Note: This errata is part of Reviews Math. Phys. 8 No. 5 (1996).
ERRATA (Reviews Math. Phys. Vol. 6, No. 4 (1994) 699–832)
SPECTRUM OF A HYDROGEN ATOM IN AN INTENSE MAGNETIC FIELD RICHARD FROESE and ROGER WAXLER
In Sec. 5 the study of the Hamiltonian (1.1) is reduced to the analysis of perturbations in l2 ⊕ L2 ([0, 1]; pdp) whose matrix elements with respect to the l2 part are given to leading order by 3 1 3 a(jk)− 2 a 2 k − 2 . ρ 1 3 a 2 j− 2 1 At the bottom of page 740 it is indicated that it follows from Lemmas 5.2 and 5.3 and Eqs. (5.12) and (5.13) that a = 1. In fact the correct choice is a = 12 (note the 1 2− 2 in (5.12)). For general a the crucial quantity (5.19) is estimated by Lemma 5.4. In the borderline case η = 0 one has Z 1 X j −3 p +2 dp = ρ ((−4a + 2) ln(k + 1) + O(1)) . ρ a εj − ε(δ) 0 ε(p) − ε(δ) j6=k
In the case a = 12 this quantity is O(ρ) for all k. This cancellation does not significantly change or simplify the analysis. It has some consequences for Sec. 7. In Lemma 7.1 condition (vii) should be replaced by 1√ 3 3 1 √ |κj − 2− 2 ρj − 2 | ≤ const ρj − 2 − 2 . As a consequence the numbers qk in part (b) of Theorem 7.2 satisfy 1 |qk + | ≤ const ρ and the vector κ(0) from part (c) should be chosen to be −1 −3 2 2j 2 (0) . κ = 0 Further, throughout Sec. 7 all factors of 2 ln(k) or 2ρ ln(k) should be set to 0. This cancellation of the logarithmic factor does, however, simplify the results (−) quoted on pages 704 through 707. In part (d) the quantity qk now satisfies (−) 1 q ≤ const + k ρ(−) (λ) 761
762
ERRATA
and the correct L2 estimate is kV˜ (0,−,k) − ψk k ≤ const ρ . (+)
In part (f) the quantity qk
now satisfies (+) qk − 2 ln(k) ≤ const
and the correct L2 estimate is kV˜ (0,+,k) − ψk k ≤ const ρ . Expanding the results of parts (d) and (f) to leading order one now finds that the behavior given in parts (c) and (e) hold for all k. However, for large k parts (d) and (f) give a sharper estimate. Finally, in part (c) of the results quoted on pages 704 through 707 (−) 1 2 α ≤ const λ2 − λ ln k λ should read
(−) |m| + 1 2 1 α λ ≤ const λ2 . − ln k 4 λ
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS AND THEIR ANALYTIC CONTINUATION TO WIGHTMAN FUNCTIONS SERGIO ALBEVERIO Fakult¨ at f¨ ur Mathematik, Ruhr–Universit¨ at Bochum, Germany SFB 237 Essen–Bochum–D¨ usseldorf, Germany BiBoS Research Centre, Bielefeld, Germany CERFIM, Locarno, Switzerland
HANNO GOTTSCHALK Fakult¨ at f¨ ur Mathematik, Ruhr–Universit¨ at Bochum, Germany
JIANG-LUN WU Fakult¨ at f¨ ur Mathematik, Ruhr–Universit¨ at Bochum, Germany SFB 237 Essen–Bochum–D¨ usseldorf, Germany Probability Laboratory, Institute of Applied Mathematics Academia Sinica, Beijing 100080, PR China Received 11 March 1996 Revised 10 April 1996 We construct Euclidean random fields X over Rd by convoluting generalized white noise F with some integral kernels G, as X = G ∗ F . We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise. Furthermore, by representing the kernels Gα of the pseudo–differential operators (−∆ + m20 )−α for α ∈ (0, 1) and m0 > 0 as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of X = Gα ∗ F , obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property. Finally we give some remarks on scattering theory for these models.
Contents 0. 1. 2. 3. 4. 5. 6.
Introduction The Generalized White Noise Euclidean Random Fields as Convoluted Generalized White Noise The Schwinger Functions of the Model and their Basic Properties Truncated Schwinger Functions and the Cluster Property On Reflection Positivity Analytic Continuation I: Laplace–Representation for the Kernel of (−∆ + m20 )−α , α ∈ (0, 1) 7. Analytic Continuation II: Continuation of the (Truncated) Schwinger Functions 7.1. Preliminary remarks 763 Review in Mathematical Physics, Vol. 8, No. 6 (1996) 763–817 c World Scientific Publishing Company
764 766 770 776 780 783 789
798 798
764
7.2. 7.3. 7.4. 7.5. 7.6.
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
The mathematical background Truncated Schwinger functions with “sharp masses” The Schwinger functions with “sharp masses” as Laplace transforms The analytic continuation of the truncated Schwinger functions Positivity in the scattering region
798 801 802 808 812
0. Introduction Since the work of Nelson in the early 70’s the problem of mathematical construction of models of interacting local relativistic quantum fields has been related to that of the construction of Markovian Euclidean generalized random fields. In models for scalar fields in space-time dimension two Markovian can be understood in the strict “global Markov sense”, as proven in [4, 35, 71, 12] (see also [20] and references therein). This is also true for a class of vector models in space-time dimensions d = 2, 4 (and 8), see [11, 13, 14, 15, 16, 17a, 55, 64, 70]. The latter models are of the gauge-type and the construction of an associated Hilbert space, in the cases d = 4 and 8, presents difficulties (these do not exist for d = 2, see [15, 8, 9, 10, 39, 65]; for a partial result for d = 4 see [14]). For results on a model of a scalar field for d = 3 (with the Markovian property replaced by reflection positivity in the sense of [37]) see references in [37] and [21], for partial (and rather negative) results for scalar fields for d = 4 see [1, 37, 33, 32, 60, 62]. For partial results on conformal fields on other types of d = 4 space-times see [58a]. A program of constructing Euclidean random fields of Markovian type by solving pseudo-stochastic partial differential equations of the form LX = F with F a Euclidean noise and L a suitable invariant pseudodifferential operator was started in [68, 69, 5, 6, 7], see also [11, 14, 15, 19, 26] in the vector case, and in [18] for the scalar case (in a recent note J. Klauder, see e.g. [49] and [50], and references therein, also advocated the use of non Gussian noises in stochastic PDEs resp. functional integrals to circumvent triviality for scalar fields, his suggestion is however different from ours). In the present case we continue the work initiated in [18], extending the study of random fields of the form X = G ∗ F (of which the above case G = L−1 is a special one), to more general G than in [18], in particular covering Gα = (−∆ + m20 )−α for α ∈ (0, 1) and m0 ≥ 0 (only the case m0 = 0, α = 1 was treated in [18]). For α = 12 , F Gaussian white noise, X is Nelson’s Euclidean free field over Rd [54]; for α = 14 , X is the time zero free field (over a space–time of dimension d + 1). The idea of extending F to be a general, not necessary Gaussian Euclidean noise (“generalized white noise”) (i.e., a generalized random field “independent at every point”, in the terminology of [34], or a “completed scattered random measure”, which is homogeneous with respect to the Euclidean group over Rd ) can be motivated from different points of view. Let us mention three of them (see also e.g. [5, 6, 7] and [11]): 1. From a general Euclidean noise F one can recover a Gaussian Euclidean (Gaussian white) noise F g as weak limit, see Remark 1.4 at the end of Sec. 1. Thus one can look at the fields X = G ∗ F constructed from F as perturbations of the free fields X = G ∗ F g constructed from F g . This perturbation is of another type
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
765
than the usual perturbations of (Euclidean) quantum field theory (given by additive Feynman–Kac type functionals, see e.g. [1, 20, 24, 66, 37]). 2. As explained in detail in Remark 5.12 below, the Schwinger functions Sn n of our model, suitably scaled by a factor λ− 2 , λ > 0, can be written in terms of the free field Schwinger functions plus a polynomial of finite order in the “coupling constant” λ−1 without a constant term, the coefficients being products of truncated Schwinger functions of order 2 ≤ l ≤ n. In this sense we have a parameter λ−1 which measures “the amount of Poisson component” which can also be seen as “the amount of interaction” present in the given Schwinger functions. 3. One can give a discrete approximation or “lattice approximation” of the models, by replacing Rd by a lattice δZd , and correspondingly L and F by discrete analogues, see e.g. for special cases, [17, 18], and [19]. One can then interpret the probabilistic law of X in a bounded region as given by a Euclidean action with a non quadratic kinetic energy part (depending essentially on the L´evy measure characterizing the distribution of F ). Let us explain this a little further, starting from the lattice approximation FδΛ of a generalized white noise F , where the superscript Λ indicates a cutoff outside a bounded region Λ ⊂ Rd . We assume that F is determined by a L´evy characteristic ψ (cf. Sec. 1) such that the convolution semigroup (µt )t>0 generated by ψ [27] is absolutely continuous w.r.t. Lebesgue measure on R. Let %t , t > 0, be the corresponding densities. Then the probability distribution of FδΛ in every lattice point δn ∈ Λδ := δZd ∩ Λ is given by dµδd (x) = %δd dx. We denote by LΛ δ the lattice discretization of a partial (pseudo-) differential operator L over the lattice Λδ . Furthermore, we assume that LΛ δ as a |Λδ | × |Λδ |-matrix is invertible. Then Λ Λ the solvable discrete stochastic equation LΛ δ Xδ = Fδ is the lattice analogue of LX = F . We set Wδ (x) := −δ −d log %δd (δ d x) ,
x ∈ R.
The lattice measure PX Λ is defined as the measure with respect to which XδΛ is the δ coordinate process. Then we have (see [18] and [19]) PXδΛ {XδΛ ∈ A} = Z −1
Z e A
−
P δn∈Λδ
Λ δ d Wδ ((LΛ δ Xδ )(δn))
Y
dXδΛ (δn) ,
δn∈Λδ
Q for Borel measurable subsets A ⊂ RΛδ . Here δn∈Λδ dXδΛ (δn) denotes the flat lattice measure and Z is a normalization constant which depends on δ, Λ and L. For (µt )t>0 the Gaussian semigroup (of mean zero and variance t) and LΛ δ the 2 12 discretization of (−∆ + m0 ) we get the usual lattice approximation of the Nelson’s free field with mass m0 > 0. If (µt )t>0 is not the Gaussian semigroup, the action Wδ is no longer quadratic and therefore contains terms which can be identified with some kind of interaction. In this sense the models can be looked upon as quantized versions of nonlinear field models (with some analogy with models like the Einstein–Infeld field model, see [5, 6, 7] and [11]).
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In this paper we construct X, as given in general by G ∗ F , study its regularity properties and the properties of the associated moment functions (Schwinger functions), proving invariance and cluster property. We also perform their analytic continuation to relativistic Wightman functions, which are shown to satisfy all Wightman axioms (possibly except for positivity), including the cluster property (a point not discussed in [18] for the special case considered there). We also provide a counterexample to the reflection positivity condition, for a particular choice of noise F (with a “sufficiently strong” Poisson component). We also indicate that despite the possible absence of reflection positivity in non Gaussian cases, one can associate scattering states which partly express a “particle structure” of the models. We also make several comments concerning the interplay of properties of G with the Markov property respectively the reflection positivity of X. One main method used in our analysis is the study of truncated Schwinger functions. This, as most of the results of the present work, is based on [38]. Some of our results have been announced in [2]. Here are some details on the single sections in this paper. In Sec. 1 we introduce the basic (white) noises F used in this work. In Sec. 2 we describe the kernels G and the random fields given by X = G ∗ F . In Sec. 3 we discuss the basic invariance properties of the moment functions Sn (Schwinger functions) of X. In Sec. 4 we discuss the cluster property of the Sn . Section 5 is devoted to a discussion of the reflection positivity property and to an example showing that it does not hold for general F . In Secs. 6 and 7 the analytic continuation of the Schwinger functions to Wightman functions is discussed, first (Sec. 6) for the two-point function and then for the general n-point Schwinger functions (Sec. 7). Section 7 also contains the discussion of “positivity properties” in a “scattering region”. 1. The Generalized White Noise In this section, we shall present some basic concepts as background for our discussions in later sections. As is well known in probability theory, an infinitely divisible probability distribution P is a probability distribution having the property that for each n ∈ N there exists a probability distribution Pn such that the n-fold convolution of Pn with itself is P , i.e., P = Pn ∗ . . . ∗ Pn (n times). By L´evy–Khinchine theorem (see e.g. Lukacs [51]) we know that the Fourier transform (or characteristic function) of P , denoted by CP , satisfies the following formula Z eist dP (s) = eψ(t) ,
CP (t) :=
R
t ∈ R,
(1)
where ψ : R → C is a continuous function, called the L´evy characteristic of P , which is uniquely represented as follows σ 2 t2 + ψ(t) = iat − 2
Z R\{0}
eist − 1 −
ist 1 + s2
dM (s) ,
t ∈ R,
(2)
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
where a ∈ R, σ ≥ 0 and the function M satisfies the following condition Z min(1, s2 )dM (s) < ∞ .
767
(3)
R\{0}
On the other hand, given a triple (a, σ, M ) with a ∈ R, σ ≥ 0 and a measure M on R \ {0} which fulfils (3), there exists a unique infinitely divisible probability distribution P such that the L´evy characteristic of P is given by (2). Let d ∈ N be a fixed space-time dimension. Let S(Rd ) (resp. SC (Rd )) be the Schwartz space of all rapidly decreasing real- (resp. complex-) valued C ∞ –functions on Rd with the Schwartz topology. Let S 0 (Rd ) (resp. SC0 (Rd )) be the topological dual of S(Rd ) (resp. SC (Rd )). We denote by h· , ·i the dual pairing between S(Rd ) (resp. SC (Rd )) and S 0 (Rd ) (resp. SC0 (Rd )). Let B be the σ-algebra generated by cylinder sets of S 0 (Rd ). Then (S 0 (Rd ), B) is a measurable space. By a characteristic functional on S(Rd ), we mean a functional C : S(Rd ) → C with the following properties 1. C is continuous on S(Rd ); 2. C is positive-definite; 3. C(0) = 1. By the well-known Bochner–Minlos theorem (see e.g. [34, 44]) there exists a one to one correspondence between characteristic functionals C and probability measures P on (S 0 (Rd ), B) given by the following relation Z eihf,ξi dP (ξ) , f ∈ S(Rd ) . (4) C(f ) = S 0 (Rd )
We have the following result Theorem 1.1. Let ψ be a L´evy characteristic defined by (1). Then there exists a unique probability measure Pψ on (S 0 (Rd ), B) such that the Fourier transform of Pψ satisfies Z Z eihf,ξi dPψ (ξ) = exp ψ(f (x))dx , f ∈ S(Rd ) . (5) S 0 (Rd )
Rd
Proof. It suffices to show that the right-hand side of (5) is a characteristic functional on S(Rd ). This is true, e.g., by Theorem 6 on p. 283 of [34]. Definition 1.2. We call Pψ in Theorem 1.1 a generalized white noise measure with L´evy characteristic ψ and (S 0 (Rd ), B, Pψ ) the generalized white noise space associated with ψ. The associated coordinate process F : S(Rd ) × (S 0 (Rd ), B, Pψ ) → R defined by F (f, ξ) = hf, ξi , is called generalized white noise.
f ∈ S(Rd ) ,
ξ ∈ S 0 (Rd )
(6)
768
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
Remark 1.3. In the terminology of [34], F is called a generalized random process with independent values at every point, i.e., the random variables hf1 , ·i and hf2 , ·i are independent whenever f1 (x)f2 (x) = 0 for f1 , f2 ∈ S(Rd ). Combining (2) and (5), we get, for f ∈ S(Rd ), that (
Z e
ihf,ξi
S 0 (Rd )
Z
dPψ (ξ) = exp ia
Rd
f (x)dx −
Z
σ2 2
[f (x)]2 dx Rd
) isf (x) eisf (x) − 1 − dM (s)dx . 1 + s2 R\{0}
Z
Z + Rd
(7)
From (7), we see that a generalized white noise F is composed by three independent parts, namely, we can give an equivalent (in law) realization of F as the following direct sum F (f, ·) = Fa (f, ·) ⊕ Fσ (f, ·) ⊕ FM (f, ·)
(8)
for f ∈ S(Rd ) with Fa , Fσ and FM the coordinate processes on the probability spaces (S 0 (Rd ), B, Pa ), (S 0 (Rd ), B, Pσ ) and (S 0 (Rd ), B, PM ), respectively, where Pa , Pσ and PM are defined, by Theorem 1.1, by the following relations Z e
ihf,ξi
S 0 (Rd )
Z dPa (ξ) = exp ia
f (x)dx
Rd
Z σ2 ihf,ξi 2 e dPσ (ξ) = exp − [f (x)] dx 2 Rd S 0 (Rd ) ) (Z Z Z isf (x) ihf,ξi isf (x) e dM (s)dx e dPM (ξ) = exp −1− 1 + s2 S 0 (Rd ) Rd R\{0} Z
for all f ∈ S(Rd ). We call Fa , Fσ and FM in order as degenerate (or constant), Gaussian and Poisson (with jumps given by M ) noises, respectively. The first two terms Fa and Fσ can be clearly understood. Let us discuss further the Poisson noise. Its characteristic functional is given by the following formula Z S 0 (Rd )
eihf,ξi dPM (ξ) (Z
) isf (x) isf (x) e dM (s)dx −1− 1 + s2 R\{0}
Z
= exp Rd
(9)
for f ∈ S(Rd ). The existence and uniqueness of the Poisson noise measure PM is assured by Theorem 1.1. In what follows, we will give a representation of Poisson noise in terms of a corresponding Poisson distribution. To do this, we first introduce some notions. Let D(Rd ) denote the Schwartz space of all (real-valued) C ∞ -functions on Rd with
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
769
compact support and D0 (Rd ) its topological dual space. Clearly, D(Rd ) ⊂ S(Rd ). As was pointed out e.g.in [34] and [45], see also e.g. [63], the Bochner–Minlos theorem (and therefore our Theorem 1.1) also holds on D(Rd ). Especially, (9) holds for PM on D0 (Rd ) and f ∈ D(Rd ). Namely, there exists a unique PM such that Z eihf,ξi dPM (ξ) D 0 (Rd )
(Z
) isf (x) isf (x) e dM (s)dx −1− 1 + s2 R\{0}
Z
= exp Λ(f )
(10)
for f ∈ D(Rd ), where Λ(f ) := supp f (⊂ Rd ) is the support of f . We assume henceforth that the first moment of M exists. In this case we can drop the third term in the exponential of the right-hand side of (10) and Theorem 1.1 assures that there exists a unique measure P˜M such that for f ∈ D(Rd ) (Z ) Z Z ihf,ξi ˜ isf (x) e e dPM (ξ) = exp − 1 dM (s)dx . (11) D 0 (Rd )
Λ(f )
R
R\{0}
R
Set κf = Λ(f ) R\{0} dM (s)dx, which is a finite and strictly positive number. Then by Taylor series expansion of the exponential and dominated convergence, we have from (11) that " #n Z Z X 1 Z ihf,ξi ˜ −κf isf (x) e dPM (ξ) = e e dM (s)dx n! Λ(f ) R\{0} D 0 (Rd ) n≥0 Z X 1 Z = e−κf ··· n! Λ(f ) R\{0} n≥0
Z
Z ···
e Λ(f )
i
Pn j=1
R\{0}
sj f (xj )
n Y
dM (sj )dxj .
(12)
j=1
Formula (12) might be interpreted as a representation of Poisson chaos and from it we get the following equivalent (in law) representation of Poisson noise * hf, ξi =
f,
Nf X
+ λj δXj
,
f ∈ D(Rd ) ,
(13)
j=1
where δx is the Dirac-distribution concentrated in x ∈ R and Nf is a compound Poisson distribution (with intensity κf ) given as follows Pr{Nf = n} =
e−κf (κf )n , n!
n = 0, 1, 2, . . .
(14)
where {(Xj , λj )}1≤j≤Nf is a family of independent, identically distributed random variables distributed according to the probability measure κ−1 f dx × dM (s) on Λ(f ) × (R \ {0}).
770
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
Concerning Poisson noise FM on S 0 (Rd ) determined by formula (9), we note that D(Rd ) is dense in S(Rd ) with respect to the topology of S(Rd ). Therefore, by the continuity of the right-hand side of (9), the chaos decomposition (12) determines the law of the coordinate process FM . Remark 1.4. It is interesting to note, in relation with the short discussion given in the introduction, that one can recover Gaussian fields from Poisson ones in a limit. In fact, let {Mn }n∈N be a certain sequence of functions satisfying (3) and n }n∈N be the sequence of Poisson noise measures determined by (10), then {PM Z Z Z σ2 n eihf,ξi dPM (ξ) −→ eia f (x)dx − [f (x)]2 dx 2 0 d d d D (R ) R R as n → ∞, which is the characteristic functional of a Gaussian law on D0 (Rd ) with mean Z E[hf, ·i] = a f (x)dx , f ∈ D(Rd ) Rd
and covariance
Z E[hf, ·ihg, ·i] = σ 2
iff
Z Rd
f (x)g(x)dx , Rd
f, g ∈ D(Rd ) ,
isf (x) isf (x) e dMn (s)dx −1− 1 + s2 R\{0} Z Z σ2 −→ ia f (x)dx − [f (x)]2 dx 2 Rd Rd
Z
as n → ∞. An example is given by Poisson laws where the left-hand side is Z Z (eisf (x) − 1)dMn (s)dx Rd
R\{0}
2 R which converges to σ2 Rd [f (x)]2 dx (i.e. to the right-hand side for a = 0) if, e.g., dMn (s) = n2 σ 2 δ n1 (s)ds.
2. Euclidean Random Fields as Convoluted Generalized White Noise Let us first give the notion of random fields. Definition 2.1. Let (Ω, E, P ) be a probability space. By a (generalized) random field X on (Ω, E, P ) with parameter space S(Rd ), we mean a system {X(f, ω), ω ∈ Ω}f ∈S(Rd ) , of random variables on (Ω, E, P ) having the following properties. 1. P {ω ∈ Ω : X(c1 f1 + c2 f2 , ω) = c1 X(f1 , ω) + c2 X(f2 , ω)} = 1, c1 , c2 ∈ R, f1 , f2 ∈ S(Rd ); 2. fn → f in S(Rd ) implies that X(fn , ·) → X(f, ·) in law.
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
771
The coordinate process F in Definition 2.1 is a random field on the generalized white noise space (S 0 (Rd ), B, Pψ ), which follows immediately from the facts that the above property 1 is fulfilled pointwise and the property 2 is implied by the pointwise convergence F (fn , ω) → F (f, ω) as n → ∞ for all ω ∈ S 0 (Rd ) and fn → f in S(Rd ) since the latter is slightly stronger than convergence in law. Let G : S(Rd ) → S(Rd ) be a linear and continuous mapping. Then by the known Schwartz theorem, there exists a distribution K ∈ S 0 (R2d ), hereafter called the kernel of G, such that Z K(x, y)f (y)dy , f ∈ S(Rd ) . (15) (Gf )(x) = Rd
It is clear that the conjugate operator G˜ : S 0 (Rd ) → S 0 (Rd ) is a measurable transformation from (S 0 (Rd ), B) into itself. Example 2.2. Let ∆ be the Laplace operator on Rd . Let Gα be the Green function (i.e., fundamental solution) of the pseudo-differential operator (−∆+ m20 )α for some arbitrary (but fixed) m0 > 0 and 0 < α. Take K(x, y) = Gα (x − y), x, y ∈ Rd . Then G = (−∆ + m20 )−α is a linear continuous mapping from S(Rd ) to S(Rd ). To see this, let F and F −1 denote the Fourier and inverse Fourier transforms, respectively. Namely, Z d e−ixy f (x)dx (Ff )(y) = (2π)− 2 Rd
(F −1 f )(x) = (2π)− 2
d
Z
eixy f (y)dy Rd
for f ∈ S(Rd ). Then we have
Z Gα (x − y)f (y)dy ,
(Gf )(x) = Rd
and
1
(FGα )(k) =
d
(2π) 2 (|k|2 + m20 )α
Thus (F(Gf ))(k) = (2π)− 2
d
Z
e−ixk
Rd
−d 2
= (2π)
e
−iyk
Rd
k ∈ Rd .
,
Z Rd
Z
f ∈ S(Rd )
Gα (x − y)f (y)dydx Z
Gα (y)dy
e−ixk f (x)dx
Rd
d
= (2π) 2 (F Gα )(k) · (F f )(k) =
Therefore Gf =
F
−1
(|k|2
1 · (F f )(k) . + m20 )α
1 2 (|k| + m20 )α
· Ff
(k) ,
f ∈ S(Rd ) .
(16)
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S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
We notice that (−∆ + m20 )−α maps real test functions to real test functions. Furthermore, by Theorem IX.4 of [61], F and F −1 are linear continuous from SC (Rd ) to SC (Rd ). Hence it suffices to verify that the multiplicative operator defined by 1 d d f (·) → (|k|2 +m 2 )α · f (·) is linear and continuous from SC (R ) to SC (R ). The lin0 earity is obvious. The continuity is derived from the fact that Mh f := hf defines a continuous multiplicative operator from SC (Rd ) to SC (Rd ) if h is C ∞ –differentiable and h itself with all its derivatives are of at most polynomial increase. This is true because in our case 1 . h(k) = (|k|2 + m20 )α In Sec. 1, we had already defined the generalized white noise measure Pψ on (S 0 (Rd ), B) associated with a L´evy characteristic ψ. Now let PK denote the image ˜ i.e., PK is a measure on (S 0 (Rd ), B) defined (probability) measure of Pψ under G, by PK (A) := Pψ (G˜−1 A) , A ∈ B . (17) Then we have the following result Proposition 2.3. The Fourier transform of PK is given by Z eihf,ξi dPK (ξ) S 0 (Rd )
Z
Z
K(x, y)f (y)dy dx ,
ψ
= exp Rd
Rd
f ∈ S(Rd ) .
(18)
Conversely, given a linear and continuous mapping G : S(Rd ) → S(Rd ) and a L´evy characteristic ψ, there exists a unique probability measure PK such that (18) is valid. Proof. For f ∈ S(Rd ), by (17) and Theorem 1.1, we derive that Z Z ˜ ihf,ξi e dPK (ξ) = eihf,Gξi dPψ (ξ) S 0 (Rd )
S 0 (Rd )
Z
eihGf,ξi dPψ (ξ)
= S 0 (Rd )
Z
Z ψ
= exp Rd
K(x, y)f (y)dy dx .
Rd
The converse statement is derived analogously to the proof of Theorem 1.1 since the operator G is continuous from S(Rd ) to S(Rd ) and thus the RHS of (18) defines a characteristic functional. Clearly, from Proposition 2.3, the associated coordinate process X : S(Rd ) × (S (Rd ), B, PK ) → R given by 0
X(f, ξ) = hf, ξi ,
f ∈ S(Rd ) ,
ξ ∈ S 0 (Rd )
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
773
˜ which is defined is a random field on (S 0 (Rd ), B, PK ). In fact, X is nothing but GF by ˜ )(f, ξ) := F (Gf, ξ) , f ∈ S(Rd ) , ξ ∈ S 0 (Rd ) . (GF Remark 2.4. Concerning Example 2.2, the above construction does not always work if m0 = 0, because (−∆)−α does not map S(Rd ) onto S(Rd ) for α > 0. In this case, if F is a Gaussian white noise, then we can obtain a random field X = (−∆)−α F for α < d4 by the following argument. Take α < d4 , then the scalar product Z (F f1 )(k) · (F f2 )(k) dk , f1 , f2 ∈ S(Rd ) (f1 , f2 )α := |k|4α Rd is continuous with respect to the (Schwartz) topology of S(Rd ), where F denotes the 2 Fourier transform as introduced previously. Thus f ∈ S(Rd ) 7→ exp{− σ2 kf k2α } ∈ [0, ∞) is a characteristic functional on S(Rd ). By Bochner–Minlos theorem, we get a unique measure PK satisfying (18) and hence the associated coordinate process is precisely X = (−∆)−α F . However, if one wants to follow a corresponding procedure with a generalized white noise, one needs an explicit calculation to show the continuity of the functional Z d −α ψ((−∆) f )(x))dx ∈ C , f ∈ S(R ) → exp Rd
where ψ is a L´evy characteristic. Then, by Bochner–Minlos theorem, one can directly construct the measure PK . In order to derive a suitable condition for the continuity of the above functional, we note that the characteristic functional of the generalized white noise (5) extends continuously from S(Rd ) to L2 (Rd , dx), provided the corresponding generalized white noise F has mean zero and finite moments of second order (see Prop. 4.3 of [25]). Furthermore, by the above considerations for the Gaussian case and the fact that F is unitary on L2C (Rd , dx), we get that (−∆)−α : S(Rd ) → L2 (Rd , dx) is continuous, if 0 < α < d4 . Thus, we can construct “mass zero” random fields X = (−∆)−α F for 0 < α < d4 and F as characterized above. In what follows, we will always assume that the continuity of G : S(Rd ) → S(Rd ) holds. Let us now turn to discuss the invariance of the random field X under Euclidean transformations. We need to introduce some notations first. A proper Euclidean transformation of Rd is, by definition, an element of the proper Euclidean group E0 (Rd ) over Rd . In fact, the proper Euclidean group E0 (Rd ) is generated by 1. all translations Ta : x ∈ Rd → Ta x := x − a ∈ Rd , a ∈ Rd ; 2. all rotations R : x ∈ Rd → Rx ∈ Rd . The (full) Euclidean group E(Rd ) over Rd is generated by all transformations in E0 (Rd ) and by all reflections. The group E(Rd ) is an inhomogeneous orthogonal
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S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
group, that is, E(Rd ) is the group of all nonsingular inhomogeneous linear transforms which preserve the Euclidean inner product. It is easy to see that among the reflections it is enough to consider the “time reflection” θ, defined by writing x ∈ Rd as x := (x0 , ~x), x0 ∈ R, ~x ∈ Rd−1 , calling x0 “time coordinate” and setting θx := (−x0 , ~x) ,
x = (x0 , ~x) ∈ R × Rd−1 .
If T is a transformation in the Euclidean group E(Rd ), the corresponding transformation on a test function f ∈ S(Rd ) is defined by (T f )(x) := f (T −1 x) ,
x ∈ Rd ;
and on S 0 (Rd ) is defined by duality as follows hf, T ξi := hT −1 f, ξi ,
f ∈ S(Rd ) ,
ξ ∈ S 0 (Rd ) .
The corresponding transformation on the random field X is defined by (T X)(f, ξ) := X(T −1 f, ξ) ,
f ∈ S(Rd ) ,
ξ ∈ S 0 (Rd ) .
By the invariance of the dualization under Euclidean transformations (which follows from the invariance of Lebesgue measure), we have (T X)(T f, ξ) = X(f, ξ) ,
f ∈ S(Rd ) ,
ξ ∈ S 0 (Rd ) .
Concerning Euclidean invariance of random fields, we have the following Definition 2.5. By Euclidean invariance of the random field X we mean that the laws of X and T X are the same, for each T ∈ E(Rd ), i.e., the probability distributions of {X(f, ·) : f ∈ S(Rd )} and {(T X)(f, ·) : f ∈ S(Rd )} coincide for each T ∈ E(Rd ). In particular, if the laws of X and θX, where θ is the “time–reflection” defined above, are the same, we say that the random field X is (time–)reflection invariant. From Bochner–Minlos Theorem, the probability distribution of {X(f, ·) : f ∈ S(Rd )} is uniquely determined by the characteristic functional CX (f ), f ∈ S(Rd ), and vice versa. Thus, the property of Euclidean invariance of random fields is also determined by means of characteristic functionals. We say that G is T -invariant, for some T ∈ E(Rd ), if GT = T G. G is called Euclidean invariant if G is invariant under all T ∈ E0 (Rd ). In case G is translation invariant its kernel K has the form K(x, y) = G(x − y) (cf. p. 39 of [67]). If the kernel G of G is also invariant under orthogonal transformations, then G is invariant under all T ∈ E(Rd ). In this case we also say for simplicity that G is the Euclidean invariant kernel of G. The action of G on test function in S(Rd ) (and by duality on
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
775
S 0 (Rd ) as well as on random fields) in the translation invariant case is by convolution Z Z K(x, y)f (y)dy = G(x − y)f (y)dy x ∈ Rd . (Gf )(x) = Rd
Rd
We then also write Gf as G ∗ f . Remark 2.6. The kernel Gα determined by formula (16) in Example 2.2 is given by Z 1 eikx dk , x ∈ Rd , Gα (x) = (2π)d Rd (|k|2 + m2 )α where the integral has to be understood in the sense of a Fourier-transform of a tempered distribution. It is invariant under all orthogonal transformations. This can be verified directly by changing integral variables in the above formula since orthogonal transforms leave |k| and dk invariant. Moreover, we have following result Proposition 2.7. Assume that the mapping G : S(Rd ) → S(Rd ) is Euclideaninvariant, then the random field X = GF is Euclidean-invariant. Proof. By Bochner–Minlos Theorem, it is sufficient to show that CX (f ) = CT X (f ) ,
f ∈ S(Rd )
for every T ∈ E(Rd ). In fact, we have i h CT X (f ) = E eiT X(f,·) i h −1 = E eiX(T f,·) Z −1 ψ(G(T f )(x))dx , = exp Rd
f ∈ S(Rd ) ,
where the last equality follows from (18). So we need only to verify that Z Z ψ((G(T f ))(x))dx = ψ((Gf )(x))dx . Rd
Rd
This holds by using the invariance of Lebesgue measure under the transformation x → T −1 x and the fact that (T −1 (Gf ))(x) = (G(T −1 f ))(x) .
Hereafter, we only deal with Euclidean invariant kernels, the derived random fields are then also Euclidean invariant. We call such random fields Euclidean random fields. From Remark 2.6, the integral kernel Gα defined in Example 2.2 is Euclidean invariant. Moreover, since the translation invariance implies that the
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integral kernels are of convolution type, the corresponding Euclidean random fields are convoluted generalized white noise. We simply denote the convoluted generalized white noise X by X := Gα ∗ F . 3. The Schwinger Functions of the Model and Their Basic Properties In 1973, E. Nelson [53] showed how to construct a relativistic quantum field theory (QFT) from a Euclidean Markov field. Inspired by this, in [56] and [57] Osterwalder and Schrader (see also [36, 42, 72]) gave a set of axioms, where Schwinger functions {Sn }n∈N0 defined on the Euclidean space–time Ed can be analytically continued to Wightman distributions, i.e. to the vacuum expectation values of a relativistic QFT. (Here and in the following we use the “sans-serif” Sn for Schwinger functions in general, whereas the Schwinger functions of our concrete model are denoted by “italic” Sn .) Apart from existence of an analytic continuation, these axioms are (E0) Temperedness, (E1) Euclidean invariance, (E2) Reflection positivity, (E3) Symmetry and (E4) cluster property. In the case of Euclidean Markov fields and also in the more general case of Euclidean reflection positive fields [33], Schwinger functions fulfilling (E0)–(E4) are obtained as the moments of the Euclidean field. In this section we will calculate the “Schwinger functions” Sn of X, which are by definition the moment functions of the convoluted generalized white noise X. We will verify (E0), (E1) and (E3). A proof of (E4) is given in Sec. 4. In Sec. 5 a partial negative result on (E2) is derived for the case of convoluted generalized white noises with a non-zero Poisson part. We now fix a L´evy characteristic ψ, such that the L´evy measure M has moments of all orders. With F we denote the generalized white noise determined by ψ. Lemma 3.1. Let CFT denote the functional which maps ϕ ∈ S(Rd ) to T Rd ψ(ϕ(x))dx. Then partial derivatives of all orders of CF exist everywhere on d d S(R ). For ϕ1 , . . . , ϕn ∈ S(R ) we get that
R
1 ∂n C T |0 = cn in ∂ϕ1 · · · ∂ϕn F
Z Rd
ϕ1 · · · ϕn dx .
(19)
Here we introduce the constants cn defined as Z c1 := a +
R\{0}
Z c2 := σ 2 + Z cn :=
s3 dM (s) 1 + s2 s2 dM (s)
R\{0}
sn dM (s) , R\{0}
n ≥ 3.
As it will be explained shortly, throughout this paper the superscript “T ” stands for the operation of “truncation” (cf. [41]).
777
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
R 2 R Proof. Since the differentiability of ϕ → ia Rd ϕdx and ϕ → σ2 Rd ϕ2 dx is immediate, we only have to deal with the Poisson part of ψ. We remark that |eiy − 1| ≤ |y| for all y ∈ R. Therefore 1 is[ϕ(x)+t1 ϕ1 (x)] |e − eisϕ(x) | ≤ |sϕ1 (x)| , t1 for all t1 > 0, s ∈ R. This shows that the LHS is uniformly bounded (in t1 ) by a function in L1 (R \ {0} × Rd , dM ⊗ dx). Analogously, for all tn > 0, 1 n−1 |s ϕ1 (x) · · · ϕn−1 (x)(eis[ϕ(x)+tn ϕn (x)] − eisϕ(x) )| ≤ |sn ϕ1 (x) · · · ϕn (x)| , tn and again the RHS is a uniform L1 (R \ {0} × Rd , dM ⊗ dx)-bound. Thus we may interchange partial derivatives and integration by the dominated convergence theorem: Z Z isϕ(x) ∂n 1 isϕ e −1− dM (s)dx in ∂ϕ1 · · · ∂ϕn Rd R\{0} 1 + s2 Z Z eisϕ ϕ1 (x) · · · ϕn (x)sn dM (s)dx = Rd
if n ≥ 2 and
R\{0}
Z ··· =
Z Rd
R\{0}
eisϕ(x) ϕ1 (x)
s3 dM (s)dx 1 + s2
if n = 1. Setting ϕ = 0 and taking into account the linear and Gaussian part, we get (19). We have CF = exp CFT . Consequently CF has partial derivatives of any order, and it follows that all moments of F , i.e., the expectation values of hϕ1 , F i · · · hϕn , F i, exist and are equal i−n times the n-th order partial derivative of CF w.r.t. ϕ1 , . . . , ϕn at the point ϕ = 0, cf. [51]. Definition 3.2. Let ϕ1 , . . . , ϕn ∈ S(Rd ). We define MnF , the n-th moment function of F , by Z MnF (ϕ1 ⊗ · · · ⊗ ϕn ) = hϕ1 , ωi · · · hϕn , ωidPψ (ω) . (20) S 0 (Rd )
By the above remark this equals =
1 ∂n CF 0 . n i ∂ϕ1 · · · ∂vpn
(21)
In order to calculate partial derivatives of CF of any order we need a generalized chain rule: Lemma 3.3. Let V be a vector space. Let g : V → C be infinitely often partial differentiable and let f : C 7→ C be analytic. Then for v1 , . . . , vn ∈ V n X X ∂n f ◦g = f (k) ◦ g ∂v1 · · · ∂vn (n) k=1
I∈Pk
Y {j1 ,...,jl }∈I
∂l g ∂vj1 · · · ∂vjl
(22)
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S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
(n)
is the collection of partitions of {1, . . . , n} into exactly k dk (x) = dxk f (x).
holds on V. Here Pk disjoint subsets, f
(k)
Proof. We proceed by induction over n. The statement for n = 1 is Leibniz’ chain rule. Observe that ∂ n+1 g◦f ∂v1 · · · ∂vn ∂vn+1 n X
∂ = ∂vn+1
=
n X
f
(k)
X
◦g
k=1
(n)
I∈Pk
(
X
f (k+1) ◦ g
k=1
(n)
+f
◦g
{j1 ,...,jl }∈I
k X X (n) I∈Pk
{j1 ,...,jl }∈I
Y
I∈Pk (k)
Y
k Y
m=1 i=1,i6=m
∂lg ∂v1 · · · ∂vl
!
∂lg ∂g ∂vj1 · · · ∂vjl ∂vn+1
∂lg ∂ l+1 g ∂vj1i · · · ∂vjli ∂vj1m · · · ∂vjlm ∂vn+1
) .
(n)
We denote I ∈ Pk by I = {I1 , . . . , Ik } and set Ii = {j1i , . . . , jli } (where, of course, (n+1) l depends on i). Using collections of partitions Pk , we can “reindex” the sums in the latter expression and get: ( n X X Y ∂lg f (k+1) ◦ g = ∂vj1 · · · ∂vjl (n+1) k=1
I∈Pk+1
+f
(k)
X
◦g
(n+1)
=
n+1 X
I∈Pk
(
Y
,{n+1}6∈I {j1 ,...,jl }∈I
X
f (k) ◦ g
k=1
,{n+1}∈I {j1 ,...,jl }∈I
(n+1)
I∈Pk
X
+ f (k) ◦ g
(n+1)
I∈Pk
∂lg ∂vj1 · · · ∂vjl
Y
,{n+1}∈I {j1 ,...,jl }∈I
Y
{n+1}6∈I {j1 ,...,jl }∈I
)
∂lg ∂vj1 · · · ∂vjl
∂lg ∂vj1 · · · ∂vjl
) .
On the RHS we have also reindexed k + 1 in the first sum in the braces to k with the (n+1) sum ranging from 2 to n + 1. Since the only partition in P1 , {{1, . . . , n + 1}}, does not contain {n + 1}, we may sum from 1 to n + 1. Similarly in the second sum (n+1) we may extend the sum from k = 1, . . . , n to k = 1, . . . , n + 1, since Pn+1 contains only {{1}, . . . , {n + 1}} and thus gives no contributions to this sum. But the last expression obtained obviously equals (22) with n replaced by (n + 1). We remark that for the case of only one variable v = v1 = · · · = vn we get Faa di Brunos expansion formula [46].
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
779
Corollary 3.4. (Cumulant formula) Let V , g as in Lemma 3.3 and let f be the exponential function. Furthermore assume g(0) = 0. Then f (k) ◦ g(0) = 1 for all k ∈ N. Let P (n) stand for the collection of all partitions I of {1, . . . , n} into disjoint subsets. It follows that for v1 , . . . , vn ∈ V we get : X ∂n exp g 0 = ∂v1 · · · ∂vn (n) I∈P
Y {j1 ,...,jl }∈I
∂l g . ∂vj1 · · · ∂vjl 0
(23)
Corollary 3.5. (Wick’s Theorem) Let f be the exponential function and g a quadratic function on a vector space V, i.e. g(tv) = t2 g(v) for all v ∈ V. Then by Corollary 3.4 X Y ∂ 2 g n f or n even ∂ 0 exp g 0 = I∈ pairings {j1 ,j2 }∈I ∂vj1 ∂vj2 (24) ∂v1 · · · ∂vn 0 f or n odd Here the pairings are those partitions I of {1, . . . , n}, where all subsets in I do contain exactly two elements. This follows from the fact that all partial derivatives of g are zero, except for partial derivatives of order 2. Proposition 3.6. Set ϕ1 , . . . , ϕn ∈ S(Rd ). Then MnF (ϕ1
⊗ · · · ⊗ ϕn ) =
X
Y
I∈P (n) {j1 ,...,jl }∈I
Z cl
Rd
ϕj1 · · · ϕjl dx .
(25)
Proof. (25) follows directly from (21), CF = exp CFT , Lemma 3.1 and Corollary 3.4. Remark 3.7. Choosing a in (20) such that c1 = 0 implies that F has mean zero. If furthermore M is a symmetric measure w.r.t. reflections at zero, then all cn vanish for n odd. For such n also MnF is zero , since in this case at least one cl with l odd appears in every summand on the RHS of (25). We now fix, as in Sec. 2, a linear continuous map G : S(Rd ) → S(Rd ) and denote ˜ Furthermore, we assume that G is Euclidean invariant. Then there its dual by G. exists a convolution kernel G which is invariant under orthogonal transformations, such that Gϕ = G ∗ ϕ for all ϕ ∈ S(Rd ) (cf. Sec. 2). Define, as explained in Sec. 2, ˜ −1 and let X be the coordinate process w.r.t. PG . PG = Pψ ◦ (G) Definition 3.8. (“Schwinger functions of X”) Set ϕ1 , . . . , ϕn ∈ S(Rd ). We define the n-th Schwinger function Sn as the n-th moment of X, i.e. Z Sn (ϕ1 ⊗ · · · ⊗ ϕn ) = hϕ1 , ωi · · · hϕn , ωidPG (ω) , n ∈ N0 . (26) S 0 (Rd )
780
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
Proposition 3.9. The Schwinger functions Sn defined above are symmetric and Euclidean invariant tempered distributions, i.e. Sn ∈ S 0 (Rdn ) for n ≥ 1. Furthermore for ϕ1 , . . . , ϕn ∈ S(Rd ) we have Z X Y Sn (ϕ1 ⊗ · · · ⊗ ϕn ) = cl G ∗ ϕj1 · · · G ∗ ϕjl dx . (27) I∈P (n) {j1 ,...,jl }∈I
Rd
Proof. The symmetry follows directly from Definition 3.8. Euclidean invariance of the moment functions follows from the Euclidean invariance in law of the random field X. Now we first prove (27). By the transformation formula, the RHS of (26) is equal to Z ˜ · · · hϕn , GωidP ˜ hϕ1 , Gωi ψ (ω) S 0 (Rd )
Z
= S 0 (Rd )
hG ∗ ϕ1 , ωi · · · hG ∗ ϕn , ωidPψ (ω) .
This together with Proposition 3.6 now implies (27). Concerning temperedness we remark that by (27) Sn isRa sum of tensor products of linear functionals, say SlT , which map ϕ1 ⊗· · ·⊗ϕl into cl Rd G∗ϕ1 · · · G∗ϕl dx. Fix Q ϕ1R, . . . , ϕj−1 , ϕj+1 , . . . , ϕl ∈ S(Rd ). Then ϕj 7→ G∗ϕi 7→ G∗ϕj m=1,m6=j G∗ϕm 7→ cl Rd G∗ϕ1 · · · G∗ϕl dx is a map composed of S(Rd )-continuous mappings and therefore is continuous in ϕj alone, provided the other ϕ’s are fixed. A use of Schwartz nuclear theorem yields the temperedness of the SlT and a second application of the nuclear theorem then implies Sn ∈ S 0 (Rd ). 4. Truncated Schwinger Functions and the Cluster Property In this section, let {Sn }n∈ N0 be a sequence of distributions, where S0 = 1 and Sn ∈ S 0 (Rdn ) for n ≥ 1. We define truncated distributions STn , n ≥ 1, in the sense of Haag [41]. The {Sn }n∈N0 determine the corresponding sequence of truncated distributions uniquely and vice versa, and we can translate many properties of one sequence into properties of the other. This is quite obvious e.g. for (E0), (E1) and (E3). Making use of arguments in the classical papers on the so-called asymptotic condition in axiomatic QFT [41, 22, 23, 48] we obtain the equivalence of the cluster property (E4) of translation invariant {Sn }n∈N0 and the cluster property of the truncated Schwinger functions {STn }n∈N . As an immediate consequence of formula (27) we have explicit formulae for the truncated Schwinger functions of our model and we can easily check their “truncated” cluster property in order to verify (E4). Definition 4.1. Let {Sn }n∈N0 be a sequence of distributions with S0 = 1 and Sn ∈ S 0 (Rdn ) for n ≥ 1. Let ϕ1 , . . . , ϕn ∈ S(Rd ). By the relation
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
Sn (ϕ1 ⊗ · · · ⊗ ϕn ) =
X
Y
STl (ϕj1 ⊗ · · · ⊗ ϕjl ) n ≥ 1
781
(28)
I∈P (n) {j1 ,...,jl }∈I
we recursively define the n-th truncated distribution STn . Here, for {j1 , . . . , jl } ∈ I we assume j1 < j2 < · · · < jl . Remark 4.2. (i) By the Schwartz nuclear theorem the sequence {Sn }n∈N0 uniquely determines the sequence {STn }n∈N and vice versa. (ii) All STn are Euclidean (translation) invariant if and only if all Sn are euclidean (translation) invariant. The same equivalence holds for temperedness and symmetry (see e.g. [22] for the symmetry). From now on we assume at least translation invariance for {Sn }n∈N0 , {STn }n∈N respectively. And we will call these distributions (truncated) Schwinger functions, even through at this level they might have little to do with QFT. Definition 4.3. Let a ∈ Rd , a 6= 0, λ ∈ R. Let Tλa denote the representation of the translation by λa on S(Rdn ), n ∈ N. Take m, n ≥ 1, ϕ1 , . . . , ϕn+m ∈ S(Rd ) (i) cluster property (E4) A sequence of Schwinger functions {Sn }n∈N0 has the cluster property if for all n, m ≥ 1 n lim Sm+n (ϕ1 ⊗ · · · ϕm ⊗ Tλa (ϕm+1 ⊗ · · · ⊗ ϕm+n )) λ→∞ o (29) − Sm (ϕ1 ⊗ · · · ϕm )Sn (ϕm+1 ⊗ · · · ⊗ ϕm+n ) = 0 . (ii) cluster property of truncated Schwinger functions (E4T) A sequence of truncated Schwinger functions {STn }n∈N has the cluster property of truncated Schwinger functions, if for all n, m ≥ 1 lim STm+n (ϕ1 ⊗ · · · ⊗ ϕm ⊗ Tλa (ϕm+1 ⊗ · · · ⊗ ϕm+n )) = 0 .
λ→∞
(30)
Remark 4.4. We could also replace limλ→∞ (·) in (29) and (30) by limλ→∞ |λ|N (·) for N arbitrary. Such conditions should be seen as cluster properties for short-range interactions. Indeed they would exclude the mass zero cases. 1 Take e.g. the free Markov field of mass zero X = (−∆)− 2 F in d dimensions, d ≥ 3, where F is a Gaussian white noise. Then (29) and (30) tend to zero only as λ−d+2 , as λ → ∞. Nevertheless, by the same proofs as in Theorem 4.5 and Corollary 4.7 below we can also show that the Schwinger functions of our model have the “shortrange” cluster property. This already indicates the existence of a “mass-gap” in the corresponding relativistic theory, obtained by the analytic continuation of the (truncated) Schwinger functions in Secs. 6 and 7. The following result is the Euclidean analogue to one proven by Araki [22, 23] for the case of (truncated) Wightman functions. Theorem 4.5. Let {Sn }n∈N , {STn }n∈N be as in Definition 4.1 and let the {Sn }n∈N0 , {STn }n∈N be translation invariant. Then {Sn }n∈N0 has the cluster prop-
782
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
erty, if and only if {STn }n∈N has the cluster property of the truncated Schwinger functions. Proof. (E4) ⇒ (E4T) Assume there exist m, n ≥ 1, such that ϕ1 , . . . , ϕn+m ∈ S(Rd ) and an a ∈ Rd \ {0}, such that the limit in (28) is not zero or does not exist. Let furthermore n + m be minimal w.r.t. this property. Define ϕλk = ϕk for k = 1, . . . , m and = Tλa ϕk for k = m + 1, . . . , m + n. By translation invariance the LHS of (29) is then equal to X
lim
λ→∞
(m+n)
I∈Pm,n
Y
STl (ϕλj1 ⊗ · · · ⊗ ϕλjl ) .
{j1 ,...,jl }∈I
(m+n)
stands for all partitions I of {1, . . . , n + m} into disjoint The symbol Pm,n subsets, such that in each partition I there is at least one subset {j1 , . . . , jl } ∈ I such that {j1 , . . . , jl } ∩ {1, . . . , m} 6= ∅, {j1 , . . . , jl } ∩ {m + 1, . . . , m + n} 6= ∅. By the assumption (E4), the above expression equals zero. Furthermore, every summand except for the one indexed by I = {{1, . . . , n+m}} tends to zero, since in each such summand at least one truncated Schwinger function in the product is evaluated on some ϕλk ’s, 1 ≤ k ≤ m and some ϕλk ’s, m + 1 ≤ k ≤ m + n, at the same time. By the minimality of n + m this factor tends to zero as λ → ∞. The other factors in the product either tend to zero (by minimality of n + m) as λ → ∞ or are constant, either by the definition of the ϕλk for k = 1, . . . , m or by the translation invariance of the STl . Consequently also the summand indexed by I = {{1, . . . , n+m}} has to converge to zero as λ → ∞, which is in contradiction with the above assumptions on n, m. (E4T) ⇒ (E4) Fix ϕ1 , . . . , ϕn+m and define ϕλk as above. Then lim Sm+n (ϕλ1 ⊗ · · · ⊗ ϕλm+n ) ( Y X = lim
λ→∞
λ→∞
(m+n) c )
I∈(Pm,n
+
(m+n)
STl (ϕλj1 ⊗ · · · ⊗ ϕλjl )
{j1 ,...,jl }∈I
X
Y
(m+n) I∈Pm,n
{j1 ,...,jl }∈I
) STl (ϕλj1
⊗ ··· ⊗
ϕλjl )
,
(m+n)
where (Pm,n )c := P (m+n) \ Pm,n . In the second term all products contain at least one factor that tends to zero as λ → ∞ by (E4T). The other factors are constant (either by the definition of ϕλk or by the translation invariance of STl ) or tend to zero by (E4T), again. Thus, the whole second sum vanishes for λ → ∞. In the first term all STl are evaluated on ϕλj1 , . . . , ϕλjl such that either {j1 , . . . , jl } ⊂ {1, . . . , m} or ⊂ {m + 1, . . . , m + n}. It follows from the definition of the ϕλk or the translation invariance of the STl , that all factors do not depend on λ. In the
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
783
first term, we may therefore omit the λ’s. The first term by this argument equals X Y STl (ϕj1 ⊗ · · · ⊗ ϕjl ) I∈P (m) {j1 ,...,jl }∈I
×
X
Y
STl (ϕj1 +m ⊗ · · · ⊗ ϕjl +m )
I∈P (n) {j1 ,...,jl }∈I
= Sm (ϕ1 ⊗ · · · ⊗ ϕm )Sn (ϕm+1 ⊗ · · · ⊗ ϕm+n ) ,
from which we get (29).
Remark 4.6. Let {Sn }n∈N0 be defined according to Definition 3.8. Let denote the sequence of truncated Schwinger functions. Then, of course, by comparison of (28) and (27) we get Z G ∗ ϕ1 · · · G ∗ ϕn dx (31) SnT (ϕ1 ⊗ · · · ⊗ ϕn ) = cn
{SnT }n∈N
Rd
for ϕ1 , . . . , ϕn ∈ S(Rd ). Corollary 4.7. Let {Sn }n∈N0 be as in Definition 3.8. Then {Sn }n∈N0 has the cluster property (E4). Proof. By Theorem 4.5 it suffices to show (E4T) for {SnT }n∈N . Fix a 6= 0 in Rd and let λ ∈ Rd , ϕ1 , . . . , ϕn+m ∈ S(Rd ), m, n ≥ 1. We have T lim Sm+n (ϕ1 ⊗ · · · ⊗ ϕm Tλa (ϕm+1 ⊗ · · · ⊗ ϕm+n )) Z = lim cm+n G ∗ ϕ1 · · · G ∗ ϕm G ∗ Tλa ϕm+1 · · · G ∗ Tλa ϕm+n dx
λ→∞
λ→∞
= lim cm+n λ→∞
Rd
Z
Rd
G ∗ ϕ1 · · · G ∗ ϕm Tλa (G ∗ ϕm+1 · · · G ∗ ϕm+n )dx
On the RHS we shift away a fast falling function G ∗ ϕm+1 · · · G ∗ ϕm+n from the fixed fast falling function G ∗ ϕ1 · · · G ∗ ϕm . The RHS therefore approaches zero faster than any negative power of λ falls to zero. 5. On Reflection Positivity In Sec. 4 we saw that many properties of Schwinger functions can be directly translated into related properties of truncated Schwinger functions. How about reflection positivity then? Let us first discuss a simple related case. If Cµ denotes the Fourier-transform of a probability measure µ on the real line, and Cµ is analytic in a neighborhood of zero, then the derivatives of Cµ fulfill the positivity condition
784
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
a1 d an dn + a0 + ···+ in dxn i dx
2 Cµ |0 ≥ 0
(32)
for all a0 , . . . , an ∈ R. This is related to (E2). Now suppose that Cµ = exp(CµT ) and onbergs theorem (see [27]) says that Cµλ is positive consider Cµλ = exp(λCµT ). Sch¨ definite for all λ ∈ R+ if and only if CµT is a L´evy characteristic (cf. Sec. 1). One can show that the derivatives of CµT at zero (the cumulants or “truncated moments of µ”) fulfill the same positivity condition (32) if we set a0 = 0, since a L´evy characteristic can be approximated by a sequence bn − Cn (0) + Cn , where for n ∈ N, Cn is a positive definite function and bn ≥ 0 (cf. [27]). On the other hand, if we can disprove one of the latter positivity conditions, we will, as a consequence of Sch¨ onbergs theorem, find some λ ∈ R+ such that Cµλ is not positive definite and therefore (32) does not hold for some λ, n, al , l = 0, . . . , n. Along these lines we now construct some counter examples of convoluted generalized white noises X = G ∗ F with nonzero Poisson part, which do not have the property of reflection positivity. It is interesting that such X do exist even in such cases where the corresponding convoluted Gaussian white noise is reflection positive. Roughly speaking, the Schwinger functions {Sn }n∈N0 belonging to X do not have the property of reflection positivity, if the terms in the Sn emerging from the “interaction” are large in comparison with the “free” terms. It remains an open question, whether reflection positivity holds or does not hold in other cases. Let us start with some definitions borrowed from the theory of infinitely divisible random distributions. Definition 5.1. Let {Sn }n∈N0 , Sn ∈ S 0 (Rdn ), n ≥ 1, S0 = 1 be a sequence of distributions and {STn }n∈N the corresponding truncated sequence. By Rd+ we denote the set of x ∈ Rd , x = (x0 , ~x) ∈ R × Rd−1 , x0 > 0. Let S((Rd+ )n ) be the Schwartz-functions on Rdn with support in (Rd+ )n . θ is the “time–reflection”, i.e. θ(x0 , ~x) = (−x0 , ~x). (i) We call {Sn }n∈N reflection infinitely divisible if for all λ ∈ R+ the sequence of Schwinger functions {Sλn }n∈N0 determined by the truncated sequence {λSTn }n∈N is reflection positive. (ii) We say, a sequence of truncated Schwinger functions {STn }n∈N is conditional reflection positive, if for all test functions ϕk ∈ S((Rd+ )k ), k = 1, . . . , n the inequality n X STk+l (θϕk ⊗ ϕl ) ≥ 0 (33) k,l=1
holds. Observe that the sum in (33) is over k, l = 1, . . . , n, which distinguishes conditional reflection positivity from reflection positivity (E2), where the sum is over k, l = 0, . . . , n and ϕ0 ∈ R. Remark 5.2. The positivity condition introduced here is a little more strict than the original positivity condition in [56]. Here we demand “positivity” of the
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
785
Schwinger functions {Sn }n∈N0 for test-functions ϕk ∈ S((Rd+ )k ), k = 1, . . . , n, ϕ0 ∈ R instead of ϕk ∈ S+< ((Rd )k ), k = 1, . . . , n, which is the original condition from [56]. Here S+< ((Rd )k ) is the space of all test functions with support in (Rd+ )< = {(x1 , . . . , xk ) ∈ Rdk : xj = (x0j , ~xj ), j = 1, . . . , 2, 0 < x01 < · · · < x0k }. Since the latter space is contained in the former, our condition is, in general, more strict. But in the present case, they are equivalent: The reason is that for a large class of convolution kernels G the Schwinger functions Sn belonging to X = G ∗ F (cf. Sec. 3), are more regular than tempered distributions namely they are locally integrable functions (cf. Lemma 7.7). One can therefore evaluate such Sn on test functions with “jumps”. Thus, by application of symmetry, we may calculate for ϕk ∈ S((Rd+ )k ), k = 1, . . . , n, ϕ0 ∈ R: n X
Sk+j (θϕk ⊗ ϕk )
k,j=0
=
n X
X
X
Sk+j (θ(1{x0
π(1)
k,j=0 π∈ Perm(k) π 0 ∈ Perm(j)
⊗ (1{x0 0
π (1)
=
n X
<···<x0π(k) } ϕk )
<···<x0π0 (j) } ϕj ))
Sk+j (θϕ˜k ⊗ ϕ˜j ) ,
k,j=0
where Perm(k) is the group of permutations of {1, . . . , k} and ϕ˜k is defined as ϕ˜k (x1 , . . . , xk ) := 1{x01 <···<x0k } (x1 , . . . , xk )
X
ϕ(xπ−1 (1) , . . . , xπ−1 (k) ) .
π∈Perm(k)
Now the functions ϕ˜k can be approximated by functions from S+< (Rdk ) and by application of the dominated convergence theorem we can derive the sharpened reflection positivity condition (E4) from the original one of [56]. Lemma 5.3. Let {Sn }n∈N0 and {STn }n∈N be as in Definition 5.1. If {Sn }n∈N0 is reflection infinitely divisible, then {STn }n∈N is conditional reflection positive. More precisely: If {STn }n∈N is not conditionally reflection positive, then there exists a λ0 > 0, such that for all λ, 0 < λ < λ0 , the sequence of Schwinger functions {Sλn }n∈N0 is not reflection positive. Proof. Suppose, {STn }n∈N is not conditionally reflection positive. Then there are test functions ϕ1 , . . . , ϕn as in Definition 5.1 such that the LHS of (33) is negative. Since n n X X 1 Sλk+l (θϕk ⊗ ϕl ) = STk+l (θϕk ⊗ ϕl ) < 0 lim λ→+0 λ k,l=1
k,l=1
786
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
there exists a λ0 > 0 such that for all λ, 0 < λ < λ0 , also the LHS of the above equation is negative. This implies the statement of the lemma. Remark 5.4. Provided a quite weak growth condition in n is fulfilled by the {STn }n∈N and the Sn , STn are symmetric, we also have: If {STn }n∈N is conditional reflection positive then {Sn }n∈N0 is reflection infinitely divisible. (see [38]). From now on we want to impose some restrictions on the kernel G. These restrictions are typical for the Green’s functions of a large class of (pseudo) differential operators. Condition 5.5. We want to consider kernels G that are continuous, real functions on Rd \ {0}, and have a singularity at the origin, i.e. lim|x|→0 G(x) = ±∞ and fall to zero as |x| → ∞. Furthermore assume that the mapping f 7→ G ∗ f is well defined and continuous from S(Rd ) to S(Rd ). Finally, G is assumed to be invariant under orthogonal transformations. Remark 5.6. Of course, the Green’s functions of the pseudo differential operators (−∆ + m2 )α , α ∈ (0, 1), fulfill Condition 5.5 (see e.g. the representation we give in Sec. 6). Definition 5.7. Let ϕ ∈ S(Rd+ ). For x ∈ Rd we write (x0 , ~x). Define ϕs (x) = for x0 > 0 and ϕs (x) = 12 ϕ(θx) for x0 < 0. Let ϕa = sign(x0 )ϕs . By q(ϕ) we denote the function 1 2 ϕ(x)
q(ϕ) = G ∗ θϕ G ∗ ϕ = (G ∗ ϕs )2 − (G ∗ ϕa )2 .
(34)
Clearly ϕs , ϕa and q(ϕ) are fast falling functions. For ϕ1 , . . . , ϕn ∈ S(Rd+ ) we get by the θ-invariance of G and the definition of q: Z T S2n (θ(ϕ1 ⊗ · · · ⊗ ϕn ) ⊗ ϕ1 · · · ⊗ ϕn ) = 2c2n q(ϕ1 ) · · · q(ϕn )dx . Rd +
We concentrate on T (θ(ϕ⊗2n ⊗ ϕ2 ) ⊗ ϕ⊗2n ⊗ ϕ2 ) = 2c4n+2 S4n+2 1 1
Z 2n
Rd +
(q(ϕ1 ))
q(ϕ2 )dx
(35)
and try to choose ϕ1 , ϕ2 ∈ S(Rd ) such that the RHS of (35) is smaller than zero, provided c4n+2 > 0. To this aim, we first need to prove some technical lemmas. Here and in the following, B (x), x ∈ Rd , shall denote the open ball of radius around x. Lemma 5.8. Fix x0 > 0 and let x = (x0 , 0) ∈ Rd . Then there exists a function ϕ2 ∈ S(Rd+ ) and an 2 > 0 such that q(ϕ2 ) < 0 on B2 (x). Proof. q(ϕ)(y) is continuous in y for all ϕ ∈ S(Rd+ ). Therefore it suffices to pick a ϕ2 s.t. q(ϕ2 )(x) < 0. The existence of an 2 follows.
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
787
Define continuous linear functionals F ± on S(Rd+ ) by Z G(x ∓ y)f (y)dy . F ± : f 7→ Rd +
F + (y) on Rd+ has a singularity in y = x, but F − (y) has none. It follows that F + and F − are linear independent and so are F s := F + + F − and F a := F + − F − . Consequently there is a function ϕ2 ∈ kernel(F s ), ϕ2 6∈ kernel(F a ). It follows from Eq. (34), that q(ϕ2 )(x) = (F s ϕ2 )2 − (F a ϕ2 )2 = −(F a ϕ2 )2 < 0 .
Lemma 5.9. Let x ∈ Rd+ , x = (x0 , 0) such that G((2x0 , 0)) 6= 0. For every 2 > 0 there exist a function ϕ1 ∈ S(Rd+ ) and an 1 , 2 > 1 > 0, in such a way that |q(ϕ1 )| > 2 on B1 (x) and |q(ϕ1 )| < 12 on Rd+ \ B2 (x). Proof. First we investigate what happens, if we take δx , the Dirac measure with mass one in x, as a “test function”: q(δx )(y) = G(y − Θx)G(y − x) . G is continuous on Rd \ {0} and G((2x0 , 0)) 6= 0. It follows that q(δx ) on Rd+ has a unique singularity in y = x, i.e. lim|x−y|→0 |q(δx )(y)| = ∞, and is continuous on Rd+ \ {y}. Now let δρ,x ∈ S(Rd+ ) be an approximation of δx , i.e. δρ,x → δx for ρ → 0. Then |q(δρ,x )| takes arbitrarily large values in a neighborhood ofx for ρ → 0. At the same time, on Rd+ \ B1 (x), |q(δρ , x)| is bounded uniformly in ρ by a positive so that |q(δρ,x )| > 2D on constant, say D 2 . Choose 1 , 0 < 1 < 2 small enough, − 12 B1 (x) for some small ρ. Fix such a ρ and let ϕ1 = D δρ,x . Then ϕ1 and 1 fulfill the conditions of the lemma. We are now able to construct very sharp maxima for the functions |q(ϕ1 )|2n on arbitrarily small neighborhoods B1 (x) of certain points x. Moreover, we can also achieve that |q(ϕ1 )|2n takes very small values outside a ball B2 (x). Of course, now we want to enforce the “negative spot” of q(ϕ2 ), detected in Lemma 5.8 through multiplication by an adequately chosen |q(ϕ1 )|2n . Lemma 5.10. It is possible to choose ϕ1 , ϕ2 ∈ S(Rd+ ) and n ∈ N, such that the integral on the RHS of (35) takes a negative value. Proof. Fix x = (x0 , 0) as in Lemma 5.9. According to Lemma 5.8 there exists a function ϕ2 ∈ S(Rd+ ) and an 2 > 0 such that q(ϕ2 ) < 0 on B2 (x). Furthermore choose ϕ1 ∈ S(Rd+ ) fulfilling the conditions of Lemma 5.9 with the above fixed 2 , x. Now ! Z Z Z Z Rd +
q(ϕ1 )2n q(ϕ2 )dx =
+ Rd \B2 (x) +
< 2−2n
Z
Rd +
q(ϕ1 )2n q(ϕ2 )dx
+ B2 (x)\B1 (x)
Z
B1 (x)
|q(ϕ2 )|dx − 22n
|q(ϕ2 )|dx B1 (x)
788
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
It is clear that for n large enough, the RHS of this inequality becomes negative. The derivation of this section’s main result is now easy. Proposition 5.11. Let G fulfill Condition 5.5 and ψ be a L´evy-characteristic with non-zero Poisson part and a L´evy measure M , such that all moments of M exist. Let F λ denote the noise determined by λψ, λ > 0, and let X λ = G ∗ F λ in the sense of Sec. 2. Then there exists a λ0 > 0 s.t. for all λ, 0 < λ < λ0 , the Schwinger functions of X λ given by Definition 3.8 are not reflection positive. Proof. Since c4n+2 > 0 for a ψ with M 6= 0, Eq. (35) together with Lemma 5.10 imply that conditional reflection positivity does not hold for the truncated Schwinger functions of X = G ∗ F 1 . Therefore, Proposition 5.11 follows from Lemma 5.3. Remark 5.12. We assume — and in the next section we will present some examples — that at least the 2-point function S2 = S2T of a convoluted generalized white noise with mean zero is reflection positive. Furthermore we may choose the L´evy measure M of ψ symmetric w.r.t the reflection at 0. In this case all Schwinger functions Sn , n odd, vanish (cf. Remark 3.7). We choose λ > 0. By scaling the test functions in the reflection positivity condition ϕ0 ∈ R 7→ ϕ0 ∈ R, ϕk ∈ S((Rd+ )k ) 7→ λk/2 ϕk ∈ S((Rd+ )k ) for k = 1, . . . , n, we get that the sequence of Schwinger functions {Snλ }n∈N0 is reflection positive (fulfills the Osterwalder–Schrader axioms) if and only if the sequence of Schwinger functions {S˜nλ }n∈N0 defined by S˜nλ := λ−n/2 Snλ is reflection positive (fulfills the Osterwalder–Schrader axioms) [56]. Take ϕ1 , . . . , ϕ2n ∈ S(Rd ) and write X
λ S˜2n (ϕ1 ⊗ · · · ⊗ ϕ2n ) =
Y
S2T (ϕj1 ⊗ ϕj2 )
I∈ pairings {j1 ,j2 }∈I
+
n−1 X k=1
λk−n
X (2n)
I∈Pk
Y
SlT (ϕj1 ⊗ · · · ⊗ ϕjl ) .
{j1 ,...,jl }∈I
λ as the 2n-point Schwinger function of a “perFor λ large we may interpret S˜2n turbed” Gaussian reflection–positive random field with covariance function S2T . The “perturbation” is a polynomial without a constant term of degree n − 1 in the “coupling constant” λ−1 . In Proposition 5.11, we have shown that the reflection positivity breaks down if the “coupling constant” λ−1 is larger than a certain thresh−1 is small. We will not old λ−1 0 . It remains an open question what happens if λ study this problem here.
On the first look, Proposition 5.11 may be discouraging. The lack of reflection positivity in the general case leads to some difficulties in the physical interpretation of the model. In the “state space” of the reconstruction theorem in [56] in general
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
789
we may find some “states” with negative norm. A straightforward probabilistic interpretation is therefore difficult in this case, since “negative probabilities” would occur. Nevertheless, even for the case of a large “coupling constant” λ−1 , some of the difficulties can be overcome, at least that is what we hope at the moment: Let us note that in the situation of Proposition 5.11 the obstructions to reflection positivity come from the higher order truncated Schwinger functions. In Secs. 6 and 7 we analytically continue the truncated Schwinger functions “by hand”. The truncated Wightman distributions obtained by this procedure fulfill the spectral condition of QFTs with a “mass-gap”, Poincar´e invariance and locality. From Haag–Ruelle theory (see e.g. p. 317 of [61] Vol. III), we know that such truncated Wightman distributions of order n ≥ 3 do not contribute to the norm of a state approaching the asymptotic resp. scattering region x0 → ±∞, because of the short range of the forces involved (in the case of a QFT with a “mass-gap”). Therefore, if stable oneparticle states exist (take e.g. our model X = Gα ∗F , where α = 12 ) the pseudonorm of the states approaching the scattering region should get positive. (For a precised discussion in a special case, see subsection 7.6.) 6. Analytic Continuation I: Laplace Representation for the Kernel of (−∆ + m20 )−α , α ∈ (0, 1) In this section, we will give a representation of the kernel (−∆ + m20 )−α , for m0 > 0 and α ∈ (0, 1), in terms of a Laplace transform (which is specified later on). In [13], an analytic continuation of the kernel associated with the pseudo differential operator (−∆)−1 (corresponding to the case that m0 = 0 and α = 1) was obtained, the starting point of which was a representation of the kernel of (∆)−1 as a Laplace transform. In [13], based on the same representation of (∆)−1 , an analytic continuation of the (vector) kernel of ∂ −1 was derived, where ∂ is the quaternionic Cauchy–Riemann operator. This section should be regarded as an introduction to the next section, where we extend the methods concerning analytic continuation of Schwinger functions of random fields in [13] and [18]. It is also interesting to extend our approach here to the case of vector kernels including the case of mass m0 > 0. We intend to investigate this problem in forthcoming papers. We notice that the kernel of (−∆ + m20 )−α is given by Gα (x) = (2π)−d
Z Rd
eikx dk + m20 )α
(|k|2
x ∈ Rd
(36)
which is the Fourier transform of a tempered distribution (see Example 2.2). The idea we will realize here is that we represent the integral (36), which is over the conjugate variable k 0 on the real time axis (i.e. , the k 0 –axis in k = (k 0 , ~k) ∈ R × Rd−1 ), by an integral over (a part of) the upper half part of the imaginary axis ik 0 (thus the k 0 –axis being replaced by an imaginary axis) so that the above Fourier transform goes over to a Laplace transform.
790
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
For simplicity of exposition, we assume first that d = 1. We want to evaluate the integral Z ∞ eikx x∈R (37) 2 α dk , 2 −∞ (k + m0 ) by some complex integral. This can be done by a contour integral around a branch point as follows. We denote by log the main branch of the complex logarithm which is holomorphic on C \ (−∞, 0]. Set (iz + m0 )−α = f1 (z) = exp{−α log(iz + m0 )} , (−iz + m0 )−α = f2 (z) = exp{−α log(−iz + m0 )} ,
z ∈ C \ i[m0 , ∞) ; z ∈ C \ i(−∞, −m0 ] .
Clearly, f1 and f2 are holomorphic functions on the indicated domains, respectively. Therefore, for arbitrarily fixed x ∈ R h(z) := eizx f1 (z)f2 (z) ikx
is a holomorphic extension of the function k ∈ R → (k2e+m2 )α ∈ C, which is defined 0 on the real line, to the domain C \ {iy : y ∈ R, |y| > m0 }. Take C ⊂ C \ {iy : y ∈ R, |y| > m0 } as indicated in Fig 1. By the well-known Cauchy integral theorem, we get Z h(z)dz = 0 . C
On the other hand Z Z h(z)dz = C
t
−t
5 Z X eikx dk + h(z)dz , (k 2 + m20 )α j=1 Cj
the curves Cj being as in Fig. 1.
Fig. 1. The closed complex contour C stays inside the domain of holomorphy of the function h. It is composed of the parts C0 , . . . , C5 .
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
791
Thus we have derived Z
t
−t
5 Z X eikx dk = − h(z)dz . (k 2 + m20 )α j=1 Cj
(38)
Moreover, we have the following result Lemma 6.1. For every α ∈ (0, 1) and x > 0, Z
∞
−∞
eikx dk = 2 sin(πα) 2 (k + m20 )α
Z
∞ m0
(r2
e−rx dr . − m20 )α
(39)
Proof. We remark at first that for any fixed x ∈ R Z
∞
−∞
eikx dk = lim 2 t→∞ (k + m20 )α
Z
t
−t
(k 2
eikx dk , + m20 )α
where by Leibniz criterion the right-hand side converges for every α > 0. Thus we should analyse each curve integral on the right-hand side of (38). We shall use polar coordinates for each integral. π (i) Using the polar coordinate representation z = reiβ , r > 0, − 3π 2 < β < 2, C1 = {(r, β) : r = t, 0 ≤ β ≤ β1 }, we have the following derivation for x ≥ 0 and β1 ∈ (0, π2 ) Z β1 iβ ixteiβ iβ iβ h(z)dz = tie e f1 (te )f2 (te )dβ 0 C1
Z
Z ≤ 0
β1
te−xt sin β dβ m2 α t2α e2iβ + t20 Z
= t
β1
1−2α 0
≤ t
1−
1−2α
≤ t =
1−2α
e−xt sin β m0 2 cos 2β + t
m 2 −α Z 0
t m 2 −α 0
t
β1
m0 4 t
i α2 dβ
e−xt sin β dβ
0
1 cos β1
1 − e−xt sin β1 h 2 iα xt2α 1 − mt0 cos β1
→0 as t → ∞.
1−
h 1+2
Z 0
sin β1
e−xtr dr
792
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
π (ii) Analogously, using the representation z = reiβ , r > 0, − 3π 2 < β < 2, π C5 = {(r, β) : r = t, −π − β1 ≤ β ≤ −π}, we have for any arbitrary fixed β1 ∈ (0, 2 )
Z
C5
h(z)dz → 0
as t → ∞. (iii) Using the polar coordinates representation z = im0 + ireiβ , r > 0, 0 ≤ β < 2π, C3 = {(r, β) : r = s, β2 ≤ β ≤ 2π − β2 }, we have for any fixed β2 ∈ (0, π2 ) and s < m0 that Z
Z 2x−β2 i(im0 +iseiβ )x e α isieiβ dβ h(z)dz = − 2 C3 β2 (im0 + iseiβ ) + m20 Z ≤
s1−α e−(m+s cos β)x α dβ (4m20 + 4ms cos β + s2 ) 2
2π−β2
β2
Z ≤ s
2π−β2
1−α β2
=
e−(m−s)x α dβ (2m − s) 2
s1−α (2π − 2β2 )e−(m−s)x α 2(m − s) 2
→0 as s → 0. (iv) Using z = im0 + ireiβ , r > 0, 0 ≤ β < 2π, C4 = {(r, β) : s ≤ r ≤ t1 , 1 β = β2 } with t1 = (t2 + m20 − 2tm sin β1 ) 2 , we have, for β2 ∈ (0, π2 ) and s < 1 < t, the following derivation Z
Z
t1
h(z)dz = C4
iβ2
ei(im0 +ire
)x f (im + ireiβ2 )f (im + ireiβ2 )ieiβ2 dr 1 0 2 0
s
Z =
=
=
=
iβ2 ei(im0 +ire )x ieiβ2 dr [i(im0 + ireiβ2 ) + m0 ]α [−i(im0 + ireiβ2 ) + m0 ]α s Z t1 iβ2 e−(m0 +re )x ieiβ2 dr (−reiβ2 )α (2m0 + reiβ2 )α s Z t1 iβ2 e−(m0 +re )x ieiβ2 dr rα (eiβ2 −π )α (2m0 + reiβ2 )α s Z t1 iβ2 e−(m0 +re )x iπα iβ2 (1−α) ie e dr rα (2m0 + reiβ2 )α s
t1
Z = ie
iπα iβ2 (1−α)
e
s
t1
iβ2 e−(m0 +re )x dr , rα (2m0 + rei β2 )α
(40)
793
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
where we had used the representation −1 = e−iπ in the fourth equality. Now we want to consider the limit of (40) as s → 0 and t1 → ∞. In order to do that, by an application of Lebesgue theorem, we estimate the integrand in (40) as follows: Z Z t1 e−(m0 +reiβ2 )x t1 e−(m0 +r cos β2 )x ≤ dr α dr s rα (2m0 + reiβ2 )α rα |2m0 + reiβ2 | s Z
t1
e−(m0 +r cos β2 )x
α dr rα (r2 + 4m0 r cos β2 + 4m20 ) 2 Z 1 Z t1 e−(m0 +r cos β2 )x + = α dr , 1 s rα (r2 + 4m0 r cos β2 + 4m20 ) 2
=
s
(41) and Z s
1
e−(m0 +r cos β2 )x α dr ≤ rα (r2 + 4m0 r cos β2 + 4m20 ) 2
Z
e−m0 x dr 2 α α 2 s r (r + 4m0 ) 2 Z 1 e−m0 x dr ≤ (2m0 )α s rα 1
=
e−m0 x 1 − s1−α (2m0 )α 1 − α
→
e−m0 x (2m0 )α (1 − α)
(42)
as s → 0. Moreover Z
t1
1
e−(m0 +r cos β2 )x α dr rα (r2 + 4m0 r cos β2 + 4m20 ) 2 Z t1 e−m0 x e−rx cos β2 dr ≤ α (1 + 4m0 cos β2 + 4m20 ) 2 1 =
e−m0 x e−x cos β2 − e−xt1 cos β2 α x cos β2 (1 + 4m0 cos β2 + 4m20 ) 2
→
e−(m0 +cos β2 )x α (1 + 4m0 cos β2 + 4m20 ) 2 x cos β2
(43)
as t1 → ∞ (or, equivalently, t → ∞). Using the above facts (40)–(43), we see that the following limit exists and is given by Z h(z)dz = ie
lim
s→0,t→∞
Z
C4
iπα i(1−α)β2
e
0
∞
e−(m0 +r cos β2 )x−irx sin β2 dr . rα (2m0 + reiβ2 )α
794
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
Now setting β2 → 0, by Lebesgue theorem again, we get Z lim
Z h(z)dz = ieiπα
lim
β2 →0 s→0,t→∞
C4
∞
e−(m0 +r)x dr rα (2m0 + r)α
∞
e−rx dr . (r2 − m20 )α
0
Z = ieαπi
m0
(v) Similarly, we get Z lim
h(z)dz = −ie
lim
β2 →0 s→0,t→∞
C2
−απi
Z
∞ m0
e−rx dr . (r2 − m20 )α
Finally, combining the above steps 1 to 5, we derive from (38) that Z
∞
−∞
Z ∞ απi eikx e−rx −απi dk = − ie − ie 2 2 α dr 2 α 2 (k + m0 ) m0 (r − m0 ) Z ∞ e−rx = 2 sin(πα) 2 α dr . 2 m0 (r − m0 )
Hence we obtain (39).
Now we give the notion of Laplace transform in one variable. The definition of Laplace transform in the multi variable case will be given in Sec. 7. Definition 6.2. Let Md denote the d-dimensional Minkowski space–time with Minkowski inner product h , iM . Choose e0 ∈ Md such that he0 , e0 iM = 1 and let {e0 , . . . , ed−1 } be an orthonormal frame in Md . Let V0+ be the forward light cone, namely, V0+ := {k ∈ Md : k 2 > 0, hk, e0 iM < 0} , and V ∗+ its closure. We recall that Rd+ := {x = (x0 , ~x) ∈ R × Rd−1 : x0 > 0}. 0 0 ~ Notice that for x = (x0 , ~x) ∈ Rd+ , the function e(x, k) := e−x k +i~xk on the forward ∗+ light cone V0 behaves as a fast falling function, i.e., there exists hx ∈ SC (Rd ) such that hx (k) = e(k, x) for k ∈ V0∗+ . Therefore, for a tempered distribution f ∈ S 0 (Rd ) with suppf ⊂ V0∗+ , we can define (Lf )(x) := he(·, x), f i := hhx , f i ,
(44)
which is well defined since by the fact that suppf ⊂ V0∗+ there is no ambiguity arising from the choice of hx . We call Lf the Laplace transform of f ∈ S 0 (Rd ) with suppf ⊂ V0∗+ . Clearly, the Laplace transform of a test function f ∈ S(Rd+ ) is given by the following formula Z 0 0 d ~ e−x k +i~xk f (k)dk , x ∈ Rd+ . (Lf )(x) = (2π)− 2 Rd +
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
795
Sometimes L (e.g. in [13]) called the Fourier–Laplace transform. Our definition here is close to the one given in [67]. The following proposition is the main result of this section, which gives a representation of Gα in terms of a Laplace transform. Proposition 6.3. For α ∈ (0, 1) and x ∈ Rd \ {0}, we have the following formula Gα (x) = 2(2π)−d sin(πα) where 1 Rd−1 :
1
Z
e−|x Rd +
0
1 0 ~ 2 2 1 (k) |k0 +i~ x~ k {k >(|k| +m0 ) 2 } dk , (k 0 2 − |~k|2 − m20 )α
(45)
(k) is the indicator of the subset {k = (k 0 , ~k) ∈ R ×
{k0 >(|~ k|2 +m20 ) 2 } 1 0 k > (|~k|2 + m20 ) 2 }
⊂ Rd+ .
Proof. By (36) and (39), for x ∈ Rd+ , we obtain (45) by Fubini theorem (cf. [38] for details on using Fubini theorem) and the following derivation Z eikx Gα (x) = (2π)−d 2 α dk 2 R (|k| + m0 ) "Z # Z 0 0 ∞ eik x −d i~ k~ x 0 e dk d~k = (2π) 2 k|2 + m20 )α Rd−1 −∞ (k 0 + |~ " # Z ∞ Z −k0 x0 e ~ = (2π)−d eik~x 2 sin(πα) dk 0 d~k 2 1 k|2 − m20 )α Rd−1 (|~ k|2 +m20 ) 2 (k 0 − |~ = 2(2π)−d sin(πα)
Z
e−k Rd +
0
1 0 ~ 2 2 1 (k) x0 +i~ k~ x {k >(|k| +m0 ) 2 } dk . (k 0 2 − |~k|2 − m20 )α
Now formula (45) results from the (full) Euclidean invariance of Gα (see Remark 2.6). 1 By changing variables in (45) as (k 0 , ~k) → ((|~k|2 + m20 ) 2 , ~k), we get a “K¨ allen–Lehmann representation” for Gα (we refer the reader to Theorem IX.33 of [61] or Theorem II.4 of [66] for the K¨ allen–Lehmann representation).
Corollary 6.4. For α ∈ (0, 1) and x ∈ Rd \ {0}, the kernel Gα of (−∆ + m20 )−α has the following representation Z ∞ Gα (x) = Cm (x)ρα (dm20 ) , (46) 0
where −d
Cm (x) = (2π)
Z Rd
e−x
k +i~ k~ x
0 0
2 1{k0 >0} (k)δ(k 0 − |~k|2 − m2 )dk ,
(47)
dm2 . (m2 − m20 )α
(48)
ρα (dm2 ) = 2 sin(πα)1{m2 >m20 }
796
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
Remark 6.5. In general, the K¨ allen–Lehmann representation characterizes the 2-point Schwinger functions of a Lorentz invariant field theory. Comparing our formula (47) with formula (6.2.6) of [37], we see that Cm is a representation of the 2-point Schwinger function (i.e. the free covariance) of the relativistic free field of mass m. From formula (7.2.2) of [37], we have for d ≥ 2 that Cm (x) = (2π)
d 2
m |x|
d−2 2 K d−2 (m|x|) , 2
x ∈ Rd \ {0} ,
(49)
where K d−2 > 0, for x ∈ Rd \ {0}, is the modified Bessel function. Thus Cm is a 2
positive function on Rd \ {0} with a singularity at the origin and exponential decay as m|x| → ∞. This implies that Gα is singular at the origin, as was assumed in Sec. 5. The consequence that Gα is of exponential decay for |x| → ∞ can be shown by using the precise estimates in Proposition 7.2.1 of [37] and the fact ρα has a “mass gap”, i.e., supp ϕα ⊂ [m20 , ∞). (For d = 1, the singularity of Gα is only due to ρα being an infinite measure.) Concerning the truncated Schwinger function S2T associated with Xα = Gα ∗ F , we have the following result, which gives a representation of the integral kernel s of S2T as a Laplace transform, while by formula (2–25) on p. 40 of [67] and translation invariance of s, we have Z Z f1 (x)s(x − y)f2 (y)dxdy , f1 , f2 ∈ S(Rd ) . S2T (f1 ⊗ f2 ) = c2 Rd
Rd
Corollary 6.6. For α ∈ (0, 12 ), −d 2
s(x − y) = 2(2π)
1 sin(π2α)L
1 (k) {k0 >(|~ k|2 +m20 ) 2 }
!
(k 2 − |~k|2 − m20 )2α
(x − y) .
(50)
Proof. By (27), we have S2T (f1 ⊗ f2 ) = c2 h(−∆ + m20 )−α f1 , (−∆ + m20 )−α f2 iL2 = c2 hf1 , (−∆ + m20 )−2α f2 iL2 ,
f1 , f2 ∈ S(Rd ) ,
where c2 is given by (27). Therefore by (36) we get s(x − y) = c2 G2α (x − y) = 2(2π)−d sin(2πα) Z ×
e
−|x0 −y 0 |k0 +i(~ x−~ y)~ k
Rd +
= 2(2π)− 2 sin(2πα)L d
1
1
{k0 >(|~ k|2 +m20 ) 2 }
(k)
dk ((k 0 )2 − |~k|2 − m20 )2α ! 1 0 ~ 2 2 12 (k) {k >(|k| +m0 ) } (x − y) . ((k 0 )2 − |~k|2 − m20 )2α
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
797
Remark 6.7. We should point out that in Sec. 7 we shall give a different derivation of s. By the above formula (47), one can analytically continue s to the kernel, w say, of the 2-point (truncated) Wightman distribution W2T = w(x − y) with ! 1 0 ~ 2 2 12 (k) {k >(|k| +m0 ) } −d −1 2 (x − y) . w(x − y) := 2(2π) c2 sin(2πα)F ((k 0 )2 − |~k|2 − m20 )2α Remark 6.8. Concerning the Schwinger function S2 and the 2-point Wightman distribution for the case α = 12 , see the discussion Sec. II.5 of [66]. Remark 6.9. Suppose F is a Gaussian white noise, then Xα = (−∆ + m20 )−α F is a generalized free field for each α ∈ (0, 12 ) and m0 > 0. We refer the reader to e.g. [42] and [66] for the notion of generalized free field. Our argument here is as follows. In this case, all Wightman distributions Wn , n ∈ N, are given symbolically by ( 0, n is odd Wn (x1 , . . . , xn ) = X T T n is even W2 (xj1 , xl1 ) · · · W2 (xj n , xl n ) , 2
2
where the sum is over {(j1 , . . . , j n2 , l1 , . . . , l n2 ) ∈ N : 1 ≤ j1 < j2 < · · · < j n2 < n and jk < lk for 1 ≤ k < n2 }. This shows that all Wn , n ∈ N, are determined by W2T . The corresponding Schwinger function Sn , n ∈ N, satisfy reflection positivity since allen–Lehmann representation, therefore the field Xα by Corollary 6.4, W2T has a K¨ is a generalized free Euclidean field. Especially Xα is reflection positive in the sense of [37]. n
Remark 6.10. Finally in this section, we should point out that concerning the kernel G1 of (−∆ + m20 )−1 , i.e., α = 1, the above procedure cannot be performed. One can apply the residue theorem instead of Cauchy integral theorem to get a Laplace transform representation of G1 . We therefore have a representation of G1 by using the following basic formula (see e.g. (II.3) of [40] or (II.4) of [66]): Z
∞
−∞
2πe−r|s| eits , dt = t2 + r 2 r
In fact, we have the following derivation Z eikx G1 (x) = (2π)−d 2 dk 2 Rd |k| + m0 Z ∞ Z −d i~ k~ x e = (2π) Rd−1
= (2π)−d+1
Z Rd−1
−∞
e
r > s ∈ (−∞, ∞) .
0
0
eik x dk 0 k 0 2 + |~k|2 + m2 0
1 −(|~ k|2 +m20 ) 2 |x0 |+i~ k~ x 1 2(|~k|2 + m20 ) 2
dk .
! d~k
798
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
The above integral representation of G1 in case m0 = 0 was used in [18] (see the corresponding formula (6.6) in Minkowski space in [18]) as a starting point for the analytic continuation of the Schwinger functions. 7. Analytic Continuation II: Continuation of the (Truncated) Schwinger Functions 7.1. Preliminary remarks In this section we present the analytic continuation of the Euclidean (truncated) Schwinger functions obtained from convoluted generalized white noise X = (−∆ + m20 )−α F , m0 > 0, α ∈ (0, 12 ], to relativistic (truncated) Wightman distributions. More generally, formal expressions are obtained for the class of random fields X given by X = G ∗ F , where G is the Laplace–transform of a Lorentz– invariant signed measure on the forward lightcone. But as we will illustrate for X = (−∆ + m20 )−α F, α ∈ ( 12 , 1), the question whether such relativistic expressions really represent the analytic continuation of the corresponding (truncated) Schwinger functions, deserves a separate discussion. We derive manifestly Poincar´e invariant formulas for the Fourier transform of the (truncated) Wightman functions. The obtained Wightman distributions fulfil the strong spectral condition of a QFT with a “mass-gap”. Locality, Hermiticity and the cluster property of the (truncated) Wightman functions follow from the (truncated) Schwinger functions’ symmetry, θ -invariance and cluster property respectively, as a result of the general procedures of axiomatic QFT. We continue the discussion, 1 started at the end of Sec. 5, for the special case X = (−∆ + m20 )− 2 F , d = 4. To the readers convenience (and to keep the length of the paper within reasonable bounds) not every step is presented with all details (for complete proofs see [38]). The methods applied here are based on and extend those of [13], [18]. 7.2. The mathematical background Let us first clarify the notations and the mathematical background. In this subsection n is a fixed integer with n ≥ 2. From Definition 6.2 recall the meaning of Md , h , iM and let {e0 , . . . , ed−1 } be an orthonormal frame in Md , by which Md is identified with R × Rd−1 ∼ = Rd . From now on we write for x ∈ Md , x = (x0 , ~x) 2 02 and x = hx, xiM = x − |~x|2 . The forward mass cone of mass m0 > 0 is defined in analogy with the forward light cone V0+ as Vm+0 := {k ∈ Md : k 2 > m20 , hk, e0 iM < 0} ,
m0 ≥ 0 .
(51)
we denote its closure. Let θ again denote the time reflection. The By Vm∗+ 0 (∗)− (∗)− backward (closed) mass cone/lightcone is defined by Vm0 \0 := θVm0 \0 . Since Md ∼ = Rd ⊂ Cd there is a complexification of the Minkowski inner product h , iM such that it is analytic in the coordinates with respect to {e0 , . . . , ed−1 }. We denote this − n d n complexification by h , iC M . Let T be the tubular domain in (C ) with base V0 , i.e. n o n (52) T n := z = (z1 , . . . , zn ) ∈ (Cd ) : zj − zj+1 ∈ Md + iV0− , 1 ≤ j ≤ n − 1
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
T n is called the backward tube. Finally define ) ( n X − dn C e(k, z) := (2π) 2 exp i < kl , zl >M ,
799
(53)
l=1 n
where k = (k1 , . . . , kn ) ∈ (Rd )n and z = (z1 , . . . , zn ) ∈ (Cd ) . We now give the definition of spectral conditions that are crucial for the theory of the Laplace transforms in n arguments as well as for the notion of causality in a relativistic QFT. ˆ fulfils the spectral ˆ ∈ S 0 (Rdn ). We say that W Definition 7.1. (i) Let W c d(n−1) ˆ ∈ SC (R ) such that condition, if there is a w ! n X ˆ 1 , . . . , kn ) = w ˆ (k1 , k1 + k2 , . . . , k1 + · · · + kn−1 )δ (54) W(k kl l=1 n−1
ˆ ⊂ (V0∗− ) and supp w . ˆ T} be a sequence of truncated distributions determined by a se(ii) Let {W l l∈N ˆ l }l∈N , W ˆ l ∈ SC (Rdl ), W ˆ 0 = 1. We say that the {W ˆ T} quence of distributions {W 0 l l∈N ˆ l} (the {W respectively) fulfil the strong spectral condition with a mass gap l∈N0 T ˆ m0 > 0 if all the distributions Wl , l ≥ 2, fulfil the spectral condition (54), where . V0∗− is replaced by Vm∗− 0 The following two theorems, taken from the theory of Laplace transforms, provide us with the necessary mathematical tools for the analytic continuation of Schwinger functions. ˆ ∈ SC (Rdn ) fulfils the spectral condition. Then Theorem 7.2. Assume that W ˆ e(·, z)i is well-defined and holomorphic in the variables ˆ = hW, (i) L(W)(z) ˆ is called the Laplace zj − zj+1 , j = 1, . . . , n − 1 on the domain z ∈ T n . L(W) ˆ transform of W. ˆ ˆ is the boundary–value of L(W)(z) for =(zj − zj+1 ) → 0 inside (ii) F −1 (W)(
=(zj −zj+1 )∈Γ→0
ˆ ˆ L(W)(z) = F −1 (W)(
holds in the sense of tempered distributions in the argument
800
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
leads to the interchange of forward and backward cones.) Theorem 7.2 is essentially equal to Theorem 2.6 and Theorem 2.9 of [67]. Let us remark that most textbooks work with other conventions on the Laplace transform as we do here, since here we mostly use the variables zj and not the difference variables ζj := zj − zj+1 . Lastly, let us remark that for distributions that depend only on one variable k ∈ Rd we keep the conventions of Sec. 6. We observe that n n n 0 (Ed )< := z ∈ (MdC ) : =(zj0 − zj+1 ) < 0 , j = 1, . . . , n − 1 , o =~zj = 0 ,
(57)
if xj − xj+1 is space-like, i.e. (xj − xj+1 )2 < 0. Furthermore, if S is a real distribution, θ–invariance of S implies Hermiticity of W, i.e. W(x1 , . . . , xn ) = W(xn , . . . , x1 ) .
(58)
Theorem 7.3 is part of the reconstruction theorem in [56] and [66]. We remark ˆ possesses a single-valued holothat for symmetric and Euclidean invariant S, L(W) n morphic extension to the so-called permuted extended tube Tp,e (see [67, 47] for definitions and proofs). Our restriction of Theorem 7.3 to tempered distributions S is motivated only by our model, where “time–coincident” Schwinger functions are 0 dn well defined. In [56] Theorem 7.3 is proved for S ∈ S6= ,C (R ), which is a larger 0 dn class of distributions than SC (R ). Remark 7.4. Let yl = (yl0 , ~yl ) = (=zl0 , <~zl ). Then (56) reads S(y1 , . . . , yn ) = (2π)−
dn 2
Z Rdn n
e−
Pn l=1
kl0 yl0 +i~ kl ~ yl
ˆ 1 , . . . , kn ) W(k
n O l=1
where y = (y1 , . . . , yn ) ∈ (Rd )< := {x ∈ Rdn : x01 < x02 < · · · < x0n }.
dkl
(59)
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
801
7.3. Truncated Schwinger functions with “sharp masses” Our goal in this subsection is to represent the truncated Schwinger functions SnT , n ≥ 2, of a convoluted generalized white noise as the restriction to the Euclidean n ˆ nT having the region (Ed )< of a Laplace transform of a tempered distribution W spectral property. In [13] and [18] this is done for the truncated Schwinger functions of X = ∆−1 F . The kernel ∆−1 (x) can be represented as the Laplace transform of the essentially (up to a multiplication with a positive constant) unique Lorentz invariant measure on the forward mass “hyperboloid” with mass m = 0. Given that, the crucial step in the analytic continuation is a change of variables, depending on the value of m. Since Sec. 6 shows that we have to deal with a continuum of masses rather than with a sharp mass, we proceed in three steps: First in this subsection we give an integral representation of SnT in terms of “truncated Schwinger functions with T , m = (m1 , . . . , mn ). Then the subsection 7.4 deals with the sharp masses” Sm,n T T ˆ m,n as the Laplace transform of a tempered distribution W representation of Sm,n n T ˆ m,n over the masses to obtain the Fourier restricted to (Ed ) . Finally, we integrate W <
transform of the truncated n-point Wightman function WnT (subsection 7.5). In that subsection we also collect the properties of the obtained (truncated) Wightman distributions arising from the main theorem of the present section. From now on, we restrict ourselves to convoluted generalized white noises X = G ∗ F with a kernel G which admits a representation of the form Z G(x) = R+
Cm (x)ρ(dm2 ) ,
x ∈ Rd \ {0} ,
(60)
where ρ is a (possibly signed) Borel measure on R+ . Furthermore we restrict ourselves to ρ’s which fulfill the following Condition 7.5. (i) There exists a mass m0 > 0, such that suppρ ⊂ [m20 , ∞). R (ii) R+ m12 |ρ|(dm2 ) < ∞. Remark 7.6. (i) Since Cm is the Laplace transform of the Lorentz invariant d d + (k) := (2π)− 2 1{k0 >0} (k)δ(k 2 − m2 ) in the sense of Sec. 6, distribution (2π)− 2 δm (60) and Condition 7.5 mean that G is the Laplace transform of a signed Lorentz . invariant measure on the forward mass-cone Vm∗+ 0 (ii) Condition 7.5 is also a sufficient condition for G : f ∈ S(Rd ) 7→ G∗f ∈ S(Rd ) to be well-defined and continuous. The proof of this statement can be verified using similar techniques as in the proof of Lemma 7.7 [38]. (iii) The ρ’s obtained in Sec. 6 obviously fulfil Condition 7.5. Lemma 7.7. Let ρ and G be as above and SnT as in (31). Then for ϕ ∈ S(Rdn ) Z hSnT , ϕi
= cn
(R+ )n
T hSm,n , ϕiρ(dm2 )
(61)
802
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
T where m = (m1 , . . . , mn ) and ρ(dm2 ) = ρ⊗n (dm21 × · · · × dm2n ). Sm,n is defined by
Z T hSm,n , ϕ1 ⊗ · · · ⊗ ϕn i :=
Rd
Cm1 ∗ ϕ1 · · · Cmn ∗ ϕn dx ,
ϕ1 , . . . , ϕn ∈ S(Rd ) . (62)
T Proof. By the nuclear theorem, (62) well-defines Sm,n ∈ S 0 (Rdn ). To prove (61) let again ϕ = ϕ1 ⊗ · · · ⊗ ϕn , with ϕ1 , . . . , ϕn ∈ S(Rd ). We remark that Cm (x) = md−2 C1 (mx) for x ∈ Rd \ {0} and C1 ∈ L1 (Rd , dx) (cf. [37, p. 126]). Therefore n Z Z Z Y Cml (yl )ϕl (x − yl ) dy1 · · · dyn dx|ρ|(dm2 ) n + d dn (R ) R R l=1 Z Z Z n n Y O yl0 0 |ρ| 0 (dm2l ) = C1 (yl )ϕl x − dy1 · · · dyn0 dx ml m2l (R+ )n Rd Rdn l=1
≤ kC1 kL1 (Rd ,dx)
Z R+
l=1
n n−1 Y 1 2 |ρ|(dm ) kϕl kL∞ (Rd ,dx) kϕn kL1 (Rd ,dx) m2 l=1
<∞
(63)
where we have applied Z Y n n−1 Y kϕl kL∞ (Rd ,dx) kϕn kL1 (Rd ,dx) , ϕl (x − zl ) dx ≤ Rd l=1
z 1 , . . . , z n ∈ Rd .
l=1
Now (63) allows us to apply Fubini’s theorem to the LHS of (61). The LHS and the RHS of (61) are therefore equal for ϕ = ϕ1 ⊗ · · · ⊗ ϕn . At the same time (63) shows, by the nuclear theorem, that both sides of (61) denote tempered distributions. These give, by the above argument, equal values if evaluated on test functions ϕ = ϕ1 ⊗ · · · ⊗ ϕn . Therefore, by the nuclear theorem again, the distributions on both sides of (61) are equal. 7.4. The Schwinger functions with “sharp masses” as Laplace transforms n
n
From now on we assume y = (y1 , . . . , yn ) ∈ (Rd )< , m = (m1 , . . . , mn ) ∈ (R+ ) . Then Z T (y1 , . . . , yn ) = Cm1 (x − y1 ) · · · Cmn (x − yn )dx Sm,n Rd
is well defined as a function, since the singularities of the functions Cml (x − yl ) are all separated from each other and are therefore all integrable and, furthermore, for large x the integrand falls off exponentially to zero (cf. [37, p. 126]). Taking into account that Z 0 0 0 ~ + e−kl |x −yl |+ikl (~x−~yl ) δm (kl )dkl , Cml (x − yl ) = (2π)−d l Rd
803
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
we get by Fubini’s theorem "Z n Y
Z T (y1 , . . . , yn ) Sm,n
d−1−dn
= (2π)
Rd
×e
−i
Pn l=1
~ kl ~ yl
# e
−kl0 |x0 −yl0 |
dx
R l=1 n Y
n X
+ δm (kl )δ l
l=1
0
! ~kl
l=1
where we have also applied the distributional identity F
√
SC0 (Rd−1 ).
n O
dkl
(64)
l=1
1 d−1 2π
= δ in the dis-
tribution space The RHS of (64) has to be considered as an integral over a submanifold of Rdn determined by the δ-distributions. If not stated otherwise, all products of distributions which occur in the following are defined in this way. The expression in the brackets [· · · ] in (64) equals n Y
1 Pn
0 l=1 kl
Z
n−1 X j−1 Y
e−kl (yl −y1 ) + 0
0
0
1
e
0
0
0
j=1 l=1
l=2
×
0 e−kl (yj −yl ) (yj+1 − yj0 )
−[(
Pj l=1
kl0 )s−(
Pn l=j+1
0 kl0 )(1−s)](yj+1 −yj0 )
ds
0 n Y
×
n−1 Y
0 0 0 1 e−kl (yl −yj+1 ) + Pn
0 l=1 kl
l=j+2
e−kl (yn −yl ) . 0
0
0
(65)
l=1
If we insert (65) into (64), then the RHS of (64) splits up into n + 1 summands, say T (y1 , . . . , yn ) = I0 (y1 , . . . , yn ) + Sm,n
n−1 X
0 (yj+1 − yj0 )Ij (y1 , . . . , yn ) + In (y1 , . . . , yn ) .
j=1
(66) We are going to write each of these summands in the form of (59). This is easy for I0 and In : Z
n Y
d−1−dn
I0 (y1 , . . . , yn ) = (2π)
Rdn n Y
1
× Pn
l=0
kl0
e−kl yl −ikl ~yl e−(− 0 0
~
l=2
+ δm (kl )δ l
l=1
n X l=1
Z
n Y
= (2π)d−1−dn
! ~kl
n O
×
ω1 +
n Y
0 l=2 kl l=2
+ δm (kl )δ l
l=2
kl0 )y10 −i~ k1 ~ y1
dkl
l=1
e−kl yl −ikl ~yl e−(− 0 0
Rd(n−1) ×Rd−1 l=2
1 Pn
Pn
n X l=1
~
! ~kl
n O l=2
Pn
dkl ⊗
l=1
kl0 )y10 −i~ k1 ~ y1
d~k1 2ω1
804
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
Z
e−
d−1−dn
= (2π)
Pn l=1
Rdn
×
n Y
n X
+ δm (kl )δ l
l=2
! kl
kl0 yl0 +i~ kl ~ yl
n O
l=1
1 2ω1 (ω1 − k10 )
dkl ,
l=1
q where we have introduced ωl = ~kl2 + m2l , l = 1, . . . , n. Thus, I0 is the Laplace transform of the tempered distribution ! n n Y X 1 d−1− dn + 2 (67) δml (kl )δ kl (2π) 2ω1 (ω1 − k10 ) l=2
l=1
n (Ed )< .
By an analogous calculation we find that In is the Laplace restricted to transform of the tempered distribution ! n−1 n Y X 1 d−1− dn − 2 (68) δml (kl )δ kl (2π) 2ωn (ωn + kn0 ) l=1
− δm l
(Ed )n< ,
l=1
+ θδm . l
where is defined as restricted to To check the temperedness of the distributions (67) and (68) we observe that for k = (k1 , . . . , kn ) in the support of these distributions the denominators are larger than 4m20 , where m0 ≤ m1 , . . . , mn . The spectral condition can be directly deduced from these formulas (cf. the proof of Proposition 7.8 below). Let us now turn to the more complicated calculations for the Ij ’s, j = 1, . . . , n−1. Z Ij (y1 , . . . , yn ) = (2π)d−1−dn Z
1
×
e 0
×e ×
−[(
[(
Rdn
Pj l=1
Pj l=1
n Y
j−1 Y
0 0
l=1
kl0 )s+(
kl0 )s+(
Pn l=j+1
Pn
~
l=j+1
e−kl yl −ikl ~yl 0 0
~
ekl yl −ikl ~yl
l=j+2
n Y
kl0 )(1−s)−(
kl0 )(1−s)−(
Pj−1 l=1
Pn l=j+2
n X
+ δm (kl )δ l
l=1
0 kl0 )]yj+1 −i~ kj+1 ~ yj+1
!
~kl
l=1
kl0 )]yj0 −i~ kj ~ yj
n O
ds
dkl .
l=1
Nn We may interchange ds and l=1 dkl integrations by Fubini’s theorem. Fur+ + 0 (ki+1 )dkj0 dkj+1 and change coordinates thermore, we integrate over δmj (kj )δm j+1 0 0 kl 7→ −kl for l = 1, . . . , j − 1, getting the RHS to be equal to Z
1
Z
j−1 Y
(2π)d−1−dn
×
˜0
×
l=1
n Y
~
e−kl yl −ikl ~yl 0
l=j j−1 Y
0 0
Rd(n−2) ×R(d−1)2 l=1
0 j+1 Y
~
e−kl yl −ikl ~yl
~
e−kl yl −ikl ~yl 0 0
l=j+1 − δm (kl ) l
n Y l=j+2
+ δm (kl ) l
n O l=1,l6=j,j+1
j+1 O d~kl dkl ds , 2ωl l=j
805
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
where we have used the following notations 0 0 , ~kj , ~kj+1 , kj+1 , . . . , kn0 ) k˜j0 = k˜j0 (k10 , . . . , kj−1 ! ! j−1 j−1 n X X X − kl0 + ωj s + ωj+1 + kl0 (1 − s) + kl0 ; := − l=1
l=j+2
l=1
0 0 0 0 = k˜j+1 (k10 , . . . , kj−1 , ~kj , ~kj+1 , kj+2 , . . . , kn0 ) k˜j+1 ! ! j−1 n n X X X − kl0 + ωj s + ωj+1 + kl0 (1 − s) − kl0 . := l=1
l=j+2
l=j+2
Pn Pj−1 0 ˜ 0 ˜ 0 0 We remark that l=1 kl + kj + kj+1 + l=j+2 kl = 0. Therefore we may 0 using the measure introduce new “integrations” over new variables kj0 , kj+1 0 0 δ(kj0 − k˜j0 (k10 , . . . , kj−1 , ~kj , ~kj+1 , kj+2 , . . . , kn0 ))δ
= δ(a(kj , kj+1 )s − b(kj+1 ))δ
n X
! kl0
0 dkj0 dkj+1
l=1
! kl0
n X
0 dkj0 dkj+1 ,
l=1
where 0 + ωj+1 a(kj , kj+1 ) = −kj0 − ωj − kj+1 0 b(kj+1 ) = −kj+1 + ωj+1 .
In this way we get Z Ij (y1 , . . . , yn ) = (2π)d−1−dn 0
1
Z Rdn
n Y
~
e−kl yl −ikl ~yl 0 0
l=1
δ(a(kj , kj+1 )s − b(kj+1 )) × 4ωj ωj+1
j−1 Y
− δm (kl ) l
l=1 n Y
+ δm (kl )δ l
l=j+2
n X l=1
! kl
n O
dkl ds .
l=1
(69) For n ≥ 3 or n = 2, m1 6= m2 , a(kj , kj+1 ) 6= 0 holds almost everywhere with Qn Pn Nn Qj−1 − + (kl ) l=j+2 δm (kl )δ ( l=1 kl ) l=1 dkl . In these respect to the measure l=1 δm l l Nn cases we may, by Fubini’s theorem again, change the order of the ds and l=1 dkl integrations. This together with Z
1
δ(as − b)ds = 0
1 1{0
for a 6= 0
inserted into (69), allows us to conclude that Ij is the Laplace transform of the distribution
806
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
Hj (k1 , . . . , kn )
1{0
d−1− dn 2
l=1
l=j+2
l=1
n
restricted to (Ed )< . The temperedness of Hj can be derived from an integral representation like that in (69), where the exponential functions have to be replaced by a test function ϕ ∈ S(Rd ): d−1− dn 2
|hHj , ϕi| ≤ (2π)
n Y l=1
Z ×
R(d−1)(n−1)
!−1 ml
! 1 d~k kϕk0,dn , (1 + |~k|2 )dn/2
(71)
where kϕk0,dn := supk∈Rdn |(1 + |k|2 )dn/2 ϕ(k)|. 0 Since Ij (y1 , . . . , yn ) is the Laplace transform of Hj (k1 , . . . , kn ), (yj+1 − yj0 )Ij 0 (y1 , . . . , yn ) is the Laplace transform of the tempered distribution ((∂j+1 − ∂j0 )Hj 0 (k1 , . . . , kn )) where ∂l0 = ∂k∂ 0 , l = 1, . . . , n. Terms that depend only on kj0 + kj+1 l
0 give a zero contribution when derived with respect to ∂j+1 − ∂j0 . This applies to Pn a(kj , kj+1 ) and δ ( l=1 kl ). Thus, only the derivatives of the characteristic functions 0 − ∂j0 )Hj . in (70) contribute to (∂j+1 Taking into account 0 1{0−ωj } 1{kj+1 >ωj+1 } ,
d dx 1{0<x} (y)
± = δ(y) and also (2ωl )−1 δ(kl0 ∓ ωl ) = δm (kl ), we calculate l
1 (∂ 0 − ∂j0 )1{0
×
1 (∂ 0 − ∂j0 )1{a(kj ,kj+1 )
+ = (2ωj )−1 1{kj0 >−ωj } (kj )δm (kj+1 ) j+1 − 0 (kj )(2ωj+1 )−1 1{kj+1 + δm <ωj+1 } (kj+1 ) j
(72)
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
807
Adding up both sides of (72) yields 0 (∂j+1 − ∂j0 )Hj (k1 , . . . , kn ) ) ( + − δm sign(kj0 + ωj )δm (kj+1 ) (kj ) sign(ωj+1 − kj0 ) j+1 j d−1− dn 2 + = (2π) 0 0 2ωj |kj0 + ωj + kj+1 − ωj+1 | 2ωj+1 |kj0 + ωj + kj+1 − ωj+1 | ! j−1 n n Y Y X − + δm (k ) δ (k )δ kl × l l m l l l=1
l=j+2
d−1− dn 2
= (2π)
j−1 Y
(
− δm (kl ) l
l=1
×
n Y
+ δm (kl )δ l
l=j+2
n X
kl
l=1 + (kj+1 ) δm j+1
2ωj (kj0 + ωj ) !
+
)
− δm (kj ) j 0 ) 2ωj+1 (ωj+1 − kj+1
.
(73)
l=1
A closer analysis shows that the singularities on the RHS of (73) have to be understood in the sense of Cauchy’s principal value. Keeping in mind that q 1 (−1) 1 ~k 2 + m2 , + = , ω = 2ω(ω + k 0 ) 2ω(ω − k 0 ) k 2 − m2 by adding up (67), (73) for j = 1, . . . , n − 1 and (68) (recall also (66)) we get the following T ˆ m,n denote the distriProposition 7.8. For m = (m1 , . . . , mn ) ∈ (R+ )n , let W bution ! n j−1 n n X Y Y X (−1) dn d−1− 2 − + (74) (2π) δml (kl ) 2 δml (kl ) δ kl kj − m2j j=1 l=1
l=j+2
l=1
for n ≥ 3 or n = 2, m1 6= m2 and − − (2π)−1 (2ω1 )−2 δm (k1 ) − (2ω1 )−1 (∂10 δm )(k1 ) δ(k1 + k2 ) 1 1
(75)
for n = 2, m1 = m2 . Then T ˆ m,n is tempered and fulfils the strong spectral condition with the mass gap (i) W m0 ≤ min{ml : l = 1, . . . , n} . n T T ˆ m,n is the restriction of L(W ) to the Euclidean region (Ed )< in the (ii) Sm,n T ˆ m,n uniquely. sense of (56). This property determines W T −1 ˆ T (iii) Wm,n = F (Wm,n ) is Poincar´e invariant.
Proof. We only deal with the case n ≥ 3 or n = 2 m1 6= m2 . Concerning the support properties, let us concentrate on the j’th summand in (74). Let k =
808
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
Pr (k1 , . . . , kn ) be in the support of this summand. For r < j, l=1 kl ∈ Vm∗− holds, P0r since each kl , l = 1, . . . , r, is in this cone. For n − 1 ≥ r ≥ j we get l=1 kl = Pn , since each kl , l = r + 1, . . . , n, is in V0∗+ and thus −kl ∈ V0∗− . − l=r+1 kl ∈ Vm∗− 0 ˆ T is tempered, and that S T is the restriction of L(W T ) The facts that W m,n m,n m,n n to (Edn )< summarizes the above discussion. We have worked in the notations of (59) rather than in that of (56), but, as already remarked, these two relations are equivalent. The other statements follow from Theorem 7.3. The derivation of (75) is relatively easy. In the following we will not use this formula and therefore leave the calculation as an exercise. Nevertheless, the formula (75) has some consequences: If we take a noise F with mean zero and X = (−∆ + m21 )−1 F = (2π)Cm1 ∗ F , then it is easy to see that (75) gives the Fourier transform of the 2-point Wightman function of the model. Since the distribution in (75) does not admit a K¨ allen–Lehmann representation, not all of the one-particle and free “states” of this model have positive norm. Thus, a good physical interpretation of such models is impossible, even if F has zero Poisson part. Two more details may be of interest: Remark 7.9. 1. Let L↑+ (Rd ) denote the proper orthochronous Lorentz group. For Λ ∈ L↑+ (Rd ) and a ∈ Rd , we recall that the Poincar´e group acts on functions ϕ(k1 , . . . , kn ) defined on the momentum space (Rd )n as follows Pn (T{Λ,a} ϕ)(k1 , . . . , kn ) = ϕ((Λ∗ )−1 k1 , . . . , (Λ∗ )−1 kn )eih l=1 kl ,aiM where the adjoint Λ∗ is w.r.t. h , iM , the Minkowski inner product. From (74) we ˆ T are “manifestly” Poincar´e invariant. know that the W m,n T to be a “local” distribution 2. Only if m1 = m2 = . . . = mn we can expect Wm,n (cf. Theorem 7.3). 7.5. The analytic continuation of the truncated Schwinger functions Proposition 7.8 immediately implies ˆ T := cn Theorem 7.10. Suppose that the distribution W n
R (R+ )n
ˆ T ρ(dm2 ), W m,n
Z
i.e. ˆ nT , ϕi = cn hW
(R+ )n
T ˆ m,n hW , ϕiρ(dm2 ) ϕ ∈ S(Rdn )
(76)
ˆ nT ∈ S 0 (Rdn ). Then is well-defined for all ϕ ∈ S(Rdn ) and furthermore W ˆ T fulfils the strong spectral condition with respect to the mass gap m0 , where (i) W n m0 is as in Condition 7.5. ˆ T ) to the Euclidean re(ii) SnT is the restriction of the Laplace transform L(W n n T ˆ n uniquely. Furthermore WnT = F −1 (W ˆ nT ) is the gion (Ed )< . This determines W ˆ T )(z), z ∈ T n , for =z → 0 as described boundary-value of the analytic function L(W n in Theorem 7.2. In this sense, we call the truncated n-point Wightman distribution WnT the analytic continuation of SnT to the Minkowski space–time.
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
809
(iii) WnT is a Poincar´e invariant, local, hermitian distribution, which fulfils, in addition, the cluster property of the truncated Wightman distributions, i.e. for ϕ1 , . . . , ϕn+m ∈ S(Rdn ) and a spacelike a ∈ Rd (a2 < 0) we have T (ϕ1 ⊗ · · · ϕm ⊗ Tλa (ϕm+1 ⊗ · · · ⊗ ϕm+n )) → 0 , Wm+n
if λ → ∞ ,
(77)
where Tλa is the translation by λa. ˆ T , m = (m1 , . . . , mn ), ml ∈ supp ρ Proof. (i) By Proposition 7.8 all the W m,n l = 1, . . . , n, fulfil the strong spectral condition with the mass gap m0 > 0 given by ˆ T , the same applies to W ˆ T. ˆ T is a superposition of such W Condition 7.5. Since W n n Pn m,n0 0 ~ − kl yl +ikl ~ yl − dn 2 l=1 (ii) As remarked before, e˜(k, y) := (2π) e on the support of ˆ T behaves like a fast falling function in k ∈ Rdn , whenever y = (y1 , . . . , yn ) ∈ W n (Rd )n< . Therefore, the following equations hold: Z ˆ T , e˜(·, y)iρ(dm2 ) ˆ T , e˜(·, y)i = cn hW hW n m,n Z
(R+ )m
= cn (R+ )m
T Sm,n (y1 , . . . , yn )ρ(dm2 )
= SnT (y1 , . . . , yn ) . The second equality is valid by Proposition 7.8, where the RHS makes sense dy a.e. by Lemma 7.3, which also implies the third equality. Theorems 7.2 and 7.3 now imply 2. (iii) Except for the cluster property, everything follows from (ii) and Theorem 7.3. For the cluster property, we refer to [56], Theorem 4.5 and Corollary 4.7 (see alternatively [61] Vol. III p. 324). Corollary 7.11. Let {Wn }n∈N0 be the Wightman distributions determined by ˆ n fulfils the spectral condition, the truncated sequence {WnT }n∈N and W0 = 1. Then W for n ∈ N. The statements 1, 2 and 3 of Theorem 7.10 hold, if the WnT , SnT are replaced by Wn , Sn , respectively and the cluster property of the truncated Wightman function is replaced by that for the Wightman functions (see [67] or [56]). For a proof of Corollary 7.11 we refer to similar discussions in Sec. 4 and to [22]. Let us now turn to the question of the temperedness of the formal expressions (76). It is for example not difficult to see that for ρ’s that have compact support in R+ , temperedness follows (c.f. (71)). Nevertheless, in these cases we cannot expect the two-point Wightman function to admit a K¨ allen–Lehmann representation. The reader is asked to convince herself/himself that there is no such representation e.g. for the case ρ(dm2 ) = f (m2 )dm2 , f > 0, where f ∈ S(R) has compact support in R+ . In this case again, we cannot give a good physical interpretation, even not for one-particle or free states. Therefore we restrict ourselves to the ρ’s obtained in Sec. 6, i.e. ρα (dm2 ) = 2 sin πα1{m2 >m20 } (m2 )
dm2 (m2 − m20 )α
α ∈ (0, 1) .
(78)
810
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
ˆ T , defined by Again it is possible to show the temperedness of the distributions W n,α (76) with ρ = ρα if α ∈ (1/2, 1) by a direct estimate, using (71). Let us therefore turn to the case α ∈ (0, 12 ]. We introduce the notations −d/2 sin πα1{k2 >m20 ,k0 >0} (k) µ+ α (k) = (2π) −d/2 µ− sin πα1{k2 >m20 ,k0 <0} (k) α (k) = (2π)
(k 2
1 − m20 )α
1 (k 2 − m20 )α
µα (k) = (2π)−d/2 cos πα1{k2 >m20 } (k) + 1{k2 <m20 } (k)
1 |k 2 − m20 |α
and we get: T ˆ n,α be defined as in (76) with ρ = ρα , α ∈ (0, 12 ). Proposition 7.12. Let W T ˆ n,α Then W is tempered and equal to
d n−1
cn (2π) 2
n j−1 X Y
n Y
µ− α (kl )µα (kj )
j=1 l=1
l=j+1
µ+ (k ) δ α j
n X
! kl
n ≥ 2.
(79)
l=1
Proof. Despite the fact that one cannot apply Fubini’s theorem because of the presence of the Cauchy principal values in (74), it can be shown by a regularization ˆ T ) and a passage to the limit, that the integration w.r.t. (of the ρα and the W m,n ρα (dm) and the evaluation with a test function ϕ ∈ S(Rdn ) can be interchanged in ˆT : (76). Therefore, we get for W n,α (
− δm (kl )dm2l l 2 2 α m20 (ml − m0 ) j=1 l=1 ! n ) X Z ∞ − n ρα (dm2j ) Y δml (kl )dm2l sin πα δ kl 2 2 α m2j − kj2 m20 (ml − m0 )
d−1− dn n−1 2
cn (2π)
2
Z ×
R+
n n Y X
Z
∞
sin πα
l=j+1
(
d−1− d 2 n−1
= cn (2π)
2
l=1
Z n j−1 X Y − µα (kl ) j=1 l=1
n ρα (dm2j ) Y R+
m2j − kj2
But (cf. [31] p. 70) Z
∞
m2 >m20
(m2
Z
∞
1 1 dm2 2 2 − k ) (m − m20 )α
1 dx (x − (k 2 − m20 ))xα Z ∞ 1 1 y = 2 2 α |k − m0 | 0 y ∓ 1 =
0
l=j+1
) µ+ α (kl )
δ
n X l=1
! kl
.
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
π cot πα =
1 |k 2 − m20 |α
π(sin πα)−1
811
if k 2 − m20 > 0 ;
1 |k 2 − m20 |α
. if k − 2
m20
<0
Thus, we get (79). It is not difficult to show the temperedness of (79) by application 2 2 − 2 of a Cauchy–Schwarz inequality, making also use of the fact that (µ+ α ) , (µα ) , (µα ) are locally integrable functions. Poincar´e invariance in (79) follows from Theorem 7.10, but is also “manifest” (cf. Remark 7.9(1)). Let us make sure that (7.9) in the case n = 2 yields the same result as Sec. 6. We have − c2 (2π)d 2 µα (k1 )µ+ α (k2 ) + µα (k1 )µα (k2 ) δ(k1 + k2 ) 2 1 δ(k1 + k2 ) = c2 4 sin πα cos πα 1{k12 >m20 ,k10 <0} (k1 ) 2 (k1 − m20 )α = c2 2 sin 2πα1{k12 >m20 ,k10 <0} (k1 )
1 δ(k1 + k2 ) (k12 − m20 )2α
(80)
for α ∈ (0, 12 ), where we have applied sin πα cos πα = 12 sin(2πα). (80) differs only by a time reflection θ from the distribution defined in Remark 6.9, which arises from different conventions in the definition of the Laplace transform in Secs. 6 and 7. Thus, we have the same result as in Sec. 6. (Note that the additional factor d (2π)− 2 in Remark 6.7 arises from the fact that the normalization factor of the d Fourier transform in one argument is (2π)− 2 while for the Fourier transform in 2 arguments it is (2π)−d .) ˆ T 1 . A technical Corollary 7.13. For n ≥ 3, Proposition 7.12 also applies to W n, 2
calculation shows that (79) is also tempered for α = 12 . Furthermore, the 2-point − (k1 )δ(k1 + k2 )] is the well-known 2-point function function W2,T 1 = (2π)c2 F −1 [δm 0 2
of the relativistic free field. Therefore, also in the case α =
1 2
Theorem 7.2 applies.
T ˆ 2,α = W ˆ 2,α admits a K¨allen– Remark 7.14. For 0 < α < 12 , c1 = 0, W Lehmann representation. Therefore the corresponding Gaussian Euclidean field with covariance function S2,α is reflection positive (but not Markov, the latter being seen from a general theorem of Pitt [59]). For α = 12 the corresponding Gaussian Euclidean field is the Markov free field of mass m0 ([54]). This can also be taken from Eq. (80) by the following considerations: For α ↑ 12 , on one hand we have that the coefficient sin(2πα) ↓ 0. This implies that ˆ T vanish on open sets in the Fourier transformed truncated 2-point functions W 2,α Rd2 which do not intersect the mass-shell {k12 = m20 , k10 < 0, k1 + k2 = 0}. On the ˆ T on this mass-shell causes non-integrability other hand, the singularity of the W 2,α 1 for α ↑ 2 . Combining these two aspects in a quantitive calculation [38], one can show that
812
S. ALBEVERIO, H. GOTTSCHALK and J.-L. WU
lim 2 sin 2πα1{k12 >m20 ,k10 <0} (k1 )
α↑ 12
1 − = 2πδm (k1 ) , 0 (k12 − m20 )2α
where the limit is the weak limit in S 0 (Rd ). Let us now have a look at (80) for 1 > α > 12 . First of all we note that (80) is no more tempered, since the exponent −2α is smaller than −1 and (80) is T ˆ 2,α . not locally integrable. Therefore, formula(80) in this case cannot represent W Nevertheless, one may speculate that (80) still holds if k1 stays away from the massshell {k12 = m20 , k10 < 0, k1 + k2 = 0}. If this were true, an interesting observation can be made: Since the function sin 2πα at α = 12 changes its sign from + to −, T for α ∈ ( 12 , 1) would the corresponding truncated 2-point Schwinger function S2,α be “reflection negative”, rather than “reflection positive”. Thus, it seems, as if the Markov property in the case of Gaussian Euclidean random fields would appear at the “boundary” of reflection-positivity. Nevertheless, a proper treatment of this problem has to be left to future work. 7.6. Positivity in the scattering region Let us concentrate on α = 12 and d = 4, c1 = 0. We thus consider the truncated 1 Wightman functions of the convoluted generalized white noise X = (−∆+m20 )− 2 F . If F is Gaussian, X is the free Markov field of mass m0 and the analytic continuation of its only nonzero Schwinger function S2T yields that S2T on (Ed )2< is the Laplace transform of − ˆ 2T (k1 , k2 ) = (2π)d+1 c2 δm W (k1 )δ(k1 + k2 ) . 0 This is the Fourier transform of the 2-point function of the relativistic free field of mass m0 [54]. Let {Wng }n∈N be the sequence of Wightman functions composed ˆ 2T ). Wng is from the truncated sequence {WnT }n∈N , WnT = 0, n 6= 2, W2T = F −1 (W T dn composed from the W2 in the way of Corollary 3.5. For ϕ ∈ SC (R ) define ϕ∗ by ϕ∗ (x1 , . . . , xn ) = ϕ(xn , . . . , x1 ). It is well known that in this case positivity holds for the {Wng }n∈N0 , i.e. for ϕ0 ∈ C, ϕl ∈ SC (Rdl ), l = 1, . . . , n, we can define a seminorm L∞ for the vector Ψ = (ϕ0 , ϕ1 , . . . , ϕn , 0, . . .) in the Borchers algebra n=0 SC (Rdn ) =: S ([29, 63]) as n X g Wl+m (ϕ∗l ⊗ ϕm ) ≥ 0 . (81) kΨk2g := l,m=0
This allows us to look at equivalence classes of vectors with norm larger than zero as the “physical states” of the theory. Let now F be a generalized white noise. We define the general non-definite (cf. Sec. 5) squared pseudo-norm k · k2 on S g in analogy to (80), where the Wl+m ’s are replaced by Wl+m ’s obtained from the T −1 ˆ T truncated Wightman distributions Wm, (Wm, 1 ) defined in Corollary 7.13. 1 = F 2 2 Let us now fix ε > 0 and define a region U in the Minkowski space–time Md as U := {k ∈ Md : k 2 ∈ [m20 − ε, m20 + ε], k 0 > 0}. Let S(U ) denote the Schwartz functions on Rd with support in U . For ϕ ∈ S(Rd ) such that ϕˆ ∈ S(U ) define i(k0 −ω)t ˆ . Let us concentrate on Ψ(t) ∈ S ϕ(x, t) := F −1 (ϕ˜t ) (x) where ϕ˜t (k) = ϕ(k)e Nr such that Ψ(t) = (ϕ0 , ϕ1 (t), . . . , ϕn (t), 0, . . . , 0, . . .) where ϕr (t) = s=1 ϕsr (t), with
CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS
813
ϕsr (t) defined as above. It is well known [43] that the “wave packet” ϕsr (t)(x) is concentrated near the plane x0 = t. In this sense, we say Ψ(t) approaches the asymptotic region x0 → ±∞ if t goes to that limit. Let us quote [61] Vol. III p. 324 ff. and [43] for the following basic results of Haag–Ruelle Theory, based only on locality, strong spectral condition and invariance of the WnT , n ≥ 3 and the special form of W2T : d T s1 (∗) W2 ϕr1 (t) ⊗ ϕrs22 (∗) (t) = 0 dt and
WnT ϕrs11 (∗) (t) ⊗ · · · ⊗ ϕrsnn (∗) (t) → 0 ,
for n ≥ 3 as t → ±∞ .
(82)
(83)
3
The limit in (83) is approached as (1+|t|) 2 (n−2) falls to zero in the general case, and faster than (1 + |t|)−N , N ∈ N falls to zero, if the ϕsrii (0)’s are non-overlapping in s velocity space, i.e. ωi−1 ki+1 6= ωj−1 kj−1 for ki ∈ supp ϕsrii kj ∈ supp ϕrjj j, i = 1, . . . , n i 6= j. Since we can expand kΨ(t)k2 into products of truncated Wightman functions of s (∗) s (∗) the type W2T (ϕr11 (t) ⊗ ϕr22 (t)) and those of the type of the LHS of (83), we get Proposition 7.15. Let Ψ(t) ∈ S, k · k2 k · k2g as above. Then kΨ(t)k2 → kΨ(0)k2g
as t → ±∞ .
(84)
The limit here is approached as (1 + |t|)− 2 → 0 in general and faster as (1 + |t|)−N → 0 for any N ∈ N for a “non-overlapping” Ψ(0). 3
Remark 7.16. 1. Let us point out, that the existence of stable one-particle respectively of free states, i.e. the holding of (82), is special for α = 12 . We do not have an analogue of these statements e.g. in the case α ∈ (0, 12 ). 2. We remark that the result of Proposition 7.15 in the non-overlapping case holds for all dimensions d ≥ 2 [43]. Proposition 7.15 can also be generalized to the dimensions d ≥ 4, but the classical literature only deals with the physical space–time d = 4. 3. If there is at least one ϕs , such that all ϕˆrs , r = 1, . . . , s take nonzero values on the mass-shell of mass m0 , then we have kΨ(0)k2g > 0 and thus also kΨ(t)k2 becomes positive for large t. 4. k · kg may also be called the norm of a free or noninteracting state. In the very vague sense of (84) we may therefore say that Ψ(t) approaches a free state. Nevertheless, we cannot define asymptotic free states as in Haag–Ruelle theory, since at the moment we have no Hilbert topology in which the Ψ(t)’s could converge. 5. For further investigations it seems therefore to be necessary to introduce a suitable auxiliary topology. A promising candidate is the Krein topology: In [3] we proved that the Wightman distributions of our model fulfill the Hilbert structure condition of [52]. The model developed here thus fulfills the modified Wightman axioms for “fields in an indefinite metric” (see e.g. [28] for a definition of such fields).
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By the general results on spaces with indefinite inner product (see again [52] and references therein) we get that there exists a Hilbert space (H, ( · , ·)) such that S ⊂ H is dense. If h · , ·i denotes the inner product on S induced by the sequence of Wightman distributions {Wn }n∈N0 , then there is a continuous self-adjoined operator η on H s.t. η 2 = 1 and (· , η ·) = h · , ·i holds on S. Finally, we would like to summarize this section as follows: In Proposition 7.12 and Corollary 7.13 we give explicit formulae for the Fourier–transformed (truncated) Wightman distributions that belong to the random field X = (−∆ + m20 )−α F , α ∈ (0, 12 ]. The sequence of Wightman distributions constructed from the former distributions fulfils all Wightman axioms (cf. [66, 67]), of relativistic QFT, except for the positivity of the square norm in the state–space, which in some cases does not hold and in others is uncertain (it is certain only for the case where F is Gaussian). Nevertheless, in the case α = 12 there exists stable one resp. free states, and therefore Haag–Ruelle theory allows us to derive a positivity condition for states Ψ(t) approaching the asymptotical respectively scattering regions as t → ±∞. Acknowledgements We thank D. Applebaum, C. Becker, P. Blanchard, R. Gielerak, Z. Haba, J. Sch¨ afer and Yu. M. Zinoviev for stimulating discussions. The financial support of D.F.G. is gratefully acknowledged. References [1] S. Albeverio, J. E. Fenstad, R. Høegh–Krohn, and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, 1986 (Russian translation, Moscow: Mir 1988). [2] S. Albeverio, H. Gottschalk, and J.-L. Wu, “Euclidean random fields, pseudodifferential operators, and Wightman functions”, Stochastic Analysis and Applications, Proc. Fifth Gregynog Symposium, Gregynog, 9–14, July, 1995, ed. I. M. Davies, A. Truman, and K. D. Elworthy, World Scientific, 1996, pp. 20–37. [3] S. Albeverio, H. Gottschalk, and J.-L. Wu, “Models of local relativistic quantum fields with indefinite metric (in all dimensions)”, SFB 237 – preprint no. 317, Bochum, 1996, to appear in Commun. Math. Phys. [4] S. Albeverio and R. Høegh–Krohn, “Uniqueness and the global Markov property for Euclidean fields: The case of trigonometric interactions”, Comm. Math. Phys. 68 (1979) 95–128. [5] S. Albeverio and R. Høegh–Krohn, “Euclidean Markov fields and relativistic quantum fields from stochastic partial differential equations”, Phys. Lett. B177 (1986) 175–179. [6] S. Albeverio and R. Høegh–Krohn, “Quaternionic non-abelian relativistic quantum fields in four space–time dimensions”, Phys. Lett. B189 (1987) 329–336. [7] S. Albeverio and R. Høegh–Krohn, “Construction of interacting local relativistic quantum fields in four space–time dimensions”, Phys. Lett. B200 (1988) 108–114, with erratum in ibid. B202 (1988) 621. [8] S. Albeverio, R. Høegh–Krohn, and H. Holden, “Markov cosurfaces and gauge fields”, Acta Phys. Austr., Suppl. XXVI, (1984) 211–231. [9] S. Albeverio, R. Høegh–Krohn, H. Holden, and T. Kolsrud, “Construction of quantised Higgs–like fields in two dimensions”, Phys. Lett. B222 (1989) 263–268.
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[10] S. Albeverio, R. Høegh–Krohn, H. Holden, and T. Kolsrud, “Representation and construction of multiplicative noise”, J. Funct. Anal. 87 (1989) 250–272. [11] S. Albeverio, R. Høegh–Krohn, and K. Iwata, “Covariant Markovian random fields in four space–time dimensions with nonlinear electromagnetic interaction”, Applications of Self–Adjoint Extensions in Quantum Physics, Proc. Dubna Conference 1987, ed. P. Exner, and P. Seba, Lecture Notes Phys. 324, Springer, 1989, pp. 69–83. [12] S. Albeverio, R. Høegh–Krohn, and B. Zegarli´ nski, “Uniqueness and global Markov property for Euclidean fields: The case of general polynomial interactions”, Commun. Math. Phys. 123 (1989) 377–424. [13] S. Albeverio, K. Iwata, and T. Kolsrud, “Random fields as solutions of the inhomogeneous quaternionic Cauchy–Riemann equation. I. Invariance and analytic continuation”, Comm. Math. Phys. 132 (1990) 555–580. [14] S. Albeverio, K. Iwata, and T. Kolsrud, “A model of four space–time dimensional gauge fields: reflection positivity for associated random currents”, Rigorous Results in Quantum Dynamics, Proc. Liblice Conference 1990, ed. J. Dittrich and P. Exner, World Scientific, 1991, pp. 257–269. [15] S. Albeverio, K. Iwata, and T. Kolsrud, “Conformally invariant and reflection positive random fields in two dimensions”, in Honor of M. Zakai, ed. E. Mayer-Wolf, E. Merzbach and A. Schwartz, Academic Press, 1991, pp. 1–14. [16] S. Albeverio, K. Iwata, and T. Kolsrud, “Homogenous Markov generalized vector fields and quantum fields over 4-dimensional space–time”, Stochastic Partial Differential Equations and Applications, Proc. Trento Conf. SPDEs, ed. G. Da Prato and L. Tubaro, Longman, Pitman Res. Notes 268, 1992, pp. 1–18. [17] S. Albeverio, K. Iwata, and M. Schmidt, “A convergent lattice approximation for nonlinear electromagnetic fields in four dimensions”, J. Math. Phys. 34 (1993) 3327– 3342. [17a] S. Albeverio and H. Tamura, “On the propagator of a scalar field in the presence of a confining nonlinear electromagnetic force”, Bochum preprint. [18] S. Albeverio and J.-L. Wu, “Euclidean random fields obtained by convolution from generalized white noise”, J. Math. Phys. 36 (1995) 5217–5245. [19] S. Albeverio and J.-L. Wu, “On the lattice approximation for certain generalized vector Markov fields in four space–time dimensions”, to appear in Acta Appl. Math. (1996), in press. [20] S. Albeverio and B. Zegarli´ nski, “A survey on the global Markov property in quantum field theory and statistical mechanics”, R. Høegh–Krohn’s Memorial Volume Ideas and Methods in Quantum and Statistical Mechanics, ed. S. Albeverio, J. E. Fenstad, H. Holden, and T. Lindstrøm, Cambridge Univ. Press, 1992, pp. 331–369. [21] S. Albeverio and X.-Y. Zhou, “A new convergent lattice approximation for the φ42 quantum field”, SFB 237 preprint-Nr. 284, Bochum, 1995. [22] H. Araki, “On the asymptotic behaviour of vacuum expectation values at large spacelike separations”, Ann. Phys. 11 (1960) 260–274. [23] H. Araki, “On the connection of spin and commutation relations between different fields”, J. Math. Phys. 2 (1961) 267–270. [24] J. C. Baez, I. E. Segal, and Z. Zhou, Introduction to Algebraic and Constructive Quantum Field Theory, Princeton Univ. Press, 1992. [25] C. Becker, “Wilson loops in two-dimensional space-time regarded as white noise”, J. Funct. Anal. 134 (1995) 321–349. [26] C. Becker, R. Gielerak, and P. Lugiewicz, “Covariant SPDES and quantum field structures”, SFB237 Preprint No. 331, Borchum, 1996. [27] C. Berg. and G. Forst, Potential Theory on Locally Compact Abelian Groups, SpringerVerlag, 1975.
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QUANTUM MECHANICS IN AF C ∗ -SYSTEMS FUMIO HIAI Department of Mathematics Ibaraki University Mito, Ibaraki 310, Japan
´ DENES PETZ Department of Mathematics Faculty of Chemical Engineering Technical University Budapest H-1521 Budapest XI. Sztoczek u.2, Hungary Received 24 November 1995 Revised 1 March 1996 Motivated from the chemical potential theory, we study quantum statistical thermodynamics in AF C ∗ -systems generalizing usual one-dimensional quantum lattice systems. Our systems are C ∗ -algebras A which have a localization {A[i,j] } of finite-dimensional subalgebras indexed by finite intervals of Z and an automorphism γ acting as a right shift on the localization. Model examples are supplied from derived towers (string algebras) for type II1 factor-subfactor pairs. Given a (γ-invariant) interaction and a specific tracial state, we formulate the Gibbs conditions and the variational principle for (γ-invariant) states on A, and investigate the relationship among these conditions and the KMS condition for the time evolution generated by the interaction. Special attention is paid to C ∗ -systems of gauge invariance (typical model in the chemical potential theory) and to C ∗ -systems considered as quantum random walks on discrete groups. The CNT-dynamical entropy for the shift automorphism γ is also discussed.
Introduction Let Zν be the simple cubic lattice of dimension ν, and Ak (k ∈ Zν ) be copies of Md (C), the d × d matrix algebra. Then the usual ν-dimensional quantum lattice system or quantum spin system is described as the infinite tensor product N ν C ∗ -algebra A = k∈Zν Ak with the space translations γk (k ∈ Z ). As is fully presented in [9], the rigorous treatment of quantum lattice systems is one of the major successes of the C ∗ -algebraic approach to quantum physics. An interaction Φ in the quantum spin C ∗ -algebra A is given when a selfadjoint element Φ(X) in N ν the local algebra AX = k∈X Ak is specifiedPfor each finite X ⊂ Z . Then the local Hamiltonian for a finite Λ ⊂ Zν is HΛ = X⊂Λ Φ(X), and the one-parameter dynamics (i.e. the time evolution) αΦ t (t ∈ R) can be introduced as the strong limit of eitHΛ · e−itHΛ as Λ → Zν under a certain decay condition for Φ. The so-called variational principle is an essential ingredient in the translation-invariant theory of quantum lattice systems, where the mean entropy of a state and the pressure of an interaction play important roles. The main results concerning KMS 819 Reviews in Mathematical Physics, Vol. 8, No. 6 (1996) 819–859 c World Scientific Publishing Company
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or Gibbs states in the above setting are summarized as follows: Given a translationinvariant interaction Φ, the αΦ -KMS condition, the Gibbs condition with respect to Φ, and the variational principle with respect to Φ are all equivalent for translationinvariant states on A [46, 32, 3]. Another important result is concerning the uniqueness of αΦ -KMS states (i.e. no phase transition) under the condition of bounded surface energies of Φ [4, 28, 47]; this is typical in the one-dimensional quantum spin case. The aim of this paper is to extend quantum statistical thermodynamics from the setting of quantum spin (UHF) C ∗ -algebras to that of AF (non-UHF) C ∗ -algebras. An earlier attempt in this direction was made by Kishimoto [29, 30]. Also, Matsui [35] recently discussed ground states on the CAR algebra and on its gauge invariant part. AF C ∗ -systems considered in this paper naturally arise from a recent development of the subfactor theory initiated by the celebrated Jones index theory [24] for type II1 subfactors. In fact, our discussions are strongly motivated from our previous study [13, 19, 20] succeeding [38, 11] on the subfactor theory from the entropic viewpoint. When N ⊂ M is an inclusion of type II1 factors with the Jones index [M : N ] < +∞, we obtain the Jones tower · · · ⊂ M−2 ⊂ M−1 = N ⊂ M0 = M ⊂ M1 ⊂ M2 ⊂ · · · by iterating the Jones basic construction both upward and downward. Then the relative commutant algebras or the string algebras 0 ∩ Mj are attached to intervals [i, j] of Z, which are finite-dimensional A[i,j] = Mi−1 S and generate the AF C ∗ -algebra A = n A[−n,n] . Furthermore, the Markov trace S τ on n Mn restricts on A and the canonical shift γ (a 2-shift on the localization {A[i,j] }) is defined on A. This way, we can get an AF C ∗ -system (A, {A[2i,2j] }, γ, τ ) which is a model example of our C ∗ -systems. Recall that the double sequence {M 0 ∩ Mn ⊂ N 0 ∩ Mn }∞ n=1 which bears the weights from the Markov trace τ is crucial in classification of type II1 subfactors; in fact, it provides a complete conjugacy invariant among strongly amenable type II1 inclusions ([42] and also [36]). Although the Jones theory for type II1 subfactors was generalized to the type III ones (see e.g. [31, 33, 23]), it is enough from the viewpoint of this paper to confine ourselves to the type II1 case because no new C ∗ -systems appear when general inclusions N ⊂ M are considered beyond type II1 ones. Also it should be mentioned that we are rather concerned with the infinite depth case in the subfactor model, while the finite depth case might arouse more interest from the subfactor theory. Indeed, when N ⊂ M has finite depth, the resulting C ∗ -algebra A has a unique tracial state so that the situation is almost the same as the quantum spin case. Another motivation of our study comes from the chemical potential theory [7, 6], where the observable algebra A is the fixed point subalgebra of the field C ∗ -algebra F by some gauge action of a compact group and a one-parameter dynamics αt on F is commuting with the gauge action so that α restricts on A. The main concern in [6] is to extend an extremal α-KMS state on A to an extremal KMS state for a modified dynamics on F. Here the notion of chemical potentials enters into the theory. A gauge action of a compact group G is sometimes given as a product action N N βg = Z Ad σg (g ∈ G) on the field algebra F = Z Md (C) where σ is a unitary representation of G on V = Cd . In this case, the observable algebra A = F β having
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Nj the localization A[i,j] = ( k=i Md (C))β and the usual shift γ is essentially the same as the C ∗ -system arising from the subfactor of Wassermann’s type [53] defined by the representation σ. It is worth noting that, in several examples, the chemical potentials in [6] bijectively corresponds to the extremal faithful (γ-invariant) tracial states on A. So it seems reasonable that we consider such tracial states on A as substitutes of chemical potentials in the abstract setup of AF C ∗ -systems. The paper is organized as follows. To begin with we fix the formalism of C ∗ systems (A, {A[i,j] }, γ) treated throughout and justify it from the subfactor model. In Sec. 2 we introduce, given an interaction Φ, the notions (in the strong and weak senses) of Gibbs condition for states on A. These notions are defined in terms of a chosen tracial state φ on A as well as the interaction Φ. Then it is proved that a state on A satisfies the KMS condition (at β = 1) for the one-parameter dynamics αΦ generated by Φ if and only if it satisfies the Gibbs condition in the strong sense for Φ with respect to some tracial state φ. In Sec. 3 we formulate the variational principle giving a γ-invariant interaction Φ in an AF C ∗ -system. In the course of doing so, we show the existence of the mean relative entropy SM (ω, φ) of a γ-invariant state ω and that of the pressure p(Φ, φ) of Φ with respect to a fixed faithful γ-invariant tracial state φ having some multiplicativity property. The variational principle was obtained in [30] in a similar but somewhat different setting of C ∗ -systems. Furthermore, as in usual quantum lattice systems, given Φ and φ as above we show the following implications for γ-invariant states on A: The Gibbs condition in the weak sense implies the variational principle and the latter implies the αΦ -KMS condition. Our final goal might be to establish the abstract version of the chemical potential theory with use of extremal faithful tracial states on A instead of chemical potentials (see Problems 2.5 and 3.12 below). Although this problem in the abstract setting seems difficult, we can get rather satisfactory results in Secs. 4 and 5 in some C ∗ -systems with additional structures. A C ∗ -system in Sec. 4 is given as the fixed point subalgebra of the product action determined by a unitary representation of a compact group, which is a typical example from the original chemical potential theory. A C ∗ -system in Sec. 5 is obtained from a discrete group with a finite number of generators, which is considered as a quantum version of random walks on groups. The final Sec. 6 is devoted to discussions on the CNT-dynamical entropy of γ in connection with its topological entropy, which are on similar lines of [11, 13, 19, 20]. Our C ∗ -systems are restricted to AF C ∗ -systems indexed to one-dimensional Z, while we can similarly consider those indexed to multi-dimensional Zν . Furthermore, index sets are restricted to intervals of Z, although we may take all finite subsets as in the usual one-dimensional spin case. These restrictions enable us to make some results considerably simpler because we have bounded surface energies. 1. Setting and Examples For i, j ∈ Z let [i, j] denote the interval {i, i + 1, . . . , j} of Z with convention [i, j] = ∅ if i > j. Assume that finite-dimensional C ∗ -algebras (or finite direct sums of matrix algebras) A[i,j] are given for all intervals [i, j] of Z, i ≤ j,
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and they satisfy the following: (i) A[i,j] ⊂ A[i0 ,j 0 ] with the common unit 1 if [i, j] ⊂ [i0 , j 0 ], i.e. i0 ≤ i ≤ j ≤ j 0 , (ii) A[i,j] and A[j+1,k] commute if i ≤ j < k. Let A denote the AF C ∗ -algebra generated by {A[i,j] }, that is, the C ∗ -compleS∞ tion of n=1 A[−n,n] . For any K ⊂ Z we denote by AK the C ∗ -subalgebra of A generated by A[i,j] with [i, j] ⊂ K. Also set A∅ = C1. We further assume that there exists an automorphism γ of A such that (iii) γ(A[i,j] ) = A[i+1,j+1] for all i ≤ j. This means that A[i,j] ∼ = A[i+k,j+k] for any i, j, k ∈ Z and γ acts on the local algebras A[i,j] as the (bilateral) right shift. So γ(A[1,∞) ) = A[2,∞) and γ|A[1,∞) is an endomorphism of A[1,∞) like the unilateral right shift. We write Sγ (A) for the set of all γ-invariant states on A. Since (A, γ) is asymptotically abelian in the norm sense, i.e. lim || [a, γ n (b)] || = 0 ,
|n|→∞
a, b ∈ A ,
it follows [9, 4.3.11] that Sγ (A) forms a (Choquet) simplex. The simplex of all tracial states on A is denoted by T (A). Also, we write Tγ (A) for the set of all γ-invariant φ ∈ T (A). The following fact might be well known and easily shown; so we omit the proof. Proposition 1.1. Tγ (A) is a face of Sγ (A) and so Tγ (A) becomes a simplex. The quantum system (A, γ) described above sometimes possesses a particular (not unique in general) trace τ having the following properties: (iv) τ is faithful and γ-invariant, (v) τ is multiplicative in the sense that τ (ab) = τ (a)τ (b) for all a ∈ A[i,j] and b ∈ A[j+1,k] , i ≤ j < k. Let Trn denote the canonical trace of A[1,n] , where the term “canonical” means that Trn (e) = 1 for all minimal projections e ∈ A[1,n] . For τ ∈ Tγ (A) let dτn /d Trn be the Radon–Nikodym derivative of τn = τ |A[1,n] with respect to Trn (so it belongs to the center of A[1,n] ). Then the following rather strong condition for τ is considered, which means that the McMillan type convergence in the uniform norm holds for τ with respect to the localization {A[1,n] }: (vi) For some constant λ ∈ (0, 1), 1 dτn − (log λ)1 = 0 . lim log n→∞ n d Trn In the rest of this section let us explain that the subfactor model provides many examples of (A, {A[i,j] }, γ) together with a distinguished trace τ . More examples will be presented in Secs. 4 and 5.
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Example 1.2. Let A0 be a finite-dimensional C ∗ -algebra, Ak (k ∈ Z) copies of Nj A0 , and A[i,j] = k=i Ak (i ≤ j). The AF C ∗ -algebra A is the infinite C ∗ -tensor N N product k∈Z Ak and γ is the usual right shift. Set τ = Z τ0 , the product of τ0 = Tr/Tr(1), where Tr is the canonical trace of A0 . Then all (i)–(vi) are satisfied. Concerning (vi) we have n−1 log dτn /d Trn = λ1 for all n with λ = Tr(1)−1 . When A0 = Md (C), the d × d matrix algebra, this A is the so-called quantum spin C ∗ algebra. Example 1.3. Let {ei }i∈Z be a two-sided sequence of the Jones projections [24, 38], that is, ei are projections such that for some 0 < λ < 1 (
ei ei±1 ei = λei , ei ej = ej ei ,
i ∈ Z, |i − j| ≥ 2 .
(1.1)
Then it is well known [24, 54] that λ−1 ∈ {4 cos2 π/(m + 1) : m ≥ 2} ∪ [4, ∞). Let A[i,j] = Alg{1, ei+1 , . . . , ej }, the algebra generated by {1, ei+1 , . . . , ej }, and A be S∞ the C ∗ -completion of n=1 A[−n,n] . We can define a shift automorphism θλ on A by θλ (ei ) = ei+1 , i ∈ Z. Moreover, the so-called λ-Markov trace φλ is defined on A, which satisfies a ∈ A[i,j] . φλ (aej+1 ) = λφλ (a) , Then (i)–(v) are satisfied. Relations (1.1) first appeared in [51] in some model of statistical physics; so the C ∗ -algebras A[i,j] as well as A are sometimes called the Temperley–Lieb algebra (see also [17, 37]). When λ−1 = 4 cos2 π/(m + 1), the Bratteli diagrams of C ⊂ A[1,1] ⊂ A[1,2] ⊂ · · ·
(1.2)
are determined by the graph Am ([24]), so that φλ is the unique tracial state on A and (vi) is satisfied (see Example 1.4 and Proposition 1.5 below). When λ ≤ 1/4, the Bratteli diagrams of (1.2) have the graph A∞ . In this case, A[i,j] and hence A are independent of 0 < λ ≤ 1/4. Furthermore, it is easy to check using the string algebra description of {A[i,j] } ([36]) that θλ is also independent of λ up to unitary equivalence on each A[i,j] . Hence all tracial states φλ (0 < λ ≤ 1/4) are invariant for γ = θ1/4 . In fact, the set of all extremal points of T (A) is {φλ : 0 ≤ λ ≤ 1/4}, where φ0 is a degenerate trace such that φ0 (ei ) = 0, i ∈ Z. So T (A) = Tγ (A). Note that φ1/4 satisfies (vi), because the φ1/4 -trace vectors (i.e. the φ1/4 -values of minimal projections) of A[1,2n−1] and A[1,2n] are respectively 2−(2n−1) (2, 4, . . . , 2n) and 2−2n (1, 3, . . . , 2n − 1) ([24]). See Examples 1.4 and 4.6 for somewhat different formulations of this example. Example 1.4. Let N ⊂ M be an inclusion of type II1 factors with the Jones index [M : N ] < +∞. Let · · · ⊂ M−3 ⊂ M−2 ⊂ M−1 = N ⊂ M0 = M ⊂ M1 ⊂ M2 ⊂ · · ·
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F. HIAI and D. PETZ
0 be the Jones tower of tunnel and basic constructions. Set A[i,j] = Mi−1 ∩ Mj for i ≤ j, which are finite-dimensional algebras. The derived tower of N ⊂ M is given as C ⊂ A[1,1] ⊂ A[1,2] ⊂ · · · , whose inclusions are described as the principal graph. Sometimes the principal graph comes from C ⊂ A[0,0] ⊂ A[0,1] ⊂ · · · ; then the one above is called the dual principal graph. (See [24, 17] on the index theory for type S∞ II1 subfactors.) Let A be the C ∗ -completion of n=1 A[−n,n] , and set a normalized S trace τ on A as the restriction of the unique trace on n Mn (called the λ-Markov trace with λ = [M : N ]−1 ). The mirrorings and the canonical shift on the derived tower were introduced in [36]. The mirroring γn of A[1,2n] is defined by
γn (x) = Jn x∗ Jn ,
x ∈ A[1,2n] ,
where Jn is the modular conjugation on L2 (Mn , τ ). Since γn+1 ◦ γn = γn ◦ γn−1 on S A[1,2n−2] (see [13] for details), we can define the canonical shift γ on n A[1,n] by γ(x) = γn+1 (γn (x)) ,
x ∈ A[1,2n] .
(1.3)
Indeed, the canonical shift γ can be defined from each M−n ⊂ M−n+1 , n ≥ 0, so that γ extends to an automorphism on A preserving τ . Then (i)–(v) are satisfied with the localization {A[2i,2j] }i≤j , because γ is a 2-shift on {A[i,j] }. Concerning (v), the following a bit stronger condition is satisfied: for any i ≤ j ≤ k A[i,j] ∪ A[i+1,j]
⊂
A[i,k] ∪ ⊂ A[i+1,k]
is a commuting square in the sense that E[i,j] E[i+1,k] = E[i+1,k] E[i,j] = E[i+1,j] ;
equivalently E[i,j] (A[i+1,k] ) = A[i+1,j] ,
where E[i,j] is the conditional expectation onto A[i,j] with respect to τ . The canonical shift γ sometimes has a natural square root [12]. For instance, when N ⊂ M is Jones’ subfactor Rλ ⊂ R ([24]) with index λ−1 ≤ 4 or Popa’s subfactor N s ⊂ M s ([41]) with index λ−1 = s > 4, {A[i,j] } coincides with Example 1.3 and γ = θλ2 . The standard matrix (or the principal graph) Γ = [akl ]k∈K,l∈L and the standard eigenvector ~s = (sk )k∈K are attached to N ⊂ M , so that ΓΓt describes the inclusions of A[1,2n] , n ≥ 0, and ΓΓt~s = λ−1~s. Furthermore, the trace vector of A[1,2n] is given as (λn sk )k∈Kn , where Kn is the set of vertices corresponding to the direct summands S of A[1,2n] and so Kn ⊂ Kn+1 , n Kn = K. Hence, denoting the minimal central projections of A[1,2n] by (fn,k )k∈Kn , we have X dτn = λn sk fn,k . d Trn k∈Kn
If N ⊂ M is extremal in the sense of [42], then we have sk = [fn,k M2n fn,k : M fn,k ]1/2 , k ∈ Kn . (See [39, 42] for details on standard invariants.) Note that (vi) holds if and only if ~s has subexponential growth, i.e. 1 log max sk = 0 , lim n→∞ n k∈Kn
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
825
and in this case N ⊂ M is extremal [20]. In particular, if N ⊂ M has finite depth (this is the case when [M : N ] < 4), i.e. |K| < +∞ (|K| denotes the cardinality of K), then (vi) holds and furthermore τ is a unique tracial state on A as is shown below for completeness. Also, note that τ is an extremal tracial state if and only if S 0 ∩ M )00 is a factor. So, a N ⊂ M has the ergodic core [42], that is, M st = ( n M−n tracial state on A is not extremal in general even though it satisfies (iv)–(vi). (The last fact will be more explicitly known in Remark 5.2.) Proposition 1.5. If N ⊂ M has finite depth, then τ is a unique tracial state on A. Proof. Choose n0 such that Kn0 = K and hence the inclusion matrix of A[1,2n] ⊂ A[1,2n+2k] is (ΓΓt )k when n ≥ n0 and k ≥ 1. Let φ be any tracial state on A and ~vn the φ-trace vector of A[1,2n] for n ≥ n0 . Then we get ~vn = (ΓΓt )k ~vn+k for k ≥ 1, so that ∞ \ (ΓΓt )k Rd+ , n ≥ n0 , ~vn ∈ k=1
where d = |K|. Hence ~vn is proportional to the Perron–Frobenius eigenvector of ΓΓt , which implies that φ|A[1,2n] = τ |A[1,2n] for all n ≥ n0 . Therefore φ = τ . 2. KMS Condition and Gibbs Condition From now on let (A, {A[i,j] }, γ) be a C ∗ -system introduced in Sec. 1, which always satisfies conditions (i)–(iii). The symbol X stands for a finite interval in Z, and let |X| = j − i + 1 for X = [i, j], i ≤ j. We say that Φ is an interaction if a selfadjoint element Φ(X) in AX is given for each X. An interaction Φ is said to be γ-invariant or translation-invariant if γ(Φ(X)) = Φ(X + 1) ,
X ⊂ Z,
where X + 1 = {k + 1 : k ∈ X}. Given an interaction Φ and a finite interval Λ ⊂ Z, the local Hamiltonian HΛ is defined by HΛ =
X
Φ(X) .
X⊂Λ
Also the surface energy WΛ is defined by WΛ =
X
{Φ(X) : X ∩ Λ 6= ∅, X ∩ Λc 6= ∅} ,
whenever the sum in the right-hand side converges in norm. We use the notion of the inner perturbation of a state on A to introduce the Gibbs condition. The theory of state perturbation was first developed in [1] (see [48] for its extension). The variational approach to state perturbation was exploited in [5, 16] by means of the relative entropy. Let ω, ψ be two states on A. For a finite
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F. HIAI and D. PETZ
interval Λ ⊂ Z, the relative entropy of ψΛ = ψ|AΛ with respect to ωΛ = ω|AΛ is given by dψΛ dωΛ dψΛ log , − log S(ψΛ , ωΛ ) = TrΛ d TrΛ d TrΛ d TrΛ where TrΛ is the canonical trace of AΛ . Then the relative entropy S(ψ, ω) is defined as S(ψ, ω) = sup S(ψΛ , ωΛ ) = lim S(ψ[−n,n] , ω[−n,n] ) . n→∞
Λ⊂Z
(See [37] on the relative entropy for states on a C ∗ -algebra and for normal states on a von Neumann algebra.) For any state ω on A and h = h∗ ∈ A, since ψ 7→ S(ψ, ω) + ψ(h) is weakly* lower semicontinuous and strictly convex on the state space of A, the perturbed state [ω h ] is defined as a unique minimizer of this functional [16, 37]. Recall [5, 16] that for selfadjoint h, k ∈ A the chain rule [[ω h ]k ] = [ω h+k ]
(2.1)
holds and |S(ψ, ω) − S(ψ, [ω h ])| ≤ 2||h|| . In particular, S(ω, [ω h ]) ≤ 2||h|| .
(2.2)
Let πω be the GNS representation of A associated with ω and Ωω the corresponding cyclic vector. When Ωω is separating for πω (A)00 and ∆ω is the modular operator for Ωω , the perturbed vector Ωhω is defined by ([1, 2]) Ωhω =
∞ X
Z (−1)n
n=0
Z
1/2
t1
dt1 0
0
Z dt2 · · ·
tn−1
dtn 0
∆tωn πω (h)∆tωn−1 −tn πω (h) · · · ∆tω1 −t2 πω (h)Ωω log ∆ω − h Ωω . = exp 2 (Note that Ωhω stands for Ωω (−h) in the notation of [1, 2].) Then another def˜ and [ω h ]˜ are the inition of [ω h ] is given as [ω h ] = hπω (·)Ωhω , Ωhω i/||Ωhω ||2 . If ω h 00 ˜ (x) = hxΩω , Ωω i and [ω h ]˜(x) = normal extensions of ω and [ω ] to πω (A) , i.e. ω h h h 2 00 h ω πω (h) ] because for a state ψ hxΩω , Ωω i/||Ωω || , x ∈ πω (A) , then we have [ω ]˜ = [˜ on A (see e.g. [37, p. 93]) ( S(ψ, ω) =
˜ ω S(ψ, ˜) +∞
if ψ has the normal extension ψ˜ to πω (A)00 , otherwise.
(2.3)
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
827
When an interaction Φ, φ ∈ T (A), and a finite interval Λ ⊂ Z are given, the canonical state or the local Gibbs state φcΛ on AΛ with respect to Φ and φ is defined by φ(e−HΛ a) , a ∈ AΛ . φcΛ (a) = φ(e−HΛ ) Under the above preparation let us introduce the notion of the Gibbs condition as follows. Definition 2.1. Let ω be a state on A, Φ an interaction, and φ ∈ T (A). Assume that WΛ is defined for any finite interval Λ ⊂ Z. (1) We say that ω satisfies the Gibbs condition in the strong sense with respect to Φ and φ if Ωω is separating for πω (A)00 and if for any Λ ⊂ Z the conditional expectation from πω (A)00 onto πω (AΛ ) ∨ πω (AΛc )00 with respect to [ω −WΛ ]˜ exists and the following holds: [ω −WΛ ](ab) = φcΛ (a)[ω −WΛ ](b) ,
a ∈ AΛ , b ∈ AΛc .
(2) We say that ω satisfies the Gibbs condition in the weak sense with respect to Φ and φ if [ω −WΛ ]|AΛ = φcΛ for any Λ ⊂ Z. Obviously (1) implies (2). When ω satisfies the Gibbs condition in the sense of 0 0 (2), φ ∈ T (A) is unique because φcΛ = φ0c Λ gives φ = φ on AΛ for φ, φ ∈ T (A). In the usual quantum spin case, the Gibbs condition (1) coincides with [9, 6.2.16], because A has the unique tracial state and φcΛ is the unique local Gibbs state on AΛ . In this section we assume that an interaction Φ is not necessarily γ-invariant but satisfies the following: P (a) supi∈Z X3i ||Φ(X)|| < +∞ , (b) supΛ⊂Z ||WΛ || < +∞ .
P Note that (a) and (b) are satisfied if Φ is γ-invariant and X30 |X| ||Φ(X)|| < +∞ (because X is restricted to finite intervals). Under assumptions (a) and (b), [27, Theorem 8] (also [9, 6.2.6]) says that Φ generates a one-parameter dynamics αΦ on A. More precisely, there exists a strongly continuous one-parameter automorphism group αΦ t (t ∈ R) on A such that itHΛ ae−itHΛ || = 0 lim ||αΦ t (a) − e
Λ→Z
for all a ∈ A and uniformly for t in finite intervals. Furthermore, the generator of S αΦ is the closure of the derivation δ0 with D(δ0 ) = Λ AΛ given by δ0 (a) = i
X
[Φ(X), a] ,
a ∈ AΛ .
X∩Λ6=∅
So we can consider the KMS condition (at β = 1) for a state on A with respect to Φ αΦ . Note that αΦ t ◦ γ = γ ◦ αt if Φ is γ-invariant. (A recent development on the
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F. HIAI and D. PETZ
existence of one-parameter dynamics in the Zν -lattice spin and CAR cases is found in [34, 35].) Our main result in this section is presented as follows. Theorem 2.2. If ω is a state on A and Φ is an interaction as above, then the following conditions are equivalent : (i) ω satisfies the KMS condition with respect to αΦ ; (ii) ω satisfies the Gibbs condition in the strong sense with respect to Φ and φ for some φ ∈ T (A). Proof. (ii) ⇒ (i). Assume that ω satisfies (ii) for some φ ∈ T (A). Fix Λ ⊂ Z arbitrarily and write M = πω (A)00 , MΛ = πω (AΛ ), and MΛc = πω (AΛc )00 . Let ω ˜ be the normal extension of ω, and let ω ˆ = [˜ ω πω (−WΛ ) ] (= [ω −WΛ ]˜, the normal −WΛ ] to M). Since ω ˆ as well as ω ˜ is faithful by [1, Corollary 4.4], we extension of [ω ˆt (t ∈ R) for ω ˜ and ω ˆ , respectively. have the modular automorphism groups σt and σ ˆ exists, Since the conditional expectation from M onto MΛ ∨ MΛc with respect to ω ˆ |MΛ ∨MΛc is it follows by [50] that σ ˆt (MΛ ∨MΛc ) = MΛ ∨MΛc , t ∈ R, and hence σ c the modular automorphism group for ω ˆ |MΛ ∨MΛc . Let σ Λ and σ Λ be the modular ˆ |MΛc , respectively. Since [MΛ , MΛc ] = 0 automorphism groups for ω ˆ |MΛ and ω ˆ (xy) = ω ˆ (x)ˆ ω (y), x ∈ MΛ , y ∈ MΛc , we see that with finite-dimensional MΛ and ω ˆ ) is naturally isomorphic to (MΛ ⊗ MΛc , ω ˆ |MΛ ⊗ ω ˆ |MΛc ). Hence (MΛ ∨ MΛc , ω σ ˆ |MΛ = σ Λ . But since ω ˆ (πω (a)) = [ω −WΛ ](a) = φcΛ (a) ,
a ∈ AΛ ,
it is easy to see that σ ˆ (x) = eitπω (HΛ ) xe−itπω (HΛ ) ,
x ∈ MΛ .
(2.4)
ˆ . Then the perturbation theory for Now let δσ and δσˆ be the generators of σ and σ the modular automorphism group says (see [9, Sec. 5.4.1]) that δσ (x) = δσˆ (x) + i[πω (WΛ ), x] ,
x ∈ D(δσ ) = D(δσˆ ) .
By (2.4) we have MΛ ⊂ D(δσˆ ) and for every a ∈ AΛ δσ (πω (a)) = i[πω (HΛ ), πω (a)] + i[πω (WΛ ), πω (a)] X = iπω [Φ(X), a] X∩Λ6=∅
= πω (δ(a)) , where δ is the generator of αΦ . Thus we have shown that [ a∈ AΛ . πω (δ(a)) = δσ (πω (a)) , Λ⊂Z
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
829
S Since Λ AΛ is a core of δ, we obtain πω ◦ δ ⊂ δσ ◦ πω and hence πω ◦ αΦ = σ ◦ πω . This implies that ω satisfies the KMS condition with respect to αΦ . To prove the converse, the next lemma is useful. Lemma 2.3. Let M be a von Neumann algebra and ϕ a faithful normal state on M. Let M0 and M1 be von Neumann subalgebras of M and assume that the conditional expectation E1 : M → M1 with respect to ϕ exists. If h ∈ M1 is selfadjoint, then (1) [ϕh ] = [(ϕ|M1 )h ] ◦ E1 , (2) [ϕh ]|M0 = ϕ|M0 whenever ϕ(xy) = ϕ(x)ϕ(y) for all x ∈ M0 and y ∈ M1 . Proof. (1) Put ϕ1 = ϕ|M1 so that ϕ = ϕ1 ◦ E1 . When h ∈ M1 is selfadjoint, we have for every state ψ on M S(ψ, ϕ) + ψ(h) ≥ S(ψ|M1 , ϕ1 ) + (ψ|M1 )(h) ≥ S([ϕh1 ], ϕ1 ) + [ϕh1 ](h) = S([ϕh1 ] ◦ E1 , ϕ) + [ϕh1 ] ◦ E1 (h) . This implies that [ϕh ] = [ϕh1 ] ◦ E1 . (2) The assumption means that E1 (x) = ϕ(x)1, x ∈ M0 . Hence for every x ∈ M0 we get by (1)
[ϕh ](x) = [ϕh1 ](E1 (x)) = ϕ(x) .
Proof of (i) ⇒ (ii) of Theorem 2.2. Assume that ω is an αΦ -KMS state. Then the cyclic vector Ωω is separating for M = πω (A)00 ([9, 5.3.9]) and the normal ˜ . For any fixed extension σ of αΦ to M is the modular automorphism group for ω Λ = [l, m], put Q = −(HΛ + WΛ ) and let αΦ,Q be the perturbation of αΦ by Q; namely the generator δ Q of αΦ,Q is given as δ Q (a) = δ(a) + i[Q, a] ,
a ∈ D(δ) = D(δ Q ) .
Furthermore define αt (a) = eit(H[−n,l−1] +H[m+1,n] ) ae−it(H[−n,l−1] +H[m+1,n] ) , (n)
a ∈ A.
Now let us prove that (n)
(a) = lim αt (a) , αΦ,Q t n→∞
a ∈ A, t ∈ R .
(2.5)
To do so, it suffices (see [48, 4.1.2]) to show that (1 ± δ (n) )−1 → (1 ± δ Q )−1 strongly where δ (n) is the generator of α(n) . For any Λ0 ⊂ Z and a ∈ AΛ0 we have ||{(1 ± δ (n) )−1 − (1 ± δ Q )−1 }(1 ± δ Q )(a)|| = ||(1 ± δ (n) )−1 {(1 ± δ Q )(a) − (1 ± δ (n) )(a)}|| ≤ ||δ Q (a) − δ (n) (a)|| .
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F. HIAI and D. PETZ
Let Ξn = Ξ0n =
X X
{Φ(X) : X ∩ Λ 6= ∅, X ∩ [−n, n]c 6= ∅} , {Φ(X) : X ∩ Λ0 6= ∅, X ∩ [−n, n]c 6= ∅} .
Then ||Ξn ||, ||Ξ0n || → 0 (n → ∞). When Λ ⊂ [−n, n], since H[−n,n] + Q = H[−n,l−1] + H[m+1,n] − Ξn , we get δ Q (a) − δ (n) (a) = i
X
Φ(X) + Q, a − i[H[−n,l−1] + H[m+1,n] , a]
X∩Λ0 6=∅
= −i[Ξn , a] + i
X X∩Λ0 6=∅
Φ(X) −
X
Φ(X), a
X⊂[−n,n]
= −i[Ξn , a] + i[Ξ0n , a] , (n)
so that ||δ Q (a) − δ (n) (a)|| → 0. Therefore (2.5) is shown. Since αt (a) = a for (n) a ∈ AΛ and αt (a) ∈ AΛc for a ∈ AΛc , it follows from (2.5) that αΦ,Q |AΛ = idAΛ , t
αΦ,Q (AΛc ) = AΛc . t
(2.6)
Moreover let αΦ,−WΛ be the perturbation of αΦ by −WΛ , which is the perturbation Λ (a) = eitHΛ αΦ,Q (a)e−itHΛ , we get by (2.6) of αΦ,Q by HΛ . Since αΦ,−W t t Λ (AΛ ) = AΛ , αΦ,−W t
Λ αΦ,−W (AΛc ) = AΛc . t
(2.7)
The normal extensions of αΦ,Q and αΦ,−WΛ are the perturbations of σ by πω (Q) and πω (−WΛ ), respectively, which are denoted by σ Q and σ −WΛ . Recall [1, Proposition 4.3] (also [9, 5.4.4]) that σ Q and σ −WΛ are the modular automorphism groups for the normal extensions [ω Q ]˜ and [ω −WΛ ]˜ of [ω Q ] and [ω −WΛ ] to M, respectively. Let MΛ and MΛc be as in the proof of (ii) ⇒ (i). Since σt−WΛ (MΛ ∨ MΛc ) = MΛ ∨ MΛc by (2.7), the conditional expectation from M onto MΛ ∨ MΛc with respect to [ω −WΛ ]˜ exists by [50]. It follows from (2.6) that φ˜Λ = [ω Q ]˜|MΛ is a trace on MΛ and the conditional expectation E1 : M → MΛc with respect to [ω Q ]˜ exists. Furthermore, as is readily seen because (MΛ ∨ MΛc , σ Q ) is naturally isomorphic to (MΛ ⊗ MΛc , idMΛ ⊗ (σ Q |MΛc )) by (2.6), we have [ω Q ]˜(xy) = [ω Q ]˜(x)[ω Q ]˜(y) = φ˜Λ (x)[ω Q ]˜(y) ,
x ∈ MΛ , y ∈ MΛc .
Now let Λ1 ⊃ Λ and Q1 = −(HΛ1 + WΛ1 ). Since by (2.1) [ω Q1 ]˜ = [˜ ω πω (Q1 ) ] = [[˜ ω πω (Q) ]πω (Q1 −Q) ]
(2.8)
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
831
and πω (Q1 −Q) ∈ MΛc , Lemma 2.3(2) together with (2.8) implies that [ω Q1 ]˜|MΛ = S [ω Q ]˜|MΛ , that is, φ˜Λ1 |MΛ = φ˜Λ . Hence there exists a tracial state φ˜ on πω ( Λ AΛ ) ˜ Λ = φ˜Λ for all Λ ⊂ Z. Thus we obtain φ ∈ T (A) which extends such that φ|M S φ˜ ◦ πω on Λ AΛ . Since σtQ (MΛ ∨ MΛc ) = MΛ ∨ MΛc , the conditional expectation E0 : M → MΛ ∨ MΛc with respect to [ω Q ]˜ exists. So we have by Lemma 2.3(1) and (2.8) [ω −WΛ ]˜= [([ω Q ]˜)πω (HΛ ) ] = [([ω Q ]˜|MΛ ∨ MΛc )πω (HΛ ) ] ◦ E0 = [(φ˜Λ ⊗ [ω Q ]˜|MΛc )πω (HΛ ) ] ◦ E0 π (H ) = ([φ˜Λω Λ ] ⊗ [ω Q ]˜|MΛc ) ◦ E0
(2.9)
under the natural isomorphism MΛ ∨ MΛc ∼ = MΛ ⊗ MΛc . Since π (HΛ )
[φ˜Λω
](πω (a)) =
φ˜Λ (e−πω (HΛ ) πω (a)) φ˜Λ (e−πω (HΛ ) )
=
φ(e−HΛ a) = φcΛ (a) , φ(e−HΛ )
a ∈ AΛ ,
it follows from (2.9) that [ω −WΛ ](ab) = φcΛ (a)[ω Q ](b) ,
a ∈ AΛ , b ∈ AΛc ,
completing the proof.
Let K(Φ) denote the set of all KMS states on A with respect to α , which becomes a simplex (see [9, 5.3.30]). Then Theorem 2.2 asserts that a correspondence ω ∈ K(Φ) 7→ φ ∈ T (A) is determined in the way that ω is a Gibbs state in the strong sense for Φ and φ. Φ
Proposition 2.4. If Φ is γ-invariant and ω ∈ K(Φ) is γ-invariant, then so is the corresponding φ ∈ T (A). Proof. By the γ-invariance of Φ and ω we have [ω −WΛ ] = [ω −WΛ+1 ] ◦ γ ,
Λ ⊂ Z.
Indeed, this is easily checked due to the definition of state perturbation through the relative entropy. So we get φcΛ = φcΛ+1 ◦ γ = (φ ◦ γ)cΛ on AΛ for every Λ ⊂ Z, which implies φ = φ ◦ γ. We here want to pose the following problems, which may be considered as an abstract version of the chemical potential theory [6] (see Sec. 4). Problems 2.5. (1) Is the correspondence ω 7→ φ bijective between K(Φ) and T (A)?
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F. HIAI and D. PETZ
(2) Under a suitable assumption, is ω an extremal αΦ -KMS state if and only if the corresponding φ is an extremal tracial state? Furthermore, when Φ is γinvariant, is ω an extremal γ-invariant αΦ -KMS state if and only if φ is extremal in Tγ (A)? The next proposition is a partial answer concerning the injectivity of ω 7→ φ in the above (1), whose proof is a slight modification of [4] (also [48, 4.7.3]); so we omit it. Proposition 2.6. Let ω, ω 0 ∈ K(Φ) and assume that ω is extremal. If ω and ω correspond to the same φ ∈ T (A), then ω = ω 0 . 0
As a consequence, when T (A) is a singleton (as in the finite depth case of Example 1.4), we obtain the uniqueness of αΦ -KMS state as a slight generalization of [4, 28, 47]. (The existence of αΦ -KMS state is due to [44].) 3. Variational Principle In this section let (A, {A[i,j] }, γ) be as in Sec. 1 and φ be a fixed tracial state on A satisfying (iv) and (v) in Sec. 1. For n ≥ 1 and ω ∈ Sγ (A), let S(ωn , φn ) be the relative entropy of ωn = ω|A[1,n] with respect to φn = φ|A[1,n] . Lemma 3.1. If ω ∈ Sγ (A), then limn→∞ n1 S(ωn , φn ) exists and lim
n→∞
1 1 S(ωn , φn ) = sup S(ωn , φn ) , n n≥1 n
(3.1)
Proof. It suffices to show the superadditivity: S(ωm+n , φm+n ) ≥ S(ωm , φm ) + S(ωn , φn ) ,
m, n ≥ 1 .
Put Dn = dωn /dφn , which is a positive element of A[1,n] with φ(Dn ) = 1. Then S(ωn , φn ) = ω(log Dn ). By (ii)–(v) in Sec. 1, Dm γ m (Dn ) is a positive element of A[1,m+n] and φ(Dm γ m (Dn )) = φ(Dm )φ(γ m (Dn )) = 1. So we have by the positivity of relative entropy 0 ≤ S(Dm+n , Dm γ m (Dn )) = φ(Dm+n (log Dm+n − log Dm − log γ m (Dn ))) = ω(log Dm+n ) − ω(log Dm ) − ω ◦ γ m (log Dn ) = S(ωm+n , φm+n ) − S(ωm , φm ) − S(ωn , φn ) . Hence {S(ωn , φn )} is a superadditive sequence.
For each ω ∈ Sγ (A), the mean relative entropy of ω with respect to φ is defined by (3.1) above and is denoted by SM (ω, φ) (∈ [0, +∞]).
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
833
Proposition 3.2. The function ω 7→ SM (ω, φ) is affine and weakly* lower semicontinuous on Sγ (A). Moreover, if ω ∈ Sγ (A), then SM (ω, φ) = 0 if and only if ω = φ. Proof. Since ω 7→ S(ωn , φn ) is weakly* continuous, the weak* lower semicontinuity of ω 7→ SM (ω, φ) follows from (3.1). The affinity is immediate from the following: For any ω, ω 0 ∈ Sγ (A) and 0 ≤ α ≤ 1 αS(ωn , φn ) + (1 − α)S(ωn0 , φn ) ≥ S(αωn + (1 − α)ωn0 , φn ) ≥ αS(ωn , φn ) + (1 − α)S(ωn0 , φn ) + α log α + (1 − α) log(1 − α) . Indeed, the first inequality is a convexity property of relative entropy and the second is seen from the operator monotonity of log t. If ω ∈ Sγ (A) and SM (ω, φ) = 0, then S(ωn , φn ) = 0 for all n ≥ 1 by (3.1), which implies ω = φ. Proposition 3.2 shows that φ is extremal in Sγ (A) or equivalently extremal in Tγ (A) by Proposition 1.1. However, φ is not necessarily extremal in T (A) (see the last sentence of Example 1.4 or Remark 5.2). For ω ∈ Sγ (A), the von Neumann entropy S(ωn ) of ωn is given as dωn dωn dωn = −ω log S(ωn ) = Trn − log dTrn dTrn d Trn dφn , = −S(ωn , φn ) − ω log d Trn
(3.2)
because dωn /d Trn = (dωn /dφn )(dφn /d Trn ) and dφn /d Trn belongs to the center of A[1,n] . We define the mean entropy s(ω) by 1 S(ωn ) , n→∞ n
s(ω) = lim whenever the limit exists.
Proposition 3.3. Assume that τ is a tracial state on A satisfying (iv)–(vi) in Sec. 1. Then for every ω ∈ Sγ (A) the mean entropy s(ω) exists and s(ω) = log λ−1 − SM (ω, τ ) .
(3.3)
The function s(ω) is affine and weakly* upper semicontinuous on Sγ (A). Moreover, s(ω) ≤ log λ−1 = s(τ ), and s(ω) = log λ−1 if and only if ω = τ . Proof. The first assertion is obvious from (3.2) and (vi). What remains is immediate from Proposition 3.2. The above proposition shows that if a tracial state τ on A satisfying (iv)–(vi) exists, then it is uniquely determined as a γ-invariant state maximizing the mean entropy.
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F. HIAI and D. PETZ
Now let Φ be an interaction. In this section we assume that Φ is γ-invariant and has relatively short range: X ||Φ(X)|| < +∞ . |X| X30
Let B denote the set of all γ-invariant interactions of relatively short range, which becomes a real Banach space with the obvious linear operations and the norm P |||Φ||| = X30 ||Φ(X)||/|X|. Given Φ ∈ B define AΦ ∈ A by AΦ =
X Φ(X) . |X|
X30
Obviously ||AΦ || ≤ |||Φ|||. Moreover, for simplicity we write Hn = H[1,n] , the local Hamiltonian, and φcn = φc[1,n] , the local Gibbs state on A[1,n] with respect to Φ and φ, i.e. φcn (a) = φ(e−Hn a)/φ(e−Hn ), a ∈ A[1,n] . Lemma 3.4. If Φ ∈ B and ω ∈ Sγ (A), then limn→∞ n1 ω(Hn ) exists and lim
n→∞
1 ω(Hn ) = ω(AΦ ) . n
Proof. Since n X ω(Hn ) Hn 1 k γ (AΦ ) n − ω(AΦ ) ≤ n − n k=1
1 ≤ n we have
1 n ω(Hn )
n X X k=1
||Φ(X)|| : X 3 k, X ∩ [1, n]c 6= ∅ |X|
→ ω(AΦ ) in the same way as [9, 6.2.39].
Theorem 3.5. If Φ ∈ B, then limn→∞
1 n
,
log φ(e−Hn ) exists and
1 log φ(e−Hn ) = sup {−SM(ω, φ) − ω(AΦ )} . n→∞ n ω∈Sγ (A) lim
Proof. For every ω ∈ Sγ (A) we get dωn e−Hn 0 ≤ S(ωn , φcn ) = ω log − log dφn φ(e−Hn ) = S(ωn , φn ) + ω(Hn ) + log φ(e−Hn ) , so that
1 ω(Hn ) 1 log φ(e−Hn ) ≥ − S(ωn , φn ) − . n n n
(3.4)
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
835
By Lemmas 3.1 and 3.4 we have lim inf n→∞
1 log φ(e−Hn ) ≥ −SM (ω, φ) − ω(AΦ ) . n
Therefore lim inf n→∞
1 log φ(e−Hn ) ≥ sup {−SM (ω, φ) − ω(AΦ )} . n ω∈Sγ (A)
(3.5)
On the other hand, thanks to (v), for each n ∈ N a state ϕ(n) on A can be defined by (n)
ϕ
Qj−1 mn −Hn (e ) m=i γ (x) = φ x , φ(e−Hn )j−i
x ∈ A[in+1,jn] , i ≤ j ,
which is periodic, i.e. ϕ(n) ◦ γ n = ϕ(n) . Then we define ω (n) ∈ Sγ (A) by 1 X (n) ϕ ◦ γk . n n
ω (n) =
k=1
Using the convexity and the monotonicity of relative entropy, we have for j ∈ N 1X S(ϕ(n) ◦ γ k |A[1,jn] , φ|A[1,jn] ) n n
(n)
S(ωjn , φjn ) ≤
k=1
1X S(ϕ(n) ◦ γ k |A[−k+1,(j+1)n−k] , φ ◦ γ k |A[−k+1,(j+1)n−k] ) n n
≤
k=1
= S(ϕ(n) |A[1,(j+1)n] , φ|A[1,(j+1)n] ) d(ϕ(n) |A[1,(j+1)n] ) = ϕ(n) log d(φ|A[1,(j+1)n] ) =
j X
ϕ(n) ◦ γ mn (−Hn ) − (j + 1) log φ(e−Hn )
m=0
= −(j + 1)ϕ(n) (Hn ) − (j + 1) log φ(e−Hn ) . Since (n) (n) n X ϕ (Hn ) ϕ (Hn ) 1 (n) (n) k − ω (AΦ ) = − ϕ (γ (AΦ )) n n n k=1 n H 1X k n ≤ − γ (AΦ ) → 0 (n → ∞) n n k=1
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F. HIAI and D. PETZ
as in the proof of Lemma 3.4, we get given ε > 0 1 (n) S(ωjn , φjn ) SM (ω (n) , φ) = lim j→∞ jn 1 ϕ(n) (Hn ) − log φ(e−Hn ) n n 1 ≤ −ω (n) (AΦ ) − log φ(e−Hn ) + ε n for every n large enough. Therefore 1 lim sup log φ(e−Hn ) ≤ sup {−SM (ω, φ) − ω(AΦ )} . n→∞ n ω∈Sγ (A) ≤−
(3.6)
The result follows from (3.5) and (3.6).
Definition 3.6. Define the thermodynamic free energy or the pressure p(Φ, φ) of Φ ∈ B with respect to φ by 1 log φ(e−Hn ) . p(Φ, φ) = lim n→∞ n The above theorem asserts the variational equality: sup {−SM (ω, φ) − ω(AΦ )} .
p(Φ, φ) =
(3.7)
ω∈Sγ (A)
Since ω 7→ −SM (ω, φ) − ω(AΦ ) is affine and weakly* upper semicontinuous by Proposition 3.2, it follows that S(Φ, φ) = {ω ∈ Sγ (A) : p(Φ, φ) = −SM (ω, φ) − ω(AΦ )} is nonempty and forms a face of Sγ (A), so that S(Φ, φ) is a simplex. When ω ∈ S(Φ, φ), we say that ω satisfies the variational principle (or it is thermodynamically stable) with respect to Φ and φ. From the variational equality (3.7) the following can be easily shown as in [9, 6.2.40]. Proposition 3.7. p(Φ, φ) is convex in Φ ∈ B and |p(Φ, φ) − p(Ψ, φ)| ≤ |||Φ − Ψ||| ,
Φ, Ψ ∈ B .
The next proposition says that when a trace τ satisfies (vi) in Sec. 1 as well, the pressure p(Φ, τ ) is identical (up to an additive constant) to the usual one defined by using Trn instead of τn . So, our variational principle reduces to the usual one when A is a spin C ∗ -algebra with φ = τ the unique tracial state. Proposition 3.8. Assume that τ is a tracial state on A satisfying (iv)–(vi) in Sec. 1. Then for every Φ ∈ B, the limit P (Φ) = limn→∞ n1 log Trn (e−Hn ) exists and P (Φ) = p(Φ, τ ) + log λ−1 . Furthermore P (Φ) =
sup {s(ω) − ω(AΦ )} . ω∈Sγ (A)
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
837
Proof. Condition (vi) means that we have with εn ↓ 0 (n → ∞) exp{n(log λ − εn )}1 ≤
dτn ≤ exp{n(log λ + εn )}1 . d Trn
Since τ (e−Hn ) = Trn ((dτn /d Trn )e−Hn ), we get exp{n(log λ − εn )}Trn (e−Hn ) ≤ τ (e−Hn ) ≤ exp{n(log λ + εn )}Trn (e−Hn ) , so that
1 log Trn (e−Hn ) − 1 log τ (e−Hn ) + log λ ≤ εn . n n
This implies that P (Φ) exists and P (Φ) = p(Φ, τ )+log λ−1 . The variational equality for P (Φ) follows from (3.7) and (3.3). The next proposition asserts that when Φ ∈ B satisfies ||W[1,n] || = o(n) (this P is the case if X30 ||Φ(X)|| < +∞), a Gibbs state in the weak sense satisfies the variational principle. Proposition 3.9. Let Φ ∈ B be such that the surface energy Wn = W[1,n] is defined for every n ≥ 1 and n1 ||Wn || → 0. If ω ∈ Sγ (A) satisfies the Gibbs condition in the weak sense with respect to Φ and φ, then it satisfies the variational principle with respect to Φ and φ. Proof. Since by (3.4) S(ωn , φcn ) = S(ωn , φn ) + ω(Hn ) + log φ(e−Hn ) , we have
1 S(ωn , φcn ) = SM (ω, φ) + ω(AΦ ) + p(Φ, φ) . n Hence it suffices to show that limn→∞ n1 S(ωn , φcn ) = 0. But we get lim
n→∞
S(ωn , φcn ) ≤ S(ω, [ω −Wn ]) ≤ 2||Wn || due to the monotonicity of relative entropy and (2.2). Hence the conclusion follows from the assumption of Φ. Now let B0 denote the set of all interactions Φ ∈ B such that X ||Φ(X)|| < +∞ and sup ||W[1,n] || < +∞ . X30
(3.8)
n≥1
Then Φ ∈ B0 generates a strongly continuous one-parameter automorphism group Φ αΦ satisfying αΦ t ◦ γ = γ ◦ αt . The next theorem means that S(Φ, φ) ⊂ K(Φ) for Φ ∈ B0 . Theorem 3.10. Let Φ ∈ B0 and ω ∈ Sγ (A). If ω satisfies the variational principle with respect to Φ and φ, then it satisfies the KMS condition with respect to αΦ . The details of this proof may be omitted because it is a slight modification of that in [46, 32] (also [9, 6.2.42]). It can be performed by use of convex analysis on
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F. HIAI and D. PETZ
the real Banach space B0 with the norm X ||Φ(X)|| + sup ||W[1,n] || (≥ |||Φ|||) . ||Φ||0 = X30
n≥1
At the end we use the following approximation property: If ||Φn − Φ||0 → 0 in B0 , Φ n then αΦ t (a) → αt (a) strongly as n → ∞ for all a ∈ A and t ∈ R, which can be seen because (1 ± δn )−1 → (1 ± δ)−1 strongly where δn and δ are the generators of αΦn and αΦ ([27], [48, 4.1.2]). The following is a consequence of Theorem 2.2, Propositions 2.6, 3.9, and Theorem 3.10 altogether, which is a complete analogue of the one-dimensional quantum spin case. Corollary 3.11. Assume that φ is a unique tracial state on A. If Φ ∈ B0 , then there exists a unique ω ∈ Sγ (A) which satisfies one (hence all ) of the following equivalent conditions: (i) (ii) (iii) (iv)
the the the the
KMS condition with respect to αΦ ; Gibbs condition in the strong sense with respect to Φ and φ; Gibbs condition in the weak sense with respect to Φ and φ; variational principle with respect to Φ and φ.
In view of Problems 2.5 we are interested in the following: Problems 3.12. Let φ, φ0 be tracial states on A satisfying (iv) and (v) in Sec. 1. (1) If ω ∈ S(Φ, φ), then does ω satisfy the Gibbs condition in the strong or weak sense with respect to Φ and φ? (2) Is S(Φ, φ) a singleton? (3) Are S(Φ, φ) and S(Φ, φ0 ) disjoint if φ 6= φ0 ? As a weak result in this direction we give: Proposition 3.13. Let φ, φ0 be different tracial states on A satisfying (iv) and (v) in Sec. 1. Then S(Φ, φ) and S(Φ, φ0 ) are disjoint for Φ ∈ B with sufficiently small |||Φ|||. Proof. Choose m ∈ N with φm 6= φ0m . Suppose that ω ∈ S(Φ, φ) ∩ S(Φ, φ0 ). Since for 0 ≤ t ≤ 1 −SM (ω, φ) − ω(AΦ ) ≥ −SM ((1 − t)ω + tφ, φ) − ((1 − t)ω + tφ)(AΦ ) = −(1 − t)SM (ω, φ) − ((1 − t)ω + tφ)(AΦ ) by Proposition 3.2, we get SM (ω, φ) ≤ (φ − ω)(AΦ ) ≤ 2||AΦ || ≤ 2|||Φ||| . An inequality for the relative entropy (see [37, 1.15]) gives ||ωm − φm || ≤ {2S(ωm , φm )}1/2 ≤ 2(m|||Φ|||)1/2
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
839
thanks to (3.1). Since the same inequality holds with φ0 instead of φ, we have ||φm − φ0m || ≤ ||ωm − φm || + ||ωm − φ0m || ≤ 4(m|||Φ|||)1/2 , so that |||Φ||| ≥
1 m (||φm
− φ0m ||/4)2 . This shows the conclusion.
4. Gauge Actions and Chemical Potentials In this section we consider a quantum system obtained as the fixed point subalgebra of a quantum spin C ∗ -algebra by a gauge action of a compact group. This kind of C ∗ -systems are typical examples of field systems discussed in the chemical potential theory [7, 6]. Such systems arise also from Wassermann’s subfactors [53] in the subfactor theory. N Let F denote a one-dimensional quantum spin C ∗ -algebra k∈Z Fk , Fk being copies of Md (C), and γ the automorphism of right shift. Let G be a compact group and σ a (possibly reducible) unitary representation of G on V = Cd . So G acts on N End V = Md (C) as Ad σ. We set a product action β of G on F by βg = Z Ad σg , g ∈ G. Let A be the fixed point subalgebra F β of F . Then A is called the observable algebra, while F is called the field algebra. The restriction of the shift γ on A is Nj denoted by the same γ. Moreover for i, j ∈ Z, i ≤ j, set F[i,j] = k=i Fk and β A[i,j] = A ∩ F[i,j] = F[i,j] , the fixed point algebra of β|F[i,j] . Then A is the AF C ∗ -algebra generated by {A[i,j] } by [45, Proposition 2.1]. We call (A, {A[i,j] }, γ) the C ∗ -system of gauge invariance for (G, σ). Now let Φ ∈ B0 , that is, Φ is a γ-invariant interaction in the observable algebra A satisfying (3.8), which is also considered as an interaction in the field algebra F . So Φ generates a one-parameter automorphism group αΦ on F , whose restriction on A is the dynamics generated on A. As in Sec. 2 let K(Φ) denote the set of αΦ -KMS states on A. We further use the following notations: edEK(Φ) = {ω ∈ K(Φ) : extremal} , EK f (Φ) = {ω ∈ EK(Φ) : faithful} , ET (A) = {φ ∈ T (A) : extremal} , ET f (A) = {φ ∈ ET (A) : faithful} . Lemma 4.1. In the above setting, T (A) = Tγ (A), and φ ∈ T (A) is extremal if and only if φ satisfies (v) in Sec. 1. Proof. Let φ ∈ T (A) be extremal. Then by [45, Theorem 3.2] φ is the restriction N of a product state Z ψ0 on A where ψ0 is a state on Md (C). This implies that φ is γ-invariant and multiplicative in the sense of (v). Hence T (A) = Tγ (A). R Each φ ∈ T (A) has the integral decomposition φ = ET (A) ρdm(ρ) where m is a S∞ probability measure on ET (A). Assume that φ satisfies (v). If a ∈ n=1 A[−n,n] is selfadjoint, then we get for j large enough
840
F. HIAI and D. PETZ
(Z
)2 ρ(a)dm(ρ)
= φ(a)2 = φ(aγ j (a))
ET (A)
Z ρ(aγ j (a))dm(ρ)
= Z
ET (A)
ρ(a)2 dm(ρ) ,
= ET (A)
which implies that ρ(a) is constant for m-a.e. ρ ∈ ET (A). Thus m is supported on a single point, that is, φ is extremal. Thus ET f (A) coincides with the set of all φ ∈ T (A) satisfying (iv) and (v) in Sec. 1. Proposition 4.2. In the above setting, every ω ∈ K(Φ) is γ-invariant. Proof. It suffices to show that EK(Φ) ⊂ Sγ (A). Let ω ∈ EK(Φ). By Theorem 2.2 there exists φ ∈ T (A) such that ω satisfies the Gibbs condition in the strong sense with respect to Φ and φ. Then, as in the proof of Proposition 2.4, we have for every Λ ⊂ Z and a ∈ AΛ [(ω ◦ γ)−WΛ ](a) = [ω −WΛ+1 ](γ(a)) = φcΛ+1 (γ(a)) = (φ ◦ γ)cΛ (a) = φcΛ (a) thanks to Lemma 4.1. This implies that ω ◦ γ satisfies the Gibbs condition in the weak sense with respect to Φ and φ. Hence ω = ω ◦ γ follows from Proposition 2.6. By the asymptotic abelianness of (A, γ), the above proposition shows that K(Φ) is a face of Sγ (A), so that if ω ∈ K(Φ) then ω ∈ EK(Φ) if and only if ω is extremal in Sγ (A). From now on the notion of chemical potentials enters into our discussions. Associated with (G, σ) above, let us introduce the set Ξ(G, σ) as follows: Ξ(G, σ) is the set of all continuous one-parameter subgroups t 7→ ξt of G, where we identify ξ, ξ 0 ∈ Ξ(G, σ) if Adσξt0 = Adσg−1 ξt g , t ∈ R, for some g ∈ G. Then Ξ(G, σ) can be regarded as the chemical potentials in the setting of this section. Also, the next proposition says that Ξ(G, σ) gives a parametrization of the elements of ET f (A). Proposition 4.3. The bijective correspondence φ ↔ ξ between ET f (A) and Ξ(G, σ) is determined in the way that φ extends to a KMS state with respect to βξt . Proof. For each ξ ∈ Ξ(G, σ), since t 7→ σξt is a continuous one-parameter unitary group, there exists a unique selfadjoint h ∈ Md (C) such that τ0 (e−h ) = 1 and Ad σξt = Ad eith , t ∈ R, where τ0 is the tracial state of Md (C). Noting that the N N product state Z τ0 (e−h · ) is a unique KMS state with respect to βξt = Z Ad σξt ,
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
we define φ as the restriction of
N
Z τ0 (e
−h
841
· ) on A. For any a ∈ A[−n,n] , since
X n n X j j exp it γ (h) a exp − it γ (h) j=−n
=
O n
j=−n
O n eith a e−ith = βξt (a) = a ,
−n
t ∈ R,
−n
we get
a,
n X
γ j (h) = 0 .
(4.1)
j=−n
Hence for any a, b ∈ A[−n,n] φ(ab) =
O n
τ0
−n
exp
γ (h) ab = φ(ba) ,
n X
−
j
j=−n
which implies that φ is tracial on A. Moreover φ is faithful and obviously satisfies (v) in Sec. 1. So φ ∈ ET f (A) by Lemma 4.1. Conversely let φ ∈ ET f (A) be given. By [6] (also the final remark of [9, Sec. 5.4.3]), φ has an extremal γ-invariant extension ψ to F and ψ is a KMS N 0 state with respect to βξt = Z Adσξt for some ξ ∈ Ξ(G, σ). Furthermore, if ψ is another extremal γ-invariant extension of φ, then ψ 0 = ψ ◦ βg for some g ∈ G by [6, Theorem II.1] (also [9, 5.4.24]) and the corresponding ξ 0 ∈ Ξ(G, σ) satisfies βξt0 = βg−1 βξt βg or Ad σξt0 = Ad σg−1 ξt g . Thus we obtain the conclusion. In particular, assume that G is a compact connected Lie group with maximal torus T of dimension N . Let F T denote the fixed point algebra of the restriction N of β = Z Adσ to T , so that A ⊂ F T ⊂ F . It is known [18] that every extremal tracial state on A extends to an extremal tracial state on F T . Moreover, every N element of ET f (F T ) is of the form Z Tr(D(r) · ) where the density matrix D(r) is explicitly written with a vector r ∈ (RN )++ . Thus the set ET f (A) in this case is parametrized by (RN )++ , more precisely, by a fundamental domain for the action of the Weyl group of G on (RN )++ (see [18] for details). Now let Φ ∈ B0 , φ ∈ ET f (A), and ξ ∈ Ξ(G, σ) with φ ↔ ξ in the sense of Proposition 4.3. As in the proof of Proposition 4.3, we have a unique selfadjoint element h of F0 = Md (C) such that τ0 (e−h ) = 1 and Ad σξt = Ad eith , t ∈ R. Define an interaction Φh in the field algebra F as follows: ( h
Φ (X) =
Φ({j}) + γ j (h) Φ(X)
if X = {j}, j ∈ Z , otherwise.
(4.2)
Then Φh is γ-invariant and generates a one-parameter automorphism group h α on F. So there exists a unique αΦ -KMS state on F , which is automatically extremal in Sγ (F) and faithful (see [4, 28]). Φh
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F. HIAI and D. PETZ
Lemma 4.4. With the above notations, Φ αΦ t βg = βg αt , h
= αΦ αΦ t βξt , t
t ∈ R, g ∈ G,
(4.3)
t ∈ R.
(4.4)
h
Hence αΦ = αΦ |A. Proof. Since βg (Φ(X)) = Φ(X) and hence βg (H[−n,n] ) = H[−n,n] , we get eitH[−n,n] βg (a)e−itH[−n,n] = βg (eitH[−n,n] ae−itH[−n,n] ) for all a ∈ A, t ∈ R, and g ∈ G. Letting n → ∞ gives (4.3). The local Hamiltonian H[−n,n] (Φh ) of Φh inside [−n, n] is given by h
H[−n,n] (Φ ) = H[−n,n] +
n X
γ j (h) .
j=−n
Since [H[−n,n] ,
Pn j=−n
γ j (h)] = 0 by (4.1), we have h
exp(itH[−n,n] (Φ )) = e
itH[−n,n]
X n j exp it γ (h) . j=−n
Hence (4.4) follows.
The above (4.3) shows that (F , A, G, αΦ , β, γ) is a field system in the chemical potential theory. For ω ∈ K(Φ) and ξ ∈ Ξ(G, σ), we say that ξ is the chemical potential of ω if there exists an extension ϕ of ω to F which satisfies the αΦ t βξt -KMS condition. By (4.4) and the fact mentioned before Lemma 4.4, we see that there exists a unique ω ∈ K(Φ) with the chemical potential ξ, which is automatically γ-invariant and faithful. Theorem 4.5. Let Φ ∈ B0 , φ ∈ ET f (A), and ξ ∈ Ξ(G, σ) with φ ↔ ξ in the sense of Proposition 4.3. Then the following conditions for a state ω on A are equivalent: (i) ω ∈ K(Φ) with the chemical potential ξ; (ii) ω satisfies the Gibbs condition in the strong sense with respect to Φ and φ; (iii) ω is γ-invariant and satisfies the Gibbs condition in the weak sense with respect to Φ and φ. Moreover, if ω satisfies the above conditions, then ω ∈ EK f (Φ) and φcΛ converges to ω weakly* as Λ → Z. Proof. (i) ⇒ (ii). Assume that ω extends to an αΦ t βξt -KMS state ϕ on F . By (4.4) and [9, 6.2.42 and its remark], ϕ satisfies the Gibbs condition with respect to
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
843
Φh . In the following let us work in the von Neumann algebra M = πϕ (F )00 via the Φ GNS representation πϕ . Since αΦ t βξt (A) = αt (A) = A and the normal extension Φ of αt βξt to M is the modular automorphism group for the normal extension ϕ˜ of ϕ, the conditional expectation E : M → πϕ (A)00 with respect to ϕ˜ exists. Set ˜ (πϕ (a)), a ∈ A. For any finite interval Λ ⊂ Z, since ω ˜ = ϕ|π ˜ ϕ (A)00 so that ω(a) = ω 00 πϕ (−WΛ ) ∈ πϕ (A) , we have by Lemma 2.3 ω πϕ (−WΛ ) ] ◦ E . [ϕ−WΛ ]˜= [ϕ˜πϕ (−WΛ ) ] = [˜
(4.5)
But it is seen that [˜ ω πϕ (−WΛ ) ] is the normal extension of [ω −WΛ ] via πϕ , i.e. ω πϕ (−WΛ ) ](πϕ (a)) , [ω −WΛ ](a) = [˜
a ∈ A,
(4.6)
because (2.3) holds with πϕ (A)00 instead of πω (A)00 . By (4.5) and (4.6) we have [ω −WΛ ] = [ϕ−WΛ ]|A. Hence, if a ∈ AΛ and b ∈ AΛc , then τ (e−HΛ (Φ ) a) −WΛ [ω ](b) , τ (e−HΛ (Φh ) ) h
[ω −WΛ ](ab) = [ϕ−WΛ ](ab) =
where τ is the tracial state of F and HΛ (Φh ) is the local Hamiltonian of Φh inside Λ. Since X γ j (h) HΛ (Φh ) = HΛ + and [HΛ ,
j∈Λ
P j∈Λ
γ j (h)] = 0 as (4.1), we get
τ (e−HΛ (Φ ) a) = τ h
exp
−
X
γ j (h) e−HΛ a = φ(e−HΛ a) ,
(4.7)
j∈Λ
so that [ω −WΛ ](ab) = φcΛ (a)[ω −WΛ ](b) . This together with (i) ⇒ (ii) of Theorem 2.2 implies (ii). (ii) ⇒ (iii). The γ-invariance of ω follows from Theorem 2.2 and Proposition 4.2. (iii) ⇒ (i). Let ϕ be a unique αΦ t βξt -KMS state on F , so that ωξ = ϕ|A is the unique state satisfying (i). Since ϕ is extremal in Sγ (F ) (i.e. γ-ergodic), it is readily seen that ωξ is extremal in Sγ (A) and so ωξ ∈ EK f (Φ). If ω satisfies (iii), then we have ω ∈ K(Φ) by Proposition 3.9 and Theorem 3.10. Hence Proposition 2.6 implies ωξ = ω. For the last assertion, ωξ ∈ EK f (Φ) was already shown. Since the local Gibbs state on FΛ for Φh converges to ϕ weakly* as Λ → Z, for every Λ0 ⊂ Λ and a ∈ AΛ0 we get by (4.7) τ (e−HΛ (Φ ) a) → ϕ(a) = ωξ (a) (Λ → Z) . = τ (e−HΛ (Φh ) ) h
φcΛ (a)
Hence φcΛ → ωξ weakly*.
844
F. HIAI and D. PETZ
Since [6] says that each ω ∈ EK f (Φ) extends to an αΦ t βξt -KMS state on F for some ξ ∈ Ξ(G, σ), we thus obtain the following bijective correspondences: φ ∈ ET f (A) ↔ ξ ∈ Ξ(G, σ) ↔ ω ∈ EK f (Φ) , where ξ is the chemical potential of ω and ω satisfies the Gibbs condition in the strong sense with respect to Φ and φ. Then we have: Theorem 4.6. The above correspondence φ 7→ ω is a weak*-homeomorphism from ET f (A) onto EK f (Φ). Before the proof we give a few preparations concerning irreducible decompoNn Nn Nn b sitions of 1 σ, n ≥ 1. Let Kn = {ρ ∈ G : ρ ≺ 1 σ} and write 1 σ = P m ρ with multiplicities m . Then the direct summands of A n,ρ n,ρ [1,n] are ρ∈Kn L indexed by Kn , that is, A[1,n] = ρ∈Kn A[1,n] fn,ρ and A[1,n] fn,ρ ∼ = Mmn,ρ (C) with minimal central projections fn,ρ . The following property will be useful in the proof below: 1 log max dim ρ = 0 . (4.8) lim n→∞ n ρ∈Kn Here we give a short proof for convenience. Consider the direct sum π = σ ⊕ σ ¯ : G → SU (2d), where σ ¯ is the representation conjugate to σ. Then Kn is included Nn in the irreducible decomposition of 1 σ0 ◦ π, where σ0 denotes the standard representation of SU (2d). Hence max dim ρ ≤
ρ∈Kn
max Nn
ζ≺
1
dim ζ . σ0
It is well known in unitary representation theory that the sequence {maxζ≺Nn σ0 dim ζ}∞ n=1 has polynomial growth (hence subexponential growth). So 1
(4.8) is shown. Proof of Theorem 4.6. We denote the correspondence in question by φ 7→ ωφ . Let φk , φ ∈ ET f (A) and corresponding ξk , ξ ∈ Ξ(G, σ) be given. Let ϕk and ϕ be Φ the KMS states with respect to αΦ t βξk,t and αt βξt , respectively. Moreover, for each ξk take a unique selfadjoint hk ∈ Md (C) as in the proof of Proposition 4.3. Pd Let λk,1 ≥ · · · ≥ λk,d be the eigenvalues of hk . Since j=1 e−λk,j = d, note that 1 ≤ e−λk,d ≤ d and so 0 ≥ λk,d ≥ − log d. First assume that φk → φ weakly*. To show that ωφk → ωφ weakly*, it is enough to show that {ωφk } has a subsequence which converges to ωφ . Now suppose {hk } is unbounded. Choosing a subsequence we may assume that λk,1 → +∞ and hk /λk,1 → h∞ , where h∞ has the largest eigenvalue 1 and the smallest 0. Then N it is easy to see by [9, 5.3.25] that { Z τ0 (e−hk · )} has a weak*-limit point ψ∞ N which is a ground state with respect to Z Ad eith∞ . Hence ψ∞ is a non-faithful N product state on F. But, since φk = Z τ0 (e−hk · )|A → φ, we have φ = ψ∞ |A, contradicting φ ∈ ET f (A) (see [6, Theorem II.1]). Thus {hk } is bounded, so that we
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
may assume ||hk − h|| → 0 for some h ∈ Md (C). Then we have φ = and hence Ad σξt = Ad eith = lim Ad eithk = lim Ad σξk,t , k→∞
845
N
Z τ0 (e
−h
· )|A
k→∞
→ strongly. This implies by [9, 5.3.25] that ϕk → ϕ and thus so that ωφk = ϕk |A → ωφ = ϕ|A. Conversely assume that ωφk → ωφ weakly*. Suppose {hk } is unbounded. Then we may assume that λk,1 → +∞ and hk /λk,1 → h∞ . As in the first half, {ϕk } N has a weak*-limit point ϕ∞ which is a ground state with respect to Z Ad eith∞ . Furthermore ωφ = ϕ∞ |A. Now let Pn be the support projection of ϕ∞ |F[1,n] , and Pn j Qn be the eigenprojection of j=1 γ (h∞ ) corresponding to the smallest eigenvalue. Then Pn ≤ Qn (see [9, 6.2.52]). For each ρ ∈ Kn , note that the inclusion A[1,n] fn,ρ ⊂ fn,ρ F[1,n] fn,ρ can be identified with the following: αΦ t βξk,t
αΦ t βξt
a ∈ Mmn,ρ (C) 7→ a ⊗ I ∈ Mmn,ρ (C) ⊗ Mdim ρ (C) . Pn Pn Since j=1 γ j (hk ) commutes with A[1,n] , so does j=1 γ j (h∞ ). Hence Qn commutes with A[1,n] . Under the above identification of A[1,n] fn,ρ ⊂ fn,ρ F[1,n] fn,ρ , this implies that Qn fn,ρ ∈ (Mmn,ρ (C) ⊗ Mdim ρ (C)) ∩ (Mmn,ρ (C) ⊗ I)0 = I ⊗ Mdim ρ (C) .
(4.9)
If Qn fn,ρ = 0, then fn,ρ Pn fn,ρ = 0 and so ωφ |A[1,n] fn,ρ = ϕ∞ |A[1,n] fn,ρ = 0, which contradicts the faithfulness of ωφ . Therefore Qn fn,ρ 6= 0, so that rank Qn fn,ρ ≥ mn,ρ by (4.9). Since this holds for each ρ ∈ Kn and 1, 0 are among the eigenvalues of h∞ , we get X mn,ρ ≤ rank Qn ≤ (d − 1)n , ρ∈Kn
which implies that dn =
X
mn,ρ dim ρ ≤ (d − 1)n max dim ρ .
ρ∈Kn
So we arrive at log
1 d ≤ log max dim ρ , ρ∈Kn d−1 n
ρ∈Kn
n ≥ 1,
contradicting (4.8). Thus {hk } is bounded. So assume ||hk − h0 || → 0 and let N φ0 = Z τ0 (e−h0 · )|A. Then, as in the first half, we have ωφk → ωφ0 = ωφ and thus φk → φ0 = φ. Under an additional assumption, we can show the equivalence between the Gibbs condition and the variational principle as follows. Proposition 4.7. In the situation of Theorem 4.5, if ξ is in the center of G (in particular, if G is abelian), then the conditions of Theorem 4.5 for a state ω on A is equivalent to the following: (iv) ω is γ-invariant and ω ∈ S(Φ, φ).
846
F. HIAI and D. PETZ
N Proof. Let ψ = Z τ0 (e−h · ) so that φ = ψ|A. The assumption of ξ being in the center of G implies that Adσg (σξt ) = σξt , g ∈ G. Hence for any Λ ⊂ RZ, we have N N ith ∈ AΛ and so Λ e−h ∈ AΛ . Define EA : F → A by EA (a) = G βg (a)dg Λe (dg is the normalized Haar measure on G), which is a τ -preserving conditional expectation and satisfies EA (FΛ ) = AΛ for all Λ ⊂ Z. For each ω ∈ Sγ (A), define ϕ = ω ◦ EA and put ωn = ω|A[1,n] , ϕn = ϕ|F[1,n] , En = EA |F[1,n] , etc. Since ωn ◦ En = ϕn and φn (En (a)) = τ
O n
e−h EA (a)
1
=τ
O n
e
−h
a = ψn (a) ,
a ∈ F[1,n] ,
1
we get by [37, 5.15] S(ωn , φn ) = S(ωn ◦ En , φn ◦ En ) = S(ϕn , ψn ) dϕn dψn dϕn log = Trn − log d Trn d Trn d Trn X n j n γ (h) − log d = −S(ϕn ) − ϕn − j=1
= −S(ϕn ) + nϕ(h) + n log d . Therefore SM (ω, φ) = −s(ϕ) + ϕ(h) + log d .
(4.10)
On the other hand, we get by (4.7) p(Φ, φ) = lim
n→∞
1 log φ(e−H[1,n] ) = P (Φh ) − log d , n
(4.11)
where P (Φh ) is the pressure of Φh in the field algebra F . Furthermore AΦh = AΦ + h .
(4.12)
By (4.10)–(4.12) altogether we see that ω ∈ S(Φ, φ) if and only if ϕ satisfies the variational principle with respect to Φh ; equivalently ϕ is an αΦ t βξt -KMS state by (4.4). This shows that (iv) implies (i) of Theorem 4.5. Proposition 3.9 gives the converse. In view of Theorems 4.5, 4.6, and Proposition 4.7 altogether, Problems 2.5 and 3.12 are to some extent solved in the case of C ∗ -systems of gauge invariance. In the rest of this section we briefly explain that C ∗ -systems treated in this section are derived from a certain model of Example 1.4. Let σ : G → End V be a finite-dimensional unitary representation of a compact group G and set σk = σ ¯ (k ∈ 2Z + 1) where σ ¯ is the representation conjugate to σ. (k ∈ 2Z), σk = σ
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
847
Nj For j ∈ Z let Rj be the type II1 factor generated by −∞ End V via the GNS representation for the tracial state, and Mj be the fixed point subalgebra Rjβ under N the product action βj = jk=−∞ Ad σk . Since (Rjβ )0 ∩ Rj = C as is well known, Mj is a type II1 factor. Then the so-called Wassermann’s subfactor [53] arising from σ is defined as M−1 ⊂ M0 , i.e. Rβ ⊂ (R ⊗ EndV )β⊗Adσ with R = R−1 , which is an extremal inclusion of type II1 factors with the index [M0 : M−1 ] = (dim V )2 . In the construction above we may take as β any minimal action on R. (In fact, minimal actions on R are unique [43].) The Jones tower of M−1 ⊂ M0 is {Mj }∞ j=−∞ defined above and the relative N 0 commutants A[i,j] = Mi−1 ∩ Mj are given by A[i,j] = ( ji End V )β , the fixed point N algebra of jk=i Ad σk . In particular, the derived tower {A[1,n] }n≥0 is C ⊂ (End V )Adσ¯ ⊂ (End V ⊗ End V )Adσ¯ ⊗Adσ ⊂ (EndV ⊗ End V ⊗ End V )Adσ¯ ⊗Adσ⊗Adσ¯ ⊂ ··· and the standard invariant of M−1 ⊂ M0 is determined by the way of irreducible decompositions of σ ¯ ⊗σ ⊗· · ·⊗σ ¯ ⊗ σ (like the Clebsch–Gordan rules). For example, the vertices of the principal graph corresponding to the direct summands of A[1,2n] is given by b:ρ≺σ Kn = {ρ ∈ G ¯ ⊗ σ ⊗ ···⊗ σ ¯ ⊗ σ (n factors of σ ¯ ⊗ σ)} S and the standard eigenvector ~s = (sρ )ρ∈K , K = n Kn , is given by sρ = dim ρ. N The C ∗ -algebra generated by {A[i,j] } is A = F β where F = Z End V . Note that the canonical shift (1.3) on A coincides with γ 2 |A where γ is the right shift on F. In this way, we observe that the C ∗ -system derived from Wassermann’s subfactor M−1 ⊂ M0 is nothing but the field system obtained by the gauge action σ ¯ ⊗ σ : G → End V ⊗ End V . In particular, when σ is self-conjugate, γ restricts on A and we arrive at the same setting as was stated at the beginning of this section. Example 4.8. According to Popa’s classification [40, 42] of hyperfinite type II1 subfactors of index 4, there are three cases having infinite depth. All of these are realized as Wassermann’s subfactors (see [17, 4.7.d]). Consider the following three unitary representations on V = C2 , all of which are self-conjugate: z 0 (1) G = T and σ(z) = , z ∈ T. 0 z −1 (2) G = SU (2) and σ(g) = g, g ∈ SU (2). 0 −w−1 z 0 : z, w ∈ T , the infinite dihedral , (3) G = D∞ = w 0 0 z −1 group, and σ(g) = g, g ∈ D∞ . Then Wassermann’s subfactors for representations (1)–(3) have the principal graphs A∞,∞ , A∞ , and D∞ , respectively. (See [19, 56] for principal graphs of
848
F. HIAI and D. PETZ
Wassermann’s subfactors from other irreducible representations of SU (2).) Wassermann’s subfactor for (2) coincides with Jones’ subfactor R1/4 ⊂ R. The derived C ∗ -algebras are the fixed point algebras F SU(2) ⊂ F D∞ ⊂ F T of the CAR alN T is sometimes gebra F = Z M2 (C) under the respective product actions. (F denote the product state called the GICAR algebra.) For 0 ≤ r ≤ 1 let ψ r N r 0 · on F . Then the set ET (A) of extremal tracial states on Z Tr 0 1−r A = F G is given as ET (A) = {φr : 0 ≤ r ≤ 1} for each G where φr = ψr |A. But, for cases (2) and (3), we must write ET (A) = {φr : 0 ≤ r ≤ 1/2} because φr = φ1−r (φλ in Example 1.3 is φr with λ = r(1 − r)). Note that φ0 and φ1 are trivial traces whose GNS representations are one-dimensional. itµ The chemical 0 e , t ∈ R, potential ξr ∈ Ξ(G, σ) corresponding to φr is given by ξr,t = 0 e−itµ √ −1 where µ ∈ R is determined by r = e−µ /(e−µ + eµ ) or µ = log r − 1. Since p 0 1 0 −1 , ξr and ξ1−r ξr,t −µ = log (1 − r)−1 − 1 and hence ξ1−r,t = −1 0 1 0 are identified in cases (2) and (3). By Theorem 4.6, given Φ ∈ B0 , the elements of EK f (Φ) are parametrized as a continuous family ωr (0 < r < 1 or 0 < r ≤ 1/2). An argument as in the proof of Theorem 4.6 shows that ωr → φ0 , φ1 weakly* as r → 0, 1, respectively. With a bit more efforts, we can show that EK(Φ) = {ωr : 0 < r < 1} ∪ {φ0 , φ1 }
in case (1) ,
EK(Φ) = {ωr : 0 < r ≤ 1/2} ∪ {φ0 } in cases (1) and (2) , and thus EK(Φ) ∼ = ET (A) ∼ = [0, 1] or [0, 1/2] (weakly* homeomorphic). It is of some interest to determine EK(Φ) \ EK f (Φ) in a more general case. 5. C ∗ -Systems of Random Walks on Groups There is another important model of subfactors obtained by finitely generated discrete groups of automorphisms [8, 42, 49], whose derived C ∗ -systems are imbedded in quantum spin C ∗ -algebras. Taking account of the description of derived towers in [8] and [42, 5.1.5], we introduce a C ∗ -system associated with a finitely generated discrete group. Let G be a discrete group with a finite number of generators g1 , . . . , gd in the sense that the semigroup generated by g1 , . . . , gd is all of G. Here, gr = gs for different r, s is allowed and the identity e of G is not necessarily contained in {gr }. Let Fk = Md (C), k ∈ Z, and {ekrs }dr,s=1 be the matrix units of Fk . For i, j ∈ Z, Nj i ≤ j, define A[i,j] ⊂ k=i Fk by O j ekrk sk : grj · · · gri = gsj · · · gsi , A[i,j] = Alg k=i
whose direct summands are indexed by g ∈ Kj−i+1 where Kn = {g ∈ G : g = grn · · · gr1 for some 1 ≤ r1 , . . . , rn ≤ d} ,
n ∈ N.
(5.1)
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
849
L Thus we have A[1,n] ∼ = g∈Kn Mdn,g (C) where (5.2) dn,g = #{(r1 , . . . , rn ) : 1 ≤ r1 , . . . , rn ≤ d, g = grn · · · gr1 } . N The C ∗ -algebra generated by {A[i,j] } is included in F = Z Fk and the right shift γ restricts on A. Then we call (A, {A[i,j] }, γ) the C ∗ -system of random walk on G. Let G∗ denote the set of all homomorphisms from G into the multiplicative m,f group R++ = (0, ∞). We use the notation Tγ (A) to mean the set of all φ ∈ T (A) Pd satisfying (iv) and (v) in Sec. 1. For each χ ∈ G∗ , set W = r=1 χ(gr ) and define D ∈ Md (C) by d d X D= χ(gr )err . (5.3) W r=1 N Furthermore, define a product state ψ = Z τ0 (D · ) on F where τ0 is the tracial state of Md (C). Proposition 5.1. For every χ ∈ G∗ define ψ as above and φ = ψ|A. Then the correspondence χ 7→ φ is bijective from G∗ onto Tγm,f (A). Proof. Let φ be defined from χ ∈ G∗ as above. Let i, j ∈ Z, i ≤ j, and rk , sk ∈ {1, . . . , d} for i ≤ k ≤ j. If (ri , . . . , rj ) 6= (si , . . . , sj ), then O Y j j φ ekrk sk = τ0 (Dekrk sk ) = 0 . k=i
Also we get
k=i
Y O j j χ(g) χ(grk ) k = j−i+1 e rk rk = φ W W k=i
k=i
with g = grj · · · gri . By definition of A[i,j] these imply that φ is tracial on A[i,j] . Hence φ is a trace. It is immediate that φ satisfies (iv) and (v) in Sec. 1. Conversely let φ ∈ Tγm,f (A) be given. Assume that e = gpl · · · gp1 = gp00 · · · gp01 l
0
where pk , p0k0 ∈ {1, . . . , d}. Since e = (gpl · · · gp1 )l = (gp00 · · · gp01 )l , it follows that l
l O l O 0
j=1
0
l O l O
0
+k ep(j−1)l k pk
and
j=1
k=1
(j−1)l+k
ep0 p0
k k
k=1
are in the same direct summand of A[1,ll0 ] . Hence by the γ-invariance and the multiplicativity property of φ we get l0 l O O l l0 k epk pk =φ ekp0 p0 . φ k k
k=1
k=1
Nl So we can define W = φ( k=1 ekpk pk )−1/l independently of the expression e = gpl · · · gp1 . Now let g = grn · · · gr1 = gr0 0 · · · gr10 and g −1 = gr0000 · · · gr100 . Since n
n
e = gr0000 · · · gr100 grn · · · gr1 = gr0000 · · · gr100 gr0 0 · · · gr10 , n
n
n
850
F. HIAI and D. PETZ
we have W
−(n+n00 )
O O n00 n k k =φ er00 r00 φ e rk rk , k
k
k=1
W
−(n0 +n00 )
k=1
O O n00 n0 k k =φ er00 r00 φ er 0 r 0 , k
k
k k
k=1
k=1
N N 0 0 so that W n φ( nk=1 ekrk rk ) = W n φ( nk=1 ekr0 r0 ). Hence χ(g) ∈ R++ is well defined k k by O n if g = grn · · · gr1 . χ(g) = W n φ ekrk rk k=1
If g = grn · · · gr1 and h = gsm · · · gs1 , then O O n m ekrk rk φ eksk sk = χ(g)χ(h) . χ(gh) = W n+m φ k=1
Therefore χ ∈ G∗ . For this χ we have the conclusion follows.
k=1
Pd r=1
χ(gr ) = W
Pd r=1
φ(e1rr ) = W . Thus
m,f
Remark 5.2. Let χ ∈ G∗ and φ ∈ Tγ (A) with χ ↔ φ in the sense of Proposition 5.1. Set a finitely supported probability measure µ on G by µ(g) = P W −1 gr =g χ(gr ) for g ∈ G. Then φ can be described in terms of the random walk (or the Markov chain) on G with the initial distribution δe and the transition probabilities p(g|h) = µ(gh−1 ), g, h ∈ G. Indeed, the φ-value of a minimal central projection of A[1,n] corresponding to g ∈ Kn is given by µn (g), where µn is the nth convolution of µ. Hence we know by [25, 26] that φ is an extremal tracial state if and only if the random walk (G, µ) has the trivial Poisson boundary (i.e. the µharmonic bounded functions on G are trivial), equivalently the entropy h(G, µ) = limn→∞ n1 H(µn ) ([26]) is equal to 0. In particular, if G has subexponential growth, m,f
then h(G, µ) = 0 for any finitely supported measure µ, so that all φ ∈ Tγ (A) are extremal in T (A). By the way, it is worth noting [26] that there is a solvable (hence amenable) group G for which h(G, µ) > 0 for any finitely supported nondegenerate measure µ on G. So, in this case, every φ ∈ Tγm,f (A) is not extremal in T (A), though extremal in Tγ (A) by the remark after Proposition 3.2. Remark 5.3. Assume that G is abelian. Then for any i, j ∈ Z, i < j, it is P obvious that U (i, j) = dr,s=1 eirs ⊗ ejsr belongs to A[i,j] . Since Ad U (i, j) acts on F as the transposition interchanging the ith and jth components, the permutation group S(∞) on Z appears in the unitary group of A. So, applying Størmer’s results in the same way as in [45] we can show that any extremal tracial state on A is the restriction of a symmetric product state on F . Here we need a conditional expectation EA : F → A, whose existence is seen from the fact that the tracepreserving conditional expectations EΛ : FΛ → AΛ satisfy the commuting square
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
851
AΛ0 ⊂ FΛ0 ∪ ∪ , Λ ⊂ Λ0 . Hence Lemma 4.1 holds true and we have AΛ ⊂ FΛ ET f (A) = Tγm,f (A). (See also Example 5.6(2) below.) property for
Lemma 5.4. For any χ ∈ G∗ set D ∈ Md (C) by (5.3). Then the center Z(AΛ ) of AΛ for every finite interval Λ ⊂ Z.
N Λ
D belongs to
Proof. The case Λ = [1, n] is enough. We get n O
D=
1
=
n X
d W
χ(grn · · · gr1 )e1r1 r1 ⊗ · · · ⊗ enrn rn
r1 ,...,rn
n X X d χ(g) W g ···g g∈Kn
e1r1 r1 ⊗ · · · ⊗ enrn rn .
r1 =g
rn
P Since grn ···gr =g e1r1 r1 ⊗ · · · ⊗ enrn rn , g ∈ Kn , are central projections of A[1,n] , we 1 have the result. m,f
Now let Φ ∈ B0 , φ ∈ Tγ tion 5.1. Set
(A), and χ ∈ G∗ with φ ↔ χ in the sense of Proposi-
h = − log D = −
d X
log
r=1
d χ(gr ) err . W
N Define a one-parameter automorphism group βt on F by βt = Z Ad eith and an interaction Φh in F by (4.2). Since by Lemma 5.4 O X j − γ (h) = log D ∈ Z(AΛ ) , Λ ⊂ Z, j∈Λ
Λ
we have as Lemma 4.4 βt (a) = a ,
a ∈ A, t ∈ R , h
Φ Φ αΦ t βt = βt αt = αt ,
t ∈ R.
Thus, the next theorem can be proved in the same way as Theorems 4.5, 4.6, and Proposition 4.7. In the same proof as (iv) ⇒ (i) in Proposition 4.7, we use a conditional expectation EA : F → A preserving the tracial state τ of F mentioned in Remark 5.3. Also, in the proof of the last assertion, it is enough to look at A[1,1] ⊂ F[1,1] because we do not have factors such as Mdim ρ (C) in the proof of Theorem 4.6. Theorem 5.5. With the above assumptions and notations, the following conditions for ω ∈ Sγ (A) are equivalent: (i) ω is an αΦ -KMS state extending to an αΦ t βt -KMS state on F ; (ii) ω satisfies the Gibbs condition in the strong sense with respect to Φ and φ;
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(iii) ω satisfies the Gibbs condition in the weak sense with respect to Φ and φ; (iv) ω ∈ S(Φ, φ). Moreover, for each φ ∈ Tγm,f (A) there exists a unique ω ∈ Sγ (A) satisfying the above conditions, and the correspondence φ 7→ ω is a weak*-homeomorphism from Tγm,f (A) into the set of faithful extremal γ-invariant αΦ -KMS states on A. Examples 5.6. (1) Assume that G is a finite group and fix a generating finite set {g1 , . . . , gd }. Then G∗ = {1} and φ = τ |A is a unique trace on A as Proposition 1.5. Hence Corollary 3.11 holds in this case. (2) Let G = ZN and {w1 , . . . , wd } be any generating set. Define a unitary b = TN on Cd by representation σ of G σz = diag(z w1 , . . . , z wd ) , w(1)
where z w = z1 ZN . Since
w(N )
· · · zN n O
for z = (z1 , . . . , zN ) ∈ TN and w = (w(1), . . . , w(N )) ∈ X
σz =
1
it follows that A[1,n] = Alg
z wr1 +···+wrn e1r1 r1 ⊗ · · · ⊗ enrn rn ,
r1 ,...,rn
O n
ekrk sk : wr1 + · · · + wrn = ws1 + · · · + wsn
k=1 β where β is the product action of Ad σz , z ∈ TN . Hence the C ∗ is equal to F[1,n] system of random walk on ZN coincides with the C ∗ -system of gauge invariance for (TN , σ). Thus (ZN )∗ is isomorphic to Ξ(TN , σ) and by Theorem 4.6 it parametrizes the elements of EK f (Φ) given Φ ∈ B0 . Here the isomorphisms χ ∈ (ZN )∗ ↔ r ∈ (RN )++ ↔ ξ ∈ Ξ(TN , σ) are given as follows:
r = (r1 , r2 , . . . , rN ) = (χ(1, 0, . . . , 0), χ(0, 1, 0, . . . , 0), . . . , χ(0, . . . , 0, 1)) , ξt = (e−itr1 , e−itr2 , . . . , e−itrN ) ,
t ∈ R.
Note that the above arguments remain valid when G is an arbitrary abelian discrete group with a generating finite set. So Theorem 5.5 for abelian G is included in the results of the previous section. (3) Let G = FN , the free group with N generators g1 , . . . , gN . Set gN +k = gk−1 , 1 ≤ k ≤ N . Then the C ∗ -system (A, {A[i,j] }, γ) of random walk on FN is a subN system of F = Z M2N (C). Since (FN )∗ ∼ = (RN )++ , Theorem 5.5 says that there are, given Φ ∈ B0 , faithful extremal γ-invariant αΦ -KMS states parametrized by (RN )++ , while it is not known whether these exhaust such αΦ -KMS states on A. 6. Dynamical Entropies In this section let us discuss dynamical entropy and topological entropy (and their relation) in C ∗ -systems introduced in Sec. 1. Noncommutative dynamical entropy was first studied in [15] for an automorphism of a finite von Neumann algebra
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
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with respect to a normal tracial state. This Connes–Størmer dynamical entropy was later on extended in [14] to the so-called CNT-dynamical entropy in a general C ∗ -algebra setup. As was clarified in [14], the CNT-dynamical entropy nicely behaves for automorphisms (or endomorphisms) of nuclear C ∗ -algebras. So we consider this entropy as the most appropriate in our AF C ∗ -systems (A, {A[i,j] }, γ). Let hω (γ) denote the CNT-dynamical entropy of γ with respect to ω ∈ Sγ (A). Since S A = n A[−n,n] , we have by [14, Corollary V.4] hω (γ) = lim hω,γ (A[−n,n] ) , n→∞
where hω,γ (A[−n,n] ) = lim
m→∞
1 Hω (A[−n,n] , γ(A[−n,n] ), . . . , γ m−1 (A[−n,n] )) . m
(See [14] or [37, Chap. 10] for the definition of Hω above.) On the other hand, noncommutative versions of topological entropy were recently studied in [21, 22, 52] in AF C ∗ -algebras (or rather “local” C ∗ -algebras). Although the definitions of topological entropy in [21] and [52] are a bit different, they are identical for our γ. In fact, the topological entropy ~(γ) of γ on the S S local C ∗ -algebra n A[−n,n] (also n A[1,n] ) is given as ([21, Theorem 3.32], [52, Lemma 3.2]) 1 log Trn (1) , ~(γ) = lim n→∞ n where Trn (1) means the tracial dimension (i.e. the number of orthogonal minimal projections) of A[1,n] . As in the classical probabilistic case we first have: Proposition 6.1. hω (γ) ≤ ~(γ) for every ω ∈ Sγ (A). Proof. For any ω ∈ Sγ (A), since by [14, Proposition III.6] Hω (A[−n,n] , γ(A[−n,n] ), . . . , γ m−1 (A[−n,n] )) = Hω (A[1,2n+1] , A[2,2n+2] , . . . , A[m,2n+m] ) ≤ Hω (A[1,2n+m] ) ≤ S(ω2n+m ) ,
(6.1)
and S(ω2n+m ) ≤ log Tr2n+m (1), we have hω,γ (A[−n,n] ) ≤ lim
m→∞
1 log Tr2n+m (1) = ~(γ) . m
Hence hω (γ) ≤ ~(γ).
The next result is on the lines of [11, Theorem 1] and [13, Proposition 4.2]. Proposition 6.2. If ω ∈ Sγ (A) is multiplicative in the sense of (v) in Sec. 1, then the mean entropy s(ω) exists and hω (γ) = hω (γ|A[1,∞) ) = s(ω) .
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Proof. The first equality is immediate because hω,γ (A[−n,n] ) = hω,γ (A[1,2n+1] ) Wm−1 by the proof of Proposition 6.1. For each n, m ∈ N, let B = k=0 γ nk (A[1,n] ) and {qj : 1 ≤ j ≤ l} be a set of minimal projections in the centralizer of ωn such Pl that j=1 qj = supp ωn , the support projection of ωn . Let qjk = γ n(k−1) (qj ) for 1 ≤ j ≤ l and 1 ≤ k ≤ m. Then by the multiplicativity of ω, {qj11 · · · qjmm : 1 ≤ j1 , . . . , jm ≤ l} is a set of minimal projections in the centralizer of ω|B such that P 1 m 1 m 1 m j1 ,...,jm qj1 · · · qjm = supp(ω|B) and ω(qj1 · · · qjm ) = ω(qj1 ) · · · ω(qjm ). Hence we get by [14, Corollary VIII.8] Hω (A[1,n] , γ n (A[1,n] ), · · · , γ n(m−1) (A[1,n] )) = S(ω|B) = mS(ωn ) , so that by [14, VII.5 ii)] 1 hω (γ n ) n 1 1 lim Hω (A[1,n] , γ n (A[1,n] ), . . . , γ n(m−1) (A[1,n] )) ≥ n m→∞ m 1 = S(ωn ) . n
hω (γ) =
Therefore
1 hω (γ) ≥ lim sup S(ωn ) . n→∞ n
On the other hand, we get by (6.1) hω,γ (A[−n,n] ) ≤ lim inf m→∞
1 S(ωm ) . m
Hence s(ω) exists and hω (γ) = s(ω).
Theorem 6.3. Assume that τ is a tracial state on A satisfying (iv)–(vi) in Sec. 1. Then: hτ (γ) = ~(γ) = log λ−1 = sup hω (γ) . ω∈Sγ (A)
Moreover, if ω ∈ Sγ (A), then hω (γ) = ~(γ) if and only if ω = τ . Proof. The equality hτ (γ) = log λ−1 follows from Propositions 3.3 and 6.2. We get ~(γ) = log λ−1 by letting Φ = 0 (so Hn = 0) in the proof of Proposition 3.8. Furthermore, for any ω ∈ Sγ (A) we have hω (γ) ≤ s(ω) ≤ log λ−1
(6.2)
by Proposition 3.3 and the proof of Proposition 6.1. Hence the first part is shown. The second is immediate from (6.2) together with the last part of Proposition 3.3. For instance, let γ be the canonical shift and τ the λ-Markov trace in the C ∗ system of Example 1.4 arising from an inclusion N ⊂ M of type II1 factors. If
QUANTUM MECHANICS IN AF C ∗ -SYSTEMS
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N ⊂ M has the standard eigenvector ~s of subexponential growth, then τ satisfies (iv)–(vi) in Sec. 1 and so we have Theorem 6.3 with λ−1 = [M : N ]. Furthermore, in this setting, it can be seen by [20, Corollary 4.9] that Theorem 6.3 remains true when N ⊂ M is extremal and strongly amenable (see [42] for strong amenability). In what remains we give a bit more detailed discussions on entropies specializing in the cases of C ∗ -systems in Secs. 4 and 5. First let σ be a unitary representation of a compact group G on V = Cd and (A, {A[i,j] }, γ) the C ∗ -system of gauge invariance for (G, σ), that is, A is the fixed point algebra of the product action N N βg = Z Ad σg , g ∈ G, on F = Z Md (C). Let τ be the restriction on A of the tracial state of F , which of course satisfies (iv) and (v) in Sec. 1. The direct b : ρ ≺ Nn σ} and the τ -trace summands of A[1,n] are indexed by Kn = {ρ ∈ G 1 vector of A[1,n] is given by (dim ρ/dn )ρ∈Kn . Hence, thanks to (4.8), τ satisfies (vi) in Sec. 1 as well with λ−1 = d = dim V ; so Theorem 6.3 holds. In the next theorem we exactly compute hω (γ) when ω is the restriction on A of a symmetric product state on F . This can be applied to extremal tracial states on A due to [45, Theorem 3.2] (also Lemma 4.1). Theorem 6.4. If ϕ = then
N Z
ϕ0 on F and ω = ϕ|A where ϕ0 is a state on Md (C),
hω (γ) = s(ω) = hϕ (γ) = S(ϕ0 ) . Proof. The first equality follows from Proposition 6.2 and the last is well known. By [6, Corollary III.2.3(iii)] we have πϕ (F β|Gϕ )00 = πϕ (A)00 for the GNS representation πϕ and the subgroup Gϕ = {g ∈ G : ϕ ◦ βg = ϕ} of G. Recall [14, γ ) where ω ˜ and γ˜ are the normal extensions of ω Theorem VII.2] that hω (γ) = hω˜ (˜ and γ via πω . Thus, to prove the theorem, we may assume that ϕ ◦ βg = ϕ for all N g ∈ G. Concerning the decompositions of n1 σ and A[1,n] , let us use the notations mentioned before the proof of Theorem 4.6. Let TrF[1,n] be the canonical trace of F[1,n] and set Dn = dϕn /d TrF[1,n] . The β-invariance of ϕ implies that Dn ∈ A[1,n] and hence Dn is written as n,ρ X m X
Dn =
µn,ρ (j)pn,ρ (j) ,
ρ∈Kn j=1
where pn,ρ (j), 1 ≤ j ≤ mn,ρ , are minimal projections in A[1,n] fn,ρ so that they are P Pmn,ρ µn,ρ (j) dim ρ = 1 of rank dim ρ as projections in F[1,n] . Therefore ρ∈Kn j=1 and S(ωn ) = −
n,ρ X m X
µn,ρ (j) dim ρ log(µn,ρ (j) dim ρ)
ρ∈Kn j=1
= S(ϕn ) −
n,ρ X m X
ρ∈Kn j=1
µn,ρ (j) dim ρ log dim ρ .
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F. HIAI and D. PETZ
Since 0≤
n,ρ X m X
µn,ρ (j) dim ρ log dim ρ ≤ log max dim ρ , ρ∈Kn
ρ∈Kn j=1
it follows from (4.8) that s(ω) = s(ϕ) = S(ϕ0 ), completing the proof.
Example 6.5. In all cases (1)–(3) of Example 4.8, Theorems 6.3 and 6.4 show that for 0 ≤ r ≤ 1 hφr (γ) = s(φr ) = −r log r − (1 − r) log(1 − r) ≤ hφ1/4 (γ) = ~(γ) = log 2 . Example 1.3 for λ ≤ 1/4 is nothing but case (2) with φλ = φr , λ = r(1 − r), and the above formula hφλ (γ) = −r log r − (1 − r) log(1 − r) as well as hφλ (γ) = 12 log λ−1 for λ > 1/4 of Example 1.3 was computed in [38, 10, 55]. Furthermore, it R is not so difficult to show the following integral formula: For any φ ∈ T (A), if φ = φr dm(r) is the extremal decomposition of φ with a probability measure m on [0, 1] or [0, 1/2], then Z hφ (γ) = s(φ) = {−r log r − (1 − r) log(1 − r)}dm(r) . We omit the details of this proof. Finally let (A, {A[i,j] }, γ) be the C ∗ -system of random walk on a discrete group G with generators g1 , . . . , gd . In this case, the restriction τ on A of the tracial state of F satisfies (iv)–(vi) in Sec. 1, because n−1 log dτn /d Trn = (log d−1 )1 for all n. Hence Theorem 6.3 holds with λ−1 = d. Proposition 6.6. Assume that G has subexponential growth. If ϕ ∈ Sγ (F ) and ω = ϕ|A, then s(ω) = s(ϕ). Proof. Let En : F[1,n] → A[1,n] be the conditional expectation with respect to TrF[1,n] , which is written as X fn,g afn,g , a ∈ F[1,n] , En (a) = g∈Kn
where Kn is given by (5.1) and fn,g are the minimal central projections of A[1,n] . Set Dn = dϕn /d TrF[1,n] . Then, since Trn = TrF[1,n] |A[1,n] , we have dωn /d Trn = En (Dn ) and S(ωn ) − S(ϕn ) = Trn (Dn (log Dn − log En (Dn ))) = S(Dn , En (Dn )) . PN Now write Dn = j=1 µj pj where µj > 0, j=1 µj = 1, and pj are projections of rank one. By the joint convexity of relative entropy we get PN
S(Dn , En (Dn )) ≤
N X j=1
µj S(pj , En (pj )) =
N X j=1
µj S(En (pj )) .
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P Since the rank of En (pj ) = g∈Kn fn,g pj fn,g is at most #Kn , we have S(En (pj )) ≤ log(#Kn ) and hence 0 ≤ S(Dn , En (Dn )) ≤ log(#Kn ). Therefore 0≤
1 1 1 S(ωn ) − S(ϕn ) ≤ log(#Kn ) → 0 (n → ∞) n n n
thanks to the assumption of subexponential growth. This gives the conclusion. By Propositions 5.1, 6.2, and 6.6 we have: m,f
Corollary 6.7. Assume that G has subexponential growth. If φ ∈ Tγ χ ∈ G∗ with φ ↔ χ as in Proposition 5.1, then hφ (γ) = s(φ) = −
d X χ(gr ) r=1
W
log
(A) and
χ(gr ) . W
Pd −1 ers , which is a rank one projection in Example 6.8. Let D = r,s=1 d N Md (C). Define ϕ = Z Tr(D · ) on F and ω = ϕ|A. Let dn,g (g ∈ Kn ) be given by P (5.2). Then it is easy to check that dωn /d Trn = g∈Kn (dn,g /dn )pn,g where pn,g is a minimal projection in the direct summand of A[1,n] corresponding to g ∈ Kn . Hence, if µ is the distribution on G given by µ(g) = #{r : gr = g}/d for g ∈ G, then X dn,g dn,g log n = H(µn ) , S(ωn ) = − dn d g∈Kn
so that by Proposition 6.2 we have hω (γ) = s(ω) = h(G, µ). On the other hand, hϕ (γ) = s(ϕ) = 0. So, in view of Remark 5.2, this example shows that the assumption of subexponential growth is essential in Proposition 6.6. When −1 } (d = 2N ), it is known [8] that the generating set is {g1 , . . . , gN , g1−1 , . . . , gN 2N −2 h(G, µ) ≤ 2N log(2N − 1) for the above µ and the equality occurs only when G = FN , the free group on N generators. References [1] H. Araki, “Relative Hamiltonian for faithful normal states of a von Neumann algebra”, Publ. Res. Inst. Math. Sci. 9 (1973) 165–209. [2] H. Araki, “Golden–Thompson and Peierls–Bogolubov inequalites for a general von Neumann algebra”, Comm. Math. Phys. 34 (1973) 167–178. [3] H. Araki, “On the equivalence of the KMS condition and the variational principle for quantum lattice systems”, Comm. Math. Phys. 38 (1974) 1–10. [4] H. Araki, “On uniqueness of KMS states of one-dimensional quantum lattice systems”, Comm. Math. Phys. 44 (1975) 1–7. [5] H. Araki, “Relative entropy for states of von Neumann algebras II”, Publ. Res. Inst. Math. Sci. 13 (1977) 173–192. [6] H. Araki, R. Haag, D. Kastler, and M. Takesaki, “Extension of KMS states and chemical potential”, Comm. Math. Phys. 53 (1977) 97–134. [7] H. Araki and A. Kishimoto, “‘Symmetry and equilibrium states”, Comm. Math. Phys. 52 (1977) 211–232. [8] D. Bisch, “Entropy of groups and subfactors”, J. Funct. Anal. 103 (1972) 190–208.
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DISCRETE SPECTRUM ASYMPTOTICS FOR ¨ THE SCHRODINGER OPERATOR WITH A SINGULAR POTENTIAL AND A MAGNETIC FIELD A. V. SOBOLEV1 School of Mathematical Sciences University of Sussex Falmer, Brighton BN1 9QH, UK Received 25 July 1995 Object of the study is the operator H = H0 (h, µ) + V in L2 (Rd ), d ≥ 2, where H0 (h, µ) is the Schr¨ odinger operator with a magnetic field of intensity µ ≥ 0 and the Planck constant h ∈ (0, h0 ]. The electric (real-valued) potential V = V (x) is assumed to be asymptotically homogeneous of order −β, β ≥ 0 as x → 0. One obtains asymptotic formulae with remainder estimates as h → 0, µh ≤ C for the trace Ms = tr{φgs (H)} where φ ∈ C0∞ (Rd ), g(λ) = λs− , s ∈ [0, 1]. Due to the condition µh ≤ C the leading term of Ms does not depend on µ. It depends on the relation between the parameters d, s and β. There are five regions, in which either leading terms or remainder estimates have different form. In one of these regions Ms admits a two-term asymptotics. In this case, for an asymptotically Coulomb potential the second term coincides with the well-known Scott correction term.
1. Introduction odinger operator We study in L2 (Rd ), d ≥ 2, the Schr¨
Ha,V = Ha,V (h, µ) = H0 (h, µ) + V , d X (−ih∂l − µal )2 . H0 = H0 (h, µ) =
(1.1)
l=1
Here H0 = Ha,0 is the unperturbed operator with a magnetic real-valued vectorpotential a = (a1 , a2 , . . . , ad ), the parameter µ ≥ 0 has the meaning of intensity of the field. The function V (electric potential) is real-valued. Sometimes for the sake of brevity we use the notation a = (a, V ). We analyse the asymptotics as2 h → 0, µ ≥ 0, µh ≤ C of traces of the form Ms (h, µ) = Ms (h, µ; ψ, a) = tr{ψgs (Ha )} .
(1.2)
Here ψ ∈ C0∞ (Rd ) and the function gs , s ≥ 0, is defined as follows: ) gs (λ) = |λ|s , λ < 0 , gs (λ) = 0, λ ≥ 0 . 1 Author
supported by EPSRC under grant B/94/AF/1793 and in what follows we denote by C and c (with or without indices) various positive constants whose precise value is of no importance. 2 Here
861 Reviews in Mathematical Physics, Vol. 8, No. 6 (1996) 861–903 c World Scientific Publishing Company
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A. V. SOBOLEV
In case µ = 0 we write HV (h) (or HV ) and Ns (h; ψ, V ) instead of Ha,V (h, µ) and Ms (h, µ; ψ, a) respectively. The condition µh ≤ C will ensure that the leading terms in the asymptotic formulae which we obtain, do not depend on the magnetic field. In this sense the magnetic field under consideration is moderate, though the inequality µh ≤C allows µ to grow as h → 0. The quantity (1.2) can be viewed as a “local version” of the sum X |λk |s , Ms (h, µ; 1, a) = k≥1
where λk = λk (h, µ; a), k ≥ 1, are negative eigenvalues of Ha enumerated in the non-decreasing order. In particular, M0 is a local counterpart of the number of all negative eigenvalues. Note that due to the truncation ψ the trace (1.2) can be finite even if negative spectrum of Ha is not discrete. The asymptotics of Ms (h, µ) has been analysed in [14] in the case a, V ∈ ∞ C0 (Rd ). It was shown there that for any s ∈ [0, 1], h → 0, 0 ≤ µ ≤ Ch−1 , the trace (1.2) obeys the formula Ms (h, µ; ψ, a) = Ws (h; ψ, V ) + hµis+1 O(h−(d−s−1) ) ,
1
hµi = (1 + µ2 ) 2 ,
(1.3)
with the standard Weylian leading term Ws (h) = Ws (h; ψ, V ) = Ξs h Ξs =
|Sd−1 | (2π)d
Z
−d
Z
s+ d2 ψ(x) V− (x) dx ,
1
td−1 (1 − t2 )s dt ,
(1.4)
0
where |S | stands for the surface area of the (d − 1)-dimensional unit sphere. Corresponding result for the case µ = 0 was obtained in [8, 9]. In the present paper the smoothness assumption is removed. More precisely, we assume that the function V is C ∞ outside x = 0 and asymptotically homogeneous at x = 0: d−1
V (x) ∼ W (x) =
Φ(ˆ x) , |x|β
x → 0;
x ˆ=
x , |x|
0 ≤ β < 2,
(1.5)
with some Φ ∈ C ∞ (Sd−1 ). In Sec. 2 we shall formulate the conditions on V in a more precise form. In fact, one could have assumed that there are several points at which the potential behaves similarly to (1.5). Our results however, are stated in a form which allows one to decouple the singularities using an appropriate partition of unity and then, by means of the translation, perform in each patch the reduction to the potential satisfying (1.5). Under condition (1.5) the formula (1.3) is not necessarily true. The answer depends on the relation between the parameters β, s and d. A natural parameter that determines the form of the asymptotics in this case is ω = ω(β, s) =
2βs . 2−β
(1.6)
DISCRETE SPECTRUM ASYMPTOTICS
863
The values d and d − s − 1 (which are the orders of h in the leading and remainder term in (1.3) respectively) serve as “thresholds” — when ω crosses either of them, the asymptotics of Ms changes its form. Note in particular, that the condition ω < d is necessary and sufficient for Ws to be finite. Let us discuss individually all possible cases. Below h → 0, 0 ≤ µ ≤ Ch−1 and 0 ≤ s ≤ 1. (1) ω > d. The leading term of Ms is completely determined by the asymptotic potential W defined in (1.5). Namely, Ms (h, µ; ψ, a) = h−ω ψ(0)Ns (1; 1, W ) + o(1) .
(1.7)
Finiteness of the trace Ns (1; 1, W ) for any d ≥ 2, ω > d will follow from the Cwickel type estimate (2.23). (2) ω = d. The asymptotics is still determined by W . However, in contrast to (1.7) the leading term can be calculated effectively. Denote 2Ξs Bs (Φ) = 2−β
Z Sd−1
d
Φ− (ϑ) 2 +s dϑ .
Then Ms (h, µ; ψ, a) = h−d | ln h|ψ(0)Bs (Φ) + (| ln h| + 1)o(h−d ) .
(1.8)
We emphasize that the remainders in (1.7), (1.8) are bounded uniformly in µ ≤ Ch−1 . (3) d − s − 1 < ω < d. The leading order is given by the classical Weyl coefficient (1.4). The singularity of the potential gives rise to a second term that occupies an intermediate position between the main term and the remainder in (1.3): Ms (h, µ; ψ, a) = Ws (h; ψ, V ) + h−ω ψ(0)Θs + o(h−ω ) + O(hµis+1 hs+1−d ) .
(1.9)
The coefficient Θs = Θs (Φ, β) depends only on W . Loosely speaking, it is defined as the difference of two infinite quantities: Θs (Φ, β) = Ns (1; 1, W ) − Ws (1; 1, W ) .
(1.10)
Theorems 2.4 and 2.40 in Sec. 2 provide two equivalent regularized versions of this definition. Presumably, in general, Θs does not admit any explicit representation in terms of W . However, for the particular case of a purely Coulomb potential, it does. Let d = 3, s = 1, β = 1, Φ = −q, q ≥ 0. Then, using precise formulae for the eigenvalues of HW (1), one can prove that Θ1 = −q 2 /8. This expression for Θ1 has been known since a long time in connection with the so-called Scott correction to the ground state energy of a large atom. We refer to [8] for details and further references. It is worth mentioning that by virtue of the condition d − s − 1 < ω < d, the choice β = 1, s = 1 implies that d = 3. In other words, the Scott correction term is meaningful only in the three-dimensional case.
864
A. V. SOBOLEV
(4) ω = d − s − 1. Then Ms (h, µ; ψ, a) = Ws (h; ψ, V ) + O(| ln h|hs+1−d ) + O(hµis+1 hs+1−d ) .
(1.11)
(5) ω < d − s − 1. The asymptotics looks like (1.3) — the contribution from the singularity gets “absorbed” by the remainder: Ms (h, µ; ψ, a) = Ws (h; ψ, V ) + O(hµis+1 hs+1−d ) .
(1.12)
In particular, the local counting function M0 satisfies (1.12). We point out that in all five cases above the leading term of Ms does not depend on µ. In cases (3)–(5) the asymptotics gets more precise if µh → 0. On the contrary, if µh = const, the formulae (1.9), (1.11), (1.12) lose their asymptotic character since the remainders have the same order as Ws in this situation. This observation is consistent with the well-known fact that the Weyl term no longer describes the behaviour of Ms (h, µ) if µh ≥ c. We refer to [12] (see also [15]), where the asymptotics of M1 (h, µ; 1, a) was studied for d = 3 and a homogeneous magnetic field for any µ ≥ 0. It was shown that for µh ≥ c the leading term is to be replaced by another coefficient that takes into account the magnetic field. To conclude the introduction we sketch main steps of the proof. As in [8], we analyse separately contributions from the regions around the origin and away from it. Precisely, we split Ms (h, µ; ψ, a) into the sum of Ms (h, µ; ψ1 , a) and Ms (h, µ; ψ2 , a), where ψ1 (resp. ψ2 ) is supported inside (resp. outside) the ball 2 B(r) = {x : |x| ≤ r} of radius r ∼ h 2−β . The share of the ball depends on the interrelation between ω and d. We explain further proof in the most interesting case d − s − 1 < ω < d. Inside the ball one can neglect the magnetic field and replace the potential V by its asymptotics W . This reduces the problem to the study of Ns (h; ψ1 , W ). Using homogeneity of W one can “scale out” the parameter h, after which the definition (1.10) yields almost automatically that Ms (h, µ; ψ1 , a) ∼ Ns (h; ψ1 , W ) = Ws (h; ψ1 , W ) + h−ω Θs + o(h−ω ) .
(1.13)
The change from W back to V affects only the error o(h−ω ). Since V is smooth outside B(r), we can use for Ms (h, µ; ψ2 , a) the asymptotics (1.3) established in [14]. However the result of [14] does not apply directly, because by (1.5) V (x) is not bounded uniformly in h for |x| ≥ r and, consequently, we cannot control the remainder estimate in (1.3). To avoid this difficulty, we use the so-called multiscale approach invented by V. Ivrii (see [9]–[11], [8]), that provides a good control of the remainder estimate under fairly general conditions on V . A version of this method adjusted to our purposes, is described in [14]. As a result, we obtain Ms (h, µ; ψ2 , a) = Ws (h; ψ2 , V ) + o(h−ω ) + O(hµis+1 hs+1−d ) . Adding up this relation with (1.13), we arrive at (1.9). We point out again that in the exterior of B(r) we need only the first term of the asymptotics with a proper
865
DISCRETE SPECTRUM ASYMPTOTICS
remainder estimate. The second term in (1.9) is produced totally by the interior of B(r) (see (1.13)). Precise definitions of objects we shall be working with, and statements of the main results are given in Sec. 2. In Secs. 3, 4 we investigate the possibility of replacing the potentials a, V with 0, W in the asymptotics of (1.2). In Sec. 5 we summarize the results from [14] on the multiscale analysis and establish existence of the limit (1.10) in a suitable regularized sense. Section 6 contains the proofs of the asymptotic formulae discussed above. Notation. As a rule, m stands for a d−tuple of non-negative integer numbers: m = (m1 , m2 , . . . , md ), |m| = m1 + m2 + · · · + md . For any measurable function f one writes f± = (|f | ± f )/2. For a domain X ⊂ Rd we denote by B ∞ (X) the set of functions f ∈ C ∞ (X) bounded along with all their derivatives. This space forms a Fr´echet space with the family of natural semi-norms ||||f ||||m = supx |∂xm f (x)|, ∀|m| ≥ 0. A constant C is said to be uniform in f ∈ B ∞ (X) (or f ∈ C0∞ (X)), if it depends only on the constants in the estimates ||||f ||||m ≤ Cm , |m| ≥ 0. A function g is said to belong to B ∞ (X) uniformly in f ∈ B ∞ (X) if the derivatives |∂xm g(x)|, |m| ≥ 0, are estimated by constants which are uniform in f ∈ B ∞ (X). For spaces of vector-valued functions a(x) = {a1 (x), . . . , ad (x)} we use the same notation as for scalar functions. This convention does not cause any confusion. For instance, the notation a ∈ L2loc (Rd ) means that each component of a belongs to L2loc (Rd ). B(z, E), z ∈ Rd , E > 0, denotes the closed ball {x ∈ Rd : |x − z| ≤ E}; ◦ B(E) = B(0, E). Sometimes we use open balls B(z, E) = {x ∈ Rd : |x − z| < E} ◦ ◦ and B(E) =B(0, E) as well. For any self-adjoint operator T , D(T ) denotes its domain and R(z; T ) = (T − z)−1 — its resolvent for z ∈ C outside the spectrum of T . If T is semibounded, T [ · , ·] and D[T ] stand for the associated quadratic form and its domain respectively. Notation Sp , p ≥ 1 stands for the Neumann–Schatten class of compact operators with the norm p 1 T p = tr{(T ∗ T ) 2 } p . Classes S1 , S2 are called the trace class and the Hilbert–Schmidt class respectively. Operators T1 ∈ Sp , T2 ∈ Sq and any bounded operator T0 satisfy the following inequalities: T1 T2 t ≤ T1 p T2 q , t−1 = p−1 + q −1 ; (1.14) T1 T0 ≤ k T0 k T1 . p
p
These and other properties of compact operators can be found in [7].
866
A. V. SOBOLEV
2. Main Results 1. Definition of the Schr¨ odinger operator with a magnetic field. Let a ∈ L2loc (Rd ), d ≥ 2 be a real-valued (vector-)function. We denote by Ql = Q∗` , Πl = Π∗` , l = 1, 2, . . . , d, closures of the differential operators −ih∂l − µal , −ih∂l , odinger operator as on C0∞ (Rd ). We define the unperturbed Schr¨ H0 = Ha,0 (h, µ) = Q∗l Ql ,
H0,0 = H0,0 (h) = Π∗l Πl = −h2 ∆ .
Here and below we assume summation over repeating indices. The operator H0 can also be interpreted as that associated with the quadratic form (Ql u, Ql u) (see [3]). Due to the condition a ∈ L2loc (Rd ), the set C0∞ (Rd ) is a form core for H0 . To define the perturbed operator we use the following estimates resulting from the diamagnetic inequality (see [3]): Proposition 2.1. Let X be multiplication by a measurable function and κ > 0. Then for any λ > 0 kXR(−λ; H0 )κ k ≤ kXR(−λ; H0,0)κ k
(2.1)
and for any positive integer n XR(−λ; H0 )κ
2n
≤ XR(−λ; H0,0 )κ
2n
.
(2.2)
It follows immediately from (2.1) with κ = 1/2 that the inequality kXuk2 ≤ (H0,0 u, u) + M (h)kuk2 , ∀u ∈ C0∞ (Rd ) ,
(2.3)
kXuk2 ≤ (H0 u, u) + M (h)kuk2 , ∀u ∈ C0∞ (Rd ) ,
(2.4)
entails with the same positive constants and M (h). Let V be a real-valued function such that the estimate (2.3) is fulfilled for the function X = |V |1/2 with some < 1 and M (h) > 0. Due to (2.4) the perturbed operator Ha = Ha,V = H0 + V is well defined in the form sense. To study the local trace (1.2) it will be sufficient to assume that the operator is of the form (1.1) only in a neighbourhood of supp ψ. Its behaviour outside is irrelevant. To distinguish it from the “true” Schr¨ odinger operator Ha we shall use for such an operator the notation Aa = Aa (h, µ) (or simply A). Assumptions on A will be stated in terms of the quadratic form A[ · , ·]. Below D ⊂ Rd denotes an open bounded domain. Assumption 2.2. The operator A is self-adjoint in L2 (Rd ), semibounded from below and for any ζ ∈ C0∞ (D) the following conditions are satisfied: (1) For any u ∈ D[A] one has uζ ∈ D[A]; there exists a function ζ1 ∈ C0∞ (D) (depending on ζ) such that A[u, ζv] = A[ζ1 u, ζv] , for all u, v ∈ D[A];
867
DISCRETE SPECTRUM ASYMPTOTICS
(2) There exist real-valued functions a ∈ L2loc (Rd ) and V with X = |V |1/2 obeying (2.3) with some ∈ (0, 1), M (h) > 0, such that for any v ∈ D[A], u ∈ D[Ha ], a = (a, V ), one has ζu ∈ D[A], ζv ∈ [Ha ] and A[ζu, ζv] = Ha [ζu, ζv] . Though this assumption may look cumbersome, it is very natural in the sense that it is fulfilled for some standard special cases. For instance, the operator Aa defined by the differential expression X (−ih∂l − al )2 + V (x) l
with the Dirichlet condition on the sphere {x : |x| = R}, R > 0, obeys Assumption ◦ 2.2 with D =B(R). Our basic tool in the study of the operator A = Aa is the following resolvent identity: Lemma 2.3. Let the operator A be as specified above. Then for any function ζ ∈ C0∞ (D) one has ζR(z; Aa ) = R(z; Ha )ζ + R(z; Ha )ZR(z; Aa ) ,
(2.5)
Z = Z(ζ) = −ih(2Q∗l ∂l ζ + ih∆ζ) .
(2.6)
Proof. Clearly, it suffices to verify that for any u ∈ D(A), v ∈ D(H), H = Ha , the equality holds: ¯ = (u, Z ∗ (ζ)v) . (2.7) (ζu, Hv) − (Au, ζv) To prove this notice that u ∈ D[A], v ∈ D[H], so that by Assumption 2.2 the l.h.s. of (2.7) equals ¯ = H[ζu, v] − H[ζ1 u, ζv] ¯ = (Ql ζu, Ql v) − (Ql ζ1 u, Ql ζv) ¯ H[ζu, v] − A[ζ1 u, ζv] ¯ = [Ql , ζ]ζ1 u, Ql v − Ql ζ1 u, [Ql , ζ]v ¯ . = −ih(∂l ζu, Ql v) − ih(Ql ζ1 u, ∂l ζv) Rewrite this as ¯ , ¯ = −ih u, (2∂l ζQ ¯ l − ih∆ζ)v −ih(∂l ζu, Ql v) − ih(ζ1 u, Ql ∂l ζv) which coincides with the r.h.s. of (2.7).
In the same way one proves the identity for powers of the resolvents: ζR(z; A)k = R(z; H)k ζ +
k X j=1
R(z; H)j Z(ζ)R(z; A)k−j+1 , ∀k ∈ N .
(2.8)
868
A. V. SOBOLEV
For tr{ψgs (A)} we keep the same notation as for tr{ψgs (H)}, i.e. Ms (h, µ; ψ, a). This will not cause any confusion in what follows. We shall use extensively the following scaling properties of the operator Aa and the trace Ms (h, µ; ψ, a). Let f , ` be some positive numbers and let z ∈ Rd . Let the unitary dilation operator U` and the translation operator Tz be defined by d
(U` u)(x) = ` 2 u(`x) ,
(Tz u)(x) = u(x + z) .
Denote Vˆ (x) = f −2 V (`x + z) ,
ˆ(x) = `−1 a(`x + z) , a
ˆ ψ(x) = ψ(`x + z) .
(2.9)
Define also two auxiliary parameters which will play the role of the Planck constant and the size of the magnetic field after the scaling: α=
h , f`
ν=
µ` . f
(2.10)
Since V obeys (2.3), we have 1 ˆ (α)kuk2 , k |Vˆ | 2 uk2 ≤ H0,0 (α)u, u + M
ˆ (α) = f −2 M (h) , M
∀u ∈ C0∞ (Rd ) . (2.11)
It is clear that the operator f −2 (U` Tz )Aa (U` Tz )∗
(2.12)
ˆ = {x ∈ Rd : `x + z ∈ D} and the operator satisfies Assumption 2.2 with the set D ˆ a = {ˆ a, V }. Therefore it is natural to denote the operator (2.12) by Aaˆ . Haˆ (α, ν), ˆ By the unitary equivalence of trace, ˆ ˆa) . Ms (h, µ; ψ, a) = f 2s Ms (α, ν; ψ,
(2.13)
Note also the scaling property of the Weyl coefficient (1.4): ˆ Vˆ ) , Ws (h; ψ, V ) = f 2s Ws (α; ψ,
(2.14)
which can be verified by direct calculation. ◦ ◦ ˆ is simply B(1). Notice that in the case D =B(z, `) the set D 2. Conditions on a, V . Let us specify conditions on the potential V and vector-potential a, for which we shall obtain asymptotic formulae described in the Introduction. Assume that V, a ∈ C ∞ (Rd \ {0}), a is continuous, and for all x 6= 0 ) |∂ m V (x)| ≤ Cm |x|−β−|m| , 0 ≤ β < 2 , |m| ≥ 0 ; |∂ m a(x)| ≤ Cm |x|1−|m| , |m| ≥ 1 .
(2.15)
Moreover,
DISCRETE SPECTRUM ASYMPTOTICS
869
V (x) = |x|−β Φ(ˆ x) + U (x) ,
(2.16)
with a function Φ ∈ C ∞ (Sd−1 ) and some U ∈ L∞ (Rd ) such that ess-sup|x|≤t |U (x)| ≤ U0 (t) ,
U0 ∈ L∞ (0, ∞) ,
U0 (t) → 0 ,
t → 0.
(2.17)
As a rule we use the notation W (x) = Φ(ˆ x)|x|−β . Notice that (2.15) contains no estimates on the function a itself, but only on its derivatives. This fact is quite natural, since the constant component of a can be chosen arbitrarily or eventually eliminated by a simple gauge transformation. 1 1 It is easy to check that the functions X = |V | 2 and X = |W | 2 obey (2.3) for any > 0 and β
M (h) = C(h2 )− 2−β ,
(2.18)
the constant C here being dependent only on C0 in (2.15). In the sequel, we usually subsume into the constant C and omit from notation. Unless otherwise stated all the functions denoted by O( · ) or o( · ) will be uniform in all C ∞ -functions involved (in the sense specified in the end of the Introduction) and in µ ∈ [0, Ch−1 ]. Moreover, they will also be uniform in the functions V , a satisfying (2.15), (2.16) and (2.17). In other words, they will depend only on the constants Cm from (2.15) and the function U0 . For instance, o(1) in (1.7) stands for a function which tends to zero as h → 0 uniformly in ψ ∈ C0∞ (Rd ), the functions V , W , U and µ ≤ Ch−1 . The symbols “lim”, “lim sup” are used in situations, when no uniformity (in the sense specified above) is claimed. ◦
3. Results. We always assume that Aa obeys Assumption 2.2 with D = B(4E) for some fixed E > 0. The function ψ is supposed to belong to C0∞ (B(E/2)). As was mentioned in the Introduction, the form of the asymptotics is governed by the parameter ω defined in (1.6). For some values of ω the leading or the second term are expressed in terms of the trace Ns for the operator HW . In the next theorem we prepare some properties of this trace needed for stating our main result. Let Ws be the Weyl coefficient defined in (1.4). Below we denote φρ (x) = φ(xρ−1 ), ρ > 0, for any function φ. Theorem 2.4. Let Φ ∈ C ∞ (Sd−1 ), s ∈ (0, 1], β ∈ (0, 2) and W (x) =
Φ(x) , Φ ∈ C ∞ (Sd−1 ) . |x|β
Let φ ∈ C0∞ (Rd ) be a function such that φ(x) = 1, |x| ≤ 1 and let φρ (x) = φ(x/ρ), ρ > 0. (1) If ω > d, then Ns (1; φρ , W ) = Ns (1; 1, W ) + o(1) ,
ρ → ∞.
(2.19)
870
A. V. SOBOLEV
(2) If d − s − 1 < ω(β, s) < d, then Ns (1; φρ , W ) − Ws (1; φρ , W ) = Θs + o(1) ,
ρ → ∞,
(2.20)
with some real number Θs = Θs (Φ, β) independent of the function φ. The relation (2.20) can be interpreted as a “regularized limit” of Ns (1; φρ , W ). A remarkable fact is that there is another choice of regularization that yields the same value of Θs : Theorem 2.40 . Let the parameters β, s and the function W be as in Theorem 2.4, and let d − s − 1 < ω(β, s) < d. Then lim Ns (1; 1, W + κ) − Ws (1; 1, W + κ) = Θs (Φ, β).
κ→+0
(2.21)
This limit may not be uniform in the function W . We stress again that the relations (2.19), (2.20) are uniform in W and φ. If one does not require any uniformity, the proof of (2.19) is fairly simple. Indeed, the finiteness of Ns in (2.19) can be easily obtained from the following bound for the number of negative eigenvalues of HW +κ , κ > 0 ( see [4]). For any V ∈ Lqw (Rd ), q > d/2, d d N0 (1; 1, V + κ) ≤ Cκ 2 −q kV kqLq = Cκ 2 −q sup tq mes{x : V− (x) > t} . w
(2.22)
t>0
This bound results from a Cwickel type estimate (see [6, 13]). Clearly, W ∈ Lqw (Rd ) with q = dβ −1 > d/2, so that (2.22) yields N0 (1; 1; W + κ) ≤ Cκ 2 − β , d
d
C = C(β, Φ) .
(2.23)
For ω > d, we have Z Ns (1; 1, W ) = −
∞
Z κs dN0 (1; 1, W + κ) = s
0
∞
κs−1 N0 (1; 1, W + κ)dκ .
0
Now one concludes from (2.23) that gs (HW ) ∈ S1 , i.e. Ns (1; 1, W ) < ∞ ,
ω(β, s) > d .
Note that for d ≥ 3 (2.23) can be obtained also from the classical Rosenblum–Lieb– Cwickel estimate: Z d W (x) + κ −2 dx . N0 (1; 1; W + κ) ≤ C Since φρ converges weakly to 1 as ρ → ∞, and gs (HW ) is trace-class, we have lim Ns (1; φρ , W ) = Ns (1; 1, W ) .
ρ→∞
DISCRETE SPECTRUM ASYMPTOTICS
871
This convergence however is not a priori uniform in W . The proof of Theorem 2.4 is more complicated, but provides the uniformity in W . The next theorem constitutes the main result of the paper. Theorem 2.5. Let s ∈ [0, 1] be a fixed number and let h ∈ (0, h0 ], µ ≥ 0, ◦ µh ≤ C. Suppose that the operator Aa satisfies Assumption 2.2 with D = B(4E) and ψ ∈ C0∞ (B(E/2)). Then the following assertions take place: (1) (2) (3) (4) (5)
If If If If If
ω > d, then (1.7) holds. ω = d, then (1.8) holds. d − s − 1 < ω < d, then (1.9) holds. ω = d − s − 1, then (1.11) holds. ω < d − s − 1, then (1.12) holds.
The remainder estimates are uniform in the functions a, V , ψ and may depend on E. It is worth pointing out that no information on Aa outside B(4E) is involved in Theorem 2.5. In particular, the lower bound of Aa is irrelevant. Our assumption that the potential has only one singularity located at the origin, has been imposed for convenience only. One can easily obtain corresponding asymptotics for Ms in the case of many singular points, by reducing the problem to Theorem 2.5 with the help of an appropriate partition of unity and a translation transformation. The partition of unity argument also permits one to extend Theorem 2.5 to arbitrary ψ ∈ C0∞ (Rd ) and D ⊃ supp ψ. To conclude this section we discuss some properties of the coefficient Θs defined in (2.20). As was mentioned in the Introduction, in general it cannot be expressed explicitly in terms of W . We can only say that it is homogeneous in Φ: 2s
Θs (qΦ, β) = q 2−β Θs (Φ, β) , ∀q > 0 .
(2.24)
This relation follows from definition (2.20), homogeneity of the function W and the equalities 2s Ns (1; φρ , qW ) = q 2−β Ns (1; φρ0 , W ) , 0 1 ρ = ρq 2−β , 2s Ws (1; φρ , qW ) = q 2−β Ws (1; φρ0 , W ) , which result from (2.13), (2.14) with f = `−1 ,
1
` = q β−2 .
Nevertheless, the limit (2.20) can be calculated explicitly for d = 3, s = 1, β = 1 and Φ(θ) = −q, q ≥ 0. In this case Θ1 (−q, 1) = −
q2 . 8
(2.25)
A proof of this result, based on the relation (2.21) and a precise formula for the eigenvalues of HW (1), was given in [8]. This proof is so simple that we reproduce it here.
872
A. V. SOBOLEV
Due to (2.24) one can assume that q = 1. The eigenvalues of HW (1) are −(4n2 )−1 , n ≥ 1, each of them having a multiplicity n2 (see e.g.[5]). Consequently, M1 (κ) = M1 (1; 1, W + κ) =
m X 1 − κ n2 , ∀κ > 0 . 2 4n n=1
(2.26)
√ Here m denotes the integer part of a = (2 κ)−1 . On the other hand, W1 (1; 1, W + κ) =
1 15π 2
Z
κ − |x|−1
52
dx = −
4 √ 15π κ
Z
1
(1 − t) 2 t− 2 dt 5
1
0
1 4 √ B(1/2, 7/2) = √ , = 15π κ 12 κ
(2.27)
where B(· , ·) denotes the beta function. It follows from (2.26) that M1 (κ) = =
m X m −κ n2 4 n=1
κ m κm m − m(2m + 1)(2m + 2) = − (2m2 + 3m + 1) . 4 12 4 6
Representing m as a + v with some v ∈ (−1, 0], we obtain that √ κa2 a κa3 v − − − κa2 v + + O( κ) . 4 3 2 4 √ −1 Taking into account that a = (2 κ) , this leads to M1 (κ) =
M1 (κ) =
√ 1 1 √ − + O( κ) . 12 κ 8
Comparing this with (2.27) and using (2.21), we arrive at (2.25). 3. Reduction to Haa The purpose of this and the next section is to show that in the asymptotics of tr{ψg(Aa )} one can replace Aa by the “asymptotic” operator HW with the potential (1.5). We shall do this in two steps. Firstly, in this section we justify the change Aa → Ha . Further reduction to HW will be done in Sec. 4. All the bounds to be obtained do not depend on the magnetic potential, so that without loss of generality one can set µ = 1. 1. First we study the resolvent R(z, Ha ). We always assume that a ∈ L2loc (Rd ) 1 and the function X = |V | 2 satisfies (2.3) with some < 1, so that the operator Ha is well-defined. Furthermore, it follows from (2.4) that inf σ(Ha ) ≥ −M (h) .
(3.1)
Sometimes for shortness we write simply H and M instead of Ha and M (h) respectively. It will be convenient to assume that M ≥ 1. We denote dM (z) =
873
DISCRETE SPECTRUM ASYMPTOTICS
dist{z, [−M, ∞)}. Everywhere below z ∈ C \ {[−M, ∞)}, so that R(z, H) exists and is bounded. The integer number l (with or without indices) takes the values 1, 2, . . . , d and the numbers m, m1 , m2 , . . . equal either 0 or 1. We begin with some straightforward estimates. Let X denote an arbitrary func1 tion satisfying (2.4) with the same , M as the function |V | 2 . It follows from the definition of H0 that 1
2 kQm l R(−λ; H0 ) k ≤ λ
m−1 2
m = 0, 1, ∀λ > 0 .
,
(3.2)
Furthermore, (2.4) entails that 1
)
1
kXR(−λ; H0 ) 2 k ≤ 2 , 1 2
− 12
1 2
k(H0 + λ) R(−λ; H) k ≤ (1 − )
∀λ ≥ −1 M .
(3.3)
,
Further, due to the resolvent identity, for any λ > M we have the equality: 1
1
R(z; H) = R(−λ; H0 ) 2 S(λ, z)R(−λ; H0 ) 2 , (3.4) 1 1 1 1 S(λ, z) = (H0 + λ) 2 R(−λ; H) 2 I + (λ + z)R(z; H) R(−λ; H) 2 (H0 + λ) 2 . By the second inequality in (3.3), kS(λ, z)k ≤ 2(1 − )−1
λ + |z| , dM (z)
λ ≥ −1 M .
(3.5)
As a rule, in what follows we omit the dependence of the coefficients on . Lemma 3.1. Let X satisfy (2.4) and let m1 , m2 = 0, 1 be such that m1 +m2 ≤ 1. Then 1
2 2 kX m1 Qm l R(−λ; H) k ≤ Cλ
2 kX m1 Qm l R(z; H)k ≤ C
m1 +m2 −1 2
, ∀λ ≥ −1 M,
m1 +m2 2
(|z| + M ) dM (z)
(3.6)
.
(3.7)
Proof. The estimate (3.6) is a consequence of (3.2) and (3.3). Let us prove (3.7). By (3.4), (3.5), and (3.6) with λ = |z| + −1 M , we have 1
1
m1 m2 2 Ql R(−λ; H0 ) 2 k kS(λ, z)k kR(−λ; H0 ) 2 k kX m1 Qm l R(z; H)k ≤ kX
≤ Cλ This provides (3.7).
m1 +m2 −1 2
λ + |z| − 1 λ 2. dM (z)
Let us proceed to the estimates of R(z, H) in the classes of compact operators. For the resolvent of the operator H0,0 , necessary bounds can be found in [14, Lemma 3.3]. Their proof is based on a simple criterion for the operators of the form
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A. V. SOBOLEV
a(x)b(−ih∂) (see, e.g. [13]). It provides for any f ∈ Lp (Rd ), κ > 0 and λ > 0 the bound f R(−λ; H0,0 )κ
≤ Cp kf kLp λ−κ+ 2p h− p , ∀p ≥ 2, p > d(2κ)−1 . d
p
d
In combination with Proposition 2.1 this yields for κ > 0 and any integer n ≥ 1: f R(−λ; H0 )κ
≤ Cn kf kL2n λ−κ+ 4n h− 2n , d
2n
2n > d(2κ)−1 .
d
(3.8)
Moreover, the following lemma holds. Lemma 3.2. Let f ∈ L2n (Rd ) for some n > d/2. Then d
f R(z; H)
≤ Ckf kL2n h− 2n d
2n
(M + |z|) 4n . dM (z)
Proof. By (3.4) for λ = |z| + −1 M , and (1.14), f R(z; H)
2n
≤
1
f R(−λ; H0 ) 2
2n kS(λ, z)k
1
kR(−λ; H0 ) 2 k .
It remains to apply (2.2), (3.8) for κ = 1/2 and (3.5).
2. Now we shall study the properties of the resolvent “sandwiched” between two functions with disjoint supports. Until the end of this section the functions χ and φ will be supposed to satisfy the conditions ) χ ∈ C0∞ (B(ρ)) ; |χ| ≤ 1 ; φ ∈ B ∞ (Rd ) , supp φ ⊂ Rd \ B(νρ) ,
(3.9)
|φ| ≤ 1 ,
with some ρ > 0 and ν > 1. All the constants in theorems below do not depend on ρ, χ, φ but may depend on ν. Lemma 3.3. Let χ and φ obey (3.9). Let m, m1 , m2 = 0, 1 be such that m + m1 ≤ 1. Then for any N ≥ 0 m2 1 kX m χQm φk ≤ CN,ν l1 R(z; H)(Ql2 )
m+m1 +m2 2
(M + |z|) dM (z)
(M + |z|)h2 ρ2 dM (z)2
N . (3.10)
Let n and k be two integers such that n > d/2 and k ≤ 2n. Then for p = 2n/k and any N ≥ 1 m2 1 X m χQm φ l1 R(z; H)(Ql2 )
p
m+m1 +m2 2
(M + |z|) ≤ CN,ν dM (z)
1
ρ(M + |z|) 2 h
dp
1
(M + |z|) 2 h ρdM (z)
N k .
(3.11)
875
DISCRETE SPECTRUM ASYMPTOTICS
In particular, for any N > d/2 m2 1 X m χQm φ l1 R(z; H)(Ql2 )
≤ CN,ν
1
m+m1 +m2 2
(M + |z|) dM (z)
1
ρ(M + |z|) 2 h
d
1
(M + |z|) 2 h ρdM (z)
2N .
(3.12)
Proof. We prove first (3.11). It suffices to do that for N = 1. The result for all N will follow if one replaces n by nN and k by kN . We start with the following simple observation: Let η ∈ C ∞ (R) be a function such that 0 ≤ η ≤ 1 and η(t) = 1, t ≤ 1/3; η(t) = 0, t ≥ 2/3. Let us define the following family of functions χ(j) ∈ C0∞ (Rd ), j = 1, 2, . . . , k: ρ(ν − 1)(j − 1) k |x| − ρ − . (x) = η ρ(ν − 1) k
(j)
χ
(3.13)
It is clear that χ = χχ(1) and χ(j) = χ(j) χ(j+1) , χ(j) φ = 0, so that χ(j) R(z; H)(Ql2 )m2 φ = −[R(z; H), χ(j) ](Ql2 )m2 φ = R(z; H)[H0 , χ(j) ]χ(j+1) R(z; H)(Ql2 )m2 φ . Therefore the representation holds: m2 1 φ X m χQm l1 R(z; H)(Ql2 ) 1 = X m χQm l1
k Y R(z; H)[H, χ(j) ] R(z; H)(Ql2 )m2 φ . j=1
According to (3.4) for any λ > M one can write 1
1
R(z; H)[H, χ(j) ] = R(−λ; H0 ) 2 STj (H0 + λ) 2 ,
S = S(λ, z) ,
where 1
1
Tj = R(−λ; H0 ) 2 [H0 , χ(j) ]R(−λ; H0 ) 2 . Therefore 1
m2 1 1 2 φ = X m χQm X m χQm l1 R(z; H)(Ql2 ) l1 R(−λ; H0 ) ) ( k Y 1 × STj SR(−λ; H0 ) 2 (Ql2 )m2 φ . j=1
Noting that [H0 , ζ] = −ih(Ql ∂l ζ + ∂l ζQl ) , ∀ζ ∈ B ∞ (Rd ) ,
(3.14)
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A. V. SOBOLEV
we conclude that Tj = −ih(Zj + Zj∗ ) , 1 1 Zj = R(−λ; H0 ) 2 Ql ∂l χ(j) R(−λ; H0 ) 2 . Consequently, (3.2), (3.8) and (3.13) yield that Zj
≤ Ck∂χ(j) kL2n λ− 2 + 4n h− 2n ≤ Ck,ν ρ 2n −1 λ− 2 + 4n h− 2n . 1
2n
d
d
d
1
d
d
Let λ = −1 M + |z|. Then by (1.14) and (3.5), k Y STj j=1
≤ kSkk 2n k
k Y
Tj
2n
≤ Chk
j=1
M + |z| dM (z)
k
ρ 2n −k λ− 2 + 4n h− 2n dk
1
ρ(M + |z|) 2 ≤C h
dk 2n
k
dk
dk
1
(M + |z|) 2 h ρdM (z)
k .
Now we get from (3.14): m2 1 X m χQm φ l1 R(z; H)(Ql2 )
2n k
k Y
1
1 2 ≤ kX m Qm l1 R(−λ; H0 ) k
1
{STj }
kSkkR(−λ; H0 ) 2 (Ql2 )m2 k . 2n k
j=1
Using (3.2), (3.3), (3.5), this yields (3.11). The estimate (3.12) is a direct consequence of (3.11) for k = 2n. The bound (3.10) can be proven analogously. Corollary 3.4. Let the functions χ, φ satisfy (3.9) and the numbers m, m1 , m2 be as in Lemma 3.3. Then for any k ≥ 1 and N > d/2 k m2 1 X m χQm φ l1 R(z; H) (Ql2 )
≤ Ck,N
1
m+m1 +m2 2
(M + |z|) dM (z)k
1
ρ(M + |z|) 2 h
d
1
(M + |z|) 2 h dM (z)ρ
2N .
(3.15)
Proof is by induction. The corollary is already proved for k = 1. Assume that (3.15) is true for some k. Weshall deduce from here that it is also true for k + 1. Let χ1 ∈ C0∞ B((1 + 2ν)ρ/3) be a function such that χ1 (x) = 1, |x| ≤ (2 + ν)ρ/3 and φ1 = 1 − χ1 . Then by (1.14) k+1 1 X m χQm (Ql2 )m2 φ l1 R(z; H)
≤ kX
m
k 1 χQm l1 R(z; H) k
+ X
m
1
χ1 R(z; H)(Ql2 )m2 φ
k 1 χQm l1 R(z; H) φ1 1
kR(z; H)(Ql2 )
1
m2
φk .
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DISCRETE SPECTRUM ASYMPTOTICS
Since supp φ1 ⊂ Rd \ B (2 + ν)ρ/3 , the pairs of functions χ1 , φ and χ, φ1 satisfy the conditions of Lemma 3.3. The desired estimate for the first summand in the r.h.s. follows from Lemmas 3.3 and 3.1. The second summand obeys the same estimate due to the inductive assumption and Lemma 3.1. In the next lemma we assume that ρ ≥ C. Lemma 3.5. Let χ obey (3.9) with ρ ≥ C and let k, n be two integers such that n > d/2, k ≤ 2n. Then 1 d p 2 2n k −k ρM . (3.16) χR(±iM ; H) p ≤ CM , p= h k Proof. By Lemma 3.2 the bound (3.16) is true for k = 1. Further proof is by induction: assuming that (3.16) is true for some k, we shall deduce (3.16) for k + 1. Let χ1 ∈ C0∞ (B(3ρ)) be a function such that χ1 (x) = 1, |x| ≤ 2ρ; |χ1 | ≤ 1, and φ = 1 − χ1 . Then supp φ ⊂ Rd \ B(2ρ). By (1.14) χR(±iM ; H)k+1
2n k+1
≤ χR(±iM ; H)φ + χR(±iM ; H)
2n k+1
kR(±iM ; H)k k
2n
χ1 R(±iM ; H)k
2n k
.
Since χ, φ obey the conditions of Lemma 3.3, by (3.11) with N = 1 the first summand is bounded by d(k+1) d(k+1) k+1 1 1 1 2n 2n 2 1 ρ(2M ) 2 (2M ) 2 h −k−1 ρM ≤ CM . C k+1 dM (±iM ) h dM (±iM )ρ h Here we used the bounds M ≥ 1, ρ ≥ C. Further, by Lemma 3.2 and the inductive assumption the second term does not exceed d(k+1) 1 dk 1 2n 2n 2 2 d d − 2n −1 −k ρM −k−1 ρM 4n M ≤ CM (2M ) M . C kχkL2n h h h This and preceding estimate provide (3.16) with k + 1. Proof is completed.
Corollary 3.6. Let χ be as in Lemma 3.5. Let g = g(λ) be a function such that g(λ) = 0, λ ≥ λ0 and |g(λ)| ≤ C|λ|s , s ≥ 0. Then 1 d 2 s ρM χg(H) 1 ≤ CM . h Proof. It follows from (3.16) with 2n = k > d that χg(H)
1
≤ χR(iM ; H)k 1 k(H − iM )k g(H)k 1 d 2 −k ρM ≤ CM k(H − iM )k g(H)k . h
For H ≥ −M , the last factor is bounded by CM s+k .
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A. V. SOBOLEV
Corollary 3.7. Let χ be as in Lemma 3.5 and let m1 , m2 = 0, 1 be such integers that m1 + m2 ≤ 1. Then for any k > d 1 d m1 +m2 2 m2 m1 k+1 −k−1 ρM 2 X χQl R(±iM ; H) . 1 ≤ CM h Proof. Let χ1 be a function introduced in the proof of Lemma 3.5. Then k+1 2 X m1 χQm l R(±iM ; H)
1
k 2 ≤ X m1 χQm l R(±iM ; H)φ 1 kR(±iM ; H) k k 2 + kX m1 χQm l R(±iM ; H)k χ1 R(±iM ; H)
1
.
It remains to apply Lemma 3.1, (3.16) and (3.12).
3. Until now the operator under consideration was supposed to have the form Ha in the entire space. As was explained in Sec. 2, we may assume that A coincides with some H = Ha on some open set only, for we are dealing with local traces of the form (1.2). From now on we work with an operator A satisfying Assumption 2.2 ◦ with D =B(4ρ), ρ > 0. Our goal is to show that in the asymptotics of tr{χg(A)}, χ ∈ C0∞ (B(4ρ)) one can replace the operator A by H = Ha . To this end we use the identities (2.5), (2.8). ◦
Lemma 3.8. Let the operator A obey Assumption 2.2 with D = B(4ρ), where ρ ≥ C. Then for any χ satisfying (3.9), any integers k ≥ 1 and N > d/2, the bound holds: χ R(z; A)k − R(z; H)k 1 1 d 1 2N +1 1 ρ(M + |z|) 2 h(M + |z|) 2 ≤ CN . (3.17) h ρdM (z) | Im z|k The constant CN does not depend on χ, V and ρ. Proof. Let η ∈ C0∞ (R) be a function such that η(t) = 1, |t| ≤ 2. Denote χ1 (x) = η(|x|ρ−1 ), φ = 1 − χ1 . Then χ1 χ = χ and the pair φ, χ satisfies conditions of Lemma 3.3. Due to the obvious identity χ R(z; A)k − R(z; H)k = χ χ1 R(z; A)k − R(z; H)k χ1 − χR(z; H)k φ the problem amounts to proving the bound (3.17) for the operators T1 = χ χ1 R(z; A)k − R(z; H)k χ1 , T2 = χR(z; H)k φ . By (3.15), the estimate (3.17) is obviously satisfied for T2 . Further, by virtue of (2.8), to prove (3.17) for T1 it suffices to establish (3.17) for each of the operators (j)
T1
= χR(z; H)j Z(χ1 )R(z; A)k−j+1 ,
j = 1, . . . , k .
879
DISCRETE SPECTRUM ASYMPTOTICS
(See (2.6) for definition of Z(·).) Now it is clear that (j) T1 1 ≤ h 2 χR(z; H)j Ql ∂l χ1 1 + h χR(z; H)j ∆χ1
1
kR(z; A)k−j+1 k .
By definition supp ∂χ1 and supp χ obey the conditions of Corollary 3.4. Taking into account that |∂χ1 | ≤ Cρ−1 and |∆χ1 | ≤ Cρ−2 , estimating the terms in the brackets by means of (3.15), and the last factor by | Im z|−k+j−1 , we get (3.17). Corollary 3.9. Let the operator A and the function χ be as in Lemma 3.8. Then for any g ∈ C0∞ (R) one has 1 d ρM 2 χg(A) 1 ≤ C . h The constant C depends on g only. Proof. For z = iM and k > d χg(A)
1
≤ χR(z; A)k
1 k(A
≤ k(A − z) g(A)k k
− z)k g(A)k
k
χR(z; H)
1
+ χ R(z; A) − R(z; H) k
k
.
1
The first factor is bounded by CM k , since g ∈ C0∞ . By (3.16) and (3.17) the second factor does not exceed 1 d 2 −k ρM . CM h
This provides the desired estimate.
4. When studying the difference g(A)−g(H) we use the following representation for a function of a selfadjoint operator in terms of its resolvent (see [2]): Proposition 3.10. Let g ∈ C0∞ (R). Then for any selfadjoint operator B the relation holds: n−1 X 1 Z ∂ j g(λ) Im[ij R(λ + i; B)]dλ g(B) = πj! R j=0 +
1 π(n − 1)!
Z
Z
1
τ n−1 dτ 0
R
∂ n g(λ) Im[in R(λ + iτ ; B)]dλ , ∀n ≥ 2 . (3.18)
Combining Lemma 3.8 and this representation, we obtain ◦
Theorem 3.11. Suppose that A satisfies Assumption 2.2 with D = B(4ρ), ρ ≥ C, and χ obeys (3.9). Let g ∈ C ∞ (R) be a function such that g(λ) = 0 if λ ≥ λ0 and for some s ≥ 0, L ≥ L0 > 0 |∂ n g(λ)| ≤ Cn Ln hλis , ∀n ∈ N.
(3.19)
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A. V. SOBOLEV
Then for any N > (d + 1)/2 + s χ[g(A) − g(H)]
1
≤ CN L
2N +3
M
s+1
1
ρM 2 h
d
1
hM 2 ρ
2N +1 ,
(3.20)
where the constant CN does not depend on a, V , χ and ρ. Proof. Let λ ∈ R and 0 < |τ | ≤ 1. Denote δ(λ, τ ) = R(λ + iτ ; A) − R(λ + iτ ; H) . Then (3.17) with k = 1 yields for any N > d/2: χδ(λ, τ )
1
χδ(λ, τ )
1
2N +1−d d+1 h ≤ CN M N + 2 |τ |−2N −2 , −2M ≤ λ ≤ λ0 , ρ 2N +1−d d−1 h ≤ CN |λ|−N + 2 |τ |−1 , λ ≤ −2M . ρ
(3.21)
(3.22)
The representation (3.18) does not apply to the function g since it is allowed to grow as λ → −∞. Instead of g we use its modification. Let ζ ∈ C ∞ (R) be a function such that ζ(t) = 0 ,
t ≤ −2 ;
ζ(t) = 1 ,
t ≥ −3/2 .
˜ = max{(inf A)− , M }. Define the function g˜ ∈ C0∞ (R) by the equality Let M ˜ )g(λ). Then because of (3.1) we have g˜(A) = g(A), g˜(H) = g(H). g˜(λ) = ζ(λ/M By virtue of (3.19) (3.23) |∂ n g˜(λ)| ≤ Cn Ln hλis , ∀n ∈ N , with some constants Cn independent of A, H. According to (3.18) (n)
g˜(A) − g˜(H) = I1 (n) I1
=
n−1 X j=0
(n)
I2
=
1 π(n − 1)!
1 πj!
Z
(n) 1
∂ j g˜(λ) Im ij δ(λ, 1) dλ ,
R
Z
1
∂ n g˜(λ) Im in δ(λ, τ ) dλ .
τ n−1 dτ R
0
Let us estimate first the integral to (3.23) and (3.21), (3.22) χI2
Z
(n)
+ I2 , ∀n ∈ N;
(n) I2 .
To that end set n = 2N + 3. Then, according
2N +1−d Z 1 Z d+1 h dτ M N + 2 hλis dλ ρ −2M≤λ≤λ0 0 Z 1 Z d−1 + τ 2N +1 dτ |λ|−N + 2 +s dλ .
≤ CN L2N +3
0
λ≤−2M
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DISCRETE SPECTRUM ASYMPTOTICS
Assuming that N > (d + 1)/2 + s (so that the second integral exists), we obtain (n)
χI2
1
≤ CN L2N +3
2N +1−d d+3 h M N +s+ 2 . ρ
(3.24)
Now, in view of (3.21), (3.22) and (3.23), (n)
χI1
1
≤ CN L2N +2 Z +
2N +1−d Z d+1 h M N + 2 hλis dλ ρ −2M≤λ≤λ0 d−1 |λ|−N + 2 +s dλ
λ≤−2M
≤ CN L
2N +2
2N +1−d d+3 h M N +s+ 2 . ρ
Combining this bound with (3.24), we obtain from here (3.20). 4. Reduction to Asymptotic Potential
Here we continue the analysis of the previous section and show that under suitable conditions one can replace Ha in the trace tr{ψg(Ha )} by the operator HW without any magnetic potential and with the asymptotic electric potential (1.5). Actually, our argument does not use the precise form of W . The following general 1 1 conditions will be sufficient. Firstly, we assume that both X = |V | 2 and X = |W | 2 satisfy the inequality (2.3) for some M = M (h) and a fixed < 1. Hence both Ha and HW are semi-bounded from below and Ha ≥ −M (h) ,
HW ≥ −M (h) .
(4.1)
Recall that M (h) ≥ 1. Secondly, instead of (2.16) we impose the condition W (x) = Y (x)Ψ(x)Y (x) ,
V (x) = Y (x)(Ψ(x) + F (x))Y (x) ,
x ∈ B(ρ) ,
(4.2)
where Y , Ψ, F are some real-valued functions such that Y obeys (2.3) and Ψ ∈ d L∞ (Rd ), F ∈ L∞ loc (R ). Moreover, throughout this section we suppose that ∞ d a ∈ Lloc (R ). 1. Let us study resolvents of the operators Ha and HW . As in the previous section we rely upon an appropriate version of the resolvent identity. Namely, for any φ ∈ B ∞ (Rd ) we have ) φR(z; Ha ) = R(z; HW )φ + R(z; HW )Z1 R(z; Ha ) , (4.3) Z1 = Z1 (φ) = −ih Πl ∂l φ + ∂l φQl + al φQl + Πl φal − φF Y 2 . Note also the identity for the difference of powers of the resolvents, similar to (2.8): φR(z; Ha )k = R(z; HW )k φ +
k X j=1
R(z; HW )j Z1 (φ)R(z; Ha )k−j+1 , ∀k ∈ N . (4.4)
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A. V. SOBOLEV
d Denote kf kρ = ess-sup|x|≤ρ |f (x)| for f ∈ L∞ loc (R ) and
1 1 K(z) = K(z, ρ) = (M + |z|) 2 kakρ + kF kρ (M + |z|) 2 .
(4.5)
As in Sec. 3 we begin with estimating difference of the resolvents. Throughout this section we assume as a rule that χ ∈ C0∞ B(ρ/2) and |χ(x)| ≤ 1. Lemma 4.1. Let the functions V , W be as specified above and let ρ ≥ C be a number from (4.2). Then for any χ ∈ C0∞ (B(ρ/2)) and N ≥ 0 kχ R(z; Ha ) − R(z; HW ) k 2N +1 1 K(z, ρ) 1 (M + |z|) 2 h × . (4.6) ≤ CN + dM (z) ρdM (z) dM (z) If k > 2d + 1 then for any N > d/2 χ R(±iM ; Ha )k − R(±iM ; HW )k 1 2N +1 1 d K(0, ρ) 1 ρM 2 h . ≤ CN k + 1 M h M M 2ρ The constant CN does not depend on V , W , χ and ρ.
(4.7)
Proof. Let η ∈ C0∞ (R) be a function such that |η| ≤ 1 and η(t) = 1, |t| ≤ 2/3; η(t) = 0, |t| ≥ 1. Denote χ1 (x) = η(|x|/ρ) and φ = 1 − χ1 . Due to the obvious identity χ R(z; Ha )k − R(z; HW )k = χ χ1 R(z; Ha )k − R(z; HW )k χ1 − χR(z; HW )k φ , the problem amounts to proving the desired bounds for the operators ) T1 = χ χ1 R(z; Ha )k − R(z; HW )k χ1 , T2 = χR(z; HW )k φ
(4.8)
(k = 1 for (4.6)). Let us prove first (4.7). For T2 the estimate (4.7) is fulfilled due to Corollary 3.4. Further, by (4.4) T1 =
k X
(j)
T1 ,
(j)
T1
= χR(z; HW )j Z1 R(z; Ha )k−j+1 ,
Z1 = Z1 (χ1 ) .
j=1 (j)
Thus it suffices to obtain (4.7) for each T1 individually. We analyse first the terms with j ≤ k/2, so that k − j > d. It is clear that Z1 = Z1 χ2 for χ2 (x) = η |x|/(2ρ) . Therefore for z = ±iM (j)
T1
1
≤ h χR(z; HW )j Πl ∂l χ1
1
kχ2 R(z; Ha )k−j+1 k
+ h χR(z; HW )j ∂l χ1
1
kχ2 Ql R(z; Ha )k−j+1 k
+ kχR(z; HW )j k χ1 al Ql R(z; Ha )k−j+1
1
+ kχR(z; HW ) Πl k χ1 al R(z; Ha )
1
j
k−j+1
+ kχR(z; HW ) Y χ1 F k Y χ2 R(z; Ha ) j
k−j+1 1
.
(4.9)
883
DISCRETE SPECTRUM ASYMPTOTICS
Since supp χ and supp ∂χ1 are separated from each other and |∂χ1 | ≤ Cρ−1 , by Corollary 3.4 and Lemma 3.1 the first and the second terms are bounded by CM
−k
h
2N +1−d , ∀N > d/2 .
1
M 2ρ
Further, by Lemma 3.1 and Corollary 3.7 the third and the fourth terms do not exceed 1 d 1 ρM 2 . Ckakρ M −k− 2 h And, finally, the fifth term is bounded by CkF kρ M
−k
1
ρM 2 h
d .
Combining the bounds above, we obtain for all j ≤ k/2 the estimate (k) T1 1
≤ CM
−k
1
ρM 2 h
d
h
2N +1
1
M 2ρ
K(0, ρ) . + M
In case j > k/2 the proof is the same except that the last three terms in (4.9) should be replaced with χR(z; HW )j
1
kχ1 al Ql R(z; Ha )k−j+1 k
+ χR(z; HW )j Πl
1
kχ1 al R(z; Ha )k−j+1 k
+ χR(z; HW )j Y χ1 F
1
kY χ2 R(z; Ha )k−j+1 k .
The trace norms here are finite since j > d + 1. Further proof goes as before. (j) Summing the bounds for T1 over j, one obtains (4.7). Proof of (4.6). Let k = 1 in (4.8). The desired bound for T2 follows from (3.10). Furthermore, as in the proof of (4.7), by (4.3) kT1 k ≤ hkχR(z; HW )Πl ∂l χ1 k kχ2 R(z; Ha )k + hkχR(z; HW )∂l χ1 k kχ2 Ql R(z; Ha )k + kχR(z; HW )k kχ1 al Ql R(z; Ha )k + kχR(z; HW )Πl k kχ1 al R(z; Ha )k + kχR(z; HW )Y χ1 F k kY χ2 R(z; Ha )k . By Lemmas 3.2 and 3.4 first two terms are bounded by 2N +1 1 1 (M + |z|) 2 h , ∀N ≥ 0 . C dM (z) ρdM (z)
884
A. V. SOBOLEV
In view of Lemma 3.1 the remaining terms do not exceed 1
kakρ
(M + |z|) 2 M + |z| + kF kρ . dM (z)2 dM (z)2
Combining the bounds above, we arrive at (4.6).
2. The next step is to study the functions of operators Ha and HW . Throughout the rest of the section the conditions of Lemma 4.1 are always assumed to be fulfilled. For functions of self-adjoint operators we use the representation (3.18). Theorem 4.2. Let g ∈ C ∞ (R) be a function satisfying the conditions of Theorem 3.11 and χ ∈ C0∞ (B(ρ/2)). Then for any N ≥ 0 1 (4.10) kχ[g(Ha ) − g(HW )]k ≤ CN L2N +3 M s+1 (hM 2 ρ−1 )2N +1 + K(0, ρ) , where the function K(z, ρ) is defined in (4.5) and the constant CN depends only on constants Cn in (3.19) and the function χ. Proof mimics the proof of Theorem 3.11. Let λ ∈ R and 0 < |τ | ≤ 1. Denote δ(λ, τ ) = R(λ + iτ ; Ha ) − R(λ + iτ ; HW ) . Then (4.6) yields for any N ≥ 0: 2N +1 1 K(λ) 1 (M + |λ|i) 2 h , ∀λ ∈ R . + kχδ(λ, τ )k ≤ τ ρτ τ
(4.11)
In order to apply the representation (3.18), instead of g we use its modification. Let ζ ∈ C ∞ (R) be the function introduced in the proof of Theorem 3.11. Define g˜ ∈ C0∞ (R) by the equality g˜(λ) = ζ(λ/M )g(λ). Then because of (4.1) we have g˜(Ha ) = g(Ha ), g˜(HW ) = g(HW ). By virtue of (3.19) |∂ n g˜(λ)| ≤ Cn Ln hλis , ∀n ∈ N ,
(4.12)
with constants Cn independent of Ha , HW . According to (3.18) (n)
g˜(Ha ) − g˜(HW ) = I1 (n) I1
=
n−1 X j=0
(n)
I2
=
1 π(n − 1)!
1 πj!
Z
Z
(n)
+ I2
, ∀n ∈ N ;
∂ j g˜(λ) Im ij δ(λ, 1) dλ ,
R
Z
1
∂ n g˜(λ) Im in δ(λ, τ ) dλ .
τ n−1 dτ R
0
Set n = 2N + 3. In view of (4.12) and (4.11) Z (n) (hM 1/2 ρ−1 )2N +1 + K(0) dλ kχI1 k ≤ CL2N +2 M s −2M≤λ≤λ0
≤ CL
2N +2
M
s+1
(hM 1/2 ρ−1 )2N +1 + K(0) .
(4.13)
885
DISCRETE SPECTRUM ASYMPTOTICS
Let us estimate the integral (4.13). According to (4.12) and (4.11) Z 1 Z 1 (n) kχI2 k ≤ CL2N +3 M s dτ (hM 2 ρ−1 )2N +1 dλ K(λ)dλ
Z
1
+
τ
2N +1
dτ −2M≤λ≤λ0
0
≤ CL
−2M≤λ≤λ0
0
Z
2N +3
M
1 (hM 2 ρ−1 )2N +1 + K(0) .
s+1
(n)
Along with the bound for I1 , this yields (4.10). Now we estimate the trace class norm of the difference g(Ha ) − g(HW ):
Theorem 4.3. Let the functions g ∈ C ∞ (R) and χ ∈ C0∞ (B(ρ/2)) be as in Theorem 4.2. Then for any N > d/2 χ g(Ha ) − g(HW ) χ 1 2N +1 1 d 1 2 M2h 2N +3 s+1 ρM ≤ CN L M + K(0, ρ) . (4.14) h ρ The constant CN depends on the function χ and does not depend on a, V , W , ρ, h. Proof. Denote g˜(λ) = (λ − iM )k g(λ), k > 2d + 1 . Then with z = iM g(Ha ) − g(HW ) = R(z; Ha )k g˜(Ha ) − R(z; HW )k g˜(HW ) = R(z; Ha )k − R(z; HW )k g˜(Ha ) + R(z; HW )k g˜(Ha ) − g˜(HW ) . Therefore χ g(Ha ) − g(HW ) χ
1
≤ χ R(z; Ha )k − R(z; HW )k 1 k˜ g(Ha )χk + χR(z; HW )k 1 k g˜(Ha ) − g˜(HW ) χk .
(4.15)
Let us estimate each factor individually. Note that K(z, ρ) ≤ CK(0, ρ). Therefore, according to (4.7) χ R(z; Ha )k − R(z; HW )k 1 2N +1 1 d 2 K(0, ρ) h −k ρM , ∀N > d/2 . (4.16) ≤ CM + 1 h M ρM 2 Further, due to (3.16) k
χR(z; HW )
1
≤ CM
−k
1
ρM 2 h
d .
(4.17)
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A. V. SOBOLEV
Since the function g˜ satisfies (3.19) with s1 = s + k, we have in view of Theorem 4.2 k g˜(Ha ) − g˜(HW ) χk
1 ≤ CN L2N +3 M s+k+1 (hM 2 ρ−1 )2N +1 + K(0, ρ) , ∀N ≥ 0 .
(4.18)
To estimate χ˜ g(Ha ) recall that Ha ≥ −M , so that k˜ g(Ha )k ≤ CM s+k . Combining this bound with (4.16), (4.17) and (4.18), we obtain from (4.15): χ g(Ha ) − g(HW ) χ
1
≤ CN M s
1
ρM 2 h
d
h
2N +1 +
1
K(0, ρ) M 1 2N +1
M2ρ 1 d 2 hM 2 2N +3 s+1 ρM + CN L M h ρ
+ K(0, ρ) .
This provides (4.14).
3. Now we combine the results of this section and Sec. 3. Precisely, let V , W be functions introduced in the beginning of the section and let the operator A ◦ d obey Assumption 2.2 with the functions a ∈ L∞ loc (R ), V and D =B(4ρ). We shall compare the traces tr{χgs (A)} and tr{χgs (HW )} for χ ∈ C0∞ (B(r/2)) with r ≤ ρ. Theorem 4.4. Let the operators A, Ha , HW be as above, and χ ∈ C0∞ (B(r/2)) with some r, ρ ≥ r ≥ C. Then for any L ≥ L0 > 0 and N > (d + 1)/2 + s, s ≥ 0
1 d ρM 2 M 1 ≤ CN L h 1 d 1 2N +1 M 2r M 2h × + K(0, ρ) + CL−s . ρ h
χ gs (A) − gs (HW ) χ
2N +3
s+1
(4.19)
Proof. Let ζ ∈ C0∞ (R) be a non-negative function such that ζ(λ) = 1, |λ| ≤ 1/2, and ζ(λ) = 0, |λ| ≥ 1. Denote g (1) (λ) = gs (λ)ζ(Lλ) and g (2) (λ) = gs (λ)(1 − ζ(Lλ)). Obviously, g (2) ∈ C ∞ and satisfies (3.19). Since χ ∈ C0∞ (B(r/2)) ⊂ C0∞ (B(ρ/2)), by Theorems 3.11 and 4.3 we have:
χ g
(2)
(A) − g
(2)
(HW ) χ
1
≤ CN L ×
2N +3
1
M
M 2h ρ
s+1
1
ρM 2 h
2N +1
+ K(0, ρ) ,
as N > (d + 1)/2 + s. On the other hand, χg (1) (A)χ
1
≤ CL−s χζ(A)
1
d
.
(4.20)
887
DISCRETE SPECTRUM ASYMPTOTICS
Therefore by Corollary 3.9 χ g (1) (A) − g (1) (HW ) χ
1
≤ CL−s
χζ(A) 1 + 1 d M 2r ≤ CL−s . h
χζ(HW )
1
Combining this estimate with (4.20), we arrive at (4.19). 5. Asymptotics in Case of Smooth Potentials. Proof of Theorems 2.4, 2.40
1. As mentioned in the Introduction, one of the basic ingredients of our method is the asymptotics of Ms (h, µ; ψ, a) for smooth functions a, V obtained in [14]. Before stating the result from [14] we specify conditions on the operator A = Aa . Let D ⊂ Rd be a bounded open domain. Assumption 5.1. (1) A is self-adjoint and semi-bounded from below. ¯ such that C0∞ (D) ∈ D(A) (2) There exist real-valued functions V , a ∈ C ∞ (D) and u = Ha u for any u ∈ C0∞ (D). ¯ ` ∈ C 1 (D) ¯ be two functions such that Let f ∈ C(D), f (x) > 0 ,
¯; x∈D
`(x) > 0 ,
|∂x `(x)| ≤ % < 1 ,
x ∈ D;
cf (y) ≤ f (x) ≤ Cf (y) , ∀x ∈ D ∩ B(y, `(y)) ,
(5.1) y ∈ D.
(5.2)
We are interested in the asymptotics of Ms for an operator Aa , satisfying ¯ ψ ∈ C0∞ (D) Assumption 5.1 with the domain D and some functions V , a ∈ C ∞ (D), which obey the bounds |∂xm a(x)| ≤ Cm `(x)1−|m| , |∂xm V (x)| ≤ Cm f (x)2 `(x)−|m| ,
|m| ≥ 1 ;
|∂xm ψ(x)| ≤ Cm `(x)−|m| ,
) |m| ≥ 0 ,
x ∈ D. (5.3)
2
One can think of f (x) as a measure of the size of V (x), while `(x) characterizes the behaviour of V (x), a(x) and ψ(x) under differentiation. Emphasize that the functions f (x), `(x) are allowed to depend on h, µ. We require only that f (x)`(x) ≥ ch ;
f (x)2 ≥ cµh ,
x ∈ D.
(5.4)
We also need the following condition on supp ψ: [
B x, 8`(x) ⊂ D ,
where the union is taken over those x ∈ D, for which B(x, `(x)) ∩ supp ψ 6= ∅.
(5.5)
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A. V. SOBOLEV
The next Proposition results from [14]: Proposition 5.2. Let the operator A obey Assumption 5.1 for an open set D with the functions V , a and ψ satisfying conditions (5.1)–(5.5) for some % < 1/8. Then Ms (h, µ; ψ, a) − Ws (h; ψ, V ) ≤ CR(h, µ) , where R(h, µ) =
Z f (x)2s g D
µ`(x) h , `(x)−d dx , `(x)f (x) f (x)
g(a, b) = (1 + bs+1 )as+1−d .
The constant C is uniform in the functions a, V , f , `, ψ satisfying (5.1)–(5.5). Using the explicit form of the function g(h, µ), one can rewrite R(h, µ): R(h, µ) = hs+1−d I1 (h, µ) + µs+1 hs+1−d I2 (h, µ) , with I1 = I1 (h, µ) =
Z
f (x)d+s−1 `(x)−s−1 dx ,
D
Z I2 = I2 (h, µ) =
f (x)d−2 dx . (5.6) D
0
2. Proofs of Theorems 2.4, 2.4 rely on the following Lemma. Lemma 5.3. Let V obey (2.15) and let ψ ∈ C0∞ (Rd ) be a function such that supp ψ ⊂ {x : r ≤ |x| ≤ ρ}, ρ ≥ r ≥ 1 and |∂ m ψ(x)| ≤ Cm |x|−|m| , ∀|m| ≥ 0 , ∀x 6= 0 . If ω > d − s − 1, then for any κ ∈ [0, 1] and ρ ∈ [r, Cκ
1 −β
] one has
|Ns (1; ψ, V + κ) − Ws (1; ψ, V + κ)| ≤ Cr−σ , with σ=
(5.7)
(5.8)
2−β (ω − d + 1 + s) > 0 , 2
uniformly in ψ and V . Proof. We apply Proposition 5.2 in the particular case a = 0, µ = 0, h = 1. It is clear that the conditions (5.1), (5.2), (5.5) are satisfied for β
f (x) = |x|− 2 , κ ≤ 1 ;
`(x) = %|x| ,
% < 1/16 ,
D = {x ∈ R : r/4 < |x| < 2ρ} . d
The condition (5.3) for V + κ and ψ is obviously fulfilled due to the inequality 1 ρ ≤ Cκ− β and conditions (2.15) and (5.7). Moreover, since β < 2, (5.4) is also fulfilled (recall that h = 1). Therefore, according to Proposition 5.2 Z β d+s−1 Ns (1; ψ, V + κ) − Ws (1; ψ, V + κ) ≤ C |x|−s−1 dx |x|− 2 D
Z
2ρ
≤C r/4
t−σ−1 dt
889
DISCRETE SPECTRUM ASYMPTOTICS
with σ defined above. For ω > d − s − 1, the number σ is positive and therefore the integral is bounded by Cr−σ , which implies (5.8). 3. Proof of Theorem 2.4. Let φ ∈ C0∞ (Rd ) be such that φ(x) = 1, |x| ≤ 1. The function ψ(x) = φρ (x) − φr (x) obeys (5.7) (recall that φρ (x) = φ(xρ−1 )). Thus the relation (5.8) with κ = 0 and V = W reads: |Ns (1; ψ, W ) − Ws (1; ψ, W )| ≤ Cr−σ .
(5.9)
This guarantees convergence of Ns (1; φρ , W ) − Ws (1; φρ , W ) as ρ → ∞, which immediately entails (2.20) as d − s − 1 < ω < d. Independence of the limit of the function φ also follows from (5.9). To prove Theorem 2.4 for ω > d, note that Z Ws (1, ψ, W ) ≤ C
2ρ
t−δ−1 dt ,
δ=
r
2−β (ω − d) > 0 . 2
In combination with (5.9) this gives: |Ns (1; ψ, W )| ≤ C(r−σ + r−δ ) , which implies the convergence of Ns (1; φρ , W ) as ρ → ∞. Furthermore, since the operator φρ gs (HW ) converges weakly to gs (HW ) as ρ → ∞, the latter operator is trace class and ρ → ∞,
tr{φρ gs (HW )} = tr gs (HW ) + o(1) ,
which is equivalent to (2.19).
4. Proof of Theorem 2.40. We remind that in Theorem 2.40 no uniformity in W is claimed, so that we use the symbols “lim”, “lim sup” in accordance with our notational convention made in Sec. 2. Let φ ∈ C0∞ (Rd ) be as in Theorem 2.4 and let ρ = Cκ−1/β , where the constant C is such that 4|W (x)| ≤ κ, |x| ≥ ρ/2. For r ∈ [1, ρ] we split Ns (1; 1, W + κ) and Ws (1; 1, W + κ) as follows: + Ns (1; φρ − φr , W + κ) + Ns (1; 1 − φρ , W + κ) , Ws (1; 1, W + κ) = Ws (1; φr , W + κ) + Ws (1; φρ − φr , W + κ) . Ns (1; 1, W + κ) = Ns (1; φr , W + κ)
(5.10)
We shall infer Theorem 2.40 from the following Lemma 5.4. Suppose that Ns (1; 1 − φρ , W + κ) → 0 ,
ρ = Cκ− β , 1
κ → 0,
(5.11)
890
A. V. SOBOLEV
and lim lim sup |Ns (1; φr , W + κ) − Ns (1; φr , W )| = 0 .
r→∞ κ→+0
(5.12)
Then (2.21) holds. Proof. Indeed, it follows from (5.10), (5.11) and Lemma 5.3 with ψ = φρ − φr that lim sup |Ns (1; 1, W + κ) − Ws (1; 1, W + κ) − Θs | κ→+0
≤ lim sup lim sup |Ns (1; φr , W + κ) − Ws (1; φr , W + κ) − Θs | . r→∞
(5.13)
κ→+0
Further, (5.12) along with the equality lim Ws (1; φr , W + κ) = Ws (1; φr , W ) , ∀r > 0 ,
κ→+0
and Theorem 2.4, yield lim lim sup Ns (1; φr , W + κ) − Ws (1; φr , W + κ) − Θs = 0 .
r→∞ κ→+0
Now (5.13) provides (2.21).
Thus it remains to establish (5.11), (5.12). Proof of (5.11). Let uk , λk be normalized eigenfunctions and associated eigenvalues of the operator HW . Denote ψρ = 1 − φρ . Then X
Ns (1; ψρ , W + κ) =
|λk + κ|s hψρ uk , uk i .
λk ≤−κ
Recall that ρ = Cκ−1/β was chosen in such a way that 4|W (x)| ≤ κ, |x| ≥ ρ/2, so that by Lemma A1 from the Appendix, |hψρ uk , uk i| ≤ C
p 1 exp −c |λk |ρ , ∀k : λk 6 −κ . |λk |
Hence Ns (1; ψρ , W + κ) ≤ C
X λk ≤−κ
|λk + κ|s
p 1 exp −c |λk |ρ |λk |
1 1 ≤ CN0 (1; 1, W + κ)κs−1 exp −cκ 2 − β . Since β < 2, this bound in combination with (2.23) provides (5.11). Proof of (5.12). The potentials W and W + κ can be presented in the form (4.2) with β Ψ(x) = Φ(ˆ x) , Y (x) = |x|− 2 , F (x) = κ|x|β , ∀x ∈ Rd .
891
DISCRETE SPECTRUM ASYMPTOTICS
Since β < 2, the functions Y , |W +κ|1/2 , |W |1/2 obey (2.3) for h = 1, any prescribed < 1 and M = M (). Therefore the quantity K(0, r) defined in (4.5) is bounded by K(0, r) ≤ CkF kr = Cκrβ . According to Theorem 4.4 with A = HW +κ and HW , we have for any L ≥ L0 , ρ ≥ r |Ns (1; φr , W + κ) − Ns (1; φr , W )| ≤ CN L2N +3 ρd ρ−2N −1 + κρβ + L−s rd . Set L = rd/s+ε (recall that s > 0 in Theorem 2.40 ), ρ = L1+ε , ε > 0. Then for sufficiently large N the equality (5.12) follows. Now Theorem 2.40 follows from Lemma 5.4. 6. Proof of Theorem 2.5 Throughout this section we assume that V , a obey (2.15), (2.16), (2.17) and all the other conditions of Theorem 2.5 are fulfilled. It will be convenient to choose for the vector-potential such a gauge that a(0) = 0, so that |a(x)| ≤ C|x| ,
x ∈ Rd .
(6.1)
Recall that the functions X = |V |1/2 and X = |W |1/2 obey (2.3) for any ∈ (0, 1) and M (h) defined in (2.18). 1. First of all we study the asymptotics of Ms (h, µ; ψ, a), where ψ is supported in a small neighbourhood of the origin, which depends on h. Precisely, let ( χ ∈ C0∞ (B(1)) ; 2 (6.2) φ(x) = χ (x) , χ(x) = 1 , |x| ≤ 1/2 . Our aim is to study Ms (h, µ; φr , a), φr (x) = φ(xr−1 ), with 2 2−β h , r = rθ (h) = θ
0 < θ ≤ 1.
(6.3)
Here θ ∈ (0, 1] plays the role of an additional parameter. Denote κ = κ(β) =
2+β . 2−β
(6.4)
We shall find the asymptotics of Ms (h, µ; φr , a) as h → 0, µhκ → 0 and θ → 0. Note that due to the inequality κ > 1 the condition µhκ → 0 is less restrictive than our usual condition µh ≤ C. To distinguish asymptotics in h and in θ we introduce the notation (t) for an arbitrary function of a parameter t ∈ [0, C], which possesses the following two properties uniformly in the functions U, a and the parameters h ∈ (0, h0 ], µhκ ≤ C: (1) (t) is bounded on [0, C]; (2) (t) → 0 as t → 0.
892
A. V. SOBOLEV
Lemma 6.1. Let the function a, V and φ be as specified above, s ≥ 0, β ∈ [0, 2), and let h ∈ (0, h0 ], µhκ ≤ C. Let r = rθ be defined in (6.3). Suppose that the ◦ operator Aa satisfies Assumption 2.2 with D = B(4E) and functions a, V , which obey (2.15)–(2.17). Then Ms (h, µ; φr , a) ≤ Cθ−σ h−ω , ∀θ ∈ (0, 1] ,
C = Cβ,s ,
σ = σ(β, s) > 0 .
(6.5)
Moreover, there exists such θ0 = θ0 (h, µ) = (h + µhκ ) that for any θ ∈ [θ0 , 1] the following relations hold: Ms (h, µ; φr , a) = h−ω Ns (1; 1, W ) + (θ) , ω(β, s) > d , (6.6) Ms (h, µ; φr , a) = Ws (h; φr , V ) + h−ω Θs + (θ) , d − s − 1 < ω(β, s) < d .
(6.7)
Proof. We reduce the problem to that in the ball B(1) by performing a scaling ˆ be defined by (2.9) with ` = r, f = `−β/2 , and α, ν by transformation. Let Vˆ , a (2.10). Then β β α = hr 2 −1 = θ , ν = µr 2 +1 = µhκ θ−κ , (6.8) and in view of (2.16), ) x) + U (`x) , Vˆ (x) = `β V (`x) = |x|−β Φ(ˆ ˆ (x) = `−1 a(`x) , a
ˆ φ(x) = φ(`x) .
(6.9)
1 1 By virtue of (2.11) and (2.18), the functions X = |Vˆ | 2 and X = |W | 2 obey (2.3) with the Planck constant α and the constant β
M = M (α) = Cα− 2−β .
(6.10)
Let Aaˆ be the operator defined in (2.12). Since r−βs = h−ω αω , the relations (2.13) and (2.14) yield that ) Ms (h, µ; φr , a) = h−ω αω Ms (α, ν; φ, aˆ) , (6.11) Ws (h; φr , V ) = h−ω αω Ws (α; φ, Vˆ ) . By (6.11) the estimate (6.5) is equivalent to Ms (α, ν; φ, aˆ) ≤ Cα−σ , ∀α ∈ (0, 1] ,
νακ ≤ C .
(6.12)
Furthermore, the relations (6.6) and (6.7) amount to proving that there exists θ0 = (h + µhκ ) such that for any α ∈ [θ0 , 1] and for ν defined in (6.8), one has Ms (α, ν; φ, aˆ) = α−ω Ns (1; 1, W ) + (α) , ω > d ; (6.13) Ms (α, ν; φ, aˆ) = Ws (α; φ, Vˆ ) + α−ω Θs + (α) , d − 1 − s < ω < d, respectively.
(6.14)
893
DISCRETE SPECTRUM ASYMPTOTICS
Proof of (6.12). Let g ∈ C ∞ (R) be a function such that g(λ) = 0, λ ≥ 1 and gs (λ) ≤ g(λ) ≤ gs (λ − 1). Then by Corollary 3.6 and Theorem 3.11 Ms (α, ν; φ, aˆ) ≤ φg(Aaˆ ) 1 ≤ φ g(Aaˆ ) − g(Haˆ ) +1 s+1+ d+2N 2
≤ CM (α)
1
+ φg(Haˆ )
−d+2N +1
α
1 0
+ C M (α)s+ 2 α−d , d
for any N > (d + 1)/2 + s. In view of (6.10), for any fixed N the r.h.s. does not exceed Cα−σ with some σ > 0, which proves (6.12). Proof of (6.13) and (6.14) breaks into two steps. Step 1. We may assume s > 0 (otherwise ω = 0). First of all we shall find such θ0 = (h + µhκ ) that |Ms (α, ν; φ, aˆ) − Ns (α; φ, W )| ≤ C , ∀α ∈ [θ0 , 1] .
(6.15)
By cyclicity of trace Ms (α, ν; φ, aˆ) = tr χgs Haˆ (α, ν) χ . ◦
The operator Aaˆ obeys Assumption 2.2 with D =B(4ρ) for any ρ ≤ Erθ−1 . Further, by (6.9) the functions W and Vˆ have the form (4.2) with Ψ(x) = Φ(ˆ x), F (x) = U (`x), Y (x) = |x|−β . Thus to justify (6.15) we can use Theorem 4.4 with r = 2. By (4.19) and (6.10), for any L ≥ L0 ≥ C, ρ ≤ Erθ−1 , we have χ gs Haˆ (α, ν) − gs HW (α) χ 1 −σ2 2N +1 α 2N +3 d −σ1 ≤ CN L ρ α + K(0, ρ) + CL−s α−σ3 , ρ 1 1 akρ + kF kρ M 2 . K(0, ρ) = M 2 νkˆ
(6.16) (6.17)
Here and below by σj , j = 1, . . . , we denote positive exponents depending only on s, d. Their precise values are of no importance. Next, we pick up the parameters ρ, L, θ0 , N so as to guarantee boundedness of the r.h.s. of (6.16) uniformly in α ∈ [θ0 , 1]. To that end set L = α−σ3 /s , so that the second summand in (6.16) is bounded. Then the r.h.s. can be estimated by C1 α−
σ3 s
−σ2 −1 2N +1 d −σ4 ρ
ρ α
+ C2 α−(2N +3)
σ3 s
ρd α−σ1 K(0, ρ) + C3 .
(6.18)
Now choose ρ = α−γ1 , γ1 > σ3 /s + σ2 and fix a sufficiently big N in such a way that the first term in (6.18) is uniformly bounded. To make sure that ρ ≤ Erθ−1 , it suffices to assume that α ≥ h γ2 ,
0 < γ2 < (1 + γ1 (2 − β)/2)−1 .
894
A. V. SOBOLEV
Actually, under this condition we have more: 2 2−β γ3 1 − γ2 1 + γ1 > 0. ρrθ ≤ h , γ3 = 2−β 2
(6.19)
By (6.10) and (6.17) the second term in (6.18) does not exceed K(α, ν) = Cα−σ5 K(0, ρ) = Cα−σ6 νkˆ akρ + α−σ7 kF kρ . Here, in view of (2.17), kF kρ ≤ U0 (rρ). Further, according to (6.1), |ˆ a(x)| ≤ C|x|. Therefore by (6.8), νkˆ akρ ≤ µhκ α−κ ρ = µhκ α−κ−γ1 . Thus, in view of (6.19), we have K(α, ν) ≤ Cα−σ6 µhκ α−κ−γ1 + α−σ7 U0 (hγ3 ) . Choosing
1 σ +σ 6 7 1 + hγ2 θ0 = (µhκ ) σ8 + U0 (hγ3 )
(6.20)
with σ8 = σ6 + κ + γ1 , one can guarantee boundedness of K uniformly in α. This completes the proof of (6.15). 2
Step 2. Study of Ns (α; φ, W ). According to (2.13), (2.14) with ` = α 2−β , we have ) Ns (α; φ, W ) = α−ω Ns (1; φρ , W ) , ρ = `−1 . (6.21) Ws (α; φ, W ) = α−ω Ws (1; φρ , W ) , Let us consider separately two cases: ω > d and ω < d. Let ω > d. According to Theorem 2.4, Ns (1; φρ , W ) = Ns (1; 1, W ) + (α) ,
α → 0.
Taking into account (6.21) and (6.15), we get from here (6.13), and, consequently (6.6). Let d − s − 1 < ω < d. According to Theorem 2.4 Ns (1; φρ , W ) = Ws (1; φρ , W ) + Θs + (α) ,
α → 0.
Along with (6.21) this yields that Ns (α; φ, W ) = Ws (α; φ, W ) + α−ω Θs + (α) ,
α → 0.
(6.22)
Let us replace here Ws (α; φ, W ) with Ws (α; φ, Vˆ ). One can check directly that |Ws (α; φ, W ) − Ws (α; φ, Vˆ )| ≤ Ws α; φ, |x|−β kF k1 ≤ Cα−d kF k1 ≤ Cα−d U0 (hγ3 ) .
895
DISCRETE SPECTRUM ASYMPTOTICS 1
Adding, if necessary, U0 (hγ3 ) d to θ0 defined in (6.20), we obtain |Ws (α; φ, W ) − Ws (α; φ, Vˆ )| ≤ C , ∀α ∈ [θ0 , 1] . Thus (6.22) transforms into Ns (α; φ, W ) = Ws (α; φ, Vˆ ) + α−ω Θs + (α) . Combining this with (6.15), we obtain (6.14) and, consequently, (6.7).
2. Proof of Theorem 2.5. We break up the trace Ms (h, µ) into two parts as follows. Let the function φ ∈ C0∞ (B(1)) and the parameter r = rθ be defined by (6.2), (6.3) respectively. Denote ψ1 (x) = ψ(x)φr (x) ,
ψ2 (x) = ψ(x) 1 − φr (x) .
(6.23)
Then, obviously, Ms (h, µ) =
2 X
Ms (h, µ; ψk ) .
k=1
We study these two summands separately. Recall that µh ≤ C and by definition (6.4) κ > 1. Consequently, the parameter θ0 defined in Lemma 6.1, is actually (h). Step 1. Asymptotics of Ms (h, µ; ψ1 ). We claim that for θ ∈ [θ0 , 1] Ms (h, µ; ψ1 , a) = ψ(0)h−ω Ns (1; 1, W ) + h−ω (θ) + (h) , Ms (h, µ; ψ1 , a) = Ws (h; ψ1 , V ) + h−ω
ω > d; ψ(0)Θs + (θ) + (h) ,
d − s − 1 < ω < d.
(6.24)
(6.25)
Moreover, Ms (h, µ; ψ1 , a) ≤ Ch−ω ,
θ = 1 , ∀ω ≥ 0 .
(6.26)
The latter bound follows from (6.5). To prove (6.24) and (6.25) notice that for θ ∈ [θ0 , 1] Ms (h, µ; ψ1 , a) = ψ(0)Ms (h, µ; φr , a) + h−ω (h) , Ws (h; ψ1 , V ) = ψ(0)Ws (h; φr , V ) + h−ω (h) ,
ω > 0; ω < d.
(6.27) (6.28)
Indeed, Ms (h, µ; ψ1 ) = ψ(0)Ms (h, µ; φr ) + Ms (h, µ; ψ3 φr ) , where ψ3 (x) = ψ(x) − ψ(0). x ∈ supp φr . Thus, by (6.5)
Since |∂ψ(x)| ≤ C, one has |ψ3 (x)| ≤ Cr for
|Ms (h, µ; ψ3 φr )| ≤ CrMs (h, µ; φr ) ≤ Crθ−σ h−ω = Ch−ω h 2−β θ−σ1 . 2
896
A. V. SOBOLEV
Increasing (if necessary) θ0 defined in Lemma 6.1, we prove that Ms (h, µ; ψ3 φr ) = h−ω (h) ,
θ ∈ [θ0 , 1] ,
which yields (6.27). The bound (6.28) can be proven similarly. Now (6.6) along with (6.27) provide (6.24). The relation (6.25) follows from (6.7) and (6.27), (6.28). Step 2. Asymptotics of Ms (h, µ; ψ2 ). Here we use the multiscale method described in Sec. 5. First we introduce functions f (x) and `(x). Let % ∈ (0, 1/32) be some number and let β f (x) = |x|− 2 , `(x) = %|x| . (Recall that the same f and ` were used in the proof of Lemma 5.3.) Obviously, the functions `(x) and f (x) satisfy (5.1), (5.2) on the open set D = {x ∈ Rd : r/4 < |x| < 4E} . Conditions (5.3) are fulfilled for V and a due to (2.15). To check (5.4) notice that due to the inequality β < 2 we have for x ∈ D f (x)`(x) = %|x|
2−β 2
≥ %hθ−1 ≥ %h , ∀θ ∈ (0, 1] .
The lower bound f (x)2 ≥ cµh in (5.4) is trivially fulfilled since f (x) ≥ c, x ∈ D and µh ≤ C. Further, by definitions (6.2), (6.23), supp ψ2 ⊂ {x ∈ Rd : r/2 < |x| < E/2} , so that for any % ∈ (0, 1/32) the condition (5.5) is satisfied. Furthermore, it is clear that ψ2 obeys (5.3). Thus the conditions of Proposition 5.2 are satisfied and therefore ) Ms (h, µ; ψ2 , a) − Ws (h; ψ2 , V ) ≤ CR(h, µ) , (6.29) R(h, µ) = hs+1−d I1 + µs+1 hs+1−d I2 , with the integrals I1 , I2 defined in (5.6). Let us estimate I1 : Z I1 = f (x)d+s−1 `(x)−s−1 dx D
Z
β
|x|− 2 (d+s−1) |x|−s−1 dx ≤ C
≤C r/4<|x|<4E
where σ=
Z
4E
tσ−1 dt , r/4
2−β (d − 1 − s − ω) . 2
Hence σ d−1−s−ω Cr ≤ C hθ−1 , ω > d−s−1; I1 ≤ C(| ln r| + 1) ≤ C | ln h| + | ln θ| + 1 , ω = d − s − 1 ; C , ω < d−s −1.
897
DISCRETE SPECTRUM ASYMPTOTICS
The integral I2 is always bounded uniformly in h, µ irrespectively of the values of β < 2 and s: Z β |x|− 2 (d−2) dx ≤ C . I2 ≤ C |x|<4E
Therefore
−ω ω−d+1+s + C2 µs+1 hs+1−d , C1 h θ R(h, µ) ≤ C(| ln h| + | ln θ| + hµis+1 )hs+1−d , Chµis+1 hs+1−d , ω < d − s − 1 .
ω > d−s −1; ω = d−s −1;
(6.30)
Step 3. End of the proof. Let us combine the results of the two previous steps. Let first ω > d. We show that the region outside the singularity does not contribute to the leading term of the asymptotics. Indeed, for µh ≤ C, we have by (6.30) R(h, µ) ≤ Ch−ω ((θ) + (h)), θ ∈ [θ0 , 1] . Further, by virtue of (2.15), Ws (h; ψ2 , V ) ≤ Ch−d
Z
|x|−β
d 2 +s
dx
r/2≤|x|≤E/2
≤ Ch−d
Z
E/2
tδ−1 dt , r/2
δ=
2−β (d − ω) . 2
Thus by the definition (6.3) Ws (h; ψ2 , V ) ≤ Ch−d rδ ≤ Ch−ω θω−d . Consequently, in view of (6.29),
Ms (h, µ; ψ2 ) ≤ Ch−ω (θ) + (h) .
Combining this with (6.24), we arrive at
Ms (h, µ; ψ) = h−ω ψ(0)Ns (1; 1, W ) + (θ) + (h) .
For θ = θ0 this yields (1.7). Let ω = d. We fix θ = 1. In view of (6.26) and (6.29), (6.30), Ms (h, µ; ψ, a) = Ws (h; ψ2 , V ) + O(h−d ) . To calculate Ws (h; ψ2 , V ) we present it for arbitrary ρ ∈ [r, E/2] as follows: Z s+ d2 −1 d ψ2 (x) V− (x) dx Ξs h Ws (h; ψ2 , V ) = r/2≤|x|≤r
Z
s+ d2 (ψ(x) − ψ(0)) V− (x) dx
+ r≤|x|≤E/2
Z
s+ d2 dx V− (x)
+ ψ(0) Z
r≤|x|≤ρ
s+ d2 dx . V− (x)
+ ψ(0) ρ≤|x|≤E/2
(6.31)
898
A. V. SOBOLEV
The condition ω = d implies that |V (x)|s+ 2 ≤ C|x|−d . Taking also into account that |ψ(x)−ψ(0)| ≤ C|x|, one sees that the first two integrals are uniformly bounded and the last one does not exceed C(| ln ρ| + 1). Hence d
Ws (h; ψ2 , V ) − h−d ψ(0)I(h, V ) ≤ Ch−d (| ln ρ| + 1) , Z s+ d2 dx . V− (x) I(h, V ) = Ξs
(6.32)
r≤|x|≤ρ
Let us analyze I = I(h, V ). By (2.17) Z I − Ξs
Z s+ d2 dx ≤ CU0 (ρ) W− (x)
r≤|x|≤ρ
|x|−d dx . r≤|x|≤ρ
The integral in the l.h.s. equals Z Ξs
Z
d
Sd−1
ρ
Φ− (ϑ) 2 +s dϑ
t−1 dt = Bs (Φ)| ln h| + O(| ln ρ|) ,
r
and the one in the r.h.s. equals C(| ln r| − | ln ρ|)U0 (ρ) ≤ C 0 | ln h|U0 (ρ) . Therefore
I − | ln h|Bs (Φ) ≤ C 1 + | ln ρ| + | ln h|U0 (ρ) .
In combination with (6.32), (6.31), this yields Ms (h, µ, ψ, a) − h−d ψ(0)| ln h|Bs (Φ) ≤ Ch−d 1 + | ln h|U0 (ρ) + | ln ρ| . Pick ρ = C(| ln h| + 1))−1 . For U0 (ρ) → 0, ρ → 0, the r.h.s. will be of order h−d | ln h|(h). This leads to (1.8). Let d − s − 1 < ω < d. According to (6.29), (6.30) Ms (h, µ; ψ2 , a) = Ws (h; ψ2 , V ) + h−ω (θ) + O(µs+1 hs+1−d ) . Adding (6.25), we get Ms (h, µ; ψ, a) = Ws (h; ψ, V ) + h−ω ψ(0)Θs + (θ) + O(µs+1 hs+1−d ) . If θ = θ0 , this provides (1.9). Let ω ≤ d − s − 1. We fix θ = 1. The region around the singularity does not contribute into the asymptotics. Indeed, in view of (6.26), Ms (h, µ; ψ1 , a) ≤ Ch−ω .
899
DISCRETE SPECTRUM ASYMPTOTICS
Besides, Ws (h; ψ1 , V ) ≤ Ch−ω , so that (6.29) entails Ms (h, µ; ψ, a) = Ms (h, µ; ψ2 , a) + O(h−ω )
= Ws (h; ψ2 , V ) + O R(h, µ) = Ws (h; ψ, V ) + O R(h, µ) .
Using the second (for ω = d − s − 1) or the third (for ω < d − s − 1) inequality from (6.30), we obtain (1.11) or (1.12). Appendix. An eigenfunction estimate for HV An important ingredient in the proof of Theorem 2.40 is an estimate for the eigenfunctions of the Schr¨ odinger operator HV (h) for large x. We shall use the method which is essentially the Agmon’s method from [1]. Lemma A1 below will be valid for any (real-valued) potential V as long as the operator HV = −h2 ∆ + V is defined as a form-sum on the form domain of ∆. Besides, we assume that the function V is semibounded from below for |x| ≥ ρ0 > 0 and denote r ≥ ρ0 .
V1 (r) = ess-sup|x|≥r V− (x) ,
Lemma A1. Let the potential V be as specified above. Let ρ ≥ ρ0 + 1 and u(x) be a normalized eigenfunction of HV corresponding to an eigenvalue λ < 0, |λ| ≥ 4V1 (ρ − 1). Then p |λ| h2 exp −c ρ , ∀ρ ≥ ρ0 + 1 , |u(x)| dx ≤ C |λ| h |x|≥2ρ
Z
2
(A1)
the constants c and C being independent of λ, V , ρ, h. Proof. Let ρ ≥ ρ0 + 1, ζ ∈ C 1 (Rd ) be a non-negative function such that ( ζ(x) =
0 , |x| ≤ ρ − 1 ,
(A2)
1 , |x| ≥ ρ ,
and let g ∈ B ∞ (Rd \ B(ρ0 )) be a function g(x) = g(r), r = |x|, such that g(r1 ) ≤ g(r2 ) , sup |∂g(x)| ≤ 1 ,
ρ0 ≤ r1 ≤ r2 ,
x
g(x) = const ,
|x| ≥ ρ˜ ≥ ρ ,
(A3)
with some ρ˜. Furthermore, denote v(x) = φ(x)u(x) ,
δ
φ(x) = ζ(x)e h g(x) ,
δ > 0.
900
A. V. SOBOLEV
More precise choice of the functions ζ, g and the parameter δ will be made later on. Note that the function v belongs to the form-domain of HV , since φ(x) ∈ C 1 and φ(x) = const for sufficiently large x. Since u is the eigenfunction, we have Z Z (−ih∂u)(x)(−ih∂v)(x)dx + V (x)|u(x)|2 φ(x)dx Z +η
2
|u(x)|2 φ(x)dx = 0 ,
λ = −η 2 .
(A4)
For (−ih∂v)(x) = (−ih∂u)(x)φ(x) − ihu(x)∂φ(x) , the first term in (A4) equals Z Z (−ih∂u)(x)(−ih∂v)(x)dx = |h∂u(x)|2 φ(x)dx + S , Z
S=h
h∂u(x)u(x)∂φ(x)dx .
In view of (A4), (A5), Z Z |h∂u(x)|2 φ(x)dx + V (x) + η 2 |u(x)|2 φ(x)dx ≤ |S| .
(A5)
(A6)
Let us estimate S. Since δ
∂e h g(x) =
δ δ ∂g(x)e h g(x) , h
and |∂g(x)| ≤ 1, we have Z δ δ |S| ≤ h |h∂u(x)||u(x)| |∂e h g(x) |ζ(x) + e h g(x) |∂ζ(x)| dx Z ≤δ
Z |h∂u(x)||u(x)|φ(x)dx + h
δ
|h∂u(x)||u(x)|e h g(x) |∂ζ(x)|dx .
Using the Young inequality for the first and the second terms separately, we get for any > 0 Z Z δ2 |h∂u(x)|2 φ(x)dx + |u(x)|2 φ(x)dx |S| ≤ 2 2 Z Z δ δ h2 2 h g(x) 2 |h∂u(x)| e + |∂ζ(x)| dx + |u(x)|2 e h g(x) dx . (A7) 2 2 supp ∂ζ Estimating the r.h.s. of (A6) by (A7) and rearranging the terms, we obtain Z δ 2 h g(x) 2 ζ(x) − |∂ζ(x)| dx 1− |h∂u(x)| e 2 2 Z Z δ2 h2 δ |u(x)|2 φ(x)dx ≤ |u(x)|2 e h g(x) dx . + V (x) + η 2 − 2 2 supp ∂ζ
(A8)
901
DISCRETE SPECTRUM ASYMPTOTICS
Let us specify now the choice of the function ζ. We shall choose ζ in such a way that the function ζ(x) − |∂ζ(x)|2 K(x) = 1 − 2 2 in the first integral in the l.h.s. of (A8) is non-negative for all x ∈ Rd . Let ζ(x) = ξ(r − ρ + 1), r = |x|, where the function ξ ∈ C 1 (R+ ) is defined by ξ(r) = r2 ,
0≤r≤
1 ; 2
1 1 , ≤ r ≤ 1, 4 2 ξ(r) = 1 , r ≥ 1 .
ξ(r) ≥
Obviously, ζ satisfies (A2). Then for the function K(x) we have the relations 0 ≤ |x| ≤ ρ − 1 ; K(x) = 1 − 5/2 (|x| − ρ + 1)2 , K(x) = 0 ,
ρ − 1 ≤ |x| ≤ ρ − 1/2 ;
1 K(x) ≥ 1 − /2 − max |∂r ξ(r)|2 , 4 2 r K(x) = (1 − /2) ,
ρ−
1 < |x| ≤ ρ ; 2
|x| > ρ .
Choosing small enough one can guarantee that K(x) ≥ 0 for all x ∈ Rd . Therefore the first integral in the l.h.s. of (A8) is non-negative. Thus (A8) provides the estimate Z Z δ2 h2 δ 2 2 |u(x)| φ(x)dx ≤ |u(x)|2 e h g(x) dx . V (x) + η − 2 2 supp ∂ζ Taking into account also that supp ∂ζ ∈ {x : ρ − 1 ≤ |x| ≤ ρ}, and that g is a monotone function (see (A3)), we get from here: Z δ2 2 |u(x)|2 φ(x)dx V (x) + η − 2 Z h2 δ g(ρ) h2 δ g(ρ) h e eh |u(x)|2 dx ≤ . (A9) ≤ 2 2 supp ∂ζ The last inequality holds because u is normalized. η 2 − δ 2 /(2) = η 2 /2. Since η 2 ≥ 4V1 (ρ − 1), we have: V (x) +
η2 η2 ≥ , 2 4
Define δ = 1/2 η so that
x ∈ supp ζ .
Therefore, by monotonicity of g, the l.h.s. of (A9) has the lower bound η2 4
Z
δ
|u(x)|2 ζ(x)e h g(x) dx ≥
η 2 δ g(2ρ) eh 4
Z |x|≥2ρ
|u(x)|2 dx .
902
A. V. SOBOLEV
Now it follows from here and (A9) that η 2 δ g(2ρ) eh 4 which leads to
Z |x|≥2ρ
|u(x)|2 dx ≤
Z |x|≥2ρ
|u(x)|2 dx ≤
2h2 hδ e η2
h2 δ g(ρ) eh , 2
g(ρ)−g(2ρ)
.
(A10)
To obtain from here (A1) we specify the choice of the function g, satisfying (A3). Let ξ1 ∈ B ∞ (R) be a non-negative function such that ∂r ξ1 (r) ≥ 0 and 1 0, r ≤ − , 2 ξ1 (r) = 1 1, r ≥ . 2 Denote ξ2 = 1 − ξ1 and define r − 4ρ r − 4ρ + 6ρξ1 , g(r) = γ rξ2 4ρ 4ρ
γ > 0.
Clearly, g(r) = γr, r ≤ 2ρ and g(r) = 6γρ, r ≥ 6ρ. In particular, the third property (A3) is fulfilled for ρ˜ = 6ρ. To prove monotonicity of g we calculate its derivative: r 3 r − 4ρ r − 4ρ r − 4ρ −1 + ∂r ξ2 + ∂r ξ1 γ ∂r g(r) = ξ2 4ρ 4ρ 4ρ 2 4ρ 1 r − 4ρ r − 4ρ + (6ρ − r)∂r ξ1 . (A11) = ξ2 4ρ 4ρ 4ρ The r.h.s. is non-negative, since ξ2 ≥ 0, ∂r ξ1 ≥ 0 and r ≤ 6ρ on the support of ∂r ξ1 . This implies that g is monotone. Notice that the r.h.s. of (A11) is bounded from above by 1+
3 max ∂r ξ1 (r) . 2 r
Let the number γ −1 be equal to this bound, so that |∂r g| ≤ 1. This completes the proof of (A3) for g. To get p (A1) from (A10) it remains to notice that g(ρ) − g(2ρ) = −γρ and δ = cη = c |λ|. Acknowledgement The final version of the present paper was written during my stay at the International Erwin Schr¨ odinger Institute for Mathematical Physics in Vienna in December 1994. I am grateful to T. Hoffmann-Ostenhof whose invitation gave me an opportunity to spend two productive weeks in the stimulating atmosphere of the Institute.
DISCRETE SPECTRUM ASYMPTOTICS
903
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