ζ)−1 = R(W )(ζ) ,
for all h ∈ H, W ∈ W and ν-almost all ζ. In the next section, we prove that the modular structure associated with pairs (R(W ), Ω0 ), W ∈ W, also decomposes in such a manner that conditions (a)–(c) are satisfied in ν-almost all sectors.
4 The rigidity of geometric modular action We maintain the standing assumptions in this section and turn our attention to the modular structures, their properties and their behavior under the central decomposition carried out above. Let 1/4
= {∆W AΩ0 | A ∈ R(W )+ } PW
(4.1)
denote the natural positive cone corresponding to the pair (R(W ), Ω0 ), where R(W )+ is the set of all positive elements in R(W ), and let P0 = PW . W ∈W
Of course, we have Ω0 ∈ P0 . As shown in [2], every vector Φ ∈ PW , which is either cyclic or separating for R(W ), is both cyclic and separating for R(W ). Moreover, Φ the modular conjugation JW corresponding to the pair (R(W ), Φ) coincides with JW [2, Thm. 4]. Hence, if Ω ∈ P0 is cyclic or separating for all R(W ), W ∈ W, Ω then JW = JW , for every W ∈ W. Thus, the pair ({R(W )}W ∈W , Ω) must also fulfill conditions (a)–(c), if ({R(W )}W ∈W , Ω0 ) does. The CGMA therefore selects state vectors which lie in P0 , and so we wish to investigate the structure of P0 and the properties of the states determined by the elements of P0 . We begin with the following lemma.
Lemma 4.1 Under the standing assumptions, P0 is a pointed, weakly closed convex cone such that (4.2) Ω, AJW AΩ0 ≥ 0 , for all Ω ∈ P0 , W ∈ W and A ∈ R(W ). Proof. It is shown in [2, Thm. 4] that PW is a pointed, weakly closed, selfdual convex cone. Since P0 is an intersection of these cones, it is clearly a weakly closed
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convex cone. Moreover, if Ω and −Ω are contained in P0 , they are also in PW ; hence, Ω = 0. In the same theorem it is shown that Ω, AJW AΩ0 ≥ 0, for all Ω ∈ PW and A ∈ R(W ). Since P0 ⊂ PW , for all W ∈ W, the final assertion follows.
This lemma enables us to prove the following result. Proposition 4.2 Under the standing assumptions, every element of P0 is invariant under U (P+ ); in particular, P0 ⊂ E0 H. In fact, E0 H is the linear span of P0 and P0 = Z+ Ω0 . Proof. Theorem 4 (2) in [2] entails that if Ω ∈ P0 , then JW Ω = Ω, for all W ∈ W. Since U (P+ ) = J , one has U (λ)Ω = Ω, for every λ ∈ P+ . Thus, in particular, P0 ⊂ E0 H. A basic result of modular theory (cf. [8, Lemma 3.2.16]) entails that every element of the center Z(W ) is left invariant by the adjoint action of the modular unitaries ∆it W , t ∈ R. Hence, Proposition 3.1 implies that for every Z ∈ Z+ 1/4 one has ∆W ZΩ0 = ZΩ0 , and thus ZΩ0 ∈ PW , for every W ∈ W, by (4.1). This entails the inclusion Z+ Ω0 ⊂ P0 . Proposition 3.2 then implies that E0 H is the linear span of P0 . Since P0 ⊂ E0 H = ZΩ0 , there exists a normal operator Z affiliated with Z such that Ω = ZΩ0 . But for any A ∈ Z+ one has A = A1/2 JW A1/2 JW , so that (4.2) yields Ω, AΩ0 ≥ 0, for all A ∈ Z+ . Setting A = B ∗ B, B ∈ Z, this implies 0 ≤ ZΩ0 , B ∗ BΩ0 = ZBΩ0 , BΩ0 . The restriction of Z to ZΩ0 is therefore positive. But Z can be decomposed into + , Z − affiliated with Z such that Z = Z+ − Z− + four positive operators Z+ , Z− , Z i(Z+ − Z− ), and since Ω0 is separating for Z, it follows that Z = Z+ . Although the modular conjugations associated with a given von Neumann algebra and different cyclic and separating vectors from P coincide, typically the corresponding modular unitaries differ from vector to vector. However, the rigidity of the structure investigated here carries through also to the modular operators. Corollary 4.3 Under the standing assumptions, if Ω ∈ P0 is cyclic or separating Ω for R(W ) and ∆Ω W is the associated modular operator, then ∆W = ∆W . Proof. By Proposition 4.2, there exists a positive operator Z affiliated with Z such that Ω = ZΩ0 . This operator, just as every positiveelement of Z ⊂ Z(W ), commutes with the antiunitary JW , the algebra R(W ) R(W ) and with any modular group associated with R(W ). Hence, for any A ∈ R(W ) one has 1/2 (∆Ω AΩ W)
= JW A∗ Ω = JW A∗ ZΩ0 = JW A∗ JW ZΩ0 1/2
1/2
= ZJW A∗ Ω0 = Z∆W AΩ0 = ∆W ZAΩ0 1/2
1/2
= ∆W AZΩ0 = ∆W AΩ ,
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where JW R(W )JW = R(W ) has also been used. Thus, one concludes ∆Ω W ⊂ ∆W . A similar argument interchanging the roles of Ω and Ω0 completes the proof. In light of the fact that any normal state on B(H), when restricted to R(W ), can be implemented on R(W ) by a suitable vector in PW [2, Theorem 6], it is noteworthy that these different implementers sit in the various natural positive cones PW in such a way that only very well-behaved states are determined by the vectors left in the intersection P0 . Proposition 4.2 also entails that under the central decomposition of R, in ν-almost all H(ζ) the corresponding set P0 (ζ) contains only vectors proportional to Ω0 (ζ). Hence, in each irreducible vacuum sector at most one state can satisfy the CGMA in the form of conditions (a)–(c). The next theorem establishes the properties under central decomposition of the various modular structures of concern to us. Theorem 4.4 Under the standing assumptions, in reference to the structures dis (ζ) represent the cussed in Theorem 3.4, let, for each W ∈ W, JW (ζ), ∆W (ζ), PW ⊕ modular objects associated with the pair (R(W )(ζ), Ω0 (ζ)), where Ω0 = S Ω0 (ζ) dν(ζ). Then for each W ∈ W, t ∈ R, the fields ζ → JW (ζ), ζ → ∆it W (ζ) and ζ → PW (ζ) are measurable and ⊕ ⊕ ⊕ it JW (ζ) dν(ζ) , ∆it = ∆ (ζ) dν(ζ) , P = PW (ζ) dν(ζ) . JW = W W W S
S
S
Conditions (a)–(c) hold in ν-almost all sectors. If, moreover, P0 (ζ) = PW (ζ), then also ζ → P0 (ζ) is measurable and ⊕ P0 (ζ) dν(ζ) . P0 =
W ∈W
S
For ν-almost all ζ, P0 (ζ) = {c Ω0 (ζ) | c ∈ [0, ∞)}. Proof. For every W ∈ W, the measurability of the fields ζ → JW (ζ), ζ → ∆it W (ζ) and the equalities ⊕ ⊕ JW = JW (ζ) dν(ζ) , ∆it = ∆it W W (ζ) dν(ζ) S
S
are assured by [24, Thm. III.2]. From Theorem 3.4 it follows that ⊕ JW = U (λW ) = U (λW )(ζ) dν(ζ) , S
for every W ∈ W. Corollary II.2.2 in [18] then yields the equality JW (ζ) = U (λW )(ζ) for ν-almost all ζ. With a possible change in the zero set N , this equal In Section 3 of [16] it was shown that for a ity may be assured for all W ∈ W.
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locally generated net satisfying Haag duality the map W → JW from the space of wedges to the topological group J is continuous, as is the map λW → JW from P+ to J . This continuity and the continuity of the representation U (P+ ) entail then that condition (c) holds in ν-almost all sectors. The isotony in ν-almost every sector was established in Theorem 3.4. From [24, Prop. II.2] it follows that for a fixed W0 ∈ W, Ω0 (ζ) is cyclic and separating for R(W0 )(ζ) for ν-almost all ζ. In view of the covariant action of the unitaries U (λ)(ζ) ↑ on the wedge algebras R(W )(ζ) proven in Theorem 3.4 and the transitivity of P+ on W, this is therefore true for all W ∈ W and the same set of ζ. Hence, conditions (a)–(c) with the possible exception of the injectivity in (a) hold in ν-almost all sectors. ↑ )(ζ) By Proposition A.1, if the map W → R(W )(ζ) is not injective, then U (P+ is trivial and R(W )(ζ) is abelian and independent of W ∈ W. But then R(ζ) is an abelian factor with cyclic vector. If this ⊕ were true for all ζ in a measurable set M ⊂ S with positive ν-measure, then M H(ζ) dν(ζ) would be a subspace of H ↑ on which the corresponding subrepresentation of U (P+ ) was trivial and of R was abelian. This degenerate situation has been excluded by the standing assumptions. Since ∆W commutes with Z(W ), Proposition 3.1 implies that it commutes with Z and hence is also decomposable. Appealing to [24, Thm. I.8, Thm. III.2], it follows that ⊕
1/4
∆W AΩ0 =
S
1/4
∆W (ζ)A(ζ)Ω0 (ζ) dν(ζ) ,
for all A ∈ R(W ), and therefore that PW
⊕
= S
PW (ζ) dν(ζ) ,
for all W ∈ W. As the standing assumptions hold in ν-almost all sectors, one may apply Proposition 4.2 in each sector to conclude P0 (ζ) = Z(ζ)+ Ω0 = Z+ (ζ)Ω0 , and ⊕ thereby also P0 = S P0 (ζ) dν(ζ). Since for ν-almost all ζ the elements of Z+ (ζ) are positive multiples of the identity operator on H(ζ), the final assertion is immediate. We remark that also all of the modular unitaries {∆it W }t∈R can be reunited in ν-almost all sectors as above by first decomposing the operators ∆it W , for t rational and then using the strong continuity to reconstruct ∆it (ζ) for all and W ∈ W, W t ∈ R. From Section 3 of [16] and [13, Prop. 4.6] one knows that also the map W → ∆it W is strongly continuous, given our standing assumptions. This is then employed to reconstruct ∆it W (ζ) for all W ∈ W. A conceptually simple and quite general criterion for stable states on general space-times is the Modular Stability Condition, proposed in [13]. We recall this condition here for the convenience of the reader.
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(d) For any W ∈ W, the elements ∆it W , t ∈ R, of the modular group corresponding to (R(W ), Ω0 ) are contained in the group J generated by all finite products of the modular involutions {JW }W ∈W . We refer the interested reader to [11, 13] for a discussion of the background of this condition and a brief account of other interesting approaches towards an algebraic characterization of ground states on general space-times. As shown in [13], if the standing assumptions of this paper and the Modular Stability Condition hold, then modular covariance obtains: ∆it W = U (λW (2πt)), for all t ∈ R and W ∈ W, where {λW (2πt) | t ∈ R} is the one-parameter subgroup of boosts leaving W invariant. In addition, the spectrum condition holds. We close this section with a theorem which summarizes the consequences of the Modular Stability Condition for the topics under consideration here. Theorem 4.5 If the standing assumptions and the Modular Stability Condition hold for Ω0 , then the conditions (a)–(d) also obtain for any Ω ∈ P0 which is cyclic or separating for all wedge algebras R(W ). In addition, R = Z = Z(W ), for every W ∈ W. Hence, the central decomposition in Theorem 3.4 results in irreducible vacuum sectors in which the Modular Stability Condition is satisfied in ν-almost every sector, as is modular covariance and the spectrum condition. In ν-almost all sectors, R(W )(ζ) is a type III1 factor, for all W ∈ W. Proof. It has already been shown that conditions (a)–(c) hold for every Ω as described. Corollary 4.3 entails that also condition (d) is satisfied by the modular unitaries associated to each wedge algebra by such vectors Ω (and, of course, they manifest modular covariance). Together, Proposition 5.1 and the proof of Theorem 5.1 in [13] entail that U (R4 ) fulfills the spectrum condition. It then follows from [19, Prop. 3.1] that Z(W ) = Z = R , for every W ∈ W. Therefore, in the central decomposition in Theorem 3.4 one has the spectrum condition for U (R4 )(ζ), for ν-almost all ζ [19, Thm. 4.1]. Furthermore, from the proof of Lemma 3.2 in [19] one may conclude that R(W )(ζ) is a type III1 factor, for all W ∈ W. From the proof of Theorem 4.4 and [18, Cor. II.2.2], it follows that for νalmost all ζ one has ∆it W (ζ) = U (λW (2πt))(ζ), for all t ∈ R and W ∈ W, i.e., modular covariance holds in ν-almost all sectors. From the proof of Theorem 4.4 it also follows that U (P+ )(ζ) = J (ζ), for ν-almost all ζ. Therefore, the Modular Stability Condition also holds in ν-almost all sectors. We mention that if the hypothesis of Theorem 4.5 holds, then one can show using [13, Thm. 5.1] and [31, Thm. 1.2] that any Ω ∈ P0 and the corresponding Ω(ζ), for ν-almost all ζ, determine passive states on their respective nets with respect to all uniformly accelerated observers. Hence, the CGMA and the Modular Stability Condition select particularly stable states.
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5 Final comments A number of different criteria [9, 21, 26, 32, 33] have been proposed to select physically relevant states for quantum field theories on curved space-times, where translation covariance and the spectrum condition are simply not applicable. However, these criteria, when they obtain, are valid for an entire folium of states and therefore beg the question of which state (or states) of the respective folium is to be regarded as fundamental, i.e., as a reference or ground state [13]. We emphasized in [13] that the CGMA is a selection criterion for states and not an entire folium. But the CGMA explicitly places constraints only on the algebras R(W ), W ∈ W, and the modular conjugations JW , W ∈ W – the algebras are state-independent and each modular conjugation JW is common to every state vector in the natural cone PW , which is itself so large that it spans the Hilbert space H. However, the CGMA is a condition on the entire set {JW | W ∈ W}, and therefore the vectors selected by the CGMA are those in P0 . We have shown in this paper that the vectors remaining in the intersection P0 share the properties one would desire of reference states, without any appeal to the spectrum condition, and that the structures associated with the CGMA and the Modular Stability Condition are gratifyingly rigid. Moreover, we have shown that these conclusions do not rely upon the more technical assumptions of the CGMA in Minkowski space [13], which were designed to assure not only the existence of the representation of the Poincar´e group discussed above, but also to derive the Poincar´e group and its action upon Minkowski space from the initial data ({R(W )}W ∈W , Ω0 ). Already the conditions (a)–(c), themselves consequences of the CGMA in Minkowski space, are sufficient to assure the above-mentioned conclusions. Acknowledgements: We are grateful to the anonymous referee for drawing our attention to Proposition 2.1, which simplified the discussion. DB wishes to thank the Institute for Fundamental Theory and the Department of Mathematics of the University of Florida and SJS wishes to thank the Institute for Theoretical Physics of the University of G¨ ottingen for hospitality and financial support which facilitated this research. This work was supported in part by a research grant of Deutsche Forschungsgemeinschaft (DFG).
A
Nets of wedge algebras
We show that in the presence of the other standing assumptions, the injectivity of the map W → R(W ) can fail only in the most extreme manner. Proposition A.1 Let all of the standing assumptions hold, except condition (a). If the map W → R(W ) is order-preserving but not injective, then the representation ↑ ) is trivial, and R(W ) is abelian and independent of W ∈ W. U (P+
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Proof. Let W1 , W2 ∈ W be distinct wedges such that R(W1 ) = R(W2 ). Then the corresponding modular conjugations must coincide, i.e., JW1 = JW2 . Since W1 = W2 , condition (c) and U (P+ ) = J entail the existence of a nontrivial ↑ element λ0 = λW1 λW2 ∈ P+ such that U (λ0 ) = JW1 JW2 = 1. With λ0 = (Λ0 , x0 ), ↑ 4 Λ0 ∈ L+ , x0 ∈ R , one would then have U (x0 )−1 = U (x0 )−1 U (λ0 ) = U (Λ0 ) . ↑ ) it then Hence, U (Λ0 ) ∈ U (R4 ), and since U (R4 ) is a normal subgroup of U (P+ follows that U (ΛΛ0 Λ−1 ) = U (Λ)U (Λ0 )U (Λ)−1 ∈ U (R4 ) ,
for every Λ ∈ L↑+ . The elements {ΛΛ0 Λ−1 | Λ ∈ L↑+ } generate a (nontrivial) normal subgroup of L↑+ . But L↑+ is a simple group, so one deduces that U (L↑+ ) ⊂ U (R4 ). The representation U (L↑+ ) is therefore abelian and hence trivial. But then for every x ∈ R4 and Λ ∈ L↑+ one has U (x) = U (Λ)U (x)U (Λ)−1 = U (Λx), so that it follows ↑ that U (x) is independent of x ∈ R4 . Thus, U (P+ ) is trivial. But the covariance ↑ of the net under the adjoint action of U (P+ ) then entails that R(W ) = R(λW ), ↑ . Thus, one must conclude, in particular, that R(W ) = R(λW W ) = for all λ ∈ P+ ↑ R(W ) = R(W ) , for all W ∈ W. As P+ acts transitively upon W, the proof is completed.
References [1] H. Araki, On the algebra of all local observables, Prog. Theor. Phys. 32, 844– 854 (1964). [2] H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon–Nikodym theorem with a chain rule, Pac. J. Math. 50, 309–354 (1974). [3] J.J. Bisognano and E.H. Wichmann, On the duality condition for a Hermitian scalar field, J. Math. Phys. 16, 985–1007 (1975). [4] J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17, 303–321 (1976). [5] H.-J. Borchers, On structure of the algebra of field operators, Nuovo Cim. 24, 214–236 (1962). [6] H.-J. Borchers, On the structure of the algebra of field operators, II, Commun. Math. Phys. 1, 49–56 (1965). [7] H.-J. Borchers and J. Yngvason, On the algebra of field operators: the weak commutant and integral decompositions of states, Commun. Math. Phys. 42, 231–252 (1975).
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[8] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, Berlin, Heidelberg, New York: Springer Verlag, 1979. [9] R. Brunetti, K. Fredenhagen and M. K¨ ohler, The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes, Commun. Math. Phys. 180, 633–652 (1996). [10] D. Buchholz and S.J. Summers, An algebraic characterization of vacuum states in Minkowski space, Commun. Math. Phys. 155, 449–458 (1993). [11] D. Buchholz, M. Florig and S.J. Summers, An algebraic characterization of vacuum states in Minkowski space, II: Continuity aspects, Lett. Math. Phys. 49, 337–350 (1999). [12] D. Buchholz, M. Florig and S.J. Summers, The second law of thermodynamics, TCP and Einstein causality in anti-de Sitter space-time, Class. Quantum Grav. 17, L31–L37 (2000). [13] D. Buchholz, O. Dreyer, M. Florig and S.J. Summers, Geometric modular action and spacetime symmetry groups, Rev. Math. Phys. 12, 475–560 (2000). [14] D. Buchholz, J. Mund and S.J. Summers, Transplantation of local nets and geometric modular action on Robertson–Walker space-times, in: Mathematical Physics in Mathematics and Physics (Siena) (R. Longo, ed.), Fields Institute Communications 30, 65–81 (2001). [15] D. Buchholz, J. Mund and S.J. Summers, Covariant and quasi-covariant quantum dynamics in Robertson–Walker space-times, Class. Quantum Grav. 19, 6417–6434 (2002). [16] D. Buchholz and S.J. Summers, An algebraic characterization of vacuum states in Minkowski space, III: Reflection maps, Commun. Math. Phys. 246, 625–641 (2004). [17] D. Buchholz and S.J. Summers, Stable quantum systems in anti-de Sitter space: Causality, independence and spectral properties, J. Math. Phys. 45, 4810-4831 (2004). [18] J. Dixmier, Les alg`ebres d’op´erateurs dans l’espace Hilbertien, Paris: GauthierVillars, 1969. [19] W. Driessler and S.J. Summers, Central decomposition of Poincar´e-invariant nets of local field algebras and absence of spontaneous breaking of the Lorentz group, Ann. Inst. Henri Poincar´e 43, 147–166 (1985). [20] W. Driessler and S.J. Summers, On the decomposition of relativistic quantum field theories into pure phases, Helv. Phys. Acta 59, 331–348 (1986).
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[36] R.F. Streater and A.S. Wightman, PCT, Spin and Statistics, and All That, Reading, Mass.: Benjamin/Cummings Publ. Co., 1964. [37] S.J. Summers and E.H. Wichmann, Concerning the condition of additivity in quantum field theory, Ann. Inst. Henri Poincar´e 47, 113–124 (1987). Detlev Buchholz Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen D-37077 G¨ ottingen Germany email:
[email protected] Stephen J. Summers Department of Mathematics University of Florida Gainesville FL 32611 USA email:
[email protected]fl.edu Communicated by Klaus Fredenhagen submitted 25/05/04, accepted 29/10/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 625 – 656 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/04625-32, Published online 28.07.2005 DOI 10.1007/s00023-005-0218-8
Annales Henri Poincar´ e
Semiclassical Propagation of Coherent States with Spin-Orbit Interaction Jens Bolte and Rainer Glaser
Abstract. We study semiclassical approximations to the time evolution of coherent states for general spin-orbit coupling problems in two different semiclassical scenarios: The limit → 0 is first taken with fixed spin quantum number s and then with s held constant. In these two cases different classical spin-orbit dynamics emerge. We prove that a coherent state propagated with a suitable classical dynamics ap√ proximates the quantum time evolution up to an error of size and identify an Ehrenfest time scale. Subsequently an improvement of the semiclassical error to an arbitrary order N/2 is achieved by a suitable deformation of the state that is propagated classically.
1 Introduction Ever since their introduction by Schr¨ odinger as early as 1926 [Sch26], coherent states have found an increasing range of applications in quantum mechanics, see, e.g., [KS85, Per86]. In a semiclassical context their virtues become particularly transparent in attempts to relate the quantum time evolution of a system to its classical trajectories. Coherent states can, e.g., even be used to identify the limiting classical dynamics of a given quantum system. However, apart from the exceptional case of the harmonic oscillator that Schr¨ odinger chose for his construction, every quantum wave packet necessarily disperses. Schr¨odinger’s original intention to mimic classical trajectories in quantum mechanics can therefore only be put into practice up to the time scale on which wave packets begin to delocalize. Beyond that the quantum time evolution looses its tight relation to classical trajectories, although coarser classical structures possibly remain to be of influence [SB02, Sch04]. More recently the notion of an Ehrenfest time was introduced [Chi79, Zas81], intended to indicate that the Ehrenfest relations can only connect quantum dynamics and classical trajectories on limited time scales. For classical dynamics with positive Lyapunov exponents it is argued that the Ehrenfest time is logarithmic in . This conclusion can be drawn from √ the observation that coherent states are localized in phase space on a scale of , and that an unstable classical dynamics expands domains in phase space with exponential rates in the unstable directions. 1 | log | a coherent state is no longer localized in directions Thus for times beyond 2λ that are expanded with an exponent λ. A finer analysis reveals that the precise value of the Ehrenfest time depends on the problem that is studied; e.g., using L2 -
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norms to measure the difference between the quantum time evolution of a coherent state and a coherent state that is propagated with the classical dynamics, a critical 1 time scale of 6λ | log | was proven to hold [CR97]. On the other hand, the same difference measured in terms of expectation values of observables can be controlled 1 | log |. For details see [CR97, BB00, BR02]. It can up to times of the order of 2λ moreover be shown that on finite time intervals a coherent state is exponentially localized around the corresponding classical trajectory [HJ00]. Except for heat kernel asymptotics in the case of particles in non-abelian gauge fields [HPS83] most of the previous work on a semiclassical control of the propagation of coherent states is concerned with systems that possess only translational degrees of freedom. In this article it is our aim to extend these investigations to systems with non-relativistic spin-orbit interactions. After having identified appropriate coherent states, we intend to compare solutions of the Schr¨ odinger equation ∂ψ ˆ (t, x) = Hψ(t, x) , i ∂t where the initial wave function ψ(0) is a coherent state, with a coherent state that is evolved along suitable classical trajectories. The quantum Hamiltonians that we wish to allow are of a general spin-orbit coupling type, ˆ = H0 (Q, ˆ Pˆ ) + C(Q, ˆ Pˆ ) · S ˆ , H
(1.1)
ˆ Pˆ , and S ˆ denoting the standard position, momentum, and spin operators, with Q, respectively. Examples of such Hamiltonians arise when the spin is coupled to an e B, or in the context of atomic spin-orbit external magnetic field, such that C = mc coupling with C being proportional to orbital angular momentum. Apart from atomic and molecular physics spin-orbit coupling also plays an important role in nuclei, where it essentially determines their shell structure [BM69], as well as in solid state physics. In the latter case recent experimental progress ˘ towards controlling the spin dynamics of electrons in semiconductors [SFHZ01] calls for a theoretical description of such set-ups. As opposed to some pure quantum calculations semiclassical considerations are often particularly transparent and provide a clear physical picture. With our work we therefore intend to improve the understanding of spin-orbit coupling by establishing mathematically rigorous statements about the quantum dynamics of localized particles with spin and their relation to appropriate classical trajectories. One issue to be settled is how the semiclassical limit should be performed in the presence of spin-orbit interactions. In principle two parameters controlling the passage to a classical description are available, which are associated with the two types of degrees of freedom: translational and spin. On the one hand, with (an effective) approaching zero the semiclassical limit is achieved in a standard way for the translational degrees of freedom. On the other hand, for an isolated spin can be eliminated from both kinematics and dynamics. The role of a semiclassical parameter is then taken over by 1/s, where s = 1/2, 1, 3/2, . . . denotes the spin
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quantum number. When both types of degrees of freedom interact through a spinorbit coupling one can therefore pass to the semiclassical limit in various ways. In the absence of a theory that is uniform in both and 1/s we subsequently focus on two important scenarios: The most straightforward approach is to view, say, an electron as a particle with fixed spin 1/2 and to employ as the only semiclassical parameter. In the ˆ 0 in (1.1) then dominates limit → 0 the energy scale of the translational part H ˆ is that of the spin-orbit coupling term, since in the latter the spin operator S proportional to . Although it might appear that thus the spin has evaded the leading order semiclassical description, it does in fact contribute in an essential way through a classical spin precession driven by the orbital motion, see [BK99a, BG00, BGK01, BG04]. E.g., in classically chaotic systems this type of spin motion is responsible for the quantum eigenvalue spectrum to possess correlations of the Gaussian symplectic ensemble of random matrix theory [BK99b]. Moreover, in this semiclassical framework the exact spectrum of the relativistic hydrogen atom is recovered [Kep03], and anomalous magneto-oscillations in semiconductor devices can be described to a good approximation [KW02]. A second option for the semiclassical limit is to keep the “classical spin” s at a fixed value S, thus performing → 0 and s → ∞ simultaneously. In this scenario the energy scale of spin-orbit interactions remains comparable to that of the purely translational part, leading to a classical spin-orbit Hamiltonian. Therefore, coupled Hamiltonian dynamics emerge with classical particle trajectories influenced by the spin. This scenario enables an immediate classical description of the Stern-Gerlach experiment, and generally corresponds to a “strong” spin-orbit coupling. In this paper we examine the propagation of coherent states under the influence of spin-orbit interactions in both of the above mentioned semiclassical scenarios. In Section 2 we first provide a precise characterization of the quantum Hamiltonians under investigation and then describe the classical dynamics that will result in due course. Section 3 is devoted to outlining the construction of coherent states for both translational and spin degrees of freedom, along with their basic properties. Our principal results are developed in Section 4. For both semiclassical scenarios separately we extend the approach devised previously [Hel75, Lit86, CR97] in systems without spin in that we first construct suitable approximate Hamiltonians that propagate coherent states exactly along classical trajectories. We then prove that, measured in Hilbert space norm, the full quantum √ dynamics differs from a classically propagated coherent state by an error of size as long as finite times are taken into account. The vanishing of this difference up to some, semiclassically infinite, Ehrenfest time is also established. Subsequently we improve the semiclassical error to O(N/2 ) for arbitrary N ∈ N by replacing the classically propagated coherent states with a suitable sum of squeezed states. Again such a procedure is possible up to the Ehrenfest time. We conclude in Section 5 with discussing some implications of our main results.
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2 Background It is our aim to investigate the time evolution of an initial coherent state in both translational and spin degrees of freedom generated by a general spin-orbit quantum Hamiltonian with an emphasis on a semiclassical description. This is the reason why we represent quantum observables as matrix-valued semiclassical pseudodifferential operators within the framework of Weyl calculus, see [Rob87, DS99] ˆ under consideration are defined on the for details. The quantum Hamiltonians H domain S (Rd ) ⊗ C2s+1 in the Hilbert space L2 (Rd ) ⊗ C2s+1 and are of the form x + y W i 1 ˆ ξ·(x−y) H Hψ (x) = op [H]ψ (x) = , ξ ψ(y) dy dξ . e (2π)d 2 T∗ Rd (2.1) Here T∗ Rd ∼ = Rd × Rd denotes the cotangent bundle over the Euclidean configuration space Rd , i.e., the phase space of the translational degrees of freedom. The spin s = 1/2, 1, 3/2, . . . is described by the matrix degrees of freedom of the Weyl symbol H and will later be represented on its phase space S2 . Spin-orbit Hamiltonians are characterized by symbols of the form H(x, ξ) = H0 (x, ξ) + C(x, ξ) · dπs (σ/2) ,
(2.2)
where H0 and the components Ck , k = 1, 2, 3, of C are real valued and smooth functions on T∗ Rd which for all multi-indices α and β satisfy the growth estimate Mx /2 M /2 1 + |ξ|2 ξ |∂xα ∂ξβ F (x, ξ)| ≤ Kαβ 1 + |x|2
(2.3)
with suitable constants Kαβ > 0 and M = (Mx , Mξ ) ∈ R2 . The spin-orbit coupling term in (2.2) contains the spin operators Sˆk := dπs (σk /2) ,
k = 1, 2, 3 ,
obeying the well-known commutation relations [Sˆk , Sˆl ] = i klm Sˆm . Here 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = 1 0 i 0 0 −1
(2.4)
are the Pauli matrices, considered as elements of the real Lie algebra su(2), and dπs denotes the (2s + 1)-dimensional representation of su(2) derived from the corresponding unitary irreducible representation πs of the Lie group SU(2) according d to dπs (X) = i dλ πs e−iλX λ=0 . ˆ ˆ (t) = e− i Ht The time evolution U generated by the quantum Hamiltonian ˆ will be unitary provided that H itself is essentially self-adjoint on the domain C0∞ (Rd ) ⊗ C2s+1 . In the present framework this is guaranteed, for sufficiently small , once the symbol H is such that H + i is elliptic, i.e., if H(x, ξ) + i −1 ≤ c 1 + |x|2 −Mx /2 1 + |ξ|2 −Mξ /2 (2.5)
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holds for all (x, ξ) ∈ T∗ Rd with some constant c > 0 and M as in (2.3); here · is an arbitrary matrix norm. Details can be found in [Rob87, DS99]. In the following ˆ we assume this condition to hold and do not notationally distinguish between H and its self-adjoint extension. In the semiclassical limit we will have to deal with two types of classical spin-orbit dynamics. In the first case only the translational degrees of freedom evolve under a Hamiltonian flow. This is defined on the phasespace T∗ Rd and is generated by the classical Hamiltonian H0 . Thus Φt0 (q, p) = q(t), p(t) satisfies Hamilton’s equations of motion, ˙ = −∂x H0 q(t), p(t) , q(t) ˙ = ∂ξ H0 q(t), p(t) and p(t) with initial conditions q(0), p(0) = (q, p). This flow then drives a classical spin through the equations of motion ˙ n(t) = C q(t), p(t) × n(t) on the sphere S2 with initial condition n(0) = n. Here n ∈ R3 with |n| = 1 is considered as a point on S2 . The curve n(t) therefore describes the Thomas pre- cession of a normalized classical spin vector on S2 along the trajectory q(t), p(t) in T∗ Rd . The combined dynamics (q, p, n) → Φt0 (q, p), n(t; q, p, n) (2.6) yield a flow on the product phase space T∗ Rd × S2 , which is a symplectic manifold whose symplectic form is composed of the natural symplectic forms of its factors. This flow has the form of a skew product, see [CFS82] for details, and thus is not Hamiltonian; however, it leaves the natural volume measure derived from the symplectic form invariant. The second flow relevant for our subsequent discussion includes a classical spin dynamics coupled to the motion of the translational part in a Hamiltonian manner and is also defined on the product phase space T∗ Rd × S2 . These dynamics are generated by the classical spin-orbit Hamiltonian Hso (x, ξ, n) := H0 (x, ξ) + Sn · C(x, ξ) ,
(2.7)
where the constant S > 0 measures the length of the classical spin vector s := Sn. The Hamiltonian flow Φtso (q, p, n) = q(t), p(t), n(t) is therefore determined by the equations of motion q(t) ˙ = ∂ξ Hso q(t), p(t), n(t) , (2.8) p(t) ˙ = −∂x Hso q(t), p(t), n(t) , ˙ n(t) = C q(t), p(t) × n(t) . The Hamiltonian coupling of the degrees of freedom prescribed by these equations imply that in contrast to the previous case the translational dynamics are affected by the spin.
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Apart from the associated classical flow in semiclassical approximations of quantum dynamics also the linear stability of the flow plays a role. Quantitatively this can be measured in terms of the Lyapunov exponents, see the appendix for a discussion. They express the rate of phase space expansion or contraction, respectively, induced by the flow in different tangent directions. Moreover, the differential of the flow is a symplectic map on the tangent bundle of phase space. Its metaplectic representation is an essential ingredient in the semiclassical propagation of coherent states.
3 Coherent states Within the setting outlined in the preceding section we wish to describe the time evolution of an initial coherent state semiclassically. The starting point therefore is the Schr¨ odinger equation ∂ψ ˆ (t, x) = Hψ(t, x) with ψ(0, x) = ϕB i (q,p) ⊗ φn (x) , ∂t whose initial condition is the product of a coherent state ϕB (q,p) of the translational degrees of freedom and a spin-coherent state φn . The principal question we then address is to what extent the quantum mechanical time evolution can be approximated by some classical dynamics, i.e., we want to estimate the difference − i Ht e ˆ ϕB ⊗ φn − eiα(t) ϕB(t) (3.1) (q,p) (q(t),p(t)) ⊗ φn(t) in terms of , where q(t), p(t), n(t) is an appropriate classical trajectory and eiα(t) is a suitable phase factor. 2 d 2s+1 For both types of coherent states, ϕB , we use (q,p) ∈ L (R ) and φn ∈ C Perelomov’s construction [Per86] that applies to a general Lie group G with unitary irreducible representation π on a Hilbert space H: Fix a non-zero vector Ψ0 ∈ H and consider Ψg := π(g)Ψ0 for every g ∈ G. Hence the vectors Ψg and Ψh define the same quantum state, i.e., Ψh = eiα Ψg , if and only if g −1 h lies in the stability subgroup H ⊂ G of the vector Ψ0 , H := {g ∈ G; π(g)Ψ0 = eiα(g) Ψ0 } .
(3.2)
The quantum states generated by the vectors Ψg , g ∈ G, can thus be labeled by the points η of the coset space G/H. A section g(η) in the bundle G → G/H then determines a choice of vectors Φη := Ψg(η) = π(g(η))Ψ0 ,
for η ∈ G/H ,
(3.3)
representing these states. The vectors Φη are called coherent state vectors for (G, π, H). The two types of coherent states that play a role in the present setting can be constructed according to this general scheme by choosing the Heisenberg group G = H(Rd ) for the translational part and the group G = SU(2) for the spin part. We now describe the two situations that emerge from this procedure separately.
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Coherent states for the Heisenberg group
The Heisenberg group H(Rd ) is a non-compact (2d + 1)-dimensional Lie group that consists of the elements (q, p, λ) with (q, p) ∈ T∗ Rd and λ ∈ R. The group multiplication is given by (q, p, λ) (q , p , λ ) = q + q , p + p , λ + λ + 12 (pq − qp ) . According to the Stone-von Neumann Theorem any unitary irreducible reprei sentation π of H(Rd ) that fulfills π(0, 0, λ) = e λ is unitarily equivalent to the 2 d Schr¨ odinger representation ρ on L (R ), i ˆ ˆ i i 1 ρ (q, p, λ)ψ (x) = e λ e (pQ−qP ) ψ (x) = e (λ+p(x− 2 q)) ψ(x − q) . ˆ k and Pˆk , k = 1, . . . , d, are the standard self-adjoint position and momenHere Q tum operators defined on suitable domains in L2 (Rd ). In order to construct coherent states for the Heisenberg group we therefore consider the Schr¨ odinger representation ρ on L2 (Rd ). One immediately sees that given any non-zero vector Ψ0 ∈ L2 (Rd ) its stability subgroup is H = {(0, 0, λ); λ ∈ R}. Thus coherent states can be labeled by points (q, p) ∈ G/H, i.e., by points in the phase space T∗ Rd . This labeling can be achieved in terms of the section g(q, p) := (q, p, − 21 qp) in G → G/H. One usually prefers a reference vector Ψ0 ∈ L2 (Rd ) that is normalized, rapidly decreasing, and satisfies ˆ 0 = 0 Ψ0 , QΨ
and
Ψ0 , Pˆ Ψ0 = 0 ,
so that any reasonable lift of this vector to the phase space T∗ Rd is concentrated at (0, 0) ∈ T∗ Rd . A convenient choice with these properties is ψ0B (x) :=
i 1 (det Im B)1/4 e 2 xBx , (π)d/4
where B is some complex symmetric d × d matrix with positive-definite imaginary part. The coherent states (3.3) that follow from the above definitions now read B 1 ϕB (q,p) (x) = ρ (q, p, − 2 qp)ψ0 (x) (3.4) i 1 1 = (det Im B)1/4 e (p(x−q)+ 2 (x−q)B(x−q)) . d/4 (π) Note that these coherent states differ slightly from more conventional choices for which B = i1d and the section g˜(q, p) = (q, p, 0) are used, leading to a different phase convention. Despite the fact that after allowing for more general matrices B the coherent states loose the minimum uncertainty property, this generalization will prove useful since the action of the metaplectic representation on them can be conveniently expressed in terms of B, see also [Sch01]. The alternative phase convention is of less consequence but simplifies the notation.
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Although from the above construction it is obvious that a coherent state ϕB (q,p) is concentrated in some neighborhood of the point (q, p) in phase space it is instructive to calculate explicit phase-space lifts. E.g., its Wigner transform is given by i B 1 1 ](x, ξ) = e− ξy ϕB W [ϕB (q,p) (q,p) (x − 2 y) ϕ(q,p) (x + 2 y) dy d R (3.5) 1 = 2d e− ((x,ξ)−(q,p))GB ((x,ξ)−(q,p)) , where GB is the positive-definite symmetric 2d × 2d matrix Im B + Re B(Im B)−1 Re B − Re B(Im B)−1 GB := . −(Im B)−1 Re B (Im B)−1 This representation reveals a concentration of the coherent state in the vicinity of the phase-space point (q, p). Moreover, since the sum of position and momentum uncertainties reads 1 tr GB , (x − q)2 + (ξ − p)2 W [ϕB (3.6) (q,p) ](x, ξ) dx dξ = (2π)d 2 T∗ Rd the spreading of the coherent state in phase space can be measured in terms of GB .
3.2
Spin-coherent states
In quantum mechanics the spin of a particle is implemented through the (2s + 1)dimensional irreducible representation πs of the compact Lie group SU(2), where s = 1/2, 1, 3/2, . . . denotes the spin quantum number. Within Perelomov’s framework spin-coherent states are hence constructed from (SU(2), πs , C2s+1 ). The reference vector Ψ0 ∈ C2s+1 can be chosen such that the coherent states possess the minimum uncertainty property; this is achieved with Ψ0 being a maximal weight vector for the irreducible representation dπs of the Lie algebra su(2). The real Lie algebra su(2) consists of the hermitian and traceless 2×2 matrices X, such that e−iX ∈ SU(2). A convenient basis of su(2) is formed by the Pauli matrices (2.4). We also consider the complexified Lie algebra su(2)C := su(2) ⊗ C with basis given by X± :=
1 (σ1 ± iσ2 ) , 2
X3 :=
1 σ3 , 2
and commutation relations [X3 , X± ] = ±X± ,
[X+ , X− ] = 2X3 .
The vector X3 spans a Cartan subalgebra, which exponentiates to a maximal torus T U(1) in SU(2), and X± ∈ su(2)C span the root spaces g± ⊂ su(2)C . Their representations dπs (X± ) are raising and lowering operators, respectively.
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More precisely, the representation space C2s+1 decomposes into a direct sum of the one-dimensional eigenspaces of dπs (X3 ) (weight spaces) Vm = {φ ∈ C2s+1 ; dπs (X3 )φ = mφ}, where m = −s, −s + 1, . . . , s. The raising and lowering operators dπs (X± ) map the weight spaces into one another, dπs (X± )Vm = Vm±1 for m = ±s. The weights m = ±s are called maximal and minimal weights, respectively. The corresponding weight vectors are annihilated by the raising or lowering operator. In the usual angular momentum notation a normalized weight vector is denoted as |s, m . For a given representation πs of SU(2) we choose a maximal weight vector |s, s as the reference vector Ψ0 . According to (3.2) the stability group of this vector is H = {g = e−iλσ3 ; λ ∈ [0, 2π)} ∼ = U(1) , which can be identified with a maximal torus T . Thus coherent states are labeled by points in the coset space G/H ∼ = SU(2)/U(1) ∼ = S2 . As in the case of the Heisenberg group this manifold is naturally symplectic and can be viewed as the corresponding classical phase space. The definition of coherent states finally requires a section in G → G/H, i.e., in the Hopf bundle SU(2) → S2 . This principal U(1)-bundle, however, is non-trivial so that no smooth global section exists. We therefore here give local constructions that, nevertheless, allow for suitable interpretations in terms of global objects. We parameterise points on S2 by n ∈ R3 with |n| = 1 and use spherical coordinates, n(θ, ϕ) = (sin θ cos ϕ, sin θ sin ϕ, cos θ) with θ ∈ [0, π) and ϕ ∈ [0, 2π). Introducing eϕ := (− sin ϕ, cos ϕ, 0) our choice of a local section reads (see also [Per86]) i cos θ2 − sin θ2 e−iϕ gn = e− 2 θeϕ ·σ = . sin θ2 eiϕ cos θ2 Under the double covering map R : SU(2) → SO(3) that is defined through (R(g)x) · σ = gx · σg −1 , the matrix gn(θ,ϕ) corresponds to the rotation R(gn(θ,ϕ)) about the axis eϕ with angle −θ, such that R(gn )e3 = n, where e3 = (0, 0, 1) represents the north pole on S2 . With these choices spin-coherent states are the normalized vectors (3.7) φn := πs (gn )|s, s . These states are conveniently represented on the phase space S2 through the Husimi transform (see [Per86]), s 1+m·n h[φn ](m) := | φm , φn | = , 2 which clearly indicates a concentration, in the semiclassical limit s → ∞, of φn at the point n ∈ S2 .
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Our next aim is to investigate the relation between the propagation of a spincoherent state (3.7) generated by a (time-dependent, linear) spin-Hamiltonian ˆ spin = C(t) · S ˆ H
(3.8)
ˆ denotes defined on C2s+1 , and a suitable classical time evolution n(t) on S2 . Here S the vector of spin operators Sˆk = dπs (σk /2). The dynamics of a coherent state φn follows from the equation i
∂φ ˆ spin φ(t) (t) = H ∂t
with
φ(0) = φn .
(3.9)
A solution of this problem can be related to the curve g(t), t ∈ R, in SU(2) determined by g(t) ˙ + through
i 2
C(t) · σ g(t) = 0
with
g(0) = idSU(2)
φ(t) = πs (g(t))φ(0) = πs g(t)gn |s, s .
(3.10) (3.11)
An associated classical time evolution then arises from the adjoint action of g(t) on n · σ ∈ su(2) via n(t) · σ := g(t)n · σg(t)−1 = (R(g(t))n) · σ. This implies ˙ n(t) = C(t) × n(t)
with
n(0) = n .
(3.12)
The corresponding coherent state vector φn(t) differs from the quantum time evolution φ(t) of φn only by a phase; both vectors therefore describe the same quantum state. Since this phase is required for later purposes, we now determine it explicitly. To this end we notice that n(t) can on the one hand be represented as n(t) = R g(t) n = R g(t)gn e3 , and on the other hand as
n(t) = R gn(t) e3 .
−1 Thus, under the double covering map, gn(t) g(t)gn ∈ SU(2) is associated with a rotation about e3 with some angle (t), such that i
−1 g(t)gn = e 2 (t)σ3 ∈ T . gn(t)
From (3.11) it now follows that i φ(t) = πs g(t)gn |s, s = πs gn(t) πs e 2 (t)σ3 |s, s = eis (t) φn(t) ,
(3.13)
thus confirming the claimed relation between the quantum and ‘classical’ propagation of the spin-coherent state φn . Due to the explicit dependence of the phase
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on s it suffices to calculate the angle (t) for s = 12 . For this one notices that in polar coordinates
i θ(t) 2 (t) i cos( ) e (t) 1 1 2 φ(t) = e 2 gn(t) | 2 , 2 = . (3.14) i(ϕ(t)+ 12 (t)) sin( θ(t) 2 )e In a standard calculation (see, e.g., [BK99a]) (t) can now be determined by using (3.14) in equation (3.9), leading to t C(t ) · n(t ) + 1 − cos θ(t ) ϕ(t (t) = − ˙ ) dt . (3.15) 0
If one introduces a classical spin vector s := Sn, with some S > 0, one can relate the angle (t) to Hamilton’s principal function of the spin. The observation that ˙ Lspin (t) = −C(t) · s(t) − S 1 − cos θ(t) ϕ(t) is the Lagrangian of the classical spin motion implies S(t) to be the spin-action Rspin (t).
4 Time evolution of coherent states In this section we discuss the time evolution of coherent states in two different semiclassical limits. In the first scenario we consider → 0 while the spin quantum number s is fixed. This will imply that primarily the translational degrees of freedom become semiclassical. The spin-orbit interaction therefore occurs on the level of the subprincipal symbol of the Hamiltonian (2.2), enforcing the skewproduct structure (2.6) of the resulting classical dynamics with the translational motion driving the spin. In the second scenario we fix the product S := s and hence consider the combined limits → 0 and s → ∞. Thus both types of degrees of freedom are treated semiclassically on equal footing. This results in a classical spin-orbit coupling with the Hamiltonian dynamics (2.8) generated by the function (2.7). We begin with the first scenario which is close to the time evolution of coherent states without spin degrees of freedom.
4.1
Semiclassics with fixed spin
In the present scenario is the only semiclassical parameter so that we consider the quantum Hamiltonian (2.1) as a Weyl operator with matrix-valued symbol (2.2) that has a scalar principal part; the subprincipal symbol then contains the spin-orbit coupling. This setting ensures that the propagation of coherent states is closely analogous to the case without spin, compare [CR97]. Guided by this analogy we first construct an approximate Hamiltonian that propagates coherent states exactly. Regarding the translational part we exploit
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the fact that the time evolution generated by a quadratic Hamiltonian preserves the form ϕB (q,p) given in (3.4) of a coherent state for the Heisenberg group. The spin part of the coherent state shall be propagated by a Hamiltonian of the form (3.8) and can hence be calculated explicitly. Using the convenient notation w := the Taylor expansion of the symbol (2.2) about (x, ξ) ∈ T∗ Rd , we now consider some smooth curve z(t) = q(t), p(t) in phase space. The Weyl quantization of the leading terms in the Taylor expansion (of different order in the principal and in the subprincipal symbol), 2 ν 1 (ν) H z(t) w − z(t) + C z(t) ·dπs (σ/2) , HQ (t, w) := ν! 0
(4.1)
|ν|=0
ˆ ˆ Q (t) that is quadratic in Q ˆ and Pˆ and linear in S. yields a quantum Hamiltonian H (ν) ν Here H0 (w) stands for the derivative ∂w H0 (w) of order |ν| in the 2d components of w = (x, ξ). The time evolution ψQ (t) ∈ L2 (Rd ) ⊗ C2s+1 of a coherent state ϕB (q,p) ⊗ φn generated by the approximate Hamiltonian, i
∂ψQ ˆ Q (t)ψQ (t) (t) = H ∂t
ψQ (0) = ϕB (q,p) ⊗ φn ,
with
(4.2)
can be expressed in terms of a coherent state: Proposition 4.1. The solution of the quadratic Schr¨ odinger equation (4.2) is a time-dependent coherent state with an additional phase,
ψQ (t) = e
R0 (t) +s (t)+ π 2 σ(t)
i
B(t)
ϕ(q(t),p(t)) ⊗ φn(t) .
(4.3)
Here q(t), p(t) = Φt0 (q, p) is the solution of Hamilton’s equations of motion generated by the principal symbol H0 , q(t) ˙ = ∂ξ H0 q(t), p(t) , p(t) ˙ = −∂x H0 q(t), p(t) , (4.4) with initial condition q(0), p(0) = (q, p) and principal function R0 (t) =
t
p(t )q(t ˙ ) − H0 (q(t ), p(t )) dt .
(4.5)
0
The complex symmetric d × d matrix B(t) is given by −1 B(t) = ∂p p(t)B + ∂q p(t) ∂p q(t)B + ∂q q(t) ,
(4.6)
where the derivatives are taken with respect to the initial conditions; it also gives rise to the Maslov phase σ(t). Moreover, n(t) of the spin precession is a solution equation (3.12) in which C(t) stands for C q(t), p(t) from (4.1); (t) then is the associated angle (3.15).
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Proof. For the proof we adapt the method of [Sch01] to the present situation and therefore introduce the ansatz i
i
1
ψQ (t, x) = (π)−d/4 γ(t) e θ(t) e (p(t)(x−q(t))+ 2 (x−q(t))B(t)(x−q(t))) φn(t) in equation (4.2). To deal with the spin contribution to the left-hand side we use the fact that according to (3.9) and (3.13) i
∂ is (t) e φn(t) = C q(t), p(t) · dπs (σ/2) eis (t) φn(t) , ∂t
(4.7)
if and only if n(t) solves (3.12). It hence remains to consider (see [Sch01]) θ(t) i 1 ∂ i (π)−d/4 γ(t) ei( −s (t)) e (p(t)(x−q(t))+ 2 (x−q(t))B(t)(x−q(t))) eis (t) φn(t)
∂t 1 = H0 + H0,x (x − q(t)) + H0,ξ · B(t)(x − q(t)) + (x − q(t)) · H0,xx (x − q(t)) 2 1 1 + (x − q(t)) · H0,ξx B(t)(x − q(t)) + (x − q(t)) · B(t)H0,ξx (x − q(t)) 2 2 1 + (x − q(t)) · B(t)H0,ξξ B(t)(x − q(t)) + tr H0,ξx + H0,ξξ B(t) ψQ (t, x) . 2 2i (4.8) Here the abbreviations H0,x = ∂x H0 and H0,ξx = ∂ξ ∂x H0 , etc. have been employed. These expressions are to be evaluated at z(t). Comparing coefficients of powers of and of x − q(t) in (4.8) then yields the conditions
θ˙ = qp ˙ − H0 , −p˙ + B q˙ = H0,x + BH0,ξ , ˙ −B = H0,xx + H0,ξx B + BH0,ξx + BH0,ξξ B, 1 γ˙ = − tr H0,ξx + H0,ξξ B + is˙ . γ 2
(4.9)
With the identification R0 = θ the first and the second equation immediately imply (4.5) and (4.4), respectively. The other two equations involve the time evolution B(t) of the complex symmetric d × d matrix B with positive-definite imaginary part; they determine the action of the metaplectic group on the vector ϕB (q,p) . At this stage we recall that the symplectic group Sp(d, R) acts on the Siegel upper half-space (see [Fol89]) Σd := {Z ∈ Md (C); Z T = Z, Im Z > 0} via S[Z] = (S11 Z + S12 )(S21 Z + S22 )−1 ,
where
S=
S11 S21
S12 S22
∈ Sp(d, R) .
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In the present context the differential of the Hamiltonian flow Φt0 generated by the classical Hamiltonian H0 is symplectic, S0,z (t) := DΦt0 (z) ∈ Sp(d, R), and hence can act on the initial value B ∈ Σd . Indeed, B(t) = S0,z (t)[B] ∈ Σd
(4.10)
yields the solution of the third equation in (4.9) and implies (4.6). The fourth equation requires the introduction of the Maslov multiplier m(S, Z) := det(S21 Z + −1/2 S22 ) for S ∈ Sp(d, R) and Z ∈ Σd . This allows us to define the Maslov phase π σ(t) through ei 2 σ(t) = m(O(t), i1), where O(t) is an orthogonal symplectic matrix that is uniquely associated with S0,z (t). One can then show (cf. [Sch01]) that 1/4 i π σ(t)+is (t) e2 . γ(t) = det Im B(t) We remark that the state (4.3) is closely analogous to the respective solution without spin-orbit coupling. It differs from the latter only by the factor eis (t) φn(t) . This observation not only means that quantum mechanically the translational part and the spin part are not entangled, but also on the classical level the translational dynamics are independent of the spin precession n(t). The combination of classical translational and spin motion rather has the structure of a skew product (2.6), indicating that only the spin dynamics depends on the translational part, and not vice versa. Our aim now is to compare the time evolution generated by the original ˆ with the one generated by the approximate Hamiltonian quantum Hamiltonian H ˆ HQ (t). For this we will follow the method devised in [CR97] for the case without spin. The presence of spin requires some modifications that, however, are modest when the spin quantum number s is fixed. But for the clarity of the presentation, and to prepare for the more involved situation to be dealt with in the second semiclassical scenario, we will now present the argument in some detail. ˆ generates a unitary and strongly As stated in Section 2 the Hamiltonian H ˆ continuous one-parameter group U(t, t0 ), if its symbol satisfies the ellipticity condition (2.5). When considering the limit → 0 and keeping s fixed this requirement need only be imposed on the principal symbol, i.e., we demand Mx /2 M /2 1 + |ξ|2 ξ . |H0 (x, ξ) + i| ≥ c 1 + |x|2
(4.11)
ˆ Q (t). Using ˆQ (t, t0 ) be the corresponding unitary group generated by H Let now U Duhamel’s principle we may then express the difference between these unitary operators as ˆQ (t, t0 ) = 1 ˆ (t, t0 ) − U U i
t
t0
ˆ (t, t ) H ˆ −H ˆ Q (t ) U ˆQ (t , t0 ) dt . U
(4.12)
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Since we are interested in the difference (3.1), we have to consider the action of (4.12) on the initial state ϕB (q,p) ⊗ φn with t0 = 0. This requires an estimate of ˆ −H ˆ Q (t ))ψQ (t ) , (H
(4.13)
where ψQ (t ) is the time-dependent coherent state (4.3). One can achieve this with the help of the following lemma, which is an immediate extension of a result given in [CR97]. Lemma 4.2. Let f, g ∈ C ∞ (T∗ Rd ) be symbols that satisfy the estimate (2.3) with M = 0 and let F : T∗ Rd → T∗ Rd be a linear map with Hilbert-Schmidt norm F HS . Fix α, β ∈ N2d with k := |α| = |β| + 2 > 2 and introduce the symbol A(w) := (F w)α f (F w) + (F w)β g(F w) . Then for any real number κ > 0 there exist C > 0 and N ∈ N such that
opW [A]ψ ≤ Ck/2
F kHS
sup |γ|≤k+N
γ |∂w f (w)| + F k−2 HS
sup |γ|≤k−2+N
γ |∂w g(w)|
√ holds for any function ψ (x) = −d/4 ψ x/ with ψ ∈ S (Rd ) and 0 < + √ F HS < κ. We intend to apply this lemma to the difference (4.13), with f corresponding to the Taylor remainder of H0 of order three and g to the Taylor remainder of C · dπs (σ/2) of order one. But first we replace (4.13) by ˆQ (t , 0)∗ H ˆ −H ˆ Q (t ) U ˆQ (t , 0)ψQ (0) U (4.14) and invoke an appropriate Egorov theorem. Since the Hamiltonian generating ˆQ (t, 0) has a symbol that is composed of a scalar and quadratic principal part as U well as a matrix-valued subprincipal part, one can combine the techniques used in [BG00] and [Sch01]. This shows that ˆ (t) := U ˆQ (0, t) H ˆ −H ˆ Q (t) U ˆQ (t, 0) W (4.15) is a Weyl operator with symbol −1 W (t, w) = d∗ z(t) H − HQ (t) z − S0,z (t)(w − z(t)) d z(t) . (4.16) Here d z(t) is the representation πs g(t) of the solution to equation (3.10) in which C(t) stands for C z(t) . Thus d z(t) φn = eis (t) φn(t) describes the transport of a spin-coherent state along the trajectory z(t). Since the principal part of the symbol H − HQ (t) is scalar it is not affected by the
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conjugation with d z(t) . In the subprincipal term this conjugation rotates the ˆ Therefore, the spin part of the Egorov ˆ = dπs (σ/2) to R g(t) S. spin operator S relation (4.16) does not contribute to an estimate of (4.14) in an essential way. If one now localizes the symbol (4.16) in w with some smooth function that is compactly supported around z(t), leading to an error of size O(∞ ) when one ˆ to a coherent state located at z(t), one can proceed to use Lemma 4.2 applies W as in [CR97]. This shows that there exists a constant K > 0 such that ˆ −H ˆ Q (t))ψQ (t) ≤ K3/2 θ(t)3 δ(t)m , (H
(4.17)
where
and δ(t) := sup 1 + |z(t )| θ(t) := max 1, sup S0,z (t )HS t ∈[0,t]
(4.18)
t ∈[0,t]
depend on the classical trajectory z(t) = (q(t), p(t)). The constant m = max {Mx , Mξ } is related to M = (Mx , Mξ ) appearing in (2.3). We then obtain: Theorem 4.3. Let the conditions imposed on the Hamiltonian in Section 2 and the ellipticity condition (4.11) hold. Then the coherent state ψQ (t) defined in (4.3) ˆ (t, 0) ϕB ⊗ φn in the following sense, semiclassically approximates ψ(t) = U (q,p) √ ψ(t) − ψQ (t) ≤ K t θ(t)3 .
(4.19)
The right-hand side vanishes in the combined limits → 0 and t → ∞ as long as t Tz (). The time scale Tz () depends on the linear stability of the trajectory z(t). If the latter possesses a positive and finite maximal Lyapunov exponent 1 λmax (z), one has Tz () = 6λmax (z) | log |. In the case of a trajectory on a (non-
degenerate) KAM-torus this time scale is Tz () = C −1/8 . Proof. Conservation of energy, H0 z(t) = E, together with the ellipticity condition (4.11) implies that δ(t) is bounded from above by some constant depending on E. Thus the estimate (4.17) immediately yields (4.19) when used in (4.12). If z(t) is a trajectory with a positive, but finite, maximal Lyapunov exponent the dominant behavior as t → ∞ comes from the term θ(t)3 . This is due to the relation 1 λmax (z) = lim sup log S0,z (t)HS , t→∞ t
1 see (A.1), which readily implies Tz () = 6λmax (z) | log |. In the appendix we also discuss sufficient conditions under which finite maximal Lyapunov exponents occur. If z(t) is a trajectory on a KAM-torus one can introduce local action-angle variables (I, φ) in a neighborhood of that torus such that in these canonical coordinates the flow reads I(t) = I and φ(t) = φ + ω(I)t, see [Laz93]. One therefore finds S0,z (t)2HS = d + f (I) t2 ,
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such that θ(t) ∼ Kt as t → ∞, which finally yields Tz () = C −1/8 . In the degenerate case, where f (I) = 0, this changes to Tz () = C −1/2 . In a next step we want to improve the semiclassical error in (4.19) to an arbitrary (half-integer) power of . This requires higher order approximations that may be achieved as in [CR97] by iterating Duhamel’s principle (4.12), resulting in the Dyson expansion ˆ (t, 0) − U ˆQ (t, 0) = U
N −1
−j
(i)
t
t
ˆQ (t, 0) W ˆ (tj ) . . . W ˆ (t1 ) dtj . . . dt1 U
... 0
j=1
tj−1
+ RN (t; ) (4.20) with remainder term t −N ... RN (t; ) = (i) 0
t
ˆ tN ) U ˆQ (tN , 0) W ˆ (tN ) . . . W ˆ (t1 ) dtN . . . dt1 . U(t,
tN −1
In order to estimate the contribution of the remainder when (4.20) is applied to the initial coherent state ψ(0) = ϕB (q,p) ⊗ φn we use the argument leading to (4.17) repeatedly. This yields the bound RN (t; )ψ(0) ≤ KN N/2 tN θ(t)3N δ(t)mN .
(4.21)
ˆ −H ˆ Q (tk ) appearing in the sum We then replace the symbol of each difference H in (4.20) by its Taylor expansion, nk ν 1 (ν) H0 z(tk ) w − z(tk ) ν!
|ν|=3
+
n k −2 |ν|=1
ν 1 w − z(tk ) C (ν) z(tk ) · dπs (σ/2) + rk (tk , w) . (4.22) ν!
The integers nk are chosen sufficiently large such that, after quantization, the contribution of the remainder rk to an application of (4.20) to ψ(0) can be absorbed in the error estimate (4.21). Similar to the case without spin treated in [CR97] the quantization of the main terms in (4.22) produces matrix-valued differential operators pˆkj (t) = opW [pkj (t)] with time-dependent coefficients acting on the coherent state ϕB (q,p) ⊗ φn . The symbols pkj (t)(x, ξ) are polynomials in (x, ξ) of degree ≤ k. Lemma 4.2 finally leads to the following result: ˆ with symbol (2.2) satisfies Theorem 4.4. Suppose that the quantum Hamiltonian H the conditions specified in Section 2 and the ellipticity condition (4.11). Then for t > 0 and any N ∈ N there exists a state ψN (t) ∈ L2 (Rd ) ⊗ C2s+1 , localized at
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ˆ (t, 0) ϕB ⊗ q(t), p(t), n(t) , that approximates the full time evolution ψ(t) = U (q,p) φn of a coherent state up to an error of order N/2 . More precisely, ψ(t) − ψN (t) ≤ CN
N −1 j=1
t
j
√ ( θ(t))2j+N .
The right-hand side vanishes in the combined limits → 0 and t → ∞ as long as t Tz (), where Tz () denotes the same time scale as in Theorem 4.3. Furthermore, ψN (t) arises from ϕB (q,p) ⊗ φn through the application of certain (time-dependent) differential operators pˆkj (t) = opW [pkj (t)] of order ≤ k, followed ˆ Q (t), according to by the time evolution generated by H ˆQ (t, 0) pˆkj (t) ψ(0) . ψN (t) = ψQ (t) + U (k,j)∈∆N
Here we have defined ∆N := {(k, j) ∈ N × N; 1 ≤ k − 2j ≤ N − 1, k ≥ 3j, 1 ≤ j ≤ N − 1}. We remark that the matrix-valued differential operators pˆkj (t) do not increase the frequency set of a semiclassical distribution such as the initial state ϕB (q,p) ⊗φn . This follows for the translational part from the respective statement without spin [Rob87], whereas the spin part is only acted upon by a matrix producing linear ˆ combinations of φn . Moreover, according to Proposition 4.1, UQ (t, 0) propagates the frequency set along the trajectory q(t), p(t), n(t) so that both ψQ (t) and ψN (t) are semiclassically localized at q(t), p(t), n(t) .
4.2
Semiclassics with s fixed
We now consider the second semiclassical scenario in which both semiclassical parameters, and s, are used. For this purpose we still represent the Hamiltonian ˆ as a matrix-valued semiclassical Weyl operator. That way appears as before, H whereas the second parameter s ∈ N/2 controls the dimension of the space C2s+1 on which the symbol operates as a linear map. As we will see, the parameter s enters relevant estimates through the expression dπs (σ/2). To leading order this will produce factors of s. Our desire to perform systematic semiclassical expansions therefore forces us to keep the combination S := s fixed in the semiclassical limit. This means that from now on we consider → 0 and s → ∞ with s = S. An inspection of Proposition 4.1 and its proof reveals that replacing s by the constant S will shift the spin-action term (t), which before was of subleading semiclassical order, to an additional contribution to the action R0 . This suggest
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that now the translational classical dynamics will be influenced by the spin, requiring a modified quadratic Hamiltonian. Not only that, revisiting the proof of Theorem 4.3 shows that we also have to estimate the application of spin operators to spin-coherent states in terms of s. This requires knowledge of the following: Lemma 4.5. For any X = x·σ/2 ∈ su(2), n ∈ S2 and N ∈ N there exist differential (j) operators Dn of degree 2j on C ∞ (S2 ) ⊗ C2s+1 and constants CN > 0 such that N 1 (j) CN dπs (X)φn − s + 1 1 + 1 x · n φ D (4.23) n ≤ N +1 . n j 2 s s s j=0 The leading order in this asymptotic expansion is determined by the constant (0) Dn = 1, dπs x · σ/2 φn = s x · n φn 1 + O(s−1 ) . (4.24) Proof. We start with expressing a linear map L on the representation space C2s+1 in terms of Berezin’s quantization, L = (2s + 1) P [L](n) Π(n) dn , (4.25) S2
where P [L] denotes the upper (or P -) symbol of L, see, e.g., [Sim80, Per86]. Furthermore, dn is the normalized area measure on S2 and Π(n) stands for the projector onto the one-dimensional subspace in C2s+1 spanned by the coherent state vector φn . In the present context the relevant linear maps are representation operators of Lie-algebra elements X = x · σ/2 ∈ su(2). Their upper symbols are simple, P [dπs (x · σ/2)](n) = (s + 1) x · n , see [Sim80, Per86], so that an application of such an operator to a coherent state reads dπs (x · σ/2)φn = (2s + 1)(s + 1) m · x φm , φn φm dm . (4.26) S2
The coherent states not being defined globally on S2 is irrelevant to this expression since these states have been defined on a set of full measure. An asymptotic expansion of the integral (4.26), as s → ∞, can be achieved with the method of steepest descent. This is a variant of the stationary phase method, with a complex phase function, and is described in detail in [H¨ or90]. The first step consists in identifying the relevant phase factor, which in the present case is given by 1+n·m φn , φm = eisϕn (m) with Im ϕn (m) = − log , (4.27) 2
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where ϕn is independent of s, see [Per86]. Outside of a neighborhood of m = −n the function Im ϕn is finite and non-negative; it has a unique minimum at m = n. The real part of the phase ϕn can be identified as the area of the spherical triangle with edges defined by the north pole, n and m. Hence m = n is the unique, nondegenerate stationary point of the phase. Up to an error of size O(e−s ) one can hence cut out a neighborhood of m = −n from the integral (4.26) and use the representation (4.27) for Im ϕn . The method of steepest descent then implies the (j) existence of differential operators Dn of order 2j on C ∞ (S2 )⊗C2s+1 and constants (0) CN > 0 such that for any N ∈ N the expansion (4.23) holds. The constant Dn fixing the leading order can be identified by choosing x = n, since dπs n · σ/2 φn = sφn . (0)
Comparing with (4.23) therefore yields Dn = 1, which implies (4.24).
When constructing a quadratic Hamiltonian we now have to take into account that an application of a spin operator to a spin-coherent state contributes to the leading semiclassical order, as (4.24) means ˆ n = Sn φn + O(s−1 ) . Sφ ˆ Q (t) = opW [HQ (t)] with We are therefore led to define a quadratic Hamiltonian H matrix-valued Weyl symbol as follows, 2 2 ν ν 1 (ν) 1 H0 z(t) w − z(t) + S n(t) · C (ν) z(t) w − z(t) ν! ν! |ν|=0 |ν|=1 + C z(t) · dπs (σ/2) . (4.28) Like in (4.1) we have introduced a yet to be determined trajectory z(t) = q(t), p(t) in T∗ Rd with initial condition z(0) = z = (q, p), as well as a curve n(t) on ˆ Pˆ ) and linear in S, ˆ S2 with n(0) = n. This Hamiltonian, being quadratic in (Q, propagates an initial coherent state exactly:
HQ (t, w) =
Proposition 4.6. The solution of the quadratic Schr¨ odinger equation i
∂ψQ ˆ Q (t)ψQ (t) (t) = H ∂t
with
ψQ (0) = ϕB (q,p) ⊗ φn
(4.29)
is, up to an additional phase, again a coherent state, Rso (t) π B(t) (4.30) ψQ (t) = ei( + 2 σ(t)) ϕ(q(t),p(t)) ⊗ φn(t) . Here q(t), p(t), n(t) = Φtso (p, q, n) is the solution of Hamilton’s equations of motion (2.8) on T∗ Rd × S2 generated by the classical spin-orbit Hamiltonian
Hso (x, ξ, n) := H0 (x, ξ) + Sn · C(x, ξ) .
(4.31)
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The phase of ψQ (t) is determined by Rso (t) =
t
p(t )q(t ˙ ) − H0 (q(t ), p(t )) dt + S(t) ,
(4.32)
0
which can be viewed as a total spin-orbit principal function, and by the Maslov phase σ(t). The latter derives from the time evolution −1 B(t) = ∂p p(t)B + ∂q p(t) ∂p q(t)B + ∂q q(t)
(4.33)
of the complex symmetric d × d matrix B ∈ Σd . Proof. The proof of this proposition parallels that of Proposition 4.1; however, a few modifications are necessary. One can again consider (4.30) as an ansatz and determine its ingredients by inserting it into (4.29), leading to equations analogous to (4.8). As opposed to (4.9) the fact that now S = s is fixed shifts the term with (t) from the last equation to the first one. Moreover, due to the modified definition of the quadratic Hamiltonian the principal symbol H0 is replaced by Hso in all places but one, yielding θ˙ = qp ˙ − H0 + S ˙ −p˙ + B q˙ = Hso,x + BHso,ξ −B˙ = Hso,xx + Hso,ξx B + BHso,ξx + BHso,ξξ B γ˙ 1 = − tr Hso,ξx + Hso,ξξ B . γ 2
The first two equations fix the translational part of the classical dynamics to be solutions of (2.8) with some n(t) and yield the spin-orbit principalfunction (4.32). In the last two equations Hso , which is evaluated at q(t), p(t), n(t) , can be viewed ˜ so (w, t) = Hso (w, n(t)), for the translational as a time-dependent Hamiltonian, H degrees of freedom, with the time-dependence introduced through n(t). These equations can be solved in the same manner as in the time-independent case, yielding B(t) = Sso,z (t)[B] as in (4.10). Here Sso,z (t) is a solution of d ˜ so Sso,z (t) = J H z(t), t Sso,z (t) dt
(4.34)
with Sso,z (0) = 12d ; it hence yields (4.33). The classical spin motion n(t) so far has remained undetermined. Since the equation for the spin-coherent state is again (4.7), it follows that n(t) must be a solution to the spin part of (2.8).
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In contrast to the previous case the classical dynamics that governs the time evolution of the coherent state ψQ (t) now is Hamiltonian on the product phase space T∗ Rd × S2 , see (2.8). This means that the spin precession is not merely following the translational motion, but there occurs a mutual influence of both dynamics. This effect is caused by the energy scales of the translational and the spin dynamics being comparable in the semiclassical limit, whereas when s is fixed the energy scale of the translational motion dominates. ˆ We now compare the time evolution generated by the full Hamiltonian H ˆ with the approximate dynamics following from the quadratic Hamiltonian HQ (t) whose symbol is given in (4.28). As opposed to the situation discussed previously, see (4.11), when keeping s fixed the ellipticity condition has to be imposed on ˆ see (2.5), which implies the full symbol of H, −Mx /2 −Mξ /2 −1 1 + |ξ|2 ≥ H(x, ξ) + i c 1 + |x|2 −1 H(x, ξ) + i ψ . ≥ ψ Here in the middle · denotes the operator norm on C2s+1 , and on the right-hand side ψ is any non-zero vector in C2s+1 . Choosing ψ = H(x, ξ) + i φn and using (4.24) we then conclude that the spin-orbit Hamiltonian (4.31) is elliptic, in the sense that Mx /2 M /2 1 + |ξ|2 ξ |Hso (x, ξ, n) + i| ≥ C 1 + |x|2 holds for all (x, ξ, n) ∈ T∗ Rd × S2 . Therefore, we can again base our further investigation of the difference between the two quantum dynamics on the Duhamel relation (4.12). This requires to estimate the analogue of (4.13), where in the ˆ −H ˆ Q (t) is the Weyl quantization of the symbol present situation H 2 ν 1 H − HQ (t) (w) = dπs (σ/2) − Sn(t) · C (ν) z(t) w − z(t) ν! (4.35) |ν|=1 [3]
+ H0 (t, w) + C [3] (t, w) · dπs (σ/2) , [3]
in which H0 and C [3] denote Taylor remainders of order three. Introducing an ˆ (t) as in (4.15), the same type of an Egorov theorem as above applies, operator W leading to the symbol −1 (t)(w − z(t)) d z(t) (4.36) W (t, w) = d∗ z(t) H − HQ (t) z − Sso,z ˆ (t). We remark that z(t) being the projection of Φt (z, n) to T∗ Rd here of W so t requires the differential Sso,z (t) of Φso with respect to z. The conjugation with d z(t) hasno effect on the scalar terms in (4.35), whereas it rotates the spin op erator to R g(t) dπs (σ/2). Hence, for the application of (4.36) to a spin-coherent
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state φn we can employ Lemma 4.5. By also converting estimates with respect to s into ones with respect to this yields to leading order R g(t) dπs (σ/2) − Sn(t) φn = S R g(t) n − n(t) φn + O(s−1 ) (4.37) = O() . Moreover, the complete asymptotic series in powers of s−1 provided by Lemma 4.5 results in a full asymptotic expansion of (4.37) in powers of . This observation now enables us to apply Lemma 4.2 in a completely analogous way to that used previously, yielding ˆ −H ˆ Q (t))ψQ (t) ≤ K3/2 θ(t)3 δ(t)m . (H Here the quantities θ(t) and δ(t) are defined as in (4.18), however, now with the differential Sso,z (t), and z(t) as given in Proposition 4.6. The stability of the trajectory z(t) is encoded in the quantity ˜ max (z) = lim sup 1 log Sso,z (t)HS . λ t→∞ t
(4.38)
˜ max (z) a Lyapunov Since z(t) is not the integral curve of a flow, rather than calling λ exponent we refer to it as a stability exponent. This can, however, be bounded by the maximal Lyapunov exponent of the flow-line (z(t), n(t)) in T∗ Rd × S2 , see the appendix. Thus, in close analogy to Theorem 4.3 we finally obtain: Theorem 4.7. Let the conditions imposed on the Hamiltonian in Section 2 hold. Then the coherent state ψQ (t) defined in (4.30) semiclassically approximates ˆ (t, 0) ϕB ⊗ φn in the following sense, ψ(t) = U (q,p) √ ψ(t) − ψQ (t) ≤ K t θ(t)3 , when s is kept fixed. The right-hand side vanishes in the combined limits → 0, s → ∞ and t → ∞ as long as t Tz (). The time scale Tz () depends on the linear stability of the trajectory z(t). If the latter possesses a positive and finite stability 1 ˜ max (z), one has Tz () = | log |. In case z(t) is a projection to exponent λ ˜ 6λ (z) max
T∗ Rd of a trajectory (z(t), n(t)) on a (non-degenerate) KAM-torus in T∗ Rd × S2 this time scale is Tz () = C −1/8 .
As in the previous case an improvement of the semiclassical error can be achieved with the Dyson expansion (4.20). The present case, however, requires an additional estimate of the spin contribution in terms of s. Concerning the error term RN (t; )ψ(0), the translational part is dealt with by a repeated application of the argument leading to Theorem 4.7. For the spin part an inspection of the relations (4.35) and (4.36) reveals the necessity to estimate the successive application of the operators Λ(tk ) := C (ν) z(tk ) · R g(tk ) dπs (σ/2) − Sn(tk )
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to the spin-coherent state φn . Representing these operators in the form (4.25), the result of their l-fold (l ≤ j) application reads l . . . P [Λ(tl )](ml ) . . . P [Λ(t1 )](m1 )× Λ(tl ) . . . Λ(t1 )φn = (2s + 1) S2
S2
× Π(ml ) . . . Π(m1 )φn dml . . . dm1 , (4.39) with the upper symbols P [Λ(tk )](mk ) = C (ν) z(tk ) · S R(g(tk ) mk − n(tk ) + R g(tk ) mk . (4.40) Starting with ml , the integral (4.39) can be successively evaluated with the method of steepest descent similar to the proof of Lemma 4.5. The relation Π(ml ) . . . Π(m1 )φn = φml , φml−1 · · · φm1 , φn φml then shows that the critical points of the phase are given by ml = ml−1 = · · · = m1 = n. At these points, however, the upper symbols P [Λ(tk )](mk ) are of order , compare (4.40). The application of the method of steepest descent therefore yields in leading order a contribution O(l ) = O(s−l ). Derivatives of total order n contribute terms of the order O(s−n l−n ) = O(s−l ), if n ≤ l, and of the order O(s−n ) otherwise. Altogether there hence exist differential operators D(κ) of order ≤ 2κ on C ∞ ((S2 )l ) ⊗ C2s+1 such that K 1 l 1 (κ) Λ(tl ) . . . Λ(t1 )φn − 1 + D 2s sκ κ=l P [Λ(tl )](ml ) . . . P [Λ(t1 )](m1 )φn m =···=m1 =n l
(4.41)
is of the order s−(K+1) for any K ≥ l. The left-hand side of (4.39) hence is of the order O(s−l ) = O(l ), meaning that every factor Λ(tk ) contributes a factor of . We therefore finally obtain an estimate of the remainder term to the Dyson series given by RN (t; )ψ(0) ≤ KN N/2 tN θ(t)3N δ(t)mN . The main terms in the Dyson expansion are treated by replacing each factor of (4.35), occurring at t = tk , with the Taylor expansions 2 ν 1 dπs (σ/2) − Sn(tk ) · C (ν) z(tk ) w − z(tk ) ν!
|ν|=1
+
nk ν 1 (ν) H0 z(tk ) + C (ν) z(tk ) dπs (σ/2) w − z(tk ) + rk (tk , w) , ν! ν=3
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where again the integers nk are chosen sufficiently large. The contribution of the translational degrees of freedom can be dealt with as in the previous semiclassical scenario, and the spin contribution follows from the expansion (4.41). Finally grouping together terms of corresponding orders in , we arrive at a statement analogous to Theorem 4.4. ˆ with symbol (2.2) satisfies Theorem 4.8. Suppose that the quantum Hamiltonian H the conditions specified in Section 2. Then for t > 0 and any N ∈ N there exists a state ψN (t) ∈ L2 (Rd ) ⊗ C2s+1 , localized at q(t), p(t), n(t) , that approximates the ˆ (t, 0) ϕB ⊗ φn of a coherent state up to an error full time evolution ψ(t) = U (q,p) of order N/2 when s is fixed. More precisely, ψ(t) − ψN (t) ≤ CN
N −1 j=1
t
j
√ ( θ(t))2j+N .
The right-hand side vanishes in the combined limits → 0, s → ∞ and t → ∞ as long as t Tz (), where Tz () denotes the same time scale as in Theorem 4.7. Furthermore, ψN (t) arises from ϕB (q,p) ⊗ φn through the application of certain (time-dependent) differential operators qˆkκj (t) = opW [pkj (t)] ⊗ rκ , ˆQ (t, 0) qˆkκj (t) ψ(0) , U ψN (t) = ψQ (t) + (k,κ,j)∈∆N
where pkj (t) is a polynomial in (x, ξ) of degree ≤ k and rκ is a differential operator of order ≤ 2κ on C ∞ (S2 ) ⊗ C2s+1 . Here we have also defined ∆N := {(k, κ, j) ∈ N3 ; 1 ≤ k + 2κ − 2j ≤ N − 1, k + 2κ ≥ 3j, 1 ≤ j ≤ N − 1} . The semiclassical localization of ψN (t) here is different from the situation covered by Theorem 4.4 in that the operators rκ act on φn . But these are differential operators and hence do not increase the frequency set. This means that ψN (t) is semiclassically localized at Φtso (q, p, n) and in in this respect is not different from the classically propagated coherent state ψQ (t).
5 Discussion In the previous section we analyzed the semiclassical behavior of coherent states in two different limits. In various places we saw that the difference between the two cases is expressed in the way the classical translational and spin motion are coupled. Otherwise the final results agree to a large extent. This includes the mechanisms of semiclassical localization in the product phase space T∗ Rd × S2 . The problem of how the localization of an initial coherent state develops with time can be made more explicit by using semiclassical phase-space lifts of the coherent states. At t = 0 the state ψ(0) = ϕB (q,p) ⊗ φn is concentrated in
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a neighborhood of the point (q, p, n) ∈ T∗ Rd × S2 . This concentration can be ˆ measured in terms of expectation values ψ(0), Aψ(0) of operators Aˆ = opW [A] ∞ that are quantizations of well-localized symbols A ∈ C0 (T∗ Rd ) ⊗ M2s+1 (C). For simplicity we also assume that A is independent of . At later times ψ(t) can in both semiclassical scenarios be approximated by an appropriate coherent state ψQ (t), such that ˆ Q (t) + o(1) , ˆ ψ(t), Aψ(t) = ψQ (t), Aψ
t Tz () .
(5.1)
The expectation value on the right-hand side has a phase-space representation 1 B(t) ˆ Q (t) = W [ϕz(t) ](w) φn(t) , A(w)φn(t) C2s+1 dw . ψQ (t), Aψ (2π)d ∗ d T R (5.2) A comparison with (3.5) clearly reveals that the state ψQ (t) is concentrated at the point (q(t), p(t), n(t)) in the semiclassical limit as long as the quadratic form GB(t) / is strictly positive definite. Either of the time evolutions (4.6) and (4.33) of B now imply [Sch01] GB(t) = (Sz (t)−1 )∗ GB Sz (t)−1 , 2 , so that with GB = γB
tr GB(t) = Sz (t)−1 γB 2HS .
(5.3)
Here Sz (t) denotes the differential of the appropriate flow with respect to (x, ξ). If z(t) now is a trajectory with maximal Lyapunov (or stability) exponent λmax (z) > 0, the quantity (5.3) grows as e2λmax t such that the requirement for the state 1 ψQ (t) to remain localized therefore is t 2λmax (z) | log |. This time scale is three times larger than Tz (), which is the estimated time in (5.1) for the coherent state ψQ (t) to still well approximate the full time evolution ψ(t). Let us remark that the limitations in (5.1), to approximate the expectation value in terms of a coherent state, derive from estimating the difference ψ(t)−ψQ (t) in L2 -norm. But the error term on the right-hand side of (5.1) measures this difference in a considerably weaker form so that one might expect it to vanish as → 0 and t → ∞ also for times Tz () ≤ t 3Tz (). In the case without spin Bouzouina and Robert [BR02] proved that this indeed holds, suggesting that the same is true in the present setting. Expectation values in coherent states such as (5.1) can also be used to obtain the leading semiclassical description of the propagation of observables. To see ˆ as above, be a bounded Weyl operator and denote its quantum time this let A, ˆ (t, 0). Here, however, we do not necessarily require ˆ =U ˆ (t, 0)∗ Aˆ U evolution by A(t) the symbol to be compactly supported. The relations (5.1) and (5.2) then remain valid so that 1 B(t) ˆ ψ(0), A(t)ψ(0) = W [ϕz(t) ](w) φn(t) , A(w)φn(t) C2s+1 dw + o(1) . (2π)d ∗ d T R
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ˆ is bounded it may also be expressed as a Weyl operator, with symbol Since A(t) A(t) such that for t Tz () equation (5.1) can be rewritten as 1 B(0) W [ϕz(0) ](w) φn(0) , A(t)(w)φn(0) C2s+1 dw (2π)d ∗ d T R 1 B(t) W [ϕz(t) ](w) φn(t) , A(w)φn(t) C2s+1 dw = o(1) . − (2π)d T∗ Rd The semiclassical localization properties of the coherent states discussed above therefore imply that in leading order the symbol of the time evolved observable ˆ can be expressed in terms of the symbol of Aˆ transported along the classical A(t) flow q(t), p(t), n(t) , φn , A(t)(q, p)φn C2s+1 − φn(t) , A q(t), p(t) φn(t) C2s+1 = o(1) . The C2s+1 -expectation values in spin-coherent states are lower (or Q-) symbols (see, e.g., [Sim80, Per86]) of the matrix-valued functions A(t) and A, respectively. In terms of this mixed phase space representation of operators, employing Weyl calculus for the translational part and Q-symbols for the spin part, this means that the quantum time evolution of observables follows the classical dynamics in leading semiclassical order. This statement represents a limited version of an Egorov theorem and again is valid for both semiclassical scenarios discussed in the preceding section, up to the time scale t Tz ().
Acknowledgments A major part of this work has been performed when both authors stayed at the Mathematical Sciences Research Institute, Berkeley. We would like to thank the MSRI for its hospitality and for the support extended to us. Financial support by the Deutsche Forschungsgemeinschaft under contract no. Ste 241/15-2 is gratefully acknowledged. R.G. was also supported through the Doktorandenstipendium D/02/47460 by Deutscher Akademischer Austauschdienst.
Appendix: Linear stability of Hamiltonian flows The flows Φt0 and Φtso introduced in Section 2 are both Hamiltonian flows on symplectic phase spaces. They are generated by smooth Hamiltonian functions H on 2n-dimensional smooth manifolds M with symplectic forms ω. In the first case the Hamiltonian is H0 (x, ξ), defined on the phase space M = T∗ Rd so that n = d and ω = dx ∧ dξ. In the situation of classical spin-orbit coupling the Hamiltonian Hso (x, ξ, n) is given on M = T∗ Rd × S2 . Thus, n = d + 1 and ω = dx ∧ dξ + dn, where dn denotes the normalized area two-form on the sphere S2 . In this appendix we want to recall the notion of Lyapunov exponents and give sufficient criteria of their existence in terms of properties of the Hamiltonian function.
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The linear stability of a flow Φt is determined by properties of the differential DΦ (α) which is a linear map from the tangent space Tα M to TΦt (α) M . It, more over, is a multiplicative cocycle over the flow Φt , i.e., DΦt+t (α) = DΦt (Φt (α)) DΦt (α). If one introduces a Euclidean scalar product in the tangent spaces, this gives rise to the adjoint DΦt (α)∗ . Then DΦt (α)∗ DΦt (α) is a non-negative symmetric linear map on Tα M whose eigenvalues we denote by t
(1)
(2n)
µt (α) ≥ · · · ≥ µt
(α) ≥ 0 .
The 2n Lyapunov exponents of the flow Φt at α ∈ M are now given by the expressions 1 (k) λk (z) := lim sup log µt (z) , t→∞ 2t if these are finite. The largest Lyapunov exponent λmax (α) provides a quantitative measure for the linear stability of Φt since it measures the leading rate of local phase space expansion; it can be obtained from the relation λmax (α) = lim sup t→∞
1 log tr DΦt (α)∗ DΦt (α) . 2t
(A.1)
Hamiltonian flows leave the energy shells ΩE := {α ∈ M ; H(α) = E} invariant. If E is a regular value of the Hamiltonian function H, the energy shell ΩE is a smooth submanifold of M of dimension 2n−1. In such a case two Lyapunov exponents are always zero. They correspond to the direction of the flow and the direction transversal to the energy shell. Of the remaining 2n − 2 Lyapunov exponents half are non-negative (if they exist) and the rest of the Lyapunov spectrum is given by minus the first half. In general it is not known whether the Lyapunov exponents are finite. If, however, an energy shell ΩE is compact, one can introduce the normalized Liouville measure as a flow invariant probability measure on ΩE . In this case one can apply Oseledec’ multiplicative ergodic theorem to the restriction of Φt to this energy shell [Ose68]; it guarantees that the Lyapunov exponents are finite for almost all points on ΩE with respect to Liouville measure. Moreover, if the flow is ergodic with respect to Liouville measure λk (α) is constant on a set of full measure. Since we want to consider also non-compact energy shells we now give alternative sufficient criteria for the finiteness of Lyapunov spectra. Proposition A.1. Let H ∈ C ∞ (M ) be a Hamiltonian function such that the HilbertSchmidt norm of D2 H is bounded on the energy shell ΩE,α that contains the point α ∈ M . Then the Lyapunov exponents λ1 (α), . . . , λ2n (α) are finite. Proof. Fix α ∈ M and introduce canonical coordinates (q, p) ∈ U ⊂ Rn × Rn in a neighborhood of α. Then in this neighborhood D2 H is represented by the
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matrix H (q, p) of second derivatives with respect to (q, p). In these coordinates ˜ t (q, p); its differential satisfies the equation we denote the flow by Φ t d ˜t ˜ t (q, p) , ˜ t (q, p)|t=0 = 12n , ˜ (q, p) DΦ DΦ (q, p) = J H Φ DΦ (A.2) dt 0 1n where J = −1 . By integrating (A.2) and taking the Hilbert-Schmidt norm n 0 one obtains t s t ˜ ˜ (q, p) HS DΦ ˜ s (q, p)HS ds . DΦ (q, p)HS ≤ 2n + J H Φ 0
For simplicity we here assume that for the points Φs (α), s ∈ [0, t], one can use the same system of canonical coordinates. Gronwall’s inequality then yields the estimate (t > 0) s ˜ (q, p) HS ≤ 2n eCt , ˜ t (q, p)HS ≤ 2n exp t sup J H Φ DΦ s∈[0,t]
with some constant C > 0. The last line follows from the boundedness of D2 H on ΩE,α . Since on the other hand ˜ t (q, p)HS = DΦ
(1) (2n) µt (α) + · · · + µt (α) ,
the bound 1 log µmax (α) ≤ K t 2t for the maximal eigenvalue µmax (α) follows. This finally implies the assertion. t An application of this Proposition to the two flows Φt0 (defined on M = T∗ Rd ) and Φtso (defined on M = T∗ Rd × S2 ) immediately yields Corollary A.2. If the norm of H0 is bounded on ΩE,(x,ξ) ⊂ T∗ Rd , the 2d Lyapunov exponents λ0,k (x, ξ) of the flow Φt0 are finite. If, in addition, the derivatives C (ν) (x , ξ ) of order |ν| ≤ 2 are bounded for all (x , ξ , n ) ∈ ΩE,(x,ξ,n) ⊂ T∗ Rd ×S2 , the 2d + 2 Lyapunov exponents λso,k (x, ξ, n) of the flow Φtso are also finite. In the second semiclassical scenario, however, rather than the Lyapunov exponent λso,k (q, p, n) of a point (q, p, n) ∈ T∗ Rd × S2 the stability exponent (4.38) of the projection to T∗ Rd entered Theorem 4.7. Revisiting the proof of Proposition A.1 shows that in view of (4.34) such a stability exponent is finite under the same conditions as stated in Corollary A.2 for λso,k . Moreover, a simple estimate yields the bound ˜ max ≤ λso,max . λ
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Jens Bolte and Rainer Glaser Abteilung Theoretische Physik Universit¨ at Ulm Albert-Einstein-Allee 11 D-89069 Ulm Germany email:
[email protected] email:
[email protected] Communicated by Klaus Fredenhagen submitted 13/04/04, accepted 24/11/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 657 – 695 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/04657-39, Published online 28.07.2005 DOI 10.1007/s00023-005-0219-7
Annales Henri Poincar´ e
Quantum Backreaction (Casimir) Effect I. What are Admissible Idealizations? Andrzej Herdegen Abstract. Casimir effect, in a broad interpretation which we adopt here, consists in a backreaction of a quantum system to adiabatically changing external conditions. Although the system is usually taken to be a quantum field, we show that this restriction rather blurs than helps to clarify the statement of the problem. We discuss the problem from the point of view of algebraic structure of quantum theory, which is most appropriate in this context. The system in question may be any quantum system, among others both finite- as infinite-dimensional canonical systems are allowed. A simple finite-dimensional model is discussed. We identify precisely the source of difficulties and infinities in most of traditional treatments of the problem for infinite-dimensional systems (such as quantum fields), which is incompatibility of algebras of observables or their representations. We formulate conditions on model idealizations which are acceptable for the discussion of the adiabatic backreaction problem. In the case of quantum field models in that class we find that the normal ordered energy density is a well-defined distribution, yielding global energy in the limit of a unit test function. Although we see the “zero point” expressions as inappropriate, we show how they can arise in the quantum field theory context as a result of uncontrollable manipulations.
1 Introduction The Casimir effect, bearing its name from the pioneer work by Casimir [1], has become in recent decades an increasingly popular topic in quantum field theory, with a new review of the subject appearing every few years, see [2, 3, 4, 5, 6]. The effect consists in the response of a quantum field, even in a ground state, to the introduction of external, usually macroscopic, bodies. Initially the effect existed as a theoretical prediction only, and a rather mysterious one, for that matter. However, increasing experimental evidence of its existence (see, e.g., [6]) has lead to attempts at better understanding of its theoretical foundation. The problem is, that the theoretical side of the phenomenon has been plagued from the beginning by divergent expressions, as well as conceptual difficulties, which have proved to be surprisingly persistent. This is the more surprising, that models usually considered in this context are linear, so the usual sources of quantum field infinities are absent here. In an earlier paper [7] I have given a diagnosis of the reasons of this state of affairs and proposed to treat the problem from the algebraic point of view. This is the most natural and fruitful framework in quantum physics, with its beginnings already in the classical book on quantum mechanics by Dirac, and modern
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developments in quantum field theory and statistical physics described, e.g., in monographs [8] and [9]. When viewed from that angle the source of difficulties is rather obvious, and can be briefly termed as uncritical use of the concept of quantum field [7]. More precisely, what we mean is this. The first step to define a quantum theory is to identify a set of quantum observables (we ignore here the question of non-observable variables) together with algebraic relations between them, such as canonical commutation relations. Once we have this, a concrete physical realization of the theory corresponds to a choice of a representation of the algebra of observables. Non-comparable physical situations are realized by nonequivalent representations [8]. Although we want to see the real world as a unity, physics, of course, is about idealizations, and various idealizations need not be compatible (take, e.g., an isolated system and a thermodynamic limit system). However, if we want to consider transitions from one physical situation to another and compare values of one and the same observable in various states, all situations taken into account must be describable in one common representation. Now, these scheme is violated in most treatments of the Casimir effect. For a typical situation of a quantum field in a region with movable sharp boundaries the difficulty arises already on the algebraic level: there is no consistent choice of an algebra of observables for all physical situations coming into play. The energy of the “free” field is an observable defined in the vacuum representation of the algebra of the field smeared with Schwartz test functions. For this algebra evolutions imposed on the field by the presence of boundaries cannot be defined. Furthermore, even if one “smooths out” the boundaries so as to make a common choice of an algebra possible, one still has to satisfy rather severe restrictions necessary to ensure the equivalence of representations. These restrictions are typically violated in usual treatments. For these reasons we have advocated in [7] the view, that the model of sharp boundaries, as well as many other insufficiently regular models, are wrong idealizations in the context of Casimir effect, and we have also proposed (and analyzed) a class of models imitating Dirichlet conditions. Let us stress this: once a model has correctly been chosen, there is no space (nor need) for further ad hoc regularizations, and the formalism should yield well-defined answers to legitimate questions. Although views on nonphysical nature of sharp boundary conditions have been also expressed elsewhere (see, e.g., [10]), it seems that the conditions for a model to be acceptable in the sense described above have not been analyzed before. For instance, in a series of recent papers Graham et al. [11] investigate a linear model imitating Dirichlet conditions. Being linear, the model should be well defined without any renormalization (except for a trivial normal ordering for quadratic quantities like energy density; for external potentials without bounded states this is our example (iii)1 at the end of Section 3 below). However, renormalization is ad hoc imposed on it by the authors in order to give meaning to a meaningless expression. The algebraic problems we have described often do not appear if one restricts attention to local quantities in quantum field theory. This fact is connected with what in algebraic formulation is called the local quasiequivalence of representations
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(see [8]). The point is as follows. In quantum field theory observables are equipped with the property of locality: each local observable carries as a label an open spacetime region with compact closure in which it may be measured. As stated above, two representations of the totality of these observables representing two different physical situations may be non-equivalent. However, physically one would expect that even if the two situations are globally non-comparable, one should be able to compare results of local measurements (think of the vacuum representation and a thermodynamic limit representation). Mathematical formulation of this expectation is this: think of states in each of the representations as density operators; restrict attention to an arbitrarily chosen compact region of spacetime; then for each state in one of the representations there is a state in the other which yields the same expectation values for observables localized in the chosen region. If the two representations have this property they are called locally quasiequivalent. As it turns out physically important representations do indeed often have this relative property, and then expectation values of local quantities may be compared. However, in the situation we want to consider in this paper this result falls short of our needs in twofold way. First, we want to calculate expectation values of global quantities, which are limits of local ones for the size of the spacetime region tending to infinity – in this case the global differences of the representations come into play. Second, in situations like fields with imposed boundary conditions, even finite regions which overlap with boundaries are not local in the above sense: for those regions even the scopes of local algebras in presence of boundaries are different than in the vacuum theory. Another important point we want to stress in our analysis of Casimir effect is the choice of the observable to be compared in various considered states. In our view the backreaction of a system perturbed by external agents is determined by the expectation value of the energy as defined by the unperturbed system, one and the same (as an operator) in all states to be considered. A more systematic discussion of this point in a wider context will be found in the next section. Here we want to note that some local, in the spirit of the last paragraph, calculations of the Casimir energy do follow similar ideology; in the gravitational context see esp. a paper by Kay [12], and for electromagnetic field with conducting boundaries a paper by Scharf and Wreszinski [13]. However, in many other local calculations, esp. those using “the Green function method”, the situation is somewhat ambiguous: it is often not clear enough what the general viewpoint is, and the result may agree with the above method in some cases, but disagree in others. We shall discuss this point more fully in Section 6 below. For the global energy, as determined by the unperturbed system, to be defined in states of the system influenced by external conditions, as required by the above ideology, we need one common algebra and globally equivalent representations, as explained earlier. This imposes restrictions on the perturbed dynamics, which are usually violated, and the transition from local Casimir energy to global one is then blocked by infinities of physical nature. Any “regularization” thereof is an ad hoc procedure, striving at this late stage to compensate for the wrong idealization in interaction with external con-
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ditions. Finally, there is a group of works explicitly comparing the expectation values of different global observables: energy with and without interaction. Here, apparently, is the place of the “zero point” ideology. We shall come back to this point later on, here we only note, that in this case infinities are even more likely to appear. In that method one subtracts expectation values of regularized “bare” energy observables; different energy observables may have different singularities, not cancelling under subtraction. The present paper is the first of the two in which we develop and describe more fully what was announced in [7] (we use notation slightly changed at some points with respect to that paper). Here we discuss more general results on the admissibility of models for the purpose of investigation of quantum backreaction. In the second paper applications to particular models are discussed. We use rigorous mathematics, and present real proofs. However, we hope that the paper is readable for a wide audience. In Section 2 we place the quantum field Casimir effects in a wider context of a backreaction of a quantum system to adiabatic changes. This section thoroughly discusses the foundation for the calculation of this backreaction in any quantum system. In Section 3 we discuss quantization of a class of linear systems, which include quantum fields under linear external perturbations. We put stress on less widely known aspects of this otherwise standard procedure which are important in the present context. Section 4 discusses an application to a finite-dimensional system. In Section 5 we treat infinite-dimensional cases, and we formulate conditions for admissibility of a model for the discussion of backreaction effects. More specifically, we consider a quantum field case in Section 6. We show that with a slight strengthening of these conditions not only global energy, but also energy density may be defined, and in the appropriate limit global energy is recovered. Section 7 contains somewhat more explicit discussion of the points made earlier in this introduction on the existing calculations of Casimir effect. We also comment there on the “zero point” expressions for the Casimir energy. We try to understand, from the point of view of the formalism presented in the present work, how such expressions may arise. We show how imposing unacceptable idealization of sharp boundaries and doing unjustified manipulations leads from our expression for the energy density to “zero point” expressions for Casimir energy. Appendix gives a simple form to a handful of mathematical facts in Fock space which are needed in the main text. These are known results, but we believe that this summary makes some of them more accessible.
2 A quantum system under external conditions Trying to put the discussed phenomenon in a broader context we shall adopt the following point of view. The Casimir-type effect consists in the backreaction of a quantum system on the adiabatically changing external conditions under which the system is placed.
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The background for this idea is this. We consider a larger closed system consisting of two subsystems Q and M . These subsystems interact with each other, but to certain degree (this will be made more precise below) maintain their separate identity. Part Q is our relatively simple quantum system under consideration (say, electromagnetic field), while M is supposed to be of much more complicated nature (say, conductor plates), and to have among its variables some of collective, macroscopic type (separation of the plates). We want to determine the effect of the evolution of the joint system on the collective variables attached to M . Because of the complicated nature of the part M of the system and its interaction with Q, to tackle the problem one has to make some simplifying assumptions. There are at least two possibilities, both of them of phenomenological nature. In both cases one simply represents part M of the system by a few collective variables (such as separation of the plates), suppressing all the details of this subsystem, and representing the interaction between M and Q by some simple effective model. The first possibility is to equip the collective variables with a fully quantum nature, and put forward a simple model for the closed system. This approach, when applied to the more specific situation of a quantum field in interaction with macroscopic bodies, is chronologically more recent one in this field, and is called the dynamic Casimir effect (see [6]). Although we admit that this forms an open possibility, we shall not take it up in this article. Firstly, not much can be said with high degree of certainty and mathematical rigor. Secondly, the apparent attractiveness of the approach does not necessarily withstand a closer scrutiny. A macroscopic body undergoes “constant observation”, so effects of decoherence play primary role, which is not taken into account in this approach. Another possibility, which we take up in this paper, has more restricted aspirations, but admits mathematically rigorous results, as we are going to argue below. We have to admit, however, that there is some confusion at its physical formulation. We hope to contribute to its removal. This second approach consists in approximating the collective quantities, which characterize a macroscopic body as a whole, by classical variables. Moreover, one considers only situations, in which the whole system changes adiabatically. The effect of the evolution on the macroscopic (classical) variables in this context is what we referred to as a Casimir-type effect at the beginning of this section. More specifically, the Casimir effect refers to a quantum field in interaction with macroscopic bodies. One should be more specific about physical assumptions and approximations involved in the situation implied in the last paragraph. This is, in our opinion, a point not clear enough in many discussions of the Casimir effect. Therefore we shall try to be systematic, even at a risk of being too detailed. (i) One considers first the isolated quantum system Q (M is absent). We give its description in the algebraic formulation of the Heisenberg picture, see, e.g., [9].
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(i)1 Basic quantum variables at a fixed time form an abstract *-algebra A, e.g., an algebra of canonical commutation relations (or, more technically, its exponentiation to the Weyl form). (i)2 This algebra is represented by operators in a Hilbert space H: π : A → π(A) ,
A → π(A) ,
π(αA + βB) = απ(A) + βπ(B) ,
π(A∗ ) = π(A)∗ , π(AB) = π(A)π(B) ,
(2.1)
where π(A) is a concrete algebra of operators in H. Vectors in that space, or, more generally, density operators acting in this space, represent states of the system Q. Representation π is assumed to be irreducible; then vectors correspond to pure states. (i)3 The intrinsic dynamics of Q is defined by an automorphism of the algebra A: (2.2) αt : A → A , A → αt A . This automorphism is implemented by a unitary evolution in the Hilbert space H: π(αt A) = U (t)π(A)U (t)∗ ,
U (t) = exp(itH) ,
(2.3)
where H has the interpretation of the energy operator of the system. This operator is supposed to have nonnegative spectrum, and usually is assumed to have a ground state, represented by a unit eigenvector to the lowest point in the spectrum. One does not perturb the above relations by adding a multiple of the identity operator to H, so the ground state may be assumed to have zero energy. By irreducibility of π the energy operator H is then uniquely determined. (ii) One introduces now part M into the system. This part is characterized by classical variables (we shall denote them by a), so no new quantum variables are added. Therefore system Q should retain its identity, and changes in its state will influence the classical variables of M . Thus various states to be considered must be physically comparable. These assumptions have mathematical consequences. (ii)1 Identity of the system Q is formed by the algebra A ((i)1 above), so this algebra must remain unaffected by M . (ii)2 Physical comparability of states demands that also the particular representation π of A ((i)2 above) remains unaffected by the introduction of M . We stress the importance of this point as it is both crucial for the scheme, as we see it, and usually overlooked. If various physical situations to be considered demanded different algebras or different (nonequivalent) representations, the approximation would break down, as one could not follow
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the change in the system Q brought about by the creation of (and changes in) M , and its reaction to that occurrence. Further support for this point will be found below. Let us note again, what was discussed in the introduction, that the local quasiequivalence of representations if Q is a quantum field system is not enough for our purposes. (iii) We consider now dynamics in presence of M , and assume at first that the variables a are frozen. In this case Q is still a closed system in interaction with conditions created by M , and for each fixed a its evolution is again given by an automorphism of the algebra A: αat : A → A ,
A → αat A .
(2.4)
One assumes implementability of new evolutions in the representation π: for each a we have π(αat A) = Ua (t)π(A)Ua (t)∗ ,
Ua (t) = exp(itHa ) .
(2.5)
For each a the generator Ha is defined by this up to the addition of a multiple of the identity operator, so we have the freedom Ha → Ha + λa id ,
(2.6)
where λa is any real function of parameters a. (iv) One allows now the coupled system Q − M to evolve. Part Q alone is not a closed system any more, so it could be too restrictive to assume that the evolution of its variables would be given at the algebraic level, as an automorphism. However, this evolution should still be describable in terms of unitary operators in the Hilbert space H (not forming a one-parameter group, in general); this corresponds to the assumption of conservation of probabilities in the subsystem Q. The use of the Schr¨odinger picture for the quantum part Q will be more convenient in the present context. State of the coupled system Q−M is specified at a given time by a vector in the Hilbert space H (describing the state of Q), and values of a and, possibly, their time derivatives. We formulate the evolution of this system. (iv)1 Suppose that a(t) is known as a function of time. We assume that this functional dependence is very slow (system M is “heavy”). It is then a justified approximation to assume that the time-dependent Hamiltonian of the evolution of the system Q is given by Ha(t) (with Ha defined in (iii) above). As a(t) is slowly varying we assume the adiabatic approximation to calculate the evolution. One is usually interested in the situations in which the initial state of Q is given by an eigenvector of Ha for the initial value of a. Suppose that for each a we have a nondegenerate, normalized eigenvector ψa of Ha : Ha ψa = Ea ψa ,
(2.7)
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and the family ψa depends continuously on a. If at t = 0 the state of Q was given by ψa(0) , then at later times in the adiabatic approximation its state is equal to ψ(t) = eiϕ(t) ψa(t) , where ϕ(t) is a real function depending functionally on Ea and ψa . If an operator B represents an observable, then the time-dependence of its expectation value is given by Bt = (ψa(t) , B ψa(t) ) ,
(2.8)
so it is a function of a in this approximation. It is important to note that the eigenvalues Ea are modified by the addition of λa under the transformation (2.6), but both the eigenvectors ψa and the mean values Bt remain unchanged. (iv)2 Finally, the evolution of the macroscopic variables a(t) must be determined. This is the most controversial part of the problem, but we believe that the foregoing discussion indicates its proper solution. The intrinsic energy stored in the quantum part Q is represented (in the Schr¨ odinger picture) by the operator H ((i)3 above), which in the coupled system is not a constant of motion any more. Under the assumptions of (iv)1 its expectation value is a function of a, depending on the choice of the continuous family of eigenvectors ψa : Ea := (ψa , H ψa ) ,
(2.9)
and the time-dependence of this expectation value is through a(t) only. Changes in Ea correspond to the energy which has been transferred from Q to the rest of the system, which (with the suppression of all microscopic details of M ) is described by the variables a. Thus Ea plays the role of a potential energy with respect to these variables. We assume that the rest of the total energy of the coupled system is supplied by the kinetic energy of M , thus we obtain a potential system, with the generalized force given by Fa = −
∂Ea . ∂a
(2.10)
With a specific form of the kinetic energy for a particular model the motion of a(t) could be determined, and with large inertial parameters (a “heavy” system) the approximation of its slow change should be confirmed. We have thus spelled out all the assumptions and arrived at the basic formulas (2.9), (2.10). In the following sections we shall take these formulas as a starting point. The derivation of the formulas was not rigorous, as this would demand more information on the underlying microscopic model of the closed system Q−M . A detailed analysis of these questions is both outside the usual discussions of Casimir
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effect, and also beyond the reach of a rigorous calculation at present. However, we believe that the proposed discussion offers more plausibility than most of the statements of the problem to be found in literature. In particular, points made by us in (ii) above are typically ignored; we shall see their consequences when Q is an infinite-dimensional system, e.g., a quantum field. Furthermore, we want to draw a closer attention to the formula (2.9) and contrast it with what one obtains by the generalization of the “zero point” method to the more general context discussed in this section. In the latter case our formulas (2.9) and (2.10) are replaced respectively by ∂Ea , (2.11) Faz.p. = − Eaz.p. = Ea − E0 , ∂a where Ea is the eigenvalue determined by (2.7), and E0 some reference eigenvalue of H. One can object to these formulas on several grounds. (a) The philosophy behind them seems to be this: the backreaction of Q on M is due to the changes in Ha , which may be interpreted as the sum of intrinsic energy H of Q and some interaction energy. However, we think that it is M which absorbs the interaction and transforms it in a phenomenological way into an effect on macroscopic variables a, while Q has a rather clear-cut identity. (b) The energy given by the “zero point” philosophy is not a quantum mechanical average of any clear-cut observable: with changing a one changes the observable Ha . Moreover, as already pointed out, Ha and their eigenvalues are subject to the gauge freedom (2.6). The usual argument runs that this is fixed by the quantization of the “proper” classical expression for Ha . We regard this argument as very unreliable. Quantum theory is the more fundamental one, so in case of doubt it should not seek a verdict from the classical theory. (c) We put forward the following “consistency check”. Suppose that for certain values of parameters a the effect of M on Q vanishes. In this case the backreaction force should vanish as well. The supposition means that for a = a0 the vector ψa0 is also an eigenvector of H, Hψa0 = Eψa0 with some eigenvalue E. Using this equation one easily shows that our formulas yield ∂(ψa , ψa ) Fa0 = −E = 0, ∂a a=a0 so they pass the check. On the other hand ∂Ea Faz.p. = − , 0 ∂a a=a0 which, in general, has no reason to vanish.
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(d) In Section 4 below we discuss an example of a Casimir-type effect in a canonical system with finite degrees of freedom. In this example the “zero point” method fails dramatically, yielding a completely unphysical result. How, then, may “zero point” expressions arise? We shall show in Section 7 below how for quantum fields problems “zero point” expressions may be related to ours by unjustified idealizations and manipulations.
3 A class of quasi-free systems We discuss in this section a general quantization scheme for a class of simple models. This class includes linear perturbations of multi-dimensional harmonic oscillators or quantum fields. Consider first the classical case. Let R be a real Hilbert space, and denote its scalar product by (. , .). Let h be a selfadjoint, strictly positive (hence invertible, with densely defined inverse h−1 ) operator in R, with the domain DR (h). We form the external direct sum L = DR (h) ⊕ R ⊂ R ⊕ R, and denote its elements by V = v ⊕ u, v ∈ DR (h), u ∈ R. With the symplectic form σ defined by σ(V1 , V2 ) = (v2 , u1 ) − (v1 , u2 )
(3.1)
space L becomes the phase space of a classical model. Let the Hamiltonian function of the model be given by H(v, u) = 12 [(u, u) + (hv, hv)] (where all mass parameters have been absorbed by momenta). The evolution determined by this Hamiltonian in L is given by (3.2) Tt (v ⊕ u) = cos(ht)v + sin(ht)h−1 u ⊕ − sin(ht)hv + cos(ht)u . The differential form of this evolution is actually valid only on a subspace of L (dense in R ⊕ R), but the evolution itself is properly defined on the whole of L. Operators Tt form a one-parameter group of symplectic transformations Tt Ts = Tt+s ,
σ(Tt V1 , Tt V2 ) = σ(V1 , V2 ) .
(3.3)
Note, also, that T−t = (id ⊕ − id) Tt (id ⊕ − id) .
(3.4)
Each V ∈ L may be identified with an element of the dual space by the rule V (V ) = (v , u) + (u , v) = σ(V , (id ⊕ − id)V ) .
(3.5)
Then using Eqs. (3.3) and (3.4) one easily shows that (Tt V )(V ) = V (Tt V ) .
(3.6)
The above model may be generalized by considering a more general subspace contained in DR (h) ⊕ R and invariant under the evolution law (3.2). We use this freedom to choose L = DR (h) ⊕ DR (h−1/2 ) (3.7)
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(the invariance under (3.2) is easily checked). The evolution law Tt may be now expressed as a unitary evolution in a complex Hilbert space. One introduces a complex Hilbert space K which is the complexification of R, K = R ⊕ iR, with scalar product (denoted by the same symbol) and complex conjugation defined by (v1 + iu1 , v2 + iu2 ) = (v1 , v2 ) + (u1 , u2 ) + i(v1 , u2 ) − i(u1 , v2 ) , K x → Kx ≡ x¯ ∈ K ,
K(v + iu) = v − iu .
(3.8) (3.9)
We shall write v = Re(v + iu), u = Im(v + iu). The operator h has a unique extension to a complex-linear operator on K, denoted by the same symbol, with the domain D(h) = DR (h) ⊕ iDR (h). This new h is again a selfadjoint, positive operator, and it commutes with the conjugation. Consider now a real-linear operator j : L → Ran j ⊂ K , j(V ) = h1/2 v − ih−1/2 u . (3.10) Its range Ran j is a real-linear subspace of K, dense in K, and j is a bijection of L onto Ran j. Then for all V ∈ L: j(Tt V ) = eiht j(V ) ,
(3.11)
so Ran j is invariant under eiht , and the evolution may be expressed as Tt V = j −1 (eiht j(V )) .
(3.12)
Space K, regarded as a real vector space, has a natural symplectic structure introduced with the symplectic form Im (f, g). Space Ran j is its symplectic subspace. One easily shows that σ(V1 , V2 ) = Im (j(V1 ), j(V2 )) ,
(3.13)
so j is a symplectic transformation of L onto Ran j. The mapping j, as well known, serves to construct the ground state representation of the quantum version of the model, and the space K is then the “one-particle space” (see below). A natural problem thus arises: to extend the construction of the space L to the largest possible space compatible with the symplectic mapping (3.10), that is to extend L and j so as for Ran j to cover the whole space K (instead of being only dense in K, as above). One defines on DR (h±1/2 ) the scalar products (v1 , v2 )+ = (h1/2 v1 , h1/2 v2 ) ,
v1 , v2 ∈ DR (h1/2 ) ,
(3.14)
(u1 , u2 )− = (h−1/2 u1 , h−1/2 u2 ) ,
u1 , u2 ∈ DR (h−1/2 ) ,
(3.15)
of and denotes by R+ and R− the Hilbert spaces obtained by the completion −1/2 (h ), respectively, with respect to the norms
v
= (v, v) DR (h1/2 ) and D R + + and u − = (u, u)− . For v ∈ DR (h1/2 ) and u ∈ DR (h−1/2 ) we have
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h1/2 v = v + and h1/2 u = u − . Therefore operators h1/2 and h−1/2 ex 1/2 and h −1/2 respectively, tend by continuity to bijective isometric operators h 1/2 : R → R , h +
−1/2 : R → R , h −
1/2 v = v ,
h +
(3.16)
−1/2 u = u .
h −
(3.17)
We note for future use that R± ∩ R = DR (h±1/2 ) .
(3.18)
This is easily seen in the spectral representation of h: if h is a multiplication by a positive, different from zero almost everywhere, function f in a space L2 (M, dµ), then R consists of functions ψ for which M |ψ(m)|2 dµ(m) < ∞, R± consists of functions for which M (f (m))±1 |ψ(m)|2 dµ(m) < ∞, and DR (h±1/2 ) – of those satisfying both conditions. For v ∈ DR (h1/2 ) and u ∈ DR (h−1/2 ) one has |(v, u)| = |(h1/2 v, h−1/2 u)| ≤ v + u − ,
(3.19)
thus (v, u) extends to a continuous pairing R+ × R− v, u → v, u ∈ R ,
|v, u| ≤ v + u − .
(3.20)
Now one can set L = R+ ⊕ R− ,
σ (V1 , V2 ) = v2 , u1 − v1 , u2 ,
1/2 v − ih −1/2 u , j : L → K , j(V ) = h Tt V = j −1 (eiht j(V )) .
(3.21) (3.22) (3.23)
As a consequence of (3.18) one has L ∩ (R ⊕ R) = DR (h1/2 ) ⊕ DR (h−1/2 ) .
(3.24)
It is easy to see that now Ran j = K, the space given by Eq. (3.7) is dense in L (in its Hilbert space structure norm), and the time evolution on L is the continuous extension of the evolution on the space (3.7). Moreover, j(V2 )) = (v1 , v2 )+ + (u1 , u2 )− + i σ (V1 , V2 ) , ( j(V1 ),
(3.25)
σ so, in particular, j is a symplectic mapping of (L, ) onto (K, Im(., .)). Relations (3.5) and (3.6) are also generalized to (V , [id ⊕(− id)]V ) , V (V ) = v , u + u , v = σ (Tt V )(V ) = V (Tt V ) .
(3.26) (3.27)
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Once we have the largest arena consistent with the scheme, particular models are defined by choosing a subspace invariant under the evolution: L ⊂ L ,
Tt L ⊂ L .
(3.28)
The maximal model is invariant under the time reversal, represented by the operator id ⊕(− id) appearing in (3.26). We want to retain this property for the model defined by L, which is equivalent to the assumption L = L+ ⊕ L− ,
L± ⊂ R± .
(3.29)
Examples of particular spaces include the class of spaces 1
L = DR (hr+1 ) ∩ DR (h−s ) ⊕ DR (hr ) ∩ DR (h−t− 2 ) , r, s, t ∈ 0, ∞) ,
s≤t+
1 2
,
t≤s+
3 2
,
(3.30)
all of which are contained in (3.7). is The quantum version of the maximal model (with the symplectic space L) now obtained by standard procedure (see, e.g., [9], vol. II). Starting with expression (3.26) one aims at replacing V by some “quantum variable” Φ. In quantum theory a concrete representation of a quantum variable is an operator in a Hilbert space. If the classical variable is real, its quantum counterpart should be represented by a selfadjoint operator. Thus one assumes that a Hilbert space H is given, and for each V ∈ L one has a selfadjoint operator Φ(V ) in that space. The functional dependence of Φ(V ) on V is assumed to be linear, and the canonical commutation relations are imposed: σ (V1 , V2 ) id , [Φ(V1 ), Φ(V2 )] = i
(3.31)
where one still has to clarify the domain problems. If V = v ⊕ u, then we shall also write Φ(V ) = Φ(v, u). The element P (v) ≡ Φ(v, 0) has the interpretation of the quantum momentum for the “test vector” v, and X(u) ≡ Φ(0, u) – of the quantum position variable for the “test vector” u. With the linearity of Φ(V ) the above commutation relations are equivalent to those in a more familiar form [X(u1 ), X(u2 )] = 0 , [P (v1 ), P (v2 )] = 0 , [P (v), X(u)] = −iv, u id .
(3.32)
It is well known that there are many different concrete representations of the above scheme, and this is why it is desirable to formulate the canonical commutation relations in an algebraic way. As there are no bounded operators satisfying these relations, it is usual to take them in an exponentiated variant. This leads to the Weyl form of these relations. The Weyl algebra over the symplectic space L is and a unit element 1, the unique C ∗ -algebra generated by elements W (V ), V ∈ L, by the relations i (V1 , V2 ) W (V + V ) , W (V1 )W (V2 ) = e− 2 σ 1 2 W (0) = 1 . W (V )∗ = W (−V ) ,
(3.33)
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One looks for representations of this algebra by bounded operators in a Hilbert space (which exist for all C ∗ -algebras). Let π be such a representation in the Hilbert space H. One says that this representation is regular, if for each V ∈ L the one-parameter group of unitary operators R s → π(W (sV ))
(3.34)
is strongly continuous. If this is the case, then there exist, by Stone’s theorem (e.g., [14]), selfadjoint operators Φ(V ) such that π(W (V )) = exp(iΦ(V )) .
(3.35)
Moreover, one shows that for each finite-dimensional subspace L ⊂ L there exists a dense subspace D ⊂ H which is contained in the domains of all operators Φ(V ), V ∈ L , is an invariant subspace and an essential domain of selfadjointness for all of them, and on which linearity of Φ(V ) in its argument V ∈ L and commutation relations (3.31) are satisfied (this follows from the Stone–von Neumann uniqueness theorem, cf. [9], vol.II). While not all canonical systems with these properties arise in this way from regular representations of the corresponding Weyl algebra, most of those needed in physics do, and one usually restricts attention to this class. DynamThe algebra A of the maximal model is thus the Weyl algebra over L. ics of the model is a “quasi-free” evolution obtained by a simple “quantization” of the classical evolution Tt . Being guided by the replacement V → Φ in Eq. (3.27) and the relation (3.35), one defines it on the algebraic level by αt (W (V )) = W (Tt V ) .
(3.36)
One looks now for a representations π of the algebra in which this evolution law may be implemented: π(W (Tt V )) = U (t) π(W (V )) U (t)∗ ,
U (t) = eitH ,
(3.37)
where H is a selfadjoint operator. The ground state representation is obtained if H is a nonnegative operator with zero energy ground state. This representation is constructed in standard way with the use of the Fock space method. Let W0 (f ) = exp[iΦ0 (f )], f ∈ K, be the Weyl system of operators in the Fock space H built on the “one-particle” space K (see Appendix A.1). We set π(W (V )) := W0 ( j(V )) ,
(3.38)
or, which is equivalent, π(W (V )) = eiΦ(V ) ,
j(V )) . Φ(V ) = Φ0 (
(3.39)
Using identity (3.25) and properties of the operators W0 (f ) one easily shows that this indeed constitutes a representation of the Weyl algebra (3.33). By the irreducibility of the Weyl system in Fock space this representation is irreducible.
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Moreover, using Eqs. (3.38) and (3.23) one rewrites the condition (3.37) as W0 (eith f ) = eitH W0 (f )e−itH ,
f ∈ H.
(3.40)
The discussion of Appendix A.1 shows now that H = dΓ(h) ,
(3.41)
where dΓ(h) is the “second quantization” of h (see Eqs. (A.19–A.21)). This energy operator has nonnegative spectrum, and a unique ground state represented by the “Fock vacuum” Ω. Consider now a restriction of this model defined by a subspace L invariant under evolution (Eq. (3.28)) and time reflection (Eq. (3.29)). The algebra of the model is the subalgebra of the Weyl algebra (3.33) obtained by restricting the test vectors to L. It is well known, that the resulting model is not identical with the maximal one if L = L (see [9], vol. II), but one can demand that its ground state representation approximates that of the maximal model. This representation may be constructed as before, but the scope of Weyl operators used in this representation is restricted to {W0 (f ) | f ∈ j(L)} – cf. Eq. (3.38). This set is irreducible in K if, and only if, the space j(L) is dense in K, or, what is the same, L is dense in R+ ⊕ R− . With the time reflection symmetry assumption (3.29) this takes the form (3.42) L+ is dense in R+ , L− is dense in R− . We restrict attention to those spaces L which satisfy this condition. This restriction can be paraphrased by saying that there are no superselection rules in the Fock space of the ground state representation.
Examples (i) Multi-dimensional harmonic oscillator In this case R is a finite-dimensional Euclidean space, and h is a positive selfadjoint operator defined on the whole of R. We choose L = R ⊕ R. Space K is the unitary space obtained by complexification of R, and Ran j = K. The more familiar simple form of the model is obtained by choosing in R an orthonormal basis (e1 , . . . , en ) of eigenvectors of h and putting Xi = Φ(0, ei ), Pi = Φ(ei , 0). The system is then the set of n independent harmonic oscillators with canonical variables {Xi , Pi }, unit masses, and frequencies ωi , where hei = ωi ei . (ii) Free scalar field Free quantum fields are usually defined as operator-valued distributions on test functions of all spacetime variables. The evolution equation (KleinGordon for the scalar field) is already encoded in this formulation, which
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is manifestly relativistically covariant. For our purposes the equivalent initial value formulation is preferable – we want to separate evolution law, as far as it is possible, from setting up of the algebra. Standard identifications for this model are as follows: R = L2R (R3 ) , K = L2 (R3 ) , √ h = −∆ , L = DR (R3 ) ⊕ DR (R3 ) , where subscript R denotes the real part of the respective function space, D(R3 ) is the space of infinitely differentiable complex functions of compact support, and ∆ is the Laplace operator. Standard solution of the initial value problem for the Klein-Gordon equation has the form of Eq. (3.2), and the assumptions (3.28), (3.29) and (3.42) are satisfied. (iii) Scalar field with external time-independent interaction Loosely speaking, the choice of h here is the square root of a selfadjoint positive operator of heuristic form “h2 = −∆ + interaction”. There are a few possibilities. (iii)1 If the interaction is given by an external field σ = σ( x) then the choice of spaces R and K remains the same as in the free case, while h2 = −∆+σ (we assume that h2 is still positive – there are no bound states). Depending on the form of σ the choice of L as in the free case may be admissible (satisfy assumptions (3.28) and (3.42)) or not. A safe choice for L is supplied by any of the cases given by Eq. (3.30). More generally, h2 may be any positive selfadjoint perturbation of −∆ in the sense of operators or forms. (iii)2 The next possibility arises from restricting the region accessible to the field to a proper subset Λ ⊂ R3 . In this case R = L2R (Λ), K = L2 (Λ), and h2 = −∆B , where ∆B is a selfadjoint extension of the Laplace operator defined on twice differentiable functions with support inside Λ, determined by some boundary conditions “B”. Here, of course, the free field choice of L is not admissible, and a safe choice is again given by the formula (3.30). (iii)3 Finally, we consider a setting usually assumed for the Casimir effect. The whole physical space R3 is divided by two-dimensional surfaces into disjoint open regions Λ1 , . . . , Λs . Position of the dividing boundaries is characterized by a set of parameters a. One chooses R and K as in the free field case. Depending on parameters a, a family of positive operators ha is given by h2a = −∆a , where ∆a is the Laplace operator in L2 (R3 ) determined by the assumed boundary conditions (Dirichlet, Neumann, etc.) at the dividing surfaces with positions given by parameters a.
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In consequence, the choice of the symplectic space L must be adjusted to the position of the boundaries. A choice of simple possibilities is again given by Eq. (3.30) (with ha replacing h). In the Casimir problem one wants to compare states of the system at different values of a. However, spaces L depend nontrivially on a, thus the respective Weyl algebras are also different, and do not define the same quantum system. This constitutes the difficulty of traditional treatments of the Casimir effect which we anticipated in Section 2. Infinities naturally appear then, and are a consequence of an uncritical use of the notion of a quantum field. In the following sections we discuss Casimir effects for some systems in the category described in the present section. We start with a simple finite-dimensional case.
4 Deformation of a finite-dimensional harmonic oscillator Our unperturbed quantum system Q is here an n-dimensional quantum oscillator described in example (i) of the last section. Thus the algebra of the model is the Weyl algebra based on a finite-dimensional symplectic space L = R ⊕ R, and its representation is given by π(W (V )) = W0 (j(V )) in the Fock space H based on the finite-dimensional one-excitation space K. The energy operator of the model is given by H = dΓ(h). We consider now a combined system Q−M , as described in Section 2, and assume that the influence of M on Q for frozen parameters a manifests itself in the change of axes and frequencies of oscillations. Thus the time evolution of the algebra for frozen parameters is given by αat (W (V )) = W (Tat ) ,
(4.1)
where Tat has the form (3.2), but with operator h replaced by an operator ha from a family {ha }. For each a the irreducible representation of the algebra in which this evolution is implemented by a unitary one-parameter group, with a nonnegative energy operator, is constructed by the same method, as in the free Q case, in the same Fock space H. Thus −1/2 u, ja (V ) = h1/2 a v − iha
πa (W (V )) = W0 (ja (V )) .
(4.2)
The Hamilton operator is given by dΓ(ha ), with ground state described by the Fock vacuum Ω. However, as discussed in Section 2, we want to describe the same physical situation with the use of the representation π. Therefore for each a we look for a unitary operator Ua which by similarity transforms representation πa onto π: Ua πa (W (V ))Ua∗ = π(W (V )) ,
Ua Φa (V )Ua∗ = Φ(V ),
V ∈ L.
(4.3)
We substitute here the definitions of the representations π and πa , and set ja (V ) ≡ f . This condition then takes the form Ua W0 (f )Ua∗ = W0 (La f ) ,
f ∈ K,
where La := j ja−1 .
(4.4)
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Both j and ja are bijective symplectic mappings, so La is a symplectic transformation of the space (K, Im(. , .)), and the above condition states that Ua implements the corresponding Bogoliubov transformation in the Fock space (see Appendix A.3). As K is finite-dimensional, such Ua exists. The explicit form of transformation La is easily obtained: La f = h1/2 h−1/2 Re f + ih−1/2 h1/2 a a Im f ,
(4.5)
and then from (A.29) one finds La = Ta + Sa , Ta complex-linear and Sa complexantilinear, 1 1 Ta = (Ba−1 + Ba∗ ) , Sa = (Ba−1 − Ba∗ )K . (4.6) 2 2 where −1/2 Ba = h1/2 , Kf = f¯ . (4.7) a h In the representation π the Hamiltonians of the new evolutions are given by Ha = Ua dΓ(ha )Ua∗ ,
(4.8)
and the ground state of Ha is given by Ωa = U a Ω .
(4.9)
Suppose now, that a is a single real parameter, and under the influence of the external conditions the state of the subsystem Q changes adiabatically over the states Ωa , as discussed in Section 2. The potential for the backreaction force is therefore, in accordance with point (iv) in Section 2, determined by Ea = (Ωa , H Ωa ) .
(4.10)
We take into account that H = dΓ(h) and use the expression for a form matrix element of dΓ(h) as given in Eqs. (A.23), (A.24): Ea =
a(h1/2 fi )Ωa 2 , i
where {fi } is an arbitrary orthonormal basis of K. We use Eqs. (A.40), (A.42) and (A.45) to find
a(h1/2 fi )Ωa 2 = a∗La (Sa ∗ h1/2 fi )Ωa 2 = (fi , h1/2 Sa Sa ∗ h1/2 fi ) . Thus we obtain
1
Ea = Tr h1/2 Sa Sa ∗ h1/2 = Tr (ha − h)h−1 a (ha − h) . 4
(4.11)
Let the eigenvalues and orthonormal eigenvectors of h and ha be given by hei = i ei ,
ha eai = ai eai .
(4.12)
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−1 Then using the spectral representation h−1 a = k ak |eak eak | and employing the basis ei for the calculation of the trace we find Ea =
( ak − i )2 i,k
4 ak
|(eak , ei )|2 .
(4.13)
We consider a simple example. Let Q be a two-dimensional oscillator in physical space, vectors e1 , e2 being its main axes, and let the effect of the external conditions be the rotation of these axes by an angle ϕ (≡ a), without a change in the frequencies. In this case we have ϕk = k , (eϕ1 , e2 ) = −(eϕ2 , e1 ) = sin ϕ, so Eϕ =
( 2 − 1 )2 −1 2 ( 1 + −1 2 ) sin ϕ . 4
(4.14)
The backreaction “force” in this case is a torque Fϕ = −
( 2 − 1 )2 −1 ( 1 + −1 2 ) sin 2ϕ . 4
(4.15)
For 1 → 0 (with 2 kept constant) the torque tends to infinity. This is what one should expect. This limiting case describes the situation in which the harmonic force in the direction of e2 extends translationally invariant in the direction of e1 ; any rotation of this picture involves an “infinite” change. Note, that the “zero point” prescription for the force gives zero in the above example, which is an utterly unphysical prediction.
5 An infinite-dimensional system Let now R be an infinite-dimensional real Hilbert space. We want to consider a situation analogous to that discussed in the last section: system Q defined by the operator h, and its perturbations by a family of operators ha . We need to take into account complications arising from the unboundedness of the operators, as explained in Section 3. We want to be able to define for our model both evolutions (that determined by h and by ha ), and both ground state representations. Thus the model has to fit into structures defined in Section 3 both by h as well as ha . In particular, its symplectic space should in a canonical way be a part of both L and La . However, the construction of these spaces is based on different, in general, parts 1/2 −1/2 of R ⊕ R (DR (h1/2 ) ⊕ DR (h−1/2 ) and DR (ha ) ⊕ DR (ha ) respectively), and without some restrictions there is no canonical way of identification of their parts. We assume that D± ≡ DR (h±1/2 ) ∩ DR (h±1/2 ) is dense in R± and in Ra± , a
(5.1)
(in this, and similar statements below, signs are either all upper, or all lower). In this case space R± is the completion of D± with respect to the norm . ± , and Ra± is the completion of the same subspace with respect to . a± .
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Suppose now that the space of a model contains at least a subspace L0 such that L0 = L0+ ⊕ L0− ,
L0± ⊂ D± ,
L0± is dense in R± and in Ra± ,
(5.2)
which is a strengthening of the condition (3.42). Note that then j(L0 ) = j(L0 ), ja (L0 ) = ja (L0 ), and both spaces are dense in K. Under these assumptions we show that the following three conditions are equivalent: (i) The symplectic mapping La := jja−1 : ja (L0 ) → j(L0 )
(5.3)
extends to a bounded operator in K, with a bounded inverse. (ii) The operators h and ha satisfy the conditions DR (h±1/2 ) = DR (h±1/2 ), a −1/2 Ba ≡ h1/2 and Ba−1 extend to bounded operators in K . a h
(5.4) (5.5)
(iii) There exists a selfadjoint, positive, bounded operator Ca in R, with bounded inverse, and such that (5.6) ha = h1/2 Ca h1/2 in the sense of forms, that is: ha is the unique selfadjoint operator defined by the closed form q(v1 , v2 ) = (h1/2 v1 , Ca h1/2 v2 ) with the form domain Q(q) = DR (h1/2 ). so both evolutions are well defined If these conditions are satisfied, then La = L, in L. Note that (i) is a necessary condition for the ground state representations defined by h and ha to be equivalent. The equivalence implies that Eq. (4.3) is satisfied in particular for all V ∈ L0 , or, equivalently, Eq. (4.4) for all f ∈ ja (L0 ). But then, as shown in the Appendix A.3, La extends to a bounded symplectic mapping in K, with a bounded inverse. Let (i) be satisfied. Then on L0 from Eq. (5.3) we have La ja = j and 0 ja = L−1 a j, which implies that for w± ∈ L± one has
h±1/2 w± ≤ const. h±1/2 w± , a
h±1/2 w± ≤ const. h±1/2 w± . a
(5.7)
This means that the norms . ± and . a± are equivalent on L0± , so they yield the same completion, on which these inequalities are preserved. But L0± is dense in R± and Ra± , so R± = Ra± as sets, hence also L = La as sets, with equivalent norms. Moreover, from Eq. (3.18) we find DR (h±1/2 ) = Ra± ∩ R = R± ∩ R = DR (h±1/2 ) . a
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The boundedness of Ba and Ba−1 follows now from (5.7), which ends the proof of (ii). Conversely, if (ii) is satisfied, then one easily shows that the formula La f = Ba−1 Re f + iBa∗ Im f gives the extension needed in (i) (cf. Eq. (4.5)). The equivalence of (ii) and (iii) follows by polar decomposition of closed 1/2 operators (e.g., [14]). If we assume (ii), then ha = Ba h1/2 with the domain DR (h1/2 ) is a selfadjoint operator, so ha = h1/2 Ba∗ Ba h1/2 , and Ca = Ba∗ Ba fulfills the conditions of (iii). Conversely, let Ca be bounded, positive selfadjoint, with 1/2 a bounded inverse. Then Ca h1/2 with the domain equal to DR (h1/2 ) is a closed 1/2 operator. Indeed, let vn ∈ DR (h1/2 ), vn −vm → 0 and Ca h1/2 (vn −vm ) → 0. −1/2 But Ca is bounded, so also h1/2 (vn − vm ) → 0. As h1/2 is closed, there exists 1/2 v ∈ DR (h ) such that vn − v → 0, h1/2 (vn − v) → 0, and by boundedness 1/2 1/2 1/2 of Ca also Ca h1/2 (vn − v) → 0, which shows that Ca h1/2 is indeed closed. Thus the form q defined in (iii) is closed, and the condition (5.6) means that 1/2 1/2 |Ca h1/2 | = ha . It follows that there exists an orthogonal operator Fa such that 1/2 1/2 1/2 1/2 Ca h1/2 = Fa ha , so ha h−1/2 extends to a bounded operator Ba = Fa∗ Ca . −1 As Ca has a bounded inverse, so the same is true for Ba , which ends the proof of equivalence of (i)–(iii). These preliminary results show that if the two ground state representations are to be equivalent in our model, we have to assume that (5.6), and then all conditions (i)–(iii), are true. Then the space L is invariant under both evolutions and forms the widest possible space of the model. Any subspace (if it exists) L = L+ ⊕ L− ⊂ L which is also invariant under both evolutions and dense in L can also be taken as the symplectic space of the model. With these assumptions the symplectic mapping La decomposes into the bounded complex-linear and complex-antilinear parts, La = Ta + Sa . The application of the results described in Appendix A.3 shows that the necessary and sufficient condition for the unitary equivalence of the ground state representations is that Sa is a Hilbert-Schmidt (HS) operator, that is
Na ≡ Tr Sa Sa∗ < ∞ . (5.8) Going through the steps (4.10–4.11) in the present infinite-dimensional context we see that Ea is finite if, and only if, Sa∗ h1/2 extends to a HS operator, and then
(5.9) Ea = Tr h1/2 Sa Sa∗ h1/2 . The quantity Na appearing in (5.8) has a clear-cut physical meaning. The results of the Appendix A.3 show that if (5.8) is satisfied, then Ωa ∈ D(N ), where N is the “particle” (excitation) number operator. A calculation analogous to that carried out for the energy yields (Ωa , N Ωa ) = Na , so Na is the mean value of the excitation number in the ground state.
(5.10)
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In the rest of this section we obtain the following criterion for admissible perturbations. Let ha be given by (5.6). The ground state representations are unitarily equivalent (Na < ∞) if, and only if, Ca = id +δa ,
(5.11)
where δa is any operator satisfying conditions δa is a HS operator ,
id +δa > 0 .
(5.12)
In this case we can write in the sense of forms ha = h + h1/2 δa h1/2 .
(5.13)
Moreover, if conditions (5.12) are satisfied, then Ea is finite if, and only if, δa h1/2 extends to a HS operator .
(5.14)
With the condition (5.12) satisfied one has 1 δa2 , Tr 4 id +δa
(5.15)
1 1/2 δa2 Tr h h1/2 . 4 id +δa
(5.16)
Na = and if in addition (5.14) holds, then Ea =
To prove these assertions note first that equations (4.6) remain in force with our assumptions in the present infinite-dimensional context, and then Sa Sa∗ =
1 (Ca − id)2 . 4 Ca
(5.17)
If the ground state representations are equivalent, then Sa∗ is a HS operator, so −1/2 1/2 (Ca − id) is HS as well. But Ca is bounded, therefore also δa = Ca − id is Ca HS. The second condition in (5.12) is satisfied by the positivity of Ca . Conversely, suppose that δa satisfies conditions (5.12). By the first of these conditions δa has a purely discrete spectrum with no other convergence points than zero, and then −1/2 by the second Ca = id +δ a ≥ b id, with b > 0. Hence Ca−1 is bounded, and Ca δa 1 is HS, so Na = Tr Sa Sa∗ = 4 Tr[δa2 (id +δa )−1 ] < ∞. If the conditions (5.12) are satisfied, then in a completely analogous way one proves the equivalence of the condition (5.14) with the finiteness of Ea , and the equation (5.16).
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6 Energy density of quantum field In this section we consider the case of a quantum field, and for definiteness we take the scalar field (massive or massless) with standard √ commutation relations and free evolution. Thus here R = L2 (R3 ), h = m2 id −∆, (m ≥ 0), as described in and we take for the test function space the largest space L, the previous section. Perturbations ha are assumed to be in the class defined by Eqs. (5.11–5.14) (in fact, a slight strengthening of these conditions will be needed). We show that in this setting the energy density in the ground states Ωa is well defined as a tempered distribution, and for the test function tending to unit function one recovers the energy expectation value Ea . We assume a slight strengthening of our assumptions and demand that for some α ∈ (0, 1) there is: h(1+α)/4 δa h(1+α)/4
is a HS operator .
(6.1)
Note that this statement with α = 0 is a consequence of our earlier assumptions. Indeed, if δa and δa h1/2 are HS, then
0 ≤ Tr(h1/4 δa h1/4 )2 = lim Tr P0,n (h)(h1/4 δa h1/4 )2 P0,n (h) n→∞
= lim Tr δa h1/2 δa h1/2 P0,n (h) = Tr(δa h1/2 )2 < ∞ , n→∞
where {PF (h)}, F a Borel set in R, is the spectral family of h. Loosely speaking, the energy density operator of the scalar field is determined by the point-splitting procedure and normal ordering with respect to the vacuum as
(6.2) H( x) = lim : H2 ( x, y ) : ≡ lim H2 ( x, y) − (Ω, H2 ( x, y )Ω) , y → x
y → x
1
x) · ∇X(
y) + m2 X( x)X( y ) , H2 ( x, y) = P ( x)P ( y ) + ∇X(
2
(6.3)
where X(u) = Φ(0, u), P (v) = Φ(v, 0) (see (3.32) and the preceding remarks), and to get X( x) and P ( x) one sets formally v and u equal to Dirac delta concentrated at x. We are interested in the energy density (Ωa , H( x)Ωa ) in the ground state Ωa . We now make this precise. The real Schwartz test function space SR is contained in DR (h1/2 ) ∩ DR (h−1/2 ), so functions from that space may be used for “smearing” both X( x) as P ( x). Let w1 , w2 ∈ SR . The precise meaning of (6.3) is H2 (w1 , w2 ) 1
1 ) · Φ(0, ∇w
2 ) + m2 Φ(0, w1 )Φ(0, w2 ) . (6.4) = Φ(w1 , 0)Φ(w2 , 0) + Φ(0, ∇w 2 To find normal-ordered expectation value (Ωa , : H2 (w1 , w2 ) : Ωa ) we need to know (Ωa , : Φ(V1 )Φ(V2 ) : Ωa ), where we assume that Vi ∈ SR ⊕ SR . We recall the definitions of the representations π and πa and their equivalence relations: Φ(V ) = Φ0 (j(V )) ,
Φa (V ) = Φ0 (ja (V )) ,
Ωa = U a Ω ,
Ua Φa (V )Ua∗ = Φ(V )
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(Eqs. (3.39), (4.2), (4.9) and (4.3) respectively). Using them one finds 1 (j(V1 ), j(V2 )) 2 1 1 = (h1/2 v1 , h1/2 v2 ) + (h−1/2 u1 , h−1/2 u2 ) + 2 2 1 (Ωa , Φ(V1 )Φ(V2 )Ωa ) = (Ω, Φa (V1 )Φa (V2 )Ω) = (ja (V1 ), ja (V2 )) 2 1 1/2 1 −1/2 1/2 = (ha v1 , ha v2 ) + (ha u1 , h−1/2 u2 ) + a 2 2 (Ω, Φ(V1 )Φ(V2 )Ω) =
(Ωa , : Φ(V1 )Φ(V2 ) : Ωa ) =
(6.5) i σ(V1 , V2 ) , 2 (6.6) i σ(V1 , V2 ) , 2
1 1/2 1 −1/2 δa (h v1 , δa h1/2 v2 ) − h u1 , h−1/2 u2 , 2 2 id +δa (6.7)
so (Ωa , : H2 (w1 , w2 ) : Ωa ) ≡ Ta (w1 , w2 ) = Ta1 (w1 , w2 ) + Ta2 (w1 , w2 ) , where Ta1 (w1 , w2 ) =
Ta2 (w1 , w2 ) =
1 1/2 δa2 h w1 , h1/2 w2 , 4 id +δa
(6.8)
(6.9)
m2 1 1/2 δa δa h w1 , h−1/2 w1 , h1/2 w2 − h−1/2 w2 4 id +δa 4 id +δa 1 −1/2
δa
2 . (6.10) ∇w1 , h − h−1/2 ∇w 4 id +δa
We have added the conjugation sign over w1 on the r.h. side to make the expression linear rather than antilinear also for complex functions. The Ta1 part and the first term in Ta2 result from splitting δa =
δa2 δa + . id +δa id +δa
(6.11)
We show that: (i) Ta (w1 , w2 ) defines a distribution Ta ( x, y) on S(R6 ). (ii) For each η ∈ R3 the expression Ta (ξ + η , ξ − η ) is a distribution on S(R3 ), and for each test function f the function
d3 ξ
η → Ea (
η , f ) = Ta (ξ + η , ξ − η )f (ξ) (6.12) is continuous and bounded (we use the “integral” notation of distributions). The energy density according to point-splitting procedure is then the distribution Ea (f ) ≡ Ea ( 0, f ) . (6.13)
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= f ( ξ),
f ( 0) = 1, f ∈ S(R). Then (iii) Let f (ξ) lim Ea (f ) = Ea .
→0
(6.14)
Before starting the proof we fix conventions for the Fourier transforms. For a, b ∈ Rn we set (6.15) fˆ(b) = (2π)−n/2 f (a)e−ib·a d3 a , fˇ(a) = fˆ(−a) . We consider the Ta1 and Ta2 parts separately. Expressions (6.12) and (6.13) for Tai replacing Ta will be denoted Eai (
η , f ) and Eai (f ) respectively. As 12 δa (id +δa )−1/2 h1/2 is a HS operator in L2 (R3 ), it is an integral operator with a kernel ka ( x, y ) ∈ L2 (R6 ) (see, e.g., [14]). Thus Ta1 is obviously a distribution on S(R6 ), determined by the ordinary function Ta1 ( x, y) = ka ( z , x)ka ( z, y ) d3 z . (6.16) As for each
η there is ka ( z, ξ + η )ka ( z, ξ − η ) ∈ L1 (R6 , d3 z d3 ξ), the distribution Ea1 (
η , f ) is indeed well defined,
a ( z, ξ − η ) d3 z d3 ξ . Ea1 (
η , f ) = ka ( z, ξ + η )f (ξ)k (6.17)
a ( z, ξ − η ) with respect to z and Now, Fourier-transforming ka ( z, ξ +
η ) and f (ξ)k
ξ one finds 1 η · (
p + q) d3 r d3 p d3 q , (6.18) η, f ) = q )fˆ(
p − q)e−i
kˆa ( r, p )kˆa ( r,
Ea1 (
(2π)3/2 as the integrand on the r.h. side is absolutely integrable. Therefore Ea1 ( η , f ) is continuous in
η . For
η = 0 we get
2 f (ξ)
d3 zd3 ξ . (6.19) Ea1 (f ) = |ka ( z, ξ)|
2 is absolutely integrable, we see immediately that for f As the function |ka ( z, ξ)| as in (iii) there is
2
2 d3 z d3 ξ = 1 Tr h1/2 δa h1/2 = Ea . lim Ea1 (f ) = |ka ( z, ξ)| (6.20) →0 4 id +δa We now turn to Ta2 and take into account our assumption (6.1). Using the identity (6.11) and the fact that δa and δa h1/2 are HS, one finds that an equivalent formulation of the assumption is that h(1+α)/4 δa (id +δa )−1 h(1+α)/4 is a HS
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operator; we denote its kernel in the momentum space by la (−
p, q ). We evaluate Ta2 (w1 , w2 ) in momentum space, making use of the identity w1 (
p) = w ˆ1 (−
p): 1 la (
Ta2 (w1 , w2 ) = p, q )t(
p, q )w ˆ1 (
p)w ˆ2 ( q) d3 p d3 q , (6.21) 4 where
2
p · q − m2 2 2 2 (1−α)/8 t(
p, q ) = (
p + m )(
q +m ) . 1+ (
p2 + m2 )( q2 + m2 )
(6.22)
As la is square integrable, and t polynomially bounded, Ta2 defines a distribution Ta2 ( x, y). Let f, g ∈ S(R3 ). Then
η ) d3 ξ d3 η Ta2 (ξ +
η , ξ −
η )f (ξ)g(
= 2 la ( r + s, r − s) t( r + s, r − s)fˆ(2 r)ˆ g(2 s) d3 r d3 s , (6.23) where on the l.h. side the integral notation is symbolic, but on the r.h. side this is the ordinary integration. Now, one shows the following estimate t( r + s, r − s) ≤
4| r|2 , (| r |2 + | s|2 + m2 )(3+α)/4
(6.24)
(note that t(
p, q ) ≥ 0). To prove this it is convenient to consider the cases | r|2 ≥ | s|2 + m2 and | r|2 < | s|2 + m2 separately. In the first region one then 2
(1−α)/8 uses the obvious bound t(
p, q ) ≤ 2 (
p + m2 )( q2 + m2 ) , while in the second one finds that for the given | r| and | s| the function on the l.h. side is the biggest for r · s = 0. Using the bound one easily shows that (6.25) [t( r + s, r − s)]2 d3 s ≤ const. | r|4−α . Therefore t( r + s, r − s)fˆ(2 r) ∈ L2 (R6 ), so la ( r + s, r − s) t( r + s, r − s)fˆ(2 r) ∈ L1 (R6 ) .
(6.26)
Using this fact in (6.23) one finds that
d3 ξ Ea2 (
η , f ) = Ta2 (ξ + η , ξ −
η )f (ξ) (6.27) 1 = √ la ( r + s, r − s) t( r + s, r − s)fˆ(2 r)e−i2 η · s d3 r d3 s 2π 3/2 indeed defines a distribution and is continuous in η . Thus 1 √ Ea2 (f ) = la ( r + s, r − s) t( r + s, r − s)fˆ(2 r) d3 r d3 s . 2π 3/2
(6.28)
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Using square-integrability of la and the estimate (6.25) we have |fˆ( r)|2 | r|4−α d3 r . |Ea2 (f )|2 ≤ const.
683
(6.29)
For f as in (iii) one easily then finds lim Ea2 (f ) = 0 ,
→0
(6.30)
which ends the proof of our claims. In our calculation of the energy density we have used the standard definition of the Wick normal ordering. As an aside, it may be of interest to mention that this definition has been recently improved for the cases where the reference state Ω depends on external fields (as, e.g., in a fixed curved classical spacetime). The problem with the usual definition in such cases is, that the scalar subtraction function depends nonlocally on the background. This may be remedied, as it turns out, by an additional subtraction of a smooth function (in “Hadamard states”; see the papers by Hollands and Wald [15], and Brunetti, Fredenhagen and Verch [16]; for an application to external field electrodynamics see also [17]). This has no immediate bearing on the discussion in the present work, but may have applications in related problems with external fields present from the start (as in a curved spacetime).
7 Remarks on relations with some other approaches In this section we make some remarks on the relation of our approach to other calculations of Casimir energy. We shall discuss a few characteristic examples of local calculations, and next comment on the “zero point” ideology. First, to make our point on local Casimir energy, we need to consider a general situation briefly sketched in the introduction, where two representations of local algebras in some open region M0 in spacetime are locally quasiequivalent. Suppose we have two representations π and π ˜ of the algebras of observables in M0 , ˜ respectively. We assume that the representations acting in Hilbert spaces H and H are locally quasiequivalent, but say nothing on their (global) equivalence. This is the expected state of affairs in many situations typically considered for Casimir problems. For instance, for a scalar field M0 may be the whole spacetime outside some 2-surfaces in 3-space, π the vacuum representation of the field, and π ˜ the representation built on the ground state of the field in presence of the boundary conditions imposed on the boundaries of M0 . We choose a state in the representa˜ If the representations are not equivalent tion π ˜ , that is a density operator ρ˜ in H. it makes no sense to ask for a state in the representation π which gives the same expectation values as ρ˜ for all observables. However, the local quasiequivalence tells us that if we restrict attention to an open subset O with a compact closure contained in M0 , then there exists a density operator ρO in H such that Tr[˜ ρπ ˜ (A)] = Tr[ρO π(A)]
for A in O .
(7.1)
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The local energy density is not one of the fundamental local observables A, but it may be locally built with the use of them. In the sequel we restrict attention to the scalar field and construct local energy density as in (6.2). Thus given a state ρ˜ the Casimir energy density in O according to the views we follow in this paper is ˜ x) = Tr[ρO H( x)] E(
(7.2)
(expectation value of a fixed, free field energy density operator). Let ρ˜ be, for sim˜ Then using (6.2) and (7.1) plicity, the projection operator onto the unit vector Ω. we can write for the Casimir energy at a given time t = 0:
˜ x)X(
˜ x ) Ω ˜ P˜ ( x)P˜ ( x ) + ∇
X(
˜ x) · ∇
X(
˜ x ) + m2 X(
˜ ˜ x) = 1 lim Ω, E(
x → x 2
− Ω, P ( x)P ( x ) + ∇X(
x) · ∇X(
x ) + m2 X( x)X( x ) Ω ,
(7.3)
˜ ˜ u), P˜ (v) = Φ(v, ˜ 0), and Φ where X(u) = Φ(0, u), P (v) = Φ(v, 0), X(u) = Φ(0, ˜ are operators representing the field under π and π and Φ ˜ respectively. Recall that ˜ = H, Ω ˜ = Ω, Φ ˜ = Φa , and one in the case discussed in Section 6 there is H recovers the formula obtained at the beginning of that section. More generally, let in each of the representations a different time evolution be given by unitary opera˜ tors: free evolution U (t) and evolution influenced by background U(t) respectively, and denote Xt ( x) = U (t)X( x)U (t)∗ ≡ ϕ(t, x) , ˜ t ( x) = U (t)X(
˜ x)U (t)∗ ≡ ϕ(t, X ˜ x) ,
Pt ( x) = U (t)P ( x)U (t)∗ , P˜t ( x) = U (t)P˜ ( x)U (t)∗ .
(7.4)
˜ x)/∂t, then one If for both evolutions there is Pt ( x) = ∂ϕ(t, x)/∂t, P˜t ( x) = ∂ ϕ(t, ˜ x) at t = 0 as can write the last formula for E(
˜ x) = 1 lim ∂t ∂t + ∇ ˜ ϕ(t,
·∇
+ m2 (Ω, ˜ − (Ω,ϕ(t, x)ϕ(t , x )Ω) E(
˜ x)ϕ(t ˜ , x )Ω) 2 t,t →0 x x → (7.5) This formula was derived along similar lines by Kay [12] in the context of the free field in a locally flat spacetime with nontrivial topology. There are no boundaries in that case, but the net of local algebras of observables in this spacetime differs from that in the globally flat Minkowski spacetime, so the notion of a global Casimir energy in the sense we use here has no application. In the context of electromagnetic field bounded by conductors in Minkowski space the opinion similar to ours, that one should compare expectation values of the fixed free field energy density, was expressed by Scharf and Wreszinski [13]. Consider a massless scalar field analogy of the setting. Then M0 is the spacetime region outside boundaries. Put m = 0 in the last formula, use the translational
→ −∇),
symmetry of the two-point functions (which enables the replacement ∇
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and the wave equation, which both correlation functions satisfy outside the boundaries. This leaves us with ˜ ϕ(t, ˜ − (Ω, ϕ(t, x)ϕ(t , x )Ω) , ˜ x) = 1 lim ∂t ∂t + ∂ 2 (Ω, ˜ x)ϕ(t ˜ , x )Ω) E(
t 2 t,t →0 x x → (7.6) which is the formula used in [13]. No global energy density may be obtained in this way (if not by an ad hoc regularization of the infinities in the density) due to the algebraic problems explained earlier. Next, we want to comment on the “Green function” method. In papers following this method one usually states that the (local) Casimir energy is the difference between the energy “in the vacuum state with the barriers present and with them absent” (see, e.g., [10]), with no further explanation on what energy is meant. Staying with the scalar field as our example, one then uses with not much comment a formula similar to (7.5), in which, however, the products of fields are replaced by time-ordered products. This brings no change of the result in this simple case, but in general has to be justified. As long as outside the barriers the field follows the same local equation (the distinct time evolutions agree locally), the ambiguity as to what energy is meant does not show up. However, this does not matter only because for sharp boundaries one cannot determine the global energy anyway. And in fact, if the barriers are replaced by external fields one has to make it clear what is being calculated. An example of such calculation is attempted in [2], where one of the sections treats on the quantum Dirac field in an external classical electromagnetic field. The authors’ intention apparently is to compare the energy of the Dirac field itself, so they keep the free field energy expression. However, they take over the form of this expression containing time derivatives, and to eliminate them they use different field equations in the two cases, which spoils the original intention (remember that the Dirac equation is first order, so the time derivative of the field is not an independent initial value variable). Another example of the external field calculations is to be found in [11]. Here the authors with the intention of finding the global Casimir energy explicitly compare expectation values of two different energy operators: energy of the field with the interaction terms included and the energy as given by the free field theory, in the ground states of the two respective evolutions. In our opinion this is one of the reasons for the appearance of infinities in their expressions, which are eliminated by adding “counterterms” to the model which does not need them (except for trivial normal ordering of quadratic observables). We note, moreover, that it does not follow from the smoothness of the external field alone that the ground state representations, with the external field present or not, are equivalent globally. In the rest of this section we try to understand, from the point of view of the formalism presented in this paper, how “zero point” expressions may arise in the context of Casimir effect for quantum fields. In our opinion their appearance is a consequence of unjustified manipulations. Accordingly, the equations and trans-
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formations to be found below are not to be taken at face value. We indicate this by putting a dot over the equality sign. The “zero point” expression for Casimir energy has the form 1 . 1 Eaz.p. = ωak − ωk , (7.7) 2 2 k
k
where ωk and ωak are appropriately discretized frequencies of free and perturbed field respectively. In our language this would be . 1 Eaz.p. = Tr(ha − h) , (7.8) 2 which usually is meaningless, but is then “regularized” to squeeze a finite result. We show how this expression may arise. We have shown in the previous section that Ea is a limit of the energy density distribution value for the test function tending to one, see Eq. (6.14). Also, it turned out that in this limit only the part Ea1 (f ), Eq. (6.19), of the density distribution contributes. Thus we can use Eq. (6.20) to calculate the total energy. Distribution Ea1 (f ) is determined by part Ta1 , Eq. (6.9), of (Ωa , : H2 (w1 , w2 ) : Ωa ). If we do not pay due attention to domains we can rewrite Eq. (6.9) by expressing it in terms of h and ha instead of h and δa . The result is . 1 −1 )h| y w1 ( x)w2 ( y ) d3 x d3 y , (7.9) Ta1 (w1 , w2 ) = x|ha − h + h(h−1 a −h 4 which implies
. 1 −1 Ea = )h| x d3 x x|ha − h + h(h−1 a −h 4 (7.10)
. 1 −1 = Tr ha − h + h(h−1 − h )h . a 4 Let us now, again ignoring difficulties, apply this to the case of sharp boundaries, where h2a = −∆B(a) with appropriate boundary conditions B(a). Suppose that the support of w1 and w2 stays outside the boundaries. Then h2a wi = h2 wi , and we have 1 . 1 −1 2 −1 )h w2 ) + (w1 , h2 (h−1 )w2 ) , (7.11) (w1 , (ha − h)w2 ) = (w1 , (h−1 a −h a −h 2 2 (written in two terms only for symmetry reasons), or, for x and y outside the boundary, . 1 −1 2 −1 x|ha − h| y = x|(h−1 )h + h2 (h−1 )| y . (7.12) a −h a −h 2 This needs regularization on the boundaries. Assuming some form of it one writes . 1 −1 2 −1 Tr(ha − h) = )h + h2 (h−1 )| x d3 x x|(h−1 a −h a −h 2 (7.13)
. 1 −1 2 2 −1 −1 = Tr (h−1 − h )h + h (h − h ) . a a 2
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Suppose that the regularization used cuts high momenta, so as to allow one to change the order of operators under the trace sign. Then
. −1 Tr(ha − h) = Tr h(h−1 )h . (7.14) a −h Using this in (7.10) one arrives at (7.8).
A
Appendix. Fock space operators and Bogoliubov transformations
In the appendix we give a brief review of some known properties of Fock space operators which are needed in the main text. The main sources of reference for Section A.1 are books [9] (vol. II) and [14]. The content of Sections A.2 and A.3 is a rather common knowledge. Precise original proofs of the criterions of equivalence of representation use rather more advanced and less common techniques [18], so we think a simple proof with the use of creation/annihilation operators is worth presenting in A.4. (The results on equivalence have been later generalized, in the widest form in [19].)
A.1 Weyl system in a Fock space Let H be the symmetric Fock space based on the “one-particle (excitation) space” K, i.e., ∞ H= Hn , H0 = C , Hn = S(K ⊗ · · · ⊗ K) (n ≥ 1) , (A.1) n=0
n times
where S is the symmetrization projection operator. The scalar product in H will be denoted by (. , .), the “Fock vacuum” vector by Ω, and the particle (excitation) number operator by N . On the domain D(N 1/2 ) the annihilation and creation operators are defined in the usual way: for each f ∈ K and ψ, χ ∈ D(N 1/2 ) one sets √ (χ, a(f )ψ) = (a∗ (f )χ, ψ) , (A.2) a∗ (f )ψ = S(f ⊗ N + 1 ψ) , and shows that
a# (f )ψ ≤ f (N + 1)1/2 ψ , and for ϕ ∈ D(N )
a# (f ) = a(f ) or a∗ (f ) ,
[a(f ), a∗ (g)]ϕ = (f, g)ϕ .
(A.3)
(A.4)
∗
Operators a(f ) and a (f ) are respectively antilinear and linear in f . Let Hf be the finite-excitation subspace (dense in H), i.e., Hf =
k ∞ k=0 n=0
Hn .
(A.5)
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Operators Φ0 (f ) are defined in the following way. One initially sets 1 Φ0 (f )ψ = √ a(f ) + a∗ (f ) ψ 2
for ψ ∈ D(N 1/2 ) .
(A.6)
Using the bounds (A.3) one shows that these operators are essentially selfadjoint on Hf , so their closures Φ0 (f ) are selfadjoint. For ψ ∈ D(N 1/2 ), ϕ ∈ D(N ), f, g, fk ∈ K and real α, β one has Φ0 (αf + βg)ψ = αΦ0 (f )ψ + βΦ0 (g)ψ , 1 1 a(f )ψ = √ Φ0 (f ) + iΦ0 (if ) ψ , a∗ (f )ψ = √ Φ0 (f ) − iΦ0 (if ) ψ , 2 2 if fk − f → 0 then Φ0 (fk )ψ − Φ0 (f )ψ → 0 (k → ∞) , [Φ0 (f ), Φ0 (g)]ϕ = i Im(f, g)ϕ .
(A.7) (A.8) (A.9) (A.10)
Using these relations one shows that the Weyl operators defined by W0 (f ) = eiΦ0 (f )
(A.11)
have the following properties i
W0 (f )W0 (g) = e− 2 Im(f,g) W0 (f + g) , W0 (f )∗ = W0 (−f ) , W0 (0) = id ; (A.12) (A.13) the set {W0 (f ) | f ∈ K} is irreducible ; 1
2
(Ω, W0 (f )Ω) = e− 4 f ;
(A.14)
if fk − f → 0 then W0 (fk )ψ − W0 (f )ψ → 0 (k → ∞) , ψ ∈ H . (A.15) Let U be a unitary operator in K. One defines a unitary operator Γ(U ) in H by which implies
Γ(U )W0 (f )Ω = W0 (U f )Ω ,
(A.16)
Γ(U )W0 (f )Γ(U )∗ = W0 (U f ) .
(A.17)
It is then easy to show that Γ(U ) : Hn → Hn ,
Γ(U )Hn = U ⊗ · · · ⊗ U .
(A.18)
n times
Let now h be a selfadjoint operator in K. Then Γ(eith ) is a one-parameter group of unitary operators. The generator of this group, denoted dΓ(h), is a selfadjoint operator, (A.19) Γ(eith ) = exp(itdΓ(h)) . Let Dh be any domain of essential selfadjointness of h and denote DdΓ(h) =
∞ k k=0 n=0
S(Dh · · · Dh ) , n times
(A.20)
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which means that DdΓ(h) is formed by finite linear combinations of symmetrized products of vectors from Dh . One shows that dΓ(h)S(D ...D ) = h ⊗ id ⊗ · · · ⊗ id + · · · + id ⊗ · · · ⊗ id ⊗h , (A.21) h
h
and that dΓ(h) is essentially selfadjoint on DdΓ(h) . We assume now that h is a nonnegative operator. Then dΓ(h) is also nonnegative and has the following representation in terms of quadratic forms. As (a(f ))∗ is densely defined (its domain contains D(N 1/2 )), the annihilation operator a(f ) is closable, we denote its closure by a ¯(f ). Let {fi } be any orthonormal basis of K formed of vectors in D(h1/2 ), and denote D(¯ a(h1/2 fi )) and
¯ a(h1/2 fi )ψ 2 < ∞} . (A.22) Q(q) = {ψ ∈ H | ψ ∈ i
i
One shows that the following form on Q(q) is closed (¯ a(h1/2 fi )ψ, a ¯(h1/2 fi )χ) . q(ψ, χ) =
(A.23)
i
It is easy to check by direct calculation that the restriction of this form to DdΓ(h) gives q(ψ, χ) = (ψ, dΓ(h)χ) . (A.24) As DdΓ(h) is a core of dΓ(h), the unique selfadjoint operator defined by the form q is identical with dΓ(h). Thus Q(q) = D(dΓ(h)
1/2
)
and q(ψ, χ) = (dΓ(h)
1/2
1/2
ψ, dΓ(h)
χ) .
(A.25)
In particular, for all ψ, χ ∈ D(dΓ(h)) identity (A.24) holds. We note that the particle number operator may be represented as a special case of this construction, N = dΓ(id) . (A.26)
A.2 Symplectic transformations of (K, Im(. , .)) Hilbert space K, as a real vector space, is a symplectic space with the form Im(. , .). Its real-liner, bijective transformation L is a symplectic transformation if for all f, g ∈ K Im(Lf, Lg) = Im(f, g) . (A.27) The inverse transformation is then also a symplectic transformation satisfying the same condition. Substituting f → L−1 f in (A.27) one has Im(f, Lg) = Im(L−1 f, g) .
(A.28)
One defines operators on K:
T =
T = 12 (L − iLi) , S = 12 (L + iLi) , L = T + S ,
−1 1 2 (L
− iL
−1
i) , S =
−1 1 2 (L
+ iL
−1
i) , L
−1
(A.29)
=T +S .
(A.30)
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Operators T and T are complex-linear, while S and S are complex-antilinear. Using their definitions and the relation (A.28) it is easy to show that operators in the two pairs T , T and S , −S are mutually adjoint, so T = T ∗∗ and S = S ∗∗ . Thus both operators are everywhere defined and closed, so they are bounded. Separating the identities L−1 L = id and LL−1 = id into linear and antilinear parts one gets T ∗ T = S ∗ S + id , ∗
∗
T T = SS + id ,
T ∗S = S ∗ T , ∗
(A.31)
∗
T S = ST .
(A.32)
Conversely, if the operators T and S satisfy the above relations on the whole Hilbert space K, they are bounded and define a symplectic transformation L = T + S. Furthermore, if the relations are satisfied, then T is a bijection of K onto K. Thus if T = UT |T | is its unique polar decomposition, then UT is a unitary operator. We set S = UT R. It follows then from the first equalities in (A.31) and (A.32) that R∗ R = RR∗ . If R = K|S| is the unique polar decomposition of R, then this condition is equivalent to K|S| = |S|K, so K is a partial antiisometry of (Ker |S|)⊥ onto itself. From the first relation in (A.31) |T |2 = id +|S|2 , so K|T | = |T |K as well. The second relation in (A.31) then gives K ∗ = K. We summarize the results: T = UT (id +|S|2 )1/2 ,
S = UT |S|K ,
[|S|, K] = 0 ,
(A.33) ⊥
|S| is bounded , UT is unitary , K is a conjugation on (Ker |S|) .
(A.34)
Conversely, if these conditions are satisfied, then T and S satisfy conditions (A.31) and (A.32), and determine a symplectic transformation by L=T +S,
L−1 = T ∗ − S ∗ .
(A.35)
If |S| has no continuous spectrum, then it follows from the above relations that its orthonormal basis of eigenvectors may be chosen such that |S|fi = λi fi ,
Kfi = fi .
(A.36)
A.3 Bogoliubov transformations in a Fock space With the notation of the foregoing subsections let L = T + S be a symplectic transformation of the space (K, Im(., .)), and let us denote W0 L (f ) = W0 (Lf ) .
(A.37)
It is easy to show that these new operators also satisfy the Weyl relations (A.12). The transformation W0 (f ) → W0 L (f ) is called a Bogoliubov transformation. Its equivalent form is Φ0 (f ) → Φ0L (f ) = Φ0 (Lf ) ,
W0L (f ) = eiΦ0 L (f ) .
(A.38)
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For ψ ∈ D(Φ0 (Lf )) ∩ D(Φ0 (Lif )) one defines 1 aL (f )ψ = √ Φ0 L (f ) + iΦ0 L (if ) ψ , 2
1 a∗L (f )ψ = √ Φ0 L (f ) − iΦ0 L (if ) ψ , 2 (A.39) and shows by a simple calculation that for ψ ∈ D(N 1/2 ): aL (f )ψ = a(T f )ψ + a∗ (Sf )ψ ,
a∗L (f )ψ = a∗ (T f )ψ + a(Sf )ψ .
(A.40)
Then using Eqs. (A.31) one also finds for ψ ∈ D(N 1/2 ) a(f )ψ = aL (T ∗ f )ψ − a∗L (S ∗ f )ψ , and for ϕ ∈ D(N )
a∗ (f )ψ = a∗L (T ∗ f )ψ − aL (S ∗ f )ψ ,
[aL (f ), a∗L (g)]ϕ = (f, g)ϕ .
(A.41) (A.42)
One says that the Bogoliubov transformation is implementable in H if there exists a unitary operator UL such that either of the following (and then both) conditions hold W0 L (f ) = UL W0 (f )UL∗ ,
Φ0 L (f ) = UL Φ0 (f )UL∗ ,
f ∈ K.
(A.43)
The necessary and sufficient condition for the implementability of the Bogoliubov transformation is that S be a Hilbert-Schmidt operator, i.e.,
(A.44) Tr S ∗ S < ∞ . If the condition is satisfied, then there exists a unique, up to a phase factor, normalized vector ΩL satisfying the conditions aL (f )ΩL = 0 , Moreover, one has ΩL ∈
∞
f ∈ K.
(A.45)
D(N l/2 ) .
(A.46)
l=1
Equations UL a∗ (f1 ) . . . a∗ (fk )Ω = a∗L (f1 ) . . . a∗L (fk )ΩL ,
k = 0, 1, . . . ,
(A.47)
with arbitrary test vectors fi , define the unique (up to a phase factor) unitary operator UL implementing the Bogoliubov transformation. For the completeness we sketch a simple proof of these statements in the next subsection. A slight generalization of the above results is needed in the main text. Let J and J be real subspaces of K, dense in K, and let L : J → J be a bijective symplectic transformation (i.e., a real-linear transformation satisfying Eq. (A.27) for f ∈ J ). Suppose that there exists a unitary operator UL such that W0 (Lf ) = UL W0 (f )UL∗ ,
f ∈J .
(A.48)
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Then L and L−1 extend to bounded symplectic transformations on K, and Eq. (A.43) is satisfied for all f ∈ K. Indeed, suppose that (A.48) is fulfilled. Then using Eq. (A.14) one finds 1
2
e− 4 Lf = (UL∗ Ω, W0 (f )UL∗ Ω) ,
f ∈J ,
(A.49)
which shows that L is a continuous transformation on its domain (if fn → 0, then by Eq. (A.15) also Lfn → 0). Thus L extends by continuity to a bounded operator on K. From Eq. (A.48) we have UL∗ W0 (f )UL = W0 (L−1 f ) for f ∈ J , thus similar reasoning shows that the extension of L is a bijective symplectic transformation of K onto itself. Equation (A.48) now extends by (A.15) to all f ∈ K.
A.4 Proof of the statements (A.44–A.47) Let the Bogoliubov transformation be implemented as in Eq. (A.43). For each pair of vectors ψ, ϕ ∈ D(N 1/2 ) one has then (a∗L (f )ψ, UL ϕ) = (ψ, UL a(f )ϕ). We substitute here ϕ = Ω, f = T −1 g, and use Eq. (A.40). This yields (a∗ (g)ψ, ΩL ) = −(a(ST −1 g)ψ, ΩL ) ,
(A.50)
where ΩL = UL Ω. Substituting here for ψ all vectors of the form a∗ (g1 ) . . . a∗ (gk )Ω for k = 0, 1, . . . recursively, it is easy to see that (Ω, ΩL ) cannot vanish, as otherwise ΩL would be orthogonal to the whole Hilbert space. Let now {fi } be an orthonormal basis and put in (A.50) g = fi and ψ = a∗ (fj )Ω, which gives (a∗ (fi )a∗ (fj )Ω, ΩL ) = −(ST −1 fi , fj )(Ω, ΩL ) .
(A.51)
Take the sum over i, j of the absolute values squared of both sides of this equation. On the l.h. side one then gets a quantity smaller or equal 2 ΩL 2 , so ST −1 is a Hilbert-Schmidt operator. As T is a bounded operator, the same is true for S. Conversely, let now S be a HS operator, so there exists an orthonormal basis {fi } satisfying (A.36), and denote gi = UT fi , which defines another orthonormal basis. Then λi Sfi = λi gi , T fi = λ2i + 1gi , so ST −1 gi = 2 gi , λi + 1 (A.52) ∞ 2 λi < ∞ . where i=1
We look for a vector ΩL which lies in the domain of all operators aL (f ) and satisfies Eq. (A.45). If such vector exists, then it must satisfy Eq. (A.50) for all possible g and ψ. It is sufficient to substitute for g all basis vectors gi and for ψ all vectors from the basis of the particle number representation {|n1 , n2 , . . .} with profiles g1 , g2 , . . .. This gives the recurrent conditions λi n1 . . . ni + 1 . . . |ΩL = − 2 λi + 1
ni n1 . . . ni − 1 . . . |ΩL ., ni + 1
(A.53)
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which are solved for numbers n1 n2 . . . |ΩL uniquely up to a common constant factor c by n1 n2 . . . |ΩL = 0 2m1 2m2 . . . |ΩL = c
if not all ni are even , mi " (2mi − 1)!! λi , − 2 (2mi )!! λi + 1
∞ ! i=1
(A.54)
Using these explicit expressions one finds that for each l = 0, 1, . . . the following sum converges: ∞ n1 ,n2 ,...
l ni
|n1 n2 . . . |ΩL |2
i=1
∞
≤ |c|2 2l
m1 ,m2 ,...
mi
l ! ∞
i=1
j=1
λ2j 1 + λ2j
mj
< ∞ . (A.55)
The first inequality is obvious, while the second bound will be shown below. For l = 0 the bound shows that the coefficients n1 n2 . . . |ΩL indeed define a vector ΩL solving the conditions (A.53). The bounds for l ∈ N show that this vector is in the domain of all operators N l/2 . In particular, ΩL is in the domain of all operators aL (f ). This completes the proof of the existence and uniqueness (up to a phase) of a normalized vector solving equation (A.45), and of the property (A.46). Statements about the operator UL are now easily proved with the use of commutation relations, and the irreducibility of the Weyl system. To prove the missing step in Eq. (A.55) we denote qi = λ2i (1+λ2i )−1 . From the ∞ finiteness of the sum λ2i it follows that also the following expressions converge: i=1
l ∞ ∞ qi = λ2l pl ≡ i 1 − qi i=1 i=1 ∞ !
∞
for all l ∈ N ,
! 1 = r≡ 1 + λ2i ≤ exp 1 − qi i=1 i=1
∞
λ2i
.
i=1
One shows by induction with respect to l that m1 ,m2 ,...
∞ i=1
mi
l ! ∞
mj
qj
= Wl (p1 , . . . , pl ) r ,
(A.56)
j=1
where Wl are polynomials. For l = 0 the l.h. side is an infinite product of geometrical series, so the equality holds with W0 = 1.
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The step from l to l + 1 is obtained by the application of the homogeneity ∞ ∂ qi to the both sides of the equation. A direct calculation yields operator ∂qi i=1 ∞ i=1
qi
∂ r = p1 r ∂qi
and
∞ i=1
qi
∂ pk = k(pk + pk+1 ), ∂qi
which confirms the inductive claim and completes the proof of the bound (A.55).
References [1] H.G.B. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948). [2] G. Plunien, B.M¨ uller, W. Greiner, Phys. Rep. 134, 87 (1986). [3] E. Elizalde, A. Romeo, Am. J. Phys. 59, 711 (1991). [4] W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics, Academic Press, San Diego, 1994. [5] K.A. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy, World Scientific, Singapore, 2001. [6] M. Bordag, U. Mohideen, V.M. Mostepanenko, Phys. Rep. 353, 1 (2001) (quant-ph/0106045). [7] A. Herdegen, Acta Phys. Pol. B 32, 55 (2001) (hep-th/0008207). [8] R. Haag, Local Quantum Physics, Springer, Berlin, 1992. [9] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vol. I & II, Springer, Berlin, 1996. [10] D. Deutsch, P. Candelas, Phys. Rev. D 20, 3063 (1979). [11] N. Graham, R. Jaffe, V. Khemani, M. Quandt, M. Scandurra, H. Weigel, Nucl. Phys. B 645, 49 (2002); Phys. Lett. B 572, 196 (2003); N. Graham, R. Jaffe, V. Khemani, M. Quandt, O. Schroeder, H. Weigel, hep-th/0309130. [12] B.S. Kay, Phys. Rev. D 20, 3052 (1979). [13] G. Scharf, W.F. Wreszinski, Found. Phys. Lett. 5, 479 (1992). [14] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. I, Academic Press, London, 1976. [15] S. Hollands, R. M. Wald, Comm. Math. Phys. 223, 289 (2001).
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[16] R. Brunetti, K. Fredenhagen, R. Verch, Comm. Math. Phys. 237, 31 (2003). [17] P. Marecki, DESY THESIS 2004-002 (hep-th/0312304). [18] D. Shale, Trans. Amer. Math. Soc. 103, 149 (1962); A. van Daele, A. Verbeure, Comm. Math. Phys. 20, 268 (1971). [19] H. Araki, S. Yamagami, Publ. RIMS Kyoto Univ. 18, 283 (1982). Andrzej Herdegen Institute of Physics Jagiellonian University Reymonta 4 PL-30 059 Cracow Poland email:
[email protected] Communicated by Klaus Fredenhagen submitted 13/04/04, accepted 24/11/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 697 – 723 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/04697-27, Published online 28.07.2005 DOI 10.1007/s00023-005-0220-1
Annales Henri Poincar´ e
The Einstein-Vlasov System with a Scalar Field Hayoung Lee Abstract. We study the Einstein-Vlasov system coupled to a nonlinear scalar field with a nonnegative potential in locally spatially homogeneous space-time, as an expanding cosmological model. It is shown that solutions of this system exist globally in time. When the potential of the scalar field is of an exponential form, the cosmological model corresponds to accelerated expansion. The Einstein-Vlasov system coupled to a nonlinear scalar field whose potential is of an exponential form shows the causal geodesic completeness of the space-time towards the future. The asymptotic behavior of solutions of this system in the future time is analyzed in various aspects, which shows power-law expansion.
1 Introduction Particle systems are modeled statistically by distribution functions, which at any time represent the probability to find a particle in a given position, with a given momentum. The distribution functions contain a wealth of information and macroscopic quantities are calculated from these functions. The models being considered here are those in which collisions between particles are sufficiently rare to be neglected. The collection of these collisionless particles is described by Vlasov equations. For this reason, matter considered in these physical models is said to be collisionless matter or Vlasov matter. The time evolutions of particle systems are determined by the interactions between the particles which rely on the physical situation. Each particle is driven by self-induced fields which are generated by all particles together. Naturally combinations of interaction processes are also considered but in many situations, one of them is strongly dominating and the weaker processes are neglected. In gravitational physics, these fields are described by the Einstein equations. The physical models concerned in this paper is described by the Vlasov equation which is coupled to the Einstein equations by means of the energy-momentum tensor. One application of the Vlasov equation coupled to this self-gravitating system is cosmology. The particles are in this case galaxies or even clusters of galaxies. The simplest cosmological models are those which are spatially homogeneous. Spatially homogeneous space-times can be classified into two types; Bianchi models and the Kantowski-Sachs models. The models with a three-dimensional group of isometries G3 acting simply transitively on spacelike hypersurfaces are Bianchi models. There are nine types I-IX, depending on the classification of the structure constants of the Lie algebra of G3 . Those admitting a group of isometries G4 which acts on spacelike hypersurfaces but no subgroup G3 which acts transitively on
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the hypersurface are Kantowski-Sachs models. In fact, G3 subgroup acts multiply transitively on two-dimensional spherically symmetric surfaces. If we take as a cosmological space-time one which admits a compact Cauchy hypersurface, the Bianchi types which can occur for a spatially homogeneous cosmological model are only type I and IX and also Kantowski-Sachs models. Because of the existence of locally spatially homogeneous cosmologies, we take a larger class of space-times possessing a compact Cauchy hypersurface so that this allows a much bigger class of Bianchi types to be included. Since the Cauchy problem for the Einstein-Vlasov system is well posed, it is enough to define the class of initial data. Here is the definition. o
o
o
Definition 1 Let g ij , k ij and F be initial data for a Riemannian metric, a second fundamental form, and a matter, respectively, on a three-dimensional manifold o o o M . Then this initial data set (g ij , k ij , F ) for the Einstein-Vlasov system is called locally spatially homogeneous if the naturally associated data set on the universal is homogeneous, i.e., invariant under a transitive group action. covering M So the space-times considered here will be Cauchy developments of locally homogeneous initial data sets on some manifolds. Note that a complete Riemannian manifold is locally homogeneous if and only if the universal cover is homogeneous. For Bianchi models the universal covering space can be identified with a Lie group G. So the natural choice for G in this case is a simply connected three-dimensional Lie group. (For a detailed discussion on this subject we refer to [7, 8].) In this paper, we discuss the dynamics of expanding cosmological models, particularly accelerated expansion. There are two subjects concerning this rapid expansion. One is the very early universe close to the big bang (inflation) and the other is the present era (quintessence) supported by the observations of supernovae of type Ia. One simple way to obtain accelerated expansion is to introduce a positive cosmological constant, which leads to exponential expansion. In homogeneous spacetimes it has been studied by Wald in [11] with general matter which satisfies the dominant and strong energy conditions. When the matter is described by the Vlasov equation, the detailed asymptotics of solutions have been analyzed in [4]. In the inhomogeneous case Vlasov matter model has been studied in [5, 6] under some symmetric conditions. In [9] by Rendall, vacuum and perfect fluid cases are handled. Another choice for accelerated expanding cosmological models, which is more sophisticated, is a nonlinear scalar field. It has been analyzed by Rendall in [10] that when the potential of the scalar field has a positive lower bound with general matter satisfying the dominant and strong energy conditions then the homogeneous models expand exponentially. In the case of an exponential potential, the models shows power-law expansion which has been studied in [2, 3] by Kitada and Maeda. Bianchi type IX and Kantowski-Sachs models have complicated features when a positive cosmological constant or a nonlinear scalar field is present. It has been
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seen that there are chaotic behaviors between expanding and recollapsing phases in these models. In the discussion of expanding cosmology, the models being concerned in this paper are all Bianchi types except IX, with a nonlinear scalar field and vanishing cosmological constant. The fields described by the Einstein equations are coupled to the Vlasov equation and a nonlinear scalar field by the energy-momentum tensor which is of form 1 (Vlasov) Tαβ = Tαβ + ∇α φ∇β φ − ∇γ φ∇γ φ + V (φ) gαβ (1.1) 2 (Vlasov)
where φ is a scalar field which represents dark energy, V is a potential and Tαβ is the energy-momentum tensor of the collisionless matter described by the Vlasov (Vlasov) equation. Tαβ satisfies the dominant and strong energy conditions given respectively by (Vlasov)
(1) Tαβ v α wβ ≥ 0 where v α and wβ are any two future pointing timelike vectors, (Vlasov) 1 (Vlasov) µν α β (2) Tαβ − 2 gαβ (Tµν g ) v v ≥ 0 for any timelike vector v α . As a consequence of the Bianchi identity in (1.1) the scalar field φ satisfies the equation ∇α ∇α φ = V (φ). To make the notation not too heavy the superscript (V lasov) will be omitted for the rest of the paper. The content of the rest of this paper is the following. Section 2 presents the detailed formulation of the system being considered. In Section 3, we prove the global existence of solutions for the Einstein-Vlasov system coupled to a nonlinear scalar field with a general potential. Also the causal geodesic completeness of the space-time towards the future will be present, in the case of an exponential potential. In Section 4, we study the asymptotic behavior of solutions at late times in various aspects, when the potential of the scalar field is of an exponential form. We observe the future asymptotic behaviors of the mean curvature, the metric, the momenta of particles along the characteristic curves as well as the generalized Kasner exponents and the deceleration parameter. Also we analyze the energymomentum tensor in an orthonormal frame on the hypersurfaces. As we will see later on, the cosmological model being considered in this paper exhibits power-law expansion.
2 Einstein-Vlasov system with a scalar field Here is the formulation of the Einstein-Vlasov system coupled to a nonlinear scalar field with a potential. Let G be a simply connected three-dimensional Lie group and {ei }, a left invariant frame and {ei }, the dual coframe. Consider the space-
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time as a manifold G × I, where I is an open interval and the space-time metric of our model has the form ds2 = −dt2 + gij (t)ei ⊗ ej .
(2.1)
The initial value problem for the Einstein-Vlasov system is investigated in the case of this special form of the metric and the distribution function f depends only on t and v i , where v i are spatial components of the momentum in the frame ei . Initial data will be given on the hypersurface G × {t0 }. Now the constraints are R − (kij k ij ) + (kij g ij )2 = 16πT00 + 8πψ 2 + 16πV (φ)
(2.2)
∇ kij = −8πT0j .
(2.3)
i
The evolution equations are d gij = −2kij dt d kij = Rij + (klm g lm )kij − 2kil kjl − 8πTij dt − 4πT00 gij + 4π(Tlm g lm )gij − 8πV (φ)gij d φ=ψ dt d ψ = (klm g lm )ψ − V (φ). dt
(2.4) (2.5)
(2.6) (2.7)
These equations are written using frame components. Here kij is the second fundamental form, R is the Ricci scalar curvature and Rij is the Ricci tensor of the three-dimensional metric. And φ is a scalar field, depending only on t, with a nonnegative potential V (φ). ψ is a function introduced by the relation (2.6). The nonnegative assumption on the potential is very natural. It implies that the dominant energy condition is satisfied and then the weak energy condition follows. Here are the components of the energy-momentum tensor of the Vlasov matter; (2.8) T00 (t) = f (t, v)(1 + grs v r v s )1/2 (det g)1/2 dv T0i (t) = f (t, v)vi (det g)1/2 dv (2.9) Tij (t) = f (t, v)vi vj (1 + grs v r v s )−1/2 (det g)1/2 dv. (2.10) Here v := (v 1 , v 2 , v 3 ) and dv := dv 1 dv 2 dv 3 . The Vlasov equation is i ∂t f + 2kji v j − (1 + grs v r v s )−1/2 γmn v m v n ∂vi f = 0.
(2.11)
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i Here the Ricci rotation coefficients γmn are defined as i γmn =
1 ik l l l g (−Cnk gml + Ckm gnl + Cmn gkl ) 2
i where Cjk are the structure constants of the Lie algebra of G. To have a complete set of equations it is necessary to compute Rij in terms of gij . In this paper, it is i enough to know that Rij is of the form (det g)−n (polynomial in gij and Cjk ). To −1 control (det g) we use
d log(det g) = −2(kij g ij ). (2.12) dt Note that in discussing expanding cosmological models, the sign convention for (kij g ij ) in the paper is negative. Also it is true that if models are initially expanding, i.e., (kij g ij )(t0 ) < 0 then (kij g ij )(t) < 0 for all time t ≥ t0 (for details see [10]). The evolution equations are in general partial differential equations, i.e., d/dt is ∂t . However due to the locally spatially homogeneous space-time, the partial differential equations are reduced to ordinary differential equations. For the rest of the paper, C denotes a positive constant which changes from line to line and may depend only on the initial data. Also Cl (l = 0, 1, 2, . . .) are positive constants.
3 Global existence of solutions and geodesic completeness In this section, we will show global existence solutions of the Einstein-Vlasov System coupled to a nonlinear scalar field with a potential. As a first step, conditions will be established under which solutions of this system exist globally in time, with the technique appeared in [7] by which the existence of solutions for the EinsteinVlasov system in the absence of a scalar field has been proved. And then eventually it will be proved that these conditions are fulfilled in the system being considered. Also we will observe the casual geodesic completeness of the space-time towards the future direction, when the potential of the scalar field is of an exponential form. Proposition 1 Let gij (t0 ), kij (t0 ), φ(t0 ), ψ(t0 ) and f (t0 , v) be an initial data set for the evolution equations (2.4)–(2.7) and the Vlasov equation (2.11) which has Bianchi symmetry and satisfies the constraints (2.2) and (2.3). Also let f (t0 , v) be a nonnegative C 1 function with compact support. And assume that the potential of the scalar field V (φ) is a nonnegative C 2 function. Then there exists a unique C 1 solution (gij , kij , φ, ψ, f ) of the Einstein-Vlasov system, on an interval [t0 , T ), for some time T . If |g|, (det g)−1 , |k|, |φ|, |ψ|, f and the diameter of supp f are bounded on [t0 , T ), then T = ∞. Proof. The characteristics of (2.11) are the solutions V i (s, t, v) of the equation dV i i = 2kji V j − (1 + grs V r V s )−1/2 γmn V mV n ds
(3.1)
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with V i (t, t, v) = v i . Let f (0) (t, v) = f (t0 , v), gij (t) = gij (t0 ), kij (t) = kij (t0 ), (n)
(n)
φ(0) (t) = φ(t0 ) and ψ (0) (t) = ψ(t0 ). If f (n) , gij , kij , φ(n) and ψ (n) are given for (n)
some n, determine V (n+1) by solving the characteristic equation (3.1) with kij and
(n) gij . Let f (n+1) (t, v) = f (t0 , V (n+1) (t0 , t, v)). Define an energy-momentum tensor (n+1) (n) (n+1) (n+1) Tαβ with f (n+1) and gij in (2.8)–(2.10). Determine gij , kij , φ(n+1) and (n+1) (n) (n) ψ (n+1) by solving (2.4)–(2.7) with Tαβ , gij , kij , φ(n) and ψ (n) in the right(n+1) (n+1) hand side of equations and with gij , kij , φ(n+1) and ψ (n+1) in the left-hand (n+1) (n+1) side. Now let [t0 , T ) be the maximal interval on which gij is positive defi(n) (n) (n) (n) (n) nite. By induction, one can see that f , gij , kij , φ and ψ are C 1 on their
domains of definition. Let |g| be the maximum modulus of any component gij and |k| for kij . Suppose that for all n ≤ N − 1 the following bounds hold: |g (n) − g (0) | ≤ A1 , (n)
|φ
(det g (n) )−1 ≤ A2 , (0)
−φ
| ≤ A4 ,
|ψ
(n)
−ψ
|k (n) − k (0) | ≤ A3 (0)
| ≤ A5 .
(3.2) (3.3)
Also suppose that |v| ≤ A6 whenever f (n) (t, v) = 0. Here Ai (i = 1, . . . , 6) are positive constants which are for the moment arbitrary. The characteristic system (3.1) implies a bound for the form |v| ≤ C0 + B6 t whenever f (N ) (t, v) = 0, where (N ) B6 depends only on Ai ’s. As a consequence (2.8)–(2.10) imply a bound for Tαβ depending only on Ai ’s. The evolution equations (2.4)–(2.7) imply bounds of the form |g (N ) − g (0) | ≤ B1 t,
|k (N ) − k (0) | ≤ B3 t
|φ(N ) − φ(0) | ≤ B4 t,
|ψ (N ) − ψ (0) | ≤ B5 t
where Bi ’s depend only on Ai ’s. If Ai ’s are fixed then the inequalities in (3.2) imply an inequality of the form (det g (N ) )−1 ≤ B2 whenever t ≤ T and T is some positive time depending only on Ai ’s. Now fix Ai ’s in such a way that A2 > (det g (0) )−1 and A6 > C0 . Next reduce the size of T if necessary so that Bi T < Ai (i=1, 3, 4, 5), B2 < A2 and C0 + B6 T < A6 . Then all iterates exist on the interval [t0 , T ) and g (n) , k (n) , φ(n) and ψ (n) are bounded on that interval independently of n. Now we need to show that these iterations converge. Consider the difference of successive iterates for n ≥ 2, |(g (n+1) − g (n) )(t)| + |(k (n+1) − k (n) )(t)| + |(φ(n+1) − φ(n) )(t)| + |(ψ (n+1) − ψ (n) )(t)| t |(g (n) − g (n−1) )(s)| + |(k (n) − k (n−1) )(s)| + |(φ(n) − φ(n−1) )(s)| ≤C t0 +|(ψ (n) − ψ (n−1) )(s)| + (f (n+1) − f (n) )(s)∞ ds. (3.4)
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For the difference of the characteristics note that d
d V (n+1) − V (n) ≤ C |V (n+1) −V (n) |+|g (n) −g (n−1) |+|k (n) −k (n−1) | . (3.5) ds ds Define α(n) (t) := |(g (n+1) − g (n) )(t)| + |(k (n+1) − k (n) )(t)|
(3.6)
+ |(φ(n+1) − φ(n) )(t)| + |(ψ (n+1) − ψ (n) )(t)| + sup{|V (n+1) − V (n) |(s, t, v) : s ∈ [t0 , t], v ∈ suppf (n+1) (t) ∪ suppf (n) (t)}. Then we get f (n+1) (t) − f (n) (t)∞ ≤ f (0) C 1 α(n) (t).
(3.7)
Therefore (3.4)–(3.7) imply that α(n) (t) ≤ C
t
(n) α (s) + α(n−1) (s) ds.
t0
Applying Gr¨ onwall’s inequality to this gives (n)
α
t
(t) ≤ C
α(n−1) (s) ds.
t0
Therefore α(n) (t) ≤ C n−2 α(2) tn−2 /(n − 2)! and so {g (n) }, {k (n) }, {φ(n) }, {ψ (n) } and {V (n) } are Cauchy sequences on the time interval [t0 , T ). Denote the limits of these sequences by g, k, φ, ψ and V∞ , respectively. Also by (2.4)–(2.7), dg (n) /dt, dk (n) /dt, dφ(n) /dt, dψ (n) /dt and dV (n) /dt are uniformly convergent. Thus (g, k, φ, ψ, f ) is a C 1 solution of the system on the interval [t0 , T ). Now let us check whether a solution exists uniquely or not. If two solutions with the same initial data are given, define a quantity α(t) in terms of their difference in the same way that α(n) (t) was defined in terms of the differences of two iterates. Applying the same argument as above leads to an estimate of the form
t
α(t) ≤ C
α(s) ds. t0
By Gr¨ onwall’s inequality we can see that α(t) is zero and hence that the two solutions agree. Therefore the solution which has been constructed is uniquely determined by the initial data. Define A := R − (kij k ij ) + (kij g ij )2 − 16πT00 − 8πψ 2 − 16πV (φ) Ai := ∇i kij + 8πT0j .
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Then after a lengthy calculation we obtain d l ij A = 2(kij g ij )A − 2γij g Al dt d Ai = (klm g lm )Ai + 2kil Al . dt That is, (2.4)–(2.11) imply a homogeneous first-order ordinary differential system for constraints (2.2) and (2.3). Therefore we can conclude that if the initial data satisfy the constraints then so does the solution of the evolution equations (2.4)– (2.7) with energy-momentum tensors (2.8)–(2.10) and the Vlasov equation (2.11). In the above argument so far, we see that the size of T is only restricted by the quantities ; |g (0) |, (det g (0) )−1 , |k (0) |, |φ(0) |, |ψ (0) |, f (0) and the diameter of supp f (0) . Thus if the quantities |g|, (det g)−1 , |k|, |φ|, |ψ|, f and the diameter of supp f are bounded on the same time interval [t0 , T ), then a solution exists on [t, t + ) for any t ∈ [t0 , T ) and some independent of t. It can be concluded that the original solution can be extended to the larger interval [t0 , T + ). Therefore this completes the proof. Let us state some properties of linear algebra which can be found in [4, 7]. We shall make use of these properties later on. Let A be a n × n matrix. Let A1 and A2 be n × n symmetric matrices with A1 positive definite. Define a norm of a matrix by A := sup{Ax/x : x = 0, x ∈ Rn }. Also define a relative norm by A2 A1 := sup{A2 x/A1 x : x = 0, x ∈ Rn }. Then from these definitions, one can see that
and also
A2 ≤ A2 A1 A1
(3.8)
1/2 −1 . A2 A1 ≤ tr(A−1 1 A2 A1 A2 )
(3.9)
Proposition 2 If (kij g ij ) is bounded on [t0 , T ), then T = ∞. Proof. Let σij be the trace free part of the second fundamental form kij . Then we have kij = 13 (kij g ij )gij + σij . By this fact, we rewrite the constraint (2.2) as 1 1 1 (kij g ij )2 = − R + (σij σ ij ) + 8πT00 + 4πψ 2 + 8πV (φ). 3 2 2
(3.10)
It has been proved by Wald in [11] that in all Bianchi models except type IX, the Ricci scalar curvature is zero or negative. Also due to the nonnegative potential, we get 1 (kij g ij )2 ≥ 4πψ 2 . 3 So if (kij g ij ) is bounded on [t0 , T ), then ψ is bounded on [t0 , T ) and so is φ.
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From evolution equations (2.4) and (2.5), we have d (kij g ij ) = R + (kij g ij )2 + 4π(Tij g ij ) − 12πT00 − 24πV (φ). dt
(3.11)
Using the constraint (2.2) we get d (kij g ij ) = (kij k ij ) + 4π(Tij g ij ) + 4πT00 + 8πψ 2 − 8πV (φ). dt Thus we obtain
d (kij g ij ) ≥ (kij k ij ) − 8πV (φ). dt
Then
t t0
Note that
t
V (φ)(s) ds ≥ kij (t0 )g ij (t0 ) +
(kij g ij ) + 8π
(kij k ij )(s) ds. t0
t
t
V (φ)(s) ds ≤ V t t0
|φ(s)| ds + C(t + 1) t0
where V t := sup{|V (φ)(s)| : for all s ∈ [t0 , t]}. Since φ is bounded on [t0 , T ), t then t0 V (φ)(s) ds is bounded for all t in [t0 , T ). Therefore with the boundedness T of (kij g ij ), we conclude that t0 (kij k ij )(s) ds < ∞. Let g and k be the norms of the matrices with entries gij and kij , respectively. Let kg be the relative norm of the matrix with entries kij with respect to the matrix with entries gij . Then using (3.9) we have t g(t) ≤ g(t0 ) + 2 k(s) ds t0 t
≤ g(t0 ) + 2
k(s)g g(s) ds t0 t
≤ g(t0 ) + 2
(kij k ij )1/2 (s)g(s) ds.
t0
By Gr¨ onwall’s inequality, we get
t ij 1/2 g(t) ≤ g(t0 ) exp 2 (kij k ) (s) ds . t
t0
Since t0 (kij k ij )1/2 (s) ds is bounded on [t0 , T ), also |g| is bounded on [t0 , T ). Using (2.12), we see that (det g)−1 is bounded on the same interval. It is known that if (det g) and its inverse are bounded then the scalar curvature R is bounded from above. Note that in (2.2) we have R + (kij g ij )2 ≥ (kij k ij ).
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Thus kij k ij is bounded on [t0 , T ). By the inequality k ≤ (kij k ij )1/2 g also |k| is bounded. The boundedness of |g| and (det g)−1 implies that g is uniformly positive definite on the interval. Hence the solutions of the characteristic equation are also bounded. Therefore by Proposition 1, the proof completes. So far it has been proved that solutions of the system exist as long as some quantities are bounded in a finite time interval [t0 , T ) for arbitrary T . These conditions are satisfied, as we will see in the following theorem. Theorem 1 Let gij (t0 ), kij (t0 ), φ(t0 ), ψ(t0 ) and f (t0 , v) be an initial data set for the evolution equations (2.4)–(2.7) and the Vlasov equation (2.11) which has Bianchi symmetry and satisfies the constraints (2.2) and (2.3). Also let f (t0 , v) be a nonnegative C 1 function with compact support. And assume that the potential of the scalar field V (φ) is a nonnegative C 2 function. Then there exists a unique C 1 solution (gij , kij , φ, ψ, f ) of the Einstein-Vlasov system for all time. Proof. Consider (3.11) with (3.10) 1 3 d (kij g ij ) = − R + (σij σ ij ) + 4π(Tij g ij ) + 12πT00 + 12πψ 2 . dt 2 2 Then d (kij g ij ) ≥ 0. dt
(3.12)
Since the cosmological models we are considering here is expanding, i.e., (kij g ij ) < 0, with (3.12) we conclude that (kij g ij ) is bounded for t ≥ t0 and the proof completes from Proposition 2.
3.1
Geodesic completeness with an exponential potential
The next result asserts the geodesic completeness of locally spatially homogeneous space-times for the Einstein-Vlasov system coupled to a nonlinear scalar field whose potential is an exponential form. Theorem 2 Suppose the hypotheses of Theorem 1 hold. And assume that the potential of the scalar field V (φ) is of form V (φ) = V0 e−λκφ √ where V0 is a positive constant, λ ∈ (0, 2) and κ2 = 8π. Then the space-time is future complete. The proof of this theorem can be founded in Subsection 4.7.
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4 Asymptotics of solutions with an exponential potential We study the asymptotic behavior of solutions in the future time with a particular form of the potential V (φ). Namely the potential is given, as in the previous √ section, by V (φ) = V0 e−λκφ where V0 is a positive constant, 0 < λ < 2 and in [1] that in order for power-law inflation to occur λ κ2 = 8π. In has been shown √ must be smaller than 2. Note that the case λ = 0 corresponds to the model with a positive cosmological constant instead of the scalar field which has been well understood in [4]. Briefly, this model exhibits exponential expansion. For detailed information, we refer to [4]. We introduce a new time coordinate τ defined by dτ = e−λκφ/2 dt.
(4.1)
¯ := Reλκφ , T¯αβ := Tαβ eλκφ and ψ¯ := ψeλκφ/2 . Then And let k¯ij := kij eλκφ/2 , R the Hamiltonian constraint (2.2) become ¯ − (k¯ij k¯ij ) + (k¯ij g ij )2 = 16π T¯00 + 8π ψ¯2 + 16πV0 . R
(4.2)
The evolution equations are d gij = −2k¯ij dτ d ¯ ¯ ij + (k¯lm g lm )k¯ij − 2k¯il k¯l − 8π T¯ij kij = R j dτ
(4.3) (4.4)
λκ ¯ ¯ kij ψ − 4π T¯00 gij + 4π(T¯lm g lm )gij − 8πV0 gij + 2 d ¯ φ = ψ, dτ d ¯ λκ ¯2 ψ = (k¯lm g lm )ψ¯ + λκV0 + ψ . dτ 2 Also the Vlasov equation becomes i ∂τ f + 2k¯ji v j − eλκφ/2 (1 + grs v r v s )−1/2 γmn v m v n ∂vi f = 0.
(4.5) (4.6)
(4.7)
Now we define two functions : ¯ k¯ij g ij ) := − 2 (k¯ij g ij ) − λκψ¯ (ψ, 3 ¯ k¯ij g ij ) := (k¯ij g ij )2 − 12π(ψ¯2 + 2V0 ). ¯ ψ, S( Note that the function S¯ will play the same roles as (kij g ij ± 3Λ) in the papers [4, 11]. The basic idea of the following proposition is from [3]. Here the computation is carried out carefully so that the error terms are explicitly determined for the future reference.
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Proposition 3 Let σij be the trace free part of the second fundamental form kij such that 1 k¯ij = (k¯lm g lm )gij + σ ¯ij (4.8) 3 where σ ¯ij := σij eλκφ/2 . Then we have ∗ S¯ = O(e− τ )
σ ¯ij σ ¯
ij
= O(e
− ∗ τ
(4.9)
)
(4.10)
¯ = O(e R ) ∗ T¯00 = O(e− τ ).
(4.11)
− ∗ τ
(4.12)
Proof. Note that using (4.8) we rewrite the constraint (4.2) as 3¯ 3 + (¯ σij σ (k¯ij g ij )2 = − R ¯ ij ) + 24π T¯00 + 12π ψ¯2 + 24πV0 . 2 2
(4.13)
So S¯ becomes
¯ k¯ij g ij ) = − 3 R ¯ + 3 (¯ ¯ ψ, σij σ S( ¯ ij ) + 24π T¯00 . (4.14) 2 2 Recall that the Ricci scalar curvature is zero or negative in all Bianchi models except type IX. So the models are allowed to evolve in the region of S¯ ≥ 0. Note that from (4.3) and (4.4) we have d ¯ ij ¯ ¯ + (k¯ij g ij )2 + 4π(T¯ij g ij ) − 12π T¯00 − 24πV0 + λκ (k¯ij g ij )ψ. (kij g ) = R dτ 2 Using the constraint (4.2) we get d ¯ ij λκ ¯ ij ¯ (kij g ) = (k¯ij k¯ij ) + 4π(T¯ij g ij ) + 4π T¯00 + 8π ψ¯2 − 8πV0 + (kij g )ψ. (4.15) dτ 2 ¯ we obtain Thus using (4.8) and the definitions of and S, d ¯ ¯ S = −S¯ + 2(k¯ij g ij )[(¯ σij σ ¯ ij ) + 4π(T¯ij g ij ) + 4π T¯00 ] ≤ −S. dτ The last step is due to the fact that considering expanding space-times implies (k¯ij g ij ) < 0. In [3] it is shown that there exists a lower bound of , say ∗ , which only depends on the initial condition of the space-times. Therefore we have ¯ ) = O(e− ∗ τ ) S(τ and as a consequence from the definition of S¯ we have ∗ (k¯ij g ij )2 − 12π(ψ¯2 + 2V0 ) = O(e− τ ).
(4.16) ∗
¯ = T¯00 = O(e− τ ). Here while ¯ ij = R By (4.14) the rest of the claims follows ; σ ¯ij σ we use the notation O(·), we lose a piece of information from above that S¯ is nonnegative. So we want to point out that the errors in (4.9)–(4.12) are non-negative.
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Asymptotic behaviors of (k¯ij gij ) and ψ¯
The estimate (4.16) is unsatisfactory in the sense that we do not have sufficient ¯ In this subinformation to say individual asymptotic behaviors of (k¯ij g ij ) and ψ. section, we obtain asymptotic behaviors of these quantities, separately. Proposition 4 ψ¯ = λγ + O(e−ητ )
(4.17)
k¯ij g ij = −3κγ + O(e−ητ )
(4.18)
√ √ where γ := 2V0 / 6 − λ2 and η := min{ 21 κγ(6 − λ2 ), ∗ /2}. Note that in the case 0 < λ < 2/3, we have η = ∗ /2. 1 Proof. With the evolution equation (4.6) consider d ¯ (ψ − λγ) = Fψ¯ (ψ¯ − λγ) dτ where λκ ¯ Fψ¯ (ψ¯ − λγ) := (k¯ij g ij + 3κγ)(ψ¯ − λγ) + (ψ − λγ)2 2 + (λ2 − 3)κγ(ψ¯ − λγ) + λγ(k¯ij g ij + 3κγ) (4.9) implies (k¯ij g ij )2 = κ2
3 ∗ (ψ¯ − λγ)2 + 3λγ(ψ¯ − λγ) + 9γ 2 + O(e− τ ). 2
When ψ¯ = λγ, we get k¯ij g ij + 3κγ = 3κγ −
∗ 9κ2 γ 2 + O(e− ∗ τ ) = O(e− τ ).
∗
Then Fψ¯ (0) = O(e− τ ) and after a lengthy elementary computation one can see ∗ that Fψ¯ (0) = − 12 κγ(6 − λ2 ) + O(e− τ ). So when ψ¯ is close to λγ, ∗ ∗ d ¯ 1 (ψ − λγ) = O(e− τ ) + [− κγ(6 − λ2 ) + O(e− τ )](ψ¯ − λγ) dτ 2 + C(ψ¯ − λγ)2 . 1
2
Define Yψ¯ (τ ) := e 2 κγ(6−λ
)τ
(ψ¯ − λγ). Then
2 ∗ ∗ 2 1 1 d Yψ¯ = e 2 κγ(6−λ )τ O(e− τ ) + O(e− τ )Yψ¯ + Ce− 2 κγ(6−λ )τ Yψ¯2 . dτ
1 ∗ , the lower bound of depends on not only the constants λ and V but also initial data. 0 When λ ∈ (0, 2/3), the trivial lower bound of , which may not be sharp, is 13 κ 6V0 (2 − 3λ2 ) (see [2, 3] for details). In this case 12 κγ(6 − λ2 ) > ∗ /2 is true.
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This yields 2 2 2 ∗ 1 1 d Yψ¯ = Ce− 2 κγ(6−λ )τ Yψ¯ + e 2 κγ(6−λ )τ O(e− τ /2 ) . dτ This implies that 1
2
−[Yψ¯ + e 2 κγ(6−λ
)τ /2
O(e−
∗
τ /2 −1
)]
2
1
= C(e− 2 κγ(6−λ
)τ
+ 1).
Consequently, 1
2
Yψ¯ = C + O(e− 2 κγ(6−λ
)τ
1
2
) + e 2 κγ(6−λ
)τ
O(e−
∗
τ /2
).
Therefore we have 2 2 ∗ 1 ψ¯ − λγ = Ce− 2 κγ(6−λ )τ + O(e−κγ(6−λ )τ ) + O(e− τ /2 )
and this gives the proof of (4.17). For (4.18), with (4.8) and (4.15) consider d ¯ ij (kij g + 3κγ) = Fk¯ (k¯ij g ij + 3κγ) dτ where Fk¯ (k¯ij g ij + 3κγ) λκ ¯ ij 1 (kij g + 3κγ)(ψ¯ − λγ) := (k¯ij g ij + 3κγ)2 + κ2 (ψ¯ − λγ)2 + 3 2 1 1 + λγκ2 (ψ¯ − λγ) + (λ2 − 4)κγ(k¯ij g ij + 3κγ) 2 2 + (¯ σij σ ¯ ij ) + 4π T¯00 + 4π(T¯ij g ij ). (4.9) implies ∗ 2 4γ ¯ ij (kij g + 3κγ) + λ2 γ 2 + O(e− τ ). ψ¯2 = 2 (k¯ij g ij + 3κγ)2 − 3κ κ ∗
When k¯ij g ij = −3κγ, one can see that Fk¯ (0) = O(e− τ ) and also Fk¯ (0) = ∗ − 12 κγ(6 − λ2 ) + O(e− τ ). Therefore ∗ ∗ d ¯ ij 1 (kij g + 3κγ) = O(e− τ ) + [− κλ(6 − λ2 ) + O(e− τ )](k¯ij g ij + 3κγ) dτ 2 + C(k¯ij g ij + 3κγ)2
where k¯ij g ij + 3κγ is small and by the same argument as above (4.18) follows.
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Relation between τ and t and asymptotics in terms of t
So far in the present section, we have obtain asymptotics of quantities, σij σ ij , R, T00 , kij and gij , in terms of the time coordinate τ after rescaled by a certain factor of the scalar field. In order to study further asymptotics, it is necessary to recover these quantities in terms of the time coordinate t. For this reason, in this subsection the relation between the two time coordinates and the rescaling factor e−λκφ/2 in terms or t will be analyzed. Proposition 5
e
2
−λ κγτ /2
O(t−2 ln t), 2eλκC1 /2 −1 t + = O(t−(ζ+1) ), λ2 κγ O(t−2 ),
if λ2 κγ/2 = η if λ2 κγ/2 > η if λ2 κγ/2 < η
where ζ := 2η/λ2 κγ and C1 is a constant. Proof. From (4.17) we have φ(τ ) = λγτ + C1 + O(e−ητ ).
(4.19)
So by (4.1) we get
τ
t(τ ) = t(τ0 ) + eλκC1 /2
exp[λ2 κγs/2 + O(e−ηs )] ds
τ0
where t(τ0 ) = t0 . Then 2
e−λ
κγτ /2
t(τ )
τ 2 t(τ0 ) + eλκC1 /2 e−λ κγτ /2 exp[λ2 κγs/2 + O(e−ηs )] ds τ0 τ −λ2 κγτ /2 λκC1 /2 λ2 κγ(s−τ )/2 t(τ0 ) + e e [1 + O(e−ηs )] ds =e 2
= e−λ
κγτ /2
τ0
2eλκC1 /2 = λ2 κγ
2 O(τ e−λ κγτ /2 ), −ητ O(e ), + −λ2 κγτ /2 O(e ),
if λ2 κγ/2 = η if λ2 κγ/2 > η if λ2 κγ/2 < η.
In all cases, we have 2
e−λ
κγτ /2
t(τ ) ≤ C.
Consequently 2
e−λ Thus the proposition follows.
κγτ /2
= O(t−1 ).
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Proposition 6
e
−λκφ/2
O(t−2 ln t), 2 −1 = 2 t + O(t−(ζ+1) ), λ κγ O(t−2 ),
if λ2 κγ/2 = η if λ2 κγ/2 > η if λ2 κγ/2 < η.
Proof. By (4.19), 2
e−λκφ/2 = e−λ
κγτ /2 −λκC1 /2
e
(1 + O(e−ητ )).
Combining this with Proposition 5 yields the conclusion of the proposition.
Proposition 7 σij σ ij = O(t−(ξ+2) ) R = O(t
−(ξ+2)
T00 = O(t
−(ξ+2)
(4.20)
)
(4.21)
)
(4.22)
where ξ := 2∗ /λ2 κγ. Proof. By Proposition 5 e−
∗
τ
= O(t−2
∗
/λ2 κγτ
).
Also Proposition 6 implies e−λκφ = O(t−2 ). Combining these with (4.10)–(4.12) in Proposition 3 concludes the proposition.
4.3
Asymptotic behaviors of kij gij , ψ and φ in terms of t
In this part, we will observe asymptotic behaviors of kij g ij , ψ and φ in terms of t using the relation between two time coordinates τ and t we have obtained in the previous subsection. Proposition 8 O(t−2 ln t), if λ2 κγ/2 = η 2 −1 t + ψ= O(t−(ζ+1) ), if λ2 κγ/2 > η λκ O(t−2 ), if λ2 κγ/2 < η O(t−2 ln t), if λ2 κγ/2 = η 6 −1 ij kij g = − 2 t + O(t−(ζ+1) ), if λ2 κγ/2 > η λ O(t−2 ), if λ2 κγ/2 < η.
(4.23)
(4.24)
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Proof. Note that from Proposition 5 e−ητ = O(t−ζ ). So combining (4.17) in Proposition 4 and Proposition 6, (4.23) follows. With (4.18) in Proposition 4 the same argument applies to prove (4.24). Proposition 9 O(t−1 ln t), 2 φ= ln t + C2 + O(t−ζ ), λκ O(t−1 ),
if λ2 κγ/2 = η if λ2 κγ/2 > η if λ2 κγ/2 < η
where C2 is a constant. Proof. The proposition follows directly from (4.23) in the previous proposition.
4.4
Asymptotic behaviors of gij and gij
Asymptotics of gij and g ij will be analyzed first in the time coordinate τ . By means of the relation between the two time coordinates τ and t in Subsection 4.2, these asymptotics will be recovered in terms of t. ∗ ¯ ij = O(e− τ ). Using the It has been observed in Proposition 3 that σ ¯ij σ following two lemmas we will identify σ ¯ij (τ ) which play a role to analyze gij (τ ). ¯ ) and ¯ Lemma 1 Let g(τ ), k(τ σ (τ ) denote the norms of the matrices with ¯ ¯ij (τ ), respectively. Then entries gij (τ ), kij (τ ) and σ ¯ σ (τ ) ≤ Ce−
∗
τ /2
g(τ ).
Proof. The lemma follows by the fact that ¯ σ (τ )g(τ ) ≤ (¯ σij σ ¯ ij )1/2 ≤ Ce−
∗
τ /2
.
The last inequality is due to (4.10). Lemma 2 |e−2κγτ gij (τ )| ≤ C |e
2κγτ ij
g (τ )| ≤ C.
(4.25) (4.26)
Proof. Let g˜ij (τ ) := e−2κγτ gij (τ ). Then using (4.8) and (4.18), we get d 2 g˜ij = −2κγ˜ gij − (k¯lm g lm )˜ gij − 2e−2κγτ σ ¯ij dτ 3 = O(e−ητ )˜ gij − 2e−2κγτ σ ¯ij .
(4.27)
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Now let us use the norms again. Let ˜ g(τ ) be a norm of the matrix with entries g˜ij (τ ). Then we get τ −ηs ˜ g(τ ) ≤ ˜ g(τ0 ) + Ce ˜ g(s) + Ce−2κγs ¯ σ(s) ds (4.28) τ0 τ −ηs g(s) + Ce−2κγs ¯ σ(s)g˜ ˜ g(s) ds. ≤ ˜ g(τ0 ) + Ce ˜ τ0
Note that
σij σ ¯ ij )1/2 ≤ Ce2κγτ e− ¯ σ (τ )g˜ ≤ e2κγτ (¯
∗
τ /2
.
So combining this with (4.28) yields
τ
˜ g (τ ) ≤ ˜ g(τ0 ) +
Ce−ηs ˜ g(s) ds.
τ0
By Gr¨ onwall’s inequality, this becomes
τ
˜ g(τ ) ≤ ˜ g(τ0 ) exp
Ce−ηs ds ≤ C.
τ0
Therefore |˜ gij (τ )| is bounded by a constant for all τ ≥ τ0 . Also (4.26) follows by the same argument. Proposition 10
¯ij = O(e− e−2κγτ σ
∗
τ /2
).
Proof. The proposition follows by Lemmas 1 and 2. Proposition 11 gij (τ ) = e2κγτ (Gij + O(e−ητ )) ij
g (τ ) = e
−2κγτ
(G + O(e ij
−ητ
)).
(4.29) (4.30)
Here Gij and G ij are independent of τ . Proof. Let us consider (4.27) again ; d g˜ij (τ ) = O(e−ητ )˜ gij (τ ) − 2e−2κγτ σ ¯ij (τ ) dτ where g˜ij (τ ) = e−2κγτ gij (τ ). Then Lemma 2 and Proposition 10 imply ∗ d g˜ij (τ ) ≤ Ce−ητ g˜ij + Ce− τ /2 ≤ Ce−ητ . dτ
d Since dτ g˜ij is decaying exponentially, there exists a limit, say Gij , of g˜ij as τ goes to infinity. Then this gives
g˜ij (τ ) = Gij + O(e−ητ ),
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715
gij (τ ) = e2κγτ Gij + O(e−ητ ) .
Here the lower-order term of gij (τ ) is of an exponential form so that it is combined with the leading order term, which makes it possible to compute g ij (τ ) explicitly. So g ij (τ ) is g ij (τ ) = e−2κγτ G ij + O(e−ητ ) . Proposition 12 O(t−1 ln t), if λ2 κγ/2 = η λ2 κγ 4/λ2 O(t−ζ ), gij (t) = t4/λ Gij + if λ2 κγ/2 > η (4.31) 2eλκC1 /2 −1 if λ2 κγ/2 < η O(t ), O(t−1 ln t), if λ2 κγ/2 = η 2eλκC1 /2 4/λ2 2 O(t−ζ ), if λ2 κγ/2 > η . g ij (t) = t−4/λ G ij + λ2 κγ −1 O(t ), if λ2 κγ/2 < η (4.32)
2
Proof. Recall that
e−ητ = O(t−ζ ).
Proposition 5 implies −4/λ2 O(t−1 ln t), if λ2 κγ/2 = η λκC1 /2 2e O(t−ζ ), if λ2 κγ/2 > η = t4/λ 2 + λ κγ −1 O(t ), if λ2 κγ/2 < η λ2 κγ 4/λ2 O(t−1 ln t), if λ2 κγ/2 = η 2 O(t−ζ ), = t4/λ + if λ2 κγ/2 > η . 2eλκC1 /2 −1 O(t ), if λ2 κγ/2 < η
e2κγτ
2
So by means of this and Proposition 11, (4.31) follows. The same argument for g ij yields (4.32).
4.5
Asymptotics of the generalized Kasner exponents and the deceleration parameter
Let λi be the eigenvalues of kij (t) with respect to gij (t), i.e., the solutions of det(kji − λδji ) = 0. Define the generalized Kasner exponents by λi λi . pi := = lm ) λ (k lm g l l The name comes from the fact that in the special case of the Kasner solutions these are the Kasner exponents. Note that while the Kasner exponents are constants, the generalized Kasner exponents are in general functions of t. The generalized Kasner
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exponents always satisfy the first of the two Kasner relations, but in general do not satisfy the second, where these two Kasner relations are pi = 1, (4.33) i
(pi )2 = 1.
(4.34)
i
The following proposition exhibits that the space-time isotropizes at late times. Proposition 13 pi (t) =
1 + O(t−ξ/2 ) 3
where ξ = 2∗ /λ2 κγ. Proof. First note that by (4.8) λi are also the solutions of i 1 det σ ¯j − [λ − (k¯lm g lm )]δji = 0. 3 So the eigenvalues of σ ¯ij (τ ) with respect to gij (τ ) are ˜ i := λi − 1 (k¯lm g lm ). λ 3
Also note that
˜ 2 ¯lm σ ¯ lm . Then (4.10) implies i (λi ) = σ ˜ i = O(e− ∗ τ /2 ). λ
Therefore using this and (4.18) we obtain pi −
1 = 3
˜
λi 1 ¯ lm ) ( k 3 lm g
= O(e−
∗
τ /2
).
Thus Proposition 5 completes the proof.
There is another quantity to be considered regarding expanding cosmological models, which is the deceleration parameter, say q. This deceleration parameter is related to the mean curvature, as follows d (kij g ij ) = −(1 + q)(kij g ij )2 . dt In accelerated expanding models, the deceleration parameter is negative. Proposition 14
O(t−1 ln t), λ2 q = −1 − O(t−ζ ), + 6 O(t−1 ),
if λ2 κγ/2 = η if λ2 κγ/2 > η if λ2 κγ/2 < η.
Proof. The proof is a straightforward computation from (4.24).
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Asymptotics of momenta
We will analyze the behavior of the momenta of the distribution function f along the characteristics where f is a constant. From the Vlasov equation (2.11) we define the characteristic curve V i (t) by dV i i = 2kji V j − (1 + grs V r V s )−1/2 γmn V mV n dt
(4.35)
for each V i (t0 ) = v0i given t0 . The characteristics Vi , rather than V i , has a simpler form, which makes analyzing the behavior of the momenta easier. So here Vi (t) satisfies dVi j = −(1 + grs V r V s )−1/2 γmn Vp Vq g pm g qn gij (4.36) dt for each Vi (t0 ) = vi0 given t0 . Also observe that Vi (τ ) satisfies dVi i Vp Vq g pm g qn gij . = −eλκφ/2 (1 + grs v r v s )−1/2 γmn dτ
(4.37)
For the rest of the paper, the capital V i and Vi indicate that v i and vi are parameterized by the coordinate time t or τ . Theorem 3 Vi (t) from (4.36) converges to a constant along the characteristics as t goes to infinity. That is O(t−ζ ), if ζ ≤ 1, Vi (t) = C3 + O(t−ω ), if ζ > 1, where ω := min{ζ, 4/λ2 − 1}. Furthermore, O(t−1 ln t), i −4/λ2 O(t−ζ ), V (t) = t C4 + O(t−1 ),
if λ2 κγ/2 = η if λ2 κγ/2 > η . if λ2 κγ/2 < η
Before the proof of this theorem, some lemmas are required. Lemma 3
(g ij Vi Vj )(τ ) = e−2κγτ V + O(e−ητ )
where V is a constant. Proof. First note that by Propositions 10 and 11, we have σ ¯ ij = O(e−(2κγ+
∗
/2)τ
).
∗
¯ ij . Then σ ˜ ij is bounded by a constant for all τ ≥ τ0 . Let σ ˜ ij := e(2κγ+ /2)τ σ ij Since G in Proposition 11 is positive definite and time independent, there exists a constant C, independent of time, such that σ ˜ ij Vi Vj ≤ CG ij Vi Vj .
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Then by means of this and (4.18), we obtain d ij (g Vi Vj ) = 2k¯ ij Vi Vj dτ 2 = (k¯lm g lm )g ij Vi Vj + 2¯ σ ij Vi Vj 3 ∗ ≤ − 2κγ + Ce−ητ g ij Vi Vj + Ce−(2κγ+ /2)τ G ij Vi Vj ≤ − 2κγ + Ce−ητ g ij Vi Vj .
(4.38)
Here to get the first equal sign, (4.37) is used. Yet the terms involved with (4.37) l combining with g ij . Now consider vanish due to the antisymmetric property of γmn 2κγτ ij g Vi Vj . Then one can see from (4.38) that Vτ := e dVτ = O(e−ητ )Vτ . dτ So there exists a limit of Vτ , say V as τ goes to infinity. Then we have Vτ = V + O(e−ητ ) .
This completes the proof. Lemma 4 (g Vi Vj )(t) = V ij
λκC1 /2
2e λ2 κγ
4/λ2 t
−4/λ2
O(t−2 ln t), + O(t−(ζ+1) ), O(t−2 ),
if λ2 κγ/2 = η if λ2 κγ/2 > η if λ2 κγ/2 < η.
Proof. Recall that e−ητ = O(t−ζ ). Then by means of Proposition 5 and Lemma 3 we have
4/λ2 4 2eλκC1 /2 −1 t (g Vi Vj )(t) = V + O(t−(ζ+ λ2 ) ) 2 λ κγ O(t−2 ln t), if λ2 κγ/2 = η + O(t−(ζ+1) ), if λ2 κγ/2 > η O(t−2 ), if λ2 κγ/2 < η. ij
Note that ζ + 4/λ2 > ζ + 2. So the lemma follows. Lemma 5 |Vi |(t) ≤ C5 + O(t−ζ ) where C5 is a constant and for all i.
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Proof. Since G ij is positive definite, there exists a constant C such that |Vi |2 (τ ) ≤ CG ij Vi (τ )Vj (τ ). Note that e−2κγτ G ij Vi Vj is the leading order term in (g ij Vi Vj )(τ ) in (4.30). So using Lemma 3, we conclude that G ij Vi Vj = V + O(e−ητ ). Combining this with the fact that e−ητ = O(t−ζ )
we complete the proof.
Proof of Theorem 3. Note that in Bianchi type I, since all structure constants are j zero, also the Ricci rotation coefficients γmm are zero. So from (4.36) it is clear that Vi (t) = vi0 for all t. More generally Lemma 5 says that all Vi are bounded by a constant when t goes to infinity. However this allows oscillating behaviors. So to rule out these i cases, it is necessary to analyze dV dt . Combining Proposition 12 and (4.36), we have dVi −4/λ2 |Vp ||Vq |. dt ≤ Ct Here the right-hand side is a summation for some p and q. Then by Lemma 5 this implies dVi −4/λ2 . (4.39) dt ≤ Ct i This leads to the conclusion that when t goes to the infinity, dV dt goes to zero, and so Vi goes to a constant. Combining Lemma 5 and (4.39) we have 2
Vi (t) = C3 + O(t−ζ ) + O(t−(4/λ
−1)
).
Since 4/λ2 − 1 > 1 Vi (t) = C3 +
O(t−ζ ), O(t−ω ),
if ζ ≤ 1, if ζ > 1,
where ω =: min{4/λ2 − 1, ζ}. Now combining this with (4.32) completes the proof.
4.7
Geodesic completeness
In this part, we will prove the completeness of future directed causal geodesics, which has been postponed in Subsection 3.1.
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Proof of Theorem 2. The geodesic equations for a metric of the form (2.1) imply that along the geodesics the variables t, v i and v 0 satisfy the following system of differential equations : dt = v0 ds dv 0 = kij v i v j ds dv i i = 2kji v j v 0 − γmn vm vn ds
(4.40)
where s is an affine parameter. For a particle with rest mass m moving forward in time, v 0 can be expressed by the remaining variables, v 0 = (m2 + gij v i v j )1/2 .
(4.41)
The geodesic completeness is decided by looking at the relation between t and the affine parameter s, along any future directed causal geodesic. This relation is clear from (4.40) and (4.41). I.e., it is given by dt = (m2 + gij v i v j )1/2 . ds To control this, it is necessary to control gij v i v j as a function of the coordinate time t. Consider first the case of a timelike geodesic. I.e., m > 0. Then V i (t) satisfy dV i i = 2kji V j − (m2 + gij V i V j )−1/2 γmn V mV n. dt In this case, the arguments presented in Subsection 4.6 is valid, in particular those in Lemmas 3 and 4 when m = 1. Therefore by Lemma 4, (gij V i V j )(t) is bounded 2 above by Ct−4/λ , and so by C, for all t ≥ t0 . Hence this gives that −1/2 ds 2 = m + (gij V i V j )(t) ≥ C. dt Therefore when s is recovered by integrating this, the integral of the right-hand side diverges as t goes to infinity. Now consider a null geodesic, i.e., m = 0. Then in this case V i (t) satisfy dV i i = 2kji V j − (gij V i V j )−1/2 γmn V mV n. dt Also Lemma 4 is valid. Therefore (gij V i V j )(t) is bounded by a constant and this gives −1/2 ds = gij (V i V j )(t) ≥ C. dt Therefore as t goes to infinity so does s.
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Asymptotics of the energy-momentum tensor
In this subsection it will be analyzed the asymptotic behavior of the energymomentum tensor in an orthonormal frame on the hypersurfaces. Proposition 15 Let {ˆ ei } be an orthonormal frame. The energy-momentum is described by ρ(t) = f (t, vˆ)(1 + |ˆ v |2 )1/2 dˆ v Ji (t) = f (t, vˆ)ˆ vi dˆ v Sij (t) = f (t, vˆ)ˆ vi vˆj (1 + |ˆ v |2 )−1/2 dˆ v
tensor
(4.42) (4.43) (4.44)
where ρ := Tˆ00 is the energy density, Ji := Tˆ0i the components of the current density and Sij := Tˆij are the spatial components of the energy-momentum tensor. Here the hats indicate that objects are written in the orthonormal frame. Furtherv = dˆ v1 dˆ v2 dˆ v3 . more vˆ := (ˆ v1 , vˆ2 , vˆ3 ) and dˆ Then ρ(t), Ji (t) and Sij (t) tend to zero as t goes to infinity. More precisely, 2
ρ(t) = O(t−6/λ ),
2
Ji (t) = O(t−8/λ ),
2
Sij (t) = O(t−10/λ ).
Furthermore 2 Ji (t) = O(t−2/λ ) ρ(t) 2 Sij (t) = O(t−4/λ ). ρ(t)
(4.45) (4.46)
Proof. Note that f (t0 , v) has compact support on v. Also observe that Theorem 3 implies that Vi (t) is uniformly bounded. Combining these two facts implies that there exists a constant C such that f (t, v) = 0,
if |vi | ≥ C
(4.47)
for all t. By (4.31) we have f (t, vˆ) = 0, So using (4.47) and (4.48) we get ρ(t) =
|ˆ vi |≤Ct−2/λ2
2
if |ˆ vi | ≥ Ct−2/λ .
(4.48)
f (t, vˆ)(1 + |ˆ v |2 )1/2 dˆ v.
Note that since f (t, vˆ) is a constant along the characteristics, |f (t, vˆ)| ≤ f0 := sup{|f (t0 , vˆ)| : for all vˆ}.
(4.49)
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So we obtain ρ(t) ≤ C
2
|ˆ vi
|≤Ct−2/λ2
f (t, vˆ) dˆ v ≤ Cf0 t−6/λ .
Also by (4.47)–(4.49) we have f (t, vˆ)ˆ vi dˆ v Ji (t) = |ˆ vi |≤Ct−2/λ2 2 2 ≤ Ct−2/λ f (t, vˆ) dˆ v ≤ Cf0 t−8/λ . |ˆ vi |≤Ct−2/λ2
Similarly Sij (t) =
|ˆ vi
|≤Ct−2/λ2
−4/λ2
f (t, vˆ)ˆ vi vˆj (1 + |ˆ v |2 )−1/2 dˆ v
≤ Ct
2
|ˆ vi
|≤Ct−2/λ2
f (t, vˆ) dˆ v ≤ Cf0 t−10/λ .
Now let us estimate the ratios Ji /ρ and Sij /ρ. By means of (4.47) and (4.48), we get f (t, vˆ)ˆ vi dˆ v Ji (t) = ρ(t) f (t, vˆ)(1 + |ˆ v |2 )1/2 dˆ v 2 2 f (t, vˆ) dˆ v ≤ Ct−2/λ ≤ Ct−2/λ . f (t, vˆ)(1 + |ˆ v |2 )1/2 dˆ v Similarly Sij (t) = ρ(t)
f (t, vˆ)ˆ vi vˆj (1 + |ˆ v |2 )−1/2 dˆ v f (t, vˆ)(1 + |ˆ v |2 )1/2 dˆ v 2
≤ Ct−4/λ .
In this proposition since all components of the energy momentum tensor in an orthonormal frame go to zero as t goes to infinity, it can be concluded that in a certain sense solutions of Einstein-Vlasov system coupled to a nonlinear scalar field with a exponential potential are approximated by vacuum Einstein solutions. In a more detailed level (4.45) and (4.46) resemble the non-tilted dust-like solutions in which Ji (t) and Sij (t) are identically zero.
Acknowledgments The author thanks Alan D. Rendall for discussions on the subject of this paper.
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References [1] J.J. Halliwell, Scalar fields in cosmology with an exponential potential, Phys. Lett. B185, 341–344 (1987). [2] Y. Kitada K. Maeda, Cosmic no-hair theorem in power-law inflation, Phys. Rev. D45, 1416–1419 (1992). [3] Y. Kitada K. Maeda, Cosmin no-hair theorem in homogeneous spacetimes: I. Bianchi models, Class. Quantum Grav. 10, 703–734 (1993). [4] H. Lee, Asymptotic behaviour of the Einstein-Vlasov system with a positive cosmological constant, Math. Proc. Camb. Phil. Soc. 137 495–509 (2004). [5] S.B. Tchapnda N. N. Noutchegueme, The surface-symmetric Einstein-Vlasov system with cosmological constant, Preprint gr-qc/0304098 (2003). [6] S.B. Tchapnda N. A.D. Rendall, Global existence and asymptotic behaviour in the future for the Einstein-Vlasov system with positive cosmological constant, Class. Quantum Grav. 20, 3037–3049 (2003). [7] A.D. Rendall, Cosmic censorship for some spatially homogeneous cosmological models, Ann. Phys. 233, 82–96 (1994). [8] A.D. Rendall, Global properties of locally spatially homogeneous cosmological models with matter, Math. Proc. Camb. Phil. Soc. 118, 511–526 (1995). [9] A.D. Rendall. Asymptotics of solutions of the Einstein equations with positive cosmological constant, Ann. Henri Poincar´e 5, 1041–1064 (2004). [10] A.D. Rendall, Accelerated cosmological expansion due to a scalar field whose potential has a positive lower bound, Class. Quantum Grav. 21, 2445–2454 (2005). [11] R. Wald, Asymptotic behaviour of homogeneous cosmological models in the presence of a positive cosmological constant, Phys. Rev. D28, 2118–2120 (1983).
Hayoung Lee Max-Planck-Institut f¨ ur Gravitationsphysik Am M¨ uhlenberg 1 D-14476 Golm bei Potsdam Germany email:
[email protected] Communicated by Sergiu Klainerman submitted 01/04/04, accepted 14/06/04
Ann. Henri Poincar´e 6 (2005) 725 – 746 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/03725-22, Published online 28.07.2005 DOI 10.1007/s00023-005-0221-0
Annales Henri Poincar´ e
Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems Ian Melbourne, Viorel Nit¸ic˘a and Andrei T¨ or¨ ok∗
Abstract. Let f : X → X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We find several criteria for transitivity of noncompact connected Lie group extensions. As a consequence, we find transitive extensions for any finitedimensional connected Lie group extension. If, in addition, the group is perfect and has an open set of elements that generate a compact subgroup, we find open sets of stably transitive extensions. In particular, we find stably transitive SL(2, R)extensions. More generally, we find stably transitive Sp(2n, R)-extensions for all n ≥ 1. For the Euclidean groups SE(n) with n ≥ 4 even, we obtain a new proof of a result of Melbourne and Nicol stating that there is an open and dense set of extensions that are transitive. For groups of the form K × Rn where K is compact, a separation condition is necessary for transitivity. Provided X is a hyperbolic attractor, we show that an open and dense set of extensions satisfying the separation condition are transitive. This generalizes a result of Nit¸ic˘ a and Pollicott for Rn -extensions.
1 Introduction This paper is part of a program to classify the obstructions to (stable) topological transitivity in various classes of partially hyperbolic transformations. We concentrate on noncompact group-extensions of hyperbolic systems. Consider a transformation f : X → X, a Lie group G, and a mapping β : X → G called a cocycle. These determine a skew product, or G-extension, fβ : X × G → X × G,
fβ (x, h) = (f x, β(x)h).
It is assumed throughout that X is a hyperbolic basic set and that G is a finitedimensional connected Lie group. The G-extension fβ is called stably transitive if β lies in the interior (usually in the H¨ older topology) of the subset of extensions that are topologically transitive. (Recall that a transformation g : Y → Y is transitive if it has a dense orbit.) The question we intend to address is whether noncompact group extensions of a hyperbolic basic set are typically stably topologically transitive. If β takes values in a proper closed sub-semigroup S of G then obviously fβ is not transitive. An example is the group G = SL(n, R) with sub-semigroup ∗ The research of V.N. was supported partly by NSF Grant DMS 99-71826. The research of A.T. was supported partly by NSF Grant DMS-0244529. The research of I.M., V.N. and A.T. was partly supported by EPSRC Grant GR/R87543/01.
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S consisting of matrices with non-negative entries. Since Int S = ∅, we can construct open sets of nontransitive SL(n, R)-extensions. Our conjecture is that this situation is the only essential obstruction to transitivity. Conjecture 1.1 Assume that X is a hyperbolic basic set for f : X → X and G a finite-dimensional connected Lie group. Among the C 0 -small H¨ older cocycles β : X → G that are not cohomologous to a cocycle with values in a maximal subsemigroup of G with non-empty interior, there is a H¨ older open and dense set for which the extension fβ is transitive. Recall the definition of cohomology: Definition 1.2 Let G be a topological group. If β1 , β2 : X → G are continuous functions, and f : X → X is a transformation, then β1 and β2 are called cohomologous (over f ) if there exists a continuous function u : X → G such that β1 = (u ◦ f )β2 u−1 . In order to simplify the language, we let e denote both the identity element e ∈ G and the constant cocycle e : X → G that takes the value e everywhere, and we introduce: Definition 1.3 Let r ≥ 0. We say that a cocycle β : X → G is C r -small if it is C r -close to the identity cocycle e : X → G. In some cases (for example, if G is nilpotent or G = K Rn is a semidirect product of a compact group K and Rn ), the Conjecture might hold even for cocycles that are not C 0 -small. This paper attempts to give certain evidence in support of Conjecture 1.1. older-openness Many of our results are “H¨ older-open and C r -dense”, where H¨ s means “C -open for any s ∈ (0, 1), s < r”. Previously studied situations where the Conjecture is known to hold are the following: • G compact: Note that in this case, semigroups coincide with subgroups and there are no proper subgroups with nonempty interior. It was proved by Brin [4] that if the fiber is a compact connected Lie group, then the transitive extensions of a transitive Anosov diffeomorphism contain a set that is open and dense in the older topology. In C 2 -topology. As observed in [17], Brin’s result also holds in the H¨ older-open and fact, for any r > 0 the C r cocycles that are transitive contain a H¨ C r -dense set, and this result generalizes to extensions of a hyperbolic attractor. The latter result does not hold for extensions of general hyperbolic basic sets when r < 1 (in particular, the result is false if X is a subshift of finite type and G is a torus). However for compact group extensions of general hyperbolic basic sets, Field et al. [6] prove that the transitive extensions contain a set that is (i) H¨ older open and dense (proving the Conjecture), and (ii) C 2 -open, C r -dense for all r ≥ 2. (See also [19, 8, 7].) • G = SE(n), n even: For all n ≥ 2, there are again no sub-semigroups with nonempty interior (Corollary 6.9). For n ≥ 4 even, Melbourne and Nicol [11] prove
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that the set of stably transitive extensions of a hyperbolic basic set is H¨ older-open and C r -dense for all r > 0. This is obtained as an application of Nit¸ic˘a [13], by verifying ε-accessibility and density of recurrent points. The argument for ε-accessibility in [11] breaks down for n = 2, but this can be recovered by assuming that the hyperbolic basic set is an attractor. So, for all r > 0, SE(2)-extensions of hyperbolic attractors are transitive for a H¨ older-open and C r -dense set of cocycles. (See Proposition 6.1.) Conjecture 1.1 remains open for SE(n)-extensions with n ≥ 3 odd, though some partial results are obtained in this paper. • G = Rn : Here, the maximal semigroups with non-empty interior are the halfspaces whose bounding hyperplane contains the origin. Hence, stable transitivity is certainly not a generic property of Rn -extensions. However, there are no further obstructions. Nit¸ic˘a and Pollicott [15] prove that an Rn -extension fβ over an infranil Anosov diffeomorphism is transitive (and hence stably transitive) if and only if β is not cohomologous to a cocycle with values in such a half-space. Moreover, the transitive H¨ older Rn -extensions are actually C 0 -stably transitive. For general hyperbolic basic sets, transitive Rn -extensions need not be stably transitive. However, let S denote the set of cocycles that are not cohomologous to a cocycle with values in a half-space. For cocycles in S, Field et al. [6] prove a result identical to that stated above for compact group extensions. Again this proves the Conjecture for Rn -extensions. Identical statements hold for general Abelian finite-dimensional Lie groups G = Rn × T d, where T d is a d-dimensional torus. Write fβk (x, g) = (f k x, β(k, x)g). For k ≥ 0 this gives β(k, x) = β(f k−1 x)β(f k−2 x) · · · β(f x)β(x). The key notion in this paper is the following: Definition 1.4 Let fβ : X × G → X × G be a skew-extension. Given x ∈ X, let Lβ (x) = {g ∈ G| there exist xk ∈ X and nk > 0 such that xk → x and fβnk (xk , e) → (x, g)}. That is, Lβ (x) consists of the possible limits limk→∞ β(nk , xk ), subject to xk → x and f nk (xk ) → x. Note that we do not require that nk → ∞ or that xk = x. Clearly Lβ (x) is a closed subset of G. In Section 3, we study the properties of Lβ (x) when f is hyperbolic. In particular, Lβ (x) is a semigroup of G. (See Lemma 3.1.) Under a center bunching condition on β , fβ is transitive provided that Lβ (x) = G for some x ∈ X. (See Theorem 3.3.) As a consequence we obtain new results about the existence of transitive and stably transitive noncompact group extensions. We note that the bunching condition is automatically satisfied for nilpotent groups and semidirect products K Rn where K is compact, as well as for sufficiently C 0 -small cocycles.
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General finite-dimensional connected Lie groups
For arbitrary finite-dimensional connected Lie groups, there always exist transitive extensions. Theorem 1.5 Let G be a finite-dimensional connected Lie group, and X a hyperbolic basic set for f : X → X. Then for any r > 0 there is a C r cocycle β : X → G such that fβ is transitive. The cocycle β can be chosen to be arbitrarily C r -small.
1.2
Groups with compact elements
For a restricted class of groups we can exhibit stably transitive skew-products. Introduce the following property: Definition 1.6 Call an element g ∈ G compact if it generates a compact subgroup. Let C ⊂ G denote the set of compact elements. Theorem 1.7 Let G be a finite-dimensional connected Lie group and let X be a hyperbolic basic set for f : X → X. Let r > 0. (a) If G is perfect and Int C = ∅, then there is a nonempty H¨ older-open set of C r cocycles β : X → G for which fβ is transitive. This set contains cocycles that are arbitrarily C r -small. (b) If G is a semidirect product of a compact connected Lie group and Rn , G is perfect, and Int C is dense in G, then there is a H¨ older-open and C r -dense set of cocycles β : X → G for which fβ is transitive. Part (a) of this theorem applies immediately to the symplectic group Sp(2n, R) (see Corollary 4.5). Part (b) applies to the Euclidean group SE(n), n ≥ 4 even (see Corollary 4.7) and so we recover by a different technique the result of [11].
1.3
The groups G = K × Rn , K compact
Let K be a compact connected Lie group and form the direct product K × Rn . As was the case for Rn , there are maximal semigroups with nonempty interior of the form K × {half-space}. We show that these are the only obstructions when X is a hyperbolic attractor. Denote by S the set of C r cocycles β : X → K × Rn for which the Rn component of β is not cohomologous to a cocycle with values in a half-space. Theorem 1.8 Suppose that X is a hyperbolic attractor and G is of the form K × Rn where K is a compact connected Lie group. Let r > 0. Then there is a H¨ older-open and C r -dense subset of cocycles in S for which fβ is transitive.
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The groups SE(n)
We now consider Euclidean group extensions, so G = SE(n) = SO(n)Rn , n ≥ 2. (When n = 1, we have an R-extension dealt with above.) The simplest situation is n ≥ 4 even. Melbourne and Nicol [11] proved that there is an open-dense set of transitive SE(n)-extensions for such n and, as mentioned above, we recover their result as a consequence of Theorem 1.7. The case n = 2 was mentioned above (see Proposition 6.1). However, the results in [11] have nothing to say about the case n ≥ 3 odd. We can prove a result about stable transitivity in a special case: Theorem 1.9 Let σ : Σ → Σ be a transitive subshift of finite type. Let n ≥ 3. Then the class of locally constant cocycles β : Σ → SE(n) contains a C 0 -open and H¨ older dense subset for which σβ is transitive.
1.5
Semigroup problem
For many groups (see [22]), it is not hard to show that there is a large open set U ⊂ Gp (p large enough) such that if F ∈ U then the family F generates G as a group (that is, the group generated by F is dense in G). To obtain the condition Lβ (x) = G, we would like to prove that for a typical family F ∈ Gp that generates G as a group, if F is not contained in a maximal semigroup with non-empty interior, then F generates G as a semigroup as well. We refer to this as the Semigroup Problem. This is true for G = Rn [15] and more generally for groups of the form K × Rn where K is compact, see Theorem 5.10. The result is also true for G = SE(n), see Theorem 6.8.
1.6
Structure of the paper
In §2 we introduce certain invariance properties for a metric on a group, and prove a few inequalities related to them and H¨ older cocycles. In §3 we prove that the invariant Lβ (x) is a semigroup and obtain a criterion for transitivity in terms of Lβ (x). In §§4, 5 and 6 we prove the transitivity results for general Lie groups, K × Rn , and SE(n). In §7 we list some open questions.
2 Inequalities 2.1
Hyperbolicity
Let M be a smooth manifold endowed with a Riemannian metric. Let f : M → M be a smooth diffeomorphism and X ⊂ M a compact and f -invariant subset of M . We say that f : X → X is hyperbolic if there exists a continuous T f -invariant splitting E s ⊕ E u of the tangent bundle TX M and constants C > 0, 0 < λ < 1,
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such that for all n ≥ 0 and x ∈ X we have:
(Df n )x v ≤ Cλn v , v ∈ E s
(Df −n )x v ≤ Cλn v , v ∈ E u .
(2.1)
We say that X is maximal and isolated if there exists an open neighborhood U of X such that every compact f -invariant set of U is contained in X. The set X is a basic set for f : M → M if: 1. f is hyperbolic on X; 2. X is maximal and isolated; 3. f : X → X is transitive. We say that a basic set X is a hyperbolic attractor if there is a forward invariant open set U ⊂ M such that X = ∩n≥0 f n (U ).
2.2
Center bunching
Let G be a connected Lie group with Lie algebra LG. Let Ad denote the adjoint action of G on LG, and choose a norm on LG. There is a metric d on G with the following properties (Pollicott and Walkden [20, p. 288]): 1. d(γ1 δ, γ2 δ) = d(γ1 , γ2 ); 2. d(δγ1 , δγ2 ) ≤ Ad(δ) d(γ1 , γ2 ); for any γ1 , γ2 , δ ∈ G. The estimates we need are related to the fact that the skew-extension can be viewed as a partially hyperbolic transformation (see, e.g., [16, 17, 20]). We are using the terminology of [20]. Definition 2.1 Given a cocycle β : X → G, define µ ≥ 1 to be µ = max lim sup Ad(β(n, x)) 1/n , lim sup Ad(β(n, x))−1 1/n . n→∞ x∈X
n→∞ x∈X
For α ∈ (0, 1), we say that a C α cocycle β is center bunched if µλα < 1. Remark Although center bunching is sufficient for some of the constructions in this paper, our main results require a strong center bunching condition of the form µ8 λα < 1. If G is compact or nilpotent, then µ = 1 so that H¨ older cocycles are automatically (strongly) center bunched. The same is true for semidirect products G = K Rn where K is compact.
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The α-H¨ older (semi)norm of β : X → G is defined by
β α = sup x=y
d(β(x), β(y)) . distX (x, y)α
The main result of this section is the following: Lemma 2.2 Let (G, d) be a connected Lie group, X a hyperbolic basic set for f : X → X, and β : X → G an α-H¨ older cocycle. Assume the center bunching condition µλα < 1. Then there is a constant C = C(f, β) > 0 with the following property. Given ε > 0 sufficiently small and any n ≥ 1, assume that there are two trajectories xk = f k x0 , yk = f k y0 , such that d(xk , yk ) < ε for 0 ≤ k ≤ n − 1. Then d(β(n, x0 ), β(n, y0 )) ≤ C Ad(β(n, x0 )) + 1 εα .
(2.2)
Proof. From the local product structure it follows (for ε sufficiently small) that s u (xk ) ∩ Wloc (yk ) consists of a single point for 0 ≤ k ≤ n − 1. the intersection Wloc s u Denote zk = Wloc (xk ) ∩ Wloc (yk ) and note that zk = f k z0 . There is a constant C0 , independent of n, x0 and y0 , such that d(xk , zk ) ≤ C0 λk d(x0 , z0 ), d(yk , zk ) ≤ C0 λn−k d(yn−1 , zn−1 ), d(x0 , z0 ) ≤ C0 d(x0 , y0 ), d(yn−1 , zn−1 ) ≤ C0 d(xn−1 , yn−1 ). By center bunching, there exists δ > 0 such that (µ + δ)λα < 1. By definition of µ, there exists a constant C1 > 0 such that Ad(β(k, x))±1 ≤ C1 (µ + δ)k for all x ∈ X and k ≥ 1. Denote: ωk = β(xk ), Ω = β(n, x0 ) = ωn−1 ωn−2 . . . ω0 , γk = β(yk ), Γ = β(n, y0 ) = γn−1 γn−2 . . . γ0 , φk = β(zk ),
Φ = β(n, z) = φn−1 φn−2 . . . φ0 .
We claim that there are constants C , C > 0 depending only on f and β such that d(Ω, Φ) ≤ C Ad(Ω)
β α d(x0 , z0 )α
(2.3)
d(Φ, Γ) ≤ C β α d(yn−1 , zn−1 )α .
(2.4)
It then follows from the triangle inequality that d(Ω, Γ) ≤ C( Ad(Ω) + 1)εα as required with C = max{C , C }C0α β α .
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Proof of (2.3). d(Ω, Φ) = d(ωn−1 ωn−2 . . . ω0 , φn−1 φn−2 . . . φ0 ) ≤ ≤
n−1 k=0 n−1
d(ωn−1 . . . ωk+1 ωk φk−1 . . . φ0 , ωn−1 . . . ωk+1 φk φk−1 . . . φ0 )
Ad(ωn−1 . . . ωk+1 ) d(ωk , φk ) =
k=0
≤ Ad(Ω)
n−1
Ad(Ωω0−1 . . . ωk−1 ) d(ωk , φk )
k=0 n−1
Ad(β(k + 1, x0 ))−1 d(ωk , φk ).
k=0
Moreover, Ad(β(k + 1, x0 ))−1 ≤ C1 (µ + δ)k+1 and d(ωk , φk ) ≤ β α d(xk , zk )α ≤ β α {C0 λk d(x0 , y0 )}α , and so d(Ω, Φ) ≤ Ad(Ω)
β α λ−α C0α C1 d(x0 , y0 )α
n−1
[(µ + δ)λα ]k+1
k=0
≤ C Ad(Ω)
β α d(x0 , y0 )α , where C = C0α λ−α C1 (1 − (µ + δ)λα )−1 . Proof of (2.4). Similarly, d(Γ, Φ) = d(γn−1 γn−2 . . . γ0 , φn−1 φn−2 . . . φ0 ) n ≤ d(γn−1 . . . γn−k+1 γn−k φn−k−1 . . . φ0 , γn−1 . . . γn−k+1 φn−k φn−k−1 . . . φ0 ) ≤ ≤
k=1 n k=1 n
Ad(γn−1 . . . γn−k+1 ) d(γn−k , φn−k )
Ad(β(k − 1, yn−k+1 ))
β α {C0 λk d(yn−1 , zn−1 )}α
k=1
≤ C β α d(yn−1 , zn−1 )α where C = C0α λα C1 (1 − (µ + δ)λα )−1 .
3 Criteria for transitivity of skew-products We introduced the closed subset Lβ (x) ⊂ G in Definition 1.4. We now show that Lβ (x) is a semigroup.
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Lemma 3.1 Let X be a hyperbolic basic set for f : X → X, and β : X → G an α-H¨ older cocycle, where G is a finite-dimensional connected Lie group. Assume the center bunching condition µλα < 1. Then, for any x ∈ X, the set Lβ (x) is a closed semigroup. Proof. Let h1 , h2 ∈ Lβ (x); we show that h2 h1 ∈ Lβ (x). It follows from the definition of Lβ (x) that for any ε > 0 there are positive integers ni and points yi ∈ X, i = 1, 2 such that: d(yi , x) < ε,
d(f ni (yi ), x) < ε,
d(β(ni , yi ), hi ) < ε.
(3.1)
We can arrange also that
Ad(β(ni , yi ) ≤ Ad(hi ) + 1,
(3.2)
for i = 1, 2. By standard shadowing techniques (see [12, page 74]), there is a K > 0 depending only on f such that one can (Kε)-shadow the pseudo-orbit {y1 , f y1 , . . . , f n1 y1 ≈ y2 , f y2 , . . . , f n2 y2 } by an orbit of length n1 + n2 of a point z ∈ X. Since d(γ2 γ1 , ω2 ω1 ) ≤ Ad(ω2 ) d(γ1 , ω1 ) + d(γ2 , ω2 ), it follows that d(β(n1 + n2 , z), h2 h1 ) = d(β(n2 , f n1 z)β(n1 , z), h2 h1 ) ≤ Ad(h2 ) d(β(n1 , z), h1 ) + d(β(n2 , f n1 z), h2 ).
(3.3)
Using Lemma 2.2 together with inequalities (3.1) and (3.2), we obtain d(β(n1 , z), h1 ) ≤ d(β(n1 , z), β(n1 , y1 )) + d(β(n1 , y1 ), h1 ) ≤ C Ad(β(n1 , y1 )) + 1 (Kε)α + ε ≤ C Ad(h1 ) + 2 (Kε)α + ε. A similar estimate holds for d(β(n2 , f n1 z), h2 ). Substituting these estimates into (3.3), gives d(β(n1 + n2 , z), h2 h1 ) ≤ C (h1 , h2 )εα , where C (h1 , h2 ) is a constant independent of the lengths of the orbits. Taking ε → 0+ , we conclude that h2 h1 ∈ Lβ (x). Remark Finite-dimensionality of G is used only to guarantee that Ad(h) : LG → LG is a bounded operator for h ∈ G. For G an infinite-dimensional connected Lie group, it remains true that if hi ∈ Lβ (x) with Ad(hi ) < ∞ for i = 1, 2, then h2 h1 ∈ Lβ (x). The next result follows from the symbolic representation of basic sets for hyperbolic diffeomorphism due to [3].
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Lemma 3.2 Let X be a hyperbolic basic set for f : X → X. Then there is a constant K > 1, such that for any ε > 0 and any x, y ∈ X there exists a trajectory of f joining B(x, ε) to B(y, ε) in at most 2 ln ε/ ln λ + K steps. Proof. There exists an ω-H¨older and onto map π : ΣA → X where ΣA is a subshift of finite type with metric dθ , 0 < θ < 1 and ω = ln λ/ ln θ (as in for example [18, Theorem III.3, p. 228]). There exists a constant C0 such that the π-image of any cylinder C−m,m has diameter less than C0 (θm )ω = C0 λm . In particular, if C0 λm < ε, then the cylinder C−m,m determined by x has the π-image in B(x, ε). For this, it suffices to take m=
ln ε ln λ
−
ln C0 +1 . ln λ
(3.4)
From the transitivity of f it follows that there exists a constant K0 such that any two symbols in ΣA can be joined by a block of length less than K0 . Consider now the blocks B1 and B2 of length 2m+1 corresponding to the cylinders determined by x and respectively y, and a block B3 of length less than K0 joining the last symbol of B1 with the first symbol of B2 . Then the block B1 B3 B2 gives a trajectory in ΣA of length less than 2m + K0 between an element in the cylinder determined by x and an element in the cylinder determined by y. Applying π we obtain a trajectory in X of length less than 2m + K0 . If m is chosen as in formula (3.4), then the lemma follows with K = K0 − 2 ln C0 / ln λ + 2. We can now state and prove our criteria for transitivity of noncompact extensions. Theorem 3.3 Let G be a connected Lie group. Assume that X is a hyperbolic basic set for f : X → X, and β : X → G is a H¨ older cocycle. Assume the strong center bunching condition µ8 λα < 1. If there exists x0 ∈ X such that Lβ (x0 ) = G, then the skew-product fβ is transitive. Proof. We need to show that for any open sets U, V ⊂ X × G there is a positive integer N such that fβN (U )∩V = ∅. Let (y, g1 ) ∈ U and (z, g2 ) ∈ V . Let h = g2 g1−1 . Let ε > 0 be fixed, smaller than the hyperbolicity constant λ, and such that B((y, g1 ), ε) ⊂ U and B((z, g2 ), ε) ⊂ V . Let ω1 be an orbit of f from B(y, ε) to B(x0 , ε), and ω2 an orbit of f from B(x0 , ε) to B(z, ε), chosen as in Lemma 3.2. The orbits ω1 , ω2 have length at most n where n ≤ 2 ln ε/ ln λ + K. Since Lβ (x0 ) = G, there exists an orbit ω of f starting and ending in B(x0 , ε) such that d(β(ω), β(ω2 )−1 hβ(ω1 )−1 ) < ε. Altogether, ω1 ωω2 gives a pseudo-orbit for fβ starting in U and ending in V . By standard shadowing techniques (see [12, page 74]), one can find an orbit ω 1 ω ω 2 of f which K ε-shadows the pseudo-orbit ω1 ωω2 . The constant K > 0 depends only on f . We obtain an orbit ( ω1 ω ω 2 , β( ω1 ω ω 2 )) for fβ starting in U , and we must show that this orbit ends in V also.
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Choose δ so that (µ + δ)λα/8 < 1. There exists C1 such that Ad(β(k, x))±1
≤ C1 (µ + δ)k for all k ≥ 1 and x ∈ X. Since ω1 , ω2 have length at most n,
Ad(β(ωi ))±1 ≤ C1 (µ + δ)n , i = 1, 2.
(3.5)
For ε sufficiently small, we can ensure that
Ad(β(ω)) ≤ 2C12 Ad(h) (µ + δ)2n , (since d(β(ω)β(ω1 )h
−1
(3.6)
β(ω2 ), e) < ε,
so by continuity Ad(β(ω)β(ω1 )h−1 β(ω2 )) < 2). By the triangle inequality, Lemma 2.2 and (3.5), (3.6), d(β(ω2 )β(ω)β(ω1 ), β( ω2 )β( ω )β( ω1 )) ω2 )) + Ad(β(ω2 )) d(β(ω), β( ω )) ≤ d(β(ω2 ), β( + Ad(β(ω2 )β(ω)) d(β(ω1 ), β( ω1 )) ≤ C Ad(β(ω2 )) + 1 + Ad(β(ω2 )) Ad(β(ω)) + 1 + Ad(β(ω2 ))
Ad(β(ω)) Ad(β(ω1 )) + 1 (K ε)α ≤ C((µ + δ)4n + 1)εα . Recall that n ≤ 2 ln ε/ ln λ + K, so that α+8 ln(µ+δ)/ ln λ (µ + δ)4n εα ≤ Cε 4n which has a positive exponent by the choice of δ. Hence C((µ+δ) +1)εα converges to 0 as ε approaches 0.
4 Transitive extensions for general Lie groups In this section we describe several applications of our transitivity criteria. Throughout this section we assume that X is a hyperbolic basic set for f : X → X. Without loss of generality we can also assume that f has fixed points (if not, take an iterate of it). Let G be a finite-dimensional connected Lie group. By β : X → G we denote a center bunched α-H¨older cocycle. The proofs depend on a way to generate elements of Lβ (x). Lemma 4.1 Let x ∈ X be a fixed point for the transformation f and y a homoclinic point to x. If there is a subsequence nk → ∞ such that β(nk , x) → e, then ωx (y) ∈ Lβ (x), where ωx (y) is the holonomy of the homoclinic loop determined by y. Let us describe the meaning of ωx (y). Consider the homoclinic path determined by the orbit of y ∈ W s (x) ∩ W u (x) (covered along W u (x) from x to y and then along W s (x) from y to x). Then, the lift to the unstable/stable foliations of fβ , with initial point (x, e), of this homoclinic path ends at (x, ωx (y)).
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Note that these holonomy values can be easily modified by changing β in an open set which contains only finitely many iterates of y. Moreover, the holonomy varies continuously with the cocycle β: if β is C α -close to β, then ω x (y) is close to ωx (y). See more details in the proof. Proof of Lemma 4.1: As in [17, Theorem A.3] and [16, Theorems 2.4 and A.1], under the standing hypothesis of this section, the stable leaf of fβ through (x, e) is the graph of the function γxs : W s (x) → G,
γxs (t) = lim β(n, t)−1 β(n, x). n→∞
(4.1)
This function is α-H¨older, and varies continuously with the cocycle β in the follows s (x), γk,x → γxs ing sense: if βk → β in C 0 and βk stay C α -bounded, then, on Wloc in C 0 . Applying the above results to f −1 , we obtain that the unstable manifold is the graph of γxu : W u (x) → G,
γxu (t) = lim β(n, t)−1 β(n, x), n→−∞
and the same continuous dependence holds. Therefore, the holonomy around the homoclinic loop determined by y ∈ W s (x) ∩ W u (x) is −1 ωx (y) = lim β(n, y)−1 β(n, x) β(−n, y)−1 β(−n, x) n→∞
= lim β(n, x)−1 β(2n, f −n y)β(−n, x). n→∞
Hence, if β(nk , x) → e, then ωx (y) ∈ Lβ (x) because lim f 2nk (f −nk y, e) k→∞ β
= lim (f nk y, β(2nk , f −nk y)) = (x, ωx (y)). k→∞
Proposition 4.2 Let G be a connected finite-dimensional Lie group. There exists k ≥ 1 (k = 2 dim G suffices) such that for any ε > 0, there exist g1 , . . . , gk with d(gi , e) < ε such that the closed sub-semigroup generated by g1 , . . . , gk is G. Proof. Choose ξ1 . . . , ξn that generate LG. For each i, choose ai , bi > 0 with ai /bi ∈ Q and set gi = exp(ai ξi ), hi = exp(−bi ξi ). Shrink ai and bi if necessary so that d(gi , e) < ε and d(hi , e) < ε. The closed sub-semigroup Si generated by gi and hi is in fact a Lie group and is the closure of the one-parameter subgroup generated by ξi . Hence, if S is the closed sub-semigroup generated by g1 , . . . , gn , h1 , . . . , hn , then S is a Lie group with Lie algebra containing ξ1 , . . . , ξn . Hence LS = LG and so S = G as required.
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Proof of Theorem 1.5: Let k be given by Proposition 4.2. Pick a fixed point x of f (or of an iterate of f ), and k homoclinic points to x, say y1 , . . . , yk , such that they have mutually disjoint orbits. By Proposition 4.2, we may choose a set of group generators {g1 , . . . , gk } of G that are as close to the identity as desired. Let β : X → G be the trivial cocycle. One can obtain ωx (yi ) = gi by setting β(yi ) = gi , while keeping β ≡ e on the remaining points in the trajectories of yi (also keeping β(x) = e). Since we only have to perturb β at finitely many points, the resulting cocycle is arbitrarily C r -close to the identity. By keeping the cocycle small, we ensure also that Theorem 3.3 and formula (4.1) hold. Since β(x) = e, Lemma 4.1 implies that all these holonomies are in Lβ (x), hence Lβ (x) = G. The conclusion follows from Theorem 3.3. Proposition 4.3 Let G be a connected finite-dimensional Lie group. Let C denote the set of compact elements in G and suppose that Int C = ∅. Then e ∈ Int C. Proof. First note that if g ∈ G and n ≥ 1, then g ∈ C if and only if g n ∈ C. Hence nth roots of elements in Int C lie in Int C. Thus it suffices to verify that there are elements in Int C of infinite order. (Such elements generate tori and hence have nth roots arbitrarily close to e.) We use the following structure theorem for finite-dimensional connected Lie groups ([5]): There is a compact connected Lie group K ⊂ G that is maximal in the sense that every compact element is conjugate to an element of K. The condition Int C = ∅ implies that dim K ≥ 1. In particular, there is a dense set of elements in K of infinite order. Hence, if g ∈ C, then g lies in a copy of K and can be perturbed to have infinite order. The following lemma appears in [10, Lemma 3], for pairs of generators. Lemma 4.4 (Kuranishi) Let G be a connected perfect Lie group. If {f1 , f2 , . . . , fk } ⊂ G is a finite set that topologically generates G as a group, then there is a neighborhood V of e such that for any fi ∈ V fi , the set {f1 , f2 , . . . , fk } topologically generates G as well.
Proof of Theorem 1.7: First we prove statement (a). As in the proof of Theorem 1.5, we start with the trivial cocycle β and make C r -small perturbations at finitely many points. Again, we pick a fixed point x and k homoclinic points y1 , . . . , yk . The main difference is that we begin by perturbing β(x) to lie in Int C (this is possible by Proposition 4.3). Since β(x) is a compact element, we are still in a position to apply Lemma 4.1. Choose k near identity elements g1 , . . . , gk as in Proposition 4.2 but with the additional property that gi ∈ Int C. (The proof of Proposition 4.2 is easily modified using the fact that dim C = dim G.) After the initial perturbation at x,
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the holonomies ωx (yi ) are arbitrarily close to the identity, so we can make C r small perturbations at yi so that ωx (yi ) = gi . In this way, we obtain a transitive extension just as in the proof of Theorem 1.5. It remains to verify that transitivity persists under H¨ older perturbations of β. The properties β(x) ∈ Int C and gi = ωx (yi ) ∈ Int C certainly persist, so the only question is whether the gi continue to topologically generate G as a semigroup. Since gi are compact elements, it is equivalent to show that they generate G as a group. But since G is perfect, it follows from Lemma 4.4 that generating G as a group is a stable property. To prove statement (b), note that strong center bunching is now automatic. If moreover the set of compact elements in G is open and dense, then we can start with any cocycle, and the proof proceeds as above. Corollary 4.5 Let X be a hyperbolic basic set for f : X → X . Then any C r neighborhood of the identity cocycle e : X → Sp(2n, R) contains a H¨ older-open set of cocycles β for which fβ is transitive. Proof. Recall that Sp(2n,
R) is the group of all matrices M ∈ GL(2n, R) satisfying 0 I n . This is a semisimple group and hence is M T JM = J where J = −In 0 perfect. It is well known that Int C consists of those M for which all eigenvalues are simple, lie on the unit circle, and are not equal to ±1 (e.g., [1, Example 3.5]). Now apply Theorem 1.7. Lemma 4.6 The group SE(n), n ≥ 3, is perfect. Proof. Recall that SE(n) = SO(n)Rn . Since SO(n) is perfect for n ≥ 3, SO(n) ⊂ [SE(n), SE(n)]. Let (k, v), (k, v ) ∈ SE(n). Then: (k, v)(k, v )(k, v)−1 (k, v )−1 = (e, (1 − k)(v − v )). For any v0 ∈ Rn we can choose v, v ∈ Rn and k ∈ SO(n) such that (1−k)(v−v ) = v0 , thus Rn ⊂ [SE(n), SE(n)]. The statement in the previous lemma is not true for SE(2). Corollary 4.7 Let X be a hyperbolic basic set for f : X → X . If n ≥ 4 even, then in the set of C r SE(n)-extensions of f there is a H¨ older-open and C r -dense subset of stably transitive transformations. Proof. The interior of the set C of compact elements is dense in SE(n) for n even. By Lemma 4.6, SE(n) is perfect for n ≥ 3. Hence the result is a corollary of Theorem 1.7. Remark The argument for SE(n), n ≥ 4 even, generalizes as follows. Suppose that Γ ⊂ GL(n) is perfect, that Int C = ∅, and that there is an open subset of Int C
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consisting of matrices with no eigenvalue equal to 1. Form the semidirect product G = Γ Rn where the automorphism is given by restriction of the action of GL(n) on Rn . Then G is perfect and Int C = ∅ for G. Hence Theorem 1.7 applies to G. In particular, there exist stably transitive Sp(2n, R) R2n -extensions for all n ≥ 1.
5 K × Rn -extensions The main aim in this section is to prove Theorem 1.8 concerning K×Rn -extensions. We start by reviewing techniques of Brin [4] and Nit¸ic˘a [13]. For the moment, G is any semidirect product K Rn where K is a compact connected Lie group. Definition 5.1 Let X be a metric space, and f : X → X a continuous map. A point x ∈ X is called nonwandering if for any neighborhood U of x there is a positive integer n such that f n (U ) intersects U . A proof of the following lemma follows from Appendix A in [17]. Lemma 5.2 Let X be a hyperbolic basic set for f : X → X, and β : X → G a H¨ older cocycle. Then there exist a pair of fβ -invariant H¨ older foliations of X × G, called stable, respectively unstable. Definition 5.3 Let X be a hyperbolic basic set for f : X → X, and β : X → G a H¨ older cocycle. Denote by W s (x) and W u (x) the leaves of the stable, respectively unstable, foliations passing through x ∈ X × G. The pair of stable and unstable foliations is called ε-accessible for any ε > 0 if for any pair of points x, y ∈ X × G and any ε > 0 there is a sequence of points x0 = x, x1 , . . . , xn ∈ X × G such that xi ∈ W s (xi−1 ) or xi ∈ W u (xi−1 ), and d(xn , y) < ε. The following lemma is proved in [13, Theorem 2.2]. Lemma 5.4 Let X be a hyperbolic basic set for f : X → X, and β : X → G a H¨ older cocycle. If the skew-product fβ has a dense set of nonwandering points and the pair of stable and unstable foliations is ε-accessible for any ε > 0, then fβ is transitive. From [4] it is easy to derive the following lemma. Lemma 5.5 Let X be a hyperbolic attractor for f : X → X. Then, the set of C r cocycles β : X → G for which the stable and unstable foliations of fβ are accessible contains a H¨ older-open and C r -dense set. Remark We conjecture that the previous lemma holds under the weaker assumption that X is a hyperbolic basic set for f : X → X. We will see in Corollary 6.4 that this is indeed the case if the fiber is SE(n), n ≥ 3. In the remainder of this section, we prove Theorem 1.8. Let π2 be the canonical projection from K × Rn onto Rn . For β : X → K × Rn denote β2 = π2 ◦ β.
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An obstruction to transitivity
We first describe a necessary condition for transitivity that follows from the Livˇsic Positive Theorem of Bousch [2, §4]. Definition 5.6 ([15]) For β2 : X → Rn , let Hβ2 = {β2 (k, x) | x ∈ X is a point with prime period k} ⊂ Rn . Lemma 5.7 Let X be a hyperbolic basic set for f : X → X, and β : X → K × Rn a H¨ older cocycle. Then β is cohomologous to a cocycle whose Rn -component takes values in a half-space if and only if Hβ2 is contained in the half-space, Proof. One direction is clear. Conversely, it follows from [15, Lemma 2(2)] that β2 is cohomologous via a function u : X → Rn to a cocycle γ : X → Rn taking values in the half-space. Define γˆ = uβ(u ◦ T )−1 . Then γˆ is cohomologous to β and the Rn -component π2 γˆ = γ takes values in the half-space. Clearly, fβ cannot be transitive in the situation described in Lemma 5.7.
5.2
Transitivity of K × Rn extensions
For the sake of completeness, we include the following well-known result. Proposition 5.8 Let G be a Lie group. Then any compact semigroup S ⊂ G is actually a subgroup, hence it contains the identity element. Proof. Let g ∈ S. We show that g −1 ∈ S. Since S is compact, there is an increasing sequence {ni } such that {g ni } converges. Then g ni −ni−1 −1 lies in S and converges to g −1 . Lemma 5.9 Let X be a hyperbolic basic set for f : X → X, and β : X → K × Rn a continuous cocycle. Suppose that there exists x ∈ X and v ∈ Rn such that (x, v) is a transitive point for fβ2 . Then every point in X × (K × Rn ) is nonwandering for fβ . Proof. Let y ∈ X. Because K is compact, the transitivity of fβ2 implies that Lβ (y) contains an element (k, 0). By Proposition 5.8, (e, 0) ∈ Lβ (x), that is, (y, e, 0) is nonwandering. Due to the skew-product structure, this is equivalent to the whole fiber {y} × K × Rn being nonwandering. Proof of Theorem 1.8: By Lemma 5.5, there is an open and dense set of cocycles β possessing the accessibility property for the pair of stable and unstable foliations. Restricting to the open subset S it follows from [6] that the Rn -extension fβ2 is transitive for an open and dense set of cocycles β ∈ S. By Lemma 5.9, the corresponding K × Rn -extensions fβ consist of nonwandering points. The result follows from Lemma 5.4.
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The semigroup problem for K × Rn
Theorem 5.10 Let S ⊂ K × Rn . Assume that the closure of the group generated by S is K × Rn , and that the projection of S onto Rn does not lie in a half-space. Then the closure of the semigroup generated by S is K × Rn . Proof. Let (k0 , v0 ) ∈ S and let T denote the closure of the semigroup generated by S. We show that (k0 , v0 )−1 ∈ T . Note that an element (h, 0) ∈ K × Rn generates a compact subgroup by Proposition 5.8. Hence, if (h, 0) ∈ T , then (h, 0)−1 ∈ T . Let π2 S be the projection of S on Rn . By assumption, π2 S does not lie in a half-space. Moreover the closure of the group generated by π2 S is Rn . It follows from [15, Lemma 5] that the closure of the semigroup generated by π2 S is Rn as well. Since K is compact, π2 T = Rn . In particular, there exists k1 ∈ K such that (k1 , −v0 ) ∈ T . Let h = k0 k1 . Then (h, 0) = (k0 , v0 )(k1 , −v0 ) ∈ T and so (h, 0)−1 ∈ T . Hence, (k0 , v0 )−1 = (k1 , −v0 )(h, 0)−1 ∈ T .
6 SE(n)-extensions Recall that SE(n) = SO(n) Rn is the group generated by rotations and translations in Rn . The multiplication in SE(n) is given by (k1 , v1 )(k2 , v2 ) = (k1 k2 , k1 v2 + v1 ).
6.1
Transitivity of SE(2)-extensions
Proposition 6.1 Let X be a hyperbolic attractor for f : X → X and let r > 0. Then there is a H¨ older-open and C r -dense set of cocycles β : X → SE(2) for which fβ is transitive. Proof. By Lemma 5.5, accessibility of the pair of stable and unstable foliations holds for an open and dense set of cocycles. By [11, Theorem 3.2], the recurrent points are dense for an open and dense set of cocycles. The result follows from Lemma 5.4.
6.2
Generating sets for SE(n)
Lemma 6.2 Let n ≥ 1. The set of (n + 1)-tuples that generate Rn as a closed group is dense in (Rn )n+1 . Proof. See Lemma 2.6 in [14].
Lemma 6.3 Let n ≥ 3. The set of (n + 3)-tuples in SE(n) that generate SE(n) as a closed group is open and dense in SE(n)n+3 .
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Proof. Choose an arbitrary (n + 3)-tuple {(ki , vi )}i ⊂ SE(n). It follows from [21] that we can find k 1 , k 2 arbitrarily close to k1 , k2 such that the closed group generated by k 1 , k 2 is SO(n). Then find k i arbitrarily close to ki , 3 ≤ i ≤ n + 3, so that the inverses of the elements k i , 3 ≤ i ≤ n + 3, are in the group generated by k1 , k2 . −1 Hence, there are elements vi ∈ Rn , 3 ≤ i ≤ n + 3, such that (k i , vi ) are in the 2 group generated by {(k i , vi )}i=1 . Therefore, the group generated by {(ki , vi )}n+3 i=1 −1 contains (k i , vi )(k i , vi ) = (e, vi + k i vi ), 3 ≤ i ≤ n + 3. From Lemma 6.2 it follows that we can find vectors v i arbitrarily close to vi , 3 ≤ i ≤ n + 3, such that n the (n + 1)-tuple {v i + ki vi )}n+3 i=3 generates a subgroup dense in R . If we denote n v 1 = v1 and v 2 = v2 , it follows that R is in the closure of the group generated by −1 the (n + 3)-tuple {(k i , v i )}n+3 i=1 . Since (k i , 0) = (k i , v i )(e, −k i v i ), 1 ≤ i ≤ 2, and k 1 , k 2 generate a dense subgroup of SO(n), it follows that the closure of the group generated by the (n + 3)-tuple {(k i , v i )}n+3 i=1 is SE(n), thus proving the density. By Lemma 4.6, SE(n) is perfect and so openness follows from Lemma 4.4. Corollary 6.4 Let X be a basic hyperbolic set for f : X → X, and n ≥ 3, r > 0. Then, those C r cocycles β : X → SE(n) for which the stable and unstable foliations of fβ are ε-accessible for any ε > 0, form a H¨ older-open and C r -dense set. Proof. Using Lemma 6.3, the proof of the lemma is similar to the proof of εaccessibility in [7, Theorem 3.1.1].
6.3
The semigroup problem for SE(n)
Lemma 6.5 Let v, w ∈ Rn . If ∠(v, w) > cos−1 (−3/4), then |v + w| < max{|v|, |w|} − min{|v|, |w|}/4. Proof. Assume that |v| ≥ |w|. Then: |v + w|2 = |v|2 + |w|2 + 2|v||w| cos ∠(v, w) ≤ |v|2 + |w|2 − 3|v||w|/2 = |v|2 + |w|(|w| − 3|v|/2) ≤ |v|2 + |w|(|w| − |w| − |v|/2) = |v|2 − |v||w|/2 ≤ (|v| − |w|/4)2 .
Lemma 6.6 Let G be a topological group and S ⊂ G. Assume that there is a compact subset K ⊂ G such that for any g ∈ G there is a word w in the semigroup generated by S with wg ∈ K. Then the closure of the semigroup generated by S in G is a group. Proof. We show that the inverse element of any element g ∈ S belongs to the closure of the semigroup generated by S. Let g ∈ S. By the assumption of the lemma there are w1 , w2 , . . . , wk , . . . words in the semigroup generated by S such that wk gwk−1 g . . . w2 gw1 g ∈ K for any k. Since K is compact, there is a subsequence Wi = wki gwki −1 g . . . w2 gw1 g that converges to an element g0 in the closure of the semigroup generated by S. Consider now the sequence Wi+1 Wi−1 = wki+1 g . . . wki +2 gwki +1 g which is included in the
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semigroup generated by S and converges to identity. It follows that the sequence Wi+1 Wi−1 g −1 = wki+1 g . . . wki +2 gwki +1 is included in the semigroup generated by S and converges to g −1 . Lemma 6.7 Let S ⊂ SE(n), n ≥ 1. Assume that the closure of the group generated by S is SE(n), and the semigroup generated by S is unbounded. Then there are constants L > 0, C > 0 such that for any (k, v) ∈ SE(n) with |v| > L, there exists ˆ vˆ) in the semigroup generated by S such that |ˆ ˆ < |v| − C. (k, v + kv| Proof. Since the closure of the group generated by S is SE(n), the projection of S on SO(n) has to generate a dense group. Since SO(n) is compact, it follows from Proposition 5.8 that the projection of S on SO(n) generates a dense semigroup. Hence we can find a finite set {(ki , vi )} ⊂ S such that for any v, w ∈ Rn there is (ki , vi ) such that ∠(ki v, w) > cos−1 (−9/10). Let N = maxi |vi | for all i. Choose now an element (k, v) of the semigroup generated by S such that |v| > M = 100N + 1. Define (k i , v i ) = (k, v)(ki , vi ) = (kki , v + kvi ), and let C = 4 mini |v i |. Note that C > 0 and |v i − v| ≤ N . Note also that for any v, v ∈ Rn there is (k i , v i ) such that ∠(k i v, v ) > cos−1 (−9/10). Assume now that L = max |v i | ≤ |v| + N , and (k, v) ∈ SE(n) with |v| > L. Choose k i pointing such that ∠(k i v, v) > cos−1 (−9/10), and consequently ∠(k i v, v i ) > cos−1 (−3/4). From Lemma 6.5 it follows now that |k i v + v i | < |v| − |v i |/4 ≤ |v| − C. Theorem 6.8 Let S ⊂ SE(n), n ≥ 1. Assume that the closure of the group generated by S is SE(n). Then the closure of the semigroup generated by S is SE(n). Proof. It follows from Proposition 5.8 that the closure of the semigroup generated by S is unbounded. Hence we can apply Lemma 6.7. Define the compact set K = SO(n) × D where D ⊂ Rn is the closed disk of radius L centered at 0 and L is the constant given in Lemma 6.7. Let g ∈ SE(n). We can apply Lemma 6.7 several times and find an element w in the semigroup generated by S such that wg ∈ K. It follows now from Lemma 6.6 that the closure of the semigroup generated by S is SE(n). Corollary 6.9 Assume n ≥ 3. The set of (n + 3)-tuples in SE(n) that generate SE(n) as a closed semigroup is open and dense in SE(n)n+3 . Proof. This follows from Lemma 6.3 and Theorem 6.8.
6.4
Locally constant SE(n)-extensions over subshifts of finite type
Let k ≥ 2, and let A be a k × k 0 − 1 matrix. Define Z ∈ {1, . . . , k} | A(ω , ω ) = 1 for all n ∈ Z . Σ = ΣA = ω = (ωn )∞ n n+1 −∞ The map σ : Σ → Σ given by (σω)n = ωn+1 is called a subshift of finite type. Fix an integer N ≥ 0 and symbols α−N , . . . , αN , and call the subset Cα−N ,...,αN = {ω ∈ Σ | ωni = αi for i = −N, . . . , N }
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a (symmetric) cylinder of rank N. For any positive integer N we define a partition of Σ given by the family of symmetric cylinders Cα−N ,...,αN of rank N . We consider cocycles which are constant over the elements of a finite partition P of Σ given by cylinders. Such cocycles are called locally constant. Note that locally constant cocycles are H¨older. Lemma 6.10 Let σ : Σ → Σ be a transitive subshift of finite type and G a connected Lie group for which there exists k ≥ 2 such that the set of k-tuples in G that generate G as a closed semigroup is open and dense in Gk . Then the class of locally constant cocycles β : Σ → G contains a C 0 -open and H¨ older-dense subset for which σβ are topologically transitive. Proof. The proof is similar to arguments in [14], and we refer the reader to that paper. Remark Note that if G contains a noncompact connected semisimple Lie group then no k as in Lemma 6.10 exists [22, Corollary 7]. Proof of Theorem 1.9: It follows from Corollary 6.9 that the set of (n + 3)-tuples in SE(n) that generate SE(n) as a closed semigroup is open and dense in SE(n)n+3 . Now apply Lemma 6.10.
7 Some open questions In this paper, we have explored the validity of Conjecture 1.1 on the stable transitivity of partially hyperbolic group extensions for various classes of Lie groups. However, the present results depend significantly on the properties of the basic set X and the group G. There are many open questions even at the level of the existence of stably transitive extensions. For instance, suppose X is a hyperbolic attractor (the simplest case). (a) Does there exist a stably transitive SE(3) extension of X (more generally, SE(n) with n ≥ 3 odd)? (b) Does there exist a stably transitive SL(3, R) extension of X (more generally, SL(n, R) with n ≥ 3)? For groups of the form K × Rn with K compact, we prove stable transitivity for extensions of a hyperbolic attractor, but the situation for general basic sets remains open: (c) If Xis a general hyperbolic basic set and K is a compact connected Lie group, does there exist a stably transitive K × Rn -extension of X? Questions (a) and (c) indicate the lack of knowledge about the relatively tractable class of groups that are semidirect products K Rn where K is compact, despite the progress in [11, 15] and in this paper. Similarly, question (b) illustrates the situation for semisimple Lie groups other than Sp(2n, R).
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References [1] P. Ashwin, I. Melbourne, Noncompact drift for relative equilibria and relative periodic solutions, Nonlinearity 10, 595–616 (1997). [2] T. Bousch, La condition de Walter, Ann. Sci. Ecole Norm. Sup. 34, 287–311 (2001). [3] R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92, 725–747 (1970). [4] M.I. Brin, Topological transitivity of a class of dynamical systems and frame flow on manifolds of negative curvature, Func. Anal. and Appl. 9, 9–19 (1975). [5] R.W. Carter, G. Segal, I.G. MacDonald, Lectures on Lie groups and Lie Algebras, London Mathematical Society Student Texts 32, Cambridge University Press, 1995. [6] M. Field, I. Melbourne, A. T¨ or¨ ok, Stable ergodicity for smooth compact Lie group extensions of hyperbolic basic sets, to appear in Ergod. Th. & Dynam. Sys. [7] M. Field, V. Nit¸ic˘a, Stable topological transitivity of skew and principal extensions, Nonlinearity 14, 1055–1069 (2001). [8] M. Field, W. Parry, Stable ergodicity of skew extensions by compact Lie groups, Topology 38, 167–187 (1999). [9] A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995. [10] M. Kuranishi, On everywhere dense imbeddings of free groups in Lie groups, Nagoya Math. J. 2, 63–71 (1951). [11] I. Melbourne, M. Nicol, Stable transitivity of Euclidean group extensions, Ergod. Th. & Dynam. Sys. 23, 611–619 (2003). [12] S. Newhouse, Lectures on dynamical systems, Dynamical Systems, Progress in Mathematics, 8, Birkh¨ auser, Basel, 1–114, 1980. [13] V. Nit¸ic˘a, A note about topologically transitive cylindrical cascades, Israel Journal of Math. 126, 141–156 (2001). [14] V. Nit¸ic˘a, Examples of topologically transitive skew-products, Discrete Contin. Dynam. Systems 6, 351–360 (2000). [15] V. Nit¸ic˘a, M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms, Ergod. Th. & Dynam. Sys. 25, 257–269 (2005). [16] V. Nit¸ic˘a, A. T¨ or¨ ok, Regularity of the transfer map for cohomologous cocycles, Ergod. Th. & Dynam. Sys. 18, 1187–1209 (1998).
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[17] V. Nit¸ic˘a, A. T¨ or¨ ok, An open and dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one, Topology 40, 259–278 (2001). [18] W. Parry, M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Ast´erique 187-188, Soci´et´e Math´ematique de France, Montrouge 1990. [19] W. Parry, M. Pollicott, Stability of mixing for toral extensions of hyperbolic systems, Proc. Steklov Inst. Math. 216, 350–359 (1997). [20] M. Pollicott, C. P. Walkden, Livˇsic theorems for connected Lie groups, Trans. Amer. Math. Soc. 353, 2879–2895 (2001). [21] J. Schreier, S. Ulam, Sur le nombre de generateurs d’un groupe topologique compact et connexe, Fund. Math. 24, 302–304 (1935). [22] J. Winkelmann, Generic subgroups of Lie groups, Topology 41, 163–181 (2002). Ian Melbourne Department of Mathematics and Statistics University of Surrey Guildford, Surrey GU2 7XH, United Kingdom email:
[email protected] Viorel Nit¸ic˘a Department of Mathematics 323 Anderson Hall West Chester University West Chester, PA 19383, USA and Institute of Mathematics of the Romanian Academy P.O. Box 1–764 RO-70700 Bucharest, Romania email:
[email protected] Andrei T¨ or¨ ok University of Houston Department of Mathematics 651 PGH, Houston, TX 77204-3008, USA and Institute of Mathematics of the Romanian Academy P.O. Box 1–764 RO-70700 Bucharest, Romania email:
[email protected] Communicated by Viviane Baladi submitted 24/05/04, accepted 11/10/04
Ann. Henri Poincar´e 6 (2005) 747 – 789 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/04747-43, Published online 28.07.2005 DOI 10.1007/s00023-005-0222-z
Annales Henri Poincar´ e
Spectral Asymptotics of the Harmonic Oscillator Perturbed by Bounded Potentials Markus Klein, Evgeny Korotyaev and Alexis Pokrovski 2
d 2 2 Abstract. Consider the operator T = − dx 2 + x + q(x) in L (R), where q is a real function with q and 0x q(s) ds bounded. The spectrum of T is purely discrete and consists of simple √ eigenvalues. We determine their asymptotics µn = (2n + π 1) + (2π)−1 −π q( 2n + 1 sin θ) dθ + O(n−1/3 ) and we extend these results for complex q.
1 Introduction and main results It is well known that the spectrum of the quantum-mechanical harmonic oscillator d2 2 on L2 (R) is purely discrete and consists of the simple eigenvalues T 0 = − dx 2 +x d2 µ0n = 2n + 1, n 0. In this paper we consider the perturbed operator T = − dx 2 + x2 +q(x) in L2 (R), where q belongs to the complex Banach space B given by x ∞ B = q ∈ L (R) : qB = q∞ + q ∞ + q1 ∞ < ∞ , q1 (x) ≡ q(t)dt. 0
(1.1) For q ∈ B the spectrum of T is purely discrete, and for real q, all eigenvalues µ0 < µ1 < µ2 < · · · of T are simple (see Theorem 1.1) and obviously µn = µ0n + O(1), n → ∞. Our goal is to determine the asymptotics of µn − µ0n as n → ∞. For decaying potentials (e.g., q , xq ∈ L2 (R)) a complete inverse spectral theory is obtained in [4], [5]. For bounded potentials, however, we did not find any results (even for the direct spectral problem) in the literature. The existing methods (e.g., those of [4], [5]) cannot be used for q ∈ B. We shall show in Theorem 1.1 that there exist fundamental solutions ψ± of Eq. −y + (x2 + q(x))y = λy, (x, λ) ∈ R × C, (1.2) which satisfy the asymptotics √ √ λ−1 λ−1 x2 x2 ψ± (x, λ) = (± 2x) 2 e− 2 (1 + o(1)), ψ± (x, λ) = −x(± 2x) 2 e− 2 (1 + o(1)) (1.3) as x → ±∞ and locally uniformly in λ. If q ≡ 0 then these solutions have the form √ 0 (x, λ) = D λ−1 (± 2x), where Dr is the Weber (parabolic cylinder) function ψ± 2 (see [1]). We introduce x the Wronskian w = {ψ− , ψ+ }, where {f, g} = f g − f g and recall q1 (x) = 0 q(t)dt.
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Theorem 1.1 Let q, q1 ∈ L∞ (R). Then i) For any λ ∈ C there exist unique solutions ψ± (x, λ) of (1.2) with the asymp (x, ·) and totics (1.3). Moreover, for each x ∈ R the functions ψ± (x, ·), ψ± w = {ψ− , ψ+ } are entire. ii) µ is an eigenvalue of T iff µ is a zero of the Wronskian w(·). Each eigenvalue µ is simple and for some constant c(µ) = 0 the following identities are fulfilled 2 ψ+ (·, µ) = c(µ)ψ− (·, µ) , ψ+ (x, µ)dx = c(µ)w (µ). (1.4) R
In particular, if q is real, then all roots of the Wronskian w(·) are real and simple. We remark that for complex q there might be double zeroes of the Wronskian w = {ψ− , ψ+ } for bounded |λ|. Due to Lemma 6.1 we deduce that for large |λ| 1 these zeroes are simple and they have asymptotics µn = µ0n + O(n− 6 ) as n → ∞, ∞ where we denote by {µn }1 the sequence of all zeros of w in C (counted with multiplicity) such that −∞ < Re µ1 Re µ2 Re µ3 . . . . Below we always label the zeroes of the Wronskian w by Re µn Re µn+1 , n 0, counting their multiplicities. The main result of our paper is Theorem 1.2 For complex q ∈ B the zeroes µn , n 0 of the Wronskian w(·) have the following asymptotics π 1 1 1 µ1n = q( µ0n sin ϑ) dϑ = qB O(n− 4 ). (1.5) µn = µ0n + µ1n + O(n− 3 ), 2π −π In particular, for n sufficiently large, all zeroes µn are simple. Remark. i) We emphasize that for real q the labeling of eigenvalues and zeroes of the Wronskian w(·) (counting multiplicity) coincides. For complex q these labelings might be different. The eigenvalues still coincide with the zeroes of the Wronskian as a set. ii) We intend to use the spectral results of this paper to study the band structure of the spectrum of the 2d Schr¨ odinger operator with a homogeneous (or periodic) magnetic field and a bounded potential. We emphasize that the main point of Theorem 1.2 is the estimate on the remainder. In some cases our formula for µ1n can be rewritten in a more explicit form. Proposition 1.3 Let q(x) = R eixt dν(t) ∈ B for some Borel measure dν on R which satisfies the condition Cq = R (1 + |t|−p ) dν(t) < ∞ for some p > 32 . Then σ( µ0n ) 3 √ 0 1 µn = J0 (t µn ) dν(t) = + Cq O(n− 4 ), 1 0 (µn ) 4 R cos(|t|s − π4 ) 2 σ(s) = dν(t), (1.6) 1 π R |t| 2
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where J0 is the Bessel function. If, in addition, q has the form q(x) = then π 2 qk cos(s|tk | − ). σ(s) = π 4 |tk |
749
k∈Z qk e
ixtk
,
(1.7)
k∈Z
Next we describe the physical motivation of our paper. Consider a 2d charged particle in a constant magnetic field and in a periodic (or almost-periodic) structure along one of the axes. The corresponding Hamiltonian has the form H = −∂x2 − (−i∂y − x)2 + q(x), in L2 (R2 ),
(1.8)
here the real function q depends on x ∈ R only. Such operators are widely discussed in the physics literature (see [2], [3], [7], [6], [9]). The Fourier transform with respect to y reduces the spectral problem for H to analysis of the eigenvalues µn (θ) of the operator −∂x2 + (θ − x)2 + q(x) in L2 (R) or H(θ) = −∂x2 + x2 + q(x + θ) in L2 (R)
(1.9)
as functions of θ ∈ R. For real q ∈ B the spectrum of H(θ) is purely discrete and consists of simple eigenvalues µ0 (θ) < µ1 (θ) < µ2 (θ) < · · · . The first step in the spectral analysis of H is the analysis of H(θ) for fixed θ. The second step is the study of µn (θ) as a function of θ ∈ R. In the physics literature during the last years several papers were devoted to systems with trapping potentials plus periodic potentials. The trapping potential typically is modeled by a harmonic one. Such systems are natural models in the investigation of Bose-Einstein condensation, vortex lattices etc. (see the ref. in [8]). Our paper gives rigorous spectral results for (a variant of) the model in [8]. Our proof is long and somewhat technical. In principle, it follows a wellknown path: We compute the asymptotics of the spectrum from the Wronskian between fundamental solutions, we use integral equations in semiclassical variables and we use Rouche’s theorem to establish the labeling of eigenvalues in Lemma 6.1. The integrals in our integral equations, however, do not converge absolutely. We estimate them by systematic integration by parts. Here it is crucial that q1 (x) = x q(t)dt is bounded. Technically, these estimates (for complex energies) are the 0 main new point in this paper. We are not aware of any paper where such techniques have been applied to study the asymptotics of solutions of ordinary differential equations. In particular, such techniques are not found in [10] (which does not even mention the result of the same author [11] on the asymptotics of the parabolic cylinder functions). Therefore, our list of technical references is quite short. The plan of the paper is as follows. In Section 2 we introduce the quasiclassical variable and the integral equation for the fundamental solutions and show their existence. In Sections 3 and 4 we solve this equation by iteration. In Section 5 we derive the asymptotics of the Wronskian (using the results of Section 3 and 4). In Section 6 we prove Theorem 1.2. In the Appendix we prove Lemmas 2.3–2.5 which are crucial for Section 3 and 4.
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2 Preliminaries In this section we introduce the integral equation (in semiclassical variables) for (1.2) and state Lemmas 2.3–2.5 which are crucial to analyze the integral equation in Sections 3 and 4. As a first step, we shall prove Theorem 1.1. Proof of Theorem 1.1. Consider the function ψ+ , the proof for ψ− is similar. In order to prove that ψ+ is an entire function of λ it is sufficient to show that it is analytic in each disc D(µ) = {λ ∈ C : |λ − µ| 1}, µ ∈ C. For λ ∈ D(µ) we have (see [1]) the uniform asymptotics
0 0 ψ+ (x, λ) = g(x) 1 + O x−2 , ψ+ (x, λ) = −xg(x) 1 + O x−2 , x → +∞, (2.1) √ λ−1 x2 1 1−λ λ − 0 0 where g(x) = ( 2x) 2 e 2 . Let h(x, λ) = 2√π Γ( 2 )(ψ− (x, λ)−sin 2 ·ψ+ (x, λ)); note that h(x, λ) =
1 (1+O(x−1 )), 2xg(x)
h (x, λ) =
1 (1+O(x−1 )), 2g(x)
x → +∞, (2.2)
0 (see [1]) uniformly for λ ∈ D(µ), so that {ψ+ , h} = 1. Define the entire function + + M (x, y) = h(x, λ)ψ0 (y, λ) − ψ0 (x, λ)h(y, λ). Then a solution of t 0 ψ(x, λ) = ψ+ (x, λ) + lim M (x, y)q(y)ψ(y, λ) dy (2.3) t→∞
x
solves (1.2). We rewrite (2.3) in the form t ψ(·, λ) , p(x, λ) = p0 (x, λ) + lim K(x, y)q(y)p(y, λ) dy, p = t→∞ x g where x > 1 and K(x, y) = U (x, y) =
M(x,y)g(y) . g(x)
p0 =
0 ψ+ (·, λ) , g (2.4)
Let h0 = 2xg(x)h(x, λ) and
h0 (x)p0 (y) g 2 (y) , 2x g 2 (x)
V (x, y) =
p0 (x)h0 (y) . 2y
(2.5)
By U, V, K we denote the Volterra integral operators with kernel U (x, y), V (x, y), K(x, y). Note that K = U − V . In order to study Eq. (2.4) introduce the spaces of functions Fα = {f ∈ C([1, ∞)) : f α ≡
sup |x|α |f (x)| < ∞}, x∈[1,∞)
Fα,β = {f ∈ Fα : f ∈ Fβ }, equipped with the norm f α,β = f α + f β , α, β ∈ R. By (2.1), for λ ∈ D(µ) we have p0 0,1 c < ∞. (2.6)
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Let f ∈ Fα for some α ∈ R and u = U qf . Then we have
∞
h0 (x)p0 (y) g 2 (y) q(y)f (y)dy 2x g 2 (x) x 2 2 h0 (x) ∞ = p0 (y)ex −y y λ−1 q(y)f (y)dy. (2.7) 2xλ x ∞ 2 2 Using (2.1), (2.2) and the estimate x e−y y γ dy Ce−x xγ−1 for x 1 and γ ∈ R we obtain u(x) =
uα+2 Cq∞ f α ,
u α+1 Cq∞ f α
(2.8)
uniformly in λ ∈ D(µ). Here and below C is some absolute constant. Let f ∈ Fα,β for some α > −1 and β > 0. Set v = v1 + v2 = V qf , where ∞ f (y) dy, v1 (x) = p0 (x) (h0 (y) − 1)q(y) 2y x and integration by parts gives t ∞ f (y) f (y) v2 (x) = lim p0 (x) q(y) dy = p0 (x) [q1 (x) − q1 (y)] dy, t→∞ 2y 2y x x where the last integral converges absolutely. Using (2.1–2.2) we obtain |v(x)| CCq (f α x−α−1 +f β x−β ),
|v (x)| CCq (f α x−α−2 +f β x−β−1 ) (2.9) for x 1, uniformly in λ ∈ D(µ), where Cq = q∞ + q1 ∞ . Thus K : Fα,β → Fα ,β ,
α = min{α + 1, β},
β = min{α + 1, β + 1}. (2.10)
Consider the iterations pn+1 = Kqpn , n 0. By (2.6), we have p0 ∈ F0,1 ; using (2.5), (2.8), (2.9) and (2.10), we conclude that pn+1 αn+1 ,βn+1 CCq pn αn ,βn , α2n = n, β2n = 1 + n, α2n+1 = 1 + n, β2n+1 = n. (2.11) and α0 = 0, β0 = 1. Using (2.6) we obtain |p2n (x)| (CCq )2n cx−n , |p2n+1 (x)| (CCq )2n+1 cx−n−1 , Hence for x x0 = (2CCq )2 the series p(x) = pn (x) and n0
|p2n (x)| (CCq )2n cx−n−1 , |p2n+1 (x)| (CCq )2n+1 cx−n−1 .
p (x) =
n0
pn (x)
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converge absolutely and uniformly in λ ∈ D(µ); p(x) gives the solution of Eq.(2.4). Moreover, p(x) = 1 + O(x−1 ), p (x) = O(x−1 ), x → ∞, (2.12) uniformly for λ ∈ D(µ). Therefore ψ = gp is a solution of (2.3). By (2.4) and (2.12), ψ satisfies (1.3). For each n 0 and fixed x x0 the iterations pn (x, ·), pn (x, ·) are analytic in D(µ). Hence p(x, ·), p (x, ·) are analytic in D(µ) for each fixed x x0 . (x, λ) are analytic in D(µ) for each fixed x x0 . Hence, By (2.4), ψ+ (x, λ) and ψ+ the solution is also analytic in λ for any fixed x (see this simple fact, e.g., in [12]). Thus ψ+ (x, λ) and ψ+ (x, λ) are entire functions of λ for any x ∈ R.
Denote by ψ+ another solution of (1.2) satisfying (1.3). Then {ψ+ , ψ + } = 0. Thus ψ+ = cψ + . By (2.2), c = 1 and thus ψ+ is unique. ii) If ψ ∈ L2 solves (1.2), then ψ, ψ ∈ L∞ . Thus {ψ, ψ+ } = 0 = {ψ, ψ− }. Hence, if µ is an eigenvalue of T with eigenfunction ψ1 , then ψ1 is proportional to both ψ+ and ψ− . This proves ψ+ (·, µ) = c(µ)ψ− (·, µ). Furthermore it follows, firstly, that all eigenvalues are simple, and secondly, that the zeroes of w(·) = {ψ− , ψ+ } coincide with the eigenvalues of T . To prove the second equality in (1.4), observe that for any solution ψ of ˙ ψ} , where ψ˙ = ∂ψ/∂λ. Then the identity (1.2) a simple calculation gives ψ 2 = {ψ, ψ+ (·, µ) = c(µ)ψ− (·, µ) yields R
2 ψ+ (x)dx =
0
2 ψ+ (x)dx + c2 (µ)
−∞
0 = {ψ˙ + , ψ+ }−∞ +
+∞
0 +∞ 2 ˙ c (µ){ψ− , ψ− } 0
2 ψ− (x)dx
= {ψ˙ + , ψ+ }(0) − c2 (µ){ψ˙ − , ψ− }(0)
= c(µ){ψ˙ + , ψ− }(0) − c(µ){ψ˙ − , ψ+ }(0) = c(µ)w, ˙ since ψ± (±∞) and ψ˙ ± (±∞) vanish. Since ψ± is real for real µ and real q, all zeroes of w are simple in this case. Theorem 1.1 gives no information on high-energy asymptotics λ → ∞. To derive those, we introduce some notations and auxiliary functions. Throughout the paper we use C± = {z ∈ C : ± Im z > 0} and the following agreements: • The functions log z and z α = eα log z for α ∈ R take their principal values on C \ R− . • For λ ∈ C+ \ {0} we set λ = |λ|e2iϑ , ϑ ∈ [0, π2 ]. 1
Define the function z = (1 + |z|2 ) 2 , z ∈ C. For any interval I ⊂ R we introduce a sector S(I) = {z ∈ C : arg z ∈ I}. Define the function ξ(t) =
t 1 2 s2 − 1 ds = t t − 1 − log(t + t2 − 1) , 2 1
π t ∈ S(− , 0), (2.13) 2
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where ξ(t) > 0 for t > 1. The function ξ is a conformal mapping from S(− π2 , 0) onto Ξ = C− ∪ {Re ξ < 0, Im ξ ∈ [0, π4 )}. The following uniform asymptotics is fulfilled: 1 2 1 π ξ(t) = |t| → ∞, t ∈ S[− , 0]. (2.14) t − − log 2t + O(|t|−2 ) , 2 2 2 We introduce the function π t ∈ S(− , 0), 2
2
k(t) = (3ξ(t)/2) 3 ,
2
k(0) = −(3π/8) 3 < 0.
(2.15)
Note that k(t) is a conformal mapping from S(− π2 , 0) onto the domain K given by 2 3 arg k 2π 2π K = S[− , 0) k ∈ S(−π, − ] : |k| sin 3 < |k(0)| . (2.16) 3 3 2 By (2.14), the following asymptotics and estimates are fulfilled: 2 4 1 π k(t) = (3/4) 3 (t 3 + O(t 3 )), t ∈ S[− , 0], 2
|t (k)| Ck
− 14
1
5
|t (k)| Ck − 4 ,
,
3
t(k) = (4/3) 2 (k 4 + O(1)), k ∈ K, k ∈ K,
(2.17) (2.18)
where t(k) is the inverse function for k(t). Here and below C is an absolute constant. For fixed λ we consider the change of variable x → k = k( √xλ ); it maps R+ 1
onto the curve γ˜λ = k(e−iϑ R+ ). Note that λ 2 t(k) is real. The domain K and the curve γ˜λ are presented in Fig. 1. √ By (1.2), the function y1 (k, λ) = y(√λt(k)) solves t (k)
y1 (k, λ) − λ2 ky1 (k, λ) = v0 (k)y1 (k, λ) + vq (k, λ)y1 (k, λ),
k ∈ γ˜λ ,
(2.19)
where 3
1
v0 (k) = −t (k) 2 [k (t)− 2 ] |t=t(k)
1
vq (k, λ) = λ t (k)2 q(λ 2 t(k)).
(2.20)
Using (2.17) and (2.18) we obtain |v0 (k)| C(1 + |k|)−2 ,
k ∈ K.
(2.21)
2
For each λ ∈ C+ \ {0} we define the basic variable z = λ 3 k, where 2
1
z(x, λ) = λ 3 k(t), t = xλ− 2 ,
λ ∈ C+ \ {0},
x 0.
(2.22)
In particular, we have 2
2
z0 ≡ z(0, λ) = −λ 3 (3π/8) 3 ,
λ ∈ C+ \ {0}.
(2.23)
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k-plane
K∗ = k S(0, π/2)
ppp.p . . . . . . . . . . . . 6. . . . . . . . . . . ppp.p . . . . . . . . . . . . . . . . . . . . . . . pp p. . . . . . . . . . . . . . . . . . . . . . . ppp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.pp . . . . . . . . . . . . . . . . . . . . . . pp.p . . . . . . . . . . . . . . . . . . . . . pp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pp. . . . . . . . . . . . . . . . . . . . . . pp . . . . . . . . . . . . . . . . . . . . . . pp . . . . . . . . . . . . . . . . . . . . . p. . . . . . . . . . . . . . . . . . . . . . k(0) ppp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ppp p pp.p p pp p p p.pppp p p p p.ppppp p p p. . . . . . . . . . . . . . . . . . ppp . . . . . . . .p ppp.p pp p.p p p.ppp p.pppp.pp pp.ppppp.pppp.p . . . . . . . . . . . . . . . . . . . . . . . . . . pp . . . . . . . . ppp.ppppppppp.pppppp . . . . . . . . . . . pp.p . . . . . . . . . . . . . . . . . . . . pp.pppp.pppp.pppp.ppppp.pppp.ppp . . . . . . . . . . . . . . . . pp . . . . . . . . . . . . . . ppp pp.pppppppp p. . . . . . . ppp p.p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .pppp.pppp.pppp.ppp p.p ppp.pp . . . . . . . . . . . . . . . . . . . . . . . . . .pp . . . p pp. . . . . . . . . . . . . . . . . . . .pppppp pp p.pppp . . p.p p p.p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ppp p.pppp.p ppp.ppp p pp pp
= k(e−iϑ R+ ) γ pp p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . λ
K = k S(−π/2, 0)
Figure 1. The curve γ
λ in the k-plane. The dotted line is the boundary of K ∪ K∗ .
6
....
z-plane -
........
........ 4 ....... ϑ......................... ........ ......................................................... 3 . . . . . . . ....... .....................
z.r....0........................................................................................− ...... . Γλ
r
.......................................................... ....................... .................. ............... ........
z∗
Γ+ λ
Figure 2. The curve Γλ in the z-plane. Each mapping z(·, λ) : R+ → Γλ = z(R+ , λ) is a real analytic isomorphism (see Fig. 2 and Lemma 7.5 about Γλ ). 2 2 3 If λ > 0, then Γλ = [−λ 3 ( 3π 8 ) , ∞) is a half-line. For any λ ∈ C+ \ {0} and 0 x1 < x2 set zn = z(xn , λ), n = 1, 2. We define the curves Γλ (z1 , z2 ) = {z : z = z(x, λ), x ∈ [x1 , x2 ]}, Γλ (z1 ) ≡ Γλ (z1 , ∞) = {z : z = z(x, λ), x x1 }. 2
By (2.19), the function u(z, λ) = y1 (zλ− 3 , λ) solves ∂z2 u(z, λ) − zu(z, λ) = V (z, λ)u(z, λ),
V = V0 + Vq ,
z ∈ Γλ ,
(2.24)
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∂ here and below ∂z = ∂z and Vq and V0 (which does not include q) are given by √ 2 2 λt(zλ− 3 ) v0 zλ− 3 ρ2 (z, λ)q − 23 zλ . V0 (z, λ) = , V (z, λ) = , ρ(z, λ) = t q 4 1 λ3 λ3 (2.25) Using (2.18), for λ ∈ C+ \ {0} and z ∈ S[−π, π3 ] we obtain 2
1
|ρ(z, λ)| Czλ− 3 − 4 ,
2
2
5
|∂z ρ(z, λ)| |λ|− 3 zλ− 3 − 4 ,
|V0 (z, λ)|
C
, |λ| + |z|2 (2.26) 4 3
where C does not depend on λ and z. To analyze (2.24) we need well-known properties [1] of the Airy functions Ai and Bi: 3 2 e− 3 z 2 − 32 ) , 1 + O(z Ai(z) = √ 24z
{Ai(z), Bi(z)} = 1,
|z| → ∞,
| arg z| < π − ε,
∀ε > 0,
2πi −i π Bi(z) = i 2e 3 Ai(ωz) − Ai(z) , ω = e 3 ,
π
π
π
Ai(z) = e−i 3 Ai(zω) + ei 3 Ai(zω),
(2.27) (2.28)
π
Bi(z) = ie−i 3 Ai(zω) − iei 3 Ai(zω). (2.29)
LetΓ ⊂ C be a smooth curve. For any continuous functionf on Γ we denote by Γ f (s) ds the usual complex line integral. We denote by Γ f (s) |ds| the line integral of f along Γ with respect to the arc length |ds| = (dx)2 + (dy)2 . For integration along the infinite curve Γλ , defined above, we use the standard notation p.v. Γλ (z) f (s) ds = lim Γλ (z,w) f (s) ds as w → ∞, w ∈ Γλ . We will study the (formal) integral equation u+ (z, λ) = u0 (z) + p.v. J0 (z, s)V (s, λ)u+ (s, λ) ds, z ∈ Γλ , (2.30) Γλ (z)
u0 (z) = Ai(z),
J0 (z, s) = Ai(s)Bi(z) − Ai(z)Bi(s),
We rewrite (2.30) in the form v+ (z) = a(z) + p.v. J(z, s)V (s, λ)v+ (s) ds, Γλ (z)
u+ (z) = v+ (z)e
,
3
2
a(z) ≡ Ai(z)e 3 z 2 ,
3
− 23 z 2
z, s ∈ C.
J(z, s) = J0 (z, s)e
3 2 2 3 (z
z ∈ Γλ , (2.32)
3
−s 2 )
(2.31)
.
(2.33)
3 2
If z < 0 and λ ∈ C+ , then z takes its values on the lower side of the cut. This agreement provides continuity as arg λ ↓ 0, since for λ ∈ C+ the curve Γλ lies in the lower half-plane. By (2.27) and (2.29), the following estimates are fulfilled: 1
|a(z)| Cz − 4 , − 54
|a (z)| Cz
,
∀z ∈ C,
| arg z| π − ε, ∀ε > 0.
(2.34) (2.35)
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We write (2.32) in the form (with the integral operator J) v+ = a + JV v+ ,
(Jf )(z) = p.v.
J(z, s)f (s)ds.
(2.36)
Γλ (z)
The next lemma (proved in the Appendix) gives a splitting of Γλ . Here and below we fix 4 1 δ ∈ (0, arccos 2− 3 ). (2.37) 3 Lemma 2.1 For any λ ∈ C+ \ {0} there exists a unique point z∗ ≡ z∗ (λ) ∈ Γλ such that 1. if 0 arg λ < δ, then |z∗ | = min |z|, z∈Γλ
2. if δ arg λ π, then z∗ = Γλ ∩ {z : arg z = − π3 }. Below for any λ ∈ C+ \ {0} we define z∗ by Lemma 2.1. We define x∗ , t∗ and the sets Γ± λ , Γλ by x∗ + − + z∗ = z(x∗ , λ), t∗ = √ , Γ− λ = Γλ (z(0, λ), z∗ ), Γλ = Γλ (z∗ , ∞), Γλ = Γλ ∪ Γλ . λ (2.38) + and Γ . On each We will use z∗ to split the contour Γλ into the two contours Γ− λ λ we will determine the asymptotics of the solutions of Eq. (2.32) as |λ| → ∞. Γ± λ + The technique for the cases Γ− and Γ is different. λ λ 3
Lemma 2.2 Let λ ∈ C+ \ {0}. Let h(x, λ) = | exp( 23 z(x, λ) 2 )| for x 0. Then 1. if λ > 0, then h(·, λ) is strictly increasing on [x∗ , ∞) and h(·, λ) ≡ 1 on [0, x∗ ], 2. if 0 < arg λ π, then h(·, λ) is strictly increasing on [0, ∞). Lemma 2.3 Let λ ∈ C+ and |λ| 1. Assume a) z ∈ Γλ , δ arg λ π or b) z ∈ Γ+ λ . Then for some constant C (independent of λ and z) the following estimates are fulfilled: 3 3 4 4 1 |e− 3 s 2 |s −α |ds| C|e− 3 z 2 |z −α− 2 , α ∈ R, (2.39) Γλ (z)
Γλ (z)
s −α |ds| Cz −α+1 ,
α > 1.
(2.40)
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Lemma 2.4 Let λ ∈ C+ and |λ| 1. Then for some constant C (independent of λ) the following estimates are fulfilled: 2 −1 (1−α) for 0 α < 1, C(1 − α) |λ| 3 −α s |ds| (2.41) C log(|λ| + 1) for α = 1, Γ− λ C(α − 1)−1 for α > 1,
|ds|
Γλ
4
|λ| 3 + |s|2
C 2
|λ| 3
.
(2.42)
3 Analysis of the integral equation In this section we consider the integral equation v+ = a + JV v+ for large |λ| in − the cases: i) (λ, z) ∈ C+ × Γ+ λ and ii) (λ, z) ∈ S[δ, π] × Γλ . The remaining case − (λ, z) ∈ C+ × Γλ is treated in Sect. 4. For λ ∈ C+ \ {0} define the Banach spaces of functions on Γλ : + λ α Fα = f ∈ C(Γλ ) : f α ≡ sup z |f (z)| < ∞ for | arg λ| < δ, (3.1) z∈Γ+ λ
Fαλ
=
f ∈ C(Γλ ) : f α ≡ sup z |f (z)| < ∞ α
for
z∈Γλ
λ Fα,β
δ | arg λ| π, (3.2)
λ λ = f ∈ Fα : f ∈ Fβ ,
f α,β = f α + f β ,
(3.3)
λ where α, β > 0. Evidently Fαλ ⊂ Fαλ and Fα,β ⊂ Fαλ ,β for α < α and β < β . Now we formulate the main result of this section; its proof is given at the end of this section.
Theorem 3.1 Let q, q1 ∈ L∞ (R), and λ ∈ C+ . Then the equation v+ = v0 + JV v+ , 1 v0 (z) ≡ a(z), has a unique solution v+ ∈ F λ1 ,1 for ε ≡ c0 |λ|− 6 (q∞ + q1 ∞ + 1) 4
12 , where c0 > 1 is an absolute constant. Moreover, the solution satisfies 1
|v+ (z)| Cz −1 ,
(3.4)
3
|v+ (z) − v0 (z)| εCz −1 ,
(3.5)
|v+ (z)| Cz − 4 , |v+ (z) − v0 (z)| εCz − 4 ,
where the constant C does not depend on λ, z. We have the identity 2
3
Ai(zω) = e 3 z 2 a(zω),
ω=e
2πi 3
,
−π arg z <
π . 3
(3.6)
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Using (2.28), (2.29) and (3.6) we rewrite the kernel J(z, s), given by (2.33), in the form 3 3 π 4 π J(z, s) = −2ie−i 3 a(z)a(sω) − e 3 (z 2 −s 2 ) a(zω)a(s) , z, s ∈ S(−π, ). 3 (3.7) Note that by Lemma 7.5.1, in both cases i) and ii) (see the first lines in Sect. 3) we have Γλ (z) ⊂ S[−π + 23 δ, 0], so (3.7) holds on Γλ (z). Following (3.7), we represent JVq as the sum of two operators. In the next two Lemmas (3.2 and 3.3) we estimate these two operators in suitable function spaces. In Lemma 3.4 we estimate JV0 (which is asymptotically small in comparison with JVq ). These estimates, combined in Lemma 3.5, give an a priori estimate for JV . In Theorem 3.1 we prove convergence of the iterations series for the equation v+ = v0 + JV v+ . This gives the estimates for v+ necessary for further analysis. For v0 (z) ≡ a(z) given by (2.32), due to (2.34), (2.35) and Lemma 7.5.1 we have v0 41 , 54 C, (3.8) uniformly in λ ∈ C \ {0}. For any fixed λ ∈ C+ \ {0} and z1 , z2 ∈ Γλ such that zj = z(xj , λ), j = 1, 2 and 0 x1 x2 we define the function z2 √ 1 2 qˆ(z, λ)ρ(z, λ) dz, qˆ(z, λ) = q( λt(zλ− 3 )), (3.9) Q(z2 , z1 ) ≡ λ− 6 z1
x where ρ is given by (2.25). We have the identity Q(z2 , z1 ) = x21 q(x) dx and an estimate |Q(z2 , z1 )| 2q1 ∞ for any z1 , z2 ∈ Γλ , (3.10) x ∞ since q1 (x) = 0 q(t)dt ∈ L (R). By Lemma 7.5.2 and Lemma 7.5.3, we have 2
2
C1 |λ| 3 |z| C2 |λ| 3
for
z ∈ Γ− λ,
δ arg λ π,
(3.11)
where C1 and C2 are independent of z and λ. We estimate the term of JVq , corresponding to the first term in the decomposition (3.7). λ Lemma 3.2 Let q, q1 ∈ L∞ (R). Assume λ ∈ C+ , |λ| 1 and f ∈ Fα,β for + 3 and on Γλ for α > 0, β > 4 (that is, f is defined on Γλ for δ arg λ π λ 0 arg λ δ). Then g(z, λ) = p.v. Γλ (z) a(sω)Vq (s)f (s) ds ∈ Fα,β and satisfies
1 1 1 |a(z)g(z, λ)| C|λ|− 6 q1 ∞ f α z −α− 2 + f β z −β+ 2 ,
(3.12)
1 3 1 |a (z)g(z, λ)| C|λ|− 6 q1 ∞ f αz −α− 2 + f β z −β− 2 .
(3.13)
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Proof. Consider the case 0 arg λ π, z ∈ Γ+ λ . By Lemma 7.5.1, we have Γλ (z) ⊂ S[−π + 23 δ, 0], so the uniform estimates (2.34) and (2.35) hold on Γλ (z) 1 for both a(z) and a(zω). Writing F (z) = λ− 6 a(zω)ρ(z, λ) ( ρ is given by (2.25)), integration by parts yields g(z, λ) = p.v. (∂s Q(s, z)) F (s)f (s)ds = −p.v. Q(s, z)(F (s)f (s)) ds, Γλ (z)
Γλ (z)
where we used Q(z, z) = 0 and
lim +
Γλ w→∞
(3.14) Q(w, z)F (w)f (w) = 0 (this holds by
(2.26), (2.34) and (3.10)). Thus using (2.26), (2.34), (2.35) and (3.10) we have 5 q1 ∞ |g(z, λ)| C f α s −α− 4 |ds| 1 6 |λ| Γλ (z) s 2 5 1 1 + f α |λ|− 3 2 − 4 s −α− 4 |ds| + f β s −β− 4 |ds| . λ3 Γλ (z) Γλ (z) Due to Lemma 7.5.4 we have |z| = inf |s|. Using also (2.40) we obtain s∈Γλ (z)
|g(z, λ)| C
q1 ∞ 1
|λ| 6 f α z −α
+ Γλ (z)
5
1
s − 4 |ds| + f α z −α− 4
2
|λ|− 3
s 2 3
5
− 4 |ds|
λ Γλ (z) 1 1 1 3 f β s −β− 4 |ds| C|λ|− 6 q1 ∞ f α z −α− 4 + f β z −β+ 4 , Γλ (z)
(3.15) which together with (2.34) and (2.35) proves (3.12) and (3.13), respectively. Consider the case 0 arg λ δ, z ∈ Γ− λ . Lemma 7.5.1 gives Γλ (z) ⊂ S[−π + 2 δ, 0], so the uniform estimates (2.34) and (2.35) hold on Γλ (z) for both a(z) and 3 a(zω). We have g(z, λ) = g− (z, λ) + g+ (λ), where g+ (λ) = p.v. a(sω)Vq (s)f (s) ds, g− (z, λ) = a(sω)Vq (s)f (s) ds. Γλ (z)
Γλ (z,z∗ )
Using (3.11) and (3.15) for z = z∗ we have f α f α q1 ∞ f β f β q1 ∞ |g+ (λ)| C C . 1 1 + 3 1 1 + 3 |λ| 6 |z∗ |α+ 4 |z∗ |β− 4 |λ| 6 |z|α+ 4 |z|β− 4 (3.16) Integration by parts and the identities Q(z∗ , z∗ ) = 0 and qˆ(s, λ)ρ(s) = −∂s Q(z∗ , s) yield Q(z∗ s) (F (s)f (s)) ds, g− (z, λ) = Q(z∗ , z)F (z)f (z) − Γλ (z,z∗ )
1
F (z) = λ− 6 a(zω)ρ(z, λ), (3.17)
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Using (2.26), (2.34), (2.35) and (3.10) we obtain |g− (z, λ)| C
q1 ∞ 1
|λ| 6 f α 1
|z|α+ 4
+
Γ− λ
f α 5
|s|α+ 4
+
f α 1
2
|s|α+ 4 |λ| 3
|ds| +
f β
Γ− λ
1
|s|β+ 4
|ds| .
By (2.41) and (3.11), we have 1 1 3 |g− (z, λ)| Cq1 ∞ |λ|− 6 f α |z|−α− 4 + f β |z|−β+ 4 .
(3.18)
Combining (3.16) and (3.18) with (2.34) gives (3.12). The estimate (3.13) follows from (3.16), (3.18) and (2.35). For the analysis of the part of JVq corresponding to the second term in (3.7) we also integrate by parts. Introduce an analogue of Q(z1 , z2 ): ∞ √ 3 1 − 43 s 2 e qˆ(s, λ)ρ(s)ds = e−2λξ(s/ λ) q(s) ds, z ∈ Γ+ P (z) = 1 λ, λ 6 Γλ (z,∞) x (3.19) √ √ where x = λ t z2 and recall qˆ(z, λ) = q( λt( z2 )). Using (2.26) and (2.39) λ3
λ3
for q ∈ L∞ (R) gives
|P (z)| q∞ C
Γλ (z)
4
3
|e− 3 s 2 | 1 6
4
1 4
(|λ| + |s| )
3
3
z ∈ Γ+ λ.
|ds| Cq∞ |e− 3 z 2 |z − 4 ,
(3.20) Due to |ρ(z)| C and (2.39), we obtain another estimate 3 4 1 C q∞ − 43 z 32 |e− 3 s 2 | |ds| C |z − 2 , |P (z)| q∞ 1 1 |e 6 6 |λ| Γλ (z) |λ|
z ∈ Γ+ λ.
(3.21) We estimate the part of JVq , corresponding to the second term in the decomposition (3.7). λ Lemma 3.3 Let q ∈ L∞ (R). Assume λ ∈ C+ , |λ| 1, and f ∈ Fα,β for α > + 0, β > 0 (that is, f is defined on Γλ for δ arg λ π and on Γλ for 0 arg λ δ). 3 4 λ and satisfies Then g(z, λ) = Γλ (z) a(s)e− 3 s 2 Vq (s)f (s) ds ∈ Fα,β
3 4 1 5 7 |e 3 z 2 a(z)g(z, λ)| Cq∞ |λ|− 6 f α z −α− 4 + f β z −β− 4 ,
(3.22)
3 4 1 3 5 |g(z, λ) a(zω)e 3 z 2 | Cq∞ |λ|− 6 f α z −α− 4 + f β z −β− 4 . (3.23)
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Proof. Assume 0 arg λ π and z ∈ Γ+ λ . By Lemma 7.5.1, we have Γλ (z) ⊂ S[−π + 23 δ, 0], so the uniform estimates (2.34) and (2.35) hold on Γλ (z) for both 1 a(z) and a(zω). Let F (z) = λ− 6 a(z)ρ(z), where ρ is given by (2.25). g −P F f = I1 +I2 , I1 = P (s)F (s)f (s)ds, I2 = P (s)F (s)f (s)ds. Γλ (z)
Γλ (z)
Using (2.26), (2.34) and (3.20) we have 1
4
3
|P (z)F (z)f (z)| C|λ|− 6 q∞ |e− 3 z 2 |f α z −α−1 .
(3.24)
In order to estimate I1 and I2 we use (2.26), (2.34), (2.35), (2.39) and (3.21). This gives 3 3 − 14 − 43 s 2 − 23 − 43 s 2 f α |e | f α |e | q∞ |λ| s + |I1 | C |ds| 1 2 7 5 3 α+ α+ 3 3 4 4 4 |λ| λ s s Γλ (z) s 2
λ3 1
|I2 | C
q∞ 1
|λ| 3
4
3
Cq∞ |λ|− 6 |e− 3 z 2 |f α z −α−2 , − 14 3 s |ds| q∞ − 4 z 32 f β − 43 s 2 3 f β |e | | β+ 3 . 2 3 C 1 |e λ3 s β+ 4 |λ| 6 z 2 Γλ (z)
The above estimates for I1 , I2 and (3.24) give 3 1 4 3 |g(z, λ)| C|λ|− 6 q∞ |e− 3 z 2 | f α z −α−1 + f β z −β− 2 .
(3.25)
The last estimate together with (2.34) and (2.35) implies (3.22) and (3.23). − + Assume δ arg λ π and z ∈ Γ− λ . Using Γλ = Γλ ∪Γλ we have g = g− + g+ , where 3 3 4 4 a(s)e− 3 s 2 Vq (s)f (s) ds, g+ (λ) = a(s)e− 3 s 2 Vq (s)f (s) ds. g− (z, λ) = Γλ (z,z∗ )
Γ+ λ
Using (3.11) and (3.25) for z = z∗ we obtain f α q∞ − 4 z∗ 23 f β 3 |g+ (λ)| C | + 1 |e 3 |z∗ |α+1 |λ| 6 |z∗ |β+ 2 f α q∞ − 4 z∗ 32 f β 3 C | + β+ 3 . (3.26) 1 |e |z|α+1 |λ| 6 |z| 2 Using (2.34), (2.25), (2.26), (2.39) and (3.11) results in 3 q∞ f α q∞ − 4 z 32 f α 4 3 |g− (z, λ)| C |e− 3 s 2 | α+ 1 |ds| C | α+1 . (3.27) 1 1 |e |z| |λ| 3 Γλ (z,z∗ ) |s| 4 |λ| 6 Now (3.22) and (3.23) follow from (3.26) and (3.27) taking into account (2.34), 3 (2.35) and the fact that, by Lemma 2.2, | exp{ 32 z(x, λ) 2 }| is strictly increasing.
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In the following Lemma we estimate the operator JV0 . We show that as λ → ∞ it is asymptotically small in comparison with JVq . Lemma 3.4 Let λ ∈ C+ , |λ| 1 and f ∈ Fαλ for some α > 0 (that is, f is defined λ on Γλ for δ arg λ π and on Γ+ λ for 0 arg λ δ). Then JV0 f ∈ Fα+ 12 and ∂ C f α C f α (JV0 f )(z, λ) |(JV0 f )(z, λ)| (3.28) 2 1 , 2 3 . α+ ∂z |λ| 3 z 2 |λ| 3 z α+ 2 Proof. By Lemma 7.5.1, we have Γλ (z) ⊂ S[−π + 23 δ, 0], so the decomposition (3.7) holds on Γλ (z). We estimate the part of JV0 , corresponding to the first term in decomposition (3.7). Lemma 7.5, (2.26), (2.34), (2.40) and the inequality 2 4 z |λ| 3 2(|λ| 3 + |z|2 ) imply f α |ds| a(sω)V0 (s)f (s) ds C 1 4 α+ Γλ (z) 4 |λ| 3 + |s|2 Γλ (z) s f α C C f α (3.29) 2 5 |ds| 2 1 α+ |λ| 3 Γλ (z) s 4 |λ| 3 z α+ 4 In order to estimate the part of JV0 , corresponding to the second term in 2 decomposition (3.7), we use (2.26), (2.34), (2.39) and the inequality z |λ| 3 4 2(|λ| 3 + |z|2 ). This gives 3 4 3 f α |e− 3 s 2 | − 43 s 2 a(s)e V0 (s)f (s) ds C |ds| 4 α+ 14 2 Γλ (z,∞) Γλ (z,∞) |λ| 3 + |s| s f α
4
3
|e− 3 s 2 |
4
3
C |e− 3 z 2 |f α
. (3.30) 2 5 |ds| 2 7 |λ| 3 Γλ (z,∞) s α+ 4 |λ| 3 z α+ 4 The first estimate in (3.28) follows from (3.29) and (3.30) taking into account (2.34), (3.7). In order to estimate ∂z g(z, λ) we note that ∂z g(z, λ) = Γλ (z) ∂z J(z, s)V0 (s) f (s) ds. Therefore the second estimate in (3.28) follows from (3.29), (3.30), (2.35) and (3.7). C
Combining the results of Lemmas 3.2-3.4 we estimate the operator JV = JVq + JV0 . λ for Lemma 3.5 Let q, q1 ∈ L∞ (R). Assume λ ∈ C+ , |λ| 1, and f ∈ Fα,β 3 α > 0, β > 4 (that is, f is defined on Γλ for δ arg λ π and on Γ+ λ for λ 0 arg λ δ). Then JV f ∈ Fα,β and f α q∞ + q1 ∞ f β C f α |(JV f )(z, λ)| C (3.31) + 1 1 + 1 2 1 , α+ β− 6 2 2 |λ| z z |λ| 3 z α+ 2 f α q∞ + q1 ∞ f β C f α |∂z (JV f )(z, λ)| C + 1 3 + 1 2 3 . (3.32) α+ β+ |λ| 6 z 4 z 2 |λ| 3 z α+ 2
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Proof. Recall that JV = JVq +JV0 . By Lemma 7.5.1, we have Γλ (z) ⊂ S[−π+ 23 δ, 0], so the decomposition (3.7) holds on Γλ (z). Taking into account this decomposition, we deduce that the combination of Lemmas 3.2 and 3.3 gives the estimate for JVq (the corresponding terms in (3.31) and (3.32) contain curved brackets). Together with the estimate (3.28) for JV0 this proves (3.31) and (3.32). Proof of Theorem 3.1. We consider the case λ ∈ C+ . The proof for λ ∈ C− is similar. Let vn+1 = JV vn , n 0. Substituting vn , g = vn+1 in (3.31), (3.32) and taking into account (3.8) for v0 we obtain |vn+1 (z)|
εvn αn αn + 12
z
+
εvn βn βn − 12
z
,
|vn+1 (z)|
εvn αn αn + 34
z
+
εvn βn 1
z βn + 2
,
where α0 =
1 5 , β0 = , 4 4
1 1 αn+1 = min{αn + , βn − }, 2 2
3 1 βn+1 = min{αn + , βn + }. 4 2 (3.33)
Therefore n
1
|v2n (z)| ε2n v0 14 , 54 z − 4 − 4 , n
n
3
|v2n+1 (z)| ε2n+1 v0 14 , 54 z − 4 − 4 , (3.34)
5
n
|v2n (z)| ε2n v0 14 , 54 z − 4 − 4 , |v2n+1 (z)| ε2n+1 v0 41 , 54 z − 4 −1 , (3.35) and for ε < 1 the series v+ (z) = ∞ n=0 vn (z) converges absolutely and is a solution of v+ = a + JV v+ . For ε < 12 (3.8), (3.34) and (3.35) give (3.4), (3.5). (1) We prove uniqueness. Suppose that there exists another solution v+ ∈ F λ1 ,1 4
(1)
and let y = v+ − v+ ∈ F λ1 ,1 . We have y = JV y and therefore y = (JV )n y for any 4 integer n 1. Lemma 3.5 yields |y(z)| Cεn y 41 ,1 . Taking n → ∞ for ε < 1 we obtain y = 0.
4 Uniform asymptotics In this section we consider the equation v+ = a + Jv+ for large |λ|, | arg λ| δ and + λ z ∈ Γ− λ (see the case z ∈ Γλ in Sect. 3). For | arg λ| δ denote by F− the class of − λ λ functions f on Γλ such that f, f ∈ L∞ (Γ− ). We also set F = F ⊕ F λ1 ,1 , where − λ 4
λ λ Fα,β is given by (3.1)–(3.3) and elements of Fα,β are functions on Γ+ λ , | arg λ| < δ and on Γλ , δ | arg λ| π. The main result of this section is
Theorem 4.1 Let q ∈ B and | arg λ| δ. Then the equation v+ = v0 + JV v+ , 1 v0 ≡ a, has a unique solution v+ ∈ F λ for |λ| 6 2c0 (qB + 1), where c0 > 1 is an absolute constant. If, in addition, (z, λ) ∈ Γ− λ × S[−δ, δ], then the following estimates are fulfilled: 1
|v+ (z)| Cz − 4 ,
1
|v+ (z)| Cz 4 ,
(4.1)
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|v1 (z)| Cεz − 4 ,
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ε = c0 |λ|− 6 (qB + 1), (4.2)
v1 = JV v0 ,
1
|v+ (z) − v0 (z) − v1 (z)| Cε2 z − 4 ,
|v+ (z) − v0 (z) − v1 (z)| ε2 Cz 4 . (4.3)
Corollary 4.2 Let q ∈ B. Then the equation u+ = Ai + J0 V u+ √ has a unique solution u+ (z, λ) such that u+ (z, λ) = Ai(z)(1 + o(1)), ∂z u+ (z, λ) + zu+ (z, λ) = 5 1 Ai (z)O(z − 4 ) as Γλ z → ∞ for |λ| 6 2c0 (qB + 1), where c0 > 1 is an 2
3
absolute constant. Moreover, u+ (z, λ) = e− 3 z 2 v+ (z, λ). The following estimates are fulfilled uniformly in z ∈ Γ− λ : 2
3
Let | arg λ| δ and let u1 (z, λ) = e− 3 z 2 v1 (z, λ)). Then 3
2
1
1
|u+ (z, λ)| Ce 3 | Re z 2 | z − 4 , 3
2
|u1 (z, λ)| Cε
e 3 | Re z 2 | z
1 4
1
3
2
|∂z u1 (z, λ)| Cεz 4 e 3 | Re z 2 | ,
,
3
2
|∂z u+ (z, λ)| Cz 4 e 3 | Re z 2 | ,
3
2
ε = c0
(4.4) qB + 1
, 1 |λ| 6 (4.5)
1
|u+ (z, λ) − Ai(z) − u1 (z, λ)| Cε2 e 3 | Re z 2 | z − 4 ,
(4.6)
3 ∂z u+ (z, λ) − Ai (z) − ∂z u1 (z, λ) Cε2 z 41 e 32 | Re z 2 | .
(4.7)
Let δ | arg λ| π. Then 3
2
3
|u+ (z, λ)−Ai(z)| Cεe 3 | Re z 2 | z − 4 , 2
3
2
1
|∂z u+ (z, λ)−Ai (z)| Cεe 3 | Re z 2 | z − 4 . (4.8)
3
Proof. Set u+ (z, λ) = e− 3 z 2 v+ (z, λ), where v+ is given by Theorem 3.1 (for δ < | arg λ| π) and Theorem 4.1 (for | arg λ| δ). By (2.32), (2.33) and (3.4), u+ is a solution of (2.30) with the required asymptotics. In order to prove uniqueness, we suppose that there exists another solution (1)
(1)
2
3
(1)
u+ with the same asymptotics as Γλ z → ∞. Then v+ (z, λ) = e 3 z 2 u+ (z, λ) is in F λ (for | arg λ| δ) or in F λ1 ,1 (for δ < | arg λ| π) and solves v+ = v0 + JV v+ . 4
(1)
(1)
Hence, by Theorems 3.1 and 4.1, v+ = v+ , implying u+ = u+ . The estimates (4.4–4.7) follow from (4.1–4.3). The estimate (3.5) implies (4.8). In order to prove Theorem 4.1 we introduce a special decomposition (4.11) of functions on Γ− λ , correlated with the decomposition (4.18), (4.19) of the integral operator JV . In terms of these decompositions we formulate and prove Lemmas 4.3 and 4.4. These Lemmas are the main ingredients for the proof of Theorem 4.1. Below we only consider the case 0 arg λ δ. The proof for −δ arg λ 0 is similar. By Lemma 7.5.1, Γ− λ is arbitrarily close to R− as arg λ → 0. Thus we
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cannot use (2.35) in order to estimate the terms in the decomposition (3.7) for J(z, s). Thus we need a different representation. We have 2
3
π z ∈ S[−π, − ), 3
Ai(zω) = e− 3 z 2 a(zω),
ω=e
2πi 3
.
Using (2.28), (2.29) and (3.6), we obtain from (2.31) and (2.33) 3 3 4 2 z − 23 s 2 3 a(zω)Ai(sω) , z ∈ Γ− J(z, s) = −2i a(zω)a(sω) − e λ,
(4.9)
s ∈ Γλ ,
(4.10) + π 5 π π where, by Lemma 7.3.1, Γ− ⊂ S[−π, − + δ] ⊂ S[−π, − ) and Γ ⊂ S[− λ λ 2 12 3 2 − δ , 0] ⊂ S(−π, π]. Due to (4.10), the operator JV has the form a(zω)×(integral 4 4
3
operator)+e 3 z 2 a(zω)×(integral operator). Thus in order to estimate JV we estimate these integral operators, introducing for functions on Γ− λ a decomposition in the spirit of (4.10). λ and a fixed decomposition For f ∈ F− 4
3
f (z) = a(zω)fp (z) + e 3 z 2 a(zω)fe (z),
z ∈ Γ− λ,
λ ∈ {λ ∈ S[0, δ] : |λ| 1}, (4.11)
we define p0 (f, λ) = sup
z∈Γ− λ
3 4 |fp (z)| + |e 3 z 2 fe (z)| , 1
p1 (f, λ) = sup z 2 z∈Γ− λ
3 4 |fp (z)| + |e 3 z 2 fe (z)| . (4.12)
We remark that this decomposition is not unique. But this is irrelevant for our λ proof. If f ∈ F− has a decomposition (4.11), then we have 1
|f (z)| Cp0 (f, λ)z − 4 ,
1 |f (z)|z − 4 C p0 (f, λ) + p1 (f, λ)z −1 ,
and 4
z ∈ Γ− λ, (4.13)
3
|fe (z)| |e− 3 z 2 |p0 (f, λ),
|fp (z)| p0 (f, λ), 1
4
|fp (z)| p1 (f, λ)z − 2 ,
3
1
|fe (z)| |e− 3 z 2 |z − 2 p1 (f, λ).
(4.14) (4.15)
Using (3.6), (4.9) and the identity (2.29) we obtain for 0 arg λ δ π
π
4
3
v0 (z) = a(z) = ei 3 a(zω)+e−i 3 e 3 z 2 a(zω),
π z ∈ Γ− λ ⊂ S[−π, − ). 3
(4.16)
For this decomposition using (2.34) and (2.35) we obtain p0 (v0 , λ) 2,
p1 (v0 , λ) = 0,
0 arg λ δ.
(4.17)
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Below we estimate (JV f )
Ann. Henri Poincar´e
in terms of p0 and p1 assuming that f ∈ F λ =
Γ− λ
λ F− ⊕ F λ1 ,1 . Using (4.10) for some function f on Γ− λ we obtain the following de4 composition: 4
3
z ∈ Γ− λ, (4.18) $ where λ ∈ S[0, δ] {λ ∈ C : |λ| 1} and the operators Jp and Je are given by 3 2 a(sω)f (s) ds, (Je f )(z) = 2i e− 3 s 2 Ai(sω)f (s) ds. (Jp f )(z) = −2i (JV f )(z) = a(zω)gp (z) + e 3 z 2 a(zω)ge (z),
gp = Jp V f,
Γλ (z)
ge = Je V f,
Γλ (z)
(4.19) exists in the prinIn the applications below the integral along the infinite curve Γ+ λ cipal value sense. In this section we will always define p0 (JV f, λ) and p1 (JV f, λ) using the decomposition (4.18) and (4.19). In order to estimate Jp V and Je V we rewrite the integrals in (4.19) in the form π (4.20) z ∈ Γ− (Jp f )(z) = (jp f )(z) + hp (f ), λ ⊂ S[−π, − ), 3 (jp f )(z) = −2i a(sω)f (s) ds, hp (f ) = −2i a(sω)f (s) ds, (4.21) Γ+ λ
Γλ (z,z∗ )
corresponding to the splitting Γλ (z) = Γλ (z, z∗ ) ∪ Γ+ λ . A similar decomposition of Je gives π (4.22) z ∈ Γ− (Je f )(z) = (je f )(z) + he (f ), λ ⊂ S[−π, − ), 3 3 3 2 2 (je f )(z) = 2i e− 3 s 2 Ai(sω)f (s) ds, he (f ) = 2i e− 3 s 2 Ai(sω)f (s) ds. Γ+ λ
Γλ (z,z∗ )
(4.23) Note that the standard asymptotics (2.27) for Ai(sω) fails in the neighborhood of arg s = − π3 . Taking into account (4.9) we obtain 3 4 (je f )(z) = 2i e− 3 s 2 a(sω)f (s) ds. (4.24) Γλ (z,z∗ )
2
3
π −3s2 By (2.29), (2.32), (3.6) and (4.9), for s ∈ Γ+ Ai(sω) = λ ⊂ S(−π, 3 ) we have e π
4
3
ei 3 a(sω) − e− 3 s 2 ωa(s). This gives 3 π 4 he (f ) = 2i ei 3 a(sω) − e− 3 s 2 ωa(s) f (s) ds. Γ+ λ
(4.25)
π Using (4.19–4.25), for z ∈ Γ− λ ⊂ S[−π, − 3 ) we have 4
3
(Jf )(z) = a(zω) ((jp f )(z) + hp (f )) + e 3 z 2 a(zω)((je f )(z) + he (f )).
(4.26)
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For zj = z(xj , λ) ∈ Γ− λ , j = 1, 2 and 1 6 P± (z1 , z2 ) = λ
767
x1 < x2 we define 4
Γλ (z1 ,z2 )
3
e± 3 s 2 ρ−1 (s)Vq (s) ds.
(4.27)
λ Lemma 4.3 Let q ∈ B. Assume 0 arg λ δ, |λ| 1. Let f Γ− ∈ F− have λ a decomposition (4.11) and f Γ+ = 0. Then (JVq f )(z) = a(zω)(jp Vq f )(z) + λ
3 4 2 3z
e a(zω)(je Vq f )(z) ∈ fulfilled:
λ F−
and for this decomposition the following estimates are
1 p0 (JVq f, λ) CqB |λ|− 6 p0 (f, λ) + p1 (f, λ) log(|λ| + 1) , 1
p1 (JVq f, λ) Cq∞ |λ|− 3 p0 (f, λ).
(4.28) (4.29)
π Proof. By Lemma 7.3.1, Γ− λ ⊂ S[−π, − 3 ), so (4.10) holds. Using (4.11), (4.18), (4.20), (4.21), (4.22), (4.23) and (4.24) we obtain
(jp Vq f )(z) = −2i(Ipp (z) + Ipe (z)), where
Ipp (z) =
a(sω)a(sω)Vq (s)fp (s) ds, Ipe (z) =
Iep (z) =
(je Vq f )(z) = 2i(Iep (z) + Iee (z)), (4.30)
Γλ (z,z∗ )
4
3
a(sω)2 e− 3 s 2 Vq (s)fp (s) ds, Iee (z) =
Γλ (z,z∗ )
4
3
a(sω)2 e 3 s 2 Vq (s)fe (s) ds,
Γλ (z,z∗ )
a(sω)a(sω)Vq (s)fe (s) ds.
Γλ (z,z∗ )
1
We estimate Ipp . Let Fj (s) = λ− 6 a(sω)a(sω)fj (s)ρ(s), j = e, p. Integration by parts yields Ipp (z) = −Q(z∗ , z)Fp (z) + Q(z∗ , s)Fp (s)ds, (4.31) Γλ (z,z∗ )
where Q is given by (3.9). Using (2.25), (2.26), (2.34), (2.35) and (3.10), we obtain & % ' ( 2 |fp (s)| q1 ∞ |fp (z)| |fp (s)| |λ|− 3 |fp (s)| + + + |ds| . |Ipp (z)| C 1 1 1 3 1 |λ| 6 z 2 s 2 s 2 s 2 Γ− λ Due to (2.41), (4.14) and (4.15) we have − 16 |Ipp (z)| Cq1 ∞ |λ| p0 (f, λ) + p1 (f, λ) log(|λ| + 1) ,
z ∈ Γ− λ.
(4.32)
We estimate Ipp (z) = −a(zω)a(zω)Vq (z)fp (z). Using (2.25), (2.26) and (2.34) we have Ipp (z) Cq∞ |fp (z)||λ|− 13 z − 12 , z ∈ Γ− . (4.33) λ
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M. Klein, E. Korotyaev and A. Pokrovski
Ann. Henri Poincar´e
The estimate of Iee is similar. Using Fe and Q we integrate by parts. Using (2.25), (2.26), (2.34), (2.35) and (3.10), we obtain & % ' ( 2 q1 ∞ |fe (z)| |fe (s)| |λ|− 3 |fe (s)| |fe (s)| |Iee (z)| C + + + |ds| . 1 1 1 3 1 |λ| 6 z 2 s 2 s 2 s 2 Γ− λ 3
4
π We have |e− 3 z 2 | 1 for z ∈ Γ− λ ⊂ S[−π, − 3 ). Using (2.41), (4.14) and (4.15) we have q1 ∞ − 4 z 32 π 3 |Iee (z)| C |e | p (f, λ) + p (f, λ) log(|λ| + 1) , z ∈ Γ− 0 1 1 λ ⊂ S[−π, − ). 3 |λ| 6 (4.34) We estimate Iee (z) = −a(zω)a(zω)Vq (z)fe (z). Using (2.25), (2.26) and (2.34) we obtain 1 1 π (4.35) (z)| Cq∞ |fe (z)||λ|− 3 z − 2 , z ∈ Γ− |Iee λ ⊂ S[−π, − ). 3
In order to estimate Ipe we use P+ (s, z), given by (4.27). Integrating by parts we have 1 P+ (s, z)F (s)ds, F (z) = λ− 6 a(zω)2 fe (z)ρ(z). Ipe (z) = −P+ (z∗ , z)F (z∗ )+ Γλ (z,z∗ )
Using (A.26), (2.34), (2.35), (2.25) and (2.26) we obtain ) 3 4 2 qB |Ipe (z)| C |fe (z∗ )| |e 3 z∗ | 1 |λ| 6 & ' ( 2 3 4 2 3 |fe (s)| |λ|− 3 s − |e 3 | + |fe (s)| s 2 + |ds| . + 1 1 s 2 s 2 Γ− λ 4
3
π We have |e 3 s 2 | 1 for s ∈ Γ− λ ⊂ S[−π, − 3 ). Using (2.41) and (4.14–4.15) we obtain − 16 |Ipe (z)| CqB |λ| (4.36) p0 (f, λ) + p1 (f, λ) log(|λ| + 1) , z ∈ Γ− λ. 4
3
We estimate Ipe (z) = −a(zω)2 e 3 z 2 Vq (z)fe (z). Lemma 2.2 and (2.25), (2.26) and (2.34) give 3 Ipe (z) Cq∞ |e 43 z∗2 fe (z)||λ|− 13 z − 12 ,
z ∈ Γ− λ.
(4.37)
In order to estimate Iep we use P− (s, z), given by (4.27). Integrating by parts we have 1 P− (z∗ , s)F (s)ds, F (z) = λ− 6 a(zω)2 fp (z)ρ(z). Iep (z) = −P− (z∗ , z)F (z)+ Γλ (z,z∗ )
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769
Using (A.25), (2.34), (2.35), (2.25) and (2.26) we obtain |Iep (z)| C
qB
%
4
3
|fp (z)e− 3 z 2 |
1
1
|λ| 6 +
z 2
Γλ (z,z∗ )
|e
3 − 43 s 2
|
|fp (s)| 1
s 2 4
3
& − 32
+ |fp (s)| s 4
2
+
|λ|− 3
'
( |ds| .
1
s 2
3
By Lemma 2.2, we have max |e− 3 s 2 | = |e− 3 z 2 |. Using (2.41), (4.14) and (4.15) Γλ (z,z∗ )
we obtain − 16
|Iep (z)| CqB |λ| We estimate (2.34) yield
Iep (z)
|e
3
− 43 z 2
| p0 (f, λ) + p1 (f, λ) log(|λ| + 1) ,
z ∈ Γ− λ. (4.38)
3
2 − 43 z 2
= −a(zω) e
Vq (z)fp (z). Lemma 2.2 and (2.25), (2.26),
3 Iep (z) Cq∞ |e− 43 z 2 fp (z)||λ|− 13 z − 12 ,
z ∈ Γ− λ.
(4.39)
Due to (4.30), (4.32), (4.34), (4.36) and (4.38) we have 4
3
p0 (JVq f, λ) = |(jp Vq f )(z)| + |e 3 z 2 (je Vq f )(z)| qB C p0 (f, λ) + p1 (f, λ) log(|λ| + 1) , 1 |λ| 6 where z ∈ Γ− λ . This proves (4.28). Due to (4.30), (4.33), (4.35), (4.37) and (4.39) we have 4 z 32 3 p1 (JVq f, λ) = |∂z (jp Vq f )(z)| + e ∂z (je Vq f )(z) C where z ∈
Γ− λ.
q∞ 1 3
|λ| z
4
1 2
3
(|fp (z)| + |e 3 z 2 fe (z)|),
This proves (4.29).
Lemma 4.4 Let q ∈ B and 0 arg λ δ, |λ| 1. Assume f = f − + f + , where λ λ f + = f Γ+ ∈ Fα,β for α > 0, β > 34 and f − = f Γ− ∈ F− . Then g = (JV f )Γ− ∈ λ
λ F−
λ
3 4 2 3z
λ
and for the decomposition g(z) = a(zω)(Jp V f )(z) + e a(zω)(Je V f )(z) the following estimates are fulfilled: qB C − − + − p0 (g, λ) C , λ), p0 (f , λ) + p1 (f , λ) log(|λ| + 1) + f α,β + 1 2 p0 (f |λ| 6 |λ| 3 (4.40) 1 (4.41) p1 (g, λ) C|λ|− 3 (q∞ + 1) p0 (f − , λ).
770
M. Klein, E. Korotyaev and A. Pokrovski
Ann. Henri Poincar´e
Proof. By (4.26), we have g = g0 + g− + g+ , where 4
3
g+ (z) = a(zω)hp (Vq f + ) + e 3 z 2 a(zω)he (Vq f + ), 4
3
g− (z) = a(zω)(jp Vq f − )(z) + e 3 z 2 a(zω)(je Vq f − )(z), 4
3
g0 (z) = a(zω)(Jp V0 f )(z) + e 3 z 2 a(zω)(Je V0 f )(z). Firstly, we estimate g+ . From (3.15) and (3.25) we have |hp (Vq f )|
C 1
|λ| 6
q1 ∞ f + α,β , |he (Vq f )| 4
3 2
1
C(q∞ |e− 3 z∗ | + q1 ∞ )|λ|− 6 f + α,β . (4.42) 3 2
3
4
By Lemma 2.2, we have |e 3 (z 2 −z∗ ) | 1 for z ∈ Γ− λ . Thus 1
p0 (g+ , λ) C(q∞ + q1 ∞ )|λ|− 6 f + α,β ,
p1 (g+ , λ) = 0.
(4.43)
Secondly, we estimate g0 . Lemma 2.2 and (4.21), (4.24), (2.25), (2.26), (2.34), (2.42) give p0 (f − , λ) |ds| p0 (f − , λ) , z ∈ Γ− (4.44) |(jp V0 f )(z)| C 4 1 C 2 λ, − |λ| 3 + |s|2 s 2 3 |λ| Γλ |(je V0 f )(z)| C|e
3
− 43 z 2
|
p0 (f − , λ) |ds|
Γ− λ
4 3
|λ| +
|s|2
s
1 2
3
4
C|e− 3 z 2 |
p0 (f − , λ) |λ|
z ∈ Γ− λ.
2 3
(4.45)
Using Lemma 2.2, (3.29) and (3.30) we have 2
2
|hp (V0 f )| C|λ|− 3 f + α , 4
4
3 2
|he (V0 f )| C|λ|− 3 |e− 3 z∗ |f + α .
(4.46)
3 2
3
By Lemma 2.2, we have |e 3 (z 2 −z∗ ) | 1 for z ∈ Γ− λ . Thus substituting (4.44), (4.45) and (4.46) in (4.26) we obtain 2
p0 (g0 , λ) C(p0 (f − , λ) + f + α )|λ|− 3 .
(4.47)
Using (4.21), (4.24), (2.34), (2.25), (2.26) and (4.13) we obtain |∂z (jp V0 f )(z)| which proves
C p0 (f − , λ) |λ|
4 3
z
1 2
,
|∂z (je V0 f )(z)|
4
p1 (g0 , λ) Cp0 (f − , λ)|λ|− 3 .
C |λ|
4
4 3
3
|e− 3 z 2 |
p0 (f − , λ) 1
z 2
,
(4.48)
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771
Finally, apply (4.28) to g− ; together with (4.43) and (4.47) this gives (4.40). Applying (4.29) to g− together with (4.43) and (4.48) gives (4.41). Proof of Theorem 4.1. We consider the case 0 arg λ δ, the proof for −δ arg λ 0 is similar. Let vn+1 = JV vn , n 0, where v0 ≡ a. Introduce vn± = vn Γ± . λ By (3.31–3.32), for some absolute constant c0 > 0 we have + αn+1 ,βn+1 εvn+ αn ,βn , vn+1
where αn and βn are given by (3.33). We estimate vn− in terms of p0 and p1 , using the decomposition (4.16) for v0− and (4.18), (4.19) for vn− , n 1. Substituting f = vn and g = vn+1 in (4.40), (4.41) and choosing c0 sufficiently large, we obtain − , λ) ε p0 (vn− , λ) + p1 (vn− , λ) log(|λ| + 1) + vn+ αn ,βn , p0 (vn+1 − )ε p1 (vn+1
p0 (vn− , λ) 1
|λ| 6
.
Using (3.8) and (4.17) (in particular, p1 (v0 , λ) = 0) we obtain for v1 and v2 p0 (v1− , λ) εL,
1
p1 (v1− , λ) ε|λ|− 6 L,
L = 2 + v0 41 , 54 ,
1 1 p0 (v2− , λ) ε εL + εL|λ|− 6 log(|λ|+ 1)+ v1+α1 ,β1 ε2 (2 + |λ|− 6 log(|λ|+ 1))L, 1
p1 (v2− , λ) ε2 |λ|− 6 L. Increasing the constant c0 we obtain by induction for each integer n 0 p0 (vn− , λ) εn L,
p1 (vn− , λ)
1 1
|λ| 6
εn L,
By Theorem 3.1, for ε < 1 the series v+
vn+ αn ,βn εn v0 14 , 54 .
Γ+ λ
=
∞
+ n=0 vn
(4.49)
converges in F 14 ,1 -
norm and gives a solution = v0 + JV v+ on Γ+ λ . By (4.49), for ε < 1 the series ∞ of v+ − ∞ − p (v , λ) and p (v , λ) converge; for ε 12 the following estimates 0 1 n n n=0 n=0 are fulfilled: p0 (v+ Γ− , λ) 2L, λ
2L p1 (v+ Γ− , λ) 1 , λ |λ| 6
p0 v+ Γ− − v0− − v1− , λ 2ε2 L,
p0 (v1− , λ) 2εL,
p1 (v1− , λ)
2εL 1 , |λ| 6 (4.50)
1 p1 v+ Γ− − v0− − v1− , λ 2|λ|− 6 ε2 L. λ λ (4.51) ∞ ∞ convergence By (4.13), convergence of n=0 p0 (vn− , λ) and n=0 p1 (vn− , λ) implies ∞ ∞ of n=0 vn− in C 1 -norm on Γ− n=0 vn converges λ . Thus for ε < 1 the series v+ =
772
M. Klein, E. Korotyaev and A. Pokrovski
Ann. Henri Poincar´e
and solves the equation v+ = v0 + JV v+ on Γλ . Using (4.13) and (4.49) we obtain (4.1) and (4.2) from (4.50); similarly we obtain (4.3) from (4.51). (1) We prove uniqueness. Suppose that there exists another solution v+ ∈ F λ = (1) λ F− ⊕ F λ1 ,1 . By Theorem 3.1, we have v+ |Γ+ = v+ |Γ+ . Consider the difference 4
λ
(1)
λ
λ y = (v+ − v+ )|Γ− ∈ F− . We have y = (JV )n y for any n > 1. Lemma 4.4 yields λ n p0 (y, λ) Cε (p0 (y, λ) + p1 (y, λ)). Taking n → ∞ for ε < 1 gives y = 0.
5 Asymptotics of the Wronskian In this section we shall determine the asymptotics of w(λ) = {ψ− , ψ+ } as λ → ∞. 2
3
To this end we find the asymptotics of u1 (z, λ) = e− 3 z 2 (JV v0 )(z) (given by Corollary 4.2) as λ → ∞ in the sector | arg λ| δ. Introduce an auxiliary function 2πi E+ (λ) = 2 Ai(sω)Ai(sω)Vq (s) ds, ω=e 3 . (5.1) Γ− λ
By (2.23), we have 2 32 π z = ±i λ, 3 0 4
0 ± arg λ π,
z0 = z(0, λ) = −λ
2 3
3π 8
23 .
(5.2)
If Im λ > 0, then we get Γλ ⊂ C− . We follow the agreement (see p.6) that noninteger powers take their principal values on C \ R− . By (2.26), z(0, λ) is in the sector S(−π, −π/3). The signs in equ. (5.2) correspond to our choice of branch. Thus the solutions u+ (z, |λ|e±iπ ) are different. This is not a problem: when we return to the x-coordinate in (5.15), multiplication by φ(λ) makes the solution analytic for sufficiently large λ on R− . Lemma 5.1 Assume q ∈ B and ν =
π 4 (λ
+ 1).
1. Let | arg λ| δ, |λ| → ∞. Then * + 1 π 1 −1 u+ (z0 , λ) = z0 4 sin ν 1 + O(λ− 6 ) + E+ (λ) cos ν + O λ− 2 e 4 | Im λ| , 1 4
*
− 16
∂z u+ (z0 , λ) = z0 − cos ν 1 + O(λ
+ 1 π (5.3) ) + E+ (λ) sin ν + O λ− 6 e 4 | Im λ| , (5.4)
2. Let δ ± arg λ π, |λ| → ∞. Then u+ (z0 , λ) =
e∓iν 1
2z04
+O
π
e 4 | Im λ| 2
λ3
1
z4 , ∂z u+ (z0 , λ) = − 0 e∓iν +O 2
π
e 4 | Im λ| 1
λ3
.
(5.5)
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773
Proof. We consider the case Im λ 0. The proof for Im λ 0 is similar. 1. Let 0 arg λ δ. We have 3
2
2
3
u1 (z, λ) = e− 3 z 2 v1 (z) = e− 3 z 2 (JV v0 )(z) = Ai(zω)(Jp V v0 )(z) + Ai(zω)(Je V v0 )(z). 2
By (4.47), (JV0 v0 )(z) = O(λ− 3 ). Since V = Vq + V0 , we have as λ → ∞ 3 3 3 − 23 z 2 − 23 z 2 − 23 − 14 32 | Re z 2 | u1 (z, λ) = e v1 (z) = e (JVq v0 )(z) + O |λ| z e .
(5.6)
Using (4.16), (2.28), (2.29), (3.6), (4.9) and (4.20–4.25) we obtain 2
3
e− 3 z 2 (JVq v0 )(z) = Ai(zω)(Jp Vq v0 )(z) + Ai(zω)(Je Vq v0 )(z) = 2iωAi(z) a(sω)a(s)Vq (s) ds −2iAi(zω)
Γ+ λ
−2iωAi(zω)
4
3
e− 3 s 2 a2 (s)Vq (s) ds + 2Bi(z) 3
4
Γ+ λ
a(sω)a(sω)Vq (s) ds
Γλ (z,z∗ )
π
e 3 s 2 a2 (sω)Vq (s) ds + 2iei 3 Ai(zω)
Γλ (z,z∗ )
4
3
e− 3 s 2 a2 (sω)Vq (s) ds,
Γλ (z,z∗ )
2πi 3
where ω = e . Now we set z = z0 and write the asymptotics of u1 in terms of E+ . By Lemma 7.7, (3.6), (4.9) and (2.34), we have 3 1 1 2 2 − u1 (z0 , λ) = 2iωAi(z0 ) a(sω)a(s)Vq (s) ds + Bi(z0 )E+ (λ) + O λ− 3 z0 4 |e− 3 z0 | . Γ+ λ
(5.7)
Using (2.35) we get
∂z u1 (z, λ) = 2iωAi (z0 )
Γ+ λ
1 1 3 2 a(sω)a(s)Vq (s)ds+Bi (z0 )E+ (λ)+O λ− 3 z 4 |e− 3 z 2 | .
(5.8) 1 Lemma 3.2, (3.6), (2.34) and (2.35) give Γ+ a(sω)a(s)Vq (s) ds = O(λ− 6 ). Thereλ fore using (4.6), (4.7), (5.2), (5.6), (5.7) and (5.8) we obtain 1 π 1 u+ (z0 , λ) = Ai(z0 )(1 + O(λ− 6 )) + Bi(z0 )E+ (λ) + O λ− 2 e 4 | Im λ| , (5.9) 1 π 1 ∂z u+ (z0 , λ) = Ai (z0 )(1 + O(λ− 6 )) + Bi (z0 )E+ (λ) + O λ− 6 e 4 | Im λ| , (5.10) where z0 is given by (2.23). Recall the standard uniform asymptotics of Airy functions [1] in the sector |arg z| < π3 − ε, ε > 0, as z → ∞: 1
Ai(−z) = z − 4 (sin η + O(F (z)),
1
Ai (−z) = −z 4 (cos η + O(F (z))),
(5.11)
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Bi(−z) = z − 4 (cos η + O(F (z)), 3
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1
Bi (−z) = z 4 (sin η + O(F (z))),
(5.12)
3 2 2 3 | Im z
| . Note that for 0 arg λ δ we have where η = 23 z 2 + π4 and F (z) = z − 2 e 2 −π arg z0 −π + 3 δ. Thus we substitute (5.11) and (5.12) into (5.9–5.10). This gives (5.3),(5.4). 2. Let δ arg λ π. By (4.8), we have 2 π 1 π u+ (z0 , λ) = Ai(z0 ) + O λ− 3 e 4 | Im λ| , u+ (z0 , λ) = Ai (z0 ) + O λ− 3 e 4 | Im λ| .
(5.13) Note that for δ arg λ π we have −π + 23 δ arg z0 0. Thus we apply (2.27) and (5.2) to (5.13), which gives (5.5). Introduce the function 3
φ(λ) = 2 4 (λ/2e)λ/4 ,
λ ∈ C \ R− ,
φ(λ) > 0 for
λ > 0.
(5.14)
In the next Lemma we write ψ+ , defined by (1.3), in terms of u+ . Lemma 5.2 Let q ∈ B. Let ψ+ be the solution of Eq. (1.2), satisfying (1.3); let u+ 1 be the solution of Eq. (2.30), given by Corollary 4.2. Then for |λ| 6 c0 (qB + 1), where c0 1 is some absolute constant, the following identity holds: ψ+ (x, λ) = φ+ (x, λ),
where
φ+ (x, λ) = φ(λ)
u+ (z(x, λ)), λ) , z (x, λ)
x 0. (5.15)
Proof. The function φ+ , given by (5.15), solves Eq.(1.2) by changing variables according to (2.22). In order to prove (5.15) it is sufficient to demonstrate that φ+ has the asymptotics (1.3). Using (2.17), (2.22), (2.27), Corollary 4.2 and (5.2), we have for x → ∞ φ+ (x, λ) = e−
x2 2
√ λ−1 1 ( 2x) 2 (1 + O(x− 3 )), 2
x λ+1 1 e− 2 √ ∂φ+ (x, λ) = − √ ( 2x) 2 (1 + O(x− 3 ), ∂x 2
which proves (5.15).
In order to obtain the asymptotics of the Wronskian w(λ) = {ψ− , ψ+ } we need also the asymptotics of the fundamental solution ψ− (see Theorem 1.1). We use our results for x > 0 and consider the reflected potential q− (x) = q(−x) for x ∈ R+ . We define √ 1 2 Vq− (z) = λ− 3 ρ2 (z, λ)ˆ q− (z, λ), qˆ− (z, λ) = q− ( λt(zλ− 3 )), where ρ is given by (2.25). Let u− (z, λ) = u+ (z, λ; Vq− ), where u+ is given by Corollary 4.2 (with Vq replaced by Vq− ). By Lemma 5.2, the fundamental solution
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ψ− is related to u− by ψ− (−x, λ) = φ(λ)
u− (z(x, λ), λ) , z (x, λ)
x 0.
Introduce E ≡ E− + E+ , where E− is given by E− (λ) = 2 Ai(sω)Ai(sω)Vq− (s) ds, Γ− λ
ω=e
(5.16)
2πi 3
.
(5.17)
By Lemma 5.1, for | arg λ| δ the solution u− (z0 , λ) has the asymptotics (5.5); for | arg λ| δ it has the asymptotics (5.3), (5.4) with E+ replaced by E− (let ν = π4 (λ + 1)): − 14
u− (z0 , λ) = z0
* + 1 π 1 sin ν 1 + O(λ− 6 ) + E− (λ) cos ν + O λ− 2 e 4 | Im λ| , (5.18)
+ 1 π * 1 1 u− (z0 , λ) = z04 − cos ν 1 + O(λ− 6 ) + E− (λ) sin ν + O λ− 6 e 4 | Im λ| . (5.19) Lemma 5.3 Let q ∈ B. Then for |λ| → ∞ the following uniform asymptotics hold: + 1 π * π π | arg λ| δ, (5.20) w(λ) = φ2 (λ) cos λ − E(λ) sin λ + O λ− 3 e 2 | Im λ| , 2 2 w(λ) =
+ 4 + O(λ− 16 ) 1 φ2 (λ) ∓i π λ * e 2 1 + O(λ− 6 ) = , 2 φ2 (−λ)
δ ± arg λ π. (5.21)
Proof. Using (2.22), (5.15) and (5.16) we obtain w(λ) = −φ (λ) u+ (z0 , λ)u− (z0 , λ) + u+ (z0 , λ)u− (z0 , λ) 2
− φ2 (λ)
u+ (z0 , λ)u− (z0 , λ) . 16λk 2 (0)
Substituting (5.3–5.4) and (5.18–5.19) in the last identity, we obtain (5.20); substituting (5.5) and the same formulae for u− , we obtain (5.21).
6 Proof of Theorem 1.2 Below we need the following asymptotics of the unperturbed Wronskian w0 from [4]
πλ · 1 + O λ−1 , 2
w0 (λ) = φ−2 (−λ) 4 + O λ−1 , w0 (λ) = φ2 (λ) cos
|arg λ| δ,
(6.1)
|arg λ| δ.
(6.2)
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Lemma 6.1 Let q ∈ B. Then there is N0 ∈ Z such that for eigenvalues. each integer N > N0 the Wronskian w(·) has exactly N zeroes in the disc {z : |z| < 2N } (counting multiplicity) and for each n > N , exactly one simple zero in the disc 1 {z : |z − µ0n | < n− 6 }. There are no other zeroes. Remark. If q is real, all zeroes of w(cot) are simple. Each simple zero corresponds to a simple eigenvalue. Thus, for real q, the multiplicity of eigenvalues and zeroes coincides. For complex q it might be different. We recall that we always label the eigenvalues according to the multiplicity of the Wronskian. Proof. Consider the contours |λ| = 2n, |λ − µ0n | = δn , n > N , n ∈ N, where 1 π 1 πλ δ π 2 |Im λ| for |λ| = 2n and 2 |Im λ| δn = n− 6 < logπ 2 . Then | cos πλ 2 | 4e π cos 2 4 e for |λ − µ0n | = δ, δ < logπ 2 . By the asymptotics (5.20), (5.21), (6.2) and (6.1), there 6 exists an integer N0 > logπ 2 such that for any integer N > N0 on these contours |w(λ) − w0 (λ)| 12 |w0 (λ)|. It follows that w(λ) does not vanish on these contours. Hence, by Rouche’s theorem, w(λ) has as many roots, counted with multiplicities, as w0 (λ) in each of the bounded regions and in the remaining unbounded region. Since w0 (λ) has only the simple roots {µ0n }∞ n=0 , the Lemma is proved. Lemma 6.2 Let q ∈ B. Then for |λ| → ∞ the following asymptotics are fulfilled: 1 E(λ + ε) − E(λ) = O λ− 3 , E(λ) = −
1 2
1
−1
1 ε = O λ− 6 ,
√ 1 q(t λ)dt √ +O qB λ− 3 , 1 − t2
δ δ arg λ , (6.3) 2 2 1 E(λ) = O qB λ− 4 , λ > 0. −
(6.4)
Proof. Consider the case 0 arg λ δ. The proof for −δ arg λ 0 is similar. x∗ , where x∗ is defined by z∗ = Consider E+ , given by (5.1). Recall that t∗ = √ λ z(x∗ , λ) (see Lemma 2.1 and below). We have λ = |λ|e2iϑ . For some c ∈ [0, 1] we set t1 = t∗ − c 2 e−iϑ , x1 = √t1λ and z1 = z(x1 , λ). The length of Γλ (z1 , z∗ ) is |λ| 3 2 |Γλ (z1 , z∗ )| = |λ| 3 [t1 ,t∗ ] |k (t)| |dt|. By Lemma 7.3.4, |t∗ | is bounded uniformly in λ. Thus using (2.17) we conclude that |Γλ (z1 , z∗ )| C. Therefore using (2.25), (2.26), (2.34), (3.6) and (4.9) gives q∞ Ai(sω)Ai(sω)Vq (s) ds C (6.5) 1 . Γλ (z1 ,z∗ ) |λ| 3 Next, using the asymptotics (2.27) and Lemma 7.5.1, we have uniformly in | arg λ| δ C π Ai(zω)Ai(zω) + i 1 (6.6) z ∈ Γ− 1 3 , λ ⊂ S[π, − ]. 3 4z 2 |z| 2 (1 + |z|) 2
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Substituting (6.5) and (6.6) in (5.1), we have
E+ (λ) = 2
Ai(sω)Ai(sω)Vq (s) ds + O
1
Γλ (z0 ,z1 )
i =− 2 2
q∞ |λ| 3
ds Vq (s) √ + O s Γλ (z0 ,z1 )
q∞ 1
|λ| 3
. (6.7)
2
Making the substitution s = k(t)λ 3 , ds = λ 3 k (t) dt and using (2.25) we obtain √ q∞ q( λt) i E+ (λ) = − dt + O . (6.8) 1 2 e−iϑ [0,t1 ] k(t)k (t) |λ| 3 √ √ √ Using (2.13) and (2.15) we have k (t) k(t) = t2 − 1 = i 1 − t2 , where 1 − t2 > 0 for −1 < t2 < 1. Hence 1 q∞ 1 d q(t|λ| 2 ) c √ E+ (λ) = − dt + O , d=1− 1 2 . 2iϑ 2 2 0 3 e −t |λ| |λ| 3 Using similar arguments for E− , we obtain E(λ) = Ec (λ) + O
qB
,
1
λ3
1 Ec (λ) = − 2
d
−d
q(t |λ|)dt √ . e2iϑ − t2
(6.9)
In order to prove (6.3) we set c > 0. We estimate the partial derivatives of Ec (λ) with respect to real and complex parts of λ = µ + iν = |λ|e2iϑ . This gives ∂ ∂ Ec (λ)| C q 2B , | ∂ν Ec (λ)| C q 2B . Therefore as λ → ∞ | ∂µ |λ| 3
|Ec (λ + ε) − Ec (λ)| |ε| ·
|λ| 3
5
sup |∇Ec (λ )| CqB |λ|− 6 ,
|λ−λ |ε
1 ε = O |λ|− 6 ,
which together with (6.9) proves (6.3). Setting c = 0 and λ > 0 in (6.9) proves the first asymptotics in (6.4). In order to prove the second one we use the decomposition E0 (λ) = E1 (λ) + I2 (λ), λ > 0, where √ √ q(t λ) + q(−t λ) 1 1 √ dt. I2 (λ) = − 2 1−λ− 21 1 − t2 λ 12 −1 1 1 q(x) dx = q1 (λ 2 − 1) − q1 (λ 2 t) we integrate the expression for E1 Using 12 λ t
by parts. This gives |E1 (λ)| C q1 1 ∞ . A straightforward estimate gives |I2 (λ)| q ∞ 1 2λ 4
λ4
. Combining these estimates proves the second relation in (6.4). 1
Proof of Theorem 1.2. By Lemma 6.1, in each disc Dn = {λ : |λ−µ0n | n− 6 } there exists exactly one simple eigenvalue µn for n sufficiently large; now we improve
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this estimate. Using the asymptotics (5.20) and |sin( π2 λ)| 12 for λ ∈ Dn , we obtain 1 πµn − E(µn ) + O(n− 3 ) = 0, cot n → ∞. (6.10) 2 1
Asymptotics (6.3) yields E(µn ) − E(µ0n ) = O(n− 3 ) and the estimate |µm − µ0n | 1 1 π(µ −µ0 ) n− 6 implies cot πµ2 n = − n2 n + O(n− 2 ). Substituting the last asymptotics into (6.10) we have 1 2 µn − µ0n = − E(µ0n ) + O(n− 3 ), π
n → ∞.
(6.11)
Substituting (6.4) into (6.11) and using the change of variable t = sin θ we have the first asymptotics in (1.5). Moreover, due to (6.4) we get the second asymptotics in (1.5). ixt Proof of Proposition 1.3. Substituting q(x) = R e dν(t) into (6.9) and using the π iz sin φ 1 identity J0 (z) = 2π e dφ (see [1]) for the Bessel function J0 we have −π π √ √ 1 1 µn = q( λ sin ϑ) dϑ = J0 (t λ) dν(t) = I1 + I2 , λ > 0, (6.12) 2π −π R where
I1 =
|t|<ε
√ J0 (t λ) dν(t),
I2 =
|t|>ε
√ 3 J0 (t λ) dν(t), ε = λ− 4p , ,
Using J0 (−z) = J0 (z) and the asymptotics J0 (z) = | arg z|
2 πz
1 3 < . 4p 2 (6.13)
cos(z − π4 ) + O
e|Im z| 3
z2
,
π 2,
(see [1]) we obtain √ O(1) 2 dν(t) 1 π cos dν(t) + 3 λ|t| − , I2 = 1 3 π|t| 4 4 4 λ |t|>ε λ |t|>ε t 2
(6.14)
3
where the last term is O(λ− 4 ). Next, p |I1 | C dν(t) Cε |t|<ε
|t|<ε
3 dν(t) = Cγεp = O(λ− 4 ). |t|p
(6.15)
Similarly we have √ 1 1 2 dν(t) π Cεp− 2 = O(λ− 2 ). (6.16) cos |t| λ − dν(t) C |t|<ε πt 4 |t| |t|<ε √ 3 σ( µ0n ) Using (6.12–6.16) and setting λ = µ0n gives µ1n = + O(n− 4 ) which implies 1 0)4 (µ n (1.6). Moreover, substituting dν(t) = k∈Z δ(t − tk )qk dt into (6.12) we obtain (1.7).
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7 Appendix For fixed arg λ = 2ϑ we have t = √xλ ∈ e−iϑ R+ . We rewrite t ∈ S[− π2 , 0] and ξ in the form ϕ ∈ [0, π], η 0, (A.1) t = re−iϑ = 1 + ηe−iϕ , η 3ϕ 1 π ξ(t) = e−i 2 (A.2) 2 + se−iϕ · s 2 ds, t ∈ S[− , 0]. 2 0 Lemma 7.1 Let t ∈ S[− π2 , 0], R(t) = |ξ(t)|, Φ(t) = arg ξ(t). Then 1. t = re−iϑ = 1 + ηe−iϕ , where r, η 0, ϑ ∈ [0, π2 ], ϕ ∈ [0, π], . ϑ / 2. Φ(t) + 3ϕ 2 ∈ −2,0 , 3
3
3. if η 1, then 23 |t − 1| 2 R(t) 2|t − 1| 2 , 3 if η 0, then 23 sin 2 ϑ R(t), 0 4. if ϑ ∈ (0, π2 ], then Φ(t) = [− 3π 2 , −2ϑ), r∈R+
5. if −π − ϑ Φ(t), then arg e2iϑ ∂r ξ(t) ∈ [− π3 + ϑ6 , ϑ2 ]. Proof. 1. Straightforward. 2. Consider the integrand in formula (A.2). For s ∈ [0, η] we have √ η√ −iϕ · s 12 ds ∈ [− ϑ ,0]. arg 2 + se−iϕ ∈ [− ϑ2 ,0], hence Φ(t) + 3ϕ = arg 2 + se 2 0 √2 −iϕ 3. Consider the integrand in (A.2). s ∈ [0, η] we have For 1 < Re 2 + se √ √ 1 3 η 2 and | 2 + se−iϕ | < 3. Using R(t) = 0 2 + se−iϕ · s 2 ds gives 3 η 2 R(t) < 3
2η 2 , η = |t − 1|. The relation
min |t − 1| = sin ϑ for ϑ ∈ [0, π2 ] finishes the proof. 0 Φ(t) 4. Fix ϑ ∈ (0, π2 ]. By direct calculation, ξ(S[− π2 , 0]) ⊂ S[− 3π 2 , 0], so r∈R+
2iϑ ξ(t) ⊂ S[− 3π 2 , 0]. Using (2.14) we have arg e 0 → 0 as r → ∞. Also we have ξ(0) = i π4 . Since ξ(t) is continuous, this gives r∈R+ Φ(t) ⊃ [− 3π 2 , −2ϑ). Thus we
2iϑ need only prove that arg e ξ(t) < 0. Consider e2iϑ ξ(t), t = re−iϑ , as
a function of the real parameter r ∈ R. Note that by Lemma 7.2.2, Im e2iϑ ξ(t) strictly decreases in r. Since ξ(S[− π2 , 0]) ⊂ S[− 23 π, 0] and ξ(0) = i π4 , as r increases e 2iϑ ξ(t) only the negative half of the
hits π 2iϑ imaginary axis. Therefore, as soon as arg e ξ(t) > − , we have Im e2iϑ ξ(t) < 2
2iϑ 0. Hence arg e ξ(t) < 0, which finishes the proof. , √ , ϕ −iϕ −iϕ 5. By (A.2), we have e2iϑ ∂r ξ(t) = e−i 2 eiϑ 2η 1 + ηe 2 , where 1 + ηe 2 arg t=−ϑ
∈ S[− ϑ2 , 0]. Therefore
ϕ ϑ ϕ arg e2iϑ ∂r ξ(t) ∈ [− + , − + ϑ]. 2 2 2
(A.3)
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Next, by hypothesis, Φ ∈ [−π − θ, −2ϑ]. Thus, using 2., we obtain −ϕ ∈ [− 32 (π + ϑ), −ϑ]. Substituting this into (A.3) proves 5. Lemma 7.2 Let t = re−iϑ ∈ S[− π2 , 0], R(t) = |ξ(t)|, Φ(t) = arg ξ(t). Then 1. arg ∂r ξ(t) ∈ [− π2 − ϑ, −2ϑ),
2. if ϑ ∈ (0, π2 ] and r 0 then Im e2iϑ ∂r ξ(t) < 0 and Re e2iϑ ∂r ξ(t) > 0,
if ϑ = 0 and r ∈ [0, 1), then Im e2iϑ ∂r ξ(t) < 0 and Re ξ(t) = 0,
if ϑ = 0 and r ∈ (1, ∞), then Im ξ(t) = 0 and Re e2iϑ ∂r ξ(t) > 0, 3. if Φ(t) ∈ (−π − 2ϑ, − π2 − ϑ), then ∂r Φ(t) > 0, 4. if Φ(t) ∈ (− 3π 2 − 2ϑ, −π − ϑ), then ∂r R(t) < 0, 5. if Φ(t) ∈ (− π2 − 2ϑ, −ϑ), then ∂r R(t) > 0. Proof. √ −iϑ −iϑ 1. By direct calculation, t2 − 1. We have (t2 −1) ∈ ∂r ξ(t) = e ξ π(t) = e 2 −2iϑ − 1 ∈ [− 2 , −ϑ). S[−π, −2ϑ), so that arg |t| e 2. For ϑ ∈ (0, π2 ] the result follows from 1. For ϑ = 0, the result follows from (2.13) by direct calculation. (t)| 3. We have ∂r Φ(t) = |ξ|ξ(t)| sin {arg ∂r ξ(t) − Φ(t)}. By 1, ∂r Φ(t) is strictly positive for Φ(t) ∈ (−π − 2ϑ, − π2 − ϑ). 4. and 5. We have ∂r R(t)2 = 2|ξ (t)||ξ(t)| cos {arg ∂r ξ(t) − Φ(t)}. By 1, ∂r R(t) is positive for Φ(t) ∈ (− π2 − 2ϑ, −ϑ) and negative for Φ(t) ∈ (− 3π 2 − 2ϑ, −π − ϑ). For λ = |λ|e2iϑ ∈ C+ \ {0} we have 3 2 z(x, λ) 2 = λξ(t), 3
x π t = √ = re−iϑ ∈ S[− , 0]. 2 λ
(A.4)
Lemma 7.3 For each λ = |λ|e2iϑ ∈ S[0, δ) \ {0} there exists a unique z∗ ∈ Γλ such that |z∗ | = minz∈Γλ |z|. Moreover, the following relations hold (x∗ is defined by z∗ = z(x∗ , λ)): √ 1. if arg λ = 0, then z∗ = 0 and x∗ = λ, if arg λ ∈ (0, δ), then z∗ ∈ S[− π2 − ϑ2 , − π2 + 56 ϑ], 2. |z(·, λ)| is strictly decreasing on [0, x∗ ) and strictly increasing on (x∗ , ∞),
3. if t = √xλ for x ∈ [0, x∗ ), then arg e2iϑ ∂r ξ(t) ∈ [− π2 , − π4 + 34 ϑ], 4. if t∗ =
x∗ √ , λ
then |t∗ |
1 . sin( π 3)
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Proof. We show uniqueness of z∗ . By (A.4), it is sufficient to show that for each λ there exists a unique solution t∗ of ∂r |ξ(t)|√= 0 for t = re−iϑ , r ∈ R+ . For arg λ = 0 direct calculation gives z∗ = 0 and x∗ = λ. Next we show the existence of t∗ in the case λ = |λ|e2iϑ ∈ S(0, δ) \ {0}. Since ξ(t) = 0, we have 2∂r |ξ(t)| = |ξ(t)|−1 Re ξ(t) ∂ξ(t) . Let us use the ∂r
representation (A.1): t = re−iϑ = 1 + ηe−iϕ . For any t ∈ S[− π2 , 0) Lemma 7.1.2 3ϕ ϑ ∂ implies arg ξ(t) ∈ [− 3ϕ 2 − 2 , − 2 ]; similarly Lemma 7.2.1 implies arg ∂r ξ(t) ∈ ϕ ϕ ϑ 3ϑ 2 [− 2 − 3ϑ 2 , − 2 − ϑ]. Thus arg ∂r |ξ(t)| ∈ [−ϕ + 2 , −ϕ + 2 ] and we have
π ϑ π 3ϑ + ,− + for ϕ ∈ , 2 2 2 2 3π ϑ π + , +ϑ . for ϕ ∈ 2 2 2
∂r |ξ(t)| > 0 ∂r |ξ(t)| < 0
(A.5) (A.6)
By (A.5) and (A.6), there exists at least one solution t∗ of ∂r |ξ(t)| = 0. Moreover, any solution satisfies ) 1 π 3ϑ π ϑ + , + ϕ∈ . (A.7) 2 2 2 2 Now we show uniqueness of t∗ . We only need to verify that for any t satisfying 2 ∂ξ(t) 2 ∂2 2 > 0. This is true if ξ(t) · ∂ ∂rξ(t) (A.7) we have 12 ∂r 2 |ξ(t)| = ∂r + Re 2 2 |∂r ξ(t)| > |ξ(t)| ∂r2 ξ(t) . ∂ξ(t) ∂r
We have to
√ = e−iϑ t2 − 1,
∂ 2 ξ(t) ∂r 2
(A.8)
= e−3iϑ √t2r−1 . Therefore (A.8) is equivalent 3
|(t2 − 1) 2 /t| > |ξ(t)|.
(A.9)
Due to (A.7), |t| 1. Therefore we have for t = 1 + ηe−iϕ 3
3
3
|t − 1| 2 |t + 1| 2 |t2 − 1| 2 3 3 = η 2 |2 + ηe−iϕ | 2 , |t| 1 Thus using 2ϑ ∈ (0, δ) ⊂ [0, 43 arccos 3
1
1
23
|t2 − 1| 2 3 η2 |t|
|2 + ηe−iϕ | 2 cos
3ϑ . (A.10) 2
) we obtain
3 3 3ϑ 2 > 2η 2 . 2 cos 2
(A.11)
Since ϑ ∈ [0, δ) ⊂ [0, π3 ], (A.7) gives η 1. Thus we apply Lemma 7.1.3, which 3 gives 2η 2 |ξ(t)|. Substituting the last estimate into (A.11) yields (A.9) and (A.8). Therefore √ there exists a unique solution t∗ of ∂r |ξ(t)| = 0. Thus z∗ = z(x∗ , λ) for x∗ = t∗ λ.
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1. and 2. Using (A.7) and Lemma 7.1.2 we obtain arg ξ(t∗ ) ∈ [− 3π 4 − − 3ϑ ]. By (A.4), this proves 1. Uniqueness of t and the relation (A.4) ∗ 4 prove 2. 3. By (A.5), for r < |t∗ | we have −π −ϕ − π2 − ϑ2 . Using Lemma 7.1.2 we obtain & ' 2iϑ
π 3ϑ π l −iϕ ϕ iϑ −i ϕ . − arg e ∂r ξ(t) = arg e e 2 1 − e ϑ− − + 2 2 2 4 4 11ϑ 3π 4 ,− 4
4. (A.1) yields r = we have
2π 3
−
ϑ 3
ϕ∗ −
sin ϕ sin(ϕ−ϑ) . By (A.7) and Lemma 7.1.2, for t∗ sin ϕ∗ ϑ π3 , so that r∗ = sin(ϕ sin1 π . ∗ −ϑ) 3
= 1 + η∗ e−iφ∗
Lemma 7.4 For each λ = |λ|e2iϑ ∈ S[δ, π] \ {0} there exists a unique z∗ ∈ Γλ such that arg z∗ = − π3 . Moreover, the following relations are valid (if x∗ is defined by z∗ = z(x∗ , λ)): 1. if x ∈ [0, x∗ ), then arg z(x, λ) < − π3 , and if x ∈ (x∗ , ∞), then arg z(x, λ) >π 3, 2. |z(·, λ)| is strictly increasing on (x∗ , ∞), 3. if t∗ =
x∗ √ , λ
then |t∗ |
1 sin π 3
uniformly in λ.
Proof. By Lemma 7.1.4 and (A.4), Γλ intersects the line {z : arg z = − π3 } at least once. By (A.4), the sector S[− π3 − ε, − π3 + ε] in the z-plane for small ε > 0 π 3ε corresponds to the sector S[− π2 − 2ϑ − 3ε 2 , − 2 − 2ϑ − 2 ] in the ξ-plane. By ∂ Lemma 7.2.3, in this sector ∂r Φ(t) > 0. Therefore, by (A.4), ∂x arg z(x, λ) > 0 for π π z ∈ S[− 3 − ε, − 3 + ε]. Hence z∗ is unique and 1 holds. 2. By (A.4), for x ∈ (x∗ , ∞) and t = √xλ the hypothesis of Lemma 7.2.5 is
∂ R(t) > 0 and |z(x, λ)| is strictly increasing in x. fulfilled. Therefore ∂r 3. By (A.4), we have arg ξ(t∗ ) = − π2 − 2ϑ. By Lemma 7.1.2, for t∗ = 1 + sin ϕ ϑ 2π η∗ e−iϕ∗ we have ϕ∗ − ϑ ∈ [ 2π 3 − 3 , 3 ]. Therefore using the relation r = sin(ϕ−ϑ) 1 we obtain |t∗ | sin π . 3
In the next Lemma we analyze the curves Γ± λ , defined in (2.38). Lemma 7.5 Let λ = |λ|e2iϑ ∈ C+ \ {0}. Then 2
2
3 1. if ϑ = 0, then Γλ = [−λ 3 ( 3π 8 ) , ∞), + π 4 π if 0 < ϑ 2 , then Γλ ⊂ S[−π + 43 ϑ, 0), Γ− λ ⊂ S[−π + 3 ϑ, − 3 ], Γλ ⊂ π ϑ S[− 2 − 2 , 0), 2
2. inf z∈Γλ |z| |λ| 3 sin ϑ, 2
2
− 3 3 3. Γ− λ ⊂ {z : |z| C|λ| }, and the length |Γλ | C|λ| ,
4. the function |z(·, λ)| is strictly increasing on [x∗ , ∞).
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Proof. 1. Direct calculation yields the result for ϑ = 0. For 0 < ϑ π2 the 2 2 assertion on Γλ follows from Lemma 7.1.4 and z = λ 3 ( 32 ξ) 3 . The assertion on Γ± λ for 0 arg λ δ follows from 1 of the present Lemma and Lemma 7.3.1. For δ arg λ π it follows from Lemma 7.4.1. 2 2 2. Follows from Lemma 7.1.3 and z = λ 3 ( 32 ξ) 3 . 3. By Lemma 7.3.4 and Lemma 7.4.3, |t∗ | sin1 π uniformly in λ ∈ C+ \ {0}. 3 Using definition (2.15) we have that k(t) and k (t) for t ∈ [0, t∗ ] are also uniformly 2 2 3 3 bounded. Now the relations |Γ− λ | = |λ| [0,t∗ ] |k (t)||dt| and |z(x, λ)| = |λ| |k(t)|, x t = √λ yield the proof. ∂ |z(x, λ)| > 0 if ∂r |ξ(t)| > 0 for t = re−iϑ = √xλ . Thus the 4. (A.4) gives ∂x result follows from Lemma 7.4.2 (for δ arg λ π) and from Lemma 7.3.2 (for 0 arg λ δ).
Proof of Lemma 2.1. Case 1 (or 2) follows from Lemma 7.3 (or Lemma 7.4). 2iϑ
Re λξ(t)
Proof of Lemma 2.2. Fix λ = |λ|e . By (A.4), we have h(x) = e for t = √xλ = re−iϑ . By Lemma 7.2.2, if either 0 < arg λ π, r 0 or arg λ = 0, r > 1, then ∂r Re (λξ(t)) > 0. It remains to consider the case ϑ = 0 and x ∈ [0, x∗ ); by Lemma 7.2.2, Re ξ(t) = 0 and therefore h = 1. In order to prove Lemma 2.3, for fixed λ = |λ|e2iϑ ∈ C \ {0} define the 3
function Ψ(z) = 23 z|λ|2 . Ψ maps Γλ (z1 , z2 ) onto γλ (u1 , u2 ), uj = Ψ(zj ), j = 1, 2. Similarly, we set γλ (u) = Ψ(Γλ (z)) for u = Ψ(z) and γλ± = Ψ(Γ± λ ). We also set u0 = Ψ(z0 ) and u∗ = Ψ(z∗ ). Consider the integral γλ (u) f (v)|dv|. By (A.4), we have the parametrization 2iϑ −iϑ γλ (u) = v ∈ C : v = v(r) = e ξ(re ), r > ru , where u = Ψ(z(ru |λ|, λ)) . (A.12) Consider the image of the part of Γλ , defined by the hypothesis a) and b) of Lemma 2.3, under the mapping Ψ. Assumption a) transforms into a ) δ arg λ π and u ∈ γλ . Similarly, b) becomes b ) 0 arg λ δ, u ∈ γλ+ . Then by Lemma 7.2.2, Re v(r) is strictly increasing in r. Thus we choose the variable κ = Re v and obtain ∞ |f (v)||dv| = |f (κ + i Im v(κ))| |∂κ v| dκ, |∂κ v| = 1 + (∂κ Re v(κ))2 . γλ (u)
Re u
Let us show that in both cases a ) and b ) the following estimate holds: ∞ |f (v)||dv| C |f (κ + i Im v(κ))| dκ. γλ (u)
Re u
(A.13)
dv . In terms of the parametrization v(r) = e2iϑ ξ(re−iϑ ) we have Let us estimate dκ
∂r Im e2iϑ ξ(t) d Im v(κ) = = tan arg ∂r e2iϑ ξ(t). dκ ∂r Re (e2iϑ ξ(t))
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Consider the two cases a ) and b ). a ) Let δ arg λ π and u ∈ γλ . For each point v(r) ∈ γλ consider t = re−iϑ π such that v(r) = e2iϑ ξ(t). By Lemma 7.2.1, arg ∂r ξ(t) ∈ [− 2 − ϑ, −2ϑ). Hence arg ∂r e2iϑ ξ(t) ∈ [− π2 + δ2 , 0) and d Imdκv(κ) sin(1 δ ) uniformly for δ arg λ π. 2 dv Therefore dκ is also uniformly bounded. + b ) Let 0 arg λ δ and u ∈ γλ+ = Ψ(Γ+ λ ). For each point v(r) ∈ γλ π −iϑ 2iϑ consider t = re such that v(r) = e ξ(t). Since 0 ϑ 7 , arg ξ(t∗ ) ∈ 3π 11ϑ 3π 3ϑ [− 4 − 4 , − 4 − 4 ], (equivalent to Lemma 7.3.1) implies that −π − ϑ arg ξ(t).
Therefore 7.1.5, for v ∈ γλ+ we have arg e2iϑ ∂r ξ(t) ∈ [− π3 + ϑ6 , ϑ2 ]. by Lemma dv is uniformly bounded. Hence d Imdκv(κ) sin(1 π ) and dκ 3 dv Thus dκ is uniformly bounded in both cases a ) and b ), implying (A.13). Proof of Lemma 2.3. Firstly we prove (2.39) for the case δ arg λ π. Lemma 7.5.2 gives dist(Γλ , {0}) >const uniformly in λ. Thus we replace · by | · |. The change of variables v = Ψ(s) in (2.39) results in the equivalent relation 3 e−2|λ| Re v C e−2|λ| Re u 2 z2 ∈ γλ . |dv| , u = (A.14) 2 (α+ 12 ) |λ| |u| 23 (α+ 12 ) 3 |λ| γλ (u) |v| 3 By (A.13), we have the auxiliary estimate e−2|λ| Re u e−2|λ| Re v |dv| C , |λ| γλ (u)
u ∈ γλ .
(A.15)
We show that (A.14) follows from (A.15). If u ∈ γλ+ , then Lemma 7.5.4 yields |u| = min |v|. Therefore (A.15) gives (A.14). If u ∈ γλ− , then dist(γλ , {0}) >const and
v∈γλ (u)
we have γλ (u)
e−2|λ| Re v 2
1
|v| 3 (α+ 2 )
|dv| C
γλ (u)
e−2|λ| Re v |dv|,
u ∈ γλ− .
(A.16)
By Lemma 7.5.3, γλ− (u) is uniformly bounded. Therefore, |u| is bounded on γλ− , so (A.16) and (A.15) imply (A.14). + b) We prove (2.39) for the case z ∈ Γ+ λ . It suffices to show that for any z ∈ Γλ we have 3 3 4 4 3 3 |e− 3 s 2 | |e− 3 z 2 | − 43 s 2 − 43 z 2 |e | |ds| C|e |, |z| ≤ 1; |ds| C 1 , |z| > 1. |s|α |z|α+ 2 Γλ (z) Γλ (z) (A.17) By the change of variable u = Ψ(z) the first estimate in (A.17) follows from (A.15). For the proof of the second estimate in (A.17) we observe that by Lemma 7.5.4, for z ∈ Γ+ min |v|. Hence, (A.17) follows from λ we have |z| = v∈Γλ (z)
(A.15). This proves (2.39).
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We shall prove (2.40). The change of variable v = Ψ(s) = 23 s|λ|2 (see (A.12)) yields − 23 3 3λ |ds| |dv| 2 (α−1) 2 z2 ε . = I, I = , ε = < 1, u = 1 2 2 α α 3 3 |λ| Γλ (z) s γλ (u) |v| 3 (ε + |v| 3 ) (A.18) Assume z ∈ Γ+ , 0 < arg λ π and let u = a + ib, a, b ∈ R. Using Lemma 7.3.1, λ Lemma 7.3.3, Lemma 7.1.5 and definition of z∗ , we have b 0. By Lemma 7.2.2, Im v is strictly decreasing on γλ+ , so that |b| = min | Im v|. Thus using (A.13)
we obtain I CJ,
v∈γλ (u)
∞
J= a
dκ 1 3
2
2
|κ| (ε + |b| 3 + |κ| 3 )α
. 2
Consider two cases: a 0 and a < 0. Firstly, let a 0. Then using |u| 3 = ε|z| we have 3 1 3 ∞ dx 2 J= = 2 a 23 (ε + |b| 32 + x)α α − 1 (ε + |b| 32 + a 23 )α−1 C 1−α = Cε1−α z 2 (ε + |u| 3 )α−1 which together with (A.18) proves (2.40). Secondly, let a < 0. Due to Lemma 7.3.1 and Lemma 7.4.1, for u = a + ib ∈ γλ+ and a < 0 we have |a| C|b| uniformly in 2 2 0 arg λ π. Therefore |b| C|u|, and using |u| 3 = ε|z| and x = κ 3 , we obtain ∞ dκ 3 C 1−α = = Cε1−α z . J 2 1 2 2 2 2 α α−1 α−1 3 3 3 3 3 κ (ε + |b| + κ ) (ε + |b| ) (ε + |u| ) 0 (A.19) which together with (A.18) proves (2.40). 3 2
2 z∗ Assume z ∈ Γ− λ and δ arg λ π. Recall that u∗ = 3 |λ| . We have I = I− + I+ , where 3 2 z2 |dv| |dv| I− = , I = , u = . + 1 2 1 2 α + |v| 3 (ε + |v| 3 )α 3 |λ| γλ (u,u∗ ) |v| 3 (ε + |v| 3 ) γλ
3 By Lemma 7.5.2, we have |z| sin δ2 for z ∈ Γλ , so |u| 23 sin δ2 2 for u ∈ γλ . By Lemma 7.5.3, the length of the curve |γλ− | < C. Therefore I− C. By (A.19) for u = u∗ , we have I+ C so that I C. Next, by Lemma 7.5.3, γλ− is uniformly 2 bounded, implying C (ε + |u| 3 )1−α and I |λ|
2 3 (α−1)
C |λ|
2 3 (α−1)
(ε + |u|
which together with (A.18) proves (2.40).
2 3
)α−1
=
C , z α−1
(A.20)
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Proof of Lemma 2.4. First we prove (2.41). The change of variable u = Ψ(z) yields |ds| |dv| 2 2 α−1 3 ε = I, I = , ε = ( |λ|)− 3 C, (A.21) 1 2 α − s α − 3 2 Γλ γλ |v| 3 (ε + |v| 3 ) where u, v is defined before (A.13). By Lemma 7.2.2, Im v is strictly decreasing on γλ− . Thus we parameterize the last integral by χ = Im v, so that v(χ) = Re v(χ) + iχ. Lemma 7.5.3 gives γλ− ⊂ {u ∈ C : |u| < c} for some c > 0 independent of λ. Therefore c , dχ I2 |∂χ v| 1 , |∂ v| = 1 + (∂χ Re v(χ))2 (A.22) χ 2 χ 3 {ε + χ 3 }α 0 Recall λ = |λ|e2iϑ and t = re−iϑ . In order to estimate d Redχv(χ) we use the parametrization (A.12). Thus we have
∂r Re e2iϑ ξ(t) d Re v(χ) = = cot arg e2iϑ ∂r ξ(t) . dχ ∂r Im (e2iϑ ξ(t)) By Lemma 7.3.3, Lemma 7.1.5 Lemma 7.4.1, and (2.37) , e2iϑ ∂r ξ(t) is in a sector dv d Re v(χ) isolated from the real axis uniformly in λ. Therefore dχ C and dχ C. 2
Substituting this estimate in (A.21) and making the change of variables x = ε−1 χ 3 we obtain C1 /ε 2 dx C 3 I α−1 , ε = ( |λ|)− 3 < 1, (A.23) α ε (1 + x) 2 0 which together with (A.21) yields (2.41). Now we prove (2.42). By the change of variable u = Ψ(z), we have |ds| |dv| I− + I+ = , I± = 4 2 1 4 . 3 3 2 ± 3 3 3 |λ| Γλ |λ| + |s| γλ ( 2 |v|) (1 + ( 2 |v|) 3 ) For I+ we use (A.13), which gives I+ C estimate I− we use the parametrization of
γλ−
∞ 0
1
dκ
4
κ 3 (1+κ 3 )
(A.24)
C. In order to
by χ = Im v. Repeating the ar dv guments used above for the proof of (2.41), we conclude that dχ C and ∞ dχ C. The estimates of I± < C and (A.24) imply I− C 0 1 4 χ 3 (1+χ 3 )
(2.42).
Lemma 7.6 Let λ = |λ|e2iϑ ∈ S[0, δ] and q, q ∈ L∞ (R). Let P± be defined by (4.27). Then for some absolute constant C the following estimates are fulfilled: 3 4 1 |P− (z, z∗ )| C|e− 3 z 2 | q ∞ + q∞ |λ|− 6 ,
z ∈ Γ− λ,
(A.25)
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3 4 2 1 |P+ (z1 , z2 )| C|e 3 z2 | q ∞ + q∞ |λ|− 6 ,
x1 x2 x∗ , z1,2 = z(x1,2 , λ) ∈ Γ− λ . (A.26) Proof. Let ξ1,2 = ξ(t1,2 ) and t1,2 = Lemma 2.2 gives |e
2λ(ξ1 −ξ2 )
x1,2 √ . λ
Then, Lemma 7.3.2 yields |ξ1 − ξ2 | 2|ξ1 |.
− 1| 2|λ||ξ1 − ξ2 | and |e2λ(ξ1 −ξ2 ) − 1| C. Therefore
|(e2λ(ξ1 −ξ2 ) − 1)/λξ1 | C(1 + |λ||ξ1 |)−1 . 2
(A.27)
2
Using z1,2 = λ 3 ( 32 ξ1,2 ) 3 we obtain the following estimate uniformly in 0 arg λ δ: 3 3 4
2
|(e 3 (z1
−z22 )
−3
3 ± 43 s 2
Introduce the functions R± (s, z) = e 3 √ 4 ±2 se± 3 s 2 . Integrating by parts, we have P− (z∗ , z) = R− (z, z∗ )f (z) +
3
− 1)z1 2 | Cz1 2 .
Γλ (z,z∗ )
−e
3 ± 43 z 2
(A.28)
. Evidently ∂s R± (s, z) =
R− (s, z∗ )f (s) ds, √ 1 f (s) = ρ(s)ˆ q (s)/(2 sλ 6 ).
(A.29)
1
d qˆ(s)| Cq ∞ |λ|− 6 we obtain Therefore using (2.25), (2.26), (A.28) and | ds 3
4
|P− (z, z∗ )| C %
q∞
|e− 3 z 2 | 1
|λ| 6 +
& |ds|
1
|λ|− 6 q ∞
2
+
|λ|− 3 q∞
q∞
+
'(
3
, (A.30)
which together with (2.40–2.41) gives (A.25). Similarly we have P+ (z1 , z2 ) = −R+ (z1 , z2 )f (z1 ) − R+ (s, z2 )f (s)ds.
(A.31)
1
z 2
1
s 2
Γλ (z,z∗ )
1
s 2
s 2
Γλ (z1 ,z2 )
1
d Again using (2.25), (2.26), (A.28) and | ds qˆ(s)| Cq ∞ |λ|− 6 we have 3 2
4
|P+ (z1 , z2 )| C
|e 3 z2 | 1
|λ| 6
q∞ (1 + |z2 |)
1 2
+ Γλ (z1 ,z2 )
&
− 16
|λ|
q ∞
s
1 2
which together with (2.40), (2.41) gives (A.26).
+
− 23
|λ|
q∞
s
1 2
+
q∞ s
3 2
'
|ds| ,
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2πi
Lemma 7.7 Let q ∈ B. Define F (z) = a2 (ze± 3 )Vq (z). Then uniformly in z ∈ Γ− λ and λ ∈ S[−δ, δ] the following estimates hold for sufficiently large |λ|: 3 3 3 qB qB 4 4 2 2 −s 2 ) (z s 3 e3 F (s) ds C , e F (s) ds C 1 1 . Γλ (z) Γλ (z,z∗ ) |λ| 3 |λ| 3 (A.32) Proof. We show the first estimate in (A.32).Using (2.25), (2.26), (2.34) and (2.39) we obtain 3 3 4 q∞ (z 2 −s 2 ) 3 F (s) ds C (A.33) 1 . +e |λ| 3 Γλ
3 − 43 s 2
4
3 2
−e− 3 z∗ we integrate by parts the integral over Γλ (z, z∗ ). Using R− (s, z∗ ) = e We have 3 3 4 e 3 (z 2 −s 2 ) F (s)ds I= Γλ (z,z∗ )
3 ) 1 4 F (s) e3z2 F (z) = R− (s, z∗ ) √ ds . (A.34) R− (z, z∗ ) √ + 2 z s Γλ (z,z∗ ) 1
d Therefore, using Lemma 2.2, | ds qˆ(s)| C|λ|− 6 q ∞ , (2.25), (2.26), (2.34), (2.35) and (A.28), we obtain from (A.34) ) 1 1 2 q∞ |λ|− 3 q∞ C q∞ |λ|− 6 q ∞ + |I| + + |ds| . (A.35) 1 1 3 1 s s |λ| 3 z 2 s 2 + 2 Γ− λ
By Lemma 2.4, for sufficiently large |λ| this gives 1 1 |I| C|λ|− 3 qB 1 + |λ|− 6 log(|λ| + 1) . Together with (A.33) this proves the first estimate in (A.32). In order to prove 4
3
the second estimate in (A.32) we use similar arguments with R+ (s,z∗ ) = e 3 s 2 − 4
3 2
e 3 z∗ .
Acknowledgments. The authors would like to thank Vladimir Geiler for discussions on the physical models and for the ref. [8] and Horst Hohberger for the Figures 1 and 2.
References [1] M. Abramowitz and A. Stegun, eds. Handbook of Mathematical Functions. N.Y.: Dover Publications Inc. [2] M. Altarelli, G. Platero, Magnetic hole levels in quantum wells in parallel magnetic field, Surf. Sci. 196, 540–544 (1988).
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[3] T. Ando, A. Fauler, F. Stern, Electronic properties of two-dimensional systems, Rev. Modern Phys. 54, 437–672 (1982). [4] D. Chelkak, P. Kargaev, E. Korotyaev, An Inverse Problem for an Harmonic Oscillator Perturbed by Potential: Uniqueness, Lett. Math. Phys. 64 (1), 7–21 (2003). [5] D. Chelkak, P. Kargaev, E. Korotyaev, Inverse problem for harmonic oscillator perturbed by potential, characterization, Comm. Math. Phys. 249, 133–196 (2004). [6] M. De Dios Leyva, V. Galindo, Interband optical absorption in superlattices in an in-plane magnetic field, Phys. Rev. B. 48, 4518–4523 (1993). [7] V.A. Geyler, I.V. Chudaev, Schr¨ odinger operators with moving point perturbations and related solvable models of quantum mechanical systems, Z. Anal. Anwendungen 17, no. 1, 37–55 (1998). [8] C. Hooley, J. Quintanilla, Single-Atom Density of states of an opticallattice, Phys. Rev. Let. 93 no 8, 080404-1-4 (1998). [9] J. Maan, Magneto-optical properties of superlattices and quantum wells, Surf. Sci. 196, 518–532 (1988). [10] F. Olver, Asymptotics and special functions, Academic Press, New YorkLondon, 1974. [11] F. Olver, Two inequalities for parabolic cylinder functions, Proc. Cambridge Philos. Soc. 57, 811–822 (1961) . [12] J. P¨ oschel, E. Trubowitz, Inverse Spectral Theory, Boston: Academic Press, 1987.
Markus Klein Institut f¨ ur Mathematik Universit¨ at Potsdam Germany email:
[email protected] Alexis Pokrovski Institute for Physics St.Petersburg State University Russia email:
[email protected] Communicated by Bernard Helffer submitted 23/04/04, accepted 26/10/04
Evgeny Korotyaev Institut f¨ ur Mathematik Humboldt Universit¨ at zu Berlin Rudower Chaussee 25 D-12489, Berlin Germany email:
[email protected]
Ann. Henri Poincar´e 6 (2005) 791 – 799 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/04791-9, Published online 28.07.2005 DOI 10.1007/s00023-005-0223-y
Annales Henri Poincar´ e
On the Energy and Helicity Conservations for the 2-D Quasi-Geostrophic Equation Yong Zhou Abstract. In this paper, we establish sufficient conditions in Besov space for weak solutions of 2-D quasi-geostrophic equation to guarantee the conservations of energy and helicity. These two conservation laws are of great interest for both mathematical theory and applications in meteorology and oceanography.
1 Introduction A fundamental equation to describe the motion of inviscid Newtonian flow is the following 3-D Euler equations in Ω × (0, T ), (∂t + u · ∇)u + ∇P = f, divu = 0, in Ω × (0, T ) (1.1) in Ω, u(x, 0) = u0 (x), where u(x, t) ∈ R3 is the velocity field, P is pressure, while f is the external force. Onsager [14] conjectured that the energy of weak solutions to 3-D Euler equations is conserved as long as it belongs to a H¨ older space C γ with γ > 13 . This α conjecture was proved by Constantin, E and Titi [5] in the Besov space B3,∞ with 1 older space. Just as stated in [5], the significance of Onsager’s α > 3 instead of H¨ conjecture can be appreciated in the context of Kolmogorov theory of turbulence. However, so far there are many fundamental questions concerning 3-D Euler equations are not clear, cf. [2, 4, 11]. Another way to understand the problems is by constructing and studying models in lower dimensions. A 1-D model was studied by Constantin, Lax and Majda [6] for the 3-D Euler vorticity equation as ωt = H(ω)ω, x where H is the Hilbert transform and the velocity is defined by u = −∞ ω(y)dy. A 2-D model of the quasi-geostrophic equation was studied by Constantin, Majda and Tabak [7]. The 2-D quasi-geostrophic equation is as follows in Ω × (0, T ), (∂t + u · ∇)θ = 0, 1 ∂Ψ ∂Ψ ⊥ u = ∇ Ψ = − ∂x2 , ∂x1 , θ = −(−∆) 2 Ψ, in Ω × [0, T ), (1.2) in Ω, θ(x, 0) = θ0 (x),
792
Y. Zhou
Ann. Henri Poincar´e
where Ω ⊂ R2 , θ = θ(x, t) ∈ R1 , u(x, t) ∈ R2 is the velocity field, while Ψ(x, t) ∈ R1 is the stream function. The operator (−∆)γ (γ > 0) is defined by [17] γ f (ξ) = |ξ|2γ fˆ, (−∆) where fˆ denotes the Fourier transform of f . In [7] and [15], θ is interpreted as a potential temperature. Just as stated in [9], θ may be regarded as a vorticity (pointed out by R. de la Llave). This important model (1.2) has been intensively studied for its similarity to the 3-D Euler equations and applications in meteorology and oceanography very recently [7, 8, 9, 10, 13, 15]. The purpose of his paper is to find sufficient conditions to guarantee the conservation of energy (Onsager’s conjecture for 2-D quasi-geostrophic equation) and helicity for weak solutions to the quasi-geostrophic equation. To avoid problems involving boundaries, we assume Ω is the whole space R2 2 or Π = [0, 1]2 with periodic boundary conditions. The first main theorem reads α (Ω)) be a weak solution to the Theorem 1.1 Let θ ∈ C(0, T ; L2 (Ω)) ∩ L3 (0, T ; B3,∞ 2-D quasi-geostrophic equation, i.e.,
0
T
Ω
θ(x, t)∂t φ(x, t)dxdt +
Ω
θ0 (x)φ(x, 0)dx
T
+ 0
Ω
u(x, t)θ(x, t) · ∇φ(x, t)dxdt = 0,
for any test function φ(x, t) ∈ C ∞ (Ω × R1+ ) with compact support. If α > 13 , then the energy is conserved, i.e., E(t) = |θ(x, t)|2 dx = |θ0 (x)|2 dx, Ω
Ω
for all t ∈ [0, T ). If we assume f = 0 in (1.1), the vorticity equation for 3-D Euler equations reads (∂t + u · ∇)ω = ∇u · ω,
(1.3)
with ω = ∇ × u. By differentiating the equation, we obtain an equation similar to (1.3) with the gradient perpendicular to θ instead of the vorticity (∂t + u · ∇)∇⊥ θ = ∇u · ∇⊥ θ. For 3-D Euler equations, one of the most important conservation laws is helicity [1, 11, 12] defined by H(t) =
Ω
v(x, t) · ω(x, t)dx.
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Very recently, Chae [3] established a regularity condition ω ∈ L3 (0, T ; B α9 ,∞ ) with 5
α > 13 , to show the conservation of helicity for weak solutions to 3-D Euler equations. For the 2-D quasi-geostrophic equation, although θ is a scalar and ∇⊥ θ is a vector, we can define the ‘helicity’ as follows ¯ Hi (t) = θ(x, t)∂i θ(x, t)dx, Ω
for i = 1, 2, where ∂i denotes ∂xi . The second main result of this paper is concerning the conservation of this ‘helicity’. 4
Theorem 1.2 Let ∇θ ∈ C(0, T ; L 3 (Ω)) ∩ L3 (0, T ; B α3 ,∞ (Ω)) be a weak solution to 2
the 2-D quasi-geostrophic equation. If α > 13 , then the ‘helicity’ is conserved, i.e., H¯i (t) = θ(x, t)∂i θ(x, t)dx = θ0 (x)∂i θ0 (x)dx, Ω
Ω
for i = 1, 2, t ∈ [0, T ). Remark 1.1 For smooth solutions to 2-D quasi-geostrophic equation (1.2), the energy and helicity are conserved.
2 Proof of the main theorems For the completeness of this paper, we recall the definition of Besov spaces. We start by recalling the Littlewood-Paley decomposition of temperate distributions. Let S be the class of Schwartz class of rapidly decreasing functions. Given f ∈ S, the Fourier transform is defined by 1 ˆ F (f ) = f = e−ix·ξ f (x)dx. (2π)N/2 RN One can extend F and F −1 to S in the usual way, where S denotes the set of all tempered distributions. Let φ ∈ S satisfying
12 5 ˆ Suppφ ⊂ ξ : ≤ |ξ| ≤ φˆ 2−j ξ = 1, and 6 5 j∈Z
ˆ −j ξ), in other words, φj (x) = 2jN φ(2j x), for any for ξ = 0. Setting φˆj = φ(2 f ∈ S , we define ∆j f = φj ∗ f and Sj f = φk ∗ f. k≤j−1
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Then the homogeneous Besov semi-norm f B˙ s is defined [16, 18] by p,q For −∞ < s < ∞, 0 < p ≤ ∞, 0 < q ≤ ∞, set 1/q q jqs 2 ∆ f if q ∈ (0, ∞), p j L j∈Z f B˙ s = p,q js if q = ∞. supj∈Z 2 ∆j f Lp , s is a quasi-normed space with the above quasi-norm. For s > 0, The space B˙ p,q s (p, q) ∈ (1, ∞) × [1, ∞], we define the inhomogeneous Besov space norm f Bp,q of f ∈ S as s f Bp,q = f Lp + f B˙ s . p,q
The inhomogeneous Besov spaces are Banach spaces equipped with the norm s . f Bp,q ∈ C0∞ (R2 ) be the standard mollifier supported in B(0, 1) and denote
φ(x) xLet 1 ε ε ε ε φ ε by φ (x). Let f (x) = (f ∗ φ ) (x). α First, we recall the following inequalities [16, 18] for functions in Bp,∞ with 1 < p < ∞ and α > 0. α , u(· + y) − u(·)Lp ≤ C|y|α uBp,∞
(2.1)
α ∇uε Lp ≤ Cεα−1 uBp,∞ ,
(2.2)
α . uε − uLp ≤ Cεα uBp,∞
(2.3)
In the above inequalities, C’s are absolute constants. Proof of Theorem 1.1. We follow the idea of Constantin, E and Titi. Due to divergence free of the velocity field u(x, t), we rewrite equation (1.2) as ∂t θ + div(uθ) = 0,
(2.4)
u(x, t) = (−R2 θ, R1 θ),
(2.5)
and
where Ri be the Riesz transform. Now, we do the regularization of the equation (2.4) to obtain ∂t θε + div(uθ)ε = 0, where (uθ)ε = uε θε − (u − uε ) (θ − θε ) + rε (u, θ)
(2.6)
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with rε (u, θ) =
Ω
φε (y) (u(x − y) − u(x)) (θ(x − y) − θ(x)) dy.
Therefore, for (2.6) directly, we have d 2 |θε (x, t)| dx = 2 (uθ)ε (x, t) · ∇θε (x, t)dx dt Ω Ω ε ε = 2 u (x, t)θ (x, t) · ∇θε (x, t)dx Ω −2 (u − uε ) (x, t) (θ − θε ) (x, t) · ∇θε (x, t)dx Ω +2 rε (u, θ)(x, t) · ∇θε (x, t)dx Ω = −2 (u − uε ) (x, t) (θ − θε ) (x, t) · ∇θε (x, t)dx Ω +2 rε (u, θ)(x, t) · ∇θε (x, t)dx Ω ≤ C (u − uε )(t)L3 (θ − θε )(t)L3 + rε (u, θ)(t)
3 L2
∇θε )(t)L3 . (2.7)
From (2.5), and thanks to the boundedness of Riesz transform Ri : Lp → Lp with 1 < p < ∞, cf. [17], we have (u − uε )(t)L3 ≤ C(θ − θε )(t)L3
(2.8)
and rε (u, θ)(t)
3
L2
≤ ≤
φε (y)(θ(· − y) − θ(·))(t)2L3 dy 2 Cθ(t)B3,∞ α |y|2α φε (y)dy ≤ C 2α θ(t)2B3,∞ α , (2.9) Ω
Ω
where we used (2.1). Substituting (2.8) and (2.9) into (2.7), taking (2.2) and (2.3) into account, and integrating with respect to time, then we have t ε |θε (x, t)|2 dx − ≤ Cε3α−1 |θ (x, 0)| dx θ(τ )3B3,∞ α dτ → 0, Ω
Ω
0
α (Ω)) with α > 13 . when ε → 0 as long as θ ∈ C(0, T ; L2 (Ω)) ∩ L3 (0, T ; B3,∞ This finishes the proof for Theorem 1.1.
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Proof of Theorem 1.2. Differentiating equation (2.4), after regularization, we have ∂t (∂i θε ) + div(u∂i θ)ε + div(∂i uθ)ε = 0, i = 1, 2,
(2.10)
with (u∂i θ)ε = uε (∂i θ)ε − (u − uε ) (θ − (∂i θ)ε ) + rε (u, θ) and
rε (∂i u, θ) =
Ω
φε (y) (u(x − y) − u(x)) (∂i θ(x − y) − ∂i θ(x)) dy,
similar version for (∂i uθ)ε . By direct computation, we have d ∂i θε (x, t)θε (x, t)dx = − div(uθ)ε (x, t)∂i θε (x, t)dx dt Ω Ω ε ε − div(u∂i θ) (x, t)θ (x, t)dx − div(∂i uθ)ε (x, t)θε (x, t)dx. (2.11) Ω
Ω
Then by integration by parts and due to divergence free of uε and ∂i uε , (2.11) reduces to d ∂i θε (x, t)θε (x, t)dx dt Ω = rε (u, θ)(x, t) · ∇∂i θε (x, t)dx + rε (u, ∂i θ)(x, t) · ∇θε (x, t)dx Ω Ω + rε (∂i u, θ)(x, t) · ∇θε (x, t)dx Ω − (u − uε )(x, t)(θ − θε )(x, t) · ∇∂i θε (x, t)dx Ω − (u − uε )(x, t)(∂i θ − ∂i θε )(x, t) · ∇θε (x, t)dx Ω − (∂i u − ∂i uε )(x, t)(θ − θε )(x, t) · ∇θε (x, t)dx Ω
=
I1 + I2 + I3 + I4 + I5 + I6 .
(2.12)
First, we do estimate for rε (u, θ) as follows. For 1 < p < ∞, we have rε (u, θ)(·, t)Lp ≤ φε (y)u(· − y, t) − u(·, t)L2p θ(· − y, t) − θ(·, t)L2p dy Ω ≤C φε (y)θ(· − y, t) − θ(·, t)2L2p dy Ω ≤C φε (y)∇(θ(· − y, t) − θ(·, t))2 2p dy L p+1 Ω ≤ C∇θ(t)2B α2p |y|2α φε (y) ≤ Cε2α ∇θ(t)2B α2p , (2.13) p+1
,∞
Ω
p+1
,∞
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where the Sobolev embedding is used. By taking p = 3 in (2.13), we have |I1 | ≤ ∇∂i θ(t)
3
L2
rε (u, θ)(t)L3 ≤ Cε3α−1 ∇θ(t)3B α3 2
,∞
,
(2.14)
where we used (2.2). We can do a similar estimate for rε (u, ∂i θ). rε (u, ∂i θ)(·, t)
6
L5
φε (y)u(· − y, t) − u(·, t)L6 ∂i (θ(· − y, t) − θ(·, t)) 32 dy L ≤ C φε (y)θ(· − y, t) − θ(·, t)L6 ∂i (θ(· − y, t) − θ(·, t)) 32 dy L Ω ≤ C φε (y)∇(θ(· − y, t) − θ(·, t))2 3 dy L2 Ω ≤ C∇θ(t)2B α3 |y|2α φε (y) ≤ Cε2α ∇θ(t)2B α3 .
≤
Ω
2
,∞
Ω
2
,∞
Therefore, |I2 | ≤ ∇θε (t)L6 rε (u, ∂i θ)(t) ≤ Cε3α−1 ∇θ(t)3B α3 2
,∞
6
L5
≤ C∇(∇θε (t))
3
L2
ε2α ∇θ(t)2B α3 2
.
,∞
(2.15)
Due to the relation (2.5) between θ and u, the estimate of I3 is similar to I2 , |I3 | ≤ Cε3α−1 ∇θ(t)3B α3 2
,∞
.
(2.16)
I4 can be estimated straightforward as follows. |I4 | ≤ (u − uε )(t)L6 (θ − θε )(t)L6 ∇∂i θε (t) ≤ C∇(θ − θε )(t)2 3 ∇∂i θε (t) L2
≤ Cε
3
L2
3
L2 3α−1
∇θ(t)3B α3
,∞
2
, (2.17)
where (2.2) and (2.3) are used. Now, we turn our attention to I5 . |I5 | ≤ (u − uε )(t)L6 ∂i (θ − θε )(t)
3
L2
≤ C∇(θ − θε )(t)2 3 ∇(∇θε )(t) L2
∇θε (t)L6 3
L2
≤ Cε3α−1 ∇θ(t)3B α3 2
,∞
. (2.18)
I6 is similar to I5 , |I6 | ≤ Cε3α−1 ∇θ(t)3B α3 2
,∞
.
(2.19)
Substituting (2.14)–(2.19) into (2.12) and integrating with respect to time, then we have ε ε θε (x, t)∂i θε (x, t)dx − θ (x, 0)∂i θ (x, 0)dx Ω
Ω
≤ Cε3α−1
0
t
∇θ(τ )3B α3 2
,∞
dτ → 0,
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4
when ε → 0 as long as ∇θ ∈ C(0, T ; L 3 (Ω)) ∩ L3 (0, T ; B α3 ,∞ (Ω)) with α > 13 . Here 2 we use θε (x, t)∂i θε (x, t)dx ≤ ∇θ(t) 43 θ(t)L4 ≤ C∇θ(t)2 4 . (2.20) L
Ω
L3
The proof for Theorem 1.2 is complete. Remark 2.1 Under the same conditions of Theorem 1.2, it follows from inequality (2.20) that we have a lower bounds for ∇θ(t) 43 as L
∇θ(t)2 4
L 3 (Ω)
˜ ≥ C θ0 (x)∂i θ0 (x)dx , Ω
˜ for i = 1, 2, t ∈ [0, T ) and some absolute positive constant C.
Acknowledgment The author would like to express sincere gratitude to his supervisor Professor Zhouping Xin for enthusiastic guidance and constant encouragement. Thanks also to Professor Pedlosky for helpful comments. This work is partially supported by Hong Kong RGC Earmarked Grants CUHK-4028-04P and Shanghai Leading Academic Discipline.
References [1] V.I. Arnold, B.A. Khesin, A. Boris, Topological methods in hydrodynamics, Applied Mathematical Sciences, 125, Springer-Verlag, New York (1998). [2] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94, 61–66 (1984). [3] D. Chae, Remarks on the helicity of the 3-D incompressible Euler equations, Comm. Math. Phys. 240, no. 3, 501–507 (2003). [4] J.Y. Chemin, Perfect incompressible fluids, Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford University Press, New York, 1998. [5] P. Constantin, W. E, E.S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation, Comm. Math. Phys. 165, no. 1, 207–209 (1994). [6] P. Constantin, P.D. Lax, A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38, no. 6, 715– 724 (1985).
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[7] P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the 2D quasigeostrophic thermal active scalar, Nonlinearity 7, no. 6, 1495–1533 (1994). [8] P. Constantin, Q. Nie, N. Schorghofer, Nonsingular surface quasi-geostrophic flow, Phys. Lett. A 241, no. 3, 168–172 (1998). [9] D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasigeostrophic equation, Ann. of Math. 2, no. 3, 148, 1135–1152 (1998). [10] D. Cordoba, C. Fefferman, Growth of solutions for QG and 2D Euler equations, J. Amer. Math. Soc. 15, no. 3, 665–670 (2002). [11] A. Majda, A. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. [12] H.K. Moffat, A. Tsinober, Helicity in Laminar and turbulent flow, Ann. Rev. Fluid Mech. 24, 281–312 (1992). [13] K. Ohkitani, M. Yamada, Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. Fluids 9, no. 4, 876–882 (1997). [14] L. Onsager, Statistical hydrodynamics, Nuovo Cimento 9, 6 (1949). Supplemento, no. 2 (Convegno Internazionale di Meccanica Statistica), 279–287. [15] J. Pedlosky, Geophysical fluid Dynamics, Springer-Verlag, New York (1987). [16] T. Runst, W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter & Co., Berlin (1996). [17] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. (1970). [18] H. Triebel, Theory of function spaces, II. Monographs in Mathematics, 84, Birkh¨auser Verlag, Basel (1992).
Yong Zhou Department of Mathematics East China Normal University Shanghai 200062 China email:
[email protected] Communicated by Rafael D. Benguria submitted 17/06/04, accepted 16/12/04
Ann. Henri Poincar´e 6 (2005) 801 – 820 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05801-20, Published online 05.10.2005 DOI 10.1007/s00023-005-0224-x
Annales Henri Poincar´ e
Existence and Stability of Even-dimensional Asymptotically de Sitter Spaces Michael T. Anderson∗
Abstract. A new proof of Friedrich’s theorem on the existence and stability of asymptotically de Sitter spaces in 3+1 dimensions is given, which extends to all even dimensions. In addition we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future.
1 Introduction Consider globally hyperbolic vacuum solutions (M n+1 , g) to the Einstein equations with cosmological constant Λ > 0, so that Ricg −
Rg g + Λg = 0. 2
(1.1)
The simplest solution is (pure) de Sitter space on M n+1 = R × S n , with metric gdS = −dt2 + cosh2 (t)gS n (1) .
(1.2)
More generally, let (N n , gN ) be any compact Riemannian manifold with metric gN satisfying the Einstein equation RicgN = (n − 1)gN . Then the (generalized) de Sitter metric N gdS = −dt2 + cosh2 (t)gN , (1.3) on R × N is also a solution of (1.1), with Λ = n(n − 1)/2. Let dS + be the space of all globally hyperbolic spacetimes (M n+1 , g) satisfying (1.1), with a spatially compact Cauchy surface, which are asymptotically de Sitter (dS) to the future, i.e., future conformally compact in the sense of Penrose; the terminology asymptotically simple is also used in this context. Thus there is a smooth function Ω such that the conformally compactified metric g¯ = Ω2 g,
(1.4)
¯ = M ∪ I + , where I + is a compact nextends to the compactified spacetime M ¯ , with Ω > 0, I + = manifold without boundary. The function Ω is smooth on M −1 + Ω (0) and dΩ = 0 on I . The boundary metric γ = g¯|I + depends on the choice ∗ Partially
supported by NSF Grant DMS 0305865
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of Ω; however the conformal class [γ] of γ is independent of Ω and is called future conformal infinity. Such spacetimes are geodesically complete to the future of an initial compact Cauchy surface Σ diffeomorphic to I + . There are no restrictions on the class [γ] or the topology of I + , and so such spacetimes are sometimes also called asymptotically locally de Sitter. Changing the time orientation gives the same notion for dS − , with future conformal infinity I + replaced by past conformal ¯ may be measured in H¨ older infinity I − . The smoothness of g¯ in (1.4) up to M m,α spaces C , but we will mostly use Sobolev spaces H s which are more natural in this context. In addition, let dS ± be the space of such globally hyperbolic spacetimes which are in both dS + and dS − ; thus such spacetimes are (completely) global, in the sense that they are geodesically complete and asymptotically simple both to the past and to the future. Mathematically, the most significant result on the structure of such spacetimes is Friedrich’s theorem [8], [9] that in 3 + 1 dimensions, the Cauchy problem with data on I + , (or I − ) is well posed, cf. also [10] for recent discussions. Thus, for arbitrary Cauchy data on I + , there is a unique spacetime (M 4 , g) which realizes this data at future infinity. Moreover, small but arbitrary variations of the Cauchy data give rise to small perturbations of the solution. It follows in particular that the space dS ± of global solutions is open; thus spaces in dS ± , in particular pure de Sitter space (M 4 , gdS ), are stable under small perturbations of the Cauchy data at I + , (or I − ). The same statement holds for perturbations of the data on a compact Cauchy surface Σ for (M 4 , g). The purpose of this paper is to extend Friedrich’s theorem to arbitrary even dimensions. Let I + be any closed n-manifold, n odd, and let γ be any H s+n smooth Riemannian metric on I + , s > n2 + 1. Next, let τ be any H s symmetric bilinear form on I + satisfying the constraints trγ τ = 0, δγ τ = 0,
(1.5)
i.e., τ is transverse-traceless with respect to γ. Define γ1 ∼ γ2 and τ1 ∼ τ2 if these data are conformally related, i.e., there exists λ : I + → R+ such that γ2 = λ2 γ1 and τ2 = f (λ)τ1 , where f is chosen so that (1.5) holds for τ2 , cf. [5] for the exact transformation formula. Let ([γ], [τ ]) be the equivalence class of (γ, τ ). Then Cauchy data for the Einstein equations (1.1) with Λ > 0 consist of triples (I + , [γ], [τ ]). The form τ corresponds to the order n behavior of the metric; roughly for g¯ as in (1.3), τ = (∂Ω )n g¯|I + ; see §2 for further details. Theorem 1.1 The Cauchy problem for the Einstein equations with Cauchy data (I + , [γ], [τ ]) at future conformal infinity is well posed in H s+n × H s , for any s > n 2 + 2. Thus, given any Cauchy data ([γ], [τ ]) ∈ H s+n (I + )×H s (I + ) satisfying (1.5), up to isometry there is a unique Einstein metric (M n+1 , g) ∈ dS + whose conformal compactification as in (1.4) induces the given data ([γ], [τ ]) on I + .
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This result has the following simple consequence: Theorem 1.2 The space dS ± is open with respect to the H s+n ×H s topology on I + , s > n2 + 2. Thus, given any dS solution (M n+1 , g0 ) ∈ dS ± , any H s+n × H s small perturbation of the Cauchy data ([γ], [τ ]) on I + (or I − ) gives rise to complete solution (M n+1 , g) ∈ dS ± globally close to (M n+1 , g0 ). In particular, the evenN in (1.3) are globally stable. dimensional pure de Sitter spaces gdS Here, globally close is taken with respect to a natural H s topology on the ¯ = M ∪ I + ∪ I − , see the proof for details. The conformal compactification M n+1 complete solution (M , g) induces H s+n × H s Cauchy data at both past and future conformal infinity I − , I + . Of course the size of the allowable perturbations in Theorem 1.2 depends on (M n+1 , g0 ). We describe briefly the main ideas in the proof of Theorem 1.1; full details are given in §2. The Einstein equations (1.1) induce a 2nd order system of equations for a compactified metric g¯ in (1.4). However, this system is degenerate at I + = {Ω = 0} and this degeneracy causes severe problems in trying to prove the well-posedness of the system. In 3 + 1 dimensions, Friedrich [8] has developed a larger and more complicated system of evolution equations, the conformal Einstein equations, for the (unphysical) metric g¯ together with other variables. This expanded system is non-degenerate and shown to be symmetric hyperbolic; then standard results on such systems lead to the well-posedness of the conformal field equations. However, it seems very unlikely that this method could succeed in higher dimensions, cf. [10], due at least in part to the special form of the Bianchi equations in 3 + 1 dimensions. The approach taken here is to replace the Einstein equation by a more complicated but conformally invariant higher-order equation for the metric alone, whose solutions include the vacuum Einstein metrics (with Λ term). In 3 + 1 dimensions, this system is the system of 4th order Bach equations, cf. (2.12) below. The Bach equations have been used in a number of contexts in connection with issues related to conformal infinity, cf. [14], [15], [16], for example. In higher even dimensions, in place of the Bach tensor, we use the ambient obstruction tensor H of Fefferman-Graham [6], which agrees with the Bach tensor in 3 + 1 dimensions; this tensor is also characterized as the stress-energy tensor of the conformal anomaly, cf. [5]. The tensor H is a symmetric bilinear form, depending on a given metric g on M n+1 and its derivatives up to order n + 1. The equation H=0 (1.6) is conformally invariant, and includes all Einstein metrics (of arbitrary signature and Λ-term). It is a system of (n + 1)st -order equations in the metric, whose n+1 leading order term in suitable coordinates is of the form 2 , where is the wave operator of the metric g. Conformal invariance implies that the system (1.6) is non-degenerate at I + = {Ω = 0}. Theorem 1.1 is then proved by showing that
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natural gauge choices for the diffeomorphism and conformal invariance of (1.6) lead again to a symmetrizable system of evolution equations. In the context of Theorem 1.2, it is of interest to understand the closure dS ± of the space dS ± , i.e., the structure of spacetimes which are limits of spacetimes in dS ± but not themselves in dS ± . A first step in this direction was taken in [2] in 3+1 dimensions, and Theorem 1.1 allows one to extend this to any even dimension. Let dS ± be the closure of dS ± with respect to the H s+n × H s topology on the Cauchy data on either I + or I − , i.e., the union of the closures with respect to data on I − and I + . Let ∂dS ± = dS ± \ dS ± be the resulting boundary consisting of limits of spaces in dS ± which are not in dS ± . Theorem 1.3 For (n + 1) even, a space in the boundary ∂dS ± of dS ± , is described by one of the following three configurations: I. A pair of solutions (M, g + ) ∈ dS + and (M, g − ) ∈ dS − , each geodesically complete and globally hyperbolic. One has I − = ∅ for (M, g + ) and I + = ∅ for (M, g − ). Both solutions (M, g + ) and (M, g − ) are “infinitely far apart”. II. A single geodesically complete and globally hyperbolic solution (M, g) ∈ dS + , either with a partial compactification at I − , or I − = ∅. III. A single geodesically complete and globally hyperbolic solution (M, g) ∈ dS − , either with a partial compactification at I + , or I + = ∅. Cases II and III have been distinguished here, but these behaviors become identical under a switch of time orientation. One of the main points here is that singularities do not form on spaces within dS ± . One does expect singularities to form “past” the boundary ∂dS ± . The most natural limits are those of type I; this behavior occurs very clearly and explicitly in the family of dS Taub-NUT metrics on R × S n , cf. [2] for further discussion. It would be very interesting to know more about the structure of dS ± ; for instance, is it compact and connected? Theorems 1.1–1.3 are proved in §2, and we close the paper with some remarks on extending these results to vacuum equations with Λ ≤ 0 and to the Einstein equations coupled to matter fields. I would like to thank the referee and Piotr Chru´sciel for very useful comments on the paper.
2 Proofs of the results Throughout the paper, we consider globally hyperbolic vacuum spacetimes (M, g) with Λ > 0 in (n + 1) dimensions. By rescaling if necessary, it is assumed that Λ is normalized to Λ = n(n − 1)/2, so that the Einstein equations read Ricg = ng.
(2.1)
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The simplest solution of (2.1) is (pure) de Sitter space on M = R×S n , with metric (1.2), or its generalization in (1.3). These de Sitter metrics gdS are geodesically complete and globally conformally compact, i.e., in dS ± . In fact, defining s ∈ 1 and letting (− π2 , π2 ) by cosh(t) = cos(s) g¯ = cos2 (s)g,
(2.2)
one has g¯dS = −ds2 + gS n (1) , which is the metric on the Einstein static spacetime in the region s ∈ [− π2 , π2 ]. ¯ = M ∪ I + ∪ I − and the loci The metric g¯dS is real analytic on the closure M π π + − I = {s = 2 } = {t = ∞}, I = {s = − 2 } = {t = −∞} represent future and past conformal infinity. The induced metric on I ± is of course the unit round metric gS n (1) on S n . The same discussion holds for (N, gN ) as in (1.3) in place of gS n (1) . Consider Einstein metrics (M n+1 , g) in dS + , so that there is a compactification (2.3) g¯ = ρ2 g as in (1.4) to future conformal infinity I + , with I + = {ρ = 0}; all of the analysis below works equally well for spaces in dS − . A compactification g¯ = ρ2 g as in (2.3) is called geodesic if ρ(x) = distg¯ + (x, I ). These are often the simplest compactifications to work with for computational purposes. Each choice of boundary metric γ ∈ [γ] on I + determines a unique geodesic defining function ρ, (and vice versa). The Gauss Lemma gives the splitting (2.4) g¯ = −dρ2 + gρ , g = ρ−2 (−dρ2 + gρ ), where gρ is a curve of metrics on I + . The asymptotic behavior of g at I + is thus determined by the behavior of gρ as ρ → 0. For example, the geodesic compactification of the de Sitter metric (1.2) with respect to the unit round metric at I + is ρ g¯dS = −dρ2 + (1 + ( )2 )2 gS n (1) , 2 for ρ ∈ [0, ∞). Now consider a Taylor series type expansion for the curve gρ on I + . This was analyzed in case of asymptotically hyperbolic or AdS metrics with Λ < 0 by Fefferman-Graham [6], and for dS metrics by Starobinsky [19] when n = 3. This idea of course has further antecedents in the Bondi-Sachs expansion and peeling properties of the Weyl tensor when Λ = 0. In any case, the FG expansion holds equally well for metrics in dS + (or dS − ) in place of asymptotically AdS metrics; in fact the two expansions are very closely related, cf. [2], [18] and further references therein. The exact form of the expansion depends on whether n is odd or even. If n is odd, then gρ ∼ g(0) + ρ2 g(2) + · · · + ρn−1 g(n−1) + ρn g(n) + ρn+1 g(n+1) + · · · , with g(0) = γ.
(2.5)
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This expansion is even in powers of ρ up to order n − 1. The coefficients g(2k) , 0 < k < n/2 are locally determined by the boundary metric γ = g(0) ; they are explicitly computable expressions in the curvature of γ and its covariant derivatives. For example for n ≥ 3, g(2) =
1 Rγ (Ricγ − γ), n−2 2(n − 1)
(2.6)
cf. also [5], [2] for formulas for g(k) for k > 2. The term g(n) is transverse-traceless, i.e., trγ g(n) = 0, δγ g(n) = 0,
(2.7)
but is otherwise undetermined by γ and the Einstein equations (2.1); thus, at least formally, it is freely specifiable. For k > n, terms g(k) occur for k both even and odd; the term g(k) depends on two boundary derivatives of g(k−2) . The main point is that all coefficients g(k) are locally computable expressions in g(0) and g(n) . Mathematically, the expansion (2.5) is formal, obtained by compactifiying the Einstein equations and taking iterated Lie derivatives of g¯ at ρ = 0. If the ¯ ), then the expansion holds up to order geodesic compactification g¯ is in C m,α (M m + α, in the sense that gρ = g(0) + ρ2 g(2) + · · · + ρm g(m) + O(ρm+α ).
(2.8)
Suppose instead n is even. Then the expansion reads gρ ∼ g(0) + ρ2 g(2) + · · · + ρn−2 g(n−2) + ρn g(n) + ρn (log ρ)H + · · ·
(2.9)
Again the terms g(2k) up to order n−2 are explicitly computable from the boundary metric γ, as is the coefficient H of the ρn (log ρ) term. The term g(n) satisfies trγ g(n) = a, δγ g(n) = b, where a and b are explicitly determined by the boundary metric γ and its derivatives, but g(n) is otherwise undetermined by γ and the Einstein equations; as before, it is formally freely specifiable. The series (2.9) is even in powers of ρ, (at all orders) and terms of the form ρ2k (log ρ)j appear at order > n. Again the coefficients g(k) and H(k) depend on two derivatives of g(k−2) and H(k−2) . Although the expressions (2.5) and (2.9) are only formal in general, Fefferman-Graham [6] showed that if the undetermined terms (g(0) , g(n) ) are analytic on the boundary I + , then the expansion (2.5) converges, (for n odd), cf. also [2]. Thus gρ is analytic in ρ for ρ small and one has a dS Einstein metric in this region given by (2.4). A similar result has recently been proved by Kichenassamy [13], (cf. also [17]), for n even; in this case the polyhomogeneous expansion (2.9) converges to gρ for ρ small.
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The term H, which appears only when n is even, has a number of important interpretations. First Fefferman-Graham [6] observed that this tensor, locally computable in terms of the boundary metric γ, is a conformal invariant of γ and is (by definition) an obstruction to the existence of a formal power series expansion of the compactified Einstein metric; in fact it is the only obstruction. The tensor H is also important in the (A)dS/CFT correspondence, in that (up a constant) it equals the stress-energy tensor (i.e., the metric variation) of the conformal anomaly of the corresponding CFT, cf. [5]. It also arises as the stress-energy or metric variation of the Q-curvature of the boundary metric γ, cf. [7]. The tensor H is transverse-traceless δH = tr H = 0,
(2.10)
= λ2−n H. and a conformal invariant of weight 2 − n, i.e., if g = λ2 g, then H Further, if g is conformal to an Einstein metric, with any value of Λ, then H = 0.
(2.11)
In addition, as observed in [6], these properties hold for metrics of any signature and so the equation (2.11) can be viewed as a conformally invariant version of the Einstein equations with an arbitrary Λ term and arbitrary signature. We are not aware of any analogue of such a tensor in odd dimensions. We will use the tensor H to study de Sitter type solutions of the Einstein equations (2.1). Although the derivation of the obstruction tensor H arises from the structure at infinity of conformally compactified odd-dimensional Einstein metrics, once it is given, one can use it to study the Einstein equations themselves in even dimensions. Thus, replacing n by n+1, a vacuum solution of the Einstein equations (M n+1 , g) with Λ > 0, (or any Λ), in even dimensions is a solution of (2.11). For (M n+1 , g) ∈ dS + , the equation (2.11), being conformally invariant, also holds for the compactified Einstein metric g¯ in (2.2); moreover it has the important advantage of being a non-degenerate system of equations in g¯. As is well-known [8], the translation of the Einstein equations for (M, g) to the compactified setting g¯ leads to a degenerate system of equations for g¯. When n = 3, so dim M = 4, up to a constant factor H is the Bach tensor B, given by R R (2.12) B = D∗ D(Ric − g) + D2 (tr(Ric − g)) + R, 6 6 where R is a term quadratic in the full curvature of g. (The specific form of R will not be of concern here.) In general, for n ≥ 3 odd, one has, again up to a constant factor, n+1 H = (D∗ D) 2 −2 [D∗ D(P ) + D2 (trP )] + L(Dn γ), (2.13) where P = P (γ) = Ricg −
Rg g, 2n
(2.14)
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cf. [7] for example. This is a system of PDE’s in the metric g, of order n + 1; L(Dn g) denotes lower-order terms involving the metric up to order n. The Bach equation B = 0 was originally developed by Bach as a conformally invariant version of the Einstein equations (with Λ = 0), and has been extensively studied in this context, cf. [14], [15], [16] for some recent work and references therein. It was also used in [3] to study regularity properties of conformally compact Riemannian Einstein metrics. While Einstein metrics, (of any signature and Λ), are solutions of (2.11), of course not all solutions of (2.11) are Einstein. In addition, for Lorentzian metrics, H is not a hyperbolic system of PDE’s in any of the usual senses; the equation (2.11) is invariant under diffeomorphisms and conformal changes of the metric, and so requires at least a choice of diffeomorphism and conformal gauge to obtain a hyperbolic system. To describe these gauge choices, suppose (M, g) ∈ dS + , so that g is an Einstein metric, satisfying (2.1), and so (2.11), which is asymptotically dS to the future. Assume that (M, g) has a geodesic compactification which is at least C n ; then (2.15) g¯ = ρ2 g = −dρ2 + gρ , and gρ has the expansion (2.8), with m = n, α = 0. One has I + = {ρ = 0} and we set γ = g(0) . By the solution to the Yamabe problem, one may assume without loss of generality that the representative γ ∈ [γ] has constant scalar curvature, i.e., Rγ = const,
(2.16)
on I + . However, closer study shows that the operator P in (2.14) is not well behaved in the coordinates adapted to (2.15), i.e., the natural geodesic coordinates (ρ, yi ), where yi are local coordinates on I + extended to coordinate functions on M to be invariant under the flow of ∇ρ. Further, with this choice of conformal ¯ of g¯. gauge, it is difficult to control the scalar curvature R It is simplest and most natural to choose a conformal gauge of constant scalar curvature, (although other choices are possible). Thus, set g = σ 2 g¯,
(2.17)
= const. In this gauge, the equation (2.13) for g where σ is chosen to make R simplifies to n+1 ∗ D) 2 −1 Ric + L(Dn H = (D g) = 0. (2.18) is not important, but it simplifies matters if one The choice of constant for R chooses = R| ¯ I + = − n(n − 2) Rγ ≡ c0 . (2.19) R n−1 The middle equality follows by taking the trace of (2.6), and combining this with the Raychaudhuri equation on g¯ and (2.28) below.
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For the diffeomorphism gauge, we choose, as usual, harmonic coordinates xα with respect to g; α = 0. x It is assumed that the Cauchy data for x0 are such that x0 is a defining function for I + near I + , and we relabel x0 = t so that the coordinates are (t, xi ), i = 1, . . . , n. As usual, Greek letters are used for spacetime indices, while Latin is used for spatial indices. Equivalently, but from a slightly different point of view, given arbitrary local coordinates xα , with x0 a defining function for the boundary, the condition that xα is harmonic with respect to g is α = ∂α gαβ + 1 gαβ gµν ∂α gµν = 0. x 2
(2.20)
In the coordinates xα , the metric g¯ in (2.15) becomes g¯ = g00 (dt2 ) + 2g0i dtdxi + gij dxi dxj ,
(2.21)
g00 = −(∂t ρ)2 , g0i = ∂t ρ∂i ρ, gij = ∂i ρ∂j ρ + (gρ )ij .
(2.22)
where Similarly, for g, one has g0i dtdxi + gij dxi dxj , g= g00 (dt2 ) + 2
(2.23)
with gαβ = σ 2 gαβ . As long as the coordinates are g-harmonic, the Ricci curvature has the form 1 µν gαβ + Qαβ ( g , ∂ g), Ricαβ = − g ∂µ ∂ν 2 ∗ D has the form Similarly at leading order, the Laplacian D g µν ∂µ ∂ν in harmonic coordinates. Thus, with these choices of gauge for the conformal and diffeomor = 0 has the rather simple form phism invariance, the equation H
( g µν ∂µ ∂ν )
n+1 2
gαβ + L(Dn g) = 0.
(2.24)
This is an N × N system of PDE’s for gαβ which is diagonal, i.e., uncoupled, at leading order, N = (n + 1)(n + 2)/2. These choices for the conformal and diffeomorphism gauges are the simplest; however, they are not necessary and other choices, for instance gauges determined by fixed gauge source functions, cf. [12], could also be used. Having discussed the equations for the metric, we have left to determine the equations for ρ in (2.15) and σ in (2.17). The fact that ρ is a geodesic defining ¯ 2 = −1, implies that function for g¯, i.e., |∇ρ| g ¯ ∂t (g αβ ∂α ρ∂β ρ) = 0, or equivalently, g αβ ∂α ρ∂β ρ) = 0. ∂t (σ 2
(2.25)
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To derive the equation for σ, the equation for the Ricci curvature relating g and g¯ is Ric = Ric + (n − 1)
2σ σ D +{ − n|d log σ|2 } g. σ σ
(2.26)
Taking the trace gives the equation relating the scalar curvatures as ¯=R + 2n σ − n(n + 1)|d log σ|2 . σ −2 R g σ
(2.27)
Using the formula analogous to (2.26) relating the Ricci curvature of g and g¯, together with the fact that g satisfies (2.1) and ρ is a geodesic defining function gives ¯ ¯ = −2n ρ = −2nRic(T, T ), R (2.28) ρ where T = ∂ρ = −∇g¯ ρ. (Observe that the middle term in (2.28) is degenerate at I + , since ρ = 0 there; however, the last term in (2.28) is non-degenerate at ρ = 0). Substituting (2.28) in (2.27) and using (2.26) gives then the equation 2σ D + n(n − 1)|d log σ|2 = −2nσ −2 [Ric(T, R (T, T )], T ) + (n − 1) g σ or equivalently, ∇ T T = − T T (σ) − ∇σ,
σ3 1 1 σ Ric(T, T) − c0 − σ|d log σ|2g , (2.29) n−1 2n(n − 1) 2
where we have also used (2.19). The equations (2.24), (2.25), and (2.29) represent a coupled system of evolution equations for the variables ( gαβ , ρ, σ) on a domain U in (Rn+1 )+ with coordinates (t, xi ); the boundary ∂0 U = U ∩ {t = 0} corresponds to a portion of I + . Written out in more detail, these are: ( g µν ∂µ ∂ν )
n+1 2
gαβ = L1 (Dn g)αβ ,
i ρ)(∂i ∂t ρ) = L2 (Dρ, Dσ, D g ), g00 ∂t ∂t ρ + 2(∇ 0 ρ)2 ∂t ∂t σ + 2(∇ 0 ρ)(∇ i ρ)∂i ∂t σ + (∇ i ρ)(∇ j ρ)∂i ∂j σ = L3 (Dσ, D2 g). (∇
(2.30) (2.31) (2.32)
α ρ denotes the Here Dk w denotes derivatives up to order k in the variable w and ∇ αβ α-component of ∇ρ, ∇ρ = g ∂β ρ∂α . The terms Li are lower-order terms. Observe that the system (2.30) for the metric gαβ is a closed sub-system, i.e., it does not involve ρ or σ. Moreover, although the equations (2.31) and (2.32) for ρ and σ are coupled to each other and to (2.30), the system (2.30)–(2.32) is uncoupled at leading order.
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Following common practice, we now reduce the system (2.30)–(2.32) to a system of 1st order equations. There is not a unique way to do this, but we will discuss perhaps the simplest method, which uses pseudodifferential operators. As usual, the domain ∂0 U ⊂ Rn is viewed as a domain in the n-torus T n and the variables ( gαβ , σ, ρ) are extended to functions on I × T n . Recall that a system of 1st order evolution equations ∂t u =
m
Bj (t, x, u)∂j u + c(t, x, u)
(2.33)
j=1
is symmetrizable in the sense of Lax, cf. [20], [21], if there is a smooth matrix valued function R(t, u, x, ξ) on R × Rp × T ∗ (T n ) \ 0, homogeneous of degree 0 in ξ, such that R is a positive definite p × p matrix with R(t, u, x, ξ) Bj (t, u, x)ξj self-adjoint, for each (t, u, x, ξ). It is well known [20], [21] that strictly hyperbolic systems of PDE, diagonal at leading order, are symmetrizable. A symmetrizer R is given by R= Pk Pk∗ , where Pk is the projection onto the k th eigenspace of the symbol Bj (t, u, x)ξj , 1 ≤ k ≤ p. Proposition 2.1 There is a reduction of the system (2.30)–(2.32) to a symmetrizable system of 1st order evolution equations on I × T n . Proof. Consider first the closed system (2.30) for g. This system is not strictly hyperbolic; the leading order symbol is diagonal and has two distinct real eigenvalues, each of multiplicity (n + 1)/2. However, the eigenspaces of the symbol of n+1 n21 is strongly hy2 vary smoothly and do not coalesce. Thus the operator perbolic, cf. [12] and references therein. In these circumstances, it is essentially k is symmetrizable, for any k; for completeness we standard that the operator sketch the proof following [20, §5.3]. Let u = gαβ be the variable in RN , N = (n + 1)(n + 2)/2. Write , (we drop the tilde here and below), in the form (g 00 )−1 = ∂t2 −
1
Aj ( u, Dx )∂tj ,
j=0
where Aj is a differential operator in x, homogeneous of order 2 − j, depending smoothly u . Then 00 −1
[(g )
]
(n+1)/2
=
∂tn+1
−
n
Bj ( u, Dx )∂tj ,
(2.34)
j=0
where Bj are differential operators in x, homogeneous of order n + 1 − j. Set , for j = 0, . . . , n, where Λ = (1 − ∆)1/2 and ∆ is the standard uj = ∂tj Λn−j u Laplacian on T n . Then (2.30) becomes ∂t uj = Λuj+1 , 0 ≤ j < n, ∂t un =
n j=0
Bj (P u, Dx )Λj−n uj + C(P u),
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where P u = Dn u involves u up to n derivatives. More precisely, for β + j ≤ n, ∂xβ ∂tj u = ∂xβ Λj−n uj ; for example Λ−n u0 = u . This is a system of 1st order pseudodifferential equations in the variables u = {uj }, j = 0, . . . , n of the form ∂t u = L(P u, Dx)u + C(P u).
(2.35)
The eigenvalues λν (w, ξ) of the matrix L(w, ξ) are the roots of the characteristic equation τ n+1 − Bj (w, ξ)τ j , (up to an overall factor of i). Hence, from (2.34), one sees that for each (w, ξ), ξ = 0, there are two distinct roots, each of multiplicity (n+ 1)/2. The eigenvalues vary smoothly with (w, ξ) and remain a bounded distance apart on the sphere |ξ| = 1. The same is true of the corresponding eigenspaces. Hence, the system (2.35) has a symmetrizer R constructed in the same way as following (2.33), cf. [20, Prop. 5.2.C] or [21, Prop. 16.2.2]. Next we show that the equation (2.31) is also symmetrizable. Let φi = i ρ/ g 00 . Introducing the vector variable v = (ρ, ρ0 , . . . , ρn ) with ρ0 = ∂t ρ, ρj = −2∇ ∂j ρ, the equation is equivalent to the system ∂t ρ = ρ0 , ∂t ρ0 = φj ∂j ρ0 + c(t, x, v), ∂t ρj = ∂j ρ0 . This has the form ∂t v =
n
Bj (x, t, v)∂j v + c(x, t, v),
(2.36)
j=1
where Bj is an (n + 2) × (n + 2) matrix with φj in the (2, 2) slot, 1 in the (j + 2, 2) slot, and 0 elsewhere. The system (2.36) is coupled at lower order to the equations (2.30) and (2.32) for g and σ respectively, in that Bj depends on g to order 0, while c depends on g and σ to order 1; for the moment, these dependencies are placed in the (x, t) dependence of Bj and c. The matrix Bj ξj has the entry φj ξj in the (2, 2) slot, ξj in the (j + 2, 2) slot for 3 ≤ j ≤ n, and 0 elsewhere. By a direct but uninteresting computation, it is straightforward to see that this matrix is symmetrizable in the sense following (2.33). Essentially the same argument shows that the equation (2.32) for σ is again symmetrizable. Thus let w = (σ, σ0 , . . . , σn ) with σ0 = ∂t σ, σj = ∂j σ. The equation (2.32) is equivalent to the system ∂t σ = σ0 , ∂t σ0 = φj ∂j σ0 + ψij ∂j σi + c(t, x, v), ∂t ρj = ∂j ρ0 , i ρ)/|∇ 0 ρ|, ψij = (∇ i ρ)(∇ j ρ)/|∇ 0 ρ|2 . This system has the form where φj = −2(∇ ∂t w =
n
Bj (x, t, w)∂j w + c(x, t, w),
(2.37)
j=1
where Bj is the (n + 2) × (n + 2) matrix with φj in the (2, 2) slot, 1 in the (j + 2, 2) slot, and ψij in the (i + 2, j + 2) slot. The system (2.37) is again coupled at
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lower order to the equations (2.30) and (2.31) for g and ρ respectively, in that c depends on g to order 2, while Bj depends on g to order 0 and ρ to order 1. Again a straightforward but longer (uninteresting) computation shows that the matrix Bj ξj is symmetrizable in the sense of (2.33). One may then combine the three systems (2.35), (2.36), and (2.37) to a single large system in the variable U = (u, v, w). The resulting system is then a symmetrizable system of 1st order pseudodifferential equations, cf. [20], [22]. Next consider the Cauchy data for the system (2.30)–(2.32). If one is interested in general solutions of this system, then the Cauchy data are essentially ¯ ·) = 0 on I + . However, arbitrary, subject only to the constraint equation H(∇ρ, as will be seen in Proposition 2.3 below, it is the specification of the Cauchy data which determines the class of conformally Einstein metrics among all solutions of (2.30)–(2.32). This is of course closely related to the FG expansion (2.8) of the dS metric g. The Cauchy data for σ are σ = 1, and ∂t σ = 0 at I + ,
(2.38)
while the Cauchy data for ρ are ρ = 0, and ∂t ρ = 1 at I + .
(2.39)
For the metric g, the closed subsystem (2.30) is of order n + 1, so Cauchy data are specified by prescribing (∂t )k gαβ , k = 0, 1, . . . , n at I + . We compute the data inductively. First, the condition (2.38) implies that gij = g¯ij at I + . Thus at order 0, set (2.40) g00 = −1, g0i = 0, gij = γij at I + , since ρi = 0 at I + . At 1st order, (2.38) and (2.39) together with (2.23) show that ∂t gij = ∂t g¯ij + at I , and the FG expansion (2.8) gives ∂t g¯ij = 0 at I + . Thus, set ∂t gij = 0 at I + .
(2.41)
(This condition, and related ones below, are necessary to obtain Einstein metrics.) The first derivatives of the mixed components g0α of g are determined by the requirement that the coordinates xα = (t, xi ) are harmonic at I + with respect to g, i.e., for each β, 1 ∂α gαβ + gαβ g µν ∂α gµν = 0. (2.42) 2 Via (2.40)–(2.41), this determines ∂t g0α at I + . nd At 2 order, the equation (2.27) implies, using the normalization (2.19), that ∂t2 σ = 0 at I + . Also, ∂t2 (ρi ρj ) = 0, and hence, from the FG expansion (2.8), we set ∂t2 gij = 2g(2) at I + . (2.43)
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The 2nd derivatives ∂t g0α are then determined by (2.43), the lower-order Cauchy data, (2.38)–(2.41), and the t-derivative of (2.42) at t = 0. At 3rd order, suppose first n = 3, so that dim M = 4. Then a straightforward computation, using (2.8), the Raychaudhuri equation on g¯ and (2.28), shows that ¯ = 6n tr g(3) = 0 at I + , where the last equality follows from (2.7). Hence, ∂t R (2.27) gives ∂t3 σ = 0. Similarly, (2.39) gives ∂t3 (ρi ρj ) = 0. Thus, set ∂t3 gij = 6g(3) at I + .
(2.44)
This term is free or unconstrained, subject to the transverse-traceless constraint (2.7). As before the mixed term at order 3, ∂t3 g0α is determined by taking two t-derivatives of (2.42) at t = 0, and using (2.44) together with the determination of the lower-order Cauchy data. Suppose instead n > 3 and hence n ≥ 5. Then g(3) = 0 and same arguments as above give (2.45) ∂t3 gij = 0 at I + , with ∂t3 g0α again determined from two t-derivatives of (2.42), (2.45) and lowerorder Cauchy data. At 4th order on I + , (assuming n ≥ 5), by (2.27) and the fact that ∂tk σ = 0, ¯ and again computations as above then k ≤ 3 on I + , one has ∂t4 σ = −2n∂t2 R, ¯ = 24n tr g(4) . Also, taking i-derivatives of (2.31) or (2.25) and using give ∂t2 R (2.38)–(2.39) shows that ∂t4 (ρi ρj ) = 0 at I + . It follows that, at t = 0, ∂t4 gij = 24(g(4) − 2n2 tr g(4) )γ.
(2.46)
Again, ∂t4 g0α is determined by taking three t-derivatives of (2.42) at t = 0, and using (2.46) with the determination of the lower-order Cauchy data. ¯ = c tr g(5) = 0 at I + At 5th order, suppose n = 5. As in the case n = 3, ∂t3 R 5 while ∂t (ρi ρj ) = 0. Hence, as in the case n = 3, ∂t5 gij = (5!)g(5) ,
(2.47)
which is freely specifiable, subject to the transverse-traceless constraint. As before the mixed term at order 5, ∂t5 g0α is determined by taking four t-derivatives of (2.42) at t = 0, and using (2.47) with the determination of the lower-order Cauchy data. If n > 5, then as before, gij = 0, (2.48) ∂t5 with ∂t5 g0α again determined from (2.42). It is clear that one can continue ingαβ up to order n. Since ductively in this way to determine the Cauchy data ∂tk ∂t6 (ρi ρj ) = 0 at t = 0, these and higher derivative terms contribute to the Cauchy data at order 6 and above. However, one sees by differentiations of (2.31) and (2.42) that these terms are all determined by lower-order Cauchy data for g. gαβ , 0 ≤ k ≤ n, are determined In sum, Cauchy data for gαβ , i.e., the data ∂tk by the Cauchy data (2.38), (2.39) for ρ and σ, the equations (2.31)–(2.32), (or
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(2.25), (2.27)) for ρ and σ, the harmonic equation (2.42), and the coefficients g(k) in the FG expansion (2.8). Thus, the Cauchy data are uniquely determined in terms of the free data (2.49) γ = g(0) , and τ = g(n) , which are arbitrary, subject to the transverse-traceless constraint (2.7) on g(n) and the constant scalar curvature constraint (2.16) or (2.19) on the representative γ ∈ [γ]. Abusing notation slightly, we will call (γ, τ ) Cauchy data for gαβ , since this data determines the rest of the Cauchy data ∂tk gαβ , 0 < k < n uniquely. The analysis above then gives: Proposition 2.2 The system (2.30)–(2.32) for ( gαβ , ρ, σ) is well posed in H s+n (I + ) n s + × H (I ), s > 2 + 2. Thus, given Cauchy data (γ, τ ) ∈ (H s+n (I + ), H s (I + )) satisfying (2.38), (2.39) and (2.49), and satisfying the constraints (2.7), (2.19), there is a unique solution ( gαβ , ρ, σ) of (2.30)–(2.32) with ( gαβ , ρ, σ) ∈ C(I, H s+n (I + )) ∩ C n (I, H s (I + )).
(2.50)
Further, if (γs , τs ) is a continuous curve in H s+n (I + )×H s (I + ), then the solutions ( gs , ρs , σs ) vary continuously with s. Proof. In the local coordinates (t, xi ), the system (2.30)–(2.32) is symmetrizable and so for given Cauchy data on T n , it has a unique solution on I × T n satisfying (2.50), with T n in place of I + , cf. [20, §5.2–5.3] or also [21], [22]. The existence of such local solutions holds for s > n2 + 1. By restriction, one thus obtains local solutions on the domain I × U ⊂ I × T n , for U ⊂ I + as preceding (2.30). To prove these local solutions obtained from domains on I×I + patch together to give a unique solution on I × I + , it is necessary and sufficient to prove that the system (2.30)–(2.32) has finite domains of dependence (or equivalently uniqueness in the local Cauchy problem). This is well known to be true for symmetric hyperbolic systems of PDE’s, (cf. [22, §IV.4] for example); however, the reduction of (2.30) is to a symmetric system of first-order pseudodifferential equations, for which finite propagation speed is not true in general. Nevertheless, standard methods show that solutions of (2.30)–(2.32) do have local uniqueness in the Cauchy problem. Thus, consider the closed subsystem (2.30) for g. A standard argument using the mean-value theorem shows that it suffices to prove local uniqueness in the Cauchy problem for the associated linear equation L( g ) = 0, where the coefficients of (2.30) are frozen at a given metric g0 , cf. [12] for instance. The linear operator L is self-adjoint at leading order, and hence by Proposition 2.1, one has existence and uniqueness of solutions to the Cauchy problem on I × T n for the adjoint equation L∗ u = φ. It is then well-known, cf. [22, Thm.IV.4.3], that this suffices to prove local uniqueness in the Cauchy problem for L, and hence for the system (2.30). Given the local uniqueness for g, exactly the same method can be applied to (2.31)–(2.32) to give local uniqueness for ρ and σ.
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It follows that the system (2.30)–(2.32) does have finite propagation speed, and hence the local solutions patch together uniquely to give a unique global solution on I × I + satisfying (2.50). As is well known in the case of Einstein metrics, (cf. [12] for example), the patching of local coordinate charts requires an extra derivative; the same analysis holds here, so that we assume s > n2 + 2. A lower bound for the time of existence I = [0, t0 ) of the solution depends only on an upper bound on the norm of the initial data in H s+n × H s . This implies that the last statement is an immediate consequence of the existence and uniqueness theorem. Given a solution ( gαβ , ρ, σ) of the Cauchy problem, one may then construct the “physical” metric g by setting g = ρ−2 σ −2 g.
(2.51)
Since σ is bounded away from 0 and ∞ near I + and ρ is a geodesic defining function, it is easy to see that the metric g is future geodesically complete, i.e., geodesically complete to the future of some Cauchy surface Σ ⊂ (M n+1 , g). However, it is not so clear that the metric g is Einstein, or equivalently that the metric g is conformally Einstein. For this, one needs to verify first that the gauge condition (2.42) that the coordinates remain harmonic, is preserved for the solution g. If this is so, it then follows that g is a solution of the equation (2.11). Secondly, the equation (2.11) admits solutions which are not conformally Einstein, and so one needs to verify that the constructed solution g actually is conformally Einstein. Both of these conditions on g can be verified by computation; however, the computations will be somewhat long and involved. Instead, we verify both conditions together by using the following simple conceptual technique, based on analyticity. Proposition 2.3 Any solution ( gαβ , ρ, σ) of the Cauchy problem in Proposition 2.2 defines an Einstein metric in dS + via (2.51). Proof. Suppose first that the free data (γ, g(n) ) are analytic on I + (or on a domain in I + ). As noted above, the expansion (2.5) then converges to gρ and gives a solution, denoted gE , to the Einstein equations (1.1); this metric has the form (2.4), with compactification g¯E , in a neighborhood of I + . In particular, both g¯E and ρ are analytic near I + . Moreover, since the coefficients of the equation (2.27) defining σ are then also analytic, and since the Cauchy data (2.38) for σ are analytic, the Cauchy-Kowalewsky theorem ([21, §16.4]) shows that σ and hence gE is a solution of (2.11), and gE = σ 2 g¯E are analytic near I + . Of course the metric hence a solution of (2.30)–(2.32). with Cauchy data determined by (2.38)–(2.39) and (γ, g(n) ). On the other hand, let ( g , ρ, σ) be the solution to the Cauchy problem (2.30)– (2.32) with Cauchy data determined by (2.38)–(2.39) and (γ, g(n) ) given by Proposition 2.2. Since ( gE , ρ, σ) is also a solution of this Cauchy problem, with the same
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initial data, it follows from the uniqueness part of Proposition 2.2 that gE = g in a neighborhood of I + . Moreover, the results in [1] show that gE remains analytic within its globally hyperbolic development, so that gE = g everywhere in the domain of g. Hence g = ρ−2 σ −2 g is an Einstein metric in dS + , realizing the given data on I + . (In particular, this shows that the harmonic gauge (2.42) must be preserved for analytic initial data.) Now analytic data (γ, g(n) ) are dense in H s+n × H s data with respect to the s+n × H s topology on I + . The Cauchy stability given by Proposition 2.2 implies H that if (γi , g(n),i ) is a sequence of analytic Cauchy data converging in H s+n ×H s to gi , ρi , σi ) also converge H s+n ×H s data (γ, g(n) ), then the corresponding solutions ( to the (unique) solution ( g , ρ, σ) with Cauchy data (γ, g(n) ). Hence the metric g in (2.51) is Einstein. Propositions 2.2–2.3 give the existence of an Einstein metric (M n+1 , g) ∈ dS + , with arbitrarily prescribed asymptotic behavior (γ, τ ) on I + , subject to the constraints (2.7) and (2.19). Suppose [γ ] = [γ] and [τ ] = [τ ] as following (1.5), so ¯ →M ¯, that γ = λ2 γ and τ = f (λ)τ . Given λ, there exists a diffeomorphism φ : M with φ|I + = id, such that limρ→0 φ∗ (ρ)/ρ = λ−1 , where ρ is the geodesic defining function determined by γ. Setting g = φ∗ g, one has g¯ = ρ2 φ∗ (g) = φ∗ (¯ g )( φ∗ρ(ρ) )2 . Hence, the boundary metric of g¯ equals γ . Similarly, by the uniqueness, any Einstein metric (M n+1 , g ) with Cauchy data (γ , τ ) satisfying (1.5) differs from (M n+1 , g) by a diffeomorphism φ equal to the identity on I + . This completes the proof of Theorem 1.1. Remark 2.4 Theorem 1.1 is formulated as a global result, in the sense that future conformal infinity I + is a compact smooth manifold. However, Propositions 2.12.3 all hold locally, by the finite propagation speed of the system (2.30)–(2.32). Hence, Theorem 1.1 also holds locally, where I + is an open manifold with a finite number of local charts. Of course the uniqueness statement then holds only within the domain of dependence of the initial data. Many of the standard solutions of the Einstein equations (2.1) have I noncompact; this is the case for instance for the dS Schwarzschild metrics. Proof of Theorem 1.2. Let (M, g0 ) be a dS Einstein metric in dS ± with Cauchy + − data ([γ + ], [g(n) ]) and ([γ − ], [g(n) ]) induced on I + and I − respectively. Thus, there ¯ → R such that g¯0 = Ω2 g0 extends to a exists a smooth defining function Ω : M + − ¯ metric on M = M ∪ I ∪ I I × Σ; here Σ is a Cauchy surface for (M, g0 ) and I is a compact time interval. We will choose Ω to be a geodesic defining function ρ in a neighborhood of I + and I − so that g¯0 is C(I × H s+n (Σ)) ∩ C n (I × H s (Σ)) up to ¯ . The choice of Ω defines representatives (γ + , g + ), (γ − , g − ) in the conformal M (n) (n) + − classes ([γ + ], [g(n) ]), ([γ − ], [g(n) ]). In the following, we work in the H s+n × H s topology. Let U + be an open neighborhood of Cauchy data on I + containing the given + + ). Then for all data (ˆ γ + , gˆ(n) ) ∈ U + , there exists T < ∞, depending data (γ + , g(n)
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on U + , such that the maximal globally hyperbolic dS Einstein metric (M n+1 , gˆ) + having Cauchy data (ˆ γ + , gˆ(n) ) given by Theorem 1.1 is defined on [T, ∞) × Σ; here the time factor is proper time t = − log ρ2 , where ρ is the geodesic defining function. The Cauchy data of such solutions gˆ at Σ = {T } × Σ then forms an open set U T in the space of Cauchy data for the Einstein equations (2.1) on Σ. By passing to an open subset V T ⊂ U T if necessary, the Cauchy stability theorem for the (standard) Einstein equations implies that the maximal globally hyperbolic development of any gˆ with data in V T contains the region [−T, T ]× Σ, and induces again an open set of Cauchy data V −T at {−T } × Σ. Then, as above with I + , there is an open set U − of Cauchy data on I − whose future development gives a non-empty open subset of V −T . Combining these three unique developments gives a global solution (M n+1 , gˆ) ∈ dS ± , which completes the proof of Theorem 1.2. Remark 2.5 The proof above shows that the space dS ± is also stable with respect to perturbations of the Cauchy data (Σ, γ, K), (satisfying the constraint equations), on a compact Cauchy surface Σ ⊂ (M n+1 , g) in the H s+n × H s+n−1 topology on Σ. It is well known, cf. [4] that this is not the case for perturbations of asymptotically flat Cauchy data when Λ = 0, in that smoothness of the resulting space-time at conformal infinity is lost for generic perturbations. Proof of Theorem 1.3. This result is proved for n = 3 in [2], and it is pointed out there that the same proof holds provided one has the Cauchy stability result of Theorem 1.1, (i.e., Friedrich’s result [8] when n = 3). Given then Theorem 1.1, the proof of Theorem 1.3 is exactly the same as that given in [2], to which we refer for details. Remark 2.6 This paper has focused on the de Sitter case Λ > 0 mainly for simplicity, but also because there are no direct analogues of Theorems 1.2 or 1.3 when Λ = 0 or Λ < 0, due to the more complicated nature of conformal infinity. Nevertheless, one expects that the analogues of Theorem 1.1 for Λ = 0 and Λ < 0, as formulated and proved by Friedrich [8] in the case n = 3, hold for all even dimensions. When Λ < 0, future space-like infinity I + is replaced by time-like infinity I, while when Λ = 0, I + is replaced by future null infinity. The case Λ = 0 has recently been worked out with P. Chrusciel, cf. [arXiv: gr-qc/0412020], (to appear in Comm. Math. Phys.). Remark 2.7 We close with a brief remark on the applicability of the methods used above to the Einstein equations coupled to other matter fields. As noted above, up to multiplicative constants, the tensor H is the metric variation, or stressenergy tensor, of the conformal anomaly or of the Q-curvature. The conformal invariance of these functionals corresponds to the conformal invariance of H. For functionals containing the metric coupled to other fields which are conformally invariant, and whose field equations are symmetric hyperbolic, it seems very likely that the methods used above will again lead to a well-posed Cauchy problem at I + , as in Theorem 1.1. In dimensions 3 + 1, this is the case for the Einstein
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equations coupled to gauge fields, i.e., Einstein-Maxwell or Einstein-Yang-Mills fields. Theorem 1.1 has already been proved in this situation by Friedrich [11], and so at best one would have a different method of proof of this result. In higher dimensions, the EM or YM action is not conformally invariant, and it is less clear if the method can be adapted to this situation.
References [1] S. Ahlinhac and G. M´etivier, Propagation de l’analyticit´e des solutions de syst`emes hyperboliques non-lin´eares, Inventiones Math. 75, 189–204 (1984). [2] M. Anderson, On the structure of asymptotically de Sitter and anti-de Sitter spaces, (preprint), [arXiv: hep-th/0407087], to appear in Adv. Theor. Math. Phys. [3] M. Anderson, Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds, Advances in Math. 179, 205–249 (2003). [4] L. Andersson and P.T. Chru´sciel, On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to the smoothness of Scri, Comm. Math. Phys. 161, 533–568 (1994). [5] S. deHaro, K. Skenderis and S.N. Solodukhin, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Comm. Math. Phys. 217, 595-622 (2001), [arXiv: hep-th/0002230]. [6] C. Fefferman and C.R. Graham, Conformal invariants, in Elie Cartan et les Math´ematiques d’Aujourd’hui, Ast´erisque, (1985), numero hors s´erie, Soc. Math. France, Paris, 95–116. [7] C.R. Graham and K. Hirachi, The ambient obstruction tensor and Qcurvature, (preprint), [arXiv: math.DG/0405068]. [8] H. Friedrich, On the existence of n-geodesically complete or future complete solutions of Einstein’s equations with smooth asymptotic structure, Comm. Math. Phys. 107, 587–609 (1986). [9] H. Friedrich, Existence and structure of past asymptotically simple solutions of Einstein’s field equations with postive cosmological constant, J. Geom. Phys. 3, 101–117 (1986). [10] H. Friedrich, Conformal Einstein evolution, in The Conformal Structure of Space-Time, J. Frauendiener and H. Friedrich, Eds., Lecture Notes in Physics, vol. 604, Springer Verlag, Berlin, 1–50 (2002). [11] H. Friedrich, On the global existence and the asymptotic behavior of solutions to the Einstein-Maxwell-Yang-Mills equations, Jour. Diff. Geom. 34, 275–345 (1991).
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[12] H. Friedrich and A. Rendall, The Cauchy problem for the Einstein equations, in Einstein’s Field Equations and Their Physical Implications, B.G. Schmidt (Ed.), Springer Lecture Notes in Physics, vol. 540, Springer Verlag, Berlin, 127–223 (2000). [13] C.N. Kozameh and E.T. Newman, A new approach to the vacuum Einstein equations, in Asymptoic Behavior of Mass and Space-Time Geometry, Lecture Notes in Physics, Vol. 202, F.J. Flaherty, (Ed.), Springer Verlag, New York, (1984). [14] S. Kichenassamy, On a conjecture of Fefferman-Graham, Advances in Math. 184, 268–288 (2004). [15] C.N. Kozameh, E.T. Newman and K.P. Tod, Conformal Einstein spaces, Gen. Relativ. and Gravit. 17, 343–352 (1985). [16] L.J. Mason, The vacuum and Bach equations in terms of light cone cuts, Jour. Math. Phys. 36(7), 3704–3721 (1995). [17] A. Rendall, Asymptotics of solutions of the Einstein equations with positive cosmological constant, Annales Henri Poincar´e 5, 1041–1064 (2004), [arXiv.org: gr-qc/0312020]. [18] K. Skenderis, Lectures notes on holographic renormalization, Class. Quantum Grav. 19, 5849–5876 (2002), [arXiv: hep-th/0209067]. [19] A. Starobinsky, Isotropization of arbitrary cosmological expansion given an effective cosmological constant, JETP Lett 37, 66–69 (1983). [20] M.E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics Series, vol. 100, Birkh¨ auser Verlag, Boston, (1991). [21] M.E. Taylor, Partial Differential Equations III, Applied Math. Sciences Series, vol. 117, Springer Verlag, New York, (1996). [22] M.E. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, (1981). Michael T. Anderson Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, NY 11794-3651 USA email:
[email protected] Communicated by Sergiu Klainerman submitted 30/08/04, accepted 27/01/05
Ann. Henri Poincar´e 6 (2005) 821 – 847 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05821-27, Published online 05.10.2005 DOI 10.1007/s00023-005-0225-9
Annales Henri Poincar´ e
On Initial Conditions and Global Existence for Accelerating Cosmologies from String Theory Makoto Narita Abstract. We construct a solution satisfying initial conditions for accelerating cosmologies from string/M-theory. Gowdy symmetric spacetimes with a positive potential are considered. Also, a global existence theorem for the spacetimes is shown.
1 Introduction It is expected that the inflation paradigm would be explained within superstring/ M-theory. The theory predicts that spacetime dimension is greater than four. Since observable spacetime dimension is four, it is thought that the extra dimensions would be compactified within Planck scale. Recently, it has been pointed out that it is possible to find cosmological solutions which exhibit a transient phase of accelerated expansion of the universe (like inflation) if the size of the compactified internal hyperbolic space depends on time and/or if they are S(pacelike)-brane solutions. In these models, exponential potential terms like V0 eaψ appear, where ψ denotes the compactification volume or effective dilaton field, a is a coupling constant and V0 is positive number. Explicitly, a typical action for the case is of the form √ 1 S = d4 x −g − 4R + (∇ψ)2 + V0 eaψ . (1) 2 Then, it is explained that if it would be supposed that, in the case of a > 0, the field ψ starts at a large negative value (i.e., the potential term can be neglected) with high kinetic energy (∂t ψ is positive and large enough)1 near cosmological initial singularities, then, the scalar field runs up the exponential potential, turn around and falls back. At the turning point, the potential term becomes dominant, i.e., the universe makes accelerated expansion. Thus, the universe starts out in a decelerated expansion phase (asymptotic past) and enters an accelerating phase (intermediate era), and after these, the expansion becomes deceleration again (asymptotic future). We call this scenario paradigm-A. We would like to investigate this interesting paradigm from viewpoint of mathematical relativity and cosmology. It is important to study rigorously whether or not the paradigm-A is acceptable. In particular, it should be shown that the assumption of the initial conditions for ψ is generic because, as indicated previously [EG], the accelerated expansion of the universe is all the result of the initial 1 In
the case of a < 0, ψ and ∂t ψ start at large positive and negative values, respectively.
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conditions. That is, (Q1): Are there singular solutions satisfying initial conditions in paradigm-A to the Einstein-matter equations in generic? Furthermore, to be complete the scenario of paradigm-A, we should show global existence theorems, i.e., (Q2): Are there global solutions to the Einsteinmatter equations with such exponential potentials? Unlike BKL [BKL] or cosmic no-hair conjectures [WR], which are problems in only asymptotic (local) regions of spacetimes, the paradigm-A is a global (in time) problem as mentioned already. In addition, it is also important as the first step to prove the strong cosmic censorship. For (Q1), to construct solutions satisfying the initial condition of paradigm-A, we will use the Fuchsian algorithm developed by Kichenassamy and Rendall [KR]. It is interpreted that the class of solutions we are looking for here is a subclass of asymptotically velocity-terms dominated (AVTD) singular solutions since potential terms are neglected near the singularities and, in addition, signature of the time derivative of the scalar field is restricted. By using the method, it has been shown that there are AVTD singularities in (non-)vacuum Gowdy, polarized T 2 -, polarized U (1)-symmetric spacetimes and the Einstein-scalar-p-from system without symmetry assumptions [AR, DHRW, IK, IM, NTM]. Also, systems with an exponential potential as given in (1) have been discussed formally in [DHRW, RA00]. Thus, our result is not only an answer for (Q1), but also it complements previous results. For (Q2), we want to analyze Gowdy symmetric spacetimes. Future global existence theorems for spatially compact, locally homogeneous spacetimes [LH03, LH04, RA04] and hyperbolic symmetric spacetimes [TR] with a positive potential (or a positive cosmological constant) have been proved. These spacetimes do not include gravitational waves. Also, although global existence theorems for Gowdy (more generally, T 2 -) symmetric spacetimes with or without matter have been shown [AH, ARW, BCIM, IW, MV, NM02, NM03, WM], it has not been prove the theorems for the spacetimes with a positive potential. Therefore, spacetimes with dynamical degrees of freedom of gravity and with the positive potential should be considered as the next step. As a model, we choose the bosonic action arising in low energy effective superstring (supergravity) theory since we have a similar action with (1) after the toroidal compactification of the extra dimensions. There are anti-symmetric two-form, Bµν , and three-form, Cµνρ fields in the action. It is known that, in general, p-form fields in n-dimensional spacetimes may violate the strong energy condition for p ≥ n − 1 and then, accelerated expansion of the universe would be expected [GG]. Here, we do not consider hyperbolic compactification of the extra higher dimensions, but the only fluxes of four-form field strengths are investigated because these have essentially the same effects to obtain the exponential potential terms as (1) [EG, TP, WMNR]. Then, our purposes are to construct singular solutions satisfying conditions of paradigm-A and to show a global existence theorem for Gowdy symmetric spacetimes with stringy matter fields.
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Action
The dimensionally reduced effective action in the Einstein frame is given by 1 1 −2λφ 2 1 −2λφ 2 4 √ 4 2 e e H + F , SIIA = d x −g − R + (∇φ) + 2 2 · 3! 2 · 4!
(2)
where g is the determinant of the metric gµν on a four-dimensional spacetime manifold M , 4R is the Ricci scalar of gµν , φ is the dilaton field, H = dB is the three-form field strength, F = dC is the four-form field strength and λ is a coupling constant. If λ = 1, we have the action for the type IIA supergravity in the absence of vector fields and the Chern-Simons term [LWC]. In four dimensions, there is a duality between the three-form field strength and a one-form, which is interpreted as the gradient of a scalar field. Then, we may define the pseudo-scalar axion field σ as follows: H µνρ = µνρκ e2λφ ∇κ σ.
(3)
∇µ e−2λφ F µνρκ = 0,
(4)
∂[α Fµνρκ] = 0,
(5)
Also, the field equation
and the Bianchi identity
for the four-form field strength can be solved by F µνρκ = Qµνρκ e2λφ ,
(6)
where Q is an arbitrary constant. Thus, after taking the dual transformation and solving the field equations for F , we have a reduced effective action for the IIA system of the form 1 4 √ 4 2 2λφ 2 2 2λφ (∇φ) + e (∇σ) + Q e . (7) SIIA∗ = d x −g − R + 2 Hereafter, we assume Q = 0. Thus, we have the action which is the same from with (1).
1.2
Field equations for Gowdy symmetric spacetimes
The Gowdy symmetric spacetimes admit a T 2 isometry group with spacelike orbits and the twists associated to the group vanish [GR]. The topology of spatial section can be accepted S 3 , S 2 × S 1 , T 3 or the lens space [CP]. In this paper, we assume T 3 spacelike topology.
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Now, we will choose a coordinate, which is the areal time one. This means that time t is proportional to the geometric area of the orbits of the isometry group. Explicitly, ds = −e2(η−U) αdt2 + e2(η−U) dθ2 + e2U (dx + Ady)2 + e−2U t2 dy 2
(8)
2
where ∂/∂x and ∂/∂y are Killing vector fields generating the T group action, and η, α, U and A are functions of t ∈ (0, ∞) and θ ∈ S 1 . It is also assumed that functions describing behavior of matter fields are ones of t and θ. Let us show the field equations obtained by varying the action (7) in the areal coordinate (8). Constraint equations η˙ = U˙ 2 t
+ +
e4U αU 2 + 2 (A˙ 2 + αA2 ) 4t 1 ˙2 φ + αφ2 + e2λφ (σ˙ 2 + ασ 2 ) + αQ2 e2λφ+2(η−U) , 4
(9)
η 1 ˙ e4U ˙ α = 2U˙ U + 2 AA + (φφ − + e2λφ σσ ˙ ), t 2t 2tα 2
(10)
α˙ = −tα2 Q2 e2λφ+2(η−U) .
(11)
Evolution equations η¨ − αη = +
η˙ α˙ α2 α η α e4U + − + − U˙ 2 + αU 2 + 2 (A˙ 2 − αA2 ) 2 2α 4α 2 4t 1 ˙2 2 2λφ 2 2 −φ + αφ + e (−σ˙ + ασ ) + αQ2 e2λφ+2(η−U) , (12) 4
4U ˙ ˙ ¨ − αU = − U + α˙ U + α U + e (A˙ 2 − αA2 ) + 1 αQ2 e2λφ+2(η−U) , U t 2α 2 2t2 4
(13)
A˙ α˙ A˙ α A A¨ − αA = + + − 4(A˙ U˙ − αA U ), t 2α 2
(14)
α˙ φ˙ α φ φ˙ φ¨ − αφ = − + + + λe2λφ (σ˙ 2 − ασ 2 ) − λαQ2 e2λφ+2(η−U) , t 2α 2
(15)
α˙ σ˙ α σ σ˙ + + − 2λ(φ˙ σ˙ − αφ σ ). (16) t 2α 2 Hereafter, dot and prime denote derivative with respect to t and θ, respectively. We will call this system of partial differential equations (PDEs) Gowdy symmetric IIA system. Note that these equations are not independent because the wave equation (12) for η can be derived from other equations. Indeed, there are only two dynamical degree of freedom (i.e., U and A) in the Gowdy symmetric spacetimes. σ ¨ − ασ = −
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2 Initial singularities Consider the problem (Q1). To begin with a brief review of the Fuchsian algorithm, which is a method to construct exact singular solutions to a PDE system near a singularity (t = 0). The algorithm is based on the following idea: near the singularity, decompose the singular formal solutions into a singular part, which depends on a number of arbitrary functions, and a regular part u. If the system can be written as a Fuchsian system of the form [D + N (x)] u = tf (t, x, u, ∂x u),
(17)
where D := t∂t and f is a vector-valued regular function, then the following theorem can be applied: Theorem 1 [KR] Assume that N is an analytic matrix near x = x0 such that there is a constant C with ΛN ≤ C for 0 < Λ < 1. In addition, suppose that f is a locally Lipschitz function of u and ∂x u which preserves analyticity in x and continuity in t. Then, the Fuchsian system (17) has a unique solution in a neighborhood of x = x0 and t = 0 which is analytic in x and continuous in t and tend to zero as t → 0. Thus, the regular part goes to zero and the singular part of the formal solution becomes an exact solution to the original PDE system near the singularity. Unlike the vacuum Gowdy case, the evolution equations (13)–(16) do not decouple from the constraint equations (9)–(11), since they contain the function α. Therefore, according to [IK], we take equations (9), (11), (13)–(16) as effective evolution ones and (10) as the only effective constraint equation. This is not a standard setup for the initial-value problem for the Einstein-matter equations (see example [TM]). Therefore, it is not clear whether the initial-value problem for our case away from the singularity at t = 0 has a unique solution or not, unless it is shown that the constraint (10) propagates. Let us show the local existence and uniqueness of our initial-value problem. We can obtain the following first-order system for z from the PDE system (9), (11), (13)–(16): ∂t z = f (t, θ, z , ∂θ z ),
(18)
˙ φ , σ, σ, ˙ A , φ, φ, ˙ σ , α, η). This means that the PDE syswhere z := (U, U˙ , U , A, A, tem is of Cauchy-Kowalewskaya type. Thus, ignoring the constraint equation (10), we have a unique solution to the effective evolution equations by prescribing the analytic initial data for t = t0 > 0 if all functions are analytic. Now, to assure the local existence and uniqueness of the initial-value problem, we must show that the constraint (10) propagates.
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Let us set N := η − 2DU U −
e4U 1 e2λφ α Dφφ Dσσ , DAA − − + 2t2 2 2 2α
(19)
Computing 0 = Dη − (Dη)
e2λφ e4U 1 α Dσσ − = DN + D 2DU U + 2 DAA + Dφφ − (Dη) , 2t 2 2 2α
(20)
we have a linear, homogeneous ordinary differential equation (ODE) for N of the form DN −
Dα N = 0. 2α
(21)
Thus, the uniqueness theorem for ODEs guarantees that N is identically zero for any time t if we set initial data for t = t0 such that N (t0 ) = 0. Thus, the local existence and uniqueness of the initial-value problem for our case has been shown in the analytic case. In appendix, we shall consider the smooth version of the initial-value problem for our non-standard setup of the Gowdy symmetric IIA system.
2.1
Application of the Fuchsian algorithm
Let us construct AVTD singular solutions to the Gowdy symmetric IIA system. First, we will consider the case that a solution has a maximum number of free functions. In this sense, the solution (given in Theorem 2) is generic. Neglecting spatial derivative and potential terms in the effective evolution equations, we have velocity-terms dominated (VTD) equations as follows: Dη = (DU )2 +
e4U 1 e2λφ 2 2 (Dφ) (Dσ)2 , (DA) + + 4t2 4 4 Dα = 0,
(23)
4U
e 1 DU Dα + 2 (DA)2 , 2α 4t 1 2 DADα − 4DU DA, D A = 2DA + 2α 1 D2 φ = DφDα + λe2λφ (Dσ)2 , 2α 1 DσDα − 2λDφDσ. D2 σ = 2α D2 U =
(22)
Solving this system of VTD equations, we have a VTD solution.
(24) (25) (26) (27)
Vol. 6, 2005
Global Properties for Accelerating Cosmologies
827
Then, the following formal solution is obtained:
κ(θ)2 η = k(θ)2 + ln t + η0 (θ) + t µ(t, θ), 4
(28)
α = α0 (θ) + t β(t, θ), U = k(θ) ln t + U0 (θ) + t V (t, θ),
(29) (30)
A = h(θ) + t2−4k (A0 (θ) + B(t, θ)) ,
(31)
φ = κ(θ) ln t + φ0 (θ) + t Φ(t, θ),
(32)
σ = ω(θ) + t−2λκ (σ0 (θ) + Σ(t, θ)) ,
(33)
where > 0,
0 < k(θ) <
1 , 2
α0 > 0
(34)
and −1 < λκ(θ) < 0.
(35)
Note that µ, β, V , B, Φ and Σ are regular parts and others are singular parts (=VTD solutions). Inserting this formal solution into the Einstein-matter equations, we obtain the following Fuchsian system: (D + N ) u = tδ f (t, θ, u, ∂θ u), where u := ui = (V , DV , t V , B, DB, t B , Φ, DΦ, i = 1, . . . , 14, f is a vector-valued regular function and 0 −1 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 2 − 4k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 N = 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − κ2 0 −2k −2k 0 0 0 0 − κ
2
(36) t Φ , Σ, DΣ, t Σ , β, µ), 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 −1 −2λκ 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
.
(37)
828
M. Narita
Ann. Henri Poincar´e
Note that δ > 0 if the condition (34), (35) and 3 1 κ2 + λκ + > 0 K := (k − )2 + 2 4 4
(38)
holds. To apply Theorem 1 to our Fuchsian system (36), we must verify that the boundedness condition for the matrix N holds. To do this, we have P −1 N P = N0 , where 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 − 4k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 , (39) N0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −2λκ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2k 0 0 0 0 0 κ2 0 0 0 0 0 and
P =
1 0 − −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 − −1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1
.
(40)
Vol. 6, 2005
Global Properties for Accelerating Cosmologies
829
Then, ΛN0 =
Λ 0 0 0 0 0 0 0 0 0 0 0 0 0
Λ ln Λ Λ 0 0 0 0 0 0 0 0 0 0 0 2kΛ ln Λ
0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0
Λ2−4k 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 Λ 0 0 0 0 0 0 0
0 0 0 0 0 0 Λ ln Λ Λ 0 0 0 0 0 κ Λ ln Λ 2
0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0
Λ−2λκ 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 Λ 0
0 0 0 0 0 0 0 0 0 0 0 0 0 Λ
(41)
hence P ΛN0 P −1 = ΛN is uniformly bounded for 0 < Λ < 1 if the condition (34) and (35) hold. Thus, there is a unique solution of the Fuchsian system (36) which goes to zero as t → 0, and which is analytic in θ and continuous in t. Note that (U, A, φ, σ, α, η) is a solution of the effective evolution equations of the Einstein-matter equations (9), (11), (13)–(16) if we construct (U, A, φ, σ, α, η) from (28)–(33) with V = u1 , B = u4 , Φ = u7 , Σ = u10 , β = u13 and µ = u14 . This fact follows from equations D(uI+2 − t uI ) = 0, where I = 1, 4, 7, 10. Now, we want to get a constraint condition to ensure that the solution obtained above is a genuine one to the full Einstein-matter equations. Since Dα/α = O(t ), N˙ α˙ = = O(t −1 ), N 2α
(42)
then, the right-hand side of the above equation is integrable. From this result, we can put a function P (t, θ) such that N ∝ exp P (t, θ).
(43)
This means that N is identically zero if we would choose the singular data such that N → 0 as t → 0, and then, the constraint equation (10) is satisfied. Inserting the formal solutions (28)–(33) into the constraint equation (19), we have N = η0 − 2kU0 − e4U0 (1 − 2k)h A0 −
κφ0 α + e2λφ0 κω σ0 + 0 + O(1), 2 2α0
(44)
830
M. Narita
Ann. Henri Poincar´e
where O(1) is some expression which tends to zero as t → 0. Thus, the constraint holds iff the singular data satisfy η0 − 2kU0 − e4U0 (1 − 2k)h A0 −
κφ0 α + e2λφ0 κω σ0 + 0 = 0. 2 2α0
(45)
To summarize, we have the following theorem: Theorem 2 Choose data such that conditions (34), (35) and (45) are satisfied. Suppose that is a positive constant less than min{4k, 2 − 4k, −2λκ, 2 + 2λκ, 2K}. For any choice of the analytic singular data η0 (θ), α0 (θ), k(θ), U0 (θ), h(θ), A0 (θ), κ(θ), φ0 (θ), ω(θ) and σ0 (θ), the Gowdy symmetric IIA system has a solution of the form (28)–(33), where µ, β, V , B, Φ and Σ tend to zero as t → 0. Although the solution given in Theorem 2 is generic in the sense that the solution has a maximum number of free functions, conditions for paradigm-A does not hold since λκ < 0, i.e., the universe starts with large potential and wrong sign of the time derivative of φ. To verify the validity of the paradigm-A we need to construct a solution allowing a condition λκ > 0. Indeed, this problem can be solved as follows. If an AVTD solution with λκ > 0 are needed, we replace expansion (33) with σ = ω(θ) + t Σ(t, θ).
(46)
In this case, −2λκ and Λ−2λκ sitting the 11th line and the 11th row in the matrices N and ΛN0 are replaced by and Λ , respectively. Also, the constraint condition for the singular data becomes η0 − 2kU0 − e4U0 (1 − 2k)h A0 −
κφ0 α + 0 = 0. 2 2α0
(47)
Thus, we have the following theorem which is consistent with conditions of paradigm-A. Theorem 3 Choose data such that conditions (34), (47) and λκ > −1/2 are satisfied. Suppose that is a positive constant such that max{0,−2λκ} < < min{4k,2− 4k}. For any choice of the analytic singular data η0 (θ), α0 (θ), k(θ), U0 (θ), h(θ), A0 (θ), κ(θ), φ0 (θ) and ω(θ), the Gowdy symmetric IIA system has a solution of the form (28)–(32) and (46), where µ, β, V , B, Φ and Σ tend to zero as t → 0. The positivity of K is automatically satisfied when 0 < k < 1/2 and λκ > −1/2 hold. Then, a solution to the Gowdy symmetric IIA system allowing the initial conditions for paradigm-A has been constructed. Note that we do not have the maximum number of free functions in this case. Thus, the solution given in Theorem 3 is restricted than generic one given in Theorem 2. The reason why we do not have the maximum number is the existence of dilaton coupling with kinetic terms of other fields (the axion field in our case). Generically, all fields arising in superstring/M-theory couple with the dilaton field. Therefore, we may not avoid such restriction for solutions to our problem unless the dilaton coupling is ignored.
Vol. 6, 2005
Global Properties for Accelerating Cosmologies
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3 Global existence Now, consider the problem (Q2). We will show the following theorem: Theorem 4 Let (M, g, φ, σ) be the maximal Cauchy development of C ∞ initial data for the Gowdy symmetric IIA system. Suppose that the timelike convergence condition (TCC), which is Rµν W µ W ν ≥ 0 for any timelike vector W µ , holds and ¯ such that |λ| ≤ λ ¯ < 1/2. Then, M can be covered there is a positive constant λ by compact Cauchy surfaces of constant areal time t with each value in the range (0, ∞). In the first place, we need a local existence theorem for the Gowdy symmetric IIA system, which is the Einstein-(minimally coupled) scalar system with a positive potential. Fortunately, there is no coupling caused by existence of such matter fields in the principal part of the PDE system. For this reason, since the local existence theorems for vacuum Gowdy (more generically, T 2 -symmetric) spacetimes have been shown [MV, CP], the same theorem for the Gowdy symmetric IIA system can be shown as vacuum case [FR]. Thus, it is enough to verify uniform bounds of functions (η, α, U, A, φ, σ) and their first and second derivatives to prove global existence [MA]. The strategy is similar with the case of T 2 -symmetric Einstein(Vlasov) system [AH, ARW, BCIM, IW, WM]. Let us define γ := η +
1 ln α. 2
(48)
By using γ we can rewrite the constraint equations as follows: Q2 2(γ+λφ−U) γ˙ =E− e , t 4 F γ =√ , t α α˙ = −tαQ2 e2(γ+λφ−U) ,
(49) (50) (51)
where 1 e4U E := U˙ 2 + αU 2 + 2 A˙ 2 + αA2 + φ˙ 2 + αφ2 + e2λφ σ˙ 2 + ασ 2 , 4t 4 and F :=
√ e4U ˙ 1 ˙ φφ + e2λφ σσ α 2U˙ U + 2 AA + ˙ . 2t 2
Define energies for the Gowdy symmetric IIA system 1 1 √ E + αQ2 e2(η+λφ−U) dθ, E(t) := α 4 S1
(52)
(53)
(54)
832
M. Narita
and
˜ := E(t) S1
Ann. Henri Poincar´e
E √ dθ, α
(55)
In our case, the TCC is as follows: 1 φ˙ 2 + e2λφ σ˙ 2 ≥ αQ2 e2(η+λφ−U) 2
(56)
First, we will show energy decay and energy inequalities (see Lemmas 1 and 3 in [IW]). ¯ < 1/2. Then, E and E ˜ Lemma 1 Suppose the TCC and the condition |λ| ≤ λ decrease monotonically along time t, that is, dE(t) <0 dt
and
˜ dE(t) < 0, dt
(57)
and E and E˜ are bounded on (T− , T+ ), where 0 < T− < ti < T+ < ∞. Further˜− , satisfying more, there exists numbers, E− and E E− = lim E(t) t→T−
and
˜ ˜− = lim E(t). E t→T−
Proof. One can calculate directly as follows: 4U 2λφ 2 ˙2 dE(t) e 1 σ ˙ e φ 2 2 √ =− + 2U˙ + 2 αA + dθ ≤ 0. dt αt 2t 2 2 S1
(58)
(59)
Thus, E(t) is controlled by E(ti ) for any t ∈ [ti , T+ ). The right-hand side of equation (59) can be controlled by E: dE 4 ≥ − E. dt t
(60)
4 ti E(t) ≤ E(ti ) . t
(61)
For any t ∈ (T− , ti ], we have
Then, E(t) ≤ E(ti )
ti T−
4
on (T− , ti ). This boundedness and the monotonicity of
E(t) assert that E(t) continuously extends to T− and then E− exists. ˜ By direct calculation, we have Next, we show the same results for E(t). ˜ e2λφ σ˙ 2 dE(t) 2 e4U φ˙ 2 2 2 ˙ U + 2 αA + = + − √ dt αt 4t 4 4 S1
E 1 α˙ √ + √ U˙ − λφ˙ dθ. + (62) 2α α t α
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Global Properties for Accelerating Cosmologies
833
˜ from the above form. Now, We cannot conclude the monotonicity for E(t)
2
2 α˙ U˙ 2 α˙ 1 2U˙ 2 α˙ √ =− √ √ U˙ − + , − √ + t α 2tα α t α 8α 32t α α
(63)
2
2 2φ˙ 2 λα˙ φ˙ 1 λ2 λα˙ α˙ ˙ √ =− √ − √ − + √ . φ+ 4t α 2tα α 2t α 2α 8t α α
(64)
and
Therefore, ˜ dE(t) = dt
S1
− +
2
2 4U 2λφ 2 α ˙ σ ˙ 2 e 1 e λ α ˙ √ U˙ − + 2 αA2 + + φ˙ + αt 8α 4t 4 2α 4
1 α˙ E λ2 √ 2 2(η+λφ−U) √ − + αQ e dθ, (65) 2α 16 4 α
where equation (11) has been used. By using the TCC and the inequality |λ| ≤ ˜ ¯ < 1/2, we have the conclusion of the monotonic nonincreasing property for E(t), λ ˜ dE ≤ dt
S1
Cλ α˙ E √ dθ ≤ 0, 2 α α
(66)
where Cλ < 1 is a positive constant depending only λ. ˜ ≤ E(t) for any time t. Therefore, one can see that Now, it follows that E(t) ˜ E(t) also extend continuously to T− by the monotonicity of it. Next two lemmas will be used to control dynamical parts (i.e., U , A, φ and σ) of the system. The method of the proof is based on the light cone estimate [MV, BCIM]. Lemma 2 If αα ˙ −1 is bounded, E is bounded on (T− , T+ ) × S 1 .
√ Proof. Differentiating quantities, E and F , along null directions ∂ζ := ∂t − α∂θ √ and ∂ξ := ∂t + α∂θ , we have 1 α˙ e4U 1 1 ∂ζ (E + F ) = (E + F ) − 2U˙ 2 + 2 αA2 + φ˙ 2 + e2λφ σ˙ 2 + F α t 2t 2 2 √ α˙ ˙ √ − =: L+ , (67) U + αU − λ φ˙ + αφ 2tα and ∂ξ (E − F) =
α˙ (E − F) − α
1 e4U 1 1 2U˙ 2 + 2 αA2 + φ˙ 2 + e2λφ σ˙ 2 − F t 2t 2 2 √ α˙ ˙ √ =: L− . (68) U − αU − λ φ˙ − αφ 2tα
834
M. Narita
Note that √ √ U˙ ± αU − λ φ˙ ± αφ
2 1 √ √ 2 αU + λ2 φ˙ ± αφ + 2 1 ≤ (1 + λ2 )(E + F ) + 2 1 (69) ≤ 2(1 + λ2 )E + , 2 ≤
Ann. Henri Poincar´e
U˙ ±
where |F | ≤ E has been used. Thus, α˙ CE 1 3E |L± | ≤ 2E + + , + α t 4t t
(70)
where C is a positive constant. Consider a point (t, θ) ∈ [ti , T+ ) × S 1 . Integrating the both sides of equations (67) and (68) along null passes, ∂ζ and ∂ξ , from points (ti , θ+ ) and (ti , θ− ) to the point (t, θ), respectively, we have (71) ∂ζ (E + F )dζ = E(t, θ) + F (t, θ) − E(ti , θ+ ) − F(ti , θ+ ) = L+ dζ, and
∂ξ (E − F)dξ = E(t, θ) − F(t, θ) − E(ti , θ− ) + F (ti , θ− ) =
L− dξ.
Adding these equations and using the inequality |F | ≤ E, 1 E(t, θ) ≤ E(ti , θ+ ) + E(ti , θ− ) + |L+ | dζ + |L− | dξ . 2
(72)
(73)
Taking supremums over all values of the space coordinate θ on the both sides of the inequality (73), we have
t α˙ 2 sup E 1 + C + 1 + 3 sup E ds sup E(t, θ) ≤ 2 sup E(ti , θ) + α s 4s s S1 S1 S1 S1 ti t = C1 (t) + C2 (s) sup E(s, θ)ds, (74) ti
S1
where Ci (t) are bounded and positive functions of t. We now apply Gronwall’s lemma to this inequality (74), we have boundedness for E on [ti , T+ ) × S 1. We can apply the same argument for t ∈ (T− , ti ] × S 1 , and then we have the conclusion of this lemma. Lemma 3 Let us define 4U 2 2 ¨ 2 + αU˙ 2 + e ¨2 + αA˙ 2 + 1 φ¨2 + αφ˙ 2 + e2λφ σ E˜ := U A ¨ , + α σ ˙ 4t2 4
(75)
Vol. 6, 2005
Global Properties for Accelerating Cosmologies
835
and F˜ :=
√ e4U ¨ ˙ 1 ¨ ˙ 2λφ ¨ ˙ φφ + e σ α 2U U + 2 AA + ¨ σ˙ . 2t 2
(76)
¨ are bounded, E˜ is bounded on If all functions and their first derivative, α˙ and α (T− , T+ ) × S 1 . Proof. Taking time derivative of the wave equations (13)–(16) for U , A, φ and σ, ˙ φ˙ and σ. we have wave equations for U˙ , A, ˙ Now, E˜ and F˜ satisfy equations of the form ˜+ ∂ζ (E˜ + F˜ ) = L
and
˜ −, ∂ξ (E˜ − F˜ ) = L
(77)
˜ ± involve nothing but controlled quantities, together with terms quadratic where L ¨ φ˙ , σ ¨ A˙ , φ, ¨ , U˙ , A, ¨ and σ˙ . Now, we can repeat the light cone argument and in U then, we have boundedness for E˜ on (T− , T+ ) × S 1 .
3.1
Past direction
Further estimates are given in each case of past and future directions, separately. First, consider the past direction. Lemma 4 For any t, the function γ satisfies the following condition, γ(t, θ) − min γ(t, θ) ≤ tE(t). max 1 1 S
S
(78)
Furthermore, for any t ∈ (T− , ti ], the functions U and φ satisfy the following conditions, max U (t, θ) − min U (t, θ) ≤ CE 1/2 (t), 1 1
(79)
φ(t, θ) − min φ(t, θ) ≤ CE 1/2 (t). max 1 1
(80)
S
S
and S
S
Proof. (cf. Step 1 of Section 5 in [AH]). For any θ1 , θ2 ∈ S 1 , we have θ2 θ2 θ2 tE ˜ ≤ tE(t), (81) √ dθ ≤ tE(t) γ dθ ≤ |γ | dθ ≤ |γ(t, θ2 ) − γ(t, θ1 )| = θ1 α θ1 θ1 where equation (50) and the fact |F | ≤ E have been used. Since θ1 and θ2 are arbitrary, the first conclusion follows.
836
M. Narita
Ann. Henri Poincar´e
Similarly, for any θ1 , θ2 ∈ S 1 and any t ∈ (T− , ti ], we have θ2 |U (t, θ2 ) − U (t, θ1 )| = U dθ θ1 1/2 1/2 θ2 θ2 √ dθ √ αU 2 dθ ≤ α θ1 θ1 1/2 θ2 dθ ˜ 1/2 E(t) ≤ α(ti ) θ1 ≤ CE(t)1/2 , where the H¨older inequality and the monotonicity of α have been used. The proof for φ is used the same argument.
(82)
Lemma 5 The function γ is bounded from above on (T− , ti ] × S 1 . Proof. (cf. Lemma 4 in [IW]). Note that 1 ˙ i )2 + e2λφ(ti ) σ(t φ(t ˙ i )2 ≥ α(ti )Q2 e2[η(ti )+λφ(ti )−U(ti )] > 0, 2
(83)
˜ i ) > 0. From since regular initial data at t = ti are supposed. This means E(t equation (66), we have ˜ C Q2 Cλ α˙ E E dE √ dθ = − λ ≤ (84) te2(γ+λφ−U) √ dθ, dt 2 α α 2 α 1 1 S S where Cλ < 1 is a positive constant depending on only the coupling constant λ. Suppose λ ≥ 0. Integrating this inequality from ti to t (0 < t < ti ), Cλ Q2 ti 2(γ+λφ−U) E ˜ ˜ √ dθ ds E(t) ≥ E(ti ) + se 2 α t S1
ti 2 Cλ Q ˜ ˜ E(s)ds ≥ E(ti ) + s exp 2 min γ + λ min φ − max U 2 S1 S1 S1 t
2 ti ˜ i ) 1 + Cλ Q ≥ E(t s exp 2 min γ + λ min φ − max U ds (85) , 2 S1 S1 S1 t where the monotonicity of E˜ has been used. From Lemma 4, min γ + λ min φ − max U 1 1 1 S
S
S
1/2 ≥ max γ + λ max φ − min U − tE(t) + (C λ − C )E(t) 1 2 S1 S1 S1 1/2 ≥ max γ + λ max φ − min U − t E(T ) + (C λ − C )E(τ ) , i − 1 2 1 1 1 S
S
S
(86)
Vol. 6, 2005
Global Properties for Accelerating Cosmologies
837
where C1 and C2 are positive constants, τ = ti if C1 λ − C2 < 0 and τ = T− if C1 λ − C2 ≥ 0. Thus, we have
ti 2 ˜ ≥ E(t ˜ i ) 1 + Cλ Q e−2(ti E(T− )+(C1 λ−C2 )E(τ )1/2 ) E(t) se2(γ+λφ−U) ds , (87) 2 t and then, ti se2(γ+λφ−U) ds ≤ t
1/2 2 e2(ti E(T− )+(C1 λ−C2 )E(τ ) ) Cλ Q2
˜ −) E(T −1 , ˜ i) E(t
(88)
where the condition (83) has been used. When one consider the case of λ < 0, we have the same results by exchanging maxS 1 φ and minS 1 φ in inequalities (85) and (86). Now, integrating equation (49), we have ti sQ2 2(γ+λφ−U) e sE − ds γ(t, θ) = γ(ti , θ) − 4 t Q2 ti 2(γ+λφ−U) se ds ≤ γ(ti , θ) + 4 t ˜ −) 1 2(ti E(T− )+(C1 λ−C2 )E(τ )1/2 ) E(T ≤ max e γ(ti , θ) + − 1 . (89) ˜ i) S1 Cλ 2 E(t Thus, the boundedness of γ from above has been shown.
Lemma 6 For any numbers a and b, and for n ≤ 12 , αn e2η+aφ−bU is bounded on (T− , ti ] × S 1 . Proof. (cf. Lemma 5 in [WM]). ∂t tk αn e2η+aφ−bU
k nα˙ ˙ ˙ + + 2η˙ + aφ − bU tk αn e2η+aφ−bU = t α
2 b t ˙ a 2 e4U ˙ φ+ = 2t U − + + 2αU 2 + 2 A˙ 2 + αA2 4t 2 t 2t
1 1 − n tαQ2 e2(η+λφ−U) tk αn e2η+aφ−bU + αφ2 + e2λφ (σ˙ 2 + ασ 2 ) + 2 2 ≥ 0, (90) where we have chosen 8k = 4a2 + b2 . Then, we have
k ti α(ti , θ)n e2η(ti ,θ)+aφ(ti ,θ)−bU(ti ,θ) , α(t, θ)n e2η(t,θ)+aφ(t,θ)−bU (t,θ) ≤ T− on (T− , ti ] × S 1 .
(91)
838
M. Narita
Ann. Henri Poincar´e
Lemma 7 α is bounded on (T− , ti ] × S 1 . Proof. Integrating the constraint equation (11), we have ti ti α˙ − ds = ln α(t) − ln α(ti ) = Q2 se2(γ+λφ−U) ds, α t t
(92)
for t ∈ (T− , ti ]. By using inequality (88), we have boundedness of ln α from above. As a result, 0 < α is also bounded. √ Lemma 8 For any numbers a and b, eγ+aφ−bU (= αeη+aφ−bU ) is bounded on (T− , ti ] × S 1 . Proof. We have already a result that e2η+aφ−bU is bounded on (T− , ti ] × S 1 (Lemma 6). Combining this and Lemma 7, the boundedness of eγ+aφ−bU on (T− , ti ] × S 1 follows directly. Corollary 1 αα ˙ −1 = ∂t (ln α) is bounded on (T− , ti ] × S 1 . Thus, ln α and α˙ are as well. Proof. Boundedness of αe2(η+aφ−bU ) is obtained by Lemma 8. From the constraint equation (51), we have αα ˙ −1 = −tαQ2 e2(λφ+η−U) . If we set a = λ and b = 1, the boundedness of the right-hand side of that equation is obtained. Thus, the conclusion of this lemma is shown. Lemma 9 The functions U , A, φ, σ and their first derivatives are bounded on (T− , ti ] × S 1 . Proof. From Lemma 2 and Corollary 1, we have the boundedness for E on ˙ ˙ 2U ˙ 2U λφ 1 (T− , ti ] × S . Then, U , |U |, φ, |φ |, e /2t A, e /2t A , e σ˙ and λφ e σ are bounded for all t ∈ (T− , T+ ). Once the boundedness on the first derivative of U and φ is obtained, it follows that U and φ are bounded for all t ∈ (T− , T+ ). ˙ A , σ˙ and σ , and consequently on A and σ. Then, we have bounds on A, Lemma 10 The functions α , α˙ and α ¨ are bounded on (T− , ti ] × S 1 . Also, η, η˙ and η are as well. Proof. (cf. Step 3 of Section 6 in [BCIM]). From the constraint equations (49) and (50), we have boundedness for γ˙ and γ directly. Then, γ is controlled. Differentiating both sides of equation (51) with respect to θ, we have α˙ = α −tQ2 e2(γ+λφ−U) − 2tQ2 e2(γ+λφ−U) α (γ + λφ − U ) . (93) Then, we have boundedness for α by integrating this differential equation for α in time since the coefficient of α and the second term in the right-hand side of the equation (93) are controlled. Thus, we have that η, η˙ and η is bounded.
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The boundedness of α˙ is obtained immediately from (93). Now, differentiating both sides of equation (51) with respect to t, we have α ¨ = −Q2 αe2(η+λφ−U) α + 2tα˙ + 2tα η˙ + λφ˙ − U˙ , (94)
which implies that α ¨ is bounded.
Lemma 11 The second derivatives of U , A, φ and σ are bounded on (T− , ti ] × S 1 . Proof. By Lemma 3 we have the boundedness for E˜ on (T− , ti ] × S 1 . Then, we have ¨ φ˙ , σ ¨ , U˙ , A, ¨ A˙ , φ, uniform bounds on U ¨ and σ˙ . Bounds on U , A , φ and σ follows from the wave equations (13)–(16) directly. Lemma 12 α , η¨, η˙ and η are bounded on (T− , ti ] × S 1 . Proof. By taking the time derivative of (49) and (50), we have bounds on γ¨ and γ˙ . Then, bounds on η¨ and η˙ are obtained by the definition of γ. Also, by taking the θ derivative of (50), we have bounds on γ . The boundedness for α follows from the same argument in the proof of Lemma 10. That is, differentiating both sides of equation (93) with respect to θ, we have α˙ = α −tQ2 e2(γ+λφ−U) − 4tQ2 α (γ + λφ − U )e2(γ+λφ−U) (95) 2 − 2tQ2 e2(γ+λφ−U) α γ + λφ − U + 2 (γ + λφ − U ) . Therefore, we have boundedness for α by integrating this differential equation for α in time since the coefficient of α and the second and third terms in the right-hand side of the equation (95) are bounded as shown already. Then, η is bounded by using the wave equation (12).
3.2
Future direction
Now, consider the future direction. We have already a monotonic decreasing property of E(t) along increasing t, dE/dt < 0 (Lemma 1). Therefore, for any t ∈ [ti , T+ ), E(t) ≤ E(ti ).
(96)
Proofs of the following two lemmas are similar with the argument in Step 1 of Section 5 in [AH]. θ Lemma 13 θ12 α−1/2 dθ is bounded on [ti , T+ ). Proof. The constraint equation (11) can be written as ∂t (α−1/2 ) =
t √ 2 2(η+λφ−U) αQ e . 2
(97)
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Then, −1/2
α
−1/2
(t, θ) − α
Q2 (ti , θ) = 2
t
√ s αe2(η+λφ−U) ds,
(98)
ti
for ant t ∈ [ti , T+ ). Integrating this equation from θ1 to θ2 in S 1 , we have θ2 θ2 θ2 √ 2(η+λφ−U) Q2 t α−1/2 (t, θ)dθ = s αe dθds + α−1/2 (ti , θ)dθ 2 ti θ1 θ1 θ1 t Q2 ≤ E(ti ) sds + 2π sup α−1/2 (ti , θ) 2 S1 ti Q2 E(ti )(t2 − t2i ) + C, 4
≤
(99)
where (96) has been used. Lemma 14 The functions U and φ are bounded on [ti , T+ ) × S 1 . Proof. For any θ1 , θ2 and for each t ∈ [ti , T+ ), θ2 |U (t, θ2 ) − U (t, θ1 )| = U dθ θ1 1/2 θ2
≤
−1/2
α
dθ
1/2 1/2
α
θ1
≤
θ2
2
U dθ
θ1
C(t),
(100)
where the H¨older inequality, energy bound (96) and Lemma 13 have been used. Now, t ˙ U(t, θ)dθds + C U (t, θ)dθ = S1
ti
≤ ≤
t ti
S1
S1
˙ U(t, θ) dθds + |C|
t
α1/2 dθ
1/2
S1
ti
S1
α−1/2 U˙ 2 dθ
1/2 ds + |C| ,(101)
where the H¨older inequality has been used. Since α is monotonically decreasing function along increasing time t, the right-hand side of the above inequality can be bounded. Thus, U (t, θ)dθ ≤ C(t), (102) S1
for some uniformly bounded function C(t).
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Finally, we obtain a uniform bound on U . We have the following identity:
U (t, θ) = U (t, θ)dθ + U (t, θ) − U (t, θ) dθ. (103) 2π max max 1 1 S
S1
S
S1
The right-hand side of this identity is bounded from above since (100) and (102) hold. For minS 1 U (t, θ), one can use the same argument and then, minS 1 U (t, θ) is bounded from below. Thus, U is uniformly bounded on [ti , T+ ) × S 1 . We can obtain uniform boundedness for φ by replacing U with φ in the above argument. Lemma 15 The functions γ is bounded on [ti , T+ ) × S 1 . Proof. (cf. Step 1 of Section 6 in [BCIM]). From the constraint equation (49) for γ, we have two inequalities: γ˙ ≤ te,
(104)
1 γ˙ ≥ − tQ2 e2(γ+λφ−U) . 4
(105)
and
From the inequality (104), we have γ(t, θ)dθ − γ(ti , θ)dθ = S1
S1
t
ti
d ds
γ(s, θ)dθ ds S1
√ ≤ sup α(ti , θ)
S1
≤ C = which controls
S1
S1
t ti
t
sE(s)ds ti
sE(ti )ds
CE(ti ) 2 (t − t2i ), 2
(106)
γ(t, θ)dθ from above. Now, we have the following identity:
γ(t, θ)dθ = 2π max γ+ γ dθ. (107) γ − max 1 1 S
S1
S
By the equation (50) of γ and a basic inequality, we have |γ | dθ ≤ tE(ti ).
(108)
S1
Then,
θ2 θ2 γ dθ ≤ |γ | dθ ≤ |γ | dθ ≤ tE(ti ). |γ(t, θ2 ) − γ(t, θ1 )| = θ1 θ1 S1
(109)
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Therefore, combining (107) and (109), we have the upper bound for γ: max γ ≤ C(t), 1 S
(110)
where C(t) is a bounded function of t ∈ [ti , T+ ). From the inequalities (105) and (110), and Lemma 14, if the coupling constant λ is non-negative,
1 2 2(γ+λφ−U) γ˙ ≥ − tQ e ≥ Ct exp 2 max γ + λ max φ − min U ≥ Ctec(t) ,(111) 4 S1 S1 S1 for some bounded function c(t) of t ∈ [ti , T+ ) and C < 0. If λ is negative, maxS 1 φ is replaced by minS 1 φ. Thus, γ˙ is controlled into the future, so we have upper and lower bounds for γ on [ti , T+ ) × S 1 . Corollary 2 αα ˙ −1 (hence ln α and α), η and α˙ are bounded on [ti , T+ ) × S 1 . Proof. The constraint equation (51) can be written as α˙ = −tQ2 e2(γ+λφ−U) . α
(112)
With boundedness of γ (Lemma 15), φ and U (Lemma 14), αα ˙ −1 = ∂t (ln α) 1 is bounded on [ti , T+ ) × S . As immediate results, ln α and α are bounded on [ti , T+ ) × S 1 . Since η = γ − 12 ln α, η is bounded on [ti , T+ ) × S 1 . Using these results to the constraint equation (51), we have a conclusion that α˙ is bounded on [ti , T+ ) × S 1 . Once the boundedness of αα ˙ −1 has been obtained, the following arguments are similar with ones of the past direction because key lemmas (Lemma 2 and Lemma 3) can be used and the arguments do not depend on time directions. Lemma 16 The functions U , A, φ, σ and their first derivatives are bounded on [ti , T+ ) × S 1 . Proof. From Lemma 2 and Corollary 2, we have the boundedness for E on [ti , T+ )× S 1 . The proof is the same with one of Lemma 9. Lemma 17 The functions η, ˙ η , α , α˙ and α ¨ are bounded on [ti , T+ ) × S 1 . Proof. Since the constraint equation (9) is described in terms of bounded functions and t, we have bounds on η. ˙ From the constraint equation (50), we have bounds ¨ can be obtained by the same arguon γ and then, boundedness for α , α˙ and α ment with the proof of Lemma 10. Combining this result, boundedness of γ and definition of γ, we have that η is bounded. Lemma 18 The second derivatives of U , A, φ and σ are bounded on [ti , T+ ) × S 1 .
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Proof. By Lemma 3 we have the boundedness for E˜ on [ti , T+ ) × S 1 . The proof is the same with one of Lemma 11. Lemma 19 α , η¨, η˙ and η are bounded on [ti , T+ ) × S 1 . Proof. The argument is the same to Lemma 12.
3.3
Proof of Theorem 4
We can continue to obtain bounds on higher derivatives of the fields by repeating the above arguments. Fortunately, to apply the global existence theorem in [MA], it is enough to get C 2 bounds of all functions. Thus, it has been shown that the functions η, α, U , A, φ and σ extend to t → 0 into the past direction and to t → ∞ into the future direction.
4 Comments We should like to comment concerning the TCC and the condition for coupling constant λ. Note that these conditions are needed to prove Theorem 4 into only the past direction. It is expected that the TCC is satisfied near initial singularities because strong focusing effect by gravity is dominant than repulsing one by a positive potential (cosmological constant) there. Note that spacetimes described by our AVTD solutions satisfy the TCC. Also, it is possible to expand in acceleration of the spacetimes into the future direction since the TCC does not hold necessarily there and the positive potential would become dominant. Thus, Theorem 4 does not deny paradigm-A. ¯ < 1/2 admits λ = 0, which means that there is a The condition |λ| ≤ λ positive cosmological constant. Thus, our theorem is applicable to not only theories with dilaton coupling but also ones with a pure cosmological constant. Now, ¯ It is known that there is a critical value λC in n-dimensional let us discuss λ. homogeneous and isotropic spacetimes [TP, WMNR]. In our notation with n = 4, |λC | = 1/2. Here, “critical” means the boundary whether late-time attractor solutions indicate accelerated expansion or not. Roughly speaking, λ describes steepness of the potential. Therefore, for λ2 > λ2C , the dilaton field falls down the potential hill soon and then decelerating expansion solutions with transient accelerating one are obtained, while we have attractor solutions with eternal accelerating expansion if λ2 < λ2C . It is believed that such critical values exist for generic spacetimes, although we do not know λC for spacetimes we considered here, in ¯ particular, our results give us no information about relation between λC and λ. Thus, it is not clear that the solution obtained in Theorem 3 is consistent with paradigm-A at the intermediate- and late-time. To answer this question, we need to analyze future asymptotic behavior (e.g. see [RA04]), which is left for future research.
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Local existence and uniqueness for smooth case
Let us consider the smooth version of the initial-value problem for our nonstandard setup formulated in Section 2. A key idea is to construct a symmetrichyperbolic system by introducing a new variable α := Z14 [IK]. Let us define ˙ φ , σ, σ, ˙ A , φ, φ,
:= Zi = (U, U˙ , U , A, A, ˙ σ , α, α , η). Here, i runs from 1 to 15. Z The system consisting in the effective evolution equations (9), (11), (13)–(16) becomes the following first-order symmetric-hyperbolic one:
= A1 ∂θ Z
+ F (t, θ, Z),
A0 ∂t Z
(113)
A0 = diag(1, 1, α, 1, 1, α, 1, 1, α, 1, 1, α, 1, 1, 1),
(114)
where
and
A1 =
0 A2 = 0 0
A2 0 0 0 0
0 A2 0 0 0
0 0 A2 0 0
0 0 0 A2 0
0 0 0 0 A3
0 0 0 α α 0
and
,
0 A3 = 0 0
0 0 0 0 . 0 0
(115)
(116)
Thus, we have a unique solution to the effective evolution equations by prescribing the smooth initial data for t = t0 > 0 if the constraint equations (10) and α = Z14 hold for any t. Now, as the analytic case, to assure the local existence and uniqueness of the initial-value problem, it is enough to show that the constraints propagate. Let us set N1 := η − 2DU U −
e4U 1 e2λφ Z14 Dσσ + , DAA − Dφφ − 2 2t 2 2 2α
(117)
and N2 := Z14 − α .
(118)
By direct calculation, we have the following linear, homogeneous ODE system:
= 0, (D + B)N
:= (N1 , N2 ) and where N B=
Dα 2α2
α 4α2
−1 −2α
(119) .
(120)
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is identically Thus, the uniqueness theorem for ODE systems guarantees that N
zero for any time t if we set initial data for t = t0 such that N (t0 ) = 0. Thus, the local existence and uniqueness of the initial-value problem for our case has been shown in the smooth case.
Acknowledgments I am grateful to Alan Rendall and Yoshio Tsutsumi for commenting on the manuscript.
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H. Andreasson, Global foliations of matter spacetimes with Gowdy symmetry, Commun. Math. Phys. 206, 337–365 (1999).
[AR]
L. Andersson, A.D. Rendall, Quiescent cosmological singularities, Commun. Math. Phys. 218, 479–511 (2001).
[ARW]
H. Andreasson, A.D. Rendall and M. Weaver, Existence of CMC and constant areal time foliations in T 2 symmetric spacetimes with Vlasov matter, Commun. Partial Differential Equations 29, 237–262 (2004).
[BCIM]
B.K. Berger, P.T. Chru´sciel, J. Isenberg and V. Moncrief, Global Foliations of Vacuum Spacetimes with T 2 Isometry, Ann. Physics, NY 260, 117–148 (1997).
[BKL]
V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, A general solution of the Einstein equations with a time singularity, Adv. Phys. 13, 639–667 (1982).
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P.T. Chru´sciel, On space-times with U (1) × U (1) symmetric compact Cauchy surfaces, Ann. Physics, NY 202, 100–150 (1990).
[DHRW] T. Damour, M. Henneaux, A.D. Rendall and M. Weaver, Kasner-like behaviour for subcritical Einstein-matter systems, Ann. Henri Poincar´e 3, 1049–1111 (2002). [EG]
R. Emparan and J. Garriga, A note on accelerating cosmologies from compactifications and S-branes, J. High Energy Phys. 05, 028 (2003).
[FR]
H. Friedrich and A.D. Rendall, The Cauchy Problem for the Einstein Equations, Lect. Notes Phys. 540, 127–224 (2000).
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G.W. Gibbons, Aspects of supergravity theories, in Supersymmetry, supergravity and related topics, edited by F. del Agulis et al., World Scientific, (1985) 123–181.
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[GR]
R.H. Gowdy, Vacuum spacetimes and compact invariant hypersurfaces: Topologies and boundary conditions, Ann. Physics, NY 83, 203–224 (1974).
[IK]
J. Isenberg and S. Kichenassamy, Asymptotic behavior in polarized T 2 symmetric vacuum space-times, J. Math. Phys. 40, 340–352 (1999).
[IM]
J. Isenberg and V. Moncrief, Asymptotic behaviour in polarized and half-polarized U(1) symmetric vacuum spacetimes, Class. Quantum Grav. 19, 5361–5386 (2002).
[IW]
J. Isenberg and M. Weaver, On the area of the symmetry orbits in T 2 symmetric spacetimes, Class. Quantum Grav. 20, 3783–3796 (2003).
[KR]
S. Kichenassamy and A.D. Rendall, Analytic description of singularities in Gowdy spacetimes, Class. Quantum Grav. 15, 1339–1355 (1998).
[LH03]
H. Lee, Asymptotic behaviour of the Einstein-Vlasov system with a positive cosmological constant, gr-qc/0308035.
[LH04]
H. Lee, The Einstein-Vlasov system with a scalar field, gr-qc/0404007.
[LWC]
J.E. Lidsey, D. Wands and E.J. Copeland, Superstring cosmology, Phys. Rep. 337, 343–492 (2000).
[MA]
A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag, (1984).
[MV]
V. Moncrief, Global properties of Gowdy spacetimes with T 3 × R topology, Ann. Physics, NY 132, 87–107 (1981).
[NM02]
M. Narita, On the existence of global solutions for T 3 -Gowdy spacetimes with stringy matter, Class. Quantum Grav. 19, 6279–6288 (2002).
[NM03]
M. Narita, Global existence problem in T 3 -Gowdy symmetric IIB superstring cosmology, Class. Quantum Grav. 20, 4983–4994 (2003).
[NTM]
M. Narita, T. Torii and K. Maeda, Asymptotic singular behaviour of Gowdy spacetimes in string theory, Class. Quantum Grav. 17, 4597– 4613 (2000).
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A.D. Rendall, Blow-up for solutions of hyperbolic PDE and spacetime singularities, gr-qc/0006060.
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A.D. Rendall, Accelerated cosmological expansion due to a scalar field whose potential has a positive lower bound, Class. Quantum Grav. 21, 2445–2454 (2004).
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[TM]
M.E. Taylor, Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, (1996).
[TP]
P.T. Townsend, Cosmic Acceleration and M-Theory, hep-th/0308149.
[TR]
S.B. Tchapnda and A.D. Rendall, Global existence and asymptotic behaviour in the future for the Einstein-Vlasov system with positive cosmological constant, Class. Quantum Grav. 20, 3037–3049 (2003).
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R.M. Wald, Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant, Phys. Rev. D28, 2118–2120 (1983).
[WM]
M. Weaver, On the area of the symmetry orbits in T 2 symmetric spacetimes with Vlasov matter, Class. Quantum Grav. 21, 1079–1097 (2004).
[WMNR] M.N.R. Wohlfarth, Inflationary cosmologies from compactification?, Phys. Rev. D69, 066002 (2004). Makoto Narita Max-Planck-Institut f¨ ur Gravitationsphysik Albert-Einstein-Institut Am M¨ uhlenberg 1 D-14476 Golm Germany email:
[email protected] Present address Center for Relativity and Geometric Physics Studies Department of Physics National Central University Jhongli 320 Taiwan email:
[email protected] Communicated by Sergiu Klainerman submitted 18/08/04, accepted 18/01/05
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 849 – 862 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05849-14, Published online 05.10.2005 DOI 10.1007/s00023-005-0226-8
Annales Henri Poincar´ e
Spin, Statistics, and Reflections I. Rotation Invariance Bernd Kuckert Abstract. The universal covering of SO(3) is modelled as a reflection group GR in a representation independent fashion. For relativistic quantum fields, the Unruh effect of vacuum states is known to imply an intrinsic form of reflection symmetry, which is referred to as modular P1 CT-symmetry [1, 2, 11]. This symmetry is used to construct a representation of GR by pairs of modular P1 CT-operators. The representation thus obtained satisfies Pauli’s spin-statistics relation.
1 Introduction A vacuum state of a quantum field theory usually exhibits the Unruh effect, i.e., a uniformly accelerated observer experiences it as a thermal state whose temperature is proportional to his acceleration [27]. This has been shown by Bisognano and Wichmann [1, 2] for finite-component quantum fields (in the Wightman setting). For general quantum fields, it has recently been derived from the mere condition that each vacuum state exhibits passivity to each inertial or uniformly accelerated observer [18], i.e., that in the observer’s rest frame, no engine can extract energy from the state by cyclic processes.1 By the theorem of Bisognano and Wichmann mentioned above, all familiar quantum fields also exhibit an intrinsic form of PCT-symmetry.2 Namely, one can assign to each Rindler wedge W, i.e., the set W1 := {x1 ≥ |x0 |} or its image under some Poincar´e transformation, an antiunitary involution JW . This assignment is an intrinsic construction using the vacuum vector and the field operators only. It is also basic to the so-called modular theory due to Tomita and Takesaki, where an operator like JW is called a modular conjugation. JW then implements a P1 CTsymmetry, i.e., a linear reflection in charge and at the edge of W. This property is called modular P1 CT-symmetry. Note as an aside that this symmetry is a typical property of 1 + 2-dimensional quantum fields as well, whereas these fields do not exhibit PCT-symmetry as a whole [23]. Modular P1 CT-symmetry is a consequence of the Unruh effect [11], but the converse implication does not hold: There are examples of P1 CT-symmetric quantum fields that do not exhibit the Unruh property [4]. 1 Two 2 cf.
related uniqueness results can be found in Refs. 16 and 17. also Refs. 11, 16, and 17.
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Guido and Longo have derived Pauli’s spin-statistics relation from the Unruh effect for general quantum fields in 1 + 3 dimensions [11].3 Independently from this, the present author derived the spin-statistics relation making use of modular P1 CT-symmetry only [15]. This symmetry was assumed for the field’s observables only, but since use of a theorem due to Doplicher and Roberts [8] was made later on, the result of Ref. 15 is confined to the massive-particle excitations of the vacuum. In Ref. 11 the Unruh effect was assumed for the whole field on the one hand. On the other hand, no use of the Doplicher-Roberts theorem was made, so a much larger class of fields and states was included; even fields that are covariant with respect to more than one representation of the universal covering group of L↑+ , among which there may be both representations satisfying and violating Pauli’s relation [24]. What one did obtain was a unique representation satisfying the Unruh effect. This representation exhibits Pauli’s spin-statistics connection. All spin-statistics theorems obtained before did not admit this extent of generality. This paper is the first of two that generalize the result of Ref. 15 in this spirit as well. Assuming P1 CT-symmetry with respect to all Rindler wedges whose edges are two-dimensional planes in a given tim-zero plane, a covariant unitary repre˜ of the rotation group’s universal covering is constructed. This repsentation W resentation satisfies Pauli’s spin-statistics relation. The argument does not make use of the Doplicher-Roberts theorem and applies to general relativistic quantum fields. Like its predecessor in Ref. 15, the argument is crucially based on the fact that each rotation in R3 can be implemented by combining two reflections at planes. This is, as such, well known for both SO(3) and L↑+ . A corresponding result for the universal coverings of these groups is, however, less elementary to obtain. of In Section 2, a model GR ∼ = SU (2) of the universal covering group SO(3) SO(3) will be constructed from nothing except pairs of “reflections along normal vectors”, i.e., from the family (ja )a∈S 2 , where ja is the reflection at the plane a⊥ . This representation-independent construction is set up according to the needs of ↑ ∼ SL(2, C) of L the spin-statistics theorem to be proved later on. A model GL = + will be constructed in a forthcoming paper. It is to be expected that the universal coverings of other Lie groups could be constructed the same way. Recently it has been shown by Buchholz, Dreyer, Florig, and Summers that this structure has a representation theoretic consequence: unitary representations of L↑+ can be constructed from a system of reflections satisfying a minimum of covariance conditions, as they are satisfied by the modular conjugations of a quantum field with modular P1 CT-symmetry [4, 9, 5]. This raises the question how to generalize these results to GR and GL , the goal being a considerable generalization of the spin-statistics analysis in Ref. 15. In Section 3, it is shown that this can, indeed, be accomplished for GR ; the group GL will be treated in the forthcoming paper. If a quantum field exhibits 3 cf.
also Refs. 10, 12, and 13.
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modular P1 CT-symmetry, then it is elementary to build a distinguished represen˜ of GR from the modular conjugations that implement P1 CT-symmetry. tation W This representation can, eventually, easily be shown to conform with Pauli’s spinstatistics principle. It is well known that not all GR -covariant quantum fields exhibit the spinstatistics relation, and it should be remarked that even for Lorentz covariant fields there are counterexamples [24]. This means that some condition specifying the representation or field under consideration is needed for whatever spin-statistics theorem. In the early spin-statistics theorems, this condition was that the number of internal degrees of freedom is finite, in this paper the condition is that the representation is constructed from modular P1 CT-operators. At the moment, such sufficient conditions are all one has in the relativistic setting; only in the setting of nonrelativistic quantum mechanics, a both sufficient and necessary condition has been established [19, 20].
as a reflection group 2 SO(3) There are many ways to model the universal covering group of the rotation group SO(3) =: R. Among topologists, “the” universal covering group is the group SO(3) of homotopy classes of curves starting at some base point, physicists are more familiar with SU (2), but these are, of course, not the only examples of simply connected covering groups. As a new model, a group GR will be constructed in this section from pairs of “reflections along normal vectors”, i.e., from the family (ja )a∈S 2 , where ja is the reflection at the plane a⊥ . Let MR be the pair groupoid of S 2 , i.e., the set S 2 × S 2 endowed with the concatenation (a, b) ◦ (b, c) := (a, c). Then the map ρ : MR → R defined by ρ(a, b) := ja jb is well known to be surjective. Namely, ρ(a, a) = 1 for all a ∈ S 2 . For σ = 1, choose τ ∈ R such that τ 2 = σ; if a ∈ S 2 is perpendicular to the axis of σ, then ρ(τ a, a) = σ. Call (a, b) and (c, d) equivalent if ρ(a, b) = ρ(c, d) and if there exists a σ ∈ R commuting with ρ(a, b) and satisfying (a, b) = (σc, σd). Let GR be the quotient space MR / ∼ associated with this equivalence relation, and let π : MR → GR denote the corresponding canonical projection. Define ρ˜ : GR → R by ρ˜(π(m)) := ρ(m) for all m ∈ MR . Then the diagram π
MR
−→
ρ↓
ρ˜
GR (1)
R commutes by construction. All maps in this diagram are continuous: π is continuous by definition, and continuity of ρ is elementary to show. The proof for ρ˜ is
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elementary as well: given any open set M ⊂ R, the pre-image ρ˜−1 (M ) is open if and only if π −1 (˜ ρ−1 (M )) is open. This set coincides with ρ−1 (M ), which is open by continuity of ρ. Defining ±1 := π(a, ±a) for arbitrary a ∈ S 2 , and −π(a, b) := π(a, −b) for (a, b) ∈ MR , one verifies that ρ˜−1 (σ) consists of two equivalence classes for each σ ∈ R. Lemma 1 (i) GR is a Hausdorff space. (ii) ρ˜ is a two-sheeted covering map. Before proving this lemma, we introduce some notation. ˙ For each σ ∈ R, ˙ let A(σ) be the rotation Notation. Denote the set R\{1} by R. axis of σ. If a ∈ A(σ) is one of the two unit vectors in A(σ), then there is a unique α ∈ (0, 2π) such that σ is the right-handed rotation around a by the angle α. The vector a and the angle α determine σ, and occasionally we use the notation [a, α] for σ. Note that [a, α] = [−a, 2π − α]. ˙ by M ˙ R . To each (a, b) ∈ M ˙ R , assign the axial unit Denote the set ρ−1 (R) a×b vector a(a, b) := |a×b| , and denote by (a, b) ∈ (0, π) the angle between a and b. ˙ R. Note that ρ(m) = [a(m), 2(m)] for all m ∈ M ˙ by G ˙ R . Since m ∼ n implies Denote the set ρ˜−1 (R) a(m) = a(n) and (m) = (n) one can define ˜ ˜ (π(m)) := a(m) and (π(m)) a := (m). ˙ R. ˜ Note that ρ˜(g) = [˜ a(g), 2(g)] for all g ∈ G Proof of Lemma 1.(i). Define B˙ π := {x ∈ R3 : |x| ∈ (0, π)}, and assign to each x ∈ B˙ π the rotation τ (x) := [x/|x|, |x|]. Choose any x ∈ B˙ π and an a ∈ S 2 ∩x⊥ , and ˙ R . One then obtains ξa (x) = ξb (x) for all b ∈ S 2 ∩ x⊥ , put ξa (x) := π(τ (x)a, a) ∈ G ˙ R is well defined by ξ(x) := ξa (x), where a ∈ S 2 ∩ x⊥ is so a map ξ : B˙ π → G arbitrary. ˙ R → B˙ π defined by η(g) := −(g)˜ ˜ a(g). Namely, ξ is inverse to the map η : G ˙ R , one has since b ⊥ a × b for all (a, b) ∈ M ξ(η(π(a, b))) = ξ ((a, b) · a(a, b)) = π ( τ (−(a, b)a(a, b)) b, b) = π ([−a(a, b), (a, b)] b, b) = π(a, b). So η is continuous, surjective, and has a continuous inverse, so η is a homeomor˙ R is a Hausdorff space. phism, and G
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It remains to construct disjoint neighborhoods of two distinct points g, h ∈ GR for the case that g = ±1 and h ∈ GR is arbitrary. ˜ If g = 1, then (h) = 0, so there exist disjoint open neighborhoods X and ˜ ˜ is Y of 0 and (h) in the topological space [0, π], respectively. Since the map ˜ −1 (Y ) are disjoint neighborhoods ˜ −1 (X) and V := continuous, the sets U := of 1 and h. If g = −1, there exist disjoint neighborhoods U and V of −g and −h, so −U and −V are disjoint neighborhoods of g and h, respectively. Proof of (ii). Define ρˆ : B˙ π → R˙ by ρˆ(x) := [x/|x|, 2|x|]. Then the diagram ˙R G ρ˜|G˙ R R˙
ρˆ
←−
↓η
(2)
B˙ π
commutes. ρˆ is a two-sheeted covering map, and η is a homeomorphism, so ρ˜|G˙ R = ρˆ ◦ η is a two-sheeted covering map. In order to prove that ρ˜ as a whole is a covering map, it remains to be shown ˙R ˙ R , but also in ±1. Since GR is Hausdorff, since G that ρ˜ is open not only on G −1 ˙ is a two-sheeted covering space of R, and since ρ˜ (1) = {±1} contains, like all other fibers of ρ˜, precisely two elements, it then follows that ρ˜ has continuous local inverses everywhere. So let (σn )n be any sequence in R˙ converging to 1, then some sequence (gn )n ˙ in GR needs to be found with ρ˜(gn ) = σn for all n and gn → 1; note that (−gn )n then satisfies ρ˜(−gn ) = σn as well and converges to −1. ˙ R , one has (g) ˜ ˜ ≤ π/2 or (−g) ≤ π/2. It follows that for For each g ∈ G −1 ˜ n ) ≤ π/2. Since [0, π/2] is each n some gn ∈ ρ˜ (σn ) can be chosen such that (g ˜ n ))n has at least one accumulation point, and since σn compact, the sequence ((g tends to 1, the only possible accumulation point in the interval [0, π/2] is zero. It ˜ n ) tends to zero and, hence, that gn tends to 1, proving that ρ˜ is follows that (g open. The reason why this proof is nontrivial is that ρ and π are not open. If this were the case, GR would directly inherit the Hausdorff property from MR , and the proof that ρ˜ is a covering map would be elementary. But neither ρ nor π is open. In order to see this, let (σn )n be any sequence of rotations around some fixed a ∈ S 2 , and suppose this sequence to converge to 1. If ρ were open, one would have to find, for each m ∈ π −1 (1) a sequence (mn )n converging to m and satisfying ρ(mn ) = σn for all n. Now choose m = (a, a). Since a ∈ A(σn ) for all n, one knows for all (bn , cn ) ∈ ρ−1 (σn ) that both bn and cn are perpendicular to a. As a consequence, no sequence (mn )n with ρ(mn ) = σn for all n can coverge to m = (a, a). π cannot be open either, since this would, by diagram 1 and the preceding ˙ are open. Lemma, imply that ρ is open. Only the restrictions of ρ and π to ρ−1 (R)
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Theorem 2 (i) GR is simply connected. (ii) There is a unique group product on GR such that the diagram MR × MR
◦
−→ MR
↓π×π GR × GR
↓π
−→
↓ ρ˜ × ρ˜ R×R
GR
(3)
↓ ρ˜ ·
−→
R
commutes. Proof of (i). GR = π(MR ) is pathwise connected because MR = S 2 × S 2 and because π is continuous. Together with Lemma 1, this implies the statement, since the fundamental group of R is Z2 . Proof of (ii). The outer arrows of the diagram commute, so it suffices to prove the existence and uniqueness of a group product conforming with the lower part. ˜ of an arbitrary But it is well known that each simply connected covering space G topological group G can be endowed with a unique group product such that G is a covering group.4
3 Spin and statistics The preceding section has provided the basis of a general spin-statistics theorem, which is the subject of this section. From an intrinsic form of symmetry under a charge conjugation combined with a time inversion and the reflection in one spatial direction, which is referred to as modular P1 CT-symmetry, a strongly continuous ˜ of GR will be constructed using the above and related unitary representation W ˜ exhibits Pauli’s spin-statistics reasoning. It is, then, elementary to show that W relation. In order to make the notion of rotation meaningful, fix a distinguished time direction by choosing a future-directed timelike unit vector e0 . The 2-sphere of 2 unit vectors in the time-zero plane e⊥ 0 will be called S . 1+3 in a Hilbert space H. The Let F be an arbitrary quantum field on R following standard properties of relativistic quantum fields will be used here. (A) Algebra of field operators. Let C be a linear space of arbitrary dimension,5 and denote by D the space C0∞ (R1+3 ) of test functions on R1+3 . The field 4 See,
e.g., Props. 5 and 6 in Sect. I.VIII. in Ref. 7. the “component space”, and its dimension equals the number of components, which may be infinite in what follows. 5 C is
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F is a linear function that assigns to each Φ ∈ C ⊗ D a linear operator F (Φ) in a separable Hilbert space H. (A.1) F is free from redundancies in C, i.e., if c, d ∈ C and if F (c ⊗ ϕ)) = F (d ⊗ ϕ) for all ϕ ∈ D, then c = d. (A.2) Each field operator F (Φ) and its adjoint F (Φ)† are densely defined. There exists a dense subspace D of H contained in the domains of F (Φ) and F (Φ)† and satisfying F (Φ)D ⊂ D and F (Φ)† D ⊂ D for all Φ ∈ C ⊗ D. Denote by F the algebra generated by all F (Φ)|D and all F (Φ)† |D . Defining an involution ∗ on F by A∗ := A† |D , the algebra F is endowed with the structure of a ∗-algebra. For each a ∈ S 2 , denote by Wa := {x ∈ R1+3 : xa > |xe0 |} the Rindler wedge associated with a,6 and let F(a) be the algebra generated by all F (c ⊗ ϕ)|D and all F (c ⊗ ϕ)† |D with supp(ϕ) ⊂ Wa . The algebra F(a) inherits the structure of a ∗-algebra from F by restriction of ∗. (A.3) F(a) is nonabelian for each a, and a = b implies F(a) = F(b). (B) Cyclic vacuum vector. There exists a vector Ω ∈ H that is cyclic with respect to each F(a). (C) Normal commutation relations. There exists a unitary and self-adjoint operator k on H with kΩ = Ω and with kF(a)k = F(a) for all a ∈ S 2 . Define F± := 12 (F ± kF k). If c and d are arbitrary elements of C and if ϕ, ψ ∈ D have spacelike separated supports, then F+ (c ⊗ ϕ)F+ (d ⊗ ψ) = F+ (d ⊗ ψ)F+ (c ⊗ ϕ), F+ (c ⊗ ϕ)F− (d ⊗ ψ) = F− (d ⊗ ψ)F+ (c ⊗ ϕ),
and
F− (c ⊗ ϕ)F− (d ⊗ ψ) = −F− (d ⊗ ψ)F− (c ⊗ ϕ). The involution k is the statistics operator, and F± are the bosonic and fermionic components of F , respectively. Defining κ := (1 + ik)/(1 + i) and F t (d ⊗ ψ) := κF (d ⊗ ψ)κ† , the normal commutation relations read [F (c ⊗ ϕ), F t (d ⊗ ψ)] = 0. This property is referred to as twisted locality. Denote F(a)t := κF(a)κ† . These properties imply that Ω is separating with respect to each algebra F(a), i.e., for each A ∈ F(a), the condition AΩ = 0 implies A = 0.7 6 An observer who is uniformly accelerated in the direction a can interact with precisely the events in Wa . 7 If AΩ = 0 and B, C ∈ F(−a)t , then 0 = BCΩ, AΩ = CΩ, AB ∗ Ω, so A = 0 by cyclicity of Ω.
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As a consequence, an antilinear operator Ra : F(a)Ω → F(a)Ω is defined by Ra AΩ := A∗ Ω. This operator is closable.8 Its closed extension Sa has a unique 1/2 polar decomposition Sa = Ja ∆a into an antiunitary operator Ja , which is called 1/2 the modular conjugation, and a positive operator ∆a , which is called the modular 1/2 operator. Ja is an involution.9 Sa , Ja , and ∆a are the objects of the so-called modular theory developed by Tomita and Takesaki.10 11 For each a ∈ S 2 , let ja be the orthogonal reflection at the plane a⊥ ∩ e⊥ 0, and for each ϕ ∈ D, define the test function ja ϕ ∈ D by ja ϕ(x) := ϕ(ja x).
(D) Modular P1 CT-symmetry. For each a ∈ S 2 , there exists an antilinear involution Ca in C such that for all c ∈ C and ϕ ∈ D, one has Ja F (c ⊗ ϕ)Ja = F t (Ca c ⊗ ja ϕ). The map a → Ja is strongly continuous.12 It will now be shown that pairs of modular P1 CT-reflections give rise to a strongly continuous representation of GR which exhibits Pauli’s spin-statistics connection. Lemma 3 Let K be a unitary or antiunitary operator in H such that KD = D and KΩ = Ω, and suppose there are a, b ∈ S 2 such that KF(a)K † = F(b). Then KJa K † = Jb , and K∆a K † = ∆b . † Proof. If A ∈ F(b), then KSa K † AΩ = KSa K AK Ω = A∗ Ω = Sb AΩ. The state ∈F(a)
ment now follows by the uniqueness of the polar decomposition.
In particular, this lemma implies kJa k = Ja ,
whence
Ja κ = κ† Ja
(4)
by definition of k. Using twisted locality, the lemma also implies κJa κ† = κ† Ja κ = J−a
(5)
8 By twisted locality, the operator κR † † −a κ is formally adjoint to Ra . Namely, if A ∈ κF(−a)κ and B ∈ F(a), then AΩ, Ra BΩ = AΩ, B ∗ Ω = BΩ, A∗ Ω = BΩ, κR−a κ† AΩ. Since κR−a κ† is densely defined, it follows that Ra is closable. 9 R2 = 1 implies S 2 = 1, so J ∆1/2 = S = S −1 = ∆−1/2 J ∗ , i.e., J 2 ∆1/2 = J ∆−1/2 J ∗ . a a a a a a a a a a a a a −1/2 ∗ Ja is positive, one obtains Ja2 = 1 and Ja ∆−1/2 Ja = ∆1/2 from the uniqueness Since Ja ∆a of the polar decomposition [3]. 10 The original work [26] directly applies to von-Neumann algebras, which are normed. But also for the present setting this structure has been applied earlier, e.g., in the classical papers of Bisognano and Wichmann [1, 2]. See, also, Ref. 14 for a monograph on the Tomita-Takesaki theory of unbounded-operator algebras. 11 i.e., the linear reflection with j a = −a, ja e0 = −e0 , and ja x = x for all x ∈ a⊥ ∩ e⊥ a 0 . 12 If one assumes covariance with respect to some strongly continuous representation of G R (which may also violate the spin-statistics connection), this is straightforward to derive; cf. Lemma 3. But covariance, as such, is not needed.
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which, in turn, implies Ja Jb Ja = J−ja b = Jja jb b = Jρ(a,b)b
(6)
by modular P1 CT-symmetry. Define a map W from MR into the unitary group of H by W (a, b) := Ja Jb . Lemma 4 (i) m ∼ n implies W (m) = W (n). (ii) W (m) = W (n) implies ρ(m) = ρ(n). Proof of (i). The proof of Lemma 2.4 in Ref. 5 can be taken without any relevant changes. Despite the fact that the Buchholz-Summers paper is confined to bosonic fields, which, in particular, implies Ja = J−a , it is straightforward to translate their proof to the present setting. This will not be spelled out here. The proof makes use of the continuous dependence of Ja from a assumed in Assumption (D). Proof of (ii). ρ(m) = ρ(n) would imply that there is some b ∈ S 2 such that ρ(m)b = ρ(n)b, so F(ρ(m)b) = F(ρ(n)b) by Assumption (A), whence W (m)F(b)W (m)∗ = W (n)F(b)W (n)∗ by Assumption (D), i.e., W (m) = W (n). ˜ : GR → W (MR ) is defined by W ˜ (π(m)) := W (m), By this lemma, a map W and another map ρW : W (MR ) → R is defined by ρW (W (m)) = ρ(m). The diagrams
(A)
π
MR
−→
ρ↓
R
←−
W
ρW
GR ˜ ↓W
and (B)
W (MR )
π
MR
−→
ρ↓
R
←−
ρ˜
ρW
GR ˜ ↓W
(7)
W (MR )
commute. Theorem 5 (i) There is a unique group product W on W (MR ) with the property that the diagram ◦ −→ MR MR × MR ↓π×π GR × GR
↓π
−→
˜ ×W ˜ ↓W W (MR ) × W (MR )
˜ ↓W W
−→ W (MR )
↓ ρW × ρW R×R ˜ is a homomorphism. commutes, i.e., W
GR
↓ ρW ·
−→
R
(8)
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(ii) W is the operator product in the algebra B(H) of bounded operators on H, ˜ is a representation. i.e., W ˜ of GR in C such that (iii) There is a representation D ˜ ˜ (g)F (c ⊗ ϕ)W ˜ (g)∗ = F (D(g)c ⊗ ρ˜(g)ϕ) W
for all g, c, ϕ,
(9)
where ρ˜(g)ϕ := ϕ(˜ ρ(g)−1 ·). Proof of (i). The diagram already commutes if the arrow representing W is omitted. For each g ∈ GR and each (a, b) ∈ π −1 (g) one has ˜ (±1)W ˜ (π(a, b)) = W ((±a, a) ◦ (a, b)) = W (±a, b) = W ˜ (±(π(a, b))), W ˜ (±1) ∼ ˜ (±g) ∼ so W = Z2 if and only if W = Z2 . ∼ ˜ ˜ If W (±1) = Z2 , then W is a bijection, so W is defined by ˜ (W ˜ −1 (U ) W ˜ −1 (V )). U W V := W ˜ (±1) ∼ If W = {1}, then ρW is a bijection, so W is defined by U W V := ρ−1 W (ρW (U ) · ρW (V )). ˙ R , the planes ˙ R . Given g, h ∈ G Proof of (ii). The statement is nontrivial only on G ˜ (g)⊥ and a ˜ (h)⊥ intersect in an at least one-dimensional subspace, so one can a choose (a, b) ∈ π −1 (g) and (c, d) ∈ π −1 (h) such that b = c is in this intersection. Then ˜ (π(a, b) π(c, d)) = W ˜ (π((a, b) ◦ (b, d))) W ˜ (π(a, d)) = W (a, d) =W = Ja Jd = Ja Jb Jb Jd ˜ (π(a, b))W ˜ (π(b, d)) = W (a, b)W (b, d) = W ˜ (π(a, b))W ˜ (π(c, d)). =W Proof of (iii). Define a map D from MR into the automorphism group Aut(C) of C by D(a, b) := Ca Cb . If (a, b) ∼ (c, d), then modular P1 CT-symmetry implies F (Ca Cb c ⊗ ja jb ϕ) = W (a, b)F (c ⊗ ϕ)W (a, b)∗ = W (c, d)F (c ⊗ ϕ)W (c, d)∗ = F (Cc Cd c ⊗ jc jd ϕ) = F (Cc Cd c ⊗ ja jb ϕ) for all c and all ϕ. Using Assumption (A.1), one obtains Ca Cb c = Cc Cd c for all c, so ˜ : GR → Aut(C) is defined by D(π(m)) ˜ D(a, b) = D(c, d), and a map D := D(m). ˜ ˜ This map D now inherits the representation property from W .
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Theorem 6 (Spin-statistics connection) F± (c ⊗ ϕ) =
1 ˜ (1 ± F (D(−1)c ⊗ ϕ)) 2
for all c and all ϕ. Proof. For each a ∈ S 2 one has ˜ (−1) = Ja J−a = Ja κJa κ† = Ja2 (κ† )2 = k, W so F (c ⊗ ϕ) = kF (c ⊗ ϕ)k ˜ (−1)F (c ⊗ ϕ)W ˜ (−1) =W ˜ (−1)F (c ⊗ ϕ)W ˜ (−1)† =W ˜ = F (D(−1)c ⊗ ϕ).
˜ is irreducible with spin s, then D(−1) ˜ If, in particular, D = e2πis , so F− = 0 for integer s and F+ = 0 for half-integer s.
4 PCT-symmetry In order to justify the term “modular P1 CT-symmetry”, one should show that this condition yields, at least in 1+3 dimensions, a full PCT-operator in a baseindependent fashion. Theorem 7 (PCT-symmetry) There exists an antiunitary involution Θ with the properties (i) Ja Jb Jc = Θ for each right-handed orthogonal basis (a, b, c) of e⊥ 0. (ii) There exists an antilinear involution C such that ΘF (c ⊗ ϕ)Θ = F (Cc ⊗ ϕ(− ·)). Proof. Let (a , b , c ) be a second right-handed orthonormal base, and define Θ := Ja Jb Jc . Then it follows from modular symmetry that Θ ΘF (c ⊗ ϕ)Ω = Θ ΘF (c ⊗ ϕ)ΘΘ Ω = F (Ca Cb Cc Ca Cb Cc c ⊗ ϕ)Ω ˜ = F (D(1)c ⊗ ϕ)Ω = F (c ⊗ ϕ)Ω. Since Ω is cyclic, this implies the statement.
˜ has to be If (a, b, c) is right-handed and (a , b , c ) is left-handed, then D(1) ˜ replaced by D(−1) in the above computation. Since J−a J−b J−c = κJa Jb Jc κ† , this is no surprise.
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Conclusion Both the classical geometry and the fundamental quantum field theoretic representations of the rotation group SO(3) and its universal covering group are based on reflection symmetries. At the classical level, the universal covering group GR can be constructed from P1 T-reflections. For a quantum field F with SO(3)-symmetry, a class of antiunitary P1 CT-operators exists that are fixed by the intrinsic structure of the respective field. Along precisely the same lines of argument used for the ˜ of GR is constructed. construction of GR , a covariant unitary representation W ˜ W exhibits Pauli’s spin-statistics connection.
Acknowledgments This work has been supported by the Stichting Fundamenteel Onderzoek der Materie and the Emmy-Noether programme of the Deutsche Forschungsgemeinschaft. I would like to thank Professor Arlt, Klaus Fredenhagen, and Reinhard Lorenzen for their critical comments and help concerning this manuscript.
Appendix. SU(2) versus GR can be described The isomorphism between the models SU (2) and GR of SO(3) as follows. 3 First recall the standard representation of SU (2) on 3R . Denote by σ1 , . . . , σ3 the Pauli matrices, and define x ˆ := x σ , x ∈ R . For each ν, the map ν ν ν x ˆ → Ad(±iσν )ˆ x is well known to implement the rotation [eν , π]. Since the parity transformation P is implemented by the map x ˆ → −ˆ x, one finds that for each ν, x implements the reflection jν . The determinants of the the map x ˆ → −Ad(±σν )ˆ Pauli matrices equal −1, and all of them are involutions. Now one can define an isomorphism J from S 2 onto the unitary matrices with determinant −1 by J(a) := a σ . The products of pairs of unitary matrices with determinant −1 yield all of SU (2).
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[21] R. Longo, On the spin-statistics relation for topological charges, in: Doplicher, S., Longo, R., Roberts, J.E., Zsido, L. (eds.): Operator Algebras and Quantum Field Theory, proceedings of the conference at the Accademia Nazionale dei Lincei, Rome 1996 (International Press). [22] G. L¨ uders, B. Zumino, Connection between Spin and Statistics, Phys. Rev. 110, 1450 (1958). [23] A.C. Manoharan, in: K.T. Mahanthappa, W.E. Brittin, (eds.) Mathematical Methods in Theoretical Physics, Gordon and Breach 1969 (New York) (conference held in Boulder 1968). [24] R.F. Streater, Local Field with the Wrong Connection Between Spin and Statistics, Commun. Math. Phys. 5, 88–98 (1967). [25] R.F. Streater, A.S. Wightman, PCT, Spin & Statistics, and All That, Benjamin, 1964. [26] M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and Its Applications, Lecture Notes in Mathematics 128, Springer 1970 (New York). [27] W.G. Unruh, Notes on black-hole evaporation, Phys. Rev. D14, 870–892 (1976).
Bernd Kuckert II. Institut f¨ ur Theoretische Physik Luruper Chaussee 149 D-22761 Hamburg Germany email:
[email protected] Communicated by Klaus Fredenhagen submitted 01/12/04, accepted 06/12/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 863 – 883 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05863-21, Published online 05.10.2005 DOI 10.1007/s00023-005-0227-7
Annales Henri Poincar´ e
A Central Limit Theorem for the Spectrum of the Modular Domain∗ Ze´ev Rudnick
The statistics of the high-lying eigenvalues of the Laplacian on a Riemannian manifold have been intensively studied in the past few years by physicists working on “quantum chaos”. A number of fundamental insights have emerged from these studies, though to date these have yet to be set on rigorous footing. In the case the manifold at hand is of arithmetic origin, these studies are related to some profound number theoretical problems and as such may be more amenable to investigation. In this note I make use of the arithmetic structure of the modular domain to establish Gaussian fluctuations in its spectrum for certain smooth counting functions.
1 Background To set the stage, I start with describing some of what is currently believed to hold for the statistics of the eigenvalues. Given a compact Riemannian surface M , Weyl’s law for the eigenvalues Ej of the Laplacian says that the number of eigenvalues below x grows linearly with x: #{Ej ≤ x} ∼
area(M ) x, 4π
as x → ∞ .
Let n(E, L) be the number of levels in a window around E for which the leading term in Weyl’s law predicts L levels: n(E, L) = #{E −
2π 2π · L < Ej < E + · L} area(M ) area(M )
and more generally for a test function f define nf (E, L) :=
j
∗A
f(
area(M ) (Ej − E) · ) 4π L
(1.1)
version of this paper was presented in the IAS/Park City Mathematics Institute summer session on Automorphic Forms and Applications in July 2002 as part of the author’s mini-course on Arithmetic Quantum Chaos. Supported by a grant from the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities and a Leverhulme Trust Linked Fellowship at Bristol University.
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4π which counts the levels lying in a “soft” window of length area(M) L about E. In the above L = L(E) depends on the location E. In what follows we will usually write n(L) for n(E, L), the dependence on E implicitly understood. To study the statistical behavior of n(L) we need to consider E as random, drawn from a certain distribution on the line. We denote by · this kind of energy 2E averaging, e.g., F = E1 E F (E )dE . Weyl’s law leads us to expect that the mean value of n(L) is L and likewise that of nf (L) is L · f (x)dx.
1.1
Number variance
The variance of nf (E, L) from its expected value is: Σ2f (E, L) = |nf (L) − nf (L) |2 It is customary to express the number variance by means of an integral kernel KE (τ ), called the “form factor”, so that as E → ∞ ∞ ∞ u Σ2f (E, L) ∼ L · (Lf(Lτ ))2 KE (τ )dτ , f(u)2 KE ( )du = L −∞ −∞ ∞ where f(y) = −∞ f (x)e−2πixy dx is the Fourier transform of f . For “generic” surfaces, Berry [3, 4] argued that as E → ∞, the behavior of Σ2f (E, L) for L in the range1 1 ≺≺ L ≺≺ Lmax =
√ E
(1.2)
is universal, depending only on the coarse dynamical nature of the geodesic flow on the surface, and follows that of one of a small number of random matrix ensembles: If the flow is integrable (as in the case of a flat torus) then Σ2f (E, L) ∼ ∞ L · −∞ f (x)2 dx for L → ∞, as in the Poisson model of uncorrelated levels. If the flow is chaotic (as in the case of negative curvature) then the behavior is as in the Gaussian Orthogonal Ensemble (GOE): For the sharp window (f = 1[−1/2,1/2] ), this is given by Σ2 (E, L) ∼ π22 log L for L → ∞. For sufficiently smooth f , in the ∞ GOE we have Σ2f (E, L) ∼ 2 −∞ f(u)2 |u|du, that is the variance of sufficiently smooth statistics tends to a finite value as L → ∞. The form factors for the random models are K pois (τ ) ≡ 1, and 2|τ | − |τ | log(1 + 2|τ |), |τ | ≤ 1 GOE K (τ ) = . | 2 − |τ | log 1+2|τ |τ | > 1 2|τ |−1 , It is to be emphasized that the √ above behavior is only valid in the universal √ well understood regime 1 ≺≺ L ≺≺ E; for L E the integrable case is fairly √ (at a rigorous level), see the survey [5]: The variance grows as E (a classical result 1 The
symbol f (x) ≺≺ g(x) means that f (x)/g(x) → 0
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[12] in the case of the standard flat torus). In the chaotic case it is believed [3, 4] that generically, the number variance continues to be small as in the universal regime.
1.2
Fluctuations
Our main interest here is in the value distribution of the normalized linear statistic nf (E, L) − nf (L) Σ2f (E, L) as E varies. In all the statistical models (Poisson and GOE/GUE), it is a standard Gaussian [22, 14, 13]. √ In the integrable case, when L E, the distribution is known ([17], [5]), and is definitely not Gaussian. Inside the universal regime (1.2), the distribution is believed to be Gaussian in both the integrable [6] and chaotic [1, 9, 25] cases. In the special case of the standard flat torus, this has been proved in a small part √ of the universal regime near E [18].
1.3
The modular domain
√ We start with the upper half-plane H = {x + −1y : y > 0} equipped with the hyperbolic metric ds2 = y −2 (dx2 + dy 2 ), which has constant curvature equal −1. ∂2 ∂2 The Laplace-Beltrami operator for this metric is given by ∆ = y 2 ( ∂x 2 + ∂y 2 ). The orientation-preserving isometries of the metric ds2 are the linear fractional transformations P SL(2, R) = SL(2, R)/{±I}. The modular domain is the Riemann surface obtained by identifying points in the upper half plane which differ by a linear fractional transformation with integer coefficients, that is by elements of the modular group Γ := P SL(2, Z) = SL(2, Z)/{±I} . The resulting surface H/Γ is non-compact and has cone points, but has finite hyperbolic area: area(H/Γ) = π/3. The spectrum of the Laplacian on the modular domain has a continuous component. Nonetheless, Selberg [24] showed that a version of Weyl’s law holds for the discrete spectrum: If we write the eigenvalues of −∆ on L2 (H/Γ) in the form Ej = 1/4 + rj2 , then: #{rj ≤ T } = 2+log
π
area(H/Γ) 2 2 T T − T log T + c1 T + O( ) 4π π log T
(1.3)
2 where c1 = . π In the case of the modular domain, deviations from generic statistics were discovered [10, 7, 11]. Although the geodesic flow is chaotic, the local statistics
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of the spectrum seem Poissonian, and Bogomolny, Leyvraz and Schmit [8] argued that the behavior of the form factor is given by √ √ c exp(c√2 Eτ ) , √1 ≺≺ τ ≺≺ log√ E 1 E E E √ KE (τ ) ∼ 1, τ log√ E E
for some constants c1 , c2 > 0, that is to say, in the universal regime we have √ √ √ ∞ 2c1 √LE 0 f(u)2 exp(c2 LEu )du, logEE ≺≺ L ≺≺ E 2 √ Σf (E, L) ∼ ∞ 1 ≺≺ L ≺≺ logEE . L · −∞ f(u)2 du, The only rigorous results known concern the closely related case where the modular group is replaced by quaternion groups: In the 1970’s Selberg [15, Chapter 2 2.18] gave a lower bound for the √ variance Σ (E, E) of the sharp counting function n(E) of the form Σ2 (E, E) E/(log E)2 . Luo and Sarnak [20] gave lower bounds for the averaged number variance of the sharp counting function n(E, L): √ √ √ 1 L 2 Σ (E, L )dL E/(log E)2 , E/ log E ≺≺ L ≺≺ E L 0 in the case of arithmetic (co-compact) groups. In the case of the hard window f = 1[−1/2,1/2] no upper bounds are currently available.
2 Formulation of results 2.1
Definition of the smooth counting functions
Let f be an even test function, whose Fourier transform f ∈ Cc∞ (R) is smooth and compactly supported, and normalized by requiring that ∞ f (x)dx = 1, −∞
and that
sup{|ξ| : f(ξ) = 0} = 1 .
The eigenvalues of the Laplacian are parametrized by Ej = 1/4 + rj2 . We define smooth counting functions by Nf,L (τ ) = f (L(rj − τ )) + f (L(−rj − τ )) (2.1) j≥0
This in essence carries the same information as (1.1). The relation between the expected number of levels L and the inverse width L of the momentum window is √ √ E area(H/Γ) E = . L= 2πL 6L
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The leading order behavior of Nf,L is given by ∞ 1 Nf,L (τ ) := {f (L(r − τ )) + f (L(−r − τ ))}M (r)dr 2π −∞ where
Γ Γ 1 area(H/Γ) r tanh(πr) − (1 + ir) − ( + ir) . 2 Γ Γ 2 (In keeping with tradition, I use the symbol Γ for both the modular group and the Gamma function.) The term Nf,L (τ ) is asymptotic to L: log τ 1τ area(H/Γ) ∞ τ )∼ . Nf,L (τ ) ∼ 2 f (x)dx + O( 4π L L 6L −∞ M (r) :=
2.2
The results
We will see that Nf,L − Nf,L has mean zero and show that the variance of Nf,L , when lim sup πL/ log T < 1, is asymptotic to 2κ ∞ 2 πLu 2 := du (2.2) f (u) e σL πL 0 p4 −2p3 +1 where κ = 1015 p=2 (1 + (p2 −1)3 ) = 1.328 . . . . Thus when the expected number 864 of levels L satisfies √ √ E area(H/Γ) < L ≺≺ T = E , log E the form factor KE (τ ) is given by
√ exp c2 Eτ √ KE (τ ) = c1 E
with c1 = 6κ/π, c2 = π/6. Our main result is that the fluctuations of Nf,L are Gaussian: Theorem 2.1 Assume that L → ∞ as T → ∞ but L = o(log T ). Then the limiting value distribution of (Nf,L − Nf,L )/σL is a standard Gaussian, that is x 2 Nf,L (τ ) − Nf,L (τ ) 1 du meas{τ ∈ [T, 2T ] : < x} = e−u /2 √ . lim T →∞ T σL 2π −∞ The reason that we need to assume that L = o(log T ) is that we prove Theorem 2.1 by the method of moments, and in computing the Kth moment we find Gaussian moments for L < cK log T , where cK → 0 as K grows. Plan of the paper: In Sections 3 and 4 we give some results on the hyperbolic conjugacy classes of the modular group. In Section 5 we use Selberg’s trace formula to express Nf,L (τ ) − Nf,L (τ ) as a sum Sf,L (τ ) over hyperbolic conjugacy classes plus a negligible term. The variance of Sf,L is computed in Section 6 and the higher moments in Section 7. We prove Theorem 2.1 in Section 8.
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3 The modular group 3.1
Conjugacy classes
To analyze Nf,L we use the Selberg trace formula, which for a discrete co-finite subgroup Γ ⊂ P SL(2, R) relates a sum over the spectrum of the Laplacian on L2 (H/Γ) with a sum over the conjugacy classes of the group Γ. We review some background material on these classes for the modular group P SL(2, Z). The conjugacy classes are divided into the class which consists of the identity element, hyperbolic, elliptic and parabolic classes. The hyperbolic conjugacy classes in Γ are represented by matrices P which are diagonalizable over the reals
λ and are conjugate to a matrix of the form with λ > 1. The norm of λ−1 2 P is defined as N (P ) = λ . The norm is therefore related to the trace of the corresponding group element by N (P )1/2 + N (P )−1/2 = | tr(P )|. We can write as each such P as P = P0k where P0 is primitive and k ≥ 1, where an element of Γ is primitive if it cannot be written as an essential power of another element. As is well known, primitive hyperbolic conjugacy classes correspond to closed geodesics on the Riemann surface H/Γ. In the case of the modular group Γ = P SL(2, Z), the traces are integers. If P is a hyperbolic class with trace | tr(P )| = n, n > 2 then its norm is √ n + n2 − 4 2 ) . (3.1) N (n) = ( 2 For the modular group, primitive hyperbolic conjugacy classes are parametrized by indefinite binary quadratic forms as follows (cf. [23]): Take a binary quadratic form Qa,b,c (x, y) := ax2 + bxy + cy 2, with a, b, c ∈ Z. The discriminant of Qa,b,c is d := b2 − 4ac. The form Qa,b,c is indefinite iff d > 0. We assume that d is not a perfect square. We say that Qa,b,c is primitive if gcd(a, b, c) = 1. Two binary quadratic forms
Q, Q are equivalent if Q (x, y) = Q(ax+by, cx+dy) for an element a b γ = of SL(2, Z); since the forms are quadratic, they are also equivalent c d under −γ and hence equivalence is over P SL(2, Z). Let h(d) be the number of equivalence classes of primitive binary quadratic forms of discriminant d. The automorphs of Qa,b,c are all of the form
1 (t − bu) −cu ±P (t, u) = ± 2 1 au 2 (t + bu) where (t, u) solve the Pellian equation t2 − du2 = 4
(3.2)
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If u = √ 0 then these are hyperbolic elements of SL(2, Z) with norm N (P ) = (t + u d)2 and trace t. √ Let d = 12 (td + ud d) (td , ud > 0) be the fundamental solution of (3.2). Then the matrix P (td , ud ) is a primitive hyperbolic matrix P0 of trace tr(P0 ) = td and norm N (P0 ) = 2d . It turns out that in this way we get a bijection between equivalence classes of primitive binary quadratic forms and conjugacy classes of primitive hyperbolic matrices in P SL(2, Z). Thus the number of primitive hyperbolic conjugacy classes of norm 2d is precisely the class number h(d).
3.2
The amplitude β(n)
We define, for n > 2, β(n) :=
1 2
tr(P )=n
log N (P0 ) N (P )1/2 − N (P )−1/2
(3.3)
the sum over all conjugacy classes {P } in P SL(2, Z) with | tr(P )| = n, equivalently with norm N (n) given by (3.1). These quantities turn out to be crucial in our analysis. The factor 1/2 in the definition is inserted among other reasons to give numbers with mean value 1 (as can be seen from the Prime Geodesic Theorem): β(n) ∼ N, as N → ∞ . n≤N
As representatives of the conjugacy classes of matrices with trace n > 2 we can take the matrices P0k = P (td , ud )k = P (n, u) where d runs over all discriminants, k ≥ 1 and n2 − du2 = 4, n > 2, u ≥ 1. Thus we see that β(n) =
d,u≥1 n2 −du2 =4
h(d) log d √ . du2
(3.4)
Dirichlet’s class number formula allows us to use (3.4) to express β(n) in terms of Dirichlet L-functions: For a discriminant d on associates the quadratic character χd given by χd (p) = dp for p an odd prime, χd (2) = 1 if d ≡ 1 mod 8, χd (2) = −1 if d ≡ 5 mod 8 and χd (−1) = 1. The associated L-function is L(s, χd ) = −s , Re(s) > 1. Dirichlet’s class number formula is n≥1 χd (n)n h(d) log d =
√ dL(1, χd ) .
Inserting the class number formula into (3.4) we find that β(n) =
d,u≥1:du2 =n2 −4
1 L(1, χd) . u
(3.5)
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As a consequence, one can get an upper bound of β(n) = O((log n)2 ) by using L(1, χd ) log d. What is more useful to us is that, in the mean square, β(n) is constant: Lemma 3.1 (M. Peter [21])
β(n)2 ∼ κN,
N →∞
(3.6)
n≤N
where κ is given by the product over primes κ=
p4 − 2p3 + 1 1015 (1 + ) = 1.328 . . . 864 (p2 − 1)3
(3.7)
p=2
This (complicated) expression was derived heuristically by Bogomolny, Leyvraz and Schmit [8] and proven by Manfred Peter [21], who uses the expression (3.5) for β(n) in terms of L(1, χ) and methods related to work on moments of class numbers [2]. For an extension to the case of congruence groups, see [19].
4 The length spectrum We will need to study alternating sums of the form K
± log N (nj ).
j=1
The first question is to when these alternating sums vanish. We say that a relation K
ηj log N (nj ) = 0,
ηj = ±1
(4.1)
j=1
is non-degenerate if no sub-sum vanishes, that is if there is no proper subset S ⊂ {1, . . . , K} for which j∈S ηj log N (nj ) = 0. The existence of non-degenerate relations (4.1) forces severe constraints. To explain these, recall that N (n) is a √ unit in the real quadratic field Q( n2 − 4). We claim that such such relations can occur only if all these units lie in the same quadratic field. K Lemma 4.1 Let j= ± log N (nj ) = 0 be a non-degenerate relation. Then all the norms N (ni ) lie in the same quadratic field, that is for some common d we have n2i − 4 = dfi2 for all i. Proof. We can write each norm as a power of the fundamental unit of the quadratic field in which it lies. Thus it will suffice to show that if F1 , . . . , FK be distinct real
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quadratic fields, then the fundamental units i of Fi are multiplicatively independent. Let E = F1 ∨ · · · ∨ FK be the compositum of the fields Fi . This is a Galois extension of the rationals with Galois group G = Gal(E/Q) an elementary Abelian 2-group (Z/2Z)s , for some s ≤ K. If we denote by UE the unit group of E, then G acts on UE and hence we get a linear representation on the vector space Q ⊗ UE . χ (σ) We claim that the i are eigenvectors of G, that is σ(i ) = i i for all σ ∈ G, where χi : G → {±1} are distinct characters. This forces them to be multiplicatively independent. Indeed, since we have an Abelian extension, all subfields are Galois and in particular Fi are preserved by G. Since the unit group is also preserved this means that under the action of any element σ ∈ G, i is taken to a unit of Fi which χi (σ) . The is necessarily ±1 i . That is we have a character χi of G with σ(i ) = i characters χi are distinct since the kernel of χi is precisely Gal(E/Fi ). K We next get a lower bound for j=1 ± log N (nj ) in the case it is non-zero. Lemma 4.2 i) If m = n then | log N (m) − log N (n)| ii) Suppose
K j=1
1 . min(m, n)
± log N (nj ) is nonzero. Then |
K j=1
± log N (nj )|
K j=1
1
2K−1 −1/2 N (nj )
Proof. i) Indeed, since log N (n) = 2 log n + O(1/n2 ), if m = n, say m > n, then log N (m) − log N (n) = 2 log 1 n,
n+1 Since log m n ≥ log n
we find
log N (m) − log N (n) ii) Let λj = |α − 1| ≤ 1/2 then
nj +
√ 2
|
n2j −4
K
m 1 + O( 2 ) . n n
1 1 = . n min(m, n)
so that N (nj ) = λ2j , and set α =
K j=1
λ±1 j . If
± log N (nj )| = 2| log α| |α − 1|.
j=1
So it suffices to give a lower bound for |α − 1|, assuming α = 1. This follows from Liouville’s theorem on Diophantine approximation of algebraic numbers by
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√ √ rationals; we give an explicit proof as follows: Let E = Q( n1 , . . . , nK ), which is a Galois extension of the rationals with Galois group G = Gal(E/Q) which is an elementary abelian 2-group of order 2s , for some s ≤ K. Moreover α ∈ E is an algebraic integer, and hence the norm NE/Q (α − 1) is a nonzero rational integer, hence has absolute value at least 1. Thus |NE/Q (α − 1)| = |α − 1| |ασ − 1| ≥ 1. id=σ∈G
for all σ ∈ G, we have Since λσj = λ±1 j K
|ασ − 1| ≤
λj + 1.
j=1
Thus
1 1 |α − 1| ≥ K K . |G|−1 ( j=1 λj + 1) ( j=1 N (nj ))(2K −1)/2
5
An expansion for Nf,L
5.1
The Selberg trace formula
We will transform Nf,L by using the Selberg trace formula [24]: Let g ∈ Cc∞ (R) be an even, smooth and compactly supported function, and let ∞ g(u)eiru du h(r) = −∞
so that g(u) =
1 2π
∞
h(r)e−iru dr .
−∞
The Selberg trace formula for a discrete co-compact sub-group Γ ⊂ P SL(2, R) with no elliptic elements is the identity [24] area(H/Γ) ∞ h(rj ) = h(r)r tanh(πr)dr 4π −∞ j≥0 (5.1) log N (P0 ) + g(log N (P )) N (P )1/2 − N (P )−1/2 {P } hyperbolic
where the sum is over all hyperbolic conjugacy classes of Γ. In the case of the modular group, the hyperbolic terms can be written as β(n)g(log N (n)) (5.2) 2· n>2
where the amplitude β(n) is given by (3.3).
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For groups with elliptic elements, there is an extra contribution to the RHS of (5.1) which is a sum over the finitely many conjugacy classes of elements E of finite order m ≥ 2: m {E} k=1
1 m sin(πk/m)
∞
h(r) −∞
e−2πkr/m dr. 1 + e−2πr
(5.3)
For discrete groups whose fundamental domain is non-compact but of finite volume, that is with cusps, there are extra terms coming from the contribution of the continuous spectrum and parabolic elements. For Γ = P SL(2, Z), these terms are given explicitly by [16]: g(0) log
1 π − 2 2π
∞
h(r) −∞
Γ Γ 1 (1 + ir) + ( + ir) dr Γ Γ 2 ∞ Λ(n) g(2 log n) (5.4) +2 n n=1
where Λ(n) is the von Mangold function.
5.2
Transforming Nf,L
We now apply the trace formula to derive an alternative expression for Nf,L . Taking h(r) = f (L(r − τ )) + f (L(−r − τ )) so that g(u) =
1 u −iτ u f( ) e + eiτ u 2πL 2πL
we find that Nf,L (τ ) = Nf,L (τ ) + Sf,L (τ ) + E
(5.5)
where: • The term Nf,L is given by the contribution of the identity class to (5.1) and part of the parabolic terms in (5.4): ∞ 1 Nf,L (τ ) = {f (L(r − τ )) + f (L(−r − τ ))}M (r)dr 2π −∞ where
Γ Γ 1 area(H/Γ) r tanh(πr) − (1 + ir) − ( + ir) . 2 Γ Γ 2 By Stirling’s formula, we have log τ 1 ∞ τ ). Nf,L (τ ) = f (x)dx + O( 6 −∞ L L M (r) =
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• The term Sf,L (τ ) is the contribution of the hyperbolic classes (5.2): 1 log N (n) −iτ log N (n) ) e Sf,L (τ ) = β(n)f( + eiτ log N (n) . πL n>2 2πL
(5.6)
The sum (5.6) contains only terms with log N (n) ≤ 2πL, that is n < eπL . As we will see below, it is the term Sf,L (τ ) which is responsible for the 2 fluctuations of Nf,L (τ ), and its variance is asymptotic to σL . As we can see from the formula (5.6), since f has compact support we have Sf,L (τ ) ≡ 0 for L 1. • E is the contribution of the elliptic classes (5.3) and the remaining part of the parabolic contribution (5.4), namely ∞ π 1 Λ(n) log n 1 f (0) log + f( )2 cos(2τ log n). πL 2 πL n=1 n πL
(5.7)
E is easily seen to be negligible, that is E = o(σL ). Indeed, the contribution of the elliptic elements is easily seen to be O(e−const.τ /L). As for (5.7), this is bounded as L → ∞ by (say) Mertens’ theorem. Moreover the mean value of (5.7) clearly vanishes as T → ∞. We thus see that the difference between the centered counting function Nf,L (τ ) − Nf,L and the sum Sf,L (τ ) over hyperbolic conjugacy classes is negligible relative to the standard deviation σL of Sf,L (τ ), and thus for our purposes we need only investigate the statistics of Sf,L (τ ).
6 The mean and variance of Sf,L 6.1
The averaging procedure
We define an averaging procedure by taking a non-negative weight ∞ function w ≥ 0, which is smooth and compactly supported in (0, ∞), with −∞ w(x)dx = 1. We then get an averaging operator: 1 ∞ τ F (τ )w( )dτ . F w,T := T −∞ T Let Pw,T be the associated probability measure:
t 1 ∞ 11A (f (t))w Pw,T (f ∈ A) = dt . T −∞ T Note that the requirement w ∈ Cc∞ (0, ∞) implies that the Fourier transform of w decays rapidly: w(x) |x|−A , as |x| → ∞ for all A > 1. In the concluding Section 8 we will relax the restrictions on w to allow other averages, e.g., w = 1[1,2] so that we take t uniformly distributed in [T, 2T ], or w(t) = 2t1[1,√2] when we take the eigenvalue λ = 1/4 + t2 uniformly distributed in [E, 2E], E = 1/4 + T 2 .
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The expected value of Sf,L
We will first show that the mean value Sf,L w,T tends to zero as T → ∞ provided L = O(log T ): Averaging (5.6) we find Sf,L w,T =
1 πL
β(n)f(
2
T log N (n) )2Rew( log N (n)) . 2πL 2π
Note that since log N (n) ∼ 2 log n and supp f ⊆ [−1, 1], the sum is over n ≤ eπL . Using w(x) x−A as x → ∞, we have Sf,L w,T
1 1 β(n) . L (T log n)A πL n<e
Since that
2 n≤x β(n)
x by Lemma 3.1, we have by the Cauchy-Schwartz inequality
eπL T A LA+1 which goes to zero since we assume L = O(log T ). Note that this argument also works when we allow straight averages (such as w = 1[1,2] ) as long as L < π1 log T . Sf,L w,T
6.3
The variance of Sf,L
Proposition 6.1 If lim sup πL/ log T < 1 then as T → ∞: 2κ ∞ 2 πLu 2 2 (Sf,L ) w,T ∼ σL =: du f (u) e πL 0
(6.1)
where κ is given by (3.7). Note that we have e(1−)πL/2 σL
eπL/2 L
for all > 0.
Proof. To compute (Sf,L )2 w,T , use (5.6) to get (Sf,L )2 w,T =
1 (πL)2
β(m)β(n)f(
m,n<eπL
×
1 ,2 =±1
log N (m) log N (n) )f ( ) 2πL 2πL
w(
T (1 log N (m) + 2 log N (n))). 2π
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We now deduce that as T → ∞, the only non-vanishing contribution is from the “diagonal terms” where 1 = −2 and N (m) = N (n) that is m = n. If m = n we may use Lemma 4.2 to get a lower bound | log N (m) ± log N (n)|
1 . min(m, n)
(6.2)
To be included in the sum, we need N (m), N (n) ≤ e2πL , that is m, n ≤ eπL , and so T min(m, n) A eπL A w( (1 log N (m) + 2 log N (n))) << ( ) ( ) . 2π T T Moreover, from n<x β(n)2 x we get by Cauchy-Schwartz that the off-diagonal contribution is dominated by 1 eπL A ) ( L2 T
β(m)β(n)
m,n<eπL
eAπL 2πL e L2 T A
for all A > 1. This goes to zero if πL ≤ (1 − δ)(log T ) for some δ > 0, which we assume. The diagonal terms m = n give log N (n) 2 1 ) β(n)2 f( (πL)2 n>2 2πL (where we used w(0) = 1). Since there are two such terms (corresponding to 1 = −2 = +1 or −1), we have the total diagonal contribution being 2
1 log N (n) 2 ) . β(n)2 f( 2 (πL) n>2 2πL
This can be evaluated asymptotically as L → ∞ using Peter’s formula (Lemma 3.1) to give 2κ ∞ 2 πLu 2 du =: σL . f (u) e πL 0 Thus we find
if lim sup πL/ log T < 1.
2 (Sf,L )2 w,T ∼ σL
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7 Higher moments We can now show that Sf,L (τ ) has Gaussian moments: Theorem 7.1 For K ≥ 3 the Kth moment of Sf,L /σL converges to that of a normal Gaussian provided that L → ∞ with T but that L = o(log T ): (2k)! k!2k , Sf,L (τ ) K lim ( ) = T →∞ σL w,T 0,
7.1
K = 2k even K odd.
Reduction to the pre-diagonal
By (5.6) the Kth moments of Sf,L is given by (Sf,L )K w,T =
1 (πL)K
K
log N (nj ) ) β(nj )f( 2πL πL j=1
n1 ,...,nK <e
×
w(
ηj =±1
T ( ηj log N (nj ))). 2π j
(7.1)
We now show that as T → ∞, the only (possibly) non-vanishing contribution to (7.1) is for terms satisfying: K
ηj log N (nj ) = 0
j=1
that is we have (Sf,L )K w,T =
1 K (πL) η =±1 j
j
K
β(nj )f(
ηj log N (nj )=0 j=1
log N (nj ) ) 2πL + O(
eαK L ) (7.2) T γK
for some αK , γK > 0. Since L = o(log T ) the remainder term vanishes as T → ∞. K To prove this, recall that by Lemma 4.2, if j=1 ηj log N (nj ) = 0 then for some δK > 0 |
K j=1
ηj log N (nj )| K
K j=1
−2K−1 +1/2 N (nj )
e−πδK L
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since only terms with N (nj ) < e2πL appear in (7.1). Thus for these terms we have w(
T eπL·δK A ( ) . ηj log N (nj ))) ( 2π j T
Replacing β(n) by log2 n L2 in (7.1) gives that the contribution of the terms with K j=1 ηj log N (nj ) = 0 is dominated by 1 LK
L2K (
n1 ,...,nK <eπL
eπL(K+AδK ) eπL·δK A ) LK . T TA
Since L = o(log T ), this vanishes as T → ∞ (in fact we need only assume that L < cK log T for this, if cK is sufficiently small). This proves (7.2).
7.2
Off-diagonal terms
In (7.2) we consider the sum of non-diagonal terms, that is terms for which there is at least one index j such that nj = ni for all i = j. To handle these, we use Lemma 4.1 which forces the relation K
N (nj )ηj = 1
j=1
to decompose into a union of such relations. Thus there is a decomposition {1, 2 . . . , K} = Sj so that in each subset Sj we have
N (ni )ηi = 1
(7.3)
i∈Sj
and the norms N (ni ) = (ni + n2i − 4)/2 lie in the same real quadratic field Q( dj ) for all i ∈ Sj . In the diagonal case there are K/2 such sets, e.g., S1 = {1, K/2 + 1}, S2 = {2, K/2 + 2}, . . . and the identities are of the form N (nj ) N (nK/2+j )−1 = 1, j = 1, . . . , K/2. In the off-diagonal case we assume that there is a subset Sj contains at least 3 elements. The number r of subsets is then at most (K − 1)/2, since K=
r
#Sj ≥ 3 + 2(r − 1).
j=1
To count such tuples of ni , we denote for each subset Sj by dj the common value of the square-free kernel of n2i − 4, i ∈ Sj and then write n2i − 4 = dj fi2 ,
i ∈ Sj .
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Let (dj ) be the fundamental unit of the field Q( dj ) and write N (ni ) = (dj )2ki , i ∈ Sj . Since log N (ni ) L we have ki L/ log (dj ), i ∈ Sj and the relation (7.3) implies i∈Sj ±ki = 0. Thus for each subset Sj there are at most O((L/ log (dj ))#Sj −1 ) solutions of (7.3) with log N (ni ) L. Recall that we are summing over log N (n) ≤ 2πL. Using β(n) (log n)2 2 L we find that the off-diagonal contribution is bounded by the sum over all partitions {1, . . . , K} = rj=1 Sj of L2
r
j=1 (dj )≤eπL
(
L )(#Sj −1)/2 LK (#{d fundamental : (d) ≤ eπL })r log (dj ) (7.4)
where r ≤ (K − 1)/2 is the total number of subsets Sj in our partition. Lemma 7.2 The number of fundamental discriminants d > 0 for which (d) < X is O(X 1+δ ) for all δ > 0. Proof. We need to bound the number of√fundamental discriminants d for which the fundamental solution (d) = (xd + dyd )/2 of x2 − dy 2 = 4 is at most X. Since (d) ∼ xd , this is equivalent to bounding the number of fundamental d’s for which xd X. In turn, this number is majored by the number ν(X) of all triples (d, x, y) of positive integers, with d ≡ 0, 1 mod 4, for which x2 − dy 2 = 4 and x < X, which is the sum ν(X) =
#{d, y ≥ 1, d ≡ 0, 1 mod 4 : dy 2 = x2 − 4} .
x<X
Since for x = 2 the number of pairs (d, y) with dy 2 = x2 − 4 is at most the number of divisors τ (x2 − 4) of x2 − 4, we find that ν(X) ≤
τ (x2 − 4)
2<x<X
xδ X 1+δ
2<x<X
for all δ > 0, by virtue of the bound τ (n) nδ for all δ > 0. Note: A more refined argument [23, Lemma 4.2] shows that ν(X) is asymp1+δ ) by O(X). totic to 35 16 X, so that one can replace the bound O(X Thus we find that (7.4) is bounded by LK e(1+δ)πLr LK e(1+δ)πL(K−1)/2 for all δ > 0. Since σL e(1−)πL/2 for all > 0, this shows that the sum of K−1+ the off-diagonal terms is O(σL ), for all > 0. To prove Theorem 7.1 it thus suffices to evaluate the diagonal contributions.
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The diagonal contribution
Assume now that there is the same of + signs as there are − signs. That number is K = 2k is even, and there are 2k k such choices of signs. For simplicity assume the first k are + and the last k are −. Thus we have to evaluate the sum 1 (πL)2k
2k
k
j=1 N (nj )=
2k
j=k+1 N (nj )
log N (nj ) ). β(nj )f( 2πL j=1
(7.5)
There are k! ways to pair off variables from the first k and the last k, such as the pairing nj = nk+j , 1 ≤ j ≤ k. Each such pairing contributes a term 1 (πL)2k
log N (n) 2 ) β(n) f( 2πL n>2 2
k
∼(
2 σL )k . 2
There are overlaps between the different ways of pairing off variables, which correspond to intersection of diagonals such as n1 = n2 = n3 = n4 . The contribution of these was already estimated in the study of the non-diagonal terms, as they correspond to relations (7.3) where some subset has two elements. more than Thus the total contribution of diagonal terms to (Sf,L )2k w,T is asymptotically
2k (2k)! 2k σ2 σ . · k! · ( L )k = k 2 k!2k L This proves Theorem 7.1.
8 Conclusion Since the Gaussian distribution is determined by its moments, Theorem 7.1 implies Theorem 8.1 Assume that L → ∞ as T → ∞ but L = o(log T ). Then x 2 du Nf,L (τ ) − Nf,L (τ ) lim Pw,T ( < x) = e−u /2 √ , T →∞ σL 2π −∞ So far we have assume that the weight function w defining the averages is in Cc∞ (0, ∞). To deduce the results for the standard averages (w = 1[1,2] ) as in Theorem 2.1, one proceeds by approximating 1[1,2] by “admissible” w’s in a standard fashion, see, e.g., [18]. The details are as follows: Fix > 0, and approximate the indicator function 11[1,2] above and below by smooth functions χ± ≥ 0 so that χ− ≤ 11[1,2] ≤ χ+ , where both χ± and their Fourier transforms are smooth and of rapid decay, and so that their total masses are within of unity: | χ± (x)dx − 1| < . Now set ω± := χ± / χ± . Then ω± are “admissible” and for all t, (1 − )ω− (t) ≤ 11[1,2] (t) ≤ (1 + )ω+ (t). (8.1)
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Now ! " ∞
Sf,L (τ ) t Sf,L (τ ) meas t ∈ [T, 2T ] : ∈A = 11A 11[1,2] dt σL σ T L −∞ and since (8.1) holds, we find ! (1 − )Pω− ,T
! " " Sf,L (τ ) Sf,L (τ ) 1 ∈ A ≤ meas t ∈ [T, 2T ] : ∈A σL T σL " ! Sf,L (τ ) ∈A . ≤ (1 + )Pω+ ,T σL
By Theorem 8.1 we find that ! " 2 1 Sf,L (τ ) 1 (1 − ) √ e−x /2 dx ≤ lim inf meas t ∈ [T, 2T ] : ∈A T →∞ T σL 2π A with a similar statement for lim sup; since > 0 is arbitrary this shows that the limit exists and equals ! " 2 Sf,L (τ ) 1 1 lim meas t ∈ [T, 2T ] : ∈A = √ e−x /2 dx T →∞ T σL 2π A which proves Theorem 2.1. The same consideration applies to other positive statistics, such as the number variance.
References [1] R. Aurich, J. Bolte and F. Steiner, Universal signatures of quantum chaos, Phys. Rev. Lett. 73, no. 10, 1356–1359 (1994). [2] M.B. Barban, The “Large Sieve” method and its applications in the theory of numbers, Russian Math. Surveys 21 , 49–103 (1966). [3] M.V. Berry, Semiclassical theory of spectral rigidity, Proc. Roy. Soc. London Ser. A 400, no. 1819, 229–251 (1985). [4] M.V. Berry, Fluctuations in numbers of energy levels, Stochastic processes in classical and quantum systems (Ascona, 1985), 47–53, Lecture Notes in Phys., 262, Springer, Berlin, 1986. [5] P. Bleher, “Trace formula for quantum integrable systems, lattice-point problems and small divisors”, Emerging Applications of Number Theory, D.A. Hejhal, J. Friedman, M.C. Gutzwiller, A.M. Odlyzko, eds. (Springer, 1999) pp. 1–38.
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[6] P. Bleher and J. Lebowitz, Energy-level statistics of model quantum systems: universality and scaling in a lattice-point problem, J. Statist. Phys. 74, 167– 217 (1994). [7] E. Bogomolny, B. Georgeot, M.-J. Giannoni and C. Schmit, Chaotic billiards generated by arithmetic groups, Phys. Rev. Lett. 69, no. 10, 1477–1480 (1992). [8] E. Bogomolny, F. Leyvraz and C. Schmit, Distribution of eigenvalues for the modular group, Comm. Math. Phys. 176, no. 3, 577–617 (1996). [9] E. Bogomolny and C. Schmit, Semiclassical computations of energy levels, Nonlinearity 6, no. 4, 523–547 (1993). [10] O. Bohigas, M.-J. Giannoni, and C. Schmit, in “Quantum Chaos and Statistical Nuclear Physics”, edited by Thomas H. Seligman and Hidetoshi Nishioka, Lecture Notes in Physics Vol. 263 (Springer-Verlag, Berlin, 1986), p. 18. [11] J. Bolte, G. Steil and F. Steiner, Arithmetic Chaos and Violation of Universality in Energy Level Statistics, Phs. Rev. Lett. 69, no. 15, 2188–2191 (1992). ¨ [12] H. Cram´er, Uber zwei S¨atze des Herrn G.H. Hardy, Math. Z. 15, 201–210 (1922). [13] O. Costin and J. Lebowitz, Gaussian fluctuations in random matrices, Phys. Rev. Lett 75, no 1, 69–72 (1995). [14] P. Diaconis and M. Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A, 49–62 (1994). [15] D.A. Hejhal, The Selberg Trace Formula for P SL(2, R), Lecture Notes in Mathematics, vol. 548. Berlin, Heidelberg, New York: Springer 1976. [16] D.A. Hejhal, The Selberg Trace Formula for P SL(2, R), Volume 2, Lecture Notes in Mathematics, vol. 1001. Berlin, Heidelberg, New York: Springer 1983. [17] D.R. Heath-Brown, The distribution and moments of the error term in the Dirichlet divisor problem, Acta Arithmetica 60, 389–415 (1992). [18] C.P. Hughes and Z. Rudnick, On the distribution of lattice points in thin annuli, IMRN 13, 637–658 (2004). [19] V. Lukianov, Ph.D. thesis, Tel Aviv university (in preparation). [20] W. Luo and P. Sarnak, Number Variance for Arithmetic Hyperbolic Surfaces, Commun. Math. Phys. 161, 419–432 (1994). [21] M. Peter, The correlation between multiplicities of closed geodesics on the modular surface, Comm. Math. Phys. 225, no. 1, 171–189 (2002).
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[22] H.D. Politzer, Random-matrix description of the distribution of mesoscopic conductance, Phys. Rev. B 40, no. 17, 11917–11919 (1989). [23] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15, no. 2, 229–247 (1982); corrigenda in J. Number Theory 16, no. 2, 284 (1983). [24] A. Selberg, Harmonic Analysis, in Collected Papers. Vol. I, 626–674 SpringerVerlag, Berlin, 1989. [25] F. Steiner, Quantum Chaos, in Universit¨ at Hamburg: Schlaglichter der Forschung zum 75. Jahrestag, edited by R. Ansorge (Reimer, Hamburg 1994), 542–564. Ze´ev Rudnick Raymond and Beverly Sackler School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel email:
[email protected] Communicated by Jens Marklof submitted 02/08/04, accepted 09/11/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 885 – 913 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05885-29, Published online 05.10.2005 DOI 10.1007/s00023-005-0228-6
Annales Henri Poincar´ e
Long Time Propagation and Control on Scarring for Perturbed Quantized Hyperbolic Toral Automorphisms Jean-Marc Bouclet and Stephan De Bi`evre Abstract. We show that on a suitable time scale, logarithmic in , the coherent states on the two-torus, evolved under a quantized perturbed hyperbolic toral automorphism, equidistribute on the torus. We then use this result to obtain control on the possible strong scarring of eigenstates of the perturbed automorphisms by periodic orbits. Our main tool is an adapted Egorov theorem, valid for logarithmically long times.
1 Introduction One of the main results in quantum chaos is the Schnirelman theorem. It states that, if a quantum system has an ergodic classical limit, then almost all sequences of its eigenfunctions converge, in the classical limit, to the Liouville measure on the relevant energy surface [7, 15, 20, 24]. It is natural to wonder if the result holds for all sequences (a statement commonly referred to as “unique quantum ergodicity”). This has been proven to be true for the (Hecke) eigenfunctions of the Laplace-Beltrami operator of a certain class of constant negative curvature surfaces [17] and has been conjectured to be true for all such surfaces [19]. It also has been proven to be wrong for quantized toral automorphisms in [11]. In that case, sequences of eigenfunctions exist with a semiclassical limit having up to half of its weight supported on a periodic orbit of the dynamics. This phenomenon is referred to as (strong) scarring. In [5, 12], it is shown that this last result is optimal: if a measure is obtained as the limit of eigenfunctions then its pure point component can carry at most half of its total weight. Except for the Schnirelman theorem, which holds in very great generality, all cited results are proven by exploiting to various degrees special algebraic or number theoretic properties of the systems studied. It is one of the major challenges in the field to device proofs and obtain results that use only assumptions on the dynamical properties of the underlying classical Hamiltonian system, such as ergodicity, mixing or exponential mixing, the Anosov property, etc. without relying on special algebraic properties. It is argued in [4, 5, 12] for example, that this will require a good control on the quantum dynamics for times that go to infinity (at least) logarithmically as the semiclassical parameter goes to zero: t ≥ k− ln for some constant k− > 0.
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It is well known that such control is in general hard to obtain especially since a good lower bound on k− is needed. In this paper, we concentrate on the quantized perturbed hyperbolic automorphisms of the 2d-torus, which are known to be Anosov systems classically. For those systems, we first prove an Egorov theorem valid for times proportional to ln , with an explicit control on the proportionality constant k− (Theorem 3.1). This result is obtained by adapting the techniques of [8]. We then combine this result with recent sharp estimates on the exponential mixing of the classical dynamics [3] to study the long time evolution of coherent states (Theorem 4.7), showing that on a sufficiently long logarithmic time scale, those evolved coherent states equidistribute on the torus. Roughly, the result is that for all f ∈ C ∞ (T2 ), t a W t a f (x) dx → 0, → 0, (1.1) Q(f, t, ) ≡ U ϕ,κ , Op (f )U ϕ,κ H (κ) −
T2
for times k− |ln | ≤ t ≤ k+ |ln | ,
0 ≤ k− ≤ k+ .
Here U is the unitary quantum dynamical evolution operator, OpW (f ) is the Weyl quantization of f , and ϕa,κ is a coherent state at the point a of the two-torus T2 . For detailed definitions, we refer to the following sections. This result generalizes results obtained in [4] for unperturbed hyperbolic automorphisms. To prove it, we prove an estimate of the type Q(f, t, ) ≤ ϕa,κ , (U−t OpW (f )Ut − OpW (f ◦ Φt ))ϕa,κ H (κ) a W t a + ϕ,κ , Op (f ◦ Φ ))ϕ,κ H (κ) − f (x) dx
≤ 1 (e
γq t
) + 2 (
−1 −γc t
e
T2
).
Here 1 and 2 are functions tending to zero when their argument does. The first term comes from the error term in the Egorov theorem, whereas the second one involves a classical mixing rate γc . It is obvious that this estimate leads to the result only if γq < γc . One therefore needs γc to be large (fast mixing) and γq to be small. Sharp results on the classical mixing rates of Anosov systems are hard to come by, but for some Anosov maps, among which the perturbed toral automorphisms that are the subject of this paper, such results have become available recently [3]. The remaining difficulty resides therefore in controlling the exponent in the error in the Egorov theorem. This is dealt with in the next section. We note that, although we prove the Egorov theorem for systems on the 2dtorus, we only prove the result above in full generality for d = 1. Indeed, denoting for arbitrary d by Γmin and Γmax the smallest and largest Lyapounov exponents of the system, we prove in Section 2 that, essentially, γq = 32 Γmax . On the other hand, the available estimates on the classical mixing rate [3] yield in our context here γc = 2Γmin. Of course, when d = 1, Γmax = Γmin and we have γq < γc as
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needed. This leads to (1.1). For d > 1, on the other hand, our proof of (1.1) still goes through, but only under an artificial “pinching” condition on the Lyapounov exponents of the type 3Γmax < 4Γmin . As an application of the above result, we finally show how to use the information obtained on the evolved coherent states in combination with the basic strategy of [5, 6] to gain some control on the scarring of eigenfunctions (Theorem 4.9, Corollary 4.10). Roughly speaking, we show that if a sequence of eigenfunctions of a quantized perturbed hyperbolic toral automorphism converges to a delta measure on a finite union of periodic orbits, then it must do so slowly. An improvement on this result (basically, on how slowly) has been announced recently in [13]. We do not expect this result to be optimal: indeed, it is expected, as in the case of unperturbed automorphisms, that sequences of eigenfunctions can not concentrate completely on periodic orbits, no matter how slowly. Proving this would involve controlling the quantum dynamics for longer times than we are currently able to do. A result somewhat analogous to our result on the evolution of coherent states was recently obtained for the long time evolution of Lagrangian states on compact Riemannian manifolds of negative curvature [21]. It should however be noted that such a result does not require any control on the proportionality constant preceding ln so that no precise control on either the mixing rate or the exponent in the error term of the Egorov theorem are needed in that case. We suspect that in situations were such control can be obtained, our present strategy will allow to control both coherent state evolution and strong scarring. A related result for the eigenfunctions of Laplace-Beltrami operators on compact, negatively curved Riemannian manifolds is proven using a different strategy in [1]: it is shown there that (under a suitable technical condition that may or may not hold) such eigenfunctions can not concentrate on sets of small topological entropy (and therefore on periodic orbits).
2 Weyl quantization and Egorov Theorem The purpose of this section is to recall (as compactly as possible) some properties of the Weyl quantization on T2d := (R/Z)2d as well as on R2d , for d ≥ 1. More specifically, we want to state a semi-classical version of the Egorov Theorem in the case of T2d . The latter is of course well known for R2d but it requires a proof for T2d all the more so as we need a rather explicit version of this theorem for the applications we have in mind in this paper. The Weyl quantization on R2d can be defined as the linear map f ∈ B(R2d ) → OpW (f ) ∈ L(L2 (Rd )) where OpW (f ) is the operator (belonging a priori to L(S(Rd ), S (Rd ))) with Schwartz kernel q1 + q2 , p dp. Kf (q1 , q2 ) = (2π)−d exp(i(q1 − q2 ) · p) f 2 Rd
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Here B(R2d ) is the set of smooth functions f on R2d such that ∂ γ f is bounded for all γ ∈ N2d , thus the above integral has to be understood in the sense of oscillatory integrals [16, 22, 18, 14] but it is of course a usual Lebesgue integral if f decays fast enough at infinity. The fact that OpW (f ) can be considered as a bounded operator on L2 (Rd ) follows from the Calder` on-Vaillancourt theorem [16, 22, 18, 14] which states the existence of C > 0 and d¯ > 0 such that W Op (f )ψ 2 d ≤ C sup ||∂ γ f ||L∞ (R2d ) ||ψ||L2 (Rd ) , L (R ) |γ|≤d¯
∀ f ∈ B(R2d ), ψ ∈ S(Rd ). (2.2) It is moreover well known that OpW (f ) maps the Schwartz space S(Rd ) continuously into itself and that OpW (f )∗ = OpW (f ), thus OpW (f ) can be considered as a continuous operator on S (Rd ) too. Note also that OpW (f ) is self-adjoint on L2 (Rd ) when f is real-valued. The Weyl quantization on T2d is obtained by restricting OpW (f ) to certain subspaces of S (Rd ) when f ∈ C ∞ (T2d ) (i.e., is Z2d periodic). The construction is as follows (see [7] for more details). For any ξ = (ξq , ξp ) ∈ R2d , the phase space translation operator U (ξ) is defined by i ξq · ξp U (ξ)ψ(q) = ψ(q − ξq ) exp ξp · q − , ψ ∈ S(Rd ) 2 and is clearly a unitary operator on L2 (Rd ). One easily checks that U (ξ) = OpW (χξ ),
i χξ (q, p) = exp (q · ξp − p · ξq ),
and that the following Weyl-Heisenberg relations hold for all ξ, η ∈ R2d U (ξ)U (η) = exp
i ω(η, ξ)U (ξ + η), 2
(2.3)
with ω the symplectic form defined by ω(ξ, η) = ξq · ηp − ξp · ηq . This relation shows in particular that, if n, m ∈ Z2d , U (n) and U (m) commute if and only if there exists N ∈ N such that 2πN = 1.
(2.4)
Since U (ξ) acts naturally on S (Rd ), we can introduce for any κ ∈ [0, 2π)2d the space H (κ) = {ψ ∈ S (Rd ) | U (n)ψ = eiω(κ,n)+i
nq ·np 2
ψ, ∀ n = (nq , np ) ∈ Z2d }
and it turns out that H (κ) is of dimension N d if (2.4) holds (0 otherwise) with the basis r κq d − , r ∈ {0, . . . , N − 1}d . ψrκ (q) = N − 2 eiκp ·k δ0 q − k − N 2πN 2d k∈Z
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The latter is proven in [7] as well as the existence of a unique scalar product on each H (κ) making the above basis orthonormal and U (n/N ) unitary for all n ∈ Z2d . The Weyl quantization on T2d is then defined by f ∈ C ∞ (T2d ) → OpW (f )|H (κ) . This is indeed a mapping from C ∞ (T2d ) to L(H (κ)), i.e., H (κ) is stable under OpW (f ), since one can easily check that for any Z2d periodic function f OpW (f ) = fn U (n/N ) n∈Z2d
if f (x) = n fn e2iπω(x,n) , x ∈ R2d . Let us emphasize that the spaces H (κ) are very natural in view of the following direct integral decomposition [7] ⊕ H (κ) dκ L2 (Rd ) (2π)−2d [0,2π)2d
in which the operators OpW (f )|H (κ) are the fibers of OpW (f ) for this decomposition. To streamline the discussion we will write both quantizations on R2d and T2d under a single form. From now on, M will denote either R2d or T2d . The Weyl quantization on M can then be defined as the map f ∈ B(M) → OpW (f ) ∈ L(H) where B(M) is either B(R2d ) or C ∞ (T2d ) and H is either L2 (Rd ) or H (κ) (we omit the , κ dependence in the notations). In order to write OpW (f ) in a unified way, we need to introduce the symplectic Fourier transform F on M defined by exp(iω(ξ, x))f (x) dx F f (ξ) = M
where ξ = (ξq , ξp )1 belongs to M∗ = R2d if M = R2d and (2πZ)2d if M = T2d . Then the following inversion formula holds exp iω(x, ξ)F f (ξ) dν(ξ) (2.5) f (x) = M∗
with dν = (2π)−2d × the Lebesgue measure (resp. Z2d δ2πn ) if M∗ = R2d (resp. (2πZ)2d ) and the Weyl quantization can easily be seen to be OpW (f ) = U (ξ)F f (ξ) dν(ξ). (2.6) M∗
1 Throughout this paper, x will denote the running point of M and ξ the one of M∗ , unlike the usual notation of microlocal analysis where (x, ξ) is the running point of R2d .
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Of course, when M = R2d , all the integrals must be understood in the weak sense d (in (2.6) we use the fact that U (ξ)ψ1 , ψ2 belongs to S(R2d ξ ) if ψ1 , ψ2 ∈ S(R )). ∞ Note also the existence of C, d¯ such that, if ||.||∞ denotes the L norm on M, W Op (f ) ≤ C sup ||∂ γ f ||∞ , ∀ f ∈ B(M). (2.7) H→H |γ|≤d¯
This comes from (2.2) if M = R2d and from the unitarity of U (n/N ) combined with the elementary estimate n |fn | ≤ C sup|γ|≤2d+1 ||∂ γ f ||∞ when M = T2d . This completes the definition of the Weyl quantization on M. Regarding the composition of the corresponding operators, we have the Proposition 2.1 There exists a bilinear map (f, g) → f #g from B(M)2 to B(M) such that OpW (f )OpW (g) = OpW (f #g). The function f #g has a full asymptotic expansion in powers of , meaning that for all integers J f #g = j f #j g + J rJ (f, g) j<J
α β β α −1 where f #j g = = (−1)α α!β! |α+β|=j Γ(α, β)∂q ∂p f ∂q ∂p g, with Γ(α, β) |α+β| 2d and for all γ ∈ N (2i)
γ ∂ rJ (f, g)
J+|γ|
∞
≤
Cd J!
sup
|γ1 |≤J+|γ|+d˜
||∂ γ1 f ||∞
sup
|γ2 |≤J+|γ|+d˜
||∂ γ2 g||∞ , 0 < ≤ 1. (2.8)
for some constants Cd , d˜ depending only on d. Note that (.#j .) is symmetric (resp. skew symmetric) for j even (resp. odd) and that g#1 f − f #1 g
= −i(∇p g · ∇q f − ∇q g · ∇p f ) = −i{g, f }.
(2.9)
Proof. This result is well known if M = R2d (see for instance the appendix of [8] for a simple proof). We briefly sketch the proof in the case M = T2d . Using (2.3) and (2.6), we have OpW (f )OpW (g) = eiω(η,ξ)/2 U ((ξ + η))F f (ξ)F g(η) dν(ξ)dν(η)(2.10) ∗ ∗ M M = U (ξ) eiω(η−ξ,η)/2 F f (ξ − η)F g(η) dν(η) dν(ξ). M∗
M∗
(2.11)
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Expanding eiω(η−ξ,η)/2 by the Taylor formula, we get the expansion of f #g with a remainder rJ = rJ (f, g) defined by its Fourier transform as follows F rJ (ξ) =
(−2i)−J (J − 1)!
1 0
(1 − t)J−1
M∗
eitω(η−ξ,η) ω(η − ξ, η)J F f (ξ − η)F g(η) dν(η) dt. (2.12)
= (γp , γq ) if γ = (γq , γp ) (γq , γp ∈ Nd ), we Since |ξ γ F rJ (ξ)| = |F∂ γ rJ (ξ)|, with γ have to consider γ! J! (η − ξ)β+γ1 η β+γ2 (−1)|βq |+|γ1 | ξ γ ω(η − ξ, η)J = γ !γ ! β! 1 2 γ +γ =γ 1
2
|β|=J
J+|γ|
where β = (βq , βp ) with βq , βp ∈ Nd . The sum contains at most Cd terms and since J! γ! ≤ (2d)J , ≤ 2|γ| β! γ1 !γ2 ! we conclude that (2.8) is now a simple consequence of the fact that α |F∂ α f (ξ)| dν(ξ) ≤ Cd sup ||∂ α1 +α f ||∞ . ξ F f (ξ) dν(ξ) = M∗
|α1 |≤2d+1
M∗
We omit the details.
Remark. The above proof can be repeated verbatim if M = R2d and B(R2d ) is replaced by S(R2d ). We now present a unified version of Egorov Theorem, that is the semiclassiW W cal analysis of eitOp (g)/ OpW (f )e−itOp (g)/ for f, g ∈ B(M), with g real-valued. This result is well known for M = R2d [10, 22, 16, 18] and the purpose of what follows is essentially to prove a similar result for M = T2d , with an explicit remainder term. The result is based on the following simple remark: if A is a bounded self-adjoint operator and B(t) is a strongly C 1 family of bounded operators, then eitA/ B(0)e−itA/ − B(t) i t i(t−s)A/ d = e i B(s) + [A, B(s)] e−i(t−s)A/ . (2.13) 0 ds We shall use this formula with A = OpW (g) and B(t) of the form B(t) = j OpW (fj (t)) j<J
with f0 (t), . . . , fJ−1 (t) ∈ B(M) such that and i
j<J
j fj (0) = f (i.e., B(0) = OpW (f ))
d B(s) + [A, B(s)] = O(J+1 ) ds
(2.14)
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where O(hJ+1 ) is to be understood the operator norm on H. Expanding [A, B(s)] in powers of by means of Proposition 2.1, (2.14) leads to the following conditions on the functions fj (s) ∂s f0 − {g, f0 }
= 0,
∂s fj − {g, fj }
= 2i
g#k fl ,
f0 (0) = f,
(2.15)
fj (0) = 0 for j ≥ 1
(2.16)
l+k=j+1
where, in the last sum, 3 ≤ k ≤ J − l is odd and l ≤ J − 1, which implies actually that l ≤ j − 2. This system is thus triangular and can be solved using the Hamiltonian flow φs of g, since the solution of ∂s a − {g, a} = b with as=0 = a0 is given by s s a(s, x) = a0 (φ (x)) + b(τ, φs−τ (x)) dτ, x ∈ M. 0
Note that if M = T2d and g is identified with a Z2d periodic function on R2d , the associated Hamiltonian flow φ˜s on R2d is easily seen to satisfy the identity φ˜s (x + n) = φ˜s (x) + n for all x ∈ R2d and n ∈ Z2d . This shows that the formulas for the fj (s) are the same for M = R2d and T2d , if f and g are Z2d periodic. Let us now define the linear operators Lsj on B(M) by Lsj f := fj (s). We have Ls0 f = f ◦ φs ,
Lsj ≡ 0 for j odd
(2.17)
the latter being a consequence of the (skew) symmetry of #k for k (odd) even. For j ≥ 2 even, an induction shows that Lsj =
j/2 1 ··· j (2i) k=1 m1 +···+mk =j/2 |α1 +β1 |=1+2m1 |αk +βk |=1+2mk sk−1 s 1 ··· Ls−s M α1 ,β1 Ls01 −s2 · · · M αk ,βk Ls0k dsk · · · ds1 0 0
(2.18)
0
where m1 ≥ 1, . . . , mk ≥ 1 in the sum and M α,β is the differential operator M α,β =
(−1)|α| β α α β ∂ ∂ g∂ ∂ . α!β! q p q p
(2.19)
Taking the remainders into account, one gets the following result: Theorem 2.2 (Egorov Theorem) For all f, g ∈ B(M) with g real-valued and all J ≥ 1 we have W W eitOp (g)/ OpW (f )e−itOp (g)/ = j OpW (Ltj f ) + J RJt (f, ) j<J
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where the operator RJt (f, ) has the following explicit form i
t
e
i(t−s)OpW (g)/
0
Op
W
(rJ−l+1 (g, Lsl f ))
− Op
W
(rJ−l+1 (Lsl f, g))
l<J
e−i(t−s)Op
W
(g)/
ds.
Note that estimates on ||RJt (f, )||H→H can then be derived from (2.7), (2.8) and estimates on the derivatives of Lsj f . This will be extensively used in the next section.
3 Perturbations of quantized hyperbolic maps In this section, we address the problem of the semi-classical approximation of U−t OpW (f )Ut as ↓ 0 in the Ehrenfest time limit t ≈ | ln |, when U is a unitary operator on H of the form U = e−iOp
W
(g)/
M (A)
with M (A) the quantization of a symplectic matrix with integer entries A ∈ Sp(d, Z). We refer to [7, 4, 5] and [14] for the definition of M (A) by mean of the metaplectic representation of Sp(d, R) and only quote the properties that we need. The operator M (A) is defined, up to a phase, as the unique operator on S (Rd ) such that M (A)−1 OpW (f )M (A) = OpW (f ◦ A),
∀ f ∈ B(R2d ).
(3.1)
If M = R2d , M (A) is unitary on L2 (Rd ), but if M = T2d and H = H (κ) one has to choose special values of κ to ensure that M (A) maps H (κ) into itself, in which case M (A) is unitary (see [7] for more details); from now on, we shall assume that such a choice, which depends on , has been made. Then, (3.1) holds on M = R2d and T2d and this is often expressed by saying that for linear evolutions ‘Egorov is exact’, meaning there is no remainder term. Let us now describe the results of this section. We will denote by φ the Hamiltonian flow associated to a fixed real-valued g ∈ B(M) and consider the discrete group (Φt )t∈Z of symplectomorphisms on M defined by Φ = φ ◦ A. Then, by setting ˜ j f = (L f ) ◦ A L j with the notations of (2.17) and (2.18), we can consider the functions ˜ l1 · · · L ˜ lt f Lt0 f = f ◦ Φt , L Ltj f = l1 +···+lt =j
(3.2)
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defined for j ≥ 1, t ≥ 0 integers and f ∈ B(M). Note that they depend on but ˜ 0 )t we omit this dependence for notational convenience. Note also that Lt0 = (L t and that Lj ≡ 0 if j is odd. Our goal is to show that U−t OpW (f )Ut ∼
hj OpW (Ltj f ),
↓ 0,
in a scale of times t described in terms of exponents ΓA , Γg that we now define. For the sake of simplicity, we shall assume that A is diagonalizable over R, meaning that there exists an invertible matrix P with real entries such that A = P −1 DP with D diagonal. Note that such a condition is of course satisfied if A is symmetric, e.g., the cat map. At the end of the section, we explain how to cope with general symplectic matrices A ∈ Sp(d, Z). Let us define ΓA ≥ 0 by eΓA = sup |λ|. σ(A)
Of course, this quantity is well defined for any invertible matrix A with real or complex spectrum. For z = (z1 , . . . , z2d ) ∈ C2d , we denote by |z| := (|z1 |2 + · · · + |z2d |2 )1/2 its standard Hermitian norm and set ||z||P := |P z|. The interest of the norm ||.||P is that we have ||Az||P ≤ eΓA ||z||P ,
||Im Az||P ≤ eΓA ||Im z||P
∀ z ∈ C2d ,
(3.3)
which we shall use extensively in the sequel. Then, inspired by [23, 8], we define the open sets Ωδ ⊂ C2d for δ > 0 by Ωδ = {z ∈ C2d | ||Im z||P < δ} and we consider the family of norms ||.||τ,δ defined for τ ∈ (0, 1) by ||f ||τ,δ = sup |f (z)| z∈Ωτ δ
for functions f which are bounded and analytic on Ωδ . We can now set 0 I 2 Γg = sup |||J ∇ g(z)|||P , J = −I 0 z∈Ωδ with ∇2 g the Hessian matrix of g and |||B|||P := supz=0 ||Bz||P /||z||P for B ∈ M2d (C). Note that Γg = 0 unless g is constant which is a trivial situation. We then define Γ = ΓA + Γg and our main result is the following:
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Theorem 3.1 Assume that f, g ∈ B(M), with g real-valued, have bounded and analytic extensions to Ωδ for some δ > 0. Then, for all 0 < ν < 2, there exists J0 > 0 such that for all J > J0 U−t OpW (f )Ut = j OpW Ltj f + J tJ (f, , ) (3.4) j<J
with a remainder such that, for all 0 ≤ ≤ 1, J t (f, , ) J
H→H
→0
as → 0
if
0≤t≤
2−ν | ln |. 3Γ
(3.5)
The reader may wonder what (3.5) means if Γ = 0. In such a case Γg = 0 thus g is constant so (3.4) becomes U−t OpW (f )Ut = OpW (f ◦ At ) by (3.1) which holds for all t ≥ 0. In Section 3, we will anyway be interested in the situation where ΓA > 0 and is small so that Γ > 0. We also emphasize that the analyticity assumption is imposed by our need to control high order derivatives of f and g in order to estimate tJ (f, , ). Similarly to [8], we could probably relax such a condition by considering quasi-analytic functions (e.g., Gevrey functions) which would allow us to consider compactly supported f . The rest of this section is now devoted to the proof of Theorem 3.1. The principle is rather simple and is the following: a straightforward application of Theorem 2.2 shows that ˜ j f ) + J M (A)−1 R (f, )M (A), j OpW (L ∀ J > 0, U−1 OpW (f )U = J j<J
hence an induction on t ≥ 1 shows that (3.4) holds with tJ (f, , ) =
tJ
t j−J OpW (Kj,J f) +
t
Us−t M (A)−1 RJ (EJs−1 f, )
s=1
j=J
M (A)Ut−s
(3.6)
t with the operators Kj,J and EJt defined by t = Kj,J
l1 +···+lt =j l1 <J,··· ,lt <J
˜ lt · · · L ˜ l1 , L
EJt =
l1 <J
···
˜ lt · · · L ˜ l1 . l1 +···+lt L
(3.7)
lt <J
t t Note that Kj,J depends on both j and J unless j < J in which case Kj,J = Ltj . 1 Note moreover that EJt depends on and that we set Kj,J = 0, EJ0 = id. Thus (2.7) reduces the proof of Theorem 3.1 essentially to estimate the deriva˜ l1 f . To that end, we shall use the following extension of a lemma ˜ lt · · · L tives of L of [8].
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Lemma 3.2 There exists a constant CP depending only on P such that, if a CP ||f ||τ,δ ≤ M , ∀ 0<τ <1 1−τ
(3.8)
for some M, a ≥ 0, then for all γ we have ||∂ γ f ||τ,δ ≤ M (a + |γ|) · · · (a + 1)δ −|γ|
CP 1−τ
a+|γ| ∀ 0 < τ < 1.
Proof. In [8], the authors show that the result holds with P = I and CP = e. Our lemma follows from their result applied to f ◦ P −1 . In order to estimate f ◦ Φ we will capitalize on two facts: on one hand, (3.3) implies that ||f ◦ A||τ,e−ΓA δ ≤ ||f ||τ,δ
(3.9)
and on the other hand we have, for any 0 ≤ s ≤ t, ||Im z||P ≤ τ δe−tΓg ⇒ ||Im φt−s (z)||P ≤ τ δe−sΓg . The latter is actually shown in [8] only for P = I but the very same method easily leads to this estimate. We therefore omit the proof and rather emphasize that it implies that ||f ◦ φt ||τ,δe−tΓg ≤ ||f ||τ,δ
(3.10)
which leads to the |γ|
Lemma 3.3 Assume that Cg > 0 is such that |∂ γ g(z)| ≤ γ!Cg for all z ∈ Ωδ and all |γ| ≥ 1. Assume moreover that (3.8) holds. Then for all j ≥ 2 even and all s ≥ 0 real, we have ||Lsj f ||τ,δe−sΓg
≤M
CP 1−τ
a+ 3j2 e
3j 2 sΓg
j/2 k s 3j , (a + 1) · · · a + (4dCg /δ)3j/2 2 k! k=1
for all τ ∈ (0, 1). Proof. We first note that, by an easy induction on k ≥ 0, the following result hods: if s0 , . . . , sk are non negative real numbers such that s0 + · · · + sk = s and α1 , β1 , . . . , αk , βk are non zero multi-indices such that |α1 +β1 |+· · ·+|αk +βk | = n, then for all τ ∈ (0, 1) ||Ls0k M αk ,βk · · · Ls01 M α1 ,β1 Ls00 f ||τ,δe−sΓg a+n CP ≤M Cgn δ −n esnΓg (a + 1) · · · (a + n). 1−τ
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This follows from Lemma 3.2 and (3.10) (recall that M α,β is defined by (2.19)). The lemma is then a consequence of (2.18) combined with the above estimate, the fact that (2d + 2m)! ≤ (2d)1+2m , (2d − 1)!(2m + 1)! j (k − 1 + j/2)! ≤ 2k−1+ 2 , #{(m1 , . . . , mk ) ∈ Nk | m1 + · · · + mk = j/2} = (k − 1)!(j/2)! s sk−1 and the fact that 0 · · · 0 dsk · · · ds1 = sk /k!. #{(α, β) ∈ N2d | |α + β| = 1 + 2m} =
We can now state the main ingredient of the proof of Theorem 3.1. Proposition 3.4 With the same assumptions as in Lemma 3.3, we have: for all j ≥ 2 and all integers l1 , . . . , lt such that l1 + · · · + lt = j, we have for all ∈ [0, 1] ˜ l1 f ||τ,δe−tΓ ≤ M ˜ lt · · · L ||L
CP 1−τ
a+ 3j2 e
3t 2 jΓ
3j (a + 1) · · · a + 2
(4dCg e /δ)3j/2 provided that (3.8) holds. In addition, if |f (z)| ≤ M on Ωδ , there exists a constant K such that, for all t ≥ 1, all γ and all ∈ [0, 1] t f ||∞ ≤ M tj K (1+)j (|γ| + 3j/2)!etΓ (|γ|+3j/2) , ||∂ γ Kj,J
0 ≤ j ≤ tJ.
(3.11)
Proof. Recall that we can assume that j is even. We obtain the first statement by induction on t ≥ 1 using lemma 3.3 with s = and (3.9) which we use through ||f ◦ A||τ,e−3ΓA /2 δ ≤ ||f ◦ A||τ,e−ΓA δ ≤ ||f ||τ,δ . This, together with (3.7) then yields the second statement since #{l1 + · · · + lt = j} ≤ tj .
tJ t f )||H→H . Using (2.7), Proof of Theorem 3.1. We first estimate || j=J hj Opw (Kj,J j w t (3.11) allows to estimate ln ||h Op (Kj,J f )||H→H from above by 3 3 d¯ d¯ j ln + t + + Γ + ln(K 2 t) + ln(d¯ + 3j/2) + ln CM 2 J 2 J using the fact that 1/j ≤ 1/J and that ∈ [0, 1]. By choosing J large enough we can assume that d¯ 3 3 + ≤ . 2 J 2 − ν/2 Since j ≤ Jt, the term ln(d¯ + 3j/2) is O(ln | ln |||) as ↓ 0 thus we get the existence of a new constant C such that for all ν ∈ (0, 2), ∈ (0, 1], ∈ [0, 1], 1 ≤ t ≤ (2 − ν)/3Γ and j ∈ [J, tJ] Cj
t ||hj Opw (Kj,J f )||H→H ≤ Cνj/2 (ln(C + | ln |))
˜ νj/4 . ≤ C
(3.12)
898
Since → 0.
J.-M. Bouclet and S. De Bi`evre
J≤j≤tJ
contains O(| ln |) terms, we see that ||
tJ j=J
Ann. Henri Poincar´e t hj Opw (Kj,J f )||H→H
Now the norm of second term of (3.6) multiplied by J can be estimated by
tJhJ
sup 0≤τ ≤t−1 s∈[0,], l<J
||OpW (rJ−l+1 (g, Lsl EJτ f ))||H→H + ||OpW (rJ+l−1 (Lsl EJτ f, g))||H→H
(3.13)
with the notations of Theorem 2.2. We proceed as before to estimate Lsl EJτ f and we obtain the theorem. Let us now briefly describe how to prove such results for a general A ∈ Sp(d, Z) with ΓA > 0. We claim that, in this case, we have the following result: for ˜ A > ΓA there exists an invertible matrix P with real entries such that any Γ ˜
|||P −1 AP ||| ≤ eΓA
(3.14)
where |||.||| is the matrix norm associated to the Hermitian norm |.| on R2 . We can prove this statement as follows. Assume first that the spectrum of A is real and let us choose a basis (e1 , . . . , e2d ) of R2d in which A is in Jordan normal form. If (ej , . . . , ej+p ) corresponds to a Jordan block J(λ) =
λ
1
0
0 .. . .. . 0
λ .. .
1 .. . ..
···
. ···
··· .. . .. . .. . 0
0 .. . , 0 1 λ
then by changing (ej , ej+1 , · · · , ej+p ) into (ej , εej+1 , · · · , εp ej+p ) with ε > 0, the above block is changed into the same one with 1 replaced by ε. Proceeding similarly for all the blocks, we obtain the existence of a basis in which A is the sum of a diagonal matrix of norm eΓA and of a nilpotent matrix of norm O(ε). This leads to the statement when the spectrum is real. For non-real eigenvalues λ = ρeiθ , using Jordan normal form over C2d , we have to consider blocks of the form J(λ) 0 . ¯ 0 J(λ) It is then standard that there exists a basis of real vectors in which the endomorphism represented by the above block has a matrix of the form N + ρR(θ) where N is nilpotent and R(θ) is block diagonal matrix of rotations (of dimension 2) of angle θ. Then, by changing this basis as in the case of a real spectrum, we can assume that N is small and we obtain (3.14) in the general case.
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4 Equirepartition of time-evolved localized states 4.1
The example of (generalized) coherent states
In this subsection, we shall prove that the generalized coherent states, defined below, when evolved over sufficiently long times, equidistribute on the torus. To define the states in question, we proceed as follows. Let q (4.1) ϕ (q) = −µ/2 ϕ µ with ϕ ∈ S(Rd ), |ϕ|2 = 1 and µ ∈ (0, 1). Then we set ϕa = U (a)ϕ
(4.2)
which defines a family of states in L2 (Rd ) indexed by a ∈ R2d . These are commonly referred to as (generalized) coherent states. The corresponding states on the torus, i.e., belonging to H (κ), are defined by ϕa,κ := S (κ)ϕa = e−iκq ·np U (0, np ) eiκp ·nq U (nq , 0) ϕa (4.3) np ∈Zd
nq ∈Zd
which converges in S (Rd ) (see [7]). The main property of these states that we shall use is a ϕ,κ , OpW (f )ϕb,κ H (κ) = (−1)N nq ·np eiω(κ,n) eiω(n,b)/2 ϕa , OpW (f )ϕb−n L2 (Rd )
(4.4)
n∈Z2d
which is proven in [4]. The best known example of such functions are obtained by 2 choosing µ = 1/2 and ϕ(q) = η(q) := π −d/4 e−q /2 . With this choice one obtains the standard coherent states. If ϕ˜ is another Schwartz function and ϕ˜ is defined similarly to (4.1), the Wigner function W (x) associated to ϕ , ϕ˜ is defined by f (x)W (x) dx ϕ , OpW (f )ϕ˜ L2 (Rd ) = R2d
for all f ∈ B(R2d ). For general ϕ , ϕ˜ in L2 (Rd ), W is a distribution, but for Schwartz functions it is a Schwartz function as well given by −d q, x = (q, p). e−i˜q·p/ ϕ (q − q˜/2)ϕ˜ (q + q˜/2) d˜ W (x) = (2π) With the simple dependence considered in (4.1), it is easy to see that the Wigner (a,b) function W (x) associated to U (a)ϕ and U (b)ϕ˜ takes the following form for
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any a, b ∈ R2d (a,b)
W
a+b (x) = e−iω(a,b)/2+iω(x,b−a)/ −d W1 Σµ x − 2
(4.5)
where Σµ is the linear map on R2d defined by Σµ (q, p) = (q/µ , p/1−µ ) and W1 the Wigner function of ϕ, ϕ. ˜ Note that, since W1 ∈ L1 (R2d ), (4.5) implies that (a,b) ||W ||L1 = ||W1 ||L1 is independent of . Note also that when ϕ(q) = ϕ(q) ˜ = η(q), one easily checks that W1 (x) = π −d e−x
2
(4.6)
which makes (4.5) completely explicit in this case. Our main result is Theorem 4.7. As explained in the introduction, its proof goes in two steps. use the Egorov theorem to establish that on a suitable time scale t aFirst we U ϕ,κ , OpW (f )Ut ϕa,κ H (κ) is equivalent to ϕa , OpW (f ◦ Φt )ϕa L2 (Rd ) (Propo sition 4.2). Then we use an estimate on the classical evolution (exponential mixing) to control this last term. As a warm up for the first step, we show for a particularly simple class of states how the Egorov expansion (3.4) can be reduced to the first term. Proposition 4.1 Let Ψ ∈ H be a family such that there exists C satisfying Ψ , OpW (f )Ψ ≤ C||f ||∞ , 0<≤1 (4.7) for all f in B(M) having a bounded and analytic continuation to some Ωδ . Then t U Ψ , OpW (f )Ut Ψ H − Ψ , OpW (f ◦ Φt )Ψ H → 0, →0 provided 0 ≤ t ≤ (2 − ν)| ln |/3Γ for some ν ∈ (0, 2). Proof. Using Theorem 3.1, we only have to show that for all 1 ≤ j < J we have j Ψ , OpW (Ltj f )Ψ H → 0, →0 in the specified range of times. This readily follows from the fact that j ||Ltj f ||∞ ≤ Cj j | ln |j e−j (1− 2 ) ln ν
t by estimate (3.11), where one should recall that Ltj = Kj,J if j < J.
The condition (4.7) is for instance satisfied by coherent states, in both cases (a,a) M = R2d and T2d . This readily follows from the independence of ||W ||L1 if M = R2d . In case of the torus, it is a simple exercise using the Poisson summation formula. Note also that, if f is periodic (in particular if M = T2d ), we can get rid of the analyticity of f since it is the uniform limit of a sequence of trigonometric polynomials.
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Nevertheless, regarding coherent states on the torus, the above result is not precise enough for our purpose since the term ϕa,κ , OpW (f ◦ Φt )ϕa,κ is not very explicit. This is why we give the next proposition whose proof will also be used in the proof of Theorem 4.9. Proposition 4.2 Fix a ∈ R2d and assume that 0 < µ < 1. Then, for all f ∈ C ∞ (T2d ), we have t a U ϕ,κ , OpW (f )Ut ϕa,κ H (κ) − ϕa , OpW (f ◦ Φt )ϕa L2 (Rd ) → 0, →0
provided 0 ≤ t ≤ (2 − ν)| ln |/3Γ for some ν ∈ (0, 2). Proof. Let us first note that, by truncating the Fourier series of f , there exists a sequence fM of Z2d periodic analytic functions such that fM → f in B(T2d ). Since ||OpW (f ) − OpW (fM )||H→H → 0 and a M → +∞ ϕ , OpW (f ◦ Φt )ϕa L2 (Rd ) − ϕa , OpW (fM ◦ Φt )ϕa L2 (Rd ) → 0, uniformly with respect to t ∈ R and ∈ (0, 1] by (4.5), we are left with the case where f is analytic. Then, by Theorem 3.1, we only have to study the difference j ϕa,κ , OpW (Ltj f )ϕa,κ H (κ) − ϕa , OpW (f ◦ Φt )ϕa L2 (Rd ) ,
j<J
thus the result will follow from (4.4) if we show that, in the specified range of times, → 0, (4.8) ϕa , OpW (f ◦ Φt )ϕa−n L2 (Rd ) n=0
j
ϕa , OpW (Ltj f )ϕa−n → 0 2 d L (R )
j ≥ 1.
(4.9)
n∈Z2d
We first note that the term corresponding to n = 0 in (4.9) has been studied in the proof of the previous proposition, and its limit is 0. We may therefore assume that n = 0 in both sums. Using (4.5), integrations by parts with 2 ∆x /|n|2 show that, for all j ≥ 0 and all M > 0 j a j+2M−m(2M−|γ|) ||∂ γ Ltj f ||∞ , ≤ Cj |n|−2M ϕ , Opw (Ltj f )ϕa−n L2 (Rd ) |γ|≤2M
where m = max(µ, 1 − µ). We get the result by the simple observation that j+2M−m(2M−|γ|) ||∂ γ Ltj f ||∞ ≤ Cf 2νj+CM → 0,
→0
for |γ| ≤ 2M , with C = 2(1 + ν)/3 if m < (2 − ν)/3 and C = 2(1 − m) otherwise. This follows from (3.11) by distinguishing both cases m ≥ (2 − ν)/3 and m < (2 − ν)/3.
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This proposition, combined with (4.5), allows us to reduce the study of the matrix elements of evolved coherent states to a problem in classical dynamics. By this, we mean that the main result of this section, Theorem 4.7, is a direct consequence of Proposition 4.2 and of the mixing estimates given in the Appendix A. Note that from now on, we shall be working with d = 1. As explained in the introduction, the reason for this is that, whereas the mixing rate is controlled by the smallest Lyapounov exponent of A, the error in the Egorov theorem is controlled by its largest Lyapounov exponent. As a warm-up, and in order to bring out the main strategy, we first prove a simplified version of the result: Theorem 4.3 Assume that ΓA > 0. Let a in R2 , f ∈ B(T2 ) and 1/3 < µ < 2/3. Then, for all ν > 0 there exists 0 small enough (independent of f ) such that for || < 0 we have t a W t a f (x) dx, → 0, U ϕ,κ , Op (f )U ϕ,κ H (κ) →
T2
provided that m+ν 2−ν | ln | ≤ t ≤ | ln |, Γ 3Γ
m = max(µ, 1 − µ).
(4.10)
Proof. We first remark that, by choosing 0 < Γ < ΓA close enough to ΓA and small enough we have 1>
1 + ν/2 Γ . > Γ 1+ν
(4.11)
Combined with (4.10), this estimate implies that t/| ln | > (m + ν/2)/Γ and thus e−tΓ ≤ m+ 2 . ν
By Proposition 4.2 and (4.5) we only have to study the limit of (a,a) (f ◦ Φt )(x)W (x) dx
(4.12)
(4.13)
R2
(a,a) for which W (x)dx = 1. Choosing a smooth cutoff function χ so that χ = 1 (a,a) near 0 and which is supported close to 0, then setting g (x) := W (x)χ(x − a), (a,a) − g ||L1 = O(h∞ ) thus we have ||W (a,a) (f ◦ Φt )(x)W (x) dx − (f ◦ Φt )(x)g (x) dx → 0, ↓0 R2
R2
uniformly with respect to t ∈ R. The last integral can obviously be interpreted as an integral over T2 since g is supported close to a and consequently we can use Corollary A.2. The result now simply follows from the fact that e−tΓ ||g ||W 1,1 = O(h−m )e−tΓ → 0 by (4.12).
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The above proof is a rather direct application of Proposition 4.2 and Corollary A.2 but it fails if m ≥ 2/3 (i.e., µ ∈ / (1/3, 2/3)) since e−tΓ h−m > 1, for, in that −tΓ 2/3 case, e > . The problem stems from the lower bound in (4.10), which arises (a,a) because ∇W behaves like −m . One expects on intuitive grounds that it should be possible to replace m by m = min(µ, 1−µ) which is of course less than 1/2 which is less than 2/3. We shall prove this is true, but for that purpose we will need to exploit some more detailed knowledge about the Anosov diffeomorphisms we study. The trick consists in applying a well known idea in the theory of Anosov systems: it is possible to replace (4.13) by an expression obtained by performing an integral (a,a) along along the stable foliation. Since the evolution stretches the function W the unstable manifold, this corresponds to smoothening out the fastest oscillations (a,a) in W , replacing the latter by a function that has a derivative controlled by −m . Let us start the proof. By Proposition 4.2, we have to study (4.13) where (a,a) W can be replaced, as in the proof of Theorem 4.3, by g which we can assume to be supported as close to a as we want. This will allow us to use the following result. Theorem 4.4 [3, 2] For all Γ < ΓA , there exists 0 small enough such that for all || < 0 the following holds: there exist σ > 0 and a C 1+σ diffeomorphism x → F (x) = (s(x), u(x)), from a neighborhood of a ∈ T2 to a neighborhood of 0 ∈ R2 such that F (a) = 0 and ∂s f ◦ Φt ◦ F−1 (u, s) ≤ Cf e−Γt (4.14) for all t ≥ 0, all (u, s) in the neighborhood of 0 and all f ∈ C 1 (T2 , R). Here C 1+σ denotes the corresponding H¨ older class. Using this result, we can perform the following change of variables
f ◦ Φt (x)g (x) dx R2 = f ◦ Φt ◦ F−1 (u, s) g ◦ F−1 (u, s)J (u, s) duds
(4.15)
where J ∈ C σ . On the right-hand side of this equation, we eventually want to use Corollary A.2, but the C σ regularity of J (u, s) is not sufficient for that purpose. Fortunately, the term J is essentially irrelevant in view of the following result. Lemma 4.5 i) (g ◦ F−1 )0<≤1 is a bounded family in L1 (R2 ). ii) For all θ ∈ (0, 1) there exists a family J such that, if ||.||∞ is the sup norm over a fixed small neighborhood of 0, ||J − J ||∞ ≤ Cσ θ ,
||∇J ||∞ ≤ C−θ .
Proof. i) follows from (4.5) and ii) from a standard convolution argument by a C0∞ function χ (u, s) = −2θ χ(u/θ , s/θ ).
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Using this lemma and (4.14), the right-hand side of (4.15) takes the form f ◦ Φt ◦ F−1 (u, 0)k (u) du + O1 (e−Γt ) + O2 (σ θ ) where O1 is uniform with respect to ∈ (0, 1], O2 is uniform with respect to t ≥ 0 and where the C 1 function k (u) is given by k (u) = g ◦ F−1 (u, s)J (u, s) ds. Note that k is bounded in L1 and that R k (u) du → 1 as → 0. The key remark is now that the derivative of this function is essentially controlled by −m rather than by −m , as a rough estimate would show. That is the content of the following proposition. Note that, in what follows, q, p are the canonical coordinates of R2 . They also define local coordinates on T2 close to any a, and this makes the following statement clear. Proposition 4.6 Assume that µ ≤ 1/2 (i.e., that max(µ, 1 − µ) = 1 − µ). Then, if the support of g is sufficiently close to a and if (4.16) ∂s p ◦ F−1 (0, 0) = 0 then, there exists k˜ ∈ C 1 (R) such that k − k˜ 1 → 0 as → 0 and ˜ dk /du
L1
L
≤ C−µ .
(4.17)
Proof. The condition (4.16) shows that, if δ1 , δ2 are small enough, s → (p ◦ F−1 )(u, s) is a diffeomorphism from (−δ1 , δ1 ) onto its range for each u ∈ (−δ2 , δ2 ). Thus, if the support of g is small enough, we can use (p ◦ F−1 )(u, s) as a new variable in the integral defining k so that it becomes q˜(u, p) − q(a) p − p(a) −1 , k (u) = W1 χ (u, p) j (u, p)dp µ 1−µ (u, p) the term with j (u, p) the C σ Jacobian of the change of variable and χ −1 corresponding to χ(F (u, s) − a)J (u, s). Changing again the variable with p˜ = (p − p(a))/1−µ , we would get the result if j was C 1 , by choosing θ = µ. We can overcome the non smoothness of j by the same principle as for Lemma 4.5: we choose j approaching j uniformly on the support of χ , such that ∇j = O(−µ ) and then q˜(u, p) − q(a) p − p(a) −1 ˜ , W1 k (u) = χ (u, p) j (u, p)dp µ 1−µ has the expected properties.
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Remark. The condition (4.16) expresses the fact that, at the point a, the submanifold {q = q(a)} is not aligned with the unstable manifold. Of course, if µ > 1/2, the same result holds if ∂s (q ◦ F−1 )(0, 0) = 0, with 1 − µ instead of µ in (4.17). We are now ready for the proof of the main theorem of this subsection. Theorem 4.7 Assume that 0 < µ ≤ 1/2 (resp. 1/2 ≤ µ < 1) and that the unstable manifold through a is not aligned with the submanifold {q = q(a)} (resp. {p = p(a)}). Assume moreover that ΓA > 0. Then, there exists 0 such that, for || < 0 and all f ∈ C ∞ (T2 ) t a U ϕ,κ , OpW (f )Ut ϕa,κ H
(κ)
→
f (x) dx, T2
→ 0,
provided that m+ν 2−ν | ln | ≤ t ≤ | ln |, Γ 3Γ
m = min(µ, 1 − µ).
(4.18)
Note that, when µ ∈ (1/3, 2/3), Theorem 4.3 holds as well, without the assumption on the stable and unstable manifolds, but with also a smaller time window than in (4.18). Proof. The above discussion shows that we only have to prove that
f ◦ Φt ◦ F−1 (u, 0)k˜ (u) du →
f (x) dx.
(4.19)
T2
Pick a smooth function (s) supported close to 0 such that (s)ds = 1. Then, using Theorem 4.4, the left-hand side of (4.19) takes the form
f ◦ Φt ◦ F−1 (u, s)k˜ (u)(s) duds + O(e−Γ t )
with O(e−Γ t ) uniform with respect to ∈ (0, 1]. This last integral is nothing but T2
f ◦ Φt (x)˜ g (x) dx
(1) where g˜ ◦ F−1 (u, s) = k˜ (u)(s)/J (u, s). Thus g˜ is of the form g˜ g˜(2) with (1) (1) (2) σ g ||L1 + m ||∇˜ g ||L1 = O(1). Note also that g˜ ∈ C independent of and ||˜ g˜ → 1 as → 0. Using Lemma 4.5 again to approach g˜(2) by C 1 functions, T2 (1)
we may assume that g˜ is C 1 and satisfies the same bound as g˜ . We can now repeat the arguments of Theorem 4.3 and the result follows.
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Semiclassical behavior of eigenstates
We now come to a more general result having applications in the description of the eigenvectors of U . Assume that Ψ,κ ∈ H (κ) satisfies, for all f ∈ C ∞ (T2d ), ↓ 0. (4.20) Ψ,κ , OpW (f )Ψ,κ H (κ) → f (0),
Rather vaguely, this condition says that Ψ,κ is concentrated at 0. This is confirmed by the following Lemma 4.8 There exists a sequence of positive numbers r → 0 and a family of functions χ ∈ C ∞ (T2d ) supported in a ball of radius r centered at 0 (in T2d ) such that 0 ≤ χ ≤ 1 and a a Ψ,κ − (2π)−d χ (a) η,κ , Ψ,κ H (κ) η,κ da → 0, ↓ 0. (4.21)
T2d
H (κ)
Conversely, if (4.21) holds and ||Ψ,κ ||H (κ) → 1 then (4.20) holds for all f ∈ C ∞ (T2d ). The proof of this lemma is given in Appendix B, where we also recall basic results on the coherent states decomposition over L2 (Rd ) and H (κ). Recall that 2 a η,κ is defined by (4.1), (4.2) and (4.3) with µ = 1/2 and η(q) = π −d/4 e−q /2 . The right-hand side in (4.20)
could of course be replaced by f (a0 ) for some a0 ∈ T2d or more generally by 0≤j≤J αj f (aJ ) for finitely many points a0 , . . . , aJ . Correspondingly, one can then define the concentration on a finite collection of points in a r neighborhood of those points. a , Ψ,κ H (κ) . The above To simplify the notation, we set λ (a) = χ (a) η,κ lemma proves that ψ,κ := (2π)−d
T2d
a λ (a)η,κ da
satisfies (4.20) as well and that Ψ,κ , U−t OpW (f )Ut Ψ,κ H (κ) − ψ,κ , U−t OpW (f )Ut ψ,κ H
(κ)
→ 0,
↓0
uniformly with respect to t ≥ 0. This is the first step of the proof of the next theorem, in which the notations ., . and ||.|| stand for ., . H (κ) and ||.||H (κ) respectively. Theorem 4.9 Assume that ||Ψ,κ || → 1 and that (4.21) holds for some sequence r such that r ≤ 1/2−σ , with σ > 0. Then, as → 0, Ψ,κ , U−t OpW (f )Ut Ψ,κ − (2π)−2d a b dadb → 0 λ (a)λ (b) η,κ , OpW (f ◦ Φt )η,κ T2d
T2d
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provided 0 ≤ Γ t ≤
1 +τ 2
| ln |,
1 − 3τ − 4dσ > 0 2
and
τ<
1 . 6
(4.22)
If moreover d = 1, ΓA > 0 and τ −5σ > 0, then there exists t → ∞ and (σ, τ ) > 0 such that for all || ≤ (σ, τ ) Ψ,κ , U−t OpW (f )Ut Ψ,κ → f (x) dx, ↓ 0. (4.23) T2
This theorem generalizes a result of [6], Section 5, where only the case = 0 is treated. The proof is then much simpler, since there is then no error term in the Egorov theorem. The theorem says that, if a sequence of states concentrates sufficiently fast on a point a in T2 , then the time evolved states equidistribute on the torus on some logarithmic time scale. Before proving this theorem, we show how it leads to a result on the semiclassical behaviour of the eigenvectors of U . Corollary 4.10 Assume that d = 1 and that ΓA > 0. For any 0 < σ < 1/38, there exists (σ) > 0 such that for all || < (σ), no family Ψ,κ of eigenvectors of U can satisfy simultaneously (4.20) for all f and (4.21) with r ≤ 1/2−σ . We note in passing that a similar result (with a worse value of σ) holds for d > 1 provided we impose a pinching condition on the Lyapounov exponents of A as mentioned in the introduction. Roughly speaking, this corollary shows that, if a family of eigenvectors of U concentrates on a single point in phase space in the semiclassical limit, then it must do so slowly. In other words, no such sequence can ‘live’ in a ball of too small a radius r . In view of the comment after Lemma (4.8), it is clear that this result holds also for a pure point measure supported on a finite number of periodic orbits. Given Theorem 4.9, the proof is very simple and identical to the case = 0 treated in [6], Section 5. We repeat it for completeness. Proof. For any 0 < σ < 1/38, one Furthermore, (4.22). can find τ > 5σ satisfying since Ψ,κ is an eigenfunction, Ψ,κ , U−t OpW (f )Ut Ψ,κ = Ψ,κ , OpW (f )Ψ,κ for all t, thus by choosing t = t and letting ↓ 0 we obtain f (x) dx f (0) = T2
for all f ∈ C ∞ (T2d ), which leads to a contradiction.
Proof of Theorem 4.9. Here again, it is sufficient to assume that f is analytic. Using Theorem 3.1 and Lemma 4.8, it is clear that, if τ < 1/6 and tΓ ≤ (1/2 + τ )| ln |, we have j ↓ 0. ψ,κ , OpW (Ltj f )ψ,κ → 0, Ψ,κ , U−t OpW (f )Ut Ψ,κ − j<J
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The first part of the theorem will thus be proven if we show that, for any j ≥ 2 (recall that Ltj ≡ 0 if j is odd), we have a b dadb → 0, ↓0 λ (a)λ (b) η,κ , OpW (Ltj f )η,κ (2π)−2d j T2d
T2d
if 1/2 − 3τ − 4dσ > 0. Using (4.4) and (4.5), integrations by parts similar to those of proposition 4.2 show easily that, for all M > 0, a b = (−1)N nq ·np eiω(κ,n)+iω(n,b)/2 η,κ , OpW (Ltj f )η,κ |n|≤C
a η , OpW (Ltj f )ηb−n L2 + O(M )
uniformly with respect to a, b ∈ [0, 1)2d and Γ t ≤ (1/2 + τ )| ln |, with τ < 1/6. The constant C involved in the sum is such that |b − n − a| ≥ C −1 |n| for all a, b ∈ [0, 1)2d and |n| > C. On the other hand, using (4.5) and (3.11), one sees that, for any n ∈ Z2d and any j ≥ 2, −2d (2π) |λ (a)λ (b)| j ηa , OpW (Ltj f )ηb−n dadb ≤ C−2d r4d 1/2−3τ T2d
T2d
since |λ (a)| ≤ ||Ψ,κ || is bounded and λ is supported in a set of volume O(r2d ). The first part of the theorem follows. We now the second part. Since χ can be chosen of the form χ (a) = prove
a+n1 (see Appendix B), it turns out that, for any M , n1 ∈Z2d χ r a b χ (a)χ (b) η,κ , OpW (f ◦ Φt )η,κ
can be written (−1)N nq ·np eiω(κ,n)+iω(n,b)/2 ηa , OpW (f ◦ Φt )ηb−n L2 + O(M ) |b−a−n|=O(r )
uniformly with respect to a, b ∈ [0, 1)2d . Now, if d = 1 and 5σ < τ , using (4.5) and proceeding similarly to the proof of Theorem 4.3, we see that for small enough and Γ < ΓA sufficiently close to ΓA a a b−n W t b−n η , Op (f ◦ Φ )η − η , η f (x)dx = e−tΓ O(−1/2 + r /) L2 L2 T2
uniformly on the set where |b − n − a| = O(r ), a, b ∈ [0, 1)2 . This shows that a 2 b dadb − ||ψ,κ || λ (a)λ (b) η,κ , OpW (f ◦ Φt )η,κ (2π)−2 T2 T2 f (x)dx = O(e−tΓ −1/2−5σ ) T2
and the result follows.
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A mixing theorem for perturbations of hyperbolic maps on T2
A
Let A be a 2 × 2 matrix with integer entries such that |trA| > 2 and detA = 1. For notational convenience, we assume that its eigenvalues are positive and we note them e±ΓA , with ΓA > 0. Let φ be a measure preserving diffeomorphsim on T2 , depending on a parameter , such that in C 3 (T2 ) as → 0.
φ → id
We define the associated Ruelle-Perron-Frobenius operator L as the map L g := g ◦ T−1 ,
T := φ ◦ A.
Using [3] (more precisely (2.1.7), Example 2.2.6 and Theorem 3) one has the following result. Theorem A.1 ([3]) For any Γ < ΓA , one can find 0 > 0 small enough such that the following property holds: for all || ≤ 0 , there exists a Banach space B of distributions of order 1, containing C 1 (T2 ), with norm ||.|| such that
(with ||g||W 1,1
||g|| ≤ C ||g||W 1,1 , ∀ g ∈ C 1 (T2 ) = T2 |g| + T2 |∇g|) and such that L = Π1 + R
with
R Π1 = Π1 R = 0
where Π1 g = g, 1 1 and R is a bounded operator on B with spectral radius lower than e−Γ . Here ., . is the pairing between distributions of order 1 and C 1 functions. As a direct consequence, we obtain Corollary A.2 For all Γ < ΓA , there exists 0 such that, for all || < 0 , one can find C,Γ satisfying
T2
f Tt (x) g(x)dx −
f
T2
T
g ≤ C,Γ e−tΓ ||f ||C 1 ||g||W 1,1 , 2 for all f, g ∈ C 1 (T2 ), t ≥ 0.
B Generalized coherent states decompositions In this appendix, we briefly recall some results on coherent states decompositions as well as some convenient tools for the proof of Lemma 4.8. As it is for instance proven in [14], it is well known that for any u ∈ S(Rd ) one has −d u = (2π)
ϕa , u L2 ϕa da (B.1) R2d
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where ϕa is defined by (4.2) with µ = 1/2. This implies in particular that, for any ϕ˜ ∈ S(Rd ), 2 2 2 −d ˜ L2 = (2π) | ϕ˜a , u L2 | da. (B.2) ||u||L2 ||ϕ|| R2d
This decomposition on L2 (Rd ), known as the coherent states decomposition espe2 cially when ϕ(q) = η(q) = π −d/4 e−q /2 , gives rise to a decomposition on H (κ) a ϕ,κ , S (κ)u H (κ) ϕa,κ da, (B.3) S (κ)u = (2π)−d
T2d
with the notation of (4.3). This is proven in [7]. Note the important consequence of that formula: for any ϕ˜ ∈ S(Rd ) 2 2 −d | S (κ)ϕ˜a , S (κ)u | da = Cϕ˜ ||S (κ)u||H (κ) , ∀ u ∈ S(Rd ). (2π) T2d
(B.4) These decompositions are particularly convenient since one knows rather precisely the action of pseudodifferential operators on functions of the form (4.2), as we shall see in Lemma B.1 below. Motivated by Lemma 4.8, we shall consider functions f depending possibly on . Let ε > 0 and assume that r is a sequence such that r ≥ 1/2−ε and let f be a family of functions in B(R2d ) such that γ ∂ f (x) ≤ Cγ r−|γ| ,
x ∈ R2d .
(B.5)
Lemma B.1 There exists a family Pγ of differential operators with polynomial coefficients (independent of ) such that for any f as above and any M > 0, there exists symbols f (,M,γ) satisfying (B.5) as well and differential operators QM γ with polynomial coefficients (independent of too) such that OpW (f )U (a)ϕ =
|γ|/2 ∂ γ f (a)U (a)(Pγ ϕ)
|γ|<M
+ Mε
OpW (f ,M,γ )U (a)(QM γ ϕ) .
|γ|≤2M −d/4 Whenever A = Pγ or QM (Aϕ)(q/1/2 ). γ , we have set (Aϕ) (q) =
Proof. It is essentially standard. Since U (−a)OpW (f )U (a) = OpW (f (. + a)), we are left with the case a = 0. Then, the result simply follows by writing the Taylor expansion of f at 0 and integrating by parts.
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Remark. The operators Pγ can be computed explicitly and in particular P0 = I. Combining this result and (4.4), it is not hard to deduce that for any f ∈ C (T2d ) satisfying (B.5), one has, for all M > 0, W a |γ|/2 γ Op (f )ϕ − ∂ f (a)S (κ)U (a)(P ϕ) ≤ CMε (B.6) γ ,κ |γ|<M ∞
H (κ)
2d
uniformly with respect to a ∈ [0, 1) . We are now ready for the proof of Lemma 4.8. Proof of Lemma 4.8. We only have to show the existence of a sequence r ≥ 1/2−ε for some ε > 0, satisfying r → 0, such that, if 0 ≤ χ ≤ 1 is supported close to 0 and ≡ 1 near 0 then a + n χ χ (a) := r 2d n∈Z
will satisfy the result. Let us fix ε > 0. Then for any sequence r ≥ 1/2−ε , using the Proposition 2.1, one has OpW (1 − χ )2 = j OpW (χj, ) + o(1) j<M
in operator norm, provided M = M (ε) is large enough. The symbols χj, are such −|γ|−2j that ∂ γ χj, = O(r ) and χ0, = (1−χ)2 , thus using (B.3), (B.4) and (B.6), one has a W 2 Op (1 − χ )Ψ,κ 2 = (2π)−d (1 − χ (a))2 η,κ , Ψ,κ da + o(1) T2d
using also the fact that ||Ψ,κ || → 1. By Taylor formula, there exists a function χ ˜ ∈ C ∞ (T2d ), independent of , such that χ(0) ˜ = 0 and (1 − χ (a))2 ≤ χ ˜2 (a)/r2 . Since a 2 −d (2π) χ(a) ˜ 2 η,κ , Ψ,κ da → 0 (B.7) T2d
2
by (4.20) applied to f = χ ˜ , we see that ||OpW (1 − χ )Ψ,κ || → 0 provided r2 → 0 more slowly than the left-hand side of (B.7). Furthermore there is no restriction to choose r ≥ 1/2−ε . Finally, we remark that a a η,κ , Ψ,κ χ (a)η,κ da + o(1) OpW (χ )Ψ,κ = (2π)−d T2d
by (B.3), (B.4) and (B.6) again which completes the proof of (4.21). For the converse, we note that a a Ψ,κ , OpW (f )Ψ,κ − (2π)−d da → 0. χ (a) η,κ , Ψ,κ Ψ,κ , OpW (f )η,κ T2d
The result follows then easily from the dominated convergence theorem using (B.6) and (B.4).
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References [1] N. Anantharaman, The eigenfunctions of the Laplacian do not concentrate on sets of small topological entropy, preprint june 2004. [2] M. Brin, G. Stuck, Introduction to dynamical systems, Cambridge Univ. Press (2002). [3] M. Blank, G. Keller, C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity 15, no. 6, 1905–1973 (2002). [4] F. Bonecchi, S. De Bi`evre, Exponential mixing and | ln | time scales in quantized hyperbolic maps on the torus, Comm. Math. Phys. 211, 659–686 (2000). [5]
, Controlling strong scarring for quantized ergodic toral automorphisms, Duke Math. J 117, No. 3, 571–587 (2003).
[6]
, Controlling strong scarring for quantized ergodic toral automorphisms, Section 5, mp arc 02-81 (2002).
[7] A. Bouzouina, S. De Bi`evre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Comm. Math. Phys. 178, 83–105 (1996). [8] A. Bouzouina, D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J. 111, No. 2, 223–252 (2002). [9] Y. Colin de Verdi`ere , Ergodicit´e et fonctions propres du Laplacien, Commun. Math. Phys. 102, 497–502 (1985). [10] Y.V. Egorov, On canonical transformations of pseudo-differential operators (in Russian), Uspekhi Mat. Nauk. 24, no. 5, 235–236 (1969). [11] F. Faure, S. Nonnenmacher, S. De Bi`evre, Scarred eigenstates for quantum cats of minimal periods, Commun. Math. Phys. 239, 449–492 (2003). [12] F. Faure, S. Nonnenmacher, On the maximal scarring for quantum cat map eigenstates, Commun. Math. Phys. 245, 201–214 (2004). [13] F. Faure, S. Nonnenmacher, contribution at the Workshop on Random Matrix theory and Arithmetic Aspects of Quantum Chaos, Newton Institute, Cambridge, june 2004. [14] G.B. Folland, Harmonic analysis in phase space, Ann. Math. Studies, Princeton Univ. Press 122, (1989). [15] B. Helffer, A. Martinez, D. Robert, Ergodicit´e et limite semi-classique, Comm. Math. Phys. 109, 313–326 (1987). [16] L. H¨ormander, The analysis of linear partial differential operators III, Springer-Verlag (1985).
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[17] E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Annals of Math., to appear. [18] D. Robert, Autour de l’approximation semi-classique, Progress in mathematics 68, Birkh¨ auser (1987). [19] Z. Rudnick, P. Sarnak, The behaviour of eigenstates of hyperbolic arithmetic manifolds, Commun. Math. Phys. 161, 1, 195–213 (1994). [20] A. Schnirelman, Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29, 181–182 (1974). [21] R. Schubert, Semiclassical behaviour of expectation values in time evolved coherent states for large times, preprint january 2004. [22] M. Taylor, Pseudo-differential operators, Princeton Mathematical Series 34, Princeton University Press (1981). [23] F. Tr`eves, Introduction to pseudo-differential and Fourier integral operators, Vol. 2: Fourier integral operators, Univ. Ser. Math., Plenum, New-York (1980). [24] Zelditch, Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55, 919–941 (1987).
Jean-Marc Bouclet and Stephan De Bi`evre Universit´e de Lille 1 UMR CNRS 8524 F-59655 Villeneuve d’Ascq France email:
[email protected] email:
[email protected] Communicated by Jens Marklof submitted 08/12/04, accepted 11/01/05
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 915 – 923 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05915-9, Published online 05.10.2005 DOI 10.1007/s00023-005-0229-5
Annales Henri Poincar´ e
Spin-Glass Stochastic Stability: a Rigorous Proof Pierluigi Contucci and Cristian Giardin` a Abstract. We prove the property of stochastic stability previously introduced as a consequence of the (unproved) continuity hypothesis in the temperature of the spinglass quenched state. We show that stochastic stability holds in β-average for both the Sherrington-Kirkpatrick model in terms of the square of the overlap function and for the Edwards-Anderson model in terms of the bond overlap. We show that the volume rate at which the property is reached in the thermodynamic limit is V −1 . As a byproduct we show that the stochastic stability identities coincide with those obtained with a different method by Ghirlanda and Guerra when applied to the thermal fluctuations only.
1 Introduction In a previous paper by Aizenman and Contucci [AC] the property of stochastic stability was introduced as the consequence of a continuity (in term of the inverse temperature β) hypothesis of the quenched state for the Sherrington-Kirkpatrick [SK] model. Stochastic stability says that a suitable class of perturbations of the spin glass Hamiltonian produces very small changes in the quenched equilibrium state and that such a change vanishes in the thermodynamic limit. This property has interesting consequences for the spin glass models: in terms of the overlap distribution it implies that the quenched measure is replica-equivalent [MPV, P] a property originally introduced within the replica symmetry breaking Parisi ansatz. The same property is also used in [FMPP1, FMPP2] to build a bridge between equilibrium and off-equilibrium properties in a spin-glass model being these last the only ones physically accessible to experimental investigation. More recently all and only the constraints that stochastic stability implies for the overlap moments have been completely classified [C, BCK]. In this paper we give a rigorous proof of stochastic stability property in βaverage. This result is achieved in an elementary way by use of the sum law for independent Gaussian variables and works in full generality for both mean-field and finite-dimensional spin glass models. We also derive the explicit form of the stochastic stability identities which first appeared in [AC] and we prove, using integration by parts in the spirit of [CDGG], that they coincide with a subset of the Ghirlanda-Guerra identities [G, GG], namely the part related to the thermal fluctuation bound (see also [T] for a nice set of rigorous results derived from those identities). The proof also provides the rate at which stochastic stability in β-average is reached with the thermodynamic limit which turns out to be V −1 . The paper
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is organized with Section 2 containing a list of the definitions and the statement of the two main theorems. Their proof is built in Section 3 while Section 4 shows how to apply the results to both the mean field models, which we illustrate for the Sherrington-Kirkpatrick model [SK], and for the finite-dimensional cases with the Edwards-Anderson model [EA]. Section 5 collects some comments.
2 Definitions and Results We consider a disordered model of Ising configurations σn = ±1, n ∈ Λ ⊂ Zd for some d-parallelepiped Λ of volume |Λ|. We denote ΣΛ the set of all σ = {σn }n∈Λ , and |ΣΛ | = 2|Λ| . In the sequel the following definitions will be used. 1. Hamiltonian. For every Λ ⊂ Zd let {HΛ (σ)}σ∈ΣN be a family of 2|Λ| translation invariant (in distribution) centered Gaussian random variables of volume-size covariance matrix Av (HΛ (σ)HΛ (τ )) = |Λ| QΛ (σ, τ ) ,
(2.1)
QΛ (σ, σ) = 1 .
(2.2)
and By the Schwarz inequality |QΛ (σ, τ )| ≤ 1 for all σ and τ . 2. Random partition function Z(β) :=
e−βHΛ (σ) .
(2.3)
σ∈ ΣΛ
3. Random free energy F (β) −βF (β) := A(β) := ln Z(β) .
(2.4)
4. Quenched free energy F (β) −βF (β) := A(β) := Av (A(β)) .
(2.5)
5. R-product random Gibbs-Boltzmann state Ω(−) :=
σ(1) ,...,σ(R)
(1)
(−)
e−β[HΛ (σ )+···+HΛ (σ [Z(β)]R
(R)
)]
.
(2.6)
6. Quenched equilibrium state − := Av (Ω(−)) .
(2.7)
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7. Observables. For any smooth bounded function G(QΛ ) (without loss of generality we consider |G| ≤ 1) of the covariance matrix entries we introduce the random (with respect to −) R × R matrix Q = {qk,l } by the formula G(Q) := Av (Ω(G(QΛ ))) .
(2.8)
E.g.: G(QΛ ) = QΛ (σ (1) , σ (2) )QΛ (σ (2) , σ (3) ) q1,2 q2,3 = Av
σ(1) ,σ(2) ,σ(3)
QΛ (σ (1) , σ (2) )QΛ (σ (2) , σ (3) ) e−β[
3 i=1
HΛ ( s(i) )]
. (2.9)
[Z(β)]3
8. Deformed quenched state. For every Λ ⊂ Zd let the {KΛ (σ)}σ∈ΣN be a translation invariant centered Gaussian random family of size one covariance matrix (2.10) Av (KΛ (σ)KΛ (τ )) = QΛ (σ, τ ) , where the families H and K are mutually independent with respect to the joint Gaussian distribution, i.e., Av (HΛ (σ)KΛ (τ )) = 0 . We consider Zλ (β) :=
e−βHΛ (σ)+
√
λKΛ (σ)
(2.11)
,
(2.12)
σ∈ ΣΛ
Aλ (β) := Av (ln Zλ (β)) ,
(2.13)
√
Ωλ (−) :=
λ [KΛ (σ(1) )+···+KΛ (σ(R) )] ) √ (1) )+···+K (σ (R) )] λ [K (σ Λ Λ Ω(e )
Ω((−) e
,
(2.14)
and the deformed quenched state −λ := Av (Ωλ (−)) .
(2.15)
9. Stochastic Stability. The quenched measure is said to be stochastically stable if for every observable G (see Def. 7) the deformed state is stationary in the thermodynamic limit: d Gλ = 0 lim (2.16) d Λ Z dλ It is possible to see (within Theorem 2) that there is a function of the overlap matrix elements: ∆G s.t. ∆Gλ :=
d Gλ . dλ
(2.17)
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A stochastically stable measure fulfills then the property lim ∆Gλ = 0
Λ Zd
(2.18)
for all the observables G. Our main result state that a spin glass model is stochastically stable β-almost everywhere (Theorem 1), characterizes the functions ∆G (Theorem 2) and establish their coincidence with the quantities obtained with the Ghirlanda-Guerra method when applied only to the thermal fluctuations. Theorem 1 (Stochastic Stability) The spin-glass quenched state is stochastically stable in β-average, i.e., for each interval [β1 , β2 ] and each observable G (as in Def. 7): 2 β2 2 2 . (2.19) ∆Gλ dβ ≤ β12 |Λ| Theorem 2 (Zero average Observables) The explicit form of the zero average quantities is 2∆G =
R
G q l, k − 2RG
k,l=1 k=l
R
q l, R+1 + R(R + 1)G q R+1, R+2 ,
(2.20)
l=1
which coincide with thermal part of the Ghirlanda-Guerra identities.
3 Proof of the results independent from H and K and distributed like Proof of Theorem 1. Since for H H we have, in distribution, that √ λ D HΛ −βHΛ + λKΛ = − β 2 + (3.21) |Λ| from Def. (8) of the deformed quenched state of the function
expecta G, all the λ λ 2 2 tions Gλ turn out to be functions of β + |Λ| : Gλ = g β + |Λ| . From the composite function derivation rule we deduce (the prime denotes derivative w.r.t. the argument): λ d 1 2 Gλ = g β + · (3.22) dλ |Λ| |Λ| and
λ d Gλ = g β 2 + · 2β , dβ |Λ|
(3.23)
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from which we have
d 1 d Gλ = Gλ . (3.24) dλ |Λ| dβ Integrating in dβ and using the fundamental theorem of calculus we obtain β22 Gλ (β2 ) − Gλ (β1 ) . (3.25) ∆Gλ dβ 2 = |Λ| β12 2β
Remembering the assumption on boundedness of function G (Def. 7) this complete the proof. Proof of Theorem 2. Let h(σ) = |Λ|−1 HΛ (σ) be the Hamiltonian per particle. From formula (3.24) and a direct computation of the derivative of Gλ with respect to the inverse temperature we have −2β∆Gλ =
R
Av Ωλ (h(σ (l) ) G) − Ωλ (h(σ (l) ))Ωλ (G) .
(3.26)
l=1
For each replica l (1 ≤ l ≤ R), we evaluate separately the two terms in the right side of Eq. (3.26) by using the integration by parts (generalized Wick formula) for correlated Gaussian random variables, x1 , x2 , . . . , xn n ∂ψ(x1 , . . . , xn ) Av (xi ψ(x1 , . . . , xn )) = Av (xi xj ) Av . (3.27) ∂xj j=1 It is convenient to denote by pλ (R) the Gibbs-Boltzmann weight of R copies of the deformed system pλ (R) =
e−β [
R k=1
HΛ (σ(k) ) ] +
√
λ[
[Zλ (β)]R
R k=1
KΛ (σ(k) ) ]
,
(3.28)
so that we have
R 1 dpλ (R) e−β[HΛ (τ )] = pλ (R) . − δσ(k) , τ − R pλ (R) β dHΛ (τ ) [Zλ (β)]
(3.29)
k=1
We obtain
1 Av Av Ωλ (h(σ (l) ) G) = G HΛ (σ (l) ) pλ (R) |Λ| (1) (r) σ ,...,σ (R) dp λ = Av G QΛ (σ (l) , τ ) dH (τ ) Λ (1) (r) τ σ ,...,σ
R = −β + G q l, k λ − RG q l, R+1 λ G λ k=1 k=l
(3.30)
(3.31)
(3.32)
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where in (3.31) we made use of the integration by parts formula and (3.32) is obtained by (3.29). Analogously, the other term reads
Av Ωλ (h(σ (l) )) Ωλ (G) 1 = Av G HΛ (σ (l) ) pλ (R + 1) (3.33) |Λ| (l) (1) (R) σ τ ,...,τ (R + 1) dp λ = Av (3.34) G QΛ (σ (l) , γ) dHΛ (γ) (l) (1) (R) γ σ
τ
,...,τ
= −β [Gλ + RG q l R+1 λ − (R + 1)G q R+1, R+2 λ ] .
(3.35)
Inserting the (3.32) and (3.35) in Eq. (3.26) we finally obtain Theorem 2. Remark. The proof of the theorems shows that the identities which follow from the stochastic stability property are included in the Ghirlanda-Guerra identities [GG]. Indeed the family of GG identities are obtained from the self-averaging of the internal energy per particle with respect to the full equilibrium quenched measure. This implies, by the use of the Cauchy-Schwartz inequality, the vanishing of the truncated correlation between internal energy per particle and a generic observable G in the thermodynamic limit: hG − hG → 0
as
|Λ| → ∞ .
(3.36)
But clearly the previous fluctuation can be decomposed as a sum of the thermal fluctuation (averaged over the Gaussian disorder) and the fluctuation with respect to the disorder itself, i.e., h G − hG = Av (Ω[h G]) − Av (Ω[h]) Av (Ω[G]) = Av (Ω[h G] − Ω[h]Ω[G])) + Av (Ω[h]Ω[G]) − Av (Ω[h]) Av (Ω[G])
(3.37) (3.38)
By formula (3.26) we see that the thermal fluctuations (Eq.(3.37)) are those controlled by the stochastic stability.
4 Models The results proved in the previous sections hold true in complete generality because they are based on the general property of Gaussian variables. Stochastic stability in particular is fulfilled by both mean field models (like the SherringtonKirkpatrick, its p-spin generalization, the REM and GREM models etc.) and by the finite-dimensional models (like the Edwards-Anderson and Random Field models in general dimension d). The main point to be observed and well stressed is that each one of these models has his own set of observables which describe the
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quenched equilibrium state, namely the Gaussian covariance matrix of their own Hamiltonians, see Eq. (2.1). To be more specific let illustrate the two main cases of the covariance matrix for the Sherrington-Kirkpatrick model and for the EdwardsAnderson. The SK model of Hamiltonian N 1 Ji,j σi σj HN (σ, J) = − √ N i,j=1
(4.39)
with {Jij } identical independent normal Gaussian variables has a covariance matrix given by the standard overlap function between two configurations: (SK) QΛ (σ, τ )
=
N 1 σi τi N i=1
2 .
(4.40)
The Edwards-Anderson Hamiltonian is
HΛ (J, σ) = −
Jn,n σn σn ,
(4.41)
(n,n )∈B(Λ)
where the Jn,n are again independent normal Gaussian variables and the sum runs over all pairs of nearest neighbors sites n, n ∈ Λ ⊂ Zd with |n − n | = 1. Using the standard identification of the space of nearest neighbors with the d-dimensional bond-lattice b ∈ Bd with b = (n, n ) and denoting B(Λ) the d-bond-parallelepiped associated to Λ (|B| = d|V |) we introduce, for two spin configurations σ and τ , the notation σb = σn σn and τb = τn τn . The covariance matrix turns out to be (EA)
QΛ
(σ, τ ) :=
1 Qb (σ, τ ) , |B|
(4.42)
b∈B
where the local bond-overlap Qb (σ, τ ) between σ and τ is Qb (σ, τ ) := σb τb .
(4.43)
The property of stochastic stability for the Edwards-Anderson model in terms of its link-overlap has been originally considered in [C2]. The theorem proved here provides the generalization to the generic observable G.
5 Comments In this paper we have proved that every Gaussian spin glass model is stochastically stable with respect to a suitable class of perturbations. The consequences of such a stability can be expressed as zero average observables in terms of the proper overlap that each model carries: the covariance of its own Hamiltonian. It is finally worth to mention that the identities that we proved for the Edwards-Anderson model
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are compatible with both the pictures of triviality and those of non-triviality for the overlap distribution at low temperature; for a discussion the reader may see the replica symmetry Breaking theory in [MPV], the Droplet theory in [FH, BM], the chaotic theory in [NS] and the trivial-non-trivial in [PY, KM]. Nevertheless the stochastic stability identities could suggest a test of triviality for the suitable overlap distribution in the same spirit of [MPRRZ]. We plan to return on these questions in a future work. Acknowledgments. We thank F. Guerra for many interesting discussions and in particular for an observation which led to a substantial improvement of this work. We also thanks A. Bovier, A. van Enter, S. Graffi, M. Talagrand and F.L. Toninelli.
References [AC]
M. Aizenman, P. Contucci, On the Stability of the Quenched state in Mean Field Spin Glass Models, J. Stat. Phys. 92, N. 5/6, 765–783 (1998).
[BCK]
A. Bianchi, P. Contucci, A. Knauf, Stochastically Stable Quenched Measures, math-ph/0404002, to appear in J. Stat. Phys. (2004).
[BM]
A.J. Bray and M.A. Moore, in Heidelberg Colloquium on Glassy Dynamics and Optimization, L. Van Hemmen and I. Morgenstern eds. Springer-Verlag, Heidelberg, (1986).
[C]
P. Contucci, Toward a classification theorem for stochastically stable measures, Markov Proc. and Rel. Fields. 9, N. 2, 167–176 (2002).
[C2]
P. Contucci, Replica Equivalence in the Edwards-Anderson Model, J. Phys. A: Math. Gen. 36, 10961–10966 (2003).
[CDGG]
P. Contucci, M. Degli Esposti, C. Giardin` a and S. Graffi, Thermodynamical Limit for Correlated Gaussian Random Energy Models, Commun. Math. Phys. 236, 55–63 (2003).
[EA]
S. Edwards and P.W. Anderson, Theory of spin glasses, J. Phys. F 5, 965–974 (1975).
[FH]
D.S. Fisher and D.A. Huse, Ordered Phase of Short-Range Ising SpinGlasses, Phys. Rev. Lett. 56, 1601–1604 (1986).
[FMPP1] S. Franz, M. Mezard, G. Parisi, L. Peliti, Measuring equilibrium properties in aging systems, Phys. Rev. Lett. 81, 1758 (1998). [FMPP2] S. Franz, M. Mezard, G. Parisi, L. Peliti, The response of glassy systems to random perturbations: A bridge between equilibrium and offequilibrium, J. Stat. Phys. 97, N. 3/4, 459–488 (1999).
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[G]
F. Guerra, About the overlap distribution in a mean field spin glass model, Int. J. Phys. B 10, 1675–1684 (1997).
[GG]
S. Ghirlanda, F. Guerra, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A: Math. Gen. 31, 9149–9155 (1998).
[KM]
F. Krzakala and O.C. Martin, Spin and Link Overlaps in ThreeDimensional Spin Glasses, Phys. Rev. Lett. 85, 3013–3016 (2000).
[MPRRZ] E. Marinari, G. Parisi, F. Ricci-Tersenghi, J. Ruiz-Lorenzo, F. Zuliani, Replica Symmetry Breaking in Short Range Spin Glasses: A Review of the Theoretical Foundations and of the Numerical Evidence, J. Stat. Phys. 98, N. 5, 973–1074 (2000). [MPV]
M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass theory and beyond, World Scientific, Singapore (1987).
[NS]
C.M. Newman and D.L. Stein, Spatial Inhomogeneity and Thermodynamic Chaos, Phys. Rev. Lett. 76, 4821–4824 (1996).
[P]
G. Parisi, On the probabilistic formulation of the replica approach to spin glasses, Int. Jou. Mod. Phys. B 18, 733–744 (2004).
[PY]
M. Palassini, A.P. Young, Nature of the Spin Glass State, Phys. Rev. Lett. 85, 3017–3021 (2000).
[SK]
D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett. 35, 1792–1796 (1975).
[T]
M. Talagrand, Spin glasses: a challenge for mathematicians, Berlin, Springer (2003).
Pierluigi Contucci and Cristian Giardin` a Dipartimento di Matematica Universit` a di Bologna I-40127 Bologna Italy email:
[email protected] email:
[email protected] Communicated by Jennifer Chayes submitted 13/10/04, accepted 22/11/04
Ann. Henri Poincar´e 6 (2005) 925 – 936 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05925-12, Published online 05.10.2005 DOI 10.1007/s00023-005-0230-z
Annales Henri Poincar´ e
Bohr-Sommerfeld Rules to All Orders Yves Colin de Verdi`ere
1 Introduction The goal of this paper is to give a rather simple algorithm which computes the Bohr-Sommerfeld quantization rules to all orders in the semi-classical parameter on the real line. The formula gives the highh for a semi-classical Hamiltonian H order terms in the expansion in powers of h of the semi-classical action using only integrals on the energy curves of quantities which are locally computable from the Weyl symbol. The recipe uses only the knowledge of the Moyal formula expressing the star product of Weyl symbols. It is important to note that our method assumes already the existence of Bohr-Sommerfeld rules to any order (which is usually shown using some precise Ansatz for the eigenfunctions, like the WKB-Maslov Ansatz) and the problem we address here is only about ways to compute these corrections. Existence of corrections to any order to Bohr-Sommerfeld rules is well known and can be found for example in [8] and [15] Section 4.5. Our way to get these high-order corrections is inspired by A. Voros’s thesis (1977) [13], [14]. The reference [1], where a very similar method is sketched, was given to us by A. Voros. We use also in an essential way the nice formula of in terms of the resolvent. Helffer-Sj¨ ostrand expressing f (H)
2 The setting and the main result Let us give a smooth classical Hamiltonian H : T R → R, where the symbol H admits the formal expansion H ∼ H0 + hH1 + · · · + hk Hk + · · · ; following [5] p. 101, we will assume that • H belongs to the space of symbols S o (m) for some order function m (for example m = (1 + |ξ|2 )p ) • H + i is elliptic 1 = Op and define H Weyl (H) with
OpWeyl (H)u(x) =
R2
e
i(x−y)ξ/h
dydξ x+y . , ξ)u(y) H( 2 2πh
1 Contrary to the usual notation, we denote by |dx · · · dx | the Lebesgue measure on Rn in n 1 order to avoid confusions related to orientations problems.
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is then essentially self-adjoint on L2 (R) with domain the Schwartz The operator H space S(R). In general, we will denote by σWeyl (A) the Weyl symbol of the operator A. The hypothesis: • We fix some compact interval I = [E− , E+ ] ⊂ R, E− < E+ , and we assume that there exists a topological ring A such that ∂A = A− ∪ A+ with A± a connected component of H0−1 (E± ). • We assume that H0 has no critical point in A • We assume that A− is included in the disk bounded by A+ . If it is not the case, we can always change H to −H. We define the well W as the disk bounded by A+ . Definition 1 Let HW : T R → R be equal to H in W , > E+ outside W and bounded. Then H W = OpWeyl (HW ) is a self-adjoint bounded operator. The semiclassical spectrum associated to the well W , denoted by σW , is defined as follows: σW = Spectrum(H W )∩] − ∞, E+ ] . The previous definition is useful because σW is independent of HW mod O(h∞ ). Moreover, if H0−1 (] − ∞, E + ]) = W1 ∪ · · · ∪ WN (connected components), then Spectrum(H)∩] − ∞, E + ] = ∪σWl + O(h∞ ) . The spectrum σW ∩ [E− , E+ ] is then given mod O(h∞ ) by the following Bohr-Sommerfeld rules Sh (En ) = 2πnh where n ∈ Z is the quantum number and the formal series Sh (E) =
∞
Sj (E)hj
j=0
is called the semi-classical action. Our goal is to give an algorithm for computing the functions Sj (E), E ∈ I. In fact exp(iSh (E)/h) is the holonomy of the WKB-Maslov microlocal soluˆ − E)u = 0 around the trajectory γE = H −1 (E) ∩ A. tions of (H
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H HW E+ I A+
A
A−
A− H = HW K
A+
E−
A
W Figure 1. The phase space.
It is well known that: • S0 (E) = γE ξdx = {H0 ≤E}∩W |dxdξ| is the action integral • S1 (E) = π − γE H1 |dt| includes the Maslov correction and the subprincipal term. Our main result is: Theorem 1 If H satisfies the previous hypothesis, we have: for j ≥ 2, (−1)l−1 d l−2 Sj (E) = Pj,l (x, ξ)|dt| (l − 1)! dE γE 2≤l≤L(j)
where • t is the parametrization of γE by the time evolution dx = (H0 )ξ dt, dξ = −(H0 )x dt
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• The Pj,l ’s are locally (in the phase space) computable quantities: more precisely each Pj,l (x, ξ) is a universal polynomial evaluated on the partial derivatives ∂ α H(x, ξ). The Pj,l ’s are given from the Weyl symbol of the resolvent (see Proposition (1)): L(j) ∞
1 Pj,l ˆ −1 = + hj . σWeyl (z − H) z − H0 j=1 (z − H0 )l l=2
If H = H0 , S2j+1 (E) = 0 for j > 0. In that case, the polynomial Pj,l (∂ α H) is homogeneous of degree l − 1 w.r. to H and the total weight of the derivatives is 2j, so that all monomials in Pj,l are of the form αk Πl−1 H k=1 ∂
with
l−1 k=1
|αk | = 2j and ∀k, |αk | ≥ 1.
Remark 1 We have also the following nice formula
2
(see also [14]): for any l ≥ 2,
hj Pj,l (x0 , ξ0 ) = (H − H0 (x0 , ξ0 ))(l−1) (x0 , ξ0 ) ,
j
where the power (l − 1) is taken w.r. to the star product. Proof. Let us denote h0 = H0 (x0 , ξ0 ). We have ˆ − h0 ) ˆ = (z − h0 ) − (H z−H and ˆ −1 = (z − H)
∞
ˆ − h0 )l−1 (z − h0 )−l (H
l=1
The formula follows then by identification of both expressions of the Weyl symbol of the resolvent at (x0 , ξ0 ). A less formal derivation is given by applying formula (3) to f (E) = (E − h0 )l−1 and computing Weyl symbols at the point (x0 , ξ0 ).
3 Moyal formula Let us define the Moyal product a b of the semi-classical symbols a and b by the rule: OpWeyl (a) ◦ OpWeyl (b) = OpWeyl(a b) 2I
learned this formula from Laurent Charles
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We have the well-known “Moyal formula” (see [5]): ab=
j ∞ 1 h {a, b}j j! 2i j=0
where {a, b}j (z) = [(∂ξ ∂x1 − ∂x ∂ξ1 )j (a(z) ⊗ b(z1 ))]|z1 =z with z = (x, ξ), z1 = (x1 , ξ1 ). In particular {a, b}0 = ab and {a, b}1 is the usual Poisson bracket. From the Moyal formula, we deduce the following (see also [14]): ˆ −1 of H ˆ is Proposition 1 The Weyl symbol j hj Rj (z) of the resolvent (z − H) given by L(j) ∞ ∞ Pj,l 1 j j h Rj (z) = + h (1) z − H0 j=1 (z − H0 )l j=0 l=2
where the Pj,l (x, ξ) are universal polynomials evaluated on the Taylor expansion of H at the point (x, ξ). If H = H0 , only even powers of j occur: R2j = 0. Proof. The proposition follows directly from the evaluation by Moyal formula of the left-hand side of ∞ (z − H) hj Rj = 1 . j=0
The important point is that the poles at z = H are at least of multiplicity 2 for j ≥ 1. Using ∞ ∞ (z − H) hj Rj = hj Rj (z − H) = 1 , j=0
j=0
and the fact that {., .}j are symmetric for even j’s and antisymmetric for odd j’s, we can prove the second statement by induction on j.
4 The method ∞ Let f ∈ Co∞ (I) and let us compute the trace D(f ) := Trace(f (H W )) mod O(h ) in 2 different ways:
1. Using the eigenvalues given by the Bohr-Sommerfeld rules we get: f (Sh−1 (2πhn)) + O(h∞ ) Trace(f (H W )) = n∈Z
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and, because f ◦ Sh−1 is a smooth function converging in the Co∞ topology to f ◦ S0−1 we can apply the Poisson summation formula and we get 1 D(f ) = f (Sh−1 (u))|du| + O(h∞ ) 2πh R 1 f (E)Sh (E)|dE| + O(h∞ ) 2πh R or using Schwartz distributions: and
D(f ) =
(a) D =
1 S (E) + O(h∞ ) 2πh h
using Helffer2. On the other hand, we compute the Weyl symbol of f (H) Sj¨ ostrand’s trick (see [5] p. 93): ∂F −1 |dxdy| = −1 (z)(z − H) (2) f (H) π Cz=x+iy ∂ z¯ where F ∈ C0∞ (C) is a quasi-analytic extension of f , i.e., F admits the Taylor expansion ∞ 1 (k) F (x + ζ) = f (x)ζ k k! k=0
at any real x. We start with the Weyl symbol of the resolvent (1). ˆ by putting Equation (1) into (2): We get then the symbol of f (H) j h ˆ = OpWeyl f (H0 ) + f (l−1) (H0 )Pj,l . f (H) (l − 1)! j≥1,l≥2
The justification of this formal step is done in [5]. We then compute the trace by using 1 Tr (OpWeyl (a)) = 2πh We get: 1 D(f ) = 2πh
T R
f (H0 ) +
j≥1,l≥2
T R
a(x, ξ)|dxdξ| .
1 f (l−1) (H0 )Pj,l |dxdξ| hj (l − 1)!
We can rewrite using |dtdE| = |dxdξ| and integrating by parts: l−1 l−1 d (−1) 1 T(E) + (b) D = hj Pj,l |dt| 2πh (l − 1)! dE γE j≥1,l≥2
(3)
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So we get, because l ≥ 2, by identification of (a) and (b), for j ≥ 1: Sj (E) −
(−1)l−1 d l−2 Pj,l |dt| = Cj (l − 1)! dE γE
(4)
l≥2
where the Cj ’s are independent of E. Proposition 2 In the previous formula (4), the Cj ’s are also independent of the operator. Proof. We can assume that (0, 0) is in the disk whose boundary is A− . Let us choose an Hamiltonian K which coincides with HW outside the disk bounded by A− and with the harmonic oscillator ˆ = OpWeyl( 1 (x2 + ξ 2 )) Ω 2 near the origin. We can assume that K has no other critical values than 0. We claim: for all j ≥ 1, ˆ = Cj (Ω) ˆ 1. Cj (K) ˆ = Cj (K) ˆ 2. Cj (H) Both claims come from the following facts: let us give 2 Hamiltonians whose Weyl symbols coincide in some ring B, then (i) The Pj,l are the same for 2 operators in the ring B where both have the same Weyl symbol, because they are locally computed from the symbols which are the same. (ii) The Sj (E)’s are the same for both operators because they have the same eigenvalues in the corresponding well modulo O(h∞ ): both operators have the same WKB-Maslov quasi-modes in B.
5 The case of the harmonic oscillator Proposition 3 For the harmonic oscillator, C1 = π and, for j ≥ 2, Cj = 0. ˆ = OpWeyl ( 1 (x2 + ξ 2 )) is the harmonic oscillator we have: Proof. If Ω 2 Sh (E) = 2πE + πh because En = (n − 12 )h for n = 1, . . . . It remains to compute the Pj,l ’s. Let us put ρ = 12 (x2 + ξ 2 ), and ∞
ˆ −1 = hj Rj σWeyl (z − Ω) j=0
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It is clear that the Rj ’s are functions fj (ρ, z) and from Moyal formula we get: fj+2 = −
1 (f + ρfj ) 4(z − ρ) j
and by induction on j: f2j+1 = 0 and f2j (ρ, z) =
l=3j+1 l=2j+1
al,j ρl−2j−1 , (z − ρ)l
with aj,l ∈ R. The result comes from
d dE
l−2 ρl−2j−1 |dt| = 0 , γE
if l ≥ 2j + 1.
6 The term S2 Let us assume first that H = H0 . From the Moyal formula, we have R2 = − with and
1 1 ∆ Γ {H0 , }2 = − − 3 z − H0 z − H0 4(z − H0 ) 4(z − H0 )4 ∆ = (H0 )xx (H0 )ξξ − ((H0 )xξ )2
Γ = (H0 )xx ((H0 )ξ )2 + (H0 )ξξ ((H0 )x )2 − 2(H0 )xξ (H0 )x (H0 )ξ .
A very similar computation can be found in [9] p. 93, formula (0.13). Using formulae (1) and (4), we get: S2 (E) = − Theorem 2
1 d 8 dE
∆|dt| + γE
1 24
• If H = H0 , we have S2 = −
1 d 24 dE
d dE
2 Γ|dt|.
(5)
γE
∆|dt|. γE
• In the general case, we have: 1 d 1 d S2 = − ∆|dt| − H2 |dt| + H 2 |dt| . 24 dE γE 2 dE γE 1 γE
(6)
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Formula (5) were obtained in [1], formula (3.12), and formula (6) by Robert Littlejohn [10, 2] using completely different methods. Proof. Γdt is the restriction to γE of the 1-form α in R2 with α = ((H0 )xx (H0 )ξ − (H0 )xξ (H0 )x )dx + ((H0 )xξ (H0 )ξ − (H0 )ξξ (H0 )x )dξ . Orienting γE along the Hamiltonian flow, we get using Stokes formula: Γ|dt| = α=− dα γE
γE
DE
where ∂DE = γE and DE is oriented by dx ∧ dξ. We have dα = −2∆dx ∧ dξ and hence:
Γ|dt| = 2 γE
∆|dxdξ| . DE
From |dtdE| = |dxdξ|, we get: d ∆|dxdξ| = ∆|dt| . dE DE γE So that:
d dE
Γ|dt| = 2 γE
∆|dt| γE
from which Theorem 2 follows easily.
7 Quantum numbers Theorem 3 The quantum number “n” in the Bohr-Sommerfeld rules corresponds exactly to the nth eigenvalue in the corresponding well, i.e., the nth eigenvalue of H W. Proof. It is clear that the labelling of the eigenvalues of H W is invariant by homoˆ topies leaving the symbol constant in A. We can then change H W to K for which the result is clear because the quantization rules give then exactly all eigenvalues.
8 Extensions 8.1
2d phase spaces
The method applies to any 2d phase space using only 3 things: • The star product
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• The fact that the trace of operators is given by (1/2πh)× (the integral of their symbols) • An example where you know enough to compute the Cj s The power of our method is that it avoids the use of any Ansatz. Maslov contributions come only from the computation of an explicit example.
8.2
The cylinder T (R/Z)
In that case, we replace the hypothesis by the following: • We fix some compact interval I = [E− , E+ ] ⊂ R, E− < E+ , and we assume there exists a topological ring A, homotopic to the zero section of T (R/Z), such that ∂A = A− ∪ A+ with A± a connected component of H −1 (E± ). • We assume that H has no critical point in A • We assume that A− is “below” A+ (see Figure 2).
A+
A
A− Figure 2. The cylinder. We will use the Weyl quantization for symbols which are of period 1 in x. Then Theorem 1 holds. The only change is S1 which is now 0. The proof is the same except that the reference operator is now hi ∂x instead of the harmonic oscillator.
8.3
Other extensions
It would be nice to extend the previous method to the case of Toeplitz operators on two-dimensional symplectic phase spaces, in the spirit of [3] and [4], and to the case of systems starting from the analysis in [6].
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As remarked by Littlejohn, our method does not obviously extend to semi1 , . . . , H d with d ≥ 2 degrees of freedom. classical completely integrable systems H The reason for that is that, using the same lines, we will get only the jacobian determinant of the d BS actions which is not enough to recover the actions even up to constants.
9 Relations with KdV ˆ = −∂ 2 + q(x) with q(x + 1) = Let us consider the periodic Schr¨odinger equation H x q(x). Let us denote by λ1 < λ2 ≤ λ3 < λ4 ≤ · · · the eigenvalues of the periodic ˆ Then the partition function problem for H. ∞
Z(t) =
e−tλn
n=1
admits, as t → 0+ , the following asymptotic expansion 1 a0 + a1 t + · · · + aj tj + · · · + O(t∞ ) Z(t) = √ 4πt where the aj ’s are of the following form aj =
0
1
Aj q(x), q (x), . . . , q (l) (x), . . . |dx|
where the Aj ’s are polynomials. The aj ’s are called the Korteweg-de Vries invariants because they are independent of u if qu (x) = Q(x, u) is a solution of the Korteweg-de Vries equation. See [11], [12] and [16]. Let us translate the previous objects in the semi-classical context: we have ˆ and h2 H ˆ is the semi-classical operator of order 0 whose Z(h2 ) = Tr exp(−h2 H) Weyl symbol is ξ 2 +h2 q(x). If we put f (E) = e−E , the partition function is exactly a trace of the form used in our method except that E → e−E is not compactly supported. Nevertheless, the similarity between both situations is rather clear.
References [1] P.N. Argyres, The Bohr-Sommerfeld Quantization Rule and the Weyl Correspondence, Physics 2, 131–199 (1965). [2] M. Cargo, A. Gracia-Saz, R. Littlejohn, M. Reinsch & P. de Rios, Moyal star product approach to the Bohr-Sommerfeld approximation, J. Phys. A: Math and Gen. 38 , 1977–2004 (2005). [3] L. Charles, Berezin-Toeplitz Operators, a Semi-classical Approach, Commun. Math. Phys. 239, 1–28 (2003).
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[4] L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Commun. Partial Differential Equations 28, 1527–1566 (2003). [5] M. Dimassi & J. Sj¨ ostrand, Spectral Asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes 268, Cambridge UP (1999). [6] C. Emmrich & A. Weinstein, Geometry of the transport equation in multicomponent WKB approximations, Commun. Math. Phys. 176 No.3, 701–711 (1996). [7] A. Gracia-Saz, The symbol of a function of a pseudo-differential operator, Ann. Inst. Fourier (to appear). [8] B. Helffer & D. Robert, Puits de potentiel g´en´eralis´es et asymptotique semiclassique, Ann. IHP (physique th´eorique) 41, 291–331 (1984). [9] B. Helffer & D. Robert, Riesz means of bound states and semiclassical limit connected with a Lieb-Thirring’s conjecture, Asymptotic Analysis 3, 91–103 (1990) . [10] R. Littlejohn, Lie Algebraic Approach to Higher-Order Terms, Preprint 17p. (June 2003). [11] H.P. McKean & P. van Moerbeke, The spectrum of Hill’s equation, Invent. Math. 30, 217–274 (1975). [12] W. Magnus & S. Winkler, Hill’s equation, New York-London-Sydney: Interscience Publishers, a division of John Wiley & Sons. VIII, 127 p. (1966). [13] A. Voros, D´eveloppements semi-classiques, Th`ese de doctorat (Orsay, 1977). [14] A. Voros, Asymptotic -expansions of stationary quantum states, Ann. Inst. H. Poincar´e Sect. A (N.S.) 26, 343–403 (1977). [15] San V˜ u Ngo.c, Sur le spectre des syst`emes compl`etement int´egrables semiclassiques avec singularit´es, Th`ese de doctorat (Grenoble, 1998), http://wwwfourier.ujf-grenoble.fr/˜ svungoc/ . [16] V. Zakharov & L. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system, Funct. Anal. Appl. 5, 280–287 (1977). Yves Colin de Verdi`ere Institut Fourier Unit´e mixte de recherche CNRS-UJF 5582 BP 74 F-38402 Saint Martin d’H`eres Cedex France email:
[email protected] Communicated by Bernard Helffer submitted 02/09/04, accepted 29/10/04
Ann. Henri Poincar´e 6 (2005) 937 – 990 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/05937-54, Published online 05.10.2005 DOI 10.1007/s00023-005-0231-y
Annales Henri Poincar´ e
Determination of Non-adiabatic Scattering Wave Functions in a Born-Oppenheimer Model George A. Hagedorn∗ and Alain Joye
Abstract. We study non-adiabatic transitions in scattering theory for the timedependent molecular Schr¨ odinger equation in the Born-Oppenheimer limit. We assume the electron Hamiltonian has finitely many levels and consider the propagation of coherent states with high enough total energy. When two of the electronic levels are isolated from the rest of the electron Hamiltonian’s spectrum and display an avoided crossing, we compute the component of the nuclear wave function associated with the non-adiabatic transition that is generated by propagation through the avoided crossing. This component is shown to be exponentially small in the square of the Born-Oppenheimer parameter, due to the Landau-Zener mechanism. It propagates asymptotically as a free Gaussian in the nuclear variables, and its momentum is shifted. The total transition probability for this transition and the momentum shift are both larger than what one would expect from a naive approximation and energy conservation.
1 Introduction We study scattering theory for the time-dependent molecular Schr¨odinger equation 4 ∂ 2 ∂ − i 2 ψ(x, t, ) = + h(x) ψ(x, t, ) in L2 (R, Cm ), (1.1) ∂t 2 ∂x2 where the electronic hamiltonian h(x) is an m×m self-adjoint matrix that depends on the nuclear position variable x ∈ R. The Born-Oppenheimer parameter > 0 denotes the fourth root of the electron mass divided by the mean nuclear mass. We compute the leading order asymptotics of nuclear wave functions associated with certain non-adiabatic transitions of the electrons. The Landau-Zener mechanism responsible for these makes them exponentially small in 1/2 as → 0. Our most general result can be found in Theorem 5.1. Describing the most general situation requires the development of a significant amount of notation and some technical hypotheses. So, in this introduction, we describe two physically interesting special cases that illustrate the main consequences of our analysis in a simple situation. Theorems 6.1 and 6.2 give precise statements of our results for these special cases. ∗ Partially Supported by National Science Foundation Grants DMS–0071692 and DMS– 0303586.
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Figure 1. A plot of typical electron energy levels involved in an avoided crossing. Energy is plotted vertically. Position is plotted horizontally, and the avoided crossing occurs in the middle of the plot where the difference between the energy levels is a minimum. Suppose h(x) is a real 2 × 2 self-adjoint matrix that depends analytically on x and has limits h(±∞) as x → ±∞ that are approached sufficiently rapidly. Denote the eigenvalues of h(x) by ej (x), and assume that e2 (x) ≥ e1 (x) + δ for all x ∈ R, where δ > 0. √ Near x = 0, assume e1 and e2 have an avoided crossing, i.e., e2 (x) − e1 (x) x2 + δ 2 close to x = 0, with δ small but positive. Such an avoided crossing corresponds to complex crossing points z0 and z0 , where the analytic continuations of e1 and e2 satisfy e1 (z0 ) = e2 (z0 ), and z0 is close to the real axis, with z0 = O(δ). Let φ1 (x) and φ2 (x) denote normalized, real eigenvectors associated with e1 (x) and e2 (x). Among the nuclear wave functions we can accommodate are Gaussian coherent states that are defined by 2
ϕ0 (A, B, , a, η, x) =
1 π 1/4 1/2 A1/2
exp
B (x − a)2 η (x − a) − +i 2 2A 2
,
where the complex numbers A and B satisfy the normalization condition Re BA = 1. These states are localized in position near x = a, and in momentum near p = η. Their position uncertainty is |A| and their momentum uncertainty is |B|. For a thorough discussion of these wave packets, see [9]. Choose E > supx∈R e2 (x). For a state incoming from the left on the upper electronic level, choose η− > 0. We assume η− is large enough so that the classical 2 energy η− /2 + e2 (−∞) > E. There exists a solution to (1.1) whose large negative
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t asymptotics are given by 2
2
ei(η− /2−e2 (−∞))t/ ϕ0 (A− + iB− t, B− , 2 , a− + η− t, η− , x) φ2 (x),
(1.2)
where the nuclear part is a free Gaussian. Since the electronic levels are isolated from one another, the large positive t asymptotics of this solution are multiples of φ2 (x), up to exponentially small errors in 1/2 . They have the leading behavior determined by the standard time-dependent Born-Oppenheimer approximation as → 0, see [8]: 2
2
eiθ1 () ei(η1 /2−e2 (∞))t/ ϕ0 (A1 + iB1 t, B1 , 2 , a1 + η1 t, η1 , x) φ2 (x), where eiθ1 () is some explicit phase, and the parameters A1 , B1 , a1 , η1 are determined by the scattering properties of the classical Hamiltonian p2 /2 + e2 (x). Our interest lies with the leading order asymptotics of the non-adiabatic component of the wave function for large positive t and → 0. We prove in Theorem 6.1 that these have the form ∗
c0 e−α
/2
2
eiθ+ () ei(η+ /2−e1 (∞))t/
2
ϕ0 (A+ + iB+ t, B+ , 2 , a+ + η+ t, η+ , x) φ1 (x), and we specify how the phase θ+ (), the -independent amplitude c0 > 0, the exponential decay rate α∗ > 0, and the parameters of the free Gaussian part A+ , B+ , a+ , and η+ > 0 are determined. As a corollary, the leading term of the transition amplitude A() (whose absolute square is the transition probability) is given by the quantity ∗
A() = c0 eiθ+ () e−α
/2
,
as → 0.
(1.3)
Let us describe the main features of this exponentially small transmitted part of the wave function. One may naively expect η+ to be determined by the energy conservation condition 2 η2 η− + e2 (−∞) = + + e1 (∞), 2 2
but this yields the wrong value. The correct value is larger. Intuitively, this is due to the faster parts of the wave function behaving less adiabatically than the slower parts. Because this dependence on the speed appears in an exponent, it leads to an O(1) change in the final momentum η+ . In other words, the higher momentum components of the incoming state are much more likely to make a transition than the lower momentum components. Hence, after the transition, there are more fast pieces of the wave function, and the final average momentum is greater than one would naively expect from an energy conservation calculation based solely on the average incoming momentum.
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This also affects the transition amplitude which is larger than what is naively expected. It is asymptotically composed of an -independent prefactor c0 times an ∗ 2 exponentially small quantity e−α / , whose decay rate α∗ is related to that of the Landau-Zener decay rate for purely adiabatic problems. Actually, α∗ consists of the sum of the imaginary part of some action integral around the complex electronic eigenvalue crossing point z0 and a contribution that depends explicitly on the nuclear part of the initial incoming state (1.2). The action integral depends only on the electronic levels and reads ζ 2(E − e2 (z)dz where ζ is a loop in the complex plane based at the origin encircling z0 . The contribution from the nuclear part of the wave packet depends on the shape of its momentum/energy density. It is that last contribution that makes the obvious candidate given by the imaginary part of the action integral taken at the classical energy E, miss the actual value of the decay rate α∗ . In that sense, (1.3), which we could call a molecular Landau-Zener formula, cannot be determined from the usual adiabatic Landau-Zener formula with just the knowledge of the electronic levels and the classical nuclear momentum close to the avoided crossing. Indeed, our analysis shows that we also need to take into account the details of the incoming wave packet to determine (1.3). This is why we resort to coherent states to get such accurate asymptotics. The way we obtain all our results is by employing a time-independent scattering theory approach that uses generalized eigenfunctions of the full Hamiltonian. We expand the wave function in terms of the generalized eigenfunctions and calculate the large |t| asymptotics. For every incoming momentum k there is classical energy conservation, but a different probability of making the non-adiabatic transition. We obtain the correct α∗ and η+ by computing the averages over k rather than by doing one calculation based on the average incoming momentum η− . Remarks 1. We obtain the analogous results when the incoming state is associated with the lower electronic level e1 , provided that we keep the average total energy above both the levels. 2. There are other components of the scattered wave function. For example, one should expect a reflected wave on the e2 electronic level and also a reflected wave on the e1 level. We prove that if the avoided crossing has a sufficiently small gap, then the other components are exponentially even smaller in 1/2 than the transmitted non-adiabatic term we compute. The second situation we describe in this introduction involves the same setup as above, but with the Gaussian incoming states replaced by more general incoming coherent states. This example illustrates the second key feature that our analysis demonstrates: even if the incoming state is not Gaussian, but is any polynomial times a Gaussian, the outgoing non-adiabatic transition state is Gaussian to leading order in .
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For m = 1, 2, . . . , we define ϕm (A, B, 2 , a, η, x) = 2−m/2 (m!)−1/2 A−m/2 (A)m/2 Hm
x−a |A|
φ0 (A, B, 2 , a, η, x),
(1.4)
where Hm is the mth order Hermite polynomial. We now replace (1.2) by 2
2
ei(η− /2−e2 (−∞))t/ ϕm (A− + iB− t, B− , 2 , a− + η− t, η− , x) φ2 (x).
(1.5)
Again, up to exponentially small errors, the large positive t asymptotics of the solution are multiples of φ2 (x). Their leading behavior is determined by the standard time-dependent Born-Oppenheimer approximation, 2
2
eiθ1 () ei(η1 /2−e2 (∞))t/ ϕm (A1 + iB1 t, B1 , 2 , a1 + η1 t, η1 , x) φ2 (x), where A1 , B1 , a1 , η1 , and θ1 () are the same as in our first example. However, our Theorem 6.2 shows that the leading order asymptotics of the non-adiabatic component of the wave function for large positive t again have the form of a freely propagating Gaussian ∗
cm −m e−α
/2
2
eiθ+ () ei(η+ /2−e1 (∞))t/
2
× ϕ0 (A+ + iB+ t, B+ , 2 , a+ + η+ t, η+ , x) φ1 (x), and display a pre-exponential factor of order −m . The values of α∗ , A+ , B+ , a+ , and η+ are the same as in our first example, and we determine the prefactor cm . The numerics presented below clearly illustrate these features. The presence of the factor of −m can be understood as follows: In momentum space, the Hermite polynomial in the wave function does not get the extra shift that the Gaussian does. For small , the scaling in these two factors causes the Hermite polynomial to behave like a constant times −m pm where the Gaussian is large. Since the Gaussian is highly localized, a zeroth order Taylor approximation can be used to see that the wave function is asymptotically a multiple of the Gaussian times −m . Our most general result, Theorem 5.1, extends these results in several ways. First, we can handle electron Hamiltonians h(x) that are m×m complex hermitian matrices which have two levels of interest that have an avoided crossing. These levels must stay well separated from the rest of the spectrum of h(x). Second, we can handle situations in which several levels display certain patterns of avoided crossings. For example, when two levels have an avoided crossing for one value of x, and one of those levels has another avoided crossing with a third level for some other value of x. However, in such cases, we can only study the non-adiabatic components for certain levels. The ones we can handle depend on the order in
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which the levels have the avoided crossings. Third, we can consider more general incoming states that do not have the form of the ϕj ’s considered above. They are characterized by an energy (or momentum) distribution which is sharply peaked around some fixed energy, so that a semiclassical analysis can be performed. In such general cases also, the nuclear part of the non-adiabatic wave function is Gaussian and exponentially small, with a decay rate sharing the properties described above. The paper is organized as follows: In the rest of the Introduction, we review the relevant literature and present numerical results for the above examples. They show excellent agreement with our analysis. (For the first example with = 0.2, our formulas give a transition probability of 1.215×10−9 and a momentum expectation after transition of 2.0516. The corresponding values from the numerical simulations are 1.217 × 10−9 and 2.0543. This agreement is remarkable, given how large our is.) In Section 2, we set up the general problem we study. We state most of our hypotheses here and make precise the notion of avoided crossing. In Section 3, we study generalized eigenvectors of the full Hamiltonian. In particular, their WKB-type analysis in the complex plane is performed here. We superimpose the generalized eigenvectors to generate solutions to the time-dependent Schr¨ odinger equation and construct asymptotic scattering states in Section 4. Non-adiabatic transition asymptotics are studied in Section 5, where our most general result is stated as Theorem 5.1. Further properties and estimates on the energy and momentum shifts are provided in Section 5. Section 6 is devoted to the special case of interest where the nuclear part of the incoming state is a Gaussian or a Gaussian times a Hermite polynomial as in (1.5). Finally, Section 7 contains the proofs of several technical results that are stated in the earlier sections. From this outline, one can see that our results depend crucially on the properties of generalized eigenvectors of the full Hamiltonian. We prove these properties by revisiting ideas and results of Joye [14], [15] that provide exponentially accurate WKB-type results in a generic avoided-crossing regime, generalizing earlier two-level adiabatic techniques from [17], [18], [19]. This step is necessary in order to control the dependence of the relevant quantities in the energy parameter. See also [21], [24] for stationary results of the same kind, obtained at fixed energy and, essentially, for two-level systems. That a complex WKB-type analysis plays an important role here should be no surprise. Indeed, in the ODE context of adiabatic-like problems dealt with in the references above, the complex WKB approach proved to be the most efficient method providing a quantitative analysis of the exponentially small leading order term of the Landau-Zener mechanism. See, however, [13] and [2] for a different successful approach of such problems, based on optimal truncation techniques. Nevertheless, the understanding of the stationary scattering process is of course not enough to get control on the time-dependent propagation of states: as our main result demonstrates, even the decay rate of the transition amplitude cannot be determined by the stationary data only. There are mathematical results on the exponentially small size of nonadiabatic transitions in the Born-Oppenheimer approximation, and for related
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problems. See, e.g., [12], [22], [1], [23]. However, to the best of our knowledge, there are no rigorous results on this topic in the literature that actually compute the leading asymptotics of non-adiabatic transitions in our time-dependent PDE setting. We have recently learned that Betz and Teufel, [3], are adapting techniques from [2] to the Born-Oppenheimer setup. They have formal and numerical results for specific electronic hamiltonians in agreement with ours. Also, rigorous results on the propagation of wave packets through avoided crossings, representing first attempts to unravel the molecular Landau-Zener mechanism, are obtained in [10], [11]. (See also [25].) In those papers, the gap δ shrinks to zero with in such a way that the transitions are of order one, so that they can be computed by perturbation theory. This is in contrast to the present situation, in which δ is small but fixed as → 0, and the transitions are exponentially small. Because of the importance of the Landau-Zener mechanism to molecular physics, there are relevant papers in the physics and chemistry literature. See, e.g., [4], [26], [27].
1.1
Numerical simulations for a Gaussian initial state
We now present graphical results of a numerical simulation in which the initial state is a Gaussian function associated with the upper energy level for a two level system. These plots are in very good agreement with the results of our analysis. We have numerically integrated equation (1.1) with = 0.2 for the Hamiltonian function 1 1 tanh(x) h(x) = . −1 2 tanh(x) 1 1 + tanh(x)2 , and there is an avoided crossing at The energy levels are ± 2 x = 0 with a minimum gap of 1. The initial state is the eigenvector associated with the upper energy level times the Gaussian φ0 (A0 +itB0 , B0 , 2 , ηt, η, x), where A0 = B0 = η = 1, with the initial time t = −10. The following two figures show the initial position and momentum probability densities, respectively. In both plots, the probability of being on the lower energy level is zero. The next two plots show the position and momentum probability densities at t = 9 after the wave function has interacted with the avoided crossing. The component associated with the lower energy level has mean momentum 2.05. It is evident from the plot that it is greater than 2. The naive energy conservation calculation predicts the following: The total energy is E = η 2 /2 + 1/2 1 + tanh(−10)2 = 1.2071. After the to the lower surface, the kinetic energy should be this value plus √ transition 2/2, so η12 /2 = 1.9142. This predicts a final momentum after the transition of η1 = 1.9566.
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0.35
0.3
0.25
0.2
0.15
0.1
0.05
0 −20
−10
−15
−5
5
0
10
20
15
25
Figure 2. Position space plot at time t = −10 of the probability density for being on the upper energy level (solid line), and 3 × 108 times the probability density for being on the lower energy level (dotted line). The dotted line cannot be seen because it coincides with the horizontal axis.
3
2.5
2
1.5
1
0.5
0 −6
−4
−2
0
2
4
6
Figure 3. Momentum space plot at time t = −10 of the probability density for being on the upper energy level (solid line), and 3 × 108 times the probability density for being on the lower energy level (dotted line). As in Figure 2, the dotted line cannot be seen.
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0.35
0.3
0.25
0.2
0.15
0.1
0.05
0 −20
−10
−15
−5
5
0
10
20
15
25
Figure 4. Position space plot at time t = 9 of the probability density for being on the upper energy level (solid line), and 3 × 108 times the probability density for being on the lower energy level (dotted line).
3
2.5
2
1.5
1
0.5
0 −6
−4
−2
0
2
4
6
Figure 5. Momentum space plot at time t = 9 of the probability density for being on the upper energy level (solid line), and 3 × 108 times the probability density for being on the lower energy level (dotted line).
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Numerical simulations for more general initial states
We next present in Figures 6 to 9 the results for the same system as above, but where the initial Gaussian φ0 has been replaced by φ3 . See (1.4). Note that the transition amplitude is significantly larger than in the example above, and that the component of the wave function that makes the transition to the lower level is approximately a Gaussian. The value of epsilon = 0.2 is not particularly small, so the component of the final state that does not make a transition is only approximately a φ3 wave packet. We have chosen this relatively large value of epsilon to avoid numerical difficulties in integrating equation (1.1). We should also note that the naive energy conservation calculation again predicts that the component of the wave function on the lower level should have mean momentum 1.9566. Since initial wave function has a greater momentum uncertainty than in the Gaussian example above, we see an even greater discrepancy between this prediction and the correct value. Our simulation yields a value of roughly 2.25. 0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0 −20
−15
−10
−5
0
5
10
15
20
25
Figure 6. Position space plot at time t = −10 of the probability density for being on the upper energy level (solid line), and 107 times the probability density for being on the lower energy level (dotted line). As in Figures 2 and 3, the dotted line cannot be seen.
2 Hypotheses for the electron Hamiltonian We begin with three general assumptions about the electron Hamiltonian h. We then impose two more assumptions that make precise the avoided crossing situations we can handle.
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1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0 −6
−4
−2
0
2
4
6
Figure 7. Momentum space plot at time t = −10 of the probability densityfor being on the upper energy level (solid line), and 107 times the probability density for being on the lower energy level (dotted line). As in Figure 2, 3, and 6, the dotted line cannot be seen.
0.25
0.2
0.15
0.1
0.05
0 −20
−15
−10
−5
0
5
10
15
20
25
Figure 8. Position space plot at time t = 9 of the probability density for being on the upper energy level (solid line), and 107 times the probability density for being on the lower energy level (dotted line).
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1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0 −6
−4
−2
0
2
4
6
Figure 9. Momentum space plot at time t = 9 of the probability density for being on the upper energy level (solid line), and 107 times the probability density for being on the lower energy level (dotted line). H1: We assume z → h(z) is a m × m matrix-valued analytic function that is analytic in z ∈ ρα = {z = x + iy : |y| ≤ α}, where α > 0. We assume h(z) is self-adjoint for z ∈ R. Since we work in a scattering framework, we further assume: H2: There exist ν > 1/2, c, and two matrices h(±∞), such that for all x ∈ R, sup h(x + iy) − h(±∞) ≤
|y|≤α
c ,
x 2+ν
where x denotes (1 + x2 )1/2 . The rate of convergence in this assumption can certainly be weakened. However, general scattering theory is not the main point of the present study. The following assumption is stronger than what one would expect physically to be necessary because it requires conditions on all eigenvalues, not just the ones of involved in the avoided crossings we consider later. However, we need this assumption for our proof. Specifically, this hypothesis for all eigenvalues plays a role in the proof of (7.29) at the end of the proof of Proposition 3.1. H3: We assume the spectrum σ(h(x)) of h(x) consists of m non-degenerate eigenvalues σ(h(x)) = {ej (x)}j=1,...,m , for any x ∈ R ∪ {±∞}.
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We let φj (x), j = 1, . . . , m, denote the corresponding eigenvectors, characterized up to constant phases by the following conditions φj (x) ≡ 1,
and
φj (x), φj (x) ≡ 0,
∀ j = 1, . . . , m,
(2.1)
where the prime denotes the derivative with respect to x. The eigenvectors are analytic in some narrow open strip containing the real axis [20]. By using the Cauchy formula, it is easy to check that our hypotheses imply
and
dn ( ej (x) − ej (±∞) ) = O( x −(2+ν) ) dxn
(2.2)
dn ( φj (x) − φj (±∞) ) = O( x −(2+ν) ), dxn
(2.3)
for any n ∈ N. We now make specific assumptions concerning avoided crossings for h. The idea is to assume h(x) belongs to a smooth family of electron Hamiltonians h(x, δ). When δ = 0, we assume there are actual crossings. When δ = 0, we assume there are no crossings for real values of x. The electron Hamiltonians we actually use have the form h(x, δ) for some small, but fixed value of δ. Our precise assumption is the following: H4: For each fixed δ ∈ [0, d], the matrix h(x, δ) satisfies H1 in a strip ρα independent of δ, and h(z, δ) is C 2 as a function of the two variables (z, δ) ∈ ρα ×[0, d]. Moreover, h(·) satisfies H2 uniformly for δ ∈ [0, d], with limiting values h(±∞, δ) that are C 2 functions of δ ∈ [0, d]. Again, some of our results hold under weaker smoothness assumptions. We can deal with multiple avoided crossings, but cannot deal with all possible patterns of avoided crossings. The following assumption describes the ones we allow. This assumption is complicated, and we recommend the reader look at the picture on page 686 of [15] to get some intuition about its meaning. H5: For each x ∈ R and each δ ∈ [0, d], σ(h(x, δ)) consists of m real eigenvalues σ(h(x, δ)) = {e1 (x, δ), e2 (x, δ), . . . , em (x, δ)} ⊂ R.
(2.4)
When δ > 0 we assume these are distinct for x ∈ [−∞, +∞] and are labeled by e1 (x, δ) < e2 (x, δ) < · · · < em (x, δ). When δ = 0, the eigenvalues are m analytic functions that have finitely many real crossings at x1 ≤ x2 ≤ · · · ≤ xp , with p ≥ 1. We assume the eigenvalues have m distinct limits as x → −∞ and as x → ∞. We label these eigenvalues ej (x, 0) in a way that is discontinuous in δ near δ = 0. This labeling is determined by the following conditions:
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i) For all x < x1 , e1 (x, 0) < e2 (x, 0) < · · · < em (x, 0). ii) For all j < l ∈ {1, 2, . . . , n}, there exists at most one xr with ej (xr , 0) − el (xr , 0) = 0, and if such an xr exists, we have ∂ (ej (xr , 0) − el (xr , 0)) > 0. ∂x
(2.5)
iii) For all j ∈ {1, 2, . . . , n}, the eigenvalue ej (x, 0) crosses eigenvalues whose indices are all superior to j or all inferior to j. Remarks i) The parameter δ can be understood as a coupling constant that controls the strength of the perturbation that lifts the degeneracies of h(x, 0) on the real axis. ii) Because h(x, 0) is self-adjoint, the eigenvalues ej (x, 0) cannot have branch points on the real axis, and they are analytic in a neighborhood of the real axis. iii) The crossings are assumed to be generic in the sense that the derivatives of ej − ek are non-zero at the crossing xr . This ensures that when δ > 0 is small, the generic behavior (3.19) holds at the corresponding complex crossing points. iv) When m = 2, H5 requires that the two eigenvalues have exactly one generic crossings when δ = 0. v) The crossing points {x1 , x2 , . . . , xp } need not be distinct, which is important when the Hamiltonian possesses symmetries. However, for each j = 1, . . . , n, the eigenvalue ej (x, δ) experiences avoided crossings with ej+1 (x, δ) and/or ej−1 (x, δ) at a subset of distinct points {xr1 , . . . , xrj } ⊆ {x1 , x2 , . . . , xp }. For certain results, we also impose the condition that these avoided crossings be generic in the sense of [7] and [14]. This condition essentially says that the low order Taylor series coefficients of certain quantities do not vanish at the crossing when δ = 0. H6: Near an avoided crossing of ej (x, δ) and en (x, δ), there exist a > 0, b > 0, and c ∈ R, such that en (x, δ) − ej (x, δ) = ± ax2 + 2cxδ + b2 δ 2 + R3 (x, δ), (2.6) where c2 < a2 b2 and R3 (x, δ) is a remainder of order 3 in (x, δ) close to (0, 0). Our final hypothesis involves both the electron Hamiltonian and an interval of energies, ∆. We ultimately consider states of the full Hamiltonian whose energy is concentrated in ∆, with ∆ high enough that scattering onto all the electron
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energy levels is possible. An energy range that satisfies this condition can always be chosen for some strip ρα , provided the minimum value in ∆ is large enough. H7: The interval ∆ ∈ R is compact and has non-empty interior. Furthermore, it is chosen so that for all j, inf
E∈∆ z∈ρα δ∈[0,δ]
|E − ej (z, δ)| > 0.
3 Generalized eigenvectors For energies E ∈ ∆, we construct generalized eigenvectors for the full Hamiltonian. For the time being, the parameter δ > 0 is fixed and we drop it in the notation. The generalized eigenvectors are solutions Ψ(x, E, ) ∈ Cm to the time-independent Schr¨ odinger equation 4 ∂ 2 + h(x) Ψ(x, E, ) = E Ψ(x, E, ). (3.1) − 2 ∂x2 For each E ∈ ∆, the set of such solutions is 2m-dimensional, and individual solutions can be characterized by their asymptotics at x = −∞ (or at x = ∞). Let Ψ(x, E, ) Φ(x, E, ) = ∈ C2m . ∂ i 2 ∂x Ψ(x, E, ) Then (3.1) is equivalent to i 2 where
∂ Φ(x, E, ) = H(x, E) Φ(x, E, ), ∂x
H(x, E) =
O
I
2 (E I − h(x))
O
E I − h(x) > 0,
(3.2)
∈ M2m (C)
for all x ∈ R ∪ {±∞}.
and (3.3)
Here, I denotes the identity matrix in Cm . Note that the matrix H(x, E) is not self-adjoint, but satisfies the relation O I ∗ H(x, E) = J H (x, E) J, where J= . (3.4) I O The small asymptotics of solutions to (3.2) are studied in [15]. Of particular importance to us is Section 7 of [15], which is devoted to the computation of exponentially small elements of the related S-matrix that we describe below. We apply the results of [15] to (3.2), keeping track of the dependence on E.
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By our hypotheses on E and h(x), the spectrum of H(E, x) consists of 2m distinct real eigenvalues =+,− σ(H(x, E)) = { kjτ (x, E) }τj=1,...,m ,
kjτ (x, E) = τ kj (x, E) = τ
with
2 (E − ej (x)) ∈ R.
(3.5)
Note that the kjτ ’s correspond to the classical momenta associated with the classical potentials ej (x). A set of corresponding eigenvectors { χτj (x, E) } is given (in block notation) by φj (x) τ χj (x, E) = (3.6) ∈ C2m . kjτ (x, E) φj (x) From these we produce new eigenvectors 1 ϕτj (x, E) = χτj (x, E) 2 kj (x, E)
(3.7)
that satisfy the normalization convention (3.10) below, that was adopted in [15]. This normalization is motivated by the following: We can write H(x, E) = kjτ (x, E) Pjτ (x, E), j,τ
where {Pjτ (x, E)} denotes a set of non-orthogonal projections onto the eigenspaces of H(x, E). If we define φ (x) j 1 θjτ (x, E) = , (3.8) τ 2 kj (x,E) φj (x) then it is easy to check that Pjτ (x, E) = | χτj (x, E) θjτ (x, E) |,
(3.9)
where we have used the bracket notation relative to the scalar product in C2m . We use the same notation for scalar products in Cm and C2m , since no confusion should arise. We now see that the eigenvectors (3.7) satisfy the normalization conditions τ ∂ Pj (x, E) ∂x ϕτj (x, E) ≡ 0, and (3.10)
ϕτj (0, E), J ϕτj (0, E) ≡ τ ∈ {−1, 1}. We note that H, kjτ , χτj , Pjτ , and ϕτj are analytic functions of x and E when these variables are in a neighborhood of R × ∆. More precisely, if ∆ = [E1 , E2 ], we
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define Dβ = {z ∈ C : dist(z, ∆) < β}, and these functions are analytic in ρα ×Dβ , for α and β small enough. Here α must be chosen small enough so that ej and φj are analytic in ρα , (see [20]), and β must be small enough so that |E − ej (x)| > 0 in ρα × Dβ . We later make use of larger values of α in order to take advantage of the generic multivaluedness of ej and φj as functions of x. From [15], we now see that any solution to (3.2) can be written as x τ 2 Φ(x, E, ) = cτj (x, E, ) e− i 0 kj (y,E) dy/ ϕτj (x, E),
(3.11)
j,τ
where the scalar coefficients cτj ∈ C satisfy the equation x 2 ∂ τ cj (x, E, ) = aτjlσ (x, E) ei 0 (τ kj (y,E)−σkl (y,E)) dy/ cσl (x, E, ), (3.12) ∂x l,σ
with aτjlσ (x, E) = −
∂
ϕτj (x, E), Pjτ (x, E) ∂x ϕσl (x, E) . ϕτj (x, E) 2
We can rewrite (3.12) as an integral equation cτj (x, E, ) = cτj (x0 , E, )
x x 2 aτjlσ (x , E) ei 0 (τ kj (y,E)−σkl (y,E)) dy/ cσl (x , E, ) dx . (3.13) + x0
l,σ
As we shall soon see, our hypotheses imply the existence of the limits lim cτj (x, E, ) = cτj (±∞, E, ),
x±∞
so that with the notation
cτ1 (x, E, ) τ c2 (x, E, ) ∈ Cm , cτ (x, E, ) = .. . cτm (x, E, )
we can define an associated S-matrix, S ∈ M2m (C), by the identity + + c (+∞, E, ) c (−∞, E, ) = . S(E, ) c− (−∞, E, ) c− (+∞, E, ) This S-matrix naturally takes the block form ++ S (E, ) S +− (E, ) S(E, ) = . S −+ (E, ) S −− (E, )
(3.14)
(3.15)
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Due to the symmetry (3.4), it also satisfies the relation (see [15]), I O −1 ∗ S (E, ) = R S (E, ) R, where R = . O −I Its elements describe transmission and reflection coefficients at fixed energy E which play key roles in our analysis. The off-diagonal elements are exponentially small and their asymptotics are determined in [15]. With this notation, the generalized eigenvectors are given by Ψ(x, E, ) = ×
j
1 φj (x) 2 kj (x, E)
−i c+ j (x, E, ) e
x 0
kj (y,E) dy/2
i + c− j (x, E, ) e
x 0
kj (y,E) dy/2
x Since 0 kj (y, E) dy x kj (±∞, E) = x 2 (E − ej (±∞)) as component of (3.16) that describes a wave traveling from the left labeled by −, and the component that describes a wave traveling to the left is labeled by +. Note also that (3.16) is simply a WKB of the generalized eigenvectors.
. (3.16)
x → ±∞, the to the right is from the right decomposition
We now state some of the general properties of the coefficients cτj (x, E, ) x 2 and of the phases ei 0 kj (y,E) dy/ that allow us to justify the scattering results described above. Lemma 3.1 Our hypotheses on h(x) imply the following, uniformly for E ∈ ∆ and all n ∈ N: ∂n ∂E n kj (x, E) − kjσ (±∞, E))
0 < C1 (n) ≤ ∂n ∂E n
(kjσ (x, E)
≤ C2 (n) < ∞, and −(2+ν)
= O( x
),
as
(3.17) x → ±∞. (3.18)
±∞ Thus, if we define ωjσ (±∞, E) = 0 (kjσ (y, E) − kjσ (±∞, E)) dy, we further have
x kjσ (y, E) dy = x kjσ (±∞, E) + ωjσ (±∞, E) + rjσ (±, x, E) 0
where, uniformly in E and for all n ∈ N, ∂n σ r (±, x, E) = O( x −(1+ν) ), ∂E n j
as
x → ±∞.
Moreover, the limits cσj (±∞, E, ) as x → ±∞ exist, and as |x| → ∞, ∂n σ c (x, E, ) = O(1), ∂E n j
for
n = 0, 1,
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uniformly in E ∈ ∆ and uniformly in ∈ (0, 0 ), where 0 is independent of E. Also, as x → ±∞ and uniformly for E ∈ ∆ and for ∈ (0, 0 ), cσj (x, E, ) − cσj (±∞, E, ) = O( x −(1+ν) ),
and
∂ σ cj (x, E, ) − cσj (±∞, E, ) = O( x −ν ). ∂E Remarks 1. This lemma is proved in Section 7. 2. The error terms in the lemma do not depend on as → 0.
3.1
Complex WKB analysis
All the information about transmissions and transitions among the asymptotic eigenstates of the electronic Hamiltonian is contained in the asymptotic values of the coefficients cσj (x, E, ±∞) defined by (3.13), and hence, in the matrix S(E, ). We extract this information by mimicking the complex WKB method of [15], while keeping track of the E dependence. The complex WKB method requires hypotheses on the behavior in the complex plane of the so-called Stokes lines for the equation (3.2) in order to provide the required asymptotics. These hypotheses are global in nature, and in general, are extremely difficult to check. However, in the physically interesting situation of “avoided crossings,” they can be easily checked. We restrict our attention to these avoided crossing situations that are described below. We consider the coefficients cj that are uniquely defined by the conditions cτj (−∞, E, ) = 1,
and cσk (−∞, E, ) = 0,
for all (k, σ) = (j, τ ).
The key to the complex WKB method lies in the multivaluedness of the eigenvalues and eigenvectors of the analytic generator H(x, E) of (3.2) in the complex x plane. For any fixed E ∈ ∆, H(·, E) is analytic in ρα , and the solutions (3.11) to (3.2) are analytic in x as well. However, the eigenvalues and eigenvectors may have branch points in ρα whose properties are inherited from those of the eigenvalues and eigenvectors of h(·). Analytic perturbation theory as described in [20] states that the eigenvalues and eigenprojections of h(x) for real x are analytic on the real axis and admit analytic multivalued extensions to ρα . The analytic continuations of the eigenvalues have branch points that are located on a set of crossing points Ω = {z0 ∈ ρα \ R : ej (z0 ) = ek (z0 ) for some j, k and some analytic continuations}. Recall that for δ = 0, the eigenvalues are analytic at any crossing points on the real axis. This follows from the self-adjointness of h(·) on the real axis. Note also that Ω = Ω by the Schwarz reflection principle.
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Generically, at a complex crossing point z0 ∈ Ω, we have the following local behavior, where c ∈ C is some constant ej (z) − ek (z) c (z − z0 )1/2 (1 + O(z − z0 )).
(3.19)
The eigenprojectors of h(x) also admit multivalued extensions in ρα \ Ω, but they diverge at generic eigenvalue crossing points. We only have to deal with generic crossing points. To see what happens to a multivalued function f in ρα \ Ω when we turn around a crossing point, we adopt the following convention: We denote by f (z) the analytic continuation of f defined in a neighborhood of the origin along some path from 0 to z. Then we perform the analytic continuation of f (z) along a negatively oriented loop that surrounds only one point z0 ∈ Ω. We denote by f˜(z) the function we get by coming back to the original point z. We define ζ0 to be a negatively oriented loop, based at the origin, that encircles only z0 when Imz0 > 0. When Imz0 < 0, we choose ζ0 to be positively oriented. We now fix z0 with Imz0 > 0. If we analytically continue the set of eigenvalues {ej (z)}m j=1 , along a negatively oriented loop around z0 ∈ Ω, we get the set { ej (z)}m j=1 with ej (z) = eπ0 (j) (z),
for j = 1, . . . , m,
where π0 : {1, 2, . . . , m} → {1, 2, . . . , m}
(3.20)
is a permutation that depends on z0 . As a consequence, the eigenvectors (2.1) possess multivalued analytic extensions in ρα \Ω. The analytic continuation φj (z) of φj (z) along a negatively oriented loop around z0 ∈ Ω, must be proportional to φπ0 (j) (z). Thus, for j = 1, 2, . . . , m, there exists θj (ζ0 ) ∈ C, such that φj (z) = e−iθj (ζ0 ) φπ0 (j) (z).
(3.21)
We now turn from h(x) to H(x, E). From Hypothesis H7, (3.5), and (3.7), we see that the set of crossing points for the eigenvalues ±kj (x, E) of H(x, E) is independent of E and coincides with Ω. Moreover, for j = 1, . . . , m, we have kjτ (z, E) = kπτ 0 (j) (z, E),
ϕ τj (z, E) = e−iθj (ζ0 ) ϕτπ0 (j) (z, E),
where the prefactor e−iθj (ζ0 ) is independent of E. The above implies a key identity for the analytic extensions of the coefficients cτj (z, E, ), z ∈ ρα \Ω. Since the solutions to (3.2) are analytic for all z ∈ ρα , the coefficients cτj must also be multi-valued. In our setting, Lemma 3.1 of [15] implies the following lemma.
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Lemma 3.2 For any j = 1, . . . , m and τ = +, −, we have cτj (z, E, ) e
−i
ζ0
kjτ (u,E) du/2
e− i θj (ζ0 ) = cτπ0 (j) (z, E, )
(3.22)
where ζ0 , θj (ζ0 ) and π0 (j) are defined as above and are independent of E ∈ ∆. Remark. Since Ω is finite, it is straightforward to generalize the study of the analytic continuations around one crossing point to analytic continuations around several crossing points. The loop ζ0 can be rewritten as a concatenation of finitely many individual loops, each encircling only one point of Ω. The permutation π0 is given by the composition of associated permutations. The factors e−iθj (ζ0 ) in (3.21) are given by the product of the factors associated withthe individual loops. The same is true for the factors exp − i ζ0 kjτ (z, E) dz/2 in Lemma 3.2. We now describe how we use the above properties. The details may be found in [15]. The idea is to integrate the integral equation corresponding to (3.13) along paths that go above (or below) one or several crossing points, and then to compare the result with the integration performed along the real axis. As z → −∞ in ρα these paths become parallel to the real axis so that the coefficients take the same asymptotic value cτm (−∞, E, ) along the real axis and the integration paths. Since the solutions to (3.2) are analytic, the results of these integrations must agree as Rez → ∞. Therefore, (3.22) taken at z = ∞ yields the asymptotics of cj (z, E, ) in the complex plane. We argue cτπ0 (j) (∞, E, ), provided we can control below that this can be done in the so-called dissipative domains (See [6], [5]), as proven in [15]. We do not go into the details of these notions because another result of [15] will enable us to get sufficient control on cj (z, E, ) in the avoided crossing situation, to which we restrict our attention. We define
x τ τσ kj (y, E) − kjσ (y, E) dy. ∆jl (x, E) = 0
By explicit computation, using formula (7.3) in (3.13), we check that (3.13) can be extended to ρα \ Ω. We next integrate by parts in (3.13), to see that (3.13) with x0 = −∞ can be rewritten as σ 2 aτml (z, E) τσ ei∆ml (z,E)/ cσl (z, E, ) cτm (z, E, ) = δjm δτ − − i2 σ τ (z, E) − k (z, E) k m l l,σ
z τ σ 2 aml (z , E) ∂ τσ 2 + i ei∆ml (z ,E)/ cσl (z , E, ) dz σ τ ∂z km (z , E) − kl (z , E) −∞ l,σ σ z aτml (z , E) aσθ 2 τθ lp (z , E) i∆ e mp (z ,E)/ + i2 cθp (z , E, ) dz , (3.23) σ τ (z , E) − k (z , E) k −∞ m l l,p,σ,θ as long as the chosen path of integration does not meet Ω. Here, denotes the analytic continuation along the chosen path of integration of the corresponding
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function defined originally on the real axis. This distinguishes cτm (∞, E, ) from τ cm (∞, E, ) computed along the real axis as x → ∞. These quantities may differ since the integration path may pass above (or below) points of Ω. If the exponentials in (3.23) are all uniformly bounded, as it is the case when the integration path coincides with the real axis, it is straightforward to get bounds of the type (3.24) c˜τm (z, E, ) = δjm δτ − + OE (2 ). In our context, all quantities depend on E ∈ ∆. However, by mimicking the proof of Proposition 4.1 of [15], it is not difficult to check that the estimate (3.24) is uniform for E ∈ ∆. cτm (z, E, ) is For later purposes, we note that by differentiating (3.23), ∂∂E uniformly bounded for 0 < < 0 and E ∈ ∆ for some fixed 0 . See the proof of Lemma 3.1 for this property on the real axis. Again, as is well known, the existence of paths from −∞ to +∞ along which the exponentials do not blow up and which pass above (or below) points in Ω is difficult to check in general. It is linked to the global behavior of the Stokes lines of the problem. See e.g., [6], [5]. This property goes under the name “existence of dissipative domains” in [15]. We avoid these complications by restricting attention to avoided crossing situations where the existence has been proven [15]. When dissipative domains exist, (3.22) and (3.24) imply cτπ0 (j) (∞, E, ) = e
−i
ζ0
kjτ (u,E) du/2
e− i θj (ζ0 ) (1 + OE (2 )),
(3.25)
where the OE (2 ) estimate is uniform for E ∈ ∆. This is the main result of Proposition 4.1 in [15] in our context, under the assumption that dissipative domains exist.
3.2
Avoided crossings
We now explore the avoided crossing situation, alluded to above, that allows us to avoid considerations of the dissipative domains. We now assume that h(x) has the form h(x, δ) and satisfies Hypotheses H4 and H5. We first check that the allowed pattern of avoided crossings for σ(h(x, δ)) can be transfered to the eigenvalues of H(x, E, δ), obtained from h(x, δ) by (3.3). From the explicit formulae (3.5), we see immediately that xc ∈ R is a real crossing point for the eigenvalues ej (x, 0) and el (x, 0) of h(x, 0) if and only if it is a real crossing point for the analytic eigenvalues kjτ (x, E, 0) and klτ (x, E, 0) of H(x, E, 0), for τ = +, −. Moreover, ∂ ∂ τ τ ∂x (el (x, 0) − ej (x, 0)) (kj (x, E) − kl (x, E)) = τ , ∂x 2 (E − ej (x, 0)) x=xc x=xc
so that the real crossings for H(x, E, 0) are also generic, in the sense of (2.5).
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Remark. Our assumption H7 on the parameter E forbids real crossings between eigenvalues kjτ (x, E, 0) and klσ (x, E, 0), with σ = τ . Regarding the ordering of the eigenvalues of H(x, E, δ), if those of h(x, δ) are ordered as in (2.4), we have −k1 (x, E, δ) < · · · < −km (x, E, δ) < 0 < km (x, E, δ) < · · · < k1 (x, E, δ). (3.26) This means that the pattern of the crossings for the group of eigenvalues {−kj (x, E, 0)}j=1,...,m is the same as that for the eigenvalues {ej (x, 0)}j=1,...,m . The pattern of the crossings for the group {kj (x, E, 0)}j=1,...,m is the reflection with respect to the horizontal axis of the one for {ej (x, 0)}j=1,...,m . Therefore, assumptions H5, i), ii), iii) are also satisfied for the eigenvalues of H(x, E, 0), for a relabeling from 1 to 2m of (3.26) with δ = 0, and x close to −∞. To any given pattern of real crossings for the eigenvalues {ej (x, 0)}j=1,...,m of h(x, 0), we associate a permutation π of {1, 2, . . . , m} as follows. Assume the eigenvalues are labeled in ascending order at x = −∞, as in property i) of H5. If ej (∞, 0) is the k th eigenvalue in ascending order at x = ∞, the permutation π is defined by π(j) = k. (3.27) We call π the permutation associated with σ(h(x, 0)). For small δ > 0, the real crossings turn into avoided crossings on the real axis and conjugate complex crossing points appear close to the real axis. Then π corresponds to the permutation π0 (3.20) for a loop ζ0 that surrounds all complex crossing points in the upper half plane that are associated with the avoided crossings. These properties of corresponding patterns of real crossings of the spectra of h(x, δ) and H(x, E, 0) immediately yield the following convenient relation between the permutation π associated with σ(h(x, 0)) and the permutation Π associated with σ(H(x, E, 0)). If we denote Π by the obvious notation Π(j, τ ) = (k, σ), then we have Π(j, τ ) = (π(j), τ ),
for all
(j, τ ) ∈ {1, . . . , m} × {−, +}.
We can now restate the main result of [15] that describes the asymptotics of the coefficients defined in (3.13), adapted to our scattering framework for incoming states entering from the left. (See (3.16).) Intuitively, this result says that for small δ > 0, dissipative domains exist, provided the pattern of real crossings satisfies H5. Therefore, estimates of the type (3.25) are true for certain indices j and n, determined by the permutation (3.27). It is not difficult to see that the permutation π describes the successive exchanges of eigenvalues one gets by following a path in the complex plane that goes above or below all complex crossing points of the eigenvalues ej (x, δ) that are associated with the avoided crossings.
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Theorem 3.1 Let h(x, δ) satisfy H4 and H5. If δ > 0 is small enough, the π(j), j elements of the matrix S −− (E, ) in (3.15), with π(j) defined in (3.27), have small asymptotics for all j = 1, . . . , m, given by −− Sπ(j),j (E, ) =
π(j)∓1
e−iθl (ζl ,δ) e
i
ζl
kl (z,E,δ) dz/2
1 + OE,δ (2 ) ,
π(j)
l=j
>j <j
where, for π(j) > j (respectively π(j) < j), ζl , l = j, . . . , π(j) − 1 (resp. l = j, . . . , π(j) + 1), denotes a negatively (resp. positively) oriented loop based at the origin which encircles the complex crossing point zr only (resp. zr ) corresponding to the avoided crossing between el (x, δ) and el+1 (x, δ) (resp. el−1 (x, δ)) at xr . The ζl kl (z, E, δ) dz denotes the integral along ζl of the analytic continuation of kl (0, E, δ), and θl (ζl ) is the corresponding factor defined by (3.21). Remarks i) Revisiting the proof of this theorem in [15], we see that we can choose δ > 0 small enough so that dissipative domains can be constructed uniformly for E ∈ ∆. This stems from the formula kj (x, E, 0) − kl (x, E, 0) =
2(el (x, 0) − ej (x, 0)) , kj (x, E, 0) + kl (x, E, 0)
whose denominator can be controlled, close to the real axis, uniformly for E ∈ ∆. ii) When there is only one avoided crossing between level j and j + 1 stemming from a real crossing at x = x0 , we have j + 1 = π(j). The theorem says −− S(j+1),j (E, ) = e−iθj (ζj ,δ) e
i
ζj
kj (z,E,δ) dz/2
1 + OE,δ (2 ) ,
where the negatively oriented loop ζj encircles the corresponding complex crossing point z0 , with Imz0 > 0. Similarly, interchanging the roles of j and j + 1, it yields with ζ¯j the conjugate of the loop ζj , ¯
−− Sj(j+1) (E, ) = e−iθj (ζj ,δ) e
i
¯ ζ j
kj (z,E,δ) dz/2
1 + OE,δ (2 ) ,
iii) Since the eigenvalues are continuous at the complex crossings, we have
lim Im kj (z, E, δ) dz = 0, for all j = 1, . . . , p. δ→0
ζj
It is shown in [14] that lim Im θj (ζj , δ) = 0
δ→0
for all
j = 1, . . . , p.
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iv) The OE,δ () errors in Theorem 3.1 depend on δ, but it should be possible to get estimates which are valid as both and δ tend to zero, in the spirit of [14], [21], and [24]. v) This result shows that at least one off-diagonal element per column of the S-matrix can be computed asymptotically. However, it is often possible to get more elements by making use of symmetries of the S-matrix. See [15] and [16]. In our avoided crossings context, transitions of the coefficients between states that correspond to electronic levels that do not display avoided crossings, i.e., that are separated by a gap of order 1 as δ → 0, are expected to be exponentially smaller than the transitions we control by means of Theorem 3.1, as δ shrinks to zero. Since the coefficients in the exponential decay rates given by the theorem vanish in the limit δ → 0, it is enough to show that the decay rates of the exponentially small transitions between well separated levels are independent of δ. That is the meaning of the following proposition, which draws heavily upon [18] and [15] and is proven in Section 7. Proposition 3.1 Let F (x, δ) be a n × n matrix that satisfies H4, except for the condition that F (·, δ) be self-adjoint. Suppose its eigenvalues {fj (x, δ)}j=1,...,m that satisfy H5. Further assume that the eigenvalues can be separated into two groups σ1 (x, δ) and σ2 (x, δ) that display no avoided crossing, i.e., such that inf
δ≥0 x∈ρα ∪{±∞}
dist(σ1 (x, δ), σ2 (x, δ)) ≥ g > 0.
Let P (x, δ) and Q(x, δ) = I − P (x, δ) be the projectors onto the spectral subspaces corresponding to σ1 (x, δ) and σ2 (x, δ) respectively, and let U (x, x0 , δ) be the evolution operator corresponding to the equation i 2
d U (x, x0 , δ) = F (x, δ) U (x, x0 , δ), dx
with
U (x0 , x0 , δ) = I. (3.28)
Then, for any δ > 0, there exists 0 (δ), C(δ) > 0 depending on δ, and Γ > 0 independent of δ, such that for all ≤ 0 (δ), lim
x→∞ x0 →−∞
2
P (x, δ) U (x, x0 , δ) Q(x0 , δ) ≤ C(δ) e−Γ/ .
Remark. This proposition implies that reflections, i.e., the transitions from wave packets traveling to the right to wave packets traveling to the left, on any electronic level, are exponentially smaller than transitions associated with the avoided crossings in which the propagation direction is not changed. This is a consequence of Hypothesis H7 which implies that complex crossings between kj+ and kl− , are far from the real axis for any j, l ∈ {1, . . . , m}. Let us investigate more closely the analytic structure of kj (z, E, δ) in our avoided crossing regime characterized by H4 and H5, in order to deduce the properties of the exponential decay rates Im ζj kj (z, E, δ) dz. We do so for the kj ’s
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that correspond to electronic eigenvalues ej (z, δ) and en (z, δ) that experience only one avoided crossing, i.e., we take n = j ± 1. We can thus drop the index j in ζj in the notation. We follow [14] where a similar analysis is performed, sometimes referring to results proven there. The general case is dealt with by adding the corresponding contributions stemming from each individual avoided crossing. We can assume that the avoided crossing takes place at x = 0, i.e., ej (0, 0) = en (0, 0) ≡ ec , where ec is the electronic eigenvalue at the crossing. We also define the momentum kc (E) at the crossing point by 2 (E − ec ) kc (E) = and the quantity Γ0 (δ) by
Γ0 (δ) = Im ej (z, δ) dz .
(3.29)
ζ
This quantity is the exponential decay rate given by the Landau-Zener Formula for a (time-dependent) adiabatic problem with hamiltonian h(t, δ). See [14]. In Section 7 we prove Lemma 3.3 With the above notation, we have the following as δ → 0, uniformly for E ∈ ∆,
Γ0 (δ) + O(δ 3 ), 2(E − ej (z, δ)) dz = Im k c (E) ζ
∂ Γ0 (δ) Im + O(δ 3 ), 2(E − ej (z, δ)) dz = − 3 and ∂E kc (E) ζ
Γ0 (δ) ∂2 + O(δ 3 ), Im 2(E − ej (z, δ)) dz = 3 5 2 ∂E kc (E) ζ where 0 < Γ0 (δ) = O(δ 2 ). Remarks i) This implies that Im ζ 2(E − ej (z, δ)) dz is a positive, decreasing, convex function of E on ∆. This remains true when the transition is mediated by several avoided crossings. ii) The first relation can be interpreted as saying that in our Born-Oppenheimer context, the (time-dependent adiabatic) Landau-Zener decay rate at fixed energy E has to be modified in order to take into account the classical velocity kc (E) at the crossing. iii) More precise estimates will be derived below, further assuming H6.
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4 Exact solutions to the time-dependent Schr¨ odinger equation We now construct solutions to (1.1) by taking time-dependent superpositions of the generalized eigenvectors Ψ(x, E, ), where E ∈ ∆. These superpositions depend on an energy density Q(E, ) that can be complex and may or may not depend on . We always assume that the following condition holds: C0: The density Q(E, ) is C 1 as a function of E ∈ ∆, for fixed > 0. In this Section, the parameter δ > 0 is kept fixed and plays no role. We therefore drop it from the notation and work under Hypotheses H1, H2, and H3. We define ψ(x, t, )
2
Q(E, ) Ψ(x, E, ) e−itE/ dE ∆
x σ 2 Q(E, ) = φj (x) cσj (x, E, )e−i 0 kj (y,E)dy/ 2 kj (x, E) ∆ j=1,...,m, σ=± =
2
≡
× e−itE/ dE
ψjσ (x, t, ).
(4.1)
j=1,...,m σ=±
Here ψjσ asymptotically describes the piece of the solution that lives on the electronic state φj and propagates in the direction characterized by σ. Since the integrand is smooth and ∆ is compact, ψ(x, t, ) is an exact solution to the timedependent Schr¨odinger equation (1.1). Note that this solution is not necessarily normalized. The following lemma, whose proof can be found in Section 7, gives a bound that we use to understand the large t behavior of ψjσ (x, t, ). It is a simple corollary that the state (4.1) belongs to L2 (R). Lemma 4.1 Assume H1, H2, H3 and C0. Let K+ =
sup j=1,...,m E∈∆, σ=±
kj (σ∞, E) < ∞
and K− =
inf
j=1,...,m E∈∆, σ=±
kj (σ∞, E) > 0.
Fix α ∈ (0, 1). Then, for either t = 0, or for any x = 0 and t = 0, such that |x/t| > K+ /(1 − α)
or
|x/t| < K− /(1 + α),
we have σ ψ (x, t, ) ≤ C /|x|, j
with C independent of
where the estimate is in the norm on Cm .
t,
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We now introduce freely propagating states ψ(x, t, , ±) ∈ L2 (R, Cm ) that describe the asymptotics of the solutions ψ(x, t, ) as t → ±∞. We use these asymptotic states when we study the scattering matrix for (1.1). We let
2 Q(E, ) ψ(x, t, , ±) = φj (x) e−itE/ cσj (±∞, E, ) 2k (±∞, E) ∆ j j=1,...,m σ=± σ
=
σ
2
× e−i(xkj (±∞,E)+ωj (±∞,E))/ dE ψjσ (x, t, , ±) (4.2)
j=1,...,m σ=±
These states are linear combinations of products of free scalar wave packets in constant scalar potentials times eigenvectors of the electronic Hamiltonian. Their propagation is thus governed by the various channel Hamiltonians. Proposition 4.1 Assume H1, H2, H3 and C0. In L2 (R) norm as t → ±∞, we have (4.3) ψ(x, t, ) − ψ(x, t, , ±) = O (1/|t|). Remarks i) The estimate (4.3) depends on . ii) By a change of variables, we immediately obtain the following corollary. Corollary 4.1 The density of the component of the asymptotic momentum space wave function on the j th electronic level as t → ±∞ is σ 2 k Q(E(k), ) cσj (±∞, E(k), ) e−iωj (±∞,E(k))/ . σ 2 Here E(k) = k 2 /2 + ej (±∞) and σ = −/+ for waves traveling in the positive/negative direction, respectively. iii) Consider a solution ψ(x, t, ) traveling in the positive direction and associated with the electronic eigenstate φj in the remote past. It is characterized by cσk (−∞, E, ) = δk,j δσ,− , and as t → −∞, it is asymptotic to
− 2 2 Q(E,) ψ(x,t,,−) = φj (x) e−itE/ ei(xkj (−∞,E)−ωj (−∞,E))/ dE. 2kj (−∞,E) ∆ (4.4) As t → +∞, the component of this state that has made the transition from state j to state n is asymptotic to the vector ψn− (x, t, , +). It is given in terms of the matrix S by
2 − 2 Q(E,) −− e−itE/ Snj φn (x) (E,)e+i(xkn (+∞,E)−ωn (+∞,E))/ dE. 2kn (+∞,E) ∆ (4.5) iv) Proposition 4.1 is proven in the Section 7.
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5 Non-adiabatic transition asymptotics 5.1
The transition integral
From now on, we assume we are in the avoided crossing situation, but we still do not make explicit the dependence in the variable δ > 0 in the notation. Section 3 gave us the semiclassical asymptotics of the elements of the Smatrix S(E, ). We now compute the small asymptotics of the integrals that describe the asymptotic states ψjσ (x, t, , ±) given by (4.2) as |t| → ∞, for the different channels. We choose our energy density Q(E, ) to be more and more sharply peaked near a specific value E0 ∈ ∆ \ ∂∆ as → 0. As a result, we obtain semiclassical Born-Oppenheimer states that are well localized in phase space. This choice is physically reasonable, and it allows us to relate the quantum scattering process to classical quantities. More precisely we consider, 2
2
Q(E, ) = e− G(E)/ e− i J(E)/ P (E, ),
(5.1)
where C1: The real-valued function G ≥ 0 is in C 3 (∆), and has a unique non-degenerate absolute minimum value of 0 at E0 in the interior of ∆. This implies that G(E) = g (E − E0 )2 /2 + O(E − E0 )3 ,
where
g > 0.
C2: The real-valued function J is in C 3 (∆). C3: The complex-valued function P (E, ) is in C 1 (∆) and satisfies n ∂ ≤ Cn , sup P (E, ) for n = 0, 1. n ∂E E∈∆
(5.2)
≥0
Remark. Typical interesting choices of Q have G = g (E − E0 )2 , J = 0, and P an -dependent multiple of a smooth function with at most polynomial growth in (E − E0 )/. In our avoided crossing situation, we have already proved the following: A wave packet incoming from the left in the remote past produces reflected waves (i.e., components that travel to the left in the remote future) that are exponentially smaller than the components that travel to the right in the remote future. We have also proved that the non-trivial transitions to electronic states that are not involved in the avoided crossing are exponentially smaller than those to electronic states that are involved in the avoided crossing. Thus, the leading non-adiabatic transitions are described by the asymptotics of those coefficients cσl (±∞, E, ) that satisfy cσk (−∞, E, ) =
δj,k δσ,−
c− n (+∞, E, ) =
e−iθj (ζ) ei
(5.3) ζ
kj (z,E)dz/2
(1 + OE (2 )),
(5.4)
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where n = π(j) = j ± 1. We recall that the error term OE (2 ) depends analytically on the energy E in a neighborhood of the compact set ∆. We have already noted in the comments after (3.24) that the term OE (2 ) satisfies (5.2). The form chosen for the energy densities should make it clear that Gaussian wave packets will play a particular role in the asymptotic analysis of (4.5). Therefore we use the specific notation introduced in (1.2) for them. Recall that a normalized free Gaussian state propagating in the constant potential e(+∞) is characterized by the classical quantities A+ (t) B+ (t)
= A+ + i t B+ , = B+ ,
a+ (t) η+ (t)
= a+ + η+ t, = η+ ,
t 2 η+ (s)/2 − e(+∞) ds, =
S+ (t)
and
0
with Re(A+ B+ ) = 1 (see [9]). The associated nuclear wave packet has the form 2
eiS+ (t)/ ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x) 2 2 eit(η /2−e(∞))/ (x − (a+ + η+ t))2 B+ = exp − 22 (A+ + itB+ ) π 1/4 (A+ + itB+ ) η+ (x − (a+ + η+ t)) . (5.5) × exp i 2 We now have everything to state our main result: Theorem 5.1 Let ψ(x, t, ) be a solution of the Schr¨ odinger equation (1.1) with electronic Hamiltonian h(x, δ) that satisfies Hypotheses H4, H5, H7. Assume that the solution is characterized asymptotically in the past by lim
t→−∞
with ψj− (x, t, , −) = φj (x)
∆
ψ(x, t, ) − ψj− (x, t, , −) = 0, − 2 2 Q(E, ) e−itE/ ei(xkj (−∞,E)−ωj (−∞,E))/ dE, 2kj (−∞, E)
where the energy density is supported on the interval ∆, and 2
2
Q(E, ) = e− G(E)/ e− i J(E)/ P (E, ) satisfies C1, C2, and C3. Let n = π(j) be given by (3.27), and let
α(E) = G(E) + Im( kj (z, E) dz), ζ
κ(E) = J(E) − Re( kj (z, E) dz) + ωn− (∞, E). ζ
(5.6) (5.7)
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Assume E ∗ is the unique absolute minimum of α(·) in Int ∆. Then, there exist δ0 > 0, p > 0 arbitrarily close to 3, and a function 0 : (0, δ0 ) → R+ , such that for all δ < δ0 and < 0 (δ), the following asymptotics hold as t → ∞: e− i θj (ζ) 3/2 π 3/4 iS+ (t)/2 ψn− (x, t, , +) = φn (x) 2 1/4 e d ∗ α(E(k))| 2 k dk
√ ∗ 2 × ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x)P (E ∗ , ) k ∗ e−α(E )/ × e−i(κ(E
∗
)−k∗ 2 κ (E ∗ ))/2
+ O(e−α(E
∗
)/2 p
) + O (1/t) ,
where ϕ0 is parametrized by η+ = k ∗ = A+ =
2(E ∗ − en (∞)),
a+ = k ∗ κ (E ∗ ),
d2 d2 ∗ + i α(E(k))| κ(E(k))|k∗ k dk 2 dk 2
B+ =
1 d2 dk2
,
α(E(k))|k∗
d2 α(E(k))|k∗ dk 2
and
S+ (t) = t(k ∗ 2 /2 − en (∞)). All error terms are estimated in the L2 (R) norm, and the estimate O(e−α(E is uniform in t, whereas O (1/t) may depend on .
(5.8) ∗
)/2 p
)
Remarks i) All quantities computed from the electronic Hamiltonian h(x, δ) depend on δ, even though that dependences is not specified in the notation. ii) The function α has a unique absolute minimum if |∆| and δ are small enough. See Proposition 5.1. However, in the case of several absolute minima, one simply adds the contributions associated with each of them. iii) The transitions to states that travel to the left in the future are excluded from our analysis because of the lack of uniformity in E in the semiclassical asymptotics of the relevant elements of the matrix S(E, ). At the price of some more technicalities, it should also be possible to accommodate this situation by our methods. iv) When several avoided-crossings are taken into account and meet the requirements of Theorem 3.1, c− n (∞, E, ) with n = π(j) is given by a product of exponentials of the same form as those in (5.4). The analysis of this situation is essentially identical to the single avoided crossing situation, mutatis mutandis. v) Further properties of ψn (x, t, , +) are given below. In particular, the characteristics of the average momentum k ∗ and its behavior as a function of δ are detailed in Section 5.2. The energy densities corresponding to specific incoming states are studied in Section 6.
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vi) The asymptotics of the incoming wave with the electrons in the state φj in the remote past are described by the same integral, with the replacements γj → 0, ωn− (∞, ·) → ωj− (−∞, ·), 2(E − en (∞)) → 2(E − ej (−∞)), θj → 0.
(5.9)
Proof of Theorem 5.1. Apart from the E-independent factor given by φn (x) − i θj (ζ) √ e , 2 the asymptotics of (4.5) are determined by the integral
P˜ (E, ) (2(E − en (∞)))1/4
T (, x, t) = ∆
×
e
−G(E)/2
2
2
e−i(tE+J(E))/ eiγj (E)/ ei(x
√
− 2(E−en (∞))−ωn (∞,E))/2
dE,
where P˜ (E, ) = P (E, ) (1 + OE (2 )) satisfies (5.2),
γj (E) = kj (z, E) dz, and ζ
ωn− (∞, E)
−
=
∞
0
2(E − en (y)) − 2(E − en (∞)) dy.
The (1 + OE (2 )) factor in P˜ (E, ) comes from Theorem 3.1. Recall that γj and ωj+ (∞, ·) are analytic in a complex neighborhood of ∆, and that Imγj (E) is a positive, decreasing, convex function of E, for δ sufficiently small. In terms of the C 3 functions (5.6) and (5.7) we can write T (, x, t) as
T (, x, t) = ∆
P˜ (E, ) (2(E − e(∞))
2
1/4
2
e−α(E)/ e−i(tE+κ(E))/ ei(x
√
2(E−e(∞))/2
dE,
where we have dropped the index in the asymptotic eigenvalue e(∞) = en (∞). In Section 7 we analyze the small asymptotics of T essentially by Laplace’s method. The result is k2 + e(∞), and 2(E − e(∞)), or equivalently, E(k) = 2 ∗ assume that α(·) has a unique absolute minimum E . For sufficiently small δ, this Lemma 5.1 Let k(E) =
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minimum is non-degenerate and satisfies E ∗ ∈ Int ∆. With k ∗ = k(E ∗ ), there exists p > 0 arbitrarily close to 3, such that as → 0, 2
d ∗ 3 k ∗ dk κ (E ∗ )) 2 α(E(k))|k∗ + i(x + k
T (, x, t) =
2
2
d d 3/2 ( dk 2 α(E(k))|k∗ + i(t + dk2 κ(E(k))|k∗ )) √ P (E ∗ , ) −α(E ∗ )/2 (tE ∗ + κ(E ∗ ) − xk ∗ ) × 2π √ e exp −i 2 k∗ ! (x − k ∗ (t + κ (E ∗ )))2 × exp − d2 d2 22 dk 2 α(E(k))|k∗ + i(t + dk2 κ(E(k))|k∗
+
O(e−α(E
∗
)/2 p
),
(5.10)
where the error estimate is in the L2 (R) norm, uniformly in t. Remarks i) If there are several absolute minima, one simply adds their contributions to get the asymptotics of T . ii) If T is associated with the incoming wave as t → −∞, the formula holds with E0 in place of E ∗ , k0 = 2(E0 − e(−∞)) in place of k ∗ , and the changes (5.9). iii) If P satisfies C3 and P (E ∗ , ) = O(d ) for some d ≥ 32 , then the above analysis yields no information. To relate the integral T to standard Born-Oppenheimer states involving normalized free Gaussian states, we must identify (5.10) with (5.5), making use of (5.8), and taking care of the x and t dependence in the non-Gaussian part of (5.10). That is the content of the next lemma which completes the proof of Theorem 5.1. With the identifications (5.8), we have Lemma 5.2 For small and 0 < p < 3, we have −1/4 d2 P (E ∗ , ) −α(E ∗ )/2 ∗ √ T (, x, t) = 2 π α(E(k))| e k dk 2 k∗ ∗ d2 ∗ 3 ∗ ∗ + i(x + k k α(E(k))| κ (E )) ∗ ∗2 ∗ 2 2 k dk × e−i(κ(E )−k κ (E ))/ d2 d2 ∗ ∗ 2 dk α(E(k))|k + i(t + dk2 κ(E(k))|k ) 3/2
×
1/2
2
3/4
eiS+ (t)/ ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x)
+
O(e−α(E
where the error is estimated in the L2 (R) norm, uniformly in t.
∗
)/2 p
),
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Furthermore, in the L2 norm, for small and large |t|, we have √ ∗ 2 ∗ ∗2 ∗ 2 k ∗ e−α(E )/ e−i(κ(E )−k κ (E ))/ 2 −1/4 d ∗ ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x) α(E(k))| k dk 2
T (, x, t) = 3/2 21/2 π 3/4 P (E ∗ , ) ×
eiS+ (t)/
2
+
O(e−α(E
∗
)/2 p
)
+
O(3/2 /|t|),
where the first error term is uniform in t. Remarks i) We note that the quantities α(·), k ∗ , and B+ depend only on the index j, while κ(·), and hence, A+ depend on both j and n. ii) More detailed computations are carried out in the next section, which is devoted to specific incoming states.
5.2
Energy and momentum shifts
When there is a single avoided crossing, we can be more precise about the energy and momentum shifts revealed by our general analysis. For the rest of this section, we assume h(x, δ) satisfies Hypothesis H6. Under this hypothesis, it is known [14] that the decay rate in the LandauZener formula (3.29) has the form c2 π b2 − 3 + O(δ 3 ) ≡ δ 2 D + O(δ 3 ), Γ0 (δ) = δ 2 4 a a and that Imθj (ζj , δ) = 0(δ). We use these formulas to get more information on E ∗ , the typical energy of the outgoing wave packet, that is determined by the relation α (E ∗ ) = G (E ∗ ) + Imγj (E ∗ ) = 0,
(5.11)
where the primes denote derivatives with respect to E. In the next proposition, we consider two cases: – In the first case, we choose the exponent G(E) in the energy density to be independent of δ. This yields less interesting momentum and energy shifts since they vanish to leading order in δ as δ → 0, in keeping with [10]. – In the second case, we choose G(E) to depend on δ in such a way that the incoming wave packet contains a sufficiently wide spectrum of energies as δ → 0. This implies non-trivial behavior of the relevant quantities to leading order in δ. For obvious reasons, we restore δ in the notation of this discussion.
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Proposition 5.1 Let G(E) = g(E − E0 )2 /2 + O(E − E0 )3 , Imγj (E, δ) =
Γ0 (δ) D δ2 + O(δ 3 ) = + O(δ 3 ), kc (E) kc (E)
and
α(E, δ) = G(E) + Imγj (E, δ), as above. i) Assume G is independent of δ. Then, for E ∗ defined by (5.11), we have E ∗ (δ) = E0 +
Γ0 (δ) + O(δ 3 ), g kc3 (E0 )
as δ → 0. In this case, α(E ∗ (δ)) = α (E ∗ (δ)) =
Γ0 (δ) + O(δ 3 ) < α(E0 ), kc (E0 )
and
g + O(δ 2 ).
If G(E) = g(E − E0 )2 /2 + g1 (E − E0 )3 /6 + O(E − E0 )4 , then α (E ∗ (δ)) = g + g1
Γ0 (δ) Γ0 (δ) +3 5 + O(δ 3 ). g kc3 (E0 ) kc (E0 )
ii) Assume G(E, δ) = L(δ(E − E0 )), for some function L, such that G(E, δ) = g0 δ 2 (E − E0 )2 /2 + O(δ 3 ), for some g0 > 0, uniformly for E ∈ ∆. Then E ∗ (δ) = E1 + O(δ), where 0 < E1 = E1 (D/g0 ) is the unique solution to the equation (E1 − E0 ) = D/(g0 kc3 (E1 )), and is independent of δ. In this case, D ∗ 2 2 + g0 (E1 − E0 ) /2 + O(δ 3 ) α(E (δ)) = δ kc (E1 ) 1/3 = δ 2 D2/3 g0 (E1 − E0 )1/3 + g0 (E1 − E0 )2 /2 + O(δ 3 )
α (E ∗ (δ))
<
α(E0 ),
=
δ 2 g0 + 3
and Γ0 (δ) + O(δ 3 ). kc5 (E0 )
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Proof. Both statements follow from application of the Implicit Function Theorem and the observation that α is a strictly convex function of E on ∆. Remarks a) The first result shows no effect to leading order in δ in the exponential decay rate of transition probability. The value of E ∗ (δ) and the width of the outgoing wave packet can be computed. Their variations with respect to the corresponding quantities in the incoming wave packet are of order δ 2 , and hence, are rather small. b) In case ii) of the proposition, the equation that determines E ∗ can be rewritten as the quintic equation kc5 (E) − kc3 (E) kc2 (E0 ) − 2D/g0 = 0. c) In the case ii), the variation of exponential decay rate in the transition probability is given by 1 1 ∗ 2 2 − α(E (δ)) − α(E0 ) = δ D + g0 (E1 − E0 ) /2 + O(δ 3 ) kc (E1 ) kc (E0 ) δ2 D 2g0 kc (E1 )5 (kc (E0 ) − kc (E1 )) + Dkc (E0 ) + O(δ 3 ). = 6 2g0 kc (E1 ) kc (E0 ) d) The results above hold provided one knows E ∗ is the unique absolute minimum of G in the set ∆, which is generically true. Again, if there are several minima, one simply adds the corresponding contributions.
6 Energy densities and transitions when the incoming state has the form ϕm In this section we study the special case in which the incoming state is asymptotically in the past on electronic level j with the nuclear wave function given by one of the functions ϕm . For simplicity, we restrict attention to wave packets that are incoming from the left. In the simplest situation, the incoming wave packet is asymptotic to 2
ei(η− /2−ej (−∞))t/ ϕ0 (A− + itB− , B− , 2 , a− + η− t, η− , x) φj (x),
(6.1)
2 /2 + ej (−∞) is in as t → −∞. Here we choose η− > 0 and the set ∆, so that η− the interior of ∆, and that the minimum of ∆ lies strictly above the spectrum of h(x) for all x. We choose a smooth cut-off function F (E) whose support is a subset of the interior of ∆, which takes the value 1 on an interval whose interior contains 2 /2 + ej (−∞), and whose length is almost as large as that of ∆. η− From our assumptions on ∆, there is a one-to-one correspondence between E ∈ ∆ and positive k, such that k 2 /2 + ej (−∞) = E. For E ∈ ∆, we make the change of variables from k to E at t = 0 in the (rescaled) Fourier transform of
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the Gaussian in (6.1) (see [9]). Taking into account the normalization (3.7) of the generalized eigenvectors, this leads to the energy density −
2
F (E) ei ωj (E,−∞)/ Q(E, ) = ϕ0 B− , A− , 2 , η− , −a− , k(E) (6.2) π k(E) − 2 F (E) ei ωj (E,−∞)/ 2 = ϕ0 B− , A− , , η− , −a− , 2(E − ej (−∞)) π 2(E − ej (−∞)) that we use in (4.2). 2 Since η− /2 + ej (−∞) is in the interior of the set where F (E) = 1, the wave functions (6.1) and ψ(x, t, , −) defined by (4.2) with the energy density defined 2 by (6.2) differ in L2 (R) norm by an O(e−C/ ) error. To be sure that this error is smaller than the non-adiabatic effect we are studying, we assume any one of the following conditions: 1. Take the avoided crossing gap δ to be small enough that the non-adiabatic effect is larger than the error we make here. 2. Choose |B− | to be sufficiently small. That increases the value of C in this error estimate. 3. Fix the minimum of ∆, but then choose η− large enough so that the cut off is farther out in the tail of the Gaussian in momentum space. This also makes the non-adiabatic effect larger since η− is larger. With Q(E, ) chosen by (6.2), we have in the notation of (5.1), ( 2 (E − ej (−∞)) − η− )2 G(E) = (Re(A− /B− )) 2 ( 2 (E − ej (−∞)) − η− )2 , = |B− |−2 2 J(E)
=
( 2(E − ej (−∞)) − η− )2 2
(Im(A− /B− ))
(6.3)
(6.4)
+ a− ( 2(E − ej (−∞)) − η− ) − ωj− (E, −∞), P (E, ) =
−1/2
π −3/4 −3/2 B−
(2(E − ej (−∞)))−1/4 F (E).
(6.5)
Also, conditions C1, and C2 are satisfied, and provided we remove the trivial normalization factor of −3/2 from P (E, ), then condition C3 is also satisfied. We already know that asymptotically in the past, the interacting wave func2 tion determined by (6.2) agrees with (6.1) up to an O(e−C/ ) error, and we observe that the density Q(E, ) is sharply peaked around the energy η2 E0 = − + ej (−∞) corresponding to η− = 2(E0 − ej (−∞)). 2
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Thus from (6.3), we see that (E − E0 )2 + O((E − E0 )3 ), 2 (η− |B− |)2
G(E) =
i.e.,
g =
1 . (η− |B− |)2
We are not particularly interested in the main component of the wave function for large time that has not made a non-adiabatic transition. However, by a similar analysis, it could be determined by our techniques. Of course, it is what one would expect from the standard time-dependent Born-Oppenheimer approximation. Our focus is on the dominant non-adiabatic component, which is determined to leading order in by Theorem 5.1. From the above calculations and Theorem 5.1, we immediately get our main result for Gaussian incoming states: Theorem 6.1 Assume Hypotheses H4, H5, and H7, and assume ∆, A− , B− , a− , η− , δ, and the levels j and n have been chosen to satisfy the requirements above. Let Ψ(x, , t) be the solution to the Schr¨ odinger equation that is asymptotic as t → −∞ to 2
2
ei(η− /2−ej (−∞))t/ ϕ0 (A− + itB− , B− , 2 , a− + η− t, η− , x) φj (x). The leading non-adiabatic component of Ψ(x, , t) as t → ∞ and → 0 in L2 norm is on electronic level φn (x) and is given by 2
2
Anj () ei(η+ /2−en (∞))t/ ϕ0 (A+ + itB+ , B+ , 2 , a+ + η+ t, η+ , x) φn (x), where the values of A+ , B+ , a+ , η+ = k ∗ are those given by (5.8) as in Theorem 5.1. The amplitude for making this transition from level j to level n is given by " B+ −i(κ(E ∗ )−k∗ a+ )/2 −iθj (ζ) −α(E ∗ )/2 Anj () = e e e . B− In particular
∗
B+ = ((G (E ) + =
Imγj (E ∗ )) k ∗ 2 )−1/2
|B− | η− k∗
+ Imγj (E ∗ )|B− |k ∗
2
=
η− + Imγj (E ∗ ) k ∗ 2 |B− |2 k ∗
−1/2
(6.6)
Remark. Depending on the relative size of |B− | with respect to δ, we can apply Proposition 5.1 to further characterize Anj . We now turn our attention to the situation where the incoming nuclear wave packet is in the state ϕm . The only change from the situation just considered is
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that we must replace the function P (E, ) in (6.5) by −(m+1)/2
P (E, ) = (−i)m 2−m/2 (m!)−1/2 π −3/4 −3/2 (2(E − ej (−∞)))−1/4 B− 2(E − e (−∞)) − η j − F (E). × (B− )m/2 Hm |B|
(6.7)
Again, this satisfies Condition C3 if we take out the trivial factor of −(m+3/2) . Theorem 6.2 Assume the Hypotheses of Theorem 6.1. Let Ψ(x, , t) be the solution to the Schr¨ odinger equation that is asymptotic as t → −∞ to 2
2
ei(η− /2−ej (−∞))t/ ϕm (A− + itB− , B− , 2 , a− + η− t, η− , x) φj (x). The leading non-adiabatic component of Ψ(x, , t) as t → ∞ and → 0 in L2 norm is on electronic level φn (x), and is given by 2
(m)
2
Anj () ei(η+ /2−en (∞))t/ ϕ0 (A+ + itB+ , B+ , 2 , a+ + η+ t, η+ , x) φn (x), where the values of A+ , B+ , a+ , η+ = k ∗ are those given by (5.8), as in Theorem 5.1. The amplitude for making the transition from level j to level n is given by " m/2 ∗ ∗ 2 B+ e−i(κ(E )−k a+ )/ B− (m) −iθj (ζ) −α(E ∗ )/2 Anj () = e e B− B− 2m/2 (m!)1/2 ∗ k − η− × Hm |B− | ∗
=e
−iθj (ζ)
e−α(E )/ m
2
"
∗
∗
B+ e−i(κ(E )−k a+ )/ B− (m!)1/2
2
√ m 2(k ∗ − η− ) B−
× (1 + O()). In particular, B+ is again given by (6.6) and the pre-exponential factor is of order −m .
7 Technicalities Proof of Lemma 3.1. We consider only the limit x → ∞ and the choice σ = +. The other cases are similar. In this proof, cn denotes a finite constant that depends only on n, but may vary from line to line. Explicitly, for any n ∈ N, ∂n ∂E n
2(E − ej (x)) = cn (2(E − ej (x)))1/2−n ,
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uniformly for E ∈ ∆. So, the first assertion is true. Moreover, ∂n ∂E n
2(E − ej (x)) − 2(E − ej (∞)) = cn (2(E − ej (x)))1/2−n − (2(E − ej (∞)))1/2−n .
For n = 0, we have by (2.2), 2 (ej (∞) − ej (x)) 2(E − ej (x)) − 2(E − ej (∞)) = 2(E − ej (x)) + 2(E − ej (∞)) = O(ej (∞) − ej (x)) = O( x −(2+ν) ). (7.1) For n > 0, we can write (2(E − ej (x)))1/2−n − (2(E − ej (∞)))1/2−n =
+
n−1 k=0
(2(E − ej (x)))1/2 − (2(E − ej (∞)))1/2 (2(E − ej (x)))n
(2(E − ej (∞)))1/2 (2(E − ej (x)))k+1 (2(E − ej (∞)))n−k
(2(ej (x) − ej (∞)),
(7.2)
to which the estimate (7.1) applies. The second assertion follows. By definition,
∞ + (2(E − ej (y)))1/2 − (2(E − ej (∞)))1/2 dy, rj (+, x, E) = − x
n
+ ∂ so that (7.2) implies the estimates on ∂E n rj (+, x, E). We now study the properties of the cτj ’s. Again, we shall consider x → +∞; the other case is similar. We first compute 1 1 τσ ajl (x, E) = −
φj (x), φl (x) (kj (x, E) + τ σkl (x, E)) 2 kj (x, E)kl (x, E) kj φj (x), φl (x) ∂ kl (x, E) + στ − kl 2 ∂x 1 1 = −
φj (x), φl (x) (kj (x, E) + τ σkl (x, E)) 2 kj (x, E)kl (x, E) kj φj (x), φl (x) el (x) . + στ − (7.3) kl 2 kl (x, E)
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The presence of the factors φj (x), φl (x) and el (x), which are independent of E and decay as 1/ x 2+ν , implies together with (3.17) that ∂ n+m aτ σ (x, E) = O( x −(2+ν) ). ∂xm ∂E n jl
(7.4)
We denote the coefficients cτj collectively by + c (x, E, ) ∈ C2m , c(x, E, ) = c− (x, E, ) and the generator of equation (3.12) by the 2m × 2m block matrix M(x, E, ) x x 2 2 ei 0 (kj (y,E)+kl (y,E))dy/ a+− ei 0 (kj (y,E)−kl (y,E))dy/ a++ jl (x, E) jl (x, E) , = x 2 i 0x (−kj (y,E)+kl (y,E))dy/2 −− ei 0 (−kj (y,E)−kl (y,E))dy/ a−+ ajl (x, E) jl (x, E) e so that (3.12) can be rewritten as ∂ c(x, E, ) = M(x, E, ) c(x, E, ). ∂x Expressing the solutions as Dyson series, we obtain c(x, E, ) =
∞
n=0
0
x
0
x1
···
xn−1 0
× M(x1 , E, )M(x2 , E, ) · · · M(xn , E, )dx1 dx2 · · · dxn c(0, E, ). (7.5) ∞ Because of (7.4), 0 M(y, E, ) dy < ∞, uniformly for E ∈ ∆ and for → 0, and we get the usual bound c(x, E, ) ≤ e
∞ 0
M(y,E,) dy
c(0, E, ) .
Thus, by the Lebesgue Dominated Convergence Theorem, we get from (7.5) that, as x → ∞ and uniformly for E ∈ ∆ and → 0, c(x, E, ) = O(1). Next we show that c(x, E, ) − c(y, E, ) is arbitrarily small for large x and y, so that limx→∞ c(x, E, ) = c(∞, E, ) exists. It is enough to consider
y M(z, E, ) c(z, E, ) dz c(x, E, ) − c(y, E, ) = − x
and (7.4). The expression above with y = ±∞, and the properties of M, c, just proven yield c(x, E, ) − c(±∞, E, ) = O( x −(1+ν) ), uniformly in E ∈ ∆ and → 0, as x → ±∞.
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To get similar bounds on the derivatives of c with respect to E which are uniform in , we resort to integration by parts in (3.13) with x0 = −∞
σ τσ 2 aτml (x, E) ei∆ml (x,E)/ cσl (x, E, ) τ (x, E) − k σ (x, E) km l l,σ
x τ σ τσ 2 ∂ aml (z , E) 2 + i ei∆ml (z ,E)/ cσl (z , E, ) dz σ τ ∂z km (z , E) − kl (z , E) l,σ −∞
σ x aτml (z , E) aσθ θ lp (z , E) i∆τmp (z ,E)/2 θ e + i2 cp (z , E, ) dz , (7.6) σ τ (z , E) − k (z , E) k −∞ m l
cτm (x, E, ) = δjm δτ − − i2
l,p,σ,θ
where
∆τjlσ (x, E)
=
x 0
(kjτ (y, E) − klσ (y, E)) dy.
Taking care of the derivatives of the phases with respect to E ∂ i x (τ kj (y,E)+σkl (y,E))dy/2 e 0 = O( x /2 ), ∂E making use of the properties of c, M and kj , of the estimates x > 1, and choosing ≤ 1, we get by differentiation of (7.6) 2 1 ∂E c(x, E, ) ≤ K0 ∂E c(x, E, ) + 2+ν
x
x 1+ν
x
x 1 ∂E c(y, E, ) 2 dy + dy , (7.7) 1+ν
y 2+ν −∞ y −∞ where K0 is a constant independent of and E. By choosing small enough so that K0 2 < 1/2, (7.7) yields
x ∂E c(y, E, ) ∂E c(x, E, ) ≤ K1 1 + 2 dy , (7.8)
y 2+ν −∞ with another constant K1 uniform in E and . Hence, by Gronwall’s Lemma, ∂E c(x, E, ) ≤ K1 eK1
∞
−∞
y −(2+ν ) dy
= O(1),
(7.9)
if < 1/(2K0 ), uniformly in E. Similar manipulations on the difference ∂E c(x, E, ) − ∂E c(±∞, E, ) yield the estimate ∂E (c(x, E, ) − c(±∞, E, ))
±∞ 1 ∂E (c(y, E, ) − c(±∞, E, )) ≤ K2 ± dy ,
x ν
y 2+ν x for smaller than a constant uniform in E, and for all x ≥ 0, respectively x ≤ 0. Gronwall’s Lemma implies in that case ∂E (c(x, E, ) − c(±∞, E, )) = O( x −ν ), as x → ±∞, uniformly in E ∈ ∆ and in → 0.
(7.10)
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Proof of Lemma 4.1. We assume t = 0 and rewrite the exponential factors in (4.1) as e
−i(
x 0
kjσ (y,E) dy + tE)/2
= i
x
e−i( 0 x t+ 0
∂ 2 ∂E
kjσ (y,E) dy + tE)/2 ∂ ∂E
kjσ (y, E) dy
.
(7.11)
Then, for each integral in (4.1), we have
∆
Q(E, ) cσj (x, E, ) −i( x 0 e kj (x, E) = i 2 e−i(
x
dE
E2 Q(E, ) cσj (x, E, ) x ∂ σ kj (x, E) t + 0 ∂E kj (y, E) dy E 1 ! σ Q(E, ) cj (x, E, ) ∂ x ∂ σ ∂E kj (x, E) t + 0 ∂E kj (y, E) dy
kjσ (y,E) dy + tE)/2
0
−
kjσ (y,E) dy + tE)/2
∆
× i 2 e−i(
x 0
kjσ (y,E) dy + tE)/2
dE.
(7.12)
The quantities cσj (x, E, ), kj (x, E), and their derivatives with respect to E are uniformly bounded in x and E. Also,
x ∂kjσ (±∞, E) ∂kjσ (y, E) dy = x + O(1) ∂E ∂E 0 σx = + O(1), kj (±∞, E) uniformly for E ∈ ∆ as x → ±∞. Thus, the boundary terms in (7.12) satisfy 2 −i(
i e
x 0
=O
kjσ (y,E)dy+tE)2
E2 Q(E, ) cσj (x, E, ) x ∂ σ kj (x, E) t + 0 ∂E kj (y, E)dy E 1
1 1 k σ (±∞, E1 )t + x + O(1) + k σ (±∞, E2 )t + x + O(1) j j
. (7.13)
We now apply the restrictions on x/t in the statement of the Lemma. For any choice of j and σ, they ensure that the denominators on the right hand side of (7.13) can be estimated, uniformly in E and for large |x|, by σ kj (±∞, E)t + x + O(1) = |x| 1 + kjσ (±∞, E)t/x + O(1/x) ≥
|x| (α + O(1/|x|)),
where α is the number that appears in the statement of the lemma. From this, we see that the boundary terms in (7.12) are O(1/|x|).
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We estimate the integral term in (7.12) in a similar way. Under the restrictions on x/t in the lemma, we obtain
2
i e
−i(
x 0
∂ ∂E
kjσ (y,E) dy +tE)/2
∆
(Q(E, ) (kjσ (x, E))−1/2 cσj (x, E, )) x ∂ σ t + 0 ∂E kj (y, E) dy
!
x ∂2 σ Q(E, ) (kjσ (x, E))−1/2 cσj (x, E, ) 0 ∂E 2 kj (y, E) dy − dE = O(1/|x|). 2 x ∂ σ t + 0 ∂E kj (y, E) dy This implies the lemma for t = 0. When t = 0, the estimate (7.13) with t = 0 yields the result in a more direct way. Proof of Proposition 4.1. We can write ψ(x, t, ) − ψ(x, t, , ±) =
φj (x)
j=1,2 σ=±
x σ 2 2 Q(E, ) dE e−itE/ (cσj (x, E, ) − cσj (±∞, E, )) e−i 0 kj (y,E)dy/ 2kj (±∞, E) ∆
x σ 2 2 Q(E, ) dE + e−itE/ cσj (±∞, E, ) × e−i 0 kj (y,E)dy/ 2kj (±∞, E) ∆ Q(E, )cσ (x, E, )dE 2 j −i(xkjσ (±∞,E)+ωjσ (±∞,E))/2 + −e e−itE/ kj (±∞, E)kj (x, E) ∆ ! kj (±∞, E) − kj (x, E) −i 0x kjσ (y,E)dy/2 . (7.14) × e 2kj (±∞, E) + 2kj (x, E)
×
The first step of the proof consists of integrating by parts to get a factor of 1/t according to
2
f (x, E, ) e−itE/ dE =
∆
E2 2 i 2 f (x, E, ) e−itE/ t E1 −
i 2 t
∆
2 ∂ f (x, E, ) e−itE/ dE. ∂E
(7.15)
We then bound the L2 (Rx ) norm of each term that arises from these integrations by parts, with bounds that are uniform in t. From the estimates in Lemma 3.1, we see that all the boundary terms in (7.15) coming from (7.14) are of order x −(1+ν) . Thus, their L2 norms are bounded, uniformly in t. The integral terms in (7.15) coming from (7.14) all have the form
x σ 2 gj (x, E, ) e−i( 0 kj (y,E)dy+tE)/ dE, j = 1, 2, 3, (7.16) ∆
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where the first integral from (7.14) contains the function g1 (x, E, ) =
∂ ∂E
Q(E, ) (cσj (x, E, ) − cσj (±∞, E, )) kj (±∞, E)
!
Q(E, ) −i (cσj (x, E, ) − cσj (±∞, E, )) kj (±∞, E)
x ∂ σ kj (y, E) dy. (7.17) × 0 ∂E With the notation of Lemma 3.1, the second integral in (7.14) contains the function
Q(E, ) σ i(rjσ (±,x,E))/2 cj (±∞, E, ) 1 − e kj (±∞, E) σ 2 Q(E, ) cσj (±∞, E, ) 1 − ei(rj (±,x,E))/ −i kj (±∞, E)
x ∂ σ kj (y, E) dy. (7.18) × ∂E 0
∂ g2 (x, E, ) = ∂E
The third integral contains ∂ g3 (x, E, ) = ∂E
Q(E, ) cσj (x, E, ) kj (±∞, E) − kj (x, E) kj (±∞, E)kj (x, E) 2kj (±∞, E) + 2kj (x, E)
Q(E, )cσj (x, E, ) kj (±∞, E) − kj (x, E) −i kj (±∞, E)kj (x, E) 2kj (±∞, E) + 2kj (x, E)
x ∂ σ kj (y, E) dy. (7.19) × 0 ∂E By Lemma 3.1, and the condition ν > 1/2, each of these functions gj (x, E, ) satisfies the following bound, uniformly in E, gj (x, E, ) = O( x −ν ) ∈ L2 (R). Therefore, we can estimate the L2 norm of the corresponding expression (7.16) by
R
∆
gj (x, E, ) e
−i(
x 0
kjσ (y,E)dy+tE)/2
2 dE dx ≤ C1 (),
where C1 () is a finite constant that is independent of t. This finishes the proof.
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Proof of Proposition 3.1. Since the argument is virtually identical to the one presented in [18] and [15], we will be rather sketchy and mainly point out the effects of the parameter δ and of the non self-adjointness of the generator F (x, δ). Expressing the projector P (x, δ) as a integral of the resolvent (F (x, δ) − z)−1 along a loop L (or a finite number of such loops) around the set σ1 (x, δ) by means of the Riesz formula, # 1 P (x, δ) = − (F (x, δ) − z)−1 dz, (7.20) 2πi L we get a bound, uniform in δ > 0 and x ∈ ρα , P (x, δ) ≤ c. Indeed, for each x ∈ ρα , we can choose the path L uniformly in δ by hypothesis. The existence of the limits F (±∞, δ) allows us actually to consider only a finite number of distinct loops a finite distance g/4 away from spectrum of F (x, δ), for all (x, δ) . Also, uniformly in δ > 0, (F (x, δ) − z)−1 ≤ c,
(7.21)
for z on the corresponding loop L, since | det(F (x, δ) − z)| ≥ (g/4)n and F (x, δ) is uniformly bounded. By a similar argument, using Hypothesis H4, we get, uniformly in δ c , P (x, δ) − P (±∞, δ) ≤
x 2+ν ∂ , we get from (7.20), as x → ±∞ in ρα . With the notation for ∂x # 1 P (x, δ) = (F (x, δ) − z)−1 F (x, δ) (F (x, δ) − z)−1 dz. 2πi L
Thus, Hypothesis H4 yields, uniformly in δ, P (x, δ) ≤
c ,
x 2+ν
and a similar uniform estimate for K(x, δ) = [P (x, δ), P (x, δ)], K(x, δ) ≤
c .
x 2+ν
(7.22)
The operator K is the generator of the intertwining operator W defined by W (x, x0 , δ) = K(x, δ) W (x, x0 , δ),
with
W (x0 , x0 , δ) = I.
It satisfies W (x, x0 , δ) P (x0 , δ) = P (x, δ) W (x, x0 , δ), for all (x, δ) (including x = ±∞).
(7.23)
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Following [19], we construct a hierarchy of generators. Let F0 (x, δ) = F (x, δ), P0 (x, δ) = P (x, δ), and K0 (x, δ) = K(x, δ). For q ∈ N∗ , we inductively define Fq (x, δ, ) = F (x, δ) − 2 Kq−1 (x, δ, ), assuming is small enough so that the spectrum of Fq is separated into two disjoint parts corresponding to those of F . We define Pq (x, δ, ) to be the spectral projector for Fq corresponding to P (x, δ) as → 0 by perturbation theory. Then, Kq (x, δ, ) = [Pq (x, δ, ), Pq (x, δ, )]. Sections II.A and II.B of [19] and (7.21) and (7.22) show that there exist constants ∗ > 0, r > 0, Γ > 0 and c > 0, all independent of δ > 0, such that for all < ∗ , all x ∈ R, and q = q ∗ = [r/2 ], Kq∗ −1 (x, δ, ) ≤
c ,
x 2+ν
(7.24) 2
e−Γ/ Kq∗ (x, δ, ) − Kq∗ −1 (x, δ, ) ≤ c .
x 2+ν We define F∗ (x, δ, ) = Fq∗ (x, δ, ) = F (x, δ) − Kq∗ −1 (x, δ, ), P∗ (x, δ, ) = Pq∗ (x, δ, ),
(7.25)
K∗ (x, δ, ) = Kq∗ (x, δ, ), and the evolution operators W∗ and Ξ∗ by W∗ (x, x0 , δ, ) = K∗ (x, δ, ) W∗ (x, x0 , δ, ), and
with
W (x0 , x0 , δ, ) = I,
i 2 Ξ∗ (x, x0 , δ, ) = W∗ (x0 , x, δ, ) F∗ (x, δ, ) W∗ (x, x0 , δ, ) Ξ∗ (x, x0 , δ, ), with Ξ∗ (x0 , x0 , δ, ) = I .
(7.26)
The intertwining property (7.23) still holds with the ∗ indices. Therefore, Ξ∗ satisfies [Ξ∗ (x, x0 , δ, ), P∗ (x0 , δ, )] ≡ 0, for all x ∈ R. It follows from the definitions that the operator V∗ (x, x0 , δ, ) = W∗ (x, x0 , δ, ) Ξ∗ (x, x0 , δ, ) satisfies i 2 V∗ (x, x0 , δ, ) = (F∗ (x, δ, ) + i 2 K∗ (x, δ, )) V∗ (x, x0 , δ, ) and V∗ (x, x0 , δ, ) P∗ (x0 , δ, ) = P∗ (x, δ, ) V∗ (x, x0 , δ, ).
(7.27)
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Moreover, U (x, x0 , δ) − V∗ (x, x0 , δ, ) (7.28)
x V∗ (x0 , y, δ, ) (Kq∗ (y, δ, ) − Kq∗ −1 (y, δ, )) U (y, x0 , δ) dy. = i x0
The proposition will follow from 2
U (x, x0 , δ) − V∗ (x, x0 , δ, ) = O(e−Γ/ ),
(7.29)
(7.27) and lim
x→±∞
P∗ (x, δ, ) = P (±∞, δ),
due to (7.24) and (7.25). To prove (7.29), we first prove that V∗ is uniformly bounded in x, x0 , and . The analysis leading to Lemma 3.1 implies that U is uniformly bounded in x, x0 , and . This property is a consequence on the fact that the eigenvalues of F are simple and real, so that the decomposition (3.11) holds and the singular exponential factors are phases. Note that the lack of orthogonality of the eigenprojectors of F (x, δ) makes the bound on U dependent on δ. Choose B(δ) > 0, such that U (x, x0 , δ) ≤ B(δ). From (7.28) we get the inequality
x 2 C e−Γ/ sup V∗ (y, y0 , δ, ) dx, V∗ (x, x0 , δ, ) ≤ B(δ) 1 +
x 2+ν x0 y0 ,y the quantity v(, δ) = sup V∗ (x, x0 , δ, ) for some C. This implies that for some C, x0 ,x
satisfies v(, δ) ≤ B(δ)
v(, δ) e−Γ/ 1+C
2
.
This implies v(, δ) ≤
B(δ) ≤ v(δ), 1 − C B(δ) e−Γ/2
where v(δ) is uniformly bounded for sufficiently small . We now use (7.28) again to see that
U (x, x0 , δ) − V∗ (x, x0 , δ, ) ≤ v(δ)
R
2
B(δ) C e−Γ/ dx
x 2+ν
1 e−Γ/2 . ≤ C This proves (7.29) and completes the proof of the proposition.
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Proof of Lemma 3.3. Degenerate perturbation theory for self-adjoint matrices and Hypothesis H5 (see [14]) show that there exist f (z, δ) and ρ(z, δ), analytic in z for fixed δ, and C 1 as functions of (z, δ), such that 1 ρ(z, δ) (7.30) ej (z, δ) = f (z, δ) − 2 1 en (z, δ) = f (z, δ) + ρ(z, δ). 2 where, as (z, δ) → (0, 0), f (z, δ) = f (0, 0) + O(|z| + δ) = ec + O(|z| + δ),
and ρ(z, δ) = O(|z|2 + δ 2 ).
Moreover, ρ(z, δ) has two simple zeros, the complex crossing points, z0 (δ) and z¯0 (δ) that have z0 (δ) = O(δ). For concreteness, we arbitrarily choose ej < en on the real axis, although this is irrelevant for the analysis. Thus, by H7, we can write 1/2 ρ(z, δ) 2 (E − ej (z, δ)) = 2 (E − f (z, δ)) 1 + , 2 (E − f (z, δ)) where (E − f (z, δ)) and its inverse are analytic in ρα , uniformly in E ∈ ∆. Moreover, 1/2 ρ(z, δ) ρ(z, δ) 1 + O(|z|2 + δ 2 ). 1+ = 1 + 2 (E − f (z, δ)) 2 2 (E − ec ) Therefore, since 2 (E − f (z, δ)) is analytic, and we can choose the loop ζ encircling z0 (δ) or z¯0 (δ) to satisfy |ζ| = O(δ), we see that
ρ(z, δ) 1 2(E − ej (z, δ)) dz = dz + O(δ 3 ), 2 2(E − ec ) ζ ζ
ρ(z, δ) dz = O(δ 2 ). and 2 ζ
ρ(z, δ) In these two expressions, dz = ej (z, δ) dz due to the analyticity 2 ζ ζ of f in (7.30). Taking the imaginary part yields the first statement of the lemma. Note that we do not
have to worry about sign issues because Theorem 3.1 ensures 2(E − ej (z, δ)) dz, is positive. the decay rate, Im ζ
The two other statements follow from similar considerations for the integrals
1 ∂ Im 2 (E − ej (z, δ)) dz = Im dz and ∂E 2 (E − ej (z, δ)) ζ ζ
∂2 1 Im 2 (E − ej (z, δ)) dz = − Im dz. 3/2 ∂E 2 ζ ζ (2 (E − ej (z, δ)))
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Ann. Henri Poincar´e
Proof of Lemma 5.1. Consider first the minimization of the negative of the real part of the exponent. Since γj (E) tends to zero with δ (absent in the notation), if δ is small enough, we must look for minima in a neighborhood of E0 that satisfy the equation α (E) = g (E − E0 ) + Imγj (E) + O(E − E0 )2 = 0. We consider the absolute minimum E ∗ of α and assume it is unique. By Lemma 3.3, Imγj (E) < 0, so E ∗ > E0 . Note that E ∗ does not depend on x or t. Also, (n)
Imγj (E) = o(δ), n = 0, 1, 2, uniformly in E. So, we can assume E ∗ is nondegenerate since α (E ∗ ) = g + Imγj (E ∗ ) + O(E ∗ − E0 ) > 0. In terms of the variable k ∈ [k1 , k2 ], we view T as the (scaled) inverse Fourier transform of the function √ √ 2 2 R(k, t, ) = 2π2 e−α(E(k))/ P˜ (E(k), ) k e−iκ(E(k))/ e−it(k
2
/2+e(∞))/2
χ[k1 ,k2 ] (k),
where χS (·) is the characteristic function of the set S. That is T (x, t, ) = (F−1 R(·, t, ))(x), where F is defined by 1 (F g)(x) = √ 2π2
2
g(k) e−ikx/ dk.
R
With the variable k ∈ [k1 , k2 ] we have ∂2 α(E(k)) = k ∗ 2 α (E ∗ ), ∂k 2 k∗ and expanding around k ∗ , ∗
2
T (, x, t) = e−α(E )/
∂ 2 α(E(k))| √ k∗ ∂k2 (k−k∗ )2 O((k−k∗ )3 )/2 −iβ(k,x,t)/2 22 × k P˜ (E(k), ) e− e e dk, [k1 ,k2 ]
where the negative of the imaginary part of the exponent is denoted by 2 k + e2 (∞) + κ(E(k)) − x k. β(k, x, t) = t 2 We now introduce µ() = s > 0, with 2/3 < s < 1. It goes to zero in such a way that µ()/ 1 and µ()3 /2 1.
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Because E ∗ is a unique absolute minimum, the behavior of α(E) close to E ∗ , and the assumption (5.2) on P , we can reduce the integration range in T to [k ∗ − µ(), k ∗ + µ()] at the expense of a relative error whose L2 norm is of order O(∞ ), uniformly t. More precisely, T (x, t, ) = ((F−1 (R1 + R2 ))(·, t, ))(x), where R1 (k, t, ) = χ[k∗ −µ(),k∗ +µ()] (k) R(k, t, ),
and
R2 (k, t, ) = χ[k∗ −µ(),k∗ +µ()]C (k) R(k, t, ). For some a∗ > 0 and r > 0, |R2 (k, t, )| ≤ r e−α(E
∗
)/2
∗
e−a
(µ()/)2
√ | k P˜ (E(k), )|.
Hence, by the Parseval identity, uniformly in t, we have F−1 (R2 )(·, t, ))
!1/2 2
= [k∗ −µ(),k∗ +µ()]C
|R2 (k, t, )| dk = O(e−α(E
∗
)/2 ∞
).
In the remaining integral containing R1 , we further estimate eO(k−k and
∗ 3
) /2
= 1 + O(µ()3 /2 ) = 1 + O(3s−2 ),
(7.31)
√ √ √ k P˜ (E(k), ) = k ∗ P˜ (E ∗ , ) + O(µ()) = k ∗ P (E ∗ , ) + O(s + 2 ).
The contribution of order 2 comes from the error in the computation of the coefficient c− n . Using the Parseval identity again with uniform bounds on the exponential factors of R1 , we see that the contribution to T coming from the error term O(s ) ∗ 2 is bounded uniformly in t in the L2 (Rx ) norm by O(e−α(E )/ 1+2s ). Similarly, 2 the error term stemming from (7.31) yields an error in the L (Rx ) norm of order ∗ 2 O(e−α(E )/ 4s−1 ). To compute the leading term, we expand β(·, x, t) around k ∗ as β(k, x, t)
= t E ∗ + κ(E ∗ ) − x k ∗ ∂ ∗ ∗ κ(E(k)) − x + (k − k ) k t + ∂k ∗ k (k − k ∗ )2 ∂2 + κ(E(k)) t+ 2 ∂k 2 k∗ ∗ 3 3 ∂ (k − k ) κ(E(k)) , + 6 ∂k 3 ˜ k
(7.32)
988
G.A. Hagedorn and A. Joye
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where k˜ lies between k and k ∗ , and the third derivative is independent of t and x. The last term in (7.32) gives rise to a contribution which is of order ∗ 2 O(e−α(E )/ 4s−1 ) in the L2 (Rx ) norm, uniformly in t, as above. Therefore, in the L2 sense, T (, x, t) = e−α(E
×
∗
)/2
e−i(tE
∗
+κ(E ∗ )−xk∗ )/2
√ (k−k∗ ) ∗ ∂ k ∗ P (E ∗ , ) e−i 2 (k t+ ∂k κ(E(k))|k∗ )−x)
[k∗ −µ(),k∗ +µ()]
×e
∗ 2
2
2
) ∂ ∂ − (k−k ( ∂k 2 α(E(k))|k∗ +i(t+ ∂k2 κ(E(k))|k∗ )) 22
dk + O( ) + O( ) , p
∞
where p = min(1 + 2s, 4s − 1) ∈ (0, 3) can be chosen arbitrarily close to 3. Again, ∗ 2 at the cost of an error whose L2 norm is O(e−α(E )/ ∞ ), uniformly in t, we can extend the interval of integration to the whole real line and compute the Gaussian integral explicitly according to the formula (for ReM > 0)
∞√ N2 ∗ 2 ∗ 2 2π − −(M(k−k ) /2+iN (k−k ))/ ∗ (k − iN/M ) . k∗ e dk = √ e 22 M M k∗ −∞ We then get the result with ∂2 ∂2 ∗ ∗ α(E(k))| + i t + κ(E(k))| , k k ∂k 2 ∂k 2 ∂ κ(E(k))|k∗ − x. N = k∗ t + ∂k
M=
and
Proof of Lemma 5.2. The first assertion is straightforward. The second follows from the identity ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x) x = ϕ0 (A+ (t), B+ (t), 2 , a+ (t), η+ (t), x) (x − a+ (t)) + ϕ0 (A+ (t), B+ (t), a+ (t), 2 , η+ (t), x, ) a+ (t). The first term is O() in L2 (R) by scaling, and the second is of order a+ (t) = k ∗ t(1 + O(1/|t|)) for |t| large. We insert this in the first part of the lemma to obtain the second part as t → ±∞.
Acknowledgments George Hagedorn wishes to thank the Institut Fourier and the City of Grenoble for their kind hospitality and support during 2003 and 2004 when this research was conducted.
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[19] A. Joye, C.-E. Pfister, Superadiabatic Evolution and Adiabatic Transition Probability between Two Non-degenerate Levels Isolated in the Spectrum, J. Math. Phys. 34, 454–479 (1993). [20] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag Berlin Heidelberg New York 1980. [21] Ph.-A. Martin and G. Nenciu, Semiclassical Inelastic S-Matrix for OneDimensional N -States Systems, Rev. Math. Phys. 7, 193–242 (1995). [22] A. Martinez and V. Sordoni, A general reduction scheme for the timedependent Born-Oppenheimer approximation, C.R.A.S. 334, 185–188 (2002). [23] G. Nenciu and V. Sordoni, Semiclassical limit for multistate Klein-Gordon systems: almost invariant subspaces and scattering theory, Preprint mp− arc 01–36. [24] T. Ramond, Semiclassical Study of Quantum Scattering on the Line, Commun. Math. Phys. 177, 221–254 (1996). [25] V. Rousse, Landau-Zener Transitions for Eigenvalue Avoided Crossings in the Adiabatic and Born-Oppenheimer Approximations, Asymptotic Analysis 37, 293–328 (2004). [26] J.C. Tully, Molecular Dynamics with Electronic Transitions, J. Chem. Phys. 93, 1061–1071 (1990). [27] F. Webster, P.J. Rossky and R.A. Friesner, Nonadiabatic Processes in Condensed Matter: Semi-Classical Theory and Implementation, Comp. Phys. Commun. 63, 494–522 (1991).
George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0123 USA email:
[email protected] Alain Joye Institut Fourier Unit´e Mixte de Recherche CNRS-UJF 5582 Universit´e de Grenoble I BP 74 F-38402 Saint Martin d’H`eres Cedex France email:
[email protected] Communicated by Yosi Avron submitted 14/10/04, accepted 18/01/05
Ann. Henri Poincar´e 6 (2005) 991 – 1023 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/060991-33, Published online 15.11.2005 DOI 10.1007/s00023-005-0232-x
Annales Henri Poincar´ e
Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds Victor Gayral Abstract. We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of Rl . Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and non-compact spaces, as well as with periodic and non-periodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the non-planar parts of the Green functions is understood simply in terms of off-diagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the non-planar part of the one-loop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing.
1 Introduction Noncommutative geometry (NCG), specially in Connes’ algebraic and operatorial formulation [4], is an attempt to free oneself from the classical differential structure framework in modeling and understanding space-time, while keeping in algebraic form geometry’s tools such as metric and spin structures, vector bundles and connection theory. The NCG framework is well adapted to deal with quantum field theory over ‘quantum’ space-time (NCQFT) [34]. However, there is a lack of computable examples crucially needed to progress in this direction. Here we present a large class of models, the isospectral deformation manifolds, in which we show the intrinsic nature of UV/IR mixing through the analysis of a scalar theory. In [6, 7] Connes, Landi and Dubois-Violette gave a method to generate noncommutative spaces based on the noncommutative torus paradigm. For any closed Riemannian spin (this last condition could be relaxed for our purpose) manifold with isometry group of rank l ≥ 2, one can build a family of noncommutative spaces, called isospectral deformations by the authors. The terminology comes from the fact that the underlying spectral triple, that is, the dual object / ) encoding all the topological, differential, metric and spin (C ∞ (MΘ ), L2 (M, S), D structures of the original manifold, and so defining the ‘quantum Riemannian’ space [5], has the same space of spinors and the same Dirac operator as the unde/ ); only the algebra is modified. formed one (C ∞ (M ), L2 (M, S), D
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More precisely, the noncommutative algebra C ∞ (MΘ ) can be defined as a fixed point algebra under a group action [7]: α⊗τ −1 lΘ C ∞ (MΘ ) := C ∞ (M )⊗T ,
(1.1)
where TlΘ is a l-dimensional NC torus(-algebra) with deformation matrix Θ ∈ Ml (R), Θt = −Θ; α is the action of Tl on M given by an Abelian part of its is a suitable tenisometry group, τ is the standard action of Tl on TlΘ and ⊗ sor product completion. By the Myers-Steenrod Theorem [26], which asserts that Isom(M, g) ⊂ SO(n) for any n-dimensional compact Riemannian manifold (M, g), one can see that the class of such manifolds whose isometry group has rank greater or equal to two is far from small. V´ arilly [33] and Sitarz [31] independently remarked that this construction fits into Rieffel’s theory of deformation quantization for actions of Rl [28]. Given a Fr´echet algebra A with seminorms {pi }i∈I and a strongly continuous isometric (with respect to each seminorms) action of Rl , one can deform the product of the subalgebra A∞ , consisting of smooth elements of A with respect to the generators X k , k ∈ {1, . . . , l} of the action α. The algebra A∞ can be canonically endowed with a new set of seminorms {˜ pi,m }i∈I,m∈N given by p˜i,m (.) := supj≤i |β|≤m pj (X β .), β ∈ Nl . Those seminorms have the property of being compatible with the deformed product defined by the A∞ -valued oscillatory integral: −l dl y dl z e−i
α 1 Θy (a)α−z (b), a, b ∈ A∞ . aΘ b := (2π) R2l
2
Here Θ is the (real, skewsymmetric) deformation l×l matrix, < y, z >= li=1 y i z i , ∞ and if we denote by A∞ Θ the algebra (A , Θ ), the deformation process verifies ∞ ∞ ∞ (AΘ )Θ = AΘ+Θ , and hence is reversible. In [16], we investigate the equivalent of (1.1) in the non-periodic case and extend the construction of isospectral non-periodic deformations (called also θ-deformations to distinguish them from q-deformations) to non-compact manifolds within Rieffel’s framework, whose paradigms are now the Moyal planes [12]. Although we will not use directly the fixed point characterization (1.1), we want to insist on its crucial importance to understand the situation. Indeed, such a characterization means that we are transferring the noncommutative structure of the NC torus or of the Moyal plane inside the commutative algebra of smooth functions, in a way compatible with the Riemannian structure. The first studied examples of NCQFT were the NC tori and the Moyal planes, in pioneer works like [3, 11, 21, 23, 24, 34] (see also [10] and [32] for reviews). In those flat space situations, the main novelty in regard to renormalization aspects is that two kinds of Feynman diagrams coexist, respectively called planar and nonplanar. The first one yields ordinary UV divergences, while the non-planar graphs, characterized by vertices which depend on external momenta through a phase, are finite except for some values of the incoming momenta. That happens in particular
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for the zero mode in λϕΘ 4 theory on the NC torus and in the limit pµ → 0 for the same theory on the Moyal plane. This is the famous UV/IR entanglement phenomenon, which gives rise to difficulties for any renormalization scheme. In this paper, we show that for any (in general non-flat) isospectral deformation, UV/IR mixing in (Euclidean) NCQFT exists as in the (flat) paradigmatic examples of the NC torus and the Moyal planes. In the next section, isospectral deformations are constructed and their basic NCG properties are reviewed. The third section is devoted to the study of the λϕΘ 4 theory. One derives a field expansion from a (modified) heat kernel asymptotics to compute the effective action up to one loop. This construction gives a simple algebraic meaning to the presence and behavior of planar and non-planar sectors in those theories. In sections 4 and 5, using off-diagonal heat-kernel estimates, we prove the inherent generic character of the divergent structures for all kinds of isospectral deformations. Fixed points for the Rl action potentially yield a new kind of UV/IR mixing.
2 Isospectral deformations As explained in the Introduction, isospectral deformations are curved noncommutative spaces generalizing Moyal planes and noncommutative torus. To construct those NC Riemannian spaces (spectral triples), we use an approach developed in [16]. Advantages of this twisted product approach ` a la Rieffel are that it allows to treat on the same footing compact and non-compact cases (unital and nonunital algebras) as well as periodic and non-periodic deformations, and that it is well adapted for Hilbertian analysis. Let (M, g) be a locally compact, complete, connected, oriented Riemannian n-dimensional manifold without boundary, and let α be a smooth isometric action of Rl , 2 ≤ l ≤ n α : Rl −→ Isom(M, g) ⊂ Diff(M ), where l is less or equal to the rank of the isometry group of (M, g). We can then define a deformed or twisted product. The isometric action α yields a group of automorphisms on C ∞ (M ) that we will again denote by α: for all z ∈ Rl αz f (p) := f (α−z (p)). For brevity we will often write z.p ≡ αz (p) to designate the action of a group element on a point of the manifold. Obviously, the group action property reads z1 .(z2 .p) = (z1 + z2 ).p The infinitesimal generators of this action ∂ α (.) , Xj (.) := z j ∂z z=0
and 0.p = p.
j = 1, . . . , l,
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are ordinary smooth vector fields, so they leave Cc∞ (M ) invariant. Hence, given a real skewsymmetric l × l matrix Θ, one defines the deformed product of any f, h ∈ Cc∞ (M ) as a bilinear product on Cc∞ (M ) with values in C ∞ (M ) ∩ L∞ (M, µg ) by the oscillatory integral −l dl y dl z e−i α 1 Θy (f )α−z (h), (2.1) f Θ h := (2π) 2
R2l
l
where < y, z >:= j=i y j z j can be viewed as the pairing between Rl and its dual group. In spite of appearances this formula is symmetric, even with a degenerate Θ matrix (see the discussion near the end of this section), as one can rewrite the deformed product: dl y dl z ei α−y (f )α 1 (h). f Θ h := (2π)−l 2 Θz
R2l
The non-locality of this product generates a non-preservation of supports. In particular, the twisted product of two functions with disjoint support turns out to be non-zero a priori. Whereas in the periodic case (ker α Zl ) the fixed point characterization gives rise to a reasonable locally convex topology on the α⊗τ −1 invariant sub-algebra of the algebraic tensor product C ∞ (M ) ⊗ TlΘ or α⊗τ −1 depending whether M is compact or not, to obtain a smooth Cc∞ (M )⊗TlΘ algebra structure in the non-periodic case one has to complete Cc∞ (M ) to a Fr´echet algebra with seminorms defined through the measure associated to the Riemannian volume form, so that the action becomes strongly continuous and isometric with respect to each seminorm. This feature is investigated in [16]. In the sequel, as we mainly work at the linear level, Cc∞ (M ) will be deemed “large enough”. The associativity of the product (2.1) can be easily checked. The ordinary integral with Riemannian volume form µg is a trace (a proof is provided in [16]): µg f Θ h = µg f h = µg hΘ f ; (2.2) M
M
M
α is still an automorphism for the deformed product: αz (f )Θ αz (h) = αz (f Θ h);
(2.3)
the complex conjugation is an involution: (f Θ h)∗ = h∗ Θ f ∗ ;
(2.4)
and the Leibniz rule is satisfied for the generators of the action X k (f Θ h) = X k (f )Θ h + f Θ X k (h), k = 1, . . . , l.
(2.5)
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In fact, the Leibniz rule is satisfied for any order one differential operator which commutes with the action α, thus for the Dirac operator when the manifold has a spin structure. We have basically two distinct situations. When the group action is effective (ker α = {0}), i.e., for a non-periodic deformation, it is seen that the good topological assumption on α in order to avoid serious difficulties is properness. That is, we assume the map (z, p) ∈ Rl × M → (p, αz (p)) ∈ M × M to be proper. Recall that a map between topological spaces is proper if the preimage of any compact set is compact as well. On the other hand, for periodic deformations the action factors through a torus action α ˜ : Rl /Zl → Isom(M, g), and the factorized action α ˜ is automatically proper. When M is compact, α must be periodic to be proper, while in the noncompact case both situations appear. We point out that the (non-compact) nonperiodic case is the most difficult one. First, when the manifold is not compact, the essential spectrum of the Laplacian is non-empty, so its negative powers are no longer compact operators. Furthermore, for periodic deformations (of compact manifolds or not) we have a spectral subspace decomposition, indexed by the dual group of Tl , which does simplify proofs and computations. We do not explicitly treat the mixed case α : Rd × Tl−d → Isom(M, g), but its general features will be clear from what follows. The hypothesis of geodesically completeness of M guarantees selfadjointness of the (closure of the) Laplace-Beltrami operator ∆ restricted to (the dense subset Cc∞ (M ) of) L2 (M, µg ), the separable Hilbert space of squared integrable functions with respect to the measure space (M, µg ). In our convention, ∆ = (d + δ)2 is positive, and reduced to 0-forms ∆ = δd = ∗H d ∗H d where ∗H is the Hodge star. Completeness (plus boundedness from below of the Ricci curvature) is needed to have conservation of probability [2, 9]: µg (p) Kt (p, p ) = 1, M
where Kt := Ke−t∆ is the heat kernel of the manifold. Recall that Kt (p, p ) for t > 0 is a smooth strictly positive symmetric function on M × M . The restriction to manifolds without boundary is required to have a simple (with vanishing of the odd terms [17]) on-diagonal expansion of the heat kernel Kt (p, p) (4πt)−n/2 tl a2l (p), t → 0, (2.6) l∈N
where al (p) are the so called Seeley-De Witt coefficients. It is proved in [16] that for non-compact non-periodic deformations (the stateΘ ment being immediate in the periodic case) Lf ≡ LΘ f (resp. Rf ≡ Rf ), the operator of left (resp. right) twisted multiplication by f , defined by Lf ψ = f Θ ψ (resp.
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Rf ψ = ψΘ f ), for ψ ∈ H := L2 (M, µg ), is bounded for any f ∈ Cc∞ (M ). This will be also true for smooth functions decreasing fast enough at infinity. Denote by Vz the induced action of Rl on L2 (M, µg ) by unitary operators Vz ψ(p) := ψ(−z.p); then one can alternatively define Lf and Rf by an operator valued integral Lf = (2π)−l dl y dl z e−i V 1 Mf V−z , (2.7) 2 Θy
R2l
Rf = (2π)−l
R2l
dl y dl z e−i V−z Mf V 1
2 Θy
,
(2.8)
where Mf denotes the operator of pointwise multiplication by f . Such integrals do not define B¨ ochner integrals in the vector space L(H). Indeed, the operatorial norm of the integrands in (2.7) and (2.8) are not integrable functions on R2l , since they depend on y and z only through unitary operators. Actually, the latter must be understood as L(H)-valued oscillatory integrals [28]. Formulas (2.7) and (2.8) can be easily derived from (2.1) using Vz Mf V−z = Mαz (f ) and the translation z → z − 12 Θy which leaves invariant the phase due to the skewsymmetry of the deformation matrix. Note that they can be used to define (left and right) ‘Moyal multiplications’ of any bounded operator on H, taking the place of Mf in the formulas. Within this presentation, it is straightforward to check that L and R are two commuting representations (in fact R is an antirepresentation): [Lf , Rh ] = 0, ∀f, h ∈ Cc∞ (M ). Thus formulas (2.7) and (2.8) provide an other way to check the associativity of the twisted product, which is equivalent to the commutativity of the left and right regular representations. Using the trace property (2.2), one can also prove that the adjoint of the left (resp. right) twisted multiplication by f equals the left (resp. right) twisted multiplication by the complex conjugate of f : (Lf )∗ = Lf ∗ , (Rf )∗ = Rf ∗ . Again, this fact can be directly checked using formulas (2.7) and (2.8). For Lf it reads ∗ −l (Lf ) = (2π) dl y dl z ei Vz Mf ∗ V 1 − 2 Θy R2l = (2π)−l dl y dl z e−i V 1 Θz Mf ∗ V−y , R2l
2
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where the changes of variable z → 12 Θz, y → 2Θ−1 y and the relation < Θ−1 y, Θz > = − < y, z > have been used. The primary example of such a space is the n-dimensional Moyal plane RnΘ . In this case, the manifold is the flat Euclidean space Rn , l = n, and Rn acts on itself by translation. Another interesting non-compact space which carries a smooth action of Rn−1 by isometry is the n-dimensional hyperbolic space Hn , that we can make into noncommutative HnΘ by the previous prescription. For periodic actions, there is a lattice L = βZl , β ∈ Ml (Z) in the kernel of α which factors through a torus Tlβ := Rl /βZl . This quotient is a compact space if and only if the rank of β equals l. In this case, we have a spectral subspace (PeterWeyl) decomposition (see [6,28,33] for details): for any bounded operator A which is α-norm smooth (the map z ∈ Tlβ → Vz AV−z is smooth for the norm topology of L(H)), one can define a l-grading by declaring A of l-degree r = (r1 , . . . , rl ) ∈ βZl when Vz AV−z = e−i(r1 z1 +···+rl zl ) A, ∀z ∈ Tlβ . Then, any α-norm smooth operator can be uniquely written as a norm convergent sum A= Ar , r∈βZl
where each Ar is of l-degree (r1 , . . . , rl ). This is in particular the case for the operator of pointwise multiplication by any function f ∈ Cc∞ (M ), since Mf lies inside the smooth domains of the derivations δj (.) := [Xj , .]. This assertion is obtained iterating the relation
[Xj , Mf ] = MXj (f ) = Xj (f ) ∞ , which is finite since f ∈ Cc∞ (M ) and because the Xj are ordinary smooth vector fields. Writing the spectral subspace decomposition of suchoperator, we find the PeterWeyl decomposition of any f ∈ Cc∞ (M ), as f = r∈βZl fr , where fr satisfies αz (fr ) = e−i(r1 z1 +···+rl zl ) fr . The twisted product of homogeneous components satisfies the noncommutative torus relation: i
fr Θ hs = e− 2 fr hs .
(2.9)
Noncommutative tori TnΘ , odd and even Connes-Landi spheres Sθ2n+1 , S2n θ [7] are examples of such compact noncommutative spaces; and the ambient space of Sn−1 θ is a non-compact periodic deformation. In summary, it is clear that the noncommutative structures of isospectral deformations are inherited from the NC tori or Moyal planes one’s, depending whether the deformation is periodic or not. When Θ is not invertible, the deformed product reduces to another twisted product associated with the restricted action σ := α|V ⊥ , where V is the null space
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of Θ – see for example [28]. Hence, one can handle non-invertible deformation matrices without any trouble. But of course, the “effective” deformation is always of even rank. Finally, in the non-periodic case only, properness of α implies that it is also free. To see that, recall that properness of any G-action is equivalent to {g ∈ G|g.X ∩ Y = ∅} is compact for any X, Y compact subset of M – see [25]. So, taking X = Y = {p0 } for any p0 ∈ M , its isotropy group Hp0 = {z ∈ Rl |z.p0 = p0 } = {z ∈ Rl |z.{p0 } ∩ {p0 } = ∅} is compact as well. But the only compact subgroup of Rl is {0}, hence the action is automatically free. This implies that the quotient map π : M → M/Rl defines a Rl -principal bundle projection. In the periodic case, the action is no longer automatically free, and the set Msing of points with non-trivial isotropy groups can give rise to additional divergences in the effective action. This will be shown to constitute a new feature of the UV/IR mixing on isospectral deformation manifolds.
3 ϕΘ 4 theory on 4-d isospectral deformations 3.1
The effective action at one-loop
For the sake of simplicity, we now restrict to the four-dimensional case; n = dim(M ) = 4. It will be clear, nevertheless, that our techniques apply to higher dimensions without essential modifications. We consider the classical functional action for a real scalar field ϕ: λ (3.1) µg 12 (∇µ ϕ)Θ (∇µ ϕ) + 12 m2 ϕΘ ϕ + ϕΘ 4 . S[ϕ] := 4! M We could add a coupling with gravitation of the type ξR(ϕΘ ϕ) (or even ξRΘ ϕΘ ϕ), where R is the scalar curvature and ξ a coupling constant, without change in our conclusions. Indeed, this term is not modified by the deformation: due to the α-invariance of the scalar curvature, we have RΘ f = R.f for any f ∈ Cc∞ (M ), thus µg R.(ϕΘ ϕ) = µg RΘ ϕΘ ϕ = µg (RΘ ϕ).ϕ = µg R .ϕ .ϕ. M
M
M
M
Similarly, thanks to the trace property (2.2), S[ϕ] can be rewritten as λ µg 12 ϕ∆ϕ + 12 m2 ϕ ϕ + (ϕΘ ϕ) (ϕΘ ϕ) , S[ϕ] = 4! M
(3.2)
so that, as in the falt cases, the kinetic part is not affected by the deformation. Recall that in our conventions the Laplacian is positive: ∆ = −∇µ ∇µ . We aim to compute the divergent part of the effective action Γ1l [ϕ] associated to S[ϕ] at one loop. This is formally given by 12 ln(det H), where H is the effective potential. In our case (as in the commutative one) it will be seen that H = ∆+m2 +
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B, where B is positive and bounded; so that when the manifold is not compact H has a non empty essential spectrum (typically the whole interval [m2 , +∞[). In order to deal with operators having pure-point spectrum (discret with finite multiplicity), we need first (independently of any regularization scheme) to redefine formally the one-loop effective action as:
Γ1l [ϕ] := 12 ln det HH0−1 , where H0−1 := (∆ + m2 )−1 is the free propagator. We are “not so far” from having a well-defined determinant since: HH0−1 = (H0 + B)H0−1 = 1 + BH0−1 , and BH0−1 is ‘small’: not trace-class in general, but compact; more precisely BH0−1 lies inside the p-th Schatten-class for all p > 2 (see below for the concrete expression of B and [16] for a proof of this claim). Physically, to replace H by HH0−1 corresponds to remove the vacuum-to-vacuum amplitudes. We then define the logarithm of the determinant by the Schwinger “proper time” representation:
1
1 ∞ dt −1 Tr e−tH − e−tH0 . (3.3) Γ1l [ϕ] = ln det(HH0 ) := − 2 2 0 t Before giving a precise meaning to the previous expression, that is to choose a regularization scheme, we go through the computation of the effective potential H. For that, the following definition will be useful. Definition 3.1. Let (X, dµ) a measure space. A kernel operator on E, a functions space on X, is a linear map A : E → E which can be written as
Af (p) = dµ(q) KA (p, q) f (q), f ∈ E, p, q ∈ X, X
where KA is the kernel of A. This definition leads to the following rules for the product of two kernel operators and for the kernel of the adjoint: dµ(u) KA (p, u) KB (u, q), and KA∗ (p, q) = KA (q, p)∗ . (3.4) KAB (p, q) = X
In our case, (X, dµ) ≡ (M, µg ) as a measure space, E ≡ Cc∞ (M ) and we will only be interested on distributional kernels, that is those KA lying on Cc∞ (M ×M ) , the space of distributions on M × M . Recall that the effective potential (see for example [36]) is the operator whose distributional kernel is given by the second functional derivative of the classical action: δ 2 S[ϕ] δ 2 S[ϕ] , K (p, p ) := , KH (p, p ) := H0 δϕ(p)δϕ(p ) δϕ(p)δϕ(p ) λ=0
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with functional derivatives defined as usual in the weak sense
δS[ϕ] dS[ϕ + tψ] , ψ := , δϕ dt t=0 where the coupling is given by the integral with Riemannian volume form f, h = µ f h. g M Using the trace property (2.2) we find out:
dS[ϕ + tψ] λ Θ 3 2 ϕ = ∆ϕ + m ϕ + , ψ . dt 3! t=0 Hence, δS[ϕ] λ S˜p [ϕ] := = ∆ϕ(p) + m2 ϕ(p) + ϕΘ 3 (p). δϕ(p) 3! The second functional derivative reads
2 δ S[ϕ] dS˜p [ϕ + tψ] , ψ := δϕ(p)δϕ dt t=0
λ 2 = ∆ + m + (LϕΘ ϕ + RϕΘ ϕ + Rϕ Lϕ ) δpg , ψ , 3! where δpg is the distribution defined by δqg , φ = M µg (p)δqg (p)φ(p) = φ(q), for any test function φ ∈ Cc∞ (M ). In conclusion, the explicit form of the operator H is: H = ∆ + m2 +
λ (LϕΘ ϕ + RϕΘ ϕ + Rϕ Lϕ ). 3!
Because ϕ is real, the operators Lϕ and Rϕ are self-adjoint, and we can check directly the strict positivity of H: ∗ LϕΘ ϕ + RϕΘ ϕ + Lϕ Rϕ = 12 (Lϕ + Rϕ )∗ (Lϕ + Rϕ ) + 12 L∗ϕ Lϕ + 12 Rϕ Rϕ .
We are come to an important point: the existence of UV/IR mixing for field theory on isospectral deformations comes from the simultaneous presence of left and right twisted multiplications in the effective potential. Precisely, we wish to illustrate the smearing nature of the product of left and right twisted multiplication operator Lf Rh . The crucial consequence, employed in subsection 2 3.3, is that the trace of Lf Rh e−t(∆+m ) is regular when t goes to zero, contrary 2 2 2 to Tr(Lf e−t(∆+m ) ), Tr(Rf e−t(∆+m ) ), Tr(Mf e−t(∆+m ) ), which in n dimensions −n/2 behave as t when t → 0 (In fact the three latter traces are identical). Remark 3.2. For a potential reads:
λ Θ 3 3! ϕ
theory on a six dimensional manifold, the effective
H = ∆ + m2 +
λ (Lϕ + Rϕ ). 2!
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Even in the lack of the ‘mixed’ term Rϕ Lϕ , those theories have a non-planar sector, but which will be present only at the level of the two-point function; the tadpole is not affected by the mixing. Consider the non-degenerate (n = 2N, Θ invertible) Moyal plane case. The operator Lf Rh turns out to be trace-class whenever f, h ∈ S(R2N ), say. This fact is known to the experts, but rarely mentioned – to the knowledge of the author, its first mention in writing is in [1]. We do a little disgression to see how it comes about. Recall [12] that there is an orthonormal basis for L2 (R2N , d2N x), the harmonic oscillator eigentransitions (2πθ)−N/2 {fmn }m,n∈NN , θ := (det Θ)1/2N which are matrix units for the Moyal product: fmn Θ fkl = δnk fml . Expanding f, h ∈ S(R2N ) in this basis: f = m,n cmn fmn , h = m,n dmn fmn , we obtain:
Tr Lf Rh = (2πθ)−N ckl dst fmn , fkl Θ fmn Θ fst m,n,k,l,s,t
−N
= (2πθ) =
ckm dnt fmn , fkt
m,n,k,t
cmm dnn
m,n
= (2πθ)−N
d2N x f (x)
d2N y h(y) < ∞.
Then in this case one can factorize HH0−1 and extract a finite part in the effective action. We have λ 1 H0 H −1 = 1 − (Lϕθ ϕ + Rϕθ ϕ ) λ 3! ∆ + m2 + 3! (Lϕθ ϕ + Rϕθ ϕ ) λ 1 × 1 − Lϕ Rϕ . (3.5) λ 3! ∆ + m2 + 3! (Lϕθ ϕ + Rϕθ ϕ + Lϕ Rϕ ) Now, 1−
λ Lϕ Rϕ 3! ∆ + M2 +
1 λ 3! (Lϕθ ϕ
+ Rϕθ ϕ + Lϕ Rϕ )
∈ 1 + L1 (H),
so that its determinant is well defined. Thus only the determinant of the first piece of (3.5) needs to be regularized. The determinant of the second piece of (3.5) contains the whole non-planar contribution to the two-point function, while for the four-point function the finite non-planar part lies in both pieces. The structure of the effective potential, i.e., the presence of mixed products of left and right twisted multiplication operators, and thus the existence of two distinct sectors in the theory is fairly general: for noncommutative scalar field theories
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whose classical field counterparts are regarded as elements of a noncommutative algebra, and a classical action built from a trace on the algebra, the effective potential will contain in general sums and mixed products of left and right regular representation operators. Let us go back to the computation of Γ1l [ϕ]. The t-integral in (3.3) is divergent because of the small-t behavior of the heat kernel on the diagonal. We thus define a one-loop regularized effective action by:
1 ∞ dt Tr e−tH − e−tH0 . Γ1l [ϕ] := − (3.6) 2 t One can invoke less rough regularization schemes, for example a ζ-function regularization
1 ∞ dt 2 σ σ,µ tµ (3.7) Tr e−tH − e−tH0 , Γ1l [ϕ] := − 2 0 t akin to dimensional regularization. However, for the purposes of this article (3.6) will do. One can think of as of the inverse square of Λ, with Λ a momentum space cutoff. To show that the expressions (3.6) and (3.7) are now well defined, we have to prove that e−tH − e−tH0 is trace-class for all t > 0. Note that for t → ∞ 2 convergence is ensured by the global e−tm factor, and that when the spectrum of the Laplacian has a strictly positive lower bound one can construct massless, IR divergence-free NCQFT. That is the case for the twisted hyperbolic planes HnΘ since the L2 -spectrum of ∆ on Hn is the whole half line [n2 /4, ∞[. Lemma 3.3. The semigroup difference e−tH − e−tH0 is trace-class for all t > 0. Proof. Using positivity of H and H0 , the semigroup property and the holomorphic functional calculus with a path γ surrounding both the spectrum sp(H) ⊂ R+ and sp(H0 ) ⊂ R+ , we have 1 e−tH − e−tH0 = (2iπ)2 dz1 dz2 e−t(z1 +z2 )/2 (RH (z1 )RH (z2 ) − RH0 (z1 )RH0 (z2 )) , γ×γ
where RA (z) = (z − A)−1 denotes the resolvent of A. But H = H0 + B where B is bounded. Using next RH (z) = RH0 (z)(1 + BRH (z)), we find RH (z1 )RH (z2 ) − RH0 (z1 )RH0 (z2 ) = RH0 (z1 )RH0 (z2 )BRH (z2 ) + RH0 (z1 )BRH (z1 )RH0 (z2 ) + RH0 (z1 )BRH (z1 )RH0 (z2 )BRH (z2 ). The first resolvent equation and the fact that Lf (z − ∆)−k , Rf (z − ∆)−k ∈ Lp (H), for p > 2/k, f ∈ Cc∞ (M ) [16], together with the H¨ older inequality for Schatten
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classes, yield
dz1 dz2 e−t(z1 +z2 )/2 RH0 (z1 )RH0 (z2 )BRH (z2 )
γ×γ
is absolutely convergent for the trace norm. Similarly for the other terms. So e−tH − e−tH0 is trace-class as required.
3.2
Field expansion
We now tackle the -behavior of Γ1l [ϕ] to describe the divergences. We will then show that, as for the Moyal planes and noncommutative tori, there exist for general isospectral deformations two kind of contributions to the Green functions, the planar one giving rise to ordinary singularities and the non-planar one exhibiting the UV/IR mixing phenomenon. Note that, since we are in a curved background, we can no longer work with Feynman diagrams in momentum space. However, by abuse of language we continue to speak about planar and non-planar contributions, because there is a splitting at the operator level which coincides with the splitting of planar and non-planar Feynman graphs in the known flat cases. This point will become clearer in subsequent subsections. As we are only interested in the -behavior of Γ1l [ϕ] (we only consider the potentially divergent part of the regularized effective action), we need a small t
expansion for Tr e−tH − e−tH0 . This expansion will be managed in the same vein as the ones obtained in [13,35]. The Baker-Campbell-Hausdorff formula is written: t2
t3
t3
e−tH = e−tB+ 2 [∆,B]− 6 [∆,[∆,B]]− 12 [B,[∆,B]]+··· e−tH0 .
(3.8)
We now expand the first exponential up to factors which, after taking the trace, give terms of order less or equal to zero in t. Only a few terms will be important: We have first to take into account that (in n dimensions) Tr(Lf ∆k e−t∆ ) t−n/2−k , t → 0, Tr(Rf ∆k e−t∆ ) t−n/2−k , t → 0.
(3.9)
Indeed, for the “left” case (the right one being similar) since, as proved in [16], one has Lf (1 + ∆)−k ∈ Lp (H) for any p > n/2k and any f ∈ Cc∞ (M ), we conclude for all > 0:
Lf ∆k e−t∆ 1 ≤ Lf (1 + ∆)−n/2− 1
∆k
(1 + ∆)n/2+k+ e−t∆ (1 + ∆)k
≤ C()t−(n/2+k+) . The last estimate follows from functional calculus. Therefore, in the field expansion we need to correct the power in t by the order of the differential operator appearing when we expand the first exponential in the equation (3.8).
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Secondly, we have to notice that the commutators [∆, Lf ], [∆, Rf ] (and also [∆, Rf Lh ]) reduce by one the order of the differential operator (cf. equation (3.10) below). To see this, we compute the commutators [∆, Lf ], [∆, Rf ] and [∆, Rf Lh ]. The simplest way is to use the formulas (2.7) and (2.8). By [Vz , ∆] = 0 for all z ∈ Rl (from the isometry property of α) and choosing a local coordinate system {xµ }, one obtains −l
[∆, Lf ] = (2π)
−l
R2l
dl y dl z e−i V 1 Θy [∆, Mf ] V− 1 Θy−z 2
l
= (2π)
l
d yd z e
−i
R2l
2
V 1 Θy (M∆f − 2M∇µ f ∇µ ) V− 1 Θy−z 2
2
= L∆f − 2L∇µ f ∇µ ,
(3.10)
and similarly, [∆, Rf ] = R∆f − 2R∇µ f ∇µ ,
(3.11)
[∆, Rf Lh ] = Rf [∆, Lh ] + [∆, Rf ]Lh
= Rf L∆(h) + R∆(f ) Lh − 2R∇µ f L∇µ h − 2 Rf L∇µ h + R∇µ f Lh ∇µ . (3.12)
The local coordinate system used must be compatible with the deformation, that is, defined on some α-invariant open neighborhood U ⊂ M . To obtain one such, I }i∈I by letting Rl act on choose any open covering {UI }i∈I of M and define {U l it: Ui := R .Ui . This implies that in n dimensions, one only needs to use the BCH formula up to order n − 2 to capture the divergent structure of the effective action. Moreover, that the commutators decrease the degree of the differential operator is a necessary condition to make the BCH expansion meaningfull: In [15], we consider a field theory on a noncommutative 4-plane with an (associative) position-dependant Moyal product (coming from a rank-2 Poisson structure on R4 ). It turns out that the commutators [∆, Lf ] and [∆, Rf ] contain now a term with an order two differential operator. This makes the BCH development useless since the k-times iterated commutator [t∆, [· · · , [t∆, tLf ] · · · ]] contains a term which gives after the exponential expansion a contribution of order t−n/2+1 , independently of k, the number of commutators involved. Thus, in this case the whole BCH serie will be needed to capture the divergences. Putting all together, we finally obtain: e−tH =
t3 t2 t2 1 − tB + [∆, B] − [∆, [∆, B]] + B 2 e−tH0 + O(t); 2 6 2
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we mean by this estimate that we have a small-t expansion:
Tr e−tH − e−tH0 =
t2 t3 t2 2 −tH0 − tB + [∆, B] − [∆, [∆, B]] + B e Tr + O(t). 2 6 2
(3.13)
We now show that in fact, the commutators in the expression (3.13) give no contribution to the effective action. Indeed, if each terms C∆e−t∆ and ∆Ce−t∆ are trace-class, with C = B or C = [∆, B], then by the cyclicity of the trace and the fact that the Laplacian commutes with the heat semigroup, one gets
Tr ∆ C e−t∆ − C ∆ e−t∆ = Tr C ∆ e−t∆ − C ∆ e−t∆ = 0 (3.14) That C∆e−t∆ is trace-class is obvious from functional calculus and using the same arguments than those used to obtain the estimate (3.9). For ∆Ce−t∆ , it is a little bit less immediate since the latter appears as a product of a trace-class operator (Ce−t∆ ) times an unbounded one (∆). Actually, using the tautological relation ∆ C e−t∆ = C ∆ e−t∆ + [∆, C] e−t∆ , and the equations (3.10) and (3.11) (iterated once more when C = [∆, B]), one sees that this term appears also as a sum of trace-class operators. Hence (3.14) is proved and we are left with
2 λ Tr e−tH − e−tH0 = − t Tr LϕΘ ϕ + RϕΘ ϕ + Rϕ Lϕ e−t(∆+m ) 3!
t2 λ2 Tr LϕΘ4 + RϕΘ 4 + 3RϕΘ ϕ LϕΘ ϕ + 2 (3!)2 2 + 2Rϕ LϕΘ3 + 2RϕΘ 3 Lϕ e−t(∆+m ) + O(t).
3.3
Planar and non-planar contributions
We split the previous expansion in two parts. In the first one, we only keep terms like Lf e−t∆ and Rf e−t∆ . Those belong to the “planar part”, since they give commutative-like contributions as easily seen from equation (3.15) below. The second contribution, corresponding to the “non-planar part”, consists of crossed terms like Lf Rh e−t∆ . The planar contribution to the effective action is λ
2 1 ∞ Tr LϕΘ ϕ + RϕΘ ϕ e−t∆ Γ1l,P [ϕ] := dt e−tm 2 3!
t λ2 + O(0 ). − Tr LϕΘ4 + RϕΘ 4 e−t∆ 2 2 (3!)
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To compute those traces, let us show that first the trace is a dequantizer for the deformed product
(3.15) Tr Lf e−t∆ = Tr Rf e−t∆ = Tr Mf e−t∆ , whenever Mf e−t∆ is trace-class. Here Mf still denotes the operator of pointwise multiplication by f . We only treat the Lf case, since for the Rf case the arguments are similar. From the definition 2.1 and the product rule (3.4) for kernel operators, a little calculation gives the following expression for the Schwartz kernel of Lf e−t∆ : −l dl y dl z e−i f (− 21 Θy.p) Kt (z.p, p ). KLf e−t∆ (p, p ) = (2π) R2l
Then
Tr Lf e−t∆ =
µg (p) KLf e−t∆ (p, p) = (2π)−l µg (p) dl y dl z e−i f (− 21 Θy.p) Kt (z.p, p). M
M
R2l
Using next the invariance of the volume form under the isometry p → 12 Θy.p and the fact that [e−t∆ , Vz ] = 0, translated in terms of invariance of its kernel Kt (z.p, z.p ) = Kt (p, p ),
(3.16)
the claim follows after a plane waves integration:
−t∆ −l = (2π) Tr Lf e µg (p) dl y dl z e−i f (p) Kt (z.p, p) M R2l = µg (p) f (p) dl z δ(z) Kt (z.p, p) l R M
= µg (p) f (p) Kt (p, p) = Tr Mf e−t∆ . M
Hence, the planar part of the one loop effective action reads: ∞ λ t λ2 2 −t∆ Θ 4 e Tr MϕΘ ϕ e−t∆ − Γ1l,P [ϕ] = dt e−tm Tr M + O(0 ). ϕ 3! 2 (3!)2 Using the on-diagonal heat kernel expansion up to order one
t Kt (x, x) = (4πt)−2 1 − R(x) + O(t0 ), 6 where R is the scalar curvature, together with the relation KMf e−t∆ (x, x) = f (x)Kt (x, x),
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one obtains at 0 order: ∞ 2 λ 1 λ dt e−tm 1 λ2 Θ 4 . Γ1l,P [ϕ] = ϕΘ ϕ − t (ϕΘ ϕ)R + µg ϕ 2 (4πt) 3! 6 3! 2 (3!)2 M (3.17) The planar part thus yields ordinary 1 and | ln | divergences. They can be substracted adding local counter-terms to the original action. The contribution for the non-planar part is λ 2 1 ∞ Γ1l,N P [ϕ] := Tr Rϕ Lϕ e−t∆ dt e−tm 2 3! 2
t λ − + O(0 ). Tr 3RϕΘ ϕ LϕΘ ϕ + 2Rϕ LϕΘ3 + 2RϕΘ 3 Lϕ e−t∆ 2 2 (3!) We now simplify this expression. By the definition of the twisted product (2.1) g and using the identity ψ(z.p) = M µg (p ) δz.p (p ) ψ(p ), one can easily derive the Schwartz kernel of the left and right twisted multiplication operators: −l g KLf (p, p ) = (2π) dl y dl z e−i f (− 21 Θy.p) δz.p (p ), R2l
and
−l
KRf (p, p ) = (2π)
dl y dl z e−i f (z.p) δ g
(p 1 − 2 Θy.p
R2l
).
By the kernel composition rule (3.4), we obtain after few changes of variables and a plane waves integration, the kernel of Lf Rh e−t∆ in term of the heat kernel Kt : KLf Rh e−t∆ (p, p ) = dl y dl z e−i f ((− 12 Θy − z).p) h(z.p) Kt(− 12 Θy.p, p ). (2π)−l R2l
Hence, the trace of Lf Rh e−t∆ reads (with a few changes of variable):
Tr Lf Rh e−t∆ = (2π)−l µg (p) dl y dl z e−i f (p) h(z.p) Kt (−Θy.p, p) 2l M R
(3.18) = Tr Rf Lh e−t∆ . To obtain the last equality, we used the fact that Kt is symmetric, its invariance under α and the isometry p → −z.p. Invoking formula (3.18), we obtain for Γ1l,N P [ϕ]: Γ1l,N P [ϕ] = (2π)−l −
1 2
∞
dt e−tm
2
µg (p) M
R2l
dl y dl z e−i
λ 3!
ϕ(p)ϕ(z.p)
t λ2 3ϕΘ ϕ(p)ϕΘ ϕ(z.p) + 4ϕ(p)ϕΘ 3 (z.p) Kt (−Θy.p, p) + O(0 ) . 2 2 (3!)
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We shall see that the better -behavior of the non-planar part and the UR/IV entanglement phenomenon come from the presence of the off-diagonal heat kernel in the previous expression. Depending on the precise geometric setup, the non-planar contributions could still be divergent. In the unfavorable cases, the divergences are non-local as shown is the next subsections. This makes the renormalization problematic.
4 Non-periodic deformations 4.1
NCQFT on the Moyal plane in configuration space
When M = R4 with the flat metric, l = 4 and R4 acting on itself by translation, isospectral deformation gives R4Θ . In this case, the heat kernel is exactly given by Kt (x, y) = (4πt)−2 e−
|x−y|2 4t
,
so we can explicitly compute Γ1l,P (ϕ) and Γ1l,N P (ϕ). For the planar part, we obtain from (3.17) ∞ 2 λ e−tm t λ2 4 2 2 ϕ dt d x (x) − (ϕ ϕ) (x) + O(0 ), Γ1l,P [ϕ] = Θ (4πt)2 R4 3! 2 (3!)2 that will give the ordinary −1 and | ln | divergences for the respectively planar two- and four-point functions. The non-planar part is given by: Γ1l,N P [ϕ] = (2π)−4
2
∞
dt
e−tm (4πt)2
d4 x d4 y d4 z e−i e−
|Θy|2 4t
R12
1 λ λ2 t
Θ 3 × ϕ(x)ϕ ϕ(x+z)+4ϕ(x)ϕ (x+z) +O(0 ). ϕ(x)ϕ(x+z)− 3ϕ Θ Θ 2 3! (3!)2 4 The Gaussian y-integration can be performed to obtain: ∞ −1 2 −4 −tm2 Γ1l,N P [ϕ] = (2πθ) dt e d4 x d4 z e−t|Θ (z−x)| R8
1 λ λ2 t
ϕ(x)ϕ(z) − × 3ϕΘ ϕ(x)ϕΘ ϕ(z) + 4ϕ(x)ϕΘ 3 (z) + O(0 ), 2 2 3! (3!) 4 where θ := (det Θ)1/4 . Finally, the t-integration gives
2
−1
2
e−(m +|Θ (z−x)| ) m2 + |Θ−1 (z − x)|2 R8 λ λ2 3ϕΘ ϕ(x)ϕΘ ϕ(z) + 4ϕ(x)ϕΘ 3 (z) ϕ(x)ϕ(z) − + O(0 ). × 2.3! (3!)2 4(m2 + |Θ−1 (z − x)|2 )
Γ1l,N P [ϕ] = (2πθ)−4
d4 x d4 z
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This expression is regular when goes to zero – we are now in the full noncommutative picture. From the previous formula one reads off the associated (non-planar) two- and four-point functions in configuration space in the limit → 0: 1 λ , 96 m2 + |Θ−1 (x − y)|2 −1 λ2 3 e2i δ(x−y+z−u) d4 v 2 G41l,N P (x, y, z, u) = −(πθ)−8 24 2 (m + |Θ−1 (z − v − x)|2 )2 −1 e2i<x−y,Θ (z−y)> . + (m2 + |Θ−1 (x − y + z − u)|2 )2 G21l,N P (x, y) = (πθ)−4
We see that the UV/IR mixing in configuration space manifests itself in the long-range behavior of the correlation functions. The slow decreasing at infinity of the two- and four-point functions is equivalent to a IR singularity in momentum space, as shown by a Fourier transform: 2 1l,N P (ξ, η) ∝ m K1 (m|Θξ|)δ(ξ + η). G θ|ξ| Here Kn (z) denotes the n-th modified Bessel function. We retrieve the known UV/IR mixing (se for example [29]): m K1 (m|Θξ|) ∼ (θ|ξ|)−2 , |ξ| → 0. θ|ξ| This last result at one loop in the Moyal (translation-invariant) context is usually obtained by means of Feynman diagrams in momentum space – see for example [29]. We just checked that the Fourier transform for the two-point function coincides with the standard calculation’s result. However, this is not the end of the story. The behavior of the amplitudes as θ ↓ 0 presents interesting differences in configuration and momentum spaces. Assume that Θ has been put in the canonical form θ −θ Θ= , θ −θ and choose θ = θ for simplicity. In effect, developing the two-point expression in terms of θ, we find 1 θ 4 m4 θ 2 m2 1 = + − · · · . 1 − θ4 m2 + θ2 |x|2 θ2 |x|2 |x|2 |x|4 First of all, we remark that the logarithmic dependence on θ of the UV/IR mixing in momentum space (in addition to its quadratic divergence) found in [29] is apparently absent here. Now, with the sole exception of the first term, the previous
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series is made of functions that are not tempered distributions, and so they have no Fourier transform. In other words, the passage to the “commutative limit” does not commute with taking Fourier transforms. The question is subtler, though. We can ask ourselves to which kind of divergences the terms of the last development are associated to. The answer is that first term is infrared divergent in configuration space; the second one is both ultraviolet and infrared divergent, and the following are all ultraviolet divergent. It is perhaps surprising that there is a way to recover the exact result from that nearly nonsensical infinite series; this involves precisely the correction to the indicated UV divergences. Indeed we can “renormalize” (in the sense of Epstein and Glaser) the 1/|x|2k+4 functions, with the result that the redefined distributions [1/|x|2k+4 ]R are tempered. Those [.]R distributions depend on a mass scale parameter. Their 2k+4 ] (making a long history short) have been calculated Fourier transforms [1/|x| R as well [18, 30], with the result (−)k+1 |ξ|2k |ξ| 2k+4 − Ψ(k + 1) − Ψ(k + 2) , ]R (ξ) = k+1 [1/|x| 2 ln 4 k!(k + 1)! 2µ Now, a natural mass scale parameter in our context is 1/θm. This is where ln θ can sneak back in. Upon substituting this for µ in the previous formula, and summing the series of Fourier transforms, we recover on the nose the exact result: ∞ 1 m m2 θ2n m2n |ξ|2n θm|ξ| − Ψ(n + 1) − Ψ(n + 2) K1 (θm|ξ|). = + ln θ2 |ξ|2 2 n=0 4n n!(n + 1)! 2 θ|ξ| For the four-point function, again in the θ ↓ 0 limit no dependence on ln(θ) is apparent in configuration space. The resulting expression is however (UV- and) IR-divergent, and its redefinition ` a la Epstein and Glaser allows one to reintroduce the ln θ. The effect of the rank of Θ becomes clearer in position space. Indeed, for a generic n-dimensional Moyal plane with a deformation matrix of rank l ≤ n, the two-point function in momentum space is always finite and behaves as |Θξ|−n+2 , when ξ → 0. However, since Θξ ∈ Im(Θ) = Rl , the IR singularity is not locally integrable if l ≤ n − 2. It follows that the two-point Green function does not have a Fourier transform since it is not a temperate distribution. Thus in the fourdimensional case, the non-planar contribution to the tadpole in position space remains infinite if l = 2! The four-point function has a Fourier transform, its IR singularity in momentum space being of the ln type, and the Green function in position space is finite whenever l = 0. For example, had we treated R2θ ×R2 instead of R4Θ , we would have found that the four-point part of Γ1l,N P [ϕ] is convergent, while the two-point part diverges as ln . This point is discussed in details in [14], where we use the ζ-regularization scheme and the Duhamel asymptotic expansion (instead of the BCH one), in order to compare our results with those present in the literature.
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These features of the UV/IR mixing phenomenon on position space reappear in the general non-periodic case, where the effective action will still be divergent for l = 2. This is shown in the next subsection.
4.2
The divergences of the general non-periodic case
Assume ϕ ∈ Cc∞ (M ). We have also to make some more precise assumptions on the behavior of the geometry at infinity in order to control the heat kernel. In [2, 9], it is proved that if M is non-compact, complete, with Ricci curvature bounded from below (plus either uniform boundness of the inverse of the volume or of the inverse of the isoperimetric constant of the Riemannian ball for some fixed radius), then the heat kernel satisfies 2
(4πt)−2 e−dg (p,p )/4t ≤ Kt (p, p ) 2
≤ C(4πt)−2 e−dg (p,p )/4(1+c)t ,
(4.1)
where dg is the Riemannian distance and C, c are strictly positive constants. In the general periodic case, we have shown that Γ1l,N P [ϕ] is given by: Γ1l,N P [ϕ] ×
λ 3!
1 = 2(2π)l
ϕ(p)ϕ(z.p) −
∞
dt e
−tm2
µg (p) M
R2l
dl y dl z e−i Kt (−Θy.p, p)
t λ2 Θ 3 3ϕ ϕ(p)ϕ ϕ(z.p) + 4ϕ(p)ϕ (z.p) + O(0 ). Θ Θ 2 (3!)2
We now show that this expression cannot produce more important divergences than the planar contribution. Again, the regularity of those integrals depends only on l (that we may call the effective noncommutative dimension), and on the metric through the Riemannian distance function. Before estimating the two-point part of Γ1l,N P [ϕ], which is our main purpose in this section, we make the following remark: in our present setting, the two-point non-planar Green function reads ∞ 2 λ l l −i g G1l,N P,2P (p, p ) = d y d z e dt e−tm Kt (−Θy.p, p) δz.p (p ). 6(2π)l R2l Now, one can qualitatively see in this distributional expression the UV/IR entanglement phenomenon: thanks to the estimate (4.1), we have 0
∞
2
2
∞
e−tm −d2g (Θy.p,p)/4(1+c)t e (4πt)2 0 √
m dg (Θy.p, p) C 4m 1 + c √ K1 = 2 16π dg (Θy.p, p) 1+c −2 ∼ C dg (Θy.p, p), y → 0,
dt e−tm Kt (Θy.p, p) ≤ C
dt
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and the reverse inequality also holds ∞ 2 dt e−tm Kt (Θy.p, p) ≥ C d−2 g (Θy.p, p) 0
l has to be interpreted as which points precisely to the UV/IR mixing, since y ∈ R a momentum. For the two-point part of Γ1l,N P [ϕ] we have l Γ1l,N P,2P [ϕ] ≤ C λ sup d z |ϕ(z.p)| 12(2π)l p∈M Rl ∞ 2 2 e−tm × dt µ (p) |ϕ(p)| dl y e−dg (−Θy.p,p)/4(1+c)t g 2 (4πt) M Rl Cλ ≤ sup dl z |ϕ(z.p)| ϕ 1 12(2π)l p∈M Rl 2 ∞ 2 e−tm × sup dt dl y e−dg (−Θy.p,p)/4(1+c)t . 2 (4πt) Rl p∈supp(ϕ) By the properness of α, Rl dl z |ϕ(z.p)| is finite for all p ∈ M since {z ∈ Rl : z.p ∈ supp(ϕ)} is compact for each p ∈ M because ϕ has compact support. Thus, ϕ(p) ˜ := Rl dl z |ϕ(z.p)| is constant and finite on each orbit of α, and if we denote π : M → M/Rl the projection on the orbit space, then ϕ˜ factors through π to give a map ϕ¯ defined by ϕ(π(p)) ¯ := ϕ(p). ˜ Finally, ϕ¯ ∈ Cc∞ (M/Rl ) l because if p ∈ / R . supp(ϕ), so that π(p) is! not in the compact set π(supp(ϕ)), " then ϕ(π(p)) ¯ = 0. This proves that supp∈M Rl dl z |ϕ(z.p)| < ∞. Furthermore, since α acts isometrically the induced metric g˜ on the orbits (which are closed submanifolds since the action is proper [25]) is constant, so d2g (y.p, p) =
l
g˜ij (p)y i y j .
i,j=1
Here, g˜ij (p) (which depend only on the the orbit of p) are strictly positive continuous functions since in the non-periodic case the action is free, and then {(0, p) ∈ Rl × M } is the only set for which F (y, p) := dg (y.p, p) vanish. Note that we can use a global coordinate system (on one orbit) given by a suitable basis of Rl in such a way that g˜ij (p) is diagonal. Thus, with θ := (det Θ)1/l , we have: l/2 4π(1 + c)t l −d2g (−Θy.p,p)/4(1+c)t d ye = (det g˜(p))−1/2 . θ2 Rl Hence, one obtains Γ1l,N P,2P [ϕ] ≤ λ C(l, g˜, ϕ, ϕ) θ−l 6
∞
2
dt tl/2−2 e−tm ,
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where C(l, g˜, ϕ1 , ϕ2 ) := C(4π)l/2−2 (1 + c)l/2
ϕ1 1 sup 2(2π)l p∈M
d z |ϕ2 (z.p)| l
Rl
sup
(det g˜(p))−1/2 .
p∈supp(ϕ1 )
Four the four-point part, similar estimates read: Γ
≤
1l,N P,4P [ϕ] λ2 −l
72
θ
3C(l, g˜, ϕΘ ϕ, ϕΘ ϕ) + 4C(l, g˜, ϕ, ϕΘ ϕΘ ϕ)
∞
2
dt tl/2−1 e−tm .
We then have proved the following: Theorem 4.1. When M is non-compact, satisfying all assumptions on the behavior of the geometry at infinity displayed above and endowed with a smooth proper isometric action of Rl , then for ϕ ∈ Cc∞ (M ) we have: for l = 4, Γ1l,N P,2P [ϕ] ≤ C1 (ϕ, Θ) i) C2 (ϕ, Θ)| ln | for l = 2, ii)
Γ1l,N P,4P [ϕ] ≤ C3 (ϕ, Θ) for l = 4 or l = 2.
The possible remaining divergence for l = 2 refers to the fact that the IR singularity might be not integrable, as illustrated previously. In this case, the two-point non-planar Green function does not define a distribution and the theory is not renormalizable by addition of local counter-terms, already in its one-loop approximation order.
5 Periodic deformations Periodic deformations (when the kernel of α is an integer lattice) behave rather differently from non-periodic ones. In the following, we consider ker α = βZl with β a l × l integer matrix of rank l, so that Rl /βZl =: Tlβ is compact. For the sake of simplicity, we will often suppress the subscript β. Momentum space (the dual group of Tlβ ) being discrete, IR problems only occur for some values of the momentum. In favorable cases one can extract the divergent field configurations in the non-planar part (which are often finite in number when (2π)−1 Θ has irrational entries) and renormalize them like the planar contributions; then there is no really UV/IR mixing. When (2π)−1 Θ has rational entries, the theory is equivalent to the undeformed one, in the sense that there are infinitely many divergent field configurations.
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Although in all periodic cases we have a Peter-Weyl decomposition for fields, only in the compact manifold case shall we be able to describe the individual behavior of non-planar “Feynman graphs”, defined through that isotypic decomposition. Both in the compact and in the non-compact case, by means of the off-diagonal heat kernel estimate (4.1), we show in the second subsection how, for periodic deformations, the arithmetical nature of the entries of Θ, more precisely, the existence or nonexistence of a Diophantine condition on Θ, plays a role in determining the analytical nature of Γ1l,N P [ϕ].
5.1
Periodic compact case and the individual behavior of non-planar graphs
Because everything is explicit, we look first at the flat compact case. Let M = T4 with the flat metric, let R4 act on it by rotation (so and l = 4 and we are in i the ‘fully noncommutative picture’). With the orthonormal basis e (2π)2 of L2 (T4 , d4 x) the heat kernel is written 2 Kt (x, y) = (2π)−4 e−t|k| ei ,
k∈Z4
k∈Z4
and we have
i
ei Θ ei = e− 2 Θ(k,q) ei ,
with Θ(k, q) := k, Θq. Expanding the background field ϕ in Fourier modes ϕ = i , with {ck }k∈Z4 ∈ S(Z4 ) whenever ϕ ∈ C ∞ (T4 ), we obtain: k∈Z4 ck e 1 1 e−(m +|k| ) λ λ2 iΘ(k,r) c c e − r −r 2 2 2 2 2 m + |k| 3! r 2(3!) m + |k|2 k i + O(0 ). × cr cs cu−s c−r−u e− 2 Θ(r+s,u) 3 eiΘ(k,r+s) + 4 eiΘ(k,r+u) 2
2
ΓN P [ϕ] =
r,s,u
We can now analyze the individual behavior of non-planar Feynman diagrams. One sees that, thanks to the phase factors, the sum over k is finite when goes to zero, whenever (2π)−1 Θ has irrational entries and r = 0 for the two-point part, or r +s = 0 and r +u = 0 for the four-point part. In effect, returning to the Schwinger parametrization (which exchanges large momentum divergences with small-t ones) and applying the Poisson summation formula with respect to the sum over k we get: ∞ e−tm2 eiΘ(k,r) 2 = dt e−|2πk−Θr| /4t . 2 2 2 m + |k| (4πt) 0 4 4 k∈Z
k∈Z
Hence, the t-integral is finite whenever r = 0 and conclusion holds for the four-point part.
Θr 2π
∈ / Ql . Essentially the same
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We now go to the general periodic compact case. In order to be able to calculate, we make explicit use of the invariance of the heat kernel under α. Let us decompose H = L2 (M, µg ) in spectral subspaces with respect to the group action: # H= Hk . k∈Zl
Each Hk is stable under Vz (recall that Vz denotes the induced action on H) for all z ∈ Rl ; and furthermore all ψ ∈ Hk satisfy Vz ψ = e−i ψ. Note that if ψ ∈ Hk then |ψ| ∈ H0 . Let Pk be the orthogonal projection on Hk . Because the Laplacian −t∆ commutes with Vz , the heat operator also commutes with is block $ Pk ; hence e diagonalizable with respect to the decomposition H = k∈Zl Hk : Pk e−t∆ Pk . e−t∆ = k∈Zl
In each Hk the operator 0 ≤ Pk e−t∆ Pk is trace-class, so it can be written as Pk e−t∆ Pk = e−tλk,n |ψk,n ψk,n |, n∈N
where {ψk,n }n∈N is an orthonormal basis of Hk consisting of eigenvectors of Pk ∆Pk with eigenvalue λk,n . The heat semigroup being Hilbert-Schmidt, its kernel can be written as a (L2 (M × M, µg × µg )-convergent) sum: Kt (p, p ) = e−tλk,n ψk,n (p)ψk,n (p ). (5.1) k∈Zl n∈N
Because each ψk,n (p) lies in Hk , the invariance property (3.16) Kt (z.p, z.p) = Kt (p, p ) is explicit. Any ϕ ∈ C ∞ (M ) has a Fourier decomposition ϕ = r∈Zl ϕr , such that { ϕr ∞ } ∈ S(Zl ) and αz (ϕr ) = e−i ϕr . Furthermore, this decomposition provides a notion of Feynman diagrams, that is of amplitude associated with a fixed field configuration. The non-planar one-loop regularized effective action reads: Γ1l,N P [ϕ] = r,s∈Zl
1 2
µg (p) M
ϕr (p) ϕs (p) e
λ e−(m2 +λk,n ) 2 |ψ | (p) k,n m2 + λk,n 3! l
k∈Z n∈N λ2 −iΘ(k,s)
−
2(3!)2
m2
1 + λk,n
ϕr (p) ϕs (p) ϕu (p) ϕv (p)
r,s,u,v∈Zl
i i + O(0 ). × 3 e− 2 (Θ(r,s)+Θ(u,v)) e−iΘ(k,u+v) + 4 e− 2 Θ(r+s,u+s) e−iΘ(k,v)
Although we do not know the explicit form of the ψk,n , we can by momentum conservation reduce the sums exactly as in the NC torus case, as shown in the following lemma.
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Lemma 5.1 (Momentum conservation). Let ψi ∈ Hki ∩ Lq (M, µg ) for i = 1, . . . , q. Then: µg ψ1 . . . ψq = C(ψ1 , . . . , ψq ) δk1 +···+kq ,0 . M
Proof. By the α-invariance of µg and with the relation αz (ψi ) = e−i ψi we have µg ψ1 . . . ψq = ei µg ψ1 . . . ψq , M
M
for all z ∈ Rl ; the result follows. Because |ψk,n |2 (p) is constant on the orbits of α and ϕr ∈ C ∞ (M ) ⊂ L (M, µg ) for all q ≥ 1, Lemma 5.1 gives q
Γ1l,N P [ϕ]
1 = 2
λ e−(m2 +λk,n ) 2 µg (p) |ψ | (p) k,n m2 + λk,n 3! M l l k∈Z n∈N 2
r∈Z
1 λ ϕr (p) ϕ−r (p) ei − ϕr (p) ϕs (p) ϕu−s (p) ϕ−r−u (p) 2 2 2(3!) m + λk,n r,s,u∈Zl i + O(0 ). (5.2) × e− 2 Θ(r+s,u) 3 eiΘ(k,r+s) + 4 eiΘ(k,r+u) To analyze the divergences when → 0 for a fixed field configuration, note that if we re-index λk,n in a standard way (λ0 ≤ · · · ≤ λn ≤ · · · ), Weyl’s estimate asserts that λn ∼ n1/2 , hence k∈Zl n∈N
|ψn (p)|2 |ψk,n |2 (p) = = K(m2 +∆)−N (p, p), (m2 + λk,n )N (m2 + λn )N n∈N
is finite if and only if N > 2. We see that the sum over n and k in (5.2) diverges in the limit → 0 for certain values of the momenta (r = 0 for the two-point part, r + s = 0 and r + u = 0 for the four-point part) if (2π)−1 Θ has irrational entries. When the entries (2π)−1 Θ are rational, there are infinitely many divergent field l configurations since e−i = 1 for infinitely many k whenever Θr 2π ∈ Q . For other configurations, convergence is guaranteed by the estimate (4.1), as shown in the next subsection. In summary, we have shown that the behavior of an individual field configuration in the non-planar sector for any periodic compact deformation reproduces the main features of the noncommutative torus. In the next paragraph, the arithmetic nature of the entries of Θ gets into the act; also we show there that the possible existence of fixed points for the action may give rise to additional divergences.
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General periodic case and the Diophantine condition
Assume now that α periodic, but M can be compact or not (within the hypothesis of section 4.2 when M is not compact). In this general setup, the Peter-Weyl decomposition still exists, but the heat operator, not being a priori compact, cannot be written as (5.1). Thus we return to the off-diagonal heat kernel estimate. In this case, using Lemma 5.1 and the α-invariance of Kt , we obtain: λ 1 ∞ −tm2 Γ1l,N P [ϕ] = dt e µg (p) Kt (Θr.p, p)ϕr (p) ϕ−r (p) 2 3! M l r∈Z
t λ2 i − ϕr (p) ϕs (p) ϕu−s (p) ϕ−r−u (p) e− 2 Θ(r+s,u) 2(3!)2 r,s,u∈Zl × 3 Kt (Θ(r + s).p, p) + 4 Kt (Θ(r + u).p, p) + O(0 ). We consider only the case (2π)−1 Θ has irrational entries, from now on. Then divergences appear when r = 0 for the two-point function and r + s = 0, r + u = 0 for the four-point functions. This leads us to introduce a reduced non-planar oneloop effective action Γ,red 1l,N P [ϕ] by subtracting the divergent field configurations; for renormalization purposes, they have to be treated together with the planar sector. λ 1 ∞ −tm2 Γ,red [ϕ] := dt e µ (p) Kt (Θr.p, p)ϕr (p) ϕ−r (p) g 1l,N P 2 3! M t λ2 i − ϕr (p) ϕs (p) ϕu−s (p) ϕ−r−u (p)e− 2 Θ(r+s,u) 2(3!)2 × 3 Kt (Θ(r + s).p, p) + 4 Kt (Θ(r + u).p, p) .
is the notation for r∈Zl , r=0 in the two-point part, r,s,u∈Zl , r+s=0 and r,s,u∈Zl , r+u=0 in respectively the first and second piece of the four-point part. Using now the estimate (4.1) and performing the t-integration, we obtain: √ λ C 4m 1 + c ≤ lim Γ,red [ϕ] µ (p) |ϕ (p)| |ϕ (p)| g r −r 1l,N P →0 32π 2 M 3! dg (Θr.p, p) 2
m dg (Θr.p, p) λ √ K1 |ϕr (p)| |ϕs (p)| |ϕu−s (p)| |ϕ−r−u (p)| + 2(3!)2 1+c
m dg (Θ(r + s).p, p)
m dg (Θ(r + u).p, p) √ √ + 4K0 . (5.3) × 3K0 1+c 1+c Here
Definition 5.2. θ ∈ Rl \ Ql satisfies a Diophantine condition if there exists C > 0, β ≥ 0 such that for all n ∈ Zl\{0} :
nθ Tl := inf |nθ + k| ≥ C|n|−(l+β) . k∈Zl
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Diophantine conditions constitute a way to characterize and classify irrational numbers which are “far from the rationals” in the sense of being badly approximated by rationals. The set of numbers satisfying a Diophantine condition is ‘big’ (of full Lebesgue measure) in the sense of measure theory, but ‘small’ (of first category) in the sense of category theory [27]. Again because the metric is constant on the orbits we have: l g˜ij (p)(y i + k i )(y j + k j ) . d2g (y.p, p) = inf k∈Zl
i,j=1
Recall also that the modified Bessel functions have the following behavior near the origin K1 (x) =
1 + O(x0 ), x
K0 (x) = −γ + ln(2) − ln(x) + O(x),
where γ is the Euler constant. Thus, in view of { ϕr ∞ } ∈ S(Zl ), and provided the integral over the manifold with the measure µg can be carried out, in (5.3) we have l convergence if and only if d−2 g (Θr.p, p) ∈ S (Z ), that is, if and only if the entries of Θ satisfy a Diophantine condition. This result seems to be new, although the pertinence of Diophantine conditions in NCQFT had been conjectured by Connes long ago. Recently, these conditions have been found to play a role in Melvin models with irrational twist parameter in conformal field theory [22]. We said above: “provided the integral over the manifold with the measure l µg can be carried out”. This because d−2 g (αy (.), .) for a non-zero y ∈ T might not be locally integrable with respect to the measure given by the Riemannian volume form. Problems may appear on a neighborhood of the set of points with non-trivial isotropy groups. In fact, by simple dimensional analysis, we expect serious trouble when the isotropy group is one dimensional. For p ∈ M let Hp its isotropy group and let Msing := {p ∈ M : Hp = {0}}. Recall that Msing is closed and of zero-measure in M since the action is proper (see [25]), and note that for a non-zero y ∈ Tl , dg (y.p, p) = 0 if and only if p ∈ Msing and y ∈ Hp . On Mreg := M \ Msing (the set of principal orbit type), since the action is free, one can define normal coordinates on a tubular neighborhood of an orbit Tl .p. Let (ˆ xµ , x ˜i ), µ = 1, . . . , n − l, i = 1, . . . , l be respectively the transverse and the torus coordinates of a point p ∈ Mreg . Because the action is isometric, in this coordinate system the metric takes the form h(ˆ x) l(ˆ x) g(ˆ x, x ˜) = , l(ˆ x) g˜(ˆ x) where g˜ is the induced (constant) metric on the orbit. Such coordinate system is xµ , x˜i ) singular with singularities located at each point of Msing , and when x ≡ (ˆ approach p0 ∈ Msing , g˜(ˆ x) collapses to a l − dim(Hp0 ) rank matrix. Since in this
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coordinate system µg (p) d−2 g (y.p, p) equals l
% det g(ˆ x)
˜ij (ˆ x)y i y j i,j=1 g
dl x ˜ dn−l x ˆ,
when dim(Hp0 ) = 1 the singularity of d−2 g (y.p, p) for p → p0 cannot be can√ celled by det g. This is a new feature of the UV/IR mixing for generic periodic isospectral deformations which needs to be investigated in detail in each model; it occurs, for instance, for the Connes-Landi spheres and their ambient spaces. Let us summarize: Theorem 5.3. For M compact or not (within the assumptions displayed in section 4.2 in the non-compact case), endowed with a smooth isometric action of the compact group Tl , l = 2 or l = 4 and with a deformation matrix whose entries satisfy a Diophantine condition, then for any external field ϕ ∈ Cc∞ (M, ) vanishing in a neighborhood Msing the one-loop non-planar reduced effective action is finite. In other words, if the Diophantine condition is not satisfied or if d−2 g (αy (.), .) ∈ / then the reduced non-planar two-point function does not define a distribution and the theory is not renormalizable, already at one-loop, by addition of local counter-terms. L1loc (M, µg )
6 Summary and perspectives We have shown the existence of the UV/IR mixing for isospectral deformations of curved spaces. For periodic deformations the entanglement only concerns (at the level of the two-point function) the 0-th component of the field in the spectral subspace decomposition induced by the torus action. In this case, the UV/IR mixing does not generate much trouble since one can treat it for renormalization purposes together with the planar sector. In the non-periodic situation, we obtain non-planar Green functions which present the mixing in a similar form to the Moyal plane paradigms. Our approach gives an algebraic way to understand the presence of the nonplanar sector for those theories: it comes from the product of left and right regular representation operators. As a byproduct of our trace computations, we obtain that the better behavior of the non-planar sector is due to the presence of the off-diagonal heat kernel in the integrals. However, its regularizing character depends highly on the geometric data. For non-periodic deformations, the conclusion is that when the noncommutative rank is equal to two, the non-planar 1PI two-point Green function does not define a distribution and the associated effective action remains divergent [14]. Only the group action of rank four gives rise to a UV divergent-free non-planar sector in the 4-dimensional manifold case. When the action is periodic, we have shown
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that it is necessary that the entries of (2π)−1 Θ satisfy a Diophantine condition to ensure finiteness of the reduced non-planar effective action, i.e., in order that the reduced non-planar 1PI two-point Green function define a distribution. Additional divergences may exist due to the possible fixed points structure of the action α. Our treatment of the generic UV/IR behavior, can be generalized to higher dimensional isospectral deformations and/or to gauge theories. Also, we have restricted ourselves to the 4-dimensional case, for the sake of simplicity and physical interest, but it is clear that the heat kernel techniques employed here apply to higher dimensional scalar theories. For gauge theory on (any dimensional) isospectral deformations manifolds, there is an intrinsic way to define noncommutative actions of the Yang-Mills type. For any ω ∈ Ωp (M ), η ∈ Ωq (M ) (say compactly supported and smooth with respect to α) one can set ω ∧Θ η := (2π)−l dl y dl z e−i (α∗ 1 ω) ∧ (α∗z η), − 2 Θy
R2l
where α∗z is the pull-back of αz on forms. Given now an associated vector bundle π : E → M with compact structure group G ⊂ U (N ), and a connection A ∈ Ω1 (M, Lie(G)) we define the NC analogue of the YM action tr(FΘ ∧Θ ∗H FΘ ), SY M (A) := M
where FΘ := dA + A ∧Θ A. In this context, one can prove a trace property, namely: ω ∧Θ ∗H η = ω ∧ ∗H η, ∀ω, η ∈ Ωp (M ). M
M
Hence SY M (A) equals M tr(FΘ ∧ ∗H FΘ ). To manage the quantization, one can once again use the background field method in the background gauge, and if we ignore the Gribov ambiguity, the one-loop effective action reduce to the computation of determinants of operators (quadratic part in A of SY M +Sgf and Faddeev-Popov determinant) which can be locally expressed as (∇µ + LAµ − RAµ )(∇µ + LAµ − RAµ ) + B, where B is bounded and contains left, right and a product of left and right twisted multiplication operators. It is then clear that UV/IR mixing will appears in the same form as in the flat situations (see [21, 23, 24]). A further interesting task is be to look at what happens for a Grosse-Wulkenhaar like model for the non-compact case. In [20] it is proved that if we add a confining potential (harmonic oscillator in their work) in the usual λϕΘ 4 theory on the four dimensional Moyal plane, i.e., the Grosse-Wulkenhaar action Ω2 SGW [ϕ] := d4 x 12 (∂µ ϕθ ∂ µ ϕ)(x) + 2 2 (xµ ϕ)θ (xµ ϕ) θ m2 λ ϕθ ϕ(x) + ϕθ ϕθ ϕθ ϕ(x) , + 2 4!
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then the theory is perturbatively renormalizable to all orders in λ. The deep meaning of this result is not yet fully understood, but some explanations can be mentioned. First, to add a confining potential is in some sense equivalent to a compactification of the Moyal plane and in the second hand, the particular choice of the potential corresponds to a Moyal-deformation of both the configuration and the momentum space. This can be seen by the invariance (up to a rescaling) of this ↔ (πθ)2 ϕ(x). This point needs to be clariaction under pµ ↔ 2(θ−1 )µν xν , ϕ(p) fied. It would be good to know whether their renormalizability conclusion (UV/IR decoupling) holds in the general context when one adds a coupling with a confining potential in the scalar theory. Last, but not least, it remains to see whether the UV/IR entanglement concerns only θ-deformations or not. Connes-Dubois-Violette [7] 3-spheres and 4planes, whose defining algebras are related to Sklyanin algebras, are good candidates to test this point.
Acknowledgments I am very grateful to J. M. Gracia-Bond´ıa, J. C. V´ arilly and my advisor B. Iochum for their help. I also would like to thank M. Grasseau, T. Krajewski, F. Ruiz Ruiz and R. Zentner for fruitful discussions and/or suggestions. Special thanks are also due to the Departamento de F´ısica Te´orica I of the Universidad Complutense de Madrid for its hospitality during the final stages of this work. I finally would like to thank the Referee for his enlightened remarks.
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[8] T. Coulhon, E. Russ and V. Tardivel-Nachef, Sobolev algebras on Lie groups and Riemannian manifolds, Amer. J. Math. 123, 283–342 (2001). [9] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. [10] M.R. Douglas and N.A. Nekrasov, Noncommutative Fields Theory, Rev. Modern Phys. 73, 977–1024 (2001). [11] T. Filk, Divergences in a Field Theory on Quantum Space, Phys. Lett. B 376, 53–58 (1996). [12] V. Gayral, J.M. Gracia-Bond´ıa, B. Iochum, T. Sch¨ ucker and J.C. V´ arilly, Moyal planes are spectral triples, Commun. Math. Phys. 246, 569–623 (2004). [13] V. Gayral and B. Iochum, The spectral action for Moyal planes, J. Math. Phys. 46, 043503 (2005). [14] V. Gayral, J.M. Gracia-Bond´ıa and F. Ruiz Ruiz, Trouble with space and nonconstant noncommutativity field theory, Phys. Lett. B 610, 141–146 (2005). [15] V. Gayral, J.M. Gracia-Bond´ıa and F. Ruiz Ruiz, Position-dependent noncommutative products: classical construction and field theory, hep-th/0504022. [16] V. Gayral, B. Iochum and J.C. V´ arilly, Dixmier trace on non-compact isospectral deformations, in preparation. [17] P.B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, 2nd edition, CRC Press, Boca Raton, FL, 1995. [18] J.M. Gracia-Bond´ıa, Improved Epstein-Glaser renormalization in coordinate space I. Euclidean framework, Math. Phys. Analysis Geom. 6, 59–88 (2003). [19] J.M. Gracia-Bond´ıa, J.C. V´ arilly and H. Figueroa, Elements of Noncommutative Geometry, Birkh¨ auser Advanced Texts, Birkh¨ auser, Boston, 2001. [20] H. Grosse and R. Wulkenhaar, Renormalisation of φ4 -Theory on Noncommutative R4 to all order, hep-th 0403232. [21] T. Krajewski and R. Wulkenhaar, Perturbative Quamtum Gauge Fields on the Noncommutative Torus, Int. J. Mod. Phys. A 15, 1011–1030 (2000). [22] D. Kurasov, J. Marklof and G.W. Moore, Melvin models and Diophantine approximation, hep-th 0407150. [23] C.P. Martin and D. Sanchez-Ruiz, The One Loop UV Divergent stucture of U (1) Yang-Mills Theory on Noncommutative R4 , Phys. Rev. Lett. 83, 476–479 (1999).
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[24] S. Minwalla, M.V. Raamsdonk and N. Seiberg, Noncommutative Perturbative Dynamics, J. High Energy Phys. 2, 20–31 (2000). [25] P.W. Michor, Isometric actions of Lie groups and invariants, Notes of a lecture course at the University of Vienna, July 1997. [26] S.B. Myers and N. Steenrod, On the group of isometries of a Riemannian manifold, Ann. Math. 40, 406–416 (1939). [27] J.C. Oxtoby, Measure and Category, Springer, Berlin, 1972. [28] M.A. Rieffel, Deformation Quantization for Actions of Rd , Memoirs Amer. Math. Soc. 506, Providence, RI, 1993. [29] F. Ruiz Ruiz, UV/IR mixing and the Goldstone theorem in noncommutative field theory, Nucl. Phys. B637, 143–167 (2002). [30] O. Schnetz, Natural renormalization, J. Math. Phys. 38, 738–758 (1997). [31] A. Sitarz, Rieffel’s deformation quantization and isospectral deformations, Int. J. Theor. Phys. 40, 1693–1696 (2001). [32] R.J. Szabo, Quantum Fields Theory on Noncommutative space, Phys. Rep. 37, 207–299 (2003). [33] J.C. V´ arilly, Quantum symmetry groups of noncommutative spheres, Commun. Math. Phys. 221, 511–523 (2001). [34] J.C. V´ arilly and J.M. Gracia-Bond´ıa, On the ultraviolet behavior of quantum fields over noncommutative manifolds, Int. J. Mod. Phys. A14, 1305–1323 (1999). [35] D.V. Vassilevich, Non-commutative heat kernel, Lett. Math. Phys. 67, 185– 194 (2004). [36] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, fourth edition, Clarendon Press, Oxford, 2002. Victor Gayral CPT-CNRS UMR 6207 Luminy Case 907 F-13288 Marseille Cedex 9 France email: [email protected] Communicated by Vincent Rivasseau submitted 22/12/04, accepted 22/03/05
Ann. Henri Poincar´e 6 (2005) 1025 – 1090 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/061025-66, Published online 15.11.2005 DOI 10.1007/s00023-005-0233-9
Annales Henri Poincar´ e
Non-Amenability and Spontaneous Symmetry Breaking – The Hyperbolic Spin-Chain – Max Niedermaier and Erhard Seiler
Abstract. The hyperbolic spin chain is used to elucidate the notion of spontaneous symmetry breaking for a non-amenable internal symmetry group, here SO(1, 2). The noncompact symmetry is shown to be spontaneously broken – something which would be forbidden for a compact group by the Mermin-Wagner theorem. Expectation functionals are defined through the L → ∞ limit of a chain of length L; the functional measure is found to have its weight mostly on configurations boosted by an amount increasing at least powerlike with L. This entails that despite the nonamenability a certain subclass of noninvariant functions is averaged to an SO(1, 2) invariant result. Outside this class symmetry breaking is generic. Performing an Osterwalder-Schrader reconstruction based on the infinite volume averages one finds that the reconstructed quantum theory is different from the original one. The reconstructed Hilbert space is nonseparable and contains a separable subspace of ground states of the reconstructed transfer operator on which SO(1, 2) acts in a continuous, unitary, and irreducible way.
1 Introduction Spontaneous symmetry breaking is typically discussed for compact internal or for Abelian translational symmetries, see, e.g., [1, 2, 3]. Both share the property of being amenable [4]; we recall the definition below but mention already that all semisimple nonabelian noncompact Lie groups are non-amenable. The goal of this note is to elucidate the notion of spontaneous symmetry breaking for a non-amenable internal group. This is motivated by the ubiquitous appearance of noncompact internal symmetries in a gravitational context, specifically in the dimensional reduction of gravitational theories [5], further in integrable sectors of QCD [6], or in ghost- or θ-sectors of gauge theories, and also in condensed matter physics [7, 8, 9, 10, 11]. The very fact that the group is non-amenable turns out to entail a number of surprising new features. In particular spontaneous symmetry breaking becomes possible in low dimensions where it is forbidden by the Mermin-Wagner theorem [1, 12, 13] in the case of compact internal symmetries. In order to have a concrete computational framework at hand we consider a definite lattice statistical system, the hyperbolic spin chain. This is a spin chain where the dynamical variables take values in a hyperbolic (Riemannian) space of constant negative curvature and the interaction is through nearest neighbors only. The lattice formulation was chosen in order to have control over the thermodynamic limit and in preparation to the quantum field theoretical case. Indeed we expect
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that many of the qualitative results generalize to generic statistical systems as well as to quantum field theories. In an accompanying paper [14] we study the nonlinear sigma-model with a hyperbolic target space in 2 or more dimensions. The systems treated always can be regarded in two different ways: either as a system of classical statistical mechanics, or as a quantum system in imaginary time. We mostly use the former interpretation, but discuss in some detail the reconstruction of the associated quantum system. Following [3], in the quantum interpretation we consider dynamical systems (C, τ ) consisting of a ∗-algebra C (“the observables”) and a one-parameter group of automorphisms (“the time evolution”), which we take to be discrete here τ x , x ∈ Z. In addition a group of automorphisms ρ(g), g ∈ G, (“the symmetry group”) is supposed to act on C and to commute with the time evolution, τ ◦ ρ = ρ ◦ τ . A state ω (positive linear functional over C) is said to be τ -invariant if ω ◦ τ = ω and extremal τ -invariant if it is not a convex combination of different invariant states. The symmetry ρ is said to be spontaneously broken (see, e.g., [1, 2, 3]) by an (extremal) τ -invariant state ω if ω ◦ ρ = ω. In the classical statistical mechanics interpretation C is a commutative C ∗ algebra (though there may be reasons to relax this condition) and the ‘time evolution’ really plays the role of space translations. A symmetry is again given by a group of automorphisms ρ(g), g ∈ G, acting on C and leaving the Hamiltonian (or action) invariant, except for possible symmetry violating boundary condition (a very precise definition of the notion of symmetry and its spontaneous breaking can be found in [15]). The definitions of states and their invariance or noninvariance are as in the quantum interpretation. Spontaneous symmetry breaking is then said to occur if there is an infinite volume Gibbs state (for instance obtained as a limit of finite volume Gibbs states) that is noninvariant. We shall be interested in the above situation when the symmetry group is a non-amenable Lie group. A Lie group G is called amenable if there exists an (left) invariant state (“a mean”) on the space Cb (G) of all continuous bounded functions on G equipped with the sup-norm. Conversely, G is called non-amenable if no such invariant mean over Cb (G) exists. All non-compact semisimple nonabelian Lie groups are known to be non-amenable. The notion of amenability has also been extended from Lie groups to homogeneous spaces (see for instance [16, 17]). Note that if in the above definition C was taken to be Cb (G), spontaneous symmetry breaking would be automatic for all non-amenable symmetries. We shall find however that the non-amenability also forces one to consider smaller algebras of observables (e.g., C ∗ -subalgebras of Cb (G)) so that the issue becomes non-trivial again. As a guideline it may be helpful to contrast the peculiar features we find in the hyperbolic spin chain with those in the corresponding compact model. Here the expectations of an observable refer to the thermodynamic limit of the chain where the number of sites goes to infinity while the lattice spacing is still finite. Moreover we require that the expectations are defined through a thermodynamic limit that does not involve the selection of ‘fine-tuned’ subsequences. This defines a subclass of ‘regular’ observables to which we mostly limit the discussion.
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quantity
spherical spin-chain
hyperbolic spin chain
ground state(s)
unique, normalizable SO(3) invariant SO(3) invariant independent of bc SO(3) invariant independent of bc
∞ set, non-normalizable not SO(1, 2) invariant SO(1,2) invariant depend on bc bc selects ground state SO(1,2) non-invariant depend on bc
reproduces original one
different from original one
expectations of selected SO(2)-invariant observables expectations of generic non-invariant observables reconstructed quantum theory
These regular observables (later called “asymptotically translation invariant”) presumably include all bounded ones, but an explicit formula for their expectations can be derived regardless of boundedness. For the hyperbolic chain it turns out that one has to impose boundary conditions (bc) at the end(s) of the chain which keep at least one spin fixed. Remarkably, we find that even the expectations of invariant observables may depend on the choice of bc, even though in the limit the ends are separated by an infinite number of sites from those where the observable is supported! The bc we are using also single out a preferred subgroup SO(2) ⊂ SO(1, 2) and the expectation functionals turn out to project any observable onto its SO(2) invariant part. Since this averaging over SO(2) does not commute with the action of the full SO(1, 2) group generic non-invariant observables will signal spontaneous symmetry breaking, i.e., their expectations are not SO(1, 2) invariant. This is accompanied by an infinite family of nonnormalizable ‘ground states’ transforming under an irreducible representation of SO(1, 2). This representation becomes unitary under a suitable change of the scalar product; such a scalar product will be produced by the Osterwalder-Schrader reconstruction described in Section 5. Somewhat different indications of spontaneous symmetry breaking in this context have been obtained in [18, 19]. In a situation of conventional symmetry breaking (say, of a compact Lie group symmetry in higher dimensions) one can always switch to invariant expectation functionals by performing a group average over the original noninvariant ones, at the expense of making the clustering properties worse. Here, due to the non-amenability of SO(1, 2) this cannot be done; the symmetry breaking is more severe, and in this respect resembles somewhat the ‘spontaneous collapse’ of supersymmetry in a spatially homogeneous state at finite temperature [20]. It is therefore remarkable that there exists a class of ‘selected’ SO(2) but not SO(1, 2) invariant observables (later called “SO(2) and asymptotically invariant”) which get averaged to yield a SO(1, 2) invariant result. One sees that the impact of the non-amenability is quite subtle: an invariant mean for all bounded
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(let alone unbounded) observables cannot exist, however an invariant mean on a subalgebra does exist and can be constructed explicitly as a thermodynamic limit of probability measures. Schematically, the mechanism behind this is that for a finite chain of length L the functional measure has support mostly at configurations which are boosted with a parameter depending on and increasing with L. So provided a limit exists at all it will be SO(1, 2) invariant as all non-invariant contributions die out. This can be paraphrased by saying that the thermodynamic limit provides a partial invariant mean, that is a mean which is invariant only on the before-mentioned class of ‘selected’ noninvariant observables. Finally we consider the counterpart of the Osterwalder-Schrader reconstruction in this context; here it is important not only to consider the regular observables but the full algebra Cb . For the compact chain one recovers (a lattice analogue of) the quantum mechanics of a particle moving on a sphere, as expected. In the hyperbolic case, however, the reconstructed quantum theory is different from that of a particle moving on H: whereas the former has purely continuous spectrum, the latter has at least some point spectrum. The reconstructed Hilbert space turns out to be nonseparable and the reconstructed quantum theory can be viewed as an interacting (though quantum mechanical) version of the “polymer representations” of the Weyl algebra studied in other contexts [21, 22, 23]. Consistent with these results we find that the symmetry breaking disappears in the limit of a flat target space, when the symmetry group R2 becomes amenable again. The rest of the article is organized as follows. In the next section we introduce the (iterated) transfer matrix and use its asymptotics in the limit of large separations to identify the ground states. Expectation values for a chain of finite length with various bc are studied in Section 3. The thermodynamic limits for the algebras of observables outlined are constructed in Section 4. Finally these infinite volume expectations are used as the basis for the Osterwalder-Schrader reconstruction.
2 The transfer matrix The hyperbolic spin chain can be regarded as a dynamical system in the sense outlined above, with the observables being operators on a Hilbert space. On the other hand, in the classical statistical interpretation the algebra of observables is a suitable algebra of functions over (direct products of) H which we detail in Section 3. We represent H as the hyperboloid H = {n ∈ R1,2 | n · n = 1 , n0 > 0}, where a · b = a0 b0 − a1 b1 − a2 b2 is the bilinear form on R1,2 . The time evolution of the spin chain is governed by the transfer matrix Tx , x ∈ N, which we study first. The symmetry group G is SO0 (1, 2) which acts unitarily via the (left) quasiregular representation ρ on L2 (H), i.e., ρ(A)ψ(n) = ψ(A−1 n), A ∈ SO0 (1, 2). Since we use the identity component exclusively we write SO(1, 2) for SO0 (1, 2). The time evolution commutes with the group action Tx ◦ ρ = ρ ◦ Tx ,
x ∈ N,
(2.1)
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as required. In the following we analyze the spectrum, the eigenfunctions, and the large x limit of Tx , x ∈ N, in terms of its integral kernel Tβ (n · n ; x). Some results from the harmonic analysis on H are needed which we have collected in Appendix A and use freely in the following.
2.1
Spectrum and integral kernel of Tx
The basic (1-step) transfer matrix acts on L2 (H) and is defined by β β(1−n·n ) (Tψ)(n) = dΩ(n ) 2π e ψ(n ) .
(2.2)
From (A.9), (A.20), one infers that the functions ω,k and ω,l defined in (A.8) and (A.10) are exact generalized eigenfunctions of T with eigenvalues 2β β (2.3) λβ (ω) = e Kiω (β) < 1 . π The eigenvalues are even functions of ω with a unique maximum at ω = 0 (but only ω ≥ will appear in the spectral resolution). In particular it follows that the operator T has absolutely continuous spectrum given by the generalized eigenvalues λβ (ω); the spectrum covers an interval [−q, λβ (0)] with 0 < q < 1 and is infinitely degenerate. It is interesting to note that, although real and bounded above by 1, the generalized eigenvalues are positive only for 0 < ω < ω+ (β), where ω+ (β) increases with β like ω+ (β) ∼ β + const β 1/3 . For ω > ω+ (β) the behavior of λβ (ω) is oscillatory with exponentially decaying amplitude λβ (ω) ∼
2ω π β − π ω+β e 2 + ω ln −1 2 sin ω 4 β
as ω → ∞ .
(2.4)
The fact that some of the spectrum of the transfer operator is negative means that there is no reflection positivity under reflections between the lattice points. However positivity of the eigenvalues is restored in the continuum limit: introducing momentarily the lattice spacing a, physical distances xphys = xa, as well as a coupling g 2 = 1/(βa) one has lim [λ
a→0
1 g2 a
(ω)]
xphys a
= exp
− xphys
g2 1 + ω2 . 2 4
(2.5) 2
These ‘eigenvalues’ are readily recognized as those of the heat kernel exp(− g2 Cxphys ), see (A.8); 1/g could be removed by rescaling the n-fields; g then parameterizes the curvature of the hyperboloid. Besides (2.4) another feature distinguishing the non-compact spin chain from the compact ones is that the iterated transfer matrix is bounded but, having continuous spectrum, is not trace class. Heuristically this is because due to the
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invariance (2.1) the infinite volume of SO(1, 2) gets “overcounted” in any trace operation. More precisely we have the following: Lemma 2.1 Let K be a self-adjoint operator on L2 (H) commuting with the unitary representation ρ. Then K has only essential spectrum, implying that K cannot be compact. In particular K cannot be trace class. Proof. Assume that K has an eigenvalue λ. The corresponding eigenspace Hλ ⊂ L2 (H) then is invariant under the action of ρ and therefore the representation ρ can be restricted to a unitary subrepresentation ρλ . Since SO(1, 2) is noncompact, ρλ is either infinite dimensional or it is a direct sum of copies of the trivial representation. But the trivial representation cannot be a subrepresentation of ρ since the only functions carrying the trivial representation are constants, and thus are not square integrable. Remark 1. There is a stronger version of the last statement in the proof: the trivial representation also is not even weakly contained in the direct integral decomposition of L2 (H) because SO(1, 2) is not amenable [4]. Remark 2. As noted above, Tx has only absolutely continuous spectrum. Since Tx is not trace class, correlators cannot be defined by the usual expressions involving traces. The obvious remedy is gauge-fixing. This could be done by introducing a damping factor at one site and by adopting twisted boundary conditions. Then analytic computations are still feasible but are not much different from those in the simpler gauge fixing approach in which one completely freezes one spin. This is the procedure we use in Section 3. Also the iterated transfer matrix acts as an integral operator on L2 (H) with kernel dΩ(n )Tβ (n · n ; x)ψ(n ) , x = 1, 2, 3, . . . , (Tx ψ)(n) =
Tβ (n · n ; x) =
0
∞
dω ω tanh πω P−1/2+iω (n · n ) [λβ (ω)]x , 2π
where the kernels have the semigroup property dΩ(n )Tβ (n · n ; x)Tβ (n · n ; y) = Tβ (n · n ; x + y) .
(2.6)
(2.7)
Manifestly the naive expression for the trace, i.e., the dΩ(n) integral over Tβ (1; x) does not exist due to the infinite volume of H. In passing we note that in terms of the Legendre functions (2.7) amounts to the following identity (“projection property”) 2πδ(ω − ω ) P−1/2+iω (n · n ) , dΩ(n ) P−1/2+iω (n · n )P−1/2+iω (n · n ) = ω tanh πω (2.8)
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which can also be verified directly from (A.12). Integral kernels of spectral projections in the proper sense are easily obtained by integrating over intervals I ω: dω ω tanh πω P−1/2+iω (n · n ) . (2.9) PI (n · n ) := ω∈I 2π Using Eq. (2.8) one easily verifies for two intervals I , J dΩ(n )PI (n · n )PJ (n · n ) = PI∩J (n · n ) ,
(2.10)
showing that the operators PI are spectral projections for an interval in ω and hence for a corresponding spectral interval for T. Absolute continuity of the spectrum follows from the completeness relation of the generalized eigenfunctions given in Appendix A. Before proceeding let us note the continuum limit of the iterated transfer matrix. Using the notation of (2.5) one has x phys Tc (ξ; g 2 xphys ) := lim T 21 ξ; a→0 g a a ∞ dω g2 1 ω tanh πω P−1/2+iω (ξ) exp − xphys + ω 2 , (2.11) = 2 4 0 2π where the limit is understood in the strong sense. With t = −ixphys this is the correct result for the Feynman kernel evolving a wave function for time t, see, e.g., [24, 25] and [26] for the propagators on other homogeneous spaces.. Most of the discussion on the large x limit of Tβ (ξ; x) below transfers directly to the large xphys limit of Tc (ξ, g 2 xphys ). Clearly for the further analysis the properties of the transfer matrix (2.6) will be crucial. By (2.2) and by iteration of the convolution property ξ → Tβ (ξ; x) is a positive function for all x ∈ N and β > 0. For small x it can be evaluated explicitly Tβ (ξ; 0) =
1 δ(ξ − 1) , 2π
β β −βξ e e , 2π √ β 2β e−β 2(1+ξ) e , Tβ (ξ; 2) = 2π 2(1 + ξ)
Tβ (ξ; 1) =
(2.12)
with ξ = n · n ≥ 1. The fact that Tβ (ξ; 2) can be given in closed form could be used to define a coarse grained action corresponding to decimation of half of the spins. Note also the strictly monotonic decay in ξ, stronger than any power, which is masked by the rapidly oscillating integrand in (2.6). Numerical evaluation of some x ≥ 3 transfer matrices suggests that these are generic features, see Fig. 1.
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0.05 0.04 0.03 0.02 0.01 0.5 1 1.5 2 2.5 3 3.5 4
lnΞ
Figure 1. x-step transfer matrix Tβ=1 (ξ; x) for x = 3, 6, 10, in order of decreasing slope. Note the non-uniformity: Tβ (ξ; x + 1) is smaller/larger than Tβ (ξ; x) for ξ smaller/larger than an intersection point ξx . By (2.26b) the enclosed area is always the same; the value at ξ = 1 is the x-site partition function. We proceed to prove these and some further properties of the kernels of Tx : Lemma 2.2 For fixed x the kernel Tβ (ξ; x) has the following properties: (i) For any integer p ≥ 0 ∞ 1 tβ (p; x) := dξ ξ p Tβ (ξ; x) < ∞ . 2π 1
(2.13)
(ii) Tβ (ξ; x) is strictly decreasing in ξ and vanishes for ξ → ∞. (iii) Tβ (ξ; x) ≤ Tβ (1; x) P−1/2 (ξ) for all ξ ≥ 1. (iv) Let f : [1, ∞) → R+ be a strictly positive locally integrable function satisfying sup n·n↑ >K
f (n · n ) ≤ C(n · n↑ )p , f (n · n↑ )
for some constants p ≥ 0 and C, K > 0. Then
1 ↑
)T (n · n ; x)f (n · n ) dΩ(n
f (n · n↑ )
≤C ,
(2.14)
(2.15)
with some constant C . Remark. Condition (2.14) holds for any function f with power-like growth or decay at ∞. This follows from the fact that the geodesic distance between two points n, n behaves aymptotically like ln(n · n ) and the globally valid triangle inequality for the geodesic distance on H.
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Proof. (i) The proof proceeds by induction in x. Note that naively exchanging the order of integrations in (2.6) would suggest a divergent answer already for the zero-th moment. The point to observe is that the convolution property (2.7) implies the recursion relation ∞ du jβ (ξ, u) Tβ (u; x) , Tβ (ξ; x + 1) = 1
jβ (ξ, u) :=
βeβ(1−ξ u) I0 (β
u2 − 1 ξ 2 − 1) ,
(2.16)
where I0 (u) is a modified Bessel function. The kernel jβ (ξ, u) has the following properties: its integral wrt to either variable equals 1; for fixed (not too small) ξ it √ is a bell-shaped function of u decaying like exp{−β(ξ − ξ 2 − 1)u}/ u for large u, and with a single maximum whose position grows linearly in ξ and whose value decays like 1/ξ, for large ξ. In particular the troublesome rapidly oscillating integrand of (2.6) is gone. So in the expression defining the ξ-moments the interchange of the ξ and u integrations is legitimate. The ξ-integral can be done by repeated differentiation with respect to α of the formula ([28], p.722) √ ∞ − β 2 +2uαβ+α2 −αξ β e dξe jβ (ξ, u) = βe =: Fβ (α, u) . (2.17) β 2 + 2uαβ + α2 1 This will be used below to obtain explicit expressions for the low moments. Note that both √ sides of the equation are holomorphic functions of α for |α| < r0 = β(u − u2 − 1), so that we may freely differentiate at the origin. By Cauchy’s estimate
p
− ∂ Fβ (α, u)
≤ p! r−p M (r, u) , (2.18)
∂α α=0 where M (r, u) is the maximum of |F | on the circle α = r, r < r0 . With the choice r1 = β(u − u2 − 1/2) it is not hard to see that the maximum is attained for α = −r – this follows from the fact that the zeros of the quadratic form Q(α) := β + 2uαβ + α2 are both real and negative, so both the real part and the modulus of Q(reiφ ) take on their minimal value β 2 /2 for φ = π. One concludes from (2.18)
p √ √
− ∂ Fβ (α, u)
≤ p! [β(u − u2 − 1/2]−p 2 eβ(1−1/ 2) . (2.19)
∂α α=0 If we finally use the fact that u − u2 − 1/2 ≥ constu−1 , and insert into the integral defining tβ (p; x + 1) the convolution formula (2.16) we obtain √
tβ (p; x + 1) ≤ p! constp tβ (p; x) eβ(1−1/
2)
.
(2.20)
Since for x = 1 all moments exist trivially, this inequality shows the existence of all moments for all x and (i) is proven.
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(ii) For x = 1 this is manifest from (2.12). For x > 1 we again proceed by induction. Assuming that Tβ (ξ; x) is already known to be strictly decreasing, we want to show ∞ ! ∂ξ Tβ (ξ; x + 1) = du ∂ξ jβ (ξ, u) Tβ (u; x) < 0 . (2.21) 1
This follows from the properties of the kernel jβ , namely
∞ < 0 for u < u0 (ξ) , du ∂ξ jβ (ξ, u) = 0 . and ∂ξ jβ (ξ, u) > 0 for u > u0 (ξ) , 1 Using (2.22) one gets for the rhs of (2.21) u0 (ξ) du ∂ξ jβ (ξ, u)Tβ (u; x) + 1
< 1
∞
u0 (ξ)
∞
(2.22)
du ∂ξ jβ (ξ, u)Tβ (u; x)
du ∂ξ jβ (ξ, u) Tβ (u0 (ξ); x) = 0 ,
(2.23)
where in the first integral Tβ (u; x) was replaced by its minimum and in the second one by its maximum over the range of integration, using the induction hypothesis. Thus ξ → Tβ (ξ; x) is strictly decreasing for all x. The vanishing for ξ → ∞ follows from (iii). (iii) This is obtained from (2.6) and the estimate |P−1/2+iω (ξ)| ≤ P−1/2 (ξ), which is manifest from (A.11). (iv) The proof is an elementary consequence of (i). Remark 1. Iteration of Eq. (2.16) provides an efficient way to compute numerically Tβ (ξ; x) for moderately large x. This was used to produce Figs. 1 and 2. Remark 2. Explicit expressions for the low moments are obtained by differentiating (2.17) and inserting the result in the recursion (2.16). This gives p=0: p=1: p=2:
tβ (0; x + 1) = tβ (0; x) , 1 tβ (1; x + 1) = 1 + tβ (1; x) , (2.24) β 1 1 tβ (2; x + 1) = − 2 (1 + β 2 ) tβ (0; x) + 2 (3 + 3β + β 2 )tβ (1; x) , β β
etc. Since for x = 1 all moments are known tβ (p; 1) = βeβ (−∂β )p (βeβ )−1 ,
(2.25)
(which is basically a Laguerre polynomial in β) the x-recursions can be solved successively for p = 0, 1, 2, . . .. The solution of the first two is trivial and gives 1 x tβ (1; x) = 1 + , ∀x ∈ N. (2.26) tβ (0; x) = 1 , β The higher ones won’t be needed explicitly.
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In summary, the qualitative properties of all the Tβ (ξ; x), x ∈ N, are very much like the ones exemplified in Fig. 1. The rate of decrease becomes softer with increasing x but remains faster than any power. The overall scale is set by the maximum Tβ (1; x), which turns out to decrease like x−3/2 for large x. (This is to be contrasted with the flat case of the Euclidean plane R2 , where the decay is only like x−1 ).
2.2
Large x asymptotics of Tβ (ξ; x)
We next determine the large x asymptotics of Tβ (ξ, x). This is of interest because in this limit the iterated transfer matrix normally becomes a (generalized) projector onto the ground state(s), which can in particular be used to identify the latter. In a compact spin chain the kernel of the iterated transfer matrix (normalized such that the largest eigenvalue is 1) tends to a constant for x → ∞, which is indeed the ground state (unique eigenstate to the highest eigenvalue) of the transfer matrix. This is a reflection of the Mermin-Wagner theorem, i.e., of the absence of spontaneous symmetry breaking. The decay in x is exponential due to the gap in the spectrum. In our noncompact model, since the spectrum is gapless, one expects the limit of large separations x to be approached power-like rather than exponentially. This is correct, but concerning the structure of the limit we are in for a surprise: the large x behavior is not invariant under the symmetry group SO(1, 2). Instead one finds lim
x→∞
Tβ (ξ; x) = P−1/2 (ξ) , Tβ (1; x)
(2.27)
as will be shown below. So in some sense P−1/2 (ξ) plays the role of a ground state, but unlike the compact case, where there is a unique, invariant and normalizable ground state, in our case we have a whole family of generalized non-normalizable ground states ψn0 (n) = P−1/2 (n0 · n), spanning a representation space of SO(1, 2). We shall explore the consequences of (2.27) in more detail below. However already at this point it is clear that in this 1D noncompact model the Mermin-Wagner theorem cannot hold in the usual sense. For later use we also introduce the SO↑(2) averaged versions of the iterated transfer matrix and the corresponding bounds. The former is given by π 1 2 T β (ξ, ξ ; x) = dϕ Tβ ξ ξ − (ξ 2 − 1)1/2 (ξ − 1)1/2 cos(ϕ − ϕ ); x , (2.28) 2π −π where n = (ξ, ξ 2 − 1 sin ϕ, ξ 2 − 1 cos ϕ), etc.. Note that T β (ξ, ξ ; 1) = jβ (ξ, ξ ) is the convolution kernel in (2.16). We collect our results on the asymptotics of the kernels Tβ (ξ; x) and T β (ξ, ξ ; x), which contain (2.27) as a special case, in the following
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Proposition 2.3 The large x asymptotics of Tβ (ξ; x) is governed by the relations: (i) P−1/2 (ξ) Tβ (ξ; x − y) = λβ (0)−y . x→∞ Tβ (ξ ; x) P−1/2 (ξ )
(2.29)
Tβ (ξ; x)
c(β) 2
Tβ (1; x) − P−1/2 (ξ) ≤ Const (ln ξ) P−1/2 (ξ) x ,
(2.30)
lim
(ii)
where c(β) is given below in (2.33) (iii) T β (ξ, ξ ; x) = P−1/2 (ξ)P−1/2 (ξ ) , x→∞ Tβ (1; x)
(2.31)
T β (ξ, ξ ; x)
≤ [ln2 ξ + ln2 ξ ] O(x−1 ) . Tβ (1; x)P−1/2 (ξ )P−1/2 (ξ)
(2.32)
lim
(iv)
1 −
The main ingredient in the proof of this proposition is contained in Lemma 2.4 Let f be an even function of ω ∈ R, which is at least twice differentiable at 0 and grows at most polynomially as ω → ∞. Then 0
∞
˜ β (ω)]x ∼ dω ω shπωf (ω) [λ
with
˜β (ω) = λβ (ω) λ λβ (0)
π [c(β)x]3/2 π f (0) + 2f (0)[c(β)x]−1/2 + O(x−1 ) × 2 ∞
and
dt t2 exp(−βcht) c(β) = 0 ∞ . dt exp(−βcht) 0
(2.33)
Proof of Lemma 2.4. The principle behind this is that the contributions of all ˜ β (ω)| for ω > 0 get exponentially suppressed, because they are less than 1, so |λ only the ω = 0 contribution survives for x → ∞. The leading power x−3/2 arises ˜ β (ω), which from the double zero of the integrand at ω = 0 and the structure of λ has a unique maximum at ω = 0. In more detail (2.33) one applies the Laplace ˜ β (ω) is expansion (see, e.g., [29]) to the kernel exp(−xhβ (ω)), where hβ (ω) = − ln λ strictly increasing in 0 < ω < ω+ (β) with hβ (0) = hβ (0) = 0 and hβ (0) = c(β) > 0. Here ω+ (β) is the position of the first zero of λβ (ω) described after Eq. (2.2). The ˜ β (ω)| < 1 also fact that λβ (ω) changes sign at ω+ (β) is inconsequential because |λ for ω ≥ ω+ (β) and the contribution of this region to the integral is exponentially suppressed.
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Proof of Proposition 2.3. We first prove (ii). To this end set ∞ dω 1+p ˜ β (ω)]x , p = 0, 1, . . . , Dp := ω tanh πω [λ 0 2π ∞ dω ˜ β (ω)]x [P−1/2+iω (ξ) − P−1/2 (ξ)] . ω tanh πω [λ N := 0 2π This is chosen such that Tβ (ξ; x)/Tβ (1; x) − P−1/2 (ξ) = N/D0 , as is manifest from (2.6) and P−1/2+iω (1) = 1. On the other hand, using the integral representation (A.11) given in Appendix A one obtains the bound
Thus
|P−1/2+iω (ξ) − P−1/2 (ξ)| ≤ Const ω 2 P−1/2 (ξ) (ln ξ)2 .
(2.34)
Tβ (ξ; x) D2 2
Tβ (1; x) − P−1/2 (ξ) ≤ Const (ln ξ) P−1/2 (ξ) D0 .
(2.35)
Using again Laplace’s theorem, one finds D2 /D0 = O((c(β)x)−1 ), which establishes (ii) (and therefore also (2.27)). (i): We apply Lemma 2.4 to D0 = Tβ (1; x) to obtain √ π Tβ (1; x) ∼ λβ (0)x + · · · . (2.36) [2c(β) x]3/2 Combining (2.27) with Tβ (ξ; x − y) Tβ (1; x − y) Tβ (1; x) Tβ (ξ; x − y) = , Tβ (ξ ; x) Tβ (1; x − y) Tβ (1; x) Tβ (ξ ; x)
(2.37)
and (2.36) gives (i). (iii): This follows from averaging (2.29) over SO↑(2) and using (A.12c). (iv): This is proven similarly as (2.35) starting from the spectral representation for the kernels T β , which is obtained from (2.6) by averaging over the angles using (A.12c). This concludes the proof of Proposition 2.3. We want to mention a stronger bound for which we do not have a complete proof. Conjecture 2.5 The following global bound holds for all x ∈ N and for all ξ ≥ 1: ln ξ (2.38) Tβ (ξ; x) ≤ Tβ (1; x) P−1/2 (ξ) E √ , x for some function E : R+ → R+ having finite moments of all orders. c1 2 t + Remark. The asymptotics of (2.35) for large x suggests that E(t) = 1 − c(β) 3 O(t ), with a constant c1 of order unity and c(β) as in (2.33). The proposal E(t) =
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c1 2 exp(− c(β) t ) is thus plausible. In the continuum limit (2.38) then reduces to a known global bound on the heat kernel (see, e.g., [27]), noting that the geodesic distance from the origin is arccoshξ ∼ ln ξ (for large ξ) and c(β) ∼ 1/β for large β. Given (2.38) a similar global bound on T β (ξ, ξ ; x) can be obtained from (2.38) 2 2 by using ξξ − (ξ 2 − 1)1/2 (ξ − 1)1/2 cos(ϕ − ϕ ) ≥ ξ[ξ − (ξ − 1)1/2 ], and then averaging over SO↑ (2).
Let us now explore the consequences of (2.27) in more detail. Consider the map ψ → P ψ (2.39) (P ψ)(n) := dΩ(n ) P−1/2 (n · n )ψ(n ) . As map from L2 (H) to itself this would have only the null vector in its domain, because it maps even strongly decreasing functions ψ into functions with a decrease so slow that they are not square integrable. But it may be regarded for instance as a map from the test function space S into its dual space S (see Appendix A). However, the range of P does not intersect its domain of definition, so the map (2.39) cannot even be iterated. This is in strong contrast to the situation in a compact model, where the corresponding operator is a well defined projection onto the 1-dimensional subspace of constant functions. Here, on the other hand, the image (P ψ)(n) is in general not even invariant under some SO(2) subgroup. The Fourier transform of P ψ can be defined nevertheless in a distributional sense. l), consistent with Using Eq. (A.14) and one finds ω tanh πω P ψ(ω, l) = 2πδ(ω)ψ(0, the picture that the limit (2.27) lowers the ‘energy’ as much as possible. There are two further important properties that encode the ground state property of P−1/2 (n · n ). The first one is Lemma 2.6 Let K be an SO(1, 2) invariant integral operator with kernel κ(n · n ), κ ∈ L1 (ξ −1/2 ln ξdξ). Then P ψ is a generalized eigenfunction of K with eigenvalue κ (0); explicitly κ(0)P−1/2 (n · n ) . (2.40) dΩ(n )κ(n · n )P−1/2 (n · n ) = Proof. Applying the Mehler-Fock transformation (A.16) for κ and the convolution formula for the Legendre functions the left-hand side becomes ∞ dω ω tanh(πω) κ(ω)P−1/2+iω (n · n )P−1/2 (n · n ) dΩ(n ) 2π 0 ∞ = dωδ(ω) κ ˆ(ω)P−1/2+iω (n · n ) = κ (0) P−1/2 (n · n ) , (2.41) 0
(where the integral and the δ function have to be interpreted suitably to include from 1 to ∞; this ω = 0). The L1 condition ensures that κP−1/2 is integrable √ follows from the global bound P−1/2 (ξ) ≤ (1 + ln ξ)/ ξ, valid for all ξ ≥ 1.
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Taking for K the iterated transfer matrix one has in particular Tx P ψ = λβ (0)x P ψ. For x = 1 this gives explicitly ∞ du jβ (ξ, u) P−1/2 (u) = λβ (0) P−1/2 (ξ) , (2.42) 1
using (A.21) and the fact that jβ is the SO↑(2) average of Tβ (n · n; 1). Thus P−1/2 is also an eigenfunction of the recursion relation (2.16) with the correct eigenvalue. The second property is the ‘cyclicity’ of the function ψ↑ (n) := P−1/2 (n · n↑ ) for the SO↑(2) invariant subspace of L2 (H) under the action of SO↑(2) invariant operators. See Appendix A for an explicit description of this subspace and the SO↑(2) invariant operators acting on it. We repeat that the SO↑(2) denotes the stability subgroup of n↑ = (1, 0, 0). The cyclicity of ψ↑ then follows trivially from the fact that P−1/2 does not vanish anywhere, so any SO↑(2) invariant element ψ ∈ L2 (H) can be obtained by acting on it with a multiplication operator. On the other hand ψ↑ has no nice properties with respect to operators that are not SO↑(2) invariant. Defining P as the integral operator with kernel P−1/2 (n↑ ·n)P−1/2 (n↑ ·n ) one has (P ψ)(n) = P−1/2 (n↑ ·n) Cψ for some constant Cψ . As with P one needs sufficiently strong falloff of ψ for this to be well defined and the image is again not an element of L2 (H). As expected, P automatically projects out the part of a wave function lying in the orthogonal complement of the SO↑(2) invariant subspace.
3 Expectation functionals for finite length Since the transfer operator is not trace class the overall SO(1, 2) invariance has to be (‘gauge-’) fixed already for a chain of finite length. We do this by keeping the spin at one end of the chain fixed and impose various boundary conditions at the other end. Expectation functionals (mapping observables, i.e., functions of the spins into complex numbers) then are always well defined. However in the limit of infinite length interesting statements can only be made about certain subalgebras of observables which we also introduce here. As before, SO↑(2) ⊂ SO(1, 2) denotes the stability group of the vector n↑ .
3.1
Boundary conditions and algebras of observables
We consider chains of length 2L + 1, with sites x = −L, L + 1, . . . , L − 1, L, and spins nx on them, in order to make the boundary go to infinity as L → ∞, so as to obtain an infinite volume Gibbs state for the chosen action. As discussed earlier, some ‘gauge fixing’ is needed, which is accomplished conveniently by fixing the spin at the left boundary of our chain: n−L = n↑ = (1, 0, 0) ∈ H. At the other end we consider the following choices: fixed (Dirichlet) bc nL = An−L , with A ∈ SO(1, 2), or free bc (integrating with the invariant measure of H over nL ). We refer to Dirichlet bc with A = 1 as ‘periodic bc’ and with A = 1 as ‘twisted bc’. The fixing of the spin n−L avoids the overcounting of the infinite
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volume of H induced by the invariance (2.1); since the associated Faddeev-Popov determinant is just 1, it is justified to refer to fixed bc with A = 1 as ‘periodic bc’. In some cases a nontrivial twist matrix would explicitly break the otherwise manifest SO↑(2) invariance. In those cases we shall average nL over an SO↑(2) orbit, thereby maintaining the SO↑(2) invariance of the bc. We consider several classes of observables, all of which consist of functions of finitely many spins. They form algebras with addition and multiplication defined pointwise. Of particular interest is the algebra Cb of bounded continuous functions on direct products of H. Equipped with the sup-norm and completed with respect to this norm, this is a commutative C ∗ -algebra, and the expectation functionals constructed later fit the usual concept of a ‘state’ ω as a normalized positive (and therefore bounded) functional on the observable algebra, see, e.g., [1]. More generally we consider the ∗-algebra Cp of polynomially bounded functions. For the construction of expectation functionals we introduce a system of subsets of Cp , closed under a suitable norm and designed such that explicit results for thermodynamic limit can be obtained. It turns out that the expectations of a multilocal observables O ∈ Cp can always be expressed in terms of a kernel K O associated with O as follows: Definition 3.1 For O ∈ Cp and ≥ 2 set −1 dΩ(ni )O(n1 , . . . , n ) Tβ (ni−1 · ni ; xi − xi−1 ) , K O (n, n ) := i=2
(3.1)
i=2
where n1 = n and n = n . For observables O depending only on one spin set K O (n, n ) := O(n) δ(n, n ),
(3.2)
where δ(n, n ) is the delta-distribution (point measure) concentrated at n = n , defined with respect to the measure dΩ. Lemma 3.2 The assignment O → K O mapping observables O ∈ Cp into integral operators K O on L2 (H) with kernel (3.1) has the following properties: (i) let A, B ∈ Cp be two observables of ordered non-overlapping ‘support’, i.e., A depends on nx1 , . . . , nxk and B on nxk+1 , . . . , nx with xk+1 ≥ xk ; then K AB = K A Txk+1 −xk K B , (3.3) K AB (n1 , n ) = dΩ(n)dΩ(n ) K A (n1 , n) Tβ (n · n ; xk+1 − xk )K B (n , n ) , where (AB)(nx1 , . . . , nx ) = A(nx1 , . . . , nxk ) B(nxk+1 , . . . , nx ), k, − k ≥ 2. If xk+1 = xk , the transfer matrix Tβ (n · n ; 0) is interpreted as δ(n, n ). (ii) The action of SO(1, 2) on Cp , i.e., ρ(A)O(nx1 , . . . , nxl ) = O(A−1 nx1 , . . . , A−1 nx ) induces an action on the kernels K ρ(A)O (n1 , n ) = K O (A−1 n1 , A−1 n ) . (iii) For the unit 1 ∈ Cp one has: K 1 (n1 , n ) = Tβ (n1 · n ; x − x1 ).
(3.4)
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Proof. This is a straightforward computation. Remark 1. The last property also implies that the correspondence O → K O is unique only for the equivalence classes obtained by inserting into a given K O extra powers of T. For example taking in (i) for A = 1 one obtains K 1B = Txk+1 −x1 K B . In the multipoint functions this just means that not all of the ‘unobstructed’ integrations have been performed. We shall therefore usually work with a reduced representative, i.e., one which cannot written in the form Ty1 K A1 Ty2 K A2 . . . with some smaller y1 , y2 , . . . ≥ 0. Remark 2. For observables depending only on one spin neither (3.1) nor (3.3) are directly applicable. However the assignment K O (n1 , n2 ) = O(n1 ) δ(n1 , n2 ) is compatible with the formulas for the 1-point functions (3.13), (3.17) and the convolution (3.3), provided we associate n1 and n2 with the same lattice point. We now introduce various classes of observables, where the SO↑ (2) average of an observable O is denoted by O. Definition 3.3 (i) An observable O = O ∈ Cp is called invariant if [K O , ρ] = 0 ,
(3.5)
i.e., O(An1 , . . . , An ) = O(n1 , . . . , n ) for all A ∈ SO(1, 2). The set of these observables is denoted by Cinv . (ii) An observable O ∈ Cp is called asymptotically invariant if lim ρ(A)[K O , ρ] = 0 .
A→∞
(3.6)
The set of these observables is denoted by Cainv . (iii) An observable O ∈ Cp is called translation invariant if [K O , T] = 0 .
(3.7)
The set of these observables is denoted by CT inv . (iv) An observable O ∈ Cp is called asymptotically translation invariant if lim ρ(A)[K O , P ] = 0 .
A→∞
(3.8)
The set of these observables is denoted by CT ainv . In (3.8) P is the integral operator (2.39). Both in (3.6) and (3.8) A → ∞ refers to a sequence of SO(1, 2) transformations such that A → ∞, and the commutator has to obey some decay condition detailed in the next section (Definitions 4.2 and 4.5).
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These subsets of Cp are related as follows: CT inv ∪ Cinv
⊂ CT ainv ∪ ⊂ Cainv
(3.9)
where all inclusions are proper. ↑ , CT↑ inv and CT↑ ainv are defined as the SO↑(2) invariant Definition 3.4 The sets Cainv subsets of Cainv , CT inv and CT ainv , respectively.
Of course the inclusion relations are preserved and the counterpart of the diagram (3.9) remains valid for the SO↑(2) invariant subsets.
3.2
Expectation functionals
The expectation functionals for finite L are defined by explicitly given measures and for the largest class of observables Cp . For states over Cb it follows from the general though not very constructive Banach-Alaoglu theorem [30] that thermodynamic limits always exist. The system of algebras (3.9) is designed to make useful and explicit statements about the limit, even for unbounded observables. Sometimes we refer to the expectation values as ‘correlators’ by a common abuse of language. With twisted bc the finite volume average of an observable O({n}) is then defined as L−1 1 β β(1−nx ·nx+1 ) dΩ(nx ) 2π e O({n}) δ(n−L , n↑ ) , (3.10) OL,β,α = Zβ,α (2L) x=−L
Here we anticipate that in the cases of interest the dependence on the twist matrix A is only through the scalar product n↑ · nL or equivalently the “twist parameter” α := arcosh n↑·nL ≥ 0. Zβ,α (2L) is the partition function normalizing the averages, 1L,β,α = 1. The technique to evaluate expressions like (3.10) is well known from the compact models: one uses the semigroup property (2.7) to perform all integrations not ‘obstructed’ by the variables in O({n}). For the partition function there are no obstructions and one readily finds Zβ,α (2L) = Tβ (chα; 2L) .
(3.11)
For the expectation value of some multilocal observable O one has Proposition 3.5 (twisted bc): For ≥ 2 O(nx1 , . . . , nx )L,β,α (3.12) 1 = dΩ(n1 )dΩ(n ) Tβ (n↑ · n1 ; L + x1 )K O (n1 , n ) Tβ (n · nL ; L − x ) , Zβ,α (2L)
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where x1 < · · · < x . For = 1 we have 1 dΩ(n) O(n)Tβ (n↑ · n; L + x)Tβ (n · nL ; L − x) . (3.13) O(nx )L,β,α = Zβ,α (2L) Proof. This is a simple consequence of (3.10) and the definition of K O .
Remark. As will become clear later for a SO↑(2) noninvariant field O one should average nL over SO↑(2), which amounts to replacing Tβ (n · nL ; L − x ) by T β (n↑ · n , n↑ · nL ; L − x ) defined in (2.28). For a field O which is SO↑(2) invariant the replacement is an identity. Since the expectation value is taken with a positive probability measure, for observables O ∈ Cb we have |O| ≤ ||O|| where O is the supremum norm, and for nonnegative O the expectation value is nonnegative. Observe also that due to the gauge fixing the functions (3.12) are in general not translation invariant; we shall later find a simple supplementary condition which restores translation invariance even at finite L. For free boundary conditions at x = L the situation is similar: First note that the partition function with free bc at L is Zβ,free (2L) = 1. (3.14) This follows from the normalization dΩ(n ) Tβ (n · n ; 2L) = 1 and the semigroup property of Tβ (n · n ; x), see Eqs. (2.12) and (2.7). Thus the expectation of an observable O({n}) with free bc at L is simply OL,β,free =
L x=−L
dΩ(nx )
L−1
β β(1−nx ·nx+1 ) 2π e
O({n}) δ(n−L , n↑ ) . (3.15)
x=−L
Again these expectation values can be rewritten similarly as in Proposition 3.5: Proposition 3.6 (free bc): For ≥ 2 O(nx1 , . . . , nx )L,β,free = dΩ(n1 )dΩ(n ) Tβ (n↑ ·n1 ; L + x1 ) K O (n1 , n ) , (3.16) where again x1 < · · · < x and K O is as in (3.1). For = 1 O(nx )L,β,free = dΩ(n) O(n)Tβ (n↑ · n; L + x) . Proof. Again a simple consequence of (3.15) and the definition of K O .
(3.17)
Remark 1. By comparing Eqs (3.16) and (3.12) one sees that the expectation values of observables with free and twisted bc are related by dΩ(nL ) Tβ (n↑ · nL ; 2L) OL,β,α = OL,β,free . (3.18)
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In other words for finite L the free expectation is some kind of weighted average over the twisted expectations. In the thermodynamic limit this is no longer true, as we will find below. Remark 2. Due to (3.4) a SO(1, 2) transformation on the observable can always be compensated by a change in the bc
. (3.19) ρ(A)OL,β,bc = OL,β,bc ↑ −1 ↑ −1 n →A
n , nL →A
nL
Of course our interest will be in the invariance or noninvariance of the expectations when the bc are kept fixed as L → ∞. For translation invariant observables the expectation values can be simplified. Recall that for O ∈ CT inv [K O , T] = 0 ⇐⇒ ∀n, n ∈ H (3.20) dΩ(n ) K O (n, n ) Tβ (n · n ; 1) = dΩ(n ) Tβ (n · n ; 1) K O (n , n ) . For these expressions to make sense, one has to impose some technical conditions; it suffices to demand that K O is a bounded operator. Using the convolution property (2.7) it is then easy to show that for translation invariant observables the expressions (3.12) and (3.16) simplify to Proposition 3.7 (translation invariant observables): For O ∈ CT inv 1 dΩ(n) K O (n↑ , n)Tβ (n · nL ; 2L + x1 − x ) , O(nx1 , . . . , nx )L,β,α = Zβ,α (2L) O(nx1 , . . . , nx )β,free = dΩ(n) K O (n↑ , n) . (3.21) Remark 1. For twisted bc also the equivalent form of the integrand K O (n, nL ) × Tβ (n · n↑ ; 2L + x1 − x ) could be used. Observe that these expectations are translation invariant already for finite L. Moreover for free bc they are L independent altogether, so that taking the thermodynamic limit becomes trivial. Remark 2. For observables whose kernels admit a Fourier expansion (A.18) a necessary and sufficient condition for (3.20) to hold is that expansion takes the form ∞ dω ω tanh πω κ l,l (ω) ω,−l (n)ω,−l (n ) . (−)l+l (3.22) K O (n, n ) = 2π 0 l,l ∈Z
It differs from the most general one in (A.18) only by the fact that it is diagonal in the energy parameter ω, as expected. An important special case is when
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the spectral weight is (up to a sign factor) independent of l1 , l2 . Due to the addition theorem (A.12c) the kernel becomes a function of n1 · n only. In this case the corresponding observables O can be characterized directly as being SO(1, 2) invariant. Remark 3. As already seen in Section 2 the ‘vacuum structure’ can be explored by taking the thermodynamic limit of the discrete system. Equivalently one can first take the continuum limit and then consider its behavior for large Euclidean times. The continuum limit of the correlators in Propositions 3.5, 3.6, and 3.7 is obtained by substituting (xi )phys = axi ,
Lphys = aL ,
β=
1 , ag 2
(3.23)
and taking the limit a → 0. In view of (2.11) this basically amounts to replacing Tβ by Tc everywhere, with the rescaled arguments. This procedure yields the additional bonus of restoring reflection positivity.
3.3
Projection onto SO↑(2) invariant observables
For SO(1, 2) non-invariant observables we did not assume special symmetry properties. It turns out, however, that one needs to consider only SO↑(2) invariant observables (bounded or unbounded) since with our gauge fixing SO↑(2) noninvariant ones are effectively projected onto SO↑(2) invariant ones. In order to see this let us apply an SO↑(2) rotation A(ϕ), A(ϕ)n↑ = n↑ (with ϕ the rotation angle) to an SO↑(2) noninvariant observable O. Using (3.4) one finds for ≥ 2 O(A(ϕ)nx1 , . . . , A(ϕ)nx )L,β,α 1 = dΩ(n1 )dΩ(n ) Tβ (n↑ · n1 ; L + x1 )K O (n1 , n ) Zβ,α (2L)
(3.24a)
Tβ (n · A(ϕ)nL ; L − x ) , O(A(ϕ)nx1 , . . . , A(ϕ)nx )L,β,free = dΩ(n1 )dΩ(n ) Tβ (n↑ · n1 ; L + x1 ) K O (n1 , n ) ,
(3.24b)
and similarly for = 1. For free bc one sees that the dependence on the rotation angle drops out, so that the expectations with these bc are SO↑(2) invariant even if the observable is not. Equivalently SO↑(2) noninvariant observables have the same expectations as their SO↑(2) averages. For twisted periodic bc this is not quite true. However the SO↑(2) noninvariance of (3.24) is evidently caused by the noninvariance of the bc. To retain the SO↑(2) invariance of the bc one can average nL over an SO↑(2) orbit. Then Tβ is replaced with T β in Eq. (2.28) and the situation is the same as with free bc. In summary, the expectations (3.10)
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(when nL is averaged over an SO↑(2) orbit) and (3.15) for finite L are already SO↑(2) invariant and hence we need not distinguish between SO↑(2) noninvariant and SO↑(2) invariant observables. In terms of the algebras introduced in Section 3.1 a projection Cp → Cp↑ , takes place upon insertion into the expectation functionals. In terms of the kernels K O the projection amounts to the replacement π 1 O dϕ K O (A(ϕ)n1 , A(ϕ)n ) =: K (n↑ · n1 , n↑ · n ) . (3.25) K O (n1 , n ) −→ 2π −π For later reference let us issue the warning K
ρ(A)O
O
(n↑ ·n1 , n↑ ·n ) = K (n↑ ·A−1 n1 , n↑ ·A−1 n ) ,
(3.26)
that is, SO↑(2) averaging does not commute with the SO(1, 2) action. In compact sigma models, where there is no need for gauge fixing, one can choose invariant bc, so that the expectation of any noninvariant observable is equal to that of its group average. By the Mermin-Wagner theorem in dimensions 1 and 2 this remains also true in the thermodynamic limit, irrespective of the bc used. Here we find an analogous situation only with respect to the maximal compact subgroup SO↑(2), singled out by the gauge fixing. In contrast, for the full SO(1, 2) group the expectations of noninvariant and of invariant observables cannot be related by group averaging. This is because – due to the amenability of SO(1, 2), such averages (invariant means) do not exist [4]. Heuristically this can be understood by viewing the group averaging as a projector onto the trivial subrepresentation in the direct integral decomposition of tensor products of L2 (H) functions. By the nonamenability the trivial representation does not occur, though. This lack of amenability is the source of many peculiarities in the vacuum structure of the noncompact model.
4 The thermodynamic limit as a partial invariant mean By the non-amenability of SO(1,2) an invariant mean on Cb cannot exist; a fortiori this holds for the unbounded functions Cp . It is known, however, that there are subspaces of the space of bounded continuous functions on any group, such as the spaces of almost periodic or weakly almost periodic functions on which a unique invariant mean exists [4]. These spaces are defined rather abstractly by relative compactness resp. weak compactness of their orbits under the group action. In ↑ ⊂ Cp for which there is the following we will introduce concretely a subspace Cainv a unique, invariant, and explicitly computable thermodynamic limit. The infinite volume averages therefore define a ‘partial invariant mean’. We presume that the ↑ ↑ ∩ Cb of our class Cainv (viewed as functions on SO(1,2)) bounded subalgebra Cainv consists of weakly almost periodic functions, but not of almost periodic functions (the latter set contains only the constant functions [31]). For the construction of the thermodynamic limit we proceed in several steps, where we first construct the
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thermodynamic limit for the algebras in the top row of the diagram (3.9). The limit is shown to be explicitly computable and unique (but different) for free and for twisted bc. The construction does not require the selection of subsequences, i.e., works without recourse to the Banach-Alaoglu theorem. In each case we then proceed to show that this limit is SO(1, 2) invariant for the described subalgebras ↑ . in the bottom row of the diagram, trivially for Cinv and nontrivially for Cainv
4.1
Thermodynamic limit for translation invariant observables
We begin by studying the thermodynamic limit of translation invariant observables. The distinction between the bounded observables and the polynomially bounded observables turns out to be inessential and we assume O ∈ CT inv throughout. With free bc, as seen in Eq. (3.21), there is no L dependence left – so no limit has to be taken. For CT inv expectations defined with twisted bc the existence of an L → ∞ limit needs to be established. First there is a slight complication that needs to be taken care of: twisted bc nL = An−L , n−L = n↑ , with A = 1 explicitly break SO↑(2) invariance. Since in this study we are interested in the spontaneous symmetry breaking for the nonamenable SO(1, 2), we restore the SO↑(2) invariance of the bc by performing an average of nL = An↑ over SO↑(2). For finite length L the expectations will then still depend on the ‘height’ n0L = chα. In a slight abuse of terminology we shall keep referring to these bc as ‘twisted’ ones and also keep the original notation . L,β,α . Only when a confusion is possible we emphasize the additional averaging by denoting the corresponding expectations by . L,β,α,av . Proposition 4.1 For O ∈ CT inv and twisted bc the thermodynamic limit is given by the equivalent expressions: ∞ O O(nx1 , . . . , nx )∞,β,α = λβ (0)x1 −x 2π dξ K (ξ, 1) P−1/2 (ξ) . (4.1) 1
O(nx1 , . . . , nx )∞,β,α = λβ (0)x1 −x 2π = λβ (0)x1 −x 2π
∞
1
O
dξ K (1, ξ) P−1/2 (ξ) O
∞
dξ 1
(4.2)
K (ξ, n↑ ·nL ) P−1/2 (ξ) . P−1/2 (n↑ ·nL )
Proof. For Eq. (4.1) we use the SO↑(2) invariance of the bc to replace the transfer matrix Tβ by Tβ (see (2.28))and then K O by its SO↑(2) average (see (3.25)). Since by assumption the integral dΩ(n)|K O (n, n )| exists one can in the first equation of (3.21) take the L → ∞ limit inside the integral. To obtain Eq. (4.2) one uses the fact that for translation invariant observables O the integral operators K O commute with P in (2.39). Remark 1. There are no elements of CT inv depending only on one spin, except constants. For free bc no thermodynamic limit has to be taken, see Proposition 3.6.
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Ann. Henri Poincar´e
Remark 2. In particular Proposition 4.1 is valid for SO(1, 2) invariant observables O ∈ Cinv ⊂ CT inv where the kernel K O (n1 , n ) depends only on the invariant distance n1 ·n . The thermodynamic limit (4.1) is then independent of the twist n−L · nL = n↑ · nL = cosh α, i.e., O(nx1 , . . . , nx )∞,β,α = O(nx1 , . . . , nx )∞,β,0 ,
O ∈ Cinv .
(4.3)
This can be verified directly using the ground state property (2.40). Indeed, if one does not take the thermodynamic limit in (3.21) with the SO↑(2) averaged transfer matrix one obtains initially an alternative version of the second Eq. in (4.2) λβ (0)x1 −x O O(nx1 , . . . , nx )∞,β,α = dΩ(n) K (n↑ ·n) P−1/2 (n·nL ) , O ∈ Cinv . P−1/2 (chα) (4.4) Averaging over nL and use of the addition theorem (A.12c) shows that the dependence on α drops out. Alternatively one can use (2.40) to verify (4.3). Remark 3. For generic translation invariant observables the infinite volume expectations are in general not SO(1, 2) invariant. Rather one finds from (3.19) the following induced action on the kernels by O → ρ(A−1 )O: O
O
K (1, ξ) −→ K (n↑ ·An↑ , n↑ ·An) , O
K (1, ξ) −→ O
P−1/2 (n↑ ·AnL ) O ↑ K (n ·An↑ , ξ) P−1/2 (n↑ ·nL ) O
K (ξ, n↑ ·nL ) −→ P−1/2 (n↑ ·An↑ ) K (ξ, n↑ ·AnL ) ,
(4.5a) (4.5b) (4.5c)
where for free bc only (4.5a) applies while for twisted bc all three (equivalent) expressions are applicable. We shall return to these formulae later but note already here that observables in CT inv \ Cinv will in general show spontaneous symmetry breaking: ρ(A)O∞,β,bc = O∞,β,bc . For the rest of this subsection we now focus on the special case of SO(1, 2) invariant observables. Then symmetry breaking is not an issue, nevertheless the result (4.1) is surprising. Besides the mere existence of a thermodynamic limit one would of course expect that the effect of the different bc is washed out. While we have found that the dependence on the twist chα actually does disappear, free bc in general give a different thermodynamic limit. In other words, even invariant observables show a dependence on the boundary conditions, even after the boundary is removed to infinity! To illustrate this consider specifically the usual ‘spin-spin’ two-point functions with the various bc. For twisted bc the thermodynamic limit is obtained from (4.1) and (3.1) (for = 2 with O(n1 , n2 ) = n1 · n2 ) as ∞ 2π dξ ξ Tβ (ξ; x)P−1/2 (ξ) . lim n0 · nx L,β,α = lim n0 · nx L,β,0 = L→∞ L→∞ λβ (0)x 1 (4.6)
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The independence of the twist angle has been seen before to be a general feature. However the same expectation with free bc at the right end of the chain gives a different result. One finds ∞ 1 x dξ ξ Tβ (ξ; x) = 1 + , (4.7) n0 · nx β,free = 2π β 1 using Eq. (2.26b) in the second step. As seen generally in Eq. (3.21) the correlator is L-independent and thus coincides with its thermodynamic limit. But this thermodynamic limit is now different from the previous one. To make sure that the analytical expressions (4.6) and (4.7) really define different functions we evaluated them numerically; the results are shown in Figure 2 below. For periodic bc also the approach to the thermodynamic limit is shown, which turns out to be nonuniform and extremely slow.
n0 nx 500
100 50
10 5
1
0
2
4
6
8
10
x
Figure 2. Spin two-point function for β = 1: for periodic bc and L = 8, 16, 32, 64, ∞, and for free bc, in order of increasing values at fixed x. For the ‘internal energy’ Eβ,bc := limL→∞ n0 · n1 L,β,bc the discrepancy can be seen immediately: Eβ,bc = 1 +
1 ∂ 1 − lim ln Zβ,bc(2L) β L→∞ 2L ∂β
1 ∂ ln λβ (0) for twisted bc, 1+ − β ∂β = 1 1+ for free bc, β using Eq. (2.36) and (3.14), respectively.
(4.8)
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Technically the discrepancy can be traced back to the fact that in Eq. (3.18) the operations ‘averaging’ and ‘taking the thermodynamic limit’ do not commute, schematically: limL→∞ dΩ(nL )(. . .) = dΩ(nL ) limL→∞ (. . .). Indeed, the lhs is L-independent and equals Oβ,free while the integrand and hence the integral on the rhs vanishes pointwise. In fact the integrand on the right-hand side behaves very nonuniformly for L → ∞: for instance the two-point function with twisted bc is unbounded as a function of α and the convergence as L → ∞ takes place more and more slowly as α increases. These features are in sharp contrast to those of the compact O(N ) spin chains where it is well known that all boundary conditions yield the same thermodynamic limit for the correlators of invariant as well as noninvariant quantities; see for instance [32]. In the compact models no gauge fixing is required, but one could fix a spin at the boundary just as we did here, and the thermodynamic limit would be insensitive to it. This is a consequence of the Mermin-Wagner theorem, which holds in this case. One might suspect that this ‘long range order’ in the non-compact model reflects the poor choice of observables, i.e., that the kernel O(n, n ) = n · n does not define an operator on L2 (H) (as explained after Eq. (A.13)). However the situation is the same for invariant kernels O(n, n ) = κ(n · n ) which obey (A.17) and which therefore do define integral operators on L2 (H). The thermodynamic limit of the corresponding two-point functions is obtained simply by replacing ξTβ (ξ; x) with κ(ξ)Tβ (ξ; x) in Eqs. (4.6) and (4.7). These two-point functions will be conventional, decreasing functions of x. Nevertheless they will in general be different for free and for periodic bc. Another potential problem could be the lack of clustering. However for SO(1, 2) invariant observables the situation turns out to be peculiar – there is perfect clustering even at finite distance. Consider two invariant observables, A(nx1 , . . . , nx )
and
B(ny1 , . . . , nyk )
such that x1 < · · · < x ≤ y1 < · · · < yk . We claim that for all bc AB∞,β,bc = A∞,β,bc B∞,β,bc .
(4.9)
For twisted periodic bc the derivation proceeds along the lines leading to Eq. (4.3) via (4.4): we define kernels K A and K B as above and use the ground state property Eq. (2.40) of P−1/2 . It turns out that both sides of Eq. (4.9) are equal to the same multiple of κ A (0) κB (0) (and thus in particular are independent of the twist parameter). For free bc the expectations of invariant observables are already Lindependent; the asserted factorization can be seen in a way similar to the step from (3.16) to (3.21). This ‘hyperclustering’ property is unpleasant, because it means that from the correlators of invariant fields one can only reconstruct a one-dimensional Hilbert space. The latter is suggested by the fact that all vectors obtained by applying
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invariant kernels to the ground state will by (2.40) be proportional to it. Technically it follows from the Osterwalder-Schrader reconstruction of the Hilbert space, as detailed in Section 5. On the other hand this feature is a peculiarity present likewise for other one-dimensional spin models, like the compact O(N ) chains or the harmonic chains. In these models, since they are based on amenable symmetries, there exists a unique thermodynamic limit also for noninvariant correlators and therefore one obtains by the reconstruction a nontrivial infinite dimensional Hilbert space. We now show that in the noncompact models the situation encountered for invariant observables persists for a class of noninvariant ones: for all observables in CT ainv for fixed bc a unique thermodynamic limit exists but is in general different for periodic and for free bc. The hyperclustering, however, does not carry over to those observables, as we will see.
4.2
TD limit for asymptotically translation invariant observables
We now relax the condition of translation invariance to “asymptotic translation invariance”. It suffices to consider SO↑(2) invariant bc (such as free, periodic, or SO↑(2) averaged twisted bc). As explained in Section 3.3 this allows one to restrict attention to SO↑(2) invariant observables. As before we denote by K O and P the integral operators with kernels K O (n, n ) in (3.1) and P−1/2 (n · n ), respectively. Similarly [K O , P ](n, n ) is the kernel of the commutator of K O with P . We give now the precise version of Definition 3.3 (iv): Definition 4.2 O ∈ Cp is called asymptotically translation invariant iff its SO↑(2) average satisfies
O
p(ξ) ∼ ξ −1/2 (ln ξ)−3 , (4.10)
[K , P ](n, n ) ≤ p(n↑ · n) p(n↑ · n ) , for some fixed n and all n ∈ H or vice versa. For observables O(n) depending on a single spin only we define asymptotic translation invariance by the condition that their SO↑(2) average O(n) has a limit as n → ∞. The function p(ξ) needs to be bounded but it is mainly the large ξ asymptotics that matters; for definiteness we take p(ξ) = p1 ξ −1/2 (1 + ln ξ)−3 , for some p1 = p(1) > 0. To motivate the terminology “asymptotically translation invariant” recall from Section 2 that P can be viewed as a weak limit of transfer operators TL /Tβ (1; L) for L → ∞. Further, for O ∈ CT ainv one has lim ρ(A)[K O , P ] = 0 .
A→∞
(4.11)
Here we assumed that the commutator acts on L1 wave functions so that β ρ(A)([K O , T]ψ)(n) can be bounded by 2π p1 p(An↑ · n)ψ1 . The thermodynamic limit for asymptotically translation invariant multi-spin observables and twisted bc is given by the same expressions as for translation invariant observables:
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Proposition 4.3 (i) Let O ∈ CT ainv be a 1-point observable, i.e., any function of one spin such that its SO↑(2) average has a limit O(∞). Then: lim O(nx )L,β,bc = O(∞) .
L→∞
(4.12)
(ii) Let O ∈ CT ainv be a multi-point observable, ≥ 2. If Conjecture 2.5 holds then: ∞ O x1 −x 2π dξ K (ξ, 1) P−1/2 (ξ) . (4.13) O(nx1 , . . . , nx )∞,β,α = λβ (0) 1
Proof. (i) By definition the SO↑ (2) average of O(n) has a limit O(∞) for n → ∞ (and therefore is a bounded function). We write (4.14) O(nx )L,β,bc = O(∞) + dµL,β,bc(n; x)[O(n) − O(∞)] . Decomposing the second term into an integral over n↑ ·n ∈ [1, Λ] and n↑ ·n ∈ [Λ, ∞[, given choose Λ so large that sup|O(∞) − O(n)| < , with the supremum over n↑ · n ∈ [Λ, ∞[. In Lemma 4.7 below is shown that the 1-spin measure of any bounded set in H goes to 0 as L → ∞, so sending L → ∞ the first integral vanishes. This shows that the total integral goes to 0 for L → ∞ and one obtains (4.12). (ii) The proof is based on a reduction to the case (i) of a one-spin observable. It is convenient to write (AB)(n, n ) for the kernel of AB, for any pair of integral operators A, B. With this notation one starts from T β (n↑ ·n , n↑ ·nL ; L − x ) OL,β,α = dΩ(n ) (TL+x1 K O )(n↑ , n ) . (4.15) Tβ (n↑ ·nL ; 2L) These multipoint averages can be written as one-point averages as follows OL,β,α = O0,L,α L,β,α , with T β (n↑ ·n, n↑ ·nL ; L − x ) . (4.16) O0,L,α (n0 ) := dΩ(n) (Tx1 K O )(n0 , n) T β (n↑ ·n0 , n↑ ·nL ; L) For the time being (4.16) is just an identity (Fubini’s theorem); later we shall put it in the context of the Osterwalder-Schrader reconstruction. Next we observe that O0,L,α (n0 ) has a L → ∞ limit, pointwise for all n0 ∈ H, which is independent of the twist parameter α defining nL modulo SO↑(2) rotations: lim O0,L,α (n0 ) = O0,∞ (n0 ) ,
L→∞
O0,∞ (n0 ) := Here we used Eqs. (2.31).
with
dΩ(n) (Tx1 K O )(n0 , n)
P−1/2 (n↑ · n) λβ (0)−x . (4.17) P−1/2 (n↑ · n0 )
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The crucial identity now is Lemma 4.4 Assume that Conjecture 2.5 holds. Then lim O0,L,α L,β,α = lim O0,∞ L,β,α ,
L→∞
L→∞
for all O ∈ Cb .
(4.18)
Proof of Lemma 4.4. We start with the bound
(4.19)
(O0,L,α − O0,∞ )L,β,α
T (n · n↑ ; L)T (n↑ ·n , n↑ ·n ; L)
β 0 β 0 L ≤ dΩ(n0 ) O0,L,α (n0 ) − O0,∞ (n0 )
, Tβ (n↑ ·nL ; 2L) To examine the difference |O0,L,α (n0 ) − O0,∞ (n0 )| we write O0,L,α (n0 ) − O0,∞ (n0 ) = D1 (n0 ) + D2 (n0 )
(4.20)
with
and
E(n0 ) :=
D1 (n0 ) :=
O0,L,α (n0 ) − E(n0 )
D2 (n0 ) :=
E(n0 ) − O0,∞ (n0 )
dΩ(n)(Tx1 K O )(n0 , n)
(4.21)
T β (n↑ ·n; n↑ ·nL ; L − x ) . P−1/2 (n↑ · n0 )P−1/2 (n↑ ·nL )Tβ (1; L) (4.22)
Using the bound
x1 O
(T K )(n0 , n) ≤ O T (n0 · n; x )
and the convolution property of the transfer matrices, a bound for D1 is
T β (n↑ ·n0 , n↑ ·nL ; L)
, |D1 (n0 )| ≤ O
1 − ↑ ↑ P−1/2 (n · n0 )P−1/2 (n ·nL )Tβ (1; L)
while for D2 one obtains simply Tβ (n0 · n; x ) |D2 (n0 )| ≤ O dΩ(n) P−1/2 (n↑ · n0 )
T β (n↑ ·n, n↑ ·nL ; L − x )
↑ −x
− × P−1/2 (n · n)λβ (0) . P (n↑ ·nL )Tβ (1; L)
−1/2
According to (4.19) we have to estimate Tβ (n0 · n↑ ; L)T β (n↑ ·n0 , n↑ ·nL ; L) , d1,2 := dΩ(n0 ) |D1,2 (n0 )| Tβ (n↑ ·nL ; 2L)
(4.23)
(4.24)
(4.25)
(4.26)
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Ann. Henri Poincar´e
In a first step we use T β (ξ0 , ξL ; x) ≤ Tβ (1; x)P−1/2 (ξ0 )P−1/2 (ξL ) and the fact that D1,2 (n0 ) depend on ξ0 = n↑ ·n0 only to write P−1/2 (ξL )Tβ (1; L) ∞ d1,2 ≤ 2π dξ0 |D1,2 (ξ0 )|Tβ (ξ0 , L)P−1/2 (ξ0 ) . (4.27) Tβ (ξL ; 2L) 1 Next we claim
|D1,2 (ξ0 )| ≤ [ln2 ξ0 + ln2 ξL ]O(1/L) .
(4.28)
For D1 (ξ0 ) this follows directly from (4.24) and Proposition 2.3(iv). For D2 (ξ0 ) we likewise use Proposition 2.3(iv) and then apply Lemma 2.2(iv) with the function f (ξ) = P−1/2 (ξ)[ln2 ξ + ln2 ξL ] (see the remark after Lemma 2.2). This gives ∞ P−1/2 (ξ) 2 [ln ξ + ln2 ξL ] dξ T β (ξ0 , ξ; x ) |D2 (n0 )| ≤ O(1/L) P−1/2 (ξ0 ) 1 ≤ O(1/L)[ln2 ξ0 + ln2 ξL ] ,
(4.29)
as asserted. On account of (4.28) the integrand I(ξ0 ) in (4.27) vanishes pointwise for L → ∞. Using (4.28), assuming Conjecture 2.5 and recalling that P−1/2 (ξ0 ) ≤ √ √ ξ0 / L) (1 + ln ξ0 )/ ξ0 we can bound I(ξ0 ) by O(1/L)Tβ (1; L)ξ0−1 (1 + ln ξ0 )2 E(ln √ [ln2 ξ0 + ln2 ξL ]. Changing now the integration variable to t := (ln ξ0 )/ L the new integrand is bounded by 1 ln2 ξL )E(t) (4.30) F (t) := const (t + √ )2 (t2 + L L and the right-hand side is bounded uniformly in L by an integrable function. By the dominated convergence theorem we can interchange the limit with the integration and conclude that d1,2 → 0 for L → ∞, completing the proof of Lemma 4.4. This lemma, combined with (4.16), reduces the computation of the thermodynamic limit for multipoint functions to that of one-point functions: from Eqs. (4.12), (4.16) it follows that the thermodynamic limit of a multipoint observable can be computed as lim OL,β,α = lim O 0,∞ (n0 ) ,
L→∞
n0 →∞
(4.31)
whenever the limit exists. We claim that for all O ∈ CT ainv the limit does exist and is given by the rhs of Eq. (4.13). To see this we return to (4.17) and swap the order of K O and P : λβ (0)−x (Tx1 K O P )(n0 , n↑ ) O0,∞ (n0 ) = P−1/2 (n↑ · n0 ) λβ (0)x1 −x = dΩ(n)P−1/2 (n0 · n)K O (n, n↑ ) P−1/2 (n↑ · n0 ) λβ (0)−x dΩ(n) Tβ (n0 · n; x1 )[K O , P ](n, n↑ ) . + (4.32) P−1/2 (n↑ · n0 )
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We now take the SO↑(2) average wrt n0 . In the first term the n0 dependence then drops out by (A.12c) and produces the announced result. For the second term we use the defining bound (4.10) and distinguish between x1 = 0 and x1 = 0. In the first case a bound on the second term is λβ (0)−x p1 p(ξ0 )/P−1/2 (ξ0 ), which vanishes for ξ0 → ∞. For x1 = 0 we bound the integral by 2πp1 dξ T β (ξ0 , ξ; x)p(ξ), using the definition (4.10). To this integral we apply Lemma 2.2(iv) to get constp(ξ0 ), which again vanishes for ξ0 → ∞. This completes the proof of Proposition 4.3(ii). Let us add a number of comments on (4.13), (4.12). First one should note that the thermodynamic limit can be computed explicitly for all of CT ainv without having to select ‘fine-tuned’ subsequences, i.e., without recourse to the BanachAlaoglu theorem. Second one observes that translation invariance is restored in the thermodynamic limit even though for O ∈ CT ainv the finite volume expectations are not translation invariant. Third, just as for translation invariant observables the expectations (4.13), (4.12) will in general not be SO(1, 2) invariant. An exception ↑ ⊂ CT ainv to be discussed below. are observables in a subclass Cainv Just as CT inv contained the SO(1, 2) invariant observables Cinv as special cases, here there is a subspace Cainv of observables which decay sufficiently fast to an SO(1, 2) invariant one after averaging over SO↑(2). We denote the limiting observable by (4.33) O∞ (n1 , . . . , n ) := lim O(An1 , . . . , An ) , A→∞
and specify the rate of approach to the limit below. Provided the limit exists it will automatically be SO(1, 2) invariant. For example one can build a large class of Cp↑ observables satisfying (4.33) by replacing in a function of ni · nj each ni · nj with ni ·nj f (ni , nj ) or with ni ·nj + f (ni , nj ), for some SO↑(2) – but not SO(1, 2) – invariant function f that goes to a constant in the limit. Note that the dependence on the invariant part may correspond to an unbounded function. In addition any dependence on the n0i is allowed, constrained only by the requirement that the limit (4.33) exists. Of course Cainv contains the SO(1, 2) invariant observables Cinv as a proper subset. For observables depending only on one spin ( = 1) asymptotic translation invariance just reduces to the existence of the limit in (4.33), as it did for CT ainv observables with = 1. For > 1 we specify the rate of approach in which the limit in (4.33) is reached in terms of the kernels K O as follows, thereby giving a technically precise version of Definition 3.3(ii): Definition 4.5 O ∈ Cp is called asymptotically invariant, O ∈ Cainv , iff after SO↑(2) averaging the associated kernel obeys
O
K (n, n ) − K O∞ (n · n ) ≤ p(n↑ ·n) p(n↑ ·n ) , with K O∞ (n1 · n ) := lim K ρ(A)O (n1 , n ) = lim K O (A−1 n1 , A−1 n ) , A→∞
where p(ξ) is as in (4.10).
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Note that in analogy with (4.11) this implies limA→∞ ρ(A)[K O , ρ] = 0, for O ∈ Cainv , which was used as the defining property in 3.3(ii). Further Cainv ⊂ CT ainv .
(4.34)
To see this one writes [K O , TL ] = [(K O − K O∞ ), TL ] + [K O∞ , TL ]. The second commutator vanishes because SO(1, 2) invariant observables are translation invariant. The kernel of the first commutator is bounded in modulus by P−1/2 (n↑ · n) p(n↑ · n )Tβ (1; L). It follows that [K O , P ] satisfies (4.10), which verifies (4.34). It follows that the formulae (4.13), (4.12) are valid also for observables in O Cainv . Moreover the kernel K can in fact be replaced with the invariant limiting kernel K O∞ . Proposition 4.6 (i) For a multi-point observable O ∈ Cainv , ≥ 2: ∞ O(nx1 , . . . , nx )∞,β,α = λβ (0)x1 −x 2π dξ K O∞ (ξ) P−1/2 (ξ) .
(4.35)
1
↑ (ii) On Cainv the expectation functional O → O∞,β,bc is an invariant mean.
Proof. (i) We decompose K O again as K O = K O∞ +(K O −K O∞ ). For the invariant limiting kernel the manipulations proceed as for the translation invariant case and yield Eq. (4.13) with the indicated replacement. The average of the remainder by Q(1; L + x1 )Q(n↑ ·nL ; L − x )/Tβ (n↑ ·nL ; 2L) with K O − K O∞ can ∞be bounded Q(ξ; x) := 2π 1 dξT β (ξ, ξ ; x)p(ξ ), for x ∈ N. The bound vanishes in the limit L → ∞. (ii) This is a direct consequence of (4.35). Remark 1. For later reference we note again that this reasoning remains valid if the lower boundary in the Q integrals was replaced with an arbitrarily large constant Λ 1. Remark 2. The reason why the expectation functionals do no not provide an invariant mean for all of Cainv is that the SO↑(2) averaging effected by the expectations does not commute with the SO(1, 2) action. As a consequence observables ↑ in Cainv \ Cainv will typically signal spontaneous symmetry breaking. See (3.26) and the examples in Section 4.3. Likewise the hyperclustering (4.9) for SO(1, 2) invari↑ ant observables trivially generalizes to the class Cainv but fails in general for Cainv : ↑ have support as in the premise of (4.9) the limit of the products if A, B ∈ Cainv equals the product of the limits, i.e., (AB)∞ = A∞ B∞ , and to the latter (4.9) applies. A counterexample to hyperclustering in Cainv will be given in Section 4.3. We can summarize these results by saying that the thermodynamic limit effectively projects CT ainv onto CT inv and Cainv onto Cinv , i.e., the top row in the diagram (3.9) is projected onto the bottom row. In the first case translation invariance emerges but SO(1, 2) invariance is in general still absent, while in the second
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case, given SO↑(2) invariance as a ‘seed’, both properties emerge. The second result is more interesting because SO(1, 2) is not amenable, so one could not ‘by hand’ switch to invariant states by group averaging of noninvariant ones. (Presumably this is still true if one adopts a distributional group averaging as in [34, 35].) Rather the thermodynamic limit itself defines a partial invariant mean, that is ↑ gets averaged to the subclass of (bounded as well as unbounded) observables Cainv yield a SO(1, 2) invariant result. An invariant mean in the proper sense would do the same for all continuous bounded observables Cb↑ , but it cannot exist on general grounds.
4.3
The support of the functional measures
Here we discuss the results (4.12) and (4.35) in more detail. Both properties express a partial symmetry restoration and are due to a remarkable concentration (actually rather dilution) property of the underlying functional measures. Roughly speaking the measures have their support concentrated at configurations that are boosted from the origin by an amount growing at least powerlike with the number of sites; the measure of any bounded set of configurations goes to zero in the thermodynamic limit. The derivation of (4.12) given below explicitly makes use of this concentration property; the previous derivation of (4.35) did for technical reasons not explicitly rely on it. We shall explain later why the underlying concentration property is nevertheless visible in the derivation. In Section 5.2 we shall also describe an alternative proof of (4.35) which links it explicitly to the concentration property of the 1-spin measures instrumental for (4.12). This concentration property is due to the large fluctuations present in D = 1, which in compact models are the ‘enforcers’ of the Mermin-Wagner theorem, but which are here insufficient to restore the symmetry. We begin with re-evaluating the thermodynamic limit for asymptotically translation invariant observables depending only on a single spin. Take some O(nx ) ∈ CT ainv (which for = 1 coincides with C ainv by definition) depending on a single spin at site x only. By definition its SO↑(2) average has a limit as nx → ∞. We claim that this limiting value coincides with the thermodynamic limits of the O(nx ) expectation. The mechanism behind this is that the relevant measures have support ‘mostly at infinity’. To make this precise, recall that in view of (3.13) and (3.17) the spin n := nx is for finite L distributed according to the probability measures dµL,β,α,av (n; x) =
Tβ (n↑ · n; L + x)T β (n↑ · n, chα; L − x) dΩ(n) , Tβ (chα; 2L) for twisted bc
dµL,β,free(n; x) = Tβ (n↑ · n; L + x) dΩ(n) , for free bc .
(4.36)
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In the first case T β is defined as in (2.28). From Eqs. (2.27), (2.36) and (2.31) one sees that the densities multiplying dΩ behave for large L as 2 L−3/2 P−1/2 (n↑ · n) , for twisted bc, λβ (0)L+x L−3/2 P−1/2 (n↑ · n) ,
for free bc ;
(4.37)
(the approach to this asymptotic form is, however, very nonuniform in n0 , as can be seen from Eqs. (2.35),(2.38)). In particular the dependence on chα drops out in the first case. Both expressions in (4.36),(4.37) vanish pointwise in the limit but are not integrable. This implies Lemma 4.7 For any bounded subset M ⊂ H and for twisted as well as free bc dµL,bc = 0 , (4.38) lim L→∞
M
where dµL,bc stands for either of the measures in (4.36). As a consequence these measures do not have a limit as L → ∞; they ‘spread out’ over H (though not evenly); Lemma 4.7 may be interpreted as saying that the measure is getting concentrated more and more near infinity. The measures dµL,β,bc form a sequence of bounded, normalized linear functionals (‘states’) on the space Cb . By the theorem of Banach-Alaoglu [30] there is therefore a subsequence convergent to such a functional – a so-called ‘mean’; see, e.g., [4]. Because SO(1, 2) is not amenable, this mean cannot be invariant. We will give below explicit examples of elements of Cb that show this non-invariance, i.e., spontaneous symmetry breaking. However 1-spin observables invariant under SO↑(2) still have a unique thermodynamic limit, which is independent of x and β, see Proposition 4.3(i). In view of Lemma 4.7 this expresses the fact that the thermodynamic limit effectively projects a one spin observable onto the ‘boundary at infinity’ of the hyperbolic plane; for SO↑(2) invariant functions we may use the one-point compactification of H so that there is only one such boundary point at infinity. It is also instructive to estimate the size of the ‘cup’ in the hyperboloid whose contribution to the functional integral is negligible. We integrate the observable under consideration with the pointwise vanishing density in (4.37) over the compact domain {n ∈ H| n↑ ·n ≤ Λ(L)}. Demanding that the contribution of this domain still vanishes in the limit L → ∞ constrains the permitted growth of Λ(L) with Λ(L) Λ(L) dξP−1/2 (ξ)2 ∼ d ln ξ(ln ξ)2 ∼ L. For twisted bc the relevant integral is 3 (ln Λ(L)) , using (4.37) and the asymptotics in (A.13). Thus any Λ(L) satisfying ln Λ(L) = o(L1/2 ) will still give a contribution vanishing in the limit L → ∞. For Λ(L) dξP−1/2 (ξ) ∼ Λ(L) ln Λ(L). Thus any growth free bc the relevant integral is 2 3/2 Λ(L) = o(L / ln L) is allowed. To conclude our discussion of 1-point functions let us consider some examples. A simple example of a bounded SO↑(2) invariant observable is O(n) = tanh(n↑ ·
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n). Then (4.12) gives 1 for the thermodynamic limit of its expectation, which in particular is SO(1, 2) invariant. The spin field nax itself is neither bounded nor SO↑(2) invariant. However by the SO↑(2) invariance of the measures (4.36) one has nax L,β,bc = δ a0 n−L · nx L,β,bc ,
(4.39)
leaving only the SO↑(2) invariant part O(nx ) = n↑ · nx = n0x to study. Slightly generalizing the above discussion it follows that the n0x expectation with both twisted periodic and free bc diverges for L → ∞: for any constant Λ, the measure of the (compact) subset of H where |n0x | ≤ Λ goes to 0; since the total weight of the measure is always 1, the expectation value will eventually be larger than Λ(1 − ) for any > 0. For the rhs in (4.39) this is also illustrated by the numerical results for the 2-point functions shown in Fig. 2. In both of these examples the limit is SO(1, 2) invariant and does not signal spontaneous symmetry breaking. As seen before the computation of the thermodynamic limit for multi-point observables can be reduced to that of 1-point functions. Nevertheless it is instructive to outline the origin of the concentration property also for the multi-point measures. For simplicity we restrict attention to twisted bc. The counterpart of the normalized measures (4.36) for > 1 are (after integrating out nx2 , . . . , nx−1 ) dµL,β,α (n1 , n ; x1 , x ) =
(4.40)
Tβ (n↑ · n1 ; L + x1 )Tβ (n1 · n ; x − x1 )T β (chα, n↑ · n ; L − x ) dΩ(n1 )dΩ(n ) . Tβ (chα; 2L) The finite volume expectation of some observable O can be written in terms of these measures as K O (n1 , n ) . (4.41) OL,β,bc = dµL,β,bc(n1 , n ) Tβ (n1 · n ; x − x1 ) The asymptotics of the density in (4.40) is λβ (0)x1 −x P−1/2 (n↑ · n1 ) Tβ (n1 · n ; x − x1 )P−1/2 (n↑ · n ) L−3/2 .
(4.42)
This density vanishes pointwise as L → ∞ and is integrable wrt one but not wrt both variables. As before the limit of the measures therefore only exists as a mean. The concentration property ensued by (4.42) is however more subtle than for the 1-point measures. This is because invariant combinations like n1 · n contribute even for highly boosted individual n1 and n . Conditions like (asymptotic) translation invariance or (asymptotic) SO(1, 2) invariance allow one to isolate the invariant contribution by swapping the order of K O and TL+x1 while implying that the commutator does not contribute to the invariant part. In order to illustrate the mechanism we set k(n↑ ·n1 ) := sup n
O
K (n↑ · n1 , n↑ · n ) . Tβ (n1 · n ; x − x1 )
(4.43)
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Clearly |k(ξ1 )| ≤ O. If O and hence K O is SO(1, 2) invariant, k(ξ1 ) equals a constant. If K O does not contain a SO(1, 2) part the function k(ξ1 ) vanishes for ξ1 → ∞. Then dµL,β,α (n1 , n ; x1 , x )
n↑ ·n1 <Λ1
≤
Λ1
dξ k(ξ) 1
K O (n1 , n ) Tβ (n1 · n ; x − x1 )
Tβ (1; L + x1 )Tβ (1; L − x1 ) Tβ (ξ; L + x1 )T β (ξ, n↑ · nL ; L − x1 ) ≤ Tβ (n↑ · nL ; 2L) Tβ (n↑ · nL ; 2L) Λ1 ln ξ . (4.44) × P−1/2 (n↑ ·nL ) dξ k(ξ) P−1/2 (ξ)2 E √ L + x1 1
In the last step we used the SO↑(2) average of the bound in Lemma 2.2(iii) and (2.38). The estimates in (4.44) capture the qualitative features of the concentration phenomenon. There are two cases to consider: (i) K O does not contain an SO(1, 2) invariant part, in which case k(ξ1 ) → 0 as ξ1 → ∞. Using the first bound in (4.44) and the argument used for 1-point functions one sees that its L → ∞ limit is given by the ξ → ∞ limit of k(ξ) and thus vanishes, both for finite Λ1 and for Λ1 → ∞. (ii) K O does contain an SO(1, 2) invariant part, in which case limξ→∞ k(ξ) = 0. In this case it is instructive to estimate the size of the ‘cup’ in the n1 hyperboloid that does not contribute significantly to the average as L becomes large. To this end we use the second bound in (4.44) and note that for fixed L one can take the Λ1 → ∞ limit at the price that the integral scales like L3/2 for large L. In other words √ one simple recovers the normalizibility of the measures in the regime √ √ ln2 Λ1 L. On the other hand for ln Λ1 L the integral scales like (ln Λ1 / L) . In particular one can allow Λ1 to grow with L according to ln Λ1 (L) = o(L1/2 ) ,
(4.45)
and still have the bound in (4.44) vanish for L → ∞. (Note that this conclusion only depends on the simple bound Lemma 2.2 (iii) and not on (2.38).) The intermediate regime can also be analyzed; a typical case is ln Λ1 = Lq with q > 1/2, for which the integral in (4.44) approaches a finite but nonzero constant as L → ∞. The upshot is that in the original (n1 , n ) integral over H × H only the region n↑ ·n1 ≥ Λ1 (L), with Λ1 (L) as in (4.45), contributes significantly to the result for the average as L becomes large. For free bc the analysis is similar, except that the change in the rate of decay also involves powers of λβ (0). We omit the details and simply state that one can likewise allow the cutoff Λ1 to grow at least powerlike in L, without affecting the limit formulas.
4.4
Examples
We begin with some examples where a finite thermodynamic limit does not necessarily exist, like for the components of the spin field or of the Noether current.
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The individual components of the energy observable a EL,β,bc := nax nax+1 L,β,bc,
a = 0, 1, 2,
can be shown to diverge for L → ∞ by an argument similar to the one used in Section 4.2. On the other hand the invariant combination (E 0 − 2E 1 )L,β,bc has a finite limit given by (4.8). Next consider the Noether current Jxa = β(nx × nx+1 )a , where n× n denotes the SO(1, 2) invariant vector product of n, n ∈ H. (Explicitly (n × m)a = η aa a bc nb mc , with abc totally antisymmetric and 012 = 1.) For the current two-point function one finds Jx0 Jy0 L,β,bc = 0 ,
for x < y ,
1 Jx1 Jy1 L,β,bc = Jx2 Jy2 L,β,bc = − Jx · Jy L,β,bc , 2
for x < y , (4.46)
so that all components have a finite L → ∞ limit. The first equation is a special case of the more general result Jx01 O(nx2 , . . . , nx )L,β,bc = 0 ,
for x1 < x2 < · · · < x ,
(4.47)
which is obtained by specializing the general formulas (3.12), (3.16) and then using
where nx · nx+1
∂ Tβ (nx · nx+1 ; 1) = −Jx0 Tβ (nx · nx+1 ; 1) , (4.48) ∂ϕx 2 = ξx ξx+1 − ξx2 − 1 ξx+1 − 1 cos(ϕx − ϕx+1 ). Since Jx0 is es-
sentially the Noether charge generating infinitesimal SO↑(2) rotations (see below) Eq. (4.47) expresses the SO↑(2) invariance of the ‘ground states’ 1 and ψ↑ (n), respectively. Conversely, the fact that correlators involving Jx1 , Jx2 are non-zero is yet another manifestation of the SO(1, 2) symmetry breaking. Ward identities expressing the invariance of the measure and of the action can be derived along the familiar lines. For example one has a ) nby L,β,bc + δx,y (ta ny )b L,β,bc , (Jxa − Jx+1
(4.49)
with (ta )dc = −η aa η dd a d c . Replacing nby with a generic (non-invariant) observa able O(nx1 , . . . , nx ) a similar identity arises where the correlator with Jxa − Jx+1 produces a sum of contact terms. As is clear from (4.49) these linear Ward identities will in general not have a non-boring thermodynamic limit. In particular no conflict, even in spirit, with Coleman’s theorem [36] arises. Ward identities where the current enters nonlinearly can likewise be derived but are hampered by the fact that the ‘response’ are in general functions which fail to be translation invariant. In 2 or more dimensions a useful quadratic Ward identity can be derived which relates the components of the longitudinal part of the current-current correlator to the energies E a ; see [37]. In one dimension only
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the longitudinal part exists and only the SO(1, 2) invariant – and hence translation invariant – combination of these component Ward identities is useful. It reads Jp · J−p L,β,bc + 2β(E 0 − 2E 1 )L,β,bc = 0 ,
∀p = 0 ,
(4.50)
where Jpa = x e−ipx Jxa , with p = 2πn/(2L + 1), n = 0, . . . , 2L. Next we consider some examples of asymptotically translation invariant observables. They also serve to highlight the significance of the SO↑ (2) averaging in the definition of the algebras in (3.9). Recall that Cainv = {O ∈ Cp | SO↑ (2) average ↑ lies in Cainv }. The point here is that in general O and ρ(A)O, A ∈ SO(1, 2), will ↑ ↑ have different SO↑ (2) invariant images in Cainv . The elements of Cainv \ Cainv will therefore typically signal spontaneous symmetry breaking although by Section 3.3 ↑ they get effectively projected back into Cainv . An instructive example of such a ‘symmetry breaking observable’ in Cainv arises as follows: given a spacelike unit vector e = ( q 2 − 1, q sin γ, q cos γ) we define Te (n)
:=
T q (ξ)
:=
tanh(n · e) ∈ Cainv , π 1 ↑ dϕ tanh ξ q 2 − 1 − q ξ 2 − 1 cos ϕ ∈ Cainv . (4.51) 2π −π
The observable Te (n) indeed enjoys the property (4.34): after SO↑(2) averaging it has a unique limit T q (∞), which can be obtained by acting with a sequence of SO(1, 2) transformations going to infinity. This limit does not depend on n any more, so in a trivial sense it is an invariant function of the spins. It does, however, depend on e or rather on the scalar product n↑ · e and is therefore not invariant under the action of SO(1, 2) on the original observable. Spontaneous symmetry breaking is shown by the following Proposition 4.8 For all bc considered Te (nx )∞,β,bc = T q (∞) = 1 −
2 arccos 1 − q −2 ; π
(4.52)
this expectation value is manifestly not invariant under SO(1, 2): for a general A ∈ SO(1, 2) one has Te (nx )∞,β,bc = Te (Anx )∞,β,bc . Proof. We use the fact that in finite volume expectations we may replace Te (nx ) by it average over SO↑(2) rotations T q (n0x ). The argument of the tanh, i.e., αξ (ϕ) := ξ( q 2 − 1 − q 1 − ξ −2 cos ϕ), then has its minimum at ϕ = 0 and its maximum at ϕ = ±π. Because e is spacelike, there is a ξ0 (q) such that for all ξ > ξ0 (q) the minimum αξ (0) is negative, the maximum αξ (±π) is positive and there are two zeros at ϕ = ± arccos[(1 − q −2 )/(1 − ξ −2 )]1/2 whose modulus converges to
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ϕ0 := arccos(1 − q −2 )1/2 . This implies that |ϕ| > ϕ0 1, lim tanh αξ (ϕ) = −1, |ϕ| < ϕ0 ξ→∞ 0, |ϕ| = ϕ0 .
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(4.53)
By the dominated convergence theorem we can pull the limit ξ → ∞ under the integral for the ϕ averaging and obtain 2 lim T q (ξ) = 1 − arccos 1 − q −2 . (4.54) ξ→∞ π The result then follows from Eq. (4.12).
A large class of observables in Cainv can now be built by algebraic operations. Of course sums and products of Te (n) at the same or different sites will lie in Cainv , but so will be algebraic combinations built from elements of Cinv . The crucial ↑ is that hyperclustering and even ordinary clustering will now difference to Cainv fail in general. This is because ‘SO↑(2) averaging’ and ‘taking the A → ∞ limit’ in (4.33) are noncommuting operations in general. A simple example is given by the product of two tanh-observables (4.51), where 1 = Te (n)2 ∞,β,bc = Te (n)2∞,β,bc = T q (∞)2 ,
(4.55)
from (4.53) and (4.54). So we have so far found observables that show hyperclustering and others that do not cluster at all. Observables showing ordinary (exponential or powerlike) clustering presumably also exist in the large space Cb , but it is more difficult to find explicit examples.
5 Reconstruction of a Hilbert space and transfer operator The Osterwalder-Schrader type reconstruction allows to reconstruct a Hilbert space and a transfer matrix from expectation values satisfying reflection positivity as well as translation invariance. The original expectation values are recovered as expectations in a genuine, i.e., normalizable ground state vector. This construction is well documented in the literature [38, 39, 40], but in our case there are peculiarities and surprises. For this reason we describe in some detail how the construction works here. First there is a rather harmless complication: reflection positivity for reflections both in lattice sites and in midpoints between lattice points is equivalent to positivity of the transfer operator; as we found in the beginning, however, this does not hold in our case. But we still have reflection positivity for reflection in lattice points, at least if we take the thermodynamic limit with periodic bc, and this is enough for the reconstruction of the Hilbert space and a positive two-step transfer matrix.
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There is a much more serious complication: as stated above, the reconstruction produces a ground state in the proper sense, whereas we know that the original transfer matrix T on L2 (H) does not have such a ground state. So it is unavoidable that there is some discrepancy between the reconstructed quantum mechanics and the one we started from. This mismatch is also related to the fact that our expectation functional in the infinite volume is not given by a measure, but only a mean on the configuration space. In this section we consider periodic bc exclusively and denote the expectation functional (the state) . L,β,0 in Eq. (3.12) by ωL ( . ). A reconstruction in the usual sense won’t work for twisted or free bc because the x ≥ 0 and the x ≤ 0 halves of the chain have to enter symmetrically. For the algebra we take Cb in order to have the usual concept of a state available. For Cb ∩ CT ainv we saw before that the thermodynamic limit is explicitly computable and translation invariant. For the rest of Cb a thermodynamic limit exists likewise, though it may be necessary to select subsequences and to average over translations in order to have it translation invariant. We denote such a weak limiting state by ω∞ ( . ) = w − limL→∞ ωL ( . ). We denote by C+ (C− ) the subalgebra of bounded observables Cb depending only on the spins nx with x ≥ 0, (x ≤ 0) and C0 = C+ ∩ C− . Our chain admits a reflection x → −x and we introduce an antilinear time reflection ϑ acting on on Cb by replacing any function O by the same function of the reflected arguments and taking the complex conjugate: (ϑO)(n−x−1 , . . . , n−x0 ) = O(n−x0 , . . . , n−x−1 )∗ ,
x0 < · · · < x −1 ,
(5.1)
where the asterisk denotes complex conjugation. To interpret this formula correctly note that on the lhs we have written the observable ϑO in the customary form as a function of the spins on which it actually depends, in the order of increasing indices. On the rhs O is to be read as a function of spins, with the displayed arguments now appearing in the order of decreasing indices. For example O = n1 ·n3 +c n↑ ·n3 gives ϑO = n−1 · n−3 + c∗ n↑ · n−3 . We discuss the reconstruction first for a finite and then for an infinite chain.
5.1
Finite chains
Recall that we adopt untwisted periodic bc, n−L = nL = n↑ , and consider a chain of total length 2L + 1. We begin by assigning to each O ∈ C+ an element O0,L ∈ C0 with the same expectation value via dΩ(ni )O(n1 , . . . , n ) Tβ (ni−1 · ni ; xi − xi−1 ) O0,L (n0 ) := i=1
=
i=1
Tβ (n · n↑ ; L − x ) × Tβ (n0 · n↑ ; L) dΩ(n) (Tx1 K O )(n0 , n)
Tβ (n↑ · n; L − x ) , Tβ (n↑ · n0 ; L)
(5.2)
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where x0 = 0 and the first transfer matrix is to be interpreted as the identity operator if x1 = 0. Note the properties |O0,L (n)| ≤ O ,
(1)0,L (n) = 1 ,
(ρ(A)O)0,L (n) = O0,L (A−1 n)
↑ −1 n →A
n↑
(5.3)
where we denote by 1 the unit element of C+ . Further we set Tβ (n · n↑ ; L) ψ O (n) := O0,L (n) . Tβ (1; 2L)
(5.4)
The expressions (5.2) and (5.4) are designed such that 1 ωL ([ϑO] O) = dΩ(n) |O0,L (n)|2 Tβ (n · n↑ ; L)2 = dΩ(n) |ψ O (n)|2 , Tβ (1; 2L) (5.5) holds, as one can verify from (3.12). In particular reflection positivity ωL ([ϑO]O) ≥ 0 ,
∀ O ∈ C+ ,
(5.6)
is manifest. With these preparations at hand the reconstruction of the Hilbert space HL for a finite chain works as usual: a positive semidefinite scalar product is introduced on C+ by (A, B)L := ωL ([ϑB] A) ; (5.7) there will be a nontrivial null space N of elements with ωL ([ϑO] O) = 0 . The Hilbert space HL is then the completion of the quotient space C+ /N with respect to the norm induced by ωL . The necessity to divide out N becomes clear if one notices that for any O ∈ C+ one can find a unique element O0,L ∈ C0 such that O − O0,L ∈ N , namely just the one given in (5.2). The uniqueness follows from (5.5), which implies C0 ∩ N = {0}. Note that the OS norm for O coincides with the L2 -norm for ψ O . The above construction makes it manifest that for a finite chain there is a natural isometry between the reconstructed Hilbert space HL and the original L2 (H): HL turned out to be the completion of C0 with respect to the norm induced by (5.7), i.e., HL = C 0 . Note that although C0 is the universal L-independent space of bounded continuous functions on H, its completion with respect to ( , )L depends on L. Of course the L-dependence is of a rather trivial nature in that by (5.4) the map VL : HL −→ L2 (H) ,
Tβ (n · n↑ ; L) (VL ψ)(n) = ψ(n) , Tβ (1; 2L)
(5.8)
defines an isometry between Hilbert spaces. Alternatively HL could be regarded as the preimage of L2 (H) with respect to VL . It is worth noting that HL is by itself
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a commutative C ∗ -algebra, so the reconstruction of the Hilbert space can be considered as an instance of the well-known Gel’fand-Na˘ımark-Segal reconstruction, see, e.g., [1, 41]. To sum up, for a finite chain the original Hilbert space L2 (H) and the reconstructed one HL can really be identified. Unsurprisingly, for a finite chain HL also carries a unitary representation ρL of SO(2, 1) (spontaneous symmetry breaking can only arise in the thermodynamic limit); it is obtained simply by conjugating the representation ρ with VL : ρL = VL−1 ρVL .
(5.9)
Explicitly for A ∈ SO(2, 1) and ψ ∈ C0 this gives (ρL (A)ψ)(n) = ψ(A−1 n)
Tβ (A−1 n · n↑ ; L) . Tβ (n · n↑ ; L)
(5.10)
For O ∈ C+ we define (ρL (A)O)(nx0 , . . . , nx−1 ) = O(A−1 nx0 , . . . , A−1 nx−1 )
Tβ (A−1 nx−1 · n↑ ; L − x −1 ) , (5.11) Tβ (nx−1 · n↑ ; L − x −1 )
which is compatible with (5.10) and induces it via (5.2) in that (ρL (A)O)0,L (n) = (ρL (A)O0,L )(n), for all A ∈ SO(1, 2). This also ensures that ρL maps elements O − O0,L of N onto other elements of zero norm. The rhs of (5.10), (5.11) is in general no longer a bounded function of n because the asymptotics of Tβ (A−1 n · n↑ ; L) and Tβ (n · n↑ ; L) do not match, but it is of course still an element of HL with the same norm as ψ. Likewise by (5.10), (5.11) the completion NL of N wrt ωL is mapped onto itself under ρL . In preparation of the thermodynamic limit let us consider the action of ρL on the function 1 (an approximate ground state for large L): (ρL (A)1)(n) = The scalar product (ρL (A)1, ρL (B)1)L
1 = Tβ (1; 2L) =
Tβ (A−1 n · n↑ ; L) . Tβ (n · n↑ ; L)
(5.12)
dΩ(n)Tβ (n · An↑ ; L)Tβ (n · Bn↑ ; L)
Tβ (An↑ · Bn↑ ; 2L) , Tβ (1; 2L)
(5.13)
then has the finite and nonzero limit P−1/2 (An↑ ·Bn↑ ), as L → ∞. For finite L the state ωL ( · ) will not be translation invariant outside the subalgebra CT inv ∩ C+ . As a consequence there is no reconstructed transfer matrix for finite L. Conversely this provides an intrinsic reason to consider the reconstruction based on the expectations of the infinite chain.
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Thermodynamic limit
Let us thus turn to the thermodynamic limit ω∞ = w − limL→∞ ωL . Reflection positivity remains true in this limit; so one can still define a scalar product and a null space N as for the finite chain. As before a Hilbert space HOS can be constructed as the completion of C+ /N = C0 /(N ∩ C0 ). In order not to clutter the notation we continue to use the same symbols for the algebra of observables and the spaces N etc., however one should keep in mind that the spaces C+ etc. for a finite and for the infinite chain cannot be identified. In particular equation (5.4) loses its meaning in the limit: the left-hand side goes to zero pointwise, even though its norm in HOS in general does not. For observables in C+ ∩ CT ainv the explicit formula (4.13) can be used to compute the inner products ( , )OS . Outside this class in general the original definition ( , )OS = limL→∞ ( , )L has to be used. On the other hand (5.2) always has a sensible limit: Proposition 5.1 (i) The limit limL→∞ O0,L (n0 ) =: O0,∞ (n0 ) exists and obeys P−1/2 (n↑ · n) λβ (0)−x . (5.14) O0,∞ (n0 ) := dΩ(n) (Tx1 K O )(n0 , n) P−1/2 (n↑ · n0 ) (ii) 10,∞ = 1 and |O0,∞ (n0 )| ≤ O, where · denotes the sup norm. (iii) If Conjecture 2.5 holds, O − O0,∞ ∈ N ,
(5.15)
with respect to ( , )OS . Proof. (i) and (ii) are straightforward. (iii), while very plausible, requires nevertheless a proof; the one given here relies on the validity of Conjecture 2.5. It suffices to show that for any A ∈ C− lim ωL (A(O − O0,∞ )) = 0 .
L→∞
(5.16)
In order to show this, we write ωL (A(O − O0,∞ )) (5.17) = ωL (A(O − O0,L )) + ωL (A(O0,L − O0,∞ )) + (ω∞ − ωL )(A(O − O0,∞ )) . The first term vanishes by construction of O0,L , the third term goes to zero as L → ∞ by definition of ω∞ , whereas the second term requires a closer look. In view of
T (n · n↑ ; L)2
β 0
,
ωL (A(O0,L − O0,∞ )) ≤ A dΩ(n0 ) O0,∞ (n0 ) − O0,L (n0 )
Tβ (1; 2L) (5.18) the difference |O0,∞ (n0 ) − O0,L (n0 )| needs to be examined. This however has been done in Section 4.2, and the proof that the right-hand side of (4.19) vanishes for L → ∞ carries over. This completes the proof of (iii).
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The relation (5.15) has several important consequences, which we discuss consecutively. For bounded observables in CT ainv a crucial consistency condition arises from (5.15) and (4.13). Since an explicit formula for the state ω∞ = ∞ is known for these observables it must come out that AO∞ = AO0,∞ ∞
∀A ∈ C− , C+ ∈ O ,
(5.19)
using directly the limiting formulae (4.13) and (5.14). This is indeed the case: a computation shows that both sides of (5.19) reduce to A O O λβ (0)x< −x> dΩ(n)dΩ(n )K A (n↑ , n)Tβ (n · n ; xA > − x< ) ×
dΩ(n )K O (n · n )P−1/2 (n↑ · n ) .
(5.20)
O Here we wrote xO < for the leftmost and x> for the rightmost site where an observable O ∈ C is supported. The consistency condition for O ∈ CT ainv is therefore satisfied. On the other hand (5.15) is valid for all bounded observables and the computation leading to (5.20) does not seem to leave much room for expressions other than (4.13) having the same property (5.19). This suggests that (4.13) is actually valid for all bounded observables though our proof is not. Next let us consider asymptotically invariant observables. For them the result (5.22) yields an alternative derivation of (4.35). Since it is based on (5.22) this derivation highlights that the origin of the result (4.35) lies in the concentration property of the measures described in Section 4.3. To this end we write K O as K O∞ + (K O − K O∞ ) and insert into the definition of O0,∞ (n0 ). Since (5.23) is trivially satisfied for the integral operators coming from SO(1, 2) invariant operators the first term gives (4.13) with K O replaced by K O∞ , which is the asserted result. Using (4.34) the modulus of the second term can be bounded by
λβ (0)−x
Q(ξ0 , x1 ) , P−1/2 (ξ0 )
(5.21)
where ξ0 = n↑·n0 and Q(ξ0 ; x1 ) is defined in after Eq. (4.35). According to (5.22) we have to analyze the limit n0 → ∞ of this expression. To this end we split the region of integration in Q into a bounded part ξ ∈ [1, Λ] and a remainder ξ ∈ [Λ, ∞[. For the unbounded part we use Tβ (ξ0 , ξ ; x1 ) ≤ P−1/2 (ξ0 )P−1/2 (ξ ) Tβ (1; x1 ) to get a n0 independent bound p1 /Λ on it. In the bounded part we use the fact that Tβ (ξ0 , ξ ; x1 ) vanishes faster than any power in ξ0 , and does so uniformly for all ξ ∈ [1, Λ]. For large enough ξ0 the supremum sup[Tβ (ξ0 , ξ ; x1 )/P−1/2 (ξ0 )] over ξ ∈ [1, Λ] can be therefore be made smaller than 1/Λ2 . The upshot is that (5.21) can be made smaller than any prescribed quantity. This completes the derivation of (4.35) based on (5.22).
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A simple consequence of (5.15) is that in contrast to the finite volume case C0 now also intersects the null space N : for instance all functions going to zero for n → ∞ (i.e., n↑ ·n → ∞) will be mapped into the null vector of HOS , according to the previous section. The same is true for all functions that go to zero for n → ∞ after averaging over SO↑(2). In fact, according to the discussion in Section 4.2, this exhausts the intersection N ∩ C0 . Likewise products of the form c(n)ψ(n) ∈ C0 , where c(n) → c for n → ∞, differ from cψ(n) only by an element of N , since their difference goes to zero as n → ∞. This means that in linear combinations of vectors constant coefficients can always be replaced with coefficients satisfying this decay condition without changing the equivalence class mod N . Setting A = 1 in (5.17) and using the results of Section 4.3 one infers ω∞ (O) = ω∞ (O0,∞ ) = w − lim O0,∞ (An↑ ) A→∞
for all O ∈ Cb .
(5.22)
The weak limit arises because for a 1-point observable only the behavior at infinity, defined through some unbounded sequence of A’s, is relevant. This limit does not necessarily exist, however as the O0,∞ (An↑ ) form a bounded sequence in R one can always select a convergent subsequence. As shown in Section 4.2 the limit does exist for all O ∈ CT ainv without taking subsequences and is given by (4.13). For the following discussion it is convenient to introduce a somewhat smaller class of observables which we call P -invariant: O ∈ CP inv
iff P K O = K O P .
(5.23)
One has CT inv ⊂ CP inv ⊂ CT ainv .
(5.24)
The second inclusion is trivial; the first inclusion follows by taking the x → ∞ limit Tβ (1; x)−1 [K O , Tx ] = 0. The condition (5.23) is chosen such that O0,∞ (n0 ) is independent of n0 , so that by (4.32) the value of O0,∞ directly coincides with the thermodynamic limit of O ∈ CP . By a computation similar to the one in (4.32) one shows from (5.22) that for separately SO↑(2) invariant A, B ∈ CP↑ inv one has the ‘hyperclustering’ relation ω∞ (A B) = ω∞ (A) ω∞ (B) .
(5.25)
Since in general AB = A B this does not extend to all of CP inv . The above properties of CP↑ inv observables render them at the same time uninteresting from the viewpoint of the OS reconstruction. More generally we have Proposition 5.2 Observables O ∈ C+ are mapped onto multiples of the ‘canonical’ ground state ψ0 in HOS if and only if the following ‘hyperclustering relation’ holds. ω∞ ([ϑO] O) = ω∞ (ϑO) ω∞ (O) .
(5.26)
Sufficient conditions for (5.26) to hold are: (i) O 0,∞ (An↑ ) in (5.22) has a unique ↑ ∪ CP↑ inv . (and hence invariant ) limit as A → ∞. (ii) O ∈ Cainv
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Proof. The relation (5.26) is equivalent to O−ω∞ (O) ∈ N being a null vector. This in turn is equivalent to O and ω∞ (O) giving rise to the same vector in HOS . But the latter is a multiple of the ground state, as asserted. The condition (i) is sufficient because the Cauchy-Schwarz inequality then implies ω∞ ([O−ω∞ (O)]A) = 0 for all A ∈ C+ , which for A = ϑ[O −ω∞ (O)] amounts to (5.26). The fact that observables ↑ in Cainv or in CP↑ inv have hyperclustering expectations has been seen before.
5.3
The action of SO(1, 2) on HOS .
Next let us consider the action of SO(1, 2) on the reconstructed Hilbert space. Both (5.10) and (5.11) have well-defined limits for L → ∞ given by (ρ∞ (A)ψ)(n) = ψ(A−1 n)
P−1/2 (A−1 n · n↑ ) , P−1/2 (n · n↑ )
(ρ∞ (A)O)(nx0 , . . . , nx−1 ) = O(A−1 nx0 , . . . , A−1 nx−1 )
ψ ∈ C0 ,
(5.27)
P−1/2 (A−1 nx−1 · n↑ ) , P−1/2 (nx−1 · n↑ ) O ∈ C+ .
Here ρ∞ (A) is a well-defined bounded linear map from C+ onto itself because the quotient P−1/2 (A−1 n · n↑ )/P−1/2 (n · n↑ ) is a bounded continuous function with a bounded inverse. One readily verifies the representation property ρ∞ (A) (ρ∞ (B)ψ)(n) = (ρ∞ (AB)ψ)(n). Further the action on C+ is again compatible with that on C0 and induces it via (5.14), namely: (ρ∞ (A)O)0 (n) = (ρ∞ (A)O0 )(n), for all A ∈ SO(1, 2). In particular this ensures that the null space N and its completion are mapped onto itself under ρ∞ . For clarity’s sake let us add the reminder that for O ∈ C+ the assignment of x −1 ≥ 0 as the index of the last argument on which O actually depends is ambiguous since one may always consider a constant dependence on further arguments; see the comment after Eq. (3.4). Proposition 5.3 (i) The representation ρ∞ of SO(1, 2) on HOS is uniformly bounded and measurable. (ii) It does not act unitarily on all of HOS . Remark 1. Uniform boundedness means that supA ρ∞ (A)ψOS < ∞, measurability of the representation means that the functions A → (ψ1 , ρ∞ (A)ψ2 )OS and A → (ρ∞ (A)ψ1 , ψ2 )OS are measurable wrt the Haar measure on SO(1, 2). Remark 2. The fact (ii) may be surprising at first sight, upon second thought it is not: the inner product ( , )OS is constructed in terms of the limiting expectation functional w−limL→∞ ωL = ω∞ , and we already know that this functional is not ρ invariant for all O ∈ Cb . Of course ρ∞ is different from ρ but it seems ‘unlikely’ that the universal ratio P−1/2 (A−1 n·n↑ )/P−1/2 (n·n↑ ) by which they differ could ‘undo’ the symmetry breaking for all of the relevant observables at the same time. As a consequence the ‘square root’ of a bounded observable signaling the ρ symmetry breaking is likely to give rise to a wave function in C0 on which the unitarity of ρ∞ is violated.
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Proof of Proposition 5.3. (i) By (C.1) in fact ρ∞ (A)ψOS ≤ |ψ|2 ∞ using (C.1). Measurability follows from the fact that for each L the functions A → ωL (ψ1∗ ρ∞ (A)ψ2 ) and A → ωL ((ρ∞ (A)ψ1∗ ) ψ2 ) are in L∞ (SO(1, 2)). By construction of the state ω∞ ( . ) = w − limL→∞ ωL ( . ) the L → ∞ limit of the above functions exists pointwise for almost all A ∈ SO(1, 2) wrt the Haar measure. On general grounds the limiting functions are therefore measurable. As a warning we should add that for generic ψ1 , ψ2 continuity in A may be lost in the limit, as we shall see later. (ii) It suffices to give examples. One class is provided by wave functions only depending on the SO↑(2) phases. Consider ψl (n) := eilϕ , with l ∈ Z and ϕ = arctan(n1 /n2 ). Then 0 = (ψl , ψl )OS = (ρ∞ (A)ψl , ρ∞ (A)ψl )OS ,
l = l .
(5.28)
An example for the square root construction mentioned in Remark 2 is Se (n) := [Te (n)]1/2 ,
(5.29)
where Te (n) is the symmetry breaking observable of Eq. (4.51) and the principal branch of the square root is taken. Then T q (∞) = (Se , Se )OS = (ρ∞ (A)Se , ρ∞ (A)Se )OS 2 P−1/2 (n·An↑ ) = lim Te (A−1 n) , P−1/2 (n·n↑ ) n↑ ·n→∞
(5.30)
for A ∈ SO(1, 2)/SO↑(2) and with the overbar referring to the SO↑(2) average. Of course one could also extend the action of ρ from L2 (H) to C0 and thereby to HOS in the obvious way. It acts, however, uninterestingly: first of all ψ0 is ↑ are mapped onto a multiple of mapped onto itself, likewise all elements of Cainv ψ0 . Thus ρ acts ‘unitarily’ on multiples of ψ0 by not acting at all, and since ↑ symmetry breaking is generic, ρ cannot be expected to outside of the class Cainv act unitarily on sizeable subspaces of HOS . On which subspaces of HOS does ρ∞ act unitarily? Let us introduce the 0 be the closed linear subspace generated by following subsets of HOS : first let HOS the ‘ground state orbit’ {ψ ∈ HOS | ψ = ρ∞ (A)ψ0 , A ∈ SO(2, 1)} ,
(5.31)
α be the closed linear subspace generated by and HOS
{ψ ∈ HOS | ψ = ρ∞ (A)ψα , A ∈ SO(2, 1)} ,
α ∈ R \ {0} ,
(5.32)
0 with ψα (n) = exp(iα n↑ ·n). HOS does not change if we allow the coefficients to be ↑ 0 α from Cainv . HOS and HOS are by construction invariant subspaces of HOS under the action of the representation ρ∞ . It is convenient to introduce the notation
ψn0 ,α (n) :=
P−1/2 (n0 · n) iαn0 ·n e , P−1/2 (n↑ · n)
n0 ∈ H , α = 0 ,
(5.33)
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for the basis vectors; then ρ∞ acts simply by ‘rotating’ n0 , i.e., ψn0 ,α → ψAn0 ,α , A ∈ SO(1, 2). Note that this action inherits the properties of the action of SO(1, 2) on H. As such it is transitive and effective but not free. It is not free because An = n for some fixed n ∈ H implies only that A is in stability group of n. The action is manifestly transitive and also effective in that An = n for all n ∈ H implies p (p being mnemonic for ‘phase’ or ‘polymer’) as the closed A = 1. We define HOS subspace generated by all the vectors ψn0 ,α , α = 0, in (5.33). p 0 and HOS : We now describe how ρ∞ acts on HOS p 0 Theorem 5.4 HOS and HOS are orthogonal subspaces of HOS . ρ∞ acts unitarily 0 on both of these subspaces; the action is continuous on HOS , but discontinuous on p 0 HOS . Furthermore, on HOS one has (ρ∞ (A)ψ0 , ρ∞ (B)ψ0 )OS = P−1/2 (An↑ · Bn↑ ) ,
(5.34)
p whereas on HOS one has
(ψn1 ,α1 , ψn2 ,α2 )OS = δn1 ,n2 δα1 ,α2 ,
∀ n1 , n2 ∈ H , α1 , α2 ∈ R , α1 α2 = 0 . (5.35)
Proof. The derivation of Eqs (5.34), (5.35) as well as the proof of the orthogonality of the two subspaces is somewhat technical and is deferred to Appendix C. 0 From (5.34) it is easy to see that ρ∞ acts continuously on HOS : By (5.34) the scalar product of any two elements in the orbit of ψ0 is a continuous function of the group elements, and this continuity trivially extends to finite linear combinations of elements of this orbit. Denote this linear space by D. This implies that for any φ ∈ D we have limA→0 ρ∞ (A)φ − φOS = 0. Now for any element 0 and any > 0 there is a φ ∈ D such that φ − ψOS < . By the ψ ∈ HOS triangle inequality therefore limA→0 ρ∞ (A)ψ − ψOS ≤ , and since was arbitrary, limA→0 ρ∞ (A)ψ − ψOS = 0 follows. The group
structure of SO(1,2) yields (strong) continuity of the whole representation ρ∞ H0 . The discontinuity of the p is obvious from (5.35). action of ρ∞ on HOS
OS
Corollary 5.5 HOS is nonseparable. Proof. The vectors (5.33) provide an explicit nondenumerable orthonormal family. Remark. The representation ρ∞ of SO(1, 2) acts as a kind of ‘nondenumerable discrete permutation group’, ρ∞ (A)ψn0 ,α = ψAn0 ,α , on the orthonormal family (5.33) (see Appendix C). The above result should be viewed in the context of an alternative described by Segal and Kunze ([46], p. 274) which characterizes measurable unitary representations π of some locally compact group G on a nonseparable Hilbert space H. Namely let Hs be the subspace of all vectors ψs in H such that for all ϕ ∈ L1 (G)
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and all ψ ∈ H one has
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dµ(A) ϕ(A)(ψ, π(A)ψs ) = 0 .
(5.36)
G
Then H is the direct sum of two invariant subspaces H = Hc ⊕ Hs . The restriction of π to Hc is continuous while the restriction to Hs is singular, in the sense that (ψs , π(A)ψs ) = 0 for almost all A ∈ G and all ψs ∈ Hs . If H is separable Hs is absent, as follows from a theorem of von Neumann (see [30], Theorem VIII.9). u In our case we denote by HOS the maximal closed subspace of HOS on u which ρ∞ is unitary and measurable. The above alternative entails that HOS u c s c decomposes into HOS = HOS ⊕ HOS , where the restriction of ρ∞ to HOS and s HOS is continuous and singular, respectively. Our results amount to the explicit construction of subspaces 0 c ⊂ HOS , HOS
p s HOS ⊂ HOS ,
(5.37)
together with a formula for the inner products. In particular the singular subspace is non-empty (for which the nonseparability of the Hilbert space is a necessary but p s ⊂ HOS not a sufficient condition). The assumption (5.36) is satisfied for ψ ∈ HOS because (5.35) projects onto a 1-dimensional submanifold of the group which has zero measure wrt the full Haar measure. The restriction of ρ∞ to Hp ⊂ Hs is indeed singular; in fact by (5.35) (ψn0 ,α , ρ∞ (A)ψn0 ,α )OS = 0 holds for all A = 1. The simple explicit action ρ∞ (A)ψn0 ,α = ψAn0 ,α as a permutation group (acting transitively and effectively for fixed α) is somewhat surprising. The continuous 0 will later be identified as the ground state sector of the reconstructed subspace HOS transfer operator. As outlined in Appendix C there are other nondenumerable orthonormal p 0 and HOS . We did not explore families in HOS which are orthogonal to both HOS the action of ρ∞ on them, but it may well be that HOS contains other invariant subspaces on which ρ∞ acts unitarily. In this case they would likewise be subject to the above continuous-discontinuous alternative and render the inclusions in (5.37) proper. We proceed with the construction of a transfer operator TOS on HOS , which 0 in particular will justify the term ‘ground state sector’ for HOS (see Proposition 5.6 below). To this end let τ be the map from C+ to C+ that shifts all variables by 1 unit to the right, i.e., τ O(nx1 , . . . , nx ) = O(nx1 +1 , . . . , nx +1 ). τ satisfies the relation τ ϑτ = ϑ; it maps N into itself as can be seen by using the Cauchy-Schwarz inequality and translation invariance ω∞ (ϑ[τ O]τ O) = ω∞ ([ϑO] [τ 2 O]) ≤ ω∞ ([ϑO] [τ 4 O])1/2 ω∞ ([ϑO] O)1/2 .
(5.38)
τ therefore induces a well-defined operator TOS on the equivalence classes modulo N , and hence on HOS . By translation invariance TOS is symmetric. Once known to be bounded it extends to a unique selfadjoint operator on HOS . The boundedness
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follows by iterating the Cauchy-Schwarz inequality (using a classic argument of Osterwalder and Schrader) ω∞ ([ϑO] τ 2 O) ≤ ω∞ ([ϑO] τ 4 O)1/2 ω∞ ([ϑO] O)1/2 ≤ · · · n+1
≤ ω∞ ([ϑO] τ 2
−n
O)2
−n
ω∞ ([ϑO] O)1−2
.
(5.39)
−n+1
The first factor is bounded by O2 , which goes to 1 as n → ∞; the second factor goes to ω∞ ([ϑO] O), which proves that T2OS ≤ 1 and thus also TOS ≤ 1. Importantly, the vector ψ0 corresponding to O = 1 is an eigenvector (of norm 1) of the reconstructed transfer operator TOS with eigenvalue 1 = TOS OS , i.e., ψ0 is a ground state of the system. Already the mere existence of at least one normalizable ground state indicates that the reconstructed quantum mechanics given by (HOS , TOS ) is very different from the original one given by (L2 (H), T). This mismatch can be traced back to the purely continuous spectrum of the original system, which in turn stems from the noncompactness of the target space H. A further drastic discrepancy is the nonseparability of HOS . Similar surprising features arise already in the much simpler model with flat target space R, on which R also acts as an amenable symmetry. This example is also instructive because it shows that in the limit of an amenable symmetry the symmetry breaking disappears. We therefore discuss this example briefly in Appendix B. Returning to the hyperbolic model, we summarize the properties of TOS : Proposition 5.6 TOS is a self-adjoint operator on HOS with following properties: (i) ||TOS || = 1 . (ii) ρ∞ ◦ TOS = T OS ◦ ρ∞ . (iii) TOS H0 = 1 H0 . OS
OS
0 (iv) HOS = {ψ ∈ HOS | TOS ψ = ψ}. (v) TOS acts on C+ /N , i.e., on the representatives (5.14) as x −x ↑ −1 (TOS ψ)(n) = λβ (0) P−1/2 (n·n ) dΩ(n )Tβ (n·n ; x)ψ(n )P−1/2 (n ·n↑ ) ,
(5.40) up to an element of N . Remark. (ii) and (iv) show that TOS , in contrast with T, has at least some point spectrum. Despite the concrete expression in (v), it seems difficult to say more 0 . about the spectrum of TOS outside the vacuum space HOS Proof. (i) has already been shown; it is a general feature of the OsterwalderSchrader reconstruction. (ii) Recall that TOS is defined in terms of the shift τ on C+ . As τ trivially commutes with the ρ∞ action (5.27) of SO(1, 2) on C+ and both τ and ρ∞ preserve the nullspace N , the same will be true for TOS induced on the equivalence classes. This gives (ii).
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0 (iii) A simple consequence of (ii) is that TOS acts like the identity on HOS , because τ acts like the identity on the constants, in particular on the unit observable O = 1, corresponding to the ‘canonical’ vacuum ψ0 . By (ii) the same must 0 . Equivalently, the eigenspace of TOS of eigenvalue 1 hold for all elements of HOS 0 contains HOS . (iv) Let O ∈ C+ be such τ O − O ∈ N . Then also τ O0 − O0 ∈ N , because τ O − O = τ O0 − O0 + τ (O − O0 ) − (O − O0 ), and the last two terms on the rhs are in N . By the remark after (5.14) therefore (τ O0 )0 − O0 ∈ N ∩ C0 , and it suffices to consider the exact identity (τ O0 )0 = O0 . From the definition of the map (5.14) one sees that all solutions ψ ∈ C0 of (τ ψ)0 = ψ are such that ψ(n)P−1/2 (n · n↑ ) is an eigenfunction of Tβ of eigenvalue λβ (0). From (2.42) one infers that the solutions lie in the closed subspace of C0 spanned by ratios of the form (5.33) with α = 0. (v) Since (τ O0 )0 − (τ O)0 ∈ N , for all O ∈ C+ , one can use (5.14) to compute the action of τ on the representative O0 as before. One finds (5.40), first for x = 1 and then by iteration for all x ∈ N. 0 . This follows In addition to acting unitarily, ρ∞ also acts irreducibly on HOS directly from the definition (5.31). Alternatively one can use the addition theorem (A.12c) to replace the generating set ρ∞ (A)ψ0 , A ∈ SO(1, 2), by the alternative l generating set P−1/2 (ξ)/P−1/2 (ξ), l ∈ Z. These functions are known to span an irreducible and unitary representation of SO(1, 2); in the Bargmann classification it corresponds to the limit of the discrete series. To sum up, we have found that the space HOS is nonseparable and that it carries a representation ρ∞ of the symmetry group. This representation acts unitarily and discontinuously on a nonseparable proper subspace of HOS , and 0 unitarily and continuously on the separable subspace of ground states HOS of the reconstructed transfer operator TOS . This ground state sector is irreducible and can be described explicitly as
l P−1/2 (n↑ ·n)
P−1/2 (n↑ ·A−1 n)
0 HOS
A ∈ SO(1, 2)
l ∈ Z ⊂ HOS , P−1/2 (n↑ ·n)
P−1/2 (n↑ ·n)
(5.41) where the symbol ‘’ denotes equality of the span of the lhs and rhs for the 0 is given by (ρ∞ (A)ψ0 , equivalence classes modulo N . The inner product on HOS ↑ ↑ ρ∞ (B)ψ0 )OS = P−1/2 (An ·Bn ).
6 Conclusions and outlook We have found that the concept of spontaneous symmetry breaking for a nonamenable continuous internal symmetry group differs in some crucial ways from the familiar situation of an amenable symmetry group: • Symmetry breaking is unavoidable, even in dimensions 1 and 2, where it is forbidden for an amenable continuous symmetry. In one dimension the (improper) ground state in the quantum mechanical interpretation is infinitely
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degenerate; in the statistical mechanics interpretation invariant states over a ‘large’ algebra cannot be defined by group averaging. These features have been worked out in some detail for an analytically solvable model, the hyperbolic spin chain with symmetry group SO(1, 2). • In this 1-dimensional model, however, there is still some vestige of the large fluctuations that are responsible for the symmetry restoration in the compact and abelian models: the sequence of functional measures defined through the thermodynamic limit becomes concentrated at configurations ‘at infinity’ of the hyperbolic plane. As a consequence a certain subclass of non-invariant observables gets averaged to yield an invariant result. The limit of the functional measures provides an invariant mean for this subclass of observables, while outside this class symmetry breaking is generic. • While the quantum mechanics described by our model can be simply interpreted as motion of a particle on the hyperbolic plane, with absolutely continuous spectrum of the transfer matrix, the Osterwalder-Schrader reconstruction based on the infinite volume expectation values yields some surprises: the reconstructed transfer matrix has at least some point spectrum, in particular it has normalizable ground states, and the full reconstructed Hilbert space is nonseparable. These features are, however, due to the noncompact nature of the symmetry group, not its nonamenability, as can be seen from the ‘flat’ analogue discussed in Appendix B. • The Osterwalder-Schrader reconstructed Hilbert space has a nonseparable proper subspace on which a unitary representation of SO(1, 2) acts discontinuously as a kind of ‘nondenumerable discrete permutation group’, not unlike the way the spatial diffeomorphism group acts on the embedded graphs in the framework of [22, 23]. In contrast, the space of ground states of the reconstructed transfer operator is separable and a nontrivial unitary representation of SO(1, 2) acts on it continuously and irreducibly. These features are specific to the case of a nonamenable symmetry and are not present in the ‘flat’ case. In a follow-up paper we study these issues in the D-dimensional (D ≥ 2) version of the model, i.e., the nonlinear sigma-model with a hyperbolic targetspace; see, e.g., [18, 42, 19] for earlier investigations. There we use a combination of analytical techniques and of numerical simulations [14]. We also expect that there will still be a marked difference between dimensions D ≤ 2 and D ≥ 3: whereas in the low dimensional case there is, as stated above, dominance of highly boosted configurations, we expect that in D ≥ 3 spontaneous symmetry breaking in the usual sense takes place, showing normal, approximately Gaussian fluctuations around a fully ordered state, in which for instance unbounded observables like n00 have finite expectation values. Some time after the first version of this paper was posted on the web, a paper by Spencer and Zirnbauer [47] appeared, which showed that indeed in dimensions D ≥ 3 at low temperature the suitable defined spin fluctuations have finite moments.
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It would be interesting to elucidate the physical meaning of the unavoidable spontaneous symmetry breaking in the context of Anderson localization, in which such nonlinear sigma models were studied for instance in [7, 8, 9, 10, 11]. In order not to blur the discussion with (further) technicalities we contrasted here only the simplest compact and noncompact symmetric spaces. However the situation would be similar if the sphere S 2 SO(3)/SO(2) and H SO(1, 2)/SO(2) were replaced with any other dual pair of compact and noncompact Riemannian symmetric spaces (see [26] for the propagators). A further generalization would be to consider a similar dynamical system where the variables take values in an arbitrary Riemannian manifold. In particular this would allow one to examine the interplay between invariant dynamics and non-invariant states for the diffeomorphism group of the target manifold. Finally we cannot resist mentioning a potential application to quantum gravity. Supposing that in a suitable topology an appropriate version of the diffeomorphism group is nonamenable, variants of the above concepts become applicable. This would suggest a scenario in which there is no diffeomorphism invariant ground state, yet a family of selected observables has invariant expectations in each of an infinite set of ground states, while outside this family spontaneous collapse of diffeomorphism invariance is generic.
Appendix A: Harmonic analysis on H Let a · b = a0 b0 − a1 b1 − a2 b2 be the bilinear form of R1,2 and let SO0 (1, 2) =: SO(1, 2) be the component of its symmetry group connected to the identity. Consider the hyperboloid H = {n ∈ R1,2 | n · n = 1 , n0 > 0}. It is isometric to the symmetric space SO(1, 2)/SO(2) and can be parameterized either by points (∆, B), ∆ > 0, B ∈ R, in the Poincar´e upper half plane, or by geodetic polar coordinates (ξ, ϕ), ξ ≥ 1, −π ≤ ϕ < π, via 1 + ∆2 + B 2 B = ξ, n1 = − = ξ 2 − 1 sin ϕ , 2∆ ∆ 2 2 −1 + ∆ + B = ξ 2 − 1 cos ϕ . = 2∆
n0 = n2
(A.1)
The (ξ, ϕ) parameterization is adapted to a preferred SO(2) subgroup of SO(1, 2) ↑ which leaves n↑ = (1, 0, 0) invariant and which we denote by SO (2). We also note the relations ∆−1 = ξ − ξ 2 − 1 cos ϕ, B = 1 − ξ −2 sin ϕ/( 1 − ξ −2 cos ϕ − 1). For the invariant distance n · n ≥ 1 of two points n, n ∈ H, one has n · n =
2
∆2 + ∆ + (B − B )2 2 = ξξ − (ξ 2 − 1)1/2 (ξ − 1)1/2 cos(ϕ − ϕ ) . (A.2) 2∆∆
Function spaces on H come naturally equipped with the inner product (ψ1 , ψ2 ) = dΩ(n) ψ1 (n)∗ ψ2 (n) ,
(A.3)
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induced by the invariant measure dΩ(n) := 2d3 n δ(n2 − 1) θ(n0 ), which translates into dΩ(∆, B) = dBd∆∆−2 and dΩ(ξ, ϕ) = dξdϕ, respectively. As indicated we shall freely switch back and forth between the different parameterizations. The Schwartz space S(H) is defined as the space of smooth functions on H decaying faster than any power of B and ∆. The space of tempered distributions S (H) on it together with L2 (H) form a Gel’fand space triple S(H) ⊂ L2 (H) ⊂ S (H) .
(A.4)
The SO(1, 2) rotations of the ‘spins’ n induce a unitary representation ρ on S(H) via ρ(A)ψ(n) = ψ(A−1 n), A ∈ SO(1, 2). On integral operators K with kernel κ(n, n ) it acts as K → ρ(A)−1 Kρ(A) and thus as κ(n, n ) → κ(An, An ) on the kernels. Invariant operators have kernels depending on the inner product n · n only. Similarly operators invariant under ρ restricted to the SO↑(2) subgroup have kernels depending on ξ, ξ and the relative angle ϕ − ϕ only. In general the representation ρ will not be irreducible. Generic functions in S(H) can be expanded into a generalized Fourier integral whose basis functions form unitary irreps of SO(1, 2). Moreover these basis functions comprise S (H) eigenfunctions of the Laplace-Beltrami operator. To make this concrete consider the Killing vectors of H, which generate the Lie algebra sl2 e = ∂B ,
h = 2(B∂B + ∆∂∆ ) ,
[h, e] = −2e , [h, f ] = 2f ,
f = (∆2 − B 2 )∂B − 2B∆∂∆ , [f , e] = h ,
(A.5)
and are anti-hermitian wrt ( , ). Up to a sign the quadratic Casimir coincides with the Laplace-Beltrami operator −C :=
1 ∂2 ∂ 2 ∂ 1 2 1 2 2 h + (ef + fe) = ∆2 (∂∆ (ξ − 1) + 2 + ∂B )= . (A.6) 4 2 ∂ξ ∂ξ ξ − 1 ∂ϕ2
If one just blindly lets the differential operators e, h, f act on the spins (A.1) (which are not elements of L2 (H)) one sees that they act as 3 × 3 matrices t(e), t(h), t(f ) with Casimir C = −213 ; the matrices are however not (anti)hermitian even though the original differential operators (multiplied by i) are essentially self-adjoint on S(H) ⊂ L2 (H). The exponentiated differential operators therefore extend to the unitary action of SO(1, 2) on L2 (H) ρ(e−st(x) )ψ(n) = esx ψ(n) = ψ(est(x) n) ,
x = e, h, f ;
s ∈ R.
(A.7)
A more explicit description of the exponentiated differential operators is possible on irreducible representations. Simultaneous eigenstates of C and e are given by ω,k (n) := ω,k (∆, B) = ∆1/2 Kiω (|k|∆) eikB , with ω,0 (n) := ω,0 (∆, B) = ∆iω+1/2 , 1 + ω 2 ω,k , e ω,k = ik ω,k , C ω,k = 4
k = 0 .
ω > 0,
(A.8)
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where Kν (x) is a modified Bessel function defined, e.g., by Kν (β) = cosh(νt)dt. The Fourier inversion on S(H) takes the form ∞ dω k) ω,k (n) ψ(n) = ω sinh πω dk ψ(ω, π3 0 R k) = ψ(ω, dΩ(n) ψ(n)ω,k (n)∗ .
∞ 0
e−β
cosht
(A.9)
Simultaneous eigenstates of C and e − f , i.e., of the SO↑(2) rotations are given by l (ξ) , l ∈ Z, ω > 0, ω,l (n) := ω,l (ξ, ϕ) = eilϕ P−1/2+iω 1 + ω 2 ω,l , C ω,l = (e − f ) ω,l = il ω,l , 4
(A.10)
where Psl (ξ) are Legendre functions, defined, e.g., by Psl (ξ) =
Γ(s + l + 1) 2πΓ(s + 1)
2π
du eilu [ξ +
0
ξ 2 − 1 cos u]s ,
ξ ≥ 1.
(A.11)
We further note the following properties Γ(s + 1 + l) −l P (ξ) , Γ(s + 1 − l) s
(A.12a)
(−)l δ(ω − ω ) , ω tanh πω
(A.12b)
l (ξ) = Psl (ξ) = P−s−1
∞
1
−l l dξ P−1/2+iω (ξ)P−1/2+iω (ξ) =
2 Ps ξξ − (ξ 2 − 1)1/2 (ξ − 1)1/2 cos ϕ = (−)l eilϕ Ps−l (ξ) Psl (ξ ) ,
(A.12c)
l∈Z
as well as the asymptotics for ξ → ∞ Γ(iω) (2ξ)−1/2+iω + c.c. , √ πΓ( 12 + iω − l)
l P−1/2+iω (ξ) ∼
ω > 0,
ln ξ 2 √ . √ 1 πΓ( 2 − l) 2ξ
l P−1/2 (ξ) ∼
(A.13)
The Fourier inversion in the basis (A.14) takes the form ∞ dω l l) ω,−l (n) , ω tanh πω ψ(ω, (−) ψ(n) = 0 2π l∈Z
l) = ψ(ω,
dΩ(n) ψ(n)ω,l (n) .
(A.14)
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In group theoretical terms the expansions (A.9), (A.14) correspond to the decomposition of the unitary representation ρ on L2 (H) into a direct integral of unitary irreducible representations, namely those of the type 0 principal series in the Bargmann classification, see, e.g., [43]. In terms of the representation spaces ⊕ 2 dµ(ω) Cω (H) , (A.15) L (H) = with the spectral weight dµ(ω) = dω 2π ω tanh ω. We shall frequently encounter SO↑(2) invariant functions ψ = ψ(ξ), for which (A.14) reduces to the Mehler-Fock transform ∞ dω ω tanh(πω) P−1/2+iω (ξ) ψ(ω) , ψ(ξ) = 0 2π ∞ 0) . ψ(ω) = 2π dξ P−1/2+iω (ξ) ψ(ξ) = ψ(ω, (A.16) 1
It holds in the classical sense provided ∞ 2 dξ|ψ(ξ)| < ∞ ⇐⇒ 1
∞ 0
2 |ψ(ω)| ω tanh πω < ∞ ,
(A.17)
see, e.g., [44]. It is possible, however, to interpret the Mehler-Fock transform in the distributional sense and therefore give it a wider range of applicability. The Fourier decomposition of a kernel κ(n, n ) defining an integral operator K makes some of its properties manifest. Subject to suitable regularity conditions the generic form of the expansion wrt the basis (A.14) is
κ(n, n ) =
l1 +l2
∞
(−)
l1 ,l2 ∈Z
0
dω1 dω2 ω1 thπω1 ω2 thπω2 κ l1 .l2 (ω1 , ω2 ) 2π 2π ω1 ,−l1 (n) ω2 ,−l2 (n ) , (A.18)
Depending on the properties of the spectral weight κ l1 ,l2 (ω1 , ω2 ) = (ω1 ,l1 , Kω2 ,l2 ) the corresponding integral operator K will enjoy certain bonus properties: κ l1 ,l2 (ω1 , ω2 ) =
2π κl1 ,l2 (ω1 ) δ(ω1 − ω2 ) ω1 tanh πω1
κ l1 ,l2 (ω1 , ω2 ) = δl1 +l2 ,0 κ l1 (ω1 , ω2 ) κ l1 ,l2 (ω1 , ω2 ) = δl1 +l2 ,0
2π κ(ω1 ) δ(ω1 − ω2 ) ω1 tanh πω1
translation inv. (A.19a) SO↑(2) inv.
(A.19b)
SO(1, 2) inv.
(A.19c)
For K itself these properties amount to a vanishing commutator with T, ρ|SO↑(2) and ρ, respectively. The fact that the spectral weights (A.19c) lead to SO(1, 2)
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invariant operators follows from (A.12c); the kernels κ(n, n ) of these operators depend on the invariant distance n · n only. As an example of a spectral decomposition consider the transfer matrix itself where the weights are just the eigenvalues (2.3). Using, e.g., [28], p.804 and the completeness relation for the Legendre functions one verifies ∞ dω β exp{β(1 − n · n )} = ω tanh πω P−1/2+iω (n · n ) λβ (ω) . (A.20) 2π 2π 0 In representation theoretical terms this expresses the exponential of a singlet wrt the non-unitary vector irrep as a superposition of singlets wrt the unitary irrep (A.15). We note that the inverse Mehler-Fock transform (A.16) gives ∞ λβ (ω) = β dξeβ(1−ξ) P−1/2+iω (ξ). (A.21) 1
Clearly the integral kernels κ(n · n ) that give rise to well-defined operators on L2 (H) must have suitable regularity and decay properties. The asymptotics in (A.13) suggests that the kernels κ(ξ) should also decay at least like ξ −1/2 . Some decay stronger than ξ −1/2 is also necessary in order for κ to be the integral kernel of a densely defined operator from L2 (H) to L2 (H). A sufficient condition seems to be more difficult to obtain, but in any case kernels like n · n do not correspond to densely defined operators on L2 (H) (they give rise only to densely defined quadratic forms). The integrands of the Legendre functions (A.11) likewise provide eigenfunctions of the Laplace-Beltrami operator (A.6). Explicitly (A.22) Eω,u (n) := Eω,u (ξ, ϕ) = [ξ − ξ 2 − 1 cos(u − ϕ)]−1/2−iω , are bounded complex solutions for all |ϕ − u(mod2π)| > > 0), decaying like ξ −1/2+iω for ξ → ∞. √ The upper bound will diverge as → 0 because for φ = u one has |Eω,u (ξ, u)| ∼ 2ξ, for ξ → ∞. The Legendre functions (A.11) are recovered as the Fourier modes of (A.22) and vice versa. The orthogonality and completeness relations take the form (2π)2 δ(ω − ω )δ(θ − θ ) , (Eω,θ , Eω ,θ ) = ω tanh πω 2π ∞ 1 dω ω tanh ω dθ Eω,θ (n)∗ Eω,θ (n ) = δ(n, n ) . (A.23) (2π)2 0 0 The main virtue of these solutions is their simple transformation law under SO(1, 2), see, e.g., [45]. For a boost A−1 = A(θ, α)−1 mapping ξ = n0 into ξchθ − shθ cos(ϕ − α) one has Eω,u (A−1 n) = [chθ + cos(ϕ − α)shθ]−1/2−iω Eω,u (n) ,
(A.24)
for some angle u = u (θ, α). This is also a convenient starting point to show that the Fourier decomposition (A.14) indeed has the representation theoretical significance (A.15), see, e.g., [43].
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Appendix B: Flat noncompact spin chain In order to elucidate the relation between the original Hilbert space and the one obtained by Osterwalder-Schrader reconstruction it is useful to consider the simplest noncompact spin chain where the target space is R. The symmetry group in this case is also R, which in contrast to SO(1,2) is amenable. Some of the unusual aspects of this model have been analyzed already in [21]. Of course all results generalize trivially to target spaces Rn , n > 1. We consider the Hilbert space L2 (R) and take as the one-step transfer matrix simply the heat kernel exp[β −1 ∆](u, v), so that ∞ x (T ψ)(u) = dv Tβ (u − v; x) ψ(v) , x ∈ N , −∞
Tβ (u; x) =
β β exp − u2 . 2πx 2x
(B.1)
The transfer operator trivially commutes with the action of R on the wave functions, i.e., T ◦ ρ = ρ ◦ T, with ρ(a)ψ(u) = ψ(u − a). It is well known that the spectrum of T is continuous and covers the interval [0, 1]; the generalized eigenfunctions are imaginary exponentials. As in the hyperbolic case, a gauge fix is necessary; we simply fix the leftmost ‘spin’ u−L to 0, which is analogous to fixing n−L = n↑ . For the purpose of the Osterwalder-Schrader reconstruction we choose again in addition the bc uL = 0, i.e., we choose 0 Dirichlet conditions. As observable algebra we take C = Cb , the algebra of continuous bounded functions of finitely many variables ux1 , . . . , ux , and we introduce the subalgebras C+ , C− and C0 = C+ ∩ C− as in Section 5. For a finite chain the reconstruction of the Hilbert space HL proceeds as in Section 5; we define for each O ∈ C+ O0,L (u0 ) =
dui O(u1 , . . . , u )
i=1
and
Tβ (ui − ui−1 ; xi − xi−1 )
i=1
Tβ (u ; L − x ) , Tβ (u0 ; L) (B.2)
Tβ (u; L) ψ O (u) = O0,L (u) . Tβ (0; 2L)
(B.3)
The reconstructed Hilbert space HL is the completion of C0 wrt ωL in (B.5). It can be identified with the original L2 (R) by the isometry VL : HL −→ L2 (R) ,
Tβ (u; L) (VL ψ)(n) = ψ(u) . Tβ (0; 2L)
(B.4)
Equivalently HL can be viewed as the preimage of L2 (R) wrt VL . T (u;L) The thermodynamic limit can be readily understood here. The ratio Tβ(0;L) approaches a constant for L → ∞, signaling a unique ground state. On elements
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O0,∞ ∈ C0 the expectation functionals becomes an invariant mean, which exists in this case. There is a subspace HAP of almost periodic functions on which this mean is unique, see [4]; this subspace consists of the completion (in the Hilbert space norm defined by the mean) of the space of trigonometric polynomials. A brief account of the theory of almost periodic functions on R, which is due to H. Bohr, can be found in [48]. For ψ ∈ HAP the mean is u2 1 ψ(u) =: lim ωL (ψ) . (B.5) ω(ψ) = lim √ du exp − L→∞ L→∞ 2L 2πL A better known expression of the invariant mean on HAP is L 1 du ψ(u) , ω(ψ) = lim L→∞ 2L −L
(B.6)
see for instance [48]. By the uniqueness these two expressions have to be the same for an almost periodic ψ and it is straightforward to verify this equivalence for the dense subspace of trigonometric polynomials. The scalar product induced by this invariant mean can be written as L 1 (ψ , ψ)OS = lim du ψ(u)∗ ψ(u) , (B.7) L→∞ 2L −L and the unitarity of ρ on HAP is manifest. It might be surprising that the Hilbert space obtained by the OS reconstruction from C0 is nonseparable; but it is well known that already the space HAP is nonseparable [48]: there is an uncountable set of mutually orthonormal functions, namely the set (B.8) {ψα (u) = eiαu | α ∈ R} . One can introduce a shift automorphism τ like the one used in the hyperbolic case. From this one obtains a reconstructed transfer operator TOS acting on HOS ; in this case it is nonnegative and has again norm 1. TOS acts on C0 simply by Eq. (B.1). This shows that the functions (B.8) are eigenvectors (in the proper 1 2 α ). sense) of TOS with eigenvalue exp(− 2β The relation between the original system (L2 (R), T) and the reconstructed one (HOS , TOS ) turns out to be simply that the spectrum as a set remains the same, namely the interval [0, 1]. However there is now pure point spectrum on every point of the spectral interval and the generalized eigenfunctions become normalizable eigenstates. With respect to the representation of symmetry group R the original L2 (R) is a direct integral of the one-dimensional irreducible representations on the imaginary exponentials (B.8), whereas HAP is (and hence HOS contains) a direct sum over the continuous parameters α: Hα , (B.9) HOS ⊃ HAP = α∈R
where Hα is the one-dimensional Hilbert space spanned by eiαu .
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Let us end this appendix with the remark that the space HAP , huge as it is, is still only a small subspace of the full space HOS . It turns out that there are uncountably many more functions orthogonal to the exponentials discussed so far, for instance the functions pα (u) = |u|iα . Using distributional Fourier transformation one can show that (pα , ψα )OS = 0 ,
∀α = 0 , α ∈ R.
(B.10)
Presumably these functions belong to the continuous spectrum overlaying the point spectrum we have found. p 0 Appendix C: Inner products on HOS and HOS 0 Here we derive the formulas (5.34) and (5.35) for the inner products on HOS and p ↑ HOS . We begin with (5.34), i.e., (ρ∞ (A)ψ0 , ρ∞ (B)ψ0 )OS = P−1/2 (An ·Bn↑ ). By (4.12) this is equivalent to
lim
n↑ ·n→∞
fA,B (n↑ · n) ↑
=
P−1/2 (An↑ ·Bn↑ ) ,
fA,B (n · n) :=
P−1/2 (n·An↑ ) P−1/2 (n·n↑ )
with P−1/2 (n·Bn↑ ) P−1/2 (n·n↑ )
,
(C.1)
↑ ↑ where the bar as before denotes the average over SO (2). Writing ξ = n · n and ↑ 2 2 momentarily n · An = ξξA − ξ − 1 ξA − 1 cos(ϕ − ϕA ), and similarly for n · Bn↑ , the SO↑(2) average evaluates by means of (A.12c) to
fA,B (ξ) =
l∈Z
l −l P−1/2 (ξ)P−1/2 (ξ) −l l e−il(ϕA −ϕB ) P−1/2 (ξA )P−1/2 (ξB ) . P−1/2 (ξ)2
(C.2)
The series converges uniformly in ξ: using the Cauchy-Schwarz inequality, the geometric-arithmetic mean inequality and the bound −l l (ξ)P−1/2 (ξ)| ≤ P−1/2 (ξ)2 |P−1/2
the rhs is bounded by 1 and likewise the tail of the sum can be bounded uniformly in ξ. Taking now the limit ξ → ∞ under the sum, which is permitted because of the uniform convergence of the series, one obtains −l l eil(ϕA −ϕB ) (−)l P−1/2 (ξA )P−1/2 (ξB ) = P−1/2 (An↑ ·Bn↑ ) , lim fA,B (ξ) = ξ→∞
l∈Z
(C.3) using (A.13) and (A.12c). This gives (5.34); note that the result coincides with the one obtained from the ‘correlated’ limit in (5.13).
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The derivation of (5.35) we break up in several steps. Recall the notation ψα (n) = exp(iαn↑ · n), α ∈ R \ {0}. We first show that these functions form an orthonormal system (ψα , ψα )OS = 1 ,
for α = α .
(ψα , ψα )OS = 0 ,
The normalization is clear. For the orthogonality consider for α = 0 ∞ Tβ (ξ; L)2 . Iα (L) := dξ eiαξ Tβ (1; 2L) 1 To analyze this expression we integrate by parts and obtain 2 ∞ T ∂ (ξ; L) Tβ (1; L)2 β Iα (L) = − dξeiαξ . eiα + iαTβ (1; 2L) ∂ξ Tβ (1; L) 1
(C.4)
(C.5)
(C.6)
The first term is O(L−3/2 ) by (2.36); the modulus of the second term can be bounded, using the monotonicity of Tβ (ξ; L) by −
Tβ (1; L)2 αTβ (1; 2L)
∞
dξ 1
∂ ∂ξ
Tβ (ξ; L) Tβ (1; L)
2 =
Tβ (1; L)2 , αTβ (1; 2L)
(C.7)
which is also O(L−3/2 ). Together, limL→∞ Iα (L) = 0 and (C.4) is proven. We remark that this construction readily generalizes to all wave functions oscillating ‘sufficiently fast’ as ξ → ∞. Consider ψp (n) = exp i
ξ
du p(u)
with
1
(ln ξ)2 = 0. ξ→∞ ξp(ξ) lim
(C.8)
Then every pair of wave functions ψp1 (n), ψp2 (n), where the difference p1 (ξ)−p2 (ξ) is strictly monotonous for sufficiently large ξ and obeys the decay condition in (C.8) is orthogonal: (ψp1 , ψp2 )OS = 0, using Lemma 2.2 (iii) to get bounds uniform in L for the ξ → ∞ limits. For example exp{iα(ln ξ)4 }, α ∈ R, provides another nondenumerable orthonormal family, each member of which is orthogonal to each of the plain exponentials in (C.4). Here we shall only pursue the plain exponentials ψα , α ∈ R, further. Repeating the above computations with the transformed exponentials ρ∞ (A)ψα one readily shows that they remain orthogonal if they were initially. For the computation of the norms the phases are irrelevant, so they remain unity if (ρ∞ (A)ψ0 , ρ∞ (A)ψ0 )OS = (ψ0 , ψ0 )OS = 1. This however is a special case of (5.34). Thus (ρ∞ (A)ψα , ρ∞ (A)ψα )OS = (ψα , ψα )OS ,
α, α ∈ R .
(C.9)
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In a last step we show ∀ A ∈ SO(1, 2) , α , α ∈ R , αα = 0 .
(ρ∞ (A)ψα , ψα )OS = 0 ,
(C.10)
By definition one has (ρ∞ (A)ψα , ψα )OS
=
lim 2π
L→∞
Jα (ξ, ξA ) :=
0
2π
1
∞
dξ eiα ξ Jα (ξ, ξA )
Tβ (ξ, L)2 , (C.11) Tβ (1, 2L)
dϕ −iαAn↑ ·n P−1/2 (An↑ · n) e , 2π P−1/2 (ξ)
2 where we view An↑ ·n = ξξA − (ξ 2 − 1)1/2 (ξA − 1)1/2 cos(ϕ − ϕA ) as a function of ξ, ξA and ϕ − ϕA . As anticipated by the notation Jα (ξ, ξA ) is independent of ϕA . Clearly Jα (ξ, 1) = e−iαξ and |Jα (ξ, ξA )| ≤ P−1/2 (ξA ) by the addition theorem (A.12c). We take now ξA > 1 and by (C.9) we may also assume that α = 0 and wlog α > 0 (while α ∈ R may be zero). By the argument familiar from Section 4.2 only the behavior of Jα (ξ, ξA ) for large ξ will be relevant for the inner product (C.11). We claim that 1 as ξ → ∞ , Jα (ξ, ξA ) ∼ √ Q+ (ξA )e−iαp+ (ξA )ξ + Q− (ξA )e−iαp− (ξA )ξ αξ 2 − 1, with p± (ξA ) = ξA ± ξA (C.12)
and some complex constants Q± (ξA ) nowhere zero for ξA > 1. Note that 1 < ξA < p+ (ξA ) and 0 < p− (ξA ) < 1. We first show that the rhs of (C.12) is the leading term in an asymptotic expansion of Jα (ξ, ξA ) for large ξ. The point to observe is that from (A.13) we have 1 2 − 1)1/2 cos ϕ P−1/2 (ξ)−1 ∼ , P−1/2 ξξA − (ξ 2 − 1)1/2 (ξA 2 [ξA − ξA − 1 cos ϕ]1/2 (C.13) with additive corrections of O(1/ ln ξ). Asymptotically the integral becomes √2 2π eiαξ ξA −1 cos ϕ dϕ −iαξξA Jα (ξ, ξA ) ∼ e . (C.14) 2 − 1 cos ϕ]1/2 2π [ξA − ξA 0 For large ξ this integral can now be evaluated by the method of stationary phase (see, e.g., [29]) with the result (C.12). The constants Q± (ξA ) come out as −1/2 2 −1 ξ ± 2 −1 Q± (ξA ) = 2±1/2 e±iπ/4 2π ξA ξA . (C.15) A Subleading terms in the asymptotic expansion of Jα (ξ, ξA ) could be worked out similarly, but are not needed. The properties relevant in the following are that
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|Jα (ξ, ξA )| vanishes for ξ → ∞, and that the phases are linear in ξ with the given frequencies. To make sure that these are properties of Jα (ξ, ξA ) and not just of its asymptotic expansion, we verified them numerically. With (C.12) at our disposal, the rest of the derivation of (C.10) is straightforward. Substituting (C.12) into (C.11) one shows for generic ξA the vanishing of the L → ∞ limit along the lines of (C.5)–(C.7). If both α and α are nonzero one of the ξ-dependent phases might cancel for the special boost parameter ξA = 12 ( αα + αα ). The modulus of this term in the asymptotics of Jα (ξ, ξA ) then is proportional to ↑ (αξ)−1/2 which is an element of Cainv , and the L → ∞ limit vanishes on account of (4.12). This establishes (C.10). The result (5.35) then follows by combining (C.4), (C.9), and (C.10).
Acknowledgments We like to thank A. Duncan for the enjoyable collaboration in [14]. M.N. also wishes to thank M. Lashkevich for contributing to another aspect of this project, and A. Ashtekar for asking about the reconstructed state space. E.S. would like to thank S. Ruijsenaars for helpful discussions. This work was supported by the EU under contract EUCLID HPRN-CT-2002-00325.
References [1] D. Ruelle, Statistical Mechanics, W.A. Benjamin, Reading, Mass. 1969. [2] G. Sewell, Quantum mechanics and its emergent macrophysics, Princeton UP, 2002. [3] H. Narnhofer and W. Thirring, Spontaneously broken symmetries, Ann. Inst. Henri Poincar´e 70, 1 (1999) . [4] A. Paterson, Amenability, American Mathematical Society, Providence, R.I. 1988. [5] M. Niedermaier, Dimensionally reduced gravity theories are asymptotically safe, Nucl. Phys. B 673, 131 (2003); M. Niedermaier and H. Samtleben, An algebraic bootstrap for dimensionally reduced gravity, Nucl. Phys. B 579, (2000). [6] L. Faddeev and G. Korchemsky, High energy QCD as a completely integrable system, Phys. Lett. B 342, 311 (1995); S. Derkachov, G. Korchemsky, and A. Manashov, Noncompact Heisenberg spin chains from high energy QCD, Nucl. Phys. B 617, 375 (2001); Nucl. Phys. B 661, 533 (2003). [7] F. Wegner, The mobility edge problem: continuous symmetry and a conjecture, Z. Phys. B 35, 207 (1979).
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[8] A. Houghten, A. Jevicki, R. Kenway, and A. Pruisken, Noncompact sigmamodels and the existence of a mobility edge in disordered electronic systems near two dimensions, Phys. Rev. Lett. 45, 394 (1980). [9] S. Hikami, Anderson localization in a nonlinear sigma-model representation, Phys. Rev. B 24, 2671 (1981). [10] K.B. Efetov, Supersymmetry and theory of disordered metals, Adv. Phys. 32, 53 (83). [11] K.B. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, Cambridge, U.K. 1997. [12] D. Mermin and H. Wagner, Absence of ferromagnetism or anti-ferromagnetism in one or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17, 1133 (1966). [13] R.L. Dobrushin and S.B. Shlosman, Absence of breakdown of continuous symmetry in two-dimensional models of statistical physics, Comm. Math. Phys. 42, 31 (1975). [14] T. Duncan, M. Niedermaier, and E. Seiler, Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models, hep-th/0405143. [15] H.O. Georgii, Gibbs measures and phase transitions, de Gruyter, Berlin and New York 1988. [16] F.P. Greenleaf, Amenable actions of locally compact groups, J. Funct. Anal. 4, 295 (1969). [17] P. Eymard, Moy´ennes invariantes et repr´esentations unitaires, Lecture Notes in Mathematics 300, Springer-Verlag, Berlin-New York 1972. [18] D. Amit and A. Davies, Symmetry breaking in the non-compact sigma model, Nucl. Phys. B 225, 221 (1983). [19] J.W. van Holten, Quantum noncompact sigma models, J. Math. Phys. 28, 1420 (1987). [20] D. Buchholz and I. Ojima, Spontaneous collapse of supersymmetry, Nucl. Phys. B 498, 228 (1997). [21] J. L¨offelholz, G. Morchio and F. Strocchi, Spectral stochastic processes arising in quantum mechanical models with a non-L2 ground state, Lett. Math. Phys. 35, 251 (1995). [22] A. Ashtekar, J. Lewandowski, and H. Sahlmann, Polymer and Fock representations for a scalar field, Class. Quant. Grav. 20, L1 (2003). [23] A. Ashtekar, S. Fairhurst, and J. Willis, Quantum gravity, shadow states, and quantum mechanics, Class. Quant. Grav. 20, 1031 (2003). [24] C. Grosche and F. Steiner, The path integral on the pseudosphere, Ann. Phys. 182, 120 (1988).
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[25] J. Schaefer, Covariant path integral on hyperbolic surfaces, J. Math. Phys. 38, 11 (1997). [26] R. Camporesi, Harmonic analysis and propagators on homgeneous spaces, Phys. Repts. 196, 1 (1990). [27] J.P. Anker and P. Ostellari, The heat kernel on noncompact symmetric spaces, in: Lie groups and symmetric spaces, pp. 27–46, Amer. Math. Soc. Transl. Ser.2, 210, AMS. Providence, RI 2003. [28] I. Gradshteyn and I. Ryzhik, Table of integrals and products, Academic Press, New York and London 1980. [29] F. Olver, Introduction to asymptotics and special functions, Academic Press, New York and London 1978. [30] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. 1, Academic Press, New York and London 1972. [31] J. Dixmier, C ∗ algebras, North Holland, Amsterdam 1977. [32] E. Seiler and K. Yildirim, Critical behavior in a quasi D-dimensional spin model, J. Statist. Phys. 112, 457 (2003); [hep-lat/0209166]. [33] K. Ziegler, Divergencies in a Vector Model with Hyperbolic Symmetry on a Chain, Z. Phys. B 43, 275 (1981). [34] D. Giulini and D. Marolf, A uniqueness theorem for constraint quantization, Class. Quant. Grav. 16, 2489 (1999); [gr-qc/9902045]. [35] A. Gomberoff and D. Marolf, On group averaging for SO(n, 1), Int. J. Mod. Phys. D8 (1999); [gr-gc/9902069]. [36] S. Coleman, There are no Goldstone bosons in two dimensions, Comm. Math. Phys. 31, 259 (1973). [37] A. Patrascioiu and E. Seiler, Continuum limit of 2D spin models with continuous symmetry and conformal field theory, Phys. Rev. E 57, 111 (1998); Does conformal quantum field theory describe the continuum limits of 2D spin models with continuous symmetry? Phys. Lett. B 417, 123 (1998). [38] K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Comm. Math. Phys. 31, 83 (1973); Axioms for Euclidean Green’s functions 2, Comm. Math. Phys. 42, 281 (1975). [39] J. Glimm and A. Jaffe, Quantum Physics, Springer-Verlag, New York etc. 1987. [40] E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Physics vol. 159, SpringerVerlag Berlin etc. 1982. [41] R. Haag, Local Quantum Physics, Springer-Verlag Berlin etc. 1992.
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[42] Y. Cohen and E. Rabinovici, A study of the non-compact non-linear sigmamodel: A search for dynamical realizations of non-compact symmetries, Phys. Lett. B124, 371 (1983). [43] N. Vilenkin and A. Klimyk, Representations of Lie groups and special functions, Kluwer, Dordrecht 1993. [44] H. Dym and H.P. McKean, Fourier Series and Integrals, Academic Press, New York and London 1972. [45] N. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Repts. 143, 109 (1986). [46] I. Segal and R. Kunze, Integrals and operators, Springer-Verlag, Berlin – New York 1978. [47] T. Spencer and M.R. Zirnbauer, Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions, Comm. Math. Phys. 252, 167 (2004), [arXiv:math-phys/0410032]. [48] N.I. Akhiezer and I.M. Glazman, Theory of linear operators in Hilbert space, Dover, New York 1993. Max Niedermaier Laboratoire de Math´ematiques et Physique Th´eorique CNRS/UMR 6083 Universit´e de Tours Parc de Grandmont F-37200 Tours France email: [email protected] Erhard Seiler Max-Planck-Institut f¨ ur Physik Werner-Heisenberg-Institut F¨ ohringer Ring 6 D-80805 M¨ unchen Germany email: [email protected] Communicated by Joel Feldman submitted 17/12/04, accepted 20/04/05
Ann. Henri Poincar´e 6 (2005) 1091 – 1135 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/061091-45, Published online 15.11.2005 DOI 10.1007/s00023-005-0234-8
Annales Henri Poincar´ e
The Translation Invariant Massive Nelson Model: I. The Bottom of the Spectrum Jacob Schach Møller Abstract. In this paper we analyze the bottom of the energy-momentum spectrum of the translation invariant Nelson model, describing one electron linearly coupled to a second quantized massive scalar field. Our results are valid for all values of the coupling constant and include an HVZ theorem, non-degeneracy of ground states, existence of isolated groundstates in dimensions 1 and 2, non-existence of ground states embedded in the bottom of the essential spectrum in dimensions 3 and 4, (i.e., at total momenta where no isolated groundstate eigenvalue exists), and we study regularity and monotonicity properties of the bottom of the essential spectrum, as a function of total momentum.
1 Introduction and results In this section we introduce the Nelson model and formulate our main results. The notation we use is standard, but for the sake of completeness we give the basic constructions in Subsection 2.1.
1.1
Non-relativistic QED: An overview
In the last decade there has been a surge of interest in non-relativistic QED, sparked by a string of papers by H¨ ubner and Spohn, and by Bach, Fr¨ ohlich, and Sigal. See, e.g., [5, 4, 39, 38]. The purpose of this subsection is to give an overview over different aspects of the problem and place the model we study, as well as the results derived, into context. The fundamental Hamiltonian in non-relativistic QED, describing one charged particle, with mass M > 0 and charge e, coupled to a radiation field, is the minimally coupled one Hmin := 1l ⊗ dΓ(|k|) +
2 1 p ⊗ 1l − e A(x) , on L2 (R3x ) ⊗ Γ(L2 (R3k )) . (1.1) 2M
Here dΓ(|k|) is the kinetic energy of the radiation field, p = i∇x is the particle momentum operator, and A is the second quantized (massless) Maxwell field in the Coulomb gauge, i.e., ∇x · A = 0. The Hilbert space Γ(L2 (R3k )) is the bosonic Fock-space. See [34] and [5, 42]. In order to make sense of this operator (a priori as a form) one must introduce an ultraviolet cutoff into A. We recall that the model is translation invariant, in the sense that it commutes with the operator
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of total momentum P := p ⊗ 1l + 1l ⊗ dΓ(k). We remark that often the second quantized Pauli operator is taken as a starting point instead of (1.1). It is defined by replacing (p − eA)2 by (σ · (p − eA))2 , where σ is the vector of Pauli matrices. This operator differs from (1.1) by a magnetic term σ · (∇x × A) (and with L2 (R3x ) replaced by L2 (R3x ) ⊗ C2 , thus taking into account the spin of the particle). The study of Hmin is a natural starting point in non-relativistic QED. In particular in the context of scattering theory, where the dynamics of Hmin is a natural choice for ”free” dynamics. Unfortunately there are not many rigorous results established for the minimally coupled model, as it is formulated in (1.1), which are valid for all values of e (viewed as a coupling constant). We refer the reader to [35, 41]. Most results obtained in the literature are for Hmin perturbed by an electric potential, and results then pertain to existence and properties of ground states for the perturbed model, or localization in L2 (R3x ) of states below an ionization threshold. See [29, 30, 42, 43]. For a recent textbook treatment of the minimally coupled model and its classical counterpart, the Abraham model, see [56]. There are a number of different ways to obtain simpler problems. Some involve passing to phenomenological Hamiltonians, which are simpler to analyze than (1.1). We list some choices typically considered in the literature: S1) Consider the problem in the weak coupling regime, i.e., for |e| small. S2) Replace the massless photons by massive photons, which amounts to√replacing the massless dispersion relation k → |k| by a massive one k → k 2 + m2 , m > 0. This removes the infrared problem. S2 ) Set the interaction between soft photons (photons with small momenta) equal to zero. S3) Replace the minimal coupling with a linear coupling to a scalar field, i.e., replace Hmin by H = 1l ⊗ dΓ(|k|) +
1 2 p ⊗ 1l + gΦ(v), 2M
where Φ(v) is a field operator and g is a coupling constant. S4) Place the system in a confining external electric potential V , that is lim V (x) = ∞.
|x|→∞
This breaks the translation invariance of the problem. An extreme version of this are the spin-boson and Wigner-Weisskopf models. S4 ) Place the system in an external potential V such that p2 + V has isolated eigenvalues below the essential spectrum. Then consider Hmin + V ⊗ 1l in a low energy regime where states are isolated bound states of p2 + V dressed with photons. S5) A combination of the above.
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In this paper we consider the massive translation invariant linearly coupled model in any dimension, which (in dimension 3) can be viewed as a simplification of the minimally coupled model, by applying S2) and S3) as mentioned above. This model was considered by Nelson in [47], and it is distinguished by being renormalizable in a Hamiltonian setting, cf. also [10, 32, 53]. This model is often referred to as the Nelson model, a convention also adopted here. The models discussed in this introduction is part of a body of models sometimes referred to as Pauli-Fierz models. In this paper we do not consider renormalized operators. In addition we √ note that we work with more general dispersion relations ω and Ω than k 2 + m2 and p2 /2M respectively. We emphasize that we are interested in results which hold for all values of the coupling constant g. See Subsection 1.2 below for a more detailed description of the model. We remark that one can formulate the model and the simplifications discussed above for multiple particles coupled to a radiation field. For confined versions of the model, cf. S4) and S4 ) above, this makes no difference. However, for translation invariant models, not much is known. We pause to remind the reader that translation invariance, the fact that [H, P ] = 0, gives a direct integral representation H(ξ)dξ of the Hamiltonian. What we study in this paper is the bottom of the spectrum and essential spectrum of H(ξ) as functions of total momentum ξ. The former function is also called the ground state mass shell, or simply the mass shell. We note that in the massive case isolated excited states could exist and would give rise to excited mass shells. We are mainly inspired by works of Fr¨ ohlich [19, 20], Spohn [55], and one of Derezi´ nski and G´erard [14]. The results proved in these papers hold for all values of the coupling constant. Fr¨ ohlich considered properties of the ground state mass shell for the massless translation invariant Nelson model. Most of his results hold (suitably translated) also for massive photons. Derezi´ nski and G´erard were concerned with confined, in the sense of S3) above, massive linearly coupled models. They give a geometric proof of a HVZ theorem, thus locating the essential spectrum. (They furthermore apply Mourre theory and time-dependent scattering theory to the model.) Spohn proved a HVZ theorem for the translation invariant model, using in part ideas of Glimm and Jaffe (via a reference to [20]). He furthermore showed, in dimension 1 and 2, that the Hamiltonian at fixed total momentum admits an isolated groundstate. The results of Spohn are for a class of massive and subadditive dispersion relations ω. The result on existence of groundstates requires an additional assumption which excludes the dispersion relation √ k 2 + m2 , m > 0. In this paper we prove the following results for the structure of the bottom of the spectrum of the massive translation invariant Nelson model: A HVZ theorem, Theorem 1.2 (valid for ω which are not necessarily subadditive). The ground state mass shell is non-degenerate, Theorem 1.3, using a Perron-Frobenius argument of [19]. Existence of an isolated groundstate for all total momenta, Theorem 1.5√i) (dimensions 1 and 2), thus extending the result of Spohn to the case ω(k) = k 2 + m2 . Non-existence of a ground state embedded in the essential
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spectrum, Theorem 1.5 ii) (dimensions 3 and 4). Analyticity of the bottom of the essential spectrum, away from a closed countable set, Theorem 1.9. Maximality of the spectral gap and analyticity at local minima for the bottom of the essential spectrum, Theorem 1.10. See Subsection 1.3 for a precise formulation of the main results. In Subsection 4.2 we discuss how to extend the results to the model with a cutoff in the photon number operator. The models considered in this paper only fails to include the so-called (optical mode) polaron model of an electron in a ionic crystal by the requirement that ω(k) → ∞, |k| → ∞. This requirement is a consequence of our use of geometric methods to prove the HVZ theorem, and an adoption of the Glimm-Jaffe approach, as used in [20], might remedy this. However, the geometric approach is important for future work on Mourre and scattering theory. For mathematical work on the polaron model see [32, 44, 53, 54, 55], and for a textbook discussion see [18]. We remark that there are not many results on the translation invariant Nelson model, other than what we have already mentioned above, which are valid for all values of g. See however [31], Lemma 4.1 in this paper. In [54] upper and lower bounds on the effective mass are obtained (the effective mass is the inverse of the Hessian of the ground state mass shell at zero total momentum). There are more complete results available if one imposes a cutoff at small photon number, cf. [23] (the massless case with at most one photon). In the case of weak coupling there are more results, cf. [11, 22, 48]. See also [33, 36, 37]. (We remark however, that although the photon dispersion relation in [22] is massless, the interaction is of the type mentioned in S2 ) above, and the model thus retains massive features.) Finally we recall that for confined massive models, cf. S4) and S4 ) above, quite strong results, valid for all coupling constants, are available. See, apart from [14] mentioned above, the papers [1, 3, 21]. As for the massless confined model we refer the reader to [7, 9, 24, 26, 38]. We end this section with an overview of the paper. The rest of this chapter is devoted to a formal definition of the model and a presentation of assumptions and our main results. In Chapter 2 we present the second quantization formalism, extended objects pertaining to partitions of unity in Fock-space, basic estimates, and the pull-through formula. The core of the paper is Chapter 3, where all the main theorems are proved. In Chapter 4 we have assembled some miscellaneous results, and formulated the main theorems in the setting of the Nelson model with a cutoff in boson number. Finally in Appendix A, we present two mathematical tools, namely the calculus of almost analytic extensions, for a vector of commuting self-adjoint operators, and an abstract Perron-Frobenius result of Faris.
1.2
The translation invariant Nelson model
We consider a particle moving in Rν and interacting with a scalar radiation field. We write x and p = −i∇x for the particle position and momentum respectively.
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The particle Hilbert space is K := L2 (Rνx ) , and the Hamiltonian for a free particle is taken to be Ω(p), where Ω : Rν → R is a smooth dispersion relation. We are primarily interested in the standard nonp2 and Ω(p) = p2 + M 2 . Here relativistic and relativistic choices, i.e., Ω(p) = 2M M > 0 is the mass of the particle. The photon coordinates will be denoted by x = i∇k and k respectively and the one-photon space is hph := L2 (Rνk ) . The Hilbert space for the radiation field is the bosonic Fock-space F ≡ Γ(hph ) :=
∞
sn F (n) , where F (n) ≡ Γ(n) (hph ) := h⊗ ph .
(1.2)
n=0
We write Ω = (1, 0, 0, . . . ) for the vacuum. The creation and annihilation operators, a∗ (k) and a(k) satisfy the canonical commutation relations (CCR for short) [a∗ (k), a∗ (k )] = [a(k), a(k )] = 0 , [a(k), a∗ (k )] = δ(k − k ) ,
(1.3)
and a(k)Ω = 0. The free photon energy is the second quantization of the onephoton dispersion relation ω ω(k) a∗ (k) a(k) dk , where ω(k) := k 2 + m2 . (1.4) dΓ(ω) := Rν
Here m > 0 is the mass of the scalar photon. Our methods do not extend to the case of massless photons, m = 0. The full Hilbert space of the combined system is H := K ⊗ F . We will make the following identification H ≡ L2 (Rνx ; F ). The interaction considered here is linear in the field operator and is given by e−ik·x v(k) 1lK ⊗ a∗ (k) + eik·x v(k) 1lK ⊗ a(k) dk , V := (1.5) Rν
where the physical form of the interaction is v(k) = χ(k)/ ω(k) and χ is an ultraviolet cutoff, which ensures that v ∈ hph = L2 (Rνk ). The free and coupled Hamiltonians for the combined system are H := H0 + V , where H0 := Ω(p) ⊗ 1lF + 1lK ⊗ dΓ(ω) .
(1.6)
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The total momentum for the combined system is given by P := p ⊗ 1lF + 1lK ⊗ dΓ(k) .
(1.7)
The Hamiltonian H is translation invariant. That is, the energy momentum vector (P, H) has mutually commuting coordinates. Similarly for H0 . Translation invariance implies that H0 and H are fibered operators. We introduce a unitary transformation Ifib := F Γ(e−ik·x ) : H → L2 (Rνξ ; F ) , (1.8) where F is the Fourier transform F : L2 (Rνx ; F ) → L2 (Rνξ ; F ) and Γ(e−ik·x ) restricted to K ⊗ F (n) is multiplication by e−i(k1 +···+kn )·x . We have ∗ ∗ = H0 (ξ) dξ and Ifib H Ifib = H(ξ) dξ . (1.9) Ifib H0 Ifib Rν
Rν
The fiber operators H0 (ξ) and H(ξ), ξ ∈ Rν , are operators on F given by H(ξ) = H0 (ξ) + Φ(v) where H0 (ξ) = dΓ(ω) + Ω(ξ − dΓ(k)) and the interaction is
Φ(v) = Rν
v(k) a∗ (k) + v(k) a(k) dk .
(1.10)
(1.11)
We will in general use the notation v ∈ hph to denote a form-factor. In this paper we study the properties of the bottom of the joint spectrum of the vector (P, H).
1.3
Main results
In this subsection we will formulate precise conditions and state our main results. Proofs will be given in Section 3. The first condition is on the particle dispersion relation. We use the standard notation t := (1 + t2 )1/2 . Condition 1.1 (The particle dispersion relation) Let Ω ∈ C ∞ (Rν ). There exists sΩ ∈ {0, 1, 2}, a constant C, and for any multi-index α, with |α| ≥ 1, constants Cα , such that Ω(η) ≥ C −1 η sΩ − C and ∀α : |∂ α Ω(η)| ≤ Cα η sΩ −|α| . 2
p We note that the standard choices Ω(p) = 2M and Ω(p) = this condition with sΩ = 2 and sΩ = 1 respectively.
p2 + M 2 satisfy
Condition 1.2 (The photon dispersion relation) Let ω ∈ C ∞ (Rν ) satisfy i) There exists m > 0, the photon mass, such that inf k∈Rν ω(k) = ω(0) = m. ii) ω(k) → ∞, in the limit |k| → ∞.
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iii) There exists sω ≥ 0, a constant Cω , and for any multi-index α, with |α| ≥ 1, constants Cα , such that: ω(k) ≥ Cω−1 k sω − Cω and ∀α : |∂kα ω(k)| ≤ Cα k sω −|α| . The condition iii) is used in connection with pseudo differential calculus. The physical choice of ω used in (1.4) satisfies this condition (with sω = 1), and so does ω(k) = k 2 + m (with sω = 2). We introduce a space of test functions C0∞ := Γfin (C0∞ (Rν )) .
(1.12)
Note that since H0 (ξ) is a bounded from below multiplication operator on each n-particle sector, we find that it is essentially self-adjoint on C0∞ . We recall the following result, cf. [47], [19], and [20]. For completeness we give a proof in the beginning of Section 3. Proposition 1.1 Let v ∈ L2 (Rν ). Assume Ω and ω, satisfy Conditions 1.1 and 1.2 i) respectively. Then i) D(H0 (ξ)) is independent of ξ and we denote it by D. ii) Φ(v) is H0 (ξ)-bounded with relative bound 0. In particular H(ξ) is bounded from below, self-adjoint on D(H(ξ)) = D(H0 (ξ)), and essentially self-adjoint on C0∞ . iii) The bottom of the spectrum of the fiber Hamiltonians, ξ → Σ0 (ξ) := inf σ(H(ξ)), is Lipschitz continuous. We introduce some notation. First the bottom of the spectrum of the full operator: Σ0 := infν Σ0 (ξ) > −∞ . ξ∈R
For n ≥ 1 and k = (k1 , . . . , kn ) ∈ Rnν we often write k (n) = k1 + · · · + kn . We now introduce the bottom of the spectrum for a composite system at total momentum ξ, consisting of an interacting system at total momentum ξ − k (n) and n non-interacting photons with momenta k: n (n) ω(kj ) . Σ0 (ξ; k) := Σ0 ξ − k (n) +
(1.13)
j=1
The following functions are thresholds due to ground states dressed by n photons, at critical momenta: (n) (n) Σ0 (ξ) := infnν Σ0 (ξ; k) . (1.14) k∈R
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The bottom of the essential spectrum (see Theorem 1.2 below) (n)
Σess (ξ) := inf Σ0 (ξ) .
(1.15)
n≥1
We have the following elementary properties of the functions introduced above. Namely 0 ≤ Σess (ξ) − Σ0 (ξ) ≤ m Σ0 (ξ) = Σ0 ⇒ Σess (ξ) = Σ0 (ξ) + m lim Σ0 (ξ) =
|ξ|→∞
lim Σess (ξ) =
|ξ|→∞
(n)
lim Σ0 (ξ) = ∞
|ξ|→∞
(n)
lim Σ0 (ξ) = ∞ .
n→∞
(1.16) (1.17) (1.18) (1.19)
Our first result is Theorem 1.2 (HVZ) Let v ∈ L2 (Rν ). Assume Conditions 1.1, and 1.2. Then i) Eigenvalues of H(ξ) below Σess (ξ) have finite multiplicity and can only accumulate at Σess (ξ). ii) σess (H(ξ)) = [Σess (ξ), ∞). The method of proof for the HVZ theorem is geometric and follows ideas of [14], cf. Subsection 3.2. See also [1, 2, 9, 15, 24]. The name ”HVZ” (Hunziker–van Winter–Zhislin) is used because the geometric idea of the proof is quite similar to that employed in the proof of the standard HVZ theorem for N -body Schr¨ odinger operators, cf. [13, Theorem 6.2.2]. We recall that there is another method, due to Glimm and Jaffe [28], one can employ to obtain an HVZ theorem. See [55, Section 4], for the case of subadditive dispersion relations ω, and in addition [8, 20]. In the following we will impose either v ∈ L2 (Rν ) , v is real-valued, and v = 0 a.e.
(1.20)
or v ∈ L2 (Rν ) , v is real-valued, and ∀R > 0 : essinf |v(k)| > 0 .
(1.21)
k:|k|≤R
We have the following result on non-degeneracy of groundstates. This type of result is not new, cf. [31, Section 6] and [20, Section 2]. Theorem 1.3 (Non-degeneracy of ground states) Let v satisfy (1.20) and assume Conditions 1.1 and 1.2. Then, if Σ0 (ξ) is an eigenvalue for H(ξ), it is nondegenerate. We note that the result of Gross [31] is for zero total momentum only, and assumed that p → exp(−tΩ(p)) is a positive definite function for all t > 0. Gross pass to the Schr¨ odinger representation of the Fock-space, where H0 (ξ) is positivity improving if and only if ξ = 0.
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For the remaining results we will need either i) or i ) in the condition below. Condition 1.3 ω ∈ C ∞ (Rν ) satisfies i) Subadditivity: For k1 , k2 ∈ Rν we have ω(k1 + k2 ) ≤ ω(k1 ) + ω(k2 ). i ) Strict subadditivity: For k1 , k2 ∈ Rν we have ω(k1 + k2 ) < ω(k1 ) + ω(k2 ). √ The standard dispersion relation ω(k) = k 2 + m2 satisfies Condition 1.3 i ), but ω(k) = k 2 + m does not. We remark that if ω is convex and satisfies: ∀k ∈ Rν : ω(k) − k · ∇ω(k)
(>) ≥
0,
(1.22)
then ω is (strictly) subadditive. If ω is (strictly) subadditive we find, for all ξ ∈ Rν , (n)
Σ0 (ξ)
(<) ≤
(n )
Σ0
(ξ) , for n < n .
(1.23)
We thus get the following supplement to the HVZ Theorem, cf. also [55, Section 4], Corollary 1.4 Let v ∈ L2 (Rν ). Assume Conditions 1.1, 1.2, and 1.3 i). Then (1) Σess (ξ) = Σ0 (ξ). We introduce the notation I0 := { η ∈ Rν : Σ0 (η) < Σess (η) } .
(1.24)
We prove the following result on existence and non-existence of ground states. Theorem 1.5 Let v satisfy (1.21) and assume Conditions 1.1, 1.2, and 1.3 i ). Then we have: i) If 1 ≤ ν ≤ 2, then I0 = Rν , that is; Σ0 (ξ) is an isolated eigenvalue of H(ξ) for any ξ ∈ Rν . ii) If 3 ≤ ν ≤ 4 and ξ ∈ I0 , then H(ξ) has no ground state; i.e., Σ0 (ξ) is not an eigenvalue. The statement i) above is an extension to the Nelson model of a result of Spohn, [55, Section 5]. We give a new proof replacing Spohn’s functional integral approach by the pull-through formula. As for ii), we show that the expectation value of the number operator in any embedded groundstate must be infinite, thus arriving at a contradiction. The remaining results are derived under the following condition Condition 1.4 The functions Ω, ω ∈ C ∞ (Rν ), v ∈ L2 (Rν ), and i) Invariance under rotations: For any ξ ∈ Rν and O ∈ O(ν) (the orthogonal group), we have: Ω(Oξ) = Ω(ξ), ω(Oξ) = ω(ξ), and v(Ok) = v(k) a.e. ii) ω is convex. iii) Ω and ω are analytic.
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We remark that under Conditions 1.2 i) and 1.3 i), iii), subadditivity and strict subadditivity of ω are equivalent. The rotation invariance of Ω, ω, and v, implies that the ground state mass (n) shell Σ0 is invariant under rotations, and hence also the Σ0 ’s and Σess . ν For ξ ∈ R and n ∈ N we define (n)
I0 (ξ) := { k ∈ Rnν : ξ − k (n) ∈ I0 } .
(1.25) (n)
The next theorem is concerned with regularity of the functions ξ → Σ0 (ξ). (n) Our strategy is to study local minima of k → Σ0 (ξ; k). The following lemma, in conjunction with (1.23), ensures that under Condition 1.3 i ), the relevant lo(n) cal minima, i.e., global minima, are located in I0 (ξ), where the bottom of the spectrum is smooth. Lemma 1.6 Let v ∈ L2 (Rν ). Assume Conditions 1.1 and 1.2 i). Let ξ ∈ Rν , n ≥ 1 (n) (n ) (n) and k ∈ Rnν . If Σ0 (ξ; k) < inf n >n Σ0 (ξ), then k ∈ I0 (ξ). The following lemma allows us to restrict the analysis to one dimension. Lemma 1.7 Assume Conditions 1.1, 1.2 i), and 1.4 i), ii). Let ξ ∈ Rν \{0} and (n) (n) n ∈ N. Any local minimum k ∈ I0 (ξ) of k → Σ0 (ξ; k) is of the form k1 = · · · = kn = θξ, for some θ ∈ R. Let u be a unit vector in Rν . We write σ(t) = Σ0 (t u), for t ∈ R. By rota(n) tion invariance, σ is independent of u. Similarly we write σ (n) (t) := Σ0 (t u) and σess (t) := Σess (t u). With a slight abuse of notation we write ω(s) = ω(s u) and I0 (n) to denote the set of t’s such that t u ∈ I0 . We furthermore use the symbol I0 (t), n > 0 (not necessarily integer), to denote the set {s ∈ R : t − ns ∈ I0 }. In light of the previous lemma, we introduce now, for n > 0 and not necessarily integer, the following functions σ (n) (t; s) = σ(t − ns) + n ω(s) and σ (n) (t) = inf σ (n) (t; s) . s∈R
(1.26)
(n)
Note that by Lemma 1.7 we have, for integer n, Σ0 (ξ) = σ (n) (|ξ|), and in particular Σess (ξ) = σ (1) (|ξ|). In this connection we mention that a local minimum (n) for Σ0 (ξ; ·) induces a local minimum for σ (n) (|ξ|; ·). Conversely however, a local minimum for σ (n) (t; ·), which is not a global minimum, could be associated with (n) a saddle point for Σ0 (t u; ·). We have, cf. also [19, Lemma 1.6], Proposition 1.8 Let v satisfy (1.20) and assume Conditions 1.1, 1.2, and 1.4. Let λ < Σ0 . The family of self adjoint operators t → (H(t u) − λ)−1 is analytic of type A. Furthermore, the map I0 t → σ(t) is analytic.
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See [40, Chapter VII] for analytic perturbation theory. The following regularity result is proved by keeping track of global minima of s → σ (n) (t; s), as functions of t. Theorem 1.9 Let v satisfy (1.20) and assume Conditions 1.1, 1.2, 1.3 i), and 1.4. Let n > 0. There exists a closed countable set T (n) ⊂ R, and an analytic map (n) R\T (n) t → Θ(n) (t) ∈ I0 (t) with the property that the maps s → σ (n) (t; s), t ∈ R\T (n) , has a unique global minimum at s = Θ(n) (t) which is non-degenerate, i.e., ∂s2 σ (n) (t; Θ(n) (t)) > 0. In particular R\T (n) t → σ (n) (t) is analytic and d (n) σ (t) = ∂ω Θ(n) (t) , for t ∈ R \ T (n) . (1.27) dt Our final main result is concerned with the structure of the spectrum near local minima of the essential spectrum Theorem 1.10 Let v satisfy (1.20) and assume Conditions 1.1, 1.2, 1.3 i), and 1.4. Let t0 be a local minimum of t → σess (t). Then the spectral gap at t0 is maximal, i.e., σess (t0 ) − σ(t0 ) = m, the map t → σ(t) has a local minimum at t0 , the map t → σess (t) is analytic near t0 , and d2 ∂ 2 ω(0) ∂ 2 σ(t0 ) . σ (t ) = ess 0 dt2 ∂ 2 ω(0) + ∂ 2 σ(t0 )
(1.28)
2 Notation and preliminaries In this section we recall known facts. The reader is urged to consult in particular [14], where most of the results pertaining to second quantization can be found.
2.1
The second quantization functor Γ
Let h be a complex Hilbert space with inner product ·, · , which is conjugate linear in the first variable and linear in the second. We use the standard notation Γ(h) for the associated bosonic Fock-space, see (1.2). For a (not necessarily dense) subspace C ⊂ h, we write Γfin (C) for the subspace of Γ(h) consisting of finite linear combinations of elements of the algebraic tensor products C ⊗s n , n ≥ 0. If C is dense in h, then Γfin (C) is dense in Γ(h). operators. We write a∗ (f ) and a(f ), f ∈ h, for the creation and annihilation √ Recall that for u ∈ Γ(n) (h) := h⊗s n , the n-particle sector; a∗ (f )u = n + 1Sn+1 f ⊗ u ∈ Γ(n+1) (h). Here Sk is the symmetrization operator on h⊗k . We furthermore recall that a∗ (f ) and a(f ) are closed and densely defined, and that D(a(f )) = D(a∗ (f )). They satisfy the CCR: [a∗ (f ), a∗ (g)] = [a(f ), a(g)] = 0 , [a(f ), a∗ (g)] = f, g
(2.1)
and a(f )Ω = 0, for f ∈ h. The field operator Φ(f ) := a∗ (f ) + a(f )
(2.2)
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is self-adjoint on D(a∗ (f )) = D(a(f )) and essentially self-adjoint on Γfin (h). In
the case h = hph we have the relation with (1.3): a∗ (f ) = Rν f (k)a∗ (k)dk and
a(f ) = Rν f (k)a(k)dk. In particular (2.2) and (1.11) coincide. We frequently write a# (k) to denote either a(k) or a∗ (k). Similarly for a# (f ). Recall that a(k) is well defined on C0∞ = Γfin (C0∞ (Rν )), but it is not closable. The domain of its adjoint (a(k))∗ equals {0}. The ”operator” a∗ (k) should be understood as a form. See the monograph by Berezin [6]. Let b be a bounded operator between Hilbert spaces h1 and h2 . We define Γ(b) : Γ(h1 ) → Γ(h2 ) by its restriction to Γ(n) (h1 ) n times
Γ(b)|Γ(n) (h1 )
:= b ⊗ · · · ⊗ b .
In particular we have Γ(b)Ω = Ω. Recall that Γ(b) is bounded if and only if bB(h1 ;h2 ) ≤ 1. We introduce dΓ(a) for operators a : h → h with domain D(a) by dΓ(a)|Γ(n) (h) := a ⊗ 1lh ⊗ · · · ⊗ 1lh + · · · + 1lh ⊗ · · · ⊗ 1lh ⊗ a ,
(2.3)
a priori on the domain Γfin (D(a)). In particular; dΓ(a)Ω = 0. The operators Γ(b) and dΓ(a) are related through the formula Γ(ea ) = edΓ(a) (suitably interpreted). It is easy to see that if a is closed (or closable) on D(a) then dΓ(a) is closable on Γfin (D(a)). See [24, Section 3.2] for a simple proof, which applies also to similar situations below. In addition, if a is self-adjoint, then dΓ(a) is essentially selfadjoint on Γfin (D(a)), cf. [51, Subsection VIII.10, Theorem VIII.33 and Example 2]. For closed a we will by dΓ(a) understand the closure of (2.3). Otherwise dΓ(a) denotes the operator in (2.3) with the a priori domain Γfin (D(a)). For a quadratic form a with form-domain Q(a) we also write dΓ(a) for the quadratic form defined on Γfin (Q(a)) by (2.3). An important operator is the number operator N := dΓ(1lh ) ,
(2.4)
which in the case h = hph can be written as N = Rν a∗ (k)a(k)dk. See also (1.4). Let a and b be densely defined operators on h and v ∈ D(a). We have the following commutation properties, which should be interpreted as forms on Γfin (D(a∗ ) ∩ D(b∗ )) × Γfin (D(a) ∩ D(b)) and Γfin (D(a∗ )) × Γfin (D(a)) respectively. i[dΓ(a), dΓ(b)] = dΓ(i[a, b]) , ∗
[a (v), dΓ(a)] = −a∗ (av) , [a(v), dΓ(a)] = a(av) , and i[Φ(v), dΓ(a)] = − Φ(iav) .
(2.5)
Let b : h1 → h2 be a contraction and a : h1 → h2 with domain D(a). We define dΓ(b, a) : Γ(h1 ) → Γ(h2 ) on Γfin (D(a)) by dΓ(b, a)|Γ(n) (h1 ) := a ⊗ b ⊗ · · · ⊗ b + · · · + b ⊗ · · · ⊗ b ⊗ a .
(2.6)
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In particular (in the case h1 = h2 = h) dΓ(1lh , a) = dΓ(a); cf. (2.3). If a is closed (or closable) we find, as above, that dΓ(b, a) is closable on Γfin (D(a)). As for dΓ(a) we use the notation dΓ(b, a) also in the case where a is a form on h2 × h1 . Let b : h1 → h2 be a contraction, a1 : h1 → h1 and a2 : h2 → h2 be densely defined. As a form on Γfin (D(a∗2 )) × Γfin (D(a1 )) we have (Γ(b) dΓ(a1 ) − dΓ(a2 ) Γ(b)) = dΓ(b, (ba1 − a2 b)) .
2.2
(2.7)
Basic estimates involving Γ
We have the following lemma, cf. [14, Lemma 2.1], Lemma 2.1 For f ∈ h and s ≥ 0, we have a# (f ) : D(N s+1/2 ) → D(N s ) and the following holds true i) Let f1 , . . . , fn ∈ h and k ≥ 0. Then (N + 1)k a# (f1 ) · · · a# (fn ) (N + 1)− n2 −k ≤ Ck,n f1 · · · fn . ii) The following map is norm-continuous n
hn (f1 , . . . , fn ) → (N + 1)k a# (f1 ) · · · a# (fn ) (N + 1)− 2 −k ∈ B(Γ(h)) . iii) Let {f1, }∈N , . . . , {fn, }∈N be uniformly bounded sequences, converging weakly to zero in h. Then n
s − lim (N + 1)k a(f1, ) · · · a(fn, ) (N + 1)− 2 −k = 0 . →∞
˜ and a2 : h2 → ˜h. Define Suppose b ∈ B(h1 ; h2 ) is a contraction, a1 : h1 → h a as a form on D(a2 ) × D(a1 ) by (f, ag) := (a2 f, a1 g). Then, for v ∈ Γfin (D(a1 )) and u ∈ Γfin (D(a2 )), 1
1
|u, dΓ(b, a)v | ≤ u, dΓ(a∗2 a2 )u 2 v, dΓ(a∗1 a1 )v 2 .
(2.8)
˜ = h2 , a2 = 1lh2 , Here a∗# a# denote the obvious forms on h# . Taking in particular h and a1 = a we get, for v ∈ Γfin (D(a)), 1
1
(N + 1)− 2 dΓ(b, a)v ≤ v, dΓ(a∗ a)v 2 .
(2.9)
In connection with this bound we also use the easy property a ≤ b
=⇒
dΓ(a) ≤ dΓ(b) ,
(2.10)
where a and b are self-adjoint operators (or symmetric forms) on h. We also make use of the following estimate, cf. [27, Lemma A.2]. Let k ∈ N and let a and b be self-adjoint operators on h. If 0 ≤ a ≤ b for all 1 ≤ ≤ k, with ∈ N. Then (dΓ(a))k ≤ (dΓ(b))k .
(2.11)
We note that there are several bounds involving powers of second quantized operators, cf., e.g., [15, Lemma 3.2] and [24, Section 3.2] for a selection.
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ˇ The extended space and Γ
Let h0 and h∞ be two Hilbert spaces. We will use the standard unitary identification U : Γ(h0 ⊕ h∞ ) → Γ(h0 ) ⊗ Γ(h∞ ), which is determined uniquely by linearity and the two properties UΩ = U a ((f, g)) = ∗
Ω⊗Ω ∗ a (f ) ⊗ 1lΓ(h∞ ) + 1lΓ(h0 ) ⊗ a∗ (g) U .
Let a0 : h0 → h0 and a∞ : h∞ → h∞ . We have the intertwining property U dΓ(a0 ⊕ a∞ ) = dΓ(a0 ) ⊗ 1lΓ(h∞ ) + 1lΓ(h0 ) ⊗ dΓ(a∞ ) U ,
(2.12) (2.13)
(2.14)
as an identity on Γfin (D(a0 ) ⊕ D(a∞ )). Let h, h0 and h∞ be Hilbert spaces and let b = (b0 , b∞ ), where b0 ∈ B(h; h0 ) and b∞ ∈ B(h; h∞ ). We view b as an element of B(h; h0 ⊕ h∞ ) and define the ˇ associated operator Γ(b) by ˇ Γ(b) := U Γ(b) : Γ(h) → Γ(h0 ) ⊗ Γ(h∞ ) .
(2.15)
In this paper we always require b∗0 b0 +b∗∞ b∞ = 1lh , which implies bB(h;h0 ⊕h∞ ) = 1 ˇ and Γ(b) is an isometry: ˇ ∗ Γ(b) ˇ Γ(b) = 1lΓ(h) . (2.16) ˇ We interpret Γ(b) as a partition of unity. Let b = (b0 , b∞ ) be as above, and let a = (a0 , a∞ ) be an operator from h to h1 ⊕ h2 , with domain D(a) = D(a0 ) ∩ D(a∞ ). We introduce the operator ˇ a) : Γfin (D(a)) → Γ(h0 ) ⊗ Γ(h∞ ) by dΓ(b, ˇ a) := U dΓ(b, a) . dΓ(b,
(2.17)
We use the same notation for forms a = (a0 , a∞ ), where a# are forms on h# × h. Let r : h → h, q0 : h0 → h0 and q∞ : h∞ → h∞ , be densely defined operators. We have the following intertwining relation, viewed as an identity between forms ∗ ))} × Γfin (D(r)): on {Γfin (D(q0∗ )) ⊗ Γfin (D(q∞ ˇ ˇ ˇ a) , (2.18) Γ(b)dΓ(r) − dΓ(q0 ) ⊗ 1lΓ(h∞ ) + 1lΓ(h0 ) ⊗ dΓ(q∞ ) Γ(b) = dΓ(b, ∗ where a = (b0 r − q0 b0 , b∞ r − q∞ b∞ ) has form-domain {D(q0∗ ) ⊕ D(q∞ )} × D(r).
2.4
ˇ Basic estimates involving Γ
˜# and a#,2 : h# → h ˜# , where Let b = (b0 , b∞ ) be as in (2.17). Let a#,1 : h → h ˜ h# are auxiliary Hilbert spaces. Here # denotes 0 and ∞. We define a form a = (a0 , a∞ ) on {D(a0,2 ) ⊕ D(a∞,2 )} × {D(a0,1 ) ∩ D(a∞,1 )} by prescribing the forms a0 and a∞ as follows: (f, a# g) := (a#,2 f, a#,1 g) on D(a#,2 ) × D(a#,1 ).
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Let u0 ∈ Γfin (D(a0,2 )), u∞ ∈ Γfin (D(a∞,2 )), v ∈ Γfin (D(a0,1 ) ∩ D(a∞,1 )). The following key estimate follows from (2.14) and (2.8) ˇ a)v | |u0 ⊗ u∞ , dΓ(b,
1 1 ≤ u0 , dΓ(a0,2 a∗0,2 )u0 2 u∞ + u0 u∞ , dΓ(a∞,2 a∗∞,2 )u∞ 2 1
×v, dΓ(a∗0,1 a0,1 + a∗∞,1 a∞,1 )v 2 .
(2.19)
Again a∗#,2 a#,2 denote the obvious forms on D(a#,2 ), and a∗0,1 a0,1 + a∗∞,1 a∞,1 is a form on D(a0,1 ) ∩ D(a∞,1 ). ˜# = h# , a#,2 = 1lh , and a#,1 = a# ) As for (2.9) this implies (here h # 1 ˇ a)v ≤ v, dΓ(a∗ a0,1 + a∗ a∞,1 )v 12 . (N0 + N∞ )− 2 dΓ(b, 0,1 ∞,1
(2.20)
Here and in the following we use the notation (cf. (2.4)) N0 = dΓ(1lh0 ) ⊗ 1lΓ(h∞ ) and N∞ = 1lΓ(h0 ) ⊗ dΓ(1lh∞ ) .
2.5
(2.21)
Auxiliary spaces and operators
In this subsection we introduce some notation which will be used in the proof of the HVZ theorem in Subsection 3.2. We introduce auxiliary Hilbert spaces for an interacting system accompanied by a fixed number ≥ 1 of auxiliary photons H() := F ⊗ F () ≡ L2sym (Rν ; F ) . Here the subscript sym indicates that functions are symmetric under permutation, i.e., f (kτ (1) , . . . , kτ () ) = f (k1 , . . . , k ) a.e., for any τ ∈ S( ) the group of permutations of the set {1, . . . , }. For ∈ N we extend the notation for second quantization as follows dΓ() (a) =
dΓ(a) ⊗ 1lF () + 1lF ⊗ dΓ(a)|F () ,
for operators a on hph . Again dΓ(a) defined on Γfin (D(a))⊗ D(a)⊗s is closable (essentially self-adjoint) if a is closable (essentially self-adjoint). For the Hamiltonian we write () (2.22) H () (ξ) := H0 (ξ) + Φ(v) ⊗ 1lF () , where We note that
() H0 (ξ) := dΓ() (ω) + Ω ξ − dΓ() (k) . () H0 (ξ)
(2.23)
is essentially self-adjoint on ∞()
C0 ()
:= C0∞ ⊗ Γ() (C0∞ (Rν )) .
(2.24)
and write D() = D(H0 (ξ)), which is independent of ξ. Observe that there is no interaction between the auxiliary photons, nor are they coupled with the
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interacting system (apart from the coupling coming from the dispersive structure). () Note that as for Proposition 1.1, Φ(v) ⊗ 1lF () is H0 (ξ)-bounded with relative ∞() and self-adjoint on D() . bound 0, so H () (ξ) is essentially self-adjoint on C0 Using a direct integral representation we can write the auxiliary Hamiltonian for each total momentum ξ as () H (ξ) = H () (ξ; k) dν k , (2.25) Rν
where H
()
(ξ; k) := H(ξ − k) +
ω(kj ) 1lF .
(2.26)
j=1 ()
Here dν k = Πj=1 dν kj . We have a similar fibration of H0 (ξ). The fiber operators, being spectral translates of a Hamiltonian at a different total momentum, are clearly self-adjoint on D and essentially self-adjoint on C0∞ . We note the following important observations
() Σ0 (ξ; k) = inf σ H () (ξ; k) , (2.27)
() () Σ0 (ξ) = inf σ H (ξ) . (2.28)
2.6
Geometric partition of unity and extended operators
In the analysis of the many-body problem, a central tool is a geometric partition of unity in the configuration space; cf. [13]. Here we will need a similar notion, made complicated by the fact that we have to partition an infinite number of particles. The type of partition of unity used here was introduced in [14] and subsequently used by many authors, cf. [1, 2, 15, 21, 24, 27]. Here h = h0 = h∞ = hph . Let j0 , j∞ ∈ C ∞ (Rν ) be non-negative functions 2 = 1. satisfying: j0 = 1 on {k : |k| ≤ 1}, j0 = 0 on {k : |k| > 2}, and finally j02 + j∞ R R By j , R > 1, we understand the operator j = (j0 (x/R), j∞ (x/R)). Recall that x = i∇k is a differential operator. We view j R as a map from hph into hph ⊕ hph ˇ R ) is an isometry, see (2.16), and the operator Γ(j ˇ R ) : F → F ext := F ⊗ F and Γ(j ˇ R ) = 1lF . ˇ R )∗ Γ(j Γ(j
(2.29)
The partition of unity is used to decouple photons at infinity from photons near the electron. In fact the reader should think of the first component as the Fock-space for interacting photons and the second component as the Fock-space for non-interacting photons at infinity. As in the previous section we extend the notation for second quantization to these extended spaces. We will in general call operators constructed this way, extended operators. The simplest extended operator is the extended number operator, already encountered in Subsection 2.4 N ext := N0 + N∞ .
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This is a particular case of the following notation, which will be used for operators a on hph , dΓext (a) = dΓ(a) ⊗ 1lF + 1lF ⊗ dΓ(a) . (2.30) As in the previous section dΓext (a) is closable (essentially self-adjoint) if a is closable (essentially self-adjoint). Using this notation we introduce the extended Hamiltonian as (2.31) H ext (ξ) := H0ext (ξ) + Φ(v) ⊗ 1lF , where
H0ext (ξ) := dΓext (ω) + Ω ξ − dΓext (k) .
(2.32) C0∞
C0∞
The free extended Hamiltonian (2.32) is essentially self-adjoint on ⊗ and we write Dext = D(H0ext (ξ)), which is independent of ξ. Note that as for Proposition 1.1, Φ(v) ⊗ 1lF is H0ext (ξ)-bounded with relative bound 0, so H ext (ξ) is essentially self-adjoint on C0∞ ⊗ C0∞ and self-adjoint on Dext . Using the notation introduced in the previous subsection we have ∞ (2.33) F ext = F ⊕ H() , =1
and H ext (ξ) = H(ξ) ⊕
∞
H () (ξ) .
(2.34)
=1
2.7
The pull-through formula
In the following we use that a(k) makes sense as an operator on C 0 = Γfin (hph ∩ C 0 (Rν )). Here C 0 (Rν ) denotes the space of continuous functions on Rν . Note that a(k) : C 0 → C 0 , a(k) : C0∞ → C0∞ , and under the assumption v ∈ L2 (Rν ) ∩ C 0 (Rν ), we have H(ξ) : C0∞ → C 0 . Note that v need not be real-valued. For the definition of C0∞ , see (1.12). The type of formula presented here has been used previously in the study of ground states of translation invariant models, cf. [19], and confined models, see, e.g., [5, 24, 26]. Proposition 2.2 Suppose v ∈ L2 (Rν ) ∩ C 0 (Rν ). Let ξ ∈ Rν , n ≥ 1, k ∈ Rnν , and z ∈ C. For ψ ∈ C0∞ we have the identity a(k1 ) · · · a(kn ) (H(ξ) − z) ψ n = H(ξ − k (n) ) + ω(ki ) − z a(k1 ) · · · a(kn ) ψ i=1
+
n
v(ki ) a(k1 ) · · · a(k i ) · · · a(kn ) ψ ,
i=1
where k
(n)
= k1 + · · · + kn .
The notation a(k i ) indicates that the term a(ki ) is omitted from the product.
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For n = 1 we formulate another pull through formula. Note that for ψ ∈ 1 D(N 2 ), the map k → a(k)ψ is in L2 (Rν ; F ). In general, for ψ ∈ F we have 1 k → a(k)ψ in L2 (Rν ; D(N 2 )∗ ). The following proposition can be proved directly as in [26, Proposition 3.4], or by using Proposition 2.2 and an approximation argument. Proposition 2.3 Suppose v ∈ L2 (Rν ). Let ξ ∈ Rν and z ∈ C, Imz = 0. For ψ ∈ D, we have the L2 (Rν ; F )-identity −1 H(ξ − k) + ω(k) − z a(k) (H(ξ) − z) ψ −1 ψ. = a(k) ψ + v(k) H(ξ − k) + ω(k) − z
3 Spectral theory We start this section by giving a proof of Proposition 1.1. First some simple observations. Since 1 ≤ m− ω(k) for any ≥ 0, we obtain from (2.11) that N k ≤ −k m dΓ(ω)k , for k ∈ N. Since 0 ≤ dΓ(ω) ≤ H0 (ξ) and they commute, we find that dΓ(ω)k ≤ H0 (ξ)k for any k ∈ N. We thus get N k ≤ m−k H0 (ξ)k , for k ∈ N. This estimate in particular shows that for k ∈ N k
k
k
k
N 2 is H0 (ξ) 2 − bounded and N ext 2 is H0ext (ξ) 2 − bounded .
(3.1)
Proof of Proposition 1.1. We begin by showing that D(H0 (ξ)) is independent of
1 ξ. We compute on C0∞ as an operator identity H(ξ) − H(0) = ξ · 0 ∇Ω(tξ − dΓ(k))dt. By Condition 1.1 and the estimate ab ≤ aq + bp , q −1 + p−1 = 1 we obtain (H(ξ) − H(0))ψ ≤ Ω(dΓ(k))ψ + C(, ξ)ψ, for any > 0 and ψ ∈ C0∞ . That the domain is independent of ξ now follows from the Kato-Rellich theorem [49, Theorem X.12]. As for ii), the observation (3.1) (applied with k = 1), together with the N 1/2 -boundedness of Φ(v), cf. Lemma 2.1 i), implies the result. The last part follows from the variational principle and an argument similar to the one given for i). We leave it to the reader. Clearly Proposition 1.1 also holds with {H0 (ξ), H(ξ)} replaced by either of () the pairs {H0ext(ξ), H ext (ξ)} or {H0 (ξ), H () (ξ)}. We note the following consequence, for k ∈ {1, 2}, k
k
k
k
N 2 is H(ξ) 2 − bounded , N ext 2 is H ext (ξ) 2 − bounded , N () Here N () := dΓ() (1lhph ).
k 2
k
is H () (ξ) 2 − bounded .
(3.2) (3.3)
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Localization errors
ˇ R) In this subsection we show that localization errors arising when we apply Γ(j are small for large R. Lemma 3.1 Let s ∈ N0 ∩ [0, sΩ ] and f ∈ C ∞ (Rν ) satisfy the bound |(∂ α f )(η)| ≤ Cα η s−|α| , for any multi-index α. Let t = 1, if s = 0, and t = (1 + sΩ − s)/2 if s ≥ 1. We have as a form on F ext × F, R ˇ R ) (H0 (ξ) − i)−1 ˇ ) f (ξ − dΓ(k)) − f (ξ − dΓext (k)) Γ(j (H0ext (ξ) − i)−1 Γ(j = (H0ext (ξ) − i)−t B1 (R) = B2 (R)(H0 (ξ) − i)−t , where B1 and B2 are families of bounded operators which satisfy B1 (R) + B2 (R) = O(R−1/2 ), as R → ∞, locally uniformly in ξ. Proof. As a first step we compute as a form on (C0∞ ⊗ C0∞ ) × C0∞ , for 1 ≤ p ≤ ν, ˇ R ) dΓ(k;p ) − dΓext (k;p ) Γ(j ˇ R ) = dΓ(j ˇ R , sR Γ(j p),
(3.4)
R R R sR p = ([j0 , k;p ], [j∞ , k;p ]). Clearly sp are bounded operators and R , k;p ] = O(R−1 ) , as R → ∞ . [j#
(3.5)
Here we used the notation k;p to denote the p’th coordinate of a vector k ∈ Rν . (This notation should not be confused with the labeling kj of a family of vectors kj ∈ Rν .) We consider first the case s = 0. Let f˜ ∈ C ∞ (Cν ) denote an almost analytic extension of f . Let χ ∈ C0∞ (Rν ) be equal to 1 near 0. Write χn (η) = χ(η/n). Then fn = χn f has almost analytic extensions f˜n satisfying that, for all z ∈ Cν : ∂¯f˜n (z) → ∂¯f˜(z), and the estimates |∂¯f˜n (z)| ≤ C z −1− |Imz|
(3.6)
hold uniformly in n, cf. (A.4). If we take for example the Borel construction (A.2), for f˜ and the f˜n ’s, then this property is easy to verify. This well-known approximation technique has been used by many authors (in the case ν = 1), see, e.g., [52, Section 5] and [46, Section 4]. We use (3.4) to compute as a form on (C0∞ ⊗ C0∞ ) × C0∞ , for Imz = 0,
=
ˇ R) ˇ R ) |ξ − dΓ(k) − z|2 − |ξ − dΓext (k) − z|2 Γ(j T (z; R) := Γ(j ν ext ˇ R , sR ˇ R , sR dΓ(j (k;p ) + z;p ) dΓ(j p ) (ξ;p − dΓ(k;p ) − z;p ) + (ξ;p − dΓ p) . p=1
˜# = h# = hph and a#,1 = [j R , k;p ]), and (3.5), we Using (2.10), (2.20) (with h = h # conclude the following estimate 1
1
(N ext + 1)− 2 |ξ − dΓext (k) − z|−1 T (z; R) |ξ − dΓ(k) − z|−1 (N + 1)− 2 (3.7) = O |Imz|−1 R−1 . The estimate is valid uniformly in ξ and Rez = {Rez1 , . . . , Rezν }.
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We proceed to compute
=
ˇ R) ˇ R ) |ξ − dΓ(k) − z|−2ν − |ξ − dΓext (k) − z|−2ν Γ(j Γ(j ˇ R ) |ξ − dΓ(k) − z|2ν −|ξ − dΓ(k) − z|−2ν Γ(j ˇ R ) |ξ − dΓ(k) − z|−2ν − |ξ − dΓext (k) − z|2ν Γ(j
=
−
ν−1
|ξ − dΓ(k) − z|−2(ν−j) T (z; R) |ξ − dΓ(k) − z|−2(j+1) .
(3.8)
j=0
Combining this identity with (3.7), we obtain the estimate 1 ˇ R ) |ξ − dΓ(k) − z|−2ν (N ext + 1)− 2 Γ(j ˇ R ) (N + 1)− 12 − |ξ − dΓext (k) − z|−2ν Γ(j = |ξ − dΓext (k) − z|−1 O |Imz|−2ν R−1 = O |Imz|−2ν R−1 |ξ − dΓ(k) − z|−1 .
(3.9)
A small calculation using (3.4) (and again the estimates (2.10), (2.20), and (3.5)) in conjunction with (3.6) and (3.9) gives the following estimate for all 1 ≤ p ≤ ν and ≥ 0 1 ˇ R ) (ξ;p − dΓ(k;p ) + z;p ) |ξ − dΓ(k) − z|−2ν ∂¯p f˜n (z)(N ext + 1)− 2 Γ(j ˇ R ) (N + 1)− 12 − (ξ;p − dΓext (k;p ) + z;p ) |ξ − dΓext (k) − z|−2ν Γ(j (3.10) = O z −−1 |Imz|−2ν R−1 . By choosing = 2ν, in order to dampen the singularity at the real axis, we get an integrable weight factor z −2ν−1 , uniformly in n. We can now invoke the Lebesgue theorem on dominated convergence, and remove the cutoff by taking n → ∞ in the representation formula (A.6). This gives finally 1 ˇ R ) f (ξ − dΓ(k)) − f (ξ − dΓext (k)) Γ(j ˇ R ) (N + 1)− 12 (N ext + 1)− 2 Γ(j = O(R−1 ) . Note that the term in the brackets above is a bounded operator with norm bounded uniformly in R and ξ. We thus get by interpolation (and since powers of N can be moved around as we please) for 0 ≤ ρ ≤ 1/2. 1 ˇ R ) f (ξ − dΓ(k)) − f (ξ − dΓext (k)) Γ(j ˇ R ) (N + 1)−ρ (N ext + 1)ρ− 2 Γ(j 1
= O(R− 2 ) . By (3.1), this concludes the proof for the case s = 0.
(3.11)
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Next we consider the case s = 1 (and hence sΩ ∈ {1, 2}). Use Taylor’s formula
1 to write f (η) = f (0) + η · F0 (η), where F0 (η) = 0 (∇f )(tη)dt. It is easy to verify that F0 ’s coordinate functions satisfy the assumption of the lemma with s = 0. From (3.4) (again combined with (2.10), (2.20), and (3.5)) and (3.11) we get, as a form estimate on (C0∞ ⊗ C0∞ ) × C0∞ , 1 ˇ R ) f (ξ − dΓ(k)) − f (ξ − dΓext (k)) Γ(j ˇ R ) (N + 1)−ρ (N ext + 1)ρ− 2 Γ(j 1
ν
1
p=1 ν
= O(R− 2 ) + = O(R− 2 ) +
1
(ξ;p − dΓext (k;p )) O(R− 2 ) 1
O(R− 2 ) (ξ;p − dΓ(k;p )) .
(3.12)
p=1
Note that if sΩ = 1 then dΓ(k) is H0 (ξ)-bounded, and if sΩ = 2 then dΓ(k) is H0 (ξ)1/2 -bounded. Corresponding relative bounds for the extended operators hold as well. This implies the lemma for s = 1. In the remaining case s = 2 (and hence sΩ = 2). We proceed in a similar fashion, writing f (η) = f (0) + η · F1 (η), where F1 ’s coordinate functions satisfy the assumptions of the lemma with s = 1. Since in this case dΓ(k) and F1 (ξ − dΓ(k)) are H0 (ξ)1/2 -bounded, the result follows (by a similar argument) from the s = 1 case. Lemma 3.2 We have as a form on F ext × F,
R ˇ ˇ R ) (H0 (ξ) − i)−1 ) H(ξ) − H ext (ξ) Γ(j (H0ext (ξ) − i)−1 Γ(j 1
1
= (H0ext (ξ) − i)− 2 B1 (R) = B2 (R)(H0 (ξ) − i)− 2 , where B1 and B2 are families of bounded operators satisfying B1 (R)+B2(R) = o(1), as R → ∞, locally uniformly in ξ. Proof. By Lemma 3.1, applied with f = Ω and s = sΩ , we only need to prove the lemma with H(ξ) replaced by dΓ(ω) and Φ(v), and H ext (ξ) replaced by dΓext (ω) and Φ(v) ⊗ 1lF respectively. We begin by computing as a form on Dext × D ˇ R ) dΓ(ω) − dΓext (ω) Γ(j ˇ R ) = dΓ(j ˇ R , rR ) , Γ(j R where rR = ([j0R , ω], [j∞ , ω]). By Condition 1.2 iii) and pseudo differential calculus, 1 R the components of r satisfies, as operators on D(ω 2 )∗ , 1
1
R , ω] ω − 2 = O(R−1 ) , for R → ∞ . ω − 2 [j#
(Alternatively one could also use here the calculus of almost analytic extensions.) The contribution to B1 and B2 coming from dΓ(ω) thus satisfies the required
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1 bounds by (2.10), (2.19), and (3.1). Here we choose h = h# = hph , ˜h# = D(ω 2 )∗ , 1 1 1 1 R a#,2 = ω 2 , and a#,1 = {ω − 2 [j# , ω]ω − 2 }ω 2 , when applying (2.19). It remains to treat the contribution from the perturbation. We compute as a form on Dext × D, using [14, Lemma 2.14 (iii)]
ˇ R ) Φ(v) − Φ(v) ⊗ 1lF Γ(j ˇ R) Γ(j 1 ∗ R ˇ R) a ((1 − j0R )v) ⊗ 1lF + 1lF ⊗ a∗ (j∞ v) Γ(j = −√ 2 ˇ R ) a((1 − j R )v) . + Γ(j 0
R Eq. (3.1) and Lemma 2.1 ii) now yield the result, since s − limR→∞ j∞ = s− R 2 ν limR→∞ (1 − j0 ) = 0 and v ∈ L (R ).
We immediately get the following two corollaries. Corollary 3.3 We have for any R > 1 ˇ R ) : D → Dext and Γ(j ˇ R )∗ : Dext → D1/2 , Γ(j 1/2 ext where D1/2 = D(H0 (ξ)1/2 ) and D1/2 = D(H0ext (ξ)1/2 ) are independent of ξ.
The first part of the following corollary follows from Lemma 3.2 while the second part follows from the first part and the calculus of almost analytic extensions (with ν = 1), as presented in Subsection A.1. Corollary 3.4 We have, in the limit R → ∞, i) The following estimate holds true locally uniformly in ξ and z ∈ C with Imz = 0 ˇ R ) (H(ξ) − z)−1 − (H ext (ξ) − z)−1 Γ(j ˇ R ) = |Imz|−2 o(1) . Γ(j
ii) For f ∈ C0∞ (R), we have uniformly in ξ ˇ R ) f (H(ξ)) − f (H ext (ξ)) Γ(j ˇ R ) = o(1) . Γ(j
3.2
The HVZ-Theorem
In this subsection we prove Theorem 1.2. Recall the abbreviations k = (k1 , . . . , kn ) ∈ Rnν and k (n) = k1 + · · · + kn . We start by establishing three lemmas (n) Proof of Lemma 1.6. Suppose to the contrary that k ∈ I0 (ξ), that is Σ0 (ξ − k (n) ) ≥ Σess (ξ − k (n) ), cf. (1.25). Then there exist ≥ 1 and kn+1 , . . . , kn+ ,
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cf. (1.15), such that (writing k (n+) = (n)
Σ0 (ξ; k) =
n+ i=1
1113
ki )
n Σ0 ξ − k (n) + ω(ki ) i=1
≥
n+ (n+) Σ0 ξ − k + ω(ki )
≥
Σ0
i=1 (n+)
(n)
(ξ) > Σ0 (ξ; k) ,
which is a contradiction. This proves the lemma.
Lemma 3.5 Let n ≥ 1, and B ∈ L2sym (Rnν ; B(F )). Suppose B(k) commute with N for almost all k ∈ Rnν . Define for ψ ∈ C0∞ the map a(B) ψ := B(k) a(k1 ) · · · a(kn ) ψ dnν k . Rnν
Then (N + 1)−n/2 a(B) extends from C0∞ to a bounded operator on F and there exists C = C(n) such that C −1 (N + 1)−n/2 a(B)B(F ) ≤ B :=
Rnν
B(k)2B(F ) dnν k
12
.
(3.13)
Proof. Let ψ ∈ C0∞ and ϕ ∈ F, with ϕ = 1. We estimate ϕ, (N + n + 1)−n/2 a(B) ψ ϕ, (N + n + 1)− n2 B(k) a(k1 ) · · · a(kn ) ψ dnν k ≤ nν R ϕ, B(k) a(k1 ) · · · a(kn ) (N + 1)− n2 ψ dnν k = nν R n ≤ B(k)B(F ) a(k1 ) · · · a(kn ) (N + 1)− 2 ψ dnν k Rnν 1 a(k1 ) · · · a(kn ) (N + 1)− n2 ψ 2 dnν k 2 ≤ B ≤
Rnν
B ψ .
Here we used the representation N = Rν a∗ (k)a(k)dν k repeatedly in the last step. This estimate yields the lemma (with C = ((n + 1)/2)n/2 ). Lemma 3.6 Let χ ∈ C0∞ (R) and ξ ∈ Rν . Then, for all k, ≥ 0, the form N k χ(H(ξ))N extends from C0∞ to a bounded form on D∗ . Remark. We employ the standard triple: D ⊂ F ⊂ D∗ continuously and densely.
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Proof. Recall from [14, Lemma 3.2] that N k χ(H(ξ))N extends to a bounded form on F . It remains to prove that it extends further by continuity to D∗ . It is sufficient to verify that H(ξ)N k χ(H(ξ)), viewed as a form on C0∞ × F, extends to a bounded form on F ⊗ F. Let ψ ∈ C0∞ and ϕ ∈ F. We compute for k ≥ 1, H(ξ) ψ, (N + 1)k χ(H(ξ)) ϕ = Φ(v) ψ, (N + 1)k χ(H(ξ)) ϕ + (N + 1)k ψ, H0 (ξ) χ(H(ξ)) ϕ 1 1 = (N + 1)− 2 Φ(v) ψ, (N + 1)k+ 2 χ(H(ξ)) ϕ + (N + 1)k ψ, H(ξ) χ(H(ξ)) ϕ 1 1 − (N + 1)−k− 2 Φ(v) (N + 1)k ψ, (N + 1)k+ 2 χ(H(ξ))ϕ .
An application of Lemma 2.1 i) now yields the result.
Proof of Theorem 1.2. We begin with i). Let ξ ∈ Rν and let f ∈ C0∞ (R) be such that supp f ⊂ (−∞, Σess (ξ)). By definition of Σess (ξ) (see (1.13–1.15)), (2.21), (2.33), (2.34), and (2.28), we observe that H
ext
(ξ) 1l(N∞ ≥ 1) =
∞
H () (ξ)
=1
≥
∞
()
Σ0 (ξ) 1lH() ≥ Σess (ξ) 1l(N∞ ≥ 1) .
=1
Here we used the identification 1lH() = 1l(N∞ = ). The lower bound above, together with (2.29) and Corollary 3.4 ii), yields f (H(ξ)) = = =
ˇ R ) + o(1) ˇ R )∗ f (H ext (ξ)) Γ(j Γ(j Γ(j0R ) f (H(ξ)) Γ(j0R ) ˇ R )∗ f (H ext (ξ)) 1l(N∞ ≥ 1) Γ(j ˇ R ) + o(1) + Γ(j Γ(j0R ) f (H(ξ)) Γ(j0R ) + o(1) , for R → ∞ .
The first term on the right-hand side is compact, by a standard argument using Condition 1.2 ii). This implies that f (H(ξ)) is a compact operator, and hence; that the spectrum of H(ξ) below Σess (ξ) is locally finite. As for ii), fix ξ ∈ Rν and λ ≥ Σess (ξ). We wish to show that there exists n0 ≥ 1 and η = (η1 , . . . , ηn0 ) ∈ Rn0 ν such that λ = Σ0 (ξ − η (n0 ) ) +
n0 i=1
where η (n0 ) =
n0 i=1
ηi .
(n0 )
ω(ηi ) and η ∈ I0
(ξ) ,
(3.14)
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Let n0 be given by n0 + 1 = min{n : λ < minn ≥n Σ0 (ξ)}. The minima exist, and n0 ≥ 1, due to (1.15) and (1.19). There exists k = (k1 , . . . , kn0 ) such n0 (n ) that Σ0 0 (ξ) = Σ0 (ξ − k (n0 ) ) + i=1 ω(ki ) ≤ λ, where k (n0 ) = k1 + · · · + kn0 . By Condition 1.2 ii), (1.18), and continuity of Σ0 (ξ), cf. Proposition 1.1, we can find η such that the first part of (3.14) is fulfilled. The choice of n0 and Lemma 1.6 implies the last part. By i); Σ0 (ξ − η (n0 ) ), given by (3.14), is an eigenvalue for H(ξ − η (n0 ) ). We write ϕ0 for a corresponding ground state; H(ξ − η (n0 ) )ϕ0 = Σ0 (ξ − η (n0 ) )ϕ0 . Let f ∈ C0∞ (Rν ) with f ≥ 0 and f (0) = 1. Write fi, (k) = ν/2 f ( (k − ηi )). Then {f1, }∈N , . . . , {fn0 , }∈N is a family of uniformly bounded sequences in hph , which all converge weakly to 0. Let ψ = a∗ (fn0 , ) · · · a∗ (f1, )ϕ0 . The rest of the proof is concerned with showing that ψ is a Weyl sequence for the energy λ. Note that by Lemma 3.6 and Lemma 2.1 i), we have ϕ0 ∈ D(a∗ (fn0 , ) · · · a∗ (f1, )). Lemma 2.1 iii) furthermore implies that {ψ }∈N converges weakly to zero in F . For ψ to be a Weyl sequence it must satisfy ψ > 0 uniformly in . Let S(n) denote the group of permutations of n elements, and write (σk)j = kσ(j) , for σ ∈ S(n) and k ∈ Rnν . (n) Let n be such that ϕ0 = 0. Pick a compact set (of non-zero measure) nν K ⊂ R with the following properties: (K1) If k ∈ K then σk ∈ K, for all σ ∈ S(n). (K2) For k ∈ K we have ki = ηj , 1 ≤ i ≤ n and 1 ≤ j ≤ n0 . (K3) (n) 1l(k ∈ K)ϕ0 = 0. (n ) (n) (n) Let ψK be defined by ψK := 0, for n = n, and ψK := 1l(k ∈ K)ϕ0 . By property (K2), there exists 0 such that a(fj, )ψK = 0, for any 1 ≤ j ≤ n0 , and
≥ 0 . By the CCR (2.1) we thus get, for ≥ 0 , ∗ 0 fj, , fσ(j), ψK , ϕ0 = Πnj=1 a (fn0 , ) · · · a∗ (f1, ) ψK , ψ σ∈S(n0 )
=
(n) (n) 0 Πnj=1 fj , fσ(j) ϕ0 , 1l(k ∈ K) ϕ0
σ∈S(n0 ) (n)
≥ f 2n0 1l(k ∈ K) ϕ0 2 . This estimate and property (K3), implies ψ > 0 uniformly in ≥ 0 . It remains to prove that (H(ξ) − λ)ψ → 0 as → ∞. for the fiber Hamiltonian with the Let v˜ ∈ L2 (Rν ) ∩ C(Rν ). Write H(ξ) interaction Φ(v) replaced by Φ(˜ v ). Compute, as an identity on D, − k (n0 ) ) − H(ξ − η (n0 ) ) H(ξ = (k (n0 ) − η (n0 ) ) · (∇Ω)(ξ − η (n0 ) − dΓ(k)) (3.15) (n0 ) (n0 ) (n0 ) (n0 ) (n0 ) (n0 ) −η ), T (k ,η ) (k −η ) + Φ(˜ v − v) , + (k
1 where T (ζ1 , ζ2 ) = 0 (1 − t)(∇2 Ω)(ξ − tζ1 − (1 − t)ζ2 − dΓ(k))dt.
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Note that this operator is continuous and bounded uniformly in ζ1 (and ζ2 ) and commutes with the number operator. Abbreviate n0 ω Σ (k, η) := ω(kj ) − ω(ηj ) . j=1
By (3.14), (3.15), and the pull-through formula, Proposition 2.2, we get for ψ ∈ C0∞ ϕ0 , a(k1 ) · · · a(kn0 ) (H(ξ) − λ) ψ
(n0 ) = H(ξ − k ) − H(ξ − η (n0 ) ) + ω Σ (k, η) ϕ0 , a(k1 ) · · · a(kn0 ) ψ +
n0
v˜(ki ) ϕ0 , a(k1 ) · · · a(k i ) · · · a(kn0 ) ψ
i=1
=
Φ(˜ v − v)ϕ0 , a(k1 ) · · · a(kn ) ψ + ω Σ (k, η) ϕ0 , a(k1 ) · · · a(kn ) ψ − (k (n0 ) − η (n0 ) ) · (∇Ω)(ξ − η (n0 ) − dΓ(k)) ϕ0 , a(k1 ) · · · a(kn0 ) ψ + (k (n0 ) − η (n0 ) ), T (k (n0 ) , η (n0 ) )(k (n0 ) − η (n0 ) ) ϕ0 , a(k1 ) · · · a(kn ) ψ +
n0
v˜(ki ) ϕ0 , a(k1 ) · · · a(k i ) · · · a(kn0 ) ψ .
i=1
Abbreviate B1 (k) := 2 (k) := Bp,
B3 (k) :=
0 ω Σ (k, η) Πnj=1 fj, (kj ) 1lF ,
(n0 ) (n0 ) 0 (k;p − η;p ) Πnj=1 fj, (kj ) 1lF , (n0 ) 0 (n0 ) (n0 ) (n0 ) (k −η ), T (k ,η )(k (n0 ) − η (n0 ) ) Πnj=1 fj, (kj ) .
By construction of the fj, ’s we find (see (3.13) for the definition of the norm) B1 +
ν
2 Bp, + B3 → 0 , for → ∞ .
p=1
Using the notation introduced in Lemma 3.5, we can now compute ψ , (H(ξ) − λ)ψ = ϕ0 , Φ(˜ v − v)a(f1, ) · · · a(fn0 , )ψ 1 + ϕ0 , a(B ) ψ + ϕ0 , a(B3 ) ψ ν 2 + ∂p Ω(ξ − η (n0 ) − dΓ(k)) ϕ0 , a(B,p )ψ p=1 n0 ∗ (f ) · · · a∗ (f )ϕ , ψ fi, , v˜ a∗ (fn0 , ) · · · a + i, 1, 0 i=1
(3.16)
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By Lemma 2.1 ii) we can take the limit v˜ → v in L2 (Rν ). This amounts to replacing v˜ by v and H(ξ) by H(ξ) in the equation above. The resulting identity together with Condition 1.1, Lemma 2.1 i), and Lemma 3.6 implies that ψ ∈ D and n (H(ξ) − λ) ψ ≤ C (N + 1) 20 ϕ0 B1 + B3 ν n ∂p Ω(ξ − η (n0 ) − dΓ(k)) (N + 1) 20 ϕ0 B 2 +C ,p p=1
+ C0,n0 −1
n0 n0 −1 max |fj, , v | (N + 1) 2 ϕ0 Πk =i fk, .
1≤j≤ν
i=1
By (3.16) and the fact that w − lim→∞ fj, = 0, we thus find (H(ξ) − λ)ψ → 0 as → ∞, and hence; ψ is a Weyl-sequence. This concludes the proof.
3.3
Uniqueness, existence, and non-existence of ground states
We begin by applying the Perron-Frobenius theorem of Faris, which is presented in Appendix A.2. See also Fr¨ ohlich [19, 20]. We write hph = hph R ⊕ ihph R , where hph R is the real Hilbert space consisting ⊗s n of the real valued functions in hph . We define HR := ⊕∞ n=0 hph R , which is also a real Hilbert space. We define the cone (n) C := ×∞ , n=0 C (n)
C
:= { f ∈
⊗s n hph R
n−times : (−1) v ⊗ · · · ⊗ v f ≥ 0 a.e. } . n
(3.17)
In this section we assume (1.20), i.e., that the coupling function v ∈ L2 (Rν ) is real valued and non-zero a.e., which implies that C is a Hilbert cone in the sense of Definition A.1. Clearly f (H0 (ξ)) is positivity preserving in the sense of Definition A.2 ii), for any bounded non-negative Borel function f . For µ > 0 sufficiently large, the Neumann series −1
(H(ξ) + µ)
=
∞
k (H0 (ξ) + µ)−1 (−Φ(v)) (H0 (ξ) + µ)−1
(3.18)
k=0 1
converge. Note that Φ(v)(H0 (ξ) + µ)−1 ≤ Cµ− 2 ; cf. Lemma 2.1 i) and (3.1). We find from this formula that for any real-valued v ∈ L2 (Rν ), that the resolvent (H(ξ)+µ)−1 is positivity preserving. In fact, we find from (3.18) that, the resolvent (H(ξ) + µ)−1 is a sum of terms of the form
k (H0 (ξ) + µ)−1 − a# (v) (H0 (ξ) + µ)−1 , where all powers k and combinations of a∗ (v) and a(v) occur. Furthermore each of these terms are positivity preserving.
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Let u ∈ C\{0}. There exists n ≥ 0 such that un ∈ hph c⊗s n , the projection onto the n-particle sector, is non-vanishing; un = 0. We wish to prove that, under the assumption (1.20) on v, (H(ξ) + µ)−1 u is strictly positive in the sense of Definition A.2 i). Let w ∈ C\{0}. There exists n ≥ 0 such that wn ∈ hph c⊗s n is non-zero; wn = 0. We estimate (H(ξ) + µ)−1 u, w ≥ (H(ξ) + µ)−1 un , wn
n
n ≥ − a(v)(H0 (ξ) + µ)−1 un , − (H0 (ξ) + µ)−1 a(v) (H0 (ξ) + µ)−1 wn ≥ µ−n−n −1 (−1)n v(k1 ) · · · v(kn )un (k)dnν k νn R × (−1)n v(k1 ) · · · v(kn )wn (k)dn ν k . Rνn
The right-hand side is strictly positive and hence; (H(ξ) + µ)−1 u is strictly positive. Since u ∈ C\{0} was arbitrary we conclude that (H(ξ) + µ)−1 is positivity improving in the sense of Definition A.2 iii). The abstract result of Faris, Theorem A.3 now implies that a ground state, if it exists, is unique and strictly positive in the sense of Definition A.2 i). This proves Theorem 1.3. We now embark on: Proof of Theorem 1.5 ii). Let ξ be such that Σ0 (ξ) = Σess (ξ). Assume Σ0 (ξ) is an eigenvalue. By Theorem 1.3, the eigenvalue is non-degenerate and we can choose an eigenfunction ψξ ∈ C which is strictly positive. (1) Recall from Corollary 1.4 that Σess (ξ) = Σ0 (ξ), under Condition 1.3. Let (1) (1) M := {k ∈ Rν : Σ0 (ξ; k) = Σ0 (ξ)} be the set of minimizers. By (1.23) and (1) Lemma 1.6, M is a compact subset of the open set I0 (ξ). There exists k0 ∈ ∂M, ν a unit vector u ∈ R , and a number r > 0, with the following property: For any δ > 0 we have (1)
Ωrδ := {k ∈ Rν : k − k0 ≤ r and (k − k0 ) · u ≥ δ } ⊂ I0 (ξ) \ M . We also use this notation with δ = 0. For any δ > 0 there exists C(δ) such that inf Σ0 (ξ − k) + ω(k) − Σ0 (ξ − k0 ) ≥ C(δ)−1 .
k∈Ωrδ
(3.19)
Recall that Σ0 (ξ − k), k ∈ Ωr0 , are isolated eigenvalues and, again by Theorem 1.3, they are non-degenerate and we can choose eigenfunctions ψξ−k ∈ C (1) which are strictly positive. Since I0 (ξ) k → ψξ−k is continuous, we find inf ψξ−k , ψξ > 0 .
k∈Ωr0
(3.20)
Let Nδ := dΓ(1l(k ∈ Ωrδ )) = Ωr a∗ (k)a(k)dν k. Note that 0 ≤ Nδ ≤ N , δ and hence ψξ ∈ D(Nδ ) with Nδ ψξ ≤ N ψξ < ∞ uniformly in δ > 0. Using
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Proposition 2.3, (3.19), and the Lebesgue theorem on dominated convergence (to replace z, Imz = 0, by z = Σ0 (ξ)), we get ψξ , Nδ ψξ −1 2 v(k)2 H(ξ − k) + ω(k) − Σ0 (ξ) ψξ dk ≥ Ωrδ
≥ ≥
Ωrδ
−2 v(k)2 Σ0 (ξ − k) + ω(k) − Σ0 (ξ) |ψξ−k , ψξ |2 dk
inf r {|ψξ−k , ψξ |2 v(k)2 }
k∈Ω0
Ωrδ
(3.21)
−2 Σ0 (ξ − k) + ω(k) − Σ0 (ξ) dk . (1)
Since Σ0 (ξ − k) is a smooth function of k in I0 (ξ) and k0 is a global minimum of the function k → Σ0 (ξ − k) + ω(k), we find that there exists C > 0 such that 0 ≤ Σ0 (ξ − k) + ω(k) − Σ0 (ξ) ≤ C |k − k0 |2 , for k ∈ Ωr0 . This estimate together with (3.20), (3.21), and the assumption 3 ≤ ν ≤ 4 implies that |ψξ , Nδ ψξ | → ∞, as δ → 0. This contradicts ψξ ∈ D(N ), and hence; Σ0 (ξ) is not an eigenvalue. The first step in the proof of Theorem 1.5 i) is the following Lemma. Lemma 3.7 Let ξ ∈ Rν and z < Σ0 (ξ). Then Ω(ξ) − z − v(k)2 Ω, (H(ξ − k) + ω(k) − z)−1 Ω dk > 0 . Rν
Proof. Let PΩ := |Ω Ω|, and P Ω := 1lF − PΩ . Using the Feshbach projection method, cf., e.g., [4, Section II], we find −1 . (3.22) Ω, (H(ξ) − z)−1 Ω = Ω(ξ) − z − v, (H(ξ) − z)−1 v Ran P Ω
Here H(ξ) = P Ω H(ξ)P Ω as an operator on Ran P Ω , and v is viewed as an element of the one-particle space which is contained in Ran P Ω . By the spectral theorem the left-hand side of (3.22) is strictly positive and hence (3.23) Ω(ξ) − z − v, (H(ξ) − z)−1 v Ran P Ω > 0 . Viewing (H(ξ) − z)−1 v as an element of F we write −1 v, (H(ξ) − z) v Ran P Ω = v(k) Ω, a(k)(H(ξ) − z)−1 v dk .
(3.24)
Rν
Applying the pull-through formula, Theorem 2.3, with ψ = (H(ξ) − z)−1 v ∈ D, yields as an L2 (Rν ; F ) identity a(k) (H(ξ) − z)−1 v = (H(ξ − k) + ω(k) − z)−1 a(k) (H(ξ) − z) (H(ξ) − z)−1 v − v(k) (H(ξ − k) + ω(k) − z)−1 (H(ξ) − z)−1 v .
(3.25)
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We now make two observations. The first is the identity a(k) (H(ξ) − z) (H(ξ) − z)−1 v = a(k) v = v(k) Ω .
(3.26)
The second observation is that (H(ξ) − z)−1 is positivity preserving, with respect to the cone C introduced in (3.3) (after extending it by zero to the vacuum sector). This follows by a Neumann expansion, as for (H(ξ) + µ)−1 in (3.18), and Lemma A.4. Since (H(ξ − k) + ω(k) − z)−1 is also positivity preserving we find that, for a.e. k ∈ Rν , (3.27) Ω, (H(ξ − k) + ω(k) − z)−1 (H(ξ) − z)−1 v ≤ 0 . Combining (3.25)–(3.27) we get the following estimate a.e. v(k) Ω, a(k) (H(ξ) − z)−1 v ≥ v(k)2 Ω, (H(ξ − k) + ω(k) − z)−1 Ω . This estimate in conjunction with (3.23) and (3.24) concludes the proof.
Proof of Theorem 1.5 i). Assume that the statement is false at ξ, i.e., Σ0 (ξ) = Σess (ξ). The aim is to show that the equation v(k)2 Ω, (H(ξ − k) + ω(k) − z)−1 Ω dk (3.28) Ω(ξ) − z = Rν
has a solution z < Σess (ξ), which would by Lemma 3.7 provide a contradiction. In the limit z → −∞ the left-hand side dominates the right-hand side. A solution to (3.28) exists (and is necessarily unique by monotonicity) if we can show that the right-hand side diverges as z approaches Σess (ξ) from below. As in the proof of Theorem 1.5 ii) we choose a minimizer k0 ∈ Rν satisfying (1) (1) (1) Σ0 (ξ; k0 ) = Σ0 (ξ) = Σess (ξ). Then, by (1.23) and Lemma 1.6, k0 ∈ I0 (ξ) and (1) there exists a neighbourhood O ⊂ I0 (ξ) of k0 satisfying inf k∈O ψξ−k , Ω > 0. Here ψξ−k ∈ C, k ∈ O, are the strictly positive ground state eigenfunctions of H(ξ − k). We thus get v(k)2 Ω, (H(ξ − k) + ω(k) − z)−1 Ω dk Rν 2 2 ≥ inf {ψξ−k , Ω v(k) } (Σ0 (ξ − k) + ω(k) − z)−1 dk . k∈O
O
Since the right-hand side diverges in dimension 1 and 2, as z → Σess (ξ) from below, we conclude the result.
3.4
Regularity of t → σess (t)
We begin with (n) (n) Proof of Lemma 1.7. Let k be a local minimum of I0 (ξ) k → Σ0 (ξ; k). That the kj ’s must be equal follows from strict convexity of ω: Assume n ≥ 2. Let
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kj,s = (1−s)kj +s 21 (k1 +k2 ), j = 1, 2 and 0 ≤ s ≤ 1. Note that k1,s +k2,s = k1 +k2 , (n) so that substituting k1,s , k2,s for k1 , k2 only changes the contribution to Σ0 (ξ; k) coming from ω. We compute
1 d
ω(k1,s ) + ω(k2,s ) = (k2 − k1 ) ∇ω(k1,s ) − ∇ω(k2,s )} . ds 2
1 2 Since ∇ω(k1 ) − ∇ω(k2 ) = ( 0 ∇ ω(tk1 + (1 − t)k2 )dt(k1 − k2 ), we find that the derivative is strictly negative at s = 0, unless k1 = k2 . Write k1 = · · · = kn = Θ. We proceed to argue that Θ is a multiple of ξ. A local minimum is in particular a critical point, i.e., it satisfies ∇j Σ(n) (ξ; k) = −∇Σ(ξ − k (n) ) + ∇ω(kj ) = 0, 1 ≤ j ≤ n. By rotation invariance, this implies that ξ − nΘ is a multiple of Θ. This completes the proof. We introduce an index for a local minimum of s → σ (n) (t; s). (n)
Definition 3.8 Let n > 0, t ∈ R and s ∈ I0 (t). Assume s is a local minimum. We define the index to be Ind(n) (t; s) = min{ ∈ N : ∂s2 σ (n) (t; s) > 0}, with the convention that the index is ∞ if ∂s2 σ (n) (t; s) = 0 for all . For simplicity we (n) define Ind(n) (t; s) = 0 if s ∈ I0 (t) is not a local minimum for s → ∂s σ (n) (t; s ). Note that index 1 means that the local minimum is non-degenerate. (n)
Proposition 3.9 Let n > 0, t ∈ R and s ∈ I0 (t) be such that Ind(n) (t; s) ≥ 1. (n) There exist neighbourhoods Ot t and Os s, with Os ⊂ ∪t ∈Ot I0 (t ), such that the following holds 1) If Ind(n) (t; s) = 1, then there exists an analytic map Θ : Ot → Os , such that: Ind(n) (t ; Θ(t )) = 1 and Ind(n) (t ; s ) = 0, if s = Θ(t ). 2) If Ind(n) (t; s) = 2, then: For t ∈ Ot , s → σ (n) (t ; s ) has either one or two local minima in Os . For t = t, they have index 1. 3) If Ind(n) (t; s) = ∈ [3, ∞), then there exists a countable set K ⊂ Ot \{t}, with K ∪ {t} closed, such that: For t ∈ Ot , s → σ (n) (t ; s ) has between 1 and local minima in Os . For t ∈ Ot \(K ∪ {t}), they all have index 1. For t ∈ K all local minima s ∈ Os satisfies Ind(n) (t ; s ) ≤ − 1. 4) If Ind(n) (t; s) = ∞, then for t ∈ Ot \{t}, we have Ind(n) (t ; s ) = 0, for all s ∈ Os . Proof. 1) follows by analyticity in t and s of ∂s2 σ (n) (t; s), and the implicit function theorem. As for 2) and 3), we write = Ind(n) (t; s). We again invoke the implicit function theorem to construct an analytic function Θ from a neighbourhood Ot of t, into a neighbourhood Os of s, with the property that ∂s2−1 σ (n) (t ; Θ(t )) = 0, t ∈ Ot . Note that by choosing Ot small enough we have t − nΘ(t ) ∈ I0 . We begin by showing that near t no local minima can disappear to the same order as at t. We note that near t we may have at most local minima, but there is
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at least one. Let Ot tj → t and Os sj → s be such that sj is a local minimum of r → σ (n) (tj , r). Assume ∂sk σ (n) (tj , sj ) = 0 for k ≤ 2 − 1. Then necessarily, we must have sj = Θ(tj ). For 1 ≤ k ≤ 2 − 2, the function t → ∂sk σ (n) (t , Θ(t )) is analytic in Ot and vanishes on the sequence {tj }, hence it is identically zero in Ot . We can now compute 0 =
d 2−2 (n) ∂s σ (t ; Θ(t )) = n2−2 ∂ 2−1 σ(t − nΘ(t )) . dt
This implies that ∂ 2−1 ω(Θ(t )) = 0. The function ∂ 2−1 ω(s) has only isolated zeroes, since it is a analytic (and not identically zero). Hence Θ(t ) = Θ0 is a constant function on Ot . Since t → σ(t − nΘ0 ) + nω(Θ0 ) is thus linear near t, we find that σ is linear near t − ns. This implies in particular that ∂s2 σ (n) (t; s) = n∂ 2 ω(s) = 0. Recalling that ω is strictly convex we arrive at a contradiction. The statement 2) is now proved. The statement 3) follows from an induction argument in , starting with = 2. As for 4) we note that we must have σ (n) (t; s ) = C, for some constant C. In other words: σ(t − ns ) = C − nω(s ), for s near s. Compute σ(t − ns ) + nω(s ) = σ(t − n(s + (t − t )/n)) + nω(s ) = C + n{ω(s ) − ω(s + (t − t )/n)}. This gives ∂s σ (n) (t ; s ) = n{∇ω(s ) − ∇ω(s + (t − t )/n)}. This expression can only vanish if t = t . Proof of Theorem 1.9. We argue first that for a given t, the set M of global minima (n) of s → σ (n) (t; s) is finite. Note that by Lemma 1.6 we have M ⊂ I0 (t). Suppose to the contrary that M is infinite. Then either M contains a connected component (n) (n) of I0 (t) or there is a sequence in M converging to ∂I0 (t). In either case, this (n) is a contradiction since M is closed and I0 (t) is bounded and open. We remark that this also implies that a global minimum has finite index. By Proposition 3.9 2)–3), we find that the set T0 of t for which at least one of the global minima for the map s → σ (n) (t; s) have index strictly larger than 1, is closed and countable. It remains to show that the set of t for which there is more than one global minimum, all with index 1, is countable and can accumulate only at T0 . Suppose t is such that the map s → σ (n) (t; s) has global minima all with index 1. Note that for t near t these minima will persist at least as local minima, and any global minima will be found amongst these. There exists analytic maps t → Θj (t ), which parameterize these local minima. they are all defined in a neighbourhood of t, and satisfies Ind(n) (t ; Θj (t )) = 1. We estimate the rate of change of the global minima near t, using twice the critical equation (∂s σ (n) )(t ; Θj (t )) = 0, d (n) σ (t ; Θj (t )) = ∂σ(t − nΘj (t )) = ∂ω(Θj (t )) . dt
(3.29)
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Since ∂ω is monotonically increasing we find that that there exists a neighbourhood Ot of t such that for t ∈ Ot \{t}, the map s → σ (n) (t ; s) has a unique global minimum, with index 1. A compactness argument now concludes the proof. Note that (1.27) is implied by (3.29) since σ (n) (t) = σ (n) (t; Θ(n) (t)), for t ∈ T (n) .
3.5
Local minima of t → σess (t)
This subsection is devoted to the following proof. Proof of Theorem 1.10. Let t0 be a local minimum of t → σess (t) and let U t0 be an open set such that σess (t) ≥ σess (t0 ), t ∈ U. (1) The function Rν s → σ (1) (t0 ; s) has finitely many global minima Θ1 (t0 ) < (1) · · · < Θ (t0 ), all in I (1) (t0 ) and with finite index, cf. the proof of Theorem 1.9. (1)
Assume there exists 1 ≤ j ≤ such that s0 := Θj (t0 ) > 0. By Proposition 3.9 there exist Ot0 , Os0 , and K, with t0 ∈ Ot0 ⊂ U, s0 ∈ Os0 ⊂ (0, ∞) ∩ (1) (∪t∈Ot0 I0 (t)), and K ⊂ U is countable with K ∪ {t0 } closed, such that: For t ∈ Ot0 \(K ∪ {t0 }) all local minima of Os0 s → σ (1) (t; s) have index 1 (and at least one such local minimum exist). Furthermore, the set Ot0 \(K ∪{t0 }) can be written as a countable union of disjoint open intervals Iλ . On each of these intervals we get from the Implicit Function Theorem, that the number of local minima λ ≥ 1, is independent of t ∈ Iλ , and the local minima, Θλ,j (t), 1 ≤ j ≤ λ , are analytic in Iλ . As for (3.29) we compute ∂t σ (1) (t; Θλ,j (t)) = ∂ω(Θλ,j (t)) , for t ∈ Iλ .
(3.30)
Let τ (1) (t) := inf s∈Os0 σ (1) (t; s). Note that τ (1) is continuous on Ot0 and on any Iλ we have τ (1) (t) = min1≤j≤λ σ (1) (t; Θλ,j (t)). Since Θλ,j (t) > 0 we conclude from (3.30) that τ (1) is monotonely strictly increasing on any Iλ and hence by continuity on Ot0 . We now arrive at a contradiction with the assumption that t0 is local minimum for σess = σ (1) as follows. Estimate for t ∈ (−∞, t0 ) ∩ Ot0 : σ (1) (t) ≤ τ (1) (t) < τ (1) (t0 ) = σ (1) (t0 ). (1) We conclude from the argument above that any global minimum Θj (t0 ) (1)
must be less than or equal to zero. Similarly one can show that Θj (t0 ) ≥ 0, thus (1)
leaving only the possibility: = 1 and Θ(1) (t0 ) ≡ Θ1 (t0 ) = 0. This implies the first part of the theorem, namely that σess (t0 ) = σ (1) (t0 ; 0) = σ(t0 ) + m. Since the gap is m at t0 , and σess has a local minimum at t0 , we find from (1.16) that σ also has a local minimum at t0 . In particular σ has a critical point at t0 , with ∂ 2 σ(t0 ) ≥ 0, which yields the bound ∂s2 σ (1) (t0 ; s)|s=0 ≥ ∂ 2 ω(0). Hence Ind(1) (t0 ; 0) = 1. By Proposition 3.9 1), this implies that σess is analytic near t0 and ∂σess (t) = ∂ω(Θ(1) (t)) near t0 , cf. (3.29). Computing 0 = ∂t (∂s σ (1) (·; Θ(1) (·))),
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near t0 , yields the formula d (1) ∂ 2 σ(t − Θ(1) (t)) Θ (t) = 2 (1) . dt ∂s σ (t; Θ(1) (t))
(3.31)
The equation (1.28) for ∂ 2 σess (t0 ) now follows by (1.27), (3.31), and the computation dd d d2 ω(Θ(1) (t)) = ∂ 2 ω(Θ(1) (t)) Θ(1) (t) . σess (t) = 2 dt dt dt dt Recall that Θ(1) (t0 ) = 0.
We end this section with a comment on jump discontinuities of the bounded function ∂σess (t) = ∂ω(Θ(1) (t)). When t increases (away from 0), global minima are a priori not monotone, but when they jump, they jump from large s to smaller s. Passing to larger s, can only happen analytically (where ∂ 2 σ(t − s) ≥ 0, and hence a local minimum has index 1). This implies that Jump discontinuities of ∂σess always decrease the derivative.
(3.32)
4 Additional results In this section we collect some additional results, most of which have appeared elsewhere in some form. They serve to give a more complete picture of the bottom of the joint energy momentum spectrum. In addition we explain how to extend the results described in this paper to models with a number cutoff in the interaction.
4.1
Complimentary results
In this section we recall some known and partly known related results on the structure of the ground state mass shell. The first is due to Gross [31, (6.30)], cf. also [56, (15.26)]. Lemma 4.1 (Gross) Let √ v ∈ L2 (Rν ) be real-valued and symmetric, i.e., v(k) = v(−k) a.e., and ω(k) = k 2 + m2 , m ≥ 0. Assume Condition 1.1 and that, for any t > 0, the map p → e−tΩ(p) is positive definite. Then for all ξ ∈ Rν Σ0 (ξ) ≥ Σ0 (0) . Gross proved this statement for m > 0, but as remarked in [20] this extends by a limiting argument to m = 0. The second result we mention is an extension of a result of Hiroshima and Spohn. See [36, Lemma 3.1] and its proof. See also [56, (15.34)].
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Lemma 4.2 Let v satisfy (1.20), and assume Conditions 1.1 and 1.2. Let ξ ∈ I0 , write ψξ for a normalized ground state eigenfunction, and P ξ := 1lF − |ψξ ψξ |. Then {∇2 Σ0 (ξ)}ij = ψξ , ∂i ∂j Ω(ξ − dΓ(k))ψξ − P ξ ∂i Ω(ξ − dΓ(k)) ψξ , (H(ξ) − Σ0 (ξ))−1 P ξ ∂j Ω(ξ − dΓ(k)) ψξ . In particular ∇2 Σ0 (ξ) ≤ supp σ(∇2 Ω(p)) 1lRν . Note that by Theorem 1.3, H(ξ) − Σ0 (ξ) is bounded invertible on the range of P ξ . If ξ ∈ I0 is a critical point for ξ → Σ0 (ξ), then ∂j Σ0 (ξ) = ψξ , ∂j Ω(ξ − dΓ(k))ψξ = 0, 1 ≤ j ≤ ν, and hence the P ξ in the formula above for the Hessian is superfluous. This is the case considered in [36] (see also [54]). We leave the proof to the reader. In the case Ω(p) = p2 /2M , Lemma 4.2 implies a lower bound Meff ≥ M −1 := ∂ 2 σ(0) (assuming rotation invariance). See on the effective mass, where Meff [56, Section 15.2] for a discussion of effective mass. In [54] an upper bound for the effective mass is derived, implying in particular that ∂ 2 σ(0) > 0. This is still an open problem for Ω(p) = p2 /2M . We note that similarly one can prove the following statement: Replace v by gv, where g ∈ R is a coupling constant. Let g and ξ be such that ξ ∈ I0 , which is a g-dependent set. Then Σ0 (ξ) is an analytic function of the coupling constant in d a neighbourhood of g, dg Σ0 (ξ) = ψξ , Φ(v)ψξ , and d2 Σ0 (ξ) = − P ξ Φ(v) ψξ , (H(ξ) − Σ0 (ξ))−1 P ξ Φ(v) ψξ . 2 d g
(4.1)
In particular, the function g → Σ0 (ξ) is concave in the set {g : ξ ∈ I0 }. Thirdly we formulate a result, which follows from the proof of [20, Theorem 3.2]. We give a short proof of the statement here because Fr¨ ohlich concentrated on the massless case, and the proof simplifies for massive bosons. We remark that the infrared cutoff σ > 0 in [20] can be viewed as a mass. Theorem 4.3 Let v ∈ L2 (Rν ). Assume Conditions 1.1, 1.2, and that the following bounds hold for all p, k ∈ Rν |∇Ω(p)| ≤ 1 and ω(k) − |k| > 0 .
(4.2)
Then I0 = Rν . Remark. This theorem the case of relativistic elec implies in particular that in √ trons, i.e., Ω(p) = p2 + M 2 (M > 0), and ω(k) = k 2 + m2 (m > 0), we have an isolated ground state mass shell for all total momenta. This type of result was an important ingredient in [22]. Proof. Suppose I0 = Rν , and let ξ ∈ R\I0 .
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Define, for ξ, k ∈ Rν with k = 0,
F (ξ, k) := |k|−1 Ω(ξ − dΓ(k)) − Ω(ξ − k − dΓ(k) . This self adjoint operator extends from C0∞ to a bounded operator on F , and by (4.2) it satisfies the bound
Let
F (ξ, k)B(F ) ≤ 1 .
(4.3)
(n ) n := max n ≥ 1 : Σ0 (ξ) = Σess (ξ) .
(4.4) (n)
By Theorem 1.2 and (1.19) this choice of n is well defined. For k ∈ I0 (ξ), we write ψξ−k(n) ∈ D for a (normalized) ground state eigenfunction at total momentum ξ − k (n) . Note that k (n) = 0. For such k we use (4.3) and the Rayleigh-Ritz variational principle to estimate Σ0 (ξ)
≤ ψξ−k(n) , H(ξ) ψξ−k(n) = Σ0 (ξ − k (n) ) + |k (n) | ψξ−k(n) , F (ξ, k (n) ) ψξ−k(n)
(4.5)
≤ Σ0 (ξ − k (n) ) + |k (n) | . (n)
Let U := I0 (ξ) ∩ {η ∈ Rnν : Σ0 (ξ − η (n) ) ≤ Σ0 (ξ)}. The bound (4.5), Lemma 1.6, and the choice (4.4) of n, implies (n)
Σ0 (ξ) = ≥
n
(n) infnν Σ0 (ξ; k) = inf Σ0 (ξ − k (n) ) + ω(kj )
k∈R
k∈U
Σ0 (ξ) + inf
k∈U
n
j=1
ω(kj ) − |k (n) | .
j=1
By (1.18) there exists CU > 0, independent of n, such that |k (n) | ≤ CU , k ∈ U. Now choose R such that ω(k) ≥ CU + 1 for |k| ≥ R. Since |k (n) | ≤ |k1 | + · · · + |kn |, we arrive at the following estimate, cf. (4.4),
(n) Σ0 (ξ) = Σess (ξ) = Σ0 (ξ) ≥ Σ0 (ξ) + min 1 , By (4.2) this is a contradiction.
inf (ω(k) − |k|) .
k:|k|≤R
In addition to Theorem 1.9 we have a complimentary result which is concerned with the regularity of σ (n) (t) as a function of n. We leave the proof, which follows closely the proof of Theorem 1.9, to the reader Proposition 4.4 Let v satisfy (1.20). Assume Conditions 1.1, 1.2, 1.3 i), and 1.4. Let t ∈ R. There exists a closed countable set T (t) ⊂ (0, ∞), and an analytic (n) map (0, ∞)\T (t) n → Θ(n) (t) ∈ I0 (t), with the property that the maps s →
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σ (n) (t; s), n ∈ (0, ∞)\T (t), has a non-degenerate global minimum at s = Θ(n) (t), i.e., ∂s2 σ (n) (t; Θ(n) (t)) > 0. Let (a, b) ⊂ (0, ∞)\T (t). The global minimum is either unique for all n ∈ (a, b), or it is accompanied by another global minimum sitting at s = −Θ(n) (t), for all n ∈ (a, b). The case of two global minima can occur if and only if σ(t−r) = σ(t+r) for r in a neighbourhood of nΘ(n) (t). We furthermore have d (n) σ (t) = ω Θ(n) (t) − ∂ω Θ(n) (t) Θ(n) (t) , for n ∈ (0, ∞) \ T (t) . (4.6) dn The function x → ω(x) − x∂ω(x) appearing on the right-hand side of (4.6), is (n+1) (n) the one from (1.22). The identity (4.6) can be used to estimate Σ0 (ξ)−Σ0 (ξ).
4.2
Interactions with a number cutoff
In this subsection and the next we consider models of the form, cf. (1.6), HN := H0 + 1lK ⊗ 1l(N ≤ N ) V 1lK ⊗ 1l(N ≤ N ) . Here N ∈ Z is the cutoff parameter. Clearly these operators also commute with the total momentum and The corresponding fiber Hamiltonians are, cf. (1.8)–(1.10), HN (ξ) := H0 (ξ) + ΦN (v), where ΦN (v) := 1l(N ≤ N ) Φ(v) 1l(N ≤ N ) . Note that the notation is consistent since Φ0 (v) = 0. For N < 0 we clearly also have HN (ξ) = H0 (ξ). We remark that for N = 1 a complete picture can be obtained, cf. [23], (mass zero case). We note that the spin-boson model has been studied in the weak coupling regime for N = 2 in [45]. See also [25, 38, 39]. We now formulate our main results from Subsection 1.3 in the context of the cutoff models. We impose for brevity of exposition (1.21), Conditions 1.1, 1.2, 1.3 i), and 1.4 throughout this subsection. Let N ≥ 1. We introduce some notation. First the bottom of the spectrum of the full operator: ΣN ,0 := infν ΣN ,0 (ξ) , where ΣN ,0 (ξ) := inf σ(HN (ξ)) . ξ∈R
For n ≥ 1 and k = (k1 , . . . , kn ) ∈ Rnν we introduce n (n) ΣN ,0 (ξ; k) := ΣN −n,0 ξ − k (n) + ω(kj )
(4.7)
j=1
and
(n)
ΣN ,0 (ξ) :=
(n)
inf ΣN ,0 (ξ; k) .
k∈Rnν
(4.8)
The bottom of the essential spectrum is (1)
(1)
Σess,N (ξ) := ΣN ,0 (ξ) = infν ΣN ,0 (ξ; k) . k∈R
(4.9)
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We furthermore write ξ ∈ Rν : ΣN ,0 (ξ) < Σess,N (ξ) ,
(n) IN ,0 (ξ) := k ∈ Rnν : ξ − k (n) ∈ IN −n,0 . IN ,0 :=
(n)
The energies ΣN ,0 (ξ), n ≥ 1, are bottoms of branches of essential spectrum corresponding to having stripped off n photons to infinity, and having the interacting systems in a groundstate. Subadditivity of ω, (1.21), the fact groundstates lie in the cone (3.3), and the Rayleigh-Ritz variational principle ensures that the thresholds are ordered: (n) (n ) (4.10) ΣN ,0 (ξ) > ΣN ,0 (ξ) , for all n > n ≥ 1. Here the assumption v = 0 a.e. comes in. It ensures that the thresholds appear in an ordered fashion as in the full model. Note that the properties (1.16) and (1.17) do not hold for the cutoff model. The gap Σess,N (ξ) − ΣN ,0 (ξ) may exceed m. However, we do have that Σess,N (ξ) − ΣN −1,0 (ξ) ≤ m (it may be negative). We introduce, as in Subsection 1.3, the following notation. Let u be a unit vector in Rν . We write σN (t) = Σ0,N (t u), for t ∈ R. By rotation invariance, σN is inde(n) pendent of u. Similarly we write, for n ∈ N, σN (t; s) := σN −n ((t−ns) u)+nω(s u), (n) (n) σN (t) := Σ0,N (t u), and σess,N (t) := Σess (t u). With a slight abuse of notation, we use the same symbol I0,N to denote the (n) set of t’s such that t u ∈ I0,N . We furthermore use the symbol I0,N (t), n ∈ N, to denote the set {s ∈ R : t − ns ∈ I0,N }. We now list a number of results, which we do not prove here. See however the following subsection. In each case the reader can readily mimic the proofs, given in Section 3, of the corresponding results for the full model. I For each N ≥ 1 and ξ ∈ Rν , ΦN (v) is H0 (ξ) bounded with relative bound zero. In particular HN (ξ) is essentially self-adjoint on C0∞ , and D(HN (ξ)) is independent of ξ. II (HVZ) The bottom of the essential spectrum of HN (ξ) is Σess,N (ξ). Eigenvalues below Σess,N (ξ) have finite multiplicity and can only accumulate at Σess,N (ξ). See also [25, 38] for the cutoff spin-boson model. III The ground state is non-degenerate, and in addition: If 1 ≤ ν ≤ 2 then IN ,0 = Rν . If 3 ≤ ν ≤ 4 then the bottom of the spectrum ΣN ,0 (ξ) is an eigenvalue if and only if ξ ∈ IN ,0 . As a consequence of the non-degeneracy, the map IN ,0 t → σN (t) is analytic. (n)
IV Let n ∈ N. There exists a closed countable set TN ⊂ R, and an analytic (n) (n) (n) map R\TN t → ΘN (t) ∈ IN ,0 (t) with the property that the maps s → (n)
(n)
(n)
σN (t; s), t ∈ R\TN , has a unique global minimum at the point s = ΘN (t),
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(n)
with index Ind(n) (t; ΘN (t)) = 1. In particular R\TN t → σN (t) is ana(n) (n) (1) (1) d (n) lytic and dt σN (t) = ∂ω(ΘN (t)), for t ∈ R\TN . Recall σN (t) = σess,N (t). V Let t0 be a local minimum of t → σess,N (t). Then the ’spectral gap’ at t0 is maximal, i.e., σess,N (t0 ) − σN −1 (t0 ) = m, the map t → σN −1 (t) has a local minimum at t0 , the map t → σess,N (t) is analytic near t0 , and ∂ 2 σess,N (t0 ) =
4.3
∂ 2 ω(0) ∂ 2 σN −1 (t0 ) . ∂ 2 ω(0) + ∂ 2 σN −1 (t0 )
Comments on proofs
The key difference between the cutoff models and the full model, lies in the selfsimilarity of the full model. By self-similarity we mean that after removing a number of bosons to infinity, the remaining interacting system has the same Hamiltonian as the original system, albeit at a different total momentum. For the cutoff model the interacting system, after removing bosons to infinity, has a different cutoff. This is manifested in two instances, in the extended Hamiltonian and in the pull-through formula. For the cutoff model(s) one should replace the extended Hamiltonian, cf. (2.31) and (2.34), by ext (ξ) HN
:= HN (ξ) ⊕
∞
() HN (ξ) ,
(4.11)
=1 ()
where HN (ξ) =
Rν
()
HN (ξ; k)dν k and ()
HN (ξ; k) = HN − (ξ − k () ) +
ω(kj ) .
(4.12)
j=1
With this choice of extended Hamiltonian, the localization estimates derived in Subsection 3.1 applies. This is one of the inputs to the HVZ theorem. The second manifestation of the lack of self-similarity is in the pull-through formula which should be replaced by a(k) (HN (ξ) − z) ψ = HN −1 (ξ − k) + ω(k) − z a(k) ψ + v(k) 1l(N ≤ N − 1) ψ .
(4.13)
It is now left as an exercise to the reader to verify that the proofs go through. We just remark that when applying the Perron Frobenius argument, as in Subsec(j) tion 3.3, one should work only in the sub Hilbert space ⊕N j=0 Γ (hph ) of F . Any eigenfunction will vanish in n-particle sectors with n > N , which is reflected in the fact that the cutoff resolvents, (HN (ξ) + µ)−1 , are not positivity improving in the full Hilbert cone.
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Mathematical tools
A.1 Almost analytic extension In this subsect. we briefly recall the functional calculus provided by almost analytic extensions. In particular we will use a version which handles functions of a vector of commuting operators. See the monographs by Davies [12] and Dimassi and Sj¨ ostrand [16] for details. Below α will denote multi-indices. Let s ∈ R and f ∈ C ∞ (Rν ) satisfy ∀α : ∃Cα such that |∂ α f (x)| ≤ Cα x s−|α| .
(A.1)
We define an almost analytic extension f˜ ∈ C ∞ (Cν ) of f , through a Borel construction. Fix a function χ ∈ C0∞ (R) to be equal to 1 in a neighbourhood of 0, and a sequence {λk }k∈N0 , going sufficiently fast to infinity. The following choice will do: λk := max{max|α|=k Cα , λk−1 + 1}, for k ≥ 1, and λ0 = C0 . Here the constants Cα are coming from (A.1). Then, writing z = u + iv ∈ Rν ⊕ iRν , f˜(z) :=
∂ α f (u) α!
α
(iv)α
ν λ|α| vj . χ u j=1
(A.2)
Note that there exists C > 0 such that supp(f˜) ⊂ {u + iv : u ∈ supp(f ), |v| ≤ Cu } .
(A.3)
We furthermore have the property that ∀ ≥ 0 : ∃C such that |∂¯f˜(z)| ≤ C z s−−1 |Imz| . Here ∂¯ = (∂¯1 , . . . , ∂¯ν ), ∂¯j := ∂uj + i∂vj , and Imz = (v1 , . . . , vν ). If s < 0 we have the following representation, (x + z) 2ν 2ν−1 −1 ∂¯f˜(z), d z, f (x) = 2 |S | |x − z|2ν ν C
(A.4)
(A.5)
where d2ν z = Πνj=1 duj dvj is the Lebesgue measure on Cν , and |S 2ν−1 | is the volume of the unit ball in R2ν . (Note that for s < 0 the integral is absolutely convergent.) For a vector of pairwise commuting self-adjoint operators A = (A1 , . . . , Aν ), and a function f satisfying (A.1) with s < 0, the almost analytic extension thus provides a functional calculus via the formula ν 2ν−1 −1 | (A.6) ∂¯j f˜(z) (Aj + zj ) |A − z|−2ν d2ν z . f (A) = 2 |S j=1
Cν
In the case ν = 1 this reduces to 1 f (A) = ∂¯f˜(z) (A − z)−1 du dv . π C
(A.7)
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A.2 Invariant cones In this subsect. we recall a result of Faris, cf. [17], which will be used to show nondegeneracy of the ground state. It is an abstract version of the Perron-Frobenius Theorem in L2 -spaces, cf. [50, Theorem XIII.43], which together with the Q-space representation of Fock-space, has been used frequently to show non-degeneracy of the ground state, cf. [5, 28, 31]. Definition A.1 Let HR be a real Hilbert space. We say C ⊂ HR , C = {0}, is a Hilbert cone if: i) u, v ∈ C implies u + v ∈ C. ii) u ∈ C, λ ≥ 0 implies λu ∈ C. iii) C ∩ (−C) = {0}. iv) C is closed. v) u, v ∈ C implies u, v ≥ 0. vi) For all w ∈ HR there exists u, v ∈ C s. t. w = u − v and u, v = 0. An important example of a Hilbert cone is, as mentioned above, the subset of real non-negative functions in L2 (Q, dµ), where Q is a measure space. Definition A.2 Let HR be a real Hilbert space, C ⊂ HR a Hilbert cone and A a bounded operator on HR . i) We say u ∈ C is strictly positive if u, v > 0 for any v ∈ C\{0}. ii) A is positive preserving if AC ⊂ C. iii) A is positivity improving if Au is strictly positive for all u ∈ C\{0}. iv) A is ergodic if for any u, v ∈ C\{0} there exists n ≥ 0 s. t. An u, v > 0. Note that a positivity improving operator is in particular ergodic. The following theorem is due to Faris Theorem A.3 (Faris) Let HR be a real Hilbert space, C ⊂ HR a Hilbert cone and A a bounded positive self-adjoint operator on HR . Suppose furthermore that A is positivity preserving and that A is an eigenvalue for A. Then A is ergodic if and only if A is an eigenvalue of multiplicity one and there exists a strictly positive u ∈ C with Au = Au. s
−1
The lemma below follows from the identities e−s = limn→∞ ( ns + 1)−n and
∞ = 0 e−ts ds, for s > 0, in conjunction with the first resolvent formula.
Lemma A.4 Let A be a bounded from below self-adjoint operator on a real Hilbert space. Assume that there exists a λ0 < inf σ(A) such that (A − λ)−1 is positivity preserving (improving) for all λ < λ0 . Then (A − λ)−1 is positivity preserving (improving) for all λ < inf σ(A).
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Acknowledgments The author thanks Z. Ammari, V. Bach, J. Fr¨ ohlich, and C. G´erard, for useful discussions, and Dokuz Eyl¨ ul University for hospitality. This work was supported in parts by Carlsbergfondet and by a Marie-Curie individual fellowship from the European Union.
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[6] F.A. Berezin, The method of second quantization, 1 ed., Academic Press, New York, San Francisco, London, 1966. [7] V. Betz, F. Hiroshima, J. L¨ orinczi, R.A. Minlos, and H. Spohn, Ground state properties of the Nelson Hamiltonian - A Gibbs measure-based approach, Rev. Math. Phys. 14, 173–198 (2002). [8] L. Bruneau, The ground state problem for a quantum Hamiltonian model describing friction, C. R. Math. Acad. Sci. Paris 339, 151–156 (2004). [9] L. Bruneau and J. Derezi´ nski, Pauli-Fierz Hamiltonians defined as quadratic forms, Rep. Math. Phys. 54, 169–199 (2004). [10] J.T. Cannon, Quantum field theoretic properties of a model of Nelson: Domain and eigenvector stability for perturbed linear operators, J. Funct. Anal. 8, 101–152 (1971). [11] T. Chen, Operator-theoretic infrared renomalization and construction of dressed 1-particle states in non-relativistic QED, Ph.D. thesis, ETH Z¨ urich, 2002, ETH-Diss 14203. [12] E.B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge. [13] J. Derezi´ nski and C. G´erard, Scattering theory of classical and quantum N -particle systems, Text and Monographs in Physics, Springer, Berlin, 1997.
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[14]
, Asymptotic completeness in quantum field theory. massive PauliFierz Hamiltonians, Rev. Math. Phys. 11, 383–450 (1999).
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, Spectral and scattering theory of spatially cut-off P (φ)2 Hamiltonians, Comm. Math. Phys. 213, 39–125 (2000).
[16] M. Dimassi and J. Sj¨ ostrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999. [17] W.G. Faris, Invariant cones and uniqueness of the ground state for fermion systems, J. Math. Phys. 13, 1285–1290 (1972). [18] R.P. Feynman, Statistical mechanics. A set of lectures, Frontiers in physics, W. A. Benjamin, Inc., Reading, Massechusets, 1972. [19] J. Fr¨ ohlich, On the infrared problem in a model of scalar electrons and massless scalar bosons, Ann. Inst. Henri Poincar´e 19, 1–103 (1973). [20]
, Existence of dressed one-electron states in a class of persistent models, Fortschr. Phys. 22, 159–198 (1974).
[21] J. Fr¨ ohlich, M. Griesemer, and B. Schlein, Asymptotic completeness for Rayleigh scattering, Ann. Henri Poincar´e 3, 107–170 (2002). [22]
, Asymptotic completeness for Compton scattering, Comm. Math. Phys. 252, 415–476 (2004).
[23] A. Galtbayar, A. Jensen, and K. Yajima, The Nelson model with less than two photons, Ann. Henri Poincar´e 4, 239–273 (2003). [24] V. Georgescu, C. G´erard, and J. S. Møller, Spectral theory of massless Pauli Fierz models, Comm. Math. Phys. 249, 29–78 (2004). [25] C. G´erard, Asymptotic completeness for the spin-boson model with a particle number cutoff, Rev. Math. Phys. 8, 549–589 (1996). [26]
, On the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. Henri Poincar´e 1, 443–445 (2000).
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, On the scattering theory of massless Nelson models, Rev. Math. Phys. 14, 1165–1280 (2002).
[28] J. Glimm and A. Jaffe, The λ(ϕ4 )2 quantum field theory without cutoffs: II. The field operators and the approximate vacuum, Ann. Math. 91, 362–401 (1970). [29] M. Griesemer, Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics, J. Funct. Anal. 210, 321–340 (2004). [30] M. Griesemer, E. Lieb, and M. Loss, Ground states in nonrelativistic quantum electrodynamics, Invent. math. 145, 557–595 (2001).
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[31] L. Gross, Existence and uniqueness of physical ground states, J. Funct. Anal. 10, 52–109 (1972). [32]
, The relativistic polaron without cutoffs, Comm. Math. Phys. 31, 25–73 (1973).
[33] C. Hainzl, One non-relativistic particle coupled to a photon field, Ann. Henri Poincar´e 4, 217–237 (2003). [34] W. Heitler, The quantum theory of radiation, 3 ed., Dover Publications, 1983. [35] F. Hiroshima, Essential self-adjointness of translation-invariant quantum field models for arbitrary coupling constant, Comm. Math. Phys. 211, 585–613 (2000). [36] F. Hiroshima and H. Spohn, Ground state degeneracy of the Pauli-Fierz Hamiltonian including spin, Adv. Theor. Math. Phys. 5, 1091–1104 (2001). [37]
, Mass renormalization in nonrelativistic QED, mp arc (2003).
[38] M. H¨ ubner and H. Spohn, Radiative decay: nonperturbative approaches, Rev. Math. Phys. 7, 363–387 (1995). [39]
, Spectral properties of the spin-boson Hamiltonian, Ann. Inst. Henri Poincar´e 62, 289–323 (1995).
[40] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin Heidelberg, 1995 (reprint of the 1980 2nd edition). [41] E.H. Lieb and M. Loss, Self-energy of electrons in non-perturbative QED, Differential equations and mathematical physics (Providence, RI) (R. Weikard, ed.), AMS/IP Stud. Adv. Math., vol. 16, American Mathematical Society, 2000, pp. 279–293. [42]
, A bound on binding energies and mass renormalization in models of quantum electrodynamics, J. Stat. Phys. 108, 1057–1069 (2002).
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, Existence of atoms and molecules in non-relativistic quantum electrodynamics, Adv. Theor. Math. Phys. 7, 667–710 (2003).
[44] E.H. Lieb and L.E. Thomas, Exact ground state energy of the strong-coupling polaron, Comm. Math. Phys. 183, 511–519 (1997), Erratum 188, 499–500 (1997). [45] R. Minlos and H. Spohn, The three-body problem in radioactive decay: The case of one atom and at most two photons., Topics in statistical and theoretical physics. F. A. Berezin memorial volume. (Providence, RI) (R.L. Dobrushin, ed.), Transl., Ser. 2, Am. Math. Soc., vol. 177, American Mathematical Society, 1996, pp. 159–193. [46] J.S. Møller, An abstract radiation condition and applications to N -body systems, Rev. Math. Phys. 12, 767–803 (2000).
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[47] E. Nelson, Interaction of non-relativistic particles with a quantized scalar field, J. Math. Phys. 5, 1190–1197 (1964). [48] A. Pizzo, One-particle (improper) states in Nelson’s massless model, Ann. Henri Poincar´e 4, 439–486 (2003). [49] M. Reed and B. Simon, Methods of modern mathematical physics: II. Fourier analysis and self-adjointness, 1 ed., Academic Press, San Diego, 1975. [50]
, Methods of modern mathematical physics: IV. Analysis of operators, 1 ed., Academic Press, San Diego, 1978.
[51]
, Methods of modern mathematical physics: I. Functional analysis, Academic Press, San Diego, 1980, Revised and Enlarged Edition.
[52] E. Skibsted, Smoothness of N-body scattering amplitudes, Rev. Math. Phys. 4, 619–658 (1992). [53] A.D. Sloan, The polaron without cutoffs in two space dimensions, J. Math. Phys. 15, 190–201 (1974). [54] H. Spohn, Effective mass of the polaron: A functional integral approach, Ann. Phys. 175, 278–318 (1987). [55]
, The polaron at large total momentum, J. Phys. A 21, 1199–1211 (1988).
[56]
, Dynamics of charged particles and their radiation field, Cambridge University Press, Cambridge, 2004.
Jacob Schach Møller Johannes Gutenberg Universit¨ at FB Mathematik (17) D-55099 Mainz Germany email: [email protected] Communicated by Joel Feldman submitted 04/11/04, accepted 17/02/05
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 6 (2005) 1137 – 1155 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/061137-19, Published online 15.11.2005 DOI 10.1007/s00023-005-0235-7
Annales Henri Poincar´ e
An Extension Principle for the Einstein-Vlasov System in Spherical Symmetry Mihalis Dafermos and Alan D. Rendall
Abstract. We prove that “first singularities” in the non-trapped region of the maximal development of spherically symmetric asymptotically flat data for the EinsteinVlasov system must necessarily emanate from the center. The notion of “first” depends only on the causal structure and can be described in the language of terminal indecomposable pasts (TIPs). This result suggests a local approach to proving weak cosmic censorship for this system. It can also be used to give the first proof of the formation of black holes by the collapse of collisionless matter from regular initial configurations.
1 Introduction A fundamental problem in mathematical relativity is to resolve the so-called weak cosmic censorship conjecture, the statement that for “reasonable” Einstein-matter systems, generic asymptotically flat data do not lead to singularities visible from infinity. The notion of “reasonable” above is of course not a precise one, and depends very much on the context one has in mind. A natural matter source for models is provided by kinetic theory. The simplest example is then a self-gravitating collisionless gas. The study of the equations describing such a gas, the Einstein-Vlasov system, was initiated by Choquet-Bruhat in [1], where the existence of a unique maximal development was proven for the Cauchy problem. The problem of weak cosmic censorship concerns the global behaviour of the maximal development for asymptotically flat initial data. Given the current state of the art in nonlinear evolution equations, symmetry must be imposed on initial data for there to be any hope of making progress. The global study of the initial value problem for the Einstein-Vlasov equations for spherically symmetric asymptotically flat initial data was begun in [7], where, in particular, it was proven that for sufficiently small initial data, the maximal development was future causally geodesically complete. The analysis took place in so-called Schwarzschild coordinates. In [8], an extension principle was proven, again in these coordinates, saying in particular that if the solution stopped existing after finite coordinate time t, there was necessarily a singularity at the center. These results were meant to provide a first step for a global existence theorem in Schwarzschild coordinates. If this coordinate system could then be shown to cover the domain of outer communi-
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cations, and if null infinity could moreover be shown to be complete, this would then imply a proof of weak cosmic censorship for this system. There is another approach to the problem of weak cosmic censorship, due to Christodoulou [3], for the problem of a self-gravitating spherically symmetric scalar field. Christodoulou showed that initial data leading to a naked singularity was codimension 1 in the space of all initial data. This was shown by embedding such exceptional data in a one-dimensional subset of the space of initial data, such that all other initial data in this subset evolved to a spacetime with the following property, which can be expressed in the language of causal sets [6]. Given a terminal indecomposable past (TIP) with compact intersection with the Cauchy surface, then the domain of dependence of any open set containing this intersection contains a trapped surface. The statement that this latter property is true for generic initial data can be termed the trapped surface conjecture. From this property, the completeness of null infinity was then inferred, proving weak cosmic censorship. It turns out that the relation between the existence of trapped surfaces and the completeness of null infinity is quite general. Specifically, in [12], it was proven that a weaker version of the trapped surface conjecture is sufficient to prove weak cosmic censorship for a wide variety of matter in spherical symmetry. In particular, the completeness of null infinity follows from the existence of a single trapped or marginally trapped surface in the maximal development. The only really restrictive hypothesis on the matter is that “first” singularities necessarily emanate from the center. Here, the notion of “first” is tied to the causal structure and can be formulated in terms of TIPs. The goal of this paper is to prove that the above mentioned hypothesis of [12] is indeed satisfied by the Einstein-Vlasov system. As noted before, extension principles similar in spirit to this one have been proven before (cf. [8, 10]). These earlier results, however, concern the portion of the development of the Einstein-Vlasov system covered by particular coordinate systems. Thus, these previous results, as far as they concern the maximal development itself, are weaker than the results presented here, and in particular, are not sufficient to deduce the assumptions of [12].1 Finally, we make the following remark: In view of [9], there do exist spherically symmetric asymptotically flat initial data for the Einstein-Vlasov system possessing a trapped surface. Thus, the results of this paper provide in particular the first proof of the existence of solutions for collisionless matter representing the formation of a black hole.
2 Initial data Initial data in this paper are always given as follows: 1. We have a C ∞ Riemannian manifold (Σ, g¯), together with an additional symmetric 2-tensor Kab , such that there do not exist closed antitrapped surfaces 1 Of course, the results of [8, 10] also say something about the behaviour of the coordinate system to which they apply, something not addressed here.
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in the data, and a compactly supported function f0 defined on the tangent bundle of Σ, such that these satisfy √ ab 2 ¯ R − Kab K + (trK) = 16π f0 (pa )pa pa /(1 + pa pa )1/2 g¯dp1 dp2 dp3 , √ ∇a K a b − ∇b (trK) = 8π f0 (pa )pa g¯dp1 dp2 dp3 . Here the metric g¯ is used to move indices and to define the trace and covari¯ is the scalar curvature of g¯ and √g¯ the square root of its ant derivative. R determinant. 2. A smooth SO(3) action on Σ such that g¯, Kab , f0 are preserved, and such that Σ/SO(3) inherits naturally the structure of a 1-dimensional manifold. Here and throughout this paper physical units are chosen so that the gravitational constant has the numerical value unity. We recall the definition of a closed antitrapped surface. Let S be a surface in Σ which is closed, i.e., compact without boundary. Suppose that there is a preferred choice na of an outward normal to this surface and let σab be the second fundamental form of S in Σ corresponding to the outward normal. Then S is said to be antitrapped if trσ < −trK + Kab na nb .
3 The maximal development The theorem of Choquet-Bruhat [1], applied to the data considered here, together with a standard argument on preservation of symmetry, yields Proposition 1. There exists a unique C ∞ collection (M, g, f ) such that 1. 2. 3. 4.
g and f satisfy the Einstein-Vlasov equations (M, g) is globally hyperbolic, (M, g, f ) induces the initial data (Σ, g¯, K, f0 ) and Σ is a Cauchy surface Any other collection (M, g, f ) with these properties 1–3 can be embedded in the given one.
Moreover, SO(3) acts smoothly by isometry on M and preserves f , and Q = M/SO(3) inherits the structure of a time-oriented 2-dimensional Lorentzian manifold, with timelike boundary Γ, the center. Let π : M → Q denote the natural projection. On Q we can define the so-called area-radius function r(p) = Area(π −1 (p))/4π . We have r(p) = 0 iff p ∈ Γ. We can always choose global future directed null coordinates on Q, i.e., such that the metric takes the from −Ω2 dudv. The metric of M then takes the form: (1) −Ω2 dudv + r2 γ
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where γ = γAB dxA dxB is the standard metric on S 2 and xA , A = 2, 3, are local ∂ ∂ coordinates on S 2 . Let u and v be chosen so that ∂u points “inwards” and ∂v “outwards”. Such definitions are meaningful in view of the assumption of asymptotic flatness. We define ν = ∂u r , λ = ∂v r . The assumption of no antitrapped surfaces initially means by definition that ν<0
(2)
holds on the initial hypersurface. It follows that it holds throughout Q as a consequence of the Einstein equations and the dominant energy condition [2]. We shall call the region where λ > 0, the regular region, and denote it R. We call the region where λ = 0 the marginally trapped region, and denote it by A, and finally, we shall can the region where λ < 0 the trapped region, and denote it by T.
4 The extension theorem The extension principle proven in this paper will apply to a region D ⊂ Q with Penrose diagram:
?
Q
D
(i.e., a subset D = [u1 , u2 ] × [v1 , v2 ] \ (u2 , v2 )) such that D ⊂R∪A . Let Cin and Cout be the parts of the boundary of D defined by v = v1 and u = u1 respectively. One can think of D as the “top” of a non-trapped non-central indecomposable past (IP) corresponding to a candidate “first” singularity. In this language, the result of this paper is that such an IP cannot be a TIP, i.e., Theorem 1. If D ⊂ Q, then D ⊂ J − (q) for a q ∈ Q. The theorem thus says that there is no singularity of this form after all! As one might expect, the proof of Theorem 1 proceeds by obtaining a priori estimates in D and then applying an appropriate local existence result. The a priori estimates make use of a certain energy flux along null hypersurfaces. This fact, together with the fact that regular null coordinates can always be chosen, makes it natural to stick to these. We give the form of the equations in local
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null coordinates in the next two sections. Then, in Section 7, we formulate a local existence theorem (Proposition 2) for a double characteristic initial value problem. The “time” of existence, in the sense of null coordinates, will depend only on the C 2 norm of the metric and the C 1 norm (and the support) of f . We obtain energy estimates in Section 8, and use these, together with the structure of the Vlasov equation, to derive in Sections 9–10 a priori estimates for the norm of Proposition 2. The proof of Theorem 1 will follow immediately in Section 11. Finally, in Section 12, we state two applications of our results, discussed already in the Introduction. The above theorem depends on having a well-behaved matter model and the analogous result must be expected to fail for dust. This is illustrated by the Penrose diagram Fig. 1 in [13].
5 The Einstein equations in null coordinates The reader should consult [2] for general facts about the initial value problem in spherical symmetry. When specialized to this case, the Einstein equations are: ∂u ∂v r = −
1 Ω2 − λν + 4πrTuv , 4r r
∂u ∂v log Ω = −4πTuv +
Ω2 1 πΩ2 + 2 λν − 2 γ AB TAB , 2 4r r r
∂v (Ω−2 ∂v r) = −4πrTvv Ω−2 , ∂u (Ω
−2
∂u r) = −4πrTuu Ω
−2
.
(3) (4) (5) (6)
The former two equations can be viewed as wave equations for r and Ω, while the latter two equations can be viewed as constraint equations on null hypersurfaces. A specific choice of matter model, such as a collisionless gas, leads to expressions for the components of the energy-momentum tensor.
6 The Vlasov equation To describe the Vlasov equation in local coordinates, we need a coordinate system on T M. Let pu , pv , and pA denote the functions on T M, defined by writing an arbitrary X ∈ T M as X = pu
∂ ∂ ∂ + pv + pA A . ∂u ∂v ∂x
Together with the pull-back of the coordinates on spacetime these functions define a local coordinate system on T M. Let P ⊂ T M be defined by P = {g(X, X) = −1} ,
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where X ranges over future-pointing vectors. We call P the mass shell. It follows that −Ω2 pu pv + r2 γAB pA pB = −1 . (7) We use pu , pA and the pull-back of the coordinates on spacetime to define coordinates on P and pv is regarded as a function of these coordinates defined by the relation (7). The Vlasov equation is an equation for a non-negative function f :P →R which, in the case that f is spherically symmetric, is given by pu
∂f ∂f + pv ∂u ∂v
= (∂u (log Ω2 )(pu )2 + 2Ω−2 rλγAB pA pB ) + 2r−1 (νpu + λpv )pA
∂f . ∂pA
∂f ∂pu (8)
In deriving this we have used the expressions for the Christoffel symbols given in Appendix A and the fact that a spherically symmetric function f on the mass shell is a function of the variables u, v, pu , and γAB pA pB . This implies the identity pA
∂f B C ∂f = ΓA BC p p ∂xA ∂pA
which has been used to simplify the Vlasov equation. Note that both the expressions γAB pA pB and pA ∂p∂A have a meaning independent of the particular choice of coordinates xA on S 2 . Finally, to close the system, we must define the energy-momentum tensor. We first note that for any point q ∈ M, it follows that Pq , as a spacelike hypersurface in Tq M, inherits a volume form from the Lorentzian metric. In local √ coordinates this volume form can be written r2 (pu )−1 dpu γdpA dpB or alterna√ 2 v −1 v√ A B tively r (p ) dp γdp dp , where γ is the square root of the determinant of γAB . We then have ∞ ∞ ∞ √ Tab = r2 pa pb f (pu )−1 γdpu dpA dpB , (9) 0
−∞
−∞
where pa = gab pb . It follows immediately that this matter model satisfies the energy conditions: (10) Tuv ≥ 0, Tvv ≥ 0, Tuu ≥ 0 .
7 A local existence theorem To prove our extension theorem, we will certainly need to appeal to some sort of local existence theorem. In particular, it is the norm in this theorem that will tell us what quantities we must bound a priori in D. In principle, one could try to
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prove estimates so as to apply the local existence result of [1]. For various reasons, however, the following local existence theorem for a characteristic initial value problem will be more convenient: Proposition 2. Let k ≥ 2. Let Ω, r be positive C k -functions defined on [0, d] × {0} ∪ {0} × [0, d], and let f be a non-negative C k−1 function defined on the part of the mass shell over [0, d] × {0} ∪ {0} × [0, d]. Suppose that equations (5), (6) hold on {0} × [0, d] and [0, d] × {0} respectively, where Tuu and Tvv are defined by (9), and suppose in addition that the C k compatibility condition holds at (0, 0). Define the norm: Nu
=
sup {|Ω|, |Ω−1 |, |∂u Ω|, |∂u2 Ω|, |r|, |r|−1 , |∂u r|, |∂u2 r| ,
[0,d]×{0}
S, |f |, |∂u f |, |∂pu f |, |∂pA f |γ } , Nv
=
sup {|Ω|, |Ω−1 |, |∂v Ω|, |∂v2 Ω|, |r|, |r|−1 , |∂v r|, |∂v2 r| ,
{0}×[0,d]
S, |f |, |∂v f |, |∂pu f |, |∂pA f |γ } , N = sup{Nu , Nv } , were S denotes the supremum of (pu )2 + (pv )2 + γAB pA pB on the support of f and |vA |γ = (γ AB vA vB )1/2 . Then there exists a δ, depending only on N , and C k functions (unique among C 2 functions) r, Ω and a C k−1 function (unique among C 1 functions) f , satisfying equations (3), (4), (5), (6), (8) in [0, δ ∗ ] × [0, δ ∗ ], where δ ∗ = min{d, δ}, such that the restriction of these functions to [0, d]×{0}∪{0}×[0, d] is as prescribed. Proof. See Appendix B. The compatibility conditions referred to in the statement of the proposition are as follows. The data includes the values of the function f on the part of the mass shell over [0, d] × {0}. All derivatives of f tangential to this manifold can be calculated by direct differentiation. By using the field equations transverse derivatives (and thus all derivatives) of f can be computed up to order k − 1. In a similar way, all derivatives up to order k − 1 can be computed on {0} × [0, d]. The condition that derivatives determined in these two different ways agree at (0, 0) is what is referred to above as the C k compatibility condition. Let us add the remark that, defining g on M by (1), the above gives rise to a solution of the Einstein-Vlasov equations upstairs, with the obvious relation to characteristic data, interpreted upstairs.
8 Energy estimates A fundamental fact about the analysis of spherically symmetric Einstein matter systems in the non-trapped region is the existence of energy estimates. To describe these, let us first settle for a particular null-coordinate description of the set D. We normalize our u-coordinate such that ν = −1 along Cin . For the
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v coordinate, we first define the quantity 1 κ = − Ω2 ν −1 . 4 and then define v such that κ = 1 along Cout . D is thus given by [0, U ] × [0, V ] \ {(U, V )}. The concept of energy in spherical symmetry is given by the so-called Hawking mass, given by: m=
r r r (1 − ∂ a r∂a r) = (1 − 2g uv ∂u r∂v r) = (1 + 4Ω−2 λν) . 2 2 2
We will also introduce the so-called mass-aspect function µ=
2m . r
Note that κ(1 − µ) = λ .
(11)
From (3)–(6), we compute the identities: ∂u m
= =
∂v m
= =
8πr2 Ω−2 (Tuv ν − Tuu λ) 1−µ 2 r Tuu , −2πκ−1 r2 Tuv + 2π ν 8πr2 Ω−2 (Tuv λ − Tvv ν) 1−µ 2 r Tuv + 2πκ−1 r2 Tvv . −2π ν
(12)
(13)
The first point to note is that the signs of (12) and (13), together with the signs of λ and ν, give a priori bounds for both r and m. Indeed, set m0 = m(U, 0) ≥ 0 , M = m(0, V ) ,
r0 = r(U, 0) > 0 , R = r(0, V ) .
By (2) and the fact that D ⊂ R ∪ A, we have that r0 ≤ r ≤ R
(14)
throughout D. On the other hand, (12), (13) and (10) give ∂u m ≤ 0, ∂v m ≥ 0, and thus m0 ≤ m ≤ M . (15) Now we make a trivial observation. In view of the fact that we have the a priori bounds (15), if we reexamine the equations (12), (13), keeping in mind that both terms on the right-hand side have the same sign, we obtain the bounds: v2 2π(1 − µ) 2 r Tuv (u, v)dv ≤ M − m0 , (16) −ν v1
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v2
2πκ−1 r2 Tvv (u, v)dv ≤ M − m0 ,
(17)
2πκ−1 r2 Tuv (u, v)du ≤ M − m0 ,
(18)
2π(1 − µ) 2 r Tuu (u, v)du ≤ M − m0 . −ν
(19)
v1 u2
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u1 u2 u1
These will be our energy estimates. As we shall see, our use of the above estimates will not quite be symmetric for u and v. The reason is this: The “constraint” equation (6) can be seen to be equivalent to the following equation for κ: ∂u κ = 4πrν −1 Tuu κ .
(20)
From (2), (20) and (10), we see immediately 0<κ≤1
(21)
throughout D, i.e., κ−1 ≥ 1. This means that a priori we control Tuu du. Finally, note that we can rewrite equation (3) as
or alternatively
Tvv dv, but not
∂v ν = 2r−2 κνm + 4πrTuv ,
(22)
∂u λ = 2r−2 κνm + 4πrTuv .
(23)
Thus, integrating (22), in view of (21), (15), (14), and (10), we have that −2
ν ≥ −e2r0
MV
˜ . = −N
(24)
9 C 1 estimates for the metric So far, we have not used the Vlasov equation, only the energy condition (10). Indeed, all estimates obtained so far are familiar from the results of [12]. To go further, we must use the Vlasov equation itself and the special structure of the energy-momentum tensor. In this section, we shall estimate the support of f and show C 1 estimates for the metric. Before proceeding, let us give names to bounds on certain quantities on the initial segments Cin ∪ Cout . Define G = max
sup [0,U]×{0}
F =
|∂u log Ω2 |,
sup {0}×[0,V ]
sup π1−1 ({0}×[0,V ]∪[0,U]×{0})
|∂v log Ω2 |
f ,
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where π1 denotes the projection from the mass shell, define Σ to be supremum of the radius of support of f in the pv and pu directions along π1−1 ({0} × [0, V ] ∪ [0, U ] × {0}), and define X be the supremum of r4 γAB pA pB over the support of f . Let us note first two easy bounds. Clearly, 0≤f ≤F throughout the mass shell over D. Moreover, by (14) and conservation of angular momentum applied to geodesics, it follows that r4γAB pA pB ≤ X
(25)
for any x ∈ P in the support of f over D. In particular, in the expressions defining energy-momentum, we can thus always replace an integral over the variables pA by the integral over the ball of radius X about the origin. We have the following: Lemma 1. The inequality −guv g AB TAB ≤ 2Tuv holds throughout D. Proof. The inequality is equivalent to the statement that the trace of the energymomentum tensor is non-positive. This holds for collisionless matter independently of symmetry assumptions. It is proved straightforwardly by taking a trace in the formula defining the energy-momentum tensor in general coordinates with the spacetime metric. We can rewrite (4) as ∂u (∂v log Ω2 ) =
−8πTuv − 4κmr−3 ν + 8πκνr−2 γ AB TAB .
(26)
Integrating (26), applying the above lemma, the energy estimate (18), and the bounds (14), (15), (21), we estimate ∂v log Ω2 : 2 −2 AB |∂v log Ω | ≤ G + 8πTuv du − 8πκνr γ TAB du − 4κmr−3 νdu ≤ G + 8κr−2 2πr2 κ−1 Tuv du − 4κmr−3 νdu ≤ G + 8r0−2 2πr2 κ−1 Tuv du − 4κmr−3 νdu −2 2 −1 2πr κ Tuv du − 4κmr−3 νdu ≤ G + 8r0 ≤ =
G + 8r0−2 (M − m0 ) + 2(r0−2 − R−2 )M
G .
(27)
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Integrating now (41), using (27), we obtain | log Ω2 (u, v)| ≤ | log Ω2 (u, 0)| + G V , and thus, since | log Ω2 (u, 0)| ≤ C for some C, we have, 0 < c ≤ Ω2 (u, v) ≤ D .
(28)
Now, we turn to estimate the projection to the pv -axis of the support of f . We proceed by considering the geodesic equation. Let γ(s) be a geodesic crossing {0} × [0, V ] ∪ [0, U ] × {0} at s = 0, such that γ (0) is in the support of f . Let pv (s) ∂ denote the ∂v component of the tangent vector of γ. We have (pv ) (s) = −Γvvv (pv )2 − ΓvAB pA pB .
(29)
using the Christoffel symbols in Appendix A. Integrating (29), we have now by (37) s s v s v v v pv (s) = pv (0)e− 0 Γvv (p )d˜s − ΓvAB pA (˜ s)pB (˜ s)e− s˜ Γvv (p )d¯s d˜ s 0 s v(s) v v(s) v − Γ dv − Γ dv ΓvAB pA pB e v(˜s) vv d˜ s = pv (0)e v(0) vv − 0
=
pv (0)e
−
v(s) v(0)
Γv vv dv
v(s)
+
2(−ν)Ω−2 rγAB pA pB e
−
v(s) v(˜ s)
Γv vv dv
(pv )−1 dv .
v(0)
Thus, for s < s, (replacing 0 with s ) we have, by (28) and (24), the inequality pv (s) ≤ pv (s )e
−
v(s) v(s )
Γv vv dv
v(s)
+ v(s )
˜ −1 rγAB pA pB e 2Nc
−
v(s) v(˜ s)
Γv vv dv
(pv )−1 dv . (30)
Suppose pv (s) > 2Σ for some 0 ≤ v(s) ≤ V , and let s be the last previous time s > s > 0 such that pv (s ) ≥ 2Σ, i.e., we have pv (s∗ ) ≥ 2Σ on [s∗ , s]. By (30), (27), the angular momentum bound (25), and (41), we have ˜ −1 eV G V (2Σ)−1 , pv (s) ≤ 2ΣeV G + r0−3 X Nc
i.e., pv (s) ≤ C˜ . We can now easily estimate Tuu pointwise: ∞ dpv √ r2 (pu )2 f v γdpA dpB Tuu = p 0 |γAB pA pB |≤Xr −4 ∞ dpv √ = (guv )2 r2 (pv )2 f v γdpA dpB p 0 |γAB pA pB |≤Xr −4 ∞ v √ dp = 4ν 2 κ2 r2 (pv )2 f v γdpA dpB p 0 |γAB pA pB |≤Xr −4
(31)
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=
4ν 2 F κ2
˜ C
0
≤ ≤
√ r2 (pv )dpv γdpA dpB
|γAB pA pB |≤Xr −4
r0−2 ν 2 F C˜ 2 X 2
16π ˜ 2E , N
Ann. Henri Poincar´e
= ν2E
in view of (31), (24), (21), (14) and the angular momentum bound (25). (Note that Tuu ν −2 ≤ E is a coordinate invariant2 bound.) Integrating (20), we obtain now Tuu ˜ κ ≥ e− 4πr ν 2 νdu ≥ e−4πRE N U . (Actually, we have in fact already estimated κ from below since κ−1 = 4(−ν)Ω−2 .) From the inequality pu pv ≤
1 2 (p + p2v ) , 2 u
we have
Tuv ≤
1 (Tuu + Tvv ) . 2
This allows us to estimate ∂u log Ω2 = Γuuu : |Γuuu | ≤ G + 8πTuv dv − 8πκνr−2 γ AB TAB dv − 4κmr−3 νdv ˜ 2 EV + 8πTvv dv − 4κmr−3 νdv ≤ C¯ . ≤ G + 8π N We can easily obtain an estimate now for Tvv . λ can be bounded by integrating (3).
10 C 2 estimates for the metric In this section, we derive C 2 estimates on the metric and C 1 estimates for f . The ideas of this section originate in [7]. It has already been shown that the following quantities are bounded: r, r−1 , m, m−1 , κ, κ−1 , ν, ν −1 , λ, Ω, Ω−1 , all first order derivatives of Ω, all components of the energy-momentum tensor, and all Christoffel symbols in (36)–(41). From these estimates and (22) and (23), it follows that ∂v ν and ∂u λ are bounded, from (12) and (13) it follows that ∂u m and ∂v m are bounded, and from (26), it follows that ∂u ∂v Ω is bounded. Writing ν = − 41 Ω2 κ−1 and differentiating in u, we see from (20) that ∂u ν is bounded, while writing κ = − 41 Ω2 ν −1 and differentiating in v, we see that ∂v κ is bounded, and thus, from (11), we see that ∂v λ is bounded. These estimates and the formulas (36)–(41) allow us to control all first order derivatives of the Christoffel symbols, except ∂u Γuuu and ∂v Γvvv . Since the components of the curvature tensor can be expressed in terms of those derivatives of the Christoffel symbols which have already been estimated, we obtain bounds for all components of the curvature tensor in our coordinates. The above estimates allow us to estimate the first derivatives of the exponential map 2 i.e.,
it does not depend on the normalization of u
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on the tangent bundle. This, in turn allows one to estimate the derivatives of f in terms of initial data. We can, however, argue more directly as follows. Let us abbreviate the Vlasov equation (8) by X(f ) = 0 where X is the Vlasov operator written in these coordinates. Note that pv is to be thought of as expressed in terms of pu and pA via the mass shell condition (7). Define f1 = ∂u f − pu ∂u log Ω2 ∂pu f . Differentiating the Vlasov equation with respect to v, pu and pA gives the following equations: X(∂v f ) = −(∂v pv )∂v f + (∂u ∂v log Ω2 (pu )2 + ∂v (−2Ω−2 rλ)γAB pA pB )∂pu f + 2(∂v (νr−1 )pu + ∂v (λr−1 )pv + λr−1 ∂v pv )pA ∂pA f ,
(32)
2 u
X(∂pu f ) = −∂u f − (∂pu p )∂v f + 2∂u log Ω p ∂pu f v
+ 2(νr−1 + λr−1 ∂pu pv )pA ∂pA f , X(p ∂pD f ) = −p (∂pD p )∂v f − 4Ω D
D
+ 2r
−1
v
D
−2
(33)
A B
rλγAB p p ∂pu f
v A
λp ∂pD p p ∂pA f .
(34)
Differentiating the Vlasov equation with respect to u gives the following equation for f1 : X(f1 ) =
−pu ∂u (log Ω2 )X(∂pu f ) − ∂u pv ∂v f
+ −
(−pu pv ∂u ∂v log Ω2 − ∂u log Ω2 (∂u log Ω2 (pu )2 + 2Ω−2 rλγAB pA pB ) 2∂u (Ω−2 rλ)γAB pA pB )∂pu f
+
2(∂u (νr−1 )pu + ∂u (λr−1 )pv + ∂u pv λr−1 )pA ∂pA f .
(35)
The quantity X(∂pu f ) can be substituted for by one of the previous equations and ∂u f may be eliminated from the equations in favour of f1 . The result is a linear system of equations for the evolution of (f1 , ∂v f, ∂pu f, pA ∂pA f ) along the characteristics of the Vlasov equation. The coefficients are known to be bounded and so we can conclude that ∂u f , ∂v f , ∂pu f and pA ∂pA f are also bounded. (Note that since pu and pv are bounded the derivative with respect to X is uniformly equivalent to a derivative along the characteristic with respect to u or v as parameter.) From this, we immediately estimate ∂u Tab and ∂v Tab pointwise. We now estimate ∂u Γuuu by differentiating (26) in u and integrating in v, and similarly, ∂v Γvvv by differentiating in v and integrating in u. Note that |∂pA |γ can also be bounded. This can be seen by passing from polar to Cartesian coordinates and noting that the resulting metric components are C 2 . As a consequence f is C 1 .
11 The Proof of Theorem 1 Let N/2 denote the sup of the norm defined in Proposition 2, where the sup is taken now in all of D. By the estimates of the previous section, we have that N/2 < ∞. Let δ be the constant of Proposition 2 corresponding to N . Consider
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the point (U − δ/2, V − δ/2). Translate the coordinates so that this point is (0, 0). Since Q is by definition open, by continuity, there exists a δ > δ ∗ > δ/2 such that {0} × [0, δ ∗ ] ∪ [0, δ ∗ ] × {0} ⊂ Q and the assumptions of Proposition 2 hold on {0} × [0, δ ∗ ] ∪ [0, δ ∗ ] × {0}, with N and δ ∗ as already defined. It follows that there exists a unique solution of in E = [0, δ ∗ ] × [0, δ ∗ ] .
q
?
E
Q D
Thus the solution coincides in E ∩ Q by uniqueness. One sees that E ∪ Q is clearly the quotient of a development of initial data. By maximality of M, we must have E ∪ Q ⊂ Q. Thus, in particular, in the old coordinates we have (U, V ) ∈ Q, and the theorem holds with q = (U, V ).
12 Applications We will say that a spherically symmetric maximal development has a black hole, if I + is complete in the sense of [4],3 and if J − (I + ) has a non-empty complement. We have shown that the results of [12] apply to our matter model. In particular, the fact that the complement of J − (I + ) is non-empty implies the completeness of null infinity. That this set is non-empty can be inferred in turn from the existence of a single trapped or marginally trapped surface. Asymptotically flat spherically symmetric solutions of the Einstein-Vlasov system possesing a trapped surface were constructed in [9]. Thus we have Corollary 1. There exist solutions of the Einstein-Vlasov system which develop from regular initial data and contain black holes. The fundamental open question in gravitational collapse is to show that generically, either the solution is future geodesically complete or a black hole forms. In view of [12] and the results of this paper we have Corollary 2. Suppose that for generic initial data, the maximal development either contains a trapped surface or marginally trapped surface, or is future causally geodesically complete. Then weak cosmic censorship is true. Thus, weak cosmic censorship can be reduced to a slightly weaker version of Christodoulou’s trapped surfaces conjecture. As remarked in the Introduction, this suggests a local approach to its proof (cf. [3]). 3 See
[12] for a definition of I + in this context.
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The Christoffel symbols
Note: 1 guv = − Ω2 , 2 g uv = −2Ω−2 , Ω2 = −4κν . The nonvanishing Christoffel symbols are given by: ΓuAB = −g uv rλγAB ,
(36)
= −g rνγAB ,
(37)
ΓvAB ΓA Bv A ΓBu Γuuu Γvvv
uv
= =
A λr−1 δB A νr−1 δB
,
(38)
,
(39)
2
= ∂u log Ω ,
(40)
2
= ∂v log Ω .
(41)
In fact the Christoffel symbols ΓC AB , which depend on a choice of coordinates on the spheres of symmetry need not vanish but the expressions for them are not needed in this paper.
B Proof of Proposition 2 The proof of local existence follows from simpler considerations than the proof of the estimates of Sections 8–10. In particular, one does not need to consider energy estimates, for one can recover naive pointwise estimates using the smallness parameter. As in Section 10, the idea of [7] again makes its appearance, to show C 1 bounds on f directly from C 0 bounds on the curvature, before bounding the C 2 norm of the metric. Since all these methods have appeared before, we will only sketch the details here. Let initial data be fixed. Define the space A ⊂ C 2 ([0, δ] × [0, δ]) × C 1 ([0, δ] × [0, δ]) , for δ to be determined later, consisting of all twice continuously differentiable nonnegative functions r, continuously differentiable nonnegative functions Ω, extending the prescribed values, such that N −1 /2 ≤ r ≤ 2N , N
−1
(42)
/2 ≤ Ω ≤ 2N ,
sup{|∂u r|, |∂v r|, |∂u2 r|, |∂v2 r|}
(43) ≤ 2N ,
sup{|∂u Ω|, |∂v Ω|} ≤ 2N .
(44) (45)
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Consider the subset B ⊂ A, consisting of those (r, Ω) for which Ω is C 2 , and for which sup{|∂u2 Ω|, |∂v2 Ω|, |∂u ∂v Ω|} ≤ 2N . (46) Note that the closure of B in A, denoted B, consists of (r, Ω) such that ∂u Ω, ∂v Ω, are Lipschitz, with Lipschitz constants given by the above. We shall define in the next few paragraphs a continuous map Φ : B → A ˜ taking (r, Ω) to (˜ r , Ω). Given r, Ω, first, let f be defined to solve the Vlasov equations on the metric defined by r and Ω, with given initial conditions. Note that since the Christoffel symbols of this metric are Lipschitz, it follows that geodesics can be defined, and thus f can be defined by the requirement that it is preserved by geodesic motion. It follows immediately that 0≤f ≤N , (47) and, after appropriately restricting to sufficiently small δ, it follows easily by integration of the geodesic equations that S ≤ 2N .
(48)
In the case where (r, Ω) ∈ B, we have that f is in fact C 1 , since the exponential map is differentiable. If δ is chosen sufficiently small, it is clear from (42)–(46) that, in this case, we can arrange for sup{|∂v f |, |∂u f |, |∂pu f |, |∂pA f |γ } ≤ 2N .
(49)
Given now f , we can define T uv , T vv , T uu in the standard way. In view of (42)–(45), (47), and (48), these terms can be estimated. Now, set ν = ∂u r, λ = ∂v r. We define r˜ by u v 1 1 − r−2 Ω2 − λν+4πrΩ4 T uv dudv . (50) r˜(u, v) = r(u, 0)+r(0, v)−r(0, 0)+ 4 r 0 0 By appropriate differentiation of (50), it is clear from our bounds thus far that ˜ = ∂v r˜, and ∂u ∂v r˜. We can retrieve the bound we can define and estimate ν˜ = ∂u r˜, λ ¯ (42) for r˜ by integration of the ν˜, after restricting to small δ. For (r, Ω) ∈ B ⊂ B, 2 2 it is clear we can also define and estimate ∂u r˜, ∂v r˜, by differentiating (50) twice in u or twice in v, in view of the fact that all other derivatives, including ∂u T uv , ∂u ν, etc., are clearly defined and bounded, in view of (49), and since these derivatives are defined initially. By appropriate choice of δ, we can clearly arrange–for (r, Ω) ∈ B– so as to retrieve the bound (44). ˜ > 0 by the relation Define now Ω ˜2 log Ω
= log Ω2 (u, 0) + log Ω2 (0, v) − log Ω2 (0, 0) (51) u v 1 2 −2 ˜ ν − 2πΩ2 r˜−2 γ AB TAB )dudv . + (−8πTuv + Ω r˜ + 2˜ r−2 λ˜ 2 0 0
˜ to satisfy (43). Again, for small enough δ, it is clear that one can arrange for Ω
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˜ it Differentiating (51) appropriately, in view of the initial conditions for Ω, 1 ˜ follows that, for (r, Ω) ∈ B, Ω is C , and for δ small enough satsfies (45), while for ˜ is C 2 , and for δ small enough, satisfies (46). (r, Ω) ∈ B, Ω Thus, we have shown that after judicious choice of δ, Φ maps B to itself. By continuity, it maps B to itself. The map Φ can easily be shown to be a contraction in B for the norm of A, i.e., we can show that ˜ 1 ), (˜ ˜ 2 )) ≤ dA ((r1 , Ω1 ), (r2 , Ω2 )) , r1 , Ω r2 , Ω dA ((˜
(52)
for an < 1 and all (ri , Ωi ) ∈ B. To see this, define first fi , corresponding to (ri , Ωi ). Let Γi denote an arbitrary Christoffel symbol for (ri , Ωi ). We clearly have |Γ1 − Γ2 | ≤ CdA ((r1 , Ω1 ), (r2 , Ω2 )) . We easily obtain |f1 − f2 | ≤ Cδ sup |Γ1 − Γ2 | sup (|∂fi | + |fi |) . Γ
i=1,2
Clearly we can also bound sup |T1uv − T2uv | ≤ C sup |f1 − f2 |. One bounds (ν1 − ν2 ) by expressing ∂v (˜ ν1 − ν˜2 ) as a linear combination of Ω1 − Ω2 , r1 − r2 , ν1 − ν2 , λ1 − λ2 and (T1uv − T2uv ) with bounded coefficients. One immediately obtains a similar bound for sup |˜ r1 − r˜2 |. The terms sup |∂u r˜1 − ∂u r˜2 |, sup |∂v r˜1 − ∂v r˜2 |, and sup |∂u ∂v r˜1 − ∂u ∂v r˜2 |, can be handled in the same way. One then obtains a bound ˜ 2 − ∂v log Ω ˜ 2 |, and similarly for sup |∂u log Ω ˜2 − of the above form for sup |∂v log Ω 1 2 1 2 2 2 ˜ ˜ ˜ ∂u log Ω2 |. Either of these bounds of course implies a bound for sup |Ω1 − Ω2 |. To bound sup |∂u2 r˜1 − ∂u2 r˜2 |, we compute
v 1 −2 2 2 2 −1 ∂u r˜ = ∂u r˜|v=0 + ∂u − r Ω − r λν 4 0 =
=
+ 4π∂u (rΩ4 )T uv + 4πrΩ4 ∂u T uv dv
v 1 −2 2 2 −1 ∂u − r Ω − r λν + 4π∂u (rΩ4 )T uv ∂u r˜|v=0 + 4 0 − 4πrΩ4 ∂v T vv + 4πrΩ4 ( T · Γ)dv
∂u2 r˜|v=0 − 4πrΩ4 T vv (u, v) + 4πrΩ4 T uv (u, 0)
v 1 ∂u − r−2 Ω2 − r−1 λν + 4π∂u (rΩ4 )T uv + 4 0 4 + 4π∂v (rΩ )T vv + 4πrΩ4 ( T · Γ)dv .
(53)
(54)
Here we have used the equation ∇a T ab = 0, which follows from the Vlasov equation, and we have integrated by parts. It is now clear that estimates for differences follow as before. We argue in an entirely analogous way for sup |∂v2 r˜1 − ∂v2 r˜2 |.
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After restricting to sufficiently small δ, all constants in the above bounds can be made small. We thus have indeed shown (52). It follows by continuity that Φ is also a contraction on B ⊂ A, and thus, since B is closed, has a fixed point in B. Given such a fixed point (r, Ω), define f as before. To show that (r, Ω, f ) corresponds to a solution of the equations, we have basically only to show that f and ∂u Ω, ∂v Ω, which a priori are Lipschitz, are in fact C 1 . (In particular, from this it will follow that the constraint equations (5)–(6) are also satisfied.) But, in view of the fact that f is initially C 1 , it follows that f is C 1 if the exponential map is C 1 . (The C 2 compatibility condition is used at the point.) But this latter fact follows from the continuity of the curvature, as shown in Exercise 6.2 of Chapter V of [5] 4 . That the curvature is continuous follows by computation, since r is C 2 , Ω is C 1 and ∂u ∂v Ω is C 0 , and ∂u2 Ω and ∂v2 Ω do not appear in the expressions for curvature. From the C 1 property of f , the C 2 property of Ω follows immediately. Similarly, higher regularity follows immediately if it is assumed.
Acknowledgment We gratefully acknowledge the support of the Erwin Schr¨ odinger Institute, Vienna, where an important part of this research was carried out.
References [1] Y. Choquet-Bruhat, Probl`eme de Cauchy pour le syst`eme int´egro-diff´erentiel d’Einstein-Liouville, Ann. Inst. Fourier 21, 181–201 (1971). [2] D. Christodoulou, Self-gravitating relativistic fluids: a two-phase model, Arch. Rat. Mech. Anal. 130, 343–400 (1995). [3] D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. Math. 149, 183–217 (1999). [4] D. Christodoulou, On the global initial value problem and the issue of singularities, Class. Quantum Grav. 16, A23–A35 (1999). [5] P. Hartman (1992) Ordinary differential equations. Birkh¨auser, Basel. [6] R.P. Geroch, E.H. Kronheimer and R. Penrose, Ideal points in spacetime, Proc. R. Soc. Lond. A 327, 545–567 (1972). 4 If the reader does want to apply to this fact, then one can argue as follows: in view of the computations above, in the space B, we have that curvature is in fact C 1 with estimates; since derivatives of the exponential map are computed by integrating curvature on geodesics, and geodesics certainly depend C 1 on their initial conditions, in view of the fact that the Christoffel symbols are C 1 with bounds in B, it follows that we have C 2 estimates for the exponential map in B, and thus by an easy compactness argument, the exponential map of the fixed point must be C 1 . There is only one catch with this argument: r and Ω2 have to be assumed to be initially C 3 to differentiate (51) and (53) three times.
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[7] G. Rein and A.D. Rendall, Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data, Commun. Math. Phys. 150, 561–583 (1992). (Erratum: Commun. Math. Phys. 176, 475–478 (1996).) [8] G. Rein, A.D. Rendall, and J. Schaeffer, A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system, Commun. Math. Phys. 168, 467–478 (1995). [9] A.D. Rendall, Cosmic censorship and the Vlasov equation, Class. Quantum Grav. 9, L99–L104 (1992). [10] A.D. Rendall, An introduction to the Einstein-Vlasov system. Banach Center Publications 41, 35–68 (1997). [11] A.D. Rendall. The Einstein-Vlasov system. In: Chru´sciel, P.T. and Friedrich, H. (eds.) (2004) The Einstein equations and the large scale behavior of gravitational fields. Birkh¨ auser, Basel. [12] M. Dafermos, Spherically symmetric spacetimes with a trapped surface, Class. Quantum Grav. 22, 2221–2232 (2005). [13] P. Yodzis, H.-J. Seifert and H. M¨ uller zum Hagen, On the occurrence of naked singularities in general relativity, Commun. Math. Phys. 34, 135–148 (1973). Mihalis Dafermos University of Cambridge Department of Pure Mathematics and Mathematical Statistics Wilberforce Road Cambridge CB3 0WB United Kingdom email: [email protected] Alan D. Rendall Max Planck Institute for Gravitational Physics Albert Einstein Institute Am Muehlenberg 1 D-14476 Golm Germany email: [email protected] Communicated by Sergiu Klainerman submitted 15/11/04, accepted 17/02/05
Ann. Henri Poincar´e 6 (2005) 1157 – 1177 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/061157-21, Published online 15.11.2005 DOI 10.1007/s00023-005-0236-6
Annales Henri Poincar´ e
Stability of Standing Waves for Nonlinear Schr¨ odinger Equations with Inhomogeneous Nonlinearities Anne De Bouard and Reika Fukuizumi
Abstract. The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves eiωt φω (x) for a nonlinear Schr¨ odinger equation with an inhomogeneous nonlinearity V (x)|u|p−1 u, where V (x) is proportional to the electron density. Here, ω > 0 and φω (x) is a ground state of the stationary problem. When V (x) behaves like |x|−b at infinity, where 0 < b < 2, we show that eiωt φω (x) is stable for p < 1 + (4 − 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x|−b . Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method.
1 Introduction The nonlinear Schr¨ odinger equations i∂t u = −∆u − g(x, |u|2 )u,
(t, x) ∈ R1+n
(1.1)
arise in various physical contexts such as nonlinear optics and plasma physics. When g(x, |u|2 ) = V (x)|u|p−1 , equation (1.1) can model beam propagation in an inhomogeneous medium where V (x) is proportional to the electron density. L. Berg´e [2] studied formally the stability condition for soliton solutions of the above type of equations, depending on the shape of g(x, |u|2 ). The real function g(x, |u|2 ) is a potential which can either stand for corrections to the nonlinear power-law response, or for some inhomogeneities in the medium. In addition, Towers and Malomed [29] recently observed by means of variational approximation and direct simulations that a certain type of time-dependent nonlinear medium gives rise to completely stable beams. Akhmediev [1], Jones [17] and Grillakis, Shatah and Strauss [13] studied the existence and stability of solitary waves of (1.1) when g(x, |u|2 ) describes three layered media where the outside two are nonlinear and the sandwiched one is linear. Also, Merle [23] investigated the existence and nonexistence of blowup solutions of (1.1) for inhomogeneities of the form g(x, |u|2 ) = V (x)|u|4/n . In this paper, we will not exactly deal with the same nonlinearity as those in [2, 29], we consider the case g(x, |u|2 ) = V (x)|u|p−1 with V (x) satisfying the following assumptions (V1) and (V2) with n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2).
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V (x) ≥ 0,
V (x) ≡ 0,
V (x) ∈ C(Rn \ {0}, R),
Ann. Henri Poincar´e ∗
V (x) ∈ Lθ (|x| ≤ 1),
where θ∗ = 2n/{(n + 2) − (n − 2)p}. There exist C > 0 and a > {(n + 2) − (n − 2)p}/2 > b such that V (x) − 1 ≤ C |x|b |x|a
for all x with |x| ≥ 1. The main purpose in this paper is to show that under the above assumptions on V (x), the standing wave solution of (1.1) is stable for p < 1 + (4 − 2b)/n and sufficiently small frequency. As an example satisfying (V1) and (V2), we keep V (x) = (1 + |x|2 )−b/2 in mind. By a standing wave, we mean a solution of (1.1) of the form uω (t, x) = eiωt φω (x), where ω > 0 and φω (x) is a ground state of the following stationary problem x ∈ Rn , −∆φ + ωφ − V (x)|φ|p−1 φ = 0, (1.2) 1 n φ ∈ H (R ), φ ≡ 0. We recall previous results. Several authors have been studying the problem of stability and instability of standing waves for (1.1) (see, e.g., [3, 6, 7, 9, 11, 13, 22, 25, 30, 32]). First, we consider the case V (x) ≡ 1, namely, i∂t u = −∆u − |u|p−1 u,
(t, x) ∈ R1+n ,
where 1 < p < ∞ if n = 1, 2, and 1 < p < 1 + 4/(n − 2) if n ≥ 3. For ω > 0, there exists a unique positive radial solution ψω (x) of −∆ψ + ωψ − |ψ|p−1 ψ = 0, x ∈ Rn , 1 n ψ ≡ 0. ψ ∈ H (R ),
(1.3)
(1.4)
(See Strauss [26] and Berestycki and Lions [4] for the existence, and Kwong [19] for the uniqueness). It is known that a positive solution of (1.4) is a ground state. In [6] Cazenave and Lions proved that if p < 1 + 4/n then the standing wave solution eiωt ψω (x) is stable for any ω > 0. On the other hand, it is shown that if p ≥ 1+4/n then the standing wave solution eiωt ψω (x) is unstable for any ω > 0 (see Berestycki and Cazenave [3] for p > 1 + 4/n, and Weinstein [30] for p = 1 + 4/n). The aim of the paper is to study, in the case where V (x) satisfies (V1) and (V2), what happens in the complementary case of the result in [11], where instability of standing waves was shown for p > 1 + (4 − 2b)/n and sufficiently small ω > 0. We define the energy functional E and the charge Q on H 1 (Rn ) by 1 1 1 2 E(v) := ∇v2 − V (x)|v(x)|p+1 dx, Q(v) := v22 . 2 p + 1 Rn 2
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We remark that by the assumptions (V1) and (V2), the functional E is well defined on H 1 (Rn ) if p < 1 + (4 − 2b)/(n − 2). The time local well-posedness for the Cauchy problem to (1.1) with g(x, |u|2 ) = V (x)|u|p−1 in H 1 (Rn ) and the conservation of energy and charge hold (see, e.g., Theorem 4.4.6 of Cazenave [5]). Exactly, we have the following proposition. Proposition 1 Let n ≥ 3 and 1 < p < 1 + (4 − 2b)/(n − 2). Assume (V1) and lim|x|→∞ V (x) = 0. Then, for any u0 ∈ H 1 (Rn ) there exist T = T (u0 H 1 ) > 0 and a unique solution u(t) ∈ C([0, T ], H 1 (Rn )) of (1.1) with u(0) = u0 satisfying E(u(t)) = E(u0 ),
Q(u(t)) = Q(u0 ),
t ∈ [0, T ].
Before we state our theorem, we give some precise definitions. Definition 1 For ω > 0, we define two functionals on H 1 (Rn ): Sω (v) := E(v) + ωQ(v)
(action), Iω (v) := ∇v22 + ωv22 − V (x)|v(x)|p+1 dx. Rn
Let Gω be the set of all non-negative minimizers for inf{Sω (v) : v ∈ H 1 (Rn ) \ {0}, Iω (v) = 0}.
(1.5)
The existence of non-negative minimizers for (1.5) was proved by the standard variational argument since V (x) vanishes as |x| → ∞ (see [26, 11]). Namely, we have Lemma 1.1 Let n ≥ 3 and 1 < p < 1 + (4 − 2b)/(n − 2). Assume (V1) and lim V (x) = 0. Then Gω is not empty for ω > 0. |x|→∞
Remark 1.1 (i) We note that
Iω (v) = ∂λ Sω (λv)|λ=1 = Sω (v), v .
(ii) Let φω ∈ Gω . Then, there exists a Lagrange multiplier Λ ∈ R such that Sω (φω ) = ΛIω (φω ). Thus, we have Sω (φω ), φω = Λ Iω (φω ), φω . Since
Sω (φω ), φω = Iω (φω ) = 0 and
Iω (φω ), φω = −(p − 1)
V (x)|φω |p+1 < 0,
we have Λ = 0. Namely, φω satisfies (1.2). Moreover, for any v ∈ H 1 (Rn )\{0} satisfying Sω (v) = 0, we have Iω (v) = 0. Thus, by the definition of Gω , we have Sω (φω ) ≤ Sω (v). Namely, φω ∈ Gω is a ground state (minimal action solution) of (1.2) in H 1 (Rn ). It is easy to see that a ground state of (1.2) in H 1 (Rn ) is a minimizer of (1.5).
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The stability and instability in this paper is defined as follows. Definition 2 For φω ∈ Gω and δ > 0, we put 1 n iθ Uδ (φω ) := v ∈ H (R ) : inf v − e φω H 1 < δ . θ∈R
We say that a standing wave solution eiωt φω (x) of (1.1) is stable in H 1 (Rn ) if for any ε > 0 there exists δ > 0 such that for any u0 ∈ Uδ (φω ), the solution u(t) of (1.1) with u(0) = u0 satisfies u(t) ∈ Uε (φω ) for any t ≥ 0. Otherwise, eiωt φω (x) is said to be unstable in H 1 (Rn ). The following theorem is our main result in this paper. Theorem 1 Let n ≥ 3 and 1 < p < 1 + (4 − 2b)/n. Assume (V1) and (V2). Let φω ∈ Gω . Then, there exists ω∗ > 0 such that eiωt φω (x) is stable in H 1 (Rn ) for any ω ∈ (0, ω∗ ). In particular, we can take ω∗ = ∞ in the case where V (x) = |x|−b with 0 < b < 2. Remark 1.2 We make use of Hardy’s type inequality to control the degree of nonlinearity in the space H 1 (Rn ). That is why the restriction on the spatial dimensions, i.e., n ≥ 3 appears in the assumption of Theorem 1. Grillakis, Shatah and Strauss [13, 14] gave an almost sufficient and necessary condition for the stability and instability of stationary states for the Hamiltonian systems under certain assumptions. By the abstract theory in Grillakis, Shatah and Strauss [13, 14], under some assumptions on the spectrum of linearized operators, eiω0 t φω0 (x) is stable (resp. unstable) if the function φω 22 is strictly increasing (resp. decreasing) at ω = ω0 . In the papers of Shatah [24], Shatah and Strauss [25], the authors used the variational characterization of ground states instead of assumptions on the spectrum√of linearized operators. In the case V (x) ≡ 1, by the scaling ψω (x) = ω 1/(p−1) ψ1 ( ωx), it is easy to check the increase and decrease of ψω 22 . However, it seems difficult to check this property of φω 22 for V (x) ≡ 1 since we do not have the scaling invariance in general. To avoid such difficulty, we apply another sufficient condition for stability. Proposition 2 Let n ≥ 3 and 1 < p < 1 + (4 − 2b)/(n − 2). Assume (V1) and lim|x|→∞ V (x) = 0. Let φω ∈ Gω . If there exists δ > 0 such that
Sω (φω )v, v ≥ δv2H 1
(1.6)
for any v ∈ H 1 (Rn ) satisfying Re(φω , v)L2 = 0 and Re(iφω , v)L2 = 0, then the standing wave solution eiωt φω (x) of (1.1) is stable in H 1 (Rn ). Remark 1.3 In Proposition 2, the condition Re(φω , v)L2 = 0 is related to the conservation of charge Q. In fact, we have Q (φω ), v = Re(φω , v)L2 . Moreover, since it follows from Sω (eiθ φω ) = 0 for θ ∈ R that Sω (φω )iφω = 0, (1.6) does not hold if we do not restrict v ∈ H 1 (Rn ) to satisfy Re(iφω , v)L2 = 0.
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To check this sufficient condition (1.6) for the case of V (x) satisfying (V1) and (V2), we first consider the case where V (x) = |x|−b with 0 < b < 2 as a limiting problem since the stability results are already known in the case V (x) = |x|−b , which simply follow from the arguments by [24] and [25]. Indeed, in [11], the authors investigated the rescaling limit of φω (x) as ω → 0. It was shown in [11] that as ω → 0, the rescaled function φ˜ω (x) defined by √ φω (x) = ω (2−b)/2(p−1) φ˜ω ( ωx),
ω>0
(1.7)
tends to the unique positive radial solution ψ1,b (x) of (1.2) with ω = 1 and V (x) = |x|−b . Using this convergence, they proved in [11] that eiωt φω (x) is unstable for p > 1 + (4 − 2b)/n and sufficiently small ω > 0. Due to the inhomogeneous medium, the standing wave solution tends to be more unstable for small ω > 0 since 1+(4−2b)/n < p < 1+4/n is the stability region in the case where V (x) ≡ 1. From known stability properties of ψ1,b (x) (see Section 2 of [11]), we would be able to prove (1.6) in the limit. However, to our knowledge, there is no verification of (1.6) even in the case V (x) = |x|−b . For that reason, in Section 2, we first study the properties of the linearized operator at standing wave solution for the case where V (x) = |x|−b in (1.1). In Section 3, we continue analyzing the linearized operator, in particular, we observe that the kernel of real part of the linearized operator is only zero, following the method of Kabeya and Tanaka [18]. We remark that their idea could not be applied directly to our case. We need to modify their perturbed functional in order that the singularity of |x|−b at the origin does not affect the linear part of the equation (1.2). The crucial part is Section 3 because uniqueness and nondegeneracy of a solution of semilinear elliptic equations often plays an essential role in stability problems. In Section 4, we check the condition (1.6) for V (x) satisfying (V1) and (V2), following Esteban and Strauss [8] (see also [10]) and we prove Theorem 1. We remark that Fibich and Wang [9] and Liu, Wang and Wang [22] treated the stability and instability problems of standing waves for (1.1) with g(x, |u|2 ) = V (εx)|u|4/n in a radial space, where ε is a small parameter. Their ways of proof are also a sort of perturbation method. However, they use (1.4) with p = 1 + 4/n as a limiting equation, their assumptions for V (x) are different from those in this paper and it is not clear whether there exists a simple relation between ε and ω.
2 The case V (x) = |x|−b We consider the stability of standing waves for i∂t u = −∆u −
1 |u|p−1 u, |x|b
(t, x) ∈ R1+n ,
where n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2).
(2.1)
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For any ω > 0 there exists a unique positive radial solution ψω,b ∈ H 1 (Rn ) of −∆ψ + ωψ −
1 |ψ|p−1 ψ = 0, |x|b
x ∈ Rn .
(2.2)
See Stuart [27] and Remark 3.1 of [11] for existence. The positivity of solutions follows from the maximum principle. Radial symmetry of solutions was showed by Gidas, Ni and Nirenberg [12] and Li [20] (see also Li and Ni [21]), and Yanagida [33] proved the uniqueness. Moreover ψω,b is in C 2 (Rn ) and vanishes as |x| → ∞, particularly decays exponentially (see [4, 5]). This unique solution is a minimizer of db (ω) := inf{Sω,b (v) : v ∈ H 1 (Rn ) \ {0}, Iω,b (v) = 0}, where
1 1 ω 1 ∇v22 + v22 − |v(x)|p+1 dx, 2 2 p + 1 Rn |x|b 1 Iω,b (v) = ∇v22 + ωv22 − |v(x)|p+1 dx. b |x| n R
Sω,b (v) =
In this section, we note the following fact as a special case of Theorem 1. Proposition 3 Let n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/n. Then the standing wave solution eiωt ψω,b (x) of (2.1) is stable in H 1 (Rn ) for any ω > 0. Actually, this fact can be proved simply by applying the method of [24, 25] to the present case. Using the variational characterization db (ω), we may check the sufficient condition for stability db (ω) > 0 in [24] and instability db (ω) < 0 in [25]. Since ψω,b (x) is a solution of Sω,b (v) = 0, we have db (ω) = Q(ψω,b ). In √ (2−b)/2(p−1) ψ1,b ( ωx), we have 2Q(ψω,b ) = this case, by the scaling ψω,b (x) = ω ψω,b 22 = ω {(2−b)/(p−1)}−n/2 ψ1,b 22 . Therefore, for any ω > 0, the standing wave solution is stable if 1 < p < 1 + (4 − 2b)/n, and unstable if 1 + (4 − 2b)/n < p < 1+(4−2b)/(n−2). We have also blow-up instability for the case p ≥ 1+(4−2b)/n, following Weinstein [30] and Berestycki and Cazenave [3]. However, stability of standing wave solution does not always seem to imply (1.6) immediately. The constraints in (1.6) depend on the negative and zero eigenvalues of the linearized operator at ψω,b . Therefore, the main aim in this section is to show the following proposition. Proposition 4 Assume n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/n. Let ψ1,b (x) be the unique positive radial solution of (2.2) with ω = 1. Then there exists δ > 0 such that (ψ1,b )v, v ≥ δv2H 1
S1,b for any v ∈ H 1 (Rn ) satisfying Re(ψ1,b , v)L2 = 0 and Re(iψ1,b , v)L2 = 0.
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Remark 2.1 By combining this proposition with Proposition 2, it follows that the standing wave solution eit ψ1,b (x) of (2.1) is stable in H 1 (Rn ), that is, Proposition 3 holds. We define two self-adjoint operators L1,b and L2,b on L2 (Rn ) by L1,b = −∆ + 1 − p
1 p−1 ψ (x), |x|b 1,b
L2,b = −∆ + 1 −
1 p−1 ψ (x) |x|b 1,b
p−1 v ∈ L2 (Rn )} for j = 1, 2. We with domain D(Lj,b ) = {v ∈ H 2 (Rn , R) : |x|−b ψ1,b 1 n remark that for v ∈ H (R ) with v1 (x) = Re v(x) and v2 (x) = Im v(x), (ψ1,b )v, v = L1,b v1 , v1 + L2,b v2 , v2 ,
S1,b 1 p−1 2
L1,b v1 , v1 = v1 H 1 − p ψ (x)|v1 (x)|2 dx, b 1,b |x| n R 1 p−1 ψ (x)|v2 (x)|2 dx,
L2,b v2 , v2 = v2 2H 1 − b 1,b |x| n R
and Re(ψ1,b , v)L2 = (ψ1,b , v1 )L2 ,
Re(iψ1,b , v)L2 = (ψ1,b , v2 )L2 .
Thus it suffices to show the following. Lemma 2.1 Assume n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2). Let ψ1,b (x) be the unique positive radial solution of (2.2) with ω = 1. (i) If p < 1 + (4 − 2b)/n, then there exists δ1 > 0 such that
L1,b v, v ≥ δ1 v2H 1 for any v ∈ H 1 (Rn , R) satisfying (v, ψ1,b )L2 = 0. (ii) There exists δ2 > 0 such that
L2,b v, v ≥ δ2 v2L2 for any v ∈ H 1 (Rn , R) satisfying (v, ψ1,b )L2 = 0. The part (ii) of Lemma 2.1 is obtained since L2,b ψ1,b = 0 and ψ1,b (x) > 0 for x ∈ Rn . Namely, ψ1,b is the first eigenfunction of L2,b corresponding to the eigenvalue 0. Moreover, by Weyl’s theorem, the essential spectrum of L2,b are in [1, ∞), since ψ1,b tends to zero at infinity. These conclude (ii). Therefore, we prove the part (i) of Lemma 2.1. For that purpose, we need to show the following two propositions. Proposition 5 Assume n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2). If v ∈ H 1 (Rn , R) satisfies L1,b v = 0, then v ≡ 0.
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Proposition 6 Assume n ≥ 3, 0 < b < 2 and 1 < p ≤ 1 + (4 − 2b)/n. Then we have inf{ L1,b v, v : v ∈ H 1 (Rn , R), (v, ψ1,b )L2 = 0} = 0.
(2.3)
We shall prove Proposition 5 in the next section. As to Proposition 6, we give a proof in the same way as Proposition 2.7 in Weinstein [31]. First, we show the following lemma. Lemma 2.2 Assume n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2). For v ∈ H 1 (Rn ), we define the functional ∇vθ2 vγ2 J(v) = 1 |v|p+1 , |x|b where θ = {n(p − 1)}/2 + b > 0 and γ = {n + 2 − (n − 2)p − 2b}/2 > 0. Then, α := inf{J(v) : v ∈ H 1 (Rn )} is attained at a positive radial function ψ ∗ (x) ∈ H 1 (Rn ) ∩ C ∞ (Rn ) such that ψ∗ (x) =
γ 1−b/2 θb/2 α(p + 1)
1/(p−1)
ψ1,b (γ 1/2 θ−1/2 x).
Proof. We follow the proof of Theorem B of [30]. Since J(v) ≥ 0, there exists a minimizing sequence {vν } ⊂ H 1 (Rn ), that is, limν→∞ J(vν ) = α. We can assume that vν is positive since ∇|v|2 ≤ ∇v2 . Now, let v λ,µ (x) = λv(µx) for λ, µ > 0. Then we have J(v λ,µ ) = J(v), ∇v λ,µ 22 = λ2 µ2−n ∇v22 , v λ,µ 22 = λ2 µ−n v22 , 1 λ,µ p+1 1 p+1 −n+b |v | = λ µ |v|p+1 . |x|b |x|b n/2−1
We choose µν = vν 2 /∇vν 2 and λν = vν 2 v λν ,µν has the following properties.
ψν (x) ∈ H 1 (Rn ), ψν (x) ≥ 0, ψν 2 = 1, ∇ψν 22 = 1,
n/2
/∇vν 2
so that ψν :=
x ∈ Rn ,
J(ψν ) → α as ν → ∞. Namely {ψν } is bounded in H 1 (Rn ). Thus there exists a subsequence {ψν } and a limit ψ∗ (x) ∈ H 1 (Rn ) such that ψν converges to ψ∗ weakly in H 1 (Rn ). It follows
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from the Sobolev embedding on a bounded domain and the smallness of |x|−b for large |x| that 1 p+1 1 p+1 ψ (x)dx → ψ (x)dx as ν → ∞ b ν b ∗ |x| |x| n n R R for 1 < p < 1 + (4 − 2b)/(n − 2) (see the argument in [27], Lemma 1.1 and Remark 3.1 of [11]). By weak convergence, ψ∗ 2 ≤ 1 and ∇ψ∗ 2 ≤ 1. Furthermore, ∇ψν θ2 ψν γ2 ∇ψ∗ θ2 ψ∗ γ2 α ≤ J(ψ∗ ) = ≤ lim inf = lim inf J(ψν ) 1 1 ν→∞ ν→∞ p+1 p+1 |ψ | |ψ | ∗ ν b b |x| |x| 1 = α. = lim inf 1 ν→∞ p+1 |ψν | |x|b It follows that ∇ψ∗ θ2 ψ∗ γ2 = 1 and therefore ∇ψ∗ 2 = ψ∗ 2 = 1, which implies that ψν → ψ∗ strongly in H 1 (Rn ). This minimizing function ψ∗ satisfies the Euler-Lagrange equation: d J(ψ∗ + εη) = 0 for any η ∈ C0∞ (Rn ). dε ε=0 Taking into account that ∇ψ∗ 2 = ψ∗ 2 = 1 and that |x|−b |ψ∗ |p+1 = 1/α, we have 1 −θ∆ψ∗ + γψ∗ − α(p + 1) b ψ∗p = 0. |x| The smoothness of ψ∗ follows from the same method as Section 8 of Cazenave [5]. 1−b/2 b/2 1/(p−1) θ ψ(γ 1/2 θ−1/2 x) makes ψ(x) be a posThe scaling ψ∗ (x) = γ α(p+1) itive solution of (2.2) with ω = 1. By the results in [12] and [20], ψ(x) is radial. Accordingly, ψ(x) is the unique solution ψ1,b (r). Proof of Proposition 6. We remark that the infimum of (2.3) is nonpositive because the value L1,b v, v is zero for v = 0. Since J(v) attains its minimum at ψ1,b , d2 J(ψ + εη) ≥0 1,b 2 dε ε=0 for all η ∈ C0∞ (Rn ). A simple calculation concludes 2θ θ
L1,b v, v ≥ 1− (∇ψ1,b , ∇v)2L2 α 2
(2.4)
for any v ∈ H 1 (Rn , R) with (v, ψ1,b )L2 = 0, where α and θ have been defined in Lemma 2.2. The result follows since the right-hand side of (2.4) is nonnegative for p ≤ 1 + (4 − 2b)/n.
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Now we are ready to give a proof of part (i) of Lemma 2.1. Proof of Lemma 2.1 (i). Let τ := inf{ L1,b v, v : v ∈ H 1 (Rn , R), (v, ψ1,b )L2 = 0, vH 1 = 1} and suppose τ = 0 under the condition 1 < p < 1 + (4 − 2b)/n. Let {vj } ⊂ H 1 (Rn ) be a minimizing sequence, that is, lim L1,b vj , vj = 0,
j→∞
vj H 1 = 1,
(vj , ψ1,b )L2 = 0.
Since {vj } is bounded in H 1 (Rn ), there exists a subsequence still denoted by {vj } ⊂ H 1 (Rn , R) which converges weakly to some f∗ ∈ H 1 . By weak convergence, f∗ satisfies (f∗ , ψ1,b )L2 = 0. We also have 1 p−1 2 1 p−1 2 ψ v → ψ f (2.5) |x|b 1,b j |x|b 1,b ∗ as j → ∞ for 1 < p < 1 + (4 − 2b)/(n − 2). Indeed, we note that vj2 converges p−1 weakly to f∗2 in Ln/(n−2) (Rn ) by the Sobolev embedding, and that |x|−b ψ1,b (x) ∈ Ln/2 (Rn ) since |x|−b vanishes at infinity and ψ1,b (x) decays exponentially for |x| ≥ p−1 C with some C > 0. For |x| ≤ C, we know that |x|−b ψ1,b (x) ∈ Ln/2 (|x| ≤ C) if p < 1 + (4 − 2b)/(n − 2). Thus, we have 0 = lim L1,b vj , vj j→∞ 1 p−1 2 = 1 − p lim ψ vj j→∞ Rn |x|b 1,b 1 p−1 2 =1−p ψ f b 1,b ∗ Rn |x| and then, f∗ ≡ 0. Moreover, by weak convergence, f∗ H 1 ≤ 1 and 0 ≤ L1,b f∗ , f∗ ≤ lim L1,b vj , vj = 0, j→∞
where the first inequality follows from Proposition 6. We define g∗ := f∗ /f∗ H 1 and then g∗ satisfies g∗ ∈ H 1 (Rn ), g∗ H 1 = 1, (g∗ , ψ1,b )L2 = 0, g∗ ≡ 0 and
L1,b g∗ , g∗ = 0. Since the minimum is attained at an admissible function g∗ ≡ 0, there exists (g∗ , λ, β) solution of the Lagrange multiplier problem L1,b g∗ = λ(−∆g∗ + g∗ ) + βψ1,b ,
λ, β ∈ R,
(2.6)
g∗ H 1 = 1,
(2.7)
(g∗ , ψ1,b )L2 = 0.
(2.8)
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By (2.6), (2.7) and (2.8), λ = L1,b g∗ , g∗ . Thus, λ = 0 since we have assumed τ = 0. Therefore, L1,b g∗ = βψ1,b . On the other hand, let g :=
b−2 2
1 1 ψ1,b + x · ∇ψ1,b . p−1 2−b
Then we have L1,b g = ψ1,b . Accordingly, L1,b (g∗ − βg) = 0. It follows from Proposition 5 that g∗ = βg. If β = 0, then g∗ = 0, which is a contradiction. Thus β = 0. Here, β 2−b n − (g∗ , ψ1,b )L2 = (βg, ψ1,b )L2 = − ψ1,b 22 , 2 p−1 2 which violates (2.8) when p < 1 + (4 − 2b)/n. Thus, g∗ ≡ 0, a contradiction. We now conclude that τ > 0 if p < 1 + (4 − 2b)/n.
3 Nondegeneracy of unique positive radial solution for (2.2) In this section, we give a proof of Proposition 5, following Kabeya and Tanaka [18]. We always assume that n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2). Let ψ1,b (r) ∈ H 1 (Rn ) be the unique positive radial solution of (2.2). ψ1,b (r) decays exponentially and can be characterized as a critical point of the C 2 functional 1 1 1 1 p+1 v dx, S1,b,+ (v) = ∇v22 + v22 − 2 2 p + 1 Rn |x|b + where v+ = max{v, 0}. Remark 3.1 We briefly explain why S1,b,+ (v) is C 2 on H 1 (Rn ) when 1 < p < 1 + (4 − 2b)/(n − 2). For v ∈ H 1 (Rn ), let s 1 1 p+1 N (v) = v dx, M (s) = m(x, τ )dτ, p + 1 Rn |x|b + 0 p where m(x, τ ) = |x|−b τ+ . For v, h ∈ H 1 (Rn ) and t ∈ (−1, 1) \ {0}, we have M (v + th) − M (v) ≤ C|x|−b (|v+ + th+ |p + |v+ |p )|h| (3.1) t p+1 is a C 2 function on R if p > 1. The right-hand side of since the function y → y+ 1 n (3.1) belongs to L (R ) if 1 < p < 1 + (4 − 2b)/(n − 2). Therefore, by Lebesgue’s convergence theorem, N (v + th) − N (v) M (v + th) − M (v) = dx lim lim t→0 t→0 t t n R t 1 p = lim m(x, v + th)hdt dx = v hdx. b + t→0 |x| n n 0 R R
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p We conclude N (v) ∈ C 1 (H 1 (Rn ), R) and N (v)h = Rn |x|−b v+ hdx, for v, h ∈ 1 n 2 H (R ). C regularity follows from the same argument. Any non-zero critical point of S1,b,+ (v) is a positive solution by the maximum principle. On the other hand, as we mentioned in Section 2, radial symmetry of a positive solution and the uniqueness of positive radial solutions follow from [12, 20] and [33]. Thus it is ψ1,b (r). For δ > 0 small, we consider the following perturbed functional: 1 1 p+1 p−1 2 v+ dx − ψ1,b v dx . Sδ (v) = S1,b,+ (v) − δ p + 1 Rn 2 Rn Critical points v(x) of Sδ (v) satisfy −∆v + (1 +
p−1 δψ1,b )v
=
1 p + δ v+ , |x|b
x ∈ Rn .
By the maximum principle, non-zero solutions are positive. Furthermore, positive solutions are radial for small δ > 0 (see [12, 20]). Thus they satisfy 1 p−1 −∆v + (1 + δψ1,b )v = + δ v p , x ∈ Rn , |x|b v(x) > 0, v(x) = v(|x|), x ∈ Rn , v ∈ H 1 (Rn ).
such
(3.2) (3.3) (3.4)
By Yanagida [33], we see that (3.2)–(3.4) has a unique positive radial solution for small δ > 0 (see Appendix). Since ψ1,b (r) satisfies (3.2)–(3.4), the unique solution of (3.2)–(3.4) is ψ1,b (r). For δ ≥ 0, we define the Morse index index Sδ (ψ1,b ) = max{dim H : H ⊂ H 1 (Rn ) is a subspace such that
Sδ (ψ1,b )h, h < 0 for all h ∈ H \ {0}}.
ψ1,b (r) has the following properties. Lemma 3.1 (i) For sufficiently small δ ≥ 0, ψ1,b is a mountain pass critical point of Sδ (v), i.e., Sδ (ψ1,b ) = inf max Sδ (γ(s)), γ∈Γ s∈[0,1]
where Γ = {γ(s) ∈ C([0, 1], H 1 (Rn )) : γ(0) = 0, γ(1) = e0 }. Here, e0 ∈ H 1 (Rn ) satisfies Sδ (e0 ) < 0. (ii) The Morse index at ψ1,b is equal to 1 for small δ ≥ 0, i.e., index Sδ (ψ1,b ) = 1.
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For the proof of Lemma 3.1, we recall Hofer’s result in [15] (see also Tanaka [28] as a related reference). Proposition 7 ([15]) Let F be a real Hilbert space and U ⊂ F be a nonempty open subset. Assume that I ∈ C 2 (U, R) satisfies Palais-Smale condition and the gradient I has the form identity−K, where K is compact. Define A, c, d by A = {a ∈ C([0, 1], F ) : a(i) = ei , i = 0, 1}, d = inf sup I(a[0, 1]), a∈A
c = max{I(e0 ), I(e1 )} and assume d > c. Let u0 ∈ U is an isolated critical point of I at the level d. Then the Morse index at u0 is at most 1. Proof of Lemma 3.1. (i) For some ρ0 > 0 and e0 ∈ H 1 (Rn ), we have inf
vH 1 =ρ0
Sδ (v) > 0,
e0 H 1 ≥ ρ0
and
Sδ (e0 ) < 0.
Therefore Sδ (v) has mountain pass geometry. Since the embedding H 1 ⊂ L2 is compact on a bounded domain and |x|−b vanishes at infinity, Sδ (v) satisfies the Palais-Smale compactness condition if p < 1 + (4 − 2b)/(n − 2) (see Lemma 1.1 and Remark 3.1 of [11]) and small δ ≥ 0. Therefore we can apply the mountain pass theorem. Since ψ1,b is the unique non-zero critical point of Sδ (v) for sufficiently small δ ≥ 0, ψ1,b is the mountain pass critical point. (ii) By Proposition 7, the Morse index is at most one at the mountain pass critical point, i.e., index Sδ (ψ1,b ) ≤ 1. Indeed, Sδ (v) satisfies the conditions in Proposition 7. For v, h ∈ H 1 (Rn ), let Sδ (v)h = v − K(v), h H 1 , where K(v) = p−1 K1 (v) + K2 (v) : H 1 (Rn ) → H 1 (Rn ) defined by K1 (v), h H 1 = Rn δψ1,b hdx, p hdx. We see that K1 is compact and that K2 is
K2 (v), h H 1 = Rn (|x|−b + δ)v+ compact for sufficiently small δ ≥ 0. Furthermore, ψ1,b is the unique mountain pass critical point for sufficiently small δ ≥ 0. On the other hand, 1 p−1 p−1 2
Sδ (ψ1,b )h, h = ∇h22 + (1 + δψ1,b )|h|2 − p + δ ψ1,b h dx. |x|b Rn Rn Setting h = ψ1,b and using Sδ (ψ1,b ), ψ1,b = 0, we have 1 p+1
Sδ (ψ1,b )ψ1,b , ψ1,b = −(p − 1) + δ ψ1,b dx < 0. b |x| n R Thus we get index Sδ (ψ1,b ) = 1.
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Using Lemma 3.1, we verify Proposition 5. Proof of Proposition 5. Suppose that there exists a non-zero solution w0 ∈ H 1 (Rn ) of L1,b w0 = 0. It satisfies (ψ1,b )w0 , ξ = 0
S1,b,+
for all ξ ∈ H 1 (Rn ).
By Lemma 3.1 (ii) with δ = 0, we may also find a w1 ∈ H 1 (Rn ) such that (ψ1,b )w1 , w1 < 0.
S1,b,+
We define a 2-dimensional subspace H of H 1 (Rn ) by H = span{w0 , w1 }. Then we have
S1,b,+ (ψ1,b )h, h ≤ 0 for all h ∈ H. On the other hand, we have for all δ > 0, p−1 2 (ψ1,b )h, h − δ(p − 1) ψ1,b h dx
Sδ (ψ1,b )h, h = S1,b,+ Rn p−1 2 ≤ −δ(p − 1) ψ1,b h dx for all h ∈ H. Rn
n
We remark that ψ1,b (x) > 0 in R and we get
Sδ (ψ1,b )h, h < 0
for all h ∈ H \ {0}.
It means that for all δ > 0, index Sδ (ψ1,b ) ≥ 2, which is a contradiction to Lemma 3.1 (ii) with sufficiently small δ ≥ 0.
4 Proof of Theorem 1 In this section, we prove the following Lemma 4.1 to show Theorem 1. For ω > 0, we define (v, w)H 1 (ω) = Re(∇v, ∇w)L2 + ω Re(v, w)L2 , 1/2
vH 1 (ω) = (v, v)H 1 (ω) ,
v, w ∈ H 1 (Rn ).
(4.1)
Then, we see that · H 1 (ω) is an equivalent norm on H 1 (Rn ) to · H 1 . We remark that for v ∈ H 1 (Rn ) with v1 (x) = Re v(x) and v2 (x) = Im v(x), we have
Sω (φω )v, v = L1,ω v1 , v1 + L2,ω v2 , v2 , 2 2
L1,ω v1 , v1 = v1 H 1 (ω) − p V (x)φp−1 ω (x)|v1 (x)| dx, Rn 2
L2,ω v2 , v2 = v2 2H 1 (ω) − V (x)φp−1 ω (x)|v2 (x)| dx,
(4.2)
Re(φω , v)L2 = (φω , v1 )L2 ,
(4.5)
(4.3) (4.4)
Rn
under the assumptions in Proposition 2.
Re(iφω , v)L2 = (φω , v2 )L2 ,
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Lemma 4.1 Let n ≥ 3, 0 < b < 2 and 1 < p < 1 + (4 − 2b)/(n − 2). Assume (V1) and (V2). Let φω ∈ Gω . (i) Let p < 1 + (4 − 2b)/n. There exists ω1 > 0 with the following property: for any ω ∈ (0, ω1 ), there exists δ1 > 0 such that
L1,ω v, v ≥ δ1 v2H 1 (ω) for any v ∈ H 1 (Rn , R) satisfying (v, φω )L2 = 0. (ii) For any ω ∈ (0, ∞), there exists δ2 > 0 such that
L2,ω v, v ≥ δ2 v2H 1 (ω) for any v ∈ H 1 (Rn , R) satisfying (v, φω )L2 = 0. Proof of Theorem 1. Since · H 1 (ω) is equivalent to · H 1 , by (4.2) and Lemma 4.1, there exists δ > 0 such that (1.6) holds for any v ∈ H 1 (Rn ) satisfying Re(φω , v)L2 = 0 and Re(iφω , v)L2 = 0. Hence, Theorem 1 follows from Proposition 2. In order to show Lemma 4.1, we use the rescaled function φ˜ω defined by (1.7). ˜ 2,ω by ˜ 1,ω and L For ω > 0, we define the rescaled operators L x 2 −b/2 2 ˜ V √
L1,ω v, v = vH 1 − pω φ˜p−1 ω (x)|v(x)| dx, ω Rn x 2 −b/2 2 ˜ φ˜p−1
L2,ω v, v = vH 1 − ω V √ ω (x)|v(x)| dx. ω Rn √ Then, for v(x) = ω (2−b)/2(p−1) v˜( ωx), we have v 2H 1 , v2H 1 (ω) = ω 1+(2−b)/(p−1)−n/2 ˜ (φω , v)L2 = ω (2−b)/(p−1)−n/2 (φ˜ω , v˜)L2 , ˜ j,ω v˜, v˜ ,
Lj,ω v, v = ω 1+(2−b)/(p−1)−n/2 L
j = 1, 2
(see (4.1), (4.3) and (4.4)). Proof of Lemma 4.1. We show (i) by contradiction. Suppose that (i) were false. Then, there would exist {ωj } and {vj } ⊂ H 1 (Rn , R) such that ωj → 0, ˜ 1,ωj vj , vj ≤ 0, lim L
(4.6)
vj 2H 1 = 1,
(4.7)
j→∞
(vj , φ˜ωj )L2 = 0.
Since {vj } is bounded in H 1 (Rn ), there exists a subsequence of {vj } (still denoted by {vj }) and v0 ∈ H 1 (Rn , R) such that vj → v0 weakly in H 1 (Rn , R). Therefore, |vj |2 → |v0 |2 weakly in Ln/(n−2) (Rn ). Further, by Proposition 3 of
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[11], we see that φ˜ωj → ψ1 strongly in H 1 (Rn ), so that φ˜p−1 → ψ1p−1 strongly ωj in L2n/{(n−2)(p−1)} (Rn ) ∩ L(p+1)/(p−1) (Rn ). Moreover, by (V1) and (V2) if p < 1 + (4 − 2b)/(n − 2),
−b/2 x 1
lim ωj V √ =0 − b
j→∞
ωj |x| θ∗ follows from Lemma 4.2 of [11]. Thus, we have x 1 p−1 2 lim ωj −b/2 V √ (x)|v (x)| dx = ψ (x)|v0 (x)|2 dx. φ˜p−1 j ωj b 1,b j→∞ ωj Rn Rn |x| (4.8) Indeed, x 1 p−1 2 p−1 2 −b/2 ˜ V √ ωj φωj vj − b ψ1,b v0 dx ωj |x| Rn 1 p−1 2 1 ˜p−1 p−1 2 2 ψ (v − v )dx + (φωj − ψ1,b )vj dx = j 0 1,b b b Rn |x| Rn |x| x 1 2 + ωj −b/2 V √ − b φ˜p−1 ωj vj dx. ωj |x| Rn p−1 The first term converges to 0 as j → ∞ since |x|−b ψ1,b ∈ Ln/2 (Rn ) (see Proof of Lemma 2.1 (i)). The two remaining terms are estimated as follows: For some R > 0 such that |x|−b ≤ ε if |x| ≥ R, 1 ˜p−1 p−1 2 (φωj − ψ1,b )vj dx b |x| n R φ˜p−1 − ψ p−1 2n/{(n−2)(p−1)} ≤ |x|−b θ∗ L
(|x|≤R)
ωj
vj 22n/(n−2)
1,b
p−1 2 + εφ˜p−1 ωj − ψ1,b (p+1)/(p−1) vj p+1 ,
1 x 2 ωj −b/2 V √ − b φ˜p−1 ωj vj dx ω |x| n j R
−b/2 x 1
2
≤ ωj V √ φ˜ωj p−1 − b
2n/(n−2) vj 2n/(n−2) , ωj |x| θ∗
which conclude (4.8). Therefore, by (4.6), (4.7) and (4.8), we have 0 ≥ = =
˜ 1,ωj vj , vj lim inf L j→∞ x 2 lim inf vj 2H 1 − pωj −b/2 V √ (x)|v (x)| dx φ˜p−1 j ωj j→∞ ωj Rn 1 p−1 1−p ψ (x)|v0 (x)|2 dx. (4.9) b 1,b |x| n R
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Again, by (4.6), (4.8), we have 0 ≥ = ≥
˜ 1,ωj vj , vj lim inf L j→∞ x 2 p−1 2 −b/2 ˜ V √ lim inf vj H 1 − pωj φωj (x)|vj (x)| dx j→∞ ωj Rn 1 p−1 v0 2H 1 − p ψ (x)|v0 (x)|2 dx = L1,b v0 , v0 . b 1,b |x| n R
Moreover, by (4.7), we have (v0 , ψ1,b )L2 = 0. Therefore, by Lemma 2.1 (i), we have v0 ≡ 0. However, this contradicts (4.9). Hence, we conclude (i). By an analogous argument as (ii) of Lemma 2.1, we can also prove (ii).
5 Appendix 5.1
Uniqueness for (3.2)–(3.4)
We have cited the uniqueness result by Yanagida [33]. Here, we briefly check the conditions to prove the uniqueness of a solution (3.2)–(3.4). The condition appeared as (C1)–(C6) in Theorem 2.2 of [33]. In the paper [33], the following type of semilinear elliptic equations was treated: u (r) +
n−1 u (r) + g(r)u(r) + h(r)u(r)p = 0, r
r > 0,
n ≥ 3,
where we denote d/dr by . p−1 As an application to our present case, we consider g(r) = −(1 + δψ1,b ) and h(r) = r−b + δ, where δ ≥ 0, n ≥ 3, 0 < b < 2 and ψ1,b (r) is the unique positive radial solution of (2.2) with ω = 1. We remark that ψ1,b (r) ∈ C 2 (Rn ) decays exponentially as r → ∞ by the standard argument for radial solutions of elliptic equations (see, for example, Berestycki and Lions [4]) and ψ1,b (r) is monotone (r) < 0 for r > 0. First, decreasing with respect to r > 0 from [12, 20, 21], i.e., ψ1,b we know that two conditions (A1) g(r) and h(r) are in C 1 ((0, ∞)), (A2) r2−σ g(r) → 0 and r2−σ h(r) → 0 as r → +0 for some σ > 0, are satisfied. Now let m ∈ [0, n − 2] be a parameter and define G(r; m)
:=
H(r; m) :=
p−2 p−1 −δ(p − 1)rm+2 ψ1,b (r)ψ1,b (r) + 2(n − 3 − m)rm+1 (1 + δψ1,b (r))
+m(n − 2 − m)(n − 2 − m/2)rm−1 , 2b − 2(m + 2) − 2(n − 2) − m + rm−b+1 p+1 2(m + 2) − 2(n − 2) − m − δrm+1 . p+1
These are related to Pohozaev identity (see Yanagida [33] for details).
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Required conditions in [33, Theorem 2.2] are following: (C1) h(r) ≥ 0 for all r ∈ (0, ∞) and h(r) > 0 for some r ∈ (0, ∞). (C2) G(r; n − 2) ≤ 0 for all r ∈ (0, ∞). (C3) For each m ∈ [0, n − 2), there exists an α(m) ∈ [0, ∞] such that G(r; m) ≥ 0 for r ∈ (0, α(m)) and G(r; m) ≤ 0 for r ∈ (α(m), ∞). (C4) H(r; 0) ≤ 0 for all r ∈ (0, ∞). (C5) For each m ∈ (0, n − 2], there exists a β(m) ∈ [0, ∞] such that H(r; m) ≥ 0 for r ∈ (0, β(m)) and H(r; m) ≤ 0 for r ∈ (β(m), ∞). (C6) When g(r) ≡ 0 for all r ≥ 0, h(r) satisfies h(r) ≡ C0 rq , where C0 > 0 is an n−2 n+2 arbitrary constant and q := p− . 2 n−2 The condition (C6) is excluded in the present case. It is clear that (C1), (C4) and (C5) hold since 2 2b −b+1 r −2 n−2− H(r; 0) = − r(r−b + δ). p+1 p+1 Also, since p−1 p−2 (r) + δr(p − 1)ψ1,b (r)ψ1,b (r)), G(r; n − 2) = {r2 g(r)} = −r(2 + 2δψ1,b
taking δ so small that the right-hand side is nonpositive for all r ≥ 0, we can conclude (C2) for sufficiently small δ ≥ 0. The condition (C3) follows for small δ ≥ 0, too. Indeed, if 0 ≤ n − 3 − m, then we have G(r, m) > 0 for all r > 0, therefore we may take α(m) = ∞. If −1 < n − 3 − m < 0 and m ≥ 1, we have that G(r, m) → −∞ as r → ∞ and that G(r, m) tends to a nonnegative constant as r → 0. For the case where −1 < n − 3 − m < 0 and m < 1, we see that G(r,m) → −∞ as r → ∞ and G(r, m) → ∞ as r → 0. Moreover, in both cases, d G (r, m) < 0 for r > 0 and sufficiently small δ ≥ 0. Thus, there exists α(m) dr rm−2 satisfying (C3) (see a similar investigation in [18, Lemma 1.3]).
5.2
Orbital stability
Next, we remark on the proof of Proposition 2. Proposition 2 implies the following lemma: Lemma 5.1 Under the assumptions in Proposition 2, there exist C > 0 and ε > 0 such that E(u) − E(φω ) ≥ C inf u − eiθ φω 2H 1 θ∈R
for u ∈ Uε (φω ) with Q(u) = Q(φω ). We can prove this lemma following Grillakis, Shatah and Strauss [13, Theorem 3.4] (see also [16, Proposition 1], Section 2 of [10]). Theorem 1 follows from Lemma 5.1 and the proof of Theorem 3.5 of [13].
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Acknowledgment This study started while one of us (R. F) stayed in Universit´e de Paris-Sud, Orsay. R. F is grateful to the staff of Laboratoire d’Analyse Num´erique for their warm hospitality. Also, the authors wish to express their sincere appreciation to Professor Kazunaga Tanaka for his helpful advice about Section 3.
References [1] N.N. Akhmediev, Novel class of nonlinear surfaces waves: asymmetric modes in a symmetric layered structure, Sov. Phys. JETP 56, 299–303 (1982). [2] L. Berg´e, Soliton stability versus collapse, Phys. Rev. E. 62, R3071–R3074 (2000). [3] H. Berestycki and T. Cazenave, Instabilit´e des ´etats stationnaires dans les ´equations de Schr¨ odinger et de Klein-Gordon non lin´eaires, C. R. Acad. Sci. Paris. 293, 489–492 (1981). [4] H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I–Existence of a ground state, Arch. Ration. Mech. Anal. 82, 313–346 (1983). [5] T. Cazenave, Semilinear Schr¨ odinger equations, Courant Lecture Notes in Mathematics 10, New York University, New York, 2003. [6] T. Cazenave and P.L. Lions, Orbital stability of standing waves for some nonlinear Schr¨ odinger equations, Comm. Math. Phys. 85, 549–561 (1982). [7] A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Comm. Pure Appl. Math. 56, 1565–1607 (2003). [8] M. Esteban and W. Strauss, Nonlinear bound states outside an insulated sphere, Comm. Partial Differential Equations 19, 177–197 (1994). [9] G. Fibich and X.P. Wang, Stability of solitary waves for nonlinear Schr¨ odinger equations with inhomogeneous nonlinearities, Physica D. 175, 96–108 (2003). [10] R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schr¨ odinger equations with potentials, Differential and Integral Equations 16, 111–128 (2003). [11] R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schr¨ odinger equations with inhomogeneous nonlinearities, Preprint. [12] B. Gidas, W-N. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn , Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies 7A, 369–402 (1981).
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[13] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 74, 160–197 (1987). [14] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal. 94, 308–348 (1990). [15] H. Hofer, A note on the topological degree at a critical point of mountain pass type, Proc. A.M.S. 90, 309–315 (1984). [16] I.D. Iliev and K.P. Kirchev, Stability and instability of solitary waves for onedimensional singular Schr¨ odinger equations, Differential and Integral Equations 6, 685–703 (1993). [17] C.K.R.T. Jones, Instability of standing waves for non-linear Schr¨ odinger-type equations, Ergodic Theory Dynam. Systems 8∗ , 119–138 (1988). [18] Y. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in RN and S´er´e’s non-degeneracy condition, Comm. Partial Differential Equations 24, 563–598 (1999). [19] M.K. Kwong, Uniqueness of positive solutions of ∆u − u − up = 0 in Rn , Arch. Ration. Mech. Anal. 105, 234–266 (1989). [20] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations 16, 585–615 (1991). [21] Y. Li and W.-M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in Rn , Comm. Partial Differential Equations 18, 1043–1054 (1991). [22] Y. Liu, X.-P. Wang and K. Wang, Instability of standing waves of the Schr¨ odinger equation with inhomogeneous nonlinearity, Preprint. [23] F. Merle, Nonexistence of minimal blow-up solutions of equations iut = −∆u − k(x)|u|4/N u in Rn , Ann. Inst. H. Poincar´e Phys. Th´eor. 64, 33–85 (1996). [24] J. Shatah, Stable standing waves for nonlinear Klein-Gordon equations, Comm. Math. Phys. 91, 313–327 (1983). [25] J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100, 173–190 (1985). [26] W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55, 149–162 (1977). [27] C.A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc. 45, 169–192 (1982).
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[28] K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. Partial Differential Equations 14, 99– 128 (1989). [29] I. Towers and B.A. Malomed, Stable (2 + 1)−dimensional solutions in a layered medium with sign-alternating Kerr nonlinearity, J. Opt. Soc. Am. B 19, 537–543 (2002). [30] M.I. Weinstein, Nonlinear Schr¨ odinger equations and sharp interpolation estimates, Comm. Math. Phys. 87, 567–576 (1983). [31] M.I. Weinstein, Modulational stability of ground states of nonlinear Schr¨ odinger equations, Siam J. Math. Anal. 16, 472–491 (1985). [32] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39, 51–68 (1986). [33] E. Yanagida, Uniqueness of positive radial solutions of ∆u+g(r)u+h(r)up = 0 in Rn , Arch. Rat. Mech. Anal. 115, 257–274 (1991). Anne De Bouard Laboratoire de Math´ematiques Universit´e de Paris-Sud F-91405 Orsay France email: [email protected] Reika Fukuizumi Department of Mathematics Hokkaido University Sapporo 060-0810 Japan email: [email protected] Communicated by Bernard Helffer submitted 14/07/04, accepted 28/02/05
Ann. Henri Poincar´e 6 (2005) 1179 – 1196 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/061179-18, Published online 15.11.2005 DOI 10.1007/s00023-005-0237-5
Annales Henri Poincar´ e
Dispersive Estimates of Solutions to the Schr¨ odinger Equation Georgi Vodev Abstract. We prove time decay L1 → L∞ estimates for the Schr¨ odinger group eit(−∆+V ) for real-valued potentials V ∈ L∞ (R3 ) satisfying V (x) = O |x|−δ , |x| 1, with δ > 5/2.
1 Introduction and statement of results Let V ∈ L∞ (R3 ) be a real-valued function satisfying |V (x)| ≤ Cx−δ ,
∀x ∈ R3 ,
(1.1)
with constants C > 0 and δ > 5/2, where x = (1 + |x|2 )1/2 . Denote by G0 and G the self-adjoint realizations of the operators −∆ and −∆ + V (x) on L2 (R3 ). By Kato’s theorem the operator G has no strictly positive eigenvalues. This implies that G has no strictly positive resonances neither. Indeed, it is possible to show that, under the assumption (1.1) with δ > 2, such a resonance is in fact an eigenvalue (e.g., see [1], [7]). It is well known that the free Schr¨ odinger group satisfies the following dispersive estimate itG e 0 1 ∞ ≤ C|t|−3/2 , t = 0. (1.2) L →L Hereafter, given 1 ≤ p ≤ +∞, Lp denotes the space Lp (R3 ). Given any a > 0 denote by χa ∈ C ∞ (R) a function supported in the interval [a, +∞), χa = 1 on [a + 1, +∞). Let also χ denote the characteristic function of the interval [0, +∞) (the absolutely continuous spectrum of G). For potentials satisfying (1.1) with δ > 7 as well as an extra technical assumption, the following analogue of (1.2) was proved in [5]: itG e χ(G) 1 ∞ ≤ C|t|−3/2 , (1.3) L →L provided that zero is neither eigenvalue nor resonance of G (i.e., 0 is a regular point for G). Note that without the assumption that 0 is a regular point, χ in (1.3) should be replaced by χa . Note also that in [5] an analogue of (1.3) in all space dimensions n ≥ 3 is proved. Later on (1.3) was proved in [8] for potentials satisfying (1.1) with δ > 5 via the properties of the wave operators. Recently, (1.3) is proved in [6] for a class of small potentials, while in [3] (1.3) is proved for potentials satisfying (1.1) with δ > 3, provided 0 is a regular point for G. The
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purpose of this work is to extend the three dimensional result in [3] to the larger class of potentials satisfying (1.1) with δ > 5/2. Our main result is the following Theorem 1.1 Assume (1.1) fulfilled. Then, for every a > 0 there exists a constant C > 0 so that the following estimate holds itG e χa (G) 1 ∞ ≤ C|t|−3/2 , |t| ≥ 1. (1.4) L →L It is worth noticing that the estimate (1.4) with O |t|−3/2+ , ∀0 < 1, in the RHS is obtained in [6] for a class of potentials including those satisfying (1.1) with δ > 2, provided the parameter a is taken big enough (independent of t). Note also that an weaker version of (1.4) is proved in [4] for potentials satisfying (1.1) with δ > 3 as well as an extra technical assumption. Our approach is quite different from those developed in the papers mentioned above. It consists of reducing the estimate (1.4) to the following semi-classical estimate on weighted L2 spaces. Theorem 1.2 Let ϕ ∈ C0∞ ((0, +∞)) and assume (1.1) fulfilled. Then, for 0 < 1, 0 ≤ s ≤ 3/2, we have −s− itG x e ϕ(h2 G)x−s− L2 →L2 ≤ Chs |t|−s , t = 0, 0 < h ≤ 1, (1.5) with a constant C > 0 independent of t and h. The fact that supp ϕ is disjoint from zero plays an essential role in our proof of (1.5). However, if the estimate (1.5) holds with h = 1 and ϕ replaced by (1 − χa )χ, works then (1.4) holds with χa (G) replaced by χ(G). Note also that our method in all space dimensions n ≥ 3 and gives an analogue of (1.4) with O |t|−n/2 in the RHS for potentials (1.1) with δ > n2 + 1, while for n = 2 it leads −1 satisfying to (1.4) with O |t| log |t| in the RHS for potentials satisfying (1.1) with δ > 2. In the general case (1.5) holds for 0 ≤ s ≤ n/2. The only thing concerning the free operator we need is the explicit formula for the kernel of the operator f (G0 ) for suitable functions f ∈ L1loc (R+ ). In the three dimensional case it is given by a very simple formula (see (2.16) below), while in the general case this kernel can be expressed in terms of the Bessel function J(n−2)/2 . It has been pointed out by the referee that (1.4) was proved in [2] for a class of potentials including those satisfying (1.1) with δ > 2 by using different methods. It seems, however, that the fact that the dimension is three plays an important role in the method developed in [2].
2 Proof of Theorem 1.1 In this section we will derive Theorem 1.1 from Theorem 1.2. Without loss of generality we may suppose that t ≥ 1. Given a parameter 0 < h ≤ 1 and a function ϕ ∈ C0∞ ((0, +∞)), denote Φ(t; h) = eitG ϕ(h2 G) − eitG0 ϕ(h2 G0 ).
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We also set
F (t) = i
t
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ei(t−τ )G0 V eiτ G0 dτ.
0
Theorem 1.1 follows easily from the following Theorem 2.1 Under the assumption (1.1), for t ≥ 1, 0 < h ≤ 1, we have Φ(t; h) + F (t)ϕ(h2 G0 ) 1 ∞ ≤ Chβ t−3/2 , L →L
(2.1)
with constants C, β > 0 independent of t and h. Indeed, writing the function χa as χa (σ) =
1
ϕ(σθ) 0
dθ , θ
where ϕ(σ) = σχa (σ) ∈ C0∞ ((0, +∞)), we obtain itG e χa (G) − eitG0 χa (G0 ) + F (t)χa (G0 ) 1 ∞ L →L 1 √ dθ ≤ Φ(t; θ) + F (t)ϕ(θG0 ) 1 ∞ θ L →L 0 1 θ−1+β/2 dθ ≤ C t−3/2 . (2.2) ≤ Ct−3/2 0
On the other hand, it is proved in Section 6 of [6] (where in fact much more is proved) that F (t) L1 →L∞ ≤ Ct−3/2 , (2.3) provided the potential V satisfies sup x∈R3
R3
|V (y)| dy < +∞, |x − y|
which in turn is fulfilled for potentials satisfying (1.1). Now (1.4) follows from combining (1.2), (2.2), (2.3) and the fact that the operator χa (G0 ) is bounded on Lp , 1 ≤ p ≤ +∞. Proof of Theorem 2.1. We will first prove the following Lemma 2.2 For 1 ≤ p ≤ +∞, 0 ≤ s ≤ 2, 0 < h ≤ 1, we have ϕ(h2 G) − ϕ(h2 G0 ) p p ≤ Ch2 , L →L
ϕ(h2 G) − ϕ(h2 G0 ) x2 2 2 ≤ Ch2 , L →L ϕ(h2 G0 )(1 + G0 )s/2 p p ≤ Ch−s , L →L
(2.4) (2.5) (2.6)
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ϕ(h2 G)(1 + G0 )s/2 p p ≤ Ch−s , L →L −s x ϕ(h2 G0 )xs 2 2 ≤ C, L →L −s x ϕ(h2 G)xs 2 2 ≤ C, L →L
(2.7) (2.8) (2.9)
with a constant C > 0 independent of h. Proof. We will take advantage of the Helffer-Sj¨ ostrand formula ∂ϕ 1 ϕ(h2 G) = (ζ)(h2 G − ζ)−1 L(dζ), π C ∂ζ
(2.10)
where L(dζ) denotes the Lebesgue measure on C, and ϕ ∈ C0∞ (C) is an almost analytic continuation of ϕ supported in a small complex neighborhood of supp ϕ and satisfying ∂ϕ (ζ) ≤ CN |Im ζ|N , ∀N ≥ 1. (2.11) ∂ζ Thus we have
ϕ(h2 G) − ϕ(h2 G0 ) p p L →L ∂ ϕ (ζ) (h2 G − ζ)−1 V (h2 G0 − ζ)−1 p p L(dζ) ≤ O(h2 ) ∂ζ L →L C N ≤ ON (h2 ) |Im ζ| (h2 G0 − ζ)−1 V (h2 G0 − ζ)−1 p p L →L
Cϕ
+h2 (h2 G0 − ζ)−1 V (h2 G − ζ)−1 V (h2 G0 − ζ)−1 Lp →Lp L(dζ) 2
≤ ON (h )
Cϕ
|Im ζ|
N
−1
1 + h2 |Im ζ|
2 (h G0 − ζ)−1 2 p (h2 G0 − ζ)−1 p 2 L(dζ), L →L L →L
Hence (2.4) follows from this and the following well-known where Cϕ = supp ϕ. estimate 2 (h G0 − ζ)−1 2 p ≤ C|Im ζ|−1−q , ζ, ζ ∈ Cϕ , Im ζ = 0, (2.12) L →L for every 1 ≤ p ≤ +∞, where q = 3 12 − p1 , with a constant C > 0 independent of h. In the same way, to prove (2.5) it suffices to show that, with 0 ≤ s ≤ 2, V (h2 G0 − ζ)−1 xs 2 2 ≤ C|Im ζ|−2 , ζ ∈ Cϕ , Im ζ = 0, (2.13) L →L with a constant C > 0 independent of h. Using the identity V (h2 G0 − ζ)−1 xs = V xs (h2 G0 − ζ)−1 + h2 V (h2 G0 − ζ)−1 [∆, xs ](h2 G0 − ζ)−1 ,
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V (h2 G0 − ζ)−1 xs 2 2 ≤ C1 (h2 G0 − ζ)−1 2 2 L →L L →L
+ C1 h2 V (h2 G0 − ζ)−1 xs−1 L2 →L2 ∇x (h2 G0 − ζ)−1 L2 →L2 + (h2 G0 − ζ)−1
L2 →L2
≤ C2 |Im ζ|−1 1 + h V (h2 G0 − ζ)−1 xs−1 L2 →L2 . Repeating this once again leads to (2.13). To prove (2.6) observe that the function ψh (ζ) = (ζ +h2 )s/2 is holomorphic in Cϕ and satisfies there the bound |ψh (ζ)| ≤ C with a constant C > 0 independent of h. By the formula (2.10) we have 1 ∂ϕ 2 ψh (ζ) (ζ)(h2 G0 − ζ)−1 L(dζ), (ϕψh )(h G0 ) = π C ∂ζ so we get (ϕψh )(h2 G0 ) p p ≤ CN L →L
Cϕ
N |Im ζ| (h2 G0 − ζ)−1 Lp →Lp L(dζ).
Therefore, using that 2 (h G0 − ζ)−1 p p ≤ C|Im ζ|−q0 , L →L
ζ ∈ Cϕ , Im ζ = 0,
for every 1 ≤ p ≤ +∞, with constants C, q0 > 0 independent of h, we deduce (ϕψh )(h2 G0 ) p p ≤ Const, L →L which is clearly equivalent to (2.6). To prove (2.7) we will make use of the fact that the operator (G0 + 1)−1+s/2 , 0 ≤ s ≤ 2, is bounded on Lp , 1 ≤ p ≤ +∞. Therefore, using (2.4) and (2.6) with s = 0, we obtain ϕ(h2 G) − ϕ(h2 G0 ) (1 + G0 )s/2 p p L →L 2 ≤ C1 ϕ(h G) − ϕ(h2 G0 ) (1 + G0 )Lp →Lp ≤ C2 h−2 ϕ1 (h2 G) − ϕ1 (h2 G0 )Lp →Lp + C2 ϕ(h2 G) − ϕ(h2 G0 )Lp →Lp + C2 ϕ(h2 G0 )Lp →Lp ≤ Const, where ϕ1 (σ) = σϕ(σ), which together with (2.6) imply (2.7). The estimate (2.8) can be proved in the same way as (2.6) with s = 0, using (2.13) with V replaced by x−s . The estimate (2.9) follows from (2.8) and (2.5).
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Using the above lemma we will prove the following Lemma 2.3 For 0 < 1, 0 ≤ s ≤ 1, 0 < h ≤ 1, t = 0, we have ≤ Ch−s |t|−3/2+s , ϕ(h2 G0 )eitG0 x−3/2+s− L2 →L∞
ϕ(h2 G)eitG0 x−3/2+s−
L2 →L∞
≤ Ch−s− |t|−3/2+s ,
(2.14) (2.15)
with a constant C > 0 independent of t and h. Proof. Without loss of generality we may suppose that t > 0. To prove (2.14) we will make use of the fact that the kernel of operator f (G0 ) is given by the formula √ ∞ 1 sin( λ|x − y|) dλ. (2.16) [f (G0 )](x, y) = f (λ) (2π)2 0 |x − y| Thus, the kernel of the operator ϕ(h2 G0 )eitG0 is of the form Kh (|x − y|; t), where ∞ 2 1 sin(σλ) λdλ. Kh (σ; t) = eitλ ϕ(h2 λ2 ) 2 2π 0 σ Let us see that |Kh (σ; t)| ≤ Ch−m t−3/2+m σ −1 (1 + σ)1−m ,
(2.17)
for all 0 ≤ m ≤ 1. Clearly, it suffices to show (2.17) only for m = 0 and m = 1. It is easy to see that for m = 1, (2.17) follows from the bound ∞ itλ2 +iaλ ≤ Ct−1/2 , e φ(λ)dλ (2.18) −∞
for every t > 0, a ∈ R, with a constant C > 0 independent of t and a, where φ ∈ C0∞ (R) is independent of t and a. Note that (2.18) is proved in the proof of Lemma 2.4 of [6]. On the other hand, integrating by parts allows to write the function Kh in the form σ −1 t−1 ∞ itλ2 ϕ(h2 λ2 )σ cos(σλ) + 2h2 λϕ (h2 λ2 ) sin(σλ) dλ, e Kh (σ; t) = 2 i(2π) 0 so (2.17) with m = 0 follows again from (2.18). Applying (2.17) with m = s, we obtain 2 |Kh (|x − y|; t)|2 y−3+2s−2 dy ϕ(h2 G0 )eitG0 x−3/2+s− 2 ∞ ≤ sup L →L
≤ Ch−2s t−3+2s sup
x∈R3
R3
x∈R3
2−2s
(1 + |x − y|) |x − y|2
R3
y−3+2s−2 dy ≤ C h−2s t−3+2s .
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The estimate (2.15) follows from (2.7) and the following estimate ≤ Ct−3/2+s . (G0 + 1)−s/2−/2 eitG0 x−3/2+s− L2 →L∞
(2.19)
We will derive (2.19) from (2.14) and the easy observation that the estimate (2.6) holds in fact for all real s. Thus we have (2.20) (G0 + 1)−s/2−/2 ϕ(h2 G0 )eitG0 x−3/2+s− 2 ∞ ≤ Ch t−3/2+s . L →L
Let η ∈ C0∞ (R) be an even function such that η(σ) = 1 for −1 ≤ σ ≤ 1. Writing the function 1 − η(σ) for σ > 0 in the form 1 − η(σ) =
1
ϕ(σθ) 0
dθ , θ
where ϕ(σ) = −ση (σ), we can use (2.20) to get (G0 + 1)−s/2−/2 (1 − η)(G0 )eitG0 x−3/2+s− 2 ∞ L →L 1 ≤ (G0 + 1)−s/2−/2 ϕ(θG0 )eitG0 x−3/2+s− 0
≤ C t−3/2+s
0
L2 →L∞
1
dθ θ
θ−1+/2 dθ ≤ Ct−3/2+s . (2.21)
On the other hand, it is easy to see that (2.14) still holds with h = 1 and ϕ replaced by the function η(σ)(σ 2 + 1)−s/2− . This observation together with (2.21) imply (2.19). Proposition 2.4 For 0 < 1, 0 ≤ s ≤ 1/2 + /2, 0 < h ≤ 1, t > 0, we have −3/2+s− Φ(t; h) ≤ Ch1−s− t−3/2+s , (2.22) x L1 →L2
with a constant C > 0 independent of t and h. Proof. Let ϕ1 ∈ C0∞ ((0, +∞)) be such that ϕ1 ϕ ≡ ϕ. By Duhamel’s formula we have 3 Φj (t; h), (2.23) Φ(t; h) = j=1
where Φ1 (t; h) = ϕ(h2 G)eitG0 ϕ1 (h2 G) − ϕ1 (h2 G0 ) + ϕ(h2 G) − ϕ(h2 G0 ) eitG0 ϕ1 (h2 G),
1186
G. Vodev
Φ2 (t; h) = i
t/2
0
Φ3 (t; h) = i
t
t/2
Ann. Henri Poincar´e
ϕ(h2 G)eiτ G V ei(t−τ )G0 ϕ1 (h2 G)dτ,
ϕ(h2 G)eiτ G V ei(t−τ )G0 ϕ1 (h2 G)dτ.
Using (2.4), (2.5) and Lemma 2.3, we get −3/2+s− Φ1 (t; h) x
L1 →L2
≤ Ch2−s− t−3/2+s .
(2.24)
By (1.1), Lemma 2.3 and (1.5) used with s = 1 − /4 and s = 1 + /4, we have −3/2+s− Φ2 (t; h) x L1 →L2 t/2 −1−/2 ≤ C1 ϕ(h2 G)eiτ G x−1−/2 x L2 →L2 0 −3/2−/2 i(t−τ )G0 × x e ϕ1 (h2 G) 1 2 dτ L →L t/2 ≤ C2 h1−s− t−3/2+s τ −1+/4 τ −/2 dτ ≤ Ch1−s− t−3/2+s . (2.25) 0
Similarly, using (1.5) with s replaced by 3/2 − s and Lemma 2.3 with s = 1 − /2 and s = 1 + /2, we have −3/2+s− Φ3 (t; h) x L1 →L2 t/2 −3/2+s− ≤ C1 ϕ(h2 G)ei(t−τ )G x−3/2− x L2 →L2 0 −1− iτ G × x e 0 ϕ1 (h2 G)L1 →L2 dτ t/2 1−s− −3/2+s ≤ C2 h t τ −1+/2 τ − dτ ≤ Ch1−s− t−3/2+s . (2.26) 0
Now (2.22) follows from (2.23)–(2.26). We are ready now to prove (2.1). By Duhamel’s formula we have Φ(t; h) + F (t)ϕ(h2 G0 ) =
4
Qj (t; h),
j=1
where
Q1 (t; h) = ϕ1 (h2 G)eitG0 ϕ(h2 G) − ϕ(h2 G0 ) + ϕ1 (h2 G) − ϕ1 (h2 G0 ) eitG0 − F (t) ϕ(h2 G0 ), t/2 ϕ1 (h2 G)ei(t−τ )G0 V Φ(τ ; h)dτ, Q2 (t; h) = −i 0
(2.27)
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Dispersive Estimates of Solutions to the Schr¨ odinger Equation
Q3 (t; h) = −i
t
t/2
1187
ϕ1 (h2 G)ei(t−τ )G0 V Φ(τ ; h)dτ,
Q4 (t; h) = (1 − ϕ1 )(h2 G0 )F (t)ϕ(h2 G0 ). By (1.2), (2.3), (2.4), (2.6) and (2.7) with s = 0, we get Q1 (t; h) L1 →L∞ ≤ Ch2 t−3/2 .
(2.28)
By (1.1), (1.2) and (2.22) used with s = 1/2 + /2 and s = 1/2 − /2, we obtain Q2 (t; h) L1 →L∞ t/2 i(t−τ )G0 −3/2− ≤ C1 x e 0
≤ C2 h1/2−3/2 t−3/2
L2 →L
t/2
0
−1−3/2 Φ(τ ; h) x ∞
L1 →L2
dτ
τ −1+/2 τ − dτ ≤ Ch1/2−3/2 t−3/2 . (2.29)
By (2.15) used with s = 1/2 + /2 and s = 1/2 − /2 and (2.22) used with s = 0, we obtain Q3 (t; h) L1 →L∞ t/2 −3/2− ϕ1 (h2 G)eiτ G0 x−1− 2 ∞ ≤ C1 Φ(t − τ ; h) x L →L 0
≤ C2 h
1/2−2 −3/2
t/2
t
0
L1 →L2
dτ
τ −1+/2 τ − dτ ≤ Ch1/2−2 t−3/2 . (2.30)
To estimate the norm of the operator Q4 observe that its kernel, Kh (x, y; t), is of the form U h−1 |x − x |, h−1 |x − y|; h−2 t V (x )dx , Kh (x, y; t) = ch−2 R3
where c is some constant and √ ∞ ∞ √ itλ itµ sin(σ1 λ) sin(σ2 µ) ψ(λ, µ) e − e dλdµ, U (σ1 , σ2 ; t) = σ1 σ2 0 0 where ψ(λ, µ) = ϕ(µ)
ϕ1 (λ) − ϕ1 (µ) , λ−µ
λ = µ,
extends to a smooth function on R2 , compactly supported in µ and satisfying α α ∂ 1 ∂ 2 ψ(λ, µ) ≤ Cα λ−1−α1 , ∀(λ, µ), (2.31) µ λ for every multi-index α = (α1 , α2 ). Write the function U as U = U1 − U2 , where √ ∞ sin(σ1 λ) itλ U1 (σ1 , σ2 ; t) = e b1 (λ, σ2 ) dλ, σ1 0
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G. Vodev
U2 (σ1 , σ2 ; t) =
∞
0
eitµ b2 (µ, σ1 )
Ann. Henri Poincar´e
√ sin(σ2 µ) dµ, σ2
√ sin(σ2 µ) b1 (λ, σ2 ) = ψ(λ, µ) dµ, σ2 0 √ ∞ sin(σ1 λ) b2 (µ, σ1 ) = ψ(λ, µ) dλ. σ1 0
where
∞
In view of (2.31), integrating by parts easily yields |∂λα b1 (λ, σ2 )| ≤ Cα λ−1−α σ2−1 (1 + σ2 )−1 ,
∀α.
(2.32)
Let us see that a similar bound holds for the function b2 , namely α ∂µ b2 (µ, σ1 ) ≤ Cα σ −1 (1 + σ1 )−1 , ∀α. 1
(2.33)
Integrating by parts we obtain ∂µα b2 (µ, σ1 ) = 2σ1−1 = 2σ1−2
∞
0
0
∞
∂µα ψ(λ2 , µ) sin(σ1 λ)λdλ
α ∂µ ψ(λ2 , µ) + 2λ2 ∂λ ∂µα ψ(λ2 , µ) cos(σ1 λ)dλ,
where the last integral is absolutely convergent in view of (2.31), which implies (2.33) for σ1 ≥ 1. To prove (2.33) for 0 < σ1 ≤ 1, observe that there exists a constant λ0 1 so that for λ ≥ λ0 the function ∂µα ψ can be written in the form ∂µα ψ(λ, µ) = λ−1 ∂µα ϕ(µ) + ψα (λ, µ), with a smooth function ψα = O(λ−2 ). Therefore, we have λ0 √ −1 α ∂µ b2 (µ, σ1 ) = σ1 ∂µα ψ(λ, µ) sin(σ1 λ)dλ 0
+σ1−1
∞
λ0
√ ψα (λ, µ) sin(σ1 λ)dλ + σ1−1 ∂µα ϕ(µ)
∞
λ0
√ sin(σ1 λ) dλ. λ
The first two integrals are absolutely convergent, while the other one is equal to 2
∞
√ λ0
sin(σ1 λ) dλ = 2 λ
0
∞
sin(σ1 λ) dλ − 2 λ
0
√
λ0
sin(σ1 λ) dλ = Const +O(σ1 ). λ
Thus we conclude that (2.33) holds for 0 < σ1 ≤ 1, too. For j = 1, 2, set x 2 2 kj (x; σj , t) = 2 eitλ sin(σj λ)λdλ = (it)−1 eitx sin(σj x) − σj (it)−1 kj0 (x; σj , t), 0
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Dispersive Estimates of Solutions to the Schr¨ odinger Equation
1189
where kj0 (x; σj , t)
2
x
=
e 0
itλ2
e−iσj /4t cos(σj λ)dλ = 2t1/2
× k(t1/2 x + σj t−1/2 ) + k(t1/2 x − σj t−1/2 ) − k(σj t−1/2 ) − k(−σj t−1/2 ) ,
where the function k(a) =
a
2
eiλ dλ
0
is known to satisfy the bound |k(a)| ≤ C,
∀a ∈ R,
with a constant C > 0 independent of a. Hence 0 kj (x; σj , t) ≤ Ct−1/2 ,
(2.34)
with a constant C > 0 independent of x, σj and t. Integrating by parts, we can write the function U1 in the form ∞ b1 (λ2 ; σ2 )dk1 (λ; σ1 , t) U1 (σ1 , σ2 ; t) = σ1−1 0 ∞ ∂b1 2 (λ ; σ2 )k1 (λ; σ1 , t)dλ λ = −2σ1−1 ∂λ 0 ∞ 2 ∂b1 2 (λ ; σ2 )eitλ sin(σ1 λ)dλ λ = −2σ1−1 (it)−1 ∂λ ∞ 0 ∂b1 2 −1 (λ ; σ2 )k10 (λ; σ1 , t)dλ +2(it) λ ∂λ 0 ∞ 2 ∂ 2 b1 −1 −2 = 2σ1 (it) λ 2 (λ2 ; σ2 )eitλ sin(σ1 λ)dλ ∂λ 0 ∞ ∂ 2 b1 −2 λ 2 (λ2 ; σ2 )k10 (λ; σ1 , t)dλ −2(it) ∂λ 0 ∞ ∂b1 2 (λ ; σ2 )k10 (λ; σ1 , t)dλ. λ +2(it)−1 ∂λ 0 By (2.32) and (2.34), since t ≥ 1, we conclude |U1 (σ1 , σ2 ; t)| ≤ Ct−3/2 σ1−2 + σ2−2 .
(2.35)
Similarly, using (2.33) instead of (2.32) and the fact that the function b2 (µ, σ1 ) is compactly supported in µ, we get |U2 (σ1 , σ2 ; t)| ≤ Ct−3/2 σ1−2 + σ2−2 . (2.36)
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Ann. Henri Poincar´e
By (2.35) and (2.36), we have |Kh (x, y; t)| ≤ Cht−3/2 B(x, y),
where B(x, y) =
R3
|V (x )|dx + |x − x |2
R3
|V (x )|dx ≤ Const . |y − x |2
Clearly, the above estimates imply Q4 (t; h) L1 →L∞ ≤ Cht−3/2 .
(2.37)
Now (2.1) follows from (2.27)–(2.30) and (2.37).
3 Proof of Theorem 1.2 Without loss of generality we may suppose that t > 0. Since (1.5) is trivial for t/h ≤ 1, we may suppose that t/h ≥ 1. We will first show that (1.5) with s = 3/2 implies (1.5) for all 0 ≤ s ≤ 3/2. The function g(z) = (t/h)z x−z− ϕ(h2 G)eitG x−z− is analytic for z ∈ C, Re z ≥ 0, with values in L(L2 ) and satisfies the trivial bounds g(z) L2→L2 ≤ C, g(z) L2 →L2 ≤ C(t/h)Re z ,
Re z = 0, 0 ≤ Re z ≤ 3/2,
(3.1) (3.2)
with a constant C > 0 independent of z, h and t. Moreover, supposing that (1.5) holds with s = 3/2 means that (3.1) holds for Re z = 3/2. Thus, in view of (3.1) and (3.2), by Phragm`en-Lindel¨ of principle we conclude g(z) L2→L2 ≤ C,
0 ≤ Re z ≤ 3/2,
which clearly implies (1.5) for 0 ≤ s ≤ 3/2. We will now show that (1.5) with s = 3/2 follows from the estimate −s− ϕ(h2 G)eitG x−3/2− ≤ Chs t−s , x 2 2 L →L
(3.3)
(3.4)
for 0 ≤ s < 1. Denote by r = |x| the radial variable and set −∆ = −r∆r−1 = −∂r2 + r−2 ∆S 2 , where ∆S 2 denotes the (positive) Laplace-Beltrami operator on S 2 = {x ∈ R3 : |x| = 1}. We have the identity −2∆ + [r∂r , ∆ ] = 0.
(3.5)
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1191
Denote by G the self-adjoint realization of the operator −∆ + V on the Hilbert space H = L2 (R+ × S 2 , drdw). Clearly, the operator G is unitary equivalent to G, so we have −s −s1 2 itG −s2 x 1 ϕ(h2 G)eitG x−s2 2 2 = ϕ(h G )e r , (3.6) r L →L H→H
for all s1 , s2 ≥ 0. We will show that −s1 r b(r)∂r ϕ(h2 G )eitG r−s2 H→H −1 −s1 ≤ Ch r ϕ(h2 G )eitG r−s2 H→H + Ch−1 r−s1 ϕ(h 2 G )eitG r−s2
H→H
, (3.7)
where b(r) = r−1 r, ϕ(σ) = σϕ(σ). Clearly, (3.7) follows from the estimate −s −s r b(r)∂r u ≤ Ch r G u + Ch−1 r−s u , ∀u ∈ D(G ). (3.8) H H H To prove (3.8), set v = r−s G u and observe that the function r−s b(r)u satisfies the equation 2 −∂r + r−2 ∆S 2 + V r−s b(r)u = v + [−∂r2 , r−s b(r)]u. Integrating by parts leads to the estimates −s ∂r r b(r)u 2 H 2
≤ C r−s b(r)uH + v + [−∂r2 , r−s b(r)]u, r−s b(r)u H 2 2 ≤ C r−s b(r)uH + h2 v 2H + h−2 r−s b(r)uH 2 2 +γ 2 b(r)[−∂r2 , r−s b(r)]uH + γ −2 r−s uH 2 2 ≤ O h−2 r−s uH + O h2 v 2H + O γ 2 r−s b(r)∂r uH , ∀ 0 < γ 1 independent of h. Hence, −s r b(r)∂r u ≤ ∂r r−s b(r)u + C r−s u H H H −1 −s r u H + O (h) v H + O (γ) r−s b(r)∂r uH , ≤O h which implies (3.8) by taking γ > 0 small enough. By Duhamel’s formula and (3.5), we obtain the identity
t r∂r , eitG = ei(t−τ )G r∂r , G eiτ G dτ 0 t ei(t−τ )G V eiτ G dτ = 2tG eitG − 2 0 t t ei(t−τ )G r∂r V eiτ G dτ − ei(t−τ )G V r∂r eiτ G dτ. + 0
0
1192
G. Vodev
Ann. Henri Poincar´e
Let ϕ1 ∈ C0∞ ((0, +∞)) be such that ϕ1 ϕ ≡ ϕ, and denote ϕ2 (σ) = σ −1 ϕ(σ). By the above identity we obtain 2th−2 ϕ(h2 G )eitG
= + + −
ϕ1 (h2 G )r∂r eitG ϕ2 (h2 G ) − ϕ1 (h2 G )eitG r∂r ϕ2 (h2 G ) t ϕ1 (h2 G )ei(t−τ )G V eiτ G ϕ2 (h2 G )dτ 2 0 t ϕ1 (h2 G )ei(t−τ )G r∂r V eiτ G ϕ2 (h2 G )dτ 0 t ϕ1 (h2 G )ei(t−τ )G V r∂r eiτ G ϕ2 (h2 G )dτ. (3.9) 0
Using (2.9), (3.6) and (3.9), we get th−2 r−1−s ϕ(h2 G )eitG r−1−s H→H −s 2 itG ≤ C r b(r)∂r ϕ2 (h G )e r−1−s H→H −1−s itG +C r e ϕ1 (h2 G )∂r b(r)r−s H→H t −1−s + C ϕ1 (h2 G )ei(t−τ )G r−δ1 r H→H 0 −δ2 iτ G ϕ2 (h2 G )r−1−s dτ r e H→H t −1−s i(t−τ )G + C e ϕ1 (h2 G )b(r)∂r r−δ1 r H→H 0 −δ2 iτ G 2 ϕ2 (h G )r−1−s dτ r e H→H t −1−s + C ϕ1 (h2 G )ei(t−τ )G r−δ1 r H→H 0 −δ2 dτ, r b(r)∂r ϕ2 (h2 G )eiτ G r−1−s H→H
(3.10) for 0 ≤ s ≤ 1, δ1 , δ2 > 0 such that δ1 + δ2 = δ − 1 > 3/2. Applying (3.10) with s = 1/2 + , δ1 = δ2 and assuming that (3.4) holds, we obtain in view of (3.6) and (3.7), th−1 r−3/2− ϕ(h2 G )eitG r−3/2− H→H t ≤ C(t/h)−1/2 + Ch τ −3/4 (t − τ )−3/4 dτ ≤ C (t/h)−1/2 . (3.11) 0
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1193
To prove (3.4) we will use the fact that the free operator satisfies the following well-known estimate −s x ϕ(h2 G0 )eitG0 x−s 2 2 ≤ Chs t−s , (3.12) L →L for all s ≥ 0. We also need the following Proposition 3.1 For 0 < 1, 0 ≤ s ≤ 1, 0 < h ≤ 1, t > 0, we have −3/2− ϕ(h2 G)eitG x−3/2− ≤ Ct−s . x 2 2 L →L
(3.13)
By Duhamel’s formula we can write ϕ(h2 G)eitG = ϕ1 (h2 G0 )eitG0 ϕ(h2 G) + ϕ1 (h2 G) − ϕ1 (h2 G0 ) eitG ϕ(h2 G) t −i ϕ1 (h2 G0 )ei(t−τ )G0 V eiτ G ϕ(h2 G)dτ. 0
Using this identity together with (2.5), (2.9), (3.12) and (3.13) with s = 1 − /2, we arrive at −s− ϕ(h2 G)eitG x−3/2− 2 2 x L →L −s− 2 itG0 ≤ C x ϕ1 (h G0 )e x−3/2− 2 2 L →L 2 −3/2− +Ch x ϕ(h2 G)eitG x−3/2− L2 →L2 t −s− + C ϕ1 (h2 G0 )ei(t−τ )G0 x−1− 2 2 x L →L 0 −3/2− 2 iτ G ϕ(h G)e x−3/2− dτ x L2 →L2 t ≤ C(t/h)−s + C(t/h)−1+/2 + Chs (t − τ )−s−/2 τ −1+/2 dτ ≤ C (t/h)−s , 0
if 0 ≤ s < 1 and > 0 is taken so that s + /2 < 1, which is the desired result.
4 Proof of Proposition 3.1 Clearly, (3.13) is trivial for s = 0, so it suffices to prove it for s = 1. We will make use of the formula ∞ 2 1 −3/2− 2 itG −3/2− x ϕ(h G)e x = eizt/h ϕ(z)T (z; h)dz, (4.1) 2πi 0 where
T (z; h) = T + (z; h) − T − (z; h),
1194
G. Vodev
Ann. Henri Poincar´e
T ± (z; h) = x−3/2− (h2 G − z ± i0)−1 x−3/2− = lim x−3/2− (h2 G − z ± iε)−1 x−3/2− . ε→0
Note that the limit exists as an operator in L(L2 ) in view of the limiting absorption principle. Integrating by parts leads to the identity
∞
2
eizt/h ϕ(z)T ± (z; h)dz
0 2 −1
= (it/h )
∞
e 0
izt/h2
dT ± ± (z; h) dz. ϕ (z)T (z; h) + ϕ(z) dz
Therefore, (3.13) with s = 1 follows from the following Lemma 4.1 Let I ⊂ (0, +∞) be a compact interval. Then, the operator-valued functions T ± (z; h) satisfy the estimates (for z ∈ I) T ± (z; h) L2 →L2 ≤ Ch−1 , ± dT −2 dz (z; h) 2 2 ≤ Ch , L →L
(4.2) (4.3)
with a constant C > 0 independent of h and z. Proof. Clearly, (4.2) follows from the estimate −s x 1 (G − λ ± iε)−1 x−s2 2 2 ≤ Cλ−1/2 , L →L
λ ≥ λ0 ,
(4.4)
for all s1 , s2 > 1/2, 0 ≤ ε ≤ 1, ∀λ0 > 0, with a constant C = C(λ0 ) > 0 independent of λ and ε. It is well known that (4.4) holds with G replaced by G0 . To prove (4.4) for G we will use the identity x−s1 (G − λ ± iε)−1 x−s2 (1 + K(λ ∓ iε)) = x−s1 (G0 − λ ± iε)−1 x−s2 , (4.5) where the operator K(λ ∓ iε) = xs2 V (G0 − λ ± iε)−1 x−s2 takes values in the compact operators in L(L2 ). Clearly, it suffices to prove (4.4) for 0 < s2 − 1/2 1. For such s2 there exists λ1 > λ0 so that K(λ ∓ iε) L2 →L2 ≤ 1/2,
λ ≥ λ1 ,
(4.6)
and hence (4.4) follows for λ ≥ λ1 from (4.5) and the fact that (4.4) holds for G0 . Furthermore, since G has no strictly positive resonances, the operator 1+K(λ∓iε) is invertible for λ0 ≤ λ ≤ λ1 , 0 ≤ ε ≤ 1, and satisfies (1 + K(λ ∓ iε))−1 2 2 ≤ C, (4.7) L →L
Vol. 6, 2005
Dispersive Estimates of Solutions to the Schr¨ odinger Equation
1195
with a constant C > 0 independent of ε. Therefore, (4.4) for λ0 ≤ λ ≤ λ1 follows again from (4.5) combined with (4.7). Clearly, (4.3) follows from the following estimate −3/2− 2 (h G − z ± iε)−2 x−3/2− 2 2 ≤ Ch−2 , z ∈ I, 0 < ε 1. (4.8) x L →L
It suffices to prove (4.8) with G and L2 replaced respectively by G and H introduced in the previous section. Using the identity (3.5) we can write 2(z ∓ iε)r−3/2− (h2 G − z ± iε)−2 r−3/2− =
2r−3/2− (h2 G − z ± iε)−1 r−3/2− −r−3/2− (h2 G − z ± iε)−1 r∂r r−3/2− +r−3/2− r∂r (h2 G − z ± iε)−1 r−3/2− −2h2 r−3/2− (h2 G − z ± iε)−1 V (h2 G − z ± iε)−1 r−3/2− +h2 r−3/2− (h2 G − z ± iε)−1 r∂r V (h2 G − z ± iε)−1 r−3/2− −h2 r−3/2− (h2 G − z ± iε)−1 V r∂r (h2 G − z ± iε)−1 r−3/2− .
Note that in view of (3.8) we have −s r 1 b(r)∂r (h2 G − z ± iε)−1 f ≤ Ch−1 r−s1 f H H + Ch−1 r−s1 (h2 G − z ± iε)−1 f H ,
∀f ∈ H, (4.9)
for all s1 ≥ 0. Taking into account (3.6) and using (4.4) with λ = z/h2 and (4.9), we obtain from the above identity −3/2− 2 (h G − z ± iε)−2 r−3/2− ≤ Ch−2 , r H→H
which clearly implies (4.8).
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[5] J.-L. Journ´e, A. Soffer and C. Sogge, Decay estimates for Schr¨odinger operators, Commun. Pure Appl. Math. 44, 573–604 (1991). [6] I. Rodnianski and W. Schlag, Time decay for solutions of Schr¨ odinger equations with rough and time-dependent potentials, Invent. Math. 155, 451–513 (2004). [7] G. Vodev, Local energy decay of solutions to the wave equation for short-range potentials, Asympt. Anal. 37, 175–187 (2004). odinger operators, [8] K. Yajima, The W k,p -continuity of wave operators for Schr¨ J. Math. Soc. Japan 47, 551–581 (1995).
Georgi Vodev Universit´e de Nantes D´epartement de Math´ematiques UMR 6629 du CNRS 2, rue de la Houssini`ere, BP 92208 F-44332 Nantes Cedex 03 France email: [email protected] Communicated by Bernard Helffer submitted 27/11/04, accepted 29/04/05
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Ann. Henri Poincar´e 6 (2005) 1197 – 1199 c 2005 Birkh¨ auser Verlag, Basel, Switzerland 1424-0637/05/061197-3 DOI 10.1007/s00023-005-0238-4
Annales Henri Poincar´ e
Erratum to “Determination of Non–Adiabatic Scattering Wave Functions in a Born–Oppenheimer Model” G.A. Hagedorn∗ and A. Joye
We correct some typographical errors and mistakes in the published paper [1]. We freely use the paper’s notation and equation numbering. In Section 4, on the scattering properties of exact solutions to the molecular Schr¨ odinger equation, we used too rough a definition of asymptotic states. We need to consider ψ σ (x, t, , ±) =
φj (x)
j=1,··· ,m
∆
2 Q(E, ) e−itE/ cσj (±∞, E, ) 2kj (±∞, E) σ
σ
2
× e−i(xkj (±∞,E)+ωj (±∞,E))/ dE, instead of ψ(x, t, , ±) = should be replaced by
σ=±
ψ σ (x, t, , ±). With this definition, Proposition 4.1
Proposition 4.1 Assume H1, H2, H3 and C0. Then, for any 0 < β < 1/2, we have the following L2 (IR)–norm estimate as t → ±∞, ψ(x, t, ) − (ψ − (x, t, , ±) + ψ + (x, t, , ∓)) = O (1/|t|β ). In eqn. (4.4), “ψ(x, t, , −) =” should be replaced by “ψ − (x, t, , −) ,” and the qualification “for negative x’s” should be added. Similarly, in the next sentence, the stipulation “for positive x’s” should be added. The proof of Proposition 4.1 should be amended as follows: Equation (7.14) should begin with ψ(x, t, ) − ψ σ (x, t, , ±) =
φ(x)×
j=1,2
instead of “ψ(x, t, ) − ψ(x, t, , ±) = j=1,2 σ=± φ(x)×,” and every L2 (IR) should be replaced by L2 (IR± ), where IR± = {x ∈ IR : ±x > 0}. ∗ Partially supported by National Science Foundation Grants DMS–0071692 and DMS– 0303586.
1198
G.A. Hagedorn and A. Joye
Ann. Henri Poincar´e
At the end of the proof, one should add the sentence: If x ∈ IR∓ and sign(t) = ∓σ, one integrates by parts as in (7.12), and one uses x σ t + kj (y, E) dy ≥ c (|t| + |x|) ≥ c |t|β |x|1−β , 0
for all 0 < β < 1, to bound the corresponding L2 (IR∓ ) norms by constants times |t|−β , with 0 < β < 1/2. A
Consequently, the statement of Theorem 5.1 should be amended as follows: The first equation should read lim
t→−∞
ψ(x, t, ) − ψj− (x, t, , −) L2 (R− ) = 0,
instead of “limt→−∞ ψ(x, t, ) − ψj− (x, t, , −) = 0,” and it should be followed by the qualification “for negative x’s.” B The sentence before equation (5.8) should begin: Then, there exist δ0 > 0, p > 0 arbitrarily close to 5/2, and a function 0 : (0, δ0 ) → IR+ , such that for all 0 < β < 1/2, δ < δ0 , and < 0 (δ), the following asymptotics hold as t → ∞: C Finally, each occurrence of O (1/t) should be replaced by O (1/tβ ). As indicated in B, p is arbitrary close to 5/2 instead of 3, as previously erroneously stated. This comes from a missing square root in the computation of the L2 (IR) norm of the error terms in the last paragraph of page 987. They should ∗ ∗ ∗ read O(e−α(E ) 1+3s/2 ) and O(e−α(E ) 7s/2−1 ) instead of O(e−α(E ) 1+2s ) and −α(E ∗ ) 4s−1 ). Keeping track of the consequences of this correction, one sees O(e that p is arbitrarily close to 5/2. This change should also be made in Lemma 5.1 and Lemma 5.2. Finally, due to a miscalculation, formula (5.10) in Lemma 5.1 must be simplified to √ ∗ 2 2πk ∗ P (E ∗ , ) e−α(E )/ ∗ ∗ )−xk∗ ) exp −i (tE +κ(E 2 × d2 d2 1/2 ( dk2 α(E(k))|k∗ + i(t + dk 2 κ(E(k))|k∗ )) ∗ 2 (x − k ∗ (t + κ (E ∗ )))2 × exp − d2
+ O(e−α(E )/ p ). d2 2 ∗ ∗ 2 dk2 α(E(k))|k + i(t + dk2 κ(E(k))|k
T (, x, t) =
This leads to a simplification of Lemma 5.2: Only the second statement should be kept, and the error term O(3/2 /|t|) should be removed. This mistake came from the incorrect computation integral at the ∞ of the Gaussian ∗ 2 ∗ 2 end of the proof of Lemma 5.1. It should read, −∞ e−(M(k−k ) /2+iN (k−k ))/ dk
2 − 2N2 M (for Re M > 0). = 2π M e
Vol. 6, 2005
Erratum to “Non-Adiabatic Wave Functions in a B.-O. Model”
1199
References [1] G.A. Hagedorn and A. Joye, Determination of Non–Adiabatic Scattering Wave Functions in a Born–Oppenheimer Model, Ann. Henri Poincar´e 6 (2005), 5, 937–990.
George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0123 USA Alain Joye Institut Fourier Unit´e Mixte de Recherche CNRS-UJF 5582 Universit´e de Grenoble I BP 74 F–38402 Saint Martin d’H`eres Cedex France Communicated by Yosi Avron received 14/10/04
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