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February 10, J070-S0129055X11004229

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Reviews in Mathematical Physics Vol. 23, No. 1 (2011) 1–51 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004229

N/V -LIMIT FOR LANGEVIN DYNAMICS IN CONTINUUM

FLORIAN CONRAD∗,†,‡ and MARTIN GROTHAUS∗,§ ∗Mathematics Department, University of Kaiserslautern, P. O. Box 3049, 67653 Kaiserslautern, Germany †Mathematics

Department, Bielefeld University, P. O. Box 100131, 33501 Bielefeld, Germany ‡[email protected] §[email protected] http://www.mathematik.uni-kl.de/∼grothaus/ Received 4 September 2009 We construct an inﬁnite particle/inﬁnite volume Langevin dynamics on the space of simple conﬁgurations in Rd having velocities as marks. The construction is done via a limiting procedure using N -particle dynamics in cubes (−λ, λ]d with periodic boundary condition. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of N -particle systems in (−λ, λ]d with periodic boundary condition. After proving tightness of the laws of the ﬁnite particle dynamics, the identiﬁcation of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on conﬁguration space fulﬁlling a uniform Ruelle bound (and their weak limits). Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for a wide class of repulsive interaction potentials φ (including, e.g., the Lennard–Jones potential) and all temperatures, densities and dimensions d ≥ 1. Keywords: Limit theorems; non-sectorial diﬀusion processes; interacting continuous particle systems; periodic boundary condition; Ruelle bound. Mathematics Subject Classiﬁcation 2010: 60B12, 82C22, 60K35, 60H10

1. Introduction The inﬁnite particle Langevin equation dxit = vti dt, 2κ i i dwt − κvti dt − dvt = ∇φ(xit − xjt )dt, β

(1.1)

i=j

where κ > 0, β > 0, describes the motion of particles at positions xit ∈ Rd having velocities vti ∈ Rd , i ∈ N, t ∈ [0, ∞). This motion is inﬂuenced by a surrounding medium causing friction (corresponding to the second summand in the second line 1

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of (1.1)) and stochastic perturbation, modeled by a sequence of independent Rd valued Brownian motions (wti )t≥0 . Moreover, the particles interact via a symmetric pair potential φ. For investigating properties like, e.g., the equilibrium ﬂuctuations of inﬁnite systems of interacting particles the ﬁrst main step is the construction of equilibrium (martingale) solutions for the corresponding model (cf. [26]). In [11], strong solutions are constructed to (1.1) in the case d = 2 for a wide class of symmetric pair potentials φ and initial conﬁgurations. In particular, for the construction given there a singularity of φ in the origin is allowed and φ is assumed to be C 1 (Rd \{0}) with derivatives fulﬁlling some local Lipschitz continuity (we do not give all the details on the conditions). Another construction for arbitrary d, but with more restrictions on the potential can be found in [26]. The potentials treated there are assumed to be positive, of ﬁnite range and C 2 , which, in particular, does not allow any singularities. There are also constructions of deterministic dynamics for inﬁnitely many particles (κ = 0), see, e.g., [24, 30, 4], some of which work in more general situations. However, note that for the above mentioned purpose of considering a scaling limit the stochastic dynamics is preferable, since, as is mentioned in [11], one can expect it to exhibit a better long-time behavior. (See, e.g., [29] for the correspondence between ergodic properties and the Boltzmann–Gibbs principle, which is crucial for the derivation of hydrodynamic limits in [29, 26].) Up to now there are no results on the construction of equilibrium Langevin dynamics covering physically realistic situations, such as e.g., the Lennard–Jones potential in dimension d = 3. Moreover, generalizations to the case of noncontinuous forces ∇φ have never been considered and are impossible when using the method from [11, 26]. Therefore, in this article we present a completely diﬀerent approach to construct for a wide class of potentials a martingale solution to (1.1) in the sense of [14], having a grand canonical Gibbs measure as initial distribution. The general method is the one used there for the construction of stochastic gradient dynamics. As assumptions on the potential we only need weak diﬀerentiability in Rd \{0}, boundedness of the weak derivatives away from 0 and some quite weak assumption on integrability of the weak derivatives. φ is not only allowed, but even supposed to be singular in the origin. This prevents particles from hitting each other and is the correct setting for applications in physics. Before describing the construction, we make the expression “martingale solution” more precise. To do so, we have to introduce some notation. Let us consider the space Γv = {γ ⊂ Rd × Rd | prx γ ∈ Γ, v = v whenever (x, v), (x, v ) ∈ γ}

(1.2)

of locally ﬁnite simple velocity marked conﬁgurations in Rd , where Γ = {ˆ γ ⊂ γ ∩ Λ) < ∞ for all compact Λ ⊂ Rd } and prx denotes the projection to Rd | (ˆ the ﬁrst d coordinates, i.e. prx γ = {x ∈ Rd | (x, v) ∈ γ}, γ ⊂ Rd × Rd . A denotes the cardinality of a set A. By F Cb∞ (Ds , Γv ) we denote the space of smooth

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cylinder functions on Γv of the form F (·) = gF (f1 , ·, . . . , fK , ·), where K ∈ N, gF ∈ Cb∞ (RK ) (which means gF is bounded and inﬁnitely often diﬀerentiable ∞ and all derivatives are bounded) and fi ∈ Ds := Csbs (Rd × Rd ). Here we deﬁne ∞ d d ∞ Csbs (R × R ) to be the space of Cb functions with spatially bounded support, i.e. the subset of Cb∞ (Rd × Rd ) of functions having support in Λ × Rd for some com pact Λ ⊂ Rd . Moreover, one deﬁnes f, γ := (x,v)∈γ f (x, v) for f having spatially bounded support (or also for more general f , e.g. f ≥ 0) and γ ∈ Γv . Now let (xit , vti )t≥0,1≤i≤N be an N -particle solution of (1.1). Via the mapping symN: RN d × RN d → Γv , given by symN (x, v) = {(x1 , v 1 ), . . . , (xN , v N )}, we map the dynamics to Γv . (Here the possibility is ignored that positions of particles coincide.) We then ﬁnd that the law of the resulting Γv -valued process solves the martingale problem for (L, F Cb∞ (Ds , Γv )), deﬁned by LF (γ) :=

K κ ∂l ∂l gF ({fi }K i=1 , γ)(∇v fl )(∇v fl ), γ β

l,l =1

+

K

  ∂l gF ({fi }K i=1 , γ)

l=1

−

κ ∆v fl − κv∇v fl + v∇x fl , γ β

 ∇φ(x − x )(∇v fl (x, v) − ∇v fl (x , v ))

(1.3)

{(x,v),(x ,v )}⊂γ

where F is as above, γ ∈ Γv and {fi }K i=1 , · := (f1 , ·, , . . . , fK , ·). We therefore call any (possible inﬁnite particle) Γv -valued process solving the martingale problem for L on F Cb∞ (Ds , Γv ) a martingale solution of (1.1) (on conﬁguration space). Due to the degeneracy in the position coordinates of the generator L as given above, there is no hope to apply the theory of symmetric or sectorial Dirichlet forms to obtain an existence result (as is done in the case of the stochastic gradient dynamics in [25, 34, 1]). We mention that, as a possible alternative, one might think of applying the theory of generalized Dirichlet forms (cf. [31]) or results from [2] instead in order to construct the dynamics directly at least on the space of multiple conﬁgurations. Such a construction is an interesting topic for further research, but up to now we did not obtain nontrivial results in this direction (but see [5], where a Langevin dynamics of a ﬁnite particle system is constructed by these methods). As starting point for the construction of the inﬁnite particle dynamics we use the ﬁnite volume N -particle Langevin dynamics constructed in [5]. We approximate Rd by cubes Λλn = (−λn , λn ]d , n ∈ N, where λn ↑ ∞ as n → ∞, and choose a Nn sequence (Nn )n∈N of natural numbers such that limn→∞ (2λ d = ρ < ∞. n) v The marked conﬁguration space Γ , when equipped with a natural topology, is a Polish space. Generalizing results from [19], we explicitly give a corresponding complete metric in Sec. 2. Thus, the results stated in [9] can be applied in order to

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obtain tightness of the sequence of the approximating dynamics. In order to obtain the corresponding tightness estimates, we establish a (uniform improved) Ruelle bound for the correlation functions of their invariant initial distributions, the ﬁnite volume canonical Gibbs measures with periodic boundary condition. In [28] one ﬁnds the (original) proof for such a bound, which works for empty boundary condition, but only in the grand canonical setting (and for tempered inﬁnite volume grand canonical Gibbs measures, see [28, Corollary 5.3]). In [14] a Ruelle bound for canonical correlation functions with empty boundary condition is shown by an adaptation of Ruelles proof using an estimate for the partition functions from [7]. In the situation of the dynamics in [5] the boundary of (−λn , λn ] is assumed to be periodic; therefore, one has to consider the canonical correlation functions with periodic boundary condition. These functions may be written down similar to the empty boundary case using summations φˆλn (cf. (3.5) below) of the potential. However, these sums are not lower regular uniformly in n. But this would be necessary to apply the proof from [14] or [28] (essentially) directly. This problem is solved here by another modiﬁcation of this proof (basically by adding a third case to the case diﬀerentiation of Ruelles proof, cf. Remark 3.13 below), allowing us to use uniformly lower regular cutoﬀs of the φˆλn , n ∈ N. Having shown tightness of the approximating laws and therefore the existence of weak accumulation points, we next need to prove that these accumulation points solve (1.1) in the sense of the martingale problem. The main problem here is to approximate LF as in (1.3) uniformly on the side of the approximations as well as on the side of the limit by bounded continuous random variables. We prove that this is indeed possible, when the approximating measures fulﬁll uniformly a Ruelle bound. Section 3.4 contains results on such approximations which we consider to be useful in general when dealing with limits of stochastic dynamics on conﬁguration space. Though they are partly based on arguments from [14], these results can be used to generalize the construction of stochastic gradient dynamics given there to the case of potentials which are only weakly diﬀerentiable in Rd \{0} instead of C 1 (Rd \{0}). For details, see Remark 4.18 below. The question arises whether one even has convergence of the dynamics, when the initial distributions converge to some measure µ. For this it would be suﬃcient to have convergence of the semigroups corresponding to the approximating dynamics. As in [14], this convergence can be shown when imposing an additional assumption. In [14] a uniqueness condition had to be imposed on the associated Dirichlet form. In the non-sectorial case, we have to make an assumption of a slightly stronger type, namely essential m-dissipativity of (L, F Cb∞ (Ds , Γv )) in L1 (Γv ; µ): By Trotter’s semigroup convergence theorem (formulated in a modiﬁcation of the framework of Kuwae and Shioya to L1 , see [21, 20, 32]) convergence of the semigroups follows immediately in this case. We give details in Sec. 4.5. Finally, by using a method from [13], where equivalence of the microcanonical and the grand canonical ensemble are shown in the periodic boundary situation, we transport this result to the case of the canonical ensemble. This shows that the

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invariant measure of the dynamics constructed in this paper is a grand canonical Gibbs measure. The considerations in [13] and the present article are not restricted to the high temperature regime, contrary to the corresponding result in [14, Sec. 6]. This may be considered as an advantage of starting with a periodic setting. Let us brieﬂy summarize the core results of this paper: • Derivation of an (improved) Ruelle bound for ﬁnite volume canonical correlation functions with periodic boundary condition. This bound is uniform for bounded particle densities. (Theorem 3.14, Corollary 3.16.) • Tightness of the laws P (n) of Nn -particle Langevin dynamics in cubes (−λn , λn ]d with periodic boundary condition for a wide class of symmetric pair potentials Nn that are weakly diﬀerentiable in Rd \{0}. Here we assume that λn ↑ ∞ and (2λ d n) converges to some ρ ∈ [0, ∞) as n → ∞. (Theorem 4.13.) • Identiﬁcation of accumulation points P of (P (n) )n∈N as above as martingale solutions of the Langevin equation on conﬁguration space. (Theorem 4.17.) • Identiﬁcation of the limit of ﬁnite volume canonical Gibbs measures with periodic boundary condition (i.e. the invariant initial distribution of P as above) as grand canonical Gibbs measure. (Theorem 5.1.) (We should mention that the hard work was done by Georgii and by Georgii and Zessin in [15, 12, 13], where the corresponding result for limits of microcanonical Gibbs measures is shown.) The above results apply to any dimension d ≥ 1. The Ruelle bound and the result on equivalence of ensembles are true for any repulsive, tempered, bounded below potentials (see conditions (RP), (T), (BB) in Sec. 3.1). The results on the dynamics require the weak diﬀerentiability condition (WD) formulated in Sec. 4.1 and additionally, as a restriction coming from the approximation with periodic dynamics, one needs to control the forces at large distances with condition (IDF). However, this condition may be rather seen as a theoretical restriction (cf. Remark 4.1(i), and also Remark 4.1(iii)). 2. A Polish Metric on Γv A natural topology for the space Γv deﬁned in (1.2) is the topology τ generated by the continuous functions with spatially bounded support, i.e. by mappings f, · with f ∈ Csbs (Rd × Rd ). In particular, using the (vague topology, i.e. the) topology generated by C0 (Rd × Rd ) functions instead, a sequence of conﬁgurations would be able to converge to the empty conﬁguration just by convergence of the marks to inﬁnity. In this section we deﬁne a Polish metric on Γv which generates τ . We use a construction similar to the one for unmarked simple conﬁgurations given in [19] and [14]. Below we consider Γv as a subset of the set Mv of Radon measures on Rd × Rd and Γ as a subset of the set M of Radon measures on Rd (in the sense that a set of points in Rd × Rd (respectively, Rd ) is identiﬁed with the sum of Dirac

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measures in these points). The notation ·, · is then extended to the dualization between continuous compactly supported functions and Radon measures. It is well known that the vague topology on Mv is generated by the metric dMv , given by d

Mv

(µ, ν) :=

∞ k=1

2−k

|fk , µ − fk , ν| , 1 + |fk , µ − fk , ν|

µ, ν ∈ Mv ,

where fk , k ∈ N, are suitable elements of C02 (Rd × Rd ) (cf. e.g. (the proof be a sequence in C02 (Rd ) such that dM (µ, ν) := of) [18, A7.7]). Let (g ) ∞ −k | gk ,µ − gk ,ν |k k∈N k=1 2 1+| gk ,µ − gk ,ν | , µ, ν ∈ M, generates the vague topology on M. For any two µ, ν ∈ Mv assigning ﬁnite mass to any Λ × Rd , Λ ⊂ Rd compact, we may deﬁne d (µ, ν) := dMv (µ, ν) + dM (prx µ, prx ν)

(2.1)

where prx µ ∈ M denotes the image measure of µ ∈ Mv with respect to the projection to the ﬁrst d coordinates. Then d : Γv × Γv → [0, ∞) generates the topology τ and prx : (Γv , d ) → (Γ, dM ) is continuous. (For the proof we refer to [6, Lemma 3.2.1].) However, d is far from being a complete metric on Γv . Firstly, consider the sequence (δ(x,vn ) )n∈N of Dirac measures, where vn → ∞. Such a sequence is a Cauchy sequence with respect to d , but it does not converge. Secondly, nothing prevents positions of particles from converging to each other. We use the idea from [19] to solve these problems. Let (Ik )k∈N be a sequence of C 1 functions on Rd such that 1{|·|≤k} ≤ Ik ≤ 1{|·|≤k+1} and choose a function h: Rd → (0, 1] such that h ∈ C 1 (Rd ) ∩ L1 (Rd ). Moreover, let Φ: (0, ∞) → [0, ∞) be a continuous decreasing function such that limt→0 Φ(t) = ∞. Then the space Γ of simple unmarked conﬁgurations is a complete (separable) metric space when equipped with the metric γ , γˆ ) := dM (ˆ γ , γˆ ) + dΦ,h (ˆ

∞

2−k rk

k=1

|S Φ,hIk (ˆ γ ) − S Φ,hIk (ˆ γ )| 1 + |S Φ,hIk (ˆ γ ) − S Φ,hIk (ˆ γ )| for γˆ , γˆ ∈ Γ,

(2.2)

where for nonnegative f ∈ C 1 (Rd ) and γˆ ∈ Γ we set S Φ,f (ˆ γ ) := eΦ(|x−y|2 ) f (x)f (y) {x,y}⊂ˆ γ

with | · |2 being the Euclidean norm on Rd , and (rk )k≥0 is any bounded sequence of positive numbers (cf. [19, Theorem 3.5]). (The topology and the completeness of the metric are invariant with respect to the weights (rk )k∈N , as long as they are positive and bounded.) Moreover, in [19, Theorem 3.5], it is shown that the metric dΦ,h generates the vague topology on Γ. This construction solves the problem of avoiding convergence to multiple conﬁgurations. It remains to keep mass away from v = ∞. Let a: [0, ∞) → [0, ∞) be an increasing surjective C 2 function and deﬁne χk (x, v) := a(v)(hIk )(x), x, v ∈ Rd .

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We deﬁne for γ, γ ∈ Γv dΦ,a,h (γ, γ ) := dMv (γ, γ ) + dΦ,h (prx (γ), prx (γ )) +

∞

qk 2−k

k=1

|χk , γ − χk , γ | 1 + |χk , γ − χk , γ |

with (qk )k∈N also being a bounded sequence of positive numbers. Then dΦ,a,h has the desired properties, i.e. the following lemma holds. (It is shown using the completeness of Γ with respect to dΦ,h and the closedness of the set of N0 -valued measures from Mv (cf. [18, A7.4]) with respect to vague convergence, for a complete proof see [6, Lemma 3.2.2].) Lemma 2.1. dΦ,a,h generates the topology τ on Γv and (Γv , dΦ,a,h ) is a complete metric space. 3. Ruelle Bound for Finite Volume Canonical Gibbs Measures with Periodic Boundary Condition In this section, we derive the Ruelle bound for correlation functions corresponding to ﬁnite volume canonical Gibbs measures with periodic boundary condition. We ﬁrst state and discuss conditions on the potential which are similar to those in [14, Sec. 3] in Sec. 3.1 and investigate properties of the periodic sum of the potential in Sec. 3.2. In particular, we prove that the important superstability property holds uniformly for these sums as well as temperedness and lower regularity in a sense suﬃcient for our purposes. We then go on with the proof of the Ruelle bound in the periodic case in Sec. 3.3. Finally, in Sec. 3.4, we show that a uniform Ruelle bound extends to weak limits of measures, and prove some approximation results which we need for the proof of Theorem 4.17 below. Though all considerations are stated for the conﬁgurational case (not including velocities) they also extend to the case of “full” measures (with independent Gaussian distributed velocities). For details on this fact, see also Sec. 3.4. 3.1. Conditions on the potential Until Sec. 3.3, we ﬁx a tempered, repulsive, bounded below pair potential φ, i.e. a function φ: Rd → R ∪ {∞} which is measurable, symmetric (i.e. φ(x) = φ(−x), x ∈ Rd ) and fulﬁlls the assumptions (BB), (RP ), (T ) which are given below. By | · | we denote the maximum norm in Rk , k ∈ N, i.e. |(y1 , . . . , yk )| := max1≤i≤k |yi |, (y1 , . . . , yk ) ∈ Rk . (BB) (bounded below ) There exists M < ∞ such that φ(x) ≥ −M for all x ∈ Rd . (RP) (repulsion) There exist R1 > 0 and a decreasing continuous function Φ: (0, ∞) → [0, ∞) with limt→0 Φ(t)td = ∞ such that φ(x) ≥ Φ(|x|)

for |x| ≤ R1 .

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Furthermore, φ is bounded from above on {x ∈ Rd |r ≤ |x| < ∞} for all r > 0. (T) (temperedness) There exist G, R2 < ∞ and ε > 0 such that |φ(x)| ≤ G|x|−d−ε

for |x| ≥ R2 .

Note that the second condition in (RP) implies that we may (and therefore we will) set R1 = R2 =: R. Moreover, R may be chosen arbitrarily small (changing, of course, the other constants). For later use in Sec. 4, we need more regularity of the function Φ, so we state the following lemma. Lemma 3.1. Let Φ: (0, ∞) → [0, ∞) be continuous, decreasing and such that ˆ (0, ∞) → [0, ∞) such that Φ ˆ ≤ Φ, Φ(t)td → ∞ as t → 0. Then there exists Φ: ˆ d −a Φ ˆ ˆ is bounded ˆ is continuously diﬀerentiable and e Φ(t)t Φ → ∞ as t → 0 and Φ for any a > 0. Proof. The proof is straightforward and therefore omitted. Let Λ ⊂ Rd . By ΓΛ we denote the set of locally ﬁnite simple conﬁgurations in Λ (i.e. locally ﬁnite subsets). In the sequel we will often denote ﬁnite or periodic conﬁgurations by Z (or similar notations) instead of γ, such that the notation looks a bit more similar to the one in [14, Sec. 3] and [28]. For a ﬁnite conﬁguration Z ∈ ΓRd = Γ we deﬁne the conﬁgurational energy φ(x − y) (3.1) Uφ (Z) := {x,y}⊂Z

and for Z , Z ∈ ΓRd being disjoint ﬁnite conﬁgurations we deﬁne the interaction energy φ(x − y). Wφ (Z , Z ) := Uφ (Z ∪ Z ) − Uφ (Z ) − Uφ (Z ) = x∈Z ,y∈Z

It is well known (cf. [28, Proposition 1.4]) that the assumptions (RP), (T) and (BB) imply (SS) (superstability) There exist A > 0, B ≥ 0 such that, if Z is a ﬁnite conﬁguration in Rd , then (Z ∩ Q1 (r))2 − BZ. Uφ (Z) ≥ A r∈Zd

(LR) (lower regularity) There exists a decreasing mapping Ψ: N0 → [0, ∞) such that r∈Zd Ψ(|r|) < ∞ and for disjoint ﬁnite conﬁgurations Z, Z it holds Ψ(|r − r |)(Z ∩ Q1 (r))(Z ∩ Q1 (r )), (3.2) Wφ (Z, Z ) ≥ − r,r ∈Zd

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where Q1 (r) := {(x1 , . . . , xd ) ∈ Rd | ri − (r1 , . . . , rd ) ∈ Zd .

1 2

< xi ≤ ri +

1 2}

9

for r =

In the case of pair interactions corresponding to a symmetric potential which is bounded from below (i.e. the case we consider here), (LR) as given above is equivalent to (LR) as given in [28] and also to the existence of a decreasing measurable ∞ ψ: [0, ∞) → [0, ∞) such that 0 ψ(t)td−1 dt < ∞ and −φ(x) ≤ ψ(|x|) for all x ∈ Rd (see also [22, Sec. 2.2.3]). 3.2. Potentials fulfilling (RP), (T), (BB) in periodic domains For λ > 0 we deﬁne Λλ := (−λ, λ]d . If Z ∈ ΓΛλ , we deﬁne Z˜ ∈ ΓRd to be the conﬁguration resulting from 2λ-periodic continuation of Z to Rd . A conﬁguration Z ∈ ΓΛλ is said to have distances < λ, if it holds {((x1 , . . . , xd ), (y1 , . . . , yd )) ∈ Z × Z | xi − yi = λ for some 1 ≤ i ≤ d} = ∅. Note that when we consider Λλ to have a periodic boundary, λ is the maximal possible distance between two particles in Λλ . Usually (e.g. in the sense of canonical Gibbs measures in continuous systems) a conﬁguration has distances < λ. In the case of periodic boundary condition we have to deal with the conﬁgurational energy of a ﬁnite conﬁguration Z ∈ ΓΛλ with periodic boundary condition, which we deﬁne by ˜φ,λ (Z) := U φ(x − y + 2λr). (3.3) {x,y}⊂Z r∈Zd

Remark 3.2. Note that in this deﬁnition the interaction between one particle and its copies is ignored. This does not have consequences for the results derived below. The corresponding canonical Gibbs measures and their correlation functions are exactly the same as if these interactions were included. Temperedness of the potential φ ensures that the above deﬁnition makes sense as well as the following. We deﬁne for λ > 0, y ∈ Rd φλ (y) := 1(−λ,λ)d (y) φ(y + 2λr). r∈Zd

˜φ,λ in terms of a ﬁnite conﬁguration (cf. Lemma 3.3 We use φλ in order to express U below). Possibly one would at ﬁrst sight prefer to use the indicator function 1(−2λ,2λ) instead of 1(−λ,λ) to simplify this, but see Remark 3.5 below. Lemma 3.3. There is a set S ⊂ Λ2λ\Λλ , such that for Z ∈ ΓΛλ having distances < λ it holds ˜φ,λ (Z) = Wφ (Z, Z˜ ∩ S) + Uφ (Z). U λ λ Remark 3.4. In order to avoid lenghty descriptions of the shape of the set S, the assertion of Lemma 3.3 looks more mysterious than necessary (see the proof below).

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Proof. First note that by assumption for any x, y ∈ Z˜ the statement x − y ∈ Λλ is equivalent to |x − y| < λ, which is symmetric in x, y. It holds ˜φ,λ (Z) = U φ(x − y + 2λr) {x,y}⊂Z r∈Zd

{x,y}⊂Z

r∈Zd

= Uφλ (Z) +

x−y ∈Λ / λ

φ(x − y + 2λr).

(3.4)

We consider the set M := {{−r, r}|r ∈ Zd , |r| = 1} and choose an arbitrary mapping χ: M → {r ∈ Zd ||r| = 1} such that χ({−r, r}) ∈ {−r, r} for any r ∈ Zd , |r| = 1, i.e. χ selects only one representative of each antipodal pair {−r, r} from M. We deﬁne S := r∈χ(M) (Λλ + 2λr) ∩ Λ2λ . Deﬁne X1 := {{x, y} ⊂ Z||x − y| > λ} and X2 := {(x, y)|x ∈ Z, y ∈ Z˜ ∩ S, |x − y| < λ}. We deﬁne θ: X1 → X2 in the following way. For {x, y} ∈ X1 there exists (uniquely) an r{x,y} ∈ χ(M) such that y − x ∈ Λλ − 2λr{x,y} (without loss of generality, possibly after interchanging x and y). Then we set θ({x, y}) := (x, y + 2λr{x,y} ), which is in X2 . θ is a bijection fulﬁlling r∈Zd φ(x−y +2λr) = φλ (x −y ) for any θ({x, y}) = (x , y ), {x, y} ∈ X1 . This and (3.4) imply the assertion. ˜φ,λ (Z), Z ∈ ΓΛ can be easier expressed in terms of φˆλ , deﬁned by Remark 3.5. U λ ˆ φ(y + 2λr) (3.5) φλ (y) := 1(−2λ,2λ)d (y) r∈Zd

but below we prove properties of φλ which cannot be obtained for φˆλ . In particular, the latter potentials are not uniformly lower regular (or tempered). ˜φ,λ with Let us now focus on properties of φλ , λ > 0, and the total energy U periodic boundary condition. We ﬁrst observe that φλ fulﬁll uniformly in λ ≥ λ0 > 0 the conditions we imposed on φ. ˜ M ˜, G ˜ in R+ and a decreasing continuous Lemma 3.6. Let λ0 > 0. There exist R, ˜ ˜ function Φ: (0, ∞) → [0, ∞) fulﬁlling lims→0 Φ(s)sd = ∞ (which, as is possible by ˜ ˜ is bounded for Lemma 3.1, shall be continuously diﬀerentiable and such that e−aΦ Φ any a > 0), such that ˜. (i) For all λ ≥ λ0 it holds φλ ≥ −M (ii) For all λ ≥ λ0 it holds ˜ −d−ε |φλ (x)| ≤ G|x|

˜ whenever |x| ≥ R.

(iii) For all λ ≥ λ0 it holds ˜ φλ (x) ≥ Φ(|x|)

˜ whenever |x| ≤ R.

(iv) For all c > 0 it holds sup sup |φλ (x)| < ∞.

λ≥λ0 |x|≥c

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Proof. We may without loss of generality assume that in (RP), (T) it holds R1 = ˜ := R. R2 =: R < λ0 . Temporarily we choose R d d For y ∈ (−λ, λ) and r ∈ Z \{0} it holds |y + 2λr| ≥ 2λ − |y| ≥ λ ≥ R, so φ(y + 2λr) ≥ −M − G |y + 2λr|−d−ε φλ (y) = r∈Zd

≥ −M − Gλ0−d−ε

r∈Zd \{0}

|r|−d−ε

r∈Zd \{0}

and the right-hand side is a constant larger than −∞, which proves (i). The same argument shows that for y ∈ (−λ, λ)d , λ ≥ λ0 , it holds |φλ (y) − φ(y)| ≤ Gλ−d−ε |r|−d−ε ≤ Gλ0−d−ε |r|−d−ε . r∈Zd \{0}

(3.6)

r∈Zd \{0}

This proves (iv). −d−ε ˜ 1 := G(1 + d ). Let λ ∈ (λ0 , ∞) and To show (ii), we deﬁne G r∈Z \{0} |r| d ˜ If |x| ≥ λ, then φλ (x) = 0 and there is nothing to prove. Therefore, x ∈ R , |x| ≥ R. let |x| < λ. It holds |x + 2λr| ≥ |λr| ≥ λ > |x| ≥ R for all r ∈ Zd \{0}. Hence ˜ 1 |x|−d−ε |φ(x + 2λr)| ≤ G|x|−d−ε + G |λr|−d−ε ≤ G |φλ (x)| ≤ r∈Zd \{0}

r∈Zd

proving (ii). −d−ε ˜ := Φ − Gλ−d−ε d Finally, (3.6) implies (iii) with Φ . Since this 0 r∈Z \{0} |r| ˜ might become negative away from 0, we may have to choose R ˜ a bit function Φ ˜ smaller. By (iv) we see that then (ii) still holds with G1 replaced by some possibly ˜ larger constant G. In Lemma 3.8 below, the above result is used to prove that the φλ are superstable and lower regular uniformly in λ ≥ λ0 and that moreover also the energy functions ˜φ,λ of conﬁgurations in Λλ with periodic boundary are uniformly superstable. As U a preparation for obtaining the latter result we need the following lemma. Lemma 3.7. Let λ0 > 0. There exists a constant cλ0 > 0 such that for all λ ≥ λ0 , Z ∈ ΓΛλ , k ∈ N it holds 2 (Z ∩ Q (r)) ≤ (Zk ∩ Q1 (r))2 (2k + 1)d c−1 1 λ0 r∈Zd

r∈Zd

≤ (2k + 1)d cλ0 where Zk :=

r∈Zd ,|r|≤k (Z

(Z ∩ Q1 (r))2

r∈Zd

+ 2λr).

Proof. We have (2k + 1)d r∈Zd (Z ∩ Q2λ (r))2 = r∈Zd (Zk ∩ Q2λ (r))2 and the same is true when 2λ is replaced by some δ ∈ [min{2λ0 , 1/3}, 1] such that 2λ is an

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odd multiple of δ. The assertion follows from repeated application of the fact that for any δ1 , δ2 > 0 such that δ2 ≥ δ1 ≥ δ2 /2 and all ﬁnite conﬁgurations Y ∈ ΓRd it holds (Y ∩ Qδ1 (r))2 ≤ (Y ∩ Qδ2 (r))2 ≤ 6d (Y ∩ Qδ1 (r))2 . 6−d r∈Zd

r∈Zd

r∈Zd

Here for δ > 0 and r ∈ Zd we set Qδ (r) := {(x1 , . . . , xd ) ∈ Rd | δ(rl − 12 ) < xl ≤ δ(rl + 12 ) for all 1 ≤ l ≤ d}. Lemma 3.8. Let λ0 > 0. It holds ˜ ˜ N0 → [0, ∞) fulﬁlling d Ψ(|r|) <∞ (i) There exists a decreasing mapping Ψ: r∈Z such that if Z, Z are disjoint ﬁnite conﬁgurations and λ ≥ λ0 it holds ˜ − r |)(Z ∩ Q1 (r))(Z ∩ Q1 (r )). Wφ (Z, Z ) ≥ − Ψ(|r λ

r,r ∈Zd

˜ ≥ 0 such that for all λ ≥ λ0 and all ﬁnite (ii) There are constants A˜ > 0, B conﬁgurations Z, it holds ˜ (Z ∩ Q1 (r))2 − BZ. Uφλ (Z) ≥ A˜ r∈Zd

˜ for all λ ≥ λ0 and Z ∈ ΓΛ having Moreover, after possibly enlarging A, λ distances < λ it holds ˜φ,λ (Z) ≥ A˜ ˜ U (Z ∩ Q1 (r))2 − BZ. r∈Zd

Proof. We deﬁne a potential φ: Rd → R ∪ {∞} by ˜ ˜ Φ(|x|) if |x| ≤ R, φ(x) := ˜ −d−ε if R ˜ ≤ |x|, −G|x| ˜ G, ˜ Φ ˜ as in Lemma 3.6. This potential fulﬁlls (RP), (BB) and (T) and is with R, therefore superstable and lower regular by [28, Proposition 1.4]. Since φλ ≥ φ for all λ ≥ λ0 this already implies (i) and the ﬁrst assertion in (ii). For r ∈ Zd we set Λλ,r := Λλ + 2λr. Let k ∈ N be a natural number and deﬁne Zk := r∈Zd ,|r|≤k (Z + 2λr). It holds  Uφλ (Z˜ ∩ Λλ,r ) Uφλ (Zk ) = r∈Zd ,|r|≤k

+

r ∈χ(M)

 Wφλ (Z˜ ∩ Λλ,r , Z˜ ∩ Λλ,r+r )η(r, r , k)

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where χ is deﬁned as in the proof of Lemma 3.3 and η(r, r , k) = 1 for |r + r | ≤ k and 0 else. It holds η(r, r , k) = 1 for |r| ≤ k − 1, thus by Lemma 3.3  ˜φ,λ (Z) + Uφλ (Z˜ ∩ Λλ,r ) U Uφλ (Zk ) = r∈Zd ,|r|
+

r∈Zd ,|r|=k

 Wφλ (Z˜ ∩ Λλ,r , Z˜ ∩ Λλ,r+r )η(r, r , k) .

r ∈χ(M)

But for Wφλ (Z˜ ∩ Λλ,r , Z˜ ∩ Λλ,r+r )η(r, r , k), r ∈ χ(M) there are only ﬁnitely many possible ﬁnite values, independently of k, hence there exists C < ∞ such that ˜φ,λ (Z)| ≤ Ck d−1 |Uφλ (Zk ) − (2k − 1)d U proving that 1 Uφλ (Zk ). k→∞ (2k − 1)d

˜φ,λ (Z) = lim U

By the ﬁrst assertion in (ii) and by Lemma 3.7, we conclude that ˜φ,λ (Z) ≥ c−1 A˜ ˜ U (Z ∩ Q1 (r))2 − BZ. λ0 r∈Zd

3.3. Ruelle bound for canonical correlation functions with periodic boundary condition Before going into the proof of the Ruelle bound we note a property of the canonical partition functions with periodic boundary stated in Lemma 3.9 below. Its proof is a slight adaptation of the proof of [7, Lemma 3 ] to the periodic boundary case (with external potential equal to 0). Note that the result of the following lemma in particular holds for the type of potentials we consider in this section. Its assumptions are obviously weaker than (RP), (T), (BB). Lemma 3.9. Let φ: Rd → R ∪ {∞} be measurable, symmetric, bounded from below

˜φ,λ := and such that for any a > 0 it holds Ca := {|x|≥a} |φ(x)| dx < ∞. We deﬁne U 1 ˆ U ˆ , where φλ := ((−2λ, 2λ)\{0}; dx). d φ(· + 2λr) is deﬁned as limit in L φλ

loc

r∈Z

Consider for λ > 0, N ∈ N0 , β > 0

˜ N,β e−β Uφ,λ ({x1 ,...,xN }) dx1 · · · dxN , Zλ := ΛN λ

which are (N ! times) the canonical partition functions with periodic boundary condition. Set Zλ0,β := 1. Let S ⊂ [0, ∞) × (0, ∞) be any compact subset. There exists a N constant kφ,S < ∞ such that for any N ∈ N0 , λ > 0, β > 0 fulﬁlling ( (2λ) d , β) ∈ S it holds kφ,S N +1,β Z . ZλN,β ≤ (2λ)d λ

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Proof. Set ρmax := sup{ρ | ∃β such that (ρ, β) ∈ S} and choose a > 0 small enough such that the volume Va = (2a)d of a | · |-ball with radius a fulﬁlls Va ρmax ≤ 12 . Then N Va ≤ 12 (2λ)d . Fix Z = (x1 , . . . , xN ) ∈ ΛN λ and consider ¯ a (x1 ) ∪ · · · ∪ B ¯a (xN )}, where B ¯a (x) := d the set Λaλ := Λλ \{B r∈Z ,|r|≤1 {y ∈ d R | |y − (x + 2λr)| ≤ a}, x ∈ Λλ . It holds

N N |φˆλ (ξ)|dξ φˆλ (ξ − xi ) dξ ≤ Λa λ i=1 i=1 Λλ \Ua (0)

|φ(x)|dx = N Ca ≤ ρmax (2λ)d Ca . ≤N Consequently, {ξ ∈ Λλ |

N i=1

Rd \Ua (0)

φˆλ (ξ − xi ) ≤ 4ρmax Ca } has volume of at least

vol(Λaλ ) − (2λ) ≥ (2λ) 4 4 . (Here and in the sequel vol(·) shall denote Lebesgue measure.) Hence

PN ˆ ˜ φ,λ ({x1 ,...,xN }) N +1,β −β U = e e−β i=1 φλ (xi −ξ) dξ dx1 · · · dxN Zλ d

d

ΛN λ

Λλ

d

(2λ) −4βρmax Ca N,β e Zλ 4 so the assertion holds with kφ,S = 4e4βρmax Ca . ≥

Now, ﬁx λ0 > 0, β > 0 and ρmax > 0. ρmax will be used below as a bound for the particle density. We choose sequences (φj )j∈N , (Vj )j∈N and (lj )j∈N and numbers ˜ A, ˜ B ˜ as in Lemma 3.8. For P ∈ N, α > 0 as in [28, Sec. 2] corresponding to Ψ, 1 ˜ −1 ln(k)). k := k[0,ρmax ]×{β} as in Lemma 3.9 we deﬁne γ(ρmax , β) := A ˜ (B + β d We deﬁne Q(j) := [−lj − 0.5, lj + 0.5] , j ∈ N, then Vj is the volume of Q(j) with respect to Lebesgue measure. The following is somehow obvious, but important. Lemma 3.10. For any g > 0 there exists λ1 (g) > 0 such that for all λ ≥ λ1 (g) it holds (2λ)d g < ψj Vj

(3.7)

for some j ≥ P such that Q(j + 1) ⊂ Λλ/2 . Proof. For any λ > 0 large enough we can ﬁx jλ ≥ P such that Q(jλ + 1) Λλ/2 ⊂ Q(jλ + 2). Then by the deﬁnition of Vj , lj (cf. [28]) Vjλ (1 + 3α)2d ≥ Vjλ +2 ≥ λd . Thus (3.7) holds as soon as ψjλ > 2d (1 + 3α)2d g. Hence our assertion follows since jλ → ∞ and consequently ψjλ → ∞ as λ → ∞. We deﬁne λ∗ := max{λ0 , λ1 (γ(ρmax , β) cλ0 ρmax 3d )}, where cλ0 is as in Lemma 3.7 and λ1 (·) is as in the above Lemma. As in [28] we write [j] := {r ∈ Zd | |r| ≤ lj }.

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Lemma 3.11. Let λ ≥ λ∗ , and let Z ∈ ΓΛλ be such that Z has distances < λ and fulﬁlls Z ≤ ρmax (2λ)d . Let Z¯ := Z˜ ∩ (S ∪ Λλ ), where S is as in Lemma 3.3. Then one of the following statements is valid: (I) For all j ≥ P it holds

(Z¯ ∩ Q1 (r))2 ≤ ψj Vj .

r∈[j]

(II) It holds ˜φ,λ (Z) ≥ ln(k)Z. βU (III) There exists a largest q ≥ P fulﬁlling (Z¯ ∩ Q1 (r))2 ≥ ψq Vq r∈[q]

and it additionally holds Q(q + 1) ⊂ Λλ/2 . Proof. Let us at ﬁrst consider the situation where γ(ρmax , β)Z. Using Lemma 3.8(ii) we ﬁnd that

r∈Zd

(Z ∩ Q1 (r))2 ≥

˜φ,λ (Z) ≥ β(Aγ(ρ ˜ max , β)Z − BZ) ˜ βU = ln(k)Z, i.e. (II) holds. Hence we may assume for the rest of the proof that (Z ∩ Q1 (r))2 ≤ γ(ρmax , β)Z. r∈Zd

Using Lemma 3.7, the notations given there and the deﬁnition of λ∗ we ﬁnd that (Z¯ ∩ Q1 (r))2 ≤ (Z1 ∩ Q1 (r))2 ≤ γ(ρmax , β)cλ0 3d Z r∈Z d

r∈Zd

≤ cλ0 γ(ρmax , β)ρmax 3d (2λ)d < ψj0 Vj0 for some j0 ≥ P fulﬁlling Q(j0 + 1) ⊂ Λλ/2 by Lemma 3.10. Consequently, for all j ≥ j0 it holds (Z¯ ∩ Q1 (r))2 < Vj0 ψj0 ≤ ψj Vj . (3.8) r∈[j]

Now, if (I) is not valid, the existence of a largest q ≥ P such that r∈[q] (Z¯ ∩ Q1 (r))2 ≥ ψq Vq is clear. But from (3.8), we ﬁnd that this number q fulﬁlls also the second condition in (III). ˜

−d−1 Let us have another look at the energy in case (III). Set C := A 4 (1 + 3α) d c (this is the constant C from [28, Proposition 2.5]). If A ⊂ R we denote by A the complement of A in Rd .

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Lemma 3.12. Let λ ≥ λ∗ . There exists a constant κ, not depending on Z and λ, such that the following holds: If in Lemma 3.11 statement (III) is valid, then ˜φ,λ (Z ∩ Q(q + 1)c ) − ˜φ,λ (Z) ≤ −U −U

A˜ (Z ∩ Q1 (r))2 4 r∈[q+1]

− Cψq+1 Vq+1 + κ(Z ∩ Q(q + 1)). Moreover, there is another constant κ such that in the same situation ˜φ,λ (Z) ≤ −U ˜φ,λ (Z ∩ Q(q + 1)c ) − (Cψq+1 − κ )Vq+1 − ln(k)(Z ∩ Q(q + 1)). −U (q+1) ∩ (S ∪ Λ ), Proof. Let Z¯ be deﬁned as in Lemma 3.11 and deﬁne Z¯ (q+1) := Z λ (q+1) where Z := Z ∩ Q(q + 1) and S is as in Lemma 3.3. It holds

˜φ,λ (Z) = −U ˜φ,λ (Z ∩ Q(q + 1)c ) − Uφ (Z (q+1) ) −U λ ¯ + 1)) − Wφλ (Z (q+1) , Z\Q(q − Wφλ (Z ∩ Q(q + 1)c , Z¯ (q+1) \(Z (q+1) )). Using [28, Proposition 2.5a] we ﬁnd that the ﬁrst assertion is shown as soon as we can prove that −Wφλ (Z ∩ Q(q + 1)c , Z¯ (q+1) \(Z (q+1) )) ≤ κ(Z (q+1) ). But this can be seen using Lemma 3.8: Note that (Z ∩ Q(q + 1)c ) ≤ Z and d (Z¯ (q+1) \(Z (q+1) )) = 3 2−1 Z (q+1) . We obtain by the uniform lower regularity (Lemma 3.8(i)) −Wφλ (Z ∩ Q(q + 1)c , Z¯q+1 \(Z ∩ Q(q + 1))) 3d − 1 3λ (q+1) ˜ Z (Z ≤ )Ψ − λ 2 2 3λ 3d − 1 ˜ ρmax (Z (q+1) )(2λ)d Ψ ≤ − λ . 2 2 ˜ 3λ − λ) is bounded ˜ we know that λd Ψ( By the summability property of Ψ 2 independently of λ. (It even tends to 0 as λ → ∞.) Hence the ﬁrst assertion follows. The second assertion is seen from the ﬁrst one using that there exists κ > 0 such that for any l ∈ N0 it holds A˜ − l2 + (ln(k) + κ)l ≤ κ 4 and Vq+1 = [q + 1]. Remark 3.13. Note that for the proofs of Ruelle bounds in [28] and [14] it is only necessary to consider the cases (I) and (III) as in Lemma 3.11. In case (III), the restriction Q(q + 1) ⊂ Λλ/2 does not occur there. For the periodic boundary case,

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however, a restriction on q like this is essential in order to estimate the interaction term −Wφλ (Z ∩ Q(q + 1)c , Z¯ (q+1) \(Z (q+1) )) in the proof of Lemma 3.12. For this reason (II) is considered as a separate case: When one chooses λ∗ large enough and for some conﬁguration (III) holds with q being too large, the total periodic energy is large enough to be estimated from below in a suitable way. The meaning of this estimate and the other estimates in the preceding Lemmas become clear in the proof of Theorem 3.14 below (which works as in [14] or [28]). We are now prepared to prove the main result of this section. For λ > 0, n, N ∈ N and n ≤ N , one deﬁnes the canonical correlation function with periodic boundary condition by

1 N! ˜ (n,N ) (x1 , . . . , xn ) := e−β Uφ,λ ({x1 ,...,xN }) dxn+1 · · · dxN , kλ −n) (N − n)! ZλN,β Λ(N λ (0,N )

x1 , . . . , xn ∈ Λλ . This deﬁnition is supposed to imply that one sets kλ = 1 ˜φ,λ ({x1 ,...,xN }) 1 (N,N ) −β U for any N ∈ N, λ > 0. Moreover k (x1 , . . . , xN ) = N ! Z N,β e , λ

x1 , . . . , xN ∈ Λλ .

Theorem 3.14. Let φ be a pair potential fulﬁlling (RP ), (BB ), (T ) given in Sec. 3.1 and let ρmax > 0, β > 0. Then there exists a constant ξ > 0 and some λ∗ > 0 such that the following holds: For all λ ≥ λ∗ and n ∈ N0 , N ∈ N fulﬁlling n ≤ N ≤ ρmax (2λ)d , and all x1 , . . . , xn ∈ Λλ having distances < λ in the above sense (and thus Lebesgue-a.e.) it holds (n,N )

kλ

(x1 , . . . , xn ) ≤ ξ n .

Remark 3.15. (i) In fact, for any ρmax , β and λ0 > 0 there exists ξ such that the estimate from Thereom 3.14 holds for all λ ≥ λ0 . This follows since for (n,N ) ≤ λ ∈ (λ0 , λ∗ ) the particle number is restricted to N ≤ ρmax (2λ∗ )d and kλ ˜ ˜ is chosen according to Lemma 3.8 for the ρnmax k N eβN B , n ≤ N , where B prescribed value of λ0 . (ii) The restriction to x1 , . . . , xn having distances < λ is a result of the more convenient symmetric choice of φλ we made above. It should be possible to drop it when doing all calculations in a nonsymmetric way, but this would be wasted eﬀort: The correlation functions are to be seen as density of the correlation measures with respect to Lebesgue–Poisson measure (see also Sec. 3.4 or [22], or [27, p. 72]), so their values on a Lebesgue null set do not matter for applications. Both remarks are also valid for Corollary 3.16 below. Proof of Theorem 3.14. Choose λ∗ , C, k etc. as above, let D < ∞ be as in [28, Proof of Proposition 2.6]. Set   ξ := ρmax 1 + keβD + e−(βCψq+1 −βκ −ρmax )Vq+1 , q≥P

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which is < ∞, since ψq+1 → ∞ as q → ∞ and Vq+1 grows at least as fast as q. The proof is done by induction on n. For n = 0 the assertion is trivially fulﬁlled. We may assume that x1 = 0. Let S I , S II and S III be the sets of tuples −n such that Z := {x1 , . . . , xN } has distances < λ and satisﬁes (xn+1 , . . . , xN ) ∈ ΛN λ III the subset of S III such that q (I)–(III) in Lemma 3.11, respectively. Denote by Sq,l is as in Lemma 3.11(III) and l = ({xn+1 , . . . , xN } ∩ Q(q + 1)). Let (xn+1 , . . . , xN ) ∈ S I . Then as in [28, Proof of Proposition 2.6] we ﬁnd that −Wφλ ({x1 }, {x2 , . . . , xN }) ≤ D. Hence, since x1 = 0 we have ˜φ,λ ({x2 , . . . , xN }) − Wφ ({x1 }, {x2 , . . . , xN }) ˜φ,λ ({x1 , . . . , xN }) = −U −U λ ˜φ,λ ({x2 , . . . , xN }) + D. ≤ −U Thus N! 1 N,β (N − n)! Zλ ≤ eβD

˜

SI

e−β Uφ,λ ({x1 ,...,xN }) dxn+1 · · · dxN

1 N! N,β (N − n)! Zλ

≤ eβD N

˜

−n ΛN λ

e−β Uφ,λ ({x2 ,...,xN }) dxn+1 · · · dxN

k (n−1,N −1) k (x2 , . . . , xn ) ≤ eβD kρmax ξ n−1 , (2λ)d λ

(3.9)

by Lemma 3.9. Now let us consider the conﬁgurations in S II . Here Lemmas 3.11 and 3.9 yield

N! 1 ˜ e−β Uφ,λ ({x1 ,...,xN }) dxn+1 · · · dxN (N − n)! ZλN,β S II ≤ Nn

1 ZλN,β

N −n −N (2λ)d k ≤ ρnmax ≤ ρmax ξ n−1 .

(3.10)

III We ﬁnally turn to Sq,l , q ≥ P , 0 ≤ l ≤ N − n. Set N (q) := ({x1 , . . . , xn } ∩ Q(q + 1)) ≥ 1 and assume without loss of generality that x1 , . . . , xN (q) ∈ Q(q + 1). We set χq := e−β(Cψq+1 −κ )Vq+1 . Lemma 3.12 shows that

1 N! ˜ e−β Uφ,λ ({x1 ,...,xN }) dxn+1 · · · dxN III (N − n)! ZλN,β Sq,l l Vq+1 1 N! N −n ≤ χq (N − n)! ZλN,β k N (q)+l N − n − l

˜ e−β Uφ,λ ({xN (q)+1 ,...,xN −l}) dxn+1 · · · dxN −l × −n−l ΛN λ

(N −N (q)−l)

= χq

l Vq+1 Zλ N! N (q)+l (N − N (q) − l)!l! k ZλN,β (N −l−N (q),n−N (q))

× kλ

(xN (q)+1 , . . . , xn )

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k (2λ)d

N (q)+l

19

ξ n−N (q)

(ρmax Vq+1 )l . l! Summing over q ≥ P and l we obtain

N! 1 ˜ e−β Uφ,λ ({x1 ,...,xN }) dxn+1 · · · dxN N,β (N − n)! Zλ S III ≤ ξ n−1 ρmax e−(βCψq+1 −βκ −ρmax )Vq+1 . ≤ χq ρmax ξ n−1

(3.11)

q≥P

The assertion is implied by our choice of ξ, (3.9)–(3.11) since the set of tuples −n (xn+1 , . . . , xN ) ∈ ΛN such that {x1 , . . . , xN } has distances < λ has full Lebesgue λ N −n measure in Λλ . As in [14, Theorem 3.2] in the case of empty boundary condition one also obtains an improved Ruelle bound. Corollary 3.16. Under the assumptions of Theorem 3.14, there exists a constant ζ ≥ ξ such that for all λ ≥ λ∗ and n ∈ N0 , N ∈ N fulﬁlling n ≤ N ≤ ρmax (2λ)d it holds (n,N )

kλ

(x1 , . . . , xn ) ≤ ζ n inf e−β

P j=i

ˆλ (xi −xj ) φ

1≤i≤n

(3.12)

all x1 , . . . , xn ∈ Λλ having distances < λ (thus Lebesgue-a.e.). It follows also that (n,N )

kλ

2β

P

2β

˜

(x1 , . . . , xn ) ≤ ζ n e− n

{i,j}⊂{1,...,N }

ˆλ (xi −xj ) φ

= ζ n e− n Uφ,λ ({x1 ,...,xn }) . Proof. The proof is a slight modiﬁcation of the proof of the second assertion of [14, Theorem 3.2]. Since the canonical ensemble with respect to φ with periodic boundary condition equals the canonical ensemble with respect to φˆλ with empty boundary condition, we have the following Kirkwood–Salsburg type equation (cf. [14] or [16, Eq. (38.16)]): ZλN −1,β −β P2≤i≤N φˆλ (x1 −xj ) (n,N ) (n−1,N −1) (x1 , . . . , xn ) = N e (x2 , . . . , xN ) kλ kλ N,β Zλ +

N −n l=1

×

l

1 l!

(e

i=1

Λlλ

(n+l−1,N −1)

kλ

ˆλ (x1 −yi ) −β φ

(x2 , . . . , xn , y1 , . . . , yl )

− 1)dy1 · · · dyl

.

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We may assume that any tuples occurring in this formula have distances < λ and by translation invariance we are allowed to assume that x1 = 0. Then under the integral sign we may replace φˆλ by φλ . But φλ fulﬁlls (T) and (BB) uniformly in λ ≥ λ0 . Thus

|e−βφλ (y) − 1|dy < ∞, Iλ := Rd

is bounded independently of λ ≥ λ0 . Therefore, by Lemma 3.9 and Theorem 3.14, we have N −n n+l−1 l P ξ I ˆ (n,N ) kλ (x1 , . . . , xn ) ≤ e−β 2≤i≤n φλ (x1 −xi ) kρmax ξ n−1 + l! l=1

≤e

−β

P

ˆ 2≤i≤n φλ (x1 −xi ) n

ζ

where ζ := max{ξ, kρmax eξI } with I = supλ≥λ0 Iλ . Symmetry of the correlation function implies the assertions. 3.4. Weak limits of measures and Ruelle bounds In this section, we prove that a uniform (improved) Ruelle bound is transported to weak limits µn → µ of measures on Γ. Moreover, we prove that for a large γ ⊂ Rd | ˆ γ < ∞} of class of functions f : Γ0 → R deﬁned on the space Γ0 := {ˆ ﬁnite conﬁgurations it holds µn (Kf ) → µ(Kf ) and that one may ﬁnd bounded continuous local functions approximating Kf uniformly in L1 (Γ; µn ), n ∈ N, and γ ) := in L1 (Γ; µ). Here Kf : Γ → R denotes the K-transform of f , given by Kf (ˆ f (ˆ η ). For further information on this mapping see [22]. These results η ˆ⊂ˆ γ ,ˆ η ∈Γ0 then also hold for Γ0 , Γ replaced by the velocity marked spaces Γv0 , Γv , when one assumes that the velocities are independently Gaussian distributed and do also not depend on the conﬁguration. This is (basically) seen with the help of Lemma 3.17 below. Let us at ﬁrst collect some more notations (cf. [22]). By ΓΛ , we denote the subset of Γ consisting of conﬁgurations contained in Λ ⊂ Rd . Let now Λ ⊂ Rd be open. The projection prΛ : Γ → ΓΛ mapping γ ∈ Γ to γ ∩ Λ ∈ ΓΛ is continuous, when ΓΛ and Γ are equipped with the vague topology, which we will always assume below. This means, we equip ΓΛ (respectively, Γ) with the vague topology on the set of Radon measures on Λ (respectively, Rd ). Moreover, these spaces shall be equipped with the corresponding Borel σ-ﬁelds. We denote by Γn ⊂ Γ0 the set of n-point conﬁgurations and by ΓΛ,n ⊂ ΓΛ the set of n-point conﬁgurations contained in Λ, Λ ⊂ Rd open, bounded. For measurable and topological structures on these spaces we refer to [22] and to the considerations around Lemma 3.21 below. We denote the Borel σ-ﬁeld on Γ0 by B(Γ0 ). Let Λ ⊂ Rd be open and bounded. Below we use the fact that when we consider ΓΛ ⊂ Γ0 as a topological, hence as a measurable space, the corresponding measurable structure

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coincides with the one on ΓΛ induced by the vague topology (cf. [22, Remark 2.1.2]). This implies that prΛ : Γ → ΓΛ ⊂ Γ0 is measurable. The Lebesgue–Poisson measure λ on Γ0 is deﬁned by

∞ 1 1A ({x1 , . . . , xn })dx1 · · · dxn for A ∈ B(Γ0 ). λ(A) := n! (Rd )n n=0 A measure µ on Γ is said to be locally absolutely continuous with respect to Lebesgue–Poisson measure if for each open bounded Λ ⊂ Rd the image measure µ ◦ pr−1 Λ is absolutely continuous with respect to the restriction of λ to ΓΛ . For any such probability measure µ one deﬁnes the correlation functional ρµ : Γ0 → R by

d(µ ◦ pr−1 Λ ) (ˆ γ ∪ γˆ )dλ(ˆ γ ) := γ ) when γˆ ∈ ΓΛ , Λ ⊂ Rd open, bounded. ρµ (ˆ dλ ΓΛ In the same manner we deﬁne ΓvΛ , prvΛ , etc., but we replace the vague topology by the topology generated by bounded continuous functions with spatially bounded support. We deﬁne λv to be the Lebesgue–Poisson measure corresponding to the 2 intensity measure √ 1 d e−βv /2 d(x, v) (cf. [22, Chap. 3.1.3]), i.e. 2π/β

λv (A) :=

∞

1 1 nd n! 2π/β n=0 2

(Rd ×Rd )n

1A ({(x1 , v1 ), . . . , (xn , vn )})

2

× e−β(v1 +···+vn )/2 dx1 · · · dvn for A ∈ B(Γv0 ). We also deﬁne for a function f : Γv0 → R similarly to the unmarked case Kf (γ) := η⊂γ,η∈Γv f (η). 0 The diﬀerence between the velocity marked situation and the unmarked situation is negligible for the sort of results we derive below, if the velocities are assumed to be distributed independent and Gaussian. This is (mainly) seen by the following lemma. Lemma 3.17. Let µ be a probability measure on Γ which is locally absolutely continuous with respect to Lebesgue–Poisson measure λ. Then there exists a unique measure µv on Γv , deﬁned via d(µ ◦ pr−1 d(µv ◦ (prvΛ )−1 ) Λ ) ({x1 , . . . , xk }) ({(x1 , v1 ), . . . , (xk , vk )}) = v dλ dλ

(3.13)

for any {(x1 , v1 ), . . . , (xk , vk )} ∈ ΓvΛ , k ∈ N0 , Λ ⊂ Rd open, bounded. Moreover, for measures µ, µn , n ∈ N, on Γ which are locally absolutely continuous with respect to Lebesgue–Poisson measure one obtains (i) µn → µ weakly iﬀ µvn → µv weakly. (ii) (µn )n is tight iﬀ (µvn )n is tight. (iii) For any nonnegative measurable f : Γv0 → R+ 0 it holds µv (Kf ) = µ(Kf∗ )

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where 1 f∗ ({x1 , . . . , xk }) := kd 2π/β

Rkd

f ({(x1 , v1 ), . . . , (xk , vk )})

2

2

× e−(β/2)(v1 +···+vk ) dv1 · · · dvk for {x1 , . . . , xk } ∈ Γ0 , k ∈ N0 . Moreover µn (Kf∗ ) → µ(Kf∗ ) as n → ∞ iﬀ µvn (Kf ) → µv (Kf ) as n → ∞. (iv) For the correlation functionals it holds ρvµv (γ) = ρµ (prx γ), when we deﬁne ρvµv analogously to ρµ . Proof. Existence and uniqueness are seen using Kolmogorov’s theorem (cf. [22, Sec. 3.1.3]). For proofs of (i), (ii) and (iv) we refer to [6, Lemma 3.3.16]. (iii) follows for f ∈ ΓΛ,m , Λ ⊂ Rd bounded, by a calculation as in the proof of [6, Lemma 3.3.16(i)] and extends to general f by monotone convergence. If a probability measure µ on Γ is locally absolutely continuous with respect to Lebesgue–Poisson measure and its correlation functional fulﬁlls ρµ (η) ≤ ξ η

for λ-a.e. η ∈ Γ0 ,

(3.14)

it is said to fulﬁll a Ruelle bound. Note that by [22, Proposition 4.2.2] (or by Proposition 3.19 below) any measure fulﬁlling a Ruelle bound possesses ﬁnite local moments, i.e. µ((· ∩ Λ)m ) < ∞ holds for any relatively compact Λ ⊂ Rd and m ∈ N. For the unmarked situation the following lemma is contained in [22, Theorem 4.2.11]. In the velocity marked case it can be shown analogously or using Lemma 3.17(iii) above. For a measure µ on Γ0 (respectively, Γv0 ) and a nonnegative measurable function f : Γ0 → R (respectively, f : Γv0 → R) we denote by f (·)µ or f µ the measure having density f with respect to µ. Lemma 3.18. Let µ be a probablity measure on Γ which is locally absolutely continuous with respect to Lebesgue–Poisson measure. Let f ∈ L1 (Γ0 ; ρµ (·)λ). Then Kf ∈ L1 (Γ; µ), Kf L1(Γ;µ) ≤ K|f | L1(Γ;µ) ≤ f L1(Γ0 ;ρµ (·)λ) and

f (η)ρµ (η)dλ(η) = (Kf )(γ)dµ(γ). Γ0

Γ

In particular, the sum deﬁning Kf converges µ-a.s. absolutely. The same holds with µ, ρµ , λ, Γ0 , Γ replaced by µv , ρvµv , λv , Γv0 , Γv , respectively. When one assumes some more integrability for f , the above integrability result may be extended also to powers of K-transforms, since these powers can be expressed as sums of K-transforms of products:

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Proposition 3.19. Let µ be a probability measure on Γ which is locally absolutely continuous with respect to Lebesgue–Poisson measure. Let K ∈ N and f : Γm → R (or equivalently f : (Rd )m → R symmetric) be measurable. Deﬁne for M ≤ mK m YM,K := {(α1 , . . . , αmK ) ∈ {1, . . . , M }2K | αlm+1 , . . . , αlm+m are pairwise

distinct for all l = 0, . . . , K − 1, {α1 , . . . , αmK } = M }. (The last condition in this deﬁnition ensures that each element of {1, . . . , M } m appears at least once.) Assume that for any M ≤ mK and (α1 , . . . , αmK ) ∈ YM,K it holds K−1 ρµ ({ξ1 , . . . , ξM }) |f (ξαlm+1 , . . . , ξαlm+m )|dξ1 · · · dξM < ∞. (Rd )M

l=0

(3.15) Then

Γ

|Kf |K dµ < ∞

and this integral may be estimated by a (ﬁnite) linear combination of the integrals in (3.15) with coeﬃcients only depending on K and M, not on µ. A similar statement holds in the velocity marked case with independent Gaussian velocities. In this case, in (3.15) one also integrates over the velocities, but with respect to the corresponding Gaussian measure instead of Lebesgue measure. Proof. See [6, Proposition 3.3.5]. Remark 3.20. (i) In fact, we only use the above proposition for m = 1 and for m = 2. If m = 1 and ρµ fulﬁlls (3.14), the situation becomes considerably easy, since (3.15) is implied by f ∈ L1 (Rd ; dx) ∩ LK (Rd ; dx) and µ(|Kf |K ) can be estimated in terms of f L1(Rd ;dx) , f LK (Rd ;dx) and ξ. In the velocity marked case the situation for m = 1 is analogous. (ii) Note that if (µn )n∈N are as in Proposition 3.19 and fulﬁll a Ruelle bound uniformly in n, one ﬁnds that the estimate for Kf LK (Γ;µn ) resulting from Proposition 3.19 and the Ruelle bound is uniform with respect to n. Before going on we need some information on the topological and measur able structure of Γ0 . Γ0 = ∞ m=0 Γm is equipped with the topology of disjoint union. Therefore, B(Γ0 ) is generated by open sets U ⊂ Γm , m ∈ N, which are bounded in the sense that U ⊂ ΓΛ,m for some open bounded Λ ⊂ Rd . The topology on Γm , m ∈ N, is deﬁned as the quotient topology with respect to the mapping symm : (Rd )m \D → Γm , where Dm = {(x1 , . . . , xm ) ∈ (Rd )m | xi = xj for some i = j}.

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We ﬁnd that the set Um of open bounded sets in Γm is closed with respect to ﬁnite intersections and that there exists a sequence (Uk )k∈N ⊂ Um increasing to Γm . Therefore, the collection U := m Um can be used to uniquely determine measures on Γ0 that are ﬁnite on elements of U (see [3, Theorem 10.3]). We use this fact in the proof of Lemma 3.22 below as well as the following lemma. Lemma 3.21. Let U ⊂ Γ0 be open and bounded, i.e. there exists M ∈ N0 and M an open bounded subset Λ ⊂ Rd such that U ⊂ m=0 ΓΛ,m . Then there exists a sequence (fk )k∈N of nonnegative bounded continuous functions on Γ0 increasing to 1U . Proof. Assume without loss of generality that U ⊂ ΓΛ,m , m ∈ N. Since d sym−1 m (U ) ⊂ R is open we may choose nonnegative bounded continuous func. By mixing tions f˜k : Rd → R, k ∈ N, with bounded support increasing to 1sym−1 m (U) over the permutations of the arguments we may assume that the f˜k are symmetric. Now deﬁne fk ({x1 , . . . , xm }) := f˜k (x1 , . . . , xm ) for {x1 , . . . , xm } ∈ Γm , k ∈ N. The desired properties of the sequence (fk )k∈N follow immediately. We now prove the result mentioned at the beginning for the case of the original (in contrast to “improved”) Ruelle bound. A measurable function F : Γv → R (respectively, F : Γ → R) is called a cylinder function, if for some bounded measurable Λ ⊂ Rd it holds F = F ◦ prvΛ (respectively, F = F ◦ prΛ ). Lemma 3.22. Let (µn )n be a sequence of probability measures on Γ such that each µn is locally absolutely continuous with respect to Lebesgue–Poisson measure and such that moreover the correlation functionals fulﬁll a Ruelle bound (3.14) uniformly in n. Let µn → µ weakly as n → ∞. Then the following holds: (i) For any f ∈ L1 (Γ0 ; ξ · λ) it holds µn (Kf ) → µ(Kf ) as n → ∞. Moreover, there exists a sequence (Gk )k∈N of bounded continuous cylinder functions Gk : Γ → R such that Gk → Kf as k → ∞ in L1 (Γ; µ) and L1 (Γ; µn ) uniformly in n ∈ N. (ii) µ is locally absolutely continuous with respect to Lebesgue–Poisson measure and its correlation functional fulﬁlls the same Ruelle bound as the µn . ρ ρµ (iii) The sequence ( ξµ|·|n )n∈N converges in weak-* sense to ξ|·| in L∞ (Γ0 ; λ) (seen 1 as dual of L (Γ0 ; λ)). Similar statements hold for µv , µvn , n ∈ N. Proof. Let f : Γm → R be any nonnegative bounded continuous function having γ ) = 0 for all γˆ ∈ Γ0 \ΓΛ . local support, i.e. there exists Λ ⊂ Rd bounded such that f (ˆ Then the mappings Kf ∧ r: Γ → R, r > 0, are bounded and continuous (cf. [22, Proposition 4.1.5(v)]). Consequently, µn (Kf ∧ r) → µ(Kf ∧ r) as n → ∞. By )2 ) Proposition 3.19 and Remark 3.20(i) we ﬁnd that µn (Kf −Kf ∧r) ≤ µn ((Kf →0 r

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as r → ∞ uniformly in n. Moreover, for each r > 0 it holds µ(Kf ∧ r) = lim µn (Kf ∧ r) ≤ sup µn (Kf ) < ∞ n→∞

n∈N

by Lemma 3.18. So, the monotone convergence theorem implies Kf ∈ L1 (Γ; µ) and Kf ∧ r → Kf in L1 (Γ; µ) as r → ∞. Therefore (i) holds for f as described above. We continue the proof of (i) after showing (ii) and (iii). ρ Relative compactness of ( ξµ·n )n∈N with respect to weak-∗ topology follows already from boundedness and the Banach–Alaoglu theorem. Let ρ˜ be an accumulation point and set ρ := ρ˜ξ · . (This convenient method for obtaining a limiting correlation functional is taken from [28, Theorem 5.5], [22, Theorem 2.7.12], where it was used to prove the existence of a grand canonical Gibbs measure.) We now prove that ρ(·)λ coincides with the correlation measure of µ (cf. [22, Sec. 4.2]). Once this is shown, we ﬁnd by [22, Propositions 4.2.2 and 4.2.16] (the conditions given there are fulﬁlled by ρ(·)λ) that µ is locally absolutely continuous with respect to Lebesgue–Poisson measure and by [22, Proposition 4.2.14] we see that ρ is indeed the correlation functional for µ. This implies (ii), and since it implies that there is at most one accumulation point ρ˜, (iii) also follows. Let U ∈ U, i.e. U ∈ Um for some m ∈ N0 . Choose a sequence (fk )k∈N increasing to 1U as in Lemma 3.21. Then by integrability of 1U with respect to ξ · λ and Lemma 3.18 we ﬁnd that Kfk → K1U in L1 (Γ; µn ) as k → ∞ uniformly in n. As above, using the monotone convergence theorem we obtain that Kfk → K1U also in L1 (Γ; µ). Therefore, µn (K1U ) → µ(K1U )

(3.16)

as n → ∞. ρµn k = ρ˜ in L∞ (Γ0 ; λ) We choose a subsequence (ρµnk )k∈N such that limk→∞ ξ· with respect to weak-∗ topology. Then

ρµnk · ρµnk1U dλ = ξ 1U dλ → ρ1U dλ (3.17) · Γ0 Γ0 ξ Γ0 as n → ∞. By Lemma 3.18 the left-hand sides of (3.16) and (3.17) coincide. As U ∈ U was chosen arbitrarily, this implies by [22, Deﬁnition 4.2.1] and the considerations preceding Lemma 3.21 that the correlation measure of µ is indeed given by ρ(·)λ. We complete the proof of (i). Let f ∈ L1 (Γ0 ; ξ · λ). We may without loss of generality assume that f is nonnegative. Choose a sequence (fk )k∈N of bounded continuous functions having local support converging to f in L1 (Γ0 ; ξ · λ). Now, since the same Ruelle bound holds uniformly for µn , n ∈ N, and also for µ, (i) follows from Lemma 3.18: It holds Kfk → Kf in L1 (Γ; µn ) uniformly in n and in L1 (Γ; µ). In the velocity marked case (ii) now follows using the deﬁnition of µv and Lemma 3.17(iv). (iii) is also seen using this lemma: For f ∈ L1 (Γv0 , λv ) it holds

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with f∗ deﬁned as in Lemma 3.17(iii)

ρvµn ρvµ ρµn ρµ v f dλ = f∗ dλ → f∗ dλ = f dλv · · · · v v v ξ ξ ξ ξ v Γ Γ0 Γ0 Γ0 as n → ∞. (i) is shown for the velocity marked case analogously as for the unmarked case. We now extend the results from Lemma 3.22 for the case of the improved Ruelle bound. Lemma 3.23. Let (µn )n be a sequence of probability measures on Γ, which are locally absolutely continuous with respect to Lebesgue–Poisson measure and converge ˜ n )n∈N ⊂ L∞ (Γ0 ; λ) be (uniformly bounded and ) weak-∗ weakly to µ. Let ζ ≥ 1 and (h ∞ ˜ convergent to some h ∈ L (Γ0 ; λ). Assume that ρµn (ˆ η ) ≤ ζ ˆη ˜hn (ˆ η) is valid for all ηˆ ∈ Γ0 and n ∈ N. Then the following holds: ˜ n replaced by h. ˜ (i) ρµ fulﬁlls the analogous bound with h ¯ For any ¯ (ii) Assume in addition that there exists a function h such that ˜hn , ˜h ≤ h. ¯ it measurable function f : Γ0 → R which is integrable with respect to ζ · h(·)λ holds µn (Kf ) → µ(Kf ). Moreover, there exists a sequence of bounded continuous cylinder functions (Gk )k≥0 such that Gk → Kf as n → ∞ uniformly in L1 (Γ; µn ), n ∈ N, and in L1 (Γ; µ). Similar statements hold for µv , µvn , n ∈ N. Proof. Since the hn , n ∈ N, are uniformly bounded and ρµ (∅) = 1, the ρµ fulﬁll ρ a uniform Ruelle bound ρµ ≤ ζ˜· with ζ˜ ≥ ζ without loss of generality. So ( ζ˜µ·n )n converges in weak-∗ sense to

A

ρµ dλ = lim n→∞ ζ˜·

ρµ ζ˜·

A

by Lemma 3.22(iii). Thus

ρµn dλ ≤ lim n→∞ ζ˜·

·

˜hn ζ dλ = ζ˜· A

·

˜ ζ dλ h ˜· A ζ

holds for any measurable set A ⊂ ΓΛ,m for some open relative compact Λ ⊂ Rd and some m ∈ N. This implies (i). We now prove (ii). Let (fk )k∈N ⊂ L1 (Γ0 ; ζ · λ) be such that fk → f in h(·)ξ · λ). Due to Lemma 3.18 it holds L1 (Γ0 ; ¯ Kf − Kfk L1 (Γ;µn ) ≤ f − fk L1 (Γ0 ;ρµn (·)λ) ≤ f − fk L1 (Γ0 ;h(·)ξ · λ) , ¯ which converges to 0 as k → ∞ uniformly in n. Analogously we see that Kfk → Kf in L1 (Γ; µ). Now (ii) follows from Lemma 3.22(i). In the velocity marked case (i) is directly seen by Lemma 3.17(iv) and (ii) is derived analogously to the unmarked case.

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We now focus on a special class of measures, the canonical Gibbs measures. For any open bounded set Λ ⊂ Rd , N ∈ N, β > 0 and a symmetric potential φ (which we assume to be bounded below and ﬁnite a.e.) one deﬁnes the canonical Gibbs measure µφ,β Λ,N by

1 µφ,β (A) := 1A (x1 , . . . , xN )e−βUφ (x1 ,...,xN ) dx1 · · · dxN , (3.18) Λ,N φ,β ZΛ,N ΛN φ,β A ⊂ ΛN measurable, where ZΛ,N is the normalization constant. Deﬁne a mapping N symΛ,N : Λ → Γ by symΛ,N (x1 , . . . , xN ) := {x1 , . . . , xN }, x1 , . . . , xN ∈ Λ. Then −1 the image measure µφ,β Λ,N ◦ symΛ,N deﬁnes the corresponding distribution of N -point φ,β conﬁgurations. (Note that µΛ,N -a.s. symΛ,N has values in ΓΛ,N , i.e. one a.s. obtains N -point conﬁgurations.) We formulate the tightness result from [14, Lemma 5.2] more generally, such that it also admits the perodic boundary case, in which, as the particle number N and the volume Λ of the system, also the potential φ varies.

Lemma 3.24. Let (φn )n∈N be a sequence of symmetric pair interactions fulﬁlling (RP ) except of the last condition and (BB ) uniformly in n. Moreover let (Nn )n∈N ⊂ N and (Λn )n∈N be a sequence of open relatively compact subsets of Rd . φn ,β ◦ sym−1 Set µn := µN Λn ,Nn . If the correlation functionals ρµn of µn , n ∈ N, fulﬁll n ,Λn the improved Ruelle bound 2β

ρµn (η) ≤ ζ η e− η

P {x,y}⊂η

φn (x−y)

for all η ∈ Γ0

uniformly in n, then the sequence (µn )n∈N is tight. As a consequence, the same holds for the sequence (µvn )n∈N . Proof. The proof is essentially the same as the proof of [14, Lemma 5.2]. Remark 3.25. (i) Let φ fulﬁll (RP), (BB), (T) as in Sec. 3.1 and let Λn := Λλn = (−λn , λn ]d , n ∈ N. Then by Lemma 3.6 and Corollary 3.16 the conditions of Lemma 3.24 are fulﬁlled for any sequence (Nn )n∈N , (λn )n∈N fulﬁlling Nn ˆ limn→∞ λn = ∞ and limn→∞ (2λ d = ρ ∈ [0, ∞) with φn := φλn , deﬁned as n) in (3.5). Hence the corresponding sequence (µn )n∈N is tight. φn ,β (ii) In the periodic case (cf. (i)) one might rather consider µΛ ◦ per−1 Λn ,Nn n ,Nn φn ,β −1 instead of µΛ ◦ sym , where per (x , . . . , x ) := {x d 1 N 1 + Λ ,N n Λn ,Nn r∈Z n n n ,Nn 2λn r, . . . , xNn + 2λn r}. But since for any cylinder function F : Γ → R one ﬁnds φn ,β φn ,β −1 ◦per−1 that µΛ Λn ,Nn (F ) = µΛn ,Nn ◦symΛn ,Nn (F ) for suﬃciently large n, weak n ,Nn convergence properties of these sequences are equivalent. (A sequence (νn )n∈N of probability measures on Γ converges weakly to a probability measure ν if νn (F ) → ν(F ) as n → ∞ holds for all bounded continuous cylinder functions, see, e.g., [6, Lemma 3.3.15]). Finally, in order to apply the result from Lemma 3.23 to the case of periodic boundary condition, we make the following remark.

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Remark 3.26. Consider again the situation from P Remark 3.25(i). For ψ: Rd → − 2β {x,y}⊂η ψ(x−y) (or b (η) η R we deﬁne := ψ P bψ : Γ0 → R by bψ (η) := e −β y∈η\{x} ψ(x−y) ˜ ˜ inf x∈η e ), η ∈ Γ0 . Setting hn := 1Λλn bφˆλ and h := bφ we ﬁnd n by uniform stability of the periodic interaction energy of conﬁgurations in Λλn (cf. Lemma 3.8(ii)) that the ˜hn are uniformly bounded. From (3.6) we ﬁnd that ˜n → h ˜ φλn → φ pointwise and hence also φˆλn → φ pointwise, which implies that h pointwise. Together with uniform boundedness we obtain weak-∗ convergence in L∞ (Γ0 ; λ). Hence Lemma 3.23(i) can be applied and µ fulﬁlls the improved Ruelle bound for φ. ¯ which fulﬁlls the assertion of Lemma 3.23(ii) and We now choose a function h is useful for the considerations in Sec. 4.4 below. By (3.6) there exists m > 0 such that |φˆλn (y) − φ(y)| = |φλn (y) − φ(y)| ≤ m

for n ∈ N, |y| < λn

and inf φˆλn (z) = inf φλn (z) ≥ −m − M

z∈Rd

z∈Rd

for all n ∈ N, where −M denotes a lower bound of φ. Hence, setting φ(y) − m if |y| < λ1 φ(y) := , −m − M else ¯ we obtain φˆλn ≥ φ for all n ∈ N and φ ≥ φ, which implies ˜hn , ˜h ≤ bφ =: h, n ∈ N. Thus the conclusion of Lemma 3.23(ii) is valid in this case. Moreover, φ + m + M + M ≥ φ, where M := sup|x|≥λ1 φ(x) < ∞, so · Lp (Rd ;e−βφ dx) and · Lp (Rd ;e−βφ dx) are equivalent norms for p ≥ 1. 4. N/V -Limit of Langevin Dynamics We now derive the main result of this article. Starting with N -particle Langevin dynamics on cuboid domains (Sec. 4.2), we go on by proving tightness of the corresponding laws on Γv (Sec. 4.3) and ﬁnally prove (Sec. 4.4) that any weak accumulation point of these laws solves (1.1) in the sense speciﬁed in Sec. 1. Throughout this section we ﬁx an inverse temperature β > 0. We assume that any function g: A → R deﬁned on some subset A ⊂ Γ0 (respectively, Γv0 ) is extended to the whole of Γ0 (respectively, Γv0 ) by being set to 0 on the complement of A (cf. Sec. 3.4 for the deﬁnition of Γ0 , Γv0 ). 4.1. Additional conditions on the potential Let φ be a (symmetric) pair potential fulﬁlling the conditions (RP), (BB), (T) given in Sec. 3.1. Consider the following additional conditions on φ: (WD) (weak diﬀerentiability) φ is continuous in Rd \{0}, φ is weakly diﬀerentiable on this set and ∇φ is bounded on each of the sets {x ∈ Rd | |x| > r}, r > 0. Moreover, ∇φ ∈ L1 (Rd ; e−βφ dx) ∩ L3 (Rd ; e−βφ dx).

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(IDF) (integrably decreasing forces) φ is weakly diﬀerentiable in Rd \{0} and there exist R3 > 0 and a decreasing function θ: [R3 , ∞) → [0, ∞) such that |∇φ(x)| ≤ θ(|x|) and

[R3 ,∞)

for all x ∈ Rd , |x| ≥ R3

rd−1 θ(r) dr < ∞.

Remark 4.1. (i) Both assumptions we suppose to be quite natural and suﬃciently weak, allowing e.g. the Lennard–Jones potential or any other potential fulﬁlling (WD) and being such that |∇φ(x)|2 decreases when |x|2 → ∞, x ∈ Rd . (ii) In order to do the construction using a limit of dynamics corresponding to φ with periodic boundary, we need uniform L3 -integrability of the ∇φˆλ with ˆ respect to e−β φλ (cf. (3.5)), at least for a sequence λn tending to ∞ as n → ∞. When (WD) holds, this is an assumption on the behavior of ∇φ away from the origin. Condition (IDF) yields an appropriate behavior, as we prove in the following lemma. Though it might be not optimal, it is suﬃcient for our purposes. (iii) We suppose that one does not need (IDF) to construct a martingale solution of (1.1). The construction for a potential φ not fulﬁlling (IDF) might be done by constructing ﬁrst the dynamics for smooth cut-oﬀs of φ by approximation with periodic potentials and then approximating φ by the cut-oﬀs. However, we do not enter into a detailed discussion of this question here. Lemma 4.2. (i) Let φ fulﬁll (RP ), (BB ), (T ), (WD ). Then for any λ > 0 the function φˆλ is weakly diﬀerentiable in (−2λ, 2λ)d \{0} and for any λ0 > 0 it holds supλ≥λ0 ∇φˆλ L1 (Λ2λ ;e−βφˆλ dx) < ∞. (ii) If φ additionally fulﬁlls (IDF ), then sup φˆλ 3 < ∞ holds for ˆ −β φ λ≥λ0

any λ0 > 0.

L (Λ2λ ;e

λ dx)

Proof. The functions φˆλ , λ > 0, are cut-oﬀs of 2λ-periodic functions, hence all the assertions on integrals over Λ2λ can be reduced to assertions on integrals over Λλ . = 2d φˆλ p p , p ∈ [1, ∞). For example, we have φˆλ p p ˆ ˆ −β φ −β φ λ λ L (Λ2λ ;e

L (Λλ ;e

dx)

dx)

ˆ (i) For the ﬁrst assertion, it suﬃces to prove that φˆλ is the L1 (Λ2λ ; e−β φλ dx)d limit of a sequence of weakly diﬀerentiable (in (−2λ, 2λ) \{0}) functions which converges with respect to the norm · W 1,1 (Λ2λ ,e−βφˆλ dx) := · L1 (Λ2λ ;e−βφˆλ dx) + ∇ · 1 . Deﬁne φˆλ,k := φ(· + 2λr), k ∈ N. Then φˆλ,k → φˆλ ˆ d −β φ L (Λ2λ ;e

λ dx)

1

as k → ∞ in L (Λ2λ ; e

r∈Z ,|r|≤k

ˆλ −β φ

dx) and moreover for k, l ∈ N, k ≥ l, it holds

∇φˆλ,k − ∇φˆλ,l L1 (Λ2λ ;e−βφˆλ dx) ≤ 2d

2lλ≤|x|

|∇φ(x)|eβ(m+M) dx → 0

as l → ∞. Here m is as in Remark 3.26. This shows the ﬁrst assertion.

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Since m can be chosen independently of λ ≥ λ0 , we ﬁnd by an easy argument similar to the above calculation that ∇φˆλ L1 (Λ2λ ;e−βφˆλ dx) ≤

2d eβ(m+M+M ) ∇φL1 (Rd ;e−βφ dx) , where M := sup|x|≥λ0 φ(x). We now prove (ii). We may assume that R3 = R1 = R2 =: R ≤ λ0 . By the considerations preceding the proof of (i) we have to estimate 3

−β φˆ (x) λ e ∇φ(x + 2λr) dx Λλ r∈Zd 

≤

Λλ

3

|∇φ(x)| +

ˆ θ(|x + 2λr|) e−β φλ (x) dx

0=r∈Zd

independently of λ ≥ λ0 . Due to L3 -integrability of ∇φ with respect to e−βφ dx, ˆ hence with respect to e−β φλ dx uniformly in λ ≥ λ0 , it suﬃces to show that (2λ)d supx∈Λλ ( r=0 θ(|x + 2λr|))3 is bounded independently of λ ≥ λ0 . This follows from monotonicity and integrability of θ: For λ ≥ λ0 , x ∈ Λλ it holds:

θ(|y + 2λr|) dy Λλ θ(|x + 2λr|) ≤ (2λ)d d d r=(r1 ,...,rd )∈Z ri ≥1∀i

r=(r1 ,...,rd )∈Z ri ≥0∀i

θ(t)td−1 dt

C ≤

[0,∞)

(2λ)d

for some C < ∞ independent of λ. Here we extend θ to [0, ∞) by setting θ(t) := θ(R3 ) for t ≤ R3 . Other parts of the sum 0=r∈Zd are treated in an analogous way. Remark 4.3. In order to have some more concrete legitimation for the introduction of the additional condition (IDF), we consider the following example: Set d = 1 and consider φ according to (RP), (BB) such that it holds ∇φ(x) = ∓1 whenever 1 1 1 2k ± 12 − (|k|+1) 2 ≤ x ≤ 2k ± 2 + (|k|+1)2 for some k ∈ Z\{0} and ∓∇φ(x) ≥ 0 when 2k ± 12 − 14 ≤ x ≤ 2k ± 12 + 14 . Then φ can be such that ∇φ ∈ L1 ∩ L3 (Rd ; e−βφ dx) and (T) is fulﬁlled, but r∈Z ∇φ(· + 2r) behaves like √ 1 1 around 12 , so one does ·− 2 not obtain L2 integrability of ∇φˆ1 . Lemma 4.4. Let φ fulﬁll (RP), (T ), (BB ), (WD ), and assume that for some λ0 > 0 it additionally holds supλ≥λ0 ∇φˆλ Lp (Λλ ;e−φˆλ dx) < ∞ for p = 1, 2, 3. Then it holds for i = 1, 2, 3 ˆ

ˆ

sup |∇φˆλ |e− 3 φλ Li (Λ2λ ;dx) = 2d sup |∇φˆλ |e− 3 φλ Li (Λλ ;dx) < ∞. β

λ≥λ0

β

λ≥λ0

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Proof. The equality is clear, cf. the proof of Lemma 4.2. Choose any a > 0. The functions φˆλ are bounded in the set {x ∈ Λλ | |x| ≥ a} and the bound is uniform in λ ≥ λ0 (cf. Lemma 3.6). Hence there exists D > 0 iβ ˆ ˆ such that e− 3 φλ ≤ De−β φλ on this set for λ ≥ λ0 . We compute for λ ≥ λ0 β ˆ |∇φˆλ |e− 3 φλ Li (Λλ ;dx) ≤ D1/i ∇φˆλ Li (Λλ ∩{|·|>a};e−βφˆλ dx) βi

ˆ

1/i

+ |∇φˆλ |i e− 3 φλ L1 (Λλ ∩{|·|≤a};dx) ≤ D1/i ∇φˆλ Li (Λλ ;e−βφˆλ dx) + (2a)

(3−i)d 3i

∇φˆλ L3 (Λλ ;e−βφˆλ dx)

by the H¨ older inequality. The assertion follows. 4.2. The finite particle dynamics on Γv Let φ fulﬁll (RP), (T), (BB), (WD) and (IDF) and let N ∈ N, λ > 0. The state space for the N -particle dynamics is given by EλN , where Eλ := Mλ × Rd , Mλ being the manifold resulting from glueing the opposite surfaces of Λλ = (−λ, λ]d together. We deﬁne the N -particle potential Ψλ,N by Ψλ,N (x1 , . . . , xN ) := i<j φˆλ (xi − xj ), (x1 , . . . , xN ) ∈ MλN . The potential Ψλ,N fulﬁlls the assumptions of [5, Theorem 3.1] (cf. [5, Example 3.4(i)]). Thus there is a law Pλ,N on C([0, ∞), EλN ) with invariant ˆ

initial distribution µλ,N = µφΛλλ,β ,N (i.e. the canonical Gibbs measure for φ with periodic boundary condition) such that the corresponding process is a Markov process solving the martingale problem for the m-dissipative L1 (EλN ; µλ,N )-closure (Lλ,N , D(Lλ,N )) of the generator (Lλ,N , C0∞ (EλN )), given by κ (4.1) Lλ,N = ∆v − κv∇v + v∇x − (∇Ψλ,N )∇v . β We do essentially not distinguish EλN and MλN × RdN : An element (x1 , v1 , . . . , xN , vN ) of EλN we sometimes denote by (x, v), x = (x1 , . . . , xN ), v = (v1 , . . . , vN ). ∆v , ∇v denote the Laplacian and the gradient respectively, in N v-direction, v denotes multiplication by the vector v, v∇v := i=1 vi ∇vi etc. We need some information on the domain of Lλ,N . Lemma 4.5. (i) Let f ∈ C(EλN ) be such that it possesses continuous partial derivatives up to order 2 in all v-directions and continuous partial derivatives of order 1 in all x-directions. Assume moreover that f and all mentioned partial derivatives are bounded in absolute value by a multiple of (x, v) → (1 + |v|)k for some k ∈ N. Then f ∈ D(Lλ,N ) and Lλ,N f is given as in (4.1). (ii) Let f : MλN \Dλ,N → R, where Dλ,N := {(x1 , . . . , xN ) ∈ MλN |∃i, j ∈ {1, . . . , N }: xi = xj , i = j}. Assume that f is once continuously diﬀerentiable and that f, ∇x f ∈ L2 (MλN ; e−βΨλ,N dx). Then when f is considered as a function on EλN which is constant in v-directions, it holds f ∈ D(Lλ,N ) and Lλ,N f is given as in (4.1), i.e. Lλ,N f = v∇x f .

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Proof. (i) is contained in [5, Lemma 3.7(ii)], (ii) follows for functions with compact support in MλN \Dλ,N from (i), for bounded functions f it follows by a similar argument as used in the proof of [5, Lemma 3.7(ii)], and ﬁnally one shows that any unbounded f as in (ii) can be represented as graph norm limit of bounded functions as in (ii). In order to simplify notation, in the sequel we do not distinguish Mλ and Λλ as well as Eλ and Λλ × Rd . Moreover, since confusion would not cause trouble, we N do not use diﬀerent notations for ΛN λ and the set Λλ \Dλ,N . (The diagonal Dλ,N , deﬁned in Lemma 4.5(ii), is not hit Pλ,N -a.s. and hence may be omitted.) We consider the mapping perλ,N : EλN → Γv deﬁned by perλ,N (x1 , . . . , vN ) := {(x1 + 2λr, v1 ), . . . , (xN + 2λr, vN )}. r∈Zd ⊗[0,∞)

Furthermore, we deﬁne the mapping perλ,N : C([0, ∞), EλN ) → C([0, ∞), Γv ) by assigning to a path ((x1 (t), . . . , vN (t)))t≥0 the path (perλ,N (x1 (t), . . . , vN (t)))t≥0 of images with respect to perλ,N . Both mappings are well-deﬁned except on the diagonal (which is negligible with respect to both µλ,N and Pλ,N ) and measurable. We set µ(λ,N ) := µλ,N ◦ per−1 λ,N ⊗[0,∞)

and deﬁne P (λ,N ) := Pλ,N ◦ (perλ,N )−1 . These probability laws are the starting point for the construction of an inﬁnite particle Langevin dynamics as a weak limit. Sometimes we also use the mappings symλ,N : EλN → Γv , given by symλ,N (x1 , . . . , vN ) := {(x1 , v1 ), . . . , (xN , vN )}. This is done for technical reasons: The measures µ(λ,N ) are not locally absolutely continuous with respect to Lebesgue– Poisson measure and in particular do not fulﬁll a Ruelle bound. Therefore, in order to apply the results from Sec. 3.4 we have to use µλ,N ◦ sym−1 λ,N instead. Remark 4.6. Note that one at best faces some technical diﬃculties trying a ⊗[0,∞) construction by dynamics given through Pλ,N ◦ (symλ,N )−1 , where one deﬁnes ⊗[0,∞)

⊗[0,∞)

symλ,N analogously to perλ,N . The paths corresponding to these laws are not even right continuous. In contrast, the laws P (λ,N ) describe diﬀusions. 4.3. Tightness In this section we prove, under the conditions and using the notations of Sec. 4.2, Nn tightness of any sequence (P (λn ,Nn ) )n∈N such that λn ↑ ∞ and (2λ d → ρ ∈ [0, ∞) n) as n → ∞. In the sequel we abbreviate subscripts λn , Nn by n, i.e. Pn := Pλn ,Nn , symn := symλn ,Nn etc. Paths in C([0, ∞), Γv ) will below always be denoted by (γt )t≥0 . We may assume that λ1 is large enough for Theorem 3.14 (and Corollary 3.16) to apply. Tightness of the sequence of distributions P (n) ◦ γt−1 (= µ(n) ), t ≥ 0, is seen from Remark 3.25 and Lemma 3.17(ii). So we go on by estimating moments of dΦ,a,h (γt , γs ), t, s ≥ 0, with dΦ,a,h as deﬁned in Sec. 2. We follow an idea from [17]

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and use semimartingale decompositions of the summands contained in dΦ,a,h (γt , γs ). Before we do so, we need some preparations. The following lemma might also be stated more generally. However, we restrict to what we are about to use later. Lemma 4.7. Let f : Rd × Rd → R have bounded spatial support and being once continuously diﬀerentiable in the x-directions and twice continuously diﬀerentiable in the v-directions. Assume moreover that all these derivatives are bounded. f itself may be unbounded. Set F := f, · = Kf . Then F ◦ pern is an element of the domain D(Ln ) and it holds sup µn (|Ln (F ◦ pern )|3 ) < ∞. n

Proof. We may without loss of generality assume that the spatial support of f is contained in an open cube of side length less than 2λ1 . (Otherwise we use an appropriate partition of unity corresponding to a suitable locally ﬁnite open cover of Rd to decompose f (cf. the proof of Lemma 4.10 below).) Moreover, we may without loss of generality assume that the spatial support of f is relatively compact in (−λ1 , λ1 )d , since Ln commutes with simultaneous spatial translations of all particles in (the manifold) Eλn and µn is invariant with respect to these translations. So we may replace pern by symn . The ﬁrst assertion is seen from Lemma 4.5(i). (Note that f (x, v), (x, v) ∈ Rd × Rd grows at most linearly as |v| → ∞.) It holds for n ∈ N Ln (F ◦ symn ) = Kg1 ◦ symn − Kg2n ◦ symn , where g1 : Γ1 → R is given by g1 ({(x, v)}) :=

κ β ∆v f (x, v)−κv∇v f (x, v)+v∇x f (x, v) n g2 ({(x, v), (x , v )}) := ∇φˆλn (x − x )(∇v f (x, v) −

and g2n : Γ2 → R is given by ∇v f (x , v )), (x, v), (x , v ) ∈ Eλn . Let us ﬁrst prove

sup µn (|Kg1 ◦ symn |3 ) < ∞.

(4.2)

n

Since g1 has (seen as a function deﬁned on Rd × Rd ) bounded spatial support and there exists C > 0 such that |g1 (x, v)| ≤ C|v| for all (x, v) ∈ Rd × Rd , it follows g ∈ Lp (Γ0 ; λv ) for each p ∈ [1, ∞). So the improved Ruelle bound (Corollary 3.16), Proposition 3.19 and Remark 3.20 imply (4.2). Concerning g2n we have that for n ∈ N (K|g2n |(·))3 ≤

sup (x,v)∈Rd ×Rd

|∇v f (x, v)|32

|K g˜2n (prx ·)|3

where g˜2n (x, x ) := |∇φˆλn (x − x )|2 (1supps (f ) (x) + 1supps (f ) (x )), x, x ∈ Λλn , | · |2 denotes Euclidan norm and supps denotes the spatial support of f . So we are left to estimate µn (|K g˜2n ◦ symn |3 ). By Proposition 3.19 and using the improved Ruelle

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bound (3.12) of the µn ◦ sym−1 n , we have to prove that for any M ∈ {2, . . . , 6} and any A, B, C, D, E, F ∈ {1, . . . , M } such that A = B, C = D, E = F and {A, B, C, D, E, F } = {1, . . . , M }, it holds

|∇φˆλn (xA − xB )|2 |∇φˆλn (xC − xD )|2 |∇φˆλn (xE − xF )|2 sup ΛM λn

n

× inf

1≤i≤M

e−β

P j=i

ˆλ (xi −xj ) φ n

1supps (f ) (xA )1supps (f ) (xC )

× 1supps (f ) (xE )dx1 · · · dxM < ∞ which by uniform boundedness of the φˆλn from below we may replace by

c(xA , xB )c(xC , xD )c(xE , xF ) sup n

ΛM λn

× 1supps (f ) (xA )1supps (f ) (xC )1supps (f ) (xE )dx1 · · · dxM < ∞

(4.3)

β ˆ with c(x, x ) := |∇φˆλn (x − x )|2 e− 3 φλn (x−x ) , x, x ∈ Λλn . ∈ To prove (4.3), we integrate successively over all xY with Y {B, D, F }\{A, C, E}. The integration yields ﬁnite values bounded independently of n even if Y appears more than once in the tuple (A, . . . , F ) due to Lemma 4.4 and H¨ older inequality. We continue to integrate over the remaining variables until there is no ∇φˆλn -term left. For every variable which then remains (there is at least one), there is a 1supps (f ) left. It follows supn µn (|Kg2 ◦ symn |3 ) < ∞ and together with (4.2) the assertion is shown.

A much simpler case than in Lemma 4.7 is considered in the following corollary. (Note that the ﬁrst estimate is immediate.) Corollary 4.8. Let f : Rd × Rd be continuous and continuously diﬀerentiable in the v-directions such that all the derivatives are bounded, whereas f may be unbounded. Then it holds sup µn (|∇v (Kf ◦ pern )|32 ) ≤ sup µn (|∇v (Kf ◦ pern )|31 ) < ∞ n

n

where | · |1 denotes norm deﬁned by |(y1 , . . . , yl )| := l ∈ N.

l j=1

|yj | for (y1 , . . . , yl ) ∈ Rl ,

Remark 4.9. For k ∈ N consider χk deﬁned as in Sec. 2 such that a is twice continuously diﬀerentiable and ∇v a, ∆v a are bounded (and, as before, a(v) → ∞ as |v| → ∞. Then Lemma 4.7 and Corollary 4.8 apply to f = χk . If a function f is only dependent on x-coordinates, Ln (Kf ◦ pern ) does not contain ∇φˆλn , n ∈ N. This enables us to deal also with S Φ,hk , deﬁned in Sec. 2. Here hk := hIk .

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˜ be as in Lemma 3.6 (corresponding to the potential φ) and Lemma 4.10. Let Φ ˜ assume (without loss of generality) that Φ(r) = 0 for r ≥ λ1 /2. Let k ∈ N. It holds ˜ β Φ/6,h k ◦ pern ∈ D(Ln ) and S ˜

sup µn (|Ln (S β Φ/6,hk ◦ pern )|3 ) < ∞. n

˜

Proof. We ﬁrst note that S β Φ/6,hk is the K-transform of g: Γ2 → R, given ˜ by g({x, x }) := hk (x)hk (x )eβ Φ(|x−x |2 )/6 , x, x ∈ Rd . g({x, x }) is equal to hk (x)hk (x ), when |x − x | ≥ λ1 /2. We choose a locally ﬁnite open cover U of Rd such that any ∆ ∈ U has diameter < λ1 /2 and we choose a corresponding partition of unity (η∆ )∆∈U consisting of C 1 -functions. Using this partition (and noting ˜ that hk has compact support) we see that we may replace S β Φ/6,hk by Kg∆1 ,∆2 , where g∆1 ,∆2 : Γ2 → R is deﬁned by g∆1 ,∆2 ({x, x }) β

˜

:= e 6 Φ(|x−x |2 ) (ϕ∆1 (x)ϕ∆2 (x ) + ϕ∆1 (x )ϕ∆2 (x)),

x, x ∈ Rd ,

(4.4)

for ∆1 , ∆2 ∈ U, where ϕ∆1/2 := η∆1/2 hk . We ﬁrst consider the case where dist(∆1 , ∆2 ) ≤ λ1 /2, where dist denotes the |·|distance of subsets of Rd . We may assume that ∆ := ∆1 ∪∆2 is relatively compact in (−λ1 , λ1 )d and replace pern by symn using spatial translations in Eλn as in the proof of Lemma 4.7 above. By Lemma 4.5(ii) we have that (Kg∆1 ,∆2 ◦ symn ) ∈ D(Ln ), n ∈ N. For 1 ≤ i ≤ Nn and (x, v) ∈ EλNnn we make the following estimate. |∇xi (Kg∆1 ,∆2 ◦ symn )(x, v)|2 = ∇xi g∆1 ,∆2 (xi , xj ) j=i

≤

2

β˜ e 6 Φ(|xi −xj |2 ) ϕ∆2 (xj ) β Φ ˜ (|xi − xj |2 ) xi − xj ϕ∆1 (xi ) + ∇ϕ∆1 (xi ) 6 |xi − xj |2 j=i

+

2

β ˜ ˜ (|xi − xj |2 ) xi − xj ϕ∆2 (xi ) + ∇ϕ∆2 (xi ) e 6 Φ(|xi −xj |2 ) ϕ∆1 (xj ) β Φ 6 |xi − xj |2

2

j=i

β˜ β Φ(|xi −xj |2 ) ˜ 6 ≤2 e 1∆ (xj ) Φ (|xi − xj |2 ) 1∆ (xi ) + C1∆ (xi ) 6 j=i

˜ e− β6 Φ˜ is bounded and Φ ˜ ≥ 0 the right-hand side is for some C < ∞. Since Φ estimated by β˜ e 3 Φ(|xi −xj |2 ) 1∆ (xi )1∆ (xj ) C j=i

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for some C < ∞ and thus |Ln (Kg∆1 ,∆2 ◦ symn )(x, v)| β ≤ (|vi |1 + |vj |1 )e 3 Φ(|xi −xj |2 ) 1∆ (xi )1∆ (xj ). {i,j}⊂{1,...,N }

By Proposition 3.19 we only have to show that for M ∈ {2, . . . , 6}, A, B, C, D, E, F ∈ {1, . . . , M }, {A, . . . , F } = {1, . . . , M }, A = B, C = D, E = F the expression

P 2 ˆ c(xA , xB )c(xC , xD )c(xE , xF )e− M β i<j φλn (xi −xj ) dx1 · · · dxM ΛM λn

β

is bounded independently of n, where c(x, x ) = e 3 Φ(|x−x |2 ) 1∆ (x)1∆ (x ), x, x ∈ ˜ and the uniform Rd . But since the integrand is bounded by the properties Φ ˆ boundedness from below of φλn , n ∈ N, the above integral is estimated by D max{vol(∆)2 , vol(∆)6 } for some D < ∞. ˜ − x |2 ) = 0 for Now assume that dist(∆1 , ∆2 ) > λ1 /2 in (4.4). In this case Φ(|x x ∈ ∆1 , x ∈ ∆2 , so we have g∆1 ,∆2 ({x, x }) = ϕ∆1 (x)ϕ∆2 (x ). Using periodicity of the image conﬁgurations with respect to pern and spatial shifts in Eλn we may assume that ∆1 and ∆2 are relatively compact in (−λ1 , λ1 )d , so pern may be replaced by symn and the case dist(∆1 , ∆2 ) > λ1 /2 is reduced to a (trivial) special ˜ ≡ 0) of the one we treated above. case (Φ Now we arrive at the concluding tightness estimate. The expectation with respect to P (n) , n ∈ N, we denote by E (n) and in the sequel we also use similar notations for expectations with respect to other probability laws. ˜ be chosen as in Lemma 4.10 and a be chosen as in Remark 4.9. Lemma 4.11. Let Φ For each T > 0 there is a constant C > 0 such that for 0 ≤ s < t ≤ T < ∞ it holds β

˜

sup E (n) [(d 6 Φ,h,a (γt , γs ))3 ] ≤ C(t − s)3/2 , n

β

˜

when fk , gk , rk and qk in the deﬁnition of d 6 Φ,h,a are chosen in a suitable way (see the proof below). Proof. By the Minkowski inequality and since

r 1+r

≤ r for r ≥ 0 it holds

˜

(E (n) [(d 6 Φ,h,a (γt , γs ))3 ])1/3 β

≤

∞

2−k (E (n) [|Kfk (γt ) − Kfk (γs )|3 ])1/3

k=1

+

∞ k=1

2−k (E (n) [|Kgk (prx γt ) − Kgk (prx γs )|3 ])1/3

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∞

37

2−k qk (E (n) [|(Kχk )(γt ) − (Kχk )(γs )|3 ])1/3

k=1

+

∞

β

˜

β

˜

2−k rk (E (n) [|S 6 Φ,hIk ◦ prx (γt ) − S 6 Φ,hIk ◦ prx (γs )|3 ])1/3 .

(4.5)

k=1

Concerning the ﬁrst three summands on the right-hand-side we need to estimate E (n) [|Kf (γt ) − Kf (γs )|3 ]

(4.6)

for f as in Lemma 4.7. It suﬃces to prove that this expression is bounded by (t−s)3/2 C(f )D(T ) where C(f ) is a constant depending only on f and D(T ) depends only on T . Then by replacing fk by C(ffkk)1/3 and gk by C(ggkk)1/3 and setting qk :=

min{C(χk )−1/3 , 1}, k ∈ N, in the deﬁnition of the metric the ﬁrst three summands in (4.5) are convergent and less or equal than (t − s)1/2 D(T )1/3 . So let f be as in Lemma 4.7. It holds E (n) [|Kf (γt ) − Kf (γs )|3 ] = En [|(Kf ) ◦ pern (Xt , Vt ) − (Kf ) ◦ pern (Xs , Vs )|3 ]. It holds Kf ◦ pern ∈ D(Ln ). Since Pn solves the martingale problem for Ln we ﬁnd that [Kf ◦pern ],n

Mt

:= Kf ◦ pern (Xt , Vt ) − Kf ◦ pern (X0 , V0 )

t − Ln (Kf ◦ pern )(Xr , Vr )dr, 0

t ≥ 0, deﬁnes a Pn -martingale. By [5, Remark 3.22] the quadratic variation process

t [Kf ◦pern ],n 2 )t≥0 is given by ( 0 2κ of (Mt β |∇v (Kf ◦ pern )|2 (Xr , Vr ) dr)t≥0 . Using the Burkholder–Davies–Gundy inequality and the H¨ older inequality, we ﬁnd [Kf ◦pern ],n

En [|Mt

≤ En

t

s

≤

2κ β

3/2

− Ms[Kf ◦pern ],n |3 ]

3/2 2κ 2 |∇v (Kf ◦ pern )|2 (Xr , Vr )dr β (t − s)3/2 µn (|∇v (Kf ◦ pern )|32 ),

which can be estimated using Corollary 4.8. Moreover, it holds by H¨ older inequality 3 t En Ln (Kf ◦ pern )(Xr , Vr )dr ≤ (t − s)3 µn (|Ln (Kf ◦ pern )|3 ) s

≤ T 3/2 (t − s)3/2 µn (|Ln (Kf ◦ pern )|3 ). This can be estimated using Lemma 4.7. Altogether we have independently of n an estimate of (4.6) by (1 + T 1/2 )3 C(f )(t − s)3/2 for some constant C(f ), concluding the consideration of the ﬁrst three summands in (4.5).

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˜

Concerning the fourth summand we ﬁrst note that (denoting S 6 Φ,hIk ◦ prx also β ˜ by S 6 Φ,hIk ) ˜

˜

E (n) [|S 6 Φ,hIk (γt ) − S 6 Φ,hIk (γs )|3 ] β

β

β

˜

β

˜

= En [|(S 6 Φ,hIk ◦ pern )(Xt , Vt ) − (S 6 Φ,hIk ◦ pern )(Xs , Vs )|3 ]. β

˜

S 6 Φ,hIk ◦ pern is an element of D(Ln ) (cf. Lemma 4.10) such that for the correβ ˜

[S 6 Φ,hIk ◦pern ],n

sponding martingale it holds Mt β

˜

= 0 Pn -a.s. for all t ≥ 0. So β

˜

En [|(S 6 Φ,hIk ◦ pern )(Xt , Vt ) − (S 6 Φ,hIk ◦ pern )(Xs , Vs )|3 ] 3 t β ˜ Φ,hI k ◦ pern ))(Xr , Vr )dr = En (Ln (S 6 s

β

˜

≤ (t − s)3 µn (|Ln (S 6 Φ,hIk ◦ pern )|3 ) ˜

≤ T 3/2 (t − s)3/2 µn (|Ln (S 6 Φ,hIk ◦ pern )|3 ), β

which can be estimated with the help of Lemma 4.10 by T 3/2 Rk (t − s)3/2 for some β ˜ −1/3 Rk ∈ R+ . Setting rk := min{Rk , 1} in the deﬁnition of d 6 Φ,a,h we have an estimate for the fourth summand in (4.5). This completes the proof. Remark 4.12. In [14], the Lyons–Zheng decomposition was used in order to obtain the estimate corresponding to the above lemma. At ﬁrst sight, using such a decomposition seems to be a signiﬁcant simpliﬁcation of the proof given above, since one avoids having to estimate the bounded variation terms. Therefore, we should mention that this is not possible here, since we are in a non-reversible situation and the method of proving tightness by a forward/backward martingale decomposition depends heavily on reversibility of the processes (cf. [14, Proof of Lemma 5.3]). We obtain the desired tightness result. Theorem 4.13. Let φ be a symmetric pair interaction fulﬁlling (RP ), (T ), (BB ), (WD ) and (IDF ). Let (Nn )n ⊂ N and (λn )n ⊂ R+ be sequences such that λn ↑ ∞ ⊗[0,∞) −1 Nn (n) := Pn ◦ (pern ) , n ∈ N, be and (2λ d → ρ ∈ [0, ∞) as n → ∞. Let P n) deﬁned as in Sec. 4.2. Then the sequence (P (n) )n∈N is a tight sequence of probability laws on C([0, ∞), Γv ).

Proof. Using standard tightness results (cf. [9, Theorems 3.8.6 and 3.8.8]) we obtain tightness of (P (n) )n∈N as probability laws on D([0, ∞), Γv ), the space of cadlag paths in Γv . By [9, Exercise 3.25(c)] we ﬁnd that any weak accumulation point of (P (n) )n∈N assigns full measure to the space C([0, ∞), Γv ) of continuous paths, hence by [9, Exercise 3.25(d)] the assertion follows.

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4.4. The martingale problem By now we know that in the situation of Theorem 4.13 we have at least one accumulation point P of (P (n) )n . Let P (nk ) → P weakly as k → ∞. Note that then also the sequence µ(nk ) converges weakly and its weak limit µ is the invariant initial distribution of P . Moreover, also µnk ◦ sym−1 nk → µ weakly as k → ∞. In this section we verify that P is the law of an inﬁnite particle Langevin dynamics. We ﬁrst prove some preliminary technical properties of the generator L (cf. (1.3)). We do not bother about the possibility of generalizing assertions in Lemma 4.14 below to all n ∈ N, since in this section we are only interested in asymptotic properties of the sequences (P (n) )n∈N , (µ(n) )n∈N . ∞ v Lemma 4.14. Let F = gF ({fi }K i=1 , ·) ∈ FCb (Ds , Γ ). Choose n0 ∈ N such that d d each fi , i = 1, . . . , K, has support in (−λn0 , λn0 ) × R . v (i) Let n ∈ N. The expression LF (γ) is well-deﬁned for µn ◦ sym−1 n -a.e. γ ∈ Γ , −1 i.e. the sums in the deﬁnition of LF converge absolutely µn ◦symn -a.s. and LF is µn ◦ sym−1 n -a.s. independent of the version one chooses for ∇φ. Moreover, < ∞. it holds supn∈N LF L1(Γv ;µn ◦sym−1 n ) (ii) Let n ≥ n0 . Then LF is well-deﬁned µ(n) -a.e. and supn≥n0 LF L1 (Γv ;µ(n) ) < ∞ with the restriction that the versions for ∇φ have to be chosen in a way such that r∈Zd ∇φ(2λn r) = 0 for all n ∈ N.

t (iii) With the restriction from (ii), for any n ≥ n0 , t ≥ 0 the integral 0 LF (γr )dr is well-deﬁned P (n) -a.s. (iv) If P is the weak limit of a sequence (P (n) )n∈N as above, then (i) holds with µ(n) replaced by µ and (iii) holds with P (n) replaced by P .

Remark 4.15. The restriction in Lemma 4.14(ii) means that the forces acting between a particle and its periodic copies sum up to 0, which is a quite natural assumption. Note that it is not an additional assumption on φ. It is only introduced for technical reasons (cf. (4.8) below) and it can be dropped when only considering the limiting process, as one sees in (the proof of) Lemma 4.14(iv). Proof. LF consists (except of multiplication by bounded continuous partial derivatives of gF ) of two types of sums. The ﬁrst (e.g., ∆v fi , ·) are K-transforms of functions in Ds . By Lemma 3.18 and the uniform Ruelle bound, the assertion is easily shown for these sums. We concentrate on the second type of sums, which are K-transforms of functions g: Γv2 → R of the form g({(x, v), (x , v )}) = ∇φ(x − x )(∇v f (x, v) − ∇v (f (x , v )), (x, v), (x , v ) ∈ Rd × Rd , with f ∈ Ds . Let ¯ and hdλv ) with ¯h as in Remark 3.26 (here we identify h us prove that g ∈ L1 (Γv2 ; ¯ ¯ h ◦ prx ). It holds

2 2 1 |g({(x, v), (x , v )})|e−(β/2)(v +v ) e−βφ(x−x ) dx dx dv dv (2π/β)d R4d ≤ 2 |∇v f | ∞ vol(supps (f ))∇φL1 (Rd ;e−βφ dx) .

(4.7)

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The proof of (i) and of the ﬁrst assertion of (iv) are now completed by Lemma 3.18, the improved Ruelle bound, (WD) and the properties of φ (cf. Remark 3.26). (Remember that the improved Ruelle bound also holds for the limiting measure µ by Lemma 3.23.) Let us now prove (ii). Let n ≥ n0 . Due to the condition (IDF) and the bounded ness of 0=r∈Zd θ(· + 2λn r) in Λλn (cf. the proof of Lemma 4.2(ii)), hence in Λ2λn , ˆ (as element of L1 (Λ2λ ; e−β φˆλn dx)) the sum d ∇φ(· + 2λn r) deﬁning ∇φλ r∈Z

n

n

converges Lebesgue-a.e. Any version of ∇φ uniquely determines by this sum a corresponding version of ∇φˆλn , when we set ∇φˆλn equal to 0 where it does not converge. Fixing a version of ∇φ we deﬁne for γ ∈ Γv K κ ∂l ∂l gF ({fi }K L(n) F (γ) := i=1 , γ)(∇v fl )(∇v fl ), γ β l,l =1  K κ K  ∆v fl − κv∇v fl + v∇x fl , γ + ∂l gF ({fi }i=1 , γ) β l=1

−

 ∇φˆλn (x − x )(∇v fl (x, v) − ∇v fl (x , v )) .

{(x,v),(x ,v )}⊂γ

It can now be shown that (i) holds with L replaced by L(n) . Then, since in particular this means that the sums deﬁning L(n) F are absolutely convergent, we may change the order of summation. Using a version of ∇φ as speciﬁed in (ii), we obtain that L(n) F ◦ symn = LF ◦ pern

µn -a.s.

(4.8)

Hence (ii) follows. (iii) follows from Fubini’s theorem, (ii) and the fact that

t (n) E |LF (γr )|dr ≤ tLF L1(Γv ;µ(n) ) . 0

The second assertion of (iv) follows in the same mannner from the ﬁrst assertion.

From the above proof and from Lemma 3.23 we can conclude the following uniform approximation result. Lemma 4.16. Let F be as in Lemma 4.14. Then there exists a sequence (Hl )l∈N of bounded continuous cylinder functions Hl : Γv → R (cf. Sec. 3.4) such that lim lim sup Hl − LF L1 (Γv ;µ(nk ) ) = 0

l→∞ k→∞

and Hl → LF as l → ∞ in L1 (Γv ; µ). Proof. When µ(nk ) is replaced by µnk ◦ sym−1 nk , the assertion is a consequence of Lemma 3.23(ii) (cf. Remark 3.26) and the proof of Lemma 4.14 (in particular (4.7)). Let (Hl )l∈N be a corresponding sequence of bounded continuous cylinder functions.

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Fix l ∈ N and let n0 ∈ N be as in the proof of Lemma 4.14. Choose k0 large enough such that nk0 ≥ n0 and such that Hl depends only on the conﬁguration in Λλnk for k ≥ k0 . By (4.8) it holds for k ≥ k0 Hl − LF L1 (Γv ;µ(nk ) ) = Hl − L(nk ) F L1 (Γv ;µn ≤ Hl − LF L1 (Γv ;µn

k

k

◦sym−1 nk )

◦sym−1 nk )

+ LF − L(nk ) F L1 (Γv ;µn

k

. ◦sym−1 nk )

= 0. This So, we are left to prove that limk→∞ LF − L(nk ) F L1 (Γv ;µn ◦sym−1 nk ) k v reduces to proving that limn→∞ KgnL1 (Γv ;µn ◦sym−1 = 0 for gn : Γ2 → R of the n ) form gn ({(x, v), (x , v )}) := (∇φ − ∇φˆλn )(x − x )(∇v f (x, v) − ∇v f (x , v )), (x, v), (x , v ) ∈ Eλ2n , with f ∈ Ds , supp(f ) ⊂ (−δ, δ)d for some 0 < δ < λn0 . Applying Lemma 3.18 and the improved Ruelle bound we ﬁnd that it suﬃces to verify that with φ as in Remark 3.26 it holds lim ∇φ − ∇φˆλn L1 (Λλ

n→∞

−βφ dx) n +δ ;e

= 0.

(4.9)

For n ≥ n0 it holds ∇φ − ∇φˆλn L1 (Λλ

−βφ dx) n +δ ;e

≤ ∇φ − ∇φˆλn L1 (Λλ + ∇φˆλn L1 (Λλ

−βφ dx) n ;e

+ ∇φL1 (Λλ

−βφ dx) n +δ \Λλn ;e

−βφ dx) n +δ \Λλn ;e

≤ 2D∇φL1 (Rd \Λλn ;dx) + 2d D∇φL1 (Rd \Λλn −δ ;dx) , where D := supx∈Rd e−βφ(x) . (4.9) follows, since ∇φ ∈ L1 (Rd ; e−βφ dx) implies ∇φ ∈ L1 ({|x| > a}; dx) for any a > 0 due to boundedness of φ in {|x| > a}. The laws P (n) , n ∈ N, behave nicely with respect to the operator L restricted to functions F = gF (f1 , ·, . . . , fK , ·) ∈ FCb∞ (Ds , Γv ) depending only on the

t [F ] particles in Λλn : We ﬁnd that Mt := F (γt ) − F (γ0 ) − 0 LF (γr )dr, t ≥ 0, deﬁnes a martingale with respect to P (n) . To see this, ﬁrst note that for such F we have by (4.8) that LF ◦ pern = Ln (F ◦ symn ) holds µn -a.s. We obtain for any 0 ≤ s ≤ t and any bounded function G: C([0, ∞), Γv ) → R which is σ(γr : 0 ≤ r ≤ s)-measurable

t (n) G F (γt ) − F (γs ) − LF (γr )dr E s

= En G ◦ pern⊗[0,∞) F ◦ symn (Xt , Vt ) − F ◦ symn (Xs , Vs )

−

t

Ln (F ◦ symn )(Xr , Vr )dr

,

s

which is equal to 0, since Pn solves the martingale problem for (Ln , D(Ln )).

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We arrive at the result completing the construction. Theorem 4.17. Assumptions as in Theorem 4.13. Let P be an accumulation point of the sequence (P (n) )n∈N . Then P solves the martingale problem for (L, F Cb∞ (Ds , Γv )). Proof. Let P (nk ) → P weakly as k → ∞ and denote the weak limit of (µ(nk ) )k∈N ∞ v by µ. Let F = gF ({fi }K i=1 , ·) ∈ FCb (Ds , Γ ). Then there exists k0 ∈ N such that (−λnk , λnk )d contains the support of all fi , 1 = 1, . . . , K, for all k ≥ k0 . We have to prove that for any 0 ≤ s < t < ∞ and for any σ(γr : 0 ≤ r ≤ s)-measurable G: C([0, ∞), Γv ) → R being bounded and continuous it holds [F ]

E[(Mt

− Ms[F ] )G] = 0,

[F ]

[F ]

where (Mt )t ≥0 is as deﬁned above. We already know that (Mt )t ≥0 is a martin[F ] gale with respect to P (nk ) , k ≥ k0 . Hence we are left to prove that E (nk ) [GMr ] → [F ] E[GMr ] as k → ∞ for r ∈ {t, s}. As G and F are continuous and bounded, this reduces to proving r E (nk ) G LF (γr )dr 0

→E G

r 0

LF (γr )dr

as k → ∞ for each r ∈ {t, s}.

(4.10)

Choosing (Hl )l≥0 according to Lemma 4.16 we ﬁnd that for any r ≥ 0, l ∈ N and k ≥ k0 it holds |E (nk ) [GLF (γr )] − E[GLF (γr )] ≤ G∞ (LF − Hl L1 (Γv ;µ(nk ) ) + LF − Hl L1 (Γv ;µ) ) + |E (nk ) [GHl (γr )] − E[GHl (γr )]|, where we used the fact that the one-dimensional distributions of P (nk ) , P are given by µ(nk ) , µ, respectively. Hence by continuity and boundedness of G and Hl and by weak convergence of P (nk ) towards P lim sup |E (nk ) [GLF (γr )] − E[GLF (γr )]| k→∞

≤ G∞ lim sup LF − Hl L1 (Γv ;µ(nk ) ) + LF − Hl L1 (Γv ;µ) , k→∞

which by Lemma 4.16 can be made arbitrarily small by choosing l large. E (n) [GLF (γr )] is bounded uniformly in r ∈ [0, ∞) by G∞ LF L1(Γv ;µ(n) ) which is bounded uniformly in n ≥ nk0 due to Lemma 4.14(ii). Hence (4.10) follows by Lebesgue’s dominated convergence theorem. Remark 4.18. The results from Sec. 3.4 which we used to prove Theorem 4.17 also generalize some of the results in [14]. In the diﬀerentiability assumption (D)

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given there we replace continuous diﬀerentiability of the potential φ on Rd \{0} by weak diﬀerentiability and continuity. (Moreover, the assumption on Φ in (D) can be dropped, cf. Lemma 3.1 above.) Then the existence of the approximating dynamics (cf. [14, Theorem 4.1] and [10]) is still ensured. Moreover, the tightness result [14, Theorem 5.1] does not depend on continuous diﬀerentiability of φ. We do not consider the question whether the results of [14, Sec. 5.2] are still valid, but focus on the martingale problem [14, Theorem 5.10]. Since the invariant initial distributions µ(N ) , N ∈ N, of the approximating dynamics P (N ) in [14] fulﬁll a uniform improved Ruelle bound, this bound extends to the invariant distribution µ of any weak accumulation point P by Lemma 3.23(i). This can be used to prove well-deﬁnedness of the expressions in [14, (5.19) and (5.20)] a.s. with respect to the µ(N ) , µ respectively, P (N ) , P similarly to Lemma 4.14(i), (iii) and (iv). Moreover, we also obtain an approximation of elements Hµ F , F ∈ FCb∞ (D, Γ), of the range of the generator (Hµ , F Cb∞ (D, Γ)) (cf. [14, (5.19)]) by bounded continuous cylinder functions in L1 (Γ; µ(N ) ) uniformly in N and in L1 (Γ; µ) (using Lemma 3.23(ii)). This replaces the only argument in the proof of [14, Theorem 5.10] making use of the continuity of the derivatives of φ (cf. [14, p. 150]). Finally, note that also [14, Theorem 6.1] is valid in the new setting. 4.5. A conditional result on convergence Here we discuss what is gained when making an essential m-dissipativity assumption for the generator L. Let µn , P (n) , Ln etc., n ∈ N, be as above. Denote by (Ttn )t≥0 the subMarkovian strongly continuous contraction semigroup in L1 (EλNnn ; µn ) generated by (Ln , D(Ln )). Let µ be a weak accumulation point of the sequence (µn ◦ sym−1 n )n∈N and let µnk ◦ sym−1 → µ weakly as k → ∞. For any bounded continuous cylinder nk → F L1 (Γv ;µ) as k → ∞. function F : Γv → R we have F ◦ symnk 1 Nnk L (Eλn ;µnk ) k

Hence, denoting the set of bounded continuous cylinder functions by C, we have Nn convergence of Xk := L1 (Eλn k , µnk ) → L1 (Γv ; µ) =: X in the sense of [21, 20, 32] k (which are formulated for Hilbert spaces, but at least the basic deﬁnitions also extend to Banach spaces). As deﬁned in these articles, a sequence (Fk )k∈N , k ∈ N, with Fk ∈ Xk is said to converge to F ∈ X, if for one (hence all) (Um )m∈N ⊂ C such that Um → F in X as m → ∞ it holds limm→∞ lim supk→∞ Um ◦symnk −Fk Xk = 0. In particular, for any F ∈ FCb∞ (Ds , Γv ), we have that Fk := F ◦ symnk converges to F in this sense. Moreover, since for suﬃciently large k we have Fk ∈ D(Lnk ) and Lnk Fk = L(nk ) F ◦ symnk , it follows (after replacing ﬁnitely many Fk by 0) that Lnk Fk converges to LF as k → ∞ by (the proof of) Lemma 4.16 above. Thus, we have generator convergence, and when one assumes that (1 − L)F Cb∞ (Ds , Γv ) is dense in L1 (Γv ; µ), Trotter’s theorem should provide convergence of the corresponding semigroups to a limiting semigroup generated by the closure of L. We make this precise: Quite similar to [32, Theorem 2.10] or [20,

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Proposition 7.2] (using the existence of a Schauder basis in L1 (Γv ; µ) or only the bounded approximation property) one can show that there exists a uniformly bounded sequence of continuous linear operators Pn : X → Xn such that Pn F Xn → F X as n → ∞ for all F ∈ X and such that Fn ∈ Xn , n ∈ N, converge to F ∈ X as n → ∞ in the above sense iﬀ Pn F − Fn Xn → 0. (There is a bit of work to do, and we refer to [6] for a complete proof.) This is precisely the setting from [33, 23], and thus the Trotter convergence theorem from these papers can be applied. More precisely, essentially [23, Theorem 2.1] implies the following result. Theorem 4.19. Let (µ(nk ) )k∈N be weakly convergent to µ. Then (L, F Cb∞ (Ds , Γv )) is dissipative. If it is even essentially m-dissipative in L1 (Γv ; µ), i.e. (1 − L)F Cb∞ (Ds , Γv ) is dense in L1 (Γv ; µ), then there exists a strongly continuous subMarkovian semigroup of contractions (Tt )t≥0 on L1 (Γv ; µ) generated by the closure of (L, F Cb∞ (Ds , Γv )) such that Ttn → Tt strongly as n → ∞, i.e. for all F ∈ C the sequence (Ttn (F ◦ symn ))n∈N converges to Tt F in the abovementioned sense. In particular, the sequence (P (nk ) )k∈N is convergent and its limiting law P solves the martingale problem even for the closure of L and has the elementary Markov property. Proof. For any F ∈ FCb∞ (Ds , Γv ) (by the proof of Lemma 4.16) it holds µ(LF ) = limk→∞ µnk (Lnk (F ◦ symnk )) = 0, since µnk is invariant for Lnk . Hence µ is invariant for L, and it follows that L is dissipative and if L is even essentially m-dissipative (i.e. its closure generates a contraction semigroup) in L1 (Γv ; µ), it follows that the semigroup (Tt )t≥0 generated by its closure is sub-Markovian (see [8, Appendix B]). The semigroup convergence follows from [23, Theorem 2.1]. From the convergence of semigroups the convergence of ﬁnite dimensional distributions follows. This implies that all weak accumulation points of (P (nk ) )k∈N necessarily coincide, thus (P (nk ) )k∈N converges weakly to some law P having ﬁnite dimensional distributions given by the semigroup (Tt )t≥0 . This implies the Markov property and also the assertion on the martingale problem, see e.g. [2, Proof of Proposition 1.4] or [5, Appendix A]. 5. The Initial Distribution Consider the situation of Theorems 4.13 and 4.17. We now focus on the initial distribution µ of the process constructed there. We already saw in Remark 3.25 that it is an accumulation point of the sequence (µn ◦ sym−1 n )n∈N or, equivalently, of the sequence (µn ◦ per−1 ) . Our aim is to prove that µ is a tempered grand n∈N n canonical Gibbs measure (cf. [13, p. 1348] for the deﬁnition). To achieve this, we adapt considerations from [13] on the equivalence of the microcanonical and the grand canonical ensemble and extend some results obtained there to the canonical ensemble. We use results and notations from [15, 12, 13]. As in [13] we restrict to the case where λn = n + 12 , n ∈ N.

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Let Pθ be the space of probability measures P on Γv which are invariant with respect to spatial translations ϑa (γ) := γ − (a, 0), γ ∈ Γv , a ∈ Rd , and which have ﬁnite density and kinetic energy density, i.e. Γv (x,v)∈γ∩C (1 + |v|2 )dP (γ) < ∞, where C := [0, 1]d ×Rd . The set of tame cylinder functions F , i.e. functions F : Γv → R such that there exist Λ ⊂ Rd and C < ∞ with F (γ) = F (γ ∩ (Λ × Rd )) and |F (γ)| ≤ C+C (x,v)∈γ∩(Λ×Rd ) (1+|v|), γ ∈ Γv , is denoted by L. On Pθ the topology τL is deﬁned as the weakest topology such that all mappings P → P (F ) = Γv F dP , F ∈ L, are continuous. This topology is ﬁner than the weak topology on the space of probability measures on Γv . We now state the result which is shown in the course of this section. Theorem 5.1. Let a symmetric measurable function φ: Rd → R ∪ {∞} fulﬁll (RP ), (T ), (BB ) as given in Sec. 3.1. Let λn := n + 12 , n ∈ N, and (Nn )n∈N ⊂ N be such Nn [n] := µn ◦ sym−1 that (2λ d → ρ ∈ (0, ∞) as n → ∞. Deﬁne µ n with µn := µλn ,Nn n) and symn := symλn ,Nn as deﬁned in Sec. 4.2. Then (µ[n] )n∈N is relatively compact with respect to τL and any accumulation point is a grand canonical Gibbs measure.

Remark 5.2. The conditions (RP), (T), (BB) imply conditions (A1) and the nonhard-core version of (A2) from [13]. The proof given below works for the latter conditions. We exclude the case of hard-core potentials for convenience and since it is not treated in the preceding sections of this article. The proof of the above theorem mainly follows the lines of arguments in [13], in particular the beginning of [13, Sec. 6]. However, there are some modiﬁcations to be made which can only be explained in the presence of some details. Note that here we only deal with the case of periodic boundary condition (this simpliﬁes the considerations). We introduce some more notations from [13]. By ρ: Pθ → [0, ∞) we denote function assigning to each P ∈ Pθ its average particle density the τL -continuous

ρ(P ) := Γv (γ ∩ C) dP (γ). Upot : Pθ → R ∪ {∞} denotes the mean potential energy, which is given by

1 φ(x − x )dP (γ). Upot (P ) := lim n→∞ (2λn )d Γv {x,x }⊂(prx γ)∩Λλn

The mean kinetic energy Ukin : Pθ → R is deﬁned by

1 Ukin (P ) := |v|2 dP (γ). Γv 2 (x,v)∈(γ∩C)

Both functions Upot , Ukin are measure aﬃne and lower semicontinuous with respect to τL (cf. [13, p. 1349] and also [12]). Set U := Ukin + Upot . Moreover, we need to make use of the mean entropy S: Pθ → R ∪ {−∞}, which is an upper semicontinuous measure aﬃne function and such that for c ∈ R, ε ≥ 0 the sets {P ∈ Pθ : S(P ) ≥ c, Ukin (P ) ≤ ε}

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are compact and sequentially compact with respect to τL (cf. [13, p. 1349 and Lemma 4.2]). We also need to consider entropy functionals Iβ : Pθ → [0, ∞]. They are deﬁned by Iβ (P ) := limn→∞

I(Pn ,Qβ n) (2λn )d ,

where Pn := P ◦ (prvΛn )−1 , Qβn := Qβ ◦ (prvΛn )−1 and 2

Qβ denotes the Poisson point random ﬁeld with intensity measure dx e−βv /2 dv (cf. [13, Sec. 4], also for the deﬁnition of I(·, ·), which denotes relative entropy). We will sometimes make use of the identity ([13, (4.3)]) S(P ) = −Iβ (P ) + βUkin (P ) + c(β) d for P ∈ Pθ . Here c(β) = 2π/β . So, S(P ) < ∞ for all P ∈ Pθ . In [13, Theorem 3.2] it is shown that the function

(5.1)

s(ρ , ε) : = sup{S(P ) | P ∈ Pθ , U (P ) ≤ ε, ρ(P ) = ρ } = sup{S(P ) | P ∈ Pθ , U (P ) = ε, ρ(P ) = ρ },

ρ ≥ 0,

ε ∈ R,

is upper semicontinuous and concave and coincides in the convex set Σ = {(ρ , ε)|ε > Mn εmin (ρ )} with the thermodynamic entropy density limn→∞ log , where (Mn )n∈N (2λn )d denotes a sequence of microcanonical partition functions in Λλn with periodic boundary such that the densities converge towards ρ and the energy densities converge towards ε. Here εmin (ρ ) = inf{U (P ) | P ∈ Pθ , ρ(P ) = ρ , S(P ) > −∞}. We now state a variational principle for the thermodynamic free energy density, which we derive below as a direct consequence of the above mentioned corresponding result from [13] on the thermodynamic entropy density and some considerations (η) := from [27]. Set U y∈η+2λn r φ(x − y), η ∈ ΓΛλn. λn x∈η r∈Zd y=x

Lemma 5.3. For β > 0, ρ > 0, let the free energy f (ρ, β) be deﬁned by βf (ρ, β) = inf ε>εmin (ρ) (βε − s(ρ, ε)). f (ρ, β) is ﬁnite and it holds βf (ρ, β) = − lim

n→∞

log Zn = inf{βU (P ) − S(P ) | P ∈ Pθ , ρ(P ) = ρ}, (2λn )d

(5.2)

where for n ∈ N

β 2 2 1 ˜ Zn = e−β Uλn (x1 ,...,xNn ) e− 2 (v1 +···+vNn ) dx1 · · · dxNn dv1 · · · dvNn n Nn ! ΛN dNn λ ×R n

is the canonical partition function with periodic boundary condition. Proof. By [13, p. 1350] for ε > εmin (ρ) it holds s(ρ, ε) > −∞ and moreover εmin (ρ) is ﬁnite, so we conclude that f (ρ, β) < ∞. Furthermore, due to [13, Lemma 4.1, Eq. (4.4)] and (5.1) the set {S(P ): P ∈ Pθ , ρ(P ) = ρ, U (P ) = ε} is bounded from above by βε + β(B 2 /4A) + c(β) with constants A, B as in (SS). This implies that for any ε > εmin (ρ) it holds βε − s(ρ, ε) ≥ −β(B 2 /4A) − c(β), hence f (ρ, β) > −∞. The arguments in [27, p. 55] also work in the case of periodic boundary condition and including velocities, which together with [13, Theorem 3.2] yields the ﬁrst equality in (5.2).

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To prove the second one, ﬁrst note that for any P ∈ Pθ with ρ(P ) = ρ and U (P ) < ∞ it holds βU (P ) − S(P ) ≥ βU (P ) − s(ρ, U (P )) ≥ βf (ρ, β). This follows from the deﬁnition of f (ρ, β), when U (P ) > εmin (ρ). Moreover, since limε↓εmin (ρ) s(ρ, ε) = s(ρ, εmin (ρ)) due to upper semicontinuity and concavity of s(·, ·), this extends also to U (P ) = εmin . For U (P ) < εmin (ρ) it is implied by the deﬁnition of εmin (ρ). Thus βf (ρ, β) ≤ inf{βU (P ) − S(P ) | P ∈ Pθ , ρ(P ) = ρ, U (P ) < ∞}. To prove the converse inequality, for δ > 0 choose ε˜ ∈ (εmin (ρ), ∞) such that βf (ρ, β) + δ ≥ β ε˜ − s(ρ, ε˜). By [13, (3.9) and Theorem 3.2(b)] we may choose P ∈ Pθ such that ρ(P ) = ρ, U (P ) = ε˜ and such that S(P ) ≥ s(ρ, ε˜) − δ. Hence βU (P ) − S(P ) ≤ β ε˜ − s(ρ, ε˜) + δ ≤ βf (ρ, β) + 2δ. Since δ may be chosen arbitrarily small, the second equality is shown. Let n ∈ N. We deﬁne the measure µ ˆ[n] on Γv such that the conﬁgurations in d d (Λλn + 2λn r) × R , r ∈ Z are independent and distributed as shifts of µ[n] by the translation invariant measure µ ˜[n] as spatial average of the 2rλn . One deﬁnes

˜[n] := Λλ µ ˆ[n] ◦ ϑ−1 ˜[n] ∈ Pθ , cf. [13, (6.3)]. µ ˆ[n] , i.e. µ x dx. It holds µ n v Deﬁne for γ ∈ ΓΛn the measure Rn,γ ∈ Pθ deﬁned by Rn,γ :=

1 (n) δ denotes the 2λn d (2λn ) Λλn ϑx γ (n) dx, where δ· denotes Dirac measure and γ periodic continuation of γ. (See [15] for details on these translation invariant We will below also have to consider the mixture µ[n] Rn :=

empirical ﬁelds.) [n] (n) := µn ◦ per−1 n due Γv Rn,γ dµ (γ), which, on the other hand, is equal to µ (n) resulting from the (spatial) translation invariance to translation invariance of µ of µn as a measure on the manifold EλNnn (deﬁned in Sec. 4.2). Keeping this in mind, we will nevertheless use the notation µ[n] Rn in the sequel. Note that by [13, (6.3)] it holds for n ∈ N

1 1 [n] βv 2 dµ[n] (γ) = Ukin (µ[n] Rn ). µ )= (5.3) Ukin (˜ (2λn )d Γvλ 2 n

(x,v)∈γ

Nn and by the deﬁnition of µ[n] this expression is equal to C(β) (2λ d for some C(β) > 0 n) not depending on n ∈ N. The proof of Theorem 5.1 is based on the inequality given in the following lemma.

Lemma 5.4. It holds for n ∈ N S(˜ µ[n] ) ≥ βU (µ[n] Rn ) +

Proof. It holds

d(µ[n] ◦(prv Λ dQβ n

λn

)−1 )

(5.4)

1 e−Fn , where Fn : ΓvΛλn → R ∪ {∞} is −Fn ) Qβ n (e ∈ ΓvΛλn ,Nn and = ∞ else. These functions might

=

˜ (γ) for γ deﬁned by Fn (γ) := β U λn

log(Zn ) . (2λn )d

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not form an asymptotic empirical functional in the sense of [15]. Nevertheless, the ˜[n] . proof of the second assertion in [15, Lemma 5.5] is valid for the measures µ[n] , µ Therefore, using also (5.1) and (5.3) we ﬁnd that µ[n] ) + βUkin (µ[n] Rn ) + c(β) S(˜ µ[n] ) = −Iβ (˜ ≥− =

1 µ[n] ◦ (prvΛλn )−1 (Fn ) + βUkin (µ[n] Rn ) + c(β) (2λn )d +

=

1 I(µ[n] ◦ (prvΛλn )−1 ; Qβn ) + βUkin (µ[n] Rn ) + c(β) (2λn )d

1 log(Qβn (e−Fn )) (2λn )d

1 1 µ[n] ◦ (prvΛλn )−1 (Fn ) + βUkin (µ[n] Rn ) + log Zn . (2λn )d (2λn )d

By [12, (2.16)] and since Upot is measure aﬃne it holds 1 µ[n] ◦ (prvΛλn )−1 (Fn ) = βµ[n] (Upot (Rn , ·)) = βUpot (µ[n] Rn ). (2λn )d This completes the proof. Nn Since µ ˜[n] = C(β) (2λ ˜[n] is bounded. d , n ∈ N, the kinetic energy density of all µ n) Therefore, boundedness of Upot from below (by −B 2 /4A) together with convergence n) as n → ∞ (cf. Lemma 5.3) imply that (S(˜ µ[n] ))n∈N is bounded from of log(Z (2λn )d below. This together with the boundedness of the kinetic energy implies relative compactness of the sequence (˜ µ[n] )n (see the properties of S mentioned above). The following lemma shows that asymptotically one can treat µ ˜[n] , µ[n] and µ[n] Rn , n ∈ N, as equal.

µ[n] )n and (µ[n] Rn )n are asymptotically Lemma 5.5. The sequences (µ[n] )n , (˜ equivalent, i.e. for any two of them, say (ν1n )n , (ν2n )n , and any f ∈ L it holds µ[nk ] )k∈N to some µ ∈ Pθ limn→∞ |ν1n (f )−ν2n (f )| = 0. In particular, convergence of (˜ with respect to τL implies that also limk→∞ µ[nk ] = limk→∞ µ[nk ] Rnk = µ. Proof. This is shown as in the proof of [13, Lemma 6.2]: The asymptotic equivalence of µ[n] and µ[n] Rn is clear. For the second asymptotic equivalence note that µn ) < ∞ by (5.4), (5.1) and (5.3), so [15, Lemma 5.7] can be applied. For supn Iβ (˜ convenience of the reader we remark that to apply the mentioned lemma a function ψ: Rd → R, deﬁned as function of velocities, has to be chosen appropriately. (This function is used in the deﬁnitions of L and Pθ in [15].) Setting ψ(v) := 1 + |v|, v ∈ Rd , is the standard choice here. Then the deﬁnition of Pθ from [15] does not coincide with the one given above, but denotes a larger space. This, however, does not aﬀect the considerations made in the proof of [15, Lemma 5.7].

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From Lemma 5.5 above, the preceding considerations, (5.4) and the properties of U , S and ρ we ﬁnd that there exists an accumulation point µ ∈ Pθ of (µ[n] )n∈N with respect to τL fulﬁlling ρ(µ) = ρ and any such accumulation point fulﬁlls βU (µ) − S(µ) ≤ − lim

n→∞

log(Zn ) . (2λn )d

(Note that since U is bounded from below and S cannot take the value +∞ in Pθ (one may see this from (5.1) and since Iβ only takes nonnegative values), both U (µ) and S(µ) are ﬁnite.) From Lemma 5.3 we see that βU (µ) − S(µ) = βf (ρ, β). µ is a minimizer of the canonical free energy density U (·) − β −1 S(·) under the constraint ρ(·) = ρ. We ﬁnally make considerations similar to those in [13, p. 1351]: For β > 0 the function f (β, ·) is convex. (This follows from its deﬁnition, the concavity of s(·, ·) and the convexity of the eﬀective domain Σ of s(·, ·) deﬁned in [13, (3.7)].) Hence we may choose some z > 0 and p ∈ R such that ρ → −p + ρ β −1 log(z) is a tangent to f (β, ·) at ρ. This and (5.2) imply for any P ∈ Pθ it holds βU (P ) − ρ(P ) log(z) − S(P ) ≥ βf (ρ(P ), β) − ρ(P ) log(z) ≥ −βp = f (ρ, β) − ρ log(z) = βU (µ) − ρ(µ) log(z) − S(µ).

(5.5)

In order to prove this inequality for ρ(P ) = 0 (i.e. P = δ∅ ), note that U (δ∅ ) = 0, S(δ∅ ) = 0 and ρ(δ∅ ) = 0. Hence we only need to verify that p ≥ 0. This, however, follows e.g. from the fact that for any measure Q ∈ Pθ fulﬁlling U (Q) < ∞, ρ(Q) > 0 and S(Q) > −∞ (such a measure exists) it holds lim βf (ρ , β) ≤ lim βU (αQ + (1 − α)δ∅ ) − S(αQ + (1 − α)δ∅ ) = 0,

ρ →0

α→0

where we used (5.2) and the fact that U and S are aﬃne functions. (5.5) implies that µ is a minimizer of the mean free energy U (·)−ρ(·) log(z)−S(·). By [13, Theorem 3.4] we conclude that µ is a tempered grand canonical Gibbs measure, and the proof of Theorem 5.1 is completed. Acknowledgement Financial support by the DFG through the project GR 1809/5-1 is gratefully acknowledged. References [1] S. Albeverio, Yu. G. Kondratiev and M. R¨ ockner, Analysis and geometry on conﬁguration spaces: The Gibbsian case, J. Funct. Anal. 157 (1998) 242–291. [2] L. Beznea, N. Boboc and M. R¨ ockner, Markov processes associated with Lp -resolvents and applications to stochastic diﬀerential equations on Hilbert space, J. Evol. Equ. 6(4) (2006) 745–772. [3] P. Billingsley, Probability and Measure (Wiley, New York, 1979).

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[4] C. Bahn, Y. M. Park and H. J. Yoo, Nonequilibrium dynamics of inﬁnite particle systems with inﬁnite range interactions, J. Math. Phys. 40(9) (1999) 4337–4358. [5] F. Conrad and M. Grothaus, Construction, ergodicity and rate of convergence of N -particle Langevin dynamics with singular interactions, J. Evol. Equ. 10(3) (2010) 623–662. [6] F. Conrad, Construction and analysis of Langevin dynamics in continuous particle systems, PhD thesis, University of Kaiserslautern (2010). [7] R. L. Dobrushin and R. A. Minlos, The existence and continuity of pressure in classical statistical physics, Theor. Probab. Appl. 12(4) (1967) 535–559. [8] A. Eberle, Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diﬀusion Operators, Lecture Notes in Mathematics, Vol. 1718 (Springer, 1999). [9] S. Ethier and T. Kurtz, Markov Processes (Wiley & Sons, 1986). [10] T. Fattler and M. Grothaus, Construction of elliptic diﬀusions with reﬂecting boundary condition and an application to continuous N -particle systems with singular interactions, Proc. Edinburgh Math. Soc. 51(2) (2008) 337–363. [11] J. Fritz, Stochastic dynamics of two-dimensional inﬁnite-particle systems, J. Stat. Phys. 20 (1979) 361–369. [12] H.-O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction, Probab. Theory Related Fields 99 (1994) 171– 195. [13] H.-O. Georgii, The equivalence of ensembles for classical systems of particles, J. Stat. Phys. 80 (1995) 1341–1378. [14] M. Grothaus, Y. Kondratiev and M. R¨ ockner, N/V -limit for stochastic dynamics in continuous particle systems, Probab. Theory Related Fields 137 (2007) 121–160. [15] H.-O. Georgii and H. Zessin, Large deviations and maximum entropy principle for marked point random ﬁelds, Probab. Theory Related Fields 96 (1993) 177–204. [16] T. L. Hill, Statistical Mechanics, McGraw-Hill Series in Advanced Chemistry (McGraw-Hill, 1956). [17] R. A. Holley and D. W. Stroock, Generalized Ornstein–Uhlenbeck processes and inﬁnite particle branching Brownian motion, Publ. Res. Inst. Math. Sci. 14 (1978) 741–788. [18] O. Kallenberg, Random Measures, 2nd edn. (Academic Press, 1976). [19] Yu. G. Kondratiev and V. Kutovij, On the metrical properties of the conﬁguration space, Math. Nachr. 279 (2006) 774–783. [20] A. V. Kolesnikov, Convergence of Dirichlet forms with changing speed measures on Rd , Forum Math. 17 (2005) 225–259. [21] K. Kuwae and T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Comm. Anal. Geom. 11(4) (2003) 599–673. [22] T. Kuna, Studies in conﬁguration space analysis and applications, PhD thesis, University of Bonn, Bonner Mathematische Schriften 324 (1999). [23] T. G. Kurtz, Extensions of Trotter’s operator semigroup approximation theorems, J. Funct. Anal. 3 (1969) 354–375. [24] C. Marchioro, A. Pellegrinotti and E. Presutti, Existence of time evolutions for ν-dimensional statistical mechanics, Comm. Math. Phys. 40 (1975) 175–185. [25] H. Osada, Dirichlet form approach to inﬁnite-dimensional Wiener process with singular interactions, Comm. Math. Phys. 176 (1996) 117–131. [26] S. Olla and C. Tremoulet, Equilibrium ﬂuctuations for interacting Ornstein– Uhlenbeck particles, Comm. Math. Phys. 233(3) (2003) 463–491. [27] D. Ruelle, Statistical Mechanics. Rigorous Results (Benjamins, 1969).

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[28] D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18 (1970) 127–159. [29] H. Spohn, Equilibrium ﬂuctuations for interacting Brownian particles, Comm. Math. Phys. 103 (1986) 1–33. [30] R. Siegmund-Schultze, On non-equilibrium dynamics of multidimensional inﬁnite particle systems in the translation invariant case, Comm. Math. Phys. 100 (1985) 245–265. [31] W. Stannat, The theory of generalized Dirichlet forms and its applications in analysis and stochastics, Mem. Amer. Math. Soc. 142 (1999), no. 678, viii + 101 pp. [32] J. M. T¨ olle, Convergence of non-symmetric forms with changing reference measure, Master’s thesis, University of Bielefeld (2006). [33] H. F. Trotter, Approximation of semi-groups of operators, Paciﬁc J. Math. 8 (1958) 887–919. [34] M. W. Yoshida, Construction of inﬁnite-dimensional interacting diﬀusion processes through Dirichlet forms, Probab. Theory Related Fields 106 (1996) 265–297.

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Reviews in Mathematical Physics Vol. 23, No. 1 (2011) 53–81 c World Scientific Publishing Company DOI: 10.1142/S0129055X11004205

SPECTRAL AND SCATTERING THEORY FOR THE AHARONOV–BOHM OPERATORS

KONSTANTIN PANKRASHKIN∗,‡ and SERGE RICHARD†,§ ∗Laboratoire

de Math´ ematiques d’Orsay, CNRS UMR 8628, Universit´ e Paris-Sud XI, Bˆ atiment 425, 91405 Orsay Cedex, France †Department

of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WB, UK ‡[email protected] §[email protected] Received 1 December 2009 Revised 3 September 2010

We review the spectral and the scattering theory for the Aharonov–Bohm model on R2 . New formulae for the wave operators and for the scattering operator are presented. The asymptotics at high energy and at low energy of the scattering operator are computed. Keywords: Spectral and scattering theory; Aharonov–Bohm model; wave and scattering operators; boundary triple. Mathematics Subject Classification 2010: 81U15, 47A40

1. Introduction The Aharonov–Bohm (A–B) model describing the motion of a charged particle in a magnetic ﬁeld concentrated at a single point is one of the few systems in mathematical physics for which the spectral and the scattering properties can be completely computed. It has been introduced in [3] and the ﬁrst rigorous treatment appeared in [26]. A more general class of models involving boundary conditions at the singularity point has then been developed in [2, 13] and further extensions or reﬁnements appeared since these simultaneous works. Being unable to list all these subsequent papers, let us simply mention few of them: [28] in which it is proved that the A–B models can be obtained as limits in a suitable sense of systems with less singular magnetic ﬁelds, [27] in which it is shown that the low energy behavior of the scattering amplitude for two-dimensional magnetic Schr¨ odinger operators is similar to the scattering amplitude of the A–B models, and the series [7, 8, 30] in which, among other results, high energy estimates are obtained for the scattering † On

leave from Universit´e de Lyon, Universit´e Lyon I, CNRS UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, 69622 Villeurbanne Cedex, France. 53

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operator. Concerning the extensions we mention the papers [15] which considers the A–B operators with an additional uniform magnetic ﬁeld and [21] which studies the A–B operators on the hyperbolic plane. The aim of the present paper is to provide the spectral and the scattering analysis of the A–B operators on R2 for all possible values of the parameters (boundary conditions). The work is motivated by the recent result of one of the authors ([25]) showing that the A–B wave operators can be rewritten in terms of explicit functions of the generator of dilations and of the Laplacian. However, the proof of this result used certain complicated expressions for the scattering operator borrowed from [2] and we have in the meanwhile found a simpler approach. For those reasons, we have decided to start again the analysis from scratch using the modern operatortheoretical machinery. For example, in contrast to [2, 13], our computations do not involve an explicit parametrization of U (2). Simultaneously, we recast this analysis in the up-to-date theory of self-adjoint extensions ([12]) and derive rigorously the expressions for the wave operators and the scattering operator from the stationary approach of scattering theory as presented in [31]. So let us now describe the content of this review paper. In Sec. 2, we introodinger operator in R2 with a duce the operator Hα which corresponds to a Schr¨ δ-type magnetic ﬁeld at the origin. The index α corresponds to the total ﬂux of the magnetic ﬁeld, and on a natural domain this operator has deﬁciency indices (2, 2). The description of this natural domain is recalled and some of its properties are exhibited. Section 3 is devoted to the description of all self-adjoint extensions of the operator Hα . More precisely, a boundary triple for the operator Hα is constructed in Proposition 4. It essentially consists in the deﬁnition of two linear maps Γ1 , Γ2 from the domain D(Hα∗ ) of the adjoint of Hα to C2 which have some speciﬁc properties with respect to Hα , as recalled at the beginning of this section. Once these maps are exhibited, all self-adjoint extensions of Hα can be labeled by two 2 × 2-matrices C and D satisfying two simple conditions presented in (7). These self-adjoint extensions are denoted by HαCD . The γ-ﬁeld and the Weyl function corresponding to the boundary triple are then constructed. By taking advantage of some general results related to the boundary triple’s approach, they allow us to explicitly mention the spectral properties of HαCD in very simple terms. At the end of the section, we add some comments about the role of the parameters C and D and discuss some of their properties. The short Sec. 4 contains formulae on the Fourier transform and on the dilation group that are going to be used subsequently. Section 5 is the main section on scattering theory. It contains the time dependent approach as well as the stationary approach of the scattering theory for the A–B models. Some calculations involving Bessel functions or hypergeometric 2 F1 -functions look rather tricky but they are necessary for a rigorous derivation of the stationary expressions. Fortunately, the ﬁnal expressions are much more easily understandable. For example, it is proved in Proposition 11 that the channel wave operators for the original A–B operator HαAB are equal to very explicit functions of the generator of dilations. These functions are

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continuous on [−∞, ∞] and take values in the set of complex numbers of modulus 1. Theorem 12 contains a similar explicit description of the wave operators for the general operator HαCD . In Sec. 6, we study the scattering operator and in particular its asymptotics at small and large energies. These properties highly depend on the parameters C and D but also on the ﬂux α of the singular magnetic ﬁeld. All the various possibilities are explicitly analyzed. The statement looks rather messy, but this simply reﬂects the richness of the model. The parametrization of the self-adjoint extensions of Hα with the pair (C, D) is highly non unique. For convenience, we introduce in the last section a one-toone parametrization of all self-adjoint extensions and explicit some of the previous results in this framework. For further investigations in the structure of the set of all self-adjoint extensions, this unique parametrization has many advantages. Finally, let us mention that this paper is essentially self-contained. Furthermore, despite the rather long and rich history of the Aharonov–Bohm model most of the our results are new or exhibited in the present form for the ﬁrst time. Remark 1. After the completion of this paper, the authors were informed about the closely related work [10]. In this paper, the diﬀerential expression −∂x2 + (m2 − 1/4)x−2 on R+ is considered and a holomorphic family of extensions for (m) > −1 is studied. Formulae for the wave operators similar to our formula (14) were independently obtained by its authors. Remark 2. In December 2009, a two-day meeting celebrated the 50th anniversary of the Aharonov–Bohm eﬀect, and 25th anniversary since the discovery of the related geometric, or Berry phase. It was pointed out to us by the referee that an interesting discussion took place in the physics literature on this occasion. We refer to the letter [9] for more information on the subject and thank the referee for drawing our attention to this reference. 2. General Setting Let H denote the Hilbert space L2 (R2 ) with its scalar product ·, · and its norm · . For any α ∈ R, we set Aα : R2 \{0} → R2 by −y x Aα (x, y) = −α , , x2 + y 2 x2 + y 2 corresponding formally to the magnetic ﬁeld B = αδ (δ is the Dirac delta function), and consider the operator Hα := (−i∇ − Aα )2 ,

∞ 2 D(Hα ) = Cc (R \{0}).

Here Cc∞ (Ξ) denotes the set of smooth functions on Ξ with compact support. The closure of this operator in H, which is denoted by the same symbol, is symmetric and has deﬁciency indices (2, 2) ([2, 13]). For further investigation we need some more information on this closure.

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So let us ﬁrst decompose the Hilbert space H with respect to polar coordinates: For any m ∈ Z, let φm be the complex function deﬁned by [0, 2π) θ → φm (θ) := e√imθ . Then, by taking the completeness of the family {φm }m∈Z in L2 (S1 ) into 2π account, one has the canonical isomorphism H∼ Hr ⊗ [φm ], (1) = m∈Z 2

where Hr := L (R+ , r dr) and [φm ] denotes the one-dimensional space spanned by φm . For shortness, we write Hm for Hr ⊗ [φm ], and often consider it as a subspace of H. Clearly, the Hilbert space Hm is isomorphic to Hr , for any m. In this representation the operator Hα is equal to [13, Sec. 2] Hα,m ⊗ 1, (2) m∈Z

with (m + α)2 d2 1 d + − , dr2 r dr r2 and with a domain which depends on m + α. It clearly follows from this representation that replacing α by α + n, n ∈ Z, corresponds to a unitary transformation of Hα . In particular, the case α ∈ Z is equivalent to the magnetic ﬁeld-free case α = 0, i.e. the Laplacian and its zero-range perturbations, see [4, Chap. 1.5]. Hence throughout the paper, we restrict our attention to the values α ∈ (0, 1). So, for α ∈ (0, 1) and m ∈ {0, −1}, the domain D(Hα,m ) is given by Hα,m = −

2,2 {f ∈ Hr ∩ Hloc (R+ ) | −f − r−1 f + (m + α)2 r−2 f ∈ Hr }. (1)

For m ∈ {0, −1}, let Hν denote the Hankel function of the ﬁrst kind and of order 2,2 let W (g, h) stand for the Wronskian ν, and for f, h ∈ Hloc W (f, h) := f h − f h. One then has

2,2 −1 2 −2 D(Hα,m ) = f ∈ Hr ∩ Hloc (R+ ) | −f − r f + (m + α) r f ∈ Hr and lim r[W (f, h±i,m )](r) = 0 , r→0+

(1)

(1)

where h+i,m (r) = H|m+α| (eiπ/4 r) and h−i,m (r) = H|m+α| (ei3π/4 r). It is known / {0, −1} are self-adjoint on the mentioned domain, that the operators Hα,m for m ∈ while Hα,0 and Hα,−1 have deﬁciency indices (1, 1). This explains the deﬁciency indices (2, 2) for the operator Hα . The problem of the description of all self-adjoint extensions of the operator Hα can be approached by two methods. On the one hand, there exists the classical description of von Neumann based on unitary operators between the deﬁciency subspaces. On the other hand, there exists the theory of boundary triples which

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has been widely developed for the last twenty years ([12,14]). Since our construction is based only on the latter approach, we shall recall it brieﬂy in the sequel. Before stating a simple result on D(Hα,m ) for m ∈ {0, −1} let us set some conventions. For a complex number z ∈ C\R+ , the branch of the square root √ √ z → z is ﬁxed by the condition z > 0. In other words, for z = reiϕ with √ √ r > 0 and ϕ ∈ (0, 2π) one has z = reiϕ/2 . On the other hand, for β ∈ R we always take the principal branch of the power z → z β by taking the principal branch of the argument arg z ∈ (−π, π). This means that for z = reiϕ with r > 0 and ϕ ∈ (−π, π) we have z β = rβ eiβϕ . Let us also recall the asymptotic behavior (1) of Hν (w) as w → 0 in C\R− and for ν ∈ Z: Hν(1) (w) = −

2ν i 2−ν ie−iπν w−ν + wν + O(w2−ν ). sin(πν)Γ(1 − ν) sin(πν)Γ(1 + ν)

(3)

Proposition 3. For any f ∈ D(Hα,m ) with m ∈ {0, −1}, the following asymptotic behavior holds: lim

r→0+

f (r) r|m+α|

= 0.

Proof. Let us set ν := |m + α| ∈ (0, 1), and recall that f ∈ D (Hα,m ) implies (1) (1) f ∈ C 1 ((0, +∞)) and that the Hankel function satisﬁes (Hν (z)) = Hν−1 (z) − (1) ν z Hν (z). By taking this and the asymptotics (3) into account, the condition limr→0+ r[W (h±i,m , f )](r) = 0 implies that lim {rν+1 f (r) − νrν f (r)} = 0

(4)

lim {r1−ν f (r) + νr−ν f (r)} = 0.

(5)

r→0+

and that r→0+

Multiplying both terms of (5) by r2ν and subtracting it from (4) one obtains that lim rν f (r) = 0.

r→0+

(6)

On the other hand, considering (5) as a linear diﬀerential equation for f : r1−ν f (r)+ νr−ν f (r) = b(r), and using the variation of constant one gets for some C ∈ C: r C 1 f (r) = ν + ν t2ν−1 b(t)dt. r r 0 Now Eq. (6) implies that C = 0, and by using l’Hˆ opital’s rule, one ﬁnally obtains: r t2ν−1 b(t)dt f (r) r2ν−1 b(r) 1 lim lim b(r) = 0. = lim 0 = lim = ν 2ν r→0+ r r→0+ r→0+ 2νr2ν−1 r 2ν r→0+

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3. Boundary Conditions and Spectral Theory In this section, we explicitly construct a boundary triple for the operator Hα and we brieﬂy exhibit some spectral results in that setting. Clearly, our construction is very closed to the one in [13], but this paper does not contain any reference to the boundary triple machinery. Our aim is thus to recast the construction in an up-to-date theory. The following presentation is strictly adapted to our setting, and as a general rule we omit to write the dependence on α on each of the objects. We refer to [12, 14] for more information on boundary triples. Let Hα be the densely deﬁned closed and symmetric operator in H previously introduced. The adjoint of Hα is denoted by Hα∗ and is deﬁned on the domain 2,2 ∗ 2 D(Hα ) = {f ∈ H ∩ Hloc (R \{0}) | Hα f ∈ H}.

Let Γ1 , Γ2 be two linear maps from D(Hα∗ ) to C2 . The triple (C2 , Γ1 , Γ2 ) is called a boundary triple for Hα if the following two conditions are satisﬁed: (1) f, Hα∗ g − Hα∗ f, g = Γ1 f, Γ2 g − Γ2 f, Γ1 g for any f, g ∈ D (Hα∗ ), (2) the map (Γ1 , Γ2 ): D(Hα∗ ) → C2 ⊕ C2 is surjective. It is proved in the references mentioned above that such a boundary triple exists, and that all self-adjoint extensions of Hα can be described in this framework. More precisely, let C, D ∈ M2 (C) be 2 × 2 matrices, and let us denote by HαCD the restriction of Hα∗ on the domain CD ∗ D(Hα ) := {f ∈ D (Hα ) | CΓ1 f = DΓ2 f}.

Then, the operator HαCD is self-adjoint if and only if the matrices C and D satisfy the following conditions: (i) CD ∗ is self-adjoint,

(ii) det(CC ∗ + DD∗ ) = 0.

(7)

Moreover, any self-adjoint extension of Hα in H is equal to one of the operator HαCD . We shall now construct explicitly a boundary triple for the operator Hα . For √ that purpose, let us consider z ∈ C\R+ and choose k = z with (k) > 0. It is easily proved that the following two functions fz,0 and fz,−1 deﬁne an orthonormal basis in ker(Hα∗ − z), namely in polar coordinates: fz,0 (r, θ) = Nz,0 Hα(1) (kr)φ0 (θ),

(1)

fz,−1 (r, θ) = Nz,−1 H1−α (kr)φ−1 (θ),

where Nz,m is the normalization such that fz,0 = fz,−1 = 1. In particular, by making use of the equality ∞ r|Hν(1) (kr)|2 dr = (π cos(πν/2))−1 0

valid for k ∈ {e

iπ/4

,e

i3π/4

N±i,0 = (π cos(πα/2))1/2

}, one has and N±i,−1 = (π cos(π(1 − α)/2))1/2 = (π sin(πα/2))1/2 .

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Let us also introduce the averaging operator P with respect to the polar angle acting on any f ∈ H and for almost every r > 0 by 2π f(r, θ)dθ. [P(f)](r) = 0

Following [13, Sec. 3], we can then deﬁne the following four linear functionals on suitable f: Φ0 (f) = lim rα [P(fφ0 )](r), r→0+

Φ−1 (f) = lim r r→0+

1−α

[P(fφ−1 )](r),

Ψ0 (f) = lim r−α ([P(fφ0 )](r) − r−α Φ0 (f)), r→0+

Ψ−1 (f) = lim rα−1 ([P(fφ−1 )](r) − rα−1 Φ−1 (f)). r→0+

For example, by taking the asymptotic behavior (3) into account one obtains Φ0 (fz,0 ) = Nz,0 aα (z),

Φ−1 (fz,0 ) = 0,

Ψ0 (fz,0 ) = Nz,0 bα (z),

Ψ−1 (fz,0 ) = 0,

Φ−1 (fz,−1 ) = Nz,−1 a1−α (z),

Φ0 (fz,−1 ) = 0,

Ψ−1 (fz,−1 ) = Nz,−1 b1−α (z),

Ψ0 (fz,−1 ) = 0,

(8)

with aν (z) = −

2ν i k −ν , sin(πν)Γ(1 − ν)

bν (z) =

2−ν ie−iπν kν . sin(πν)Γ(1 + ν)

(9)

The main result of this section is: Proposition 4. The triple (C2 , Γ1 , Γ2 ), with Γ1 , Γ2 defined on f ∈ D (Hα∗ ) by Φ0 (f) αΨ0 (f) Γ1 f := , Γ2 f := 2 , Φ−1 (f) (1 − α)Ψ−1 (f) is a boundary triple for Hα . Proof. We use the schema from [11, Lemma 5]. For any f, g ∈ D (Hα∗ ) let us deﬁne the sesquilinear forms B1 (f, g) := f, Hα∗ g − Hα∗ f, g and B2 (f, g) := Γ1 f, Γ2 g − Γ2 f, Γ1 g. We are going to show that these expressions are well deﬁned and that B1 = B2 . (i) Clearly, B1 is well deﬁned. For B2 , let us ﬁrst recall that D (Hα∗ ) = D(Hα ) + ker(Hα∗ − i) + ker(Hα∗ + i). It has already been proved above that the four maps Φ0 , Φ−1 , Ψ0 and Ψ−1 are well deﬁned on the elements of ker(Hα∗ − i) and ker(Hα∗ + i). We shall now prove that Γ1 f = Γ2 f = 0 for f ∈ D(Hα ), which shows that B2 is also well deﬁned on D (Hα∗ ). In view of the decomposition (2)

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it is suﬃcient to consider functions f of the form f(r, θ) = fm (r)φm (θ) for any m ∈ Z and with fm ∈ D(Hα,m ). Obviously, for such a function f with m∈ / {0, −1} one has [P(f)](r) = 0 for almost every r, and thus Γ1 f = Γ2 f = 0. For m ∈ {0, 1} the equalities Γ1 f = Γ2 f = 0 follow directly from Proposition 3. (ii) Now, since Γ1 f = Γ2 g = 0 for all f, g ∈ D (Hα ), the only non trivial contributions to the sesquilinear form B2 come from f, g ∈ ker(Hα∗ − i) + ker(Hα∗ + i). On the other hand one also has B1 (f, g) = 0 for f, g ∈ D(Hα ). Thus, we are reduced in proving the equalities B1 (fz,m , fz ,n ) = B2 (fz,m , fz ,n ) for any z, z ∈ {−i, i} and m, n ∈ {0, −1}. Observe ﬁrst that for z = z and arbitrary m, n one has B1 (fz,m , fz ,n ) = fz,m , z fz ,n − zfz,m , fz ,n = 0 since z = z¯. Now, for m = n one has Γ1 fz,m ⊥ Γ2 fz ,n , and hence B2 (fz,m , fz ,n ) = 0 = B1 (fz,m , fz ,n ). For m = n one easily calculates with ν := |m − α| that B2 (fz,m , fz ,m ) = 2νNz,m Nz ,m (aν (z)bν (z ) − bν (z)aν (z )) = 0, and then B2 (fz,m , fz ,m ) = 0 = B1 (fz,m , fz ,m ). We now consider z = z and m = n. One has B1 (fz,m , fz,n ) = fz,m , zfz,n − zfz,m , fz,n = 2zfz,m , fz,n = 0 and again Γ1 fz,m ⊥ Γ2 fz,n . It then follows that B2 (fz,m , fz,n ) = 0 = B1 (fz,m , fz,n ). So it only remains to show that B1 (fz,m , fz,m ) = B2 (fz,m , fz,m ). For that purpose, observe ﬁrst that B1 (fz,m , fz,m ) = 2zfz,m , fz,m = 2z. On the other hand, one has B2 (fz,m , fz,m ) = 2i(Γ1 fz,m , Γ2 fz,m ) = 2i(2ν|Nz,m |2 aν (z)bν (z)) with ν = |m − α|. By inserting (9) into this expression, one obtains (with √ k = z and (k) > 0) −(k ν )2 e−iπν B2 (fz,m , fz,m ) = 4iν|Nz,m |2 sin2 (πν)Γ(1 − ν)Γ(1 + ν) = 4zν|Nz,m |2

sin(πν/2) . sin2 (πν)Γ(1 − ν)Γ(1 + ν)

Finally, by taking the equality Γ(1 − ν)Γ(1 + ν) =

πν sin(πν)

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into account, one obtains B2 (fz,m , fz,m ) = 4z|Nz,m|2

sin(πν/2) sin(πν/2) = 4zπ cos(πν/2) = 2z, sin(πν)π sin(πν)π

which implies B2 (fz,m , fz,m ) = 2z = B1 (fz,m , fz,m ). (iii) The surjectivity of the map (Γ1 , Γ2 ): D(Hα∗ ) → C2 ⊕ C2 follows from the equalities (8). Let us now construct the Weyl function corresponding to the above boundary triple. The presentation is again adapted to our setting, and we refer to [12] for general deﬁnitions. As already mentioned, all self-adjoint extensions of Hα can be characterized by the 2 × 2-matrices C and D satisfying two simple conditions, and these extensions are denoted by HαCD . In the special case (C, D) = (1, 0), the operator Hα10 is equal to the original Aharonov–Bohm operator HαAB . Recall that this operator corresponds to the Friedrichs extension of Hα and that its spectrum is equal to R+ . This operator is going to play a special role in the sequel. Let us consider ξ = (ξ0 , ξ−1 ) ∈ C2 and z ∈ C\R+ . It is proved in [12] that there exists a unique f ∈ ker(Hα∗ − z) with Γ1 f = ξ. This solution is explicitly given by the formula: f := γ(z)ξ with γ(z)ξ =

ξ0 ξ−1 fz,0 + fz,−1 Nz,0 aα (z) Nz,−1 a1−α (z)

The Weyl function M (z) is then deﬁned by the relation M (z) := Γ2 γ(z). In view of the previous calculations one has

M (z) = 2

α bα (z)/aα (z)

0

(1 − α) b1−α (z)/a1−α (z)

0





Γ(1 − α)2 e−iπα α 2 (k )  4α 2  = − sin(πα)  π  0

0 Γ(α)2 e−iπ(1−α) 1−α 2 (k ) 41−α

  . 

In particular, one observes that for z ∈ C\R+ one has M (0) := limz→0 M (z) = 0. In terms of the Weyl function and of the γ-field γ the Krein resolvent formula has the simple form: z )∗ (HαCD − z)−1 − (HαAB − z)−1 = −γ(z)(DM (z) − C)−1 Dγ(¯ = −γ(z)D∗ (M (z)D∗ − C ∗ )−1 γ(¯ z )∗

(10)

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for z ∈ ρ(HαAB )∩ρ(HαCD ). The following result is also derived within this formalism, see [5] for (i), [14, Theorem 5] and the matrix reformulation [16, Theorem 3] for (ii). In the statement, the equality M (0) = 0 has already been taken into account. Lemma 5. (i) The value z ∈ R− is an eigenvalue of HαCD if and only if det(DM (z) − C) = 0, and in that case one has ker(HαCD − z) = γ(z) ker(DM (z) − C). (ii) The number of negative eigenvalues of HαCD coincides with the number of negative eigenvalues of the matrix CD∗ . We stress that the number of eigenvalues does not depend on α ∈ (0, 1), but only on the choice of C and D. Let us now add some comments about the role of the parameters C and D and discuss some of their properties. Two pairs of matrices (C, D) and (C , D ) satisfying (7) deﬁne the same boundary condition (i.e. the same self-adjoint extension) if and only if there exists some invertible matrix L ∈ M2 (C) such that C = LC and D = LD ([23, Proposition 3]). In particular, if (C, D) satisﬁes (7) and if det(D) = 0, then the pair (D−1 C, 1) deﬁnes the same boundary condition (and D−1 C is selfadjoint). Hence there is an arbitrariness in the choice of these parameters. This can avoided in several ways. First, one can establish a bijection between all boundary conditions and the set U (2) of the unitary 2 × 2-matrices U by setting C = C(U ) :=

i 1 (1 − U ) and D = D(U ) = (1 + U ), 2 2

(11)

see a detailed discussion in [17]. We shall comment more on this in the last section. Another possibility is as follows (cf. [24] for details): There is a bijection between the set of all boundary conditions and the set of triples (L, I, L), where L ∈ {{0}, C, C2}, I: L → C2 is an identiﬁcation map (identiﬁcation of L as a linear subspace of C2 ) and L is a self-adjoint operator in L. For example, given such a triple (L, I, L) the corresponding boundary condition is obtained by setting C = C(L, I, L) := L ⊕ 1

and D = D(L, I, L) := 1 ⊕ 0

with respect to the decomposition C2 = [IL] ⊕ [IL]⊥ . On the other hand, for a pair (C, D) satisfying (7), one can set L := Cd with d := 2 − dim[ker(D)], I: L → C2 is the identiﬁcation map of L with ker(D)⊥ and L := (DI)−1 CI. In this framework, one can check by a direct calculation that for any K ∈ M2 (C) such that DK − C is invertible, one has (DK − C)−1 D = I(PKI − L)−1 P,

(12)

where P : C2 → L is the adjoint of I, i.e. the composition of the orthogonal projection onto IL together with the identiﬁcation of IL with L.

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Let us ﬁnally note that the conditions (7) imply some speciﬁc properties related to commutativity and adjointness. We shall need in particular: Lemma 6. Let (C, D) satisfies (7) and K ∈ M2 (C) with K > 0. Then (i) The matrices DK − C and DK ∗ − C are invertible, (ii) The equality [(DK − C)−1 D]∗ = (DK ∗ − C)−1 D holds. Proof. (i) By contraposition, let us assume that det(DK − C) = 0. Passing to the adjoint, one also has det(K ∗ D∗ − C ∗ ) = 0, i.e. there exists f ∈ C2 such that K ∗ D∗ f = C ∗ f . By taking the scalar product with D∗ f , one obtains that D∗ f, KD∗ f = f, CD∗ f . The right-hand side is real due to (i) in (7). But since K > 0, the equality is possible if and only if D∗ f = 0. It then follows that C ∗ f = K ∗ D∗ f = 0, which contradicts (ii) in (7). The invertibility of DK ∗ − C can be proved similarly. (ii) If det(D) = 0, then the matrix A := D−1 C is self-adjoint and it follows that [(DK − C)−1 D]∗ = [(K − A)−1 ]∗ = (K ∗ − A)−1 = (DK ∗ − C)−1 D. If D = 0, then the equality is trivially satisﬁed. Finally, if det(D) = 0 but D = 0 one has L := C. Furthermore, let us deﬁne I: C → C2 by IL := ker(D)⊥ and let P : C2 → C be its adjoint map. Then, by the above construction there exists ∈ R such that (DK − C)−1 D = I(P KI − )−1 P . It is also easily observed that P KI is just the multiplication by some k ∈ C with k > 0, and hence (DK − C)−1 D = I(k − )−1 P . Similarly one has (DK ∗ − C)−1 D = I(k¯ −

)−1 P . Taking the adjoint of the ﬁrst expression leads directly to the expected equality. 4. Fourier Transform and the Dilation Group Before starting with the scattering theory, we recall some properties of the Fourier transform and of the dilation group in relation with the decomposition (1). Let F be the usual Fourier transform, explicitly given on any f ∈ H and y ∈ R2 by 1 l.i.m. f(x)e−ix·y dx [F f](y) = 2π 2 R where l.i.m. denotes the convergence in the mean. Its inverse is denoted by F ∗ . Since the Fourier transform maps the subspace Hm of H onto itself, we naturally set Fm : Hr → Hr by the relation F (f φm ) = Fm (f )φm for any f ∈ Hr . More explicitly, the application Fm is the unitary map from Hr to Hr given on any f ∈ Hr and almost every κ ∈ R+ by |m| ˆ f (κ) := [Fm f ](κ) = (−i) l.i.m. rJ|m| (rκ)f (r)dr, R+

where J|m| denotes the Bessel function of the ﬁrst kind and of order |m|. The inverse ∗ is given by the same formula, with (−i)|m| replaced by i|m| . Fourier transform Fm

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Now, let us recall that the unitary dilation group {Uτ }τ ∈R is deﬁned on any f ∈ H and x ∈ R2 by [Uτ f](x) = eτ f(eτ x). Its self-adjoint generator A is formally given by 12 (X · (−i∇) + (−i∇) · X), where X is the position operator and −i∇ is its conjugate operator. All these operators are essentially self-adjoint on the Schwartz space on R2 . An important property of the operator A is that it leaves each subspace Hm invariant. For simplicity, we shall keep the same notation for the restriction of A to each subspace Hm . So, for any m ∈ Z, let ϕm be an essentially bounded function on R. Assume furthermore that the family {ϕm }m∈Z is bounded. Then the operator ϕ(A): H → H deﬁned on Hm by ϕm (A) is a bounded operator in H. Let us ﬁnally recall a general formula about the Mellin transform. Lemma 7. Let ϕ be an essentially bounded function on R such that its inverse Fourier transform is a distribution on R. Then, for any f ∈ Cc∞ (R2 \{0}) one has ∞ ds ϕ(−ln(s/r))f(s, ˇ θ) , [ϕ(A)f](r, θ) = (2π)−1/2 r 0 where the right-hand side has to be understood in the sense of distributions. Proof. The proof is a simple application for n = 2 of the general formulae developed in [18, p. 439]. Let us however mention that the convention of this reference on the minus sign for the operator A in its spectral representation has not been adopted. As already mentioned ϕ(A) leaves Hm invariant. More precisely, if f = f φm for some f ∈ Cc∞ (R+ ), then ϕ(A)f = [ϕ(A)f ]φm with ∞ ds [ϕ(A)f ](r) = (2π)−1/2 (13) ϕ(−ln(s/r))f ˇ (s) , r 0 where the right-hand side has again to be understood in the sense of distributions. 5. Scattering Theory In this section, we brieﬂy recall the main deﬁnitions of the scattering theory, and then give explicit formulae for the wave operators. The scattering operator will be studied in the following section. Let H1 , H2 be two self-adjoint operators in H, and assume that the operator H1 is purely absolutely continuous. Then the (time dependent) wave operators are deﬁned by the strong limits Ω± (H2 , H1 ) := s-lim eitH2 e−itH1 t→±∞

whenever these limits exist. In this case, these operators are isometries, and they are said complete if their ranges are equal to the absolutely continuous subspace

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Hac (H2 ) of H with respect to H2 . In such a situation, the (time dependent) scattering operator for the system (H2 , H1 ) is deﬁned by the product S(H2 , H1 ) := Ω∗+ (H2 , H1 )Ω− (H2 , H1 ) and is a unitary operator in H. Furthermore, it commutes with the operator H1 , and thus is unitarily equivalent to a family of unitary operators in the spectral representation of H1 . We shall now prove that the wave operators exist for our model and that they are complete. For that purpose, let us denote by H0 := −∆ the Laplace operator on R2 . Lemma 8. For any self-adjoint extension HαCD , the wave operators Ω± (HαCD , H0 ) exist and are complete. Proof. On the one hand, the existence and the completeness of the operators Ω± (HαAB , H0 ) has been proved in [26]. On another hand, the existence and the completeness of the operator Ω± (HαCD , HαAB ) is well known since the diﬀerence of the resolvents is a ﬁnite rank operator, see for example [19, Sec. X.4.4]. The statement of the lemma follows then by taking the chain rule [31, Theorem 2.1.7] and [31, Theorem 2.3.3] on completeness into account. The derivation of the explicit formulae for the wave operators is based on the stationary approach, as presented in [31, Secs. 2.7 and 5.2]. For simplicity, we shall CD consider only ΩCD − := Ω− (Hα , H0 ). For that purpose, let λ ∈ R+ and ε > 0. We ﬁrst study the expression ε (H0 − λ + iε)−1 f, (HαCD − λ + iε)−1 g π and its limit as ε → 0+ for suitable f, g ∈ H speciﬁed later on. By taking Krein resolvent formula into account, one can consider separately the two expressions: ε (H0 − λ + iε)−1 f, (HαAB − λ + iε)−1 g π and ε − (H0 − λ + iε)−1 f, γ(λ − iε)(DM (λ − iε) − C)−1 Dγ(λ + iε)∗ g. π The ﬁrst term will lead to the wave operator for the original Aharonov–Bohm system, as shown below. So let us now concentrate on the second expression. For simplicity, we set z = λ + iε and observe that ε z )(DM (¯ z ) − C)−1 Dγ(z)∗ g − (H0 − z¯)−1 f, γ(¯ π ε z ) − C)−1 D]∗ γ(¯ z )∗ (H0 − z¯)−1 f, g. = − γ(z)[(DM (¯ π

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Then, for every r ∈ R+ and θ ∈ [0, 2π) one has ε z ) − C)−1 D]∗ γ(¯ z )∗ (H0 − z¯)−1 f](r, θ) − [γ(z)[(DM (¯ π (1) √ T Hα ( zr)φ0 (θ) ξ (z, f) ε 0 · A(z)[(DM (¯ z ) − C)−1 D]∗ A(¯ z )∗ =− π H (1) (√zr)φ−1 (θ) ξ−1 (z, f) 1−α

with

A(z) =

and

ξ0 (z, f)

ξ−1 (z, f)

aα (z)−1

0

0

a1−α (z)−1

(1) √ Hα ( z¯·)φ0 , (H0 − z¯)−1 f (1) √ H1−α ( z¯·)φ−1 , (H0 − z¯)−1 f (1) √ F(H0 − z)−1 Hα ( z¯·)φ0 , ˆf (1) √ F(H0 − z)−1 H1−α ( z¯·)φ−1 , ˆf (1) √ F0 (H0 − z)−1 Hα ( z¯·), fˆ0 R+ (1) √ F−1 (H0 − z)−1 H1−α ( z¯·), fˆ−1 R+ (1) √ (X 2 − z)−1 F0 Hα ( z¯·), fˆ0 R+ (1) √ (X 2 − z)−1 F−1 H ( z¯·), fˆ−1 R

=

=

=

=

1−α

+

2

where ·, ·R+ denotes the scalar product in L (R+ , r dr). We shall now calculate separately the limit as ε → 0 of the diﬀerent terms. We √ recall the convention that √ for z ∈ C\R+ on choose k = z with (z) > 0. For λ ∈ R+ one sets limε→0+ λ + iε =: κ with κ ∈ R+ . We ﬁrst observe that for ν ∈ (0, 1) one has aν (λ+ ) := lim aν (λ + iε) = − ε→0+

2ν i κ−ν sin(πν)Γ(1 − ν)

but aν (λ− ) := lim aν (λ − iε) = − ε→0+

2ν ie−iπν κ−ν . sin(πν)Γ(1 − ν)

Similarly, one observes that M (λ± ) := lim M (λ ± iε) ε→0+



Γ(1 − α)2 e∓iπα 2α κ  4α 2  = − sin(πα)  π  0

 0 Γ(α)2 e∓iπ(1−α) 2(1−α) κ 41−α

  . 

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Note that M (λ+ ) = M (λ− )∗ . Finally, the most elaborated limit is calculated in the next lemma. Lemma 9. For m ∈ Z, ν ∈ (0, 1) and f ∈ Cc∞ (R+ ) one has √ lim ε(X 2 − z)−1 Fm Hν(1) ( z¯·), f R+ = ieiπν/2 (−1)|m| f (κ).

ε→0+

Proof. Let us start by recalling that for w ∈ C satisfying − π2 < arg(w) ≤ π one has [1, Eq. (9.6.4)]: Hν(1) (w) =

2 −iπν/2 e Kν (−iw), iπ

where Kν is the modiﬁed Bessel function of the second kind and of order ν. Then, for r ∈ R+ it follows that (by using [29, Sec. 13.45] for the last equality) √ [Fm Hν(1) ( z¯·)](r) = (−i)|m| l.i.m.

R+

√ ρJ|m| (rρ)Hν(1) ( z¯ρ)dρ

2 = (−i)|m| e−iπν/2 l.i.m. iπ

R+

√ ρJ|m| (rρ)Kν (−i z¯ρ)dρ

√ 2 −iπν/2 1 z¯ e ρ dρ = (−i) l.i.m. ρJ|m| (ρ)Kν −i iπ r2 r R+ √ −2 √ −2−|m| 1 |m| − ν z¯ z¯ |m| + ν = c 2 −i + 1, + 1; |m| + 1; − −i 2 F1 r r 2 2 r |m|

where 2 F1 is the Gauss hypergeometric function [1, Chap. 15] and c is given by |m| − ν |m| + ν +1 Γ +1 Γ 2 2 2 . c := (−i)|m| e−iπν/2 iπ Γ(|m| + 1) √

Now, observe that (−i rz¯ )−2−|m| = −(−i)−|m|( Thus, one has obtained √ 1 z¯·)](r) = d 2 r

[Fm Hν(1) (

√

z¯ −2−|m| r )

and −(−i

=

r2 z¯ .

√ −2−|m| |m| + ν |m| − ν r2 z¯ + 1, + 1; |m| + 1; 2 F1 r 2 2 z¯

with d=−

√ z¯ −2 r )

2 −iπν/2 e iπ

Γ

|m| − ν |m| + ν +1 Γ +1 2 2 . Γ(|m| + 1)

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By taking into account [1, Equality (15.3.3)] one can isolate from the 2 F1 -function a factor which is singular when the variable goes to 1: 2 F1

|m| − ν r2 |m| + ν + 1, + 1; |m| + 1; 2 2 z¯ |m| + ν |m| − ν r2 1 , ; |m| + 1; F = 2 1 1 − r2 z¯−1 2 2 z¯ |m| + ν |m| − ν r2 z¯ , ; |m| + 1; F =− 2 . 2 1 r − z¯ 2 2 z¯

Altogether, one has thus obtained: √ ε[(X 2 − z)−1 Fm Hν(1) ( z¯·)](r) √ −2−|m| z¯ z¯ ε = −d 2 (r − z¯)(r2 − z) r2 r

2 F1

|m| + ν |m| − ν r2 , ; |m| + 1; . 2 2 z¯

Now, observe that ε ε = 2 (r2 − z¯)(r2 − z) (r − λ + iε)(r2 − λ − iε) ε =: πδε (r2 − λ) = 2 (r − λ)2 + ε2 which converges to πδ(r2 − λ) in the sense of distributions on R as ε goes to 0. Furthermore, the map R+ r → 2 F1

|m| + ν |m| − ν r2 , ; |m| + 1; 2 2 λ − iε

∈C

is locally uniformly convergent as ε → 0 to a continuous function which is equal √ + 1)Γ( |m|−ν + 1)]−1 . By considering trivial for r = κ = λ to Γ(|m| + 1)[Γ( |m|+ν 2 2 extensions on R, it follows that √ lim ε(X 2 − z)−1 Fm Hν(1) ( z¯·), f R+

ε→0+

Z ¯ lim = −dπ

ε→0+

=−

R+

rδε (r 2 − λ)

¯ dπ κ(−1)−|m| 2κ

„ Γ

= ieiπν/2 (−1)|m| f (κ).

z¯ r2

« „ √ «−2−|m| „ r2 |m| + ν |m| − ν z¯ , ; |m| + 1; f (r)dr 2 F1 r 2 2 z¯

Γ(|m| + 1) « „ « f (κ) |m| − ν |m| + ν +1 Γ +1 2 2

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By adding these diﬀerent results and by taking Lemma 6 into account, one has thus proved: Lemma 10. For any f of the form m∈Z fm φm with fm = 0 except for a finite number of m for which fˆm ∈ Cc∞ (R+ ) and for any g ∈ Cc∞ (R2 \{0}), one has ε lim − (H0 − λ + iε)−1 f, γ(λ − iε)(DM (λ − iε) − C)−1 Dγ(λ + iε)∗ g π (1) T Hα (κ·)φ0 1 A(λ+ )(DM (λ+ ) − C)−1 DA(λ− )∗ =− (1) π H1−α (κ·)φ−1 ieiπα/2 fˆ0 (κ) ,g . × −ieiπ(1−α)/2 fˆ−1 (κ)

ε→0+

Before stating the main result on ΩCD − , let us ﬁrst present the explicit form of AB the stationary wave operator Ω− . Note that for this operator the equality between the time dependent approach and the stationary approach is known [2, 13, 26], and that a preliminary version of the following result has been given in [25]. So, let us observe that since the operator HαAB leaves each subspace Hm invariant [26], it AB gives rise to a sequence of channel operators Hα,m acting on Hm . The Laplacian AB H0 admitting a similar decomposition, the stationary wave operators Ω ± can be deﬁned in each channel, i.e. separately for each m ∈ Z. Let us immediately observe that the angular part does not play any role for deﬁning such operators. Therefore, we shall omit it as long as it does not lead to any confusion, and consider the AB from Hr to Hr . channel wave operators Ω ±,m The following notation will be useful: T := {z ∈ C | |z| = 1} and  1   − 2 πα if m ≥ 0 1 α . δm = π(|m| − |m + α|) =  2   1 πα if m < 0 2 Proposition 11. For each m ∈ Z, one has ± AB ΩAB ±,m = Ω±,m = ϕm (A),

with ϕ± m ∈ C([−∞, +∞], T) given explicitly by 1 1 (|m| + 1 + ix) Γ (|m + α| + 1 − ix) Γ α 2 2 ∓iδm ϕ± m (x) := e 1 1 (|m| + 1 − ix) Γ (|m + α| + 1 + ix) Γ 2 2 α

± ∓2iδm and satisfying ϕ± . m (±∞) = 1 and ϕm (∓∞) = e

(14)

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Proof. As already mentioned, the ﬁrst equality in proved in [26]. Furthermore it is also proved there that for any f ∈ Hr and r ∈ R+ one has α |m| [ΩAB f ](r) = i l.i.m. κ, J|m+α| (κr)e∓iδm [Fm f ](κ)dκ. ±,m R+

In particular, if f ∈

Cc∞ (R+ ), α

s-lim e∓iδm

N →∞

0

= s-lim e

this expression can be rewritten as ∞ N κJ|m+α| (κr) sJ|m| (sκ)f (s)ds dκ

α ∓iδm

N →∞

α ∓iδm

N →∞

=e

0

∞

0

∞

0

∞ 0

s r

0

s f (s) r

∞

N

sf (s) 0

= s-lim e α ∓iδm

0

κJ|m| (sκ)J|m+α| (κr)dκ ds Nr

κJ|m| ( rs κ)J|m+α| (κ)dκ

ds r

s ds κ J|m+α| (κ)dκ f (s) , κJ|m| r r

(15)

where the last term has to be understood in the sense of distributions on R+ . The distribution between square brackets has been computed in [20, Proposition 2] but we shall not use here its explicit form. Now, by comparing (15) with (13), one observes that the channel wave operator ± ± ΩAB ±,m is equal on a dense set in Hr to ϕm (A) for a function ϕm whose inverse Fourier transform is the distribution which satisﬁes for y ∈ R: ∞ √ α ± ∓iδm −y −y e κJ|m| (e κ)J|m+α| (κ)dκ . ϕˇm (y) = 2πe 0

The Fourier transform of this distribution can be computed. Explicitly one has (in the sense of distributions): α ∓iδm e−ixy e−y κJ|m| (e−y κ)J|m+α| (κ)dκ dy ϕ± m (x) = e α

= e∓iδm

R

R+

R+

κ(1−ix)−1 J|m+α| (κ)dκ

R+

s(1+ix)−1 J|m| (s)ds

which is the product of two Mellin transforms. The explicit form of these transforms are presented in [22, Eq. (10.1)] and a straightforward computation leads directly to the expression (14). The second equality of the statement follows then by a density argument. The additional properties of ϕ± m can easily be obtained by taking into account the equality Γ(¯ z ) = Γ(z) valid for any z ∈ C as well as the asymptotic development of the function Γ as presented in [1, Eq. (6.1.39)]. Since the wave operators ΩAB admit a decomposition into channel wave ± operators, so does the scattering operator. The channel scattering operator

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AB ∗ AB Sm := (ΩAB +,m ) Ω−,m , acting from Hr to Hr , is simply given by α

AB − 2iδ Sm = ϕ+ m (A)ϕm (A) = e m .

Now, let us set Hint := H0 ⊕ H−1 which is clearly isomorphic to Hr ⊗ C2 , and ⊥ . It follows from the considerations of consider the decomposition H = Hint ⊕ Hint CD Sec. 2 that for any pair (C, D) the operator Ω ± is reduced by this decomposition CD | ⊥ = ΩCD | ⊥ = ΩAB | ⊥ . Since the form of ΩAB has been exposed and that Ω − Hint − Hint − Hint − CD to Hint . For that purpose, above, we shall concentrate only of the restriction of Ω −

let us deﬁne a matrix valued function which is closely related to the scattering operator. For κ ∈ R+ we set   Γ(1 − α)e−iπα/2 α κ 0   2α   SαCD (κ) := 2i sin(πα)   Γ(α)e−iπ(1−α)/2 (1−α) 0 κ 21−α    −1 Γ(1 − α)2 e−iπα 2α κ 0     4α π     · D C D + 2 sin(πα)     2 −iπ(1−α) Γ(α) e 2(1−α) 0 κ 41−α   Γ(1 − α)e−iπα/2 α κ 0   2α   (16) · .   −iπ(1−α)/2 Γ(α)e 0 − κ(1−α) 21−α Theorem 12. For any pair (C, D) satisfying (7), the restriction of the wave operator ΩCD − to Hint satisfies the equality − ϕ0 (A) ϕ˜0 (A) 0 0 CD CD − |H = SαCD ( H0 ), Ω− |Hint = Ω + int − 0 ϕ−1 (A) 0 ϕ˜−1 (A) (17) where ϕ˜m ∈ C([−∞, +∞], C) for m ∈ {0, −1}. Explicitly, for every x ∈ R, ϕ˜m (x) is given by 1 (|m| + 1 + ix) Γ 1 1 −iπ|m|/2 πx/2 2 Γ e (1 + |m + α| − ix) e 1 2π 2 Γ (|m| + 1 − ix) 2 1 ×Γ (1 − |m + α| − ix) 2 and satisfies ϕ˜m (−∞) = 0 and ϕ˜m (+∞) = 1.

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CD is deﬁned by the formula [31, DeﬁProof. (a) The stationary representation Ω − nition 2.7.2]: ∞ ε CD f, g = lim (H0 − λ + iε)−1 f, (HαCD − λ + iε)−1 gdλ Ω − ε→0+ π 0 for any f of the form m∈Z fm φm with fm = 0 except for a ﬁnite number of m for which fˆm ∈ Cc∞ (R+ ) and g ∈ Cc∞ (R2 \{0}). By taking Krein resolvent formula into account, we can ﬁrst consider the expression ∞ ε lim (H0 − λ + iε)−1 f, (HαAB − λ + iε)−1 gdλ ε→0+ π 0 which converges to [2, 13, 26]:

∞

0

i

α |m| iδm

e

ˆ J|m+α| (κ·)fm (κ)φm , g κ dκ.

m∈Z

This expression was the starting point for the formulae derived in Proposition 11. This leads to the ﬁrst term in the right-hand side of (17). (b) The second term to analyze is −

∞

ε (H0 − λ + iε)−1 f, γ(λ − iε)(DM (λ − iε) − C)−1 Dγ(λ + iε)∗ gdλ. ε→0+ π (18) lim

0

By using Lemma 10 and by performing some simple calculations, one obtains that (18) is equal to 0

 ∞

  1 2

T

1 α (1) i Hα (κ·)φ0 2 (1)

i1−α H1−α (κ·)φ−1

 CD fˆ0 (κ)  S (κ) , g κ dκ.  α fˆ−1 (κ)

Now, it will be proved below that the operator Tm deﬁned for m ∈ {0, −1} on F ∗ [Cc∞ (R+ )] by 1 [Tm f ](r) := i|m+α| 2

0

∞

(1)

H|m+α| (κr)[Fm f ](κ)κ dκ

(19)

satisﬁes the equality Tm = ϕ˜m (A) with ϕ˜m given in the above statement. The stationary expression is then obtained by observing that F ∗ SαCD (k)F = √ SαCD ( H0 ), where SαCD (k) is the operator of multiplication by the function SαCD (·). Finally, the equality between the time dependent wave operator and the stationary wave operator is a consequence of Lemma 8 and of [31, Theorem 5.2.4].

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(c) By comparing (19) with (13), one observes that the operator Tm is equal on a dense set in Hr to ϕ˜m (A) for a function ϕ˜m whose inverse Fourier transform is the distribution which satisﬁes for y ∈ R: √ α (1) ˇ˜m (y) = 1 2πe−iδm ϕ ey κH|m+α| (ey κ)J|m| (κ)dκ. 2 R+ As before, the Fourier transform of this distribution can be computed. Explicitly one has (in the sense of distributions): α 1 −iδm (1) −ixy y y ϕ˜m (x) = e e e κH|m+α| (e κ)J|m| (κ)dκ dy 2 R R+ α 1 −iδm (1) (1+ix)−1 = e κ J|m| (κ)dκ s(1−ix)−1 H|m+α| (s)ds 2 R+ R+ 1 (|m| + 1 + ix) Γ 1 −iπ|m|/2 2 e = (−i)ix 1 2π Γ (|m| + 1 − ix) 2 1 1 (1 + |m + α| − ix) Γ (1 − |m + α| − ix) . ·Γ 2 2 The last equality is obtained by taking into account the relation between the (1) Hankel function Hν and the Bessel function Kν of the second kind as well as the Mellin transform of the functions Jν and the function Kν as presented in [22, Eqs. (10.1) and (11.1)]. The additional properties of ϕ˜m can easily be obtained by using the asymptotic development of the function Γ as presented in [1, Eq. (6.1.39)]. 6. Scattering Operator In this section, we concentrate on the scattering operator and on its asymptotic values for large and small energies. Proposition 13. The restriction of the scattering operator S(HαCD , H0 ) to Hint is explicitly given by −iπα 0 e + SαCD (κ). S(HαCD , H0 )|Hint = SαCD ( H0 ) with SαCD (κ) := 0 eiπα Proof. Let us ﬁrst recall that the scattering operator can be obtained from ΩCD − by the formula [6, Proposition (4.2)]: CD s-lim eitH0 e−itH ΩCD − = S(Hα , H0 ). t→+∞

We stress that the completeness has been taken into account for this equality. Now, let us set U (t) := e−it ln(H0 )/2 , where ln(H0 ) is the self-adjoint operator obtained

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by functional calculus. By the intertwining property of the wave operators and by the invariance principle, one also has CD s-lim U (−t)ΩCD − U (t) = S(Hα , H0 ). t→+∞

On the other hand, the operator ln(H0 )/2 is the generator of translations in the spectrum of A, i.e. U (−t)ϕ(A)U (t) = ϕ(A + t) for any ϕ: R → C. Since {U (t)}t∈R is also reduced by the decomposition (1), it follows that s-lim U (−t) ΩCD − |Hint U (t) t→+∞

− ϕ0 (A) = s-lim U (−t) t→+∞ 0

0

ϕ˜0 (A)

0

SαCD (

H0 ) U (t) + ϕ− 0 ϕ˜−1 (A) −1 (A) − 0 ϕ˜0 (+∞) ϕ0 (+∞) 0 SαCD ( H0 ). = + − 0 ϕ−1 (+∞) 0 ϕ˜−1 (+∞) The initial statement is then obtained by taking the asymptotic values mentioned in Proposition 11 and Theorem 12 into account. Even if the unitarity of the scattering operator follows from the general theory we give below a direct veriﬁcation in order to better understand its structure. In the next statement, we only give the value of the scattering matrix at energy 0 and energy equal to +∞. However, more explicit expressions for SαCD (κ) are exhibited in the proof. Proposition 14. The map R+ κ → SαCD (κ) ∈ M2 (C)

(20)

is continuous, takes values in the set U (2) and has explicit asymptotic values for κ = 0 and κ = +∞. More explicitly, depending on C, D or α one has: 0 " , eiπα ! iπα 0 " then SαCD (+∞) = e 0 e−iπα , CD = 1 and α = 1/2, then Sα (+∞) ⊥

(i) If D = 0, then SαCD (κ) = (ii) If det(D) = 0, (iii) If dim[ker(D)]

!e−iπα 0

! = (2P − 1) 0i

0" −i ,

where P

is the orthogonal !0" ! " projection onto ker(D) ,

(iv) If ker(D) = C0 or if dim[ker(D)] = 1, α < 1/2 and ker(D) = C , then " ! −iπα e 0 SαCD (+∞) = , 0 e−iπα !0" !C" (v) If ker(D) = C or if dim[ker(D)] = 1, α > 1/2 and ker(D) =

0 , then " ! iπα e 0 SαCD (+∞) = 0 eiπα .

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Furthermore, 0 " , e−iπα ! −iπα 0 " , SαCD (0) = e 0 eiπα and α = 1/2, then SαCD (0) ⊥

(a) If C = 0, then SαCD (0) = (b) If det(C) = 0, then

!eiπα 0

! 0" , where Π is = (1 − 2Π) 0i −i (c) If dim[ker(C)] = 1 the orthogonal! projection on ker(C) . " ! " (d) If ker(C) = C0 or if dim[ker(C)] = 1, α > 1/2 and ker(C) = C0 , then ! −iπα 0 " SαCD (0) = e 0 , e−iπα ! " !C" (e) If ker(C) = 0 or if dim[ker(C)] = 1, α < 1/2 and ker(C) = C0 , then ! iπα 0 " SαCD (0) = e 0 eiπα . Proof. Let us ﬁx κ > 0 and set S := SαCD (κ). For shortness, we also set L := π 2 sin(πα) C and 



Γ(1 − α) α κ  2α B = B(κ) :=   0

Φ :=

e−iπα/2 0

0

 , 

Γ(α) (1−α) κ 21−α 0 1 0 , J := . 0 −1 e−iπ(1−α)/2

Note that the matrices B, Φ and J commute with each other, that the matrix B is self-adjoint and invertible, and that J and Φ are unitary. (I) It is trivially checked that if D = 0 the statement (i) is satisﬁed. (II) Let us assume det(D) = 0, i.e. D is invertible. Without loss of generality and as explained at the end of Sec. 3, we assume than that D = 1 and that the matrix C is self-adjoint. Then one has S = Φ2 J + 2i sin(πα)BΦ(B 2 Φ2 + L)−1 BΦJ = BΦ(B 2 Φ2 + L)−1 [B(Φ2 + 2i sin(πα)) + LB −1 ]ΦJ. By taking the equality Φ2 + 2i sin(πα) = Φ−2 into account, it follows that S = BΦ(B 2 Φ2 + L)−1 (BΦ−2 + LB −1 )ΦJ = Φ(Φ2 + B −1 LB −1 )−1 (Φ−2 + B −1 LB −1 )ΦJ = Φ(B −1 LB −1 + cos(πα)J − i sin(πα))−1 × (B −1 LB −1 + cos(πα)J + i sin(πα))ΦJ.

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Since the matrix B −1 LB −1 + cos(πα)J is self-adjoint, the above expression can be rewritten as S=Φ

B −1 LB −1 + cos(πα)J + i sin(πα) ΦJ B −1 LB −1 + cos(πα)J − i sin(πα)

(21)

which is clearly a unitary operator. The only dependence on κ in the terms B is continuous and one has iπα cos(πα)J + i sin(πα) e 0 CD ΦJ = lim S (κ) = Φ κ→+∞ α 0 e−iπα cos(πα)J − i sin(πα) which proves the statement (ii) (III) We shall now consider the situation det(D) = 0 but D = 0. Obviously, ker(D) is of dimension 1. So let p = (p1 , p2 ) be a vector in ker(D) with p = 1. By (12) and by using the notation introduced in that section one has S = Φ2 J + 2i sin(πα)BΦI(P B 2 Φ2 I + )−1 P BΦJ.

(22)

Note that the matrix of P := IP : C2 → C2 , i.e. the orthogonal projection onto p⊥ , is given by |p2 |2 −p1 p¯2 P= −¯ p1 p2 |p1 |2 and that PB 2 Φ2I is just the multiplication by the number c(κ) = b21 (κ)|p2 |2 e−iπα − b22 (κ)|p1 |2 eiπα ,

(23)

Γ(α) (1−α) α . with b1 (κ) = Γ(1−α) 2α κ and b2 (κ) = 21−α κ In the special case α = 1/2, the matrices B and Φ have the special form B = π2 κ1/2 and φ = e−iπ/4 . Clearly, one also has b1 = b2 = π2 κ1/2 := b and c(κ) = −ib2 . In that case, the expression (22) can be rewritten as πκ/2 − i P + (P − 1) J (24) S=i πκ/2 + i

which is the product of unitary operators and thus is unitary. Furthermore, the dependence in κ is continuous and the asymptotic value is easily determined. This proves statement (iii) If α = 1/2, let us rewrite S as S = Φ(c(κ) + )−1 [2i sin(πα)BPB + c(κ) + ]ΦJ.

(25)

Furthermore, by setting X− := (b21 |p2 |2 −b22 |p1 |2 ) and X+ := (b21 |p2 |2 +b22 |p1 |2 ) one has c(κ) + = cos(πα)X− + − i sin(πα)X+

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and M := 2i sin(πα)BPB + c(κ) + −2i sin(πα)b1 b2 p1 p¯2 eiπα X− + = . − 2i sin(πα)b1 b2 p¯1 p2 e−iπα X− + With these notations, the unitary of S easily follows from the equality det(M ) = |c(κ)+ |2 . The continuity in κ of all the expressions also implies the expected continuity of the map (20). Finally, by taking (23) and the explicit form of M into account, the asymptotic values of SαCD (κ) for the cases (iv) and (v) can readily be obtained. (IV) Let us now consider the behavior of the scattering matrix near the zero energy. If C = 0, then det(D) = 0 and one can use (21) with L = 0. The statement (a) follows easily. (V) Assume that det(C) = 0. In this case, it directly follows from (16) that ! −iπα 0 " which proves (b). SαCD (0) = 0, and then S(0) = e 0 eiπα (VI) We now assume that dim[ker(C)] = 1 and consider two cases. Firstly, if det(D) = 0 we can assume as in (II) that D = 1 and that C is self-adjoint and use again (21). Introducing the entries of L, l11 l12 L= l12 l22 one obtains B −1 LB −1 + cos(πα)J + i sin(πα) B −1 LB −1 + cos(πα)J − i sin(πα) =

b21 l22 e−iπα ·

1 − b22 l11 eiπα − b21 b22

!

b21 l22 eiπα − b22 l11 eiπα − b21 b22 e2iπα

b1 b2 l12 (e−iπα − eiπα )

b1 b2 l12 (e−iπα − eiπα )

b21 l22 e−iπα − b22 l11 e−iπα − b21 b22 e−2iπα

.

For α = 1/2 one easily obtains the result stated in (d) and (e). For α = 1/2, it follows that B −1 LB −1 + cos(πα)J + i sin(πα) 2 L − 1, = κ→0+ B −1 LB −1 + cos(πα)J − i sin(πα) tr(L) lim

and it only remains to observe that L = tr(L)Π, where Π is the orthogonal projection on ker(L)⊥ = ker(C)⊥ . This proves (c). Secondly, let us assume that dim[ker(D)] = 1. By (11), there exists U ∈ U (2) such that ker(C) = ker(1 − U ) and ker(D) = ker(1 + U ). As a consequence, one has ker(C) = ker(D)⊥ and then P = 1 − Π. On the other hand, we can use the expressions for the scattering operator obtained in (III).

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However, observe that CI = C|ker(D)⊥ = C|ker(C) = 0 so we only have to consider these expressions in the special case = 0. The asymptotic at 0 energy are then easily deduced from these expressions. By summing the results obtained for det(D) = 0 and for dim[ker(D)] = 1, and since D = 0 is not allowed if det(C) = 0, one proves the cases (c), (d) and (e). Remark 15. As can be seen from the proof, the scattering matrix is independent of the energy in the following cases only: ! −iπα 0 " , • D = 0, then SαCD (κ) = e 0 eiπα " ! iπα e 0 • C = 0, then SαCD (κ) = 0 e−iπα , see (21), ! " ! iπα 0 " , see (25), • ker(C) = ker(D)⊥ = C0 , then SαCD (κ) = e 0 eiπα " " ! ! −iπα 0 e 0 • ker(C) = ker(D)⊥ = C , then SαCD (κ) = , see (25), 0 e−iπα ! 0" CD • α = 1/2 and det(C) = det(D) = 0, then Sα (κ) = (2P − 1) 0i −i , where P is the orthogonal projection on ker(D)⊥ ≡ ker(C), see (24).

7. Final Remarks As mentioned before, the parametrization of the self-adjoint extensions of Hα with the pair (C, D) satisfying (7) is highly non unique. For the sake of convenience, we recall here a one-to-one parametrization of all self-adjoint extensions and reinterpret a part of the results obtained before in this framework. So, let U ∈ U (2) and set C = C(U ) :=

i 1 (1 − U ) and D = D(U ) = (1 + U ). 2 2

(26)

It is easy to check that C and D satisfy both conditions (7). In addition, two diﬀerent elements U, U of U (2) lead to two diﬀerent self-adjoint operators HαCD and HαC D with C = C(U ), D = D(U ), C = C(U ) and D = D(U ), cf. [17]. Thus, without ambiguity we can write HαU for the operator HαCD with C, D given by (26). Moreover, the set {HαU | U ∈ U (2)} describes all self-adjoint extensions of Hα , and, by (10), the map U → HαU is continuous in the norm resolvent topology. Let us ﬁnally mention that the normalization of the above map has been chosen such that Hα−1 ≡ Hα10 = HαAB . Obviously, we could use various parametrizations for the set U (2). For example, one could set a −b U = U (η, a, b) = eiη b a with η ∈ [0, 2π) and a, b ∈ C satisfying |a|2 + |b|2 = 1, which is the parametrization used in [2] (note nevertheless that the role of the unitary parameter was quite

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diﬀerent). We could also use the parametrization inspired by [13]: −(1 − q 2 )1/2 e−ib qeia iω U = U (ω, a, b, q) = e (1 − q 2 )1/2 eib qe−ia with ω, a, b ∈ [0, 2π) and q ∈ [0, 1]. However, the following formulae look much simpler without such an arbitrary choice, and such a particularization can always be performed later on. We can now rewrite part of the previous results in terms of U : Lemma 16. Let U ∈ U (2). Then, (i) For z ∈ ρ(HαAB ) ∩ ρ(HαU ) the resolvent equation holds: z )∗ , (HαU − z)−1 − (HαAB − z)−1 = −γ(z)[(1 + U )M (z) + i(1 − U )]−1 (1 + U )γ(¯ (ii) The number of negative eigenvalues of HαU coincides with the number of negative eigenvalues of the matrix i(U − U ∗ ), (iii) The value z ∈ R− is an eigenvalue of HαU if and only if det((1 + U )M (z) + i(1 − U )) = 0, and in that case one has ker(HαU − z) = γ(z) ker((1 + U )M (z) + i(1 − U )). The wave operators can also be rewritten in terms of the single parameter U . We shall not do it here but simply express the asymptotic values of the scattering operator SαU := S(HαU , H0 ) in terms of U . If λ ∈ C is an eigenvalue of U , we denote by Vλ the corresponding eigenspace. Proposition 17. One has: ! −iπα 0 " (i) If U = −1, then SαU (κ) ≡ SαAB = e 0 , eiπα !eiπα 0 " U (ii) If −1 ∈ σ(U ), then Sα (+∞) = 0 e−iπα ,

(iii) !If −1 "∈ σ(U ) with multiplicity one and α = 1/2, then SαU (+∞) = (2P − 1) i 0 ⊥ 0 −i , where P is the orthogonal projection onto V−1 , ! " ! " (iv) If V−1 = C0 or if −1 ∈ σ(U ) with multiplicity one, α < 1/2 and V−1 = C0 , ! −iπα 0 " then SαU (+∞) = e 0 , e−iπα !0" ! " (v) If V−1 = C or if −1 ∈ σ(U ) with multiplicity one, α > 1/2 and V−1 = C0 , ! iπα 0 " . then SαU (+∞) = e 0 eiπα Furthermore, (a) If U = 1, then SαU (0) =

!eiπα

(b) If 1 ∈ σ(U ), then SαU (0) =

0 " , e−iπα !e−iπα 0 " , 0 eiπα

0

! (c) If 1 ∈ σ(U ) with multiplicity one and α = 1/2, then SαU (0) = (1 − 2Π) 0i where Π is the orthogonal projection on V1⊥ .

0" −i ,

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(d) If V1 = (e)

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!0"

C or if 1 ∈ σ(U ) ! −iπα 0 " , SαU (0) = e 0 e−iπα !C" If V1 = 0 or if 1 ∈ σ(U ) ! iπα 0 " SαU (0) = e 0 eiπα .

with multiplicity one, α > 1/2 and V1 =

!C" 0 , then

with multiplicity one, α < 1/2 and V1 =

!0" C , then

Remark 18. The scattering matrix is independent of the energy in the following cases only: ! −iπα 0 " , • U = −1, then SαU (κ) ≡ SαAB = e 0 eiπα " ! iπα 0 • U = 1, then SαU (κ) = e 0 e−iπα , see (21), !1 0 " ! iπα e 0 " , see (25), • U = 0 −1 , then SαU (κ) = 0 eiπα !−1 0" !e−iπα 0 " U • U = 0 1 , then Sα (κ) = , see (25), 0 e−iπα ! 0" • α = 1/2 and σ(U ) = {−1, 1}, then SαU = (2P − 1) 0i −i , where P is the orthogonal projection on V1 , see (24). Acknowledgment S. Richard is supported by the Swiss National Science Foundation. References [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, Vol. 55 (U.S. Government Printing Oﬃce, Washington, D.C., 1964). [2] R. Adami and A. Teta, On the Aharonov–Bohm Hamiltonian, Lett. Math. Phys. 43 (1998) 43–54. [3] Y. Aharonov and D. Bohm, Signiﬁcance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959) 485–491. [4] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, with an Appendix by P. Exner, 2nd edn. (AMS, Providence, Rhode Island, 2005). [5] S. Albeverio and K. Pankrashkin, A remark on Krein’s resolvent formula and boundary conditions, J. Phys. A 38 (2005) 4859–4864. [6] W. O. Amrein, J. M. Jauch and K. B. Sinha, Scattering Theory in Quantum Mechanics, Lecture Notes and Supplements in Physics, Vol. 16 (W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1977). [7] M. Ballesteros and R. Weder, High-velocity estimates for the scattering operator and Aharonov–Bohm eﬀect in three dimensions, Comm. Math. Phys. 285 (2009) 345–398. [8] M. Ballesteros and R. Weder, Aharonov–Bohm eﬀect and Tonomura et al. experiments, rigorous results, J. Math. Phys. 50 (2009) 122108, 54 pp. [9] M. V. Berry, Aptly named Aharonov–Bohm eﬀect has classical analogue, long history, Phys. Today 63 (2010) 8 pp. [10] L. Bruneau, J. Derezi´ nski and V. Georgescu, Homogeneous Schr¨ odinger operators on half-line, preprint; arXiv: 0911.5569.

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[11] J. Br¨ uning and V. Geyler, Scattering on compact manifolds with inﬁnitely thin horns, J. Math. Phys. 44 (2003) 371–405. [12] J. Br¨ uning, V. Geyler and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schr¨ odinger operators, Rev. Math. Phys. 20 (2008) 1–70. ˇˇtov´ıˇcek, Aharonov–Bohm eﬀect with δ-type interaction, J. [13] L. D¸abrowski and P. S Math. Phys. 39 (1998) 47–62. [14] V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991) 1–95. ˇˇtov´ıˇcek and P. Vytˇras, Generalised boundary conditions for the [15] P. Exner, P. S Aharonov–Bohm eﬀect combined with a homogeneous magnetic ﬁeld, J. Math. Phys. 43 (2002) 2151–2168. [16] N. Goloshchapova and L. Oridoroga, The one-dimensional Schr¨ odinger operator with point δ- and δ -interactions, Math. Notes 84 (2008) 125–129. [17] M. Harmer, Hermitian symplectic geometry and extension theory, J. Phys. A 33 (2000) 9193–9203. [18] A. Jensen, Time-delay in potential scattering theory, some “geometric” results, Comm. Math. Phys. 82(3) (1981/82) 435–456. [19] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, reprint of the 1980 edition (Springer-Verlag, Berlin, 1995). [20] J. Kellendonk and S. Richard, Weber–Schafheitlin type integrals with exponent 1, Int. Transf. Spec. Funct. 20 (2009) 147–153. [21] O. Lisovyy, Aharonov–Bohm eﬀect on the Poincar´e disk, J. Math. Phys. 48 (2007) 052112, 17 p. [22] F. Oberhettinger, Tables of Mellin Transforms (Springer-Verlag, 1974). [23] K. Pankrashkin, Resolvents of self-adjoint extensions with mixed boundary conditions, Rep. Math. Phys. 58 (2006) 207–221. [24] A. Posilicano, Self-adjoint extensions of restrictions, Oper. Matrices 2 (2008) 483–506. [25] S. Richard, New formulae for the Aharonov–Bohm wave operators, in Spectral and Scattering Theory for Quantum Magnetic Systems, Contemporary Mathematics, Vol. 500 (AMS, Providence, Rhode Island, 2009), pp. 159–168. [26] S. N. M. Ruijsenaars, The Aharonov–Bohm eﬀect and scattering theory, Ann. Phys. 146(1) (1983) 1–34. [27] H. Tamura, Magnetic scattering at low energy in two-dimensions, Nagoya Math. J. 155 (1999) 95–151. [28] H. Tamura, Norm resolvent convergence to magnetic Schr¨ odinger operators with point interactions, Rev. Math. Phys. 13(4) (2001) 465–511. [29] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge, England, 1966). [30] R. Weder, The Aharonov–Bohm eﬀect and time-dependent inverse scattering theory, Inverse Problems 18 (2002) 1041–1056. [31] D. R. Yafaev, Mathematical Scattering Theory. General Theory, Translations of Mathematical Monographs, Vol. 105 (American Mathematical Society, Providence, RI, 1992).

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Reviews in Mathematical Physics Vol. 23, No. 1 (2011) 83–125 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004230

PERTURBATION OF NEAR THRESHOLD EIGENVALUES: CROSSOVER FROM EXPONENTIAL TO NON-EXPONENTIAL DECAY LAWS

VICTOR DINU CAQP, Faculty of Physics, University of Bucharest, P. O. Box MG 11, RO-077125 Bucharest, Romania ARNE JENSEN Department of Mathematical Sciences, Aalborg University, Fr. Bajers Vej 7G, DK-9220 Aalborg Ø, Denmark [email protected] GHEORGHE NENCIU Institute of Mathematics of the Romanian Academy, P. O. Box 1-764, RO-014700 Bucharest, Romania and Department of Mathematical Sciences, Aalborg University, Fr. Bajers Vej 7G, DK-9220 Aalborg Ø, Denmark Received 6 June 2010 For a two-channel model of the form " # " Hop 0 0 +ε Hε = 0 E0 W21

W12 0

# on H = Hop ⊕ C,

appearing in the study of Feshbach resonances, we continue the rigorous study, begun in our paper (J. Math. Phys. 50 (2009) 013516), of the decay laws for resonances produced by perturbation of unstable bound states close to a threshold. The operator Hop is assumed to have the properties of a Schr¨ odinger operator in odd dimensions, with a threshold at zero. We consider for ε small the survival probability |Ψ0 , e−itHε Ψ0 |2 , where Ψ0 is the eigenfunction corresponding to E0 for ε = 0. For E0 in a small neighborhood of the origin independent of ε, the survival probability amplitude is expressed in terms of some special functions related to the error function, up to error terms vanishing as ε → 0. This allows for a detailed study of the crossover from exponential to non-exponential decay laws, and then to the bound state regime, as the position of the resonance is tuned across the threshold. Keywords: Decay laws; exponential decay; non-exponential decay; Fermi Golden Rule. Mathematics Subject Classiﬁcation 2010: 81Q10, 35B25, 35J10, 47A55, 47N50, 81U05

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1. Introduction The problem of the decay laws for resonances produced by perturbation of unstable bound states has a long and distinguished history in quantum mechanics. There is an extensive body of literature about decay laws for resonances in general, both at the level of theoretical physics (see e.g. [4, 10, 11, 22, 27–29] and references therein), and at the level of rigorous mathematical physics (see e.g. [3, 5–7, 9, 12, 16–19, 24, 31, 32] and references therein). It started with the computation by Dirac of the decay rate in second order time-dependent perturbation theory, leading to the well-known exponential decay law, e−Γt . Here Γ is given by the famous “Fermi Golden Rule” (FGR), Γ ∼ |Ψ0 , εW Ψcont,E0 |2 , where Ψ0 , E0 are the unperturbed bound state eigenfunction and energy, respectively, and Ψcont,E0 is the continuum “eigenfunction” degenerate in energy with the bound state. The FGR formula met with a fabulous success, and as a consequence, the common wisdom is that the decay law for the resonances produced by perturbation of non-degenerate bound states is exponential, at least in the leading non-trivial order in the perturbation strength (for degenerate bound states Rabi type exponentially decaying oscillations can appear). However, it has been known for a long time, at least for semi-bounded Hamiltonians, that the decay law cannot be purely exponential; there must be deviations at least at short and long times. This implies that, in more precise terms, the question is whether the decay law is exponential up to errors vanishing as the perturbation strength tends to zero. So at the rigorous level the crucial problem is the estimation of the errors. This proved to be a hard problem, and only during the past decades consistent rigorous results have been obtained. The generic result is that (see [3, 5, 12, 16, 24] and references therein) the decay law is indeed (quasi)exponential, i.e. exponential up to error terms vanishing in the limit ε → 0, as long as the resolvent of the unperturbed Hamiltonian is suﬃciently smooth, when projected onto the subspace orthogonal to the eigenvalue under consideration. For most cases of physical interest, this turns out to be the case, as long as the unperturbed eigenvalue lies in the continuum, far away from the energetic thresholds, and this explain the tremendous success of the FGR formula. The problem with the exponential decay law appears for bound states situated near a threshold, since, in this case, the projected resolvent might not be smooth, or may even blow up, when there is a zero resonancea at the threshold, see, e.g., [14–16] and references therein. As it has been pointed out in [2], at threshold the FGR formula does not apply. Moreover, the fact that the non-smoothness of the resolvent opens the possibility of a non-exponential decay at all times has been mentioned at the heuristic level [20, 22] (although this possibility for the non-degenerate case has been sometimes denied [25]). a To

clarify the terminology, H = −∆ + V on L2 (Rd ), d = 1, 2, 3, with V (x) = O(|x|−2−δ |) as |x| → ∞, is said to have a zero resonance, if HΦ = 0 has a solution, which is not in L2 (Rd ), but in a slightly larger space, see, e.g., [14, 15]. A zero resonance is also called a half-bound state.

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Let us mention that the question of the decay law for near threshold bound states is more than an academic one. While having the bound state in the very neighborhood of a threshold is a non-generic situation, recent advances in experimental technique have made it possible to realize this case for the so-called Feshbach resonances, where (with the aid of a magnetic ﬁeld) it is possible to tune the energy of the bound state (and then the resonance position) throughout a neighborhood of the threshold energy. The decay law for the case, when the resonance position is close to the threshold, has been considered at the rigorous level in [16–19]. More precisely, in [16–18], the threshold bound states were considered, but under the condition that the shift in the energy due to perturbation (see [16, (3.1)]) is suﬃciently large, such that the resonance position is at a distance of order ε from the threshold. In this case it turns out that the decay law is still exponential, but the FGR has to be modiﬁed. It is interesting to note that since in this case the ε dependence of the decay rate is a fractional power, the modiﬁed FGR cannot be obtained by naive perturbation theory. The other case, when the resonance position is very close to the threshold (in a neighborhood of the threshold, shrinking as ε → 0), has been considered in [8] for a two-channel model Hamiltonian with the structure used in Feshbach resonance theory [21, 30]. The main result is the proof at the rigorous level that for some energy ranges the decay law is deﬁnitely non-exponential. More precisely, we proved that the survival probability, up to some error terms vanishing in the limit ε → 0, can be written as an explicit integral, which has been analyzed numerically. The numerical study revealed that a remarkable variety, depending upon the values of the parameters involved, of diﬀerent decay laws appear: Close to an exponential one, deﬁnitely non-exponential, or bound state like. The present paper is a continuation of [8]. The setting is the same, but we add two important things. First, using an appropriate ansatz, close in the spirit to the well-known Lorentzian (Breit–Wigner) approximation for perturbed eigenvalues far from the threshold, but with a functional form taking into account the threshold behavior of the resolvent near threshold, we are able to cover a small ε-independent neighborhood of the threshold, improving at the same time the error term. Secondly, we express the approximated survival probability amplitude in terms of some special functions, related to the error function, replacing the exponential function in the decay law. As a result, we are able to obtain a rigorous and detailed description of the crossover of the decay law, as the resonance position is tuned through the threshold from positive to negative energies via tuning of E0 : Exponential decay with the usual FGR decay rate, to exponential decay with the modiﬁed FGR decay rate, then to non-exponential decay, and ﬁnally to bound state behavior. The contents of the paper is as follows. In Sec. 2, we recall from [8] the model Hamiltonian and its properties. Section 3 contains the guiding heuristics discussion, and a detailed description of the results. Section 4 contains the proofs. In the Appendix, we discuss the properties of the functions appearing in the expression

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for the approximate survival probability amplitude, and their relations with error function and related special functions. 2. Notation and Assumptions The setting is the same as in [8]. We repeat it below for the reader’s convenience. We develop the theory in a somewhat abstract setting, which is applicable to two channel Schr¨ odinger operators in odd dimensions, as they appear for example in the theory of Feshbach resonances (see, e.g., [21, 30], and references therein). Consider 0 Hop on H = Hop ⊕ Hcl . H= 0 Hcl In concrete cases Hop = L2 (R3 ) (or L2 (R+ ) in the spherically symmetric case), and Hop = −∆ + Vop with lim|x|→∞ Vop (x) = 0. Hop describes the “open” channel. As for the “closed” channel, one starts again with a Schr¨ odinger operator, but with lim|x|→∞ Vcl (x) = Vcl,∞ > 0. One assumes that Hcl has bound states below Vcl,∞ , which may be embedded in the continuum spectrum of Hop . Only these bound states are relevant for the problem at hand. Thus one can retain only one isolated eigenvalue (or a group of almost degenerate eigenvalues isolated from the rest of the spectrum); the inclusion of the rest of the spectrum of Hcl merely “renormalizes” the values of some coeﬃcients, without changing the qualitative picture. In this paper, we shall consider only non-degenerate eigenvalues, i.e. we shall take Hcl = E0 in Hcl = C, such that Hop 0 H= , (2.1) 0 E0 on

H = Hop ⊕ C =

ψ Ψ= ψ ∈ Hop , β ∈ C . β

Apart from the spectrum of Hop , H has a bound state 0 0 0 Ψ0 = , such that H = E0 . 1 1 1 The problem is to study the fate of E0 , when an interchannel perturbation 0 W12 εW = ε W21 0

(2.2)

(2.3)

is added to H, i.e. the total Hamiltonian is Hε = H + εW.

(2.4)

Throughout the paper we assume, without loss of generality, that ε > 0. For simplicity, we assume that W is a bounded self-adjoint operator on H.

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As already mentioned in the Introduction, the quantity to be studied is the so-called survival probability amplitude Aε (t) = Ψ0 , e−itHε Ψ0 .

(2.5)

As in [16, 8], we shall use the stationary approach to write down a workable formula for Aε (t). For this purpose, we use the Stone formula to express the compressed evolution in terms of the compressed resolvent, and then we use the Schur–Livsic– Feshbach–Grushin (SLFG) partition formula to express the compressed resolvent as an inverse (for details, further references, and historical remarks about the SLFG formula, the reader should refer to [16]). More precisely, by using the Stone formula and the SLFG formula, one arrives at the following basic formula for Aε (t), which often appears in the physics literature and is a particular case of the general formula in [16]. 1 ∞ −itx e Im F (x + iη, ε)−1 dx (2.6) Aε (t) = lim η0 π −∞ with F (z, ε) = E0 − z − ε2 g(z),

(2.7)

g(z) = Ψ0 , W Q∗ (Hop − z)−1 QW Ψ0 ,

(2.8)

where

and Q is the orthogonal projection on Hop , considered as a map from H to Hop . Since we are interested in the form of Aε (t), when E0 is near a threshold of Hop , we shall assume that 0 is a threshold of Hop , and that E0 is close to zero. The following assumption is imposed in the sequel and will not be repeated. Condition (iii) is imposed to exclude the trivial case. Assumption 2.1. (i) There exists a > 0, such that (−a, 0) ⊂ ρ(Hop ) (the resolvent set) and [0, a] ⊂ σess (Hop ). (ii) |E0 | ≤ 12 . (iii) We have QW Ψ0 = 0. From Assumption 2.1 and (2.8) we get the following result. Proposition 2.2. (i) g(z) is analytic in C\{(−∞, −a] ∪ [0, ∞)}. (ii) g(z) = g(z). (iii) g(z) is strictly increasing on (−a, 0). (iv) Im g(z) > 0 for Im z > 0. The aim of this paper is to consider at the rigorous mathematical physics level the problem of the decay law, for the case that E0 is tuned past the threshold. For that purpose we need assumptions about the behavior of the function g(z)

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in the neighborhood of the origin. In stating this assumption we use the notation from [8, 16] to facilitate reference to those papers. Assumption 2.3. For Re κ ≥ 0 and z ∈ C\[0, ∞) we let √ κ = −i z, z = −κ2 .

(2.9)

Let for a > 0 Da = {z ∈ C\[0, ∞) | |z| < a}.

(2.10)

Then for z ∈ Da g(z) =

4

κj gj + κ5 r(κ),

(2.11)

j=−1 4 1 d g(z) = − jκj−1 gj + κ3 s(κ), dz 2κ j=−1

(2.12)

sup {|r(κ)|, |s(κ)|} < ∞.

(2.13)

z∈Da

Furthermore, we assume that limIm z0 (g(z) − g−1 κ−1 ) exists and is continuous on (−a, a). As already explained, Assumption 2.3 includes the case, when Hop = −∆ + Vop in odd dimensions. The expansions for the resolvent of −∆ + Vop leading to (2.11) are provided in [13–17, 26]. Taking into account that (at least formally) d g(z) = Ψ0 , W Q∗ (Hop − z)−2 QW Ψ0 , dz the result (2.12) can be derived in the same manner. More precisely, it can be shown that the expansion (2.11) is diﬀerentiable, see [14, 26, 33]. Examples of expansions with the corresponding explicit expressions for coeﬃcients gj are given in [16, Appendix], with references to the literature. Since the form of the decay law depends strongly upon the behavior of g(z) near 0, we divide the considerations into three cases. odinger case this corresponds (i) The singular case, in which g−1 = 0. In the Schr¨ to the situation, when Hop has a zero resonance at the threshold (see, e.g., [14, 16]). Let us recall that the free particle in one dimension belongs to this class. From Proposition 2.2(iv) follows that g−1 > 0.

(2.14)

(ii) The regular case, in which g−1 = 0 and g1 = 0. We note that g−1 = 0 is the generic case for Schr¨ odinger operators in one and three dimensions. Again, from Proposition 2.2(iv), one has g1 < 0.

(2.15)

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Let us remark that the behavior Im g(x + i0) ∼ x1/2 as x → 0 is nothing but the famous Wigner threshold law [30, 23]. (iii) The smooth case, in which g−1 = g1 = 0. This case occurs for free Schr¨ odinger operators in odd dimensions larger that three, and in the spherical symmetric d g(z) is case for partial waves ≥ 1, see [16, 17]. Notice that in this case dz uniformly bounded in Da . Let for x ∈ (0, a), lim F (x + iη, ε) = F (x + i0, ε) = R(x, E0 , ε) + iI(x, E0 , ε).

η0

(2.16)

Due to (2.7) and Assumption 2.3, R(x, E0 , ε) is continuous and strictly decreasing, for suﬃciently small ε. Thus for ε and E0 small enough, with E1 = E0 − g0 ε2 > 0 the equation R(x, E0 , ε) = 0 has a unique solution x0 (E0 , ε) on (0, a): R(x0 (E0 , ε), E0 , ε) = 0.

(2.17)

To simplify the notation we omit the dependence of R(x, E0 , ε), I(x, E0 , ε), and x0 (E0 , ε) on E0 and ε. Throughout the paper, Hop and W are kept ﬁxed, while E0 and ε are parameters; ε is positive and small, and E0 is tuned past the threshold, i.e. takes values in a neighborhood of the origin. A ﬁnite number of constants will appear; they are strictly positive, ﬁnite and independent of the parameters ε and E0 . We introduce the following notation: Notation. (i) A B means that there exists a constant c such that A ≤ cB. An analogous deﬁnition holds for A B. (ii) A B means that both A B and A B hold. (iii) A ∼ = B means that A and B are equal to leading order in a parameter, e.g. A = B + δ(ε) with limε0 δ(ε) = 0. 3. Heuristics and the Results We ﬁrst give the heuristics, and then we state our results. 3.1. Heuristics For E0 outside a small (possibly ε-dependent) neighborhood of the origin, the situation is well understood, both at the heuristic level, and at the rigorous level. Indeed, for negative E0 , using the analytic perturbation theory, one can show that |Aε (t) − e−itEε | ε2 ,

(3.1)

where Eε is the perturbed eigenvalue, which coincides with E0 in the limit ε → 0. As a consequence, the survival probability remains close to one uniformly in time. On heuristic grounds, if E0 is positive, i.e. embedded in the essential spectrum of Hop , Ψ0 turns into a metastable decaying state. The main problem is to compute the “decay law”, i.e. |Aε (t)|2 , up to error terms vanishing in the limit ε → 0.

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For eigenvalues embedded in the continuum spectrum the heuristics for the exponential decay lawb |Aε (t)|2 ∼ = e−2Γ(ε)t runs as follows. Suppose F (z, ε) is suﬃciently smooth, as z approaches the real line from above, F (x + i0, ε), for x in a neighborhood of E0 . Let F (x + i0, ε) = R(x) + iI(x). Then the equation R(x) = 0 has a solution x0 nearby E0 . The idea is that the main contribution to the integral in (2.6) comes from the neighborhood of x0 , and in this neighborhood F (x + i0, ε) ∼ = x0 − x + iI(x0 ),

(3.2)

and then Im F (x, ε)−1 ∼ =

−I(x0 ) , (x − x0 )2 + I(x0 )2

(3.3)

i.e. it has a Lorentzian peak shape leading to |Aε (t)|2 ∼ = e−2|I(x0 )|t .

(3.4)

For mathematical substantiation of this heuristics in the case, where either E0 > 0 (embedded eigenvalues) or E0 = 0 (threshold eigenvalues) but the perturbation “pushes” the eigenvalue suﬃciently “far” into the continuum spectrum such that x0 ε, we send the reader to [16] and references therein. In the cases where the resolvent has an analytic continuation through the positive semi-axis, z0 ∼ = x0 + iI(x0 ) is nothing but the position of the “resonance pole” and x0 and −I(x0 ) are called resonance position and width, respectively. In this setting, the exponential decay law comes from the resonance pole contribution, while the error term comes from the contribution of the “background” integral, see [12]. It is well known that irrespective of the approach the main technical diﬃculty is to estimate the error term. The problem with the energies near the threshold is that F (x + i0, ε) might not be smooth and can even blow up (see Assumption 2.3), if the open channel has a zero resonance at the threshold. Then a Lorentzian approximation might break down. For the case at hand, elaborating on a heuristic argument in [20], one can quantify at the heuristic level how far from the origin x0 > 0 must be in order to have a chance for an exponential decay law: The contribution of the tail at negative x of the Lorentzian must be negligible. Since 0 |I(x0 )| |I(x0 )| dx , (3.5) 2 + I(x )2 (x − x ) x0 0 0 −∞ one gets the condition |I(x0 )| x0 . bA

(3.6)

better name is probably quasi-exponential decay law, in order to emphasize the fact that the equality is up to errors vanishing as ε → 0.

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Consider ﬁrst the condition (3.6) in the singular case. For x > 0 small enough I(x) ∼ = −g−1 ε2 x−1/2 , −1/2

and the condition (3.6) gives g−1 ε2 x0

x0 , i.e.

x0 ε4/3 .

(3.7)

x0 = bεp ,

(3.8)

If we take (by adjusting E0 !)

then one obtains, for 0 ≤ p < 4/3, the exponential decay law (see (3.4)) −1/2 2−p/2 ε t |Aε (t)|2 ∼ . = e−2g−1 b

(3.9)

Notice that for p = 0 (i.e. the resonance stays away from the threshold as ε → 0), (3.9) is nothing but the usual Fermi Golden Rule (FGR) formula. However, for p > 0 but not very large (i.e. the resonance position approaches zero as ε → 0, but not too fast) one gets a “modiﬁed FGR formula ” for which the ε-dependence of the resonance width is ε2−p/2 instead of the usual ε2 -dependence. For the regular case, a similar argument leads to the condition x0 ε4 ,

(3.10)

1/2 2+p/2 t |Aε (t)|2 ∼ . = e−2|g1 |b ε

(3.11)

and a decay law

Finally, in the smooth case the condition (3.6) reads 1/2

ε2 x0

1,

(3.12)

which holds true irrespective of how close to zero x0 is. In other words, in the smooth case, one observes an exponential decay law (with a resonance width vanishing as x0 → 0), as the resonance position is tuned past the threshold, via the tuning of the eigenvalue E0 . 3.2. Reduction to the case g2 = 0 We argue that it is suﬃcient to consider the case, when g2 = 0, i.e. (2.11) is replaced with g(z) =

N

κj gj + κN +1 r(κ),

(3.13)

j≥−1 j=2

which leads to a signiﬁcant simpliﬁcation of the proofs. Indeed, let F (z, ε) F˜ (z, ε) = . 1 − ε2 g2

(3.14)

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On the one hand, notice that if 1 A˜ε (t) = lim η0 π

∞

e−itx Im F˜ (x + iη, ε)−1 dx,

(3.15)

−∞

then for suﬃciently small ε (e.g., |ε2 g2 | ≤ 12 ) we have |A˜ε (t) − Aε (t)| ε2 .

(3.16)

˜1 − z − ε2 (˜ F˜ (z, ε) = E g−1 κ−1 + g˜1 κ + g˜3 κ3/2 + · · ·),

(3.17)

On the other hand,

˜1 = E0 −ε22 g0 , and g˜j = gj2 , which is exactly of the same form as F (z, ε), with E 1−ε g2 1−ε g2 but without the linear term in the expansion of g(z), and the other coeﬃcients slightly “renormalized”. In the sequel we consider only F (z, ε) with g(z) satisfying (3.13). 3.3. Resonance and bound state positions Summing up, the heuristics predicts that if the resonance position is outside an energy window of size ε4/3 and ε4 in the singular and regular case, respectively, then the decay law is exponential with a decay rate depending on, how rapidly the resonance position approaches zero as ε → 0. Moreover, it suggests that for the resonance position inside the above energy windows, the decay law is not exponential, but gives no hint about its actual form. We proceed to the rigorous substantiation of the above heuristics, and to the derivation of decay laws. As the heuristics suggests, the zeroes of R(x) (which for x < 0 coincide with those of F (x+i0, ε)) play a central role. The zero on the positive semi-axis, x0 , gives the resonance position, while the zero on negative semi-axis, xb , gives the position of the bound state. The propositions below give estimates on x0 and xb in terms of the parameters appearing in F (x + i0, ε), as given by (2.7) and (2.8). In the remainder of this paper we shall take a > 0 small enough, such that Assumption 2.1 holds true, and in addition the terms in g(z) with j ≥ 3 can be treated as perturbations. Let a (3.18) E1 = E0 − ε2 g0 with |E1 | ≤ . 2 Proposition 3.1. For E1 > 0 and ε > 0 suﬃciently small, the equation R(x) = 0 on (0, a) has a unique solution x0 , and x0 = E1 + O(ε2 x20 ).

(3.19)

lim x0 = 0.

(3.20)

In particular, E1 0

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Proposition 3.2. (i) Assume g−1 = 0. Then for ε > 0 suﬃciently small (and irrespective of the value of E1 ) the equation F (x, ε) = 0 has a unique solution on (−a, 0) and  4/3 4/3  ε , if 0 E1 ε , (3.21) |xb | ε4  , if E1 ε4/3 ,  2 E1 |xb | ε4/3 + |E1 |,

for E1 ≤ 0.

(3.22)

(ii) Assume g−1 = 0, g1 = 0. Then for E1 ≥ 0, the equation F (x, ε) = 0 has no solutions on (−∞, 0). For −a/2 ≤ E1 < 0 and ε suﬃciently small the equation F (x, ε) = 0 has a unique solution xb on (−a, 0) and |xb | ≤ |E1 |.

(3.23)

3.4. Some previous results In the case, where F (x + i0, ε) is suﬃciently smooth in a neighborhood of x0 , the mathematical substantiation of the quasi-exponential decay law (i.e. exponential decay up to errors vanishing as ε → 0) follows from the results in [16] (see further references in this paper). In particular, for the smooth case, as well as for x0 ε (i.e. p = 1 in (3.8)), in the singular and regular case, one still has (quasi)-exponential decay. Let us stress here that in these cases the ansatz (3.2) is nothing but the approximation of F (z, ε) with a linear function L(z) = α + iβ − z, where the constants α and β are ﬁxed by the condition that F and L coincide at x0 (ε): F (x0 + i0, ε) = L(x0 + i0).

(3.24)

In the (non-smooth) threshold case it has been proved in [8] that indeed in some energy windows, which depend on the spectral properties of the unperturbed Hamiltonian at the threshold, the decay law is deﬁnitely non-exponential for all times. The main idea in [8] is that in the neighborhood of z = 0 one can replace F (z, ε) by the following model function F (z, ε) ∼ = E0 − z − ε2

N

κj gj ≡ H(z, ε),

(3.25)

j=−1

which leads to non-exponential decay laws. As an example we reproduce below the main result in [8] for the regular case. In this case the model function is (3.25) with N = 2 (and g−1 = 0), √ √ ˜ − z + i˜ Hr (z, ε) = E0 − z − ε2 (g0 − ig1 z − g2 z) = d(E g1 z), (3.26) where 2 ˜ = E0 − ε g0 E 1 − ε2 g2

and g˜1 =

g1 . 1 − ε2 g2

It is assumed that ε is suﬃciently small, such that d is close to one.

(3.27)

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˜ ∈ [−a/2, (c/2)ε3/4] for some Theorem 3.3 ([8, Theorem 2.8]). Suppose E c > 0. Then for all t ≥ 0, and for suﬃciently small ε, we have the following results. ˜ ≥ 0 we have (i) For E ∞ y 1/2 −i˜ sy ε4/3 , Aε (t) − 1 e dy π 0 (f˜ − y)2 + y

(3.28)

where s˜ = (ε2 g˜1 )2 t

and

˜ f˜ = (ε2 g˜1 )−2 E.

˜ ≤ 0 we have (ii) For E 1 + 4|f˜| − 1 −itxb 1 ∞ y 1/2 −i˜ sy e e − dy ε4/3 . Aε (t) − 2 ˜ π 0 (f − y) + y 1 + 4|f˜|

(3.29)

(3.30)

Remark 3.4. In terms of f˜ the scaling in (3.8) can be written as x0 ∼ = f˜εp−4 ,

(3.31)

so according to the heuristics f˜ = const. is just the borderline between exponential and non-exponential decay laws. This is substantiated by the numerical computations presented in [8], as well as by the rigorous results in [16] for p = 1 i.e. f˜ ε−3 . Remark 3.5. Again in terms of the scaling (3.8), the interval p ∈ (0, 3/4) is not covered by the results in [8]. One of the main goals here is to ﬁll this gap, in order to have a complete picture of the crossover from exponential to non-exponential decay laws. 3.5. The model functions We recall ﬁrst that in the case of embedded eigenvalues (i.e. p = 0) the “model function” approximating F (z, ε) is the linear approximation (3.2), determined by the condition (3.24). The ansatz we shall adopt in this paper for the “model function” approximating F (z, ε) for all p ∈ (0, ∞), see (3.8), is to replace F (z, ε) by a function, H(z), resembling the expansion of F (z, ε) around the threshold, whose free parameters are ﬁxed by a condition similar to (3.24). More precisely, in the singular case, g−1 = 0, Hs (z) = α − z − ε2 βκ−1 ,

(3.32)

and in the regular case, g−1 = 0, g1 = 0, Hr (z) = α − z + ε2 βκ.

(3.33)

The signs in (3.32) and (3.33) are chosen to ensure that in all cases β ≥ 0. The condition β > 0 should be compared with the Fermi Golden Rule condition, imposed in the case of an embedded eigenvalue (p = 0 case).

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Thus in both cases there are two real parameters α and β, to be determined. In the case E1 ≥ 0 the condition used is F (x0 , ε) = Hι (x0 ), ι ∈ {s, r}. Equality of the real and imaginary parts gives the two equations used to determine α and β. In the case E1 < 0 the functions F (x) and H(x) are real-valued. The conditions used are F (xb , ε) = Hι (xb ), ι ∈ {s, r}, together with d d F (xb , ε) = Hι (xb ), dx dx

ι ∈ {s, r}.

(3.34)

Thus in the case E1 < 0 our conditions determining α and β give as a result that 1 and Hι1(x) , ι ∈ {s, r}, are equal. the residues at the pole xb of F (x,ε) It is clear from this discussion that the parameters α and β take values depending on which case is being considered. These values will be given in connection with the proofs, since they are only needed there. Remark 3.6. Let us note that the determination of α leads to x0 , for E1 ≥ 0, α= xb + small term, for E1 < 0.

(3.35)

The precise form of the small term for E1 < 0 depends on the case being considered. 3.6. Main results; error analysis We are now in a position to formulate the main technical results of this paper: For ε suﬃciently small and E0 in an ε-independent neighborhood of the threshold, the error in Aε (t) due to the replacement of F (z, ε) with the model functions Hι (z), ι ∈ {s, r} (as given by (3.32)–(3.35) vanishes in the limit ε → 0. In other words, we

∞ have to control |Aε (t) − limη0 π1 −∞ e−itx Im Hι (x + iη)−1 dx|

∞as ε → 0. The contribution of the negative semi-axis in limη0 π1 −∞ e−itx Im Hι (x + ˜b , of Hι (z) (when it exists) and equals iη)−1 dx is just the residue at the zero, x 1 −it˜ xb . Accordingly − d H (˜x ) e dx

ι

b

1 η0 π

∞

lim

−∞

e−itx Im Hι (x + iη)−1 dx = −

1 e−it˜xb + Aˆε,ι (t) d Hι (˜ xb ) dx

(3.36)

with 1 Aˆε,ι (t) ≡ π

0

∞

e−itx Im Hι (x + i0)−1 dx,

ι ∈ {s, r}.

(3.37)

Notice that by deﬁnition of Hι (z), for E1 < 0, we have x ˜b = xb , and the ∞ contribution of the negative semi-axis in limη0 π1 −∞ e−itx Im Hι (x+ iη)−1 dx and ∞ in limη0 π1 −∞ e−itx Im F (x + iη, ε)−1 dx are equal. For ι = s, xb exists also for

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E1 ≥ 0, and in this case x ˜b = xb , but still as ε → 0, we have (see the proof of Theorem 3.8).

1

d xb ) dx Hs (˜

Theorem 3.7 (Regular Case). Let g−1 = 0, g1 = 0. There exists c, such that for suﬃciently small ε, |E0 | ≤ c, and all t ≥ 0, we have ∞ −itx −1 Aε (t) − lim 1 e Im H (x + iη) dx r η0 π −∞ 1/2 ε2 (1 + x0 |ln ε|), for E1 > 0, for E1 ≤ 0. ε2 , Theorem 3.8 (Singular Case). Let g−1 = 0. There exists c, that for suﬃciently small ε, |E0 | ≤ c, and all t ≥ 0, we have

a 2

∼ =

a 2

1

d dx F (xb ,ε)

≥ c > 0,

(3.38)

≥ c > 0, such

(i) Let E1 ≥ 0, and let x ˜b be the unique solution of Hs (x) = 0 on the negative semi-axis. Then ∞ 1 1 −itxb −itx −1 e − lim e Im Hs (x + iη) dx Aε (t) + d η0 π 0 Hs (˜ xb ) dx  2 ε   | ln ε|, for x0 ε4/3 , 1/2 x (3.39) 0   4/3 4/3 for x0 ε . ε , (ii) Let E1 < 0. Then ∞ 1 1 e−itx Im Hs (x + i0)−1 dx e−itxb − lim Aε (t) + d η0 π 0 Hs (xb ) dx  2   ε , for |E1 | ε4/3 , 1/2 |E | 1 (3.40)   4/3 4/3 for |E1 | ε . ε , Remark 3.9. For the smooth case (g−1 = g1 = 0), see [8, Theorem 2.10]. This result gives an exponential decay law irrespective of the value of E0 . Remark 3.10. We compare Theorem 3.3 with Theorem 3.7. The latter result is valid in an ε-independent neighborhood of zero, and has a better error estimate. This is due to the choice of α and β. The disadvantage is that these coeﬃcients are not given in terms of expansion coeﬃcients in g(z), but are solutions to equations, which can be solved by perturbative methods.

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3.7. Error function analysis: Crossover from exponential to non-exponential decay laws As shown in the previous section, the bound state contribution has a simple form. Thus it remains to compute Aˆε,ι (t) as given by (3.37). Since Hι , ι ∈ {s, r}, have a simple functional form with only two free parameters, the integral in the right-hand side of (3.37) can be evaluated numerically or expressed in closed form in terms of some special functions. Some examples of numerical computations substantiating the heuristics presented in previous subsections have been presented in [8], but a detailed analytical study of the asymptotics was postponed. One of the main goals of this paper is to perform this analysis. The main point is that Aˆε (t), as well as its asymptotics, can be expressed in terms of some special functions 2 2z ∞ e−ix dx, (3.41) I1 (z) = iπ 0 x2 − z 2 Ip (z) =

1 dp−1 I1 (z), (p − 1)! dz p−1

p = 2, 3, . . . ,

(3.42)

closely related to the error function (see the Appendix for a detailed study of Ip (z)). 3.7.1. The regular case We begin with the (simpler) regular case. Here Hr (z) = α − z + ε2 βκ, where α lies in a small neighborhood of the origin and β > 0. Let 1 ∞ −itx ˆ e Im Hr (x + i0)−1 dx. (3.43) Aε,r (t) ≡ π 0 √ Passing to the variable k = z = iκ we get ∞ −itk2 e 1 ˆ kdk, (3.44) Aε,r (t) = iπ −∞ Pr (κ) where Pr (κ) = κ2 + ε2 βκ + α.

(3.45)

The integral on the right-hand side of (3.44) is to be understood as

A −itk2 limA→∞ −A ePr (κ) kdk. When the zeroes of Pr are distinct, a partial fraction decomposition yields the following result. Proposition 3.11 (Regular Case). Aˆε,r (t) = −

2

√ qj I1 (iκj t),

(3.46)

j=1

where κj =

1 (−ε2 β − (−1)j ε4 β 2 − 4α), 2

j = 1, 2,

(3.47)

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are the roots of Pr (κ), and 2 ε β 1 1 + (−1)j , qj = 2 ε4 β 2 − 4α are the corresponding residues of Remark 3.12. For α =

β 2 ε4 4

j = 1, 2,

(3.48)

κ Pr (κ) .

, the κj , j = 1, 2, coincide and the coeﬃcients qj 2 4

become inﬁnite. One can show that as α → β 4ε . the formula (3.46) has a limit βε2 √ Aˆε,r (t) = iI3 i t . (3.49) 2 2 4

We shall not make use of (3.49) in what follows, since the case α = β 4ε belongs to the crossover regime (see below), when Aˆε,r is given also by (3.67), which has been analyzed in [8]. As a consequence of Proposition 3.11, we can now discuss the various regimes. 3.7.1.1. The exponential regime According to the heuristics, if we set α = bεp ,

b > 0,

(3.50)

then for p ∈ [0, 4)

(3.51)

the decay law is still exponential. Fix p ∈ [0, 4). Notice that as ε → 0 p ε2 √ ε2− 2 α

(3.52)

and then (see Remark 3.12) one can use Proposition 3.11. In this case there is no bound state contribution. Using the properties of Ip (z) (see the Appendix), one obtains Proposition 3.13 (Regular Case). (i) Using (3.47) and (3.48), we have √ 2 β ε2 Aˆε,r (t) = 2q2 eiκ2 t − √ I3 (i αt) + O(ε4−p ), 2 α

(3.53)

and up to error terms as in Theorem 3.7, we have Aε (t) = Aˆε,r (t).

(3.54)

(ii) For p ∈ [0, 4) we have p 1/2 2+ 2 ε

|Aε (t)|2 = e−2βb

t

,

(3.55)

and for p ∈ (0, 4) we have p 1/2 2+ 2

|Aε (t)|2 = e−2|g1 |b

ε

in both cases up to errors vanishing as ε → 0.

t

,

(3.56)

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The formula (3.56) agrees with the heuristic formula (3.11), as well as with the rigorous result in [16] for p = 1. Notice that, as expected, as p approaches 4 from below, the exponential decay law becomes less and less accurate, so one needs to compute corrections. Proposition 3.13 gives only the ﬁrst order correction in ε2 / |α|, but the method of proof provides also the higher order corrections. 3.7.1.2. The bound state regime If α = −bεp ,

b > 0,

(3.57)

then for p ∈ [0, 4)

(3.58)

one expects (see the heuristics) a bound state regime i.e. to leading order the contribution comes from the bound state. The result below provides the mathematical substantiation as well as the ﬁrst order correction. Again the proof gives the means to compute higher order corrections. As in the previous case, as ε → 0, we have p ε2 ε2− 2 . |α|

Note that κ1 > 0, and there is a contribution from the pole of The analogue of Proposition 3.13 reads

(3.59) 1 F (z,ε)

at xb = −κ21 .

Proposition 3.14 (Regular Case). β ε2 Aˆε,r (t) = −i I3 (i |α|t) + O(ε(4−p) ), 2 |α| and up to error terms as in Theorem 3.7 we have 2 2 ε β Aε (t) = Aˆε,r (t) + 1 − i eiκ1 t . 2 |α|

(3.60)

(3.61)

To leading order one obtains the bound state behavior |Aε (t)|2 = 1.

(3.62)

3.7.1.3. The non-exponential regime We come now to the most interesting part of our analysis, when |α| = bεp ,

b > 0,

with p ≥ 4.

(3.63)

According to the heuristics, for these values of p the decay law is neither (quasi)exponential nor bound state like. We consider two cases separately. Case 1. The threshold regime given by p > 4.

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In this case the survival probability amplitude is given by Proposition 3.15 (Regular Case). Up to errors as in Theorem 3.7 we have √ Aε (t) = I1 (iε2 β t) + O(εp−4 ). (3.64) The result (3.64) implies that the decay law is non-exponential for all p > 4. Remark 3.16. The error term becomes more and more important, as p approaches the critical value p = 4. As before, higher order corrections to the leading term can be computed in terms of Ip . The leading term is independent of α and equals the threshold case x0 = E1 = 0. Since I1 can be expressed (see the Appendix) in terms of the error function, one can rewrite (3.64) as follows: √ Aε (t) = w(ei3π/4 ε2 β t) + O(εp−4 ) = eis (1 − erf(eiπ/4 s1/2 )) + O(εp−4 ), (3.65) where s = β 2 ε4 t.

(3.66)

In particular, (3.66) implies that the threshold decay time scale in the regular case is t ∼ ε−4 . Case 2. The “crossover regime”, which is given by p = 4. This case has been considered in [8]. Indeed, in this case in scaled variables s = ε4 β 2 t, f = ε4αβ 2 (note that for p = 4, f = const.) we have directly from (3.43) and (3.33): 1 Aˆε (t) = π

∞ 0

y 1/2 e−isy dy, (f − y)2 + y

(3.67)

and the integral has been analyzed numerically in [8]. The decay law is nonexponential for ﬁnite f , while as f → ±∞, one reaches the exponential and bound state behavior, respectively. 3.7.2. The singular case We now turn to the singular case, where the function approximating F (z, ε) is given by Hs (z) = α − z − ε2 β/κ, with α lying in a small neighborhood of the origin, and β > 0. The results are similar to those in the regular case, but a bit more complicated, due to the singular behavior of Hs (z). We recall (3.37) that 1 ∞ −itx e Im Hs (x + i0)−1 dx. (3.68) Aˆε,s (t) = π 0

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√ z = iκ, we can write it as 2 −1 ∞ k 2 e−itk Aˆε,s (t) = dk, π −∞ Ps (κ)

101

Passing to the variable k =

(3.69)

where Ps (κ) = κ3 + ακ − ε2 β,

(3.70)

and the integral on the r.h.s. of (3.69) is to be understood as the improper integral

A −itk2 limA→∞ −A ePs (κ) k 2 dk. If the zeroes of Ps are distinct, the partial fraction decomposition leads to the following result. Proposition 3.17 (Singular Case). Aˆε,s (t) = −

3

qj I1 (iκj t)

(3.71)

j=1

where κj , j = 1, 2, 3, are the roots of Ps (κ) (as given by the Cardano formula, see (4.82) below ), and

are the corresponding residues of

qj =

κ2j 3κ2j + α

κ2 Ps (κ)

at κ = κj , j = 1, 2, 3.

(3.72)

Without using the explicit formulae for κj we can get the following properties of the roots of Ps (κ): (i) Ps (κ) = 0 always has a positive solution, which we label κ3 . It corresponds to the bound state at x ˜b = −κ23 . (ii) For α = α0 (ε) ≡ −3(ε2 β/2)2/3

(3.73)

all κj are distinct. Note again that the case α = α0 (ε) belongs to the crossover regime. (iii) For α > α0 (ε) the other two solutions κ1 and κ2 are complex conjugates with real part equal to − κ23 . We label by κ1 the one with positive imaginary part. (iv) For α < α0 (ε), κ1 and κ2 are real, κ1 , κ2 < 0, and κ1 + κ2 = −κ3 . We recall that we take |α| = bεp . For p = 43 the somewhat complicated expressions for κj and qj have simple expansions in the limit ε → 0. Combining these expansions with the properties of Ip , one arrives at results, which are very similar to those in the regular case. As before, we give only “ﬁrst order” corrections, but higher order corrections can be computed. These expressions are much more complicated.

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3.7.2.1. The exponential regime If we have α = bεp ,

b > 0;

p ∈ [0, 4/3),

(3.74)

the decay law is still exponential at small ε. Notice that as ε → 0 3p ε2 √ ε2− 2 . 3 α

(3.75)

Using the properties of Ip (z) one gets the following result. Proposition 3.18 (Singular Case). (i) We have √ √ √ 2 βε2 Aˆε,s (t) = 2q2 eiκ2 t − i 3/2 (I1 (i αt) − i αtI2 (i αt)) 2α + O(ε2(2−3p/2) )

(3.76)

(ii) For p ∈ [0, 4/3), we have p 1/2 2− 2

|Aε (t)|2 = e−2βb

ε

t

,

(3.77)

and for p ∈ (0, 4) we have p 1/2 2− 2 ε

|Aε (t)|2 = e−2g−1 b

t

,

(3.78)

in both cases up to errors vanishing as ε → 0. A remark is in order here. In spite of the fact that there is a bound state at x ˜b = −κ23 for “large” positive α, its contribution is small and can be absorbed in the error term. 3.7.2.2. The bound state regime In the case α = −bεp ,

b > 0;

p ∈ [0, 4/3),

(3.79)

the heuristics predicts a bound state regime: The leading order contribution comes only from the bound state at x ˜b = −κ23 . The result below, similar to Proposition 3.14, provides the mathematical substantiation, as well as the ﬁrst order correction. Proposition 3.19 (Singular Case). Aˆε,s (t) =

p βε2 (I (i |α|t) − i |α|tI (i |α|t)) + O(ε2(2− 2 ) ), 1 2 2|α|3/2

(3.80)

and up to error terms as in Theorem 3.8 2 Aε (t) = Aˆε,s (t) + 2q3 eiκ3 t .

(3.81)

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To leading order one obtains the bound state behavior |Aε (t)|2 = 1.

(3.82)

3.7.2.3. The non-exponential regime As in the regular case, according to the heuristics for α = bεp , b ∈ R, p ≥ 4/3, the decay is neither exponential nor bound state like. As in the regular case, we shall distinguish between two cases. Case 1. The threshold regime, which is p > 4/3. Up to errors as in Theorem 3.8 we have the result. Proposition 3.20 (Singular Case). Assume p > 4/3. Then we have 3

Aε (t) =

√ 4 2 iκ23 t 1 e − I1 (iρj β 1/3 ε2/3 t) + O(εp− 3 ). 3 3 j=1

(3.83)

Notice that in contrast to the regular case, there is a non-vanishing contribution from the bound state. Case 2. The “crossover regime”, which in the singular case takes place at the value p = 4/3. When p = 4/3, f = β −2/3 b, b ∈ R does not depend on ε, so there is no useful expansions for κj , qj , j = 1, 2, 3. Accordingly (it has been done in [8] using the scaled variable s = ε4/3 β 2/3 t) one writes (3.37) as 1 ∞ y 1/2 ˆ Aε,s (t) = e−isy dy, π 0 y(f − y)2 + 1 and the integral can be analyzed numerically. In accordance with the above results, as f → ±∞ one reaches the exponential and bound state behavior respectively; we refer to [8] for details. 4. The Proofs We now give the proofs of the results stated in the previous section. 4.1. Proof of Proposition 3.1 Assumption 2.3 and (2.7) (see also (3.13)) imply that for 0 < x < a, we can write R in the form R(x) = E1 − x − ε2 x2 f1 (x)

with

sup |f1 (x)| < ∞.

0<x
(4.1)

We have R(0) = E1 . For a suﬃciently small ε we have R(a) < 0, and also that R(x) is strictly decreasing. Thus it follows that the equation R(x) = 0 has a unique solution, x0 ∈ (0, a), which satisﬁes x0 = E1 − ε2 x20 f1 (x0 ), which together with (4.1) ﬁnishes the proof.

(4.2)

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4.2. Proof of Proposition 3.2 Part (i). One can obtain (3.21) and (3.22) by a rather tedious perturbation procedure for solving the equation for xb ; instead we shall give below a simple geometric argument. Consider ﬁrst, for m ≥ 0 and n 1, the (unique) positive solution y˜ of fm,n (y) ≡ m + y −

nε2 = 0. y 1/2

Using Cardano’s formulae one can see that  n2/3 4/3   ε , ε4/3 , for 0 ≤ m ≤  2 y˜  4    ε , for m ε4/3 . m2

(4.3)

(4.4)

A simpler argument for (4.4) is to argue as follows. Let y˜˜ = (nε2 )2/3 , such that fm,n (y˜˜) = m. Then d fm,n (y) ≥ 1 dy

(4.5)

implies that y˜˜ − m ≤ y˜ ≤ y˜˜ ε4/3 , which gives the ﬁrst part of (4.4). Assume m ε4/3 . Then for y ε4/3 we have m−

nε2 nε2 nε2 ≤ m + y − ≤ const. m − , y 1/2 y 1/2 y 1/2

which implies

nε2 const. m

2 ≤ y˜ ≤

nε2 m

2 ,

and thus the second part of (4.4). Consider now F (x, ε) for x < 0 and |x| suﬃciently small. Recall that for x < 0 we have κ = |x|1/2 . Then E1 − x − ε2

2g−1 g−1 ≤ F (x, ε) ≤ E1 − x − ε2 , |x|1/2 2|x|1/2

(4.6)

and the estimates (4.4) lead to (3.21). The argument for (3.22) is similar. Consider fm,n (y) for m ≤ 0. Notice that (4.5) still holds true and implies y˜˜ ≤ y˜ ≤ |m| + y˜˜. Use again (4.6). Part (ii). Note that in this case F (0, ε) = E1 , and as always (see (2.7) and Propod F (x, ε) ≤ −1. This implies the non-existence of sition 2.2) for x < 0 we have dx bound states for E1 ≥ 0, the existence and uniqueness of the solution for E1 ≤ 0

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(recall that for ε suﬃciently small and − a2 ≤ E1 we have F (−a, ε) > 0), as well as (3.23).

4.3. Proof of Theorem 3.7 Consider ﬁrst the case E1 ≥ 0, such that x0 ≥ 0 exists. Lemma 4.1. Assume E1 ≥ 0. For c > 0 suﬃciently small, and 0 ≤ x0 ≤ 2c we have  1 c  c ε2 ln , for 0 ≤ x0 ≤ 1 1 ε 2 F (x + i0, ε) − Hr (x + i0) dx  0  2 for 0 ≤ x0 εp , any p > 0. ε , (4.7) Proof. We recall that we always have c ≤ a/2. Furthermore, recall also that g1 < 0. Then Assumption 2.3 and (3.13) imply R(x) ≡ E1 − x − ε2 x2 f1 (x),

√ √ I(x) = ε2 x(g1 + O(x)) ≡ −ε2 xf2 (x), (4.8)

with fj uniformly Lipschitz on [0, c], j = 1, 2. Taking c small enough, (4.8) implies |R(x)| ≥

1 |x − x0 |, 2

|I(x)| ≥

1 2√ ε x|g1 |. 2

(4.9)

By deﬁnition of the model function (see Sec. 3.5) we have α = E1 − ε2 x20 f1 (x0 ),

β = f2 (x0 ).

(4.10)

Since both f1 and f2 are uniformly Lipschitz, we get |F (x + i0, ε) − Hr (x + i0)| ≤ ε2 (|x2 f1 (x) − x20 f1 (x0 )| + √ x + x), ε2 |x − x0 |(ˆ

√ x|f2 (x) − f2 (x0 )|) (4.11)

where x ˆ lies between x and x0 . Putting together (4.9) and (4.11) we get 0

c

√ c 1 1 |x − x0 |(ˆ x + x) dx ε2 − dx. F (x + i0, ε) Hr (x + i0) |x − x0 |2 + ε4 x 0

(4.12)

We estimate side in (4.12) on three subintervals.

cthe integral on the right-hand √ Consider ﬁrst 2x0 . In this case x ˆ ≤ x x, |x − x0 | < x, |x − x0 | > x/2, and then ε

2

c

2x0

√ c √ |x − x0 |(ˆ x + x) x 2 dx ε dx ε2 . 2 4 4 |x − x0 | + ε x 2x0 x + ε

(4.13)

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Consider now

x0

. Here we have x ˆ < x0 , such that √ x0 x0 √ |x − x0 |(ˆ x + x) x0 − x dx x0 dx 2 4 |x − x0 | + ε x |x − x0 |2 + ε4 x 0 0 1 √ y = x0 dy. 4 ε 0 2 y + (1 − y) x0

0

In estimating the last integral in (4.14), we use the notation m =

1/2

0

1

1/2

y dy y 2 + m(1 − y) y dy 2 y + m(1 − y)

Since supu>0 u ln(1 +

1 2u )

ε4 . We get

1 1 y dy = ln 1 + , y2 + m 2 4m 1 m 1 dy ln 1 + . 1 + m(1 + y) m 2

1/2

0

ε4 x0

(4.14)

1

1/2

(4.15) (4.16)

< ∞, the estimates (4.15) and (4.16) imply 0

1

y 1 dy ln . 4 ε ε y 2 + (1 − y) x0

(4.17)

Finally consider √ 2x0 2x0 √ |x − x0 |(ˆ x + x) x − x0 dx x0 dx 2 + ε4 x |x − x | |x − x0 |2 + ε4 x 0 x0 x0 x0 √ √ u x0 ≤ x0 du = x0 ln 1 + 4 u 2 + ε4 u ε 0

√ 1 x0 ln ε

(4.18)

Putting together (4.13), (4.14), (4.17), and (4.18), one obtains (4.7) and the proof of Lemma 4.1 is ﬁnished. Consider now E1 < 0. In this case we claim the following result. Lemma 4.2. Assume E1 < 0. For c > 0 suﬃciently small, and − 2c < E1 < 0 we have c 1 1 2 (4.19) F (x + i0, ε) − Hr (x + i0) dx ε . 0 Proof. In this case, we write for x < 0 F (x, ε) = E1 − x − g1 ε2

|x| + ε2 |x|3/2 f (x).

(4.20)

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107

From Assumption 2.3 (recall that for F (x, ε) is analytic for x < 0) follows that d (|x|3/2 f (x))| 1. The equation for β (the equality of the derivatives of ||x|−1/2 dx F and Hr at xb ) leads to d β = −g1 + 2 |xb | (|x|3/2 f (x))|xb , dx which implies |β + g1 | |xb |. The equation for α is α − xb + ε2 Using (4.21), we get α = xb − ε2

(4.21)

|xb |β = F (xb , ε) = 0.

|xb |β = xb + ε2 |xb |g1 + ε2 O(|xb |3/2 ).

On the other hand, F (xb , ε) = 0 and (4.20) give E1 = xb + ε2 |xb |g1 + ε2 O(|xb |3/2 ),

(4.22)

(4.23)

which together with (4.21) yields |α − E1 | ε2 |xb |3/2 .

(4.24)

As in the proof of the previous lemma, we need estimates on |F | and |Hr |. We claim that 1 min{|F (x + i0, ε)|, |Hr (x + i0)|} ≥ ((|E1 | + x)2 + g12 ε4 x)1/2 . (4.25) 2 Concerning |F |, the estimate for the imaginary part is the same as in the previous √ lemma, i.e. |I(x)| ≥ 12 |g1 |ε2 x. As for |R(x)|, (4.8) is written as R(x) = E1 − x(1 + ε2 xf1 (x)), such that R(x) ≤ E1 − x2 , and then |R(x)| ≥ 12 (|E1 | + x). As for |Hr |, from its deﬁnition follows |Hr (x + i0)|2 = |α − x|2 + ε4 β 2 x.

(4.26)

For a suﬃciently small ε (and using E1 < 0), the results (4.24) and (3.23) imply α ≤ E1 + |α − E1 | = E1 + O(ε2 |E1 |3/2 ) ≤

E1 . 2

(4.27)

On the other hand, for |xb | suﬃciently small, we get from (4.21) the estimate β = −g1 + β + g1 ≥ |g1 | − O(|xb |) ≥

|g1 | . 2

(4.28)

Putting together (4.26)–(4.28), one obtains (4.25). Furthermore, from (4.20), (3.33), (4.21), and (4.24), we get |F (x + i0) − Hr (x + i0)| = |E1 − α + iε2 |x|(g1 + β)| + O(ε2 |x|3/2 ) (4.29) ε2 (|xb |3/2 + |xb | |x| + |x|3/2 ),

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108

which together with (4.25) gives 0

c

c 3/2 |x | + |x | |x| + |x|3/2 1 1 b b 2 . F (x + i0, ε) − Hr (x + i0) dx ε 2 (|E1 | + x) + g12 ε4 x 0

Consider now various terms in (4.30). estimates. c 1 |xb |3/2 2 4 dx ≤ 2 0 (|E1 | + x) + g1 ε x √ c x |xb | 2 4 dx ≤ 2 0 (|E1 | + x) + g1 ε x c x3/2 2 4 dx ≤ 2 0 (|E1 | + x) + g1 ε x

(4.30)

Due to (3.23) we have the following three

|xb |3/2 |xb | 0

c

c

0

1 |xb |3/2 1, dx 2 |E1 | 0 (|E1 | + x) √ x |xb | 1, dx 2 (|E1 | + x) |E1 | c

1 √ dx 1, x

which together with (4.30) gives (4.19). Proof of Theorem 3.7. From this point onwards, the proof of Theorem 3.7 follows closely the proof of [8, Theorem 3.3]. For the convenience of the reader we outline it for the case E1 ≥ 0. We take E1 suﬃciently small (e.g. E1 ≤ 4c ), such that α ≤ 2c . Then for x ≥ c we have |Hr (x + i0)|2 x2 + ε4 x. Using (3.33) we get √ |Im Hr (x + i0)| ε2 x. Thus c

∞

|Im Hr (x + i0)| dx ε2 |Hr (x + i0)|2

c

∞

√ x dx ε2 . x2 + ε4 x

(4.31)

Let now 1 Ac,ε (t) = lim η0 π

c

e−itx Im F (x + iη, ε)−1 dx.

0

Due to Assumption 2.3, the limit η 0 can be taken, such that Ac,ε (t) =

1 π

c

e−itx Im F (x + i0, ε)−1 dx.

(4.32)

0

Lemma 4.1 and (4.31) imply that ∞ −itx −1 Ac,ε (t) − 1 e Im Hr (x + i0) dx π 0  1 c  ε2 ln , for 0 ≤ x0 ≤ , ε 2   2 ε , for 0 ≤ x0 εp , any p > 0.

(4.33)

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We ﬁnish the proof by using “Hunziker’s trick”, see [12]. More precisely, observe that |Ac,ε (t) − Aε (t)| ≤ |Ac,ε (0) − Aε (0)| = |Ac,ε (0) − 1|.

(4.34)

From [8, Lemma 3.4(i)] we get 1 π

∞

0

Im Hr (x + i0)−1 dx = 1.

(4.35)

Putting together (4.33)–(4.35), and (4.33) for t = 0, ﬁnishes the proof of Theorem 3.7 for the case E1 ≥ 0. The proof for the case is E1 < 0 similar: use Lemma 4.2 and (see [8, Lemma 3.4(ii)]) −

1

+

d Hr (xb ) dx

1 π

∞ 0

Im Hr (x + i0)−1 dx = 1.

(4.36)

4.4. Proof of Theorem 3.8 The proof consists of the same steps as in the proof of Theorem 3.7. In this case, I(x) = ε2 x−1/2 (g−1 + O(x)) ≡ ε2 x−1/2 f2 (x) (not the same function that was also denoted by f2 above). Consider ﬁrst the case E1 ≥ 0, such that x0 ≥ 0 exists. Lemma 4.3. For c > 0 suﬃciently small and 0 ≤ x0 ≤  2 ε 1    1/2 ln , c 1 1 ε x F (x + i0, ε) − Hs (x + i0) dx  0 0  ε4/3 ,

c 2

we have

for ε4/3 x0 ≤ for 0 ≤ x0 ε

c , 2

4/3

(4.37)

.

A computation similar to the one in the proof of Lemma 4.1 leads to 0

c

c 1 |x − x0 |(ˆ x + x−1/2 ) 1 2 dx, F (x + i0, ε) − Hs (x + i0) dx ε |x − x0 |2 + ε4 x−1 0

(4.38)

where xˆ lies between x and x0 . The term containing x ˆ is the same as in the regular

−1/2 2 c |x−x0 |x case, so we have to consider only ε 0 |x−x0 |2 +ε4 x−1 dx. Consider ﬁrst the case x0 ε4/3 . Observe that ε2

|x − x0 |x−1/2 1 ≤ , 2 4 −1 |x − x0 | + ε x 2

(4.39)

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and then ε

2

ε2

0

|x − x0 |x−1/2 dx ≤ ε2 . |x − x0 |2 + ε4 x−1

(4.40)

We use the following estimate −1/2

x−1/2 ≤ |x−1/2 − x0

−1/2

| + x0

= ≤

1 x−1/2

+

|x − x0 | 1/2 xx0

−1/2 x0

|x − x0 | −1/2 + x0 x0 x

−1/2

+ x0

.

We then get ε

2

c

ε2

|x − x0 |x−1/2 ε2 dx ≤ 1/2 |x − x0 |2 + ε4 x−1 x0 +

1/2 2

1/2 x0

|x − x0 | dx |x − x0 |2 + ε4 x−1 c c |x − x0 | ε2 x−1 + 1/2 dx 2 4 ε2 ε2 |x − x0 | + ε x

1/2 x0

x0

|x − x0 |2 dx x|x − x0 |2 + ε4

c

ε2

ε2

ε2

ε

c

ε2

0

1 ln . ε

Consider now the case x0 ε4/3 . Due to (4.39) it remains to estimate Here one has x2 ≤ x − x0 ≤ x and then

c

const. ε4/3

c |x − x0 |x−1/2 x1/2 2 dx ≤ ε dx 2 4 −1 |x − x0 |2 + ε4 x−1 const. ε4/3 x + ε x c x−3/2 dx ε4/3 ,

c

const. ε4/3

.

(4.41)

const. ε4/3

and the proof of Lemma 4.3 is ﬁnished. The next step (still for the case E1 ≥ 0) is to control the error when F is replaced by Hs in the bound state contribution. Lemma 4.4. Assume E1 ≥ 0. Let x˜b be the unique solution on (−∞, 0) of the equation Hs (x) = 0. Then  4 ε    2 , for x0 ε4/3 1 1 x0 − (4.42) d d   F (xb ) xb ) ε4/3 , for x ε4/3 . Hs (˜ 0 dx dx

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Proof. 1 1 1 1 1 1 − − − ≤ + . d d d d d d F (xb ) xb ) F (xb ) xb ) Hs (˜ Hs (xb ) H(xb ) Hs (˜ dx dx dx dx dx dx (4.43) Consider the ﬁrst term in (4.43). Assumption 2.3 and (3.32) follows that 2 d ε2 F (xb ) − d Hs (xb ) |β − g−1 | ε + , dx 3/2 dx |xb | |xb |1/2 2 d F (xb ) | d Hs (xb )| 1 + ε . dx dx |xb |3/2

(4.44) (4.45)

Furthermore, the equation for β is βε2 g−1 ε2 1/2 = + ε2 O(x0 ), 1/2 |x0 | |x0 |1/2 which implies β = g−1 + O(x0 ).

(4.46)

−2 x0 ε2 1 ε2 ε2 1 + . 1+ − d d |xb |3/2 |xb |1/2 |xb |3/2 F (xb ) Hs (xb ) dx dx

(4.47)

Using (4.44)–(4.46), we get

From (3.21) and (2.17), one has

|xb |

  ε4/3 , if 0 ≤ x0 ε4/3 ,   4

ε    2, E1

if x0 ε4/3 .

(4.48)

Combining (4.47) with (4.48), one obtains  4/3  ε , for 0 ≤ x0 ε4/3 ,   1 1 − ε4 d d  F (xb )  2 , for x0 ε4/3 . Hs (xb )  x0 dx dx

(4.49)

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1 The last step in proving Lemma 4.4 is to estimate d H(x − d H1 (˜x ) . Taylor’s s b) b dx dx theorem implies d2 dx2 Hs (u) 1 1 − ˜b |, (4.50) 2 |xb − x d d H(xb ) Hs (˜ xb ) d Hs (u) dx dx dx d where u lies between x ˜b and xb . Since Hs (˜ xb ) = 0 and | dx H(x)| ≥ 1, we get

|xb − x ˜b | ≤ |Hs (xb )|. Since F (xb ) = 0, one has from (2.7), (3.32), (3.35), (3.19), and (4.46) that |Hs (xb )| = |Hs (xb ) − F (xb )| |x0 − E1 | + ε2 |g−1 − β||xb |−1/2 + ε2 |xb |1/2 ε2 (x20 + x0 |xb |−1/2 + |xb |1/2 ).

(4.51)

Again we have to consider two cases separately. First we consider the case x0 ε4/3 . The result (3.35) and the proof of Proposition 3.2, together with (4.48) leads to xb | ε4/3 . |xb | |˜ 2

d d −4/3 Since u lies between xb and x ˜b , we get | dx H(u)| 1 and | dx . 2 H(u)| ε Inserting these results into (4.51) one gets from (4.50) the result 1 1 − (4.52) ε4/3 . d d H(xb ) Hs (˜ xb ) dx dx

Consider now the case x0 ε4/3 . By the same argument as before |xb | |˜ xb | |u| x3

2

ε4 . x20

x5

d d 0 Then | dx H(u)| 1 + ε40 and | dx 2 H(u)| ε8 , which together with (4.51) and (4.50) leads to ε4 1 1 (4.53) − 2. d x0 d H(xb ) xb ) Hs (˜ dx dx

Putting together (4.49), (4.52), and (4.53) ﬁnishes the proof of the Lemma 4.4. We are left with the estimate of the error for the case E1 < 0. Since the pole positions and the residues for F and Hs coincide by the deﬁnition of Hs (see Sec. 3.5) the error comes only from the positive semi-axis integral.

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Lemma 4.5. For c > 0 suﬃciently small and − 2c < E1 < 0,  ε2   c  , for |E1 | ε4/3 1/2 1 1 |E | 1 F (x + i0, ε) − Hs (x + i0) dx  0  ε4/3 , for |E1 | ε4/3 .

113

(4.54)

Proof. The arguments that lead to (4.21), (4.24) and (4.11), can be applied in this case and yield |β − g−1 | |xb | and |α − E1 | ε2 |xb |1/2 ,

(4.55)

and furthermore |F (x + i0) − Hs (x + i0)| = |E1 − α + iε2

|x|(g−1 − β)| + ε2 O(|x|3/2 ).

(4.56)

Using (4.55) in (4.56), one gets |F (x + i0) − Hs (x)| ε2 (|xb |1/2 + |xb ||x|−1/2 + |x|1/2 ).

(4.57)

Since E1 < 0, one has |R(x)| ≥ (|E1 | + x)/2. Furthermore, |I(x)| = ε2 x−1/2 (g−1 + O(x)), and then |F (x + i0)|

1/2 ε4 (|E1 | + x)2 + . x

(4.58)

1/2 ε4 (α − x)2 + . x

(4.59)

The result (3.32) leads to |Hs (x + i0)|

The problem with |Hs | is that α − x might vanish for some x > 0. However, for x ε2 , we can use (4.55) to get x−α =

x x x x + |E1 | − E1 + + E1 − α ≥ − E1 ≥ , 2 2 2 2

and then from (4.57)–(4.59), one has c 1 1 dx − Hs (x + i0) const. ε2 F (x + i0, ε) c |xb |1/2 + |xb |x−1/2 + x1/2 dx. ε2 ε4 const. ε2 2 (|E1 | + x) + x

(4.60)

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We estimate (4.60) for |E1 | ε4/3 . In this case we get Proposition 3.2(i) |xb | ε4/3 , and then

ε2

|xb |1/2 + |xb |x−1/2 + x1/2 dx ε4 (|E1 | + x)2 + x

c

const. ε2

ε2

c

x 0

ε4/3

ε2/3 + ε4/3 x−1/2 + x1/2 dx x3 + ε4

∞

0

1 + y 1/2 + y −1/2 dy. 1 + y3

(4.61)

Using Proposition 3.2(i) once more, we get for |E1 | ε4/3 that |xb | |E1 |, and then that

ε

2

|xb |1/2 + |xb |x−1/2 + x1/2 dx ε4 (|E1 | + x)2 + x

c

const. ε2

|E1 |1/2 + |E1 |x−1/2 + x1/2 dx (x + |E1 |)2 0 ∞ ε2 1 + y 1/2 + y −1/2 dy ε4/3 . (1 + y)2 |E1 |1/2 0

ε

2

c

(4.62)

The last step is to estimate

ε2

const. ε2

0

|xb |1/2 + |xb |x−1/2 + x1/2 dx. ε4 2 (|E1 | + x) + x

As before, Proposition 3.2(i) implies for |E1 | ε4/3 that |xb | ε4/3 , and then

ε

2

const. ε2

0

ε2 ε

|xb |1/2 + |xb |x−1/2 + x1/2 dx ε4 (|E1 | + x)2 + x const. ε2

0

4/3

0

ε2/3 + ε4/3 x−1/2 + x1/2 x3 + ε4

const. ε2/3

1/2

1 + y 1/2 + y −1/2 dy · v. (1 + y 3 )1/2

xdx

(4.63)

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For |E1 | ε4/3 , again from Proposition 3.2(i), |xb | |E1 |, and then ε2

const. ε2

|xb |1/2 + |xb |x−1/2 + x1/2 dx 4 (|E1 | + x)2 + εx

0

ε

2

const. ε2

0

const.

=

ε2 |E1 |

|E1 |1/2 + |E1 |x−1/2 + x1/2 dx 2 (x + |E1 |) xε1/2 1 + y 1/2 + y −1/2 |E1 |dy ε2 . 1+y

0

(4.64)

Now (4.54) follows from (4.61) and (4.64), and the proof of the lemma is ﬁnished. Proof of Theorem 3.8. From this point, the proof is the same as the proof of Theorem 3.7: use Lemmas 4.4 and 4.5 and (see the proof of [8, Lemma 3.2]) −

1 d Hs (xb ) dx

e−itxb +

1 π

∞

0

e−itx Im Hs (x + i0)−1 dx = 1.

(4.65)

4.5. Proof of Proposition 3.11 Straightforward computation, which we omit.

4.6. Proof of Proposition 3.13 With the notation r=

α−

ε4 β 2 4

(4.66)

we can write (3.47) and (3.48) as follows. κj = (−1)j+1 ir − 2

ε2 β , 2

qj =

1 ε2 β + (−1)j+1 i , 2 4r

j = 1, 2.

(4.67)

p

Note that εr ε2− 2 . we have κ1 = κ2 . Thus iκ1 and iκ2 lie in the third and fourth quadrants respectively. Using (A.8) and (4.67), one obtains by direct computation √ √ √ √ 2 1 βε2 Aˆε,r = 2q2 eitκ2 − (I1 (κ1 t) + I1 (κ1 t) + i (I1 (κ1 t) − I1 (κ1 t). 2 4r

(4.68)

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√ √ Now using (4.67) and the Taylor expansion for I1 (κ1 t) around ir t one gets (with a slight abuse of notation) 2 2 2 √ βε √ d βε2 √ d I1 (κ1 t) = I1 (ir t) − t I1 (ir t) + t I1 (˜ z ), 2 dz 2 dz 2 √ √ where Im z˜ = r t. This expansion (and the corresponding expansion for I1 (κ1 t)) gives, together with (A.5)–(A.7), and Lemma A.4 the result 4 √ 2 ε βε2 Aˆε,r = 2q2 eitκ2 − I3 (ir t) + O . (4.69) 2r r2 √ √ Expand once again I3 (ir t), this time around i αt, see (4.66), and the proof of the ﬁrst part of the proposition is concluded. For the second part of the proposition, we ﬁrst note that Ip are uniformly bounded on the imaginary axis (see Lemma A.5). In order to compute |Aε (t)|2 up to errors vanishing as ε → 0, one need only to consider the ﬁrst term on the right-hand side of (3.53). By (3.47) we have Im κ22 = βε2 r, such that neglecting the terms vanishing as ε → 0 in (3.48), we get √

√

2

|Aε (t)|2 = e−2βrε t .

(4.70)

To further simplify (4.70), we employ the following elementary inequality. For a > 0, |b| ≤ a2 we have sup |e−(a+b)t − e−at | ≥ t≥0

2|b| sup yey . a y≥0

(4.71)

Now (3.55) follows from (4.70), (4.71), and the fact that r=

√ α(1 + O(ε4−p )).

(4.72)

Finally, the deﬁnition of β implies that β = −g1 + O(εp ),

(4.73)

which together with (3.55) and (4.71) gives (3.56), and the proof is ﬁnished. 4.7. Proof of Proposition 3.14 For the computation of Aˆε,r (t) we proceed exactly as in the proof of Proposition 3.13, with the diﬀerence that here there is no need to use Lemma A.3. In this case κ1 and κ2 are real, βε2 β 2 ε4 , r = |α| + , j = 1, 2, κj = ±r − 2 4

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and a computation similar to the one leading to (4.69) gives 4 √ ε βε2 Aˆε,r = −i I3 (ir t) + O . 2r r2

117

(4.74)

The contribution of the bound state at xb = −κ21 (see (3.36)) reads after a straightforward computation 2 βε2 1 −it˜ xb e = 1 − eitκ1 . (4.75) − d 2r Hr (˜ xb ) dx Recall that |α| = εp . Thus 4 ε r = |α| 1 + O = |α|(1 + O(ε4−p )). |α| √ Expanding in (4.74) I3 (ir t) around i |α|t, and using an argument similar to the one leading to (4.69), one obtains (3.61). Finally (3.62) is a direct consequence of (3.61).

4.8. Proof of Proposition 3.15 Recall that |α| = bεp , p > 4. As ε → 0, and |α| << in (3.47) and (3.48), one gets κ1 = −

α (1 + O(εp−4 )), ε2 β q1 = O(εp−4 ),

ε4 β 2 2 ,

we get κj = κj . Expanding

κ2 = −ε2 β(1 + O(εp−4 )), q2 = 1 + O(εp−4 ).

(4.76) (4.77)

An argument similar to the one leading to (4.69) implies that (see (3.46) and (A.7)) √ Aˆε,r = I1 (iε2 β t) + O(εp−4 ). (4.78) Thus (4.77) implies that the contribution (when it exists) of the bound state is also of the order O(εp−4 ), which together with (4.78) ﬁnishes the proof of (3.64). 4.9. Proof of Proposition 3.17 As the proof of Proposition 3.11 this is a straightforward computation. 4.10. Proof of Proposition 3.18 In what follows we need the formulae for κj . Using the scaled quantities f = (ε2 β)−2/3 α,

y = (ε2 β)−1/3 κ,

(4.79)

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one has κj = (ε2 β)1/3 yj

(4.80)

y 3 + f y − 1.

(4.81)

where yj are the solutions of

The Cardano formula gives (for the labeling of κj see the discussion following Proposition 3.17). f y j = ρj r − (4.82) 3r where ρj = e

2πi 3 (j−3)

,

r=

1 + (1 + (4f 3 /27))1/2 2

1/3 .

(4.83)

In (4.83) the principal determination of fractional powers is taken. We recall that we are assuming α = bεp and 0 < p < 4/3. Thus f = −2/3 −( 43 −p) bβ ε → ∞ as ε → ∞. Expanding in (4.82) one obtains √ βε2 −3 κ1 = i α 1 + i 3/2 + O(f ) 2α (4.84)

κ2 = κ1 κ3 =

βε2 (1 + O(f −3/2 )) α

and q1 =

βε2 1 − i 3/2 + O(f −3 ) 2 4α (4.85)

q2 = q1 q3 = O(f −3 ). Now (4.85) and Lemma A.5 imply that √ q3 I1 (iκ3 t) = O(f −3 ) uniformly in t > 0, such that up to errors of order ε4−3p , we have Aˆε,s (t) = −

2

√ qj I1 (iκj t).

j=1

From this point the proof of Proposition 3.18 is a repetition of the proof of Proposition 3.13.

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4.11. Proof of Proposition 3.19 In this case all κj , qj are real, and as ε → 0 we have βε2 (1 + O(|f |−3/2 )) |α| βε2 −3 + O(|f | ) κ1 = |α|1/2 −1 + 2|α|3/2 βε2 −3 κ3 = |α|1/2 1 + + O(|f | ) 2|α|3/2

κ2 =

(4.86)

and q2 = O(|f |−3 )

Thus we have

q3 =

βε2 1 − + O(|f |−3 ) 2 4|α|3/2

q1 =

1 βε2 + O(|f |−3 ). + 2 4|α|3/2

(4.87)

√ q2 I1 (iκ2 t) = O(|f |−3 ),

uniformly in t > 0, such that up to errors of order ε4−3p , √ Aˆε,s (t) = − qj I1 (iκj t). j=1,3

Repetition of the proof of (3.60) leads to (3.80). Adding the contribution of the bound state one obtains (3.81). Finally, taking into account (4.87) one obtains (3.82) and the proof of Proposition 3.19 is ﬁnished. 4.12. Proof of Proposition 3.20 In this case we have f → 0 as ε → 0. Expanding in (4.82) one obtains 4 4 1 κj = ρj β 1/3 ε2/3 (1 + O(εp− 3 ), qj = + O(εp− 3 ). (4.88) 3 √ √ Using the expansion of I1 (iκj t) around iρj β 1/3 ε2/3 t, one obtains from (3.71) (by an argument similar to the one leading to (4.69))

1 Aˆε,s (t) = − 3

3

√ 4 I1 (iρj β 1/3 ε2/3 t) + O(εp− 3 ).

j=1

Adding the bound state contribution and using (4.88) in the form 2q3 eiκ3 t = one obtains (3.83).

4 2 iκ3 t e + O(εp− 3 ), 3

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Acknowledgment Arne Jensen was partially supported by the grant “Mathematical Physics” from the Danish Natural Science Research Council. Most of the work was carried out, while Gheorghe Nenciu was visiting professor at Department of Mathematical Sciences, Aalborg University. The support of the Department is gratefully acknowledged. Appendix. The Function Ip Deﬁne for integers p ≥ 1, and for complex z with Im z = 0, the function ∞ 2 1 e−ix Ip (z) = dx. iπ −∞ (x − z)p

(A.1)

The integral in (A.1) is absolutely convergent, if p ≥ 2. For p = 1 one can deﬁne it by A −ix2 1 e dx. (A.2) I1 (z) = lim A→∞ iπ −A x − z One can also deﬁne

1 A→∞ iπ

I0 (z) = lim

A

2

e−ix dx.

(A.3)

−A

As an alternative to (A.2) one can use the formula 2 2z ∞ e−ix I1 (z) = dx. iπ 0 x2 − z 2

(A.4)

The functions Ip for p ≥ 2 are up to a multiplicative constant the derivatives of I1 , since we have Ip (z) =

dp−1 1 I1 (z). (p − 1)! dz p−1

Lemma A.1. For p ≥ 2 we have |Ip (z)| ≤

1 π|Im z|p−1

∞

(1 + x2 )−p/2 dx.

(A.5)

(A.6)

−∞

Proof. The result follows from a simple computation, which we omit. Lemma A.2. For p ≥ 1 and b ∈ R\{0} we have Ip (ib) = (−1)p Ip (−ib). Proof. Using (A.4) we get I1 (ib) =

2b π

0

∞

2

e−ix dx = (−1)I1 (−ib). x2 + b2

This result, together with (A.5), implies (A.7).

(A.7)

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Fig. A.1.

121

The integration path ΓA .

Lemma A.3. Assume that Re z = 0 and Im z = 0.Then we have the result 2

I1 (z) = I1 (−iz) − (sign(Re z) − sign(Im z))e−iz .

(A.8)

Proof. This result follows from the calculus of residues. We sketch the details. Consider the contour ΓA in the complex plane, as shown in Fig. A.1. We assume A > |z|. The residue theorem implies that we have 2 2 e−iζ dζ = −(sign(Re z) − sign(Im z))iπe−iz . (A.9) ΓA ζ − z The contributions from the two circular arcs vanish as A → ∞, since sin(2t) is negative for t satisfying π/2 < t < π and 3π/2 < t < 2π. Thus we have A iy2 A −ix2 e e dx + dy x − z y + iz −A −A 2

= −(sign(Re z) − sign(Im z))iπe−iz + o(1).

(A.10)

The result now follows by rewriting the second term on the left using complex conjugation, and by taking the limit A → ∞. Lemma A.4. For Im z = 0 and p ≥ 2 we have ip Ip−1 (z) + zIp (z) = Ip+1 (z). 2

(A.11)

Proof. Follows from an integration by parts. The details are omitted. Lemma A.5. We have for all p ≥ 1 that sup|Ip (ib)| < ∞. b>0

(A.12)

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Proof. For p = 1 we use (A.4) to get 2b ∞ e−it2 x 2 ∞ dt ≤ dt = 1. |I1 (ib)| = π 0 t2 + b2 π 0 t2 + x2 For p ≥ 2 we get a uniform estimate for b ≥ 1 from (A.6). For 0 < b < 1 we can deform the integration contour in Ip (ib) into the lower half plane, e.g. follow the real axis from −∞ to −1, the unit circle in the lower half plane, and then the real axis from 1 to ∞. Then we get a uniform estimate also in this case. We now establish the connections with the error function. To this end we recall some deﬁnitions, see [1, p. 297]. z 2 2 erf(z) = √ e−t dt, (A.13) π 0 erfc(z) = 1 − erf(z),

(A.14)

The integral in (A.13) can be taken along any path in the complex plane connecting 0 and z. We also need 2

w(z) = e−z erfc(−iz) for Im z > 0, see [1, 7.1.3]. Note that all these functions are entire functions. We deﬁne the function 2 2 2iz ∞ e−t i ∞ e−t dt = dt for Im z = 0. w(z) ˆ = π −∞ z − t π 0 z 2 − t2

(A.15)

(A.16)

We have the result [1, (7.1.3)] w(z) = w(z) ˆ for Im z > 0.

(A.17)

ˆ we use In the lemma below, which gives the relation between I1 (z) and w(z), Arg z to denote the determination of the argument taking values in (0, 2π). Lemma A.6. Assume that Im z = 0 and Im(eiπ/4 z) = 0. Then we have 2

ˆ iπ/4 z), I1 (z) = σ(z)e−iz + w(e where

   2,     σ(z) =

−2,        0,

(A.18)

3π < Arg z < π, 4 7π < Arg z < 2π, 4

(A.19)

otherwise.

Proof. Let z be ﬁxed and satisfying the assumption in the lemma. We denote by γA the path shown in Fig. A.2, where A > |z|.

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Fig. A.2.

123

The integration path γA .

The calculus of residues yields that 2 2 e−iζ dζ = σ(z)πie−iz , γA ζ − z where σ(z) is deﬁned in the lemma. On the other hand, using the explicit parametrization and the fact that the contributions from the two circular arcs tend to zero as A → ∞, cf. the proof of Lemma A.3, we also have ∞ −ix2 ∞ 2 2 e−iζ e e−t dζ = dx − dt + o(1). iπ/4 z −∞ x − z −∞ t − e γA ζ − z Thus the result follows by taking the limit A → ∞ and using the deﬁnitions of ˆ noting the choice of signs in these deﬁnitions. I1 (z) and w(z), We ﬁnish by giving also the relation between I1 (z) and w(z). From (A.17) and Lemma A.3, one has the result Lemma A.7. Let for Im z = 0, Σ(z) = sign(Im z). Then I1 (z) = Σ(z)w(Σ(z)eiπ/4 z).

(A.20)

Proof. For 0 < Arg z < 3π/4 (A.20) follows from (A.17) and Lemma A.3. By analytic continuation one obtains (A.20) for Σ(z) > 0. Similarly, for 7π/4 < 2 Arg z < 2π, (A.20) follows from (A.17), Lemma A.3, and w(−z) = 2e−z − w(z), see [1, 7.1.11]. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). [2] B. Baumgartner, Interchannel resonances at a threshold, J. Math. Phys. 37 (1996) 5928–5938.

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[3] L. Cattaneo, G. M. Graf and W. Hunziker, A general resonance theory based on Mourre’s inequality, Ann. H. Poincar´e 7 (2006) 583–614. [4] C. B. Chiu, E. C. G. Sudarshan and B. Misra, The evolution of unstable quantum states and a resolution of Zeno’s paradox, Phys. Rev. D 16 (1976) 520–529. [5] O. Costin and A. Soﬀer, Resonance theory for Schr¨ odinger operators, Comm. Math. Phys. 224 (2001) 133–152. [6] E. B. Davies, Resonances, spectral concentration and exponential decay, Lett. Math. Phys. 1 (1975) 31–35. [7] M. Demuth, Pole approximation and spectral concentration, Math. Nachr. 73 (1976) 65–72. [8] V. Dinu, A. Jensen and G. Nenciu, Non-exponential decay laws in perturbation theory of near threshold eigenvalues, J. Math. Phys. 50 (2009) 013516, 20 pp. [9] P. Exner, Open Quantum Systems and Feynman Integrals (Reidel, Dordrecht, 2002). [10] L. Fonda, G. C. Ghirardi and A. Rimini, Decay theory of unstable quantum systems, Rep. Prog. Phys. 41 (1978) 587–631. [11] P. T. Greenland, Seeking the non-exponential decay, Nature 335 (1988) 298–299. [12] W. Hunziker, Resonances, metastable states and exponential decay laws in perturbation theory, Comm. Math. Phys. 132 (1990) 177–188. [13] A. Jensen, Spectral properties of Schr¨ odinger operators and time-decay of the wave functions. Results in L2 (Rm ), m ≥ 5, Duke Math. J. 47 (1980) 57–80. [14] A. Jensen and T. Kato, Spectral properties of Schr¨ odinger operators and time-decay of the wave functions, Duke Math. J. 46 (1979) 583–611. [15] A. Jensen and G. Nenciu, A uniﬁed approach to resolvent expansions at thresholds, Rev. Math. Phys. 13(6) (2001) 717–754. [16] A. Jensen and G. Nenciu, The Fermi golden rule and its form at thresholds in odd dimensions, Comm. Math. Phys. 261 (2006) 693–727. [17] A. Jensen and G. Nenciu, Schr¨ odinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds, Proc. Indian Acad. Sci. (Math. Sci.) 116 (2006) 375–392. [18] A. Jensen and G. Nenciu, On the Fermi golden rule: Degenerate eigenvalues, in Perspectives in Operator Algebras and Mathematical Physics, Proc. Conf. Operator Theory and Mathematical Physics, Bucharest (August, 2005) (Theta, Bucharest, 2008), pp. 91–103. [19] A. Jensen and G. Nenciu, Exponential decay laws in perturbation theory of threshold and embedded eigenvalues, in New Trends in Mathematical Physics, Proceedings of 15th International Congress on Mathematical Physics, Rio de Janeiro, 2006 (Springer, 2009), pp. 525–539. [20] T. Jittoh, S. Masumoto, J. Sato, Y. Sato and K. Takeda, Non-exponential decay of an unstable quantum system: Small Q-value s-wave decay, Phys. Rev. A 71 (2005) 012109, 7 pp. [21] T. K¨ ohler, K. G` oral and P. Julienne, Production of cold molecules via magnetically tunable Feshbach resonances, Rev. Mod. Phys. 78 (2006) 1311–1361. [22] M. Lewenstein, J. Zakrewski, T. Mossberg and J. Mostowski, Non-exponential spontaneous decay in cavities and waveguides, J. Phys. B 21 (1988) L9–L14. [23] B. Marcelis, E. van Kempen, B. Verhaar and J. Kokkelmans, Feshbach resonances with large background scattering length: Interplay with open-channel resonances, Phys. Rev. A 70 (2004) 012701, 15 pp. [24] M. Merkli and I. M. Sigal, A time-dependent theory of quantum resonances, Comm. Math. Phys. 201 (1999) 549–576.

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[25] M. Miyamoto, Bound-state eigenenergy outside and inside the continuum for unstable multilevel systems, Phys. Rev. A 72 (2005) 063405, 9 pp. [26] M. Murata, Asymptotic expansions in time for solutions of Schr¨ odinger-type equations, J. Funct. Anal. 49(1) (1982) 10–56. [27] H. Nakazato, M. Namiki and S. Pascazio, Temporal behavior of quantum mechanical systems, Int. J. Mod. Phys. B 3 (1996) 247–295. [28] R. G. Newton, Quantum Physics (Springer-Verlag, 2002). [29] C. Nicolaides, Physical constraints on nonstationary states and nonexponential decay, Phys. Rev. A 66 (2002) 022118, 7 pp. [30] N. Nygaard, B. Schneider and P. Julienne, Two-channel R-matrix analysis of magnetic-ﬁeld-induced Feshbach resonances, Phys. Rev. A 73 (2006) 042705, 10 pp. [31] A. Orth, Quantum mechanical resonance and limiting absorption: The many body problem, Comm. Math. Phys. 126 (1990) 559–573. [32] X. P. Wang, Embedded eigenvalues and resonances of Schr¨ odinger operators with two channels, Ann. Fac. Sci. Toulouse Math. (6 ) 16(1) (2007) 179–214. odinger operators with [33] K. Yajima, The Lp boundedness of wave operators for Schr¨ threshold singularities I, Odd dimensional case, J. Math. Sci. Univ. Tokyo 13 (2006) 43–93.

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Reviews in Mathematical Physics Vol. 23, No. 2 (2011) 127–154 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004242

ON THE STOCHASTIC DEPENDENCE STRUCTURE OF THE LIMIT LOGNORMAL PROCESS

DMITRY OSTROVSKY 125 Field Point Rd. #3, Greenwich, CT 06830, USA dm [email protected] Received 2 February 2010 Revised 8 September 2010

The distribution of a single increment of the limit lognormal process of Mandelbrot, several representations of its Mellin transform, and an explicit analytic continuation of the Selberg integral are reviewed. The intermittency invariance of the limit lognormal construction is used to establish a functional Feynman–Kac equation that captures the entire stochastic dependence structure of the limit process. This equation is a general rule of intermittency diﬀerentiation that quantiﬁes how the joint distribution of an arbitrary number of increments of the limit process evolves as a function of intermittency. The solution is represented by means of a formal intermittency expansion and shown to be an exactly renormalized expansion in the joint centered moments of the limit process. The expansion coeﬃcients are related to a novel extension of the Selberg integral. Keywords: Multifractal stochastic processes; intermittency; Selberg integral; stochastic dependence. Mathematics Subject Classiﬁcation 2010: 60G57, 76F55, 81T40

1. Introduction In this paper we study the stochastic dependence structure of the limit lognormal process of Mandelbrot [10, 11] and Bacry et. al. [1]. The interest in this multifractal stochastic process derives primarily from its remarkable stochastic self-similarity property and its connection to the famous Selberg integral, confer [7, 17], which gives the positive integral moments of the limit distribution as shown in [2]. The mathematical structure that is behind these properties is the stochastic dependence structure of increments of the limit process. The key feature of the limit lognormal distribution is that it is deﬁned as the limit of an inﬁnite sum of lognormal, logarithmically correlated random variables, which are built from a two-dimensional Gaussian free ﬁeld. Such constructions have recently attracted substantial interest as they occur in a wide spectrum of problems in mathematical physics ranging from statistical modeling of fully developed turbulence [3], to conformal ﬁeld theory and quantum gravity [5, 6], to random energy models [8]. The dependence structure of 127

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increments of the limit lognormal process is thus of fundamental interest for our understanding of the mathematics of multifractality. The review part of this paper deals with the distribution of a single increment of the limit lognormal process. The limit lognormal process is deﬁned as the zeroscale limit of the exponential functional of the underlying stationary normal process with strongly dependent increments. In our previous work [13], we introduced the technique of functional Feynman–Kac equations, which translates the invariances of this underlying process with respect to scale, decorrelation length, and intermittency parameters into the corresponding functional equations for the limit lognormal process. We used the latter invariance in [14] to derive the general rule of intermittency diﬀerentiation for a single increment of the limit process. This rule is a functional equation for the derivative of the expectation of an arbitrary smooth function of the limit distribution with respect to the intermittency parameter. By iterating this rule ad inﬁnitum, we obtained a formal power series expansion of any such functional with certain universal coeﬃcients that are determined uniquely by the Selberg integral. The resulting intermittency expansion can thus be interpreted as an exactly renormalized expansion in the moments of the limit distribution. We applied it successfully in [15] to compute the expansion of the Mellin transform of a single increment of the limit process. We summed it by a moment constant method and thus obtained an explicit closed-form formula for the Mellin transform. We then veriﬁed that the resulting formula reproduces the known values of the integral moments of the limit lognormal distribution and is in fact the Mellin transform of a valid positive probability distribution. Hence, we eﬀectively introduced a new probability distribution with the properties that its integral moments at arbitrary intermittency and Mellin transform asymptotic in the limit of small intermittency coincide with the corresponding quantities of the limit lognormal distribution. It is our conjecture that the two are one and the same. In [16], we extended our solution for the Mellin transform to the general transform of the limit distribution and presented the solution in an operator form. We also computed in [16] the cumulants of the logarithm of the limit distribution, which capture the distribution uniquely as the associated moment problem is determinate, unlike the moment problems for the distribution and its reciprocal as shown in [15]. The fact that the positive integral moments of the limit lognormal distribution are given by the Selberg integral was ﬁrst established in [2]. Hence, the problem of computing the Mellin transform of this distribution is the same as that of ﬁnding an analytic continuation of the Selberg integral as a function of its dimension into a domain in the complex plane. We solved this problem in [15] and found an explicit extension of Selberg’s ﬁnite product formula in the form of an inﬁnite product of gamma factors. In this paper, we give a novel proof of this fact that is more direct than the original. The objective of the research part of this paper is to quantify the dependence structure of increments of the limit lognormal process. The main technical difﬁculty in solving this problem lies in the strongly non-markovian nature of the

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process and, more speciﬁcally, in the divergence of its positive integral moments, i.e. the Selberg integral, at any level of intermittency, however small, which renders the naive expansion in the moments to be useless. The main result of this paper is an extension of the rule of intermittency diﬀerentiation for a single increment to the case of multiple increments. We show how the intermittency invariance of the underlying normal process can be translated into a functional Feynman–Kac equation for an arbitrary number of increments to capture the entire dependence structure of the limit process. This equation is solved exactly by means of a formal intermittency expansion, whose coeﬃcients are determined by positive integral moments and vanish beyond a ﬁnite range thereby providing a type of exact renormalization as they do in the single-increment case. The main innovation is that the moments are now the joint moments and given by a nontrivial generalization of the Selberg integral. For example, in the simplest case of two adjacent increments of size 1/2, the integral involved is

k1

[0,1]k1 +k2 i<j

|xi − xj |−µ

+k2 k1

|xi + xj |−µ

i=1···k1 j=k1 +1···k1 +k2

|xi − xj |−µ dx.

(1.1)

k1 +1≤i<j

We will show how the collection of such integrals for all k1 and k2 determines the stochastic dependence of the two increments, in other words, how this dependence can be reconstructed from the dependence of the multiple integral in Eq. (1.1) on the intermittency parameter µ. The plan of this paper is as follows. In Sec. 2, we present a detailed review of the known results in the case of a single increment. In Sec. 3, we state our main result, describe some of the properties of the ensuing intermittency expansions, and relate them to an extension of the Selberg integral. In Sec. 4, we study the basic properties of the new integral in the case of two adjacent increments. In Sec. 5, we give all the proofs. Conclusions are presented in Sec. 6.

2. Review of Intermittency Diﬀerentiation and Intermittency Expansions for a Single Increment We begin by giving the necessary deﬁnitions and refer the reader to [1–3, 12] for the original presentation. Let ωε (s) ≡ ωµ,L,ε (s) be a Gaussian process in s, whose mean and covariance are functions of a ﬁnite scale ε > 0. We consider the process Mµ,L,ε (t) =

t

eωε (s) ds.

(2.1)

0

Bacry and Muzy [12] deﬁned the mean and covariance of ωε (s) to be µ L E[ωε (t)] = − 1 + log , 2 ε

(2.2)

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Cov[ωε (t), ωε (s)] = µ log

L , |t − s|

ε ≤ |t − s| ≤ L,

L |t − s| Cov[ωε (t), ωε (s)] = µ 1 + log − , ε ε

(2.3)

if |t − s| < ε, and covariance is zero in the remaining case of |t − s| ≥ L. Thus, ε is used as a truncation scale. L is the fundamental decorrelation length of the process that regulates the extent of long-range dependence. Indeed, nonoverlapping increments of Mµ,L,ε (t) are dependent only if they are within L apart. µ is the intermittency parameter.a The two key properties of this construction are, ﬁrst, that E[ωε (t)] = −Var[ωε (t)]/2 so that E[exp(ωε (s))] = 1 and, second, that Var[ωε (t)] is logarithmically divergent as ε → 0. The ﬁrst property is essential for convergence, the second is responsible for multifractality, and both are originally due to Mandelbrot [10]. The limit lognormal process Mµ,L (t) is deﬁned to be the zero scale limit ε → 0 of ﬁnite scale processes Mµ,L,ε (t). Strictly speaking, Mµ,L,ε (dt) is a random measure on the real line, whose weak a.s. convergence to a nondegenerate limit measure Mµ,L (dt) was established in [3] based on the theory of convergence of a certain class of positive martingales developed by Kahane [9] and the work of Barral and Mandelbrot [4] on log-Poisson cascades. Nondegeneracy means that E[Mµ,L (dt)] = dt. The fundamental property of the limit process, ﬁrst established in [12], is its stochastic self-similarity, also known as continuous dilation invariance. Given t < L, let Wt denote a lognormal random variable that is independent of Mµ,L (L) such that E[log Wt ] = −(1 + µ/2) log L/t, Var[log Wt ] = µ log L/t. Then, there holds the following equality in law Mµ,L (t) = Wt Mµ,L (L).

(2.4)

It must be emphasized that this equality is strictly in law, that is, Wt is not a stochastic process and Wt and Ws for s = t are not deﬁned on the same probability space. In particular, Eq. (2.4) determines the distribution of Mµ,L (t) in terms of that of Mµ,L (L) but says nothing about Mµ,L (L) or their joint distribution. Throughout this section, the decorrelation length is set at L = 1 without loss of generality because Mµ,L (t = L) = L Mµ,1 (1) in law as shown in [13]. We will abbreviate Mµ,1,ε (t) as Mµ,ε (t), Mµ,1 (t) as Mµ (t), and Mµ,1 (1) as Mµ when no confusion can arise. The positive integral moments of Mµ were shown in [2] to be given by the celebrated Selberg integral, confer [17]. Given integral l such that 2 ≤ l < 2/µ, 1 1 l ··· |si − sj |−µ ds (l) E[Mµl ] = 0

=

l−1 k=0

a The

0 i<j

Γ(1 − (k + 1)µ/2)Γ2 (1 − kµ/2) , Γ(1 − µ/2)Γ(2 − (l + k − 1)µ/2)

intermittency parameter is denoted by λ2 in Bacry and Muzy [12] so that µ = λ2 .

(2.5)

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which from now on we will denote by Sl (µ). In general, it is shown in [3] that for q > 0 we have E[Mµq ] < ∞ if q < 2/µ and, conversely, E[Mµq ] < ∞ implies that q ≤ 2/µ. Let F (x) be an arbitrary smooth function that does not involve the intermittency parameter 0 ≤ µ < 1b and let F (k) (x) denote its kth derivative. Our results on general intermittency expansions for a single increment established in [14] are summarized in the following propositions. Consider the expectation of a general functional of the limit lognormal process 1 µf (s) e dMµ (s) , (2.6) v(µ, f, F ) E F 0

where f (s) is an arbitrary continuous function that does not involve µ. This functional is path-dependent unless f ≡ 0, its somewhat peculiar functional form is motivated by the fact that this functional form is invariant under intermittency diﬀerentiation. The integration with respect to the limit measure dMµ (s) is understood in the sense of ε → 0 limit so that v(µ, f, F ) = limε→0 vε (µ, f, F ) and 1 vε (µ, f, F ) E[F ( 0 eµf (s) dMµ,ε (s))] with dMµ,ε (s) as in Eq. (2.1) with L = 1. Also, let g(s1 , s2 ) be deﬁned by g(s1 , s2 ) −log|s1 − s2 |.

(2.7)

Its signiﬁcance is that limε→0 Cov(ωµ,1,ε (s1 ), ωµ,1,ε (s2 )) = µ g(s1 , s2 ) on 0 < |s1 − s2 | < 1. Finally, we will use [0, 1]k to denote the k-dimensional unit interval [0, 1] × · · · × [0, 1]. Then, we have the following general rule of intermittency differentiation for a single increment. Theorem 2.1. The expectation v(µ, f, F ) is invariant under intermittency diﬀerentiation and satisﬁes ∂ v(µ, f, F ) = v(µ, f + g(·, s), F (1) )eµf (s) f (s)ds ∂µ [0, 1] +

1 2

[0, 1]2

v(µ, f + g(·, s1 ) + g(·, s2 ), F (2) )

× eµ(f (s1 )+f (s2 )+g(s1 ,s2 )) g(s1 , s2 )ds (2) .

(2.8)

The mathematical content of Eq. (2.8) is that diﬀerentiation with respect to the intermittency parameter µ is equivalent to a combination of two functional shifts induced by the g function. The single-integral term corresponds to the exponential prefactor in Eq. (2.6), whereas the double-integral term corresponds to the intrinsic dependence of Mµ (t) on µ. It is clear that both terms in Eq. (2.8) are of the same shown in [3], nondegeneracy is guaranteed by µ < 2, the restriction of µ < 1 ensures the ﬁniteness of the 2nd moment. Many of our results remain valid for µ < 2 but we maintain µ < 1 in this paper for simplicity.

b As

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functional form as the original functional in Eq. (2.6) so that Theorem 2.1 allows us to compute derivatives of all orders. Proposition 2.1. The expectation E[F (Mµ )] has the formal expansion 2n

∞ µn (k) E[F (Mµ )] = F (1) + F (1)Hn,k . n! n=1

(2.9)

k=2

The universal expansion coeﬃcients Hn,k , n = 1, 2, 3, . . . , are given by the binomial transform of the derivatives of the Selberg integral Hn,k =

n k k ∂ Sl (−1)k (−1)l . l ∂µn µ=0 k!

(2.10)

l=2

Proposition 2.2. The expansion coeﬃcients Hn,k satisfy Hn,k = 0

∀ k > 2n.

(2.11)

Corollary 2.1. There holds the following formal expansion in terms of the derivatives of moment-related expectations of the process 2n

∞ µn F (k) (1) ∂ n k E[(Mµ − 1) ] . (2.12) E[F (Mµ )] = F (1) + n! k! ∂µn µ=0 n=1 k=2

The representation in Eq. (2.12) reveals an essential feature of the structure of our expansions. We see that Eq. (2.12) is an exactly renormalized expansion in the centered moments of Mµ . Indeed, it is easy to see that if the positive moments of all orders were ﬁnite and Taylor expandable in µ, then Eq. (2.12) would be the same ∞ as the naive expansionc in the moments E[F (Mµ )] = F (1) + k=1 F (k) (1)E[(Mµ − 1)k ]/k!. Unlike the naive expansion, however, all the coeﬃcients in Eq. (2.12) are ﬁnite because the derivatives are taken at µ = 0, and Sl (µ) is ﬁnite so long as l < 2/µ. Moreover, our renormalization is exact as the expansion in Eq. (2.12) is not ad hoc but is rather derived from the exact functional equation in Theorem 2.1. The main result of [15] was to explicitly calculate the intermittency expansion for the Mellin transform (complex moments) of Mµ . The moments correspond to F (x) = xq for some given q ∈ C. By Proposition 2.1, the intermittency expansion for the moments is E[Mµq ]

∞ µn fn (q), = 1+ n! n=1

(2.13)

c To see this, note that the k sum in Eq. (2.12) can be extended to inﬁnity due to ∂ n /∂µn |µ=0 E[(Mµ − 1)k ] ≡ k! Hn,k = 0 ∀ k > 2n and n ≥ 1 by Proposition 2.2.

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fn (q) =

2n

(q)k Hn,k ,

n = 1, 2, 3, . . . .

133

(2.14)

k=2

As usual, (q)k denotes the “falling factorial” (q)k q(q − 1)(q − 2) · · · (q − k + 1), ζ(s)d denotes the Riemann zeta function, Bn (s) the nth Bernoulli polynomial, and Yn (x1 , . . . , xn ) the complete exponential Bell polynomial of order n. Theorem 2.2. Let f0 (q) = 1 and deﬁne the polynomials br (q), r = 0, 1, 2 . . . Br+2 (q + 1) + 2Br+2 (q) − 3Br+2 1 −q br (q) = r+1 ζ(r + 1) 2 r+2 Br+2 (q − 1) − Br+2 (2q − 1) + (ζ(r + 1) − 1) . (2.15) r+2 Then, fn (q) satisﬁes the recurrence fn+1 (q) = n!

n fn−r (q) br (q) (n − r)! r=0

(2.16)

and is given explicitly in terms of Yn by fn (q) = Yn (b0 (q)0!, b1 (q)1!, . . . , bn−1 (q)(n − 1)!). The moments have the following exact formal representation

∞ µr+1 q br (q) , q ∈ C. E[Mµ ] = exp r+1 r=0

(2.17)

(2.18)

r+1 The series ∞ br (q)/(r + 1) is divergent in general with the exception r=0 µ of a ﬁnite range of positive and negative integral q, confer Theorem 2.4 below. This means that the Mellin transform of Mµ is not analytic in the intermittency parameter, and Eq. (2.18) ought to be interpreted as its asymptotic expansion. Consider next the general transform of the form E[G(s + log Mµ )] for some ﬁxed constant s. The corresponding intermittency expansion is E[G(s + log Mµ )] =

∞ n=0

Gn (s)

µn . n!

(2.19)

The following theorem and its corollary established in [16] characterize this expansion completely. d We will write ζ(1) to denote Euler’s constant. It never enters any of the ﬁnal formulas as the coeﬃcient it multiplies is identically zero throughout this paper.

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Theorem 2.3. The coeﬃcients Gn (s) of the general intermittency expansion satisfy the recurrence n d n! br (2.20) Gn+1 (s) = Gn−r (s), G0 (s) = G(s). (n − r)! ds r=0 Corollary 2.2.

d d Gn (s) = Yn 0!b0 , . . . , (n − 1)!bn−1 G(s), ds ds

∞ µr+1 d br E[G(s + log Mµ )] = exp G(s). r+1 ds r=0

(2.21) (2.22)

The signiﬁcance of Theorem 2.3 is that it generalizes the recurrence in Theorem 2.2 and shows that the polynomials br (q) play a fundamental role not only for the Mellin transform but in fact for the general transform. The analogy of the general and Mellin transforms extends much further in that the solution for the general transform in Corollary 2.2 is obtained by replacing q with d/ds in the solution for the Mellin transform in Eqs. (2.17) and (2.18). Thus, Eq. (2.22) is an exact operator solution for the general intermittency expansion in Eq. (2.19). It is obvious that Eq. (2.22) reproduces the Mellin transform expansion in Eq. (2.18) by taking G(s) = exp(qs) for some ﬁxed q ∈ C and using d (2.23) eqs = br (q) eqs . br ds The solution for the general intermittency expansion in Corollary 2.2 is formal and needs to be regularized, i.e. we need to sum the divergent series in Eq. (2.22). The regularized solution must correspond to a valid positive probability distribution having the correct integral moments given by the Selberg integral in Eq. (2.5). In addition, it must reproduce Eq. (2.22) as its asymptotic expansion in the small intermittency limit. A solution with these properties is given below. The question µ to distinguish our of uniqueness of such a solution is open, hence we will write M construction from the limit lognormal distribution as deﬁned in the beginning of this section. We begin by regularizing the expansion for the Mellin transform in Eq. (2.18). In [15] we proposed to sum it by means of ∞ µr+1 br (q) r+1 r=0 µx (q+1) ∞ µx µx µx 1 e2 + 2e 2 q − 3 + e 2 (q−1) − e 2 (2q−1) ∼ µx ex − 1 e 2 −1 0 µx (2q−1) µx µx dx e2 − e 2 (q−1) −x − (1 + q + qe 2 ) + e Dµ (q). −q µx x e 2 −1

(2.24)

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∞ The meaning of “∼” is that the series r=0 µr+1 br (q)/(r + 1) is the asymptotic expansion in µ of the integral on the right-hand side of Eq. (2.24) that we denote by Dµ (q). The integral is convergent for (q) < 2/µ. Hence, we deﬁne the Mellin transform by q ] exp(Dµ (q)), E[M µ

(q) <

2 . µ

(2.25)

The motivation for this particular way of summing the divergent series in Eq. (2.18) is explained in the following theorems that were established in [15]. Theorem 2.4. For real integral values of q such that −2/µ + 1/2 < q < 2/µ the series in Eq. (2.18) is convergent and satisﬁes ∞ µr+1 br (q) = Dµ (q). r+1 r=0

(2.26)

In addition, for positive integral q = l, l = 1, 2, 3, . . . , such that l < 2/µ we have l−1

l ] = E[M µ

k=0

Γ(1 − (k + 1)µ/2)Γ2 (1 − kµ/2) . Γ(1 − µ/2)Γ(2 − (l + k − 1)µ/2)

(2.27)

Theorem 2.5. Given 0 < µ < 1, the function q → exp(Dµ (iq)), q ∈ R, is the characteristic function of an inﬁnitely divisible distribution. In summary, Theorem 2.5 says that for every 0 < µ < 1 exp(Dµ (iq)) is µ . Hence the characteristic function of some random variable that we call log M exp(Dµ (q)) is the Mellin transform of the random variable Mµ , whose positive integral moments coincide with the known moments of the limit lognormal distribution by Theorem 2.4 for every µ, confer Eq. (2.5). Finally, the small intermittency expansion of this Mellin transform coincides with the intermittency expansion of the limit q ] ∼ exp( ∞ µr+1 br (q)/(r + 1)) as lognormal distribution by construction E[M µ r=0 µ . The primary diﬃµ → +0. We conjecture that the two are the same Mµ = M culty in verifying this statement stems from the indeterminacy of the corresponding moment problems. µ µ is determinate and those for M Theorem 2.6. The moment problem for log M −1 are indeterminate. The left tail of M µ is lognormal, the right tail is a and M µ power law and its asymptotic behavior is P[Mµ > u] ∼ const u−2/µ as u → +∞. µ can also be computed exactly and are The negative integral moments of M given by a formula that is similar to Selberg’s ﬁnite product in Eq. (2.27). −l ] = E[M µ

l−1 j=0

Γ(2 + (l + 2 + j)µ/2)Γ(1 − µ/2) . Γ2 (1 + (j + 1)µ/2)Γ(1 + jµ/2)

(2.28)

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Unlike the positive moments, the negative moments do not become inﬁnite. In fact, the ﬁniteness of the negative moments plays an essential role in the thermodynamic formalism as developed in [11] and has long been conjectured. It is not too diﬃcult −l ] grows as l2 as l → +∞. This is the same to see from Eq. (2.28) that log E[M µ rate of growth as that of the moments of the lognormal distribution. The regularization of the general transform was done in [16] in an operator form. µ be the probability distribution deﬁned by Eq. (2.25). Then, Theorem 2.7. Let M µ )] = exp Dµ d E[G(s + log M G(s). (2.29) ds µ )] coincides with The small intermittency asymptotic expansion of E[G(s + log M the intermittency expansion in Eq. (2.22) µ )] ∼ E[G(s + log M

∞ n=0

Gn (s)

µn n!

as µ → +0.

(2.30)

The action of the operator Dµ (d/ds) is given by d Dµ f (s) ds µx ∞ 1 e 2 f (s + µx/2) + 2f (s + µx/2) − 3f (s) dx = µx x x e −1 e 2 −1 0 µx µx µx df df e− 2 f (s + µx/2) − e− 2 f (s + 2µx/2) 2 +e − 1+ + µx ds ds e 2 −1 − µx µx − e 2 f (s + 2µx/2) − e 2 f (s + µx/2) df − + e−x . (2.31) µx ds e 2 −1 We conclude our review of the single-increment distribution by giving an alternative representation of the Mellin transform. It is clear that by virtue of Eq. (2.5) the Mellin transform of the limit lognormal distribution is an analytic continuation of the Selberg integral to the complex domain. We know from Theorem 2.4 that Eq. (2.25) gives an explicit continuation to the half-plane (q) < 2/µ. The following formula for the Mellin transform that we established in [15] extends Selberg’s ﬁnite product to an inﬁnite product of gamma factors. Theorem 2.8. Let Dµ (q) be as in Eq. (2.24) and (q) < 2/µ. Then, q 2 Γ(2 + 2/µ − 2q) Γ(1 − µq/2)Γ−q (1 − µ/2) exp(Dµ (q)) = µ Γ(2 + 2/µ − q) ∞ 2q 2n Γ3 (1 − q + 2n/µ) Γ(2 − q + 2n/µ) . × µ Γ3 (1 + 2n/µ) Γ(2 − 2q + 2n/µ) n=1

(2.32)

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A direct proof that Eq. (2.32) coincides with Eq. (2.27) in the special case of integral q is given in Sec. 5. Corollary 2.3. The Mellin transform satisﬁes the following functional equations 2 2 µq+ µ ] = (2π) µ2 −1 Γ− µ2 1 − µ 2Γ(−q) Γ (1 − q) Γ(2 − q + 2/µ) E[M µq ], E[M 2 µ Γ(2 − 2q) Γ(2 − 2q + 2/µ) (2.33) q−1 ] = E[M µ

Γ(1 − µ/2) Γ(2 − (2q − 2)µ/2) Γ(2 − (2q − 3)µ/2) q E[Mµ ]. Γ(1 − µq/2) Γ2 (1 − (q − 1)µ/2) Γ(2 − (q − 2)µ/2)

(2.34)

Eq. (2.33) holds for (q) < 0 and Eq. (2.34) for (q) < 2/µ. These functional equations are interesting for at least two reasons. First, they indicate that the underlying mathematical structure is hypergeometric for they are equivalent to integral equations of the Mellin convolution type involving the Appell µ . The other F3 function in the kernel for the probability density function of M reason is that Eq. (2.34) follows directly for integral q from Selberg’s formula in Eq. (2.5). If we postulate that it must hold for complex q, then it can be used to derive the formula for the Mellin transform in Eq. (2.32). This is the route that was taken in [8], where the authors established a formula that is equivalent to our Eq. (2.32). µ is determinate, the In view of the fact that the moment problem of log M (p) p p µ uniquely. cumulants Dµ (0) ≡ ∂ /∂q |q=0 Dµ (q) capture the distribution of log M The following result was obtained in [16]. µ are Corollary 2.4. The cumulants of log M 2 µ µ 2 µ µ Dµ(1) (0) = log − log Γ 1 − −ψ 2+ − ψ 1+ − ψ(1) µ 2 µ 2 2 2 ∞ −t 1 2 e 1 +µ + t − dt, (2.35) µt/2 2t e −1 e − 1 µt 0 p µ 2 µ (p) (2p − 1)ζ p, 1 + Dµ (0) = (p − 1)! (2p − 1)ζ p, 2 + + µ 2 2

p ∞ µ 2k ζ(p) + 4(1 − 2p−2 ) ζ p, 1 + + , p = 2, 3, 4, . . . . 2 µ k=1

(2.36) Hence, the Mellin transform has nontrivial connections with both the gamma and Hurwitz zeta functions. We conclude our review by emphasizing that the uniqueness of our solution for the distribution of the single increment of the limit lognormal distribution remains an open problem.

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3. Intermittency Diﬀerentiation for Multiple Increments In this section, we will formulate the rule of intermittency diﬀerentiation for multiple increments of the limit lognormal process. We will then state basic properties of the resulting intermittency expansions with the ultimate goal of reconstructing the dependence structure of the limit process from the dependence of its joint integral moments on the intermittency parameter. The proofs of all results in this section are deferred to Sec. 5 for the convenience of the reader. Throughout this section, we continue to assume that the decorrelation length is set at L = 1. We will abbreviate Mµ,1,ε (t) as Mµ,ε (t) and Mµ,1 (t) as Mµ (t) when no confusion can arise. We will write Ij , j = 1 · · · N, for a sequence of possibly overlapping subintervals of [0, 1]. In what follows, we let Fj (x), j = 1 · · · N, denote smooth functions that equal their Taylor series on x ∈ [0, 1] and do not involve the intermittency parameter 0 ≤ µ < 1. Consider a general functional of the limit lognormal process  

N Fj eµfj (s) dMµ (s) , (3.1) v(µ, f, F , I ) E  j=1

Ij

where fj (s), j = 1 · · · N, are arbitrary continuous functions that do not involve µ. The integration with respect to the limit measure dMµ (s) is understood in the sense of ε → 0 limit so that v(µ, f, F , I ) = limε→0 vε (µ, f, F , I ) and  

N vε (µ, f, F , I ) E Fj eµfj (s) dMµ,ε (s) , (3.2) j=1

Ij

with dMµ,ε (s) as in Eq. (2.1) with L = 1. Deﬁne the function g(s1 , s2 ) by g(s1 , s2 ) −log|s1 − s2 |.

(3.3)

We will use |I| to denote the length of I, I k to denote the k-dimensional product I × · · · × I, and F (k) (x) to denote the kth derivative of F (x). Finally, we will write f+g(·, s) to denote the vector function with components fj (u) + g(u, s), j = 1 · · · N. Then, we have the following general rule of intermittency diﬀerentiation for multiple increments. Theorem 3.1. N

∂ v(µ, f, F , I ) = ∂µ j=1

N

+

fj (s)eµfj (s) v(µ, f + g (·, s), F1 · · · Fj

(1)

Ij

1 2 j=1

· · · FN , I )ds

Ij2

exp(µ(fj (s1 ) + fj (s2 ) + g(s1 , s2 ))) g(s1 , s2 )

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(2) × v(µ, f + g(·, s1 ) + g (·, s2 ), F1 · · · Fj · · · FN , I )ds1 ds2

+

N Ij ×Ik

j
exp(µ(fj (s1 ) + fk (s2 ) + g(s1 , s2 ))g(s1 , s2 ) (1)

× v(µ, f + g(·, s1 ) + g (·, s2 ), F1 · · · Fj

(1)

· · · Fk · · · FN , I )ds1 ds2 . (3.4)

The structure of the intermittency derivative thus amounts to certain functional shifts that are induced by the g function. In the special case of N = 1 and I1 = [0, 1] the last term drops out and we recover the rule of intermittency diﬀerentiation for a single increment, confer Theorem 2.1. It is worth emphasizing that Theorem 3.1 indicates that the functional in Eq. (3.1) is invariant under intermittency diﬀerentiation so that the rule can be iterated ad inﬁnitum. We are primarily inter ested in the joint distribution of the increments I1 dMµ , . . . , IN dMµ of the limit process over the intervals I1 , . . . , IN , which corresponds to the case of fj = 0 ∀ j. This joint distribution is determined by the following formal intermittency expansion. Proposition 3.1. There exist universal expansion coeﬃcients Hn;k1 ···kN such that  

N E Fj dMµ (s)  Ij

j=1

=

N

Fj (|Ij |) +

j=1

∞ µn n! n=1

2≤

PN

Hn;k1 ···kN

j=1 kj ≤2n

N

(kj )

Fj

(|Ij |).

(3.5)

j=1

The coeﬃcients Hn;k1 ···kN are universal in the sense of being the same for all can be determined from the joint positive integral moments of the F1 · · · FN . They increments I1 dMµ , . . . , IN dMµ . In fact, consider the special case of Fj (x) = xqj for arbitrary complex numbers qj , j = 1 · · · N, so that F (k) (x) = (q)k xq−k and (q)k denotes the “falling factorial” (q)k q(q − 1)(q − 2) ·· · (q − k + 1). Then, the intermittency expansion for the joint Mellin transform of I1 dMµ , . . . , IN dMµ is 

E

N

j=1

=

Ij N j=1

qj  dMµ 

(|Ij |)qj +

∞ µn n! n=1

2≤

PN

j=1

kj ≤2n

Hn;k1 ···kN

N

(qj )kj |Ij |qj −kj .

(3.6)

j=1

This object is of particular interest as it determines the joint distribution uniquely. If we restrict our attention to the case of positive integral qj ∈ N, we obtain an

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explicit formula for the expansion coeﬃcients. Denote 

qj  N dMµ . Sq1 ···qN (µ, I ) E 

(3.7)

Ij

j=1

Corollary 3.1. Given 2 ≤ k1 + · · · + kN ≤ 2n, the expansion coeﬃcients satisfy N (−1)k1 +···+kN q1 +···+qN kj −qj kj Hn;k1 ···kN = (−1) |Ij | k1 ! · · · kN ! qj j=1

qj ≤kj j=1···N

∂ n × n Sq ···q (µ, I ). ∂µ µ=0 1 N

(3.8)

Finally, the joint integral moments Sq1 ···qN (µ, I ) that enter Eq. (3.8) are given by a generalized Selberg integral. Proposition 3.2. Given non-negative integers k1 · · · kN , let sj , j = 1 · · · N, denote the vector sj ≡ (sj1 , . . . , sjkj ) such that sjl ∈ Ij ∀ l. Then, Sk1 ···kN (µ, I ) =

···

N k

k

I1 1 ×···×INN

−µ

[|sip − siq |

]

i=1 p
N

[|sip − sjq |−µ ]ds1 · · · dsN .

i<j p≤ki q≤kj

(3.9)

The integrand in Eq. (3.9) is the discriminant, i.e. the product i<j |xi −xj |2 , on the k1 + · · · + kN variables sj1 , . . . , sjkj for j = 1 · · · N that is raised to the power −µ/2. Clearly, the integral coincides with the Selberg integral, confer Eq. (2.5) above, when N = 1 and I1 = [0, 1]. Proposition 3.3. The expansion coeﬃcients as deﬁned by Eq. (3.8) for all indices k1 , . . . , kN satisfy Hn;k1 ···kN = 0

if

N

kj > 2n.

(3.10)

j=1

It is important to emphasize that the intermittency expansion in Eq. (3.5) is an exactly renormalized expansion in the joint centered moments. Indeed, the expansion can also be written in the form  

(k ) N N ∞ N Fj j (|Ij |) µn Fj dMµ (s)  = Fj (|Ij |) + E n! PN kj ! Ij n=1 j=1 j=1 j=1 ∂ × n ∂µ



n

E µ=0

2≤

N

j=1

Ij

j=1

kj ≤2n

kj   dMµ (s) − |Ij |

(3.11)

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due to the identity 

kj  N ∂ n  = k1 ! · · · kN ! Hn;k1 ···kN E dMµ (s) − |Ij | ∂µn µ=0 Ij j=1

141

(3.12)

that holds for all k1 · · · kN . As E[ I dMµ (s)] = |I|, this identity also shows that the expansion coeﬃcients Hn;k1 ···kN are essentially intermittency derivatives of the joint centered moments. Now, if all the positive joint moments were ﬁnite, then our expansion would coincide with the naive expansion  

N N N ∞ ∞ (k ) Fj dMµ (s)  = Fj (|Ij |) + ··· Fj j (|Ij |)/kj ! E Ij

j=1

j=1

 × E

k1 =1

N

j=1

Ij

kN =1 j=1

kj 

dMµ (s) − |Ij |

.

In fact, the k sum in Eq. (3.11) can be extended to inﬁnity as the derivative in N Eq. (3.12) vanishes for j=1 kj > 2n and n ≥ 1 by Proposition 3.3. Unlike the naive expansion, however, all the coeﬃcients in Eq. (3.11) are ﬁnite because the derivatives are taken at µ = 0, and Sk1 ···kN (µ, I ) is ﬁnite so long as µ is suﬃciently small. Moreover, our renormalization is exact as the expansion in Eq. (3.5) is derived from the exact functional equation in Theorem 3.1. The integral in Proposition 3.2 is unknown for N ≥ 2. In the next section we will examine some of its basic properties in relation to the Selberg integral in the special case of N = 2. 4. Properties of the Joint Integral Moments In this section, we will consider the integral in Eq. (3.9) in the case of two adjacent increments: N = 2, I1 = [a, b], and I2 = [b, c], a < b < c. From now on, it will be understood that all products involving two indices are over the set {i < j}, unless stated otherwise. Then, the joint moments take the form k1 +k2 |xi − xj |−µ dx. (4.1) Sk1 , k2 (µ, [a, b], [b, c]) = [a,b]k1

[b,c]k2

1

Denote α b − a,

β c − b,

m k1 + k2 ;

µ λ− . 2

(4.2)

j = k1 + 1 · · · m,

(4.3)

Changing variables xi = 1 −

xi − a , α

i = 1 · · · k1 ,

xj =

xj − b , β

we obtain the equality Sk1 , k2 (µ, [a, b], [b, c]) = αλk1 (k1 −1)+k1 β λk2 (k2 −1)+k2 Ik1 , k2 (−µ/2, α, β),

(4.4)

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where Ik1 , k2 (λ, α, β) is the integral Ik1 ,k2 (λ, α, β)

k1

[0,1]m 1

|xi −xj |2λ

|αxi +βxj |2λ

i=1···k1 j=k1 +1···m

m

|xi −xj |2λ dx.

(4.5)

k1 +1

In the special case of adjacent increments of equal size, we recover the integral in Eq. (1.1). The value of Ik1 , k2 (λ, α, β) is unknown for arbitrary k1 and k2 . In the rest of this section, we will establish some of its basic properties. We begin by recalling the full Selberg integral. Sm (λ, λ1 , λ2 )

m

[0,1]m 1

|xi − xj |2λ

m

xλi 1 (1 − xi )λ2 dx.

(4.6)

1

It has the essential property that its dependence on λ1 and λ2 determines the full integral due to the identity m+1

|xi − xj |2λ = y λm(m+1)

1

m

|ti − tj |2λ

1

m

|1 − ti |2λ ,

(4.7)

1

where xi = ti y, i = 1 · · · m and xm+1 = y, which implies the recurrence Sm+1 (λ, 0, 0) =

(m + 1) Sm (λ, 0, 2λ). 1 + λm(m + 1) + m

(4.8)

Our ﬁrst result is that this property of the Selberg integral extends to our integral. In fact, letting xi = ti y, i = 1 · · · k1 , xk1 +1 = y, xj = tj−1 z, j = k1 + 2 · · · m + 1, xm+2 = z, we obtain k 1 +1

|xi − xj |2λ

|αxi + βxj |2λ

i=1···k1 +1 j=k1 +2···m+2

1

k1 1

m

|ti − tj |2λ |αy + βz|2λ

m

|ti − tj |2λ

|αyti + βztj |2λ

i=1···k1 j=k1 +1···m

k1 [|1 − ti |2λ |αyti + βz|2λ ] i=1

k1 +1

×

|xi − xj |2λ

k1 +2

= y λk1 (k1 −1)+2λk1 z λk2 (k2 −1)+2λk2

×

m+2

[|αy + βztj |2λ |1 − tj |2λ ].

j=k1 +1

(4.9)

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We are thus naturally led to generalize the integral in Eq. (4.5) to Ik1 ,k2 (λ, λ1 , λ2 , λ3 , α, β) k1 |xi − xj |2λ [0,1]m

×

k1

|αxi + βxj |2λ

i=1···k1 j=k1 +1···m

1

|xi − xj |2λ

k1 +1 m

[|1 − xi |λ1 |αxi + β|λ2 |xi |λ3 ]

i=1

m

[|α + βxj |λ1 |1 − xj |λ2 |xj |λ3 ]dx.

j=k1 +1

(4.10) To simplify notation, let us introduce an auxiliary quantity Jk1 ,k2 (λ, λ1 , λ2 , λ3 , α, β) |α|λk1 (k1 −1)+k1 +k1 (λ1 +λ3 ) |β|λk2 (k2 −1)+k2 +k2 (λ2 +λ3 ) × Ik1 ,k2 (λ, λ1 , λ2 , λ3 , α, β).

(4.11)

There results the following recurrence, which generalizes Eq. (4.8). Proposition 4.1. Jk1 +1,k2 +1 (λ, 0, 0, λ3 , α, β) = |α|

1+λ3

|β|

1+λ3

(k1 + 1)(k2 + 1)

[0,1]2

|αy + βz|2λ y λ3 z λ3

× Jk1 ,k2 (λ, 2λ, 2λ, λ3 , αy, βz)dydz.

(4.12)

Hence, as in the case of the Selberg integral, it is suﬃcient to determine the dependence of the new integral on λ1 and λ2 . The quantity Jk1 ,k2 (λ, λ1 , λ2 , λ3 , α, β) has the properties that it is symmetric and recovers the joint moments by Eq. (4.4). Jk1 ,k2 (λ, λ1 , λ2 , λ3 , α, β) = Jk2 ,k1 (λ, λ2 , λ1 , λ3 , β, α),

(4.13)

Sk1 , k2 (µ, [a, b], [b, c]) = Jk1 ,k2 (−µ/2, 0, 0, 0, α, β).

(4.14)

Jk1 ,k2 (λ, λ1 , λ2 , λ3 , α, β) is also of interest as its values for positive and negative β are related by a functional equation. Consider the integral identity that follows from the symmetry of the integrand m m |xi − xj |2λ |xi − a|λ1 |xi − c|λ2 |xi − b|λ3 dx [a,c]k1

=

[b,c]k2

k1 p=0

k1 p

i=1

1

[a,b]p

[b,c]m−p

m

|xi − xj |2λ

m

|xi − a|λ1 |xi − c|λ2 |xi − b|λ3 dx.

i=1

1

(4.15) Applying the change of variables xi = 1 −

xi − a , α+β

i = 1 · · · k1 ,

xj = 1 −

xj − b , β

j = k1 + 1 · · · m

(4.16)

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to the left-hand side of Eq. (4.15), we obtain

m

[a,c]k1 [b,c]k2

m

|xi − xj |2λ

|xi − a|λ1 |xi − c|λ2 |xi − b|λ3 dx

i=1

1

= Jk1 ,k2 (λ, λ1 , λ3 , λ2 , α + β, −β).

(4.17)

Finally, applying the change of variables in Eq. (4.3) to the right-hand side of Eq. (4.15) and recalling Eq. (4.11), we get

[a,b]k1

m [b,c]k2

|xi − xj |2λ

m

|xi − a|λ1 |xi − c|λ2 |xi − b|λ3 dx

i=1

1

= Jk1 ,k2 (λ, λ1 , λ2 , λ3 , α, β),

(4.18)

which generalizes Eqs. (4.4) and (4.14). There results the following functional equation. Proposition 4.2. For any k = 0 · · · m there holds the identity Jk,m−k (λ, λ1 , λ3 , λ2 , α + β, −β) =

k k p=0

p

Jp,m−p (λ, λ1 , λ2 , λ3 , α, β).

(4.19)

If we let k = m and λ3 = 0 in Eq. (4.19), we obtain Corollary 4.1. λm(m−1)+m+m(λ1 +λ2 )

(α + β)

Sm (λ, λ1 , λ2 ) =

k1 +k2 =m

m Jk1 ,k2 (λ, λ1 , λ2 , 0, α, β). k1 (4.20)

In particular, using Eqs. (4.13) and (4.14) it is easy to calculate the moments for m = 2 and m = 3 and α = β = 1/2 in terms of the Selberg integral. 1 (21−µ − 1) 1 S1, 1 µ, 0, , , 1 = S2 (−µ/2, 0, 0), (4.21) 2 2 22−µ 1 1 1 1 1 (22−3µ − 1) S3 (−µ/2, 0, 0). S1, 2 µ, 0, , , 1 = S2, 1 µ, 0, , , 1 = 2 2 2 2 3 23−3µ (4.22) 5. The Proofs In this section we will give the proofs of all results that were stated in Sec. 3. We begin with a direct proof of the fact that our inﬁnite product formula in Theorem 2.8 coincides with Selberg’s ﬁnite product in the special case of q = l, l = 1, 2, 3, . . . .

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Proof of Eq. (2.27) from Eq. (2.32). Recall the identity, confer [18], Sec. 12.13,   ∞ l l n − a Γ(1 − bj ) j  (5.1) = n − bj Γ(1 − aj ) n=1 j=1 j=1 that holds for arbitrary constants aj and bj so long as j aj = j bj . Hence, in the special case of q = l, we obtain 2q 3 ∞ l−1 2n Γ (1 − q + 2n/µ) Γ(2 − q + 2n/µ) Γ3 (1 − µj/2) = . 3 µ Γ (1 + 2n/µ) Γ(2 − 2q + 2n/µ) j=0 Γ(1 − µ(l + j − 1)/2) n=1 (5.2) On the other hand, using the functional equation of the gamma function, we have q l−1 2 Γ(2 + 2/µ − 2q) 1 = . (5.3) µ Γ(2 + 2/µ − q) 1 − µ(l + j − 1)/2 j=0 Thus, exp(Dµ (l)) =

l−1 Γ3 (1 − µj/2) Γ(1 − µl/2) . Γl (1 − µ/2) j=0 Γ(2 − µ(l + j − 1)/2)

(5.4)

It is elementary now to see that this expression is the same as Eq. (2.27). We remark in passing that the proof of Eq. (2.28) from Eq. (2.32) goes through verbatim. We now proceed to the proof of Theorem 3.1. In the proof, we work with the general decorrelation length L and set L = 1 at the end. Throughout this section, s → ωµ,L,ε (s) refers to the Gaussian process deﬁned in Eqs. (2.2) and (2.3). As a slight abuse of notation, we write dMε (s) as an abbreviation for dMµ,L,ε (s) exp(ωµ,L,ε (s))ds. We also deﬁne µ gL,ε (s1 , s2 ) Cov(ωµ,L,ε (s1 ), ωµ,L,ε (s2 )), then limε→0 gL=1,ε (s1 , s2 ) = g(s1 , s2 ) as in Eq. (3.3). The functions Fj (x) and fj (x) are as in Sec. 3: Fj (x) is smooth, fj (x) is continuous, and neither involves µ as a parameter. We need the statements of several lemmas, whose proofs the reader can readily ﬁnd in [13]. Lemma 5.1. Let ω(s) be a Gaussian process with respect to some probability measure P that is deﬁned on an interval s ∈ [0, L], has continuous sample paths, and satisﬁes EP [exp(ω(s))] = 1 for all s. Let s1 and s2 be any two distinct times, s1 , s2 ∈ [0, L], and let C(s, t) CovP (ω(s), ω(t)) denote the covariance function of ω(s), which is assumed to be continuous. Then, the measure dQ eω(s1 ) dP is a probability measure that is equivalent to P, and the law of the process s → ω(s) + C(s, s1 ) with respect to P equals the law of the original process s → ω(s) with respect to Q on the interval [0, L]. Similarly, the measure dQ eω(s1 )+ω(s2 )−C(s1 ,s2 ) dP is a probability measure that is equivalent to P, and

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the law of the process s → ω(s) + C(s, s1 ) + C(s, s2 ) with respect to P equals the law of the original process s → ω(s) with respect to Q on the interval [0, L]. Lemma 5.2. The process ωµ,L,ε (s) has continuous sample paths. Lemma 5.3. Let f(δ, s) be an arbitrary continuous function that vanishes as δ → 0. Let B(s) exp(f(δ, s) + ωδ,L,ε (s)) − 1. Then, given any distinct s1 , . . . , sk ∈ [0, L], as δ → 0, E[B(s1 )B(s2 )] = (ef(δ,s1 ) − 1)(ef(δ,s2 ) − 1) + δ gL,ε (s1 , s2 ) + o(δ), E[B(s1 ) · · · B(sk )] = (ef(δ,s1 ) − 1) · · · (ef(δ,sk ) − 1) + o(δ),

k = 2.

(5.5) (5.6)

µf (s) N j dMε (s)) Proof of Theorem 3.1. Let uε (z, µ, f, F , I ) j=1 Fj (z Ij e and let vε (z, µ, f , F , I ) E[uε (z, µ, f , F , I )] so that v(µ, f , F , I ) in Eq. (3.1) satisﬁes v(µ, f, F , I ) = limε→0 vε (z = 1, µ, f, F , I ). The starting point is the limit A

∗ ∂ E∗ [vε (zeB (δ) , µ, f, F , I )] ∂δ δ=0

(5.7)

where B ∗ (δ) is the standard Brownian motion starting at zero that is independent of the process s → ωµ,L,ε (s). We will evaluate this limit in two diﬀerent ways. The ﬁrst is based on the intermittency parameter invariance δ ¯ δ,eL,ε (s) + , B ∗ (δ) + ωµ,L,ε (s) = ωµ−δ,L,ε (s) + ω 2

(5.8)

that we ﬁrst introduced in [13]. This equality is understood as the equality in law of stochastic processes viewed as random functions of s on the interval [0, L] at ﬁxed 0 < δ < µ and ε. The process ω ¯ δ,eL,ε (s) denotes an independent copy of the ωε (s) process at the intermittency parameter δ and rescaled decorrelation length eL, where e denotes the base of natural logarithm. Now, we need to deﬁne three auxiliary quantities. Aδ,ε (s) ωµ−δ,L,ε (s) − ωµ,L,ε (s), A¯δ,ε (s) ω ¯ δ,eL,ε (s), ¯ Cε (δ, f, I) eµf (s) (eAδ,ε (s)+Aδ,ε (s) − 1)dMε (s).

(5.9) (5.10) (5.11)

I

The key observation is that the intermittency parameter invariance in Eq. (5.8) implies the following identity in law

N ∗ δ ¯ Fj ze 2 eµfj (s)+Aδ,ε (s)+Aδ,ε (s) dMε (s) . (5.12) uε (zeB (δ) , µ, f, F , I ) = j=1

Ij

Thus, to compute the limit in Eq. (5.7), we need to expand the right-hand side of Eq. (5.12) in δ up to o(δ) terms. While we do not know how to expand either

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Aδ,ε (s) or A¯δ,ε (s) in δ, they both clearly vanish as δ → 0 and, therefore, so does Cε (δ, f, I). It follows that we can write

ze

Fj

δ 2

e

¯δ,ε (s) µfj (s)+Aδ,ε (s)+A

Ij

z (1) = δ Fj 2

dMε (s)

µfj (s) e dMε (s) eµfj (s) dMε (s) z Ij

Ij

∞ zk (k) µfj (s) Fj e dMε (s) Cεk (δ, fj , Ij ) + o(δ). z + k! Ij

(5.13)

k=0

There results the expansion of N

z (1) F δ 2 j=1 j +

N

δ

j=1

Fj (ze 2

Ij

¯

eµfj (s)+Aδ,ε (s)+Aδ,ε (s) dMε (s)).

N µfj (s) e dMε (s) eµfj (s) dMε (s) Fl z eµfl (s) dMε (s) z Ij

Ij

l=j

Il

∞ N z k1 +···+kN (kl ) Fl z eµfl (s) dMε (s) Cεkl (δ, fl , Il ) + o(δ). k1 ! · · · kN ! Il

k1 ···kN

(5.14)

l=1

The next step in the proof is to compute the expectation of this expression with the goal of eventually evaluating the limit in Eq. (5.7) by means of the identity in Eq. (5.12). There are two expectations involved: the E with respect to ωε process that is inherited from the deﬁnition of vε (z, µ, f, F , I ) and the E∗ expectation with respect to ω ¯ ε process. Interchanging their order, it follows from Eq. (5.14) that N computing the E∗ expectation is now reduced to computing E∗ [ l=1 Cεkl (δ, fl , Il )]. As Aδ,ε (s) and A¯δ,ε (s) are independent processes, it follows from Lemma 5.3 applied to B(s) = exp(Aδ,ε (s) + A¯δ,ε (s)) − 1 with f(δ, s) Aδ,ε (s) that the E∗ expectation equals E

∗

N

Cεkl (δ,

fl , Il )

l=1

=

N l=1

+δ

Il N

kl eµfl (s) [eAδ,ε (s) − 1]dMε (s)

δkl ,1 δkj ,1

l<j

+δ

N l=1

N

δkp ,0

p=l,j

δkl ,2

N p=l

δkp ,0

Il2

Il Ij

eµ(fl (s1 )+fj (s2 )) geL,ε (s1 , s2 )dMε (s1 )dMε (s2 )

eµ(fl (s1 )+fl (s2 )) geL,ε (s1 , s2 )dMε (s1 )dMε (s2 ) + o(δ). (5.15)

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It follows that the expectation of the second term in Eq. (5.14) satisﬁes

∞ N z k1 +···+kN (kl ) ∗ µfl (s) kl E Fl e dMε (s) Cε (δ, fl , Il ) z k1 ! · · · kN ! Il k1 ···kN

l=1

∞ N z k1 +···+kN (kl ) µfl (s) = Fl e dMε (s) z k1 ! · · · kN ! Il k1 ···kN l=1 kl × eµfl (s) [eAδ,ε (s) − 1]dMε (s) Il

+ δz

2

N

(1) Fl

(1) µfl (s) µfj (s) z e dMε (s) Fj e dMε (s) z Il

l<j

× Fk z k=l,j

×

Il

Ij

Ik

Ij

eµfk (s) dMε (s)

eµ(fl (s1 )+fj (s2 )) geL,ε (s1 , s2 )dMε (s1 )dMε (s2 )

N z 2 (2) µfl (s) µfj (s) Fl e dMε (s) Fj z e dMε (s) +δ z 2 Il Ij l=1 j=l × eµ(fl (s1 )+fl (s2 )) geL,ε (s1 , s2 )dMε (s1 )dMε (s2 ) + o(δ). (5.16) Il2

The ﬁrst term on the right-hand side of Eq. (5.16) can be reduced further by means of an elementary identity  

N ∂  Fj z eµfj (s) dMε (s)  ∂µ j=1 Ij =z

N

(1)

Fl

eµfl (s) dMε (s) z Il

l=1

×

Il

eµfl (s) fl (s)dMε (s)

δ→0

Ij

∞ N 1  z k1 +···+kN (kl ) Fl eµfl (s) dMε (s) z δ k1 ! · · · kN ! Il

×

Fj z eµfj (s) dMε (s)

j=l



− lim

Il

k1 ···kN

l=1



kl N eµfl (s) [eAδ,ε (s) − 1]dMε (s) − Fj z eµfj (s) dMε (s) . j=1

Ij

(5.17)

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Putting Eqs. (5.14), (5.16), and (5.17) together, we obtain for the limit in Eq. (5.7)  N ∂ z  A = − E[uε (z, µ, f , F1 · · · FN , I )] + E eµfj (s) dMε (s) ∂µ 2 I j j=1  × uε (z, µ, f,

(1) F1 · · · Fj

· · · FN , I ) + zE

N l=1

Il

eµfl (s) fl (s)dMε (s)



 N (1) eµ(fl (s1 )+fj (s2 )) × uε (z, µ, f, F1 · · · Fl · · · FN , I ) + z 2 E l<j

Il

Ij

 (1) (1) × geL,ε (s1 , s2 )dMε (s1 )dMε (s2 )uε (z, µ, f, F1 · · · Fl · · · Fj · · · FN , I )

N z2 + E eµ(fl (s1 )+fl (s2 )) geL,ε (s1 , s2 )dMε (s1 )dMε (s2 ) 2 2 I l l=1

(2) × uε (z, µ, f, F1 · · · F · · · FN , I ) . l

(5.18)

Finally, recalling the deﬁnition vε (z, µ, f, F , I ) E[uε (z, µ, f, F , I )], the expression in Eq. (5.18) can be simpliﬁed by means of Lemma 5.1. ∂ z ∂ vε (z, µ, f, F1 · · · FN , I ) + vε (z, µ, f, F1 · · · FN , I ) ∂µ 2 ∂z N (1) +z eµfl (s) fl (s)vε (z, µ, f + g(·, s), F1 · · · Fl · · · FN , I )ds

A=−

l=1

+ z2

Il

N l<j

Il

Ij

exp(µ(fl (s1 ) + fj (s2 ) + gL,ε (s1 , s2 )))geL,ε (s1 , s2 )

(1) (1) × vε (z, µ, f + g(·, s1 ) + g(·, s2 ), F1 · · · Fl · · · Fj · · · FN , I )ds1 ds2 N z2 + exp(µ(fl (s1 ) + fl (s2 ) + gL,ε (s1 , s2 )))geL,ε (s1 , s2 ) 2 Il2 l=1

(2) × vε (z, µ, f + g(·, s1 ) + g(·, s2 ), F1 · · · Fl · · · FN , I )ds1 ds2 .

(5.19)

Now, we compute the limit in Eq. (5.7) in a diﬀerent way. By the backward Kolmogorov equation we have 2 1 ∂ 2 ∂ A= +z (5.20) vε (z, µ, f, F1 · · · FN , I ). z 2 ∂z ∂z 2

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It remains to equate Eqs. (5.19) and (5.20) and make use of the identity geL,ε (s1 , s2 ) = 1 + gL,ε (s1 , s2 ).

(5.21)

The result follows by setting z = 1 and letting ε → 0. Proof of Proposition 3.1. We work with general fj . Given non-negative integers kj , j = 1 · · · N, let sj , j = 1 · · · N, denote the vector sj ≡ (sj1 , . . . , sjkj ) such that sjl ∈ Ij ∀ l. For a scalar function of a scalar we will write f (sj ) to mean the kj sum f (sj ) l=1 f (sjl ). Similarly, for a scalar function of two variables we will ki g(t, sip ), and g (t, si ) write g (s, t) (g(s, t), . . . , g(s, t)) N times, g(t, si ) p=1 (g(t, si ), . . . , g(t, si )) N times. Finally, we set

g(si , sj )

 kj ki     g(sip , sjq )  

if i = j

p=1 q=1

(5.22)

ki     g(sip , sjq )  

if i = j.

p
Then, repeated application of the diﬀerentiation rule in Theorem 3.1 shows that there exist functions hn;k1 ···kN (s1 , . . . , sN ) of vectors sj = (sj1 , . . . , sjkj ) such that ∂n v(µ, f, F , I ) = ∂µn

1≤

PN

j=1 kj ≤2n

k

I1 1

···

k

INN

   N N expµ fl (sl ) + g(sl , sj ) l=1

l≤j

N (k ) (k ) × v µ, f + g(·, sl ), F1 1 · · · FN N , I l=1

× hn;k1 ···kN (s1 , . . . , sN )ds1 · · · dsN .

(5.23)

The functions hn;k1 ···kN (s1 , . . . , sN ) are determined by the recurrence  hn+1;k1 ···kN = 

N

fl (sl ) +

l=1

+

N

N

 g(sl , sj ) hn;k1 ···kN

l≤j

i=1

fi (siki ) +

N

g(siki , sl ) hn;k1 ···ki −1···kN

l=1

N

1 g(siki −1 , siki )hn;k1 ···ki −2···kN 2 i=1 + g(siki , sjkj )hn;k1 ···ki −1···kj −1···kN +

i<j

(5.24)

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subject to the initial conditions h1; 0···ki =1···0 (0 · · · si = (s) · · · 0) = fi (s), 1 h1; 0···ki =2···0 (0 · · · si = (s1 , s2 ) · · · 0) = g(s1 , s2 ), 2 h1; 0···ki =1···kj =1···0 (0 · · · si = (s1 ) · · · sj = (s2 ) · · · 0) = g(s1 , s2 ).

(5.25) (5.26) (5.27)

This is all veriﬁed by a straightforward induction argument. Finally, we deﬁne the expansion coeﬃcients ··· hn;k1 ···kN (s1 , . . . , sN )ds1 · · · dsN . (5.28) Hn;k1 ···kN k

I1 1

k

INN

The intermittency expansion in Eq. (3.5) is then the formal Taylor series that follows from Eq. (5.23) by setting µ = 0 and recalling that dM (s) = ds at zero intermittency. Note that we have established the expansion in Eq. (3.5) for general functions fj , in which case the expansion coeﬃcients Hn;k1 ···kN depend on fj , and N the sum is over the range 1 ≤ j=1 kj ≤ 2n. If fj = 0, then this range becomes N 2 ≤ j=1 kj ≤ 2n because of the structure of the recurrence in Eq. (5.24) and the initial condition in Eq. (5.25). Proof of Corollary 3.1. The argument is based on binomial inversion. Consider the joint positive integral moments Sq1 ···qN (µ, I ) of I1 dMµ , . . . , IN dMµ as in Eq. (3.7). Let N j=1 qj ≤ 2n. The corresponding intermittency expansion given in Eq. (3.6) implies N ∂ n ) = S (µ, I H (qj )kj |Ij |qj −kj . (5.29) q ···q n;k ···k 1 N 1 N ∂µn µ=0 j=1 kj ≤qj j=1···N

The result now follows from binomial inversion. Proof of Proposition 3.2. The proof requires a routine calculation using that {ωµ,L,ε (sj )} is jointly Gaussian for any sequence {sj }, and for any Gaussian random variable ω there holds the identity E[exp(ω)] = exp(E[ω] + (1/2) Var[ω]). Proof of Proposition 3.3. The argument is based on symmetrization. Introduce the following notation. We will write Sqk to denote the set of all subsets of {1 · · · k} consisting of exactly q elements, q = 1 · · · k. Given the indices qj ≤ kj , j = 1 · · · N, k and subsets σj ∈ Sqjj of the sets {1 · · · kj } each consisting of qj indices, respectively, we will abbreviate the sum over all the possible pairs of indices as σl l=1···N

log|si − sj |

N i<j p∈σi l∈σj

log|sip − sjl | +

N i=1 p
log|sip − sil |.

(5.30)

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Then, by Proposition 3.2, we can write ∂ n Sq ···q (µ, I ) = (−1)n ∂µn µ=0 1 N ×

···

q q I1 1 ×···×INN

n



 ds1 · · · dsN 

σl ={1···ql } l=1···N

 log |si − sj | .

(5.31)

The integrand is clearly symmetric in (si1 , . . . , siqi ) for every i = 1 · · · N. We can kN by symmetrizing the integrand. Given qj ≤ kj , extend the integral to I1k1 × · · ·× IN j = 1 · · · N, it is easy to see from Eq. (5.31) that there holds the identity N l=1

|Il |kl −ql

n kl ∂ Sq ···q (µ, I ) ql ∂µn µ=0 1 N

= (−1)n



···

k

k

I1 1 ×···×INN

k σl ∈Sqll

n   ds1 · · · dsN  log |si − sj | .

(5.32)

σl l=1···N

l=1···N

We now substitute this identity into the right-hand side of Eq. (3.8), take the sum over q1 · · · qN , and obtain a new representation of the expansion coeﬃcients Hn;k1 ···kN = (−1)n 

(−1)k1 +···+kN k1 ! · · · kN !

···

k

k

I1 1 ×···×INN



ds1 · · · dsN n

    × (−1)q1 +···+qN log|si − sj |   σl k l=1···N ql ≤kl

σl ∈Sqll

   . 

(5.33)

l=1···N

l=1···N

It remains to observe that the integrand in Eq. (5.33) coincides, up to a multiplicative constant, with the terms in the multinomial expansion of    

n

σl ={1···kl } l=1···N

 log|si − sj | 

(5.34)

that involve all the indices (11) · · · (1k1 ) · · · (N 1) · · · (N kN ). The total number of these indices is k1 + · · · + kN . As each term in the multinomial expansion has exactly n factors and each factor has two indices, 2n is the greatest number of indices that can be involved in any of the expansion terms. Hence the result in Eq. (3.10).

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6. Conclusions We have presented a thorough review of the single-increment distribution of the limit lognormal process. In particular, we have reviewed our work on the Mellin transform of the limit distribution and its connection with the Selberg integral. We have also characterized the general transform and its operator formulation. Despite substantial progress in our understanding of the limit distribution, several problems remain outstanding. The problem of uniqueness of our solution for the Mellin transform is open mainly due to the fact that the corresponding moment problem is indeterminate. The problem of computing transforms other than the Mellin transform is open as the action of our operator on functions other than the exponential is unknown. The problem of computing the probability density function of the limit distribution is also open. We have derived the rule of intermittency diﬀerentiation for multiple increments of the limit lognormal process from the intermittency parameter invariance of the underlying normal process. This rule is an exact functional equation that expresses the intermittency derivative of a general class of functionals of the joint distribution of multiple increments in terms of certain expectations over the paths of the limit lognormal process. We have solved this equation by means of formal intermittency expansions and thereby showed how to reconstruct the whole dependence structure of the limit process from the dependence of its joint integral moments on the intermittency parameter. Our solution is a type of exact renormalization because the naive expansion in the joint moments is divergent. Our intermittency expansion bypasses this divergence in a systematic way as it is an expansion involving derivatives of the joint moments at zero intermittency as opposed to the moments themselves so that all the expansion coeﬃcients are ﬁnite. The task of computing particular functionals of the joint distribution of multiple increments is left to future research. Any such calculation requires the knowledge of the joint integral moments as a function of the intermittency parameter. The moments are given by a nontrivial extension of the Selberg integral that is derived in this paper. The value of this integral is unknown in general at the present time. Our results pave the ﬁrst step in the direction of addressing the very practical and equally diﬃcult problem of computing the likelihood of future values of the limit lognormal process given an observation of its past, i.e. the problem of multifractal prediction.

References [1] E. Bacry, J. Delour and J.-F. Muzy, Multifractal random walk, Phys. Rev. E 64 (2001) 026103. [2] E. Bacry, J. Delour and J.-F. Muzy, Modelling ﬁnancial time series using multifractal random walks, Phys. A 299 (2001) 84–92. [3] E. Bacry and J.-F. Muzy, Log-inﬁnitely divisible multifractal random walks, Comm. Math. Phys. 236 (2003) 449–475.

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[4] J. Barral and B. B. Mandelbrot, Multifractal products of cylindrical pulses, Probab. Theory Related Fields 124 (2002) 409–430. [5] I. Benjamini and O. Schramm, KPZ in one dimensional random geometry of multiplicative cascades, Comm. Math. Phys. 289 (2009) 653–662. [6] B. Duplantier and S. Sheﬃeld, Liouville quantum gravity and KPZ (2008); arXiv:0808.1560v1 [math.PR]. [7] P. J. Forrester and S. O. Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. 45 (2008) 489–534. [8] Y. V. Fyodorov, P. Le Doussal and A. Rosso, Statistical mechanics of logarithmic REM: Duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free ﬁelds, J. Stat. Mech. 2009 (2009) P10005; doi: 10.1088/17425468/2009/10/P10005. [9] J.-P. Kahane, Positive martingales and random measures, Chinese Ann. Math. Ser. A 8(1) (1987) 1–12. [10] B. B. Mandelbrot, Possible reﬁnement of the log-normal hypothesis concerning the distribution of energy dissipation in intermittent turbulence, in Statistical Models and Turbulence, eds. M. Rosenblatt and C. Van Atta, Lecture Notes in Physics, Vol. 12 (Springer, New York, 1972), pp. 333–351. [11] B. B. Mandelbrot, Limit lognormal multifractal measures, in Frontiers of Physics: Landau Memorial Conference, eds. E. A. Gotsman et al. (Pergamon, New York, 1990), pp. 309–340. [12] J.-F. Muzy and E. Bacry, Multifractal stationary random measures and multifractal random walks with log-inﬁnitely divisible scaling laws, Phys. Rev. E 66 (2002) 056121. [13] D. Ostrovsky, Functional Feynman–Kac equations for limit lognormal multifractals, J. Stat. Phys. 127 (2007) 935–965. [14] D. Ostrovsky, Intermittency expansions for limit lognormal multifractals, Lett. Math. Phys. 83 (2008) 265–280. [15] D. Ostrovsky, Mellin transform of the limit lognormal distribution, Comm. Math. Phys. 288 (2009) 287–310. [16] D. Ostrovsky, On the limit lognormal and other limit log-inﬁnitely divisible laws, J. Stat. Phys. 138 (2010) 890–911. [17] A. Selberg, Remarks on a multiple integral, Norske Mat. Tidsskr. 26 (1944) 71–78. [18] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th edn. (Cambridge University Press, London, 1958).

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Reviews in Mathematical Physics Vol. 23, No. 2 (2011) 155–178 c World Scientific Publishing Company DOI: 10.1142/S0129055X11004254

EIGENFUNCTIONS AT THE THRESHOLD ENERGIES OF MAGNETIC DIRAC OPERATORS

¯ ∗ and TOMIO UMEDA† YOSHIMI SAITO ∗Department

of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA

†Department

of Mathematical Sciences, University of Hyogo, Himeji 671-2201, Japan ∗[email protected] †[email protected] Received 1 October 2009 Revised 6 December 2010

Discussed are ±m modes and ±m resonances of Dirac operators with vector potentials HA = α · (D − A(x)) + mβ. Asymptotic limits of ±m modes at infinity are derived when |A(x)| ≤ Cx−ρ , ρ > 1, provided that HA has ±m modes. In wider classes of vector potentials, sparseness of the vector potentials which give rise to the ±m modes of HA are established. It is proved that no HA has ±m resonances if |A(x)| ≤ Cx−ρ , ρ > 3/2. Keywords: Dirac operators; magnetic potentials; threshold energies; threshold resonances; threshold eigenfunctions; zero modes. Mathematics Subject Classification 2010: 35Q40, 35P99, 81Q10

1. Introduction The introduction is devoted to exhibiting our results as well as to reviewing previous contributions in connection with the results in the present paper. This paper is concerned with eigenfunctions and resonances at the threshold energies of Dirac operators with vector potentials HA = α · (D − A(x)) + mβ,

D=

1 ∇x , x ∈ R3 . i

Here α = (α1 , α2 , α3 ) is the triple of 4 × 4 Dirac matrices 0 σj αj = (j = 1, 2, 3) σj 0 with the 2 × 2 zero matrix 0 and the triple of 2 × 2 Pauli matrices 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = , 1 0 i 0 0 −1 155

(1.1)

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and β=

I2 0

0 . −I2

The constant m is assumed to be positive. Throughout the present paper, we assume that each component of the vector potential A(x) = (A1 (x), A2 (x), A3 (x)) is a real-valued measurable function. In addition to this, we shall later impose four diﬀerent sets of assumptions on A(x) under which the operator −α · A(x) is relatively compact with respect to the free Dirac operator H0 = α · D + mβ. Therefore, under any set of assumptions to be made, the magnetic Dirac operator HA is a self-adjoint operator in the Hilbert space [L2 (R3 )]4 , and the essential spectrum of HA is given by the union of the intervals (−∞, −m] and [m, +∞): σess (HA ) = (−∞, −m] ∪ [m, +∞).

(1.2)

By the threshold energies of HA , we mean the values ±m, the edges of the essential spectrum σess (HA ). We shall see in Secs. 3–5 that the discrete spectrum of HA in the gap (−m, m) is empty, although we should like to mention that this fact is well known by the result of Thaller [31, Theorem 7.1, p. 195] where smoothness of vector potentials is assumed though. In other words, there are no isolated eigenvalues with ﬁnite multiplicity in the spectral gap (−m, m). In the present paper, this fact will be obtained as a by-product of Theorem 2.3 in Sec. 2, where we shall deal with an abstract Dirac operator, i.e. a supersymmetric Dirac operator. As a result, we shall have σ(HA ) = σess (HA ) = (−∞, −m] ∪ [m, +∞) under any set of the assumptions on A(x) of the present paper. In relation with the relative compactness of −α · A(x) with respect to H0 , it is worthwhile to mention a work by Thaller [30], where he showed that (1.2) is true under the assumption that |B(x)| → 0 as |x| → ∞. Here B(x) denotes the magnetic ﬁeld: B(x) = ∇ × A(x). It is clear that the assumption that |B(x)| → 0 does not necessarily imply the relative compactness of −α·A(x) with respect to H0 . In Helﬀer et al. [16], they showed that (1.2) is true under much weaker assumptions on B(x), which do not even need the requirement that |B(x)| → 0 as |x| → ∞; see also [31, §7.3.2]. It is generally expected that eigenfunctions corresponding to a discrete eigenvalue of HA decay exponentially at inﬁnity (describing bound states), and that (generalized) eigenfunctions corresponding to an energy inside the continuous spectrum (−∞, −m] ∪ [m, +∞) behave like the sum of a plane and a spherical waves at inﬁnity (describing scattering states). At the energies ±m, on which we shall focus in the present paper, (generalized) eigenfunctions are expected to behave like C0 + C1 |x|−1 + C2 |x|−2 at inﬁnity, where Cj , j = 1, 2, 3, are constant vectors in C4 . If C0 = 0 and C1 = 0, then the (generalized) eigenfunctions become either of

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±m resonances, and if C0 = 0 and C1 = 0, then the (generalized) eigenfunctions become either of ±m modes. For the precise deﬁnitions of ±m resonances and ±m modes, see Deﬁnition 6.1 in Sec. 6, and Deﬁnition 3.1 in Sec. 3, respectively. As for the exponential decay of eigenfunctions, we refer the reader to works by Helﬀer and Parisse [17], Wang [32], and a recent work by Yafaev [34]. As for the generalized eigenfunctions corresponding to an energy in (−∞, −m) ∪ (m, +∞), we refer the reader to Yamada [36]. As mentioned above, our main concern is the threshold energies ±m of the magnetic Dirac operator HA . These energies are of particular importance and of interest from the physics point of view. We should like to mention Pickl and D¨ urr [21], and Pickl [20], where they investigate generalized eigenfunctions not only at the energies ±m but also at the energies near ±m, with the emphasis on the famous relativistic eﬀect of the pair creation of an electron and a positron. It is worthwhile to note that ±m modes and ±m resonances play decisive roles in their results. In the same spirit as in [20, 21], Pickl and D¨ urr [22] mention the possibility of experimental veriﬁcations of the pair creation by combining lasers and heavy ions ﬁelds. Therefore, it is obvious that results on ±m modes and ±m resonances of magnetic Dirac operators are useful to understand the physics of the pair creations in such laser ﬁelds; see [22] for details. The goal of the present paper is to derive a series of new results on ±m modes and ±m resonances of the magnetic Dirac operators HA . Precisely speaking, we shall study asymptotic behaviors at inﬁnity of the ±m modes, show sparseness of vector potentials which give rise to the ±m modes, and establish non-existence of ±m resonances. According to Pickl [20, Theorems 3.4 and 3.5], the behavior of the generalized eigenfuntions of Dirac operators near criticality largely depends on whether Dirac operators with critical potentials have ±m resonances or not. Since the modulus of their cirtical potentials are less than or equal to C(1 + |x|)−2 , we can actually conclude from Theorems 6.1 and 6.2 in Sec. 6 that the magnetic Dirac operators with the critical potentials have no ±m resonances. However, one has to pay attention to a slight diﬀerence between our deﬁnition of the threshold resonances (cf. Deﬁnition 6.1) and theirs (cf. [20, Deﬁnition 2.3 and the paragraph after it]). Finally, we would like to mention that there is a striking diﬀerence between twoand three-dimensional Dirac operators with magnetic ﬁelds at the threshold energies. Compare the results in Sec. 6 of the present paper with those of Aharonov and Casher [1], where their arguments indicate that one can ﬁnd magnetic Dirac operators in dimension two which possess threshold resonances. See also [33, Sec. 10]. The plan of the paper is as follows. In Sec. 2, we shall prepare a few results on a supersymmetric Dirac operator, which will be used in all the later sections. In Sec. 3, we shall investigate asymptotic behaviors at inﬁnity of ±m modes of HA , provided that it has ±m modes. Sparseness of the set of vector potentials A(x) which yield ±m modes of HA will be discussed in Secs. 4 and 5 in diﬀerent regimes. In Sec. 6, we shall prove that any HA does not have ±m resonances under a stronger

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assumption than those made in the previous sections. Finally in Sec. 7, we shall give examples of vector potentials A(x) which yield ±m modes of magnetic Dirac operators HA , and shall show that these operators HA do not have ±m resonances. Then we shall propose an open question in relation with ±m resonances. 2. Supersymmetric Dirac Operators This section is devoted to a discussion about spectral properties of a class of supersymmetric Dirac operators. We should like to remark that our approach appears to be in the reverse direction in the sense that we start with two Hilbert spaces, and introduce a supersymmetric Dirac operator on the direct sum of the two Hilbert spaces. We ﬁnd this approach convenient for our purpose; see [31, Chap. 5] for the standard theory of the supersymmetric Dirac operator. The supersymmetric Dirac operator H which we shall consider in the present paper is deﬁned as follows: I 0 0 T∗ (2.1) +m on K = H+ ⊕ H− , H := T 0 0 −I where T is a densely deﬁned operator from a Hilbert space H+ to another Hilbert space H− , m is a positive constant, and the identity operators in H+ and H− are both denoted by I with an abuse of notation. We recall that the domain of H is given by D(H) = D(T ) ⊕ D(T ∗ ), and that the inner product of the Hilbert space K is deﬁned by (f, g)K := (ϕ+ , ψ + )H+ + (ϕ− , ψ − )H− for f=

+ ϕ , ϕ−

g=

ψ+ ψ−

(2.2)

∈ K.

(2.3)

The ﬁrst term on the right-hand side of (2.1) is called the supercharge and the second term the involution, and denoted by Q and τ respectively: I 0 0 T∗ , τ= Q= . (2.4) T 0 0 −I We now state the main results in this section, which are about the nature of eigenvectors of the supersymmetric Dirac operator (2.1) at the eigenvalues ±m. We mention that T does not need to be a closed operator, and that T ∗ does not need to be densely deﬁned, because we only focus on the eigenvalues ±m of H and the corresponding eigenspaces. In the standard theory of the supersymmetric Dirac operator, T is assumed to be a densely deﬁned closed operator and T ∗ needs to be densely deﬁned; cf. [31, §5.2.2]. We should like to draw attention to the fact that Theorems 2.1 and 2.2 below are simply abstract restatements of Thaller [31, Theorem 7.1, p. 195], where he dealt with the magnetic Dirac operators under the assumption that Aj ∈ C ∞ .

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From the mathematically rigorous point of view, it is not appropriate to apply [31, Theorem 7.1] to the magnetic Dirac operators with non-smooth vector potentials. However, the vector potentials we shall treat in Secs. 3–5 are not smooth. In particular we shall deal, in Sec. 4, with vector potentials which can have local singularities. In this case, even self-adjointness of the magnetic Dirac operators is not trivial. Hence [31, Theorem 7.1] is not applicable to this case. These are the reasons why we need to generalize and restate [31, Theorem 7.1] in an abstract setting. Theorem 2.1. Suppose that T is a densely defined operator from H+ to H− . Let H be a supersymmetric Dirac operator defined by (2.1). (i) If f = t (ϕ+ , ϕ− ) ∈ Ker(H − m), then ϕ+ ∈ Ker(T ) and ϕ− = 0. (ii) Conversely, if ϕ+ ∈ Ker(T ), then f = t (ϕ+ , 0) ∈ Ker(H − m). Theorem 2.2. Assume that T and H are the same as in Theorem 2.1. (i) If f = t (ϕ+ , ϕ− ) ∈ Ker(H + m), then ϕ+ = 0 and ϕ− ∈ Ker(T ∗ ). (ii) Conversely, if ϕ− ∈ Ker(T ∗ ), then f = t (0, ϕ− ) ∈ Ker(H + m). As immediate consequences, we have Corollary 2.1. Assume that T and H are the same as in Theorem 2.1. Then (i) Ker(H − m) = Ker(T ) ⊕ {0}, dim(Ker(H − m)) = dim(Ker(T )). (ii) Ker(H + m) = {0} ⊕ Ker(T ∗ ), dim(Ker(H + m)) = dim(Ker(T ∗ )). The eigenspaces corresponding to the eigenvalues ±m of supersymmetric Dirac operators do not seem to have been explicitly formulated in the literature as in the form of Corollary 2.1. It is straightforward from this formulation that the eigenspaces of H corresponding to the eigenvalue ±m are independent of m. Proof of Theorem 2.1. We ﬁrst prove Assertion (i). Let f = t (ϕ+ , ϕ− ) ∈ Ker (H − m). We then have + + + 0 T∗ I 0 ϕ ϕ ϕ +m =m , (2.5) T 0 ϕ− ϕ− ϕ− 0 −I hence

T ∗ ϕ− + mϕ+ = mϕ+

(2.6)

T ϕ+ − mϕ− = mϕ− , which immediately implies that T ∗ ϕ− = 0 and T ϕ+ = 2mϕ− . It follows that T ϕ+ 2H− = (T ϕ+ , T ϕ+ )H− = (T ϕ+ , 2mϕ− )H− = (ϕ+ , 2mT ∗ϕ− )H+ = 0. +

(2.7) −

−1

Thus we see that ϕ ∈ Ker(T ), and that ϕ = (2m)

+

T ϕ = 0.

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We next prove Assertion (ii). Let ϕ+ ∈ Ker(T ) and put f := t (ϕ+ , 0). Then it follows that Hf = t (mϕ+ , T ϕ+ ) = mt (ϕ+ , 0) = mf . We omit the proof of Theorem 2.2, which is quite similar to that of Theorem 2.1. In connection with our applications to the magnetic Dirac operator HA in later sections, we should like to consider the case where the Hilbert space H+ coincides with H− and T is self-adjoint (T ∗ = T ). In this case, the supersymmetric Dirac operator H becomes of the form 0 T I 0 H= +m (2.8) T 0 0 −I in the Hilbert space K = H ⊕ H, and it follows from Theorems 2.1 and 2.2 that the operator H of the form (2.8) possesses of a simple but important equivalence: T has a zero mode ⇔ H has an m mode ⇔ H has a −m mode,

(2.9)

which is actually a well-known fact: see [31, Corollary 5.14, p. 155]. Here we say that T has a zero mode if 0 is an eigenvalue of T . In a similar manner, we say that H has an m mode (respectively, a −m mode) if m (respectively, −m) is an eigenvalue of H. Furthermore, Theorems 2.1 and 2.2 imply the following equivalence for a zero mode ϕ of T : ϕ ϕ 0 0 Tϕ = 0 ⇔ H =m ⇔H = −m . (2.10) 0 0 ϕ ϕ We shall show in Theorem 2.3 below that a suﬃcient condition for the fact that σ(H) = σess (H) = (−∞, −m] ∪ [m, ∞)

(2.11)

is given by the inclusion σ(T ) ⊃ (0, ∞). Therefore ±m are always threshold energies of the supersymmetric Dirac operator H, provided that σ(T ) ⊃ (0, ∞). Theorem 2.3. Let T be a self-adjoint operator in the Hilbert space H. Suppose that σ(T ) ⊃ [0, +∞). Then σ(H) = (−∞, −m] ∪ [m, +∞). In particular, σd (H) = ∅, i.e. the set of discrete eigenvalues of H with finite multiplicity is empty. Proof. It follows from (2.8) that D(H 2 ) = D(T 2 ) ⊕ D(T 2 ) and that 2 T + m2 I 0 0 2 2 I H = ≥m . 0 T 2 + m2 I 0 I This inequality implies that σ(H) ⊂ (−∞, −m] ∪ [m, +∞).

(2.12)

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To complete the proof, we shall prove the fact that σ(H) ⊃ (−∞, −m]∪[m, +∞). 2 2 To this end, suppose λ0 ∈ (−∞, −m] ∪ [m, +∞) be given. Since λ0 − m ≥ 0, we 2 2 see, by the assumption of the theorem, that λ0 − m ∈ σ(T ). Therefore, we can ﬁnd a sequence {ψn }∞ n=1 ⊂ H such that 1 1 (2.13) ψn H = 1, ψn ∈ Ran ET ν0 − , ν0 + , ν0 := λ20 − m2 n n for each n, where ET (·) is the spectral measure associated with T : ∞ T = λ dET (λ).

(2.14)

−∞

Here we have used a basic property of the spectral measure: see, for example [25, Proposition, p. 236]. It is straightforward to see that (T − ν0 )ψn H → 0 as n → ∞.

(2.15)

We shall construct a sequence {fn } ⊂ D(H) = D(T ) ⊕ D(T ) satisfying fn K = 1 and (H − λ0 )fn K → 0 as n → ∞. To this end, we choose a pair of real numbers a and b so that a2 + b 2 = 1 and that

m ν0

ν0 −m

(2.16)

a a . = λ0 b b

(2.17)

This is possible because the 2 × 2 symmetric matrix in (2.17) has eigenvalues ±λ0 . We now put aψn . (2.18) fn := bψn It is easy to see that fn K = 1. By using (2.16) and (2.17), we can show that (H − λ0 )fn 2K = (m − λ0 )aψn + bT ψn 2H + aT ψn − (m + λ0 )bψn 2H = b(−ν0 + T )ψn 2H + a(T − ν0 )ψn 2H = (T − ν0 )ψn 2H → 0

as n → ∞.

We thus have shown that λ0 ∈ σ(H). Here we brieﬂy mention of the abstract Fouldy–Wouthuysen transformation UFW in connection with Theorem 2.3. The transformation UFW is a unitary operator in K, and transforms the supersymmetric Dirac operator H of the form (2.8) into the diagonal form: √ 2 T + m2 0 ∗ √ . UFW HUFW = 0 − T 2 + m2

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Note that it is possible to prove (2.11) based on this unitary equivalence. For the abstract Fouldy–Wouthuysen transformation for the supersymmetric Dirac operator of the form (2.1), we refer the reader to Thaller [31, Chap. 5, §5.6]. In all the later sections, we shall apply the obtained results on the supersymmetric Dirac operator to the magnetic Dirac operator HA of the form (1.1) in the Hilbert space K = [L2 (R3 )]4 , where we take T to be the Weyl–Dirac operator TA = σ · (D − A(x))

(2.19)

acting in the Hilbert space H = [L2 (R3 )]2 . As was mentioned above (cf. (2.9) and (2.10)), the investigations of properties of ±m modes of the magnetic Dirac operator HA are reduced to the investigations of the corresponding properties of zero modes of the Weyl–Dirac operator TA = σ · (D − A(x)). We have to emphasize the broad applicability of the supersymmetric Dirac operator in the context of the present paper. Namely, thanks to the generality of Theorems 2.1 and 2.2, we are able to utilize most of the existing works on the zero modes of the Weyl–Dirac operator TA (cf. [2–8, 10–14, 19]) for the purpose of investigating ±m modes of the magnetic Dirac operator HA . 3. Asymptotic Limits of ±m Modes In this section, we consider a class of magnetic Dirac operators HA under Assumption(SU) below, and will focus on the asymptotic behaviors at inﬁnity of ±m modes of HA , assuming that ±m are the eigenvalues of HA . In Sec. 7, we shall see that there exists inﬁnitely many A’s such that the corresponding magnetic Dirac operators HA have the threshold eigenvalues ±m. We now introduce the terminology of ±m modes for the magnetic Dirac operator HA . Definition 3.1 (Following [18]). By an m mode (respectively, a −m mode), we mean an eigenfunction corresponding to the eigenvalue m (respectively, −m) of HA , provided that the threshold energy m (respectively −m) is an eigenvalue of HA . Assumption(SU). Each element Aj (x) (j = 1, 2, 3) of A(x) is a measurable function satisfying |Aj (x)| ≤ Cx−ρ

(ρ > 1),

(3.1)

where C is a positive constant. It is easy to see that under Assumption(SU) the Dirac operator HA is a selfadjoint operator in the Hilbert space K = [L2 (R3 )]4 with Dom(HA ) = [H 1 (R3 )]4 , where H 1 (R3 ) denotes the Sobolev space of order 1. Also it is easy to see that under Assumption(SU) the Weyl–Dirac operator TA is a self-adjoint operator in the Hilbert space H = [L2 (R3 )]2 with Dom(TA ) = [H 1 (R3 )]2 . Since the operator

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−σ · A(x) is relatively compact with respect to the operator T0 := σ · D, and since σ(T0 ) = R, it follows that σ(TA ) = R. Recalling that 0 TA I 0 HA = +m , (3.2) TA 0 0 −I we can apply Theorem 2.3 to HA , and get σ(HA ) = σess (HA ) = (−∞, −m] ∪ [m, +∞). Hence ±m are the threshold energies of the operator HA . Assuming that ±m are the eigenvalues of HA , we ﬁnd that the eigenspaces corresponding to the eigenvalues ±m of HA are given as the direct sum of Ker(TA ) and the zero space {0} (cf. Corollary 2.1 in Sec. 2), and that these two eigenspaces themselves as well as their dimensions are independent of m. Theorem 3.1. Suppose that Assumption(SU) is verified, and that m (respectively, −m) is an eigenvalue of HA . Let f be an m mode (respectively, a −m modes) of HA . Then there exists a zero mode ϕ+ (respectively, ϕ− ) of TA such that for any ω ∈ S2 + u (ω) 0 2 lim r f (rω) = , (3.3) respectively, − r→∞ 0 u (ω) where u± (ω) =

i 4π

R3

{(ω · A(y))I2 + iσ · (ω × A(y))}ϕ± (y)dy,

(3.4)

and the convergence is uniform with respect to ω. Theorem 3.1 is a direct consequence of Corollary 2.1, together with [27, Theorem 1.2]. Note that under Assumption(SU) every eigenfunction of HA corresponding to either one of eigenvalues ±m is a continuous function of x (cf. [28, Theorem 2.1]), therefore the expression f (rω) in (3.3) makes sense for each ω. 4. Sparseness of Vector Potentials Yielding ±m Modes In this section, we shall discuss the sparseness of the set of vector potentials A which give rise to ±m modes of magnetic Dirac operators HA , in the sprit of Balinsky and Evans [5,6], where they investigated Pauli operators and Weyl–Dirac operators respectively. We shall make the following assumption: Assumption(BE). Aj ∈ L3 (R3 ) for j = 1, 2, 3. Under Assumption(BE) Balinsky and Evans [6, Lemma 2] showed that −σ · A is inﬁnitesimally small with respect to T0 = σ · D with Dom(T0 ) = [H 1 (R3 )]2 (see (4.5) below). This fact enables us to deﬁne the self-adjoint realization TA in the Hilbert space H = [L2 (R3 )]2 as the operator sum of T0 and −σ ·A, thus Dom(TA ) = [H 1 (R3 )]2 . It turns out that under Assumption(BE), −α · A is inﬁnitesimally small

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with respect to H0 := α·D+mβ, and hence we can deﬁne the self-adjoint realization HA in the Hilbert space K = [L2 (R3 )]4 as the operator sum of H0 and −α · A, thus Dom(HA ) = [H 1 (R3 )]4 . Therefore we can regard HA as a supersymmetric Dirac operator, and shall apply the results in Sec. 2 to HA . (Recall (3.2).) Proposition 4.1. Let Assumption(BE) be satisfied. Then σ(TA ) = R. We shall prepare a few lemmas for the proof of Proposition 4.1. Lemma 4.1. Let z ∈ C\R. Then D1/2 (T0 − z)−1 is a bounded operator in H. Moreover we have Ran(D1/2 (T0 − z)−1 ) ⊂ [H 1/2 (R3 )]2 .

(4.1)

Proof. It is suﬃcient to show the conclusions of the lemma for z = −i. Let ϕ ∈ Dom(T0 ). Then we have 2 |((σ · ξ) + iI2 )ϕ(ξ)| 2C2 dξ (T0 + i)ϕH = R3

=

R3

(|ξ|2 + 1)|ϕ(ξ)| 2C2 dξ

= Dϕ2H ,

(4.2)

where we have used the anti-commutation relation σj σk + σk σj = 2δjk I2 in the second equality. It follows from (4.2) that ϕH = D(T0 + i)−1 ϕH

(4.3)

for all ϕ ∈ H. Furthermore, we see that D1/2 (T0 + i)−1 ϕH ≤ D1/2 (T0 + i)−1 ϕ[H 1/2 (R3 )]2 = D(T0 + i)−1 ϕH = ϕH .

(4.4)

It is evident that (4.4) proves the conclusions of the lemma for z = −i. Lemma 4.2. If ϕ ∈ [H 1/2 (R3 )]2 , then (σ · A)D−1/2 ϕ ∈ H. Proof. By [6, Lemma 2], we see that for any > 0, there exists a constant k > 0 such that for all ϕ ∈ Dom(T0 ) (σ · A)ϕH ≤ T0 ϕH + k ϕH . −1/2

By virtue of the fact that D (4.5) that

ϕ ∈ Dom(T0 ) for ϕ ∈ [H

(4.5) 1/2

3

2

(R )] , it follows from

(σ · A)D−1/2 ϕH ≤ (T0 + i)D−1/2 ϕH + k D−1/2 ϕH ≤ D1/2 ϕH + k ϕH < +∞, where we have used (4.2) and the fact that D1/2 ϕH = ϕ[H 1/2 (R3 )]2 .

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Lemma 4.3. D−1 (σ · A)D−1/2 is a compact operator in H. Proof. One can make a factorization |D|1/2 1 |D|1/2 1 −1 −1/2 D (σ · A)D = (σ · A) . D |D|1/2 |D|1/2 D1/2

(4.6)

It is obvious that the ﬁrst term and the last term on the right-hand side of (4.6) are bounded operators in H. Then it follows from (4.6) and [6, Lemma 1] that the conclusion of the lemma holds true. Lemma 4.4. Let z ∈ C\R. Then (TA − z)−1 D|[H 1 (R3 )]2 can be extended to a A (z) in H. Moreover bounded operator R A (z)D−1 ϕ (TA − z)−1 ϕ = R

for ∀ ϕ ∈ H.

(4.7)

Proof. We ﬁrst show that D(T − z)−1 is a closed operator in H. To this end, suppose that {ϕj } is a sequence in H such that ϕj → 0 in H and D(T − z)−1 ϕj → ψ in H as j → ∞. Then {(T − z)−1 ϕj } is a Cauchy sequence in [H 1 (R3 )]2 , hence there exists a ψ ∈ [H 1 (R3 )]2 such that (T − z)−1 ϕj → ψ in [H 1 (R3 )]2

as j → ∞.

(4.8)

Since the topology of [H 1 (R3 )]2 is stronger than that of H, (4.8) implies that (T − z)−1 ϕj → ψ in H

as j → ∞.

(4.9)

On the other hand, since ϕj → 0 in H, and since (T − z)−1 is a bounded operator in H, we see that (T − z)−1 ϕj → 0 in H

(4.10)

as j → ∞. Combining (4.9) and (4.10), we see that ψ = 0. This fact, together with (4.8), D(T − z)−1 ϕj → 0 in H as j → ∞. Hence ψ = 0. We have thus shown that D(T − z)−1 is a closed operator. Noting that Dom(D(T − z)−1 ) = H, we can conclude from the Banach closed graph theorem that D(T − z)−1 is a bounded operator in H, which will be denoted by QA (z). A (z) := QA (z)∗ , where QA (z)∗ denotes the adjoint operator of We now put R QA (z). Then for any ϕ ∈ H and any ψ ∈ [H 1 (R3 )]2 , we have A (z)ψ)H = (QA (z)ϕ, ψ)H (ϕ, R = (D(T − z)−1 ϕ, ψ)H = (ϕ, (T − z)−1 Dψ)H .

(4.11)

It follows from (4.11) that A (z)ψ = (T − z)−1 Dψ R

(4.12)

for all ψ ∈ [H 1 (R3 )]2 . Replacing ψ in (4.12) with D−1 ϕ, ϕ ∈ H, we get (4.7).

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Proof of Proposition 4.1. Since σ(T0 ) = σess (T0 ) = R, it is suﬃcient to show that σess (TA ) = σess (T0 ).

(4.13)

To this end, we shall prove that the diﬀerence (TA + i)−1 − (T0 + i)−1 is a compact operator in H. Then, this fact implies (4.13); see [26, Corollary 1, p. 113]. We see that (TA + i)−1 − (T0 + i)−1 = (TA + i)−1 (σ · A)(T0 + i)−1 A (−i){D−1 (σ · A)D−1/2 }{D1/2 (T0 + i)−1 }, =R

(4.14)

where we have used Lemma 4.4 in (4.14). It follows from Lemmas 4.1–4.4 that (4.14) makes sense as a product of three bounded operators in H and that the product is a compact operator in H. Proposition 4.1, together with Theorem 2.3, gives the following result on the spectrum of the magnetic Dirac operator HA . Theorem 4.1. Let Assumption(BE) be satisfied. Then σ(HA ) = σess (HA ) = (−∞, −m] ∪ [m, ∞). We now state the main results in this section, which are concerned with the eigenspaces corresponding to the threshold eigenvalues of the magnetic Dirac operator HA . Theorem 4.2. Let Assumption(BE) be satisfied. Then (i) Ker(HA − m) is non-trivial if and only if Ker(HA + m) is non-trivial; in other words, {A ∈ [L3 (R3 )]3 | Ker(HA −m) = {0}} = {A ∈ [L3 (R3 )]3 | Ker(HA + m) = {0}}. (ii) There exists a constant c such that dim(Ker(HA − m)) = dim(Ker(HA + m)) ≤ c

R3

|A(x)|3 dx.

(4.15)

Moreover, the dimension of Ker(HA ∓ m) is independent of m. (iii) The set {A ∈ [L3 (R3 )]3 | Ker(HA ∓ m) = {0}} contains an open dense subset of [L3 (R3 )]3 . Proof. By Corollary 2.1, we see that Ker(TA ) is trivial ⇔ Ker(HA − m) is trivial ⇔ Ker(HA + m) is trivial.

(4.16)

Assertion (i) is equivalent to (4.16). Assertion (ii) follows from Corollary 2.1 and [6, Theorem 3]. Assertion (iii) follows from Corollary 2.1 and [6, Theorem 2].

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Remark 4.1. Assertions (i) and (ii) of Theorem 4.2 mean the following facts: The threshold energy m is an eigenvalue of HA if and only if the threshold energy −m is an eigenvalue of HA . If this is the case, their multiplicity are the same. Remark 4.2. As for the best constant in the inequality (4.15), see [6, Theorem 3]. 5. The Structure of the Set of Vector Potentials Yielding ±m Modes In this section, we shall discuss a property of non-locality of magnetic vector potentials as well as the sparseness of the set of vector potentials A which give rise to ±m modes of HA in the spirit of Elton [11], where he investigated Weyl–Dirac operators. We make the following assumption: Assumption(E). Each Aj (j = 1, 2, 3) is a real-valued continuous function such that Aj (x) = o(|x|−1 ) as |x| → ∞. It is straightforward to see that under Assumption(E), −σ · A is a bounded self-adjoint operator in the Hilbert space H = [L2 (R3 )]2 . Hence we can deﬁne the self-adjoint realization TA with Dom(TA ) = [H 1 (R3 )]2 as the operator sum of T0 and −σ · A. Also, it is straightforward to see that −α·A is a bounded self-adjoint operator in the Hilbert space K = [L2 (R3 )]4 , hence we can deﬁne the self-adjoint realization HA with Dom(HA ) = [H 1 (R3 )]4 in K as the operator sum of H0 and −α · A. Therefore, in the same way as in Sec. 5, we can regard HA as a supersymmetric Dirac operator, and apply the results in Sec. 2 to HA . We note that under Assumption(E), (−σ · A)(T0 + i)−1 is a compact operator in H. Hence, in the same way as in the proof of Proposition 4.1, we can show that σ(TA ) = R. This fact, together with Theorem 2.3, implies the following result. Theorem 5.1. Let Assumption(E) be satisfied. Then σ(HA ) = σess (HA ) = (−∞, −m] ∪ [m, ∞). To state the main results in this section, we need to introduce the following notation: A := {A | A satisﬁes Assumption(E)}. We regard A as a Banach space with the norm AA = sup{x|A(x)|}. x

Theorem 5.2. Let Assumption(E) be satisfied. Define Zk± = {A ∈ A | dim(Ker(H ∓ m)) = k} for k = 0, 1, 2, . . . . Then (i) Zk+ = Zk− for all k.

(5.1)

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(ii) Z0± is an open and dense subset of A. (iii) For any k and any open subset Ω(= ∅) of R3 there exists an A ∈ Zk± such that A ∈ [C0∞ (Ω)]3 . Proof. Assertion (i) is a direct consequence of Corollary 2.1. Assertions (ii) and (iii) follows from Corollary 2.1 and [11, Theorem 1]. It is of some interest to point out a conclusion following from Theorem 3.1 and Assertion (iii) of Theorem 5.2. Namely, there are (at least) countably inﬁnite number of vector potentials A with compact support such that the corresponding Dirac operators HA have ±m modes f ± with the property (3.4). The ±m modes f ± behave like |f ± (x)| |x|−2 for |x| → ∞, in spite of the fact that the vector potentials and the corresponding magnetic ﬁelds vanish outside bounded regions. It is obvious that this phenomenon describes a certain kind of non-locality. Also, it is of some interest to mention that HA does not have ±m resonances if the support of vector potential A is compact. This is an immediate consequence of Theorem 6.1 in the next section. 6. Non-Existence of ±m Resonances In this section, we will work in bigger Hilbert spaces than H = [L2 (R3 )]2 and K = [L2 (R3 )]4 . Therefore, the results on the supersymmetric Dirac operators in Sec. 2 are not applicable in this section. In this section, we shall occasionally write the inner product of H as (ϕ(x), ψ(x))C2 dx (ϕ, ψ)H = R3

for ϕ, ψ ∈ H, where (·, ·)C2 denotes the inner product of C2 . We need to introduce weighted L2 spaces in order to deal with ±m resonances, which do not belong to the Hilbert space K. By L2,s (R3 ), we mean the weighted L2 space deﬁned by L2,s (R3 ) := {u | xs u ∈ L2 (R3 )}

where x = 1 + |x|2 , and we set

(s ∈ R)

L2,s = [L2,s (R3 )]4 . Definition 6.1. By an m resonance (respectively, a −m resonance), we mean a function f ∈ L2,−s \K, 0 < s ≤ 3/2, such that HA f = mf (respectively, HA f = −mf ) in the distributional sense. We would like to caution that in Deﬁnition 6.1 one has to take the meaning of HA f = ±mf in the distributional sense, because of the reason that ±m resonances do not belong to the Hilbert space K, hence do not belong to the domain of the selfadjoint realization of HA . For this reason, we let HA stand for the formal diﬀerential

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operator throughout this section, in spite of the fact that HA has the unique selfadjoint realization in K under the assumption of Theorem 6.1 below. We hope this will not cause any confusion. Theorem 6.1. Assume that each element Aj (x) (j = 1, 2, 3) of A(x) is a measurable function satisfying |Aj (x)| ≤ Cx−ρ

(ρ > 3/2),

(6.1)

where C is a positive constant. Suppose that f = t (ϕ+ , ϕ− ) belongs to L2,−s for some s with 0 < s < min(1, ρ − 1) and satisfies HA f = mf (respectively, HA f = −mf ) in the distributional sense. Then f ∈ [H 1 (R3 )]4 and ϕ− = 0 (respectively, ϕ+ = 0). Theorem 6.1 implies the non-existence of ±m resonances in the sense of Deﬁnition 6.1, as well as in the sense described in the following theorem. Theorem 6.2. Let A(x) satisfy the same assumption as in Theorem 6.1. Suppose that f belongs to [L2loc (R3 )]4 and satisfies either equation of HA f = ±mf in the distributional sense. In addition, suppose that f has the asymptotic expansion f (x) = C1 |x|−1 + C2 |x|−2 + o(|x|−2 )

(6.2)

as |x| → ∞, where C1 and C2 are constant vectors in C4 . Then C1 = 0. Proof. It follows from (6.2) that f ∈ L2,−s for any s with 1/2 < s < 1. This fact, together with the assumptions of the theorem, enables us to apply Theorem 6.1 and to conclude that f ∈ [H 1 (R3 )]4 . In particular, f ∈ K, which leads to the fact that C1 = 0. We shall give a proof of Theorem 6.1 only for m resonances, since the proof for −m resonances is similar. Roughly speaking, we will mimick the idea of the proof of Assertion (i) of Theorem 2.1. Therefore we need the Weyl–Dirac operator TA = σ · (D − A(x)) again. However, we are not allowed to use the Weyl–Dirac operator as a self-adjoint operator in the Hilbert space H, but only allowed to use it as a formal diﬀerential operator instead. This is because ±m resonances do not belong to K. This fact causes complication, in a certain extent, in the proof of Theorem 6.1. We begin the proof of Theorem 6.1 with a lemma whose proof will be given after the proof of the theorem. The proof of the lemma is lengthy. Lemma 6.1. Under the hypotheses of Theorem 6.1, ϕ± have the following properties: (i) (σ · D)ϕ+ ∈ H, (σ · A)ϕ+ ∈ H and ϕ− ∈ [H 1 (R3 )]2 . (ii) ((σ · D)ϕ+ , ϕ− )H = ((σ · A)ϕ+ , ϕ− )H .

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Proof of Theorem 6.1. Let f satisfy HA f = mf in the distributional sense. Then we have mϕ+ + σ · (D − A(x))ϕ− = mϕ+ (6.3) σ · (D − A(x))ϕ+ − mϕ− = mϕ− in the distributional sense, which immediately implies σ · (D − A(x))ϕ− = 0

(6.4)

σ · (D − A(x))ϕ+ = 2mϕ− .

(6.5)

and

In view of Lemma 6.1, it follows from (6.5) that 4m2 ϕ− 2H = 2m(2mϕ− , ϕ− )H = 2m(σ · (D − A)ϕ+ , ϕ− )H = 2m{((σ · D)ϕ+ , ϕ− )H − ((σ · A)ϕ+ , ϕ− )H } = 0.

(6.6)

Hence ϕ− = 0. This fact, together with (6.5), means that σ · (D − A(x))ϕ+ = 0

(6.7)

in the distributional sense. It follows from [28, Theorem 2.2] that ϕ+ ∈ [H 1 (R3 )]2 . (Note that the hypothesis 0 < s < min(1, ρ − 1) is stronger than the one imposed in [28, Theorem 2.2].) This implies that f ∈ [H 1 (R3 )]4 , because ϕ− = 0 as was shown above. Before proving Lemma 6.1, we should like to remark that (6.4) and (6.5) follow directly from the hypothesis that HA f = mf in the distributional sense. Therefore we are allowed to use (6.4) and (6.5) in the proof of Lemma 6.1 below. Proof of Lemma 6.1. Since ρ − s > 1 by assumption, we see that (σ · A)ϕ+ ∈ [L2,ρ−s (R3 )]2 ⊂ H.

(6.8)

It follows from (6.4) and [28, Theorem 2.2] that ϕ− ∈ [H 1 (R3 )]2 . This fact, together with (6.5) and (6.8), implies that (σ · D)ϕ+ = 2mϕ− + (σ · A)ϕ+ ∈ H. Thus Assertion (i) is proved. In order to prove Assertion (ii), we need to introduce a cutoﬀ function. Let χ be a function in C ∞ (R) such that 0 ≤ χ ≤ 1, χ(r) = 1 (r ≤ 1), and χ(r) = 0 (r ≥ 2). Set χn (x) = χ(|x|/n) (n = 1, 2, 3, . . .).

(6.9)

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It is evident that ((σ · D)ϕ+ , ϕ− )H = lim ((σ · D)ϕ+ , χn ϕ− )H . n→∞

(6.10)

Let {jε }0<ε<1 be Friedrichs’ molliﬁer, i.e. j (x) := ε−3 j(x/ε), where j ∈ C0∞ (R3 ) and jL1 = 1. Since χn ϕ− ∈ H, we see that jε ∗ (χn ϕ− ) converges to χn ϕ− in H as ε ↓ 0. Hence, for each n, we have ((σ · D)ϕ+ , χn ϕ− )H = lim((σ · D)ϕ+ , jε ∗ (χn ϕ− ))H . ε↓0

(6.11)

It is straightforward that jε ∗ (χn ϕ− ) ∈ [C0∞ (R3 )]2 and that supp[jε ∗ (χn ϕ− )] ⊂ {x | |x| ≤ 2n + 1}.

(6.12)

Appealing to the deﬁnition of the distributional derivatives, we get ((σ · D)ϕ+ , jε ∗ (χn ϕ− ))H =

R3

(ϕ+ (x), (σ · D)(jε ∗ (χn ϕ− ))(x))C2 dx.

(6.13)

For each n and ε, we have

−

(σ · D)(jε ∗ (χn ϕ ))(x) =

R3

= R3

= R3

(σ · Dx )(jε (x − y))χn (y)ϕ− (y)dy −(σ · Dy )(jε (x − y))χn (y)ϕ− (y)dy jε (x − y)(σ · Dy )(χn (y)ϕ− (y))dy

= jε ∗ {(σ · D)(χn ϕ− )}(x).

(6.14)

In the third equality of (6.14), we have regarded jε (x − ·) as a function in C0∞ (R3y ) and have appealed to the deﬁnition of the destributional derivatives with respect to y variable. Note that (σ · D)(χn ϕ− ) = {(σ · D)χn }ϕ− + χn (σ · D)ϕ− .

(6.15)

Combining (6.13)–(6.15), we obtain ((σ · D)ϕ+ , jε ∗ (χn ϕ− ))H =

R3

(ϕ+ (x), jε ∗ [{(σ · D)χn }ϕ− ](x))C2 dx

+ R3

(ϕ+ (x), jε ∗ [χn (σ · D)ϕ− ](x))C2 dx.

(6.16)

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We examine the limit of each integral on the right-hand side of (6.16) as ε ↓ 0. As for the ﬁrst integral, we have + − 3 (ϕ (x), jε ∗ [{(σ · D)χn }ϕ ](x))C2 dx R + − − (ϕ (x), {(σ · D)χn }(x)ϕ (x))C2 dx ≤

R3

|x|≤2n+1

|ϕ+ (x)|C2

× |jε ∗ [{(σ · D)χn }ϕ− ](x) − {(σ · D)χn }(x)ϕ− (x)|C2 dx ≤ |ϕ+ |C2 L2 (|x|≤2n+1) jε ∗ [{(σ · D)χn }ϕ− ] − {(σ · D)χn }ϕ− H →0

(ε ↓ 0),

(6.17)

since {(σ · D)χn }ϕ− ∈ H. In the ﬁrst inequality (6.17) we have used the Schwarz inequality in C2 , and in the second inequality the Schwarz inequality in L2 . Therefore (ϕ+ (x), jε ∗ [{(σ · D)χn }ϕ− ](x))C2 dx lim ε↓0

R3

= R3

(ϕ+ (x), {(σ · D)χn }(x)ϕ− (x))C2 dx.

(6.18)

In a similar manner, we see that lim ε↓0

R3

(ϕ+ (x), jε ∗ [χn (σ · D)ϕ− ](x))C2 dx

= R3

(ϕ+ (x), χn (x)(σ · D)ϕ− (x))C2 dx,

(6.19)

where we have used the fact that ϕ− ∈ [H 1 (R3 )]2 . (Recall that this fact was shown in Assertion (i) of the lemma.) It follows from (6.11), (6.16), (6.18) and (6.19) that +

−

((σ · D)ϕ , χn ϕ )H =

R3

(ϕ+ (x), {(σ · D)χn }(x)ϕ− (x))C2 dx

+ R3

(ϕ+ (x), χn (x)(σ · D)ϕ− (x))C2 dx.

(6.20)

To estimate the ﬁrst integral on the right-hand side of (6.20), we need the fact that 1 |x| 1 (σ · ω) (ω = x/|x|). (6.21) {(σ · D)χn }(x) = χ n n i

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Note that supp[(σ · D)χn ] ⊂ {x | n ≤ |x| ≤ 2n} and that σ · ω is a unitary matrix. Hence we have (ϕ+ (x), {(σ · D)χn }(x)ϕ− (x))C2 dx 3 R 1 |ϕ+ (x)|C2 |ϕ− (x)|C2 dx sup |χ (r)| ≤ n r>0 n≤|x|≤2n

1/2 1 −2s + 2 x |ϕ (x)|C2 dx ≤ sup |χ (r)| n r>0 n≤|x|≤2n ×

1/2 −

x |ϕ n≤|x|≤2n

1 ≤ n

2s

(x)|2C2 dx

sup |χ (r)| | ϕ+ |C2 L2,−s × (1 + 4n2 )s/2 ϕ− H r>0

≤ const. n−1+s | ϕ+ |C2 L2,−s ϕ− H → 0 (n → ∞),

(6.22)

since s < 1 by assumption of Theorem 6.1. Thus the ﬁrst integral on the right-hand side of (6.20) tends to 0 as n → ∞. We now investigate the limit of the second integral on the right-hand side of (6.20) as n → ∞. It follows from (6.4) that + − (ϕ (x), χn (x)(σ · D)ϕ (x))C2 dx − ((σ · A)(x)ϕ+ (x), ϕ− (x))C2 dx R3

R3

= R3

((χn (x) − 1)(σ · A)(x)ϕ+ (x), ϕ− (x))C2 dx,

(6.23)

where we have used the fact that (σ · A)(x) is a Hermitian matrix for each x. Noting (6.8), we ﬁnd that the absolute value of the right-hand side of (6.23) is less than or equal to (χn − 1)(σ · A)ϕ+ H ϕ− H ,

(6.24)

which obviously tends to 0 as n → ∞. Combining this fact with (6.20), (6.22) and (6.23), we obtain lim ((σ · D)ϕ+ , χn ϕ− )H = ((σ · A)ϕ+ , ϕ− )H .

n→∞

(6.25)

Assertion (ii) of the lemma is a direct consequence of (6.10) and (6.25). 7. Examples, Concluding Remarks and an Open Question We shall give examples of vector potentials A(x) which yield ±m modes but do not give rise to ±m resonances. The basic idea in this section is to exploit the equivalences (2.9), (2.10), and to apply Theorem 6.1. It turns out that beautiful spectral

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properties are in common to all the examples of the magnetic Dirac operators in this section. See Properties (i)–(iv) of Examples 7.1 and 7.2. Example 7.1 ([19]). Let ALY (x) = 3x−4 {(1 − |x|2 )w0 + 2(w0 · x)x + 2w0 × x}

(7.1)

where φ0 = t (1, 0) (φ0 can be any unit vector in C2 ) and w0 = φ0 · (σφ0 ) := ((φ0 , σ1 φ0 )C2 , (φ0 , σ2 φ0 )C2 , (φ0 , σ3 φ0 )C2 ).

(7.2)

Here w0 · x and w0 × x denotes the inner product and the exterior product of R3 respectively. Then the magnetic Dirac operator HLY := HALY = α · (D − ALY (x)) + mβ has the following properties: (i) σ(HLY ) = σess (HLY ) = (−∞, −m] ∪ [m, ∞); (ii) HLY has ±m modes. Moreover, the point spectrum of HLY consists only of ±m, i.e. σp (HLY ) = {−m, m}; (iii) HLY does not have ±m resonances; (iv) HLY is absolutely continuous on (−∞, −m) ∪ (m, ∞). We shall show these properties one-by-one. It is easy to see that −σ · ALY (x) is relatively compact perturbation of T0 = σ · D, hence the Weyl–Dirac operator TLY := TALY = σ · (D − ALY (x)) is a self-adjoint operator in the Hilbert space H = [L2 (R3 )]2 with the domain [H 1 (R3 )]2 . Since the spectrum of the operator T0 equals the whole real line, we see that σ(TLY ) = R. Property (i) immediately follows from Theorem 2.3. We shall show Property (ii). According to Loss and Yau [19, Sec. II], the Weyl– Dirac operator TLY has a zero mode ϕLY deﬁned by ϕLY (x) = x−3 (I2 + iσ · x)φ0 .

(7.3)

It follows from (2.9) and (2.10) that t (ϕLY , 0) (respectively, t (0, ϕLY )) is an m mode (respectively, a −m mode) of HLY . Hence σp (HLY ) ⊃ {−m, m}. On the other hand, it follows from Yamada [35] that HLY has no eigenvalue in (−∞, −m) ∪ (m, ∞). (Note that the vector potential ALY satisﬁes the assumption of [35, Proposition 2.5].) This fact, together with Property (i), implies that σp (HLY ) ⊂ {−m, m}. Summing up, we get Property (ii). Since |ALY (x)| ≤ Cx−2 , Property (iii) follows from Theorem 6.1. Property (iv) is a direct consequence of [35, Corollary 4.2]. As for absolutely continuity and limiting absorption principle for Dirac operators, see also [36, 9, 24]. Remark 7.1. Since ALY is C ∞ , one can apply Thaller [31, Theorem 7.1, p. 195] to conclude that t (ϕLY , 0) (respectively, t (0, ϕLY )) is an m mode (respectively a −m mode) of HLY . This fact is also mentioned in [30].

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Remark 7.2. As was pointed out in [19, Sec. II], one sees that div ALY = 0, and one can ﬁnd, by a gauge transformation, a vector potential A˜LY which satisﬁes div A˜LY = 0 and rot A˜LY = rot ALY and yields a zero mode ϕ˜LY of σ·(D−A˜LY (x)). In fact, deﬁning A˜LY := ALY + ∇χLY , with χLY (x) :=

1 4π

R3

ϕ˜LY := eiχLY ϕLY

1 (div ALY )(y)dy, |x − y|

we observe that A˜LY and ϕ˜LY have the desired properties mentioned above. Moreover, we can show that |A˜LY (x)| ≤ Cx−2 (∈ L6 (R3 )). Hence, the magnetic Dirac operator HA˜LY = α·(D−A˜LY (x))+mβ shares the Properties (i)–(iv) of Example 7.1 with HALY = α · (D − ALY (x)) + mβ. This same idea is applicable to the vector potentials A() in Example 7.2 below. Example 7.2 ([2]). In the same spirit as in Example 7.1, we can show the existence of countably inﬁnite number of vector potentials with which the magnetic Dirac operators have the Properties (i)–(iv) in Example 7.1. In fact, we shall exploit a result on the Weyl–Dirac operator by Adam, Muratori and Nash [2], where they construct a series of vector potentials A() ( = 0, 1, 2, . . .), each of which gives rise a zero mode ψ () of the Weyl–Dirac operator T () := σ · (D − A() (x)). The idea of [2] is an extension of that of [19, Sec. II]. Indeed A(0) and ψ (0) give the same vector potential and zero mode as in (7.1) and (7.3). For ≥ 1, the construction of the zero mode ψ () (x) is based on an anzatz (see [2, Sec. II], (7)) and the deﬁnition of A() is given by A() (x) =

h() (x) {ψ () (x) · (σψ () (x))}, |ψ () (x)|2

where h() (x) is a real valued function deﬁned as c h() (x) = (c a real constant depending only on ) x2

(7.4)

(7.5)

and ψ () (x) · (σψ () (x)) is deﬁned in the same way as in (7.2). (For the deﬁnition of h() (x), see [29].) By the same arguments as in Example 7.1, we can deduce that the magnetic Dirac operator H () := α · (D − A() (x)) + mβ, = 0, 1, 2, . . . , has the Properties (i)–(iv) of Example 7.1. Section 3 was based upon our results on supersymmetric Dirac operators in Sec. 2 of the present paper and those of [27]. It turned out that all ±m mode have the same asymptotic limit at inﬁnity, i.e. |x|−2 as |x| → ∞. This means that the asymptotic limits of ±m modes of the mangetic Dirac operator are the same as those of zero modes of the Weyl–Dirac operator. Section 4 was based upon our results on supersymmetric Dirac operators in Sec. 2 and those of Balinsky and Evans [6] on the Weyl–Dirac operator. Section 5 was based upon our results on

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supersymmetric Dirac operators in Sec. 2 and those of Elton [11] on the Weyl– Dirac operator. In each section from Secs. 3–5, we made a diﬀerent assumption on the vector potentials. It is meaningful to compare these assumptions with each other. To this end, imitating (5.1), we introduce the following notation ASU := {A | A satisﬁes Assumption(SU)}, ABE := {A | A satisﬁes Assumption(BE)}. We then have ASU ABE , A\ASU = ∅,

ASU \A = ∅,

A\ABE = ∅,

ABE \A = ∅.

In Secs. 4 and 5, it was shown that the set of vector potentials which give rise to ±m modes is scarce in each regime. The non-existence of ±m resonances was proved in Sec. 6 under the assumption that |Aj (x)| ≤ Cx−ρ , ρ > 3/2. Based on the results in Sec. 6 , it follows that all the examples of vector potentials in this section do not have ±m resonances. A natural question arises: Is there a vector potential A which satisfies |Aj (x)| ≤ Cx−ρ , ρ > 0, and yields ±m resonances of the magnetic Dirac operator HA ? Acknowledgments The authors would like to thank the referees for their valuable comments and constructive suggestions which led them to the consideration of threshold resonances and had them improve the paper. The second author was supported by Grant-inAid for Scientiﬁc Research (C) No. 21540193, Japan Society for the Promotion of Science. References [1] Y. Aharonov and A. Casher, Ground state of a spin-1/2 charged particle in a twodimensional magnetic ﬁeld, Phys. Rev. A 19 (1979) 2461–2462. [2] C. Adam, B. Muratori and C. Nash, Zero modes of the Dirac operator in three dimensions, Phys. Rev. D 60 (1999) 125001-1–125001-8. [3] C. Adam, B. Muratori and C. Nash, Degeneracy of zero modes of the Dirac operator in three dimensions, Phys. Lett. B 485 (2000) 314–318. [4] C. Adam, B. Muratori and C. Nash, Multiple zero modes of the Dirac operator in three dimensions, Phys. Rev. D 62 (2000) 085026-1–085026-9. [5] A. A. Balinsky and W. D. Evans, On the zero modes of Pauli operators, J. Funct. Anal. 179 (2001) 120–135. [6] A. A. Balinsky and W. D. Evans, On the zero modes of Weyl–Dirac operators and their multiplicity, Bull. London Math. Soc. 34 (2002) 236–242. [7] A. A. Balinsky and W. D. Evans, Zero modes of Pauli and Weyl–Dirac operators, in Advances in Diﬀerential Equations and Mathematical Physics (Birmingham, AL, 2002), Contemp. Math., Vol. 327 (Amer. Math. Soc., Providence, Rhode Island, 2003), pp. 1–9.

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[8] A. A. Balinsky, W. D. Evans and Y. Sait¯ o, Dirac–Sobolev inequalities and estimates for the zero modes of massless Dirac operators, J. Math. Phys. 49 (2008) 043514-1– 043514-10. [9] E. Balslev and B. Helﬀer, Limiting absorption principle and resonances for the Dirac operators, Adv. Appl. Math. 13 (1992) 186–215. [10] L. Bugliaro, C. Feﬀerman and G. M. Graf, A Lieb–Thirring bound for a magnetic Pauli Hamiltonian, II, Rev. Mat. Iberoamericana 15 (1999) 593–619. [11] D. M. Elton, The local structure of zero mode producing magnetic potentials, Comm. Math. Phys. 229 (2002) 121–139. [12] L. Erd¨ os and J. P. Solovej, The kernel of Dirac operators on S3 and R3 , Rev. Math. Phys. 13 (2001) 1247–1280. [13] L. Erd¨ os and J. P. Solovej, Uniform Lieb–Thirring inequality for the threedimensional Pauli operator with a strong non-homogeneous magnetic ﬁeld, Ann. Henri Poincar´e 5 (2004) 671–741. [14] L. Erd¨ os and J. P. Solovej, Magnetic Lieb–Thirring inequalities with optimal dependence on the ﬁeld strength, J. Stat. Phys. 116 (2004) 475–506. [15] J. Fr¨ ohlich, E. H. Lieb and M. Loss, Stability of Coulomb systems with magnetic ﬁelds. I. The one-electron Atom, Comm. Math. Phys. 104 (1986) 251–270. [16] B. Helﬀer, J. Nourrigat and X. P. Wang, Sur le spectre de l’´equation de Dirac (dans ´ Norm. Sup. (4) 22 (1989) 515– R3 ou R2 ) avec champ magn´etique, Ann. Sci. Ecole 533. [17] B. Helﬀer and B. Parisse, Comparaison entre la d´ecroissance de fonctions propres pour les op´erateurs de Dirac et de Klein–Gordon. Application ` a l’´etude de l’eﬀect tunnel, Ann. Inst. H. Poincar´ e Phys. Th´eor. 60 (1994) 147–187. [18] E. Lieb, Private communication (2009). [19] M. Loss and H. T. Yau, Stability of Coulomb systems with magnetic ﬁelds. III. Zero energy bound states of the Pauli operators, Comm. Math. Phys. 104 (1986) 283–290. [20] P. Pickl, Generalized eigenfunctions for critical potentials with small perturbations, J. Math. Phys. 48 (2007) 123505-1–123505-31. [21] P. Pickl and D. D¨ urr, On adiabatic pair creation, Comm. Math. Phys. 282 (2008) 161–198. [22] P. Pickl and D. D¨ urr, Adiabatic pair creation in heavy ion and laser ﬁelds, Europhys. Lett. 81 (2008) 40001–40007. [23] C. Pladdy, Asymptotics of the resolvent of the Dirac operator with a scalar shortrange potential, Analysis 21 (2001) 79–97. [24] C. Pladdy, Y. Sait¯ o and T. Umeda, Radiation condition for Dirac operators, J. Math. Kyoto Univ. 37 (1998) 567–584. [25] M. Reed and B. Simon, Methods of Modern Mathematical Analysis. I, Functional Analysis, Revised and enlarged edition (Academic Press, London, 1980). [26] M. Reed and B. Simon, Methods of Modern Mathematical Analysis. IV, Analysis of Operators (Academic Press, New York, 1978). [27] Y. Sait¯ o and T. Umeda, The asymptotic limits of zero modes of massless Dirac operators, Lett. Math. Phys. 83 (2008) 97–106. [28] Y. Sait¯ o and T. Umeda, The zero modes and zero resonances of massless Dirac operators, Hokkaido Math. J. 37 (2008) 363–388. [29] Y. Sait¯ o and T. Umeda, A sequence of zero modes of Weyl–Dirac operators and an associated sequence of solvable polynomials, to appear in Spectral Theory, Function Spaces and Inequalities — New Techniques and Recent Trends, Operator Theory: Advances and Applications (Birkh¨ auser).

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[30] B. Thaller, Dirac particles in magnetic ﬁelds, in Recent Developments in Quantum Mechanics, eds. A. Boutet de Monvel et al. (Kluwer Acadimic Publishers, Dordrecht, 1991), pp. 351–366. [31] B. Thaller, The Dirac Equation (Springer-Verlag, Berlin, 1992). [32] X. P. Wang, Puits multiples pour l’op´erateur de Dirac, Ann. Inst. H. Poincar´ e Phys. Th´eor. 43 (1985) 269–319. [33] T. Weidl, On the virtual bound states for semi-bounded operators, Comm. Partial Diﬀerential Equations 24 (1999) 25–60. [34] D. R. Yafaev, Exponential decay of eigenfunctions of ﬁrst order systems, in Adventures in Mathematical Physics, Contemporary Mathematics, Vol. 447 (American Mathematical Society, 2007), pp. 249–256. [35] O. Yamada, On the principle of limiting absorption for the Dirac operators, Publ. Res. Inst. Math. Sci. Kyoto Univ. 8 (1972/73) 557–577. [36] O. Yamada, Eigenfunction expansions and scattering theory for Dirac operators, Publ. Res. Inst. Math. Sci. Kyoto Univ. 11 (1976) 651–689.

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Reviews in Mathematical Physics Vol. 23, No. 2 (2011) 179–209 c World Scientific Publishing Company DOI: 10.1142/S0129055X11004266

SPECTRAL RENORMALIZATION GROUP AND LOCAL DECAY IN THE STANDARD MODEL OF NON-RELATIVISTIC QUANTUM ELECTRODYNAMICS

∗ , MARCEL GRIESEMER† and ¨ ¨ JURG FROHLICH ISRAEL MICHAEL SIGAL‡ ∗Institute

for Theoretical Physics, ETH Z¨ urich, CH-8093 Z¨ urich, Switzerland and IHES, Bures-sur-Yvette, France

†Department

of Mathematics, University of Stuttgart, Pfaﬀenwaldring 57, D-70569, Stuttgart, Germany

‡Department

of Mathematics, University of Toronto, 40, St. George Street, Bahen Centre, Toronto, ON M5S 2E4 Canada ∗[email protected] †[email protected] ‡[email protected] Received 28 July 2010 Revised 16 January 2011

We prove a limiting absorption principle for the standard model of non-relativistic quantum electrodynamics (QED) and for Nelson’s model describing interactions of electrons with phonons. To this end, we use the spectral renormalization group technique on the continuous spectrum in conjunction with Mourre theory. Keywords: Non-relativistic quantum electrodynamics; renormalization group; local decay; spectral theory; scattering theory; theory of radiation; emission and absorption of light. Mathematics Subject Classification 2010: 81Q10, 81T17, 84U99, 81V10

1. Introduction The mathematical framework for the theory of non-relativistic matter interacting with the quantized electromagnetic ﬁeld (non-relativistic quantum electrodynamics) is well established. It is based on the quantum Hamiltonian, which in notation and units explained below, is of the form Hg =

n 1 (i∇xj − gA(xj ))2 + V (x) + Hf 2m j j=1

179

(1.1)

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acting on the Hilbert space H = Hp ⊗ Hf , the tensor product of the state spaces of the particle system and the quantized electromagnetic ﬁeld. Here the coupling constant g is related to the charge of the electron. An ultraviolet cutoﬀ is imposed on the electromagnetic vector potential A, with the eﬀect that photons of very high frequency do not interact with the electrons. This model provides the foundation of the physical theory that describes phenomena of emission and absorption of radiation by systems of matter, such as atoms and molecules, as well as other processes of quantum radiation interacting with matter. It has been extensively studied during the last decade; (see the books [47, 26] and references therein, for a partial list of contributions). For “reasonable” potentials V (x), the operator Hg is self-adjoint and its spectral and resonance structure — and therefore the dynamics for long, but ﬁnite timeintervals — is well understood (see, e.g., [1, 2, 15, 25, 27–29, 32, 38, 39, 45], and references therein, for recent results). However, we still know little about the asymptotic dynamics, as time tends to ∞. In particular, a complete scattering theory for this operator does not, at present, exist; (see, however, [16–18, 11, 12]). A key notion connected to the asymptotic dynamics is that of local decay. This notion also lies at the foundation of modern quantum scattering theory. It states that a quantum system of the type considered here is either in a bound state, or, as time tends to inﬁnity, it breaks apart, i.e. the probability to occupy any bounded region of physical space tends to zero and, consequently, the average distance between particles diverges to inﬁnity. In our case, this means that photons leave the region of physical space occupied by the particle system. Until recently, local decay for the Hamiltonian Hg has been proven only for energies away from O(g 2 )-neighborhoods of the ground state energy, eg , and the ionization threshold. However, starting from a state with energy below the ionization threshold, the part of the system in any bounded space region eventually winds up in a state with energy close to the ground state energy. Indeed, while the total energy is conserved, photons carry away energy from regions of space where matter is concentrated. Understanding the dynamics in this energy interval is an important matter. Recently, local decay was proven for states in the spectral interval (p) (eg , eg + gap /12) for the Hamiltonian Hg [19]. Here (p)

(p)

(p) gap := 1 − 0 , (p)

(p)

where 0 and 1 are the energies of the ground state and the ﬁrst excited state of the particle system. In this paper, we present another proof of this result. The main goal of this paper is to develop a new approach to time-dependent problems in non-relativistic QED that combines the spectral renormalization group (RG), developed in [6, 7, 3] (see also [20]), with more traditional spectral techniques such as the Mourre estimates. Our key result is that the stronger property of the limiting absorption principle (LAP) “propagates” along the renormalization ﬂow. Next, we explain units and notation employed in (1.1). We use units in which the Planck constant divided by 2π, the speed of light and the electron mass are

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equal to 1( = 1, c = 1 and m = 1). In these units the electron charge is equal to √ √ e2 1 ≈ 137 is the ﬁne-structure constant; length, time − α(e = − α), where α = 4πc and energy are measured in units of /mc = 3.86·10−11 cm, /mc2 = 1.29·10−21 sec and mc2 = 0.511 MeV, respectively, (natural units). We show below that one can set g := α3/2 . Our particle system consists of n particles of masses mj (the ratio of the mass of the jth particle to the mass of an electron) and positions xj , where j = 1, . . . , n. We write x = (x1 , . . . , xn ). The total potential of the particle system is denoted by V (x). The Hamilton operator of the particle system alone is given by n 1 Hp := − ∆xj + V (x), 2m j j=1

(1.2)

where ∆xj is the Laplacian in the variable xj . This operator acts on the Hilbert space of the particle system, denoted by Hp , which is either L2 (R3n ) or a subspace of this space determined by the symmetry of the particle system under permutations. The spin of particles will be neglected for simplicity. The electromagnetic ﬁeld is described by the quantized vector potential in the Coulomb gauge χ(k)d3 k , (1.3) A(y) = (eiky a(k) + e−iky a∗ (k)) 2|k| where χ is an ultraviolet cut-oﬀ: χ(k) = 1, in a neighborhood of k = 0, and vanishing suﬃciently fast at inﬁnity, and its dynamics is described by the quantum Hamiltonian (1.4) Hf = d3 ka∗ (k)ω(k)a(k). These operators act on the Fock space Hf ≡ F. Above, ω(k) = |k| is the dispersion law, i.e. the energy of a ﬁeld quantum with wave vector k, a∗ (k) and a(k) denote creation and annihilation operators on F , and the right-hand side can be understood as a weak integral. The symbols a∗ (k) and a(k) denote operator-valued generalized transverse vector ﬁeld: eλ (k)a# a# (k) := λ (k), λ∈{0,1}

where eλ (k), are polarization vectors, i.e. orthonormal vectors in R3 satisfying k · eλ (k) = 0, and a# λ (k) are ordinary creation and annihilation operators satisfying canonical commutation relations. The subscript λ labels the helicity of the ﬁeld quantum. (See the supplement for a brief review of the deﬁnitions of Fock space, creation and annihilation operators and the deﬁnition of the operator Hf .) First, we consider (1.1) for an atom or molecule. Then, in the natural units, g = √ α and V (x), the total Coulomb potential of the particle system, is proportional

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to α. Rescaling x → α−1 x and k → α2 k we arrive at (1.1) with g := α3/2 , V (x) of the order O(1)a and A(x) replaced by A (x), where A (x) = A(αx)|χ(k)→χ (k) , and where χ (k) := χ(α2 k) (see [8]). After that we drop the prime in the vector potential A (x) and the ultraviolet cut-oﬀ χ (x) (see a discussion of the latter below). Finally, we relax the restriction on V (x) we relax the restriction on V (x) and consider standard generalized n-body potentials (see, e.g., [43, 44, 40]): V (x) = i Wi (πi x), where πi are linear maps from R3n to Rνi , νi ≤ 3n, and Wi are Kato–Rellich potentials (i.e. Wi ∈ Lpi (Rνi ) + (L∞ (Rνi ))ε with pi = 2, for νi ≤ 3, pi > 2, for νi = 4, and pi ≥ νi /2, for νi > 4. In order not to deal with the problem of center-ofmass motion, we assume that either some of the particles (the nuclei) are inﬁnitely heavy, or the system is placed in an external potential ﬁeld. It is well known that Hf deﬁnes a positive, self-adjoint operator on F with purely absolutely continuous spectrum, except for a simple eigenvalue 0 corresponding to the eigenvector Ω (the vacuum vector, see Supplement). Thus, for g = 0, the (p) low energy spectrum of the operator H0 consists of branches [i , ∞) of abso(p) lutely continuous spectrum, where i are the isolated eigenvalues of Hp , and of (p) the eigenvalues i sitting at the tip of the branches (“thresholds”) of the continuous spectrum. The absence of gaps between the eigenvalues and the thresholds is a consequence of the fact that the photons (and phonons) are massless. This leads to subtle problems in perturbation theory, known collectively as the infrared problem. In this paper, we prove the local decay property for the Hamiltonian Hg . In fact, we prove a slightly stronger property — the limiting absorption principle — which states that the resolvent sandwiched between appropriate weighting operators has H¨ older continuous limits on the spectrum. To be speciﬁc, let B denote the selfadjoint generator of dilatations on Fock space F . It can be expressed in terms of creation and annihilation operators as i d3 ka∗ (k){k · ∇k + ∇k · k}a(k). (1.5) B= 2 We extend it to the Hilbert space H = Hp ⊗ F. Let B := (1 + B 2 )1/2 . Our goal is to prove the following (p)

1 (p) 12 gap ),

where g is the ground

B −θ (Hg − λ ± i0)−1 B −θ ∈ C ν (∆)

(1.6)

Theorem 1.1. Let g gap and let ∆ ⊂ (g , g + state energy of H. Then, as a function of λ ∈ ∆,

for θ >

a In

1 2

and 0 < ν < θ − 12 .

the case of a molecule in the Born–Oppenheimer approximation, the resulting V (x) also depends on the rescaled coordinates of the nuclei.

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Deﬁning functions of self-adjoint operators by functional calculus, we derive from this theorem the following consequence: Corollary 1.2. For ∆ as above, for any function f (λ) with supp f ⊆ ∆, and for ν < θ − 12 , we have that B −θ e−iHt f (H)B −θ ≤ Ct−ν .

(1.7)

This corollary follows from (1.6) and the formula B −θ e−iHt f (H)B −θ =

∞

−∞

dλf (λ)e−iλt ImB −θ (H − λ − i0)−1 B −θ

(see, e.g., [44] and the detailed discussion in [19]). Remark 1.3. Let Σp := inf σ(Hp ). We expect that the method of this paper can be extended to the energy interval (σ(H)\σpp (H)) ∩ (−∞, Σp − ε], for some ε > 0, for QED and Nelson’s models. Previously, the limiting absorption principle and local decay estimates were proven in [7, 9] for the standard model of non-relativistic QED and for the Nelson model away from neighborhoods of the ground state energy and ionization threshold. In [21, 22], they were proven for the Nelson model near the ground state energy and for all values of the coupling constant, but under rather stringent assumptions, including one on the infra-red behavior of the coupling functions; (see also [6, 8, 42, 46] for earlier works). Finally, as was mentioned above, Theorem 1.1 has been proven in a neighborhood of the ground state energy in [19]. The approach followed in the present paper consists of three steps. First, following [45], we use a generalized Pauli–Fierz transform to map the QED Hamiltonian Eq. (1.1) to a new Hamiltonian HgP F whose interaction has improved infrared behavior. To this new Hamiltonian we apply a suitable power of the renormalization map (depending on ∆, obtaining, in the end, a rather simple Hamiltonian whose spectral properties can be analyzed with the help of the Mourre estimate. This proves the LAP for this particular Hamiltonian. Since, as we prove in this paper, the renormalization map preserves the LAP property we conclude from this that the Hamiltonian HgP F enjoys the LAP property as well. The size of the spectral interval on which the LAP holds and the number of iterations of the RG map depend on the distance of that interval to the ground state energy. We will also consider Nelson’s model. The Hamiltonian of this model is given by HgN = H0N + IgN .

(1.8)

It acts on the state space H = Hp ⊗ F, where now F is the Fock space for phonons, i.e. spinless, massless bosons. Here g is a positive parameter — a coupling

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constant — that we assume to be small, and H0N = Hp + HfN ,

(1.9)

where Hp and HfN are given in (1.2) and (1.4), respectively, but, in Nelson’s model, a∗ and a are scalar creation and annihilation operators, and the interaction Hamiltonian is 3 d kκ(k) −ikx ∗ N Ig ≡ Ig = g {e a (k) + eikx a(k)}. (1.10) |k|1/2 (We can also treat terms quadratic in a and a∗ , but, for the sake of simplicity, we do not consider such terms.) Here, κ is a real-valued function with the property that 1/2 d3 k 2 |κ(k)| < ∞, (1.11) κµ := |k|3+2µ for some (arbitrarily small, but) strictly positive µ > 0. In the following, we ﬁx some κ with κµ = 1 and vary g. It is easy to see that the operator Ig is symmetric and bounded relative to H0N , in the sense of Kato [44, 40], with an arbitrarily small constant. Thus HgN is self-adjoint on the domain of H0N for arbitrary g. Of course, for Nelson’s model the dimension of space may be arbitrary, d ≥ 1. All the results mentioned above for the standard model Hamiltonian Hg (or more precisely for HgP F ) also hold for the Hamiltonian of Nelson’s model, HgN , with µ > 0. To simplify our exposition we prove Theorem 1.1 only for the case 0 < θ ≤ 1 leaving the case θ > 1 to the interested reader. 2. Generalized Pauli–Fierz Transform Here we describe the generalized Pauli–Fierz transform mentioned in the introduction; (see [45]). We deﬁne the Hamiltonian HgP F := e−igF HgSM eigF ,

(2.1)

which we call the generalized Pauli–Fierz Hamiltonian. In order to keep our notation simple, we deﬁne this transformation only for a single charged particle: d3 k , (2.2) (fx,λ (k)aλ (k) + fx,λ (k)a∗λ (k)) F (x) = 2|k| λ with the test function fx,λ (k) chosen as fx,λ (k) :=

e−ikx χ(k) 1 ϕ(|k| 2 eλ (k) · x). |k|

The function ϕ is assumed to be C 2 , real-valued, bounded, with bounded second derivative and satisfying ϕ (0) = 1. A straightforward calculation, using

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2

the commutator expansion e−igF (x) Hf eigF (x) = Hf − ig[F (x), Hf ] − g2 [F (x), [F (x), Hf ]], yields 1 (p + gA1 (x))2 + Vg (x) + Hf + gG(x) 2 2 where A1 (x) = A(x) − ∇F (x), Vg (x) := V (x) + g2 |fx,λ (k)|2 d3 k and λ d3 k G(x) := −i . |k|(fx,λ (k)aλ (k) − fx,λ (k)a∗λ (k)) 2|k| λ HgP F =

(2.3)

(2.4)

(The terms gG and Vg − V come from the last two terms of the expansion of e−igF (x) Hf eigF (x) .) Observe that the operator family A1 (x) is of the form d3 k , (2.5) A1 (x) = (χx,λ (k)aλ (k) + χx,λ (k)a∗λ (k)) 2|k| λ where χλ,x (k) is given by χλ,x (k) := eλ (k)e−ikx χ(k) − ∇x fx,λ (k). It satisﬁes the estimates |χλ,x (k)| ≤ const min(1, with x := (1 + |x|2 )1/2 , and

|k|x ),

d3 k |χλ,x (k)|2 < ∞. |k|

(2.6)

(2.7)

Using the fact that the operators A1 and G have much better infrared behavior than the original vector potential A, we can use our approach and prove the limiting absorption principle for HgP F and B: B −θ (HgP F − z)−1 B −θ is H¨older continuous in z,

(2.8)

where the ranges of the parameters g, z and θ and the type of H¨ older continuity are quantiﬁed in Theorem 1.1. Now we show that (2.8) and an additional restriction on the spectral interval imply the limiting absorption principle for HgSM . Let B1 := eigF (x) Be−igF (x) . We compute B1 = B + gC

(2.9)

where C := i[F (x), B]. Note that the operator C contains a term proportional to x. Let f be a real-valued function supported in (−∞, Σp ). Then, using that (Hg − z)−1 = eigF (HgP F − z)−1 e−igF , we obtain B −θ f (Hg )2 (Hg − z)−1 B −θ = DE(z)D∗ , where D := B −θ f (Hg )B1 θ eigF (x) and E(z) := B −θ (HgP F − z)−1 B −θ . The operator D is bounded by standard operator calculus estimates and the

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fact that eδx f (Hg ) is bounded, for δ > 0 suﬃciently small. Furthermore, the operator-family E(z) is H¨older continuous by result (2.8). Now, for z ∈ (−∞, Σp −ε], for some ε > 0, the previous conclusion remains true even if we remove the cut-oﬀ function f (Hg ). For further reference, we mention that the operator (2.3) can be written as PF HgP F = H0g + IgP F ,

where PF H0g

g2 |χλ,x (k)|2 2 = H0 + |fx,λ (k)| + d3 k, 2 2|k|

(2.10)

(2.11)

λ

SM with H0 := Hg=0 = Hp + Hf (see (1.1)), and IgP F is deﬁned by this relation. (The 2 g2 |χλ,x (k)| 3 d k comes from requiring IgP F to have the normal form.) term 2 λ 2|k| Note that the operator IgP F contains linear and quadratic terms in creation and annihilation operators, with coupling functions (form-factors) in the linear terms satisfying estimate (2.6) and with coupling functions in the quadratic terms satisfyPF PF PF is of the form H0g = Hpg +Hf ing a similar estimate. Moreover, the operator H0g where g2 |χλ,x (k)|2 PF Hpg := Hp + (2.12) |fx,λ (k)|2 + d3 k, 2 2|k| λ

where Hp is given in (1.2). 3. The Smooth Feshbach–Schur Map In this section, we review and extend, in a simple but important way, the method of isospectral decimations or Feshbach–Schur maps introduced in [6, 7] and reﬁned in [3]. For further extensions, see [23]. At the root of this method is the isospectral smooth Feshbach–Schur map acting on a set of closed operators and mapping a given operator to one acting on a much smaller space expected to be easier to handle. Let χ, χ be a partition of unity on a separable Hilbert space H, i.e. χ and χ are positive operators on H whose norms are bounded by one, 0 ≤ χ, χ ≤ 1, and χ2 + χ2 = 1. We assume that χ and χ are nonzero. Let τ be a (linear) projection acting from (a subspace of) the space of closed operators on H to itself, mapping selfadjoint operators into self-adjoint and having the property that operators from its image commute with χ and χ. It is also convenient to assume that τ (1) = 1. Assume that τ and χ (and therefore also χ) leave D(H) invariant, D(τ (H)) ⊇ D(H), and χD(H) ⊂ D(H). Let τ := 1 − τ and deﬁne Hτ,χ# := τ (H) + χ# τ (H)χ# , where χ# stands for either χ or χ.

(3.1)

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Given χ and τ as above, we denote by Dτ,χ the space of closed operators, H, on H which belong to the domain of τ and satisfy the following conditions: τ (H), Hτ,χ are (bounded) invertible on Ran χ, τ (H)χ and χτ (H) extend to bounded operators on H.

(3.2)

(For more general conditions see [3, 23].) We deﬁne H0 := τ (H) and W := τ (H). Then H0 and W are closed operators on H s.t. D(H0 ), D(W ) ⊇ D(H), and H = H0 + W . We remark that the domains of χW χ, χW χ, Hτ,χ , and Hτ,χ all contain D(H). The smooth Feshbach–Schur map (FSM ) maps operators on H to operators on H by H → Fτ,χ (H), where −1 Fτ,χ (H) := H0 + χW χ − χW χHτ,χ χW χ.

(3.3)

Clearly, it is deﬁned for all H ∈ Dτ,χ . Remarks. (i) The deﬁnition of the smooth Feshbach–Schur map given above is the same as in [20] and diﬀers from the one given in [3]. In [3], the map Fτ,χ (H) is denoted by Fχ (H, τ (H)) and the pair of operators (H, τ (H)) is referred to as a Feshbach pair. (ii) The Feshbach–Schur map is obtained from the smooth Feshbach–Schur map by specifying χ = projection and, usually, τ = 0. Next we deﬁne two operators entering identities involving the Feshbach–Schur map: −1 χW χ, Qτ,χ (H) := χ − χHτ,χ

(3.4)

−1 Q# τ,χ (H) := χ − χW χHτ,χ χ.

(3.5)

Note that Qτ,χ (H) ∈ B(Ran χ, H) and Q# τ,χ (H) ∈ B(H, Ran χ); (see (3.2)). The smooth Feshbach map Fτ,χ (H) of H is isospectral to H in the sense of the following theorem. Theorem 3.1. Let χ and τ be as above, with properties as speciﬁed above and let H ∈ Dτ,χ . Then Fτ,χ (H) has the following properties: (i) 0 ∈ ρ(H) ⇔ 0 ∈ ρ(Fτ,χ (H)), i.e. H is bounded invertible on H if and only if Fτ,χ (H) is bounded invertible on Ran χ. (ii) If ψ ∈ H\{0} solves Hψ = 0 then ϕ := χψ ∈ Ran χ\{0} solves Fτ,χ (H)ϕ = 0. (iii) If ϕ ∈ Ran χ\{0} solves Fτ,χ (H)ϕ = 0 then ψ := Qτ,χ (H)ϕ ∈ H\{0} solves Hψ = 0. (iv) The multiplicity of the spectral value {0} is preserved in the sense that dim Ker H = dim Ker Fτ,χ (H).

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(v) If one of the inverses, H −1 or Fτ,χ (H)−1 , exists then so does the other, and they are related as −1 χ, H −1 = Qτ,χ (H)Fτ,χ (H)−1 Qτ,χ (H)# + χ Hτ,χ

(3.6)

and Fτ,χ (H)−1 = χH −1 χ + χ τ (H)−1 χ. This theorem is proven in [3] (see [23] for a more general result). Another property of used below is H is self-adjoint ⇒ Fτ,χ (H) is self-adjoint.

(3.7)

Next, we establish a key result relating smoothness of the resolvent of an operator on its continuous spectrum with smoothness of the resolvent of its image under a smooth Feshbach–Schur map. In what follows ∆ stands for an open interval in R. Theorem 3.2. Let ∆ ⊂ R and, ∀ λ ∈ ∆, H(λ) be a C 1 family of self-adjoint operators, such that H(λ) ∈ Dτ,χ (see (3.2)). Assume that there is a self-adjoint operator B such that adjB (A) is bounded and diﬀerentiable in λ,

∀ j ≤ 2,

(3.8)

τ (H(λ)), τ¯(H(λ))χ, ∂λk (χHτ,χ (λ)−1 χ), k = where A is one of the operators χ, χ, χ¯ 0, 1. Then, for any 0 ≤ ν ≤ 1 and 0 < θ ≤ 1 and in the operator norm, we have lim B −θ (Fτ,χ (H(λ)) − iε)−1 B −θ

ε→0+

⇒ lim B −θ (H(λ) − iε)−1 B −θ ε→0+

exists and ∈ C ν (∆) exists and ∈ C ν (∆).

(3.9) (3.10)

Proof. Let Bθ := B −θ . Since τ (1) = 1, we have that (H(λ) − i)τ,χ# = H(λ)τ,χ# − i, where χ# is either χ or χ. Furthermore, since H(λ) ∈ Dτ,χ the operator family [(H(λ) − iε)τ,χ ]−1 , on Ran χ, is diﬀerentiable in λ and analytic in ε and can be expanded as [(H(λ) − iε)τ,χ ]−1 = [H(λ)τ,χ ]−1 + iε[H(λ)τ,χ ]−1 χ2 [H(λ)τ,χ ]−1 + O(ε2 ). This together with (3.3) implies that Fτ,χ (H(λ) − iε) = Fτ,χ (H(λ)) − iεκ + O(ε2 ), where κ := 1+ χW (λ)χ[H(λ)τ,χ ]−1 χ2 [H(λ)τ,χ ]−1 χW (λ)χ, which, in turn, together with (3.9), gives, in the norm topology, lim Bθ [Fτ,χ (H(λ) − iε)]−1 Bθ

ε→0+

exists and ∈ C ν (∆).

∞ = C Next, conditions (3.8) and the formula B θ θ 0 ∞ −1 −1 ] , imply that the operators Cθ := [ 0 ωdω θ/2 (ω + 1) Bθ χBθ−1 ,

Bθ χBθ−1 ,

dω (ω ω θ/2

Bθ [H(λ)τ,χ ]−1 Bθ−1

(3.11)

+ 1 + B 2 )−1 , where (3.12)

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and the transposed operators (i.e. Bθ−1 χBθ , etc.) are bounded and diﬀerentiable −1 in λ. Let Q := Qτ χ and Q# := Q# τ χ . The above property shows that Bθ QBθ −1 # and Bθ Q Bθ are bounded and diﬀerentiable in λ ∈ ∆. This, together with (3.6), implies Bθ (H(λ) − iε)−1 Bθ = (Bθ QBθ−1 )Bθ (Fτ,χ (H(λ)) − iε)−1 Bθ (Bθ−1 Q# Bθ ) + Bθ χH(λ)−1 χ χBθ . The last relation, together with (3.11), implies (3.10). Remark. Another way to relate the boundary values of the resolvents of Fτ,χ (H) and H (also starting with (3.6)) is given in [3, Theorem 2.1(v)]. 4. A Banach Space of Hamiltonians We construct a Banach space of Hamiltonians on which a renormalization transformation, involving the smooth Feshbach–Schur map, is deﬁned. To simplify matters we will think of the creation and annihilation operators used below as scalar operators. We explain at the end of a supplement how to reinterpret the corresponding expressions for the creation and annihilation operators of the electromagnetic ﬁeld. Let B1 denote the unit ball in R3 and B1d , the d-fold product of B1 . I := [0, 1] and m, n ≥ 0. Given functions w0,0 : [0, ∞) → C and wm,n : I × B1m+n → C, m + n > 0, we consider monomials, Wm,n ≡ Wm,n [wm,n ], in creation and annihilation operators of the form W0,0 := w0,0 [Hf ] (deﬁned by operator calculus), and dk(m,n) ∗ Wm,n [wm,n ] := a (k(m) )wm,n [Hf ; k(m,n) ]a(k˜(n) ), (4.1) 1/2 m+n |k (m,n) | B1 for m + n > 0. Here we use the notation k(m) := (k1 , . . . , km ) ∈ R3m ,

a∗ (k(m) ) :=

m

a∗ (ki ),

(4.2)

i=1

k(m,n) := (k(m) , k˜(n) ),

dk(m,n) :=

m i=1

|k(m,n) | := |k(m) | · |k˜(n) |,

d3 ki

n

d3 k˜i ,

(4.3)

i=1

|k(m) | := |k1 | · · · |km |.

(4.4)

We assume that, for every m and n with m + n > 0, the function wm,n [r, k(m,n) ] is 1 ≤ σ ≤ 2 times continuously diﬀerentiable in r ∈ I, for almost every k(m,n) ∈ B1m+n , and weakly diﬀerentiable in k(m,n) ∈ B1m+n , for almost every r in I. As a function of k(m,n) , it is totally symmetric with respect to the variables k(m) = (k1 , . . . , km ) and k˜(n) = (k˜1 , . . . , k˜n ) and obeys the norm bound wm,n µ,σ := ∂rn (k∂k )q wm,n µ < ∞, (4.5) n+|q|≤σ

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m+n where q := (q1 , . . . , qm+n ), (k∂k )q := 1 (kj ·∇kj )qj , with km+j := k˜j , and where the sum is taken over the indices n and q satisfying 0 ≤ n + |q| ≤ σ, and wm,n µ := max j

sup r∈I,k(m,n) ∈B1m+n

||kj |−µ wm,n [r; k(m,n) ]|,

(4.6)

for µ ≥ 0. Here, and in what follows, kj is the jth (3-dimensional) entry of the k-vector k(m,n) over which we take the supremum. For m + n = 0, the variable r ranges in [0, ∞) and we assume that the following norm is ﬁnite: sup |∂rn w0,0 (r)|; (4.7) w0,0 µ,σ := |w0,0 (0)| + 1≤n≤σ

r∈I

(for σ = 0 we drop the sum on the right-hand side). This norm is independent of µ, but we keep this index for notational convenience. The Banach space of functions µ,σ . Moreover, Wm,n [wm,n ] stresses wm,n , with wm,n µ,σ < ∞, is denoted by Wm,n the dependence of Wm,n on wm,n . In particular, W0,0 [w0,0 ] := w0,0 [Hf ]. We ﬁx three numbers µ, 0 < ξ < 1 and σ ≥ 0 and deﬁne the Banach space

µ,σ Wm,n , (4.8) Wξµ,σ := m+n≥0

with the norm wµ,σ,ξ :=

ξ −(m+n) wm,n µ,σ < ∞.

(4.9)

m+n≥0

Clearly, Wξµ ,σ ⊂ Wξµ,σ if µ ≥ µ, σ ≥ σ and ξ ≤ ξ. Remarks. (1) Though we use the same notation, the Banach spaces, Wξµ,σ , etc. introduced above diﬀer from the ones used in [45, 20]. The latter are obtained from the former by setting q = 0 in (4.5). To extend estimates of [45, 20] to the present setting one has to estimate the eﬀect of the derivatives (k∂k )q , which is straightforward. (2) Actually, one can consider smaller Banach spaces, with B1d in (4.1) replaced by S1d := {k = (k1 , . . . , kd ∈ R3d | di=1 |ki | ≤ 1}. Let χρ (r) be a smooth cut-oﬀ function such that supp χρ (r) ⊂ [0, ρ] and 0 ≤ χρ (r) ≤ 1 and let χρ := χρ (Hf ). The following basic bound, proven in [3] (see also [26], links the norm deﬁned in (4.6) to the operator norm on B[F ], where B[F ] is the algebra of all bounded operators on Fock space F . µ,0 , Theorem 4.1. Fix m, n ∈ N0 such that m + n ≥ 1. Suppose that wm,n ∈ Wm,n and let Wm,n ≡ Wm,n [wm,n ] be as deﬁned in (4.1). Then ∀ ν > 0

(Hf + ν)−m/2 Wm,n (Hf + ν)−n/2 ≤ wm,n 0 ,

(4.10)

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and (as a consequence) χρ Wm,n χρ ≤

ρ(m+n)(1+µ) √ wm,n µ , m!n!

(4.11)

where · denotes the operator norm on B[F ]. Theorem 4.1 says that the ﬁniteness of wm,n µ ensures that χρ Wm,n χρ deﬁnes a bounded operator on F . With a sequence w := (wm,n )m+n≥0 in Wξµ,σ we associate an operator by setting H(w) := W0,0 [w] +

χ1 Wm,n [w]χ1 ,

(4.12)

m+n≥1

where we write Wm,n [w] := Wm,n [wm,n ]. This form of operators on Fock space will be called generalized normal (or Wick) form. Theorem 4.1 shows that the series in (4.12) converges in the operator norm and obeys the estimate H(w) − W0,0 (w) ≤ ξw1 µ,0,ξ ,

(4.13)

for any w = (wm,n )m+n≥0 ∈ Wξµ,0 . Here w1 = (wm,n )m+n≥1 . Hence we have the linear map H : w → H(w)

(4.14)

from W µ,0 into the set of closed operators on Fock space F . Furthermore the following result was proven in [3]. Theorem 4.2. For any µ ≥ 0 and 0 < ξ < 1, the map H : w → H(w), given in (4.12), is one-to-one. µ,σ µ,σ µ,σ := H(Wξµ,σ ) and Wmn,op := H(Wmn ). Theorem 4.2 Deﬁne the spaces Wop µ,σ implies that Wop is a Banach space under the norm H(w)µ,σ,ξ := wµ,σ,ξ . Similarly, the other spaces deﬁned above are Banach spaces in the corresponding norms. Finally, we mention the following elementary result, which follows from [3, Eq. (3.33)]:

Proposition 4.3. If H(w) is self-adjoint, m+n≥1 χ1 Wm,n [w]χ1 (see (4.12)).

then

so

are

W0,0 [w]

and

5. The Renormalization Transformation Rρ In this section, we present an operator-theoretic renormalization transformation based on the smooth Feshbach–Schur map closely related to the one deﬁned in [3, 6, 7]. We ﬁx the parameter µ of our Banach spaces at some positive value, µ > 0.

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The renormalization transformation is homothetic to an isospectral map deﬁned on a subset of a suitable Banach space of Hamiltonians. It has a certain contraction property that ensures that (upon an appropriate tuning of the spectral parameter) its iteration converges to a ﬁxed-point (limiting) Hamiltonian, whose spectral analysis is particularly simple. Thanks to the isospectrality of the renormalization map, certain properties of the spectrum of the initial Hamiltonian can be studied by analyzing the limiting Hamiltonian. The renormalization map is deﬁned below as a composition of a decimation map, Fρ , and a rescaling map, Sρ . Here ρ is a positive parameter — the photon energy scale — to be chosen later. The decimation of degrees of freedom is accomplished by the smooth Feshbach map, Fτ,χ . Except in the ﬁrst step, the decimation map will act on the Banach µ,σ . We deﬁne a smooth partition of unity, {χρ , χρ }, 0 < ρ ≤ 1, χ2ρ +χ2ρ = 1, space Wop consisting of operators on F , given by χρ ≡ χ1 (Hf /ρ) and χρ ≡ χ1 (Hf /ρ),

(5.1)

where {χ1 (r), χ1 (r)} form a smooth partition of unity, χ1 (r)2 +χ1 (r)2 = 1, with the properties χ1 (r) = 1 for r ≤ 1, = 0 for r ≥ 11/10, 0 ≤ χ1 (r) ≤ 1 and sup |∂rn χ1 (r)| ≤ 30, ∀ r and for n = 1, 2. The operators τ and χ will be chosen as τ (H) = W00 := w00 (Hf )

and χ = χρ ,

(5.2)

where H = H(w) is given in Eq. (4.12). Note that by Proposition 4.3, τ maps self-adjoint opetators into self-adjoint operators. With τ and χ identiﬁed in this way, we will use the notation Fρ ≡ Fτ,χρ .

(5.3)

Let w 1 := (wm,n )m+n≥1 . The following lemma shows that the domain of this map µ,σ : contains the following polydisc in Wop µ,σ Dµ,σ (α, β, γ) := H(w) ∈ Wop ||w0,0 [0]| ≤ α, sup

|w0,0 [r]

− 1| ≤ β, w1 µ,σ,ξ

≤γ ,

(5.4)

r∈[0,∞)

for appropriate numbers α, β, γ > 0. Lemma 5.1. Fix 0 < ρ < 1, µ > 0, and 0 < ξ < 1. Then it follows that the polydisc Dµ,1 (ρ/8, 1/8, ρ/8) is in the domain of the Feshbach–Schur map Fρ . Proof. Let H(w) ∈ Dµ,1 (ρ/8, 1/8, ρ/8). We observe that W := H[w] − W0,0 [w] deﬁnes a bounded operator on F , and we only need to check the invertibility of H(w)τ,χρ on Ran χρ . Now, the operator W0,0 [w] is invertible on Ran χρ , since, for

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all r ∈ [3ρ/4, ∞), Re w0,0 [r] ≥ r − |w0,0 [r] − r| [r] − 1| − |w0,0 [0]| ≥ r 1 − sup |w0,0 r

ρ 3ρ 1 ρ ≥ 1− − ≥ . 4 8 8 2 Furthermore, by (4.11), W ≤ ξρ/8 ≤ ρ/8. Hence Re(W0,0 [w] + W ) ≥ Ran χρ , i.e. H(w)τ,χρ is invertible on Ran χρ .

(5.5) ρ 3

on

We introduce the scaling transformation Sρ : B[F ] → B[F ], by Sρ (a# (k)) := ρ−3/2 a# (ρ−1 k),

Sρ (1) := 1,

(5.6)

where a# (k) is either a(k) or a∗ (k), and k ∈ R3 . On the domain of the decimation map Fρ we deﬁne the renormalization map Rρ as Rρ := ρ−1 Sρ ◦ Fρ .

(5.7)

Remark 5.2. The renormalization map above is diﬀerent from the one deﬁned in [3]. The map in [3] contains an additional change of the spectral parameter λ := −Ω, HΩ , where, recall, Ω is the vacuum in our Fock space. We mention here some properties of the scaling transformation. It is easy to check that Sρ (Hf ) = ρHf , and hence Sρ (χρ ) = χ1

and ρ−1 Sρ (Hf ) = Hf ,

(5.8)

which shows that the operator Hf is a ﬁxed point of ρ−1 Sρ . Further, note that E · 1, E ∈ C is expanded under the scaling map, ρ−1 Sρ (E · 1) = ρ−1 E · 1, at a rate ρ−1 . Next, we show that the interaction W contracts under the scaling transformaµ,s , induces a tion. To this end we note that the scaling map Sρ , restricted to Wopξ µ,s scaling map, sρ , on Wξ by ρ−1 Sρ (H(w)) =: H(sρ (w)),

(5.9)

where sρ (w) := (sρ (wm,n ))m+n≥0 , and, using (5.6) and the change of variables of integration in the expression for Wm,n , it is easy to verify that, for all (m, n) ∈ N20 , sρ (wm,n )[r, k(m,n) ] = ρm+n−1 wm,n [ρr, ρk(m,n) ].

(5.10)

We note that, by Theorem 4.1, the operator norm of Wm,n [sρ (wm,n )] is controlled by the norm sρ (wm,n )µ = max j

sup

ρm+n−1

r∈I,k∈B1m+n

≤ ρm+n+µ−1 wm,n µ .

|wm,n [ρr, ρk(m,n) ]| |kj |µ

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Hence, for m + n ≥ 1, we have that sρ (wm,n )µ ≤ ρµ wm,n µ .

(5.11)

Since µ > 0, this estimate shows that Sρ is a contraction on the Banach spaces µ,s , m + n ≥ 1, with contraction rate given by ρµ < 1. Our next result shows Wm,n that contraction is actually a key property of the renormalization map Rρ along “stable” directions. Recall that χ1 is the cut-oﬀ function introduced at the beginning of Sec. 3. We deﬁne a constant Cχ by

2 4 n 2 Cχ := sup |∂r χ1 | + sup |∂r χ1 | ≤ 200. (5.12) 3 n=0 Theorem 5.3. Let 0 : H → H Ω , and µ > 0. Then, for any σ ≥ 1, 0 < ρ < 1/2, α, β ≤ ρ8 and γ ≤ 8Cρ χ , we have that Rρ − ρ−1 0 : Dµ,σ (α, β, γ) → Dµ,σ (α , β , γ ), √

(5.13)

ρ

continuously, with ξ := 4Cχ in the deﬁnition of the corresponding norms, and (5.14) α = 3Cχ γ 2 /2ρ , β = β + 3Cχ γ 2 /2ρ , γ = 128Cχ2 ρµ γ. Moreover, if H is self-adjoint, then so is Rρ (H). With minor modiﬁcations, this theorem follows from [3, Theorem 3.8 and its proof], especially [Eqs. (3.104), (3.107) and (3.109)]. For the norms (4.5), with q = 0, it is presented in [45, Appendix I]. The generalization to the case with q > 0 is straightforward. Remark 5.4. Subtracting the term ρ−1 0 from Rρ allows us to control the expanding direction during the iteration of the map Rρ . In [3], such control was achieved by changing the spectral parameter λ, which controls H Ω ; (see [3, Appendix I]). Recall that B denotes the dilatation generator on Fock space F (see (1.5)). Proposition 5.5. Let ∆ be an open interval in R, µ > 0, ρ and ξ be as in Theorem 5.3 and let H(λ) ∈ Dµ,1 (α, β, γ), with α, γ < ρ8 , β ≤ 18 , and be analytic in λ ∈ ∆ and self-adjoint ∀ λ ∈ ∆. Then, for 1 ≥ θ > 0 and ν ≥ 0, B −θ (Rρ (H(λ)) − i0)−1 B −θ ∈ C ν (∆)

implies that

B −θ (H(λ) − i0)−1 B −θ ∈ C ν (∆).

(5.15)

Proof. We deduce the proof of this proposition from the following: µ,1 Lemma 5.6. Let χ# ρ be either χρ or χρ . If H ∈ Wop , then the operators j j −1 adjB (χ# ρ ), Hf adB (W00 ), adB (H − W00 ),

j = 0, 1,

are bounded.

(5.16)

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Proof. The j = 0 part of the statement follows from the deﬁnitions. The j = 1 part follows from the following relations 3 (5.17) a# (k), B, a# (k) = ±i k · ∇k + 2 i[B, Hf ] = Hf , i[B, f (Hf )] = Hf f (Hf ).

(5.18)

Indeed, using Eq. (5.18) and the Helﬀer–Sj¨ ostrand operator calculus (see, e.g., [43]) one can easily derive the ﬁrst two statements in (5.16) as well as the inequalities adB (wmn )µ,0 ≤ cwmn µ,1 ,

m + n ≥ 1.

(5.19)

These bounds, together with (5.17), imply that, for m + n ≥ 1, µ,0 µ,1 adB (Wmn )Wmn,op ≤ c(m + n + 1)Wmn Wmn,op .

(5.20)

The latter inequalities, together with (4.11) and the relation H − W00 = m+n≥1 χ1 Wmn χ1 , imply the last inequality in (5.16). By Theorem 5.3, ∀ λ ∈ ∆, H(λ) ∈ D(Rρ ). Moreover, by [20, Proposition C.1] inherited from a result of [24], and by Proposition 4.3, W (λ) := τ¯(H(λ)) is analytic in λ ∈ ∆ and self-adjoint ∀ λ ∈ ∆. Lemma 5.6 implies that condition (3.8) of Theorem 3.2, with τ (H) := W00 is satisﬁed. Therefore the property (3.9) and (3.10), µ,1 ) ∩ D(Fρ ). This and the invariance of with χ = χρ , holds for H(λ) ∈ C 1 (∆, Wop −θ the operator B under rescaling (Sρ ) yield the result. 6. Renormalization Group In this section, we describe some dynamical properties of the “renormalization group” Rnρ , ∀ n ≥ 1 (more precisely, semi-group), generated by the renormalization map Rρ . A closely related iteration scheme is used in [3]. First, we observe that, ∀ w ∈ C, Rρ (wHf ) = wHf and Rρ (w1) = 1ρ w1. This motivates us to deﬁne Mf p := CHf and Mu := C1 as candidates for the manifold of ﬁxed points of Rρ and the unstable manifold, emerging from Mf p := CHf . The next theorem identiﬁes the stable manifold passing through Mf p , which turns out to be of (complex) codimension 1 and is foliated locally by (complex) co-dimension- 2 stable manifolds passing through the ﬁxed points in Mf p . This implies in particular that, in a vicinity of Mf p , there are no other ﬁxed points and that Mu is the entire unstable manifold of Mf p . We require some further deﬁnitions. As an initial set of operators we take D := Dµ,2 (α0 , β0 , γ0 ),

with α0 , β0 , γ0 1.

(6.1)

(The choice σ = 2 of the smoothness index in the deﬁnition of the polydiscs is dictated by the needs of Mourre theory, which is applied in the next section.) We

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also let Ds := Dµ,2 (0, β0 , γ0 )

(6.2)

(the subindex s stands for “stable”). For H ∈ D, we denote Hu := H Ω and Hs := H − H Ω 1 (the unstable- and stable-central-space components of H, respectively). Note that Hs ∈ Ds . We ﬁx the scale ρ such that 1 2 α0 , β0 , γ0 ρ ≤ min , Cχ , (6.3) 2 where, we recall, the constant Cχ is deﬁned in (5.12). Below, we will use the nth iteration of the numbers α0 , β0 and γ0 under the map (5.14): ∀ n ≥ 1 2 αn := c ρ−1 (cρµ )n−1 γ0 , β n = β0 +

n−1

2 c ρ−1 (cρµ )j γ0 ρ ,

j=1 µ n

γn = (cρ ) γ0 . Recall that a complex function f on an open set Ω in a complex Banach space W is said to be analytic if, for H ∈ Ω and ∀ ξ ∈ W, f (H + τ ξ) is analytic in the complex variable τ , for |τ | suﬃciently small, see [10]. Our analysis uses the following result from ([20, Theorems 5.1 and 5.2, and the proof of Proposition 5.3]): Theorem 6.1. Let δn := νn ρn with 4αn ≤ νn ≤ e : Ds → C s.t.

1 18 .

There is an analytic map

(i) e(H) ∈ R, for H = H ∗ ; (ii) If Uδ := {H ∈ D||e(Hs ) + Hu | ≤ δ}, then Uδn ⊂ D(Rnρ )

and

Rnρ (Uδn ) ⊂ Dµ,2 (2νn , βn , γn );

(6.4)

(iii) Rnρ (H) is analytic in λ = −Hu ∈ D(e(H), δn ); (iv) For H ∈ D, the number E := e(Hs ) + Hu is the eigenvalue of the operator H with the smallest real part. (In particular, if H is self-adjoint, then it is the ground state energy of H.) Theorem 6.1 is proven in [20] for somewhat simpler Banach spaces (not containing the derivatives (k∂k )q in the deﬁnition of norms). However, an extension to the Banach spaces used in this paper is straightforward and is not dwelled upon here. Theorem 6.1 implies that Mf p := CHf is (locally) a manifold of ﬁxed points of Rρ , and Mu := C1 is the unstable manifold. Moreover (see [20]), Uδn = {H ∈ D|e(Hs ) = −Hu } (6.5) Ms := n

is a local stable manifold for the ﬁxed point manifold Mf p , in the sense that, ∀ H ∈ Ms , ∃w ∈ C such that Rnρ (H) → wHf ,

µ,2 in the sense of convergence in Wop ,

(6.6)

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as n → ∞. Moreover, Ms is an invariant manifold for Rρ : Ms ⊂ D(Rρ ) and Rρ (Ms ) ⊂ Ms , though we do not need this property here (and therefore we do not prove it). 7. Mourre Estimate In this section, we prove the Mourre estimate for the operator-family H (n) (λ) := Rnρ (Hs − λ), where Hs = Hs∗ ∈ Ds , where Ds has been deﬁned in (6.2). This will yield the limiting absorption principle for H (n) (λ). The latter is then transferred with the help of Theorem 3.2 to the limiting absorption principle for the operator H = Hs − λ. In Sec. 8, this limiting absorption principle will be connected to the limiting absorption principle for the family Hg − λ, where Hg is either HgP F or HgN . Denote ∆δ := [δ, ∞). Theorem 7.1. Let H(λ) = H(λ)∗ ∈ C 1 (∆, Dµ,2 (α, β, γ)), where ∆ is an open interval in R. If δ γ and β ≤ 13 , then B −θ (H(λ) − i0)−1 B −θ ∈ C ν (∆ ∩ E −1 (∆δ )),

(7.1)

where E : λ → E(λ) with E(λ) := H(λ) Ω , for any 1/2 < θ ≤ 1 and ν < θ − 12 . Proof. In what follows we omit the argument λ. Let E := w0,0 [0], T := w0,0 [Hf ] − w0,0 [0] and W := m+n≥1 χ1 Wm,n [w]χ1 , so that H = E1 + T + W . Let H1 := H − E = T + W. , where T := i [T, B] = T (Hf )Hf and W := i [W, B]. We write i [H1 , B] = T + W µ,σ Let W ∈ Wop,ξ , 1 ≤ σ ≤ 2. (In this argument, we display the index ξ in the space µ,σ := H(Wξµ,σ ).) By relation (5.17) we have that Wop,ξ µ,σ−1 ≤ cγ, W W op,ξ

with ξ < ξ.

The shift in the smoothness index from σ to σ − 1 is due to the fact that the = i [W, B] are (k · ∇k + 3(m+n) )wm,n (r, k), coupling functions for the operator W 2 (m,n) , and one therefore looses one derivative, as compared to the where k := k coupling functions, wm,n (r, k), of W . We write i [H1 , B] =

1 1 − 1 W. H1 + T − T + W 2 2 2

µ,0 Recalling that the operator norm is dominated by the Wop -norm, we see that the last two terms are bounded as − 1 W ≤ Cγ. W (7.2) 2

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Furthermore, using the estimate |T (r) − 1| < β and the deﬁnition of T, we ﬁnd that 1 1 1 T(r) − T (r) ≥ (1 − β)r − (1 + β)r = (1 − 3β)r 2 2 2 and therefore 1 T − T ≥ 2

1 T(r) − T (r) 0≤r≤∞ 2 inf

1 (1 − 3β)r = 0. 2 This gives [H1 , B] ≥ 12 H1 − cγ, and therefore, for ∆ := 12 δ, ∞ , δ γ, ≥

inf

0≤r≤∞

E∆ (H1 )i [H1 , B] E∆ (H1 ) ≥

1 δE∆ (H1 )2 . 4

(7.3)

This proves the Mourre estimate for the operator H1 ≡ H1 (λ). Moreover, since H(λ) ∈ C ∞ (∆, Dµ,2 (α, β, γ)), we conclude that the commutators [H1 , B] and [[H1 , B], B] are bounded relative to the operator H1 ; (this is guaranteed by choosing the index σ = 2 for the polydisc Dµ,σ (α, β, γ)). Hence the standard Mourre theory is applicable and gives H¨ older continuity of the resolvent, sandwiched between factors B −θ , in the spectral parameter as well as in the “operator” (see [43]): B −θ R1 (λ, ω)E∆ (H1 (λ))B −θ ∈ C ν (∆ × R),

(7.4)

where R1 (λ, ω) := (H1 (λ) − ω)−1 and ν < θ − 1/2. Here the argument λ has been reintroduced in our notation, and we have transferred the H¨ older continuity from the operator H1 (λ) to its argument λ. Since B −θ R1 (λ, ω)B −θ = B −θ R1 (λ, ω)E∆ (H1 (λ))B −θ + B −θ R1 (λ, ω)(1 − E∆ (H1 (λ)))B −θ

(7.5)

and since the last term on the right-hand side is C ν (∆) in λ and C ∞ (∆δ ) in σ, we conclude from (7.4) that B −θ R1 (λ, ω)B −θ ∈ C ν (∆ × ∆δ ).

(7.6)

Now take σ = E(λ) + i0. Since by the condition of the theorem E(λ) := H(λ) Ω ∈ C ∞ (∆), we conclude that (7.1) holds. In the previous section the parameter δn was allowed to vary over a certain range (see Theorem 6.1). In this section we make a particular choice of δn , namely 1 n ρ . δn := 18 The deﬁnitions of the sets Ds and Uδ appearing below have been given at the beginning of Sec. 6 and in Theorem 6.1, respectively.

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1 n Theorem 7.2. Assume (6.3). Let n ≥ 1, δn := 18 ρ and let Hs = Hs∗ ∈ Ds and ρ ∆δn := e(Hs ) + δn , e(Hs ) + δn . 8 Then

B −θ (Hs − λ − i0)−1 B −θ ∈ C ν (∆δn )

(7.7)

for any 1/2 < θ ≤ 1 and ν < θ − 12 . Proof. Let Bθ := B −θ and let Dn be the disc of the radius δn centered at e(Hs ). Since ∆δn ⊂ {z ∈ R||e(Hs ) − z| ≤ δn }, we have that Hs − λ ∈ Uδn , for λ ∈ ∆δn . By Theorem 6.1, we have that the operator H (n) (λ) := Rnρ (Hs − λ) is well deﬁned and 1 (n) µ,2 ρ, βn−1 , γn−1 H (λ) ∈ D is analytic in λ ∈ ∆δn , 8 H (n) (λ) = H (n) (λ)∗ ,

∀ λ ∈ ∆δn ∩ R.

Hence, since ρ γn−1 , βn−1 ≤ 13 , due to (6.3), we have, by Theorem 7.1, that Bθ (H (n) (λ) − i0)−1 Bθ ∈ C ν (∆δn ∩ En−1 (∆δ )),

(7.8)

where 0 ≤ ν < θ − 1/2δ γn−1 , and, as before, En (λ) ≡ En (λ, Hs ) := (H (n) (λ))u , which, by the above conclusion, is analytic. We need the following proposition to describe the set En−1 (∆δ ). 1 n Proposition 7.3. Let n ≥ 0, δn := 18 ρ and Aδn := {λ ∈ C| ρ8 δn ≤ |λ − e(Hs )| ≤ δn }. For H ∈ Uδn , we denote En (λ, Hs ) := (Rnρ (H))u ≡ Rnρ (H) Ω , λ = −Hu . Then

|En (λ, Hs )| ≥ 2−8 ρ,

for λ ∈ Aδn .

(7.9)

Proof. In this proof, we do not display the dependence of various quantities on Hs . Let λ ∈ Aδn , with δn as speciﬁed above. Deﬁne E0n (λ) by the equation En (λ) = ρ−n (E0n (λ) − λ).

(7.10)

The following estimate is shown in ([45, Eq. (V.27)] of the latter paper): 1 |λ − e| + (1 − ρ)−1 ρn+1 αn+1 . 5 This inequality and the deﬁnition of αn imply that |E0n (λ) − e| ≤

(7.11)

|E0n (λ) − λ| ≥ |λ − e| − |E0n (λ) − e| ≥

4 |λ − e| − 2γ02 c(c2 ρ2µ+1 )n ρ−1 . 5

(7.12)

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Since 2γ02 cρ−1 (c2 ρ2µ+1 )n ρn+1 and since, for λ ∈ Aδn , |λ − e| ≥

ρn+1 8·18

(7.12) gives

|E0n (λ) − λ| ≥ 2−8 ρn+1 .

(7.13)

Due to (7.10), this implies the statement of the proposition. We now conclude the proof of Theorem 7.2. Proposition 7.3 says that −8

En : ∆δn λ → En (λ) ∈ ∆2 −8

Hence En−1 (∆2

ρ

ρ

.

(7.14)

) ⊃ ∆δn and therefore by (7.8) we have that B −θ (H (n) (λ) − i0)−1 B −θ ∈ C α (∆δn ),

(7.15)

which, due to Proposition 5.5, gives (7.7). 8. Initial Conditions for the Renormalization Group Now we turn to the operators (2.10) and (1.8). Let Hg stand for either the Pauli– Fierz Hamiltonian, (2.10), or the Nelson Hamiltonian, (1.8). (Not to be confused with the standard model Hamiltonian (1.1), which is not used in the remainder of this paper.) The operators Hg − λ do not belong to the Banach spaces deﬁned above. We deﬁne an additional renormalization transformation which acts on such operators and maps them into the disc Dµ,s (α0 , β0 , γ0 ) for an appropriate choice of α0 , β0 , γ0 . PF PF or for H0N and Hpg , for either Hpg = Hp + O(g 2 ) Let H0g stand either for H0g or for Hp (see (2.10)–(2.12) and (1.8)–(1.10)), so that Hg = H0g + Ig , (p)

(p)

H0g = Hpg + Hf (p)

(p)

and Hpg = Hp + O(g 2 ).

(p)

(p)

Let e0 < e1 < · · · < gap := 1 − 0 be the eigenvalues of Hpg , with e0 (p) its ground state energy. (For the Nelson model, ej coincide with the eigenvalues (p)

j

of the particle Hamiltonian Hp .) Let Pp be the orthogonal projection onto the (p)

eigenspace corresponding to e0 . On Hamiltonians acting on Hp ⊗ Hf , which were described above, we deﬁne the map −1 R(0) ρ0 = ρ0 Sρ0 ◦ Fτ0 π0 , (p)

(8.1) (p)

(p)

(p)

where ρ0 ∈ (0, gap ] is an initial photon energy scale (recall that gap := 1 − 0 ) and where τ0 (Hg − λ) = H0g − λ and π0 ≡ π0 [Hf ] := Pp ⊗ χHf ≤ρ0 ,

(8.2)

PF given after Eq. (2.10) (see above) and for any λ ∈ C. Deﬁne the for H0g := H0g set 1 (p) (8.3) I0 := z ∈ C|Re z ≤ e0 + ρ0 . 2

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We assume ρ0 g 2 . To simplify the notation we assume that the ground state (p) energy, 0 , of the operator Hp is simple (otherwise we would have to deal with matrix-valued operators on Hf ). We have Theorem 8.1. Let Hg be the Hamiltonian given either by (2.10) or by (1.8), and (p) let gap ≥ ρ0 g 2 , µ > −1/2 and λ ∈ I0 . Then: Hg − λ ∈ D(R(0) ρ0 ). (0)

(8.4)

(0)

The operators Hλ := Rρ0 (Hg − λ)|RanPp ⊗1 , λ ∈ I0 ∩ R, are self-adjoint; (0)

(p)

µ,2 (α0 , β0 , γ0 ), Hλ − ρ−1 0 (e0 − λ) ∈ D

(8.5)

µ 2 with λ ∈ I0 , µ as in Eq. (1.11), α0 = O(g 2 ρ−1 0 ), β0 = O(g ), and γ0 = O(gρ0 ); (0)

Hλ

is analytic in λ ∈ I0 .

(8.6)

For the Pauli–Fierz and Nelson Hamiltonians, µ = 1/2 and µ > 0, respectively. (p)

Note that if ψ (p) is the ground state of Hpg with the energy e0 (p) ψ ⊗ Ω, then (p)

e0 − λ = ψ0 , (Hg − λ)ψ0 .

and ψ0 = (8.7)

Theorem 8.1 is proven in [45, Appendix II], for somewhat simpler Banach spaces that do not contain the derivatives (k∂k )q . However, an extension to the Banach spaces which are used in the present paper is straightforward. (0) Let, as before, P¯p := 1 − Pp and K := Rρ0 (Hg − λ)|Ran(P¯p ⊗1) . Note that K = (p)

(H0g − λ)|Ran(P¯p ⊗1) and therefore, ∀ λ ∈ I0 ∩ R, σ(K) = σ(Hpg )\{e0 } + [0, ∞) − λ. Hence, ∀ λ ∈ I0 ∩ R,

1 1 (p) (p) (p) (p) K ≥ e1 − e0 − ρ0 ≥ (e1 − e0 ). 2 2 (0)

(0)

Therefore 0 ∈ / σ(K). This, the relation σ(Rρ0 (Hg − λ)) = σ(Hλ ) ∪ σ(K) and Theorem 3.1 imply that (0)

Hλ

is isospectral to Hg − λ

in the sense of Theorem 3.1. Moreover, similarly to Proposition 5.5 and using the relation (0)−1

−1 = Hλ R(0) ρ0 (Hg − λ)

(Pp ⊗ 1) + (H0g − λ)−1 (P¯p ⊗ 1),

(8.8)

one shows the following result Proposition 8.2. Let µ > 0, ρ0 g 2 and ∆0 ⊆ I0 ∩ R. If Hg is given in either (2.10) or (1.8), then, for any 0 ≤ θ, ν ≤ 1, (0)

B −θ (Hλ − i0)−1 B −θ ∈ C ν (∆0 ) ⇒ B −θ (Hg − λ − i0)−1 B −θ ∈ C ν (∆0 ).

(8.9)

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9. LAP on Small Sets In this section, we prove the limiting absorption principle (LAP) for Hg on small subsets of (g , g + 12 ρ0 ), where g is the ground state energy of Hg and, as in the previous section, Hg stands for either (2.10) or (1.8). In the next section, we show 1 ρ0 ) completing the proof of Theorem 1.1. Recall that these subsets cover (g , g + 18 the deﬁnition (0)

Hλ := R(0) ρ0 (Hg − λ)|RanPp ⊗1 ,

λ ∈ I0 .

(9.1)

The right-hand side is well deﬁned according to Theorem 8.1. By Eq. (8.5), if λ ∈ I0 , then (0)

(p)

µ,2 Hλ − ρ−1 (α0 , β0 , γ0 ), 0 (e0 − λ) ∈ D

(9.2)

where α0 , β0 and γ0 are given in Theorem 8.1. Since, by our assumption, g 1 we (p) can choose ρ0 in the interval (0, gap ] so that g 2 ρ−1 0 ,

gρµ0 1.

(9.3)

Let H ψ := ψ, Hψ . We isolate the stable-central and unstable components of (0) the operator Hλ : (0)

(0)

(0)

Hλs := (Hλ )s = Hλ − Hλ Ω 1

(0)

(0)

and Hλu := (Hλ )u = Hλ Ω

(9.4)

(see Sec. 6), and let e : Ds → C be the map introduced in Theorem 6.1. We pick the parameter ρ so that 1 . (9.5) 2 Then the conditions of Theorem 7.2 are satisﬁed by Hs = Hλs , µ ∈ I0 ∩ R and this theorem implies that gρµ0 ρ ≤

B −θ (Hλs − ω − i0)−1 B −θ ∈ C ν ({(λ, ω) ∈ (I0 ∩ R) × ∆λδn }), where

1 2

1 n < θ ≤ 1, 0 < ν < θ − 12 , recall, δn = 18 ρ for n ≥ 0 and 1 ∆λδ := e(Hλs ) + δ, e(Hλs ) + δ . 8

(9.6)

(9.7)

(0)

The relations (9.6) and Hλ = Hλs + Hλu yield, in turn, that (0)

B −θ (Hλ − i0)B −θ ∈ C ν (Fδn ), where Fδ := {λ ∈ I0 ∩ R| − Hλu ∈ ∆λδ },

(9.8)

which, due to Proposition 7.4, yields B −θ (Hg − λ − i0)−1 B −θ ∈ C ν (Fδn ). This is the main result of this section.

(9.9)

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10. Covering Lemma. End of Proof of Theorem 1.1 Let g be the solution to the equation e(Hλs ) = −Hλu for λ (see the deﬁnitions of Hλs and Hλu in (9.4)). By Theorem 6.1, 0 = e(Hg s ) + Hg u is the ground state (0) energy of the operator Hλ=g and therefore, by Theorem 3.1, g is the ground state energy of the operator Hg . In the lemma below we show that for g suﬃciently small 3ρ0 ρ0 ρδn , g + δn ⊂ Fδn . (10.1) g + 16 4 Since

ρ0 4 δn

>

3ρ0 16 ρδn−1 ,

this embedding implies 1 Fδn covers g , g + ρ0 . 18

(10.2)

n≥0

This together with (9.9) implies the statement of Theorem 1.1. Lemma 10.1. For g suﬃciently small, (10.1) holds. Proof. Recall that δn = obtain

1 n 18 ρ .

We ﬁrst use the relation e(Hg s ) + Hg u = 0 to

e(Hλs ) + Hλu = Hλu − Hg u − e(Hg s ) + e(Hλs ).

(10.3)

Our next goal is to estimate |Hg u − Hλu | and |e(Hλs ) − e(Hg s )|. In order to use the Cauchy bound on derivatives, we pass to complex domains. We deﬁne the set Dδ := {λ ∈ I0 ||e(Hλs ) + Hλu | ≤ δ}.

(10.4)

Recall that D(, δ) denotes the disc centered at of a radius δ. We claim that, for g suﬃciently small and for n ≥ 0, ρ 0 (10.5) D g , δn ⊂ Dδn . 4 We prove this claim by induction in n and obtain needed estimates on |Hg u − Hλu | and |e(Hλs ) − e(Hg s )| in the process. We assume it is true for n ≤ j − 1 and prove it for n = j. For j = 0, the induction assumption is absent and so our proof of the induction step yields also the ﬁrst step. We estimate |Hg u − Hλu |. Let ∆0 E(λ) be deﬁned by the relation Hλu =: ρ−1 0 (e0 − λ) + ∆0 E(λ). Then Hλu − Hg u = ρ−1 0 (g − λ) + ∆0 E(λ) − ∆0 E(g ).

(10.6)

We have by (9.2) and analyticity of ∆0 E(λ) on I0 , |∂λ ∆0 E(λ)| ≤ α0 /ρ0 . This gives |∆0 E(g ) − ∆0 E(λ)| ≤ ρ−1 0 α0 |g − λ|.

(10.7)

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The last two relations imply that in D(g , ρ40 δj ), |Hg u − Hλu | ≤

1 (1 + α0 )δj . 4

(10.8)

Now we estimate |e(Hλs ) − e(Hg s )|. Recall the deﬁnition En (ω, Hλs ) := (Rnρ (Hλs − ω))u ≡ Rnρ (Hλs − ω) Ω . Deﬁne ∆n E(ω, Hλs ) := En (ω, Hµs ) − ρ−1 En−1 (ω, Hλs ).

(10.9)

It is shown in [45, Eqs. (V.24)–(V.25)] that e(Hs ) satisﬁes the (ﬁxed point) equation e(Hs ) =

∞

ρi ∆i E(e(Hs ), Hs )

(10.10)

i=1

(the eigenvalue is equal to the sum of the corrections due to individual steps of the RG). Here the series on the right-hand side converges absolutely by the estimate −m 1 i+1 m ρ |∂ω ∆i E(ω, Hs )| ≤ αi for i ≤ j and m = 0, 1, (10.11) 12 shown in [20]. The relation (10.10) implies e(Hλs ) − e(Hg s ) =

∞

ρi (∆i E(e(Hλs ), Hλs ) − ∆i E(e(g ), Hg s )). (10.12)

i=1

We estimate this series. It follows from the analyticity of En (ω, Hs ) in Hs , see [20, Proposition 5.3], that ∆i E(ω, Hλs ) are analytic in λ ∈ Dδi , i ≤ j − 1. By the induction assumption Dδi ⊃ D(g , ρ40 δi ) for i ≤ j − 1. Hence using the Cauchy formula we conclude from (10.11) that ρ 4αi 0 on D g , δi |∂λ ∆i E(ω, Hλs )| ≤ (1 − ρ)ρ0 δi 4 for i ≤ j − 1. The latter estimate together with (10.11) and the deﬁnition δj = gives ∞

1 j 18 ρ

ρi |∆i E(e(Hλs ), Hλs ) − ∆i E(e(Hg s ), Hg s )|

i=1

≤

j−1 i=1

≤

ρ

i

∞ 12αi 4αi |e(Hλs ) − e(Hg s )| + |λ − g | + 2 ρi αi ρi+1 (1 − ρ)ρ0 δi i=j

20α1 80α1 |e(Hλs ) − e(Hg s )| + |λ − g | + 4αj ρj ρ (1 − ρ)ρ0

in D(g , ρ40 δj ). This estimate, together with the relation (10.12), gives |e(Hλs ) − e(Hg s )| ≤

40α1 δj + 160αj δj 1−ρ

(10.13)

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ρ in D(g , ρ40 δj ), provided α1 ≤ 40 . This estimate, together with (10.3) and (10.8) and the deﬁnition of Dδn , implies (10.5) with n = j, provided 1 + α0 40α1 + + 160α0 ≤ 1. (10.14) 4 1−ρ Remembering the deﬁnition of αj , we see that the latter conditions can be easily arranged by taking g suﬃciently small. This proves (10.5). Now we proceed to the proof of (10.1). Notice that Fδ can be written in the form ρ (10.15) Fδ := λ ∈ I0 ∩ R| − δ ≤ e(Hλs ) + Hλu < − δ . 8 Hence we estimate e(Hλs ) + Hλu from above and below. ρ0 ρ0 0 Since (g + 3ρ 16 ρδn , g + 4 δn ] ⊂ D(g , 4 δn ) the estimates (10.8) and (10.13), ρ0 0 with j = n, on |Hg u − Hλu | and |e(Hλs ) − e(Hg s )| hold on (g + 3ρ 16 ρδn , g + 4 δn ] and together with the relation (10.3) they give

e(Hλs ) + Hλu ≥ −|Hλu − Hg u | − |e(Hg s ) − e(Hλs )| 1 40α1 ≥ − (1 + α0 )δn − δn − 160αnδn ≥ −δn , (10.16) 4 1−ρ provided (10.14) holds. Furthermore, (10.3), (10.6) and (10.7) and (10.13), with 0 j = n, and the relation g + 3ρ 16 ρδn ≤ λ yield e(Hλs ) + Hλu ≤ ρ−1 0 (g − λ) + |∆0 E(λ) − ∆0 E(g )| + |e(g ) − e(λ)| 3 α0 40α1 ρ0 ρ0 ρδn + δn + δn + 160αn δn ≤ − ρδn , 16 4 1−ρ 8 provided ρ and ρ0 are chosen in such a way that 1 40α1 α0 ρ0 ρ ≥ + + 160αn 16 4 1−ρ ≤−

(10.17)

(10.18)

(the latter is possible, since by the deﬁnition of αj ≤ α0 = O(g 2 ρ−1 0 ) (see Theorem 8.1), if one chooses g suﬃciently small). The last two inequalities and (10.15) imply (10.1). 11. Supplement: Background on Fock Space Let h be either L2 (R3 , C, d3 k) or L2 (R3 , C2 , d3 k). In the ﬁrst case, we consider h to be the Hilbert space of one-particle states of a scalar boson or phonon, and in the second case, of a photon. The variable k ∈ R3 is the wave vector or momentum of the particle. (Recall that throughout this paper, the propagation speed c, of photon or photons and Planck’s constant, , are set equal to 1.) The Bosonic Fock space, F , over h is deﬁned by ∞

Sn h⊗n , (11.1) F := n=0

where Sn is the orthogonal projection onto the subspace of totally symmetric n-particle wave functions contained in the n-fold tensor product h⊗n of h; and

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S0 h⊗0 := C. The vector Ω := (1, 0, . . .) is called the vacuum vector in F . Vectors Ψ ∈ F can be identiﬁed with sequences (ψn )∞ n=0 of n-particle wave functions, which are totally symmetric in their n arguments, and ψ0 ∈ C. In the ﬁrst case these functions are of the form, ψn (k1 , . . . , kn ), while in the second case, of the form ψn (k1 , λ1 , . . . , kn , λn ), where λj ∈ {−1, 1} are the polarization variables. In what follows, we present some key deﬁnitions in the ﬁrst case, limiting ourselves to remarks at the end of this appendix on how these deﬁnitions have to be modiﬁed for the second case. The scalar product of two vectors Ψ and Φ is given by ∞ n d3 kj ψn (k1 , . . . , kn )ϕn (k1 , . . . , kn ). (11.2) Ψ, Φ := n=0

j=1

Given a one particle dispersion law ω(k) = |k|, the energy of a conﬁguration of n non-interacting ﬁeld particles with wave vectors k1 , . . . , kn is given by nj=1 ω(kj ). We deﬁne the free-ﬁeld Hamiltonian, Hf , by   n (Hf Ψ)n (k1 , . . . , kn ) =  ω(kj ) ψn (k1 , . . . , kn ), (11.3) j=1

for n ≥ 1 and (Hf Ω)n = 0. Here Ψ = (ψn )∞ n=0 . To ensure that the r.h.s. makes sense, we assume that ψn = 0, except for ﬁnitely many n, for which ψn (k1 , . . . , kn ) decrease rapidly at inﬁnity. Clearly, the operator Hf has a single eigenvalue, 0, with eigenvector Ω, and the rest of its spectrum is absolutely continuous. With each function ϕ ∈ h one associates an annihilation operator a(ϕ) deﬁned as follows. For Ψ = (ψn )∞ n=0 ∈ F with the property that ψn = 0, for all but ﬁnitely many n, the vector a(ϕ)Ψ is deﬁned by √ (11.4) (a(ϕ)Ψ)n (k1 , . . . , kn ) := n + 1 d3 kϕ(k)ψn+1 (k, k1 , . . . , kn ). These equations deﬁne a closable operator a(ϕ) whose closure is also denoted by a(ϕ). We also deﬁne a(ϕ)Ω = 0.

(11.5)

The creation operator a∗ (ϕ) is deﬁned to be the adjoint of a(ϕ) with respect to the scalar product deﬁned in Eq. (11.2). Since a(ϕ) is anti-linear, and a∗ (ϕ) is linear in ϕ, we write 3 ∗ (11.6) a(ϕ) = d kϕ(k)a(k), a (ϕ) = d3 kϕ(k)a∗ (k), where a(k) and a∗ (k) are unbounded, operator-valued distributions. The latter are well known to obey the canonical commutation relations (CCR): [a# (k), a# (k )] = 0, #

∗

where a = a or a .

[a(k), a∗ (k )] = δ 3 (k − k ),

(11.7)

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One can rewrite the quantum Hamiltonian Hf in terms of creation and annihilation operators, a and a∗ , as (11.8) Hf = d3 ka∗ (k)ω(k)a(k). More generally, for any operator, t, on the one-particle space h, we deﬁne operator, T , on Fock space F by the following formal expression T := a∗ (k)(ta)(k)dk, where the operator t acts on the k-variable; (T ≡ dΓ(t) is the second quantization of t.) The precise meaning of the latter expression can obtained by using a basis {φj } in the space h to rewrite it as T := j a∗ (φj )a(t∗ φj )dk. To modify the above deﬁnitions to the case of photons, one replaces the variable k by the pair (k, λ) and adds to the integrals in k sums over λ. In particular, the # creation and annihilation operators have now two variables: a# λ (k) ≡ a (k, λ); they satisfy the commutation relations # [a# λ (k), aλ (k )] = 0,

[aλ (k), a∗λ (k )] = δλ,λ δ 3 (k − k ).

(11.9)

One can introduce the operator-valued transverse vector ﬁelds by eλ (k)a# a# (k) := λ (k), λ∈{−1,1}

where eλ (k) ≡ e(k, λ) are polarization vectors, i.e. orthonormal vectors in R3 satisfying k · eλ (k) = 0. Then, in order to reinterpret the expressions in this paper for photons, one either adds the variable λ, as was mentioned above, or replaces, in appropriate places, the usual product of scalar functions or scalar functions and scalar operators by the dot product of vector-functions or vector-functions and operator-valued vector-functions. Acknowledgments Part of this work was done while the third author was staying at IAS Princeton and visiting ETH Z¨ urich and ESI Vienna. He is grateful to these institutions for hospitality. He was supported by NSERC Grant No. NA7901. The authors are grateful to anonymous referee for several useful remarks. References [1] W. Abou Salem, J. Faupin, J. Fr¨ ohlich and I. M. Sigal, On theory of resonances in non-relativisitc QED, Adv. Appl. Math. 43(3) (2009) 201–230. [2] A. Arai and M. Hirokawa, Ground states of a general class of quantum ﬁeld Hamiltonians, Rev. Math. Phys. 12(8) (2000) 1085–1135. [3] V. Bach, Th. Chen, J. Fr¨ ohlich and I. M. Sigal, Smooth Feshbach map and operatortheoretic renormalization group methods, J. Funct. Anal. 203 (2003) 44–92. [4] V. Bach, Th. Chen, J. Fr¨ ohlich and I. M. Sigal, The renormalized electron mass in non-relativistic quantum electrodynamics, J. Funct. Anal. 243(2) (2007) 426–535. [5] V. Bach, J. Fr¨ ohlich and A. Pizzo, Infrared-ﬁnite algorithms in QED: The groundstate of an atom interacting with the quantized radiation ﬁeld, Comm. Math. Phys. 264(1) (2006) 145–165.

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Reviews in Mathematical Physics Vol. 23, No. 2 (2011) 211–232 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004291

RUELLE–LANFORD FUNCTIONS AND LARGE DEVIATIONS FOR ASYMPTOTICALLY DECOUPLED QUANTUM SYSTEMS

YOSHIKO OGATA Graduate School of Mathematics, University of Tokyo, Japan [email protected] LUC REY-BELLET Department of Mathematics and Statistics, University of Massachusetts Amherst, USA [email protected] Received 22 September 2010 Revised 2 February 2011 We recover, expand, and unify quantum (and classical) large deviation results for lattice Gibbs states. The main new ingredient in this paper is a control on the overlap of spectral projections for non-commutative observables. Our proof of large deviations is based on Ruelle–Lanford functions [20, 34] which establishes the existence of a rate function directly by subadditivity arguments, as done in the classical case in [23, 32], instead of relying on G¨ artner–Ellis theorem, and cluster expansion or transfer operators as done in the quantum case in [21, 13, 27, 22, 16, 28]. We assume that the Gibbs states are asymptotically decoupled [23, 32], which controls the dependence of observables localized at diﬀerent spatial locations. In the companion paper [29], we discuss the characterization of rate functions in terms of relative entropies. Keywords: Quantum spin systems; large deviations; thermodynamics formalism; quantum statistical mechanics. Mathematics Subject Classiﬁcation 2010: 82B10, 60F10

1. Introduction Consider a spin system on the lattice Zd in thermal equilibrium described by a state ω. For a region Λ ⊂ Zd , let KΛ be a macroscopic observable, for example the total energy or the total magnetization in the region Λ. One expects, as a rule, that such observables have a distribution which is very sharply concentrated around the equilibrium mean value and that the ﬂuctuations of such observables are exponentially small in the volume |Λ| of the domain, except at a ﬁrst order phase transition where coexisting phases can induce macroscopically large ﬂuctuations. This property is expressed by a large deviation principle: For a Borel set A ⊂ R 211

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let IA (KΛ /|Λ|) denote the spectral projection onto the eigenspace of KΛ /|Λ| corresponding to eigenvalues of KΛ /|Λ| in the set A. A large deviation principle holds if there exists a rate function s(x) such that ω(IA (KΛ /|Λ|)) exp |Λ| sup s(x) . x∈A

In classical mechanics systems this problem is mathematically very wellunderstood and very general large deviations theorems have been proved both for systems on a lattice or in the continuum see [20, 34, 30, 12, 6, 10, 14, 15, 23, 32, 33]. For quantum mechanical systems, the problem of large deviations has, in comparison, received little attention and is only partially understood. The diﬃculty lies, partly, in the non-commutativity of quantum mechanical observables but also at a deeper level, in the lack of control on the boundary eﬀects in quantum mechanics. Known bulk/boundary estimates are suﬃcient to prove the existence of thermodynamic functions such as entropy and free energy, see e.g., [35, 19, 5, 36] but they are, so far, not suﬃcient to prove general large deviation results, especially at low temperatures for spatial dimension more than one. A number of quantum large deviation results have been proved in the past few years [31, 21, 13, 22, 27, 16, 17, 7, 28, 8], (see also [4] for an information-theoretic interpretation of relative entropy). Common to all these papers is that the large deviation results are obtained by an application of G¨ artner–Ellis theorem, in particular the smoothness of the logarithmic moment generating functions (i.e. a suitable free energy functional) is necessary and is proved by cluster expansion or using a transfer operator. For classical Gibbs states there are several diﬀerent proofs of the large deviation principle and we follow here the approach by Lewis, Pﬁster and Sullivan [23, 32]. In this approach, the state ω of the inﬁnite system is assumed to be asymptotically decoupled (see Sec. 3.2 for a formal and more general deﬁnition): Given a ﬁnite region V ⊂ Zd there exists a function c(V ) such that if A is a nonnegative observable supported in the region V and B is a nonnegative observable supported in the complement Zd \V one has e−c(V ) ω(A)ω(B) ≤ ω(AB) ≤ ω(A)ω(B)ec(V ) , with lim V

Zd

c(V ) = 0. |V |

In the classical case, this property is a fairly easy consequence of the DLR equation for Gibbs states. Using this property and subadditivity arguments one proves then directly the existence of a rate function s(x) for the observable of interest. This general approach to large deviation (summarized in Sec. 2) was pioneered by Lanford and Ruelle papers [20, 34] and we follow here the terminology of [23, 32]. To use this strategy for quantum systems we face two obstacles. The ﬁrst one is that it is not known whether a Gibbs-KMS state for a quantum spin system is asymptotically decoupled in general. This property is only known to hold in spatial

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dimension one (proved by Araki in [1]) and at high temperatures in arbitrary dimension (see, e.g., [2]). We believe that it is an important open problem to determine whether this property holds for general quantum spin systems or not but there is no new result in this direction in this paper. The second obstacle lies in the generalization of the subadditivity argument of [23, 32] to general non-commutative observables. The new ingredient needed here is a control on the overlap of spectral projections for KΛ and the spectral projections for KΛ1 + KΛ2 with Λ = Λ1 ∪ Λ2 , which diﬀers from KΛ by a boundary term. This new estimate is proved in Sec. 4.4, see Proposition 4.10. Using this approach we are able to recover, unify, and extend slightly the known large deviation results for quantum (and classical) spin systems. In addition the proofs given here are quite short and self-contained. This paper is organized as follows. In Sec. 2, we give a brief exposition of the road to large deviation via proving the existence of the Ruelle–Lanford function which is an Boltzmann entropy-like functional. In Sec. 3, we recall the elements of the quantum spin system formalism needed in the paper and we introduce the asymptotic decoupling condition for states of quantum systems which is central in our analysis. In Sec. 4, we prove large deviation theorems for three diﬀerent cases: (a) Commuting observables, (b) Classical observables, (c) General ﬁniterange observables in dimension 1. The discussion of the rate functions and their characterization in terms of relative entropies is in the companion paper [29]. 2. Ruelle–Lanford Functions Let X be a complete metric space, let {µn } be a sequence of Borel probability measures on X, and let {vn } an increasing sequence of positive numbers with limn→∞ vn = +∞. We say that µn satisﬁes a large deviation principle (LDP) on the scale vn if there exists a function I : X → [0, ∞], lower semicontinuous and with compact level sets, such that for any closed set C lim sup n→∞

1 log µn (C) ≤ − inf I(x), x∈C vn

(2.1)

1 log µn (O). vn

(2.2)

and for any open set O − inf I(x) ≤ lim inf x∈O

n→∞

The function I is called the rate function for the LDP. In statistical mechanics applications the measures µn are often distributions of sums of R- or Rd - valued weakly dependent random variables. One standard approach to prove an LDP is to combine the exponential Markov inequality for the upper bound (2.1) and a change of measure and ergodicity argument for the lower bound (2.2) (see, e.g., the proofs of Cramer and G¨ artner–Ellis theorem in [9]). In the presence of phase transitions, i.e. lack of ergodicity with respect to spatial translation, additional arguments are needed to provide a lower bound. For example,

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in [12], the lower bound for the LDP for classical lattice Gibbs states is obtained by using the Shannon–McMillan theorem and an approximation argument by ergodic states. Another route to LDP’s using subadditivity arguments, much in the spirit of statistical mechanics, was pioneered in a remarkable paper by Lanford [20], itself based on earlier work by Ruelle [35]. We follow closely here the presentation in [23], see also [32]. For Borel sets B let us deﬁne the set functions 1 1 m(B) = lim sup log µn (B), m(B) = lim inf log µn (B). (2.3) n→∞ vn n→∞ vn One has the elementary properties (1) For any Borel set B, we have −∞ ≤ m(B) ≤ m(B) ≤ 0. (2) If B1 ⊂ B2 , then m(B1 ) ≤ m(B2 ) and m(B1 ) ≤ m(B2 ). (3) For all B1 , B2 , we have m(B1 ∪ B2 ) = max{m(B1 ), m(B2 )}. Property (3) is an key property in large deviations and is usually refereed to as the principle of the largest term: large deviations occur in the least unlikely way of all possible ways. Let Bε (x) denote the ball of radius ε centered at x and let us deﬁne s(x) = inf m(Bε (x)),

s(x) = inf m(Bε (x)).

ε

ε

(2.4)

Definition 2.1. The pair (µn , vn ) has a Ruelle–Lanford function (RL-function) s(x) if s(x) = s(x), for all x ∈ X. In this case we set s(x) = s(x) = s(x). The next proposition is standard and shows that the existence of RL-function (almost) implies the existence of a LDP. Proposition 2.2. The Ruelle–Lanford function s(x) is upper semicontinuous and m(O) ≥ sup s(x),

O open,

(2.5)

K compact .

(2.6)

x∈O

m(K) ≤ sup s(x), x∈K

Proof (Sketch). The upper semicontinuity follows from the deﬁnition. The lower bound is immediate: For any x ∈ O and ε suﬃciently small we have m(O) ≥ m(Bε (x)) and thus m(O) ≥ s(x) = s(x) for all x ∈ O. To prove the upper bound, given ε > 0 we cover the compact set K by N = N (ε) balls Bε (xl ) with centers in xl ∈ K. Using Properties (2) and (3) we have N m(K) ≤ m Bε (xl ) ≤ max m(Bε (xl )) ≤ sup m(Bε (x)). l=1

l

Since ε is arbitrary the upper bound follows.

x∈K

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The statement in Proposition 2.2 is usually referred to as a weak large deviation principle since the upper bound holds only for compact sets. In the problems discussed in this paper, the probability measures µn are supported uniformly on compact sets and the previous lemma yields immediately a large deviation principle with rate function −s(x). More generally one obtains a large deviation principle by combining Proposition 2.2 with a proof that the sequence of probability measures µn is exponentially tight (see, e.g., [9, Sec. 1.2]). To identify the rate function we use a standard large deviation result. Proposition 2.3 (Laplace–Varadhan’s Lemma). Suppose that µn satisfies a large deviation principle on the scale vn with rate function I(x). Let f be any continuous function and suppose that for some γ > 1 we have the moment condition lim supn→∞ v1n log µn (eγvn f (x) ) < ∞. Then 1 log µn (evn f (x) ) = sup(f (x) − I(x)). n→∞ vn x lim

If X = Rn and f (x) = α · x, we obtain 1 log µn (evn α·x ) = sup(α · x + s(x)), n→∞ vn x

e(α) ≡ lim

i.e. the moment generating function of µn is the Legendre transform of −s(x). If, in addition, we know, a priori, that the rate function s(x) is concave then by convex duality we obtain that s(x) = inf (e(α) − α · x), α

that is, the rate function is the Legendre transform of the logarithmic moment generating function. Note that in our examples the moment condition will be trivially satisﬁed. 3. Quantum Lattice Systems 3.1. Interactions and states We introduce some notations and brieﬂy recall the mathematical framework for quantum spin systems, [19, 36, 5, 3]. C ∗ -algebras. Let A be a ﬁnite-dimensional C ∗ -algebra. For any ﬁnite subset Λ ⊂ Zd , let OΛ = x∈Λ Ox where Ox is isomorphic to A. If Λ ⊂ Λ , there is a natural embedding OΛ into OΛ and the algebras {OΛ }Λ⊂Zd ,ﬁnite form a partially ordered family of matrix algebras. The algebra of observables for the inﬁnite system is given by the C ∗ -inductive limit O of Λ⊂Zd ,ﬁnite OΛ . States. Let ω be a state on O, i.e. ω is a positive, normalized linear functional on O. Let {τx }x∈Zd denote the group of spatial translations. A state ω is called translation invariant if ω(τx A) = ω(A) for all x ∈ Zd and all A ∈ O. The action of Zd on O is asymptotically abelian [5] and thus the set of translation invariant states is a simplex. We say that a state is ergodic if it is an extremal point of this simplex.

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Classical subalgebras and states. A standard probabilistic setting is recovered by considering commutative (sub)algebras. Let A(cl) be an abelian subalgebra of A (cl) (cl) (cl) with N = dim A(cl) . For ﬁnite subsets Λ of Zd let OΛ = x∈Λ Ox with Ox (cl) is isomorphic to A(cl) . We denote by O(cl) the inductive limit of Λ⊂Zd ,ﬁnite OΛ . The commutative algebra O(cl) can be identiﬁed with C(L) where L = {1, . . . , N }Z with product topology is called a classical C ∗ -algebra. The restriction of any state ω on O gives a normalized linear functional ω (cl) on O(cl) . By Riesz Markov Theorem there exists a probability measure dω (cl) such that for any A ∈ O(cl) (cl) ω(A) = ω (A) = A(l)dω(l). d

L

Interactions and Hamiltonians. An interaction Ψ = {ψX }X⊂Zd,ﬁnite is a map from the the ﬁnite subsets of Zd to selfadjoint elements ψX in OX . We will assume throughout this paper that Ψ is translation invariant, i.e. τx (ψX ) = ψX+x for any X ⊂ Zd and any x ∈ Zd . An interaction Ψ is classical if there exists a classical C ∗ -subalgebra O(cl) such that ψX ∈ O(cl) for all X ⊂ Zd . We equip translation invariant interactions Ψ with the norm |X|−1 ψX ,

Ψ ≡ X0

where |X| is the cardinality of the set X and denote by B the corresponding Banach space. To any interaction Ψ ∈ B we associate Hamiltonians (or macroscopic observables) KΛ = KΛ (Ψ): For Λ ⊂ Zd ﬁnite we deﬁne KΛ = ψX . X⊂Λ

Furthermore, to any Ψ ∈ B, we associate an observable in O by 1 AΨ = ψX . |X| X0

When we consider Gibbs state, two kinds of interactions Ψ and Φ will be considered. The interaction Ψ corresponds to the observables while Φ deﬁnes the Gibbs state. We denote by KΛ the local Hamiltonian associated with Ψ and by HΛ associated with Φ. Large deviations. For n ∈ N let Λ(n) = {z ∈ Zd ; 0 ≤ zi ≤ n − 1} denote the cube with |Λ(n)| = nd lattice points and left hand corner at the origin. If ω is an ergodic state then the von Neumann ergodic theorem implies that lim

n→∞

1 KΛ(n) = ω(AΨ ) |Λ(n)|

strongly in the GNS representation and it is natural to investigate the large deviation properties, on the scale vn = |Λ(n)|, of the sequence of Borel measures on R µn (A) ≡ ω(IA (|Λ(n)|−1 KΛ(n) )),

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where A is a Borel set and IA (H) denotes the spectral projection onto the eigenspace of H spanned by the eigenvalues contained in the set A. We interpret the µn (A) as the probability that the observables |Λ(n)|−1 KΛ(n) takes value in A if the system is in the state ω. 3.2. Asymptotically decoupled states The states we consider in this paper obey a property of weak dependence between disjoint regions of the lattice. We follow here the terminology used in [32] for the classical case. Let C(m) be an arbitrary cube of side length m and let us denote by C r (m) the cube of side length m + 2r centered at the same point of Zd as C(m). Definition 3.1. A state ω on O is asymptotically decoupled with parameters g and c if (1) There exist a function g : N → N with limm→∞ g(m)/m = 0 and a function c : N → [0, ∞) with limm→∞ c(m)/|C(m)| = 0. (2) For any cube C(m), m ∈ N, any nonegative A ∈ OC(m) , any nonnegative B ∈ OC g(m) (m)c we have e−c(m) ω(A)ω(B) ≤ ω(AB) ≤ ec(m) ω(A)ω(B). Examples of asymptotically decoupled states are (a) Product states. Any product state ω0 is asymptotically decoupled with parameters c = g = 0. (b) Classical Gibbs states. Let O(cl) be a classical C ∗ -algebra and let Φ be a

classical translation invariant interaction such that Φ 0 ≡ X0 φx is ﬁnite. A Gibbs state for the interaction Φ is a probability measure ω (Φ) which satisﬁes the DLR equation (see, e.g., [35, 36]). Using the DLR equation one proves easily (see, e.g., [23, Sec. 9]) that for any positive A ∈ OC(m) we have e−c(m) ω (Φ) (A) ≤

tr(Ae−HΛ ) ≤ ec(m) ω (Φ) (A), tr(e−HΛ )

(3.1)

with c(m) = WC(m) where WC(m) is the boundary interaction WC(m) =

(Φ) is asymptotically decoupled if X∩C(m) =∅ φX . This implies easily that ω X∩C(m)c =∅

Φ 0 < ∞. (c) Quantum KMS states. Let Φ be a translation invariant interaction. A KMS state for the interaction Φ is a state which satisﬁes the KMS condition or equivalently the Gibbs condition which is a quantum analog of the DLR equation (see, e.g., [5, 36, 3] for an up-to-date presentation). It is not known if KMSGibbs states are asymptotically decoupled, in general. Let us assume however that [1, 2] either

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(i) d = 1 and Φ ﬁnite range (i.e. for some R > 0 diamX > R implies φX = 0), or

(ii) d arbitrary and Φ λ ≡ X0 eλ|X| φX is suﬃciently small, then one can show that for a Gibbs-KMS state ω (Φ) and A ∈ OC(m) we have the

bound (3.1) where c(m) = C(Φ) X∩C(m) =∅ φX . Contrary to the classical X∩C(m)c =∅

case the bound is highly nontrivial to prove and relies on the Gibbs condition, Araki perturbation theory, and control of imaginary-time dynamics. This bound implies that ω (Φ) is asymptotically decoupled. (d) Markov measures. Let ω be a stationary Markov chain on a ﬁnite state space with transition matrix Q and invariant probability q. Then ω is asymptotically decoupled if and only if Q is irreducible and aperiodic (i.e. mixing). If m is the smallest integer such that Qm has strictly positive entries then the parameters are Qm (σ1 , σ2 ) g(m) = m − 1, c(n) = sup log . q(σ2 ) σ1 ,σ2 (e) Finitely correlated states. These states are a non-commutative generalization of Markov measures and are asymptotically decoupled if and only if they are mixing which occur under suitable conditions similar to the aperiodicity condition for Markov measures. See [18, 11, 28] for details. 4. Quantum Large Deviations Theorems We prove several large deviations theorems for quantum states (in order of increasing diﬃculty) by showing the existence of concave RL-functions. This uniﬁes, simpliﬁes and extend a number of quantum large deviation results which have been proved with diﬀerent techniques (G¨ artner–Ellis Theorem via transfer operators, cluster expansions, etc.). Our proof have the advantage of being fairly short, selfcontained, to apply in some situations where the rate function is not smooth. 4.1. Preliminaries In this section we prove an energy estimate used throughout the paper and explain the strategy (after [23]) used to prove the existence of a concave Ruelle–Lanford function. The ﬁrst fact is a very slight variation on standard bulk/boundary energy estimate, see, e.g., [36, 5, 32]. Given integers n and m and a function g(m) such that limm→∞ g(m)/m = 0 we choose k to be largest even integer such that n = k(m + 2g(m)) + r,

0 ≤ r < 2(m + 2g(m)),

(having k even will be convenient in the sequel). We next decompose the cube Λ(k(m + 2g(m)) into k d pairwise disjoint and contiguous cubes C˜j , each of which are each translates of Λ(m + 2g(m)) and then further divide each cube C˜j into a

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cube Cj which is centered at the same point as C˜j and is a translate of Λ(m) and a “corridor” C˜j \ Cj of width g(m). We shall need estimates on the diﬀerence between the Hamiltonian KΛ (n) and the “decoupled” Hamiltonian for the collection of cubes

d Cj , i.e. kj=1 KCj .

Lemma 4.1. Let Ψ be an interaction with Ψ ≡ X0 |X|−1 ψx < ∞. Then there exists a function F (m) = F (m, Ψ) with limm→∞ F (m) = 0, such that kd 1 (4.1) KΛ(n) − lim sup KCj ≤ F (m). |Λ(n)| n→∞ j=1 We will also use an immediate consequence of Lemma 4.1. Corollary 4.2. Let Ψ be an interaction with Ψ < ∞. Then there exists a function F (m) = F (m, Ψ) with limm→∞ F (m) = 0, such that kd 1 1 (4.2) lim sup K − K Cj ≤ F (m). Λ(n) |Λ(km)| j=1 n→∞ |Λ(n)| Proof of Lemma 4.1. To simplify notation we set l = m + 2g(m) in the proof. If D = {x ∈ Zd ; ai ≤ xi < ai + l} is a cube of side length l and r ∈ N such that r < l/2 we denote Dr = {x ∈ Zd ; ai + r ≤ xi < ai + l − r} the cube of side length l − 2r centered at the same point as D. Let us consider two cubes D ⊂ D ⊂ Zd . We have 1

KD − KD ≤

ψX

ψX ≤ |X| x∈D Xx X ⊂D

X⊂D X ⊂D

≤

1 1

ψX +

ψX

|X| |X|

x∈D \Dr Xx

≤ |D \Dr | Ψ + |Dr |

x∈Dr Xx X ⊂D

X0 diam(X)>r

1

ψX . |X|

(4.3)

Using (4.3), we have for any r, lim sup n→∞

1

KΛ(n) − KΛ(kl)

|Λ(n)| 

 |Λ(n)\Λr (kl)| |Λr (kl)|

Ψ + ≤ lim  n→∞ |Λ(n)| |Λ(n)| =

X0 diam(X)>r

1

ψX . |X|

 X0 diam(X)>r

 1

ψX   |X|

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Since r is arbitrary, we have lim sup n→∞

1

KΛ(n) − KΛ(kl) = 0. |Λ(n)|

(4.4)

Using (4.3) again, we have kd 1 (KC˜j − KCj ) lim sup n→∞ |Λ(n)| j=1 



k d |Λ(l)|   |Λ(l)\Λr (m)| Ψ + |Λr (m)| n→∞ |Λ(n)|  |Λ(l)| |Λ(l)|

≤ lim

X0 diam(X)>r

 1

ψX  . |X|

If r = h(m) with limm→∞ h(m) = ∞ and limm→∞ h(m)/m = 0, we get d k 1 lim sup (KC˜j − KCj ) = o(m). n→∞ |Λ(n)| j=1 Finally kd KΛ(kl) − KC˜j ≤ j=1

ψX =

X⊂Λ(kl) ˜j X ⊂someC

X⊂Λ(kl) ˜j X ⊂someC

(4.5)

k |X ∩ C˜j |

ψX

|X| j=1 d

k 1 1 |X ∩ C˜j |

ψX = |Λ(kl)|d(Ψ, l) ≤ |Λ(kl)| d k j=1 |C˜j | |X| ˜ d

X ⊂Cj

(4.6) with d(Ψ, l) =

|X ∩ Λ(l)| 1

ψX = |Λ(l)| |X| X ⊂Λ(l)

≤

|Λr (l)| |Λ(l)|

X0 diam(X)>r

x∈Λ(l) Xx X ⊂Λ(l)

1

ψX

|X||Λ(l)|

1 |Λ(l)| − |Λr (l)|

ψX +

Ψ . |X| |Λ(l)|

Since l = m + 2g(m) if we pick r = h(m) as above we get kd 1 lim sup − K K ˜j = o(m). Λ(kl) C n→∞ |Λ(n)| j=1

(4.7)

(4.8)

Combining the bounds (4.4), (4.5) and (4.8) concludes the proof of Lemma 4.1.

Proof of Corollary 4.2. An easy estimate shows that the diﬀerence between

kd

k d

|Λ(n)|−1 KΛ(n) − |Λ(km)|−1 j=1 KCj and |Λ(n)|−1 KΛ(n) − j=1 KCj is O(g(m)/m) Ψ .

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The second fact is a general remark on the strategy to prove the existence of a concave RL function [23]. Remark 4.3. Let x, x1 , x2 such that 12 (x1 + x2 ) = x and let 0 < ε < ε. To prove the existence of a concave RL-function it is enough to prove that m(Bε (x)) ≥

m(Bε (x1 )) + m(Bε (x2 )) . 2

(4.9)

Indeed if we set x1 = x2 = x in (4.9), then we obtain s(x) ≥ s(x), and therefore the Ruelle–Lanford function s(x) exists. Using then (4.9) again, we obtain that s(x) ≥

s(x1 ) + s(x2 ) . 2

Since s(x) is upper-semicontinuous, this implies that s(x) is concave. 4.2. Tracial state and conserved quantities In this section, we prove a quantum large deviation theorem in the simplest possible case. We bypass a number of issue associated to taking thermodynamic limits for the states by considering ﬁrst the finite volume Gibbs states ωΛ(n) (A) =

tr(Ae−HΛ(n) ) . tr(e−HΛ(n) )

In addition, we assume that the Hamiltonian and that the macroscopic observables KΛ is a conserved quantity, i.e. the commutators [KΛ , HΛ ] vanish for all Λ. Note that, although very restrictive, this condition is, in general, satisﬁed for thermodynamic quantities such as magnetization, density, energy, etc. The following theorem provides a (weak) justiﬁcation that macroscopic conserved quantities are exponentially concentrated in equilibrium. An important special case is the case where HΛ = 0, that is one consider the tracial state tr. In this case any observable KΛ can be chosen arbitrarily and the rate function s(x) is the microcanonical entropy whose existence is of course well known. The large deviation statement for the tracial state can be found, e.g., in [36]; the only novelty here, maybe, is a very simple proof. Theorem 4.4. Let Φ and Ψ be interaction with Φ < ∞ and Ψ < ∞. Suppose that the commutators [KΛ(n) , HΛ(n) ] commute for all n. Then the probability measures µn (A) =

tr(IA (|Λ(n)|−1 KΛ(n) )e−HΛ (n) ) , tr(e−HΛ (n) )

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satisfies a large deviation principle on the scale |Λ(n)| with a concave rate function s(x). We have s(x) = inf (P (α) − αx),

sup(αx + s(x)) = P (α),

α

x −1

where P (α) = limn→∞ |Λ(n)|

log tr(e

−HΛ(n) +αKΛ(n)

) is the translated free energy.

Proof. Let us choose x, x1 , x2 and ε, ε as in Remark 4.3. Given n > m let k be the even integer such that n = km + r with 0 ≤ r < 2m − 1 (having k even is useful later). Divide the cube Λ(km) into k d disjoint contiguous cube Cj , j = 1, . . . , k d each of which is a translate of the cube Λ(m). (n) (n) Let us denote by λj the eigenvalues of HΛ(n) and by µj the eigenvalues of KΛ(n) . Since HΛ(n) and KΛ(n) commute we have (n) e−λj (n) µ

µn (Bε (x)) =

j j; |Λ(n)| ∈Bε (x)

.

(n)

e−λj

(4.10)

j

By Corollary 4.2, we can choose M and N = Nm so that for m > M and n > N we have kd |Λ(n)|−1 KΛ(n) − |Λ(km)|−1 KCj ≤ (ε − ε ). j=1 ˆ(m) be an Let µ(m) be an eigenvalue of KΛ(m) with µ(m) /|Λ(m)| ∈ Bε (x1 ) and let µ ˆ (m) /|Λ(m)| ∈ Bε (x2 ). Let us assign µ(m) to each cube eigenvalue of KΛ(m) with µ d kd ˆ(m) to the each cube Cj with j = k2 + 1, . . . , k d . Then Cj with j = 1, . . . , 2 and µ

d ˆ(m) ) is an eigenvalue of j KCj such that µ ˜(km) /|Λ(km)| ∈ µ ˜(km) ≡ k2 (µ(m) + µ Bε (x). For m > M and n ≥ N = Nm , by Weyl’s perturbation theorem, for ˆ(m) there exists an eigenvalue µ(n) of KΛ (n) such that any choice of µ(m) and µ (n) µ /|Λ(n)| ∈ Bε (x). (n) Assume that the eigenvalues λi of HΛ(n) are listed in increasing order, counting

(n) ˜ be the eigenvalues of multiplicity. Let λ i j HCj ⊗ 1Λ(n)\Λ(km) also listed in increasing order. By Weyl’s perturbation theorem, and Lemma 4.1, there exists M such that for m > M there exists N = Nm such that n ≥ N we have ˜ (n) + |Λ(n)|F (m). ˜(n) − |Λ(n)|F (m) ≤ λ(n) ≤ λ λ i i i Using the formula (4.10), we obtain that µn (Bε (x)) ≥ µm (Bε (x1 )) and thus log µn (Bε (x)) ≥ |Λ(n)|

kd 2

µm (Bε (x2 ))

kd 2

e−2|Λ(n)|F (m)

log µm (Bε (x1 )) log µm (Bε (x2 )) k d |Λ(m)| + − 2F (m). 2|Λ(m)| 2|Λ(m)| |Λ(n)|

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To conclude we take ﬁrst a lim inf over n keeping m ﬁxed and then choose a subsequence ml such that liml→∞ |Λ(ml )|−1 log µml (Bε (x1 )) = m(Bε (x1 )). Together with Remark 4.3 this concludes the proof of Theorem 4.4. Theorem 4.5. Let Φ and Ψ be interactions with Φ < ∞ and Ψ < ∞. Suppose that the commutators [KΛ(n) , HΛ(n) ] vanish for all n. Suppose ω (Φ) satisfies the condition (3.1). Then the probability measure µn (A) = ω (Φ) (IA (|Λ(n)|−1 KΛ(n) )) satisfies a large deviation principle on the scale |Λ(n)| with a concave rate function s(x). We have sup(αx + s(x)) = P (α), x −1

where P (α) = limn→∞ |Λ(n)|

log tr(e

s(x) = inf (P (α) − αx), α

−HΛ(n) +αKΛ(n)

) is the translated free energy.

Proof. Since ω (Φ) (IA (|Λ(n)|−1 KΛ(n) )) ≥ e−c(n)

tr(IA (|Λ(n)|−1 KΛ(n) )e−HΛ (n) ) , tr(e−HΛ (n) )

the theorem follows immediately from Theorem 4.4. Remark 4.6 (Equivalence of Ensembles). For the tracial case it is not diﬃcult [36] to show the variational formula s(x) = sup{s(ω); ω(AΨ ) = x} where s(ω) is the speciﬁc entropy of the state ω and that the supremum is attained exactly if ω = ω βΦ is a Gibbs-KMS state at temperature β = β(x) with β chosen in such a way that ω βΨ (AΨ ) = x. This is the equivalence of ensemble: the thermodynamic function entropy can be computed via microcanonical or canonical prescriptions. Furthermore, the LDP can be used to prove that suitable microcanonical states are equivalent to canonical states, see [36] for the classical case and [24, 25] for the quantum case. Non-commutative versions of equivalence of ensembles are considered in [7]. 4.3. Classical subalgebras In this section, we assume that ω is an asymptotically decoupled state and that Ψ ∈ B is a classical interaction, i.e. there exists a classical subalgebra O(cl) ⊂ O such that, for all X, ψX ∈ O(cl) . For example if Ψ = {ψx }x∈Zd consists of only of “one-site” interactions then Ψ is classical. More generally any classical spin system is described by a classical interaction. Note that we do not assume any relation between the interaction Ψ and the state ω; if ω = ω Φ is a Gibbs state for the interaction Φ then Φ and Ψ need not commute. As noted in Sec. 3.1 the restriction of ω on O(cl) can be identiﬁed with a probability measure dω (cl) on the conﬁguration space L. Furthermore, it is easy to see that the state ω (cl) on the C ∗ -algebra O(cl) C(L) is asymptotically decoupled whenever the state ω on O is asymptoticallly decoupled.

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We have Theorem 4.7. Let Ψ be a classical interaction with Ψ < ∞ and let ω be an asymptotically decoupled state. Then the sequence of probability measures µn (A) = ω(IA (|Λ(n)|−1 KΛ(n) )), satisfies a large deviation principle on the scale |Λ(n)| with a concave rate function s(x). Moreover s(x) = inf (f (α) − αx), α

where f (α) = lim

n→∞

1 log ω(exp(αKΛ(n) )). |Λ(n)|

Proof. The proof reduces to the classical case (see [23]) since the measures µn can be written as µn (A) = ω (cl) (IA (|Λ(n)|−1 KΛ(n) )) = I{|Λ(n)|−1 KΛ(n) ∈A} (l)dω (cl) (l) and the restriction of ω (cl) on O(cl) is asymptotically decoupled. Following Remark 4.3 we choose arbitrary x, x1 , x2 such that x21 + x22 = x and 0 < ε < ε. We divide the cube Λ(n) as explained before Lemma 4.1. We choose M and N = Nm such that for m > M and n > N kd 1 1 K − K (4.11) Cj ≤ ε − ε . |Λ(n)| Λ(n) |Λ(km)| j=1 d

Let lCj be conﬁgurations such that KCj (lCj )/|Cj | ∈ Bε (x1 ) for 1 ≤ j ≤ k2 and d KCj (lCj )/|Cj | ∈ Bε (x2 ) for k2 + 1 ≤ j ≤ k d . By (4.11) any conﬁguration lΛ(n) which coincides with lCj on all Cj satisﬁes KΛ(n) (lΛ(n) )/|Λ(n)| ∈ Bε (x). Therefore using the fact that ω (cl) is asymptotically decoupled we have the bound KΛ(n) ω IBε (x) |Λ(n)| ﬀ dω (cl) = I KΛ(n) |Λ(n)|

k 2

∈Bε (x)

d

≥

j=1

k d

I KCj |Cj

ﬀ

∈Bε (x1 ) |

kd 2

I KCj

+1

≥

|Cj

∈Bε (x2 ) |

ﬀ dω (cl)

k2d I KΛ(m)

∈Bε (x1 ) |Λ(m)|

ﬀ dω (cl)

k2d I KΛ(m) |Λ(m)|

∈Bε (x2 )

ﬀ dω (cl)

e−c(m)k . d

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Thus we obtain log µm (Bε (x1 )) log µm (Bε (x2 )) k d |Λ(m)| log µn (Bε (x)) ≥ + |Λ(n)| 2|Λ(m)| 2|Λ(m)| |Λ(n)| −

1 c(m)k d . |Λ(n)|

We conclude by taking the lim inf over n and then choosing a subsequence ml such that liml→∞ (|Λ(ml )|)−1 log µml (Bε (x1 )) = m(Bε (x1 )). The identiﬁcation of the rate function follows from Varadhan’s lemma. Remark 4.8. One can show (see [32, 29] for more details) that the rate function satisﬁes the following variational characterization: s(x) = sup{−hcl (ν, ω (cl) ); ν(AΨ ) = x}, where hcl is the classical relative entropy per unit volume, and the supremum is taken over all classical translation invariant states. 4.4. Dimension 1 Throughout this section we assume that d = 1 (so we write |Λ(n)| = n) and that ω is an asymptotically decoupled state, for example we may assume that ω a KMSGibbs state for a ﬁnite range interaction. We also assume that Ψ is a finite range interaction. The crucial estimate needed to control the eﬀect of non-commutativity is an estimate on the diﬀerence between the spectral projections associated to KΛ(n) and

k j=1 KCj (see Sec. 4.1). To prove this we relies on a “cocycle estimate” proved in [1], which follows from the fact that the time-evolution τt (A) of any local observable A for a ﬁnite-range quantum spin system can be extended to a entire analytic function of t. This allows to prove the following “exponential version” of Lemma 4.1. Proposition 4.9. Let Ψ be a finite range interaction of range R and let β ∈ R. Then there exists a function Fβ (m) = Fβ (m, R, Ψ) with lim Fβ (m) = 0.

m→∞

such that lim sup n→∞

Pk 1 log eβKΛ(n) e−β j=1 KCj ≤ |β|Fβ (m). n

(4.12)

(4.13)

Proof. The proof is an application of the results in [1], see in particular Secs. 4 and 5. The basic bound in [1, Sec. 5], is that if AX ∈ OX with diam(X) ≤ R then there exists a constant D(β, R, Ψ) such that

eβKΛ (n) e−β(KΛ (n)−AX ) ≤ e|β|D(β,R,Ψ) AX .

(4.14)

The bound (4.14) follows from Dyson formula and estimates (uniform in n) on the dynamics in imaginary time generated by the Hamiltonian KΛ(n) . To apply these

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results here we write KΛ(n) =

k

KCj +

j=1

ψX .

X⊂Λ(n) X ⊂someCj

Let tX ∈ {0, 1} and let us deﬁne the family of interpolating Hamiltonians KΛ(n) ({tX }) =

k

KCj +

j=1

tX ψX .

X⊂Λ(n) X ⊂someCj

The estimates on the dynamics in [1] are easily seen to be uniform in {tX } and so we can apply the bound (4.14) iteratively, changing at each step one tX from 1 to 0. Using that Ψ has a ﬁnite range R we obtain the bound

e

βKΛ(n)

e

−β

|β|D(β,R,Ψ)

Pk

j=1 KCj

P

≤e

X⊂Λ(n) ψX X ⊂someCj .

But the sum over X is now treated exactly as Lemma 4.1 and we ﬁnd Fβ (m) = F (m)D(β, R, Ψ). We use this bound to prove an exponential estimates which control how the

k spectral projections change when we replace KΛ(n) by j=1 KCj . Proposition 4.10. Let ε > ε > 0. Then for any α > 0 there exists a function F˜α (m) with limm→∞ F˜α (m) = 0 such that   k 1 −1 −1   lim sup log K C (n K ) (mk) I I Cj Bε (x) Λ(n) Bε (x) n→∞ n j=1 ≤ −α(ε − ε − F˜α (m)).

(4.15)

Proof. Let us write KΛ(n) =

µi Pi ,

i

k

KCj =

j=1

λl Ql ,

(4.16)

l

where Pi and Ql are rank-one projections and µi and λl are the eigenvalues of KΛ(n)

and j KCj . For any β ∈ R Pi IBδ (y) (n−1 KΛ(n) ) = i;

=e

µi n

∈Bδ (y)

β(KΛ(n) −ny) µ i; ni

e−β(µi −ny) Pi

∈Bδ (y)

≡ eβ(KΛ(n) −ny) Vβ,y,δ

(4.17)

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and  IBε (x) (mk)−1

k



KCj  =

j=1

Ql

λ

l ∈B (x) l; mk ε

=

eβ(λl −xn) Ql e−β(

P j

KCj −xn)

λ

l ∈B (x) l; mk ε

≡ Wβ,x,ε e−β(

P j

KCj −xn)

,

(4.18)

with the bounds

Vβ,y,δ ≤ e|β|nδ ,

Wβ,x,ε ≤ e|β|mk(ε +( mk −1)|x|) . n

(4.19)

If y > x we choose β = α > 0 and using Eqs. (4.17) and (4.18) as well as the bounds (4.13) and (4.19), we obtain   k 1 −1 −1   I lim sup log (mk) I K (n K ) Cj Bε (x) Bδ (y) Λ(n) n→∞ n j=1 P 1 log Wα,x,ε e−α( j KCj −nx) eα(KΛ(n) −ny) Vα,y,δ

n→∞ n P 1 ≤ lim sup −α(y − x) + log e−α j KCj eαKΛ(n)

n n→∞ α + (nδ + mkε + (n − mk)|x|) n

= lim sup

≤ −α(y − x) + αFα (m) + α(δ + ε ) + α

g(m) |x|. m

Similarly, for y < x, we choose β = −α and obtain a similar bound and ﬁnally   k 1 −1 −1   lim sup log IBε (x) (mk) KCj IBδ (y) (n KΛ(n) ) n→∞ n j=1 ≤ −α|y − x| + αFα (m) + α(δ + ε ) + α

g(m) |x|. m

(4.20)

Next we choose δ be such that ε > 2δ + ε and choose ﬁnitely many intervals Tl and xl ∈ Tl , l = 1, . . . , L such that Bε (x)C ∩ [− Ψ , Ψ ] =

l

Tl ,

Tl ⊂ Bδ (xl ).

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By the principle of the largest term, and using the bound (4.20), we obtain   k 1 −1 −1   (mk) I K C (n K ) lim sup log I Cj Bε (x) Λ(n) Bε (x) n→∞ n j=1   L k 1 −1 −1   I K I (n K ) (mk) ≤ lim sup log Cj Tl Bε (x) Λ(n) n→∞ n j=1 l=1   k 1 −1 −1   ≤ max lim sup log (mk) I K (n K ) I Cj Bε (x) Bδ (xl ) Λ(n) l n→∞ n j=1 g(m) |x| . (4.21) ≤ −α(ε − ε − δ) + α Fα (m) + m Since δ is arbitrary this concludes the proof with F˜α (m) = Fα (m) +

g(m) m |x|.

With this estimate we can now prove Theorem 4.11. Let d = 1, let ω be an asymptotically decoupled translation invariant state, and let Ψ be a finite range interaction. Then the sequence of probability measures µn (A) = ω(IA (n−1 KΛ(n) )), satisfies a large deviation principle with a concave rate function s(x). Moreover s(x) = inf (f (α) − αx), α

where f (α) = lim n−1 log ω(exp(αKΛ(n) )). n→∞

Proof. Let ω be an asymptotically decoupled state with parameters g and c. Let x, x1 , x2 be such that x21 + x22 = x and 0 < ε < ε. For any n > m we decompose Λ(n) as in Sec. 4.1. Note that k/2 j=1

IBε (x1 )

KCj m

k j=k/2+1

IBε (x2 )

KCj m

≤ IBε (x)

j

KCj

mk

,

(4.22)

and that for any projections P and Q and a state ω we have ω(P ) = ω(QP Q) + ω((1 − Q)P Q + QP (1 − Q)) + ω((1 − Q)P (1 − Q)) ≤ ω(Q) + 2 (1 − Q)P Q + (1 − Q)P (1 − Q)

≤ ω(Q) + 3 (1 − Q)P .

(4.23)

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Using that ω is asymptotically decoupled, and estimates (4.22) and (4.23), we obtain KΛ(m) KΛ(m) 1 1 log ω IBε (x1 ) log ω IBε (x2 ) + 2m m 2m m   k/2 k KCj KCj 1  + c(m)k log ω  IBε (x1 ) IBε (x2 ) ≤ mk m m mk j=1 j=k/2+1    KCj   j  c(m) 1   log ω  ≤ IBε (x)  mk  + m mk    KΛ(n) 1  logω IBε (x) ≤  mk n     k KCj       KΛ(n)  c(m)   j=1 . + 3 IBε (x)   +  IBε (x)C   mk  n m    Keeping m ﬁxed we take a lim inf over n and using Proposition 4.10 we obtain KΛ(m) KΛ(m) 1 1 log ω IBε (x1 ) log ω IBε (x2 ) + 2m m 2m m g(m) c(m) . (4.24) ≤ 1+ max{m(Bε (x)), −α(ε − ε − F˜α (m))} + m m To conclude we will use the bound (4.24) repeteadly. (a) Assume ﬁrst x = x1 = x2 and assume that s(x) > −∞. Choose ﬁrst α so large that 1 − α(ε − ε ) < m(Bε (x)), 2 and then M = M (α) so that F˜α (m) ≤ 12 (ε − ε ) for m > M . By (4.24) we have then KΛ(m) c(m) 1 g(m) log ω IBε (x) , ≤ 1+ m(Bε (x)) + m m m m and thus m(Bε (x)) ≤ m(Bε (x)). This implies that the Ruelle function s(x) exists and is ﬁnite.

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(b) Assume that s(x) > −∞ and x = 12 (x1 + x2 ). Repeating the same argument as in (a) one obtains, for m large enough, KΛ(m) KΛ(m) 1 1 log ω IBε (x1 ) log ω IBε (x2 ) + 2m m 2m m g(m) c(m) , ≤ 1+ m(Bε (x)) + m m and this implies that 12 m(Bε (x1 )) + 12 m(Bε (x2 )) ≤ m(Bε (x)). Thus the rate function s(x) is concave wherever it is ﬁnite. (c) Let us assume that s(x) = −∞. Then for any t > 0 we can ﬁnd εt such that for ε < εt we have m(Bε (x)) ≤ −t. By (4.24) we have KΛ(m) 1 log ω IBε (x) m m g(m) c(m) , ≤ 1+ max{−t, −α(ε − ε − F˜α (m))} + m m and thus taking m → ∞ we obtain m(Bε (x)) ≤ max{−t, −α(ε − ε )}, and so s(x) ≤ max{−t, −αε}. Since α and t are arbitrary we have s(x) = −∞. (d) Assume that s(x) = −∞ and x = 12 (x1 + x2 ). Repeating the same argument as in (c) for any t > 0 there exists εt > 0 such that for all α > 0, KΛ(m) KΛ(m) 1 1 log ω IBε (x1 ) log ω IBε (x2 ) + 2m m 2m m c(m) g(m) ≤ 1+ max{−t, −α(εt − ε − F˜α (m))} + m m and this implies that Hence we obtain

1 2 m(Bε (x1 ))

+ 12 m(Bε (x2 )) ≤ max{−t, −α(εt − ε )}.

1 1 1 1 s(x1 ) + s(x2 ) = s(x1 ) + s(x2 ) = −∞ ≤ s(x). 2 2 2 2 Combining (a)–(d) shows the existence of a concave RL-function and this concludes the proof of Theorem 4.11. Remark 4.12. A characterization of the rate function using classical relative entropies is proved in [29].

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Acknowledgments The ﬁrst author was supported by JSPS Grant-in-Aid for Young Scientists (B), Hayashi Memorial Foundation for Female Natural Scientists, Sumitomo Foundation, and Inoue Foundation. The second author was supported by NSF, Grant DMS-0605058.

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[19] R. B. Israel, Convexity in the Theory of Lattice Gases, Princeton Series in Physics (Princeton University Press, 1979). [20] O. E. Lanford III, Entropy and equilibrium states in classical statistical mechanics, in Statistical Mechanics and Mathematical Problems, Lecture Notes in Physics, Vol. 20 (Springer, 1973), pp. 1–113. [21] J. L. Lebowitz, M. Lenci and H. Spohn, Large deviations for ideal quantum systems, J. Math. Phys. 41 (2000) 1224–1243. [22] M. Lenci and L. Rey-Bellet, Large deviations in quantum lattice systems: One phase region, J. Stat. Phys. 119 (2005) 715–746. [23] J. T. Lewis, C.-E. Pﬁster and W. G. Sullivan, Entropy, concentration of probability and conditional limit theorems, Markov Process. Related Fields 1 (1995) 319–386. [24] R. Lima, Equivalence of ensembles in quantum lattice systems, Ann. Inst. H. Poincar´e Sect. A 15 (1971) 61–68. [25] R. Lima, Equivalence of ensembles in quantum lattice systems: States, Comm. Math. Phys. 24 (1972) 180–192. [26] T. Matsui, On non-commutative Ruelle transfer operator, Rev. Math. Phys. 13 (2001) 1183–1201. [27] K. Netoˇcny and F. Redig, Large deviations for quantum spin systems, J. Stat. Phys. 117 (2004) 521–547. [28] Y. Ogata, Large deviations in quantum spin chains, Comm. Math. Phys. 296 (2010) 35–68. [29] Y. Ogata and L. Rey-Bellet, The rate function for quantum large deviations, in preparation. [30] S. Olla, Large deviations for Gibbs random ﬁelds, Probab. Theory Related Fields 77 (1988) 343–357. [31] D. Petz, G. A. Raggio and A. Verbeure, Asymptotics of Varadhan-type and the Gibbs variational principle, Comm. Math. Phys. 121 (1989) 271–282. [32] C.-E. Pﬁster, Thermodynamical aspects of classical lattice systems, in In and Out of Equilibrium (Mambucaba, 2000), Progr. Probab., Vol. 51 (Birkh¨ auser, 2002), pp. 393–472. [33] S. Roelly and H. Zessin, The equivalence of equilibrium principles in statistical mechanics and some applications to large particle systems, Expo. Math. 11 (1993) 385–405. [34] D. Ruelle, Correlation functionals, J. Math. Phys. 6 (1965) 201–220. [35] D. Ruelle, Statistical Mechanics: Rigorous Results (World Scientiﬁc, 1999). [36] B. Simon, The Statistical Mechanics of Lattice Gases, Vol. I, Princeton Series in Physics (Princeton University Press, 1993).

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Reviews in Mathematical Physics Vol. 23, No. 3 (2011) 233–260 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004278

APPLICATIONS OF MAGNETIC ΨDO TECHNIQUES TO SAPT

GIUSEPPE DE NITTIS SISSA, via Bonomea, 265 34136 Trieste TS, Italy [email protected] MAX LEIN Technische Universit¨ at M¨ unchen, Zentrum Mathematik, Boltzmannstraße 3, 85747 Garching, Germany [email protected] Received 15 September 2010 Revised 24 December 2010 In this review, we show how advances in the theory of magnetic pseudodiﬀerential operators (magnetic ΨDO) can be put to good use in space-adiabatic perturbation theory (SAPT). As a particular example, we extend results of [24] to a more general class of magnetic ﬁelds: we consider a single particle moving in a periodic potential which is subjected to a weak and slowly-varying electromagnetic ﬁeld. In addition to the semiclassical parameter ε 1 which quantiﬁes the separation of spatial scales, we explore the inﬂuence of an additional parameter λ that allows us to selectively switch oﬀ the magnetic ﬁeld. We ﬁnd that even in the case of magnetic ﬁelds with components in Cb∞ (Rd ), e.g., for constant magnetic ﬁelds, the results of Panati, Spohn and Teufel hold, i.e. to each isolated family of Bloch bands, there exists an associated almost invariant subspace of L2 (Rd ) and an eﬀective hamiltonian which generates the dynamics within this almost invariant subspace. In case of an isolated non-degenerate Bloch band, the full quantum dynamics can be approximated by the hamiltonian ﬂow associated to the semiclassical equations of motion found in [24]. Keywords: Magnetic ﬁeld; pseudodiﬀerential operators; Weyl calculus; Bloch electron. Mathematics Subject Classiﬁcation 2010: 81Q15, 81Q20, 81S10

1. Introduction A fundamental and well-studied problem is that of a Bloch electron subjected to an electric ﬁeld and a constant magnetic ﬁeld where the dynamics is generated by ˆ ≡ H(ε, ˆ λ) := 1 −i∇x − λA(εx) 2 + VΓ (x) + φ(εx) H (1.1) 2 acting on L2 (Rdx ). Here, ε and λ are dimensionless non-negative parameters whose signiﬁcance will be discussed momentarily. A is assumed to be a smooth 233

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polynomially bounded vector potential to the magnetic ﬁeld B = dA which is constant in time and uniform in space. Since the magnetic ﬁeld is uniform, A needs to grow at least linearly. The potential generated by the nuclei and all other electrons VΓ is periodic with respect to the crystal lattice [3, 4] Γ := γ ∈ Rd γ = dj=1 αj ej , αj ∈ Z (1.2) where the family of vectors {e1 , . . . , ed } which deﬁnes the lattice forms a basis of Rd . The potential assumed to be inﬁnitesimally bounded with respect to − 21 ∆x . By [26, Theorem XIII.96], this is ensured by the following Assumption 1.1 (Periodic Potential). We assume that VΓ is Γ-periodic, i.e. VΓ (· + γ) = VΓ (·) for all γ ∈ Γ, and M dy |VΓ (y)| < ∞. The dual lattice Γ∗ is spanned by the dual basis {e∗1 , . . . , e∗d }, i.e. the set of vectors which satisfy ej ·e∗k = 2πδkj . The assumption on VΓ ensures the unperturbed periodic hamiltonian ˆ per = 1 (−i∇x )2 + VΓ H 2

(1.3)

deﬁnes a selfadjoint operator on the second Sobolev space H 2 (Rd ) and gives rise to Bloch bands in the usual manner (cf. Sec. 2.1): the unitary Bloch–Floquet–Zak ˆ per into the ﬁbered operator transform Z deﬁned by Eq. (2.1) decomposes H ⊕ ⊕

2 Z 1 ˆ Z := Z H ˆ per Z −1 = −i∇ H dk H (k) := dk + k + V (y) y Γ per per 2 M∗

where we have introduced the M ∗ := k ∈ Rd

M∗

Brillouin zone d k = j=1 αj e∗j , αj ∈ [−1/2, +1/2]

(1.4)

as fundamental cell in reciprocal space. For each k ∈ M ∗ , the eigenvalue equation Hper (k)ϕn (k) = En (k) ϕn (k),

ϕn (k) ∈ L2 (Tdy ),

where Tdy := Rd /Γ, is solved by the Bloch function associated to the nth band. Assume for simplicity we are given a band E∗ which does not intersect or merge with other bands (i.e. there is a local gap in the sense of Assumption 3.1). Then common lore is that transitions to other bands are exponentially suppressed and the eﬀective dynamics for an initial state localized in the eigenspace associated to E∗ is generated by E∗ (−i∇x ) [6, 1]. If we switch on a constant magnetic ﬁeld, no matter how weak, the Bloch bands are gone as there is no BFZ decomposition with respect to Γ for hamiltonian (1.1). ˆ is a Cantor set [7] if the ﬂux through the As a matter of fact, the spectrum of H Wigner–Seitz cell d (1.5) M := y ∈ Rd y = j=1 αj ej , αj ∈ [−1/2, +1/2]

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is irrational. Even if the ﬂux through the unit cell is rational, we recover only magnetic Bloch bands that are associated to a larger lattice Γ ⊃ Γ. A natural question is if it is at all possible to see signatures of non-magnetic Bloch bands if the applied magnetic ﬁeld is weak? Our main result, Theorem 4.1, answers this question in the positive in the following sense: if the electromagnetic ﬁeld varies on the macroscopic level, i.e. ε 1, then to leading order the dynamics is still generated by the so-called Peierls substitution E∗ −i∇x − λA(εx) + φ εx (deﬁned as a magnetic pseudodiﬀerential operator through Eq. (2.15), cf. Sec. 2.2). Hence, the dynamics are dominated by the Bloch bands even in the presence of a weak, but constant magnetic ﬁeld. Furthermore, we can derive corrections to any order in ε in terms of Bloch bands, Bloch functions, the magnetic ﬁeld and the electric potential. We do not need to choose a “nice” vector potential for B, in fact, in all of the calculations only the magnetic ﬁeld B enters. Existing theory is ill-equipped to deal with constant or even more general magnetic ﬁelds. We tackle this obstacle by incorporating the rich theory of magnetic Weyl calculus [20, 14] with two small parameters [17] into space-adiabatic perturbation theory [25, 24, 27]. Magnetic Weyl calculus does not single out constant magnetic ﬁelds, in fact we will only assume that the components of the magnetic ﬁeld B are bounded with bounded derivatives to any order. Assumption 1.2 (Electromagnetic Fields). We assume that the components of the external (macroscopic) magnetic ﬁelds B and the electric potential φ are Cb∞ (Rd ) functions, i.e. smooth, bounded functions with bounded derivatives to any order. Remark 1.1. All vector potentials A associated to magnetic ﬁelds B = dA with ∞ (Rd ). This components in Cb∞ (Rd ) are always assumed to have components in Cpol is always possible as one could pick the transversal gauge, n 1 Ak (x) := − ds Bkj (sx)sxj . j=1

0

This is to be contrasted with the original work of Panati, Spohn and Teufel where the vector potential had to have components in Cb∞ (Rd ). ˆ deﬁnes an essentially selfadjoint operator on Under Assumptions 1.1 and 1.2, H 2 d ⊂ L (Rx ). The extension of the rigorous derivation of the Peierls substitution to the case of constant magnetic ﬁeld is not only an “accademic result”, it is crucial for modeling the quantum Hall eﬀect (QHE): the Peierls substitution can be used to link the QHE to the well-studied Harper equation [2, 8–11]. In particular the Peierls substitution led Hofstadter to study a simpliﬁed tight binding model for the QHE today called the Hofstadter model [12]. The Hofstadter model is a paradigm in the study of fractal spectra (Hofstadter butterﬂy) and was used by Thouless et al. in the C0∞ (Rdx )

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seminal paper [28] to give the ﬁrst theoretical explanation of the topological quantization of the QHE. More recently, Avron [22] interpreted the results by Thouless et al. from the viewpoint of thermodynamics and connected the QHE to anomalous phase transition diagrams (colored quantum butterﬂies). Even though this list of publications is very much incomplete, it shows the signiﬁcance of the Peierls substitution in the study of the QHE. The main merit of this paper is to provide the ﬁrst rigorous proof of the Peierls substitution under the conditions relevant to the QHE and consequently the ﬁrst rigorous justiﬁcation for the use of the Hofstadter model as weak ﬁeld limit for the analysis of the QHE. Let us now explain why we have chosen to include the additional parameter λ ˆ ≡ H(ε, ˆ λ). Our proposal is to model an experimental setup in the hamiltonian H that applies an external, i.e. macroscopic electric and magnetic ﬁeld. The parameter ε 1 relates the microscopic scale as given by the crystal lattice to the scale on which the external ﬁelds vary. We always assume ε to be small. It is quite easy to fathom an apparaturs where electric and magnetic ﬁeld can be regulated separately by, say, two dials. We are interested in the case where we can selectively switch oﬀ the magnetic ﬁeld. If we regulate the strength of the magnetic ﬁeld by varying the relative amplitude λ ≤ 1 which quantiﬁes the ratio between scaled electric and magnetic ﬁeld, B ε,λ (x) := ελB(εx),

Eε (x) := εE(εx).

We emphasize that λ need not be small, as a matter of fact, λ = 1 is perfectly admissible. In this sense, λ is a perturbative parameter which allows us to take the limit B ε,λ → 0 without changing the external electric ﬁeld Eε . This is very much relevant to experiments since magnetic ﬁelds are typically much weaker than electric ﬁelds and thus both, from a physics and a mathematics perspective, the study of the dynamics under the λ → 0 limit is an interesting problem which merits further research. The aim of this review is to show how recent advances in the theory of magnetic pseudodiﬀerential operators (magnetic ΨDOs) (see [14, 17] as well as Sec. 2.2) can be used to extend the range of validity of the results of Panati, Spohn and Teufel derived via space-adiabatic perturbation theory (SAPT) [24] to magnetic ﬁelds with components in Cb∞ (Rd ). The original proof uses standard pseudodiﬀerential techniques and thus is limited to magnetic vector potentials of class Cb∞ (Rd ). In a recent work by one of the authors with Panati [5], adiabatic decoupling for the Bloch electron has been proven in the case of constant magnetic ﬁeld. Their proof rests on a particular choice of gauge, namely the symmetric gauge. We take a diﬀerent route: according to the philosophy of magnetic Weyl calculus, it is the properties of the magnetic ﬁeld and not those of the vector potentials which enter the hypotheses of theorems. As the proofs in [24] carry over mutatis mutandis, we feel it is more appropriate to elucidate the structure of the problem and mention the necessary modiﬁcations in proofs when necessary.

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Our paper is divided in ﬁve, sections: in Sec. 2, we will decompose the Hamiltonian using the Bloch–Floquet–Zak transform and rewrite it as magnetic Weyl quantization of an operator-valued function. Section 3 contains a comprehensive description of our technique of choice, SAPT. The main results, adiabatic decoupling to all orders and a semiclassical limit, will be stated and proven in Secs. 4 and 5. 2. Rewriting the Problem As a preliminary step, we will rewrite the problem: ﬁrst, we extract the Bloch band picture via the BFZ transform and then we reinterpret the BFZ-transformed hamiltonian as magnetic quantization of an operator-valued symbol. We insist we only rephrase the problem, no additional assumptions are introduced. 2.1. The Bloch–Floquet–Zak transform Usually, one would exploit lattice periodicity by going to the Fourier basis: each Ψ ∈ S(Rdx ) ⊂ L2 (Rdx ) is mapped onto e−ik·y Ψ(y + γ) (F Ψ)(k, y) := γ∈Γ

and the corresponding representation is usually called Bloch–Floquet representation (see, e.g., [16]). It is easily checked that (F Ψ)(k − γ ∗ , y) = (F Ψ)(k, y), (F Ψ)(k, y − γ) = e−ik·γ (F Ψ)(k, y),

∀ γ ∗ ∈ Γ∗ , ∀ γ ∈ Γ,

holds and F Ψ can be written as (F Ψ)(k, y) = eik·y u(k, y) where u(k, y) is Γ-periodic in y and Γ∗ -periodic up to a phase in k. For technical reasons, we prefer to use a variant of the Bloch–Floquet transform introduced by Zak [29] which maps Ψ ∈ S(Rdx ) onto u, e−ik·(y+γ) Ψ(y + γ). (2.1) (ZΨ)(k, y) := γ∈Γ

The BFZ transform has the following periodicity properties: (ZΨ)(k − γ ∗ , y) = e+iγ

∗

·y

(ZΨ)(k, y) =: τ (γ ∗ ) (ZΨ)(k, y), ∀ γ ∗ ∈ Γ∗ ,

(ZΨ)(k, y − γ) = (ZΨ)(k, y),

∀ γ ∈ Γ,

(2.2)

τ is a unitary representation of the group of dual lattice translations Γ∗ . By density, Z immediately extends to L2 (Rdx ) and it maps it unitarily onto Hτ := ψ ∈ L2loc Rdk , L2 (Tdy ) ψ(k − γ ∗ ) = τ (γ ∗ ) ψ(k) a.e. ∀γ ∗ ∈ Γ∗ , (2.3)

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which is equipped with the scalar product

ϕ, ψ τ := dk ϕ(k), ψ(k) L2 (Td ) . M∗

y

It is obvious from the deﬁnition that the left-hand side does not depend on the choice of the unit cell M ∗ in reciprocal space. The BFZ representation of momentum −i∇x and position operator xˆ on L2 (Rdx ), equipped with the obvious domains, can be computed directly, Z(−i∇x )Z −1 = idL2 (M ∗ ) ⊗ (−i∇y ) + kˆ ⊗ idL2 (Tdy ) ≡ −i∇y + k, Zx ˆZ −1 = i∇τk ,

(2.4)

where we have used the identiﬁcation Hτ ∼ = L2 (M ∗ ) ⊗ L2 (Tdy ). The superscript 1 Rdk , L2 (Tdy ) consists of τ on i∇τk indicates that the operator’s domain Hτ ∩ Hloc τ -equivariant functions. The BFZ transformed domain for momentum −i∇y + k is ˆ L2 (M ∗ ) ⊗ H 1 (Tdy ). Since the phase factor τ depends on y, the BFZ transform of x does not factor — unless we consider Γ-periodic functions, then we have x)Z −1 = idL2 (M ∗ ) ⊗ VΓ (ˆ y ) ≡ VΓ (ˆ y ). ZVΓ (ˆ ˆ namely Equations (2.4) immediately give us the BFZ transform of H, ˆ Z := Z HZ ˆ −1 = 1 −i∇y + k − λA(iε∇τ ) 2 + VΓ (ˆ H y ) + φ(iε∇τk ), k 2

(2.5)

which deﬁnes an essentially selfadjoint operator on ZC0∞ (Rdx ). If the external electromagnetic ﬁeld vanishes, the hamiltonian ⊕ Z Z ˆ per ˆ per Z −1 = H := Z H dk Hper (k) (2.6) M∗

ﬁbers into a family of operators on L2 (Tdy ) indexed by crystal momentum k ∈ M ∗ . Z Z (k − γ ∗ ) and Hper (k) via τ -equivariance relates Hper Z Z Hper (k − γ ∗ ) = τ (γ ∗ )Hper (k) τ (γ ∗ )−1 ,

∀ γ ∗ ∈ Γ∗

which, among other things, ensures that Bloch bands {En }n∈N , i.e. the solutions to the eigenvalue equation Z (k)ϕn (k) = En (k)ϕn (k), Hper

ϕn (k) ∈ L2 (Tdy ),

Z are Γ∗ -periodic functions. Standard arguments show that Hper (k) has purely dis∗ crete spectrum for all k ∈ M and if Bloch bands are ordered by magnitude, they are smooth functions away from band crossings. Similarly, the Bloch functions k → ϕn (k) are smooth if the associated energy band En does not intersect with or touch others [26]. The next subsection shows that the eﬀect of introducing an external electromagnetic ﬁeld can be interpreted as “replacing” the direct integral with the magnetic Z + φ. quantization of Hper

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2.2. Magnetic ΨDO and Weyl calculus Instead of using regular Weyl calculus, we use a more sophisticated Weyl calculus that is adapted to magnetic problems. It has ﬁrst been proposed by M¨ uller in 1999 [19] in a non-rigorous fashion. Independently, M˘ antoiu and Purice [20] as well as Iftimie, M˘ antoiu and Purice [14] have laid the mathematical foundation. All of the main results of ordinary Weyl calculus have been transcribed to the magnetic context; for details, we refer to the two aforementioned publications and those we give in the remainder of this section. 2.2.1. Ordinary magnetic Weyl calculus The basic building blocks of magnetic pseudodiﬀerential operators are PA ≡ PA ε,λ := −i∇x − λA(Q), Q ≡ Qε := εˆ x.

(2.7)

ˆ can be written in terms of PA and Q as the quantization of With this notation, H H(x, ξ) := 12 ξ 2 + VΓ (x/ε) + φ(x),

(2.8)

ˆ = H(Q, PA ). As H is the sum of a contribution quadratic in momentum and i.e. H a contribution depending only on x, this prescription is unambiguous. However, not all objects (e.g., resolvents and projections) we will encounter are of this type. We need a functional calculus for the non-commuting family of operators Q and PA of non-commutative observables that are characterized by the commutation relations A i Ql , PA i PA (2.9) i Ql , Qj = 0 j = ε δlj l , Pj = ελBlj (Q) The commutation relations can be rigorously implemented via the Weyl system W A (x, ξ) := e−i(ξ·Q−x·P

A

)

=: e−iσ((x,ξ),(Q,P

A

))

where A is a smooth, polynomially bounded vector potentialassociated to a magnetic ﬁeld with components in Cb∞ (Rd ) and σ (x, ξ), (y, η) := ξ · y − x · η is the non-magnetic symplectic form. We will also introduce the symplectic Fourier transform 1 dy dη eiσ((x,ξ),(y,η)) f (y, η), f ∈ S(T ∗ Rdx ), (Fσ f )(x, ξ) := (2π)d Rdy Rd η which is also its own inverse on S(T ∗ Rdx ) and extends to a continuous bijection on the space of tempered distributions S (T ∗ Rdx ). The Weyl quantization of h ∈ S(T ∗ Rdx ) given by 1 dx dξ (Fσ h)(x, ξ) W A (x, ξ) (2.10) OpA (h) := (2π)d Rdx Rd ξ deﬁnes a magnetic ΨDO. Associated to this, we have a product B akin to the usual Moyal product which emulates the product of magnetic operators on the level of

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functions on phase space, OpA f B g = OpA (f ) OpA (g). For suitable functions ormander-class symbols, their magnetic product is given f, g : T ∗ Rdx → C, e.g., H¨ by the oscillatory integral ε 1 dy dη dz dζ eiσ((x,ξ),(y,η)+(z,ζ)) ei 2 σ((y,η),(z,ζ)) · (f B g)(x, ξ) = 2d (2π) Rd Rd Rd Rd y η z ζ · e−iεγε

B

(x,y,z)

(Fσ f )(y, η) (Fσ g)(z, ζ)

(2.11)

where γεB (x, y, z) is the scaled magnetic ﬂux through a triangle whose corners depend on x, y and z. In [17], it was shown that for H¨ ormander-class symbols, this product has an asymptotic development in ε and λ, f B g = f g − ε 2i {f, g}λB + O(ε2 ),

(2.12)

where {f, g}λB :=

d d ∂ξl f ∂xl g − ∂xl f ∂ξl g − λ Blj ∂ξl f ∂ξj g l=1

l,j=1

is the magnetic Poisson bracket. In the limit λ → 0 it reduces to the standard Poisson bracket. The crucial fact that this product depends on the magnetic ﬁeld B rather than the vector potential A can be traced back to the gauge-covariance of magnetic Weyl quantization: if A = A + dχ is an equivalent vector potential, dA = B = dA , then the quantizations with respect to either vector potential are unitarily equivalent, (2.13) OpA+dχ (h) = e+iλχ(Q) OpA (h)e−iλχ(Q) . This is generically false if we quantize hA (x, ξ) := h x, ξ − λA(x) via usual, nonmagnetic Weyl quantization, OpA (h) := Op(hA ).a Otherwise, the properties of the relevant pseudodiﬀerential operators were to depend on the choice of gauge. Fortunately, the diﬀerence starts to appear at second order in ε [17, Sec. 1.1.2], so we generically expect results derived by non-magnetic Weyl calculus to agree to ﬁrst order. However, there is a second advantage of magnetic Weyl calculus (beyond being more natural): we can treat more general magnetic ﬁelds as properties of B enter rather than those of A — and associated vector potentials are always worse behaved than the magnetic ﬁeld. With some eﬀort, one can treat magnetic ﬁelds which admit a vector potentials whose derivatives are all bounded, ∂xa A(x) ≤ Ca for all a ∈ Nd0 , |a| ≥ 1, see [14, Sec. 5], for instance, but with magnetic Weyl calculus we are instantly able to treat magnetic ﬁelds whose components are Cb∞ (Rd ) functions with zero extra eﬀort. The limitation to this class of ﬁelds is due to the fact that we are interested in H¨ ormander class symbols which need to be bounded in the x variable. In fact, the extension of the results of Panati, Spohn and Teufel a Coincidentally,

the quantization of polynomials of degree ≤ 2 in momentum with respect to OpA are covariant. Bloch bands, however, are not quadratic functions — as are all the other objects (terms of the development of the projection and the unitary) involved in this paper.

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to magnetic ﬁelds of class Cb∞ is our main motivation as explained in the introduction. Covariance ensures that our results do not depend on the choice of a nice or symmetric gauge, any vector potential that is a smooth, polynomially bounded function will do. Many standard results of pseudodiﬀerential theory have been transcribed to the magnetic case for a large class of magnetic ﬁelds: typically, it is either assumed that ∞ (Rd ) or Cb∞ (Rd ) functions, although we shall always the components of B are Cpol assume the latter. The quantization and dequantization have been extended to tempered distributions and it was shown that smooth, uniformly polynomially bounded ∞ ∗ d functions on phase space Cpol u (T R ) are among those with good composition properties [20, Proposition 23]. H¨ ormander symbols are preserved under the magnetic Weyl product and quantizations of real-valued, elliptic H¨ ormander symbols of positive order m deﬁne selfadjoint operators on the mth magnetic Sobolev space [14]. A magnetic version of the Cald´eron–Vaillancourt theorem [14] and commutator criteria [15] show the interplay between properties of magnetic pseudodiﬀerential operators and their associated symbols. Lastly, we mention something that will be important in the next section: the magnetic Weyl quantization is a position representation for a magnetic pseudodifferential operator. However, equivalently, we can use the momentum representation A where x is quantized to Q := iε∇ξ and ξ to P := ξˆ − λA(iε∇ξ ). The commutation relations of the building block operators in momentum representation are again encoded into the Weyl system

A

W A (x, ξ) := e−iσ((x,ξ),(Q ,P

))

= FW A (x, ξ)F−1

which is related to the Weyl system in momentum representation via the Fourier transform F : L2 (Rdx ) → L2 (Rdξ ). If OpA is the quantization associated to the Weyl system W A , then OpA and OpA are related via F as well. An important consequence is that the Weyl product is independent of the choice of representation: for suitable distributions f and g, we conclude that the Weyl products must agree: A A OpA (f B ε,λ g) = Op (f ) Op (g)

= F OpA (f ) F−1 F OpA (g) F−1 −1 = F OpA (f B ε,λ g) F

= OpA (f B ε,λ g). This is related to an algebraic point of view proposed in [21] and elaborated upon in [18]: the choice of Hilbert space in this case can be seen as a choice of (equivalent) representation of more fundamental C ∗ -algebras of distributions. Properties such as boundedness of the quantization of certain classes of distributions and the form of the composition law are preserved if we choose a unitarily equivalent representation. In fact, we could have replaced F in the above argument by any other unitary operator U : L2 (Rdx ) → H where the target space H is again a separable Hilbert

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space. Gauge transformations are another particular example of unitarities which connect equivalent representations. 2.2.2. Equivariant magnetic Weyl calculus For technical reasons, we must adapt magnetic Weyl calculus to deal with equivariant, unbounded operator-valued functions. We follow the general strategy outlined in [24], but we need to be more careful as the roles of Q and PA are not interchangeable if B = 0. We would like to reuse results for Weyl calculus on T ∗ Rd — in particular, the two-parameter expansion of the product (Eq. (2.12)). Consider the building block kinetic operators macroscopic position R and magnetic crystal momentum KA , R = iε∇k ⊗ idL2 (Tdy ) ≡ iε∇k , K = kˆ − λA(R), A

(2.14)

in momentum they deﬁne selfadjoint operators whose domains are representation: ∗ dense in L2τ Rdk , L2 (Tdy ) where τ stands for either τ : γ ∗ → e−iγ ·ˆy or 1 : γ ∗ → 1. The elements of this Hilbert space can be as vector-valued tempered dis considered 2 d 2 d , L (T ) can be continuously embedded R tributions with special properties as L y τ k simplicity, let us ignore questions of domains and assume into S Rdk , L2(Tdy ) . For that h ∈ Cb∞ T ∗ Rdx , B L2 (Tdy ) is a bounded operator -valued function. Then its magnetic Weyl quantization 1 dr dk(Fσ h)(k, r)W A (k, r) (2.15) OpA (h) := (2π)d Rdr Rd k d deﬁnes a continuous operator from S Rdk , L2 (T y ) to itself which has a continuous extension as an operator from S Rdk , L2 (Tdy ) to itself [20, Proposition 21]. Here, the corresponding Weyl system W A (k, r) := e−iσ((r,k),(R,K

A

))

⊗ idL2 (Tdy ) ≡ e−i(k·R−r·K

A

)

is deﬁned in terms of the building block operators KA and R and acts trivially on L2 (Tdy ). The Weyl product f B g of two suitable distributions associated to the quantization OpA is also given by a suitable reinterpretation of Eq. (2.11) as f and g are now operator-valued functions. Furthermore, we can also develop f B g asymptotically in ε and λ [17, Theorem 1.1]. To see this, we remark that the diﬀerence between the products associated to OpA and OpA is two-fold: ﬁrst of all, OpA is a position representation while OpA is a momentum representation. Let OpA be the magnetic Weyl quantization deﬁned with respect to R := F−1 RF = εˆ r r ), i.e. the position representation. As explained and KA := F−1 KA F = −i∇r −λA(εˆ at the end of the previous subsection, unitarily equivalent representations, here OpA and OpA , have the same Weyl product. Secondly, the functions which are to be quantized by OpA and OpA take values in C and the bounded operators on L2 (Tdy ), respectively. The interested reader may

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check the proofs regarding the various properties of the product B in [20, 14, 17] can be generalized to accommodate operator-valued functions, including H¨ ormander symbols. Deﬁnition 2.1 (H¨ ormander Symbols S m ρ (B(H1 ,H2 ))). Let m ∈ R, ρ ∈ [0, 1] separable Hilbert spaces. Then a function f is said to be in and H1 , H2 be Sρm B(H1 , H2 ) if and only if for all a, α ∈ Nd0 the seminorms f := sup ( 1 + ξ 2 )|α|ρ−m ∂xa ∂ξα f (x, ξ)B(H1 ,H2 ) < ∞ m,aα (x,ξ)∈Ξ

are ﬁnite where ·B(H1 ,H2 ) denotes the operator norm on B(H1 , H2 ). In case ρ = 1, one also writes S m := S1m H¨ ormander symbols which have an expansion in ε that is uniform in the small parameter are called semiclassical. Deﬁnition 2.2 (Semiclassical Symbols AS m ρ (B(H1 ,H2 ))). A map f : m [0, ε0 ) → Sρ , ε → fε is called a semiclassical symbol of order m ∈ R and weight ρ ∈ [0, 1], that is f ∈ ASρm , if there exists a sequence {fn }n∈N0 , fn ∈ Sρm−nρ , such that for all N ∈ N0 , one has N −1 −N n ε ε fn ∈ Sρm−N ρ fε − n=0

uniformly in ε in the sense that for any N ∈ N0 and a, α ∈ Nd0 , there exist constants CN aα > 0 such that N −1 n ε fn ≤ CN aα εN fε − n=0

m−N ρ,aα

holds for all ε ∈ [0, ε0 ). If ρ = 1, then one abbreviates AS1m with AS m . Lastly, we will need the notion of τ -equivariant symbols. ∗ Deﬁnition 2.3 (τ -Equivariant Symbols AC ∞ τ (B(H1 ,H2 ))). Let τj : Γ → ofthe group Γ∗ . Then f ∈ AS00 is U(Hj ), j = 1, 2, be unitary ∗-representations ∞ τ -equivariant, i.e. an element of ACτ B(H1 , H2 ) , if and only if

f (k − γ ∗ , r) = τ2 (γ ∗ )f (k, r)τ1 (γ ∗ )−1 holds for all k ∈ Rdk , r ∈ Rdr and γ ∗ ∈ Γ∗ . Now the reader is in a position to translate the results derived in [27, Appendix B] to the context of magnetic Weyl calculus, the modiﬁcations are straightforward and all the necessary references have been given in this section.

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3. The Magnetic Bloch Electron as a Space-Adiabatic Problem Our tool of choice to derive eﬀective dynamics is space-adiabatic perturbation theory [25, 24, 27] which uses pseudodiﬀerential techniques to derive perturbation expansions order-by-order in a systematic fashion. We adapt their results by replacing ordinary Weyl calculus with magnetic Weyl calculus. Adiabatic decoupling only hinges on ε 1 and does not rely on λ to be small. 3.1. The three ingredients of space-adiabatic problems The insight of [24] was that the slow variation of the external electromagnetic ﬁeld (quantiﬁed by ε 1) leads to a decoupling into slow (macroscopic) and fast (microscopic) degrees of freedom. This is characteristic of adiabatic systems whose three main features are (i) A distinction between slow and fast degrees of freedom: the original (physical) Hilbert space H = L2 (Rdx ) is decomposed unitarily into Hslow ⊗ Hfast := L2 (M ∗ ) ⊗ L2 (Tdy ) in which the unperturbed hamiltonian is block diagonal (see diagram (3.1)). The names slow and fast Hilbert space are due to the operators deﬁned on them: on the fast Hilbert space, the two conjugate observables are −i∇y and yˆ acting on Hfast = L2 (Tdy ), the Hilbert space associated to the Wigner Seitz cell M∼ = Tdy (cf. Eq. (1.5)); their commutator is of O(1). The operators R and KA (cf. Eq. (2.14)) deﬁned on Hslow = L2 (M ∗ ) are considered slow, because their commutator is of O(ε). Since Hslow is the Hilbert space over the Brillouin zone M ∗ (cf. Eq. (1.4)), the dynamics of the slow variables (R, KA ) describes the motion across unit cells in momentum representation whereas the dynamics of the fast variables (ˆ y , −i∇y ) describe what happens within the Wigner–Seitz cell M . (ii) A small, dimensionless parameter ε that quantiﬁes the separation of spatial scales. In our situation, ε 1 relates the variation of the external electromagnetic ﬁeld to the microscopic scale as given by the lattice constant. In addition, we have the parameter λ. However, only the semiclassical parameter ε is crucial for adiabatic decoupling. (iii) A relevant part of the spectrum, i.e. a subset of the spectrum which is separated from the remainder by a gap. We are interested in the dynamics associated to a family of Bloch bands {En }n∈I that does not intersect or merge with bands from the remainder of the spectrum. ˆ Z satisﬁes the gap conAssumption 3.1 (Gap Condition). The spectrum of H per dition, namely there exists a family of Bloch bands {En }n∈I , I = [I− , I+ ] ∩ N0 such that

{En (k)}, {Ej (k)} =: Cg > 0. inf ∗ dist k∈M

n∈I

j∈I

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The spectral gap ensures that transitions from and to the relevant part of the spectrum are exponentially suppressed. Band crossings within the relevant part of the spectrum are admissible, though. In the original publication, an additional assumption was made on the existence of a smooth, τ -equivariant basis, a condition that is equivalent to the triviality of a certain U (N ) bundle over the torus Tdk where N := |I| is the number of bands including multiplicity. At least for the physically relevant cases, i.e. d ≤ 3, Panati has shown that this is always possible for non-magnetic Bloch bands [23]. For d ≥ 4, our results still hold if we add Assumption 3.2 (Smooth Frame (d ≥ 4)). If d ≥ 4, we assume there exists an orthonormal basis (called smooth frame) {ϕj (·)}j=1,...,|I| of whose elements are smooth and τ -equivariant with respect to k, i.e. ϕj (· − γ ∗ ) = τ (γ ∗ )ϕj (·) for all γ ∗ ∈ Γ∗ and for all j ∈ {1, . . . , |I|}. 3.2. Rewriting the unperturbed problem: An adiabatic point of view Let us consider the unperturbed case, i.e. in the absence of anexternal electromag⊕ Z Z ˆ per = M ∗ dk Hper (k). Each netic ﬁeld. Then the dynamics on Hτ is generated by H Z ﬁber hamiltonian Hper (k) is an operator on the fast Hilbert space Hfast = L2 (Tdy ). ⊕ Then π ˆ0 = M ∗ dk π0 (k) is the projection onto the relevant part of the spectrum, where |ϕn (k)ϕn (k)|. π0 (k) := n∈I

Even though the ϕn (k) may not be continuous at eigenvalue crossings, the projection k → π0 (k) is due to ⊕the spectral gap. Associated to the relevant band is a (nonunique) unitary u ˆ0 = M ∗ dk u0 (k) which “straightens” Hτ into L2 (Mk∗ ) ⊗ L2 (Tdy ): for each k ∈ M ∗ , we deﬁne u0 (k) := |χn ϕn (k)| + u⊥ 0 (k) n∈I

where χn ∈ L2 (Tdy ), n ∈ I, are ﬁxed vectors independent of k and u⊥ 0 (k) (also nonunique) acts on the complement of ran π0 (k) and is such that u ˆ0 is a proper unitary. Even though this means u0 is not unique, the speciﬁc choices of the {χn }n∈I and u⊥ 0 will not enter the derivation. Then we can put all parts of the puzzle into a diagram: t ˆ

e−i ε H

t ˆZ

e−i ε H

u ˆ0 Z / Hτ / L2 (M ∗ ) ⊗ L2 (Tdy ) L2 (Rdx ) π ˆ0 Πref Z −1 π ˆ0 Z −1 Z π ˆ0 ZL2 (Rdx ) _ _ _ _/ π ˆ0 Hτ _ _ _ _ _/ L2 (M ∗ ) ⊗ CN G tˆ

e−i ε heff 0

(3.1)

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The reference projection Πref = idL2 (M ∗ ) ⊗ πref acts trivially on the ﬁrst factor, L2 (M ∗ ), and projects via πref =

N

|χj χj | = u0 (k) π0 (k)u∗0 (k)

(3.2)

j=1

onto an N -dimensional subspace of L2 (Tdy ). We will identify πref L2 (Tdy ) with CN when convenient and in this sense, we identify the range of Πref with the reference space Href := L2 (M ∗ ) ⊗ CN . The dynamics in the lower-right corner is generated by the eﬀective hamiltonian ∗ ˆ eﬀ 0 := Πref u ˆZ u ˆ0 H h per ˆ0 Πref

ˆ if the relevant part of the spectrum consists of an isolated which reduces to En (k) Bloch band. 3.3. Adiabatic decoupling in the presence of external ﬁelds Now the question is whether a similar diagram exists even if the perturbation is present, i.e. if there exist a tilted projection Π, an intertwining unitary U and ˆ eﬀ 0 ? This has ˆ eﬀ that take the place of π ˆ0 , u ˆ0 and h an eﬀective hamiltonian h ∞ been answered in the positive for magnetic ﬁelds that admit Cb (Rd , Rd ) vector potentials in [24] where these objects are explicitly constructed by recursion. We replace standard Weyl calculus used in the original publication with its magnetic variant (cf. Sec. 2.2) which naturally allows for the treatment of more general magnetic ﬁelds with components in Cb∞ . The construction of Π and U detailed in the next section is a “defect construction” where recursion relations derived from ˆ Z ] = 0, [Π, H

Π2 = Π,

U ∗ U = idHτ , U U ∗ = idL2per (M ∗ )⊗L2per (M) , U Π U ∗ = Πref , relate the nth term to all previous terms. These four conditions merely characterize that Π and U are still a projection and a unitary (ﬁrst column) and adapted to the problem (second column). These equations can be translated via magnetic Weyl calculus to π, H Z B = O(ε∞ ), πB π = π + O(ε∞ ), (3.3) uB u∗ = 1 + O(ε∞ ) = u∗ B u, uB πB u∗ = πeﬀ + O(ε∞ ), where H Z (k, r) :=

1 2

2 −i∇y + k + VΓ (ˆ y ) + φ(r)

(3.4)

is the operator-valued symbol associated to the magnetic pseudodiﬀerential operaˆ Z deﬁned by Eq. (2.5). Note that the magnetic vector potential A does not tor H enter the deﬁnition of the symbol H Z and we do not need to impose conditions on

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A to ensure the symbol H Z is well-behaved after minimal substitution. This is why magnetic Weyl calculus can be used to treat much more general magnetic ﬁelds. For technical reasons, OpA (u) and U , for instance, agree only up to an error that is arbitrarily small in ε with respect to the operator norm, U = OpA (u) + O · (ε∞ ). The tilted projection and intertwining unitary are now used to deﬁne the eﬀective hamiltonian as the magnetic quantization of heﬀ := πref uB H Z B u∗ πref which generates eﬀective dynamics, i.e. for initial states in ΠHτ we can approximate A t the full time evolution in terms of e−i ε Op (heff ) . In turn, the eﬀective quantum evolution can be approximated by semiclassical dynamics. Theorem 5.2, the main result of the next section, will make this statement precise. 4. Derivation of eﬀective quantum dynamics The aforementioned “defect construction” yields the tilted projection π and the intertwining unitary u as asymptotic expansion in ε and λ. It is important that the decoupling is solely due to the separation of spatial scales quantiﬁed by ε and independent of λ which regulates the strength of the magnetic ﬁeld. 4.1. The dynamics in the Almost Invariant Subspace We will quickly explain how Π and U are computed order-by-order in ε and λ. We adapt the general recipe explained in [27] to incorporate two parameters: since the decoupling is due to the separation of spatial scales quantiﬁed by ε 1, we will order corrections in powers of ε ﬁrst. Expanding the magnetic Weyl product to zeroth order, we can check π0 , H Z B = O(ε), π0 B π0 = π0 + O(ε), u0 B u0 ∗ = 1 + O(ε) = u0 ∗ B u0 , u0 B π0 B u∗0 = πeﬀ + O(ε). Here, π0 , H Z B := π0 B H Z − H Z B π0 denotes the magnetic Weyl commutator. The asymptotic expansion of the product is key to deriving corrections in a systematic manner: the O(ε) terms can be used to infer π1 and u1 , the subprincipal symn n bols. Then, one proceeds by recursion: if π (n) := l=0 εl πl and u(n) := l=0 εl ul satisfy Eqs. (3.3) up to errors of order εn+1 , then we can compute πn+1 and un+1 . The construction of π and u follows exactly from [27, Lemmas 3.8 and 3.15]; it is purely algeraic and only uses that we have a recipe to expand the Moyal product in terms of the semiclassical parameter ε. Let us deﬁne π (n) B π (n) − π (n) =: εn+1 Gn+1 + O(εn+2 ) Z (n) d =: εn+1 Fn+1 + O(εn+2 ) H , π + εn+1 πn+1

B

(4.1)

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as projection and commutation defects and ∗

u(n) B u(n) − 1 =: εn+1 An+1 + O(εn+2 ), ∗ (n) u + εn+1 an+1 u0 B π (n+1) B u(n) + εn+1 an+1 u0 =: εn+1 Bn+1 + O(εn+2 ), (4.2) d as unitarity and intertwining defects. The diagonal part of the projection πn+1 can be computed from Gn+1 via D := −π0 Gn+1 π0 + (1 − π0 )Gn+1 (1 − π0 ). πn+1

(4.3)

an+1 = − 12 An+1

(4.4)

The term

stems from the ansatz un+1 = (an+1 + bn+1 )u0 where an+1 and bn+1 are symmetric and antisymmetric, respectively. One can solve the second equation for bn+1 = πref , Bn+1 (4.5) where πref is the reference projection on L2 (Tdy ) given by Eq. (3.2). This equation ﬁxes only the oﬀ-diagonal part of bn+1 as πref Bn+1 πref = 0 = (1−πref )Bn+1 (1−πref ) and in principle one is free to choose the diagonal part of bn+1 . This means, there is a freedom that allows arbitrary unitary transformations within πref L2 (Tdy ) as well as its orthogonal complement. In general, it is not possible to solve Z OD H , πn+1 = −Fn+1 (4.6) explicitly since Bloch functions at band crossings within the relevant part of the spectrum (which are admissible) are no longer diﬀerentiable. In any case, π can be constructed locally around (k0 , r0 ) by asymptotically expanding the Moyal resolvent (H Z − z)(−1)B , i.e. the symbol deﬁned through the relations (H Z − z)B (H Z − z)(−1)B = 1 = (H Z − z)(−1)B B (H Z − z), and setting i π(k, r) = 2π

dz(H Z − z)(−1)B (k, r) + O(ε∞ )

(4.7)

C(k0 ,r0 )

in a neighborhood of (k0 , r0 ). A recent result by Iftime et al. [15] suggests that under these circumstances (H Z is elliptic and selfadjoint operator-valued) (H Z − z)(−1)B always exists and is a H¨ ormander symbol even in the presence of a magnetic ﬁeld. We reckon their result extends to the case of operator-valued symbols, but seeing how tedious the proof is, we simply stick to the procedure used by Panati et al. [27, Lemma 5.17]. This construction uniquely ﬁxes the tilted Moyal projection π, but not the Moyal unitary u.

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As the two-parameter expansion of the product f g B

∞ n

εn λk (f B g)(n,k) ,

n=0 k=0

contributes only ﬁnitely many terms in λ for ﬁxed power of n of ε [17], we can order the terms of the expansion of π and u in powers of λ, e.g., πn =

n

λk π(n,k) .

k=0

The magnetic Weyl product as well as its asymptotic expansion are deﬁned in terms of oscillatory integrals, i.e. integrals which exist in the distributional sense. If we take the limit λ → 0 of f B g, we can interchange oscillatory integration and limit procedure [13, p. 90] and conclude limλ→0 f B g = f g where is the usual Moyal product. Similarly, we can apply this reasoning to the asymptotic expansion: for any ﬁxed N ∈ N0 , we may write the product as n N B n k B B ε λ (f g)(n,k) + εN +1 RN f g = +1 (f, g) n=0

k=0

and taking the limit λ → 0 means only the non-magnetic terms (f B g)(n,0) survive. The remainder also behaves nicely when taking the limit as it is also just another B oscillatory integral and limλ→0 RN +1 (f, g) is exactly the remainder of the nonmagnetic Weyl product. Hence, we can now prove the main result of this paper: Theorem 4.1 (Eﬀective Quantum Dynamics). Let Assumptions 1.1, 1.2 and 3.1 be satisﬁed. Furthermore, if d ≥ 4, we add Assumption 3.2. Then there exist (i) an orthogonal projection Π ∈ B(Hτ ), (ii) a unitary map U which intertwines Hτ and L2 (M ∗ ) ⊗ L2 (Tdy ), and (iii) a selfadjoint operator OpA (heﬀ ) ∈ B L2 (M ∗ ) ⊗ CN , N := |I| such that

and

Z H ˆ , Π = O(ε∞ ) −isHˆ Z A e − U ∗ e−isOp (heff ) U ΠB(H

τ)

(4.8) = O ε∞ (1 + |s|) .

(4.9)

∗

The eﬀective hamiltonian is the magnetic quantization of the Γ -periodic symbol Z B ∗

heﬀ := πref u H u πref B

∞

εn heﬀ n ∈ ASτ0≡1 B(CN )

(4.10)

n=0

whose asymptotic expansion can be computed to any order in ε and λ. To each order n in ε, only ﬁnitely many terms in λ contribute, heﬀ n = k=0 λk heﬀ (n,k) .

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One deduces from Eq. (4.8) and a Duhamel argument that the unitary time ˆ Z and Π almost commute even for macroscopic times evolution generated by H t = εs and hence, up to an error of arbitrarily large order in ε, the space Πref Hτ is left invariant by the dynamics, ˆZ

(1 − Π)e−i ε H Π = O · (ε∞ |t|). t

Thus we call ΠHτ the almost invariant subspace and in a sense, it is the tilted “eigenspace” associated to the family of relevant bands. The proof of the above theorem amounts to showing (i)–(iii) separately. Proposition 4.1 (Tilted Projection). Under the assumptions of Theorem 4.1 there exists and orthogonal projection Π ∈ B(Hτ ) such that Z ˆ , Π = O · (ε∞ ) H (4.11) and Π = OpA (π) + O · (ε∞ ) where OpA (π) is the magnetic Weyl quantization of a τ -equivariant semiclassical symbol π

∞

εn πn ∈ ACτ∞ B(Hfast )

n=0

whose principal part π0 (k, r) coincides with the spectral projection of H Z (k, r) onto the subspace corresponding to the given isolated family of Bloch bands {En }n∈I . Each term in the expansion can be written as a ﬁnite sum πn =

n

λk π(n,k) ∈ ACτ∞ B(Hfast )

k=0

ordered by powers of λ. For λ → 0, the projection π reduces to the non-magnetic ∞ projection π 0 n=0 εn π(n,0) . Proof [Sketch]. The proof relies on a well-developed magnetic Weyl calculus adapted to operator-valued symbols (cf. Sec. 2.2) and the gap condition. In particular, one needs a magnetic Cald´eron–Vaillancourt theorem, composition and quantization of H¨ ormander symbols [14] and ﬁnally, an asymptotic two-parameter expansion of the magnetic Weyl product B [17]. The interested reader may check line-by-line that the original proof [27, Proposition 5.16] can be transliterated to the magnetic context with obvious modiﬁcations. If we were using standard Weyl calculus, the major obstacle would be to control derivatives of π since vector potentials may be unbounded. In magnetic Weyl calculus the vector potential at no point enters the and the assumptions on the magnetic ﬁeld assure calculuations ∞ that π ∈ ACτ B(Hfast ) is a proper τ -equivariant semiclassical H¨ormander-class symbol (cf. Deﬁnition 2.3). The fact that we can write all of the πn as ﬁnite sum of terms ordered by powers of λ stems from the fact that calculating πn involves the expansion of the product

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up to nth power in ε, e.g., for the projection defect, we ﬁnd π (n−1) B π (n−1) − π (n−1) = εn (πa B πb (c) + O(εn+1 ) a+b+c=n

= εn

a

b

c

a+b+c=n a =0 b =0 c =0

λa +b +c π(a,a ) B π(b,b ) (c,c ) + O(εn+1 ).

Certainly, the exponent of λ is always bounded by n ≥ a +b +c . And since the sum is ﬁnite, this clearly deﬁnes a semiclassical symbol in ε is shown (cf. Deﬁnition 2.2). Similar arguments for the commutation defect in conjunction with the comments in the beginning of this section show π to be a semiclassical symbol. It is well-behaved under the λ → 0 limit and reduces to the projection associated to the case B = 0. Lastly, to make the almost projection OpA (π) into a true projection, we deﬁne Π to be the spectral projection onto the spectrum in the vicinity of 1, −1 dz OpA (π) − z . Π := 1 |z− 2 |=1 This concludes the proof. Similarly, one can modify [27, proof of Proposition 5.18] to show the existence of the intertwining unitary. Proposition 4.2 (Intertwining Unitary). Let {En }n∈I be a family of bands separated by a gap from the others and let Assumption 1.1 be satisﬁed. If d > 3, assume u0 ∈ S00 B(Hfast ) . Then there exists a unitary operator U : Hτ → L2 (M ∗ ) ⊗ L2 (Tdy ) such that U = OpA (u) + O · (ε∞ ) where u

∞

εn un ∈ AS 0 B(Hfast )

n=0

is right-τ -covariant at any order and has principal symbol u0 . Each term in the expansion can be written as a ﬁnite sum un =

n

λk u(n,k)

k=0

ordered by powers of λ. For λ → 0, the unitary u reduces to the non-magnetic ∞ unitary u0 n=0 εn u(n,0) . Proof [Sketch]. Equations (4.4) and (4.5) give us an+1 and bn+1 which combine to un+1 = (an+1 + bn+1 )u0 ; by [17, Theorem 1.1] it is also in the correct symbol class, namely S00 B(Hfast ) . The right τ -covariance is also obvious from the ansatz. Lastly, the true unitary U is obtained via the Nagy formula as described in [27]. Proof of Theorem 4.1 [Sketch]. The existence of Π and U have been the subject symof Propositions 4.1 and 4.2. By right-τ -covariance of u, heﬀ is a Γ∗ -periodic bol and since it is the magnetic Weyl product of Cb∞ T ∗ Rd , B L2 (Tdy ) functions,

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[17, Theorem 1.1] ensures and its asymptotic two-parameter 2thed product ∗ d that ∞ expansion are in Cb T R , B L (Ty ) as well. Equation (4.9) follows as usual from a Duhamel argument. The crucial statement of Theorem 4.1 is Eq. (4.9) and it is worthwhile to discuss its implications to applications: in practice, one only computes ﬁnitely many terms of the asymptotic expansions of Π, U and heﬀ . Let us call their ﬁnite resummations u(l) := ln=0 εn un , π (l) := ln=0 εn πn and heﬀ (l) := ln=0 εn heﬀ n . Assume we are interested in times t = εk s where |t| ≤ τ . Then a closer inspection of the Duhamel argument in the proof of Theorem 4.1 yields (n+k) Z A t t ∗ ) Πref OpA u(n) + O · εn−k+1 . e−i εk H Π = OpA u(n) e−i εk Op (heff Hence, if we would like to consider macroscopic times t = εs and make an error in the propagation of order O(ε2 ), then we need to expand the tilted projection, intertwining unitary and the eﬀective hamiltonian to second order. However, if the relevant band consists of a single band, computing the ﬁrst-order correction to the eﬀective hamiltonian suﬃces as we shall see in Sec. 5. 4.2. Eﬀective dynamics for a single band: The Peierls substitution In case the relevant part of the spectrum consists of a single non-degenerate band E∗ , we can calculate the ﬁrst-order correction to heﬀ explicitly: the magnetic Weyl product reduces to the pointwise product to zeroth order in ε. Thus, we can directly compute heﬀ 0 = πref u0 H0 u∗0 πref =: πref h0 πref = E∗ + φ. For the ﬁrst order, we use the recursion formula [27, Eq. (3.35)] and the fact that heﬀ 0 is a scalar-valued symbol:

heﬀ 1 = u1 H0 − h0 u1 + (u0 B H0 )(1) − (h0 B u0 )(1) u∗0 = πref u1 u∗0 , h0 πref + (u0 B H0 )(1) u∗0 − (h0 B u0 )(1) u∗0

= − 2i πref u0 , H0 λB − h0 , u0 λB πref . The term with the magnetic Poisson bracket can be easily computed:

πref u0 ,H0 λB − h0 , u0 λB u∗0 πref = πref ∂kl u0 ∂rl H0 − λBlj ∂kl u0 u∗0 ∂kj h0 u0 − ∂kj u0 u∗0 h0 − h0 u0 ∂kj u∗0

+ ∂rl h0 ∂kl u0 + λBlj ∂kl h0 ∂kj u0 u∗0 πref = 2i ∂rl φ − λBlj ∂kj E∗ Al + λBlj πref ∂kl u0 u∗0 ∂kj u0 u∗0 h0 + h0 u0 ∂kj u∗0 πref

= 2i ∂rl φ − λBlj ∂kj E∗ Al + λBlj ∂kl ϕb , Hper − E∗ ∂kj ϕb The ﬁrst term combines to a Lorentz force term, the second one — which is purely imaginary — yields the Rammal–Wilkinson term. The components of the magnetic

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ﬁeld are Cb∞ (Rd ) functions and hence, principal and subprincipal symbol are in Cb∞ (T ∗ Rd ) as well. This means, we have proven the following corollary to Theorem 4.1: Corollary 4.1. Under the assumptions of Theorem 4.1, the principal and subprincipal symbol of the eﬀective hamiltonian for a single non-degenerate Bloch band E∗ are given by heﬀ 0 = E∗ + φ heﬀ 1 = − −∂rl φ + λBlj ∂kj E∗ Al − λBlj Mlj

(4.12)

=: −FLor l Al − λBlj Mlj where Al (k) := iϕb (k), ∇k ϕb (k) and Mlj (k) = Re

i 2 ∂kl ϕb , Hper (k) − E∗ (k) ∂kj ϕb

are the Berry connection and the so-called Rammal–Wilkinson term, respectively. To leading order, this is the well-known Peierls substitution. In particular, for zero electric ﬁeld, φ ≡ 0, and constant magnetic ﬁeld (and the equations written in symmetric gauge), the ansatz E∗ (k) =

d

cos kj

j=1

yields the celebrated Hofstadter model [12]. 5. Derivation of Semiclassical Equations of Motion In the preceding section, we have approximated the full quantum evolution by a simpler eﬀective quantum evolution on a smaller reference space Href = L2 (M ∗ ) ⊗ t ˆZ CN . Now, we will link the time evolution generated by heﬀ – and thus e−i ε H – to a semiclassical ﬂow which contains quantum corrections. Conceptually, this is a two-step process: if we reconsider the diagram of spaces, t ˆ

e−i ε H

t ˆZ

e−i ε H

U Z / Hτ / L2 (M ∗ ) ⊗ L2 (Tdy ) L2 (Rdx ) −1 Πref Π Z ΠZ Z −1 ΠZL2 (Rdx ) _ _ _ _ _/ ΠHτ _ _ _ _ _/ L2 (M ∗ ) ⊗ CN G t

A (h eff )

e−i ε Op

(5.1)

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we notice that our physical observables live on the upper-left space L2 (Rdx ) — or equivalently on Hτ . The eﬀective evolution generated by heﬀ approximates the dynamics if the initial states are localized in the almost invariant subspace associated to the relevant bands. In this section, we always assume the relevant part of the spectrum consists of a single non-degenerate band E∗ and thus L2 (M ∗ ) ⊗ C1 ∼ = L2 (M ∗ ). In a ﬁrst step, we need to connect the semiclassical dynamics in the left column of the diagram with those in the lower-right corner. The second, much simpler step is to establish an Egorov-type theorem on the reference space Href = L2 (M ∗ ) ⊗ CN . 5.1. Relation between dynamics for macroscopic and eﬀective observables Since we are concerned with the semiclassical dynamics of a particle in an electromagnetic ﬁeld, the magnetic ﬁeld must enter in the classical equations of motion. There are two ways: either one uses minimal coupling, i.e. one writes down the equations of motion for position r and kinetic momentum k = k − λA(r) with respect symplectic form for the variables (k , r) and the hamiltonian to the usual Z H k − λA(r), r . Then the classical ﬂow which enters the Egorov theorem is generated by r˙ λB(r) −id ∇r (5.2) H Z (k, r) = ∇k +id 0 k˙ where the appearance of B in the matrix representation of the symplectic form is due to the fact that k is kinetic momentum. What constitutes a suitable observable? Physically, we are interested in measurements on the macroscopic scale, i.e. the observable should be independent of the microscopic degrees of freedom. On the level of symbols, this means f (k, r) has to commute pointwise with the Hamiltonian H Z (k, r) for all k and r. Hence, such an observable is a constant of motion with respect to the fast dynamics. In the simplest case, the observables are scalarvalued. This also ensures we are able to “separate” the contributions to the full dynamics band-by-band. Note that this by no means implies OpA (f ) commutes with OpA (H Z ), but rather that all of the non-commutativity is contained in the slow variables (k, r). Deﬁnition 5.1 (Macroscopic Semiclassical Observable). A macropscopic observable f is a scalar-valued semiclassical symbol (cf. Deﬁnition 2.2) AS00 (C) which is Γ∗ -periodic in k, f (k + γ ∗ , r) = f (k, r) for all (k, r) ∈ T ∗ Rd , γ ∗ ∈ Γ∗ . Our assumption that our dynamics lives on the almost-invariant subspace ΠHτ modiﬁes the classical dynamics to ﬁrst order in ε as well: instead of using KA and R as building block observables, the proper observables should be ΠKA Π and ΠRΠ. Equivalently, we can switch to the reference space representation and use the

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magnetic quantization of keﬀ := πref uB kB u∗ πref = k + ελB(r)A(k) + O(ε2 ), reﬀ := πref uB rB u∗ πref = r + εA(k) + O(ε2 ).

(5.3)

The crucial proposition we will prove next says that for suitable observables f , the eﬀect of going to the eﬀective representation is, up to errors of order ε2 at least, equivalent to replacing the arguments k and r by keﬀ and reﬀ , Πref U OpA (f )U ∗ Πref = OpA πref uB f B u∗ πref + O · (ε∞ ) =: OpA (feﬀ ) + O · (ε∞ ).

(5.4)

Then it follows that the eﬀective observable feﬀ coincides with the original observable f after a change of variables up to errors of order O(ε2 ), feﬀ = πref uB f B u∗ πref = f ◦ Teﬀ + O(ε2 ),

(5.5)

where the map Teﬀ : (k, r) → (keﬀ , reﬀ ) maps the observables k and r onto the eﬀective observables keﬀ and reﬀ deﬁned via Eqs. (5.3). Proposition 5.1. Let f be a macroscopic semiclassical observable. Then up to errors of order ε2 Eq. (5.5) holds and consequently, we have Πref U OpA (f ) U ∗ Πref = OpA feﬀ + O · (ε∞ ) = OpA f ◦ Teﬀ + O · (ε2 ). (5.6) Proof. The equivalence of the left-hand sides of Eqs. (5.6) and (5.5) follows from U = OpA (u) + O · (ε∞ ) and the fact that we only need to consider the ﬁrst two terms in the ε expansion. With the help of [17, Theorem 1.1], we conclude feﬀ ∈ AS 0 is also a semiclassical symbol of order 0. The left-hand side of (5.5) can be computed explicitly: to zeroth order, nothing changes as f commutes pointwise with u and u∗ , feﬀ 0 = πref u0 f u∗0 πref = f0 . To ﬁrst order, we have

feﬀ 1 = πref u0 f1 + u1 f0 − feﬀ 0 u1 + (u0 B f0 )(1) − (feﬀ 0 B u0 )(1) u∗0 πref

= πref u0 f1 u∗0 πref − 2i u0 , f0 λB − feﬀ 0 , u0 λB = f1 − i ∂rj f0 + λBlj (r)∂kl f0 πref ∂kj u0 u∗0 πref = f1 + ∂rj f0 + λBlj (r)∂kl f0 Aj . On the other hand, if we Taylor expand f ◦ Teﬀ = f keﬀ , reﬀ to ﬁrst order in ε, we get f keﬀ , reﬀ = f0 k + ελB(r)A(k) + O(ε2 ), r + εA(k) + O(ε2 ) + εf1 k + ελB(r)A(k) + O(ε2 ), r + εA(k) + O(ε2 ) + O(ε2 ) = f0 (k, r) + ε f1 (k, r) + λ∂kl f0 (k, r) Blj (r) Aj (k)

+ ∂rj f0 (k, r) Aj (k) + O(ε2 ) which coincides with feﬀ up to O(ε2 ).

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Now if the equations of motion (5.2) are an approximation of the full quantum dynamics, what are the equations of motion with respect to the eﬀective variables? The classical ﬂow Φeﬀ t generated by heﬀ with respect to the magnetic symplectic form (Eq. (5.2)) can be rewritten in terms of eﬀective variables, −1 2 = Teﬀ ◦ Φeﬀ Φmacro t t ◦ Teﬀ + O(ε ).

(5.7)

The right-hand side does not serve as a deﬁnition for the ﬂow of the macroscopic is the ﬂow associated to a modiﬁed observables, but it is a consequence: Φmacro t symplectic form and a modiﬁed hamiltonian. The modiﬁed symplectic form includes the Berry curvature associated to E∗ acting as a pseudo-magnetic ﬁeld on the position variables. be the ﬂow on T ∗ Rd generated by Proposition 5.2. Let Φmacro t ∇reff r˙eﬀ −id λB(reﬀ ) = hsc (keﬀ , reﬀ ) ∇keff +id εΩ(keﬀ ) k˙ eﬀ

(5.8)

where the semiclassical hamiltonian is given by −1 hsc := heﬀ ◦ Teﬀ .

(5.9)

−1 2 and Teﬀ ◦ Φeﬀ Then Eq. (5.7) holds, i.e. Φmacro t t ◦ Teﬀ agree up to errors of order ε .

Proof. We express k and r in terms of keﬀ and reﬀ in (5.2) since, for ε small enough, Teﬀ : (k, r) → (keﬀ , reﬀ ) is a bijection. For instance, the semiclassical hamiltonian −1 simpliﬁes to heﬀ ◦ Teﬀ hsc (keﬀ , reﬀ ) := heﬀ keﬀ − ελB(reﬀ )A(keﬀ ), reﬀ − εA(keﬀ ) + O(ε2 ) = E∗ (keﬀ ) + φ(reﬀ ) − ελB(reﬀ ) · M(keﬀ ) + O(ε2 ). The symplectic form can be easily expanded to λB(reﬀ − εA(keﬀ )) −id +id 0 =

λB(reﬀ ) −id λ∂reff l B(reﬀ ) A(keﬀ ) 0 −ε + O(ε2 ). +id 0 0 0

The other two terms, the time derivatives and gradients of keﬀ and reﬀ have slightly more complicated expansions, but they can be worked out explicitly. Then if we put all of them together, we arrive at the modiﬁed symplectic form (5.8). This proves the ﬁrst claim. Hence, the hamiltonian vector ﬁelds agree up to O(ε2 ) and Lemma 5.24 in [27] implies that also the ﬂows diﬀer only by O(ε2 ). Remark 5.1. These equations of motion have ﬁrst been proposed in [24, Appendix] and we have derived them in a more systematic fashion. The eﬀective coordinates

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reﬀ and keﬀ are associated to a non-standard Poisson structure on T ∗ Rdk : from Eq. (5.8), one can read oﬀ that the Poisson bracket with respect to reﬀ and keﬀ is given by {f, g}λB,εΩ =

d d ∂ξl f ∂xl g − ∂xl f ∂ξl g − λBlj ∂ξl f ∂ξj g − εΩlj ∂xl f ∂xj g l=1

l,j=1

and thus diﬀerent components of position reﬀ no longer commute, {reﬀ l , reﬀ j }λB,εΩ = −ε Ωlj . Hence, Ω acts as a pseudomagnetic ﬁeld that is due to quantum eﬀects.

5.2. An Egorov-type theorem The semiclassical approximation hinges on an Egorov-type theorem which we ﬁrst prove on the level of eﬀective dynamics: Theorem 5.1. Let heﬀ be the eﬀective hamiltonian as given by Theorem 4.1 associated to an isolated, non-degenerate Bloch band E∗ . Then for any Γ∗ -periodic semigenerated by heﬀ with classical observable f ∈ AS 0 , f = f0 + εf1 , the ﬂow Φeﬀ t respect to the magnetic symplectic form (Eq. (5.2)) approximates the quantum evolution uniformly for all t ∈ [−T, +T ], A t +i εt OpA (heff ) A Op (f )e−i ε Op (heff ) − OpA f ◦ Φeﬀ e t

B(L2 (M ∗ ))

≤ Cε2 .

(5.10)

d Proof. Since heﬀ ∈ Cb∞ (T ∗ Rd ), the ﬂow inherits the smoothness and f ◦Φeﬀ t , dt f ◦ ∈ AS 0 (C) remain also Γ∗ -periodic in the momentum variable. To compare the Φeﬀ t two time-evolutions, we use the usual Duhammel trick which yields A t OpA (f )e−i ε Op (heff ) − OpA f ◦ Φeﬀ t t −i s OpA (h )

d +i s OpA (heff ) A eff ε = e ε ds Op f ◦ Φeﬀ t−s e ds 0 t A s = dse+i ε Op (heff ) εi OpA (heﬀ ), OpA f ◦ Φeﬀ t−s 0 d −i s OpA (h ) eff f ◦ Φeﬀ e ε − OpA ds t−s t A s = dse+i ε Op (heff ) OpA εi heﬀ , f ◦ Φeﬀ t−s B 0 −i s OpA (h ) eff ε − heﬀ , f ◦ Φeﬀ . (5.11) t−s λB e

e+i ε Op t

A

(heff )

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The magnetic Moyal commutator — to ﬁrst order — agrees with the magnetic Poisson bracket, eﬀ eﬀ 2 i ε heﬀ , f ◦ Φt−s B = heﬀ , f ◦ Φt−s λB + O(ε ) =

d d Blj ∂kl heﬀ ∂kj f + O(ε2 ). ∂kl heﬀ ∂rl f − ∂rl heﬀ ∂kl f − l=1

l,j=1

Hence, the term to be quantized in Eq. (5.11) vanishes up to ﬁrst order in ε, r.h.s. of (5.11) t A A s s = dse+i ε Op (heff ) OpA 0 + O(ε2 ) e−i ε Op (heff ) = O · (ε2 ). 0

This ﬁnishes the proof. The main result combines Proposition 5.1 with the Egorov theorem we have just proven: Theorem 5.2 (Semiclassical Limit). Let Assumptions 1.1, 1.2 and 3.1 be satisﬁed; if d ≥ 4, assume in addition that Assumption 3.2 holds true. Furthermore, let us assume the relevant part of the spectrum consists of a single non-degenerate Bloch band E∗ . Then for all macroscopic semiclassical observables f (Deﬁnition 5.1) as the full quantum evolution can be approximated by the hamiltonian ﬂow Φmacro t given in Proposition 5.2 if the initial state is localized in the corresponding almost invariant subspace Z −1 ΠZL2 (Rdx ), −1 t ˆ t ˆ −1 Z ΠZ Z ΠZ e+i ε H OpA (f )e−i ε H − OpA f ◦ Φmacro 2 d ≤ CT ε2 . t B(L (Rx ))

(5.12) Proof. We now combine all of these results to approximate the dynamics: let f be a macroscopic observable. Then if we start with a state in the range of Z −1 ΠZ, the time-evolved observable can be written as ˆε

ˆε

Z −1 ΠZe−i ε H Z −1 OpA (f )Ze+i ε H Z −1 ΠZ t

t

ˆZ

ˆZ

= Z −1 Πe−i ε H OpA (f )e+i ε H ΠZ t

t

ˆ

ˆ

= Z −1 U −1 Πref U U −1 e−i ε h U OpA (f )U −1 e+i ε h U U −1 Πref UZ + O · (ε∞ ) t

ˆ

t

ˆ

= Z −1 U −1 Πref e−i ε h U OpA (f )U −1 e+i ε h Πref UZ + O · (ε∞ ) t

ˆ

t

ˆ

= Z −1 U −1 Πref e−i ε heff Πref U OpA (f )U −1 Πref e+i ε heff Πref UZ + O · (ε∞ ). t

t

After replacing U with OpA (u) (which adds another O · (ε∞ ) error) and Πref with OpA (πref ), the term in the middle combines to the quantization of feﬀ = πref uB f B u∗ πref . We apply Proposition 5.1 and the Egorov theorem

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involving heﬀ and obtain

tˆ tˆ . . . = Z −1 U −1 Πref e−i ε heff OpA πref uB f B u∗ πref e+i ε heff Πref UZ + O · (ε∞ ) tˆ tˆ = Z −1 U −1 Πref e−i ε heff OpA feﬀ e+i ε heff Πref UZ + O · (ε∞ ) U ΠZ + O · (ε2 ). = Z −1 ΠU −1 OpA f ◦ Teﬀ ◦ Φeﬀ t

Since two ﬂows are O(ε2 ) close if the corresponding hamiltonian vector ﬁelds are [27, Lemma 5.24], we conclude −1 2 . . . = Z −1 ΠU −1 OpA f ◦ Teﬀ ◦ Φeﬀ t ◦ Teﬀ ◦ Teﬀ U ΠZ + O · (ε ) = Z −1 ΠU −1 OpA f ◦ Φmacro ◦ Teﬀ U ΠZ + O · (ε∞ ) t = Z −1 ΠOpA f ◦ Φmacro ΠZ + O · (ε2 ). t This ﬁnishes the proof. Acknowledgments The authors would like to thank M. M˘ antoiu, G. Panati and H. Spohn for useful discussions. M. L. thanks G. Panati for initiating the scientiﬁc collaboration with M. M˘ antoiu. Furthermore, the authors have found the suggestions by one of the referees very useful. References [1] N. W. Ashcroft and N. D. Mermin, Solid State Phys. (Sunders College Publishing, 2001). [2] J. Bellissard. C ∗ -Algebras in Solid State Physics. 2D Electrons in a Uniform Magnetic Field, Operator Algebras and Applications, Vol. 2 (University Press, 1988). [3] E. Canc´es, A. Deleurence and M. Lewin. A new approach to the modelling of local defects in crystals: The reduced Hartree–Fock case, Comm. Math. Phys. 281(1) (2008) 129–177. [4] E. Canc´es, A. Deleurence and M. Lewin, Non-perturbative embedding of local defects in crystalline materials, J. Phys.: Condens. Matt. 20(29) (2008) 294213. [5] G. De Nittis and G. Panati. Eﬀective models for conductance in magnetic ﬁelds: derivation of Harper and Hofstadter models (2010); arXiv:0809.3199. [6] G. Grosso and G. P. Parravicini, Solid State Physics (Academic Press, 2003). [7] M. J. Gruber, Noncommutative Bloch theory, J. Math. Phys. 42(6) (2001) 2438– 2465. [8] P. G. Harper, Single band motion of conduction electrons in a uniform magnetic ﬁeld, Proc. Phys. Soc. A 68 (1955) 874–892. [9] B. Helﬀer and J. Sj¨ ostrand, Analyse semi-classique pour l’´equation de Harper (avec application ` a l’´etude de Schr¨ odinger avec champ magn´etique), M´em. Soc. Math. France 34 (1988) 1761–1771. [10] B. Helﬀer and J. Sj¨ ostrand, Analyse semi-classique pour l’´equation de Harper III, M´em. Soc. Math. France 39 (1989) 1–124. [11] B. Helﬀer and J. Sj¨ ostrand, Analyse semi-classique pour l’´equation de Harper II (comportement semi-classique pr`es d’un rationnel), M´em. Soc. Math. France 40 (1990) 1–139.

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[12] D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic ﬁelds, Phys. Rev. B 14 (1976) 2239–2249. [13] L. H¨ ormander, Fourier integral operators 1, Acta Math. 127(1) (1972) 79–183. [14] V. Iftimie, M. Mˇ antoiu and R. Purice, Magnetic pseudodiﬀerential operators. Publ. Res. Inst. Math. Sci. 44(3) (2007) 585–623. [15] V. Iftimie, M. Mˇ antoiu and R. Purice, Commutator criteria for magnetic pseudodifferential operators, Comm. Partial Diﬀerential Equations 35(6) (2010) 1058–1094. [16] P. Kuchment, Floquet Theory for Partial Diﬀerential Equations (Operator Theory: Advances and Applications) (Birkh¨ auser, 1993). [17] M. Lein, Two-parameter asymptotics in magnetic Weyl calculus, J. Math. Phys. 51 (2010) 123519. [18] M. Lein, M. Mˇ antoiu and S. Richard, Magnetic pseudodiﬀerential operators with coeﬃcients in C ∗ -algebras (2009). [19] M. M¨ uller, Product rule for gauge invariant Weyl symbols and its application to the semiclassical description of guiding centre motion, J. Phys. A: Math. Gen. 32 (1999) 1035–1052. [20] M. Mˇ antoiu and R. Purice, The magnetic Weyl calculus, J. Math. Phys. 45(4) (2004) 1394–1417. [21] M. Mˇ antoiu, R. Purice and S. Richard, Twisted crossed products and magnetic pseudodiﬀerential operators, in Advances in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., Vol. 5 (Theta, Burcharest, 2005), pp. 137–172. [22] D. Osadchy and J. E. Avron. Hofstadter butterﬂy as quantum phase diagram, J. Math. Phys. 42 (2001) 5665–5671. [23] G. Panati, Triviality of Bloch and Bloch–Dirac bundles, Ann. Henri Poincar´e 8 (2007) 995–1011. [24] G. Panati, H. Spohn and S. Teufel. Eﬀective dynamics for Bloch electrons: Peierls substitution, Comm. Math. Phys. 242 (2003) 547–578. [25] G. Panati, H. Spohn and S. Teufel, Space adiabatic perturbation theory, Adv. Theor. Math. Phys. 7 (2003) 145–204. [26] M. Reed and B. Simon, Methods of Mathematical Physics IV: Analysis of Operators (Academic Press, 1978). [27] S. Teufel, Adiabatic Perturbation Theory in Quantum Dynamics (Springer Verlag, 2003). [28] D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. D. Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49(6) (1982) 405–408. [29] J. Zak, Dynamics of electrons in solids in external ﬁelds, Phys. Rev. 168(3) (1968) 686–695.

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Reviews in Mathematical Physics Vol. 23, No. 3 (2011) 261–307 c World Scientific Publishing Company DOI: 10.1142/S0129055X1100428X

THE ADHM CONSTRUCTION OF INSTANTONS ON NONCOMMUTATIVE SPACES

SIMON BRAIN∗ and WALTER D. VAN SUIJLEKOM† Institute for Mathematics, Astrophysics and Particle Physics, Faculty of Science, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands ∗[email protected] †[email protected] Received 29 September 2010 Revised 13 January 2011 We present an account of the ADHM construction of instantons on Euclidean space-time R4 from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parametrized by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal–Groenewold plane R4 and the Connes–Landi plane R4θ . Keywords: Yang–Mills; instantons; noncommutative geometry; quantum groups. Mathematics Subject Classification 2010: 58B34, 16T05, 53C28, 81T13

1. Introduction There has been a great deal of interest in recent years in the construction of instanton gauge ﬁelds on space-times whose algebra of coordinate functions is noncommutative. In classical geometry, an instanton is a connection with anti-self-dual curvature on a smooth vector bundle over a four-dimensional manifold. The moduli space of instantons on a classical four-manifold is an important invariant of its differential structure [13] and it is only natural to try to generalize this idea to study the diﬀerential geometry of noncommutative four-manifolds. In this article, we study the construction of SU(2) instantons on the Euclidean four-plane R4 and its various noncommutative generalizations. The problem of constructing instantons on classical space-time was solved by the ADHM method of [2] and consequently it is known that the moduli space of (framed) SU(2) instantons on R4 with topological charge k ∈ Z is a manifold of dimension 8k − 3. In what follows, we review the ADHM construction of instantons on classical R4 and its extension to noncommutative geometry. In particular, we study the construction 261

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of instantons on two explicit examples of noncommutative Euclidean space-time: the Moyal–Groenewold plane R4 , whose algebra of coordinate functions has commutation relations of the Heisenberg form, and the Connes–Landi plane R4θ which arises as a localization of the noncommutative four-sphere Sθ4 constructed in [10] (cf. also [11]). In order to deform the ADHM construction, we adopt the technique of “functorial cocycle twisting”, a very general method which can be used in particular to deform the coordinate algebra of any space carrying an action of a locally compact Abelian group. Both of the above examples of noncommutative space-times are obtained in this way as deformations of classical Euclidean space-time: the Moyal plane R4 as a twist along a group of translational symmetries, the Connes–Landi plane R4θ as a twist along a group of rotational symmetries. Crucially, the twisting technique does not just deform space-time alone: its functorial nature means that it simultaneously deforms any and all constructions which are covariant under the chosen group of symmetries. In particular, the parameter spaces which occur in the ADHM construction also carry canonical actions of the relevant symmetry groups, whence their coordinate algebras are also deformed by the quantization procedure. In this way, we obtain a method for constructing families of instantons which are parametrized by noncommutative spaces. As natural as this may seem, it immediately leads to the conceptual problem of how to interpret these spaces of noncommutative parameters. Indeed, our quest in each case is for a moduli space of instantons, which is necessarily modeled on the space of all connections on a given vector bundle over space-time [3]. Even for noncommutative space-times, the set of such connections is an aﬃne space and is therefore commutative [8]. To solve this problem, we adopt the strategy of [5] and incorporate into the ADHM construction the “internal gauge symmetries” of noncommutative space-time [9], in order to “gauge away” the noncommutativity and arrive at a classical space of parameters. The paper is organized as follows. The remainder of Sec. 1 is dedicated to a brief review of the algebraic structures that we shall need, including in particular the notions of Hopf algebras and their comodules, together with the cocycle twisting construction itself. Following this, in Secs. 2 and 3, we review the diﬀerential geometry of instantons from the point of view of coordinate algebras. We recall how to generalize these structures to incorporate the notions of noncommutative families of instantons and their gauge theory. In Sec. 4, we sketch the ADHM construction of instantons from the point of view of coordinate algebras, stressing how the method is covariant under the group of isometries of Euclidean space-time. It is precisely this covariance that we use in later sections to deform the ADHM construction by cocycle twisting. This coordinate-algebraic version of the ADHM construction has in fact already been studied in some detail [4,5]. In this sense, the ﬁrst four sections of the present article consist mainly of review material. However, those earlier works studied the

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geometry of the ADHM construction from a somewhat abstract categorical point of view: in the present exposition, our goal is to give an explanation of the ADHM method in which we keep things as concrete as we are able. Moreover, the focus of those papers was on the construction of instantons on the Euclidean four-sphere S 4 . In what follows, we show how to adapt this technique to construct instantons on the local coordinate chart R4 . We illustrate the deformation procedure by giving the noncommutative ADHM construction in two special cases. In Sec. 5, we look at how the method behaves under quantization to give an ADHM construction of instantons on the Moyal plane R4 . On the other hand, Sec. 6 addresses the issue of deforming the ADHM method to obtain a construction of instantons on the Connes–Landi plane R4θ . It is these latter two sections which contain the majority of our new results. Indeed, an algorithm for the construction of instantons on the Moyal plane R4 has been known for some time [24]. However, a good understanding of the noncommutative-geometric origins of this construction has so far been lacking and our goal is to shed some light upon this subject (although it is worth pointing out that a diﬀerent approach to the twistor theory of R4 , from the point of view of noncommutative algebraic geometry, was carried out in [17]). Using the noncommutative twistor theory developed in [6], we give an explicit construction of families of instantons on Moyal space-time, from which we recover the well-known noncommutative ADHM equations of Nekrasov and Schwarz. Our methods are intended to be similar in spirit to those of [1] for the classical case; for some alternative approaches to the construction of noncommutative instantons, we refer, for example, to [7, 15, 16, 25, 28, 29] and references therein. On the other hand, the noncommutative geometry of instantons and the ADHM construction on the Connes–Landi plane R4θ was investigated in [18, 19, 21, 4, 5]. As already mentioned, however, this abstract geometric characterization of the instanton construction is in need of a more concrete description. In the present paper we derive an explicit set of ADHM equations whose solutions parametrize instantons on the Connes–Landi plane (although we do not claim at this stage that all such instantons arise in this way). We stress that our intention throughout the following is to present the various geometrical aspects of the ADHM construction in concrete terms. This means that, throughout the paper, we work purely at the algebraic level, i.e. with algebras of coordinate functions on all relevant spaces. A more detailed approach, which in particular addresses all of the analytic aspects of the construction, will be presented elsewhere. 1.1. Hopf algebraic structures Let H be a unital Hopf algebra. We denote its structure maps by ∆ : H → H ⊗ H,

: H → C,

S:H → H

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for the coproduct, counit and antipode, respectively. We use Sweedler notation for the coproduct, ∆h = h(1) ⊗ h(2) , as well as (∆ ⊗ id) ◦ ∆h = (id ⊗ ∆) ◦ ∆h = h(1) ⊗ h(2) ⊗ h(3) and so on, with summation inferred. We say that H is coquasitriangular if it is equipped with a convolution-invertible Hopf bicharacter R : H ⊗ H → C obeying g (1) h(1) R(h(2) , g (2) ) = R(h(1) , g (1) )h(2) g (2)

(1.1)

for all h, g ∈ H. Convolution-invertibility means that there exists a map R−1 : H ⊗ H → C such that R(h(1) , g (1) )R−1 (h(2) , g (2) ) = R−1 (h(1) , g (1) )R(h(2) , g (2) ) = (g)(h)

(1.2)

for all g, h ∈ H. Being a Hopf bicharacter means that R(f g, h) = R(f, h(1) )R(g, h(2) ),

R(f, gh) = R(f (1) , h)R(f (2) , g),

(1.3)

for all f, g, h ∈ H. If R also has the property that R(h(1) , g (1) )R(g (2) , h(2) ) = (g)(h) for all g, h ∈ H, then we say that H is cotriangular. A vector space V is said to be a left H-comodule if it is equipped with a linear map ∆V : V → H ⊗ V such that (id ⊗ ∆V ) ◦ ∆V = (∆ ⊗ id) ◦ ∆V ,

( ⊗ id) ◦ ∆V = id.

We shall often use the Sweedler notation ∆V (v) = v (−1) ⊗ v (0) for v ∈ V , again with summation inferred. If V , W are left H-comodules, a linear transformation σ : V → W is said to be a left H-comodule map if it satisﬁes ∆W ◦ σ = (id ⊗ σ) ◦ ∆V . Given a pair V, W of left H-comodules, the vector space V ⊗ W is also a left H-comodule when equipped with the tensor product H-coaction ∆V ⊗W (v ⊗ w) = v (−1) w(−1) ⊗ (v (0) ⊗ w(0) )

(1.4)

for each v ∈ V , w ∈ W . An algebra A is said to be a left H-comodule algebra if it is a left H-comodule equipped with a product m : A ⊗ A → A which is an H-comodule map, meaning in this case that (id ⊗ m) ◦ ∆A⊗A = ∆A ◦ m. Dually, a vector space V is said to be a left H-module if there is a linear map : H ⊗ V → V , denoted h ⊗ v → h v, such that h (g v) = (hg) v,

1v =v

for all v ∈ V . An algebra A is said to be a left H-module algebra if it a left H-module equipped with a product m : A ⊗ A which obeys h (ab) = (h(1) a)(h(2) b) for all a, b ∈ A.

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In the special case where H is coquasitriangular, every left H-comodule V is also a left H-module when equipped with the canonical left H-action : H ⊗ V → V,

h v = R(v (−1) , h)v (0)

(1.5)

for each h ∈ H, v ∈ V . The action (1.5) does not commute with the H-coaction on V ; rather it obeys the “crossed module” condition h(1) v (−1) ⊗ h(2) v (0) = (h(1) v)(−1) h(2) ⊗ (h(1) v)(0)

(1.6)

for each h ∈ H and v ∈ V . In particular, if A is a left H-comodule algebra then it is also a left H-module algebra when equipped with the canonical action (1.5). To each left H-module algebra A there is an associated smash product algebra A > H, which is nothing other than the vector space A ⊗ H equipped with the product (a ⊗ h)(b ⊗ g) = a(h(1) b) ⊗ h(2) g

(1.7)

for each a, b ∈ A, h, g ∈ H. The main example of this construction relevant to the present paper is the following. Example 1.1. Let G be a locally compact Abelian Lie group with Pontryagin and let γ : G → Aut A be an action of G on a unital ∗-algebra A dual group G by ∗-automorphisms. Then associated to this action there is the crossed prod On the other hand, the G-action gives A the structure of a uct algebra A >γ G. gives an identiﬁcation of C0 (G)-module algebra and the Fourier transform on G with the smash product A > C0 (G). A >γ G In this paper, our strategy is to keep things as explicit as possible, avoiding the technical details of analytic arguments and working purely at the algebraic level. For this reason, instead of using the function algebra C0 (G), we prefer to work with the bialgebra A[G] of representative functions on G equipped with pointwise multiplication. In this setting, we can still make sense of Example 1.1: although and A > C0 (G) are deﬁned analytically using completions the algebras A >γ G of appropriate function algebras, we think of the smash product A > A[G] as an “algebraic version” of the crossed product algebra A >γ G. This also explains our use throughout the paper of coactions of Hopf algebras in place of group actions: the construction of crossed products by group actions is not deﬁned at the algebraic level, whence we need to replace it by the smash product algebra instead. 1.2. Quantization by cocycle twist Following [22], in this section we give a brief review of the deformation procedure that we shall use later in the paper to “quantize” the ADHM construction of instantons.

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Let H be a unital Hopf algebra whose antipode we assume to be invertible. A two-cocycle on H is a linear map F : H ⊗ H → C which is unital, convolutioninvertible in the sense of Eq. (1.2) and obeys the condition ∂F = 1, i.e. F (g (1) , f (1) )F (h(1) , g (1) f (2) )F −1 (h(2) g (3) , f (3) )F −1 (h(3) , g (4) ) = (f )(h)(g) (1.8) for all f, g, h ∈ H. Given such an F , there is a twisted Hopf algebra HF with the same coalgebra structure as H, but with modiﬁed product •F and antipode SF given respectively by h •F g = F (h(1) , g (1) )h(2) g (2) F −1 (h(3) , g (3) ),

(1.9)

SF (h) := U (h(1) )S(h(2) )U −1 (h(3) ),

(1.10)

for each h, g ∈ HF , where on the right-hand sides we use the product and antipode of H and deﬁne U (h) := F (h(1) , Sh(2) ). The cocycle condition (1.8) is suﬃcient to ensure that the product (1.9) is associative. In the case where H is a Hopf ∗-algebra we also need to impose upon the cocycle F the reality condition F (h, g) = F ((S 2 g)∗ , (S 2 h)∗ ).

(1.11)

In this situation the twisted Hopf algebra HF acquires a deformed ∗-structure h∗F := V −1 (S −1 h(1) )(h(2) )∗ V (S −1 h(3) )

(1.12)

for each h ∈ HF , where V (h) := U −1 (h(1) )U (S −1 h(2) ). If H is a coquasitriangular Hopf algebra, then so is HF . In particular, if H is commutative then HF is cotriangular with “universal R-matrix” given by R(h, g) := F (g (1) , h(1) )F −1 (h(2) , g (2) )

(1.13)

for all h, g ∈ H. Since as coalgebras H and HF are the same, every left Hcomodule is a left HF -comodule and every H-comodule map is an HF -comodule map. This means that there is an invertible functor which “functorially quantizes” any H-covariant construction to give an HF -covariant one. As already mentioned, our strategy will be to apply this idea to the construction of instantons. In passing from H to HF , from each left H-comodule algebra A we automatically obtain a left HF -comodule algebra AF which as a vector space is the same as A but has the modiﬁed product AF ⊗ AF → AF ,

a ⊗ b → a ·F b := F (a(−1) , b(−1) )a(0) b(0) .

(1.14)

If A is a left H-comodule algebra and a ∗-algebra such that the coaction is a ∗-algebra map, the twisted algebra AF also has a new ∗-structure deﬁned by a∗F := V −1 (S −1 a(−1) )(a(0) )∗ for each a ∈ AF .

(1.15)

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2. The Twistor Fibration The Penrose ﬁbration CP3 → S 4 is an essential component of the ADHM construction of instantons, since it encodes in its geometry the very nature of the anti-selfduality equations on S 4 [1]. Following [6, 4], in this section we sketch the details of the Penrose ﬁbration from the point of view of coordinate algebras, then investigate what happens to the ﬁbration upon passing to a local coordinate chart. 2.1. The Penrose ﬁbration The ∗-algebra A[C4 ] of coordinate functions on the classical space C4 is the commutative unital ∗-algebra generated by the elements {zj , zl∗ | j, l = 1, . . . , 4}. The coordinate algebra A[S 7 ] of the seven-sphere S 7 is the quotient of A[C4 ] by the sphere relation z1∗ z1 + z2∗ z2 + z3∗ z3 + z4∗ z4 = 1.

(2.1)

On the other hand, the coordinate algebra A[S ] of the four-sphere S is the commutative unital ∗-algebra generated by the elements x1 , x∗1 , x2 , x∗2 and x0 = x∗0 subject to the sphere relation 4

4

x∗1 x1 + x∗2 x2 + x20 = 1. There is a canonical inclusion of algebras A[S 4 ] → A[S 7 ] deﬁned on generators by x1 = 2(z1 z3∗ + z2∗ z4 ),

x2 = 2(z2 z3∗ − z1∗ z4 ),

x0 = z1 z1∗ + z2 z2∗ − z3 z3∗ − z4 z4∗ (2.2)

and extended as a ∗-algebra map. Clearly one has x∗1 x1 + x∗2 x2 + x20 = (z1∗ z1 + z2∗ z2 + z3∗ z3 + z4∗ z4 )2 = 1,

(2.3)

so that the algebra inclusion is well-deﬁned. This is just a coordinate algebra description of the principal bundle S 7 → S 4 with structure group SU(2) (cf. [18] for further details of this construction). The twistor space of the Euclidean four-sphere S 4 is nothing other than the complex projective space CP3 . As a real six-dimensional manifold, twistor space CP3 may be identiﬁed with the set of all 4 × 4 Hermitian projector matrices of rank one, since each such matrix uniquely determines and is uniquely determined by a one-dimensional subspace of C4 . Thus the coordinate ∗-algebra A[CP3 ] of CP3 has a deﬁning matrix of (commuting) generators   a1 u 1 u 2 u 3   ∗ u1 a2 v3 v2  ,  q :=  ∗ (2.4)  u2 v3∗ a3 v1  u∗3

v2∗

v1∗

a4

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with a∗j = aj for j = 1, . . . , 4 and Tr q = to the relations j aj = 1, subject 2 coming from the projection condition q = q, that is to say r qjr qrl = qjl for each j, l = 1, . . . , 4. There is a canonical inclusion of algebras A[CP3 ] → A[S 7 ] deﬁned on generators by qjl = zj zl∗ ,

j, l = 1, . . . , 4,

(2.5)

with the relation q11 + q22 + q33 + q44 = 1 coming from the sphere relation (2.1). To determine the twistor ﬁbration CP3 → S 4 in our coordinate algebra framework, we need the map J : A[C4 ] → A[C4 ] deﬁned on generators by J(z1 , z2 , z3 , z4 ) := (−z2∗ , z1∗ , −z4∗ , z3∗ )

(2.6)

and extended as a ∗-anti-algebra map. Equipping the algebra A[C4 ] with the map J identiﬁes the underlying space C4 with the quaternionic vector space H2 [21]. Accordingly, we deﬁne A[H2 ] to be the ∗-algebra A[C4 ] equipped with the quaternionic structure J. Using the identiﬁcation of generators (2.5), the map J extends to an automorphism of the algebra A[CP3 ] given by J(a1 ) = a2 ,

J(a2 ) = a1 ,

J(v1 ) = −v1 ,

J(u2 ) = v2∗ , J(u3 ) = −v3∗ ,

J(a3 ) = a4 ,

J(a4 ) = a3 ,

J(u1 ) = −u1 ,

J(v2 ) = u∗2 ,

J(v3 ) = −u∗3

and extended as a ∗-anti-algebra map. It is straightforward to check that the subalgebra of A[CP3 ] ﬁxed by this automorphism is precisely the four-sphere algebra A[S 4 ]. Indeed, there is an algebra inclusion A[S 4 ] → A[CP3 ] deﬁned on generators by x0 → 2(a1 + a2 − 1),

x1 → 2(u2 + v2∗ ),

x2 → 2(v3 − u∗3 ),

(2.7)

which is just a coordinate-algebraic description of the Penrose ﬁbration CP3 → S 4 . 2.2. Localization of the twistor bundle Next we look at what happens to the ﬁbration A[S 4 ] → A[CP3 ] when we pass to a local chart of S 4 by removing a point. Indeed, it is well known that in making such a localization the twistor bundle CP3 → S 4 becomes isomorphic to the trivial ﬁbration R4 × CP1 over R4 . In this section we demonstrate this fact using the langauge of coordinate algebras. By deﬁnition, the localization A0 [S 4 ] of A[S 4 ] is the commutative unital ∗-algebra x1 , x ˜∗1 , x ˜2 , x ˜∗2 , x˜0 , A0 [S 4 ] := A[˜ (1 + x ˜0 )−1 | x ˜∗1 x˜1 + x ˜∗2 x˜2 + x˜20 = 1, (1 + x ˜0 )(1 + x˜0 )−1 = 1]. It is the algebra obtained from A[S 4 ] by adjoining an inverse (1 + x0 )−1 to the function 1+x0 ; geometrically this corresponds to “deleting” the point (x1 , x2 , x0 ) = (0, 0, −1) from the spectrum of the (smooth completion of the) algebra A[S 4 ], with

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A0 [S 4 ] being the algebra of coordinate functions on the resulting space. On the other hand, the coordinate algebra A[R4 ] of the Euclidean four-plane R4 is the commutative unital ∗-algebra generated by the elements {ζj , ζl∗ | j, l = 1, 2}.

(2.8)

Deﬁning |ζ|2 := ζ1∗ ζ1 + ζ2∗ ζ2 , the element (1 + |ζ|2 )−1 clearly belongs to the (smooth completion of the) algebra A[R4 ] and so we have the following result. Lemma 2.1. The map A0 [S 4 ] → A[R4 ] deﬁned on generators by x ˜1 → 2ζ1 (1 + |ζ|2 )−1 ,

x˜2 → 2ζ2 (1 + |ζ|2 )−1 ,

x ˜0 → (1 − |ζ|2 )(1 + |ζ|2 )−1 (2.9)

is a ∗-algebra isomorphism. Proof. The inverse of (2.9) is the map A[R4 ] → A0 [S 4 ] given on generators by ζ1 → x˜1 (1 + x˜0 )−1 ,

ζ2 → x ˜2 (1 + x ˜0 )−1

(2.10)

and extended as a ∗-algebra map. Thus we have an isomorphism of vector spaces. ˜2 , x˜0 satisfy the same relation as the generators One checks that the elements x ˜1 , x 4 x1 , x2 , x0 of the algebra A[S ]. The diﬀerence is that the point determined by the coordinate values (x1 , x2 , x0 ) = (0, 0, −1) is not in the spectrum of the (smooth) ˜2 , x ˜0 . In this way, we obtain R4 as a local chart of algebra generated by the x˜1 , x 4 the four-sphere S , with the identiﬁcation (2.9) deﬁning the “inverse stereographic projection”. The point (0, 0, −1) will henceforth be called the point at inﬁnity. At the level of twistor space CP3 , passing to the local chart R4 by removing the point at inﬁnity corresponds to removing the ﬁber CP1 over that point: we refer to this copy of CP1 as the line at inﬁnity and denote it by ∞ . Under the algebra inclusion (2.7), inverting the element 1 + x0 in A[S 4 ] is equivalent to inverting the element a1 +a2 in A[CP3 ]. We denote by A0 [CP3 ] the resulting localized algebra, i.e. A0 [CP3 ]

−1

:= A qjl , (a1 + a2 )

−1 qjr qrl = qjl , Tr q = 1, (a1 + a2 )(a1 + a2 ) = 1 .

r

We now show that this algebra is isomorphic to the algebra of coordinate functions on the Cartesian product R4 × CP1 . The coordinate algebra A[CP1 ] is the commutative unital ∗-algebra generated by the entries of the matrix

a ˜1 u ˜1 ˜ := q (2.11) u ˜∗1 a ˜2

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˜2 = q ˜∗ = q ˜ and Tr ˜q = 1, that is to say a subject to the relations q ˜1 a ˜2 = u ˜∗1 u˜1 , ∗ ∗ 4 ˜1 , a ˜2 = a ˜2 and a ˜1 + a ˜2 = 1. With the coordinate algebra A[R ] of (2.8), we a ˜1 = a have the following result. Lemma 2.2. There is a ∗-algebra isomorphism A[R4 ] ⊗ A[CP1 ] ∼ = A0 [CP3 ] deﬁned on generators by ζ1 ⊗ 1 → (a1 + a2 )−1 (u2 + v2∗ ), 1⊗a ˜1 → (a1 + a2 )−1 a1 ,

ζ2 ⊗ 1 → (a1 + a2 )−1 (v3 − u∗3 ),

1 ⊗ u˜1 → (a1 + a2 )−1 u1 ,

1⊗a ˜2 → (a1 + a2 )−1 a2

and extended as a ∗-algebra map. Proof. We need to show that this map is an isomorphism of vector spaces which respects the algebra relations in A[R4 ] ⊗ A[CP1 ]. Using the expressions (2.2) and (2.5), we ﬁnd in A[S 7 ] the identities 2(a1 + a2 )z3 = x∗1 z1 + x∗2 z2 ,

2(a1 + a2 )z3∗ = x1 z1∗ + x2 z2∗ ,

2(a1 + a2 )z4 = x1 z2 − x2 z1 ,

2(a1 + a2 )z4∗ = x∗1 z2∗ − x∗2 z1∗ .

In the localization where 2(a1 + a2 ) = 1 + x0 is invertible, these expressions combined with the identiﬁcations qij = zi zj∗ deﬁne the inverse of the map stated in the lemma, so that we have a vector space isomorphism. The algebra A[CP1 ] generated ˜2 , u ˜1 and u ˜∗1 is identiﬁed with the subalgebra of A[CP3 ] generated by the by a ˜1 , a localized upper left 2 × 2 block of the matrix (2.4), i.e. the subalgebra generated by the elements (a1 + a2 )−1 qij for i, j = 1, 2. It is easy to check that the relations in A[CP1 ] are automatically preserved by this identiﬁcation. To check that the trace 3 4 1 relation Tr q = j qjj = 1 in A0 [CP ] also holds in A[R ] ⊗ A[CP ], one ﬁrst computes that (a1 + a2 )−1 (z1∗ z1 + z2∗ z2 ) → 1 ⊗ 1,

(a1 + a2 )−1 (z3∗ z3 + z4∗ z4 ) → (ζ1∗ ζ1 + ζ2∗ ζ2 ) ⊗ 1,

so that the trace relation holds if and only if (a1 + a2 )−1 → (1 + |ζ|2 ), which is certainly true. Moreover, in A[CP3 ] there are relations of the form qij qkl = zi zj∗ zk zl∗ = zi zl∗ zk zj∗ = qil qkj

(2.12)

for i, j, k, l = 1, . . . , 4. By adding together various linear combinations and using the trace relation Tr q = 1, one ﬁnds that the relations (2.12) are equivalent to the projector relations q2 = q. Hence it follows that the projector relations in A0 [CP3 ] are equivalent to the remaining relations in A[R4 ] ⊗ A[CP1 ]. In this way, we see that there is a canonical inclusion of algebras A[R4 ] → A0 [CP3 ] in the obvious way; this is a coordinate algebra description of the localized twistor ﬁbration R4 × CP1 → R4 . Moreover, using the isomorphism in Lemma 2.2, the quaternionic structure J of Eq. (2.6) is well-deﬁned on the algebra A0 [CP3 ], with A[R4 ] being the J-invariant subalgebra.

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2.3. Symmetries of the twistor ﬁbration In later sections we shall obtain deformations of the twistor ﬁbration by the cocycle twisting of Sec. 1.2; for this we need a group of symmetries acting upon the twistor bundle. Here we describe the general strategy that we shall adopt. We write M(2, H) for the algebra of 2 × 2 matrices with quaternion entries. The algebra A[M(2, H)] of coordinate functions on M(2, H) is the commutative unital ∗-algebra generated by the entries of the 4 × 4 matrix   α1 −α∗2 β1 −β2∗ 

 α2 α∗1 β2 β1∗  aij bij . = A= (2.13)   cij dij  γ1 −γ2∗ δ1 −δ2∗  γ2

γ1∗

δ1∗

δ2

We think of this matrix as being generated by a set of quaternion-valued functions, writing

α1 −α∗2 a = (aij ) = α2 α∗1 and similarly for the other entries b, c, d. The ∗-structure on this algebra is evident from the matrix (2.13). We equip A[M(2, H)] with the matrix coalgebra structure ∆(Aij ) = Air ⊗ Arj , (Aij ) = δij for i, j = 1, . . . , 4. r

Dual to the canonical action of M(2, H) on C4 H2 there is a left coaction deﬁned by Ajr ⊗ zr , (2.14) ∆L : A[C4 ] → A[M(2, H)] ⊗ A[C4 ], zj → r

extended as a ∗-algebra map. This coaction commutes with the quaternionic structure (2.6), in the sense that (id ⊗ J) ◦ ∆L = ∆L ◦ J, so that we have a coaction ∆L : A[H2 ] → A[M(2, H)] ⊗ A[H2 ] (cf. [21]). The Hopf algebra A[GL(2, H)] of coordinate functions on the group GL(2, H) is obtained by adjoining to A[M(2, H)] an invertible group-like element D obeying the relation D−1 = det A, where det A is the determinant of the matrix (2.13). This yields a left coaction ∆L : A[H2 ] → A[GL(2, H)] ⊗ A[H2 ], also deﬁned by the formula (2.14) and extended as a ∗-algebra map. The group GL(2, H) is the group of conformal symmetries of the twistor ﬁbration CP3 → S 4 [27, 23]. However, since we are interested in the localized twistor bundle described in Lemma 2.2, we work instead with the localized group of symmetries

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GL+ (2, H), which is just the “coordinate patch” of GL(2, H) in which the 2 × 2 block a is assumed to be invertible. The coordinate algebra A[GL+ (2, H)] of this which localization is obtained by adjoining to A[GL(2, H)] an invertible element D −1 = det a. The coaction of A[GL+ (2, H)] on A[H2 ] is once obeys the relation D again deﬁned by the formula (2.14). We refer to [6] for further details of this construction. Throughout the paper, our strategy will be to deform the localized twistor ﬁbration and its associated geometry using the action of certain subgroups of GL+ (2, H). In dual terms, we suppose H to be a commutative Hopf ∗-algebra obtained via a Hopf algebra projection π : A[GL+ (2, H)] → H.

(2.15)

This determines a left coaction of H on A[H2 ] by projection of the coaction (2.14), namely ∆π : A[H2 ] → H ⊗ A[H2 ],

∆π := (π ⊗ id) ◦ ∆L ,

(2.16)

which makes A[H2 ] into a left H-comodule ∗-algebra. Moreover, we assume that this coaction respects the deﬁning relations of the localized twistor algebra A0 [CP3 ] given in Lemma 2.2, whence it makes A0 [CP3 ] and A[R4 ] into left H-comodule ∗-algebras in such a way that the algebra inclusion A[R4 ] → A0 [CP3 ] is a left H-comodule map. 3. Families of Instantons and Gauge Theory We are now ready to study diﬀerential structures on the twistor ﬁbration. In this section we recall the basic theory of anti-self-dual connections on Euclidean space R4 from the point of view of noncommutative geometry. Following [21, 5], we then generalize this by recalling what it means to have a family of anti-self-dual connections on R4 and when such families are gauge equivalent. These notions will pave the way for the algebraic formulation of the ADHM construction to follow. 3.1. Diﬀerential structures and instantons As discussed, our intention is to present the construction of connections and gauge ﬁelds in an entirely H-covariant framework, from which all of our deformed versions will immediately follow by functorial cocycle twisting. First of all we discuss the various diﬀerential structures that we shall need. We write Ω(C4 ) for the canonical diﬀerential calculus on A[C4 ]. It is the graded diﬀerential algebra generated by the degree zero elements zj , zl∗ , j, l = 1, . . . , 4, and the degree one elements dzj , dzl∗ , j, l = 1, . . . , 4, subject to the relations dzj ∧ dzl + dzl ∧ dzj = 0,

dzj ∧ dzl∗ + dzl∗ ∧ dzj = 0

for j, l = 1, . . . , 4. The exterior derivative d on Ω(C4 ) is deﬁned by d : zj → dzj and extended uniquely using a graded Leibniz rule. There is also an involution on Ω(C4 ) given by graded extension of the map zj → zj∗ .

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The story is similar for the canonical diﬀerential calculus Ω(R4 ). It is generated by the degree zero elements ζj , ζl∗ and the degree one elements dζj , dζl∗ , j, l = 1, 2, subject to the relations dζj ∧ dζl + dζl ∧ dζj = 0,

dζj ∧ dζl∗ + dζl∗ ∧ dζj = 0.

With π : A[GL+ (2, H)] → H a choice of Hopf algebra projection as in Eq. (2.15), we assume throughout that the diﬀerential calculi Ω(C4 ) and Ω(S 4 ) are (graded) left H-comodule algebras such that the exterior derivative d is a left H-comodule map, i.e. the coaction (2.16) obeys ∆π (dzj ) = (id ⊗ d)∆π (zj ),

j = 1, . . . , 4.

In this way, the H-coactions on Ω(C4 ) and on Ω(R4 ) are given by extending the coaction on A[C4 ]. Next we come to discuss vector bundles over R4 . Of course, the fact that R4 is contractible means that the K-theory of the algebra A[R4 ] is trivial, i.e. all ﬁnitely generated projective (let us say right) modules E over A[R4 ] have the form E = A[R4 ]N for N a positive integer and are equipped with a canonical A[R4 ]-valued Hermitian structure · | · . A connection on E is a linear map ∇ : E → E ⊗A[R4 ] Ω1 (R4 ) satisfying the Leibniz rule ∇(ξx) = (∇ξ)x + ξ ⊗ dx for all ξ ∈ E, x ∈ A[R4 ]. The connection ∇ is said to be compatible with the Hermitian structure on E if it obeys

∇ξ | η + ξ | ∇η = d ξ | η for all ξ, η ∈ E, x ∈ A[R4 ]. Since E is necessarily free as an A[R4 ]-module, any compatible connection ∇ can be written ∇ = d + α, where α is a skew-adjoint element of HomA[R4 ] (E, E ⊗A[R4 ] Ω1 (R4 )). The curvature of ∇ is the EndA[R4 ] (E)-valued two-form F := ∇2 = dα + α2 . The Euclidean metric on R4 determines the Hodge ∗-operator on Ω(R4 ), which on two-forms is a linear map ∗ : Ω2 (R4 ) → Ω2 (R4 ) such that ∗2 = id. Since H coacts by conformal transformations on A[R4 ], there is an H-covariant splitting of two-forms Ω2 (R4 ) = Ω2+ (R4 ) ⊕ Ω2− (R4 ) into self-dual and anti-self-dual components, i.e. the ±1 eigenspaces of the Hodge operator. The curvature ∇2 of a connection ∇ is said to be anti-self-dual if it satisﬁes the equation ∗F = −F . Definition 3.1. A compatible connection ∇ on E is said to be an instanton if its curvature F = ∇2 is an anti-self-dual two-form.

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It is worth pointing out here that the usual deﬁnition of an instanton imposes the additional constraint that the Yang–Mills “energy” deﬁned by the integral tr(F ∧ F ) YM(∇) := R4

should be ﬁnite or, equivalently, that the instanton connection extends to one on the conformal compactiﬁcation S 4 of R4 . Indeed, it is known that the ADHM construction of instantons on R4 (to be described in Sec. 4) produces instantons whose Yang–Mills energy is ﬁnite and, conversely, that all such instantons arise via the ADHM construction. We shall address the corresponding issue for instantons on noncommutative spaces later on in the paper. The gauge group of E is deﬁned to be U(E) := {U ∈ EndA[R4 ] (E) | U ξ | U η = ξ | η for all ξ, η ∈ E}. It acts upon the space of compatible connections by ∇ → ∇U := U ∇U ∗ for each compatible connection ∇ and each element U of U(E). We say that a pair of connections ∇1 , ∇2 on E are gauge equivalent if they are related by such a gauge transformation U . The curvatures of gauge equivalent connections are related by F U = (∇U )2 = UFU ∗ . Note in particular that if ∇ has anti-self-dual curvature then so does the gauge-transformed connection ∇U . We observe a posteriori that the above deﬁnitions do not depend on the commutativity of the algebras A[R4 ] and Ω(R4 ), so that they continue to make sense even if we allow for deformations of the algebras A[R4 ] and Ω(R4 ). 3.2. Noncommutative families of instantons Having given the deﬁnition of an instanton on R4 , we now come to discuss what it means to have a family of instantons over R4 . In the following, we let A be an arbitrary (possibly noncommutative) unital ∗-algebra. Definition 3.2. A family of Hermitian vector bundles over R4 parametrized by the algebra A is a ﬁnitely generated projective right module E over the algebra A ⊗ A[R4 ] equipped with an A ⊗ A[R4 ]-valued Hermitian structure · | · . By deﬁnition, any such module E is given by a self-adjoint idempotent P ∈ MN (A ⊗ A[R4 ]), i.e. an N × N matrix with entries in A ⊗ A[R4 ] satisfying P2 = P = P∗ ; the corresponding module is E := P(A ⊗ A[R4 ])N . Although Deﬁnition 3.2 is given in terms of an arbitrary algebra A, it is motivated by the case where A is the (commutative) coordinate algebra of some underlying classical space X. In this situation, for each point x ∈ X there is an evaluation map evx : A → C and

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the object Ex := (evx ⊗ id)P((A ⊗ A[R4 ])N ) is a ﬁnitely generated projective right A[R4 ]-module corresponding to a vector bundle over R4 . In this way, the projection P deﬁnes a family of Hermitian vector bundles parametrized by the space X. When the algebra A is noncommutative, there need not be enough evaluation maps available, but we may nevertheless work with the whole family at once. Next we come to say what it means to have a family of connections over R4 . We write A ⊗ Ω1 (R4 ) for the tensor product bimodule over the algebra A ⊗ A[R4 ] and extend the exterior derivative d on A[R4 ] to A ⊗ A[R4 ] as id ⊗ d. Definition 3.3. A family of connections parametrized by the algebra A consists of a family of Hermitian vector bundles E := P(A ⊗ A[R4 ])N over A[R4 ], together with a linear map ∇ : E → E ⊗A⊗A[R4 ] (A ⊗ Ω1 (R4 )) E ⊗A[R4 ] Ω1 (R4 ) obeying the Leibniz rule ∇(ξx) = (∇ξ)x + ξ ⊗ (id ⊗ d)x for all ξ ∈ E and x ∈ A ⊗ A[R4 ]. The family is said to be compatible with the Hermitian structure if it obeys ∇ξ | η + ξ | ∇η = (id ⊗ d) ξ | η for all ξ ∈ E and x ∈ A ⊗ A[R4 ]. It is clear that a family of connections parametrized by A = C (i.e. by a onepoint space) is just a connection in the usual sense. In general, a given family of Hermitian vector bundles E = P(A ⊗ A[R4 ])N always carries the family of Grassmann connections deﬁned by ∇0 = P ◦ (id ⊗ d). It follows that any family of connections can be written in the form ∇ = ∇0 + α, where α is a skew-adjoint element of EndA⊗A[R4 ] (E, E ⊗A⊗A[R4 ] (A ⊗ Ω1 (R4 ))) EndA⊗A[R4 ] (E, E ⊗A[R4 ] Ω1 (R4 )). Definition 3.4. Let E := P(A ⊗ A[R4 ])N be a family of Hermitian vector bundles parametrized by the algebra A. The gauge group of E is U(E) := {U ∈ EndA⊗A[R4 ] (E) | U ξ | U η = ξ | η for all ξ, η ∈ E}. We say that two families of compatible connections ∇1 , ∇2 on E are equivalent families and write ∇1 ∼ ∇2 if they are related by the action of the unitary group, i.e. there exists U ∈ U(E) such that ∇2 = U ∇1 U ∗ . More generally, if ∇1 and ∇2 are two families of connections on E parametrized by ∗-algebras A1 and A2 respectively, we say that ∇1 ∼ ∇2 if there exists a ∗-algebra

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B and ∗-algebra maps φ1 : A1 → B and φ2 : A2 → B such that φ∗1 ∇1 ∼ φ∗2 ∇2 in the above sense. Here φ∗i ∇i is the connection on E ⊗Ai B naturally induced by ∇i , where i = 1, 2 (cf. [5] for more details). In the case where A = C, i.e. for a family parametrized by a one-point space, the above equivalence relation reduces to the usual deﬁnition of gauge equivalence of connections. In the case where the families ∇1 , ∇2 are Grassmann families associated to projections P1 , P2 ∈ MN (A⊗A[R4 ]), equivalence means that P2 = U P1 U ∗ for some unitary U . Lemma 3.5. With E = P(A ⊗ A[R4 ])N , there exists PA ∈ MN (A) such that there is an algebra isomorphism EndA⊗A[R4 ] (E) EndA (PA (AN )) ⊗ A[R4 ] and hence an isomorphism U(E) U(EndA (PA (AN )) ⊗ A[R4 ]) of gauge groups. Proof. Since R4 is topologically trivial, there is an isomorphism of K-groups K0 (A ⊗ A[R4 ]) ∼ = K0 (A). It follows that for each projection P ∈ MN (A ⊗ A[R4 ]) there exists a projection PA ∈ MN (A) such that P and PA ⊗ 1 are equivalent projections in MN (A ⊗ A[R4 ]). This implies that there is an isomorphism E = P(A ⊗ A[R4 ])N PA (AN ) ⊗ A[R4 ]

(3.1)

of right A ⊗ A[R4 ]-modules, from which the result follows immediately. With these ideas in mind, ﬁnally we arrive at the following deﬁnition of a family of instantons over R4 parametrized by a (possibly noncommutative) ∗-algebra A. Definition 3.6. A family of instantons over R4 is a family of compatible connections ∇ over R4 whose curvature F := ∇2 obeys the anti-self-duality equation (id ⊗ ∗)F = −F, where ∗ is the Hodge operator on Ω2 (R4 ). Once again we stress that the “usual” deﬁnition of a family of instantons over R4 would require all members of the family to have ﬁnite Yang–Mills energy, in the sense that the A-valued integral YM(∇) := id ⊗ ((id ⊗ tr) F ∧ F ) (3.2) R4

should be well-deﬁned (i.e. ﬁnite). We shall comment on this requirement for noncommutative instantons later on in the paper. 4. The ADHM Construction Next we review the ADHM construction of instantons on the classical Euclidean four-plane R4 . We present the construction in a coordinate algebra format which

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is covariant under the coaction of a given Hopf algebra of symmetries, paving the way for a deformation by the cocycle twisting of Sec. 1.2. 4.1. The space of classical monads We begin by describing the input data for the ADHM construction of instantons. Although the ADHM construction is capable of constructing instanton bundles of arbitrary rank, in this paper we restrict our attention to the construction of vector bundles with rank two. Definition 4.1. Let k ∈ Z be a ﬁxed positive integer. A monad over A[C4 ] is a complex of free right A[C4 ]-modules, σ

τ

z z 0 → H ⊗ A[C4 ] −→ K ⊗ A[C4 ] −→ L ⊗ A[C4 ] → 0,

(4.1)

where H, K and L are complex vector spaces of dimensions k, 2k + 2 and k respectively, such that the maps σz , τz are linear in the generators z1 , . . . , z4 of A[C4 ]. The ﬁrst and last terms of the sequence are required to be exact, so that the only non-trivial cohomology is in the middle term. Given a monad (4.1), its cohomology E := Ker τz /Im σz is a ﬁnitely-generated projective right A[C4 ]-module and hence deﬁnes a vector bundle over C4 . In fact, since the maps σz , τz are assumed linear in the coordinate functions z1 , . . . , z4 , this vector bundle is well-deﬁned over the projective space CP3 [26]. With respect to ordered bases (u1 , . . . , uk ), (v1 , . . . , v2k+2 ) and (w1 , . . . , wk ) for the vector spaces H, K and L respectively, the maps σz and τz have the form j j σz : ub ⊗ Z → Mab ⊗ va ⊗ zj Z, τz : vc ⊗ Z → Ndc ⊗ wd ⊗ zj , (4.2) a,j

d,j

j j where Z ∈ A[C4 ] and the quantities M j := (Mab ) and N j := (Ndc ), j = 1, . . . , 4, are complex matrices with a, c = 1, . . . , 2k + 2 and b, d = 1, . . . , k. In more compact notation, σz and τz may be written σz = M j ⊗ zj , τz = N j ⊗ zj . (4.3) j

j

It is immediate from the formulæ (4.2) that the composition τz σz is given by j l τz σz : H ⊗ A[C4 ] → L ⊗ A[C4 ], ub ⊗ Z → Ndc Mcb ⊗ wd ⊗ zj zl Z, j,l,c,d

with respect to the bases (u1 , . . . , uk ) and (w1 , . . . , wk ) of H and L. It follows that the condition τz σz = 0 is equivalent to requiring that j j l l (Ndr Mrb + Ndr Mrb )=0 (4.4) r

for all j, l = 1, . . . , 4 and b, d = 1, . . . , k.

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j ∗ l ∗ We now introduce the conjugate matrix elements Mab and Ndc . Just as we did j ∗ j l for the matrices M and N above, we use the compact notation (M j † )ab = Mba l† l ∗ and (N )cd = Ndc . Given a monad (4.1), the corresponding conjugate monad is deﬁned to be ∗ τJ(z)

∗ σJ(z)

0 → L∗ ⊗ J(A(C4θ ))∗ −−−→ K∗ ⊗ J(A(C4θ ))∗ −−−→ H∗ ⊗ J(A(C4θ ))∗ ,

(4.5)

∗ ∗ and σJ(z) are the “adjoint” maps deﬁned by where τJ(z) ∗ ∗ σJ(z) = M j † ⊗ J(zj )∗ , τJ(z) = N j † ⊗ J(zj )∗ j

j

and J is the quaternionic involution deﬁned in Eq. (2.6) (cf. [4] for further explanation). If a given monad (4.1) is isomorphic to its conjugate (4.5) then we say that it is self-conjugate. A necessary and suﬃcient condition for a monad to be ∗ ∗ = −˜ σz and σ ˜J(z) = τ˜z , self-conjugate is that the maps σz and τz should obey τ˜J(z) j l equivalently that the matrices M , N should satisfy the reality conditions N 1 = M 2† ,

N 2 = −M 1† ,

N 3 = M 4† ,

N 4 = −M 3† .

(4.6)

This is for ﬁxed maps σz , τz . In dual terms, by allowing σz , τz to vary, we think j j , Ndc as coordinate functions on the space of all possible pairs of the elements Mab of A[C4 ]-module maps σz : H ⊗ A[C4 ] → K ⊗ A[C4 ],

τz : K ⊗ A[C4 ] → L ⊗ A[C4 ].

(4.7)

Imposing the conditions (4.4) and (4.6), we obtain coordinate functions on the space Mk of all self-conjugate monads with index k ∈ Z. Definition 4.2. We write A[Mk ] for the commutative ∗-algebra generated j j , Ndc subject to the relations (4.4) and the by the coordinate functions Mab ∗-structure (4.6). Remark 4.3. For each point x ∈ Mk there is an evaluation map x : A[Mk ] → C and the complex matrices (x ⊗ id)σz and (x ⊗ id)τz deﬁne a self-conjugate monad over C4 , (evx ⊗id)σz

(evx ⊗id)τz

0 → H ⊗ A[C4 ] −−−−−−−→ K ⊗ A[C4 ] −−−−−−−→ L ⊗ A[C4 ] → 0.

(4.8)

As already remarked, the cohomology E = Ker, τz /Im σz of a monad (4.1) deﬁnes a vector bundle E over CP3 ; the self-conjugacy condition (4.6) ensures that E arises via pull-back along the Penrose ﬁbration CP3 → S 4 . This means that the bundle E is trivial upon restriction to each of the ﬁbers of the Penrose ﬁbration and, in particular, to the ﬁber ∞ over the “point at inﬁnity”. As already mentioned, we wish to view our monads as being covariant under a certain coaction of a Hopf algebra H. Recall that A[C4 ] is already a left H-comodule

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algebra, with H-coaction deﬁned using the projection π : A[GL+ (2, H)] → H and the formula (2.16). It automatically follows that the free modules H ⊗ A[C4 ], K ⊗ A[C4 ] and L ⊗ A[C4 ] are also left H-comodules whose A[C4 ]-module structures are H-equivariant. It remains to address the requirement that the module maps σz and τz should be H-equivariant as well. Lemma 4.4. The maps σz := j M j ⊗ zj and τz := j N j ⊗ zj are H-comodule j l maps if and only if the coordinate functions Mab , Ndc carry the left H-coaction j j r r Mab → π(S(Arj )) ⊗ Mab , Ndc → π(S(Arj )) ⊗ Ndc (4.9) r

r

for each j = 1, . . . , 4 and a, c = 1, . . . , 2k + 2, b, d = 1, . . . , k. Proof. Upon inspection of Eq. (4.2) we see that σz cannot possibly be an intertwiner for the H-coactions on H ⊗ A[C4 ] and K ⊗ A[C4 ] unless we also allow for a coaction of H on the algebra A[Mk ] as well. Since the deﬁnition of σz depends j only upon the generators Mab , it is enough to check equivariance only on these generators. It is immediate that, for ﬁxed a, b, the four-dimensional H-comodule j , j = 1, . . . , 4, must be conjugate to the fourspanned by the generators Mab dimensional comodule spanned by the generators z1 , . . . , z4 , giving the coaction as stated. Indeed, we verify that M r ⊗ zr → π(S(Asr )Ars ) ⊗ M s ⊗ zs r

r,s

=

π((Ass )) ⊗ M s ⊗ zs

s

=

1 ⊗ M s ⊗ zs ,

s

as required. The same analysis applies to the map τz . By extending it as a ∗-algebra map, the formula (4.9) equips A[Mk ] with the structure of a left H-comodule ∗-algebra. This will be of paramount importance in later sections when we come to deform the ADHM construction. 4.2. The construction of instantons on R4 For self-conjugate monads, the maps of interest are the (2k + 2) × k algebra-valued matrices σz = M 1 ⊗ z1 + M 2 ⊗ z2 + M 3 ⊗ z3 + M 4 ⊗ z4 , σJ(z) = −M 1 ⊗ z2∗ + M 2 ⊗ z1∗ − M 3 ⊗ z4∗ + M 4 ⊗ z3∗ . ∗ In terms of these generators, the monad condition τz σz = 0 becomes σJ(z) σz = 0. By ∗ ∗ polarisation of this identity, one also ﬁnds that σJ(z) σJ(z) = σz σz . The identiﬁcation

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of the vector space K with its dual K∗ means that the module K ⊗ A[C4 ] acquires a bilinear form given by (Jξ)∗a ηa (4.10) (ξ, η) := Jξ | η = a

for ξ = (ξa ) and η = (ηa ) ∈ K ⊗ A[C ], where · | · is the canonical Hermitian ∗ σz = 0 implies that the columns structure on K ⊗ A[C4 ]. The monad condition σJ(z) of the matrix σz are orthogonal with respect to the form ( · , · ). Let us introduce the notation 4

∗ σJ(z) , ρ2 := σz∗ σz = σJ(z)

(4.11)

a k × k matrix with entries in the algebra A[Mk ] ⊗ A[C4 ]. In order to proceed, we need this matrix ρ2 to be invertible, although of course this is not the case in general. Thus we need to slightly enlarge the matrix algebra Mk (C)⊗ A[Mk ]⊗ A[C4 ] by adjoining an inverse element ρ−2 for ρ2 . Doing so is equivalent to deleting a collection of points from the parameter space Mk , corresponding to the so-called “instantons of zero-size” [13]. We henceforth assume that this has been done, although we do not change our notation. We collect together the matrices σz , σJ(z) into the (2k + 2) × 2k matrix V := (σz which by the deﬁnition of ρ2 obeys ∗

V V=ρ

2

σJ(z) ),

1k

0

0

1k

(4.12)

,

where 1k denotes the k × k identity matrix. We form the matrix ∗ Q := Vρ−2 V∗ = σz ρ−2 σz∗ + σJ(z) ρ−2 σJ(z)

(4.13)

and for convenience we denote Qz := σz ρ−2 σz∗ ,

∗ QJ(z) := σJ(z) ρ−2 σJ(z) .

(4.14)

Immediately we have the following result. Proposition 4.5. The quantity Q := Vρ−2 V∗ is a (2k + 2) × (2k + 2) projection, Q2 = Q = Q∗ , with entries in the algebra A[Mk ] ⊗ A[R4 ] and trace equal to 2k. Proof. That Q is a projection is a direct consequence of the fact that V∗ V = ρ2 . The matrices Qz and QJ(z) are also projections: in fact they are orthogonal projections, since Qz QJ(z) = 0. Moreover, both matrices Qz and QJ(z) have entries whose A[C4 ]-components have the form zj∗ zl for j, l = 1, . . . , 4. From the proof of Lemma 2.2, we know that we can rewrite each of these expressions in terms of generators of the algebra A[R4 ]⊗ A[CP1 ]. Since the matrix sum Q has entries which are J-invariant, it follows that the A[C4 ]-components of these entries must lie in

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the J-invariant subalgebra of A[R4 ] ⊗ A[CP1 ], which is just A[R4 ]. For the trace, we compute that (σz ρ−2 σz∗ )µµ = (σz )µr (ρ−2 )rs (σz∗ )sµ = (ρ−2 )rs (σz )µr (σz )∗sµ Tr Qz = µ

=

µ,r,s

(ρ−2 )rs (σz )∗sµ (σz )µr =

µ,r,s

µ,r,s

(ρ−2 )rs (σz∗ σz )sr = Tr 1k = k.

r,s

A similar computation establishes that QJ(z) also has trace equal to k, whence the trace of Q is 2k by linearity. From the projection Q, we construct the complementary projection P := 12k+2 − Q, also having entries in the algebra A[Mk ] ⊗ A[R4 ]. It is immediate that the trace of P is equal to two, so it follows that the ﬁnitely generated projective right A[Mk ] ⊗ A[R4 ]-module E := P(A[Mk ] ⊗ A[R4 ])2k+2 deﬁnes a family of rank two vector bundles over R4 parametrized by the space Mk of self-conjugate monads. We equip this family of vector bundles with the family of Grassmann connections ∇ := P ◦ (id ⊗ d), obtaining the following result. Proposition 4.6. The curvature F = P((id ⊗ d)P)2 of the family of Grassmann connections ∇ is anti-self-dual, that is to say (id ⊗ ∗)F = −F . Proof. By applying id ⊗ d to the relation ρ−2 ρ2 = 1k and using the Leibniz rule, one ﬁnds that (id ⊗ d)ρ−2 = −ρ−2 ((id ⊗ d)ρ2 )ρ−2 (this is a standard formula for calculating the derivative of a matrix-valued function). Using this, one ﬁnds that (id ⊗ d)(Vρ−2 V∗ ) = P((id ⊗ d)V)ρ−2 V∗ + Vρ−2 ((id ⊗ d)V∗ )P, and hence in turn that ((id ⊗ d)P) ∧ ((id ⊗ d)P) = P((id ⊗ d)V)ρ−2 ((id ⊗ d)V∗ )P + Vρ−2 ((id ⊗ d)V∗ )P((id ⊗ d)V)ρ−2 V∗ , where we have used the facts that ρ−2 V∗ P = 0 = PVρ−2 . The second term in the above expression is identically zero when acting on any element in the image E of P, whence the curvature F of the family ∇ works out to be F = P((id ⊗ d)P)2 = P((id ⊗ d)V)ρ−2 ((id ⊗ d)V∗ )P ∗ ))P. = P(((id ⊗ d)σz )ρ−2 ((id ⊗ d)σz∗ ) + ((id ⊗ d)σJ(z) )ρ−2 ((id ⊗ d)σJ(z)

It is clear by inspection that on twistor space A0 [CP3 ] this F is a horizontal twoform of type (1, 1) and it is known [1] that such a two-form is necessarily the pull-back of an anti-self-dual two-form on R4 .

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Thus we have reproduced the ADHM construction of instantons on R4 in our coordinate algebra framework: as usual, we must now address the question of the extent to which the construction depends on the choice of bases for the vector spaces H, K, L that we made in Sec. 4.1. It is clear that we are free to act on the A[C4 ]-module K ⊗ A[C4 ] by a unitary element of the matrix algebra M2k+2 (C) ⊗ A[C4 ]. In order to preserve the instanton construction, we must do so in a way which preserves the bilinear form (· , ·) of Eq. (4.10) determined by the identiﬁcation of K with its dual K∗ . It follows that the map σz in Eq. (4.2) is deﬁned up to a unitary transformation U ∈ EndA[C4 ] (K ⊗ A[C4 ]) which commutes with the quaternion structure J, namely, the elements of the group Sp(K ⊗ A[C4 ]) := {U ∈ EndA(C4θ ) (K ⊗ A[C4 ]) | U ξ | U ξ = ξ | ξ, J(U ξ) = U J(ξ)}. Similarly, we are free to change basis in the module H ⊗ A[C4 ], whence the map τz in Eq. (4.2) is deﬁned up to an invertible transformation W ∈ GL(H ⊗ A[C4 ]). Given U ∈ Sp(K ⊗ A[C4 ]) and W ∈ GL(H ⊗ A[C4 ]), the available freedom in the ADHM construction is to map σz → U σz W . Proposition 4.7. For all W ∈ GL(H) the projection P = 12k+2 − Q is invariant under the transformation σz → σz W . For all U ∈ Sp(K), under the transformation σz → U σz the projection P of transforms as P → U PU ∗ . Proof. One ﬁrst checks that ρ2 → (σz W )∗ (σz W ) = W ∗ ρ2 W , so that Qz → σz W (W ∗ ρ2 W )−1 W ∗ σz∗ = σz W (W −1 ρ−2 (W ∗ )−1 )W ∗ σz∗ = Qz , whence the projection P is unchanged. Replacing σz by U σz leaves ρ2 invariant (since U is unitary) and so has the eﬀect that Qz → U σz ρ−2 σz∗ U ∗ = U Qz U ∗ , whence it follows that P is mapped to U PU ∗ . In this way, such changes of module bases result in gauge equivalent families of instantons. However, from the point of view of constructing equivalence classes of connections it is in fact suﬃcient to consider the eﬀect of the subgroups of “constant” module automorphisms, i.e. those generated by changes of basis in the vector spaces H, K and L, described by the group Sp(K) = Sp(k + 1) ⊂ Sp(K ⊗ A[C4 ]) and the group GL(k, R) ⊂ GL(H ⊗ A[C4 ]) [2], although it is beyond our scope to prove this here. However, it is not diﬃcult to see that the algebra A[Mk ] has a total of 4k(2k +2) generators and 5k(k − 1) constraints (determined by the orthogonality relations ∗ σz = 0); the Sp(k + 1) symmetries impose a further (k + 1)(2(k + 1) + 1) σJ(z)

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constraints and the GL(k, R) a further k 2 constraints. This elementary argument yields that the construction has (8k 2 + 8k) − 5k(k − 1) − (3k 2 + 5k + 3) = 8k − 3 degrees of freedom, in precise agreement with the dimension of the moduli space computed in [3]. Definition 4.8. We say that a pair of self-conjugate monads are equivalent if they are related by a change of bases of the vector spaces H, K, L of the above form, i.e. by a pair of linear transformations U ∈ Sp(k + 1) and W ∈ GL(k, R). We denote by ∼ the resulting equivalence relation on the space Mk of self-conjugate monads. 5. The Moyal–Groenewold Noncommutative Plane R4 The Moyal noncommutative space-time R4 is arguably one of the best known and most widely-studied examples of a noncommutative space. In this section we analyze the construction of instantons on this space from the point of view of cocycle twisting. In this section we show how to deform Euclidean space-time R4 and its associated geometric structure into that of the Moyal–Groenewold space-time, then we look at what happens to the ADHM construction of instantons under the deformation procedure. 5.1. A Moyal-deformed family of monads In order to deform the ADHM construction of instantons, we need to choose a Hopf algebra H of symmetries together with a two-cocycle F by which to perform the twisting. For our twisting Hopf algebra we take H = A[R4 ], the algebra of coordinate functions on the additive group R4 . It is the commutative unital ∗-algebra A[R4 ] = A[tj , t∗j | j = 1, 2]

(5.1)

equipped with the Hopf algebra structure ∆(tj ) = 1 ⊗ tj + tj ⊗ 1,

(tj ) = 0,

S(tj ) = −tj ,

(5.2)

with ∆, extended as ∗-algebra maps and S extended as a ∗-anti-algebra map. In order to deform the twistor ﬁbration, we have to equip our various algebras with left H-comodule algebra structures, which we achieve using the discussion of Sec. 2.3. There is a Hopf algebra projection from A[GL+ (2, H)] onto H, deﬁned on generators by     α1 −α∗2 β1 −β2∗ 1 0 0 0     α∗1 β2 β1∗  α2  0 1 0 0 +     π : A[GL (2, H)] → H,   →  t∗ t∗ 1 0 (5.3)  γ1 −γ2∗ δ1 −δ2∗    1 2 ∗ ∗ −t2 t1 0 1 γ2 γ1 δ2 δ1

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and extended as a ∗-algebra map. Using Eq. (2.16), this projection determines a left H-coaction ∆π : A[C4 ] → H ⊗ A[C4 ] by z1 → 1 ⊗ z1 ,

z3 → t∗1 ⊗ z1 + t∗2 ⊗ z2 + 1 ⊗ z3 ,

(5.4)

z2 → 1 ⊗ z2 ,

z4 → −t2 ⊗ z1 + t1 ⊗ z2 + 1 ⊗ z4 ,

(5.5)

extended as a ∗-algebra map. Using the identiﬁcation of generators (2.10), the coordinate algebra A[R4 ] of Euclidean space therefore carries the coaction A[R4 ] → H ⊗ A[R4 ],

ζ1 → 1 ⊗ ζ1 + t1 ⊗ 1,

ζ2 → 1 ⊗ ζ2 + t2 ⊗ 1,

(5.6)

making A[R4 ] into a left H-comodule ∗-algebra. Let (∂j l ), j, l = 1, 2, be the Lie algebra of translation generators dual to H. Writing

t∗1 t∗2 s τ := (τr ) = , r, s = 1, 2, −t2 t1 this means that there is a non-degenerate pairing

∂j l , τr s = δjs δrl ,

j, l, r, s = 1, 2,

which extends to an action on products of the generators tj , t∗l by diﬀerentiation and evaluation at zero. Using this pairing, we deﬁne a twisting two-cocycle by 1 iΘr r s s ∂r r ⊗ ∂s s , h ⊗ g F : H ⊗ H → C, F (h, g) = exp 2

for h, g ∈ H, where Θ = (Θr r s s ), r, r , s, s = 1, 2, is a real 4 × 4 anti-symmetric matrix with rows rr and columns ss , which we choose to have the canonical form   0 0 0 α    0 0 β 0  Θ =   0 −β 0 0    −α 0 0 0 for non-zero real constants α, β and > 0 a deformation parameter. We assume for simplicity that α + β = 0. This F is multiplicative (i.e. it is a Hopf bicharacter in the sense of Eq. (1.3)) and so it is determined by its values on the generators (τr s ), r, s = 1, 2. One computes in particular that F (t∗1 , t1 ) =

1 αi, 2

1 F (t∗2 , t2 ) = − βi, 2

with F evaluating as zero on all other pairs of generators. From the formulæ (1.9), (1.10) and (1.12), one immediately ﬁnds that H = HF as a Hopf ∗-algebra. However,

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the eﬀect of the twisting on the H-comodule algebras A[C4 ] and A[R4 ] is not trivial, as shown by the following lemmata. Lemma 5.1. The algebra relations in the H-comodule algebra A[C4 ] are twisted into [z3 , z4 ] = i(α + β)z1 z2 ,

[z3∗ , z4∗ ] = i(α + β)z1∗ z2∗ ,

(5.7)

[z3 , z3∗ ] = iαz1 z1∗ − iβz2 z2∗ ,

[z4 , z4∗ ] = iβz1 z1∗ − iαz2 z2∗ ,

(5.8)

with all other relations left unchanged. In particular, the generators z1 , z2 and their conjugates remain central in the deformed algebra. Proof. The cocycle-twisted product on the H-comodule algebra A[C4 ] is deﬁned by the formula (1.14); the corresponding algebra relations can be expressed using the “universal R-matrix” of Eq. (1.13), namely zj ·F zl = R(zl (−1) , zj (−1) )zl (0) ·F zj (0) , zj ·F zl∗ = R(zl∗ (−1) , zj (−1) )zl∗ (0) ·F zj (0) .

(5.9)

One ﬁnds in particular that the R-matrix has the values R(t∗1 , t1 ) = 2F −1 (t∗1 , t1 ) = −iα,

R(t∗2 , t2 ) = 2F −1 (t∗2 , t2 ) = iβ,

(5.10)

and gives zero when evaluated on all other pairs of generators. By explicitly computing Eqs. (5.9) (and omitting the product symbol ·F ), one ﬁnds the relations as stated in the lemma. We denote by A[C4 ] the ∗-algebra generated by {zj , zj∗ | j = 1, . . . , 4} modulo the relations (5.7) and (5.8). This makes A[C4 ] into a left HF -comodule ∗-algebra. Lemma 5.2. The algebra relations in the H-comodule algebra A[R4 ] are twisted into [ζ1∗ , ζ1 ] = iα,

[ζ2∗ , ζ2 ] = −iβ,

j, l = 1, 2,

(5.11)

with vanishing commutators between all other pairs of generators. Proof. The product in A[R4 ] is twisted using the formula (1.14). Once again omitting the product symbol ·F , the corresponding algebra relations are computed to be those as stated. We denote by A[R4 ] the algebra generated by ζ1 , ζ2 , ζ1∗ , ζ2∗ , modulo the relations (5.11). This makes A[R4 ] into a left HF -comodule ∗-algebra. Remark 5.3. Since the generators z1 , z2 and their conjugates z1∗ , z2∗ remain central in the algebra A[C4 ], we see immediately from Lemma 2.2 that the localized twistor algebra has the form A[R4 ] ⊗ A[CP1 ]. Only the base R4 of the twistor ﬁbration is deformed; the typical ﬁber CP1 remains classical.

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The canonical diﬀerential calculi described in Sec. 3.1 are also deformed using this cocycle twisting procedure. The relations in the deformed calculi are given in the following lemmata. Lemma 5.4. The twisted diﬀerential calculus Ω(C4 ) is generated by the degree zero elements zj , zl∗ and the degree one elements dzj , dzl∗ for j, l = 1, . . . , 4, subject to the bimodule relations between functions and one-forms [z3 , dz4 ] = i(α + β)z1 dz2 ,

[z3∗ , dz4∗ ] = i(α + β)z1∗ dz2∗ ,

[z4 , dz3 ] = −i(α + β)z2 dz1 ,

[z4∗ , dz3∗ ] = −i(α + β)z2∗ dz1∗ ,

[z3 , dz3∗ ] = iαz1 dz1∗ − iβz2 dz2∗ ,

[z4 , dz4∗ ] = iβz1 dz1∗ − iαz2 dz2∗

and the anti-commutation relations between one-forms {dz3 , dz4 } = i(α + β)dz1 dz2 ,

{dz3∗ , dz4∗ } = i(α + β)dz1∗ dz2∗ ,

{dz3 , dz3∗ } = iαdz1 dz1∗ − iβdz2 dz2∗ ,

{dz4 , dz4∗ } = iβdz1 dz1∗ − iαdz2 dz2∗ ,

with all other relations undeformed. Proof. One views the classical calculus Ω(C4 ) as a left H-comodule algebra and accordingly computes the deformed product using the twisting cocycle F . Since the exterior derivative d commutes with the H-coaction (5.4), it is straightforward to observe that the (anti-)commutation relations in the deformed calculus Ω(C4 ) are just the same as the algebra relations in A[C4 ] but with d inserted appropriately. Lemma 5.5. The twisted diﬀerential calculus Ω(R4 ) is generated by the degree zero elements ζ1 , ζ1∗ , ζ2 , ζ2∗ and the degree one elements dζ1 , dζ1∗ , dζ2 , dζ2∗ . The relations in the calculus are not deformed. Proof. Once again, the classical calculus Ω(R4 ) is deformed as a twisted left Hcomodule algebra. Although the products of functions and diﬀerential forms in the calculus are indeed twisted, one ﬁnds that the extra terms which appear in the twisted product all vanish in the expressions for the (anti-)commutators (cf. [6] for full details). In particular, we see that the vector space Ω2 (R4 ) is the same as it is classically. Since the coaction of H on A[R4 ] is by isometries, the Hodge ∗-operator ∗ : Ω2 (R4 ) → Ω2 (R4 ) commutes with the H-coaction in the sense that ∆π (∗ω) = (id ⊗ ∗)∆π (ω),

ω ∈ Ω2 (R4 ),

so there is also a Hodge operator ∗ : Ω2 (R4 ) → Ω2 (R4 ) deﬁned by the same formula as in the classical case. In particular, this means that the decomposition of Ω2 (R4 ) into self-dual and anti-self-dual two-forms, Ω2 (R4 ) = Ω2+ (R4 ) ⊕ Ω2− (R4 ),

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is identical at the level of vector spaces to the corresponding decomposition in the classical case. The above lemmata are really just special cases of the cocycle twisting procedure: recall that in fact our “quantization map” applies to every suitable H-covariant construction. Since the algebra A[Mk ] is a left H-comodule algebra according to Lemma 4.4, it is also deformed by the quantization. We write A[Mk; ] for the resulting cocycle-twisted HF -comodule algebra. Proposition 5.6. The coordinate ∗-algebra A[Mk; ] is generated by the matrix j l , Ndc for a, c = 1, . . . , k and b, d = 1, . . . , 2k + 2, modulo the relations elements Mab 1 2 3 4 , Mrs ] = i(α − β)Mab Mrs , [Mab 1 ∗ 2 ∗ 3 ∗ 4 ∗ [Mab , Mrs ] = i(α − β)Mab Mrs ,

1 1 ∗ 3 3 ∗ 4 4 ∗ [Mab , Mrs ] = iαMab Mrs + iβMab Mrs , 2 2 ∗ 3 3 ∗ 4 4 ∗ [Mab , Mrs ] = −iβMab Mrs − iαMab Mrs

and the ∗-structure (4.6). The generators M 3 , M 4 , M 3∗ , M 4∗ are central in the algebra. Proof. From Lemma 4.4 we read oﬀ the H-coaction on generators M j , j = 1, . . . , 4, obtaining M 1 → 1 ⊗ M 1 − t∗1 ⊗ M 3 + t2 ⊗ M 4 , M 3 → 1 ⊗ M 3 , M 2 → 1 ⊗ M 2 − t∗2 ⊗ M 3 − t1 ⊗ M 4 , M 4 → 1 ⊗ M 4 , which we extend as a ∗-algebra map. The deformed relations follow immediately from an application of the twisting formula (1.14). The coaction of H on A[Mk ] does not depend on the matrix indices of the generators M j , N l , j, l = 1, . . . , 4, hence neither do the twisted commutation relations. Similar computations yield the other commutation relations as stated. In terms of the deformed product, the relations (4.4) are twisted into the relations j j l l Ndr Mrb + Ndr Mrb + i(α + β)(δ j1 δ l2 − δ j2 δ l1 ) = 0 r

for each b, d = 1, . . . , k, where δ rs is the Kronecker delta symbol. We think of A[Mk; ] as the coordinate algebra of a noncommutative space Mk; of monads on C4 . Although we do not have as many evaluation maps on A[Mk; ] as we did in the classical case, we can nevertheless work with the whole family Mk; at once. 5.2. The construction of instantons on R4 Using the noncommutative space of monads Mk; , we proceed as in Sec. 4.2 to construct families of instantons, but now on the Moyal space R4 . 4 denote the Pontryagin dual to the additive group R4 used in Eq. (5.1). Let R 4 we deﬁne unitary elements Given a pair of complex numbers c := (c1 , c2 ) ∈ C2 R

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u = (u1 , u2 ) of the algebra HF by u1 = exp(i(c1 t1 + c∗1 t∗1 )),

u2 = exp(i(c2 t2 + c∗2 t∗2 )).

(5.12)

It is straightforward to check that u1 , u2 are group-like elements of (the smooth completion of) the Hopf algebra HF , i.e. they transform as ∆(uj ) = uj ⊗ uj under the coproduct ∆ : HF → HF ⊗ HF . Lemma 5.7. There is a canonical left action of HF on the algebra A[Mk; ] given by u1 M 1 = M 1 − αc1 M 3 ,

u1 M 2 = M 2 + αc∗1 M 4 ,

u1 M 1∗ = M 1∗ − αc∗1 M 3∗ ,

u1 M 2∗ = M 2∗ + αc1 M 4∗ ,

u2 M 1 = M 1 + βc∗2 M 4 ,

u2 M 2 = M 2 + βc2 M 3 ,

u2 M 1∗ = M 1∗ + βc2 M 4∗ ,

u2 M 2∗ = M 2 + βc∗2 M 3∗ ,

with uj M l = M l and uj M l∗ = M l∗ for l = 3, 4. Proof. Recall from the proof of Proposition 5.6 that A[Mk; ] is a left HF -comodule algebra: it is therefore also a left HF -module algebra according to the formula (1.5). Evaluating the R-matrix by expanding the exponentials as power series, one ﬁnds that R(t1 , u1 ) = R(t1 , ic∗1 t∗1 ) = −αc∗1 , R(t∗1 , u1 ) = R(t∗1 , ic1 t1 ) = αc1 , R(t2 , u2 ) = R(t1 , ic∗2 t∗2 ) = −βc∗2 ,

R(t∗2 , u2 ) = R(t∗2 , ic2 t2 ) = βc2 ,

with all other combinations evaluating as zero. Using the fact that the unitaries uj are group-like elements of the Hopf algebra HF , one ﬁnds the actions to be as stated. In turn, there is an inﬁnitesimal version of the HF -action on A[Mk; ], given by t1 M 1 = iαM 3 ,

t∗1 M 1∗ = −iαM 3∗ ,

t∗1 M 2 = −iαM 4 ,

t1 M 2∗ = iαM 4∗ ,

t∗2 M 1 = −iβM 4 ,

t2 M 1∗ = iβM 4∗ ,

t2 M 2 = −iβM 3 ,

t∗2 M 2∗ = iβM 3∗ ,

with tj M l = 0 and tj M l∗ = 0 for all other possible combinations of generators. Either way, we obtain a group action 4 → Aut A[Mk; ] γ:R

(5.13)

4 on the coordinate algebra A[Mk; ] by ∗-automorphisms. of the Pontryagin dual R We also form the smash product algebra A[Mk; ] > HF associated to the above HF action, whose multiplication is deﬁned by the formula (1.7). With the coproduct

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∆(tj ) = 1 ⊗ tj + tj ⊗ 1 on HF , we ﬁnd in particular the formula j j j l l l ⊗ tr )(Mcd ⊗ ts ) = Mab Mcd ⊗ tr ts + Mab (tr Mcd ) ⊗ ts , (Mab

with similar expressions for products involving the conjugate generators M j ∗ . The corresponding algebra relations between such elements are given by j l ⊗ tr , Mcd ⊗ ts ] [Mab j j j l l l , Mcd ] ⊗ tr ts + Mab (tr Mcd ) ⊗ ts − Mcd (ts Mab ) ⊗ tr = [Mab

(5.14)

for j, l = 1, . . . , 4 and r, s = 1, 2, with similar formulæ occurring when the generators M j and tr are replaced by their conjugates. Of course, these relations are just a small part of the full algebra structure in the smash product A[Mk; ] > HF , but those in Eq. (5.14) are the ones that we will need later on in the paper. Remark 5.8. This situation is a special case of Example 1.1. Recall that, upon making suitable completions of our algebras, we think of the smash product algebra A[Mk; ] > HF = A[Mk; ] > A[R4 ] 4 determined as an algebraic version of the crossed product algebra A[Mk; ] >γ R by the action (5.13). Thanks to the functorial nature of the cocycle twisting, mutatis mutandis the ADHM construction goes through as described in Sec. 4.2. In the following, we highlight the main diﬀerences which arise as a consequence of the quantization procedure. The next lemma takes care of an important technical point: as well as twisting the relations in the algebras A[Mk ] and A[C4 ], we also have to deform the cross-relations in the tensor product algebra A[Mk ] ⊗ A[C4 ]. Lemma 5.9. The algebra structure of the twisted tensor product algebra A[Mk; ] ⊗ A[C4 ] is given by the relations in the respective subalgebras A[Mk; ] and A[C4 ] determined above, together with the cross-relations z3 M 1 = M 1 z3 − iβM 4 z2 ,

z3 M 2 = M 2 z3 − iαM 4 z1 ,

z4 M 1 = M 1 z4 + iαM 3 z2 ,

z4 M 2 = M 2 z4 + iβM 3 z1 ,

z3∗ M 1 = M 1 z3∗ + iαM 3 z1∗ ,

z3∗ M 2 = M 2 z3∗ − iβM 3 z2∗ ,

z4∗ M 1 = M 1 z4∗ + iβM 4 z1∗ ,

z4∗ M 2 = M 2 z4∗ − iαM 4 z2∗

and their conjugates. The generators z1 , z2 , M 3 , M 4 are central. Proof. The classical algebra A[Mk ] ⊗ A[C4 ] is a left comodule ∗-algebra under the tensor product HF -coaction deﬁned by Eq. (1.4). The twisted product is determined by the formula (1.14), with the non-trivial cross-terms in the deformed algebra being the ones stated in the lemma. We denote the deformed algebra by A[Mk; ] ⊗ A[C4 ], with the symbol ⊗ to remind us that the tensor product algebra structure is not the usual one, but has been twisted as well.

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Just as in the classical situation, we have a pair of matrices σz and τz , σz = M j ⊗ zj , τz = N j ⊗ zj , j

j

whose entries this time live in the twisted algebra A[Mk; ] ⊗ A[C4 ]. The resulting matrix V := (σz σJ(z) ) is a 2k×(2k+2) matrix with entries in A[Mk; ] ⊗ A[C4 ]. We set ρ2 := V∗ V. From the projection Q := Vρ−2 V∗ we construct the complementary matrix P := 12k+2 − Q, which has entries in the algebra A[Mk; ] ⊗ A[R4 ]. It is clear that this matrix P is a self-adjoint idempotent, P2 = P = P∗ , but it does not deﬁne an honest family of projections in the sense of Deﬁnition 3.2. Recall that, to deﬁne such a family, we need a matrix with entries in an algebra of the form A ⊗ A[R4 ] for some “parameter algebra” A, whereas the quantization procedure has produced a projection Q with entries in a twisted tensor product A[Mk; ] ⊗ A[R4 ]. We may nevertheless recover a genuine family of projections using the following lemma, in which we use the Sweedler notation Z → Z (−1) ⊗ Z (0) for the left coaction A[C4 ] → HF ⊗ A[C4 ] deﬁned in Eqs. (5.4) and (5.5). Lemma 5.10. There is a canonical ∗-algebra map µ : A[Mk; ] ⊗ A[C4 ] → (A[Mk; ] > HF ) ⊗ A[C4 ] deﬁned by µ(M ⊗ Z) = M ⊗ Z (−1) ⊗ Z (0) for each M ∈ A[Mk; ] and Z ∈ A[C4 ]. Proof. This follows from a straightforward veriﬁcation. One checks that µ(M ⊗ Z)µ(M ⊗ Z ) = (M ⊗ Z (−1) ⊗ Z (0) )(M ⊗ Z (−1) ⊗ Z (0) ) = M (Z (−1) (1) M ) ⊗ Z (−1) (2) Z (−1) ⊗ Z (0) Z (0) = R(M (−1) , Z (−1) (1) )M M (0) ⊗ Z (−1) (2) Z (−1) ⊗ Z (0) Z (0) = R(M (−1) , Z (−1) )M M (0) ⊗ Z(Z (0)(−1) )Z (−1) ⊗ (Z (0)(0) )Z (0) = µ(R(M (−1) , Z (−1) )M M (0) ⊗ Z (0) Z ) = µ((M ⊗ Z)(M ⊗ Z )), so that µ is an algebra map, as well as (µ(M ⊗ Z))∗ = (M ⊗ Z (−1) ⊗ Z (0) )∗ = (M ⊗ Z (−1) )∗ ⊗ Z (0)∗ = R(M (−1)∗ , (Z (−1) (1) )∗ )(M (0)∗ ⊗ (Z (−1) (2) )∗ ) ⊗ Z (0)∗ = µ(R(M (−1)∗ , Z (−1)∗ )(M (0)∗ ⊗ Z (0)∗ )) = µ((M ⊗ Z)∗ ), so that µ respects the ∗-structures as well.

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Remark 5.11. Lemma 5.10 is an example of Majid’s “bosonization” construction [22], which converts noncommutative “braid statistics” (in our case described by the twisted tensor product ⊗ ) into commutative “ordinary statistics” (described by the usual tensor product ⊗). As a consequence of Lemma 5.10, we ﬁnd that there are maps σ ˜z : H ⊗ A[C4 ] → (A[Mk; ] > HF ) ⊗ K ⊗ A[C4 ],

(5.15)

τ˜z : K ⊗ A[C4 ] → (A[Mk; ] > HF ) ⊗ L ⊗ A[C4 ],

(5.16)

deﬁned by composing σz and τz with the map µ. Explicitly, these maps are given by σ ˜z := M r ⊗ zr (−1) ⊗ zr (0) , τ˜z := N r ⊗ zr (−1) ⊗ zr (0) , r

r

which are, respectively, k × (2k + 2) and (2k + 2) × k matrices with entries in the noncommutative algebra (A[Mk; ] > HF ) ⊗ A[C4 ]. With this in mind, we form the := (˜ ˜J(z) ). (2k + 2) × 2k matrix V σz σ ∗ is a projection, = V ρ−2 V Proposition 5.12. The (2k + 2) × (2k + 2) matrix Q 2 ∗ Q = Q = Q , with entries in the algebra (A[Mk; ] > HF ) ⊗ A[C4 ] and trace equal to 2k. is a projection follows from the fact that Q is a proProof. The fact that Q jection and that µ : A[Mk; ] ⊗ A[C4 ] → (A[Mk; ] > HF ) ⊗ A[C4 ] is a ∗-algebra map. By construction, the entries of the matrix ρ2 are central in the algebra (A[Mk; ] > HF ) ⊗ A[C4 ] (this follows from the fact that its matrix entries are coinvariant under the left HF -coaction), from which it follows that the trace computation in Proposition 4.5 is valid in the noncommutative case as well [5]. we construct as before the complementary projection From the projection Q := 12k+2 − Q. It has entries in the algebra (A[Mk; ] > HF ) ⊗ A[R4 ] and has P trace equal to two. In analogy with Deﬁnition 3.2, the ﬁnitely generated projective module E := P((A[Mk; ] > HF ) ⊗ A[R4 ])2k+2 deﬁnes a family of rank two vector bundles over R4 parametrized by the noncommutative algebra A[Mk; ] > HF . We equip this family of vector bundles with the family of Grassmann connections associated to the projection P. Proposition 5.13. The curvature F = P((id ⊗ d)P)2 of the Grassmann family of connections ∇ := (id ⊗ d) ◦ P is anti-self-dual. Proof. From Lemma 5.5, we know that the space of two-forms Ω2 (R4 ) and the Hodge ∗-operator ∗ : Ω2 (R4 ) → Ω2 (R4 ) are undeformed and equal to their classical counterparts; similarly for the decomposition Ω2 (R4 ) = Ω2+ (R4 ) ⊕ Ω2− (R4 ) into

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self-dual and anti-self-dual two-forms. This identiﬁcation of the “quantum” with the “classical” spaces of two-forms survives the tensoring with the parameter space A[Mk; ] > HF , which yields that (A[Mk; ] > HF ) ⊗ Ω2± (R4 ) and (A[Mk ] ⊗ H) ⊗ Ω2± (R4 ) are isomorphic as vector spaces. Computing the curvature F in exactly the same way as in Proposition 4.6, we see that it must be anti-self-dual, since the same is true in the classical case. This Grassmann family of connections is a noncommutative deformation of the family of instantons with topological charge k constructed by the classical ADHM construction. In Proposition 5.13 we only show that the deformed ADHM construction produces famiilies of connections with anti-self-dual curvature (which are instantons in the sense of Deﬁnition 3.6) and make no claim about whether or not they have ﬁnite energy. Since it is mathematically quite diﬃcult to make sense of the ﬁnite energy condition (3.2) for instantons on the Moyal plane R4 and of its conformal compactiﬁcation, we choose to avoid the subject altogether and refer instead to [24], for example. 5.3. The Moyal-deformed ADHM equations The noncommutative ADHM construction of the previous section produced families of instantons on R4 parametrized by the noncommutative algebra A[Mk; ] > HF . We interpret the latter as an algebra of coordinate functions on some underlying “quantum” parameter space, within which we shall seek a space of classical parameters. To this end, we introduce elements of A[Mk; ] > HF deﬁned by 1 4 1 := M 1 ⊗ 1 + M 3 ⊗ 1 t∗1 − Mab 3 := M 3 ⊗ 1, M ⊗ t2 , M ab ab ab ab ab 2 2 2 := M 2 ⊗ 1 + M 3 ⊗ 1 t∗ + M 4 ⊗ 1 t1 , M 4 := M 4 ⊗ 1 M ab ab ab ab ab ab 2 2 2 j ∗ , for each a = 1, . . . , k and b = 1, . . . , 2k + 2, together with their conjugates M ab j = 1, . . . , 4. Definition 5.14. We write A[M(k; )] for the subalgebra of A[Mk; ] > HF generl ∗ , j, l = 1, . . . , 4. j , M ated by the elements M dc ab Proposition 5.15. The algebra A[M(k; )] is a commutative ∗-subalgebra of the smash product A[Mk; ] > HF . 4 are 3 and M Proof. This follows from direct computation. The generators M clearly central. On the other hand, we also have 2 = M 1 M 2 ⊗ 1 + M 1 M 3 ⊗ 1 t∗2 + M 1 M 4 ⊗ 1 t1 + M 3 M 2 ⊗ 1 t∗1 1 M M 2 2 2 1 1 1 − M 4 M 2 ⊗ t2 − iαM 3 M 4 ⊗ 1 + iβM 4 M 3 ⊗ 1, 2 2 2

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2 M 1 = M 2 M 1 ⊗ 1 + M 3 M 1 ⊗ 1 t∗ + M 4 M 1 ⊗ 1 t1 + M 2 M 3 ⊗ 1 t∗ M 2 2 2 2 1 1 1 1 − M 2 M 4 ⊗ t2 + iαM 4 M 3 ⊗ 1 − iβM 3 M 4 ⊗ 1, 2 2 2 from which it follows that the commutator is given by 2 ] = [M 1 , M 2 ] ⊗ 1 − i(α − β)M 3 M 4 ⊗ 1 = 0. 1 , M [M All other commutators are shown to vanish in the same way. Although we have made a change of generators, this does not aﬀect the family of instantons constructed in the previous section. In order to show this, let : HF ⊗ A[M(k; )] → A[M(k; )] be the left action of HF on A[M(k; )] deﬁned on generators by 1 = iαM 3 , t1 M

1∗ = −iαM 3∗ , t∗1 M

2 = −iαM 4 , t∗1 M

2∗ = iαM 4∗ , t1 M

1 = −iβ M 4 , t∗2 M

1∗ = iβ M 4∗ , t2 M

2 = −iβ M 3 , t2 M

2∗ = iβ M 3∗ , t∗2 M

l = 0 and tj M l∗ = 0 for l = 3, 4. Let us write together with tj M A[M(k; )] > HF for the smash product algebra associated to the action . Theorem 5.16. There is a ∗-algebra isomorphism φ : A[M(k; )] > HF A[Mk; ] > HF deﬁned for each h ∈ HF by 1 ⊗ h → M 1 ⊗ h + M 3 ⊗ 1 t∗ h − M 4 ⊗ 1 t2 h, M ab ab ab ab 2 1 2

3 ⊗ h → M 3 ⊗ h, M ab ab

1 4 2 ⊗ h → M 2 ⊗ h + M 3 ⊗ 1 t∗2 h − Mab M ⊗ t1 h, ab ab ab 2 2

4 ⊗ h → M 4 ⊗ h M ab ab

→

and extended as a ∗-algebra map. Proof. It is clear that this map is an isomorphism of vector spaces with inverse 1 4 1 3 1 ⊗ h − M 3 ⊗ 1 t∗1 h + M ab 3 ⊗ h, Mab ⊗ h → M ⊗ t2 h, Mab ⊗ h → M ab ab ab 2 2 2 2 ⊗ h − M 3 ⊗ 1 t∗ h − M 4 ⊗ 1 t1 h, M 4 ⊗ h → M 4 ⊗ h. ⊗ h → M Mab 2 ab ab ab ab ab 2 2

By deﬁnition, the map φ is a ∗-algebra homomorphism on the subalgebra A[M(k; )], so we just have to check that it preserves the cross-relations between

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A[M(k; )] and the subalgebra HF . This is straightforward to verify: one has for example that 1 ∗ 1 1 1 3 4 φ(1 ⊗ t1 )φ(M ⊗ 1) = (1 ⊗ t1 ) Mab ⊗ 1 + Mab ⊗ t1 − Mab ⊗ t2 2 2 1 1 1 3 4 ⊗ t1 + Mab ⊗ t∗1 t1 − Mab ⊗ t2 t1 + (t1 M 1 ) ⊗ 1 = Mab 2 2 1 ) ⊗ 1) 1 ⊗ t1 + (t1 M = φ(M 1 ⊗ 1)). = φ((1 ⊗ t1 )(M The remaining relations are veriﬁed in the same way. Our goal is now to see that the parameters corresponding to the subalgebra HF can be removed and that there is a family of instantons parametrized by the commutative algebra A[M(k; )]. This follows from the fact that there is a right coaction δR : A[M(k; )] > HF → (A[M(k; )] > HF ) ⊗ HF ,

δR := id ⊗ ∆,

(5.17)

where ∆ : HF → HF ⊗ HF is the coproduct on the Hopf algebra HF . This coaction ⊗ 1 and δR (P) is by “gauge transformations”, in the sense that the projections P are unitarily equivalent in the matrix algebra M2k+2 ((A[M(k; )] > HF ) ⊗ HF ) and so they deﬁne gauge equivalent families of instantons [5]. This means that the parameters determined by the subalgebra HF in A[M(k; )] > HF are just gauge parameters and so they may be removed. Indeed, by passing to the subalgebra of A[M(k; )] > HF consisting of coinvariant elements under the coaction (5.17), viz. A[M(k; )] ∼ = {a ∈ A[M(k; )] > HF | δR (a) = a ⊗ 1}, we obtain a projection Pk; with entries in A[M(k; )] ⊗ A[R4 ]. The precise construction of the projection Pk; goes exactly as in [5], as does the proof of the fact that the Grassmann family of connections ∇ = Pk; ◦ (id ⊗ d) has anti-self-dual curvature and hence deﬁnes a family of instantons on R4 . The commutative algebra A[M(k; )] is the algebra of coordinate funtions on a classical space of monads M(k; ). For each point x ∈ M(k; ) there is an evaluation map evx ⊗ id : A[M(k; )] ⊗ A[C4 ] → A[C4 ], which in turn deﬁnes a self-conjugate monad over the noncommutative space C4 . σz and (evx ⊗ id) τz determine In analogy with Remark 4.3, the matrices (evx ⊗ id) a complex of free right A[C4 ]-modules (evx ⊗id)˜ σz

(evx ⊗id)˜ τz

0 → H ⊗ A[C4 ] −−−−−−−→ K ⊗ A[C4 ] −−−−−−−→ L ⊗ A[C4 ] → 0.

(5.18)

The same evaluation map determines a projection (evx ⊗ id)Pk; and hence an instanton connection on R4 .

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As described in Proposition 4.7, the gauge freedom in the classical ADHM construction is determined by the choice of bases of the vector spaces H, K, L. Clearly we also have this freedom in the noncommutative construction as well: we write ∼ for the equivalence relation induced on the space M(k; ) by such changes of basis (cf. Deﬁnition 4.8). This leads to the following explicit description of the parameter space M(k; ) (cf. [17]). Theorem 5.17. For each positive integer k ∈ Z, the space M(k; )/∼ of equivalence classes of self-conjugate monads over C4 is the quotient of the set of complex matrices B1 , B2 ∈ Mk (C), I ∈ M2×k (C), J ∈ Mk×2 (C) satisfying the equations (i) [B1 , B2 ] + IJ = 0, (ii) [B1 , B1∗ ] + [B2 , B2∗ ] + II ∗ − J ∗ J = −i(α + β)1k by the action of U(k) given by B1 → gB1 g −1 ,

B2 → gB2 g −1 ,

I → gI,

J → Jg −1

for each g ∈ U(k). Proof. Recall that we write the monad maps σ ˜z , τ˜z as 1 z1 + M 2 z2 + M 3 z3 + M 4 z4 , σ ˜z = M

1 z1 + N 2 z2 + N 3 z3 + N 4 z4 τ˜z = N

j , N l , where j, l = 1, . . . , 4. Upon expanding out the confor constant matrices M ˜z = 0 and using the commutation relations in Lemma 5.1, we ﬁnd the dition τ˜z σ conditions l + N j + i(α + β)(δ j1 δ l2 − δ j2 δ l1 ) = 0 jM lM N

(5.19)

for j, l = 1, . . . , 4. Recall from Lemma 2.2 that the typical ﬁber CP1 of the twistor ﬁbration R4 × CP1 has homogeneous coordinates z1 , z1∗ , z2 , z2∗ ; it follows that the “line at inﬁnity” ∞ is recovered by setting z1 = z2 = 0. On this line, the monad ˜z = 0 becomes condition τ˜z σ 4 + N 3 = 0, 3M 4M N

3 = 0, 3M N

4 = 0. 4M N

(5.20)

Moreover, when z1 = z2 = 0 we see from the relations (5.7) and (5.8) that the coordinates z3 , z4 and their conjugates are mutually commuting, so that the line ∞ is classical. The self-conjugacy of the monad implies that the restricted bundle 4 = −N 3 3M 4M over ∞ is trivial, so we can argue as in [26] to show that the map N is an isomorphism. Using these conditions we choose bases for H, K, L such that 3M 4 = 1k and N       tr tr 1k×k 0k×k 0k×k −1k×k         3 =  0k×k  , M 4 =  0k×k  . 4 = 1k×k  , N 3 = 1k×k  , N M         02×k 02×k 0k×2 0k×2

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Now invoking conditions (5.19) for j = 3, 4 and l = 1, 2, the remaining matrices are necessarily of the form tr       tr B1 B1 −B2 −B2         1 = B2 , M 2 = B , N 2 =  B  . 1 =  B1  , N M     2   1  J J I I ∗ ∗ Using the conditions τ˜J(z) = −˜ σz and σ ˜J(z) = τ˜z , which correspond to the requirement that the monad be self-conjugate, we ﬁnd that

B1 = −B2∗ ,

B2 = B1∗ ,

J = I∗,

I = −J ∗ .

˜z = 0 it remains only to impose the Thus in order to satisfy the condition τ˜z σ conditions (5.19) in the cases j = l = 1 and j = 1, l = 2. The ﬁrst of these is condition (i) in the theorem; the second case is equivalent to requiring [B1 , B1∗ ] + [B2 , B2∗ ] + II ∗ − J ∗ J + i(α + β) = 0, giving condition (ii) in the theorem. Just as in the classical case [12], it is evident that the remaining gauge freedom in this calculation is given by the stated action of U(k), whence the result. Finally, we comment on a signiﬁcant diﬀerence between the parameter space Mk of instantons on the classical space R4 and the parameter space M(k; ) of instantons on the Moyal-deformed version R4 . Recall that, in the classical ADHM construction of Sec. 4.2, we needed to assume that the algebra-valued matrix ρ2 of Eq. (4.11) is invertible. Formally adjoining an inverse ρ−2 to the algebra Mk (C) ⊗ A[Mk ] ⊗ A[R4 ] resulted in the deletion of a collection of points from the parameter space Mk . In contrast, this noncommutative ADHM construction does not require this. Using the Moyal ADHM equations themselves one shows that the matrix ρ2 , which now having passed to the commutative parameter space has entries in the algebra A[M(k; )] ⊗ A[R4 ], is automatically invertible (we refer to [24, 14] for a proof). 6. The Connes–Landi Noncommutative Plane R4θ Next we turn to the construction of instantons on the noncommutative plane R4θ , which is an example of a toric noncommutative manifold (or isospectral deformation) in the sense of [10]. In particular, R4θ is obtained as a localization of the Connes–Landi quantum four-sphere Sθ4 , just as in Lemma 2.1 (cf. [20]), although here we shall obtain it directly from classical R4 by cocycle twisting. 6.1. Toric deformation of the space of monads Whereas the Moyal space-time R4 was obtained by cocycle twisting along an action of the group of translation sysmmetries of space-time, the noncommutative

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space-time R4θ is constructed by deforming the classical coordinate algebra A[R4 ] along an action of a group of rotational symmetries. Indeed, for the twisting Hopf algebra we take H = A[T2 ], the algebra of coordinate functions on the two-torus T2 . It is the commutative unital algebra A[T2 ] := A[sj , s−1 j | j = 1, 2] equipped with the Hopf ∗-algebra structure s∗j = s−1 j ,

∆(sj ) = sj ⊗ sj ,

(sj ) = 1,

S(sj ) = s−1 j

(6.1)

for j = 1, 2, with ∆ and extended as ∗-algebra maps and S extended as a ∗-antialgebra map. In order to deform the twistor ﬁbration, we need to equip the various coordinate algebras with left H-comodule structures. There is a Hopf algebra projection from A[GL+ (2, H)] onto H, deﬁned on generators by     α1 −α∗2 β1 −β2∗ s1 0 0 0     α2  0 s∗1 0 0  α∗1 β2 β1∗  +   (6.2)   π : A[GL (2, H)] → H,   →    γ1 −γ2∗ δ1 −δ2∗   0 0 s2 0  γ1∗

γ2

δ2

δ1∗

0

0

0

s∗2

and extended as a ∗-algebra map. Using Eq. (2.16), this projection determines a left H-coaction ∆π : A[C4 ] → H ⊗ A[C4 ] by A[C4 ] → H ⊗ A[C4 ],

zj → ςj ⊗ zj ,

(6.3)

extended as a ∗-algebra map, where we use the shorthand notation (ςj ) = (s1 , s∗1 , s2 , s∗2 ) for the generators of H. Using the identiﬁcation of generators in Eq. (2.10), this induces a coaction on the space-time algebra, A[R4 ] → H ⊗ A[R4 ],

ζ1 → ς1 ς4 ⊗ ζ1 ,

ζ2 → ς2 ς4 ⊗ ζ2 ,

(6.4)

and extended as a ∗-algebra map, making A[R4 ] into a left H-comodule ∗-algebra. As a twisting cocycle on H, we take the linear map deﬁned on generators by F : H ⊗ H → C,

F (sj , sl ) = exp(iπΘjl )

(6.5)

and extended as a Hopf bicharacter in the sense of Eq. (1.3). Here the deformation matrix Θ is the 2 × 2 real anti-symmetric matrix

0 θ 1 Θ = (Θjl ) = 2 −θ 0 for 0 < θ < 1 a real parameter. It is straightforward to check using the formulæ (1.9), (1.10) and (1.12) that the product, antipode and ∗-structure on H are in fact undeformed by F , so that H = HF as a Hopf ∗-algebra. However, the eﬀect of the twisting on the H-comodule algebras A[C4 ] and A[R4 ] is non-trivial. In what

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follows we write ηjl := F −2 (ςj , ςl ),  1  1 (ηjl ) =   ¯ µ

namely  µ µ ¯  1 µ ¯ µ ,  µ 1 1 1

µ µ ¯

1

µ = eiπθ .

(6.6)

1

Lemma 6.1. The relations in the H-comodule algebra A[C4 ] are twisted into zj zl = ηlj zl zj ,

zj zl∗ = ηjl zl∗ zj ,

zj∗ zl = ηjl zl zj∗ ,

zj∗ zl∗ = ηlj zl∗ zj∗

(6.7)

for each j, l = 1, . . . , 4. Proof. The cocycle-twisted product on the H-comodule algebra A[C4 ] is deﬁned by the formula (1.14). Just as in Lemma 5.1, the corresponding algebra relations can be expressed using the R-matrix (1.13): in this case one ﬁnds that the R-matrix takes the values R(ςj , ςl ) = F −2 (ςj , ςl ) = ηjl ,

R(ςj , ςl∗ ) = F −2 (ςj , ςl∗ ) = ηlj .

(6.8)

By explicitly computing Eqs. (5.9) (and omitting the product symbol ·F ), one obtains the relations stated in the lemma. We denote by A[C4θ ] the algebra generated by {zj , zj∗ | j = 1, . . . , 4} modulo the relations (6.7). In this way, we have that A[C4θ ] is a left HF -comodule ∗-algebra. Lemma 6.2. The algebra relations in the H-comodule algebra A[R4 ] are twisted into ζ1 ζ2 = λζ2 ζ1 ,

ζ1∗ ζ2∗ = λζ2∗ ζ1∗ ,

ζ2∗ ζ1 = λζ1 ζ2∗ , 2

where the deformation parameter is λ := µ = e

2πiθ

ζ2 ζ1∗ = λζ1∗ ζ2 ,

(6.9)

.

Proof. The product on A[R4 ] is once again twisted using the formula (1.14). Again omitting the product symbol ·F , the relations are computed to be as stated. We denote by A[R4θ ] the algebra generated by ζ1 , ζ2 and their conjugates, subject to these relations. They make A[R4θ ] into a left HF -comodule ∗-algebra. Remark 6.3. Since the generators z1 , z2 and their conjugates generate a commutative subalgebra of A[C4θ ], it is easy to see using Lemma 2.2 that it is only the base space R4θ of the localized twistor bundle that is deformed. The typical ﬁber CP1 remains classical and the localized twistor algebra is isomorphic to the tensor product A[R4θ ] ⊗ A[CP1 ]. The canonical diﬀerential calculi described in Sec. 3.1 are also deformed. The relations in the quantized calculi are given in the following lemmata. Lemma 6.4. The twisted diﬀerential calculus Ω(C4θ ) is generated by the degree zero elements zj , zl∗ and the degree one elements dzj , dzl∗ for j, l = 1, . . . , 4, subject to

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the bimodule relations between functions and one-forms zj dzl = ηlj (dzl )zj ,

zj dzl∗ = ηjl (dzl∗ )zj

for j, l = 1, . . . , 4 and the anti-commutation relations between one-forms dzj ∧ dzl + ηlj dzl ∧ dzj = 0,

dzj ∧ dzl∗ + ηjl dzl∗ ∧ dzj = 0

for j, l = 1, . . . , 4. Proof. One views the classical calculus Ω(C4 ) as a left H-comodule algebra and accordingly computes the deformed product using the twisting cocycle F . Since the exterior derivative d is H-equivariant, the (anti-)commutation relations in the deformed calculus Ω(C4θ ) are exactly the same as the algebra relations in A[C4θ ] but with d inserted appropriately. Lemma 6.5. The twisted diﬀerential calculus Ω(R4θ ) is generated by the degree zero elements ζ1 , ζ1∗ , ζ2 , ζ2∗ and the degree one elements dζ1 , dζ1∗ , dζ2 , dζ2∗ , subject to the relations ζ1 dζ2 − λdζ2 ζ1 = 0,

ζ2∗ dζ1 − λdζ1 ζ2∗ = 0,

dζ1 ∧ dζ2 + λdζ2 ∧ dζ1 = 0,

dζ2∗ ∧ dζ1 + λdζ1 ∧ dζ2∗ = 0.

Proof. Once again, the classical calculus Ω(R4 ) is deformed as a twisted left H-comodule algebra, with the relations working out to be as stated. In particular, it is clear that the vector space Ω2 (R4θ ) is the same as it is classically. The Hodge operator ∗ : Ω2 (R4 ) → Ω2 (R4 ) commutes with the H-coaction in the sense that ∆π (∗ω) = (id ⊗ ∗)∆π (ω),

ω ∈ Ω2 (R4 ),

so that there is also a Hodge operator ∗θ : Ω2 (R4θ ) → Ω2 (R4θ ) deﬁned by the same formula as it is classically. There is a decomposition of Ω2 (R4θ ) into self-dual and anti-self-dual two-forms Ω2 (R4θ ) = Ω2+ (R4θ ) ⊕ Ω2− (R4θ ) which, at the level of vector spaces, is identical to the corresponding decomposition in the classical case. We also apply the cocycle deformation the coordinate algebra A[Mk ] of the space of self-conjugate monads by viewing it as a left H-comodule algebra. We write A[Mk;θ ] for the resulting cocycle-twisted left HF -comodule algebra. Proposition 6.6. The noncommutative ∗-algebra A[Mk;θ ] is generated by the j l , Ndc for a, c = 1, . . . , k and b, d = 1, . . . , 2k + 2, modulo matrix elements Mab the relations j j l l Mcd = ηlj Mcd Mab , Mab

together with the ∗-structure (4.6).

j j l l Nba Ndc = ηlj Ndc Nba ,

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Proof. From Lemma 4.4, we read oﬀ the H-coaction on generators M j , j = 1, . . . , 4, obtaining j j → ςj∗ ⊗ Mab , Mab

l l Ndc → ςl∗ ⊗ Ndc ,

which we extend as a ∗-algebra map. The deformed relations follow immediately from an application of the twisting formula (1.14). The coaction of H on A[Mk ] does not depend on the matrix indices of the generators M j , N l , j, l = 1, . . . , 4, hence neither do the twisted commutation relations. In terms of the deformed product, the relations (4.4) are twisted into the relations j j l l (Ndr Mrb + ηjl Ndr Mrb )=0 r

for each j, l = 1, . . . , 4 and b, d = 1, . . . , k. 6.2. The construction of instantons on R4θ Just as we did for the Moyal plane, we now use the noncommutative space of monads Mk;θ to construct families of instantons on the Connes–Landi space-time R4θ . 2 Z2 . Given a The Pontryagin dual of the torus T2 is the discrete group T pair of integers (r1 , r2 ) ∈ Z2 we deﬁne unitary elements u = (u1 , u2 , u3 , u4 ) of the algebra HF by u = (u1 , u2 , u3 , u4 ) = (ς1m1 , ς2m2 , ς3m3 , ς4m4 ),

(6.10)

where (mj ) = (r1 , r1 , r2 , r2 ). It is clear that u∗1 = u2 and u∗3 = u4 , and that each uj is a group-like element of the Hopf algebra HF , i.e. it transforms as ∆(uj ) = uj ⊗ uj under the coproduct ∆ : HF → HF ⊗ HF . Lemma 6.7. There is a canonical left action of HF on the algebra A[Mk;θ ] deﬁned on generators by j j j ml = R(ςj∗ , ςlml )Mab = ηlj Mab , ul Mab

j ∗ j ∗ j ∗ ml ul Mab = R(ςj , ςlml )Mab = ηjl Mab

for j, l = 1, . . . , 4. Proof. From Proposition 5.6 we know that A[Mk;θ ] is a left HF -comodule algebra; it is therefore also a left HF -module algebra according to the formula (1.5), which works out to be as stated. This gives us an action of the group Z2 on the algebra A[Mk;θ ] by ∗-automorphisms, γ : Z2 → Aut A[Mk;θ ].

(6.11)

The smash product algebra A[Mk;θ ] > HF corresponding the to HF -action of Lemma 6.7 works out using the coproduct ∆(ςj ) = ςj ⊗ ςj on HF and the

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formula (1.7) to have relations of the form j j ml ms r r (Mab ⊗ ul )(Mcd ⊗ us ) = ηlr ηrj ηjs (Mcd ⊗ us )(Mab ⊗ ul ) j j ml ms r ∗ r ∗ ⊗ ul )(Mcd ⊗ us ) = ηrl ηrj ηsj (Mcd ⊗ us )(Mab ⊗ ul ) (Mab

for j, l, r, s = 1, . . . , 4, together with their conjugates. This is another special case of Example 1.1. We think of this smash product A[Mk;θ ] > HF as an algebraic version the crossed product algebra A[Mk;θ ] >γ Z2 determined by the group action (6.11). Lemma 6.8. The algebra structure of the tensor product A[Mk;θ ] ⊗ A[C4θ ] is determined by the relations in the respective subalgebras A[Mk;θ ] and A[C4θ ] given above, together with the cross-relations M j zl = ηjl zl M j ,

M j zl∗ = ηlj zl∗ M j ,

j, l = 1, . . . , 4,

as well as their conjugates. Proof. The classical algebra A[Mk ] ⊗ A[C4 ] is equipped with the tensor product HF -coaction of Eq. (1.4). We deform the product in this algebra using the formula (1.14). The cross-terms in the resulting algebra A[Mk;θ ] ⊗ A[C4θ ] are computed to be as stated [5]. Once again, the symbol ⊗ is to remind us that the algebra structure on the tensor product is not the standard one and has been twisted by the deformation procedure. Once again we have a pair of matrices σz and τz , σz = M j ⊗ zj , τz = N j ⊗ zj , j

j

but whose entries live in the twisted algebra A[Mk;θ ] ⊗ A[C4θ ]. The matrix V := (σz σJ(z) ) is a 2k × (2k + 2) matrix with entries in A[Mk;θ ] ⊗ A[C4θ ], using which we deﬁne ρ2 := V∗ V. From the projection Q := Vρ−2 V∗ we construct the complementary matrix P := 12k+2 − Q, which has entries in the algebra A[Mk;θ ] ⊗ A[R4θ ]. It is clear that this matrix P is a self-adjoint idempotent, P2 = P = P∗ . However, just as was the case for the Moyal plane, it does not deﬁne an honest family of projections in the sense of Deﬁnition 3.2, since it has values in the twisted tensor product algebra. We recover a genuine family of projections using the following lemma, in which we use the Sweedler notation Z → Z (−1) ⊗ Z (0) for the left coaction A[C4θ ] → HF ⊗ A[C4θ ] deﬁned in Eq. (6.3). Lemma 6.9. There is a canonical ∗-algebra map µ : A[Mk;θ ] ⊗ A[C4θ ] → (A[Mk;θ ] > HF ) ⊗ A[C4θ ] deﬁned by µ(M ⊗ Z) = M ⊗ Z (−1) ⊗ Z (0) for each M ∈ A[Mk;θ ] and Z ∈ A[C4θ ]. Proof. The proof is identical to that of Lemma 5.10, save for the replacement of the coaction (5.4) by the coaction (6.3).

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As a consequence, we ﬁnd that there are maps σ ˜z : H ⊗ A[C4θ ] → (A[Mk;θ ] > HF ) ⊗ K ⊗ A[C4θ ],

(6.12)

τ˜z : K ⊗ A[C4θ ] → (A[Mk;θ ] > HF ) ⊗ L ⊗ A[C4θ ]

(6.13)

deﬁned by composing σz and τz with the map µ. With the coaction (6.3), they work out to be M r ⊗ ςr ⊗ zr , τ˜z := N r ⊗ ςr ⊗ zr , σ ˜z := r

r

which are, respectively, k × (2k + 2) and (2k + 2) × k matrices with entries in the noncommutative algebra (A[Mk;θ ] > HF ) ⊗ A[C4θ ]. With this in mind, we form the := (˜ ˜J(z) ), this time yielding a 2k × (2k + 2) matrix (2k + 2) × 2k matrix V σz σ ∗ V. Just as in the with entries in (A[Mk;θ ] > HF ) ⊗ A[C4θ ], and deﬁne ρ 2 := V classical case, in order to proceed we need to slightly enlarge the matrix algebra Mk (C) ⊗ A[Mk;θ ] ⊗ A[C4θ ] by adjoining an inverse element ρ−2 for ρ 2 . = V ρ−2 V ∗ is a projection, Proposition 6.10. The (2k + 2) × (2k + 2) matrix Q =Q ∗ , with entries in the algebra (A[Mk;θ ] > HF ) ⊗ A[C4 ] and trace equal 2 = Q Q θ to 2k. is a projection follows from the fact that Q is a projecProof. The fact that Q tion and µ is a ∗-algebra map. By construction, the entries of the matrix ρ2 are central in the algebra (A[Mk;θ ] > HF ) ⊗ A[C4θ ] (this follows from the fact that the corresponding classical matrix elements are coinvariant under the left H-coaction), from which it follows that the trace computation in Proposition 4.5 is valid in the noncommutative case as well [5]. we construct the complementary projection P := 12k+2 − From the projection Q 4 Q. It has entries in the algebra (A[Mk;θ ] > HF ) ⊗ A[Rθ ] and has trace equal to two. In analogy with Deﬁnition 3.2, the ﬁnitely generated projective module 4 2k+2 E := P((A[M k;θ ] > HF ) ⊗ A[Rθ ])

deﬁnes a family of rank two vector bundles over R4θ parametrized by the noncommutative algebra A[Mk;θ ] > HF . We equip this family of vector bundles with the family of Grassmann connections associated to the projection P. 2 of the Grassmann family of Proposition 6.11. The curvature F = P((id ⊗ d)P) connections ∇ := (id ⊗ d) ◦ P is anti-self-dual. Proof. From Lemma 5.5, we know that the space of two-forms Ω2 (R4θ ) and the Hodge ∗-operator ∗θ : Ω2 (R4θ ) → Ω2 (R4θ ) are undeformed and equal to their classical counterparts; similarly for the decomposition Ω2 (R4θ ) = Ω2+ (R4θ ) ⊕ Ω2− (R4θ ) into selfdual and anti-self-dual two-forms. This identiﬁcation of the “quantum” with the “classical” spaces of two-forms survives the tensoring with the parameter space

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A[Mk;θ ] > HF , which yields that (A[Mk;θ ] > HF ) ⊗ Ω2± (R4θ ) and (A[Mk ] ⊗ H) ⊗ Ω2± (R4 ) are isomorphic as vector spaces. Computing the curvature F in exactly the same way as in Proposition 4.6, we see that it must be anti-self-dual, since the same is true in the classical case. Recall that, in the case of instantons on the Moyal plane R4 discussed in Sec. 5.2, we chose to avoid altogether the issue of whether or not the families of instantons constructed there have ﬁnite energy. However, in the present case of instantons on the toric manifold R4θ , it is clear that the instantons produced by the noncommutative ADHM construction are indeed localizations of the noncommutative families instantons on the conformal compactiﬁcation Sθ4 constructed in [5], whence they must have ﬁnite Yang–Mills energy. 6.3. The toric ADHM equations The previous section produced a family of instantons parametrized by the noncommutative algebra A[Mk;θ ] > HF . Just as we did for the Moyal space-time, we would like to ﬁnd a suitable commutative subalgebra of A[Mk;θ ] > HF and hence a family of instantons parametrized by a classical space. In order to do this, we introduce elements of A[Mk;θ ] > HF deﬁned by 1 1 ab M := Mab ⊗ ς1 ,

2 ab M := M 2 ⊗ ς2 ,

3 3 ab M := Mab ⊗ 1,

4 4 ab M := Mab ⊗1

j ∗ , for each a = 1, . . . , k and b = 1, . . . , 2k + 2, together with their conjugates M ab j = 1, . . . , 4. Definition 6.12. We write A[M(k; θ)] for the ∗-subalgebra of A[Mk;θ ] > HF genl ∗ . j , M erated by the elements M ab ab Proposition 6.13. The algebra A[M(k; θ)] is a commutative ∗-subalgebra of the smash product A[Mk;θ ] > HF . 4 and their conjugates are obviously central. The 3 , M Proof. The generators M ab ab 2 and their conjugates are also easily seen to commute amongst 1 , M generators M ab ab themselves. We check the case j ∈ {1, 2}, l ∈ {3, 4}, yielding j j j l l l j M l M ab cd = (Mab ⊗ ςj )(Mab ⊗ 1) = ηjl Mab Mcd ⊗ ςj = ηjl ηlj Mcd Mab ⊗ ςj j l j l M = (Mcd ⊗ 1)(Mab ⊗ ςj ) = M cd ab .

All other pairs of generators are shown to commute using similar computations. Although we have changed our set of generators, we nevertheless combine them with the Hopf algebra HF using a smash product construction. Let : HF ⊗ A[M(k; θ)] → A[M(k; θ)] be the left HF -action deﬁned by j = ηlj M j , ςl M ab ab

j ∗ = ηjl M j ∗ ςl M ab ab

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for j, l = 1, . . . , 4 and let A[M(k; θ)] > HF be the corresponding smash product algebra. The next proposition relates the parameter space A[M(k; θ)] to the parameter space A[Mk;θ ]. Theorem 6.14. There is a ∗-algebra isomorphism φ : A[M(k; θ)] > HF A[Mk;θ ] > HF deﬁned for each h ∈ HF by 1 ⊗ h → M 1 ⊗ ς1 h, M ab ab

2 ⊗ h → M 2 ⊗ ς2 h, M ab ab

3 ⊗ h → M 3 ⊗ h, M ab ab

4 ⊗ h → M 4 ⊗ h M ab ab

→

and extended as a ∗-algebra map. Proof. The given map is clearly a vector space isomorphism with inverse 1 1 ⊗ ς ∗ h, M 2 ⊗ h → M 2 ⊗ ς ∗ h, Mab ⊗ h → M 1 2 ab ab ab 3 3 ⊗ h, Mab ⊗ h → M ab

4 4 ⊗ h, Mab ⊗ h → M ab

extended as a ∗-algebra map. By deﬁnition, the map φ is a ∗-algebra homomorphism on the subalgebra A[M(k; θ)], so it remains to check that it preserves the crossrelations with the subalgebra HF . This is easy to verify: one has for example that 1 ⊗ 1) = (1 ⊗ ςj )(M 1 ⊗ ς1 ) = ηj1 M 1 ⊗ ςj ς1 φ(1 ⊗ ςj )φ(M 1 ⊗ 1)φ(1 ⊗ ςj ). = ηj1 (M 1 ⊗ ς1 )(1 ⊗ ςj ) = ηj1 φ(M The remaining relations are checked in exactly the same way. Next we focus on the task of seeing how the parameters corresponding to the subalgebra HF can be removed in order to leave a family of instantons parametrized by the commutative algebra A[M(k; θ)]. There is a right coaction δR : A[M(k; θ)] > HF → (A[M(k; θ)] > HF ) ⊗ HF ,

δR := id ⊗ ∆,

(6.14)

where ∆ : HF → HF ⊗ HF is the coproduct on the Hopf algebra HF . This coaction are ⊗ 1 and δR (P) is by gauge transformations, meaning that the projections P unitarily equivalent in the matrix algebra M2k+2 ((A[M(k; θ)] > HF ) ⊗ HF ) and so they deﬁne gauge equivalent families of instantons [5]. The parameters determined by the subalgebra HF in A[M(k; θ)] > HF are therefore just gauge parameters and so they may be removed by passing to the subalgebra of A[M(k; θ)] > HF consisting of coinvariant elements under the coaction (6.14), viz. A[M(k; θ)] ∼ = {a ∈ A[M(k; θ)] > HF | δR (a) = a ⊗ 1}. In this way we obtain a projection Pk;θ with entries in A[M(k; θ)] ⊗ A[R4θ ]. The explicit details of the construction of the projection Pk;θ are given in [5], together with a proof of the fact that the Grassmann family of connections ∇ = Pk;θ ◦(id⊗d) also has anti-self-dual curvature and hence deﬁnes a family of instantons on R4θ .

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Moreover, the commutative algebra A[M(k; θ)] is the algebra of coordinate functions on a classical space of monads M(k; θ). For each point x ∈ M(k; θ) there is an evaluation map evx ⊗ id : A[M(k; θ)] ⊗ A[C4θ ] → A[C4θ ], which in turn determines monad over the noncommutative space C4θ in terms of the σz and (evx ⊗ id) τz , i.e. a sequence of free right A[C4θ ]-modules matrices (evx ⊗ id) (evx ⊗id)˜ σz

(evx ⊗id)˜ τz

0 → H ⊗ A[C4θ ] −−−−−−−→ K ⊗ A[C4θ ] −−−−−−−→ L ⊗ A[C4θ ] → 0.

(6.15)

Recall from Proposition 4.7 that the gauge freedom in the classical ADHM construction is precisely the freedom to choose linear bases of the vector spaces H, K, L. Clearly we also have this freedom in the noncommutative construction and so we write ∼ for the resulting equivalence relation on the space M(k; θ) (cf. Definition 4.8). This yields the following description of the space M(k; θ) of classical parameters in the ADHM construction on R4θ . Theorem 6.15. For k ∈ Z a positive integer, the space M(k; θ)/∼ of equivalence classes of self-conjugate monads over C4θ is the quotient of the set of complex matrices B1 , B2 ∈ Mk (C), I ∈ M2×k (C), J ∈ Mk×2 (C) satisfying the equations (i) µ ¯B1 B2 − µB2 B1 + IJ = 0, (ii) [B1 , B1∗ ] + [B2 , B2∗ ] + II ∗ − J ∗ J = 0 by the action of U(k) given by B1 → gB1 g −1 ,

B2 → gB2 g −1 ,

I → gI,

J → Jg −1

for each g ∈ U(k). Proof. We express the monad maps σ ˜z , τ˜z as 1 z1 + M 2 z2 + M 3 z3 + M 4 z4 , σ ˜z = M

1 z1 + N 2 z2 + N 3 z3 + N 4 z4 τ˜z = N

j , N l , j, l = 1, . . . , 4. Upon expanding out the condition for constant matrices M ˜z = 0 and using the commutation relations in Lemma 6.1, we ﬁnd the conditions τ˜z σ l + ηjl N lM j = 0 jM N

(6.16)

for j, l = 1, . . . , 4. The typical ﬁber CP1 of the twistor ﬁbration R4 × CP1 has homogeneous coordinates z1 , z1∗ , z2 , z2∗ and the “line at inﬁnity” ∞ is recovered by setting z1 = z2 = 0. On this line, the monad condition τ˜z σ ˜z = 0 becomes 3M 4 + N 3 = 0, 4M N

3 = 0, 3M N

4 = 0. 4M N

(6.17)

Moreover, when z1 = z2 = 0 we see from the relations (6.7) that the coordinates z3 , z4 and their conjugates are mutually commuting, so that the line ∞ is classical. Self-conjugacy of the monad once again implies that the restricted bundle over ∞

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4 = −N 3 3M 4M is trivial, whence we can argue as in [26] to show that the map N 3 4 = 1k and M is an isomorphism. We choose bases for H, K, L such that N       tr tr 1k×k 0k×k 0k×k −1k×k         3 =  0k×k , M 4 =  0k×k  . 4 = 1k×k , N 3 = 1k×k  , N M         02×k 02×k 0k×2 0k×2 Using the conditions (6.16) for j = 3, 4 and l = 1, 2, the remaining matrices are necessarily of the form tr tr       B1 B1 −µB2 −¯ µB2     2 =  1 =  2 =  1 =  M ¯B1  , N B2 , M B2 , N µ  µB1  . J

J

I

I

∗ ∗ Invoking the relations τ˜J(z) = −˜ σz and σ ˜J(z) = τ˜z corresponding to the fact that the monad is self-conjugate, we ﬁnd that

µB2∗ , B1 = −¯

B2 = µB1∗ ,

J = I∗,

I = −J ∗ .

Thus in order to fulﬁl the condition τ˜z σ ˜z = 0 it remains to impose conditions (6.16) in the cases j = l = 1 and j = 1, l = 2. These are precisely conditions (i) and (ii) in the theorem. It is evident just as in the classical case [12] that the remaining freedom in this set-up is given by the stated action of U(k), whence the result. Acknowledgments SJB gratefully acknowledges support from the ESF network “Quantum Geometry and Quantum Gravity” and the NWO grant 040.11.163. WvS acknowledges support from NWO under VENI-project 639.031.827. Both authors thank the Institut des ´ Hautes Etudes Scientiﬁques for hospitality during a short visit in 2010. References [1] M. F. Atiyah, Geometry of Yang–Mills Fields, Fermi Lectures (Scuola Normale Pisa, 1979). [2] M. F. Atiyah, V. G. Drinfel’d, N. J. Hitchin and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978) 185–187. [3] M. F. Atiyah, N. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. London A 362 (1978) 425–461. [4] S. Brain and G. Landi, Families of monads and instantons from a noncommutative ADHM construction, in Quanta of Maths, Clay Math. Proc., Vol. 11 (AMS Providence, RI, 2010), pp. 55–84. [5] S. Brain and G. Landi, Moduli spaces of instantons: Gauging away noncommutative parameters, to appear in Quart. J. Math. (2010); DOI:10.1093/qmath/haq036. [6] S. Brain and S. Majid, Quantisation of twistor theory by cocycle twist, Comm. Math. Phys. 284 (2008) 713–774. [7] C.-S. Chu, V. V. Khoze and G. Travagliani, Notes on noncommutative instantons, Nucl. Phys. B 621 (2002) 101–130.

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[8] A. Connes, Noncommutative Geometry (Academic Press, New York, 1994). [9] A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry, Comm. Math. Phys. 182 (1996) 155–176. [10] A. Connes and G. Landi, Noncommutative manifolds, the instanton algebra and isospectral deformations, Comm. Math. Phys. 221 (2001) 141–159. [11] A. Connes and M. Dubois-Violette, Noncommutative finite-dimensional manifolds I: Spherical manifolds and related examples, Comm. Math. Phys. 230 (2002) 539–579. [12] S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984) 435–460. [13] S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds (Oxford University Press, 1990). [14] K. Furuuchi, Instantons on noncommutative R4 and projection operators, Prog. Theor. Phys. 103 (2000) 1043–1068. [15] C. R. Gilson, M. Hamanaka and J. C. Nimmo, B¨ acklund transformations for noncommutative anti-self-dual Yang–Mills equations, Glasgow Math. J. 51 (2009) 83–93. [16] Z. Horv´ ath, O. Lechtenfeld and M. Wolf, Noncommutative instantons via dressing and splitting approaches, J. High Energy Phys. 12 (2002) 060. [17] A. Kapustin, A. Kuznetsov and D. Orlov, Noncommutative instantons and twistor transform, Comm. Math. Phys. 221 (2001) 385–432. [18] G. Landi and W. D. van Suijlekom, Principal fibrations from noncommutative spheres, Comm. Math. Phys. 260 (2005) 203–225. [19] G. Landi and W. D. van Suijlekom, Noncommutative instantons from twisted conformal symmetries, Comm. Math. Phys. 271 (2007) 591–634. [20] G. Landi and W. D. van Suijlekom, Noncommutative instantons in Tehran, in An Invitation to Noncommutative Geometry (World Sci. Publ., Hackensack, NJ, 2008), pp. 275–353. [21] G. Landi, C. Pagani, C. Reina and W. D. van Suijlekom, Noncommutative families of instantons, Int. Math. Res. Not. 12 (2008) Art. ID rnn038, 32 p. [22] S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, 1995). [23] L. J. Mason and N. M. J. Woodhouse, Integrability, Self-Duality and Twistor Theory (Oxford University Press, 1996). [24] N. A. Nekrasov and A. Schwarz, Instantons on noncommutative R4 and (2, 0) superconformal six-dimensional theory, Comm. Math. Phys. 198 (1998) 689–703. [25] N. A. Nekrasov, Trieste lectures on solitons in noncommutative gauge theories, in Superstrings and Related Matters (Trieste, 2000) (World Sci. Publ., River Edge, NJ, 2001), pp. 141–205. [26] C. Okonek, M. Schneider and H. Spindler, Vector Bundles on Complex Projective Spaces (Birkhauser, Boston, 1980). [27] R. Penrose and W. Rindler, Spinors and Space-Time, Vol. 2 (Cambridge University Press, 1986). [28] Y. Tian and C.-J. Zhu, Instantons on general noncommutative R4 , Commun. Theor. Phys. 38 (2002) 691–697. [29] Y. Tian and C.-J. Zhu, Remarks on the noncommutative ADHM construction, Phys. Rev. D 67 (2003) 045016.

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Reviews in Mathematical Physics Vol. 23, No. 3 (2011) 309–346 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004308

ON THE STATIC SPACETIME OF A SINGLE POINT CHARGE

A. SHADI TAHVILDAR-ZADEH Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA [email protected] Received 6 December 2010 Revised 7 February 2011 Among all electromagnetic theories which (a) are derivable from a Lagrangian, (b) satisfy the dominant energy condition, and (c) in the weak ﬁeld limit coincide with classical linear electromagnetics, we identify a certain subclass with the property that the corresponding spherically symmetric, asymptotically ﬂat, electrostatic spacetime metric has the mildest possible singularity at its center, namely, a conical singularity on the time axis. The electric ﬁeld moreover has a point defect on the time axis, its total energy is ﬁnite, and is equal to the ADM mass of the spacetime. By an appropriate scaling of the Lagrangian, one can arrange the total mass and total charge of these spacetimes to have any chosen values. For small enough mass-to-charge ratio, these spacetimes have no horizons and no trapped null geodesics. We also prove the uniqueness of these solutions in the spherically symmetric class, and we conclude by performing a qualitative study of the geodesics and test-charge trajectories of these spacetimes. Keywords: Nonlinear electromagnetism; Einstein equations; point charges; Born–Infeld theory; conical singularity; ADM mass; point defects; self-energy; naked singularity; Birkhoﬀ theorem; test-charge orbits. Mathematics Subject Classiﬁcation 2010: 83C50, 35Q60, 78A30, 78A35

1. Introduction In this paper, we study static, spherically symmetric solutions of the Einstein– Maxwell system featuring a single spinless point charge.a Our main results are summarized in a theorem at the end of this introduction. The Einstein–Maxwell system of PDEs reads: 1 (1.1) Rµν [g] − gµν R[g] = κTµν , 2 dF = 0, (1.2) dM = 0. a In

a sequel to this paper, we will study solutions with spin. 309

(1.3)

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It describes the geometry of a spacetime (M, g) endowed with an electromagnetic ﬁeld. Here R[g] is the Ricci curvature tensor and R[g] the scalar curvature of the metric g of a four-dimensional Lorentzian manifold M. Moreover, F is the Faraday tensor of the electromagnetic ﬁeld and M is the Maxwell tensor corresponding to F(for which we will consider a whole family of choices), while T is the electromagnetic energy(density)-momentum (density)-stress tensor associated to F. Finally, κ = 8πG/c4 , with G being Newton’s constant of universal gravitation and c the “speed of light in vacuum”. (In the following we will work with units in which κ = 1.) Before we discuss the options of choosing the relationship between M and F, we should recall that an open domain M in a Lorentzian manifold M is (somewhat inappropriately) called static if it has a hypersurface-orthogonal Killing ﬁeld K whose orbits are complete and everywhere timelike in M . Such a domain possesses a time-function t, i.e. a function deﬁned on it whose gradient is timelike, and the vectorﬁeld dual to its gradient is future-directed everywhere. It can always be chosen such that Kt = 1. One can use t as a coordinate function on M , and the level sets of t provide a foliation of M into spacelike leaves Σt . It follows that the induced metric on the leaves is independent of t, i.e. the space Σt is static. Furthermore, for a “static spacetime” to be spherically symmetric means that the rotation group SO(3) acts as a continuous group of isometries on the manifold, with orbits that are spacelike 2-spheres, and its action commutes with that of the group generated by the timelike Killing ﬁeld K. For p ∈ M , let A(p) be the area of the spherical orbit that goes through p, and let r(p) = A(p)/4π be the radius of a Euclidean sphere with area A(p). As long as g(∇r, ∇r) > 0 one can use r as a spacelike coordinate function on the manifold, and in that case the metric of the spacetime can be put in the form gµν dxµ dxν = −eξ dt2 + e−ξ dr2 + r2 (dθ2 + sin2 θ dφ2 ),

(1.4)

where (θ, φ) are spherical coordinates on the orbit spheres, and ξ = ξ(r) is a smooth function which depends on the choice of relationship between F and M, here called an “aether law” (for short, and for historical reasons.) The simplest choice of the aether law is Maxwell’s M = − ∗ F, in which case the Einstein–Maxwell system will be called the Einstein–Maxwell–Maxwell system (EMM). It is well known that the Reissner–Weyl–Nordstr¨ om spacetimeb (RWN c for short) is the unique spherically symmetric, asymptotically ﬂat solution of the EMM system. For the RWN solution, one has eξ = 1 −

b Generally

q2 2m0 + 02 , r r

known as having been discovered independently by Reissner [1] and Nordstr¨ om [2], this spacetime is also a member of a whole class of electrovacuum solutions found by Weyl [3]. c See Sec. 6 for a precise statement and proof of uniqueness.

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where q0 and m0 are two real parameters. They are in fact integration constants that come from solving the radial Liouville-type equation that arises as the reduction of the EMM system under the stated symmetry assumptions. Since the spacetime is asymptotically ﬂat, its ADM mass [5] is well deﬁned, and it is seen to be equal to the parameter m0 . Also, one has a formula for the Faraday tensor in the case of RWN: q0 F = dA, A = ϕ(r)dt, ϕ = , r which suggests, via the divergence theorem, that q0 is the total charge of the spacetime. As discussed in detail in [6], the causal structure of the RWN spacetime depends in a crucial way on the ratio |q0 |/m0 : When this is less than one, which is referred ξ to as the subextremal case, the metric coeﬃcient e , in addition to being singular at 2 r = 0, has two zeros, namely, at r± = m0 ± m0 − q02 . It can be shown that r = r+ is the event horizon, the boundary of the past of the spacetime’s future null inﬁnity, and therefore the spacetime has a nonempty black hole region. It is worth noting that the causal structure of the maximal analytic extension of the subextremal RWN spacetime is quite rich and complicated, comprising an inﬁnitude of regions, and is plagued by the breakdown of determinism due to the presence of Cauchy horizons (cf. [6].) By contrast, the RWN metric in the superextremal regime, corresponding to |q0 | > m0 has a very simple causal structure. The metric coeﬃcient eξ is always positive, (t, r, θ, φ) is a global coordinate system for the manifold, and the only singularity present is the naked one, on the timelike axis r = 0. The topology of the manifold is that of R4 minus a line. In view of the fact that the empirical charge-to-mass ratio of charged particles such as the electron and the proton are huge (1018 and 1022 respectively) many researchers have been tempted by the prospect that the superextremal RWN solution is but the simplest example of spacetimes featuring one or more point charges.d Such a spacetime we shall refer to as a “charged-particle spacetime”. One of the key questions that needs to be addressed in this regard is the following: According to relativity theory’s E = mc2 , the proper mass of a charged-particle spacetime should equal its energy, which for a static spacetime is expressed as the integral of the time-time component of T over a static spacelike hypersurface. Is it then possible to attribute some, or all, of the mass of a charged-particle spacetime to the energy of the electromagnetic ﬁeld that permeates ite ? One diﬃculty with such an attribution is that in classical Maxwell–Lorentz electrodynamicsf the self-energy of a point charge is inﬁnite, and that remains to [7, §21.1] for references, and for a catalog of such solutions. about the origin of the mass of charged particles are as old as the theory of electromagnetism itself, beginning with Heaviside, and continuing with Abraham, Lorentz, Poincar´e, Mie, Einstein, Fermi, Born, Dirac, Wheeler, Feynman, Schwinger, Rohrlich and many others. f That is to say, Maxwell’s equations with “point-particle-like” sources whose formal law of motion is that of Newton’s, with the formal force being the Lorentz force. The model is ill-deﬁned unless regularized. d See

e Questions

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be the case even when the electromagnetic Maxwell–Lorentz ﬁeld is coupled to gravity via the Einstein equations. This can be clearly seen in the case of RWN because the total electrostatic energy carried by a time-slice would be ∞ 2 ∞ q0 |dϕ|2 r2 dr = dr, r2 0 0 which is inﬁniteg unless q0 = 0. In general relativity, the principle of equivalence states that it is the total energy of a system that interacts gravitationally, i.e. unlike the Newtonian theory, there is no distinction between active and passive “gravitational” masses. Thus, the total electrostatic energy will always make a contribution to the ADM mass of the spacetime, which in the case of the superextremal RWN is clearly an inﬁnite contribution. On the other hand, the ADM mass is usually interpreted as representing the total energy content of the spacetime (recall that in relativity there is no distinction between mass and energy (with c = 1)). Ingenious proposals have been made to explain the ﬁniteness of the ADM mass. Such ideas have been pursued using renormalization techniques [11], but these techniques are usually very diﬃcult or even impossible to justify rigorously.h Another diﬃculty with taking the superextremal RWN too seriously is the presence of a strong naked singularity on the time axis. Such singularities are expected to be “subject to cosmic censorship”, in the sense that they are believed to be nongeneric, and therefore unstable under small perturbations. Furthermore, the presence of a strong “eternal” singularity means that such a spacetime cannot arise as a solution of a classically-posed initial value problem. For the RWN metric the worst part of the singular behavior at r = 0 stems from the contribution of the charge q0 to the metric coeﬃcient, as can be seen for example from the Kretschmann scalar, which is a curvature invariant equal to the norm of the Riemann tensor: 7q04 48 2m0 q02 2 abcd 2 + 2 . = 6 m0 − K = Rabcd R r r 6r Clearly, K ∼ r−4 ∞ as r 0. To overcome the ﬁrst of these two divergence problems while retaining the concept of a point charge, Born [15] proposed to make the Maxwell equations g Indeed,

this is the same inﬁnity that turns up in the absence of gravity, in ﬂat spacetime, for the self-energy of a point charge, and which led Abraham and Lorentz, and later Mie, to look for alternatives to the point-charge description. We also note that in the current approach to quantum electrodynamics, the corresponding energy integral to the above is still divergent, although less violently [8], thus even in the absence of gravity, the problem of inﬁnite self-energy of point charges is not solved by going over to the quantum description [9, 10]. h For example, Dirac [12] proposed that the point charge possesses a compensating “bare mass” equal to −∞. However, it is known that a rigorous removal of the regularization using mass renormalization mbare → −∞ is impossible in the case of Lorentz electrodynamics, since the renormalization ﬂow terminates at mbare = 0 with regularization still in place [13]. Similar diﬃculties exist for the renormalization of RWN [14].

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nonlinear.i This is done by choosing a Lagrangian density L for the electromag netic action S[A] = M L(dA) in such a way that, in addition to fulﬁlling the basic requirement of generating a Lorentz- and Weyl- (gauge) invariant theory, it coincides in the weak ﬁeld limit with the Lagrangian of the Maxwell–Maxwell system,j L0 = −

1 F ∧ ∗F, 8π

while its behavior in the strong ﬁeld limit is such that it leads to ﬁnite total energies. One example of such a Lagrangian density is the well-known one-parameter family proposed by Born and Infeld [18]: Lβ = ∗

1 [1 − 1 − β 4 ∗ (F ∧ ∗F) − β 8 (∗(F ∧ F))2 ], 4 4πβ

for β > 0 (in the notation of [20]), which even leads to ﬁnite limits of the ﬁeld strengths at the location of the charge. What is perhaps less common knowledge — though it should be equally wellknown — is that a nonlinear aether law also has the power to reduce the strength of the spacetime singularity that is present when the electromagnetic ﬁeld is coupled to gravity.k This phenomenon was ﬁrst noticed by Hoﬀmann [22], who initially claimed that a solution of the Einstein–Maxwell–Born–Infeld system that he had found, was free of all singularities.l In the years since the publication of Hoﬀmann’s paper there have been several attempts at ﬁnding static electrovacuum spacetimes which are free of all singularities [23–31], either in the ﬁelds or in the metric, while at the same time various “no-go” results have been announced [32, 33], that seem to show that such solutions cannot exist. One of the goals of this paper is to take steps towards dispelling the confusion that seems to persist about this subject. Our approach here is to characterize aether laws that not only feature ﬁnite self-energies for point charges, but also lead to electrostaticm spacetimes with the mildest form of singularity possible.n Thus we will initially allow the aether law to have the most general form possible, and let the above requirements, of ﬁniteness of the self-energy and mildness of the singularity, as well as other criteria which i Born picked up on the program initiated by Mie [16], who however did not want point charges. See [20, 21] for an excellent account of the development of these ideas, and for their author’s key contribution to this program. j Which seems to have been discovered by Schwarzschild [17]. k Even though, as we will prove in this paper, it cannot quite eliminate the singularity. l Hoﬀmann’s enthusiasm for nonlinear electrodynamics was not dampened even after it was pointed out to him by Einstein and Rosen (see [23, fn. 15]), that a mild singularity remains at the center of his spacetime. m See Sec. 3 for a precise deﬁnition. n In fact, we will see that the remaining mild singularity in the ﬁeld and the metric is of the point-defect type. There are indications that this kind of singularity may be “just right” for the Hamilton–Jacobi equations to provide a law of motion for those defects [20, 21]. In this way it may turn out then that the “underwater stone” of nonlinear electrodynamics (in the words of [33]) is a gem after all!

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may arise during the course of analyzing the corresponding solutions, dictate the ﬁnal form that it should have. In particular we will prove the following: Theorem 1.1 (Informal Version). For any aether law which (a) agrees with that of Maxwell in the weak ﬁeld limit, (b) is derivable from a Lagrangian, with an energy tensor that satisﬁes the Dominant Energy Condition, and for which (c) the corresponding Hamiltonian satisﬁes certain growth conditions (to be made precise below), the following hold: • There exists a unique electrostatic, spherically symmetric, asymptotically ﬂat solution of the Einstein–Maxwell system (1.1)–(1.3) with that aether law, the maximal analytical extension of which is homeomorphic to R4 minus a line. It has a conical singularity on the time axis,o which is the mildest possible singularity for any spherically symmetric electrostatic spacetime whose aether law satisﬁes (a) and (b). No other singularities are present in the spacetime. • A generalization of Birkhoﬀ ’s Theorem shows that this solution is unique in the spherically symmetric class. • The electrostatic potential is ﬁnite on the axis of symmetry, which can be identiﬁed with the world line of a point charge. The electric ﬁeld has a point defect at the location of the charge. The ﬁeld has ﬁnite total electrostatic energy, which is equal to the ADM mass of the spacetime. The mass of the point charge is thus entirely of electromagnetic origin, i.e. it has no bare mass. • By scaling the Hamiltonian appropriately, one can arrange the total mass and the total charge of this solution to have any chosen values. The mass-to-charge ratio of the spacetime enters as a natural small parameter, measuring the departure from the Minkowski spacetime. • For small enough mass-to-charge ratio, there are no horizons of any kind and no trapped null geodesics in this spacetime. • The analysis of geodesics and test-charge trajectories shows that the naked singularity at the center of this spacetime is gravitationally attractive (unlike the case of superextremal RWN). The rest of this paper is organized as follows: In Sec. 2, we introduce the Lagrangian formulation of electromagnetics. Section 3 gives the equations satisﬁed by an electrostatic solution of (1.1)–(1.3), with an arbitrary aether law. In Sec. 4, we assume spherical symmetry as well, and obtain the general solution to the equations. Section 5 is devoted to the study of the singularities of this solution. In Sec. 6, we state and prove a Birkhoﬀ-type uniqueness result for these spacetimes. In Sec. 7, we give the precise version of our main result, and in Sec. 8, we carry out the qualitative analysis of geodesics and test-charge orbits. o That is to say, the limit as the radius goes to zero of the ratio of the circumference to the radius of a small spacelike circle going around the axis, exists but is not equal to 2π.

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2. Nonlinear Electrodynamics

p Let (M, g) be a four-dimensional Lorentzian manifold. Let (M) denote the bundle of p-forms on M. By an electromagnetic Lagrangian (density) we mean a 1 2 (M) ×M (M) which mapping Lem deﬁned on sections of the vector bundle 4 (M) (see [41] for details.) Thus if a is a 1-form on M and takes its values in f a 2-form, then Lem (a, f ) is a 4-form on M. The electromagnetic action is by deﬁnition S[a; D] := Lem (a, da), D

where D is a domain in M. A critical point of S with respect to variations of a that are compactly supported in D is called an electromagnetic potential A in D, δS = 0, δa a=A and the exterior derivative of it is the electromagnetic Faraday tensor F = dA. The Maxwell tensor M is by deﬁnition ∂Lem (2.1) M= ∂f a=A,f =F in the sense of evaluation, i.e. the object on the right, when evaluated on a vari2 2 (Tp M) as a 4-form is equal to M ∧ f˙ . The source-free ation f˙ ∈ Tp ( (M)) = Maxwell equations are the Euler–Lagrange equations for stationary points of the electromagnetic action S, and are equivalent to the system dF = 0,

dM = 0.

It can be shown [41] that the only Lorentz-invariant gauge-invariant source-free electromagnetic Lagrangians possible are those of the form Lem (a, f ) = − (x(f ), y(f )) [g],

(2.2)

where [g] is the volume form on M induced by the metric g, is a real-valued function of two variables, and x and y are the electromagnetic invariants 1 1 1 1 x(f ) := − ∗ (f ∧ ∗f ) = fµν f µν , y(f ) := ∗ (f ∧ f ) = fµν ∗ f µν . 2 4 2 4 k 4−k Here ∗ is the Hodge star operator ∗ : (M) → (M) with respect to the g metric, deﬁned by ∗σµ1 ···µ4−k =

1 ν1 ···νk σ

[g]ν1 ···νk µ1 ···µ4−k . k!

Note that for k-forms on a Lorentzian 4-manifold, ∗∗ = (−1)k+1 . Furthermore, conservation of parity implies that (x, y) = (x, −y), so that if we assume that is a C 1 function of its variables, then y (x, 0) = 0.

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By an aether law we simply mean a particular function (x, y) as the Lagrangian density function, which determines the way the electromagnetic vacuum interacts with the spacetime geometry.p For example, conventional (linear) electromagnetics corresponds to the choice = x made ﬁrst by Maxwell. Using * one deﬁnes a dot product on k-forms, by σ · τ = − ∗ (σ ∧ ∗τ ) =

1 µ1 ···µk σ τµ1 ···µk . k!

We also set |σ|2 := σ · σ even though this is not necessarily positive. It follows from (2.2) that ∗Lem = and ∗M =

∂ = x F + y ∗F. ∂F

The energy tensor T corresponding to the Lagrangian density function is a symmetric 2-covariant tensor ﬁeld on M deﬁned by Tµν = 2

∂ − gµν , ∂gµν

which in the case of an electromagnetic Lagrangian yields 1 1 λ λ Tµν = 2 x Fµλ Fν + y Fµλ ∗Fν − gµν = Fµλ ∗Mλν − gµν . 2 2

(2.3)

Recall that if both of the following hold, the energy tensor T is said to satisfy the Dominant Energy Condition: • Tµν Yµ Yν ≥ 0 for every future-directed timelike vector Y. • The vector −Tµν Yν is future directed causal when Y is a future-directed causal vector. The ﬁrst of these two is called the Weak Energy Condition. There is also a Strong Energy Condition: 1 Tµν − gµν tr T Yµ Yν ≥ 0, 2 for all future-directed timelike Y. One can prove [42] that the Dominant Energy Condition is satisﬁed for this ﬁeld theory if and only if x > 0,

− x x − y y ≥ 0.

Note that under the above assumption, tr T = −4( − x x − y y ) ≤ 0. We also note that it is possible for the Strong Energy Condition to be violated in nonlinear electrodynamics (even though, as we shall see, it can hold for nonlinear electrostatics). p Traditionally, aether law referred to the relationship between tensors M and F, similar to the constitutive relations of elastodynamics relating stresses to strains for the medium. In case of a Lagrangian theory, this is given by (2.1), and thus the choice is that of a particular Lagrangian.

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Next we deﬁne the following 1-forms, which provide a useful decomposition of the Faraday and Maxwell tensors. Let K be an arbitraryq non-null vectorﬁeld on M, and letr E := iK F

(2.4)

B := iK ∗F

(2.5)

D := iK ∗M = x E + y B

(2.6)

H := iK ∗ ∗M = −iK M = − y E + x B,

(2.7)

where iK denotes the interior product with the vectorﬁeld K, e.g. (iK F)ν = Kµ Fµν . For the Maxwell Lagrangian = x one has D = E and B = H. A general aether law will specify D and B as functions of E and H, or the other way around. Let X := g(K, K). Thus X > 0 wherever K is spacelike, X = 0 where K is null, and X < 0 where K is timelike. From the decomposition of F in terms of E and B it follows that x=

|E|2 − |B|2 , 2X

y=

E·B . X

(2.8)

We also obtain that, by (2.3) T(K, K) = E · D − X , while from (1.1) it follows that X R(K, K) = T(K, K) − tr T = B · H + X . 2

(2.9)

(2.10)

For K a timelike vectorﬁeld, we can deﬁne two electromagnetic Hamiltonians (partial Legendre transforms of the Lagrangian with respect to either E or B)s : ∂ − = |X|−1 B · H − , ∂B ∂ ˜ + = |X|−1 E · D + . H(D, B) := −|X|−1 E · ∂E ˜ we need to think of B (respectively E) as In the deﬁnition of H (respectively H) given by the aether law. Also, the factors of X in these will disappear in the next section, once the inner products are re-expressed in terms of a diﬀerent metric on spacelike slices which is conformal to the one induced by g. H(E, H) := |X|−1 B ·

q In the next section we will assume that K is a timelike Killing ﬁeld for (M, g), but the deﬁnitions in this section are independent of that. r Note that strictly speaking, only if K were the unit tangent vectorﬁeld to a timelike curve in M (the world-line of an observer) would we be justiﬁed in calling these the (ﬂattened) electric ﬁeld, magnetic induction, electric displacement, and magnetic ﬁeld, respectively. s These can also be deﬁned for a spacelike vectorﬁeld, but there will be some sign changes.

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We then have ˜ T(K, K) = |X|H(B, D),

R(K, K) = |X|H(E, H).

The Weak, Dominant, and Strong Energy Conditions, respectively, take the following form [43] in terms of the Hamiltonians ˜ ≥ 0, H

˜ ≥ |H|, H

H ≥ 0.

(2.11)

It is easy to see [44] that H is actually only a function of the three scalar invariants that one can form out of E and H using the metric. More precisely, H(E, H) = h(ν, ω, τ ), where ν :=

1 (E · E + H · H), 2|X|

ω :=

1 E · H, |X|

τ :=

1 (E · E − H · H), 2|X|

and that h ∈ C 1 (R3 ) is such that its gradient lies on the future unit hyperboloid in R3 , i.e. h2ν − h2ω − h2τ = 1,

hν > 0.

˜ in terms of the invariants µ, , σ deﬁned analogously in Similar results hold for H, terms of D, B: ˜ H(B, D) = ˜h(µ, , σ), where µ :=

1 (D · D + B · B), 2|X|

:=

1 D · B, |X|

σ :=

1 (D · D − B · B), 2|X|

˜ ∈ C 1 (R3 ) with gradient lying on the future unit hyperboloid. and likewise h 3. Electrostatic Spacetimes Let K be a timelike hypersurface-orthogonal Killing ﬁeld for the spacetime (M, g). Let X = g(K, K) < 0 and deﬁne e = dX. Let 1 Kµ Kν X denote the metric induced on the quotient Q of M under the symmetry generated by K, and let hµν = gµν −

γ = |X|h, so that γ is also a Riemannian metric on Q, conformal to h. Since K is assumed to be twist-free, the quotient Q can be identiﬁed with a spacelike hypersurface in M.

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Let (xi ), i = 1, 2, 3 be an arbitrary coordinate system on Q. Then the line element of g is given by ds2 = Xdt2 + |X|−1 γij dxi dxj . g

γ

It follows in particular that ∇ · V = X ∇ · X1 V for any vectorﬁeld V such that LK V = 0. The Einstein–Maxwell system in the static case reduces [43] to the following set of equations: γ 1 −2 e = H(E, H) (3.1) ∇· X X γ 1 · D =0 (3.2) ∇ X γ 1 B =0 (3.3) ∇· X 1 1 1 Rij [γ] − γij R[γ] = ei ej + (Ei Dj + Bi Hj ) 2 2X 2 X 1 1 2 |e| + H(E, H) , − γij 4X 2 γ X

(3.4)

where to close the system one has to remember that D=

∂H ∂E

,

B=

∂H ∂H

,

(3.5)

with being the operation of lowering indices with respect to the γ metric. Equation (3.4) conﬁrms that we can interpret the above as a system of Einstein equations for the manifold Q coupled to the ﬁelds (e, E, H). We further recall that Maxwell’s equations for F and M also imply that E and H are exact 1-forms E = dϕ,

H = dψ.

Thus the above is a system of equations for the three potentials X, ϕ, ψ, and the metric γ of the quotient manifold. We also observe that the parity conservation assumption about the Lagrangian y (x, 0) = 0 implies that, away from singularities, D = 0 whenever E = 0, and likewise H = 0 whenever B = 0. This fact, together with the invariance of the equations under interchanging D with B and E with H, implies that the system of equations in the magnetostatic case E ≡ 0 is formally the same as that in the electrostatic case H ≡ 0, even though (it turns out) the restrictions on the Hamiltonian under which meaningful solutions can be obtained are diﬀerent.t t For

examples of regular magnetostatic (magnetic monopole) solutions, see [28].

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In this paper we are conﬁning ourselves to the electrostatic case, where ω = = 0, τ = ν and σ = µ. If we deﬁne η(ν) = h(ν, 0, ν),

˜ 0, µ), ζ(µ) = h(µ, ˜

∂H then in the electrostatic case D = ∂H ∂E = η (ν)E and E = ∂D = ζ (µ)D. We can moreover express η in terms of ζ. This is because in the electrostatic case B = H = 0, and thus ˜ ∂H dζ ˜ = 2µ , H + H = D, Eγ = D, ∂D dµ γ

so that we have η(ν) = 2µζ (µ) − ζ(µ).

(3.6)

In terms of the reduced Hamiltonian ζ the Dominant Energy Condition (2.11) takes the following simple form ζ (µ) > 0,

ζ − µζ (µ) ≥ 0,

∀µ ≥ 0,

(3.7)

while the Strong Energy Condition reads 2µζ (µ) − ζ(µ) ≥ 0,

∀µ ≥ 0.

(3.8)

We note that in the electrostatic case, ν = −x and y = 0, thus η(ν) = − (x, 0) = − (−ν, 0). On the other hand the function ζ can be obtained from via a Legendre transform: Given = (x, y) let f (t) := − (− 21 t2 , 0) and let f∗ (s) = supt [st − f (t)] √ be the Legendre transform of f . Then it is easy to see that ζ(µ) = f∗ ( 2µ). As an example, here are the Lagrangian density function originally proposed by Born [15], and its two reduced Hamiltonians: √ √ B (x) = 1 + 2x − 1, ηB (ν) = 1 − 1 − 2ν, ζB (µ) = 1 + 2µ − 1. Let ξ := log(−X). With ν = 12 |dϕ|2γ and D = η (ν)dϕ, the electrostatic Einstein–Maxwell system then becomes γ

−ξ ∇ · dξ = 2e η(ν), γ

∇ · (e

−ξ

D) = 0,

1 1 Rij [γ] − γij R[γ] = ∂i ξ∂j ξ − e−ξ η (ν)∂i ϕ∂j ϕ − γij 2 2

(3.9)

(3.10) 1 |dξ|2γ − e−ξ η(ν) . 4 (3.11)

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4. Spherical Symmetry If we assume that the spacetime (and the electromagnetic ﬁeld) in addition to being static is also spherically symmetric, there will be a further reduction in the Einstein–Maxwell system. In particular, using the area-radius coordinate r we may rewrite the metric γ as follows γij dxi dxj = dr2 + eξ r2 (dθ2 + sin2 θ dφ2 ), where now ξ = ξ(r), and also ϕ = ϕ(r), D = D(r)dr. From (3.10), γ 1 1 0 =∇ · (e−ξ D) = ξ 2 ∂r (eξ r2 e−ξ D) = ξ 2 ∂r (r2 D), e r e r and thus one obtains that c D = 2 dr, r where c is an arbitrary constant. On the other hand, since |dr|2γ = 1, µ= From (3.9), we now obtain 1 d r2 eξ dr Change variable to u =

1 r

c2 1 2 |D|γ = 4 . 2 2r

(4.1)

dξ eξ r 2 = 2e−ξ η(ν). dr

to obtain d2 ξ η(ν) (e ) = c2 . du2 µ

(4.2)

We can now use (3.6), integrate (4.2) twice, change the order if integration and recompute the kernel to obtain a formula for ξ as a function of r: c2 ∞ ζ(µ) dr c ξ(r) =c + + , (4.3) e r r r µ r 2 2

where c, c , c are arbitrary constants, and µ = 2rc 4 . The requirement that the solution be asymptotically ﬂat now forces c = 1. On the other hand, setting ζ ≡ 0 should give the Schwarzschild solution, hence c = −2m0 where m0 is the mass parameter in the Schwarzschild metric. We can also ﬁnd an expression for the electrostatic potential ϕ as a function of r. Recall that dr ϕ (r)dr = dϕ = E = ζ (µ)D = cζ (µ) 2 , r and thus ∞ dr ζ (µ) 2 . ϕ(r) = c r r Since in the Maxwell–Maxwell case ζ(µ) = µ, comparison with the RWN solution shows that c = q0 , the charge parameter in RWN. Finally, a direct computation shows [43] that (3.11) is identically satisﬁed.

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Thus we have the following generalization of RWN to nonlinear gravitoelectrostatics: 2 ∞ 2m0 + eξ(r) = 1 − ζ(µ)r2 dr , (4.4) r r r ∞ dr q2 ϕ(r) = q0 ζ (µ) 2 ; µ = 0 4 . (4.5) r 2r r These simple and elegant formulae seem to have ﬁrst appeared in [25], with a diﬀerent derivation, although special cases of it were known long before (see below). Note that only the reduced Hamiltonian ζ makes an appearance in them. The prospect of generating exact solutions to the Einstein–Maxwell system with interesting and desirable properties, just by inserting a judiciously chosen ζ into these formulae has proven to be irresistible to many theoreticians. In particular we should mention the solution found by Hoﬀmann [45] to the Einstein–Maxwell– Born–Infeld system, which corresponds to the following Hamiltonian: (4.6) ζBI (µ) = 1 + 2µ − 1. Another early example is the solution found by Infeld and Hoﬀmann [23], where they made the following choice ζIH (µ) = log(1 + µ),

(4.7)

and obtained a metric which was completely smooth and free of all singularities! Their work was followed up by Rao [24], who attempted to ﬁnd a large family of actions leading to such solutions. Infeld and Hoﬀmann however may also be the ﬁrst of many investigators who have made the mistake of picking a Hamiltonian that is not admissible because it cannot arise from a Lagrangian: One has to remember that ζ(µ) is the electrostatic ˜ reduction of the Hamiltonian H(B, D), which in turn is subject to the following restrictions: (1) It must in the weak ﬁeld limit agree with Maxwell’s choice for the aether law. (2) It must correspond to an energy tensor that satisﬁes the dominant energy condition. (3) It must be the Legendre–Fenchel transform in E of a Lagrangian density = (E, B), i.e. it must be convex in D. It follows from the above that the function ζ is subject to the following requirements: (R1) limµ→0 ζ(µ) µ = 1. (R2) ζ (µ) > 0 and ζ − µζ (µ) ≥ 0 for all µ > 0. (R3) ζ (µ) + 2µζ (µ) ≥ 0 for all µ > 0. The Hamiltonian (4.7) proposed by Infeld and Hoﬀmann violates the third condition above. Therefore it cannot arise from a single-valued Lagrangian.

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The condition (R3) is equivalent to insisting that for the above solution (4.5), ν = − 21 (ϕ (r))2 be monotone decreasing in r (note that µ = q02 /(2r4 ) is always monotone, independent of the choice of a Lagrangian). It was shown by Bronnikov et al. [32] that this monotonicity requirement of ν rules out the possibility of having electrostatic solutions with a regular center (i.e. no curvature blowup at r = 0) that are at the same time Maxwellian in the weak ﬁeld limit. The same argument applies to show that the three conditions above are incompatible with having a regular center.u Therefore spacetimes corresponding to (4.4) and (4.5) must have some kind of a singularity at r = 0. In view of this fact, fantastic claims about existence of singularity-free point-charge metrics in nonlinear electrodynamics, which every now and then appear in the literature, should be viewed with a healthy dose of skepticism, and the “Hamiltonian” involved should be examined carefully, for it may violate one or more of the above conditions. We note also that (R3) implies in particular that the Strong Energy Condition, which we mentioned before can be violated in nonlinear electrodynamics, nevertheless holds in the electrostatic case, since 0 ≤ 2µζ + ζ = (2µζ − ζ) , which upon integration on [0, µ] and using (R1) implies that (3.8) holds. This will have important consequences, as we will see below. Interestingly, a question that should have been addressed long ago, but was not, is this: Since the above requirements rule out solutions that are everywhere regular, what is then the mildest singular behavior allowed by them? In the next section we characterize those static spherically symmetric point-charge metrics that have the mildest form of singularity possible at their center. 5. Singularities 5.1. Singularities in the metric We begin by calculating the curvature tensor of a static spherically symmetric metric of the form ds2 = f 2 dt2 − f −2 dr2 + r2 dΩ2 , with f = f (r). The nonzero components of the Riemann tensor are R0101 = f f + f , 2

R0202 = R0303 = −R1212 = −R1313 = r−1 f f , R2323 = r−2 (1 − f 2 ). The indices here refer to the rigid frame {ω (µ) } deﬁned as follows: ω (0) = f dt,

ω (1) = f −1 dr,

ω (2) = rdθ,

ω (3) = r sin θ dφ.

u Another possibility is of course, not to have a center at all [28]. The topology of such a spacetime however, does not seem to lend itself to the point-charge concept.

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Thus the Kretschman scalar is, with X = −f 2 , K2 = X + r−2 X + r−4 (1 + X)2 . 2

2

(5.1)

It is evident from (5.1) that for there to be no spacetime curvature singularity at r = 0 it is necessary and suﬃcient that 1+X(r) = O2 (r2 ) as r → 0.v More generally, the Kretschman scalar K will blow up like r−α if and only if 1 + X(r) = O2 (r2−α ). For spherically symmetric, electrostatic spacetimes, 2 X = −1 + m(r), r where

m(r) := m0 −

∞

ζ r

q02 2r4

r2 dr

is the mass function. Thus we see that K will blow up at least like r−3 if m(0) = 0, as it is for example in the case of the “negative mass” Schwarzschild solution, where m0 < 0 and q0 = 0. For the superextremal RWN solution, the situation is much worse since m(0) = −∞. Since our goal here is to characterize solutions which are as mildly singular as possible at the location of the charge, we may start by requiring m(0) = 0. Now, since |q0 |3/2 ∞ −7/4 y ζ(y)dy = m0 − |q0 |3/2 Iζ m(0) = m0 − 11/4 2 0 and m0 is an integration constant, it is always possible to meet this requirement as long as ∞ y −7/4 ζ(y)dy < ∞. (5.2) Iζ := 2−11/4 0

From now on we will add this to the list of requirements that the reduced Hamiltonian ζ must satisfy: ∞ µ−7/4 ζ(µ)dµ < ∞. (R4) 0

Note that this new restriction implies the following: If ζ(µ) is assumed to grow like a power µα , then we must have α < 3/4, which of course rules out the Maxwellian case. Having made such a choice of the integration constant m0 , we now observe that mADM = m(∞) = |q0 |3/2 Iζ . This means that the mass of this spacetime is entirely of electrical nature. Moreover, by an appropriate scaling of the aether law, namely β (x, y) := β −4 (β 4 x, β 4 y), a continuous function f : R+ → R, integer k and α > 0 we say that f = Ok (r α ) if limr→0 r j−α dj f /dr j exists and is ﬁnite for j = 0, . . . , k.

v For

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which for the reduced Hamiltonian amounts to using ζβ (µ) = β −4 ζ(β 4 µ) in place of ζ, it is possible to “ﬁt” the ADM mass and total charge of this solution to the empirical mass and charge of any particle. This is because Iζβ = β −1 Iζ , so for a given pair of numbers (q, m) setting q0 = q and β :=

q3/2 Iζ m

(5.3)

will result in m(∞) = m, while at the same time the scaled version of an admissible Hamiltonian function ζ will remain admissible. In this connection it is worth mentioning that if one carries out this procedure for the Born–Infeld Lagrangian, with m0 and q0 set to the mass and charge of the electron, the value for the scaling parameter β thus obtained coincides with the value originally proposed by Born [20]. Once m0 is chosen as above, the mass function can be rewritten as follows: ∞ r 2 q q 2 2 ζβ ζβ (5.4) r dr = m − r dr . m(r) = 2r4 2r 4 0 r We can now obtain some estimates for m(r). Recall the second part of (R2): ζ(µ) ≥ µζ (µ). Integrating this inequality on [0, µ] and using (R1) we easily obtain ζ(µ) ≤ µ,

∀µ > 0.

This gives the following lower bound for the mass function: m(r) ≥ m −

q2 , 2r

which in turn gives the following bound on the metric coeﬃcient eξ : eξ ≤ 1 −

2m q2 + 2. r r

These two estimates are clearly only useful for large r. In fact, assuming slightly more than (R1), one can turn these into large-r asymptotics for m and eξ . Namely, let us assume that (R1)

ζ(µ) = µ + O(µ5/4 ),

as µ → 0.

Substituting into the second expression for the mass function in (5.4) we obtain 1 1 q2 2m q2 ξ +O + 2 +O m(r) = m − , e =1− . 2 2r r r r r3 As advertised, these asymptotics are identical to those of the RWN solution.

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We now need to establish the small-r behavior of the mass function. Clearly, m(r) = O2 (r3 ) if and only if ζ goes to a constant as µ → ∞. On the other hand, we recall (R3) on ζ, i.e. the convexity requirement. It is equivalent to (µ1/2 ζ (µ)) > 0.

(5.5)

Integrating this on an interval [µ0 , µ] for a ﬁxed µ0 > 0, we obtain √ ζ(µ) ≥ C1 + C2 µ, with C1 , C2 = 0 constants depending on µ0 . Therefore the reduced Hamiltonian must grow at least like µ1/2 in order to satisfy this requirement. In particular, it cannot go to a constant at inﬁnity, hence there will be curvature blowup at r = 0 no matter what aether law is chosen, as anticipated in [32]. Assuming now that the reduced Hamiltonian grows at the slowest possible rate, namely like µ1/2 , we obtain that there must be a conical singularity at r = 0 where K blows up like r−2 , and that there are no horizons for this metric. In order to do this rigorously, we need to make the growth condition more precise: (R5) There exists positive constants Jζ , Kζ , Lζ depending only on the proﬁle ζ such that √ √ Jζ Lζ Jζ ≤ ζ (µ) ≤ Jζ µ − Kζ ≤ ζ(µ) ≤ Jζ µ, − . µ 2µ1/2 2µ1/2 We should here pause to mention that an example of a reduced Hamiltonian satisfying all ﬁve requirements (R1)–(R5) is the one of Born–Infeld (4.6). Many other examples can easily be constructed by considering smooth, monotone increasing, √ concave functions of µ that behave like µ for small µ and like c µ for large µ. Assuming (R5), from (5.4) we have, √ 2Jζ m A 2 . (5.6) , := m(r) ≤ r, A := 2 2 Iζ |q| This right away implies that there will be no horizons as long as is small enough, since 2m(r) ≥ 1 − A 2 > 0. −X = 1 − r Combining (R5) with our previously obtained bounds for the reduced Hamiltonian, √ √ max{0, Jζ µ − Kζ } ≤ ζ(µ) ≤ min{µ, Jζ µ},

Jζ Lζ Jζ max 0, 1/2 − ≤ ζ (µ) ≤ min 1, 1/2 . µ 2µ 2µ

(5.7)

We can now use this to obtain a lower bound for the mass function that does not degenerate near r = 0, namely m(r) ≥

Kζ A 2 r − 4 r3 , 2 3β

(5.8)

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as well as the following small r asymptotics for m: m(r) =

B 2 6 3 A 2 r− r + o2 (r3 ), 2 2m2

B 2 :=

2Kζ . 3Iζ4

Consequently, −X = eξ = a2 + b2 r2 + o2 (r2 )

√ 3 with a = 1 − A 2 and b = B m . This in turn implies that there is a conical singularity at r = 0, because the line element of the spacetime metric is Xdt2 − X −1 dr2 +r2 (dθ2 +sin2 θ dφ2 ), and the coeﬃcient of dr2 at r = 0 is a1 , which is greater than one. In fact, introducing standard Cartesian space coordinates (x1 , x2 , x3 ) near 3 r = 0, with r = ( i=1 (xi )2 )1/2 we see that the line element in these coordinates is i j

1 xx 2 −1 + δij dxi dxj , Xdt + 2 |X| r i,j thus the metric has no continuous extension at r = 0 unless a = 1. We therefore need to take the axis r = 0 out of the spacetime manifold M, which gives it the topology of R4 minus a line. We also note that, given a proﬁle ζ, the deﬁcit angle of the conical singularity is proportional to = m/|q|. This will be quite small precisely when the empirical charge-to-mass ratio of the particle to which this solution is being ﬁtted is large, which happens to be the case for the electron and the proton, etc. This means that in the study of these metrics it is permissible to consider the small regime. The last observation we would like to make about the metric before moving on to discussing the electric ﬁeld, is that the metric coeﬃcient eξ = −X is monotone increasing. The easiest way to see this is from (4.2). Recall that the assumption (R3) implies that the Strong Energy Condition is satisﬁed, and thus η(ν) ≥ 0, so that eξ is convex as a function of u = 1r . We have already established that a2 ≤ eξ ≤ 1, eξ |u=0 = 1, limu→∞ eξ = a2 < 1. Thus eξ cannot have a local maximum or a local minimum at a ﬁnite u, and must therefore be monotone decreasing in u, hence monotone increasing in r. We will see that this fact, which is in great contrast to the behavior of the same metric coeﬃcient in the RWN spacetimes, has important consequences, in particular for the trajectories of test particles. 5.2. Singularities in the electric field In the spherically symmetric, electrostatic case, the only nonzero component of the Faraday tensor is Frt = ϕ (r) and the 1-form E = iK F = dϕ. We have ∞ 2 4 q β dr ϕ(r) = q ζ . 4 2r r2 r One easily computes that sgn(q)ϕ(0) =

3

, 2

sgn(q)ϕ (0) = −

A 3

. 2m

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Moreover, using (5.7), we have

|q| 3 A 3 C 7 3 3 A 3 − r ≤ sgn(q)ϕ(r) ≤ min , − r+ r , max 0, 2 2m r 2 2m 3m3

C 7 |q| A 3 A 3 + 3 r2 , max − 2 , − ≤ sgn(q)ϕ (r) ≤ min 0, − r 2m 2m m with C := 2Lζ /Iζ4 . In particular, ϕ is monotone decreasing, and is asymptotic to the Coulomb potential as r → ∞. Moreover, ϕ (0) = 0, and thus E becomes undeﬁned at r = 0. More precisely, since E = ϕ (r)dr and dr is a unit covector whose direction is undeﬁned at r = 0, it follows that the covectorﬁeld E is of ﬁnite magnitude and undeﬁned direction, i.e. has a point defect at r = 0. E is otherwise smooth. We now check that the total electrostatic energy is ﬁnite, and in fact is equal to the ADM mass: By virtue of (1.1), the energy tensor T is divergence free, i.e. ∇µ Tµν = 0. Let Pµ := −Tµν Kν where K = ∂t is the timelike Killing ﬁeld of (M, g). It follows that δP = ∗d ∗ P = 0 and thus by the divergence theorem ∗P is a conserved current, i.e. Σt ∗P is independent of t. With coordinates (t, r, θ, φ) as before, K = (1, 0, 0, 0) and we calculate that the only nonzero component of ∗P is −1 T(K, K) − det g (∗P)123 = −g00 P0 − det g = X ˜ = H(B, D) − det g = ζ(µ)r2 sin θ. Thus, deﬁning the total electromagnetic energy carried by the slice Σt to be 1 E := ∗P, 4π Σt we see that for the particle-spacetimes under study ∞ 2π π ∞ 1 (∗P)123 drdθdφ = ζ(µ)r2 dr = m, E= 4π 0 0 0 0 as promised. 6. Uniqueness in the Spherically Symmetric Class Birkhoﬀ’s celebrated theorem [34] states that any spherically symmetric solution of vacuum Einstein’s equations Rµν = 0 is locally isometric to a region in Schwarzschild spacetime.w This was generalized to the electrovacuum case w This result is often paraphrased inaccurately as “any spherically symmetric vacuum solution of Einstein’s equations must be static”, which makes it no longer true in general [36]. The theorem was discovered ﬁrst by Jebsen [35], whose proof appeared two years before Birkhoﬀ’s. Jebsen’s proof however, contains an error [36]. There is a more general result, by Eiesland [37], on necessary and suﬃcient conditions for the existence of an extra Killing ﬁeld for spherically symmetric (not necessarily vacuum) spacetimes, a preliminary version of which was also announced [38] before Birkhoﬀ’s book but the ﬁnal paper appeared two years after it. Birkhoﬀ’s theorem is a corollary of Eiesland’s result.

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by Hoﬀmann [39], thereby proving the uniqueness of the RWN solution in the spherically symmetric class. Many other extensions and generalizations have followed since, see [40] and references therein. Here we prove that the charged-particle spacetime whose existence we have established in the above, enjoys the same uniqueness property. To begin, we recall the deﬁnition of spherical symmetry: A Lorentzian manifold (M, g) is locally spherically symmetric if every point in M has an open neighborhood V such that (1) There is a linearly independent set of vector ﬁelds {Ui }3i=1 on V which generate a faithful representation of the Lie algebra of the rotation group SO(3), and whose orbits are two-dimensional and spacelike. (Note that the orbits do not need to be contained in V .) (2) LUi g = 0 in V for i = 1, 2, 3, i.e. Ui are Killing vector ﬁelds for the metric. Next we recall the following (see [37] for a proof): Proposition 6.1. The necessary and suﬃcient condition for local spherical symmetry of a smooth Lorentzian manifold (M, g) is that every point in M has a neighborhood in which there exists a system of local coordinates (t, r, θ, φ) such that the line element of the metric in those coordinates has the form ds2 = −A2 (t, r)dt2 + B 2 (t, r)dr2 + C 2 (t, r)(dθ2 + sin2 θ dφ2 ),

(6.1)

with A, B, C smooth functions of (t, r).x In this case, the three Killing ﬁelds generating spherical symmetry are U1 = sin φ ∂θ + cot θ cos φ ∂φ ,

U2 = cos φ ∂θ − cot θ sin φ ∂φ ,

U3 = ∂φ ,

and we can check that these vectorﬁelds satisfy the commutation relations of the Lie algebra of the rotation group: [Ui , Uj ] = kij Uk , where ijk is the completely antisymmetric 3-symbol. We also note that [Ui , Z] = 0 if Z is either ∂t or ∂r . Moreover, we can express ∂θ and ∂φ in terms of (U1 , U2 ): ∂θ = sin φ U1 + cos φ U2 ,

∂φ = tan θ(cos φ U1 − sin φ U2 ).

(6.2)

Suppose now that F = Fµν dxµ ∧ dxν is a spherically symmetric 2-form deﬁned in a neighborhood V of a spherically symmetric manifold (M, g), i.e. LUi F = 0 for i = 1, 2, 3. Let X, Y, Z be any three vectorﬁelds on M. For any 2-form F we have the identity (LX F)(Y, Z) = XF(Y, Z) − F([X, Y], Z) − F(Y, [X, Z]). error was that he had assumed C(t, r) ≡ r. This can only be achieved if ∇C is spacelike, thus for a complete proof one had to consider three other cases as well, of ∇C being timelike, null, or zero.

x Jebsen’s

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Thus, for a spherically symmetric F, we obtain the following (1) Ui F(∂t , ∂r ) = 0, i = 1, 2, 3. (2) Ui F(Uj , Z) = kij F(Uk , Z) for i, j = 1, 2, 3 and Z ∈ {∂t , ∂r }. (3) Ui F (Uj , Uk ) = 0 for i, j, k distinct, and Ui F(Ui , Uj ) = kij F(Ui , Uk ) for i = 1, 2, 3. Thus, from item (1), ∂θ F(∂t , ∂r ) = ∂φ F(∂t , ∂r ) = 0 which implies that F(∂t , ∂r ) = f (t, r).

(6.3)

From item (2) we have U1 F (U3 , Z) = −F(U2 , Z), U2 F(U3 , Z) = F(U1 , Z) and U3 F(U3 , Z) = 0, giving a diﬀerential equation for F(U3 , Z), and solving that we obtain that for each choice of Z there is a function a(t, r) such that F(∂φ , Z) = a(t, r) sin θ.

(6.4)

On the other hand, item (2) also implies U3 F(U1 , Z) = F(U2 , Z), U3 F(U2 , Z) = −F(U1 , Z), which gives a diﬀerential system for F(Ui , Z), for i = 1, 2. Solving it, making use of (6.2) and (6.4) one obtains that for each choice of Z there is a function b = b(t, r) such that F(∂θ , Z) = b(t, r).

(6.5)

Finally, in a similar manner item (3) gives rise to a diﬀerential system for F(U3 , Ui ), i = 1, 2, which upon solving yields that there is a function c(t, r) such that F(∂θ , ∂φ ) = c(t, r) sin θ. We have thus shown that a spherically symmetric 2-form F on a spherically symmetric manifold, with coordinates (t, r, θ, φ) as above, must have the following form Ftr = f (t, r), Ftφ = a1 (t, r) sin θ,

Ftθ = b1 (t, r),

Fr,θ = b2 (t, r),

Frφ = a2 (t, r) sin θ,

Fθφ = c(t, r) sin θ.

(6.6)

Now, assume that F is a closed 2-form: dF = 0. Using the identity LX F = iX dF + diX F valid for any vectorﬁeld X and any tensor F we obtain that for a spherically symmetric closed 2-form F, diUj F = 0,

j = 1, 2, 3.

Writing the resulting diﬀerential equations for the components of F as found in (6.6) one then sees that the only solution is a1 = a2 = b1 = b2 = 0,

c(t, r) = c,

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with c an arbitrary constant. Thus we have proven the following Lemma (stated without proof in [39]): Lemma 6.1. On a spherically symmetric manifold M with coordinates (t, r, θ, φ) the only nonzero components of a spherically symmetric Faraday tensor F are Ftr = f (t, r),

Fθφ = c sin θ,

(6.7)

where f is an arbitrary function and c an arbitrary constant. Note that the constant c here is the total magnetic charge: −1 c= F, 4π S where S is any spacelike 2-sphere in M that contains the origin. We can now compute the energy tensor T. Recall that the line element of the metric g has the form (6.1). The nonzero components of the dual tensor ∗F are therefore ∗Ftr =

AB c, C2

∗Fθφ =

−C 2 f (t, r) sin θ, AB

and the dual to the Maxwell tensor ∗M = x F + y ∗ F is computed to have components AB C2 f sin θ. (6.8) ∗Mtr = x f + c y 2 , ∗Mθφ = c x − y C AB We now quote the main theorem in [37] (slightly reworded to match our notation): Theorem 6.1 ([37]). The necessary and suﬃcient conditions that a locally spherically symmetric Lorentzian manifold with line element ds2 = −A2 dt2 + B 2 dr2 + C 2 (dθ2 + sin2 θ dφ2 ), A, B and C being arbitrary functions of t and r, and C not a constant, shall admit a one-parameter group of isometries generated by a vectorﬁeld K = k0 (t, r)∂t + k1 (t, r)∂r are A2 Gtr = Ψ ∂r C∂t C,

(6.9)

AB(Grr − Gtt ) = Ψ C[A2 (∂r C)2 + B 2 (∂t C)2 ],

(6.10)

where Gνµ = Rνµ − 12 Rδµν is the Einstein tensor and Ψ = Ψ(C) is an arbitrary function of C, or a constant. If the above conditions are satisﬁed, then the Killing ﬁeld K is, up to a constant multiple, given by k0 (t, r) = −

1 ∂r C, eΨ AB

k1 (t, r) =

1 ∂t C. eΨ AB

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We are now in a position to state our uniqueness result: Theorem 6.2. Let (M, g, F) be a locally spherically symmetric solution of the Einstein–Maxwell system (1.1)–(1.3). There exists a system of local coordinates (t, r, θ, φ) on M in which the line element of the metric g takes the form (6.1). If ∇C is spacelike in M, then there exists a Killing vectorﬁeld K which is hypersurface-orthogonal and timelike everywhere in M, i.e. the solution is static as well, and in the case of no magnetic charge it is thus isomorphic to a region in the electrostatic charged-particle spacetime found in our Sec. 4. Proof. The existence of the coordinate system is guaranteed by Proposition 6.1. By Lemma 6.1 the Faraday tensor F in these coordinates has the form (6.7). On the other hand, by virtue of (1.1) we must have Gνµ = Tνµ . The relevant components of the energy tensor T are easily computed from (6.7) and (6.8) to be: AB −f Ttt = Trr = 2 2 x f + c y 2 − , Trt = 0. A B C We thus see that the solution satisﬁes the conditions of Eiesland’s theorem, with Ψ a constant. We can take the components of the Killing ﬁeld to be k0 = −∂r C/(AB) and k1 = ∂t C/(AB). In that case g(K, K) = −(∂r C/B)2 + (∂t C/A)2 = −g −1 (∇C, ∇C), and thus K is timelike if and only if ∇C is spacelike. Since dK is easily seen to be proportional to dt ∧ dr, the twist of K vanishes: K ∧ dK = 0, and therefore K is hypersurface-orthogonal, i.e. the solution is static. It would then be a matter of changing the coordinates (t, r) into new coordinates (t , r ) that satisfy Kt = k0

∂t ∂t + k1 = 1, ∂t ∂r

Kr = k0

∂r ∂r + k1 = 0, ∂t ∂r

(6.11)

which is solvable by quadratures, in order for the line element of g in the new coordinates to have the form ds2 = −P 2 dt2 + Q2 dr2 + R2 (dθ2 + sin2 θ dφ2 ), where P, Q, R are functions of r only, and R is not a constant. At this point we may again use that G00 − G11 = 0, where the indices now refer to the new coordinates x0 = t , x1 = r . From the deﬁnition of Gνµ we compute (see, e.g., [37]) that G00 =

1 (Q3 + 2Q RR − QR2 − 2QRR ), Q3 R 2

G11 =

1 (P Q2 − P R2 − 2RR P ). P Q2 R 2

The equality of the two is now easily seen to imply that there exists a constant λ such that P Q = λR . Thus letting τ = λt and taking (τ, R, θ, φ) as the new

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coordinates, the metric takes the form P2 2 λ2 dτ + 2 dR2 + R2 (dθ2 + sin2 θ dφ2 ), 2 λ P which is the form we assumed in Sec. 4 in order to derive the static spherically symmetric solution we found. ds2 = −

Remark 6.1. The above uniqueness result can most likely be strengthened by considering the remaining cases, of ∇C being timelike, null, or zero, ﬁnding in each case what the solution reduces to, similar to the treatment of the cosmological vacuum solutions in [40]. We do not pursue this here, however. 7. Precise Statement of the Main Result We are now in a position to give the precise version of the main result: Theorem 7.1. Let ζ : R+ → R+ be any C 2 function satisfying the following conditions: (1) (2) (3) (4) (5)

ζ(µ) = µ + O(µ5/4 ) as µ → 0. ζ (µ) > 0 and ζ − µζ ≥ 0 for all µ > 0. ζ (µ) + 2µζ (µ) ≥ 0 for all µ > 0. ∞ Iζ := 0 µ−7/4 ζ(µ)dµ < ∞. √ There exists positive constants µ0 , Jζ , Kζ , Lζ such that Jζ µ − Kζ ≤ ζ(µ) ≤ √ Jζ L Jζ − µζ ≤ ζ (µ) ≤ 2µ1/2 hold for µ > µ0 . Jζ µ, and 2µ1/2

Let m > 0 and q = 0 be two ﬁxed real numbers, and let := m/|q|. Deﬁne ζβ (µ) :=

1 ζ(β 4 µ), β4

β :=

q3/2 Iζ . m

Let be any electromagnetic Lagrangian density function with the property that the electrostatic reduction of its Hamiltonian is the function ζβ . Then the following hold: (1) The Einstein–Maxwell system (1.1)–(1.3) with the electromagnetic Lagrangian density L = − ∗ has a unique electrostatic, spherically symmetric, asymptotically ﬂat solution, the maximal analytic extension of which is a Lorentzian manifold (M, g), called a charged-particle spacetime. M is topologically equivalent to R4 minus a line. There exists a global coordinate system (t, r, θ, φ) on M with the property that K := ∂t is an everywhere-timelike Killing ﬁeld, r is the area-radius of rotation group orbits, r = 0 is invariant under K, and (θ, φ) are standard spherical coordinates on S2 . (2) Any region in a spherically symmetric electrovacuum spacetime where the arearadius of the rotation group orbits has a spacelike gradient, is necessarily static as well, and in the case of no magnetic charge is isometric to a region in the corresponding charged-particle spacetime (M, g) deﬁned above.

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(3) The line element of the metric g has the form ds2 = −eξ dt2 + e−ξ dr2 + r2 (dθ2 + sin2 θ dφ2 ), where 2m(r) , e := 1 − r ξ

m(r) :=

r

ζβ 0

q2 2s4

s2 ds,

m(r) is the mass function of the spacetime M. In particular, m(0) = 0, m is increasing in r, and the ADM mass of M is m(∞) = m. The mass function moreover has the following asymptotics: B 2 6 3 A 2 r− r + o2 (r3 ) 2 2m2 1 q2 +O m(r) = m − 2r r2 m(r) =

√

where A :=

2Jζ Iζ2

and B 2 :=

2Kζ . 3Iζ4

as r → 0, as r → ∞,

Consequently, grr |r=0 = (1 − A 2 )−1 > 1 and

thus the metric has a conical singularity at r = 0. (4) The mass function satisﬁes the following bounds

2

2 A

B 2 6 3 A

q2 max r− r, m . r , m − ≤ m(r) ≤ min 2 2m2 2r 2 Consequently, for small enough, gtt < 0 everywhere, which implies that there are no horizons in M, hence the conical singularity at r = 0 is naked. (5) r = 0 is a curvature singularity, where the Kretschman scalar blows up like r−2 . Any spherically symmetric electrostatic spacetime homeomorphic to R4 minus a line, whose reduced Hamiltonian satisﬁes the ﬁrst three conditions listed above for ζ will necessarily have a singularity at r = 0 which is at least this strong. (6) The spacetime M is endowed with an electromagnetic ﬁeld F = dA, where A = −ϕ(r)dt, ∞ 2 q ds ζβ ϕ=q 2s4 s2 r is the electrostatic potential, and E := e−ξ/2 dϕ is the (ﬂattened) electric ﬁeld.y The potential ϕ is smooth, monotone decreasing, and has the following asymptotics 3 A 3 − r + O(r3 ) 2 2m 1 q ϕ(r) = + O r r3

sgn(q)ϕ(r) =

as r → 0, as r → ∞.

Thus the total charge of the spacetime is q. The electric ﬁeld E is smooth everywhere except at r = 0 where it has a point-defect, i.e. its magnitude has a ﬁnite limit but its direction is undeﬁned. y That

ˆ is, E = iK ˆ F where K is the unit vectorﬁeld in the direction of K.

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(7) The total electrostatic energy carried by a time slice is ﬁnite and is equal to m, therefore the mass of the spacetime is entirely of electric origin. (8) Radial null geodesics fall into the singularity and are thus incomplete in one direction, while null geodesics with nonzero angular momentum are inﬁnitely extendible in both directions. For small, these are deﬂected by the singularity by an amount that is proportional to 2 , and there are no trapped null geodesics. (9) Similarly, timelike geodesics and test-charge trajectories can only reach the singularity if they are radial, while non-radial ones have either bound orbits or escape orbits. The singularity at r = 0, in contrast to the one in the superextremal RWN solution, is gravitationally attractive. The only items in the above theorem that we have not yet proved are the last two, regarding geodesics. These will be established below: 8. Geodesics and Test-Charge Trajectories Consider the following Lagrangian density, deﬁned on a velocity bundle (see [41] for deﬁnitions): 1 gαβ (q)vα vβ + eAλ (q)vλ , 2 where A is an electromagnetic vector potential deﬁned on M, q ∈ M and v ∈ Tq M. The corresponding action ˙ L(q, q)ds A[q] = L(q, v) =

R

is deﬁned on curves q(s) = (q (s)) in M, where the dot represents diﬀerentiation with respect to an aﬃne parameter s, which is related to arclength parametrization (“proper time” for timelike curves) τ by τ = ms. The stationary points of this action are, for e = 0 and m2 = 1, 0, −1 respectively timelike, null, and spacelike geodesics of the spacetime. Moreover, stationary points of the action with e = 0 and m2 = 1 represent possible worldlines of a “test charge”z of mass m and charge e in the spacetime (M, g) permeated by the electromagnetic ﬁeld F. The Euler–Lagrange equations for these geodesics are α

e dqβ Ddqα , = Fα β (q) 2 dτ m dτ z This is a fairly standard treatment of test charges, cf. [46–50]. Notice however that it is not without conceptual problems: If one thinks, e.g., as in [49, 50], that the charged-particle spacetime in question is the spacetime of an electron, then one cannot possibly “test” an electron with an electron, i.e. a particle which has charge comparable to that of the charged-particle spacetime can by no stretch of imagination be considered a test particle, since its eﬀect on the geometry of spacetime cannot be ignored, while on the other hand if the spacetime is to represent an elementary particle then of course no particle of smaller charge exists to test it with! The situation considered here is thus merely a cartoon, and proper treatment of this subject is postponed to a future paper, where we plan to consider charged-particle spacetimes featuring two point charges.

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where D/dτ is covariant diﬀerentiation operation on tangent vectors, and F is the electromagnetic ﬁeld tensor. Let ∂L = gαβ (q)vβ + eAα (q) pα = ∂qα be the canonical momenta, and H the corresponding Hamiltonian to L deﬁned by 1 H = pα vα − L = gαβ (q)(pα − eAα (q))(pβ − eAβ (q)) 2 1 2 e2 = |p| − eA · p + |A|2 . 2 2 ∂H ˙ The geodesic equations in Hamiltonian form are q˙ = ∂H ∂p and p = − ∂q . The ﬁrst constant of motion is H itself, and from the normalization condition discussed above we have that along solutions H = − 12 m2 . Moreover, J (p, q) is a constant of motion iﬀ {J , H} = 0. Let us now take g to be a point-charge metric, with empirical charge q and mass m (that is entirely of electric origin). g is the spherically symmetric electrostatic solution to the Einstein–Maxwell system with a nonlinear aether law, corresponding to a reduced Hamiltonian ζβ that is subject to the restrictions (R1)–(R5) discussed in the previous sections but otherwise arbitrary, and let F be the corresponding electromagnetic ﬁeld permeating this spacetime. The line element of g thus has the form

ds2g = −eξ dt2 + e−ξ dr2 + r2 (dθ2 + sin2 θ dφ2 ), where (θ, φ) are spherical coordinates on the standard unit sphere S2 . Moreover F = dA where A = ϕ(r)dt, and the functions ξ(r), ϕ(r) are given in terms of ζ by r 2 q 2m(r) ξ ; m(r) = ζβ (8.1) e = 1− s2 ds; 4 r 2s 0 ∞ 2 q ds ϕ=q ζβ ; ζβ (x) = β −4 ζ(β 4 x), (8.2) 4 2s s2 r with β as in (5.3). With coordinates (xα ) = (t, r, θ, φ) thus chosen, we have pt = −eξ t˙ + eϕ; and H=

pr = e−ξ r; ˙

˙ pθ = r2 θ;

˙ pφ = r2 sin2 θ φ,

1 1 1 −e−ξ (pt − eϕ)2 + eξ p2r + 2 p2θ + 2 2 p2φ . 2 r r sin θ

(8.3)

Since ∂H/∂t = ∂H/∂φ = 0 we get two more constants of motion, pt and pφ , and we call the values that they take on a solution the energy E and angular momentum about the φ axis Φ: pt ≡ E;

pφ ≡ Φ.

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A fourth constant is also found, by observing that d 2˙ 2Φ2 ∂H d r θ = pθ = − = 2 3 cos θ. ds ds ∂θ r sin θ Thus, if one can arrange it so that θ = π/2 when θ˙ = 0, then θ¨ = 0 as well, so that θ will stay constant at π/2. But since we still have the freedom of choosing the axes for the two spherical coordinates, we can always choose the φ axis in such a way that the plane θ = π/2 is the plane through the origin that contains the geodesic’s initial velocity vector, so that θ˙ = 0 initially. We have thus shown that all geodesics of these particle-spacetimes are planar, and for any single geodesic we can always assume that it is contained in the equatorial plane θ = π/2. Now, evaluating (8.3) along a solution we obtain that (−e−ξ (E − eϕ)2 + e−ξ r˙ 2 + ˙ m2 )r2 ≡ −Φ2 , which can be used to ﬁnd an equation for r: 2 Φ + m2 eξ . r˙ 2 = (E − eϕ)2 − r2 With four constants of motion the equations are reduced to a ﬁrst-order system of ODEs. We proceed to study this system, ﬁrst for timelike and null geodesics (e = 0) and then for test-charges (e = 0). 8.1. Geodesics For timelike (respectively null) geodesics, e = 0 and m = 1 (respectively m = 0). The equations are 2 ˙t = −e−ξ E, r˙ = ± E 2 − Φ + m2 eξ , φ˙ = 1 Φ. (8.4) r2 r2 √ We observe that for Φ = 0 the geodesic is radial, with r˙ = ± E 2 − m2 eξ . Thus, since a2 ≤ eξ < 1, for E < ma there is no solution. For ma ≤ E ≤ m the geodesic does not have enough energy to escape, and will fall radially towards the singularity r = 0 where curvature blows up. The geodesic reaches the singularity at a ﬁnite parameter value, since the function under the square root has only a simple zero. This is also the case for geodesics with E > m (and therefore for all null geodesics), the only diﬀerence being that they can reach inﬁnity, and in fact have a well-deﬁned asymptotic velocity: (dr/dt)|r=∞ = 1 − m2 /E 2 . All radial geodesics are therefore inextendible in one direction and thus incomplete. Let Φ > 0 now. It is more convenient to reformulate the equations in terms of a reciprocal radial variable, and to measure length in units of m. Let m x := . r From the r˙ equation in (8.4) one easily obtains dφ ±1 = x , dx δ 2 − (x2 + γ 2 ) 1 − 2xM 2

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where γ :=

mm , Φ

δ :=

mE , Φ

:=

m , |q|

and M (y) is the normalized mass function ∞ 1 ζ(µ)µ−7/4 dµ M (y) := 11/4 2 Iζ Iζ4 y4 /2 = 1−

1

211/4 Iζ

Iζ4 y 4 /2

ζ(µ)µ−7/4 dµ,

0

which, in view of (5.7), satisﬁes the following bounds

1 A B2 A max 1 − y, − , ≤ M (y) ≤ min 1, 2 2y 2y 3 2y

1 −A −A 3B 2 max − , 2 ≤ M (y) ≤ min 0, 2 + , 2 2y 2y 2y 4 where

√ 2Jζ ≥ 1, A := Iζ2

B 2 :=

2Kζ , 3Iζ4

are constants that depend only on the proﬁle ζ. Note that for the RWN metric, MRWN (y) = 1 − 12 y. Let x 2 2 fγ,(x) := (x + γ ) 1 − 2xM 2 .

We note that for null geodesics, γ = 0. In that case, it is not hard to see that f0, (x) = x2 eξ(m/x) is monotone increasing, for small enough, thus there is a unique x0 > 0 such that δ = f0, (x0 ), and we must have x < x0 along any null geodesic. Hence r0 := m/x0 is the perihelion for the geodesic, i.e. the closest it can get to the singularity. Furthermore, since f0, has no critical points, it follows that there are no bounded orbits for null geodesics, so that all null geodesics with Φ > 0 can be extended in both directions to an inﬁnite value for the parameter, and the events corresponding to those inﬁnite values are points at inﬁnity. This can be easily seen from the r˙ equation ±dr ds = 2 2 E − Φr2 eξ

for r0 < r < ∞,

where s is any aﬃne parameter along the null geodesic. Since under the square-root has only a simple zero at r = r0 and goes to a constant as r → ∞ we see that the integral of the right-hand side of the above is ﬁnite over any ﬁnite subinterval of [r0 , ∞); and that this integral diverges only when the upper limit is inﬁnite. We thus have (dφ/dx)2 = 1/(f0,(x0 ) − f0, (x)). For the Minkowski space, f0, (x) = x2 and x0 = δ, thus the solution is φ(x) − φ(0) = sin−1 (x/δ), which

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as expected is a straight line at a distance 1/δ from the origin. Since the particlespacetime is always asymptotically Euclidean, Φ/(mE) = 1/δ is the impact parameter, the distance from the central singularity of the initial asymptote of the geodesic. Geodesics with 1/δ = 0 go straight to the central singularity, while geodesics with 1/δ > 0 will be deﬂected by it, as we will see below: We ﬁnd it more convenient to parametrize the family of null geodesics in the equatorial plane by their reciprocal perihelion x0 . Let φ0 denote the corresponding angle to x0 . We then have x0 ±dx φ − φ0 = . f0, (x0 ) − f0, (x ) x The integral is convergent at the upper limit since as we said, the above denominator has only a simple zero there. Letting φ± denote the two values for φ obtained at x = 0, we have x0 dx =: π + d(x0 ). φ+ − φ− = 2 f0, (x0 ) − f0, (x) 0 The quantity d(x0 ) represents the total deﬂexion of null geodesics due to spacetime curvature. For the Minkowski space, one easily computes that d(x0 ) = 0, ∀x0 > 0. By contrast for a static, spherically symmetric spacetime whose line element is −a2 dt2 + (1/a2 )dr2 + r2 (dθ2 + sin2 θ dφ2 ) with a2 < 1, so that it has a conical singularity at r = 0, we have f (x) = a2 x2 and as a result d(x0 ) = π( 1a − 1) > 0. One can thus say that the null geodesics are “bent” by the gravitational attraction of the conical singularity. Below we will establish the lower bound d(x0 ) ≥ c 2 when x0 is suﬃciently large, or equivalently, when r0 /m is small. For large r0 /m on the other hand, the geodesic stays far away from the singularity and thus its qualitative behavior is the same as in the RWN spacetime, as analyzed in [48] and more extensively in [51, §40]. As to the lower bound, we have, using that a2 x2 ≤ f0, (x) ≤ a2 x2 + b2 , with 2 a = 1 − A 2 and b2 = 2B 2 6 , x0 dx 1 (π + d(x0 )) ≥ 2 2 2 f (x 0 0) − a x =

ax0 1 1 1 sin−1 . ≥ sin−1 a a f (x0 ) 1 + (b/ax0 )2

B

. We then have Let 0 < c < Aπ/4 be ﬁxed and assume x0 ≥ Aπ/4−c 2 1 b 1 1 −1 4 (π + d(x0 )) ≥ √ sin + O( ) 1− 2 2 ax0 1 − A 2 π B A − 3 + O( 4 ) = 1 + 2 + O( 4 ) 2 2 x0 Aπ π B 2 ≥ + −

+ O( 4 ), 2 4 x0

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and thus d(x0 ) ≥ c 2 , which is the desired lower bound. Next we consider time-like geodesics with nonzero angular momentum. One sees (0) = −γ 2 < 0 and that the bounds for M (y) obtained that fγ,(0) = γ 2 > 0, fγ, above imply that fγ, (x) grows like x2 for large x. Thus once again geodesics with δ ≥ γ (i.e. those with E ≥ m) will have a perihelion and are extendible to inﬁnity in either direction. In contrast to the null geodesics, however, there are bounded orbits for time-like geodesics. This is because fγ, will have at least one critical point. Let f∗ := minx fγ,(x) = fγ, (x∗ ). It then follows that there will be no solution with √ √ δ < f∗ , and that for f∗ < δ < γ the equation fγ,(x) = δ 2 has at least two solutions, corresponding to the perihelion and the aphelion of a bound orbit. There √ will also be at least one stable circular orbit, for δ = f∗ . 8.2. Trajectories of test charges For the RWN metric, a detailed study of test-charge trajectories was done in [50]. We are not aware of a comparable study done for the generalized RWN metrics with a nonlinear aether law, other than the particular case of the Born–Infeld law [52] in the black hole regime, and some preliminary discussion of the general case in [53]. Here we are going to analyze the qualitative behavior of all the testcharge trajectories in a given particle-spacetime with a small enough mass-to-charge ratio . Recall that the orbit of a single test particle of mass m and charge e, in an electrostatic, spherically symmetric particle-spacetime of mass m and charge q satisﬁes the following system: 2 1 Φ −ξ 2 eξ , + m φ˙ = 2 Φ. t˙ = −e (E − eϕ), r˙ = ± (E − eϕ)2 − 2 r r In addition, the orbit lies in a plane, which is taken to be the plane θ = π/2. The system is clearly invariant under the simultaneous sign change of E, e, Φ and the independent variable s. Therefore, it is enough to consider the case E ≥ 0. Let x := m/r as before. It then follows that 2 x x x2 2 2 − (1 + λ x ) 1 − 2xM 2 κ − ρN 2 , x˙ = ± mm

where =

m |q|

as before and we have introduced new parameters ρ :=

e/m , q/m

κ :=

E , m

λ :=

Φ , mm

and where N (y) is the normalized electrostatic potential: Iζ4 y4 /2 1 ζ (µ)µ−3/4 dµ, N (y) := 7/4 2 Iζ 0

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which in view of (5.7) satisﬁes the bound

A A C 3 3 + max 0, − . ≤ N (y) ≤ min y, − 2 2y 2 2y 3y 3 Moreover, N (y) = ζ (Iζ4 y 4 /2), so that

A C A max 0, 2 − 4 ≤ N (y) ≤ min 1, 2 . y y y Note that for the RWN metric, NRWN (y) = y. κ and λ are proportional to the energy and angular momentum of the test charge. The ratio ρ is positive in the case where the particle-spacetime and the test particle have charges of the same sign, and negative if they have the opposite sign. Setting ρ = 0 will reproduce the results obtained above for timelike geodesics. Our goal is to obtain a bifurcation diagram in the κ, ρ, λ parameter space for the above system, for a ﬁxed small value of . It is clear that the diagram will be invariant under the simultaneous sign change of κ and ρ, and thus it is enough to restrict our attention to κ ≥ 0. Let x x gρ,κ, (x) := κ − ρN 2 , hλ, := (1 + λ2 x2 ) 1 − 2xM 2 .

Consider ﬁrst the case of orbits with zero angular momentum: Φ = λ = 0. The motion of the test charge is then radial. We recall that h0, (x) = eξ(m/x) . Thus, h0, is convex, decreasing, h0, (0) = 1, h0, (0) = −2 and limx→∞ h0, (x) = a2 = 1 − A 2 . (0) = −ρ/ 2 and that limx→∞ gρ,κ, (x) = On the other hand, gρ,κ, (0) = κ, gρ,κ, κ − 3ρ/2. Accordingly, there are three main parameter regimes to consider: Case 1. ρ < 0. The two charges q and e thus have the opposite sign. The function 2 must have the same gρ,κ, is increasing, and has a horizontal asymptote. Thus gρ,κ, properties, and in particular it will be concave and increasing, while we have already established that h0, is convex and decreasing. It then follows that there will be no trajectory with κ ≤ a + 3ρ/2, for a + 3ρ/2 < κ < 1 the trajectory is bounded, and corresponds to a test charge that cannot escape and falls radially inward and into the singularity, and for κ ≥ 1 the test charge has enough energy to escape to inﬁnity. Case 2. ρ > 0, κ > 3ρ/2. The two charges thus have the same sign, and the test charge has relatively high energy. The function gρ,κ, is decreasing, has a positive 2 will have these same three properties asymptotic value, and is convex. Thus gρ,κ, 2 and h0, , as well. In order to ﬁnd the number of intersections of the graphs of gρ,κ, one needs to compare the values of the derivatives of these two functions at x = 0 and near x = ∞. we have κρ (0) − h0, (0) = 1 − 2 , 2gρ,κ, (0)gρ,κ,

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while, using the large y asymptotics established above for the two functions M (y) and N (y), we have 1 1 3 2gρ,κ, (x)gρ,κ, (x) − h0, (x) = −ρ 2 κ − ρ + O < 0 as x → ∞. 2 x2 x3 Thus we need to distinguish two subcases: . (Note that this requires ρ2 < 2 2 /3, or equivalently, Case 2a. κ < 2 /ρ = mm eq e/m < 2/3, which does not allow the test charge to have a charge-to-mass ratio typical of actual elementary particles.) In this case there will be a critical value of 2 and κ, say κc ∈ (3ρ/2, a + 3ρ/2), such that the graphs of the two functions gρ,κ c , h0, are tangent to each other at a positive x = xc . Thus for κ < κc there will be no trajectories, for κ = κc there will be a stable equilibrium, corresponding to the test charge remaining at rest with respect to the singularity, for κc < k < a + 3ρ/2 the test charge shuttles back and forth between a perihelion and an aphelion, which grow further apart as κ is increased, for a + 3ρ/2 ≤ κ < 1 the perihelion coincides with the singularity, and for κ ≥ 1 the aphelion is at r = ∞, i.e. the test charge is allowed to escape. Case 2b. κ ≥ 2 /ρ. The two above-mentioned graphs can have no point of tangency, irrespective of κ. Moreover, there is no intersection at a positive x, and hence no trajectory, with κ ≤ 1 (i.e. for E ≤ m). For 1 < κ < a + 3ρ/2 there is a single intersection, which corresponds to the perihelion of a hyperbolic trajectory that avoids the singularity, and for κ > a + 3ρ/2 the test charge in one direction falls into the singularity and in the other direction is allowed to escape to inﬁnity. Case 3. ρ > 0 and κ < 3ρ/2. The charges have the same sign and the test charge has relatively low energy. The function gρ,κ, has a negative asymptotic value, thus 2 will be tangent to the x-axis at some point x = x∗ , which will be the graph of gρ,κ, where the global minimum is achieved, and is asymptotic to (κ − 3ρ/2)2. There are no trajectories with 3ρ/2 − a < κ < 1. For κ < min{1, 3ρ/2 − a} the trajectories correspond to falling-in test particles. For each 1 < κ < 3ρ/2−a there is an avoiding trajectory as well as one that falls in. For 3ρ/2 − a < κ < 3ρ/2 all trajectories avoid the singularity. Based on the above analysis, the κ-ρ parameter plane is divided into several regions as in Fig. 1. The shaded region in this ﬁgure is a forbidden zone: there are no trajectories for (κ, ρ) in this region. Region I corresponds to shuttling orbits, Region II to test particles that are trapped and fall into the singularity. Particles with parameters in Region III can also fall into the center, but have enough energy so that in the opposite direction they are allowed to escape to inﬁnity. Region IV corresponds to hyperbolic trajectories that avoid the singularity altogether, while each point in Region V corresponds to two trajectories, one that falls into the center and another that avoids it. Next, we consider charged trajectories with nonzero angular momentum, i.e. λ > 0. The only thing that changes in the above analysis is the behavior of hλ, ,

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κ

III

IV V 1

a

I II

II

ρ

0

Fig. 1.

Bifurcation diagram for trajectories with no angular momentum.

κ

H E,H 1

E

a

0

Fig. 2.

E

ρ

Bifurcation diagram for trajectories with positive angular momentum.

which now grows like x2 for large x, and has a global minimum at a positive x. As a result, there will be no trajectories falling into the singularity. The trajectories in fact will be similar to classical Keplerian trajectories, divided into bound and escape (scattering) orbits. The following diagram (Fig. 2) shows the various parameter regimes for each type of orbit. The shaded region in Fig. 2 is a forbidden one. The parameters in regions labeled E give rise to bound orbits, while those in regions labeled H give rise to escape (scattering) ones. One observes that, in sharp contrast to the case of superextremal RWN studied in [50], there is no evidence here of gravity being repulsive in the

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vicinity of the naked conical singularity at the center of these particle-spacetimes. On the contrary, the singularity appears to be attractive.

Acknowledgments I was introduced to nonlinear electrodynamics by my dear friend and colleague Michael Kiessling, and I also owe him the initial impetus for studying solutions with no horizon. I have beneﬁtted greatly from his help and encouragement throughout this project, and I am indebted to him for his critical reading of many drafts. I would also like to thank the anonymous referee for constructive comments, and the Institute for Advanced Study for their hospitality while I was a member there during the ﬁnal stage of this work. References ¨ [1] H. Reissner, Uber die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie, Ann. Phys. (Berlin) 50 (1916) 106–120. [2] G. Nordstr¨ om, On the energy of the gravitational ﬁeld in Einstein’s theory, Proc. Kon. Ned. Akad. Wet. 20 (1918) 1238–1245. [3] H. Weyl, Zur Gravitationstheorie, Ann. Phys. (Berlin) 54 (1917) 117–145. [4] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (W. H. Freeman & Co, New York, 1973). [5] R. Arnowitt, S. Deser and C. W. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. 122 (1961) 997–1006. [6] S. Hawking and G. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973). [7] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact Solutions of Einstein’s Field Equations (Cambridge University Press, Cambridge, 2003). [8] V. F. Weisskopf, On the self-energy and the electromagnetic ﬁeld of the electron, Phys. Rev. 56 (1939) 72–85. [9] R. P. Feynman, Lectures in Physics, Vol. 2, Chap. 28 (Addison-Wesley, Reading, Mass., 1964). [10] S. Weinberg, The Quantum Theory of Fields, Vol. I (Cambridge University Press, Cambridge, 1995), p. 31. [11] R. Arnowitt, S. Deser and C. W. Misner, Gravitational-electromagnetic coupling and the classical self-energy problem, Phys. Rev. 120 (1960) 313–319. [12] P. A. M. Dirac, Classical theory of radiating electrons, Proc. R. Soc. London A 167 (1938) 148–169. [13] W. Appel and M. K.-H. Kiessling, Mass and spin renormalization in Lorentz electrodynamics, Ann. Phys. (N.Y.) 289 (2001) 24–83. [14] M. K.-H. Kiessling, personal communication. [15] M. Born, Modiﬁed ﬁeld equations with a ﬁnite radius of the electron, Nature 132 (1933) 1004. [16] G. Mie, Grundlagen einer Theorie der Materie, Ann. Phys. 37 (1912) 511–534; ibid. 39 (1912) 1–40; ibid. 40 (1913) 1–66. [17] K. Schwarzschild, Zur Elektrodynamik, I. Zwei Formen des Princips der Action in der Elektronentheorie, in Nachrichten von der Gesellschaft der Wissenschaften zu Goettingen (1903), pp. 126–131.

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[18] M. Born and L. Infeld, Foundation of the new ﬁeld theory, Nature 132 (1933) 1004. [19] M. Born and L. Infeld, Foundation of the new ﬁeld theory, Proc. R. Soc. London A 144 (1934) 425–451. [20] M. H.-K. Kiessling, Electromagnetic ﬁeld theory without divergence problems 1. The Born legacy, J. Stat. Phys. 116 (2004) 1057–1120. [21] M. H.-K. Kiessling, On the motion of point defects in relativistic ﬁelds, preprint (2011). [22] B. Hoﬀmann, Gravitational and electromagnetic mass in the Born–Infeld electrodynamics, Phys. Rev. 47 (1935) 877–880. [23] B. Hoﬀmann and L. Infeld, On the choice of the action function in the new ﬁeld theory, Phys. Rev. 51 (1937) 765–773. [24] B. S. Madhava Rao, Generalized action-functions in Born’s electro-dynamics, Proc. Indian Acad. Sci. Sec. A 6 (1937) 158–173. [25] R. Pellicer and R. J. Torrence, Nonlinear electrodynamics and general relativity, J. Math. Phys. 10 (1969) 1718–1723. [26] M. Demianski, Static electromagnetic geon, Found. Phys. 16 (1986) 187–190. [27] E. Ay´ on-Beato and A. Garc´ıa, Regular black hole in general relativity coupled to nonlinear electrodynamics, Phys. Rev. Lett. 80 (1998) 5056–5059. [28] K. A. Bronnikov, Regular magnetic black holes and monopoles from nonlinear electrodynamics, Phys. Rev. D 63 (2001) 044005, 6 pp. [29] I. Dymnikova, Regular electrically charged vacuum structures with de Sitter centre in nonlinear electrodynamics coupled to general relativity, Classical Quant. Grav. 21 (2004) 4417. [30] D. J. Cirilo-Lombardo, New spherically symmetric monopole and regular solutions in Einstein–Born–Infeld theories, J. Math. Phys. 46 (2005) 042501. [31] D. J. Cirilo-Lombardo, Rotating charged black holes in Einstein–Born–Infeld theories and their ADM mass, Gen. Relativ. Gravit. 37 (2005) 847–856. [32] K. A. Bronnikov, V. N. Melnikov, G. N. Shikin and K. P. Staniukovich, Scalar, electromagnetic, and gravitational ﬁelds interaction: Particlelike solutions, Ann. Phys. (N.Y.) 118 (1979) 84. [33] K. A. Bronnikov, Comment on “Regular black hole in general relativity coupled to nonlinear electrodynamics”, Phys. Rev. Lett. 85 (2000) 4641. [34] G. D. Birkhoﬀ, Relativity and Modern Physics (Harvard University Press, Cambridge MA, 1923), p. 253. ¨ [35] J. T. Jebsen, Uber die allgemeinen kugelsymmetrischen L¨ osungen der Einsteinschen Gravitationsgleichungen im Vakuum, Arkiv f¨ or Matematik, Astronomi och Fysik 15 (1921) 1–9; English translation in Gen. Relativ. Gravit. 37 (2005) 2253–2259. [36] J. Ehlers and A. Krasi´ nski, Comment on the paper by J. T. Jebsen [reprinted in Gen. Relativ. Grav. 37 (2005) 2253–2259], Gen. Relativ. Gravit. 38 (2006) 1329–1330. [37] J. A. Eiesland, The group of motions of an Einstein space, Trans. Amer. Math. Soc. 27 (1925) 213–245. [38] J. A. Eiesland, Bull. Amer. Math. Soc. 27 (1921) 410, paragraph. [39] B. Hoﬀmann, On the spherically symmetric ﬁeld in relativity, Quart. J. Math. 3 (1932) 226–237. [40] K. Schleich and D. M. Witt, A simple proof of Birkhoﬀ’s theorem for cosmological constant, J. Math. Phys. 51 (2010) 112502. [41] D. Christodoulou, The Action Principle and Partial Diﬀerential Equations, Chap. 6 (Princeton University Press, Princeton, NJ, 1999). [42] J. Plebanski, Lectures on Nonlinear Electrodynamics (NORDITA, Copenhagen, 1968).

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[43] A. S. Tahvildar-Zadeh, One- and two-Killing ﬁeld reductions of the Einstein–Maxwell system with arbitrary constitutive laws, in preparation. [44] I. Bialinicki-Birula, Nonlinear electrodynamics: Variations on a theme by Born and Infeld, in Quantum Theory of Particles and Fields, eds. B. Jancewicz and J. Lukierski (World Scientiﬁc, Singapore, 1983), pp. 31–48. [45] B. Hoﬀmann, On the new ﬁeld theory, Proc. R. Soc. London A 148 (1935) 353–364. [46] C. Darwin, The gravity ﬁeld of a particle, Proc. R. Soc. London A 249 (1959) 180– 194. [47] C. Darwin, The gravity ﬁeld of a particle, II, Proc. R. Soc. London A 263 (1961) 39–50. [48] J. C. Graves and D. R. Brill, Oscillatory character of Reissner–N¨ordstrom metric for an ideal charged wormhole, Phys. Rev. 120 (1960) 1507–1513. [49] B. Carter, Global structure of the Kerr family of gravitational ﬁelds, Phys. Rev. 174 (1968) 1559–157. [50] W. B. Bonnor, The equilibrium of a charged test particle in the ﬁeld of a spherical charged mass in general relativity, Classical Quant. Grav. 10 (1993) 2077–2082. [51] S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1983). [52] N. Bret´ on, Geodesic structure of the Born–Infeld black hole, Classical Quant. Grav. 19 (2002) 601–612. [53] J. Diaz-Alonso and D. Rubiera-Garcia, Electrostatic spherically symmetric conﬁgurations in gravitating nonlinear-electrodynamics, Phys. Rev. D 81 (2010) 064021.

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Reviews in Mathematical Physics Vol. 23, No. 4 (2011) 347–373 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X1100431X

LOCALIZED ENDOMORPHISMS IN KITAEV’S TORIC CODE ON THE PLANE

PIETER NAAIJKENS Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands [email protected] Received 20 December 2010 Revised 28 March 2011 We consider various aspects of Kitaev’s toric code model on a plane in the C ∗ -algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher–Haag–Roberts program (speciﬁcally, through its generalization to inﬁnite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of D(Z2 ), i.e. Drinfel’d’s quantum double of the group algebra of Z2 . Keywords: Localized endomorphisms; superselection sectors; anyons; toric code. Mathematics Subject Classiﬁcation 2010: 81T05, 81T25, 18D10

1. Introduction Kitaev’s quantum double model [1] has attracted much interest in recent years. One of its interesting features is that the model has anyonic excitations. Such models may be relevant to a new approach to quantum computing, where topological properties of a system are used to do computations (see [2, 3] for reviews). Here we consider the simplest case of this model, corresponding to the group Z2 . This model is often called the toric code, although we will consider it on the plane instead of on a torus. This model is not powerful enough for applications to quantum computing, but it has interesting properties nonetheless. In particular, it has anyonic excitations (albeit abelian anyons). The toric code has been studied by many authors by now, for example [1, 4, 5]. We take a diﬀerent viewpoint, namely that of local quantum physics. Indeed, the model can be discussed in the C ∗ -algebraic approach to quantum spin 347

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systems [6, 7]. We show that single excitations can be described by states that cannot be distinguished from the ground state when restricted to measurements outside a cone extending to inﬁnity. This structure is familiar from the algebraic approach to quantum ﬁeld theory [8], in particular when massive particles are considered [9]. The states describing these single excitations lead, via the GNS construction, to inequivalent representations (superselection sectors) of the observable algebra. In fact, these states fulﬁll a certain selection criterion, pertaining to the fact that they are localized and transportable. The analysis of such representations is central to the Doplicher–Haag–Roberts (DHR) program in algebraic quantum ﬁeld theory [10, 11]. In particular, it turns out that these representations can equivalently be described by endomorphisms of the observable algebra. This description leads in a natural way to the notion of composition of excitations and to statistics of (quasi)particles from ﬁrst principles. This analysis can be carried out completely for the toric code on the plane. A related approach is taken for example in [12, 13], where the authors consider G-spin (or, more generally, Hopf-C ∗ ) chains. There, excitations localized in bounded regions (satisfying the so-called DHR criterion) are considered. Since every injective endomorphism of a ﬁnite dimensional algebra is in fact an automorphism, the authors consider amplimorphisms to obtain non-abelian charges. Here, we take a diﬀerent approach, and look instead at endomorphisms localized in certain inﬁnite “cone” regions. In our model the irreducible endomorphisms are all automorphisms, but since we consider excitations localized in infinite regions, ﬁnite dimensionality of the algebras is not an obstruction any more. The idea of construction charged sectors localized in inﬁnite regions is not new: it is used, for example, in the work of Fredenhagen and Marcu [14]. Discrete gauge theories in d = 2 + 1 show similar algebraic features (i.e. fusion and braiding) of anyons [15]. Similar models have been studied in the constructive approach to quantum ﬁelds in lattice gauge theory, in particular for the gauge group Z2 in [14, 16]. These results have been generalized to the group ZN in [17, 18]. Although the setting considered here is diﬀerent, some of the methods used are similar. A ﬁeld theoretic interpretation of the model discussed here can be found in [1, Sec. 4]. The paper is organized as follows. In Sec. 2, we recall the model and discuss the ground state in the C ∗ -algebraic setting. In Sec. 3 localized automorphisms describing excitations are described. Section 4 is devoted to fusion and statistics of excitations. Then follows a discussion of operator-algebraic aspects of von Neumann algebras generated by observables localized in cones. Finally, in the last section we prove that the excitations are described by the representation theory of the quantum double D(Z2 ). 2. The Model We describe Kitaev’s model in the C ∗ -algebraic framework for quantum lattice systems [4]. Consider a square Z2 lattice. On each bond of the lattice, i.e. an edge

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between two vertices of distance 1, there is a spin-1/2 particle. That is, at each bond b the local state space is H{b} = C2 , with observables A({b}) = M2 (C). The set of bonds will be denoted by B. If Λ ⊂ B is a ﬁnite set, A(Λ) is the algebra of observables living on the bonds of Λ. It is the tensor product of the observable algebras acting on the individual bonds of Λ. If Λ1 ⊂ Λ2 there is an obvious inclusion of corresponding algebras, by identifying HΛ2 ∼ = HΛ1 ⊗ HΛ2 \Λ1 . This deﬁnes a local net of algebras, with respect to the inclusion A(Λ1 ) → A(Λ2 ) for Λ1 ⊂ Λ2 . Deﬁne Aloc = A(Λf ), Λf ⊂B

the algebra of local observables. The union is over the ﬁnite subsets Λf of B. The algebra A of quasi-local observables is the completion of Aloc in the norm topology, turning it into a C ∗ -algebra. Alternatively, one can see it as the inductive limit of the net Λ → A(Λ) in the category of C ∗ -algebras. Note that A is a uniformly hyperﬁnite (UHF) algebra [6]. The algebra of observables localized in an arbitrary subset Λ of B is deﬁned as · A(Λf ) , A(Λ) = Λf ⊂Λ

where the union is again over ﬁnite subsets. An operator A is said to have support in Λ, or to be localized in Λ, if A ∈ A(Λ). The set supp(A) ⊂ B is the smallest subset in which A is localized. The Hamiltonian of Kitaev’s model is deﬁned in terms of plaquette and star operators, each supported on four bonds (see Fig. 1). If s is a point on the lattice, star(s) denotes the star based at s. Similarly, plaq(p) are the bonds enclosing a plaquette p. The corresponding star and plaquette operators are given by σjx , Bp = σjz , As = j∈star(s)

j∈plaq(p)

Fig. 1. The Z2 lattice. The gray bonds each carry a spin-1/2 particle. A star (dashed lines) and plaquette (thick lines) are shown.

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where the tensor product is understood as having Pauli matrices σ x (respectively σ z ) in places j, and unit operators in all other positions. It is then straightforward to check that for all stars s and plaquettes p, we have [As , Bp ] = 0. These operators are used to deﬁne the local Hamiltonians. If Λf ⊂ B is ﬁnite, the associated local Hamiltonian is As − Bp . HΛf = − star(s)⊂Λf

plaq(p)⊂Λf

There is a natural action of Z2 on the quasi-local algebra, acting by translations. Denote this action by τx for x ∈ Z2 . Note that the interactions are of ﬁnite range, and moreover, they are translation invariant. Hence, there exists an action αt of R on A describing the dynamics of the system [7], as well as a derivation δ that is the generator of the dynamics. For observables localized in a ﬁnite set Λ, the action of this derivation is given bya δ(A) = i[HΛ , A],

A ∈ A(Λ).

By deﬁnition, ground states for these dynamics are states ω of A such that −iω(X ∗ δ(X)) ≥ 0 for all X ∈ Aloc . In [4] it is shown that the model admits a unique ground state, which can be computed explicitly. Since we will need the argument later, for the convenience of the reader we summarize the results. The following lemma is crucial in the computation of the ground state. The proof is a straightforward application of the Cauchy–Schwartz inequality and the fact that for A positive, ω(A) = 0 implies that ω(A2 ) = 0. Lemma 2.1. Let ω be a state on a C ∗ -algebra A, and suppose X = X ∗ such that X ≤ I and ω(X) = 1. Then ω(XY ) = ω(Y X) = ω(Y ) for any Y ∈ A. Consider now the abelian algebra AXZ generated by the star and plaquette operators. This algebra is in fact maximal abelian: AXZ ∩ A = AXZ [4]. Let ω be the state on AXZ such that ω(As ) = ω(Bp ) = 1 for all plaquette and star operators.b With help of the lemma, this completely determines the state on AXZ . Moreover, it minimizes the local Hamiltonians, hence any ground state of the system must be equal to ω if restricted to AXZ . The goal is then to show that this state has a unique extension to A. be a bit more precise: the derivation δ deﬁned here is norm-closable and it is the closure δ that generates the dynamics [7, Theorem 6.2.4]. By a density argument, it is often enough to consider δ instead of its closure. b That such a state exists can be seen by mapping the model to an Ising spin model. a To

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Let ω0 be an extension of ω to the algebra A.c Using the lemma one can show that for X, Y ∈ Aloc , (ω0 (X ∗ Y ) − ω0 (X ∗ As Y )) − iω0 (X ∗ δ(Y )) = s

+

(ω0 (X ∗ Y ) − ω0 (X ∗ Bp Y )),

(2.1)

p

where the variable s runs over all stars in the lattice, and p over all plaquettes. If one takes X = Y , an application of the Cauchy–Schwartz inequality shows that the right hand side is positive, hence ω0 is a ground state. As mentioned before, in the model at hand this extension is actually unique. In fact, let X be a monomial in the Pauli matrices, say X = i∈Λ σiki where Λ ⊂ B is ﬁnite and ki = x, y or z. Then ω0 (X) is non-zero if and only if X is a product of star and plaquette operators, in which case it is 1. This completely determines the state ω0 , since the value of ω0 (X) can be computed by a repeated application of Lemma 2.1. For example, to make plausible why ω0 is zero if X is not a product of star and plaquette operators, consider an operator of the form A = σjx for some bond j. Then there is a plaquette p such that j ∈ plaq(p). But then ω0 (A) = ω0 (Bp σjx Bp ) = −ω0 (A). In particular, for a local observable A that is a monomial in the Pauli matrices, the set of bonds where A has a σ x component should have the property that the intersection with each plaquette plaq(p) has an even number of elements. Continuing in this manner, one can show that indeed only products of star and plaquette operators lead to non-zero expectation values [4]. Proposition 2.1. There is a unique (hence pure) ground state ω0 . This state is translation invariant. The self-adjoint H0 generating the dynamics in the GNS representation (π0 , H0 , Ω), when normalized such that H0 Ω = 0, satisfies Sp(H0 ) ⊂ {0} ∪ [4, ∞). Proof. We have already discussed existence and uniqueness of ω0 . Translations map star operators into star operators, and plaquette operators into plaquette operators, hence the ground state is translation invariant. Since ω0 is a ground state, it is invariant under the dynamics and the time evolution can be implemented by a strongly continuous group t → Ut of unitaries. We can choose Ut such that Ut Ω = Ω. It follows that there is an (unbounded) self-adjoint H0 such that Ut = eitH0 and H0 Ω = 0. We claim that Sp H0 ⊂ {0} ∪ [M, ∞) is equivalent to −iω0 (X ∗ δ(X)) ≥ M (ω0 (X ∗ X) − |ω0 (X)|2 ),

(2.2)

for all X ∈ Aloc , because the ground state is non-degenerate. Indeed, since H0 Ω = 0 with Ω the GNS vector, the inequality can equivalently be written as c By

the Hahn–Banach theorem an extension ω0 of ω to A always exists.

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XΩ, H0 XΩ ≥ M (XΩ2 − | Ω, XΩ |2 ) because XΩ, H0 XΩ = ω0 (X ∗ δ(X)). Here we have identiﬁed X with its image π0 (X), which is possible since π0 is a representation of a UHF (hence simple) algebra. On the other hand, the spectrum condition is equivalent to H0 + MP Ω ≥ MI , where PΩ is the projection on the subspace spanned by Ω (by non-degeneracy, this is the spectral projection corresponding to {0}). This is equivalent to the condition Ψ, (H0 + MP Ω )Ψ = Ψ, H0 Ψ + M | Ω, Ψ |2 ≥ M Ψ2 for all Ψ in the domain D(H0 ) of H0 . But π(Aloc )Ω is a core for H0 (compare with the proof of [7, Proposition 5.3.19]), hence it is enough to check the inequality for Ψ = XΩ with X ∈ Aloc . This shows that inequality (2.2) is equivalent to the assertion on the spectrum of H0 . We now show that inequality (2.2) indeed holds for M = 4. As a ﬁrst step, we claim that if either X or Y is a local operator in AXZ , −iω0 (X ∗ δ(Y )) = 4(ω0 (X ∗ Y ) − ω0 (X)ω0 (Y )) = 0.

(2.3)

Under these assumptions, the left-hand side can be seen to vanish by Eq. (2.1) and Lemma 2.1. As for the right-hand side, consider the case where X ∈ AXZ (the other case is proved similarly). In this case, X = i λi Xi where each Xi is a product of star and plaquette operators. Using Lemma 2.1 again, it follows that ω0 (X ∗ Y ) = i λi ω0 (Y ) = ω0 (X)ω0 (Y ), proving the claim. Now consider the general case, with a local operator X = XXZ + i∈I λi Xi , where XXZ ∈ AXZ and each Xi (with i in some ﬁnite set I) is a monomial in / AXZ . Since Xi ∈ / AXZ , there is some As or Bp the Pauli matrices such that Xi ∈ that does not commute with Xi . Suppose this is As . Since Xi is a monomial in the Pauli matrices, this actually implies that {As , Xi } = 0, in other words, they anti-commute. Note that this implies that ω0 (Xi ) is zero for each i = 0, since by the same trick as used before it follows that ω0 (Xi ) = −ω0 (Xi ). By the remarks above, Eq. (2.2) reduces to −i ω0 (Xi∗ δ(Xj )) ≥ 4 ω0 (Xi∗ Xj ). (2.4) i,j∈I

i,j∈I

Note that for each Xi , there is a ﬁnite number ni of plaquette and star operators that anti-commute with Xi . In fact, ni ≥ 2, since if there is for example one star operator that does not commute with Xi , there must necessarily be another one with this property.d Note that if ni = nj , there is a star or a plaquette operator that commutes with Xi and anti-commutes with Xj (or vice versa). Consequently, ω0 (Xi∗ Xj ) = 0. Now deﬁne for each integer k the ﬁnite set Ik = {i ∈ I : ni = k} and the opera k = tors X i∈Ik Xi , with the understanding that Xk = 0 if Ik is the empty set. By d This

amounts to saying that excitations always exist in pairs in ﬁnite regions in Kitaev’s model [1].

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∗X the considerations above, it then follows that i,j∈I ω0 (Xi∗ Xj ) = k≥2 ω0 (X k k ), since ni ≥ 2 for each i ∈ I. On the other hand, from Eq. (2.1) it follows that −iω0 (Xi∗ δ(Xj )) = 2ni ω(Xi∗ Xj ). It then follows that the left-hand side of the ∗X inequality (2.4) is equal to 2 k≥2 kω0 (X k k ). From this it easily follows that inequality (2.4) holds. The spectrum condition has far-reaching consequences for the correlation functions; for example, it implies that ground state correlations decay exponentially [19]. 3. Localized Endomorphisms In this section we describe localized excitations of the system. In his model, Kitaev associates certain string operators to paths on the lattice (or the dual lattice). These string operators create excitations at the endpoints of the paths [1]. The idea is to consider a single excitation by moving one of the excitations to inﬁnity, as is done, for example, in [14]. Before this construction is introduced, we give some preliminary deﬁnitions. By a site, we mean either a point on the lattice, a plaquette, or a pair of a plaquette with one of its vertices (i.e. a combined site). Sites can be seen as the places where excitations can be introduced. Between two sites of the same type, we can consider paths. A path between two points on the lattice is just a path consisting of bonds of the lattice. A path between plaquettes can be viewed as a path on the dual lattice. A path between combined sites is called a ribbon (see Fig. 2). One can think of a ribbon as being composed by a path on the lattice and one on the dual lattice. Definition 3.1. Let γ be a ﬁnite path between two sites. If γ is a path on the lattice, deﬁne the corresponding string operator as ΓγZ = i∈γ σiz . If it is a path x on the dual lattice, the string operator is deﬁned as ΓγX = i∈γ σi . Here i ∈ γ

Fig. 2. A path on the lattice (left black line) and a ribbon. The dots on the ribbon indicate a combined site, i.e. a plaquette with one of its vertices.

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means that i is a bond that intersects the path on the dual lattice. Finally, a string operator corresponding to a ribbon is a combination of these constructions. That is, ΓγY = ΓγX1 ΓγZ2 , where γ1 is the path on the lattice and γ2 the path on the dual lattice, corresponding to the ribbon. It should be clear from the context whether we consider paths on the lattice, paths on the dual lattice, or ribbons. We say that a path or the corresponding string operator is of type X, Y or Z, corresponding to the subscripts used in the deﬁnition. We ﬁrst make some observations that will be used later. Consider a plaquette p. The corresponding plaquette operator Bp is just the string operator ΓγZ , where γ is the closed path consisting of the edges of the plaquette. If p is, for example, a plaquette adjacent to p, Bp Bp is the string operator corresponding to the closed path on the outer edges of the two plaquettes. Continuing this way, it follows that the string operator corresponding to a closed path on the lattice is the product of plaquette operators corresponding to the plaquettes enclosed by the path. The reader will have no trouble checking that similarly a string operator corresponding to a closed path on the dual lattice is the product of all star operators corresponding to the stars enclosed by the path. The idea now is to study “elementary” excitations by ﬁrst considering a pair of excitations (created by a string operator), and then move one of the excitations to inﬁnity. This technique is also used in, for instance, lattice gauge theory [14, 18]. We show that in Kitaev’s model such excitations can be described by localized automorphisms of A. Definition 3.2. Let ρ be a ∗-endomorphism of A. Let Λ ⊂ B be arbitrary. Then ρ is said to be localized in Λ if ρ(A) = A for all A ∈ A(Λc ). Here Λc denotes the complement of any subset Λ of B. We will primarily be interested in cone regions, although in fact the speciﬁc shape of the regions is not important (see also Remark 3.1 below). Definition 3.3. Consider a point on the lattice Z2 , with two semi-inﬁnite lines emanating from it, such that the angle between those lines is positive but smaller than π. A cone Λ ⊂ B consists of all bonds that are in the area bounded by the two lines, or intersected by one of the lines. See Fig. 3 for an example. Remark that for x ∈ Z2 there is a translated cone Λ+x. Furthermore, ∪x∈Z2 (Λ+ x) is the set of all bonds. Finally, τx (A(Λ)) = A(Λ + x) for any Λ ⊂ B. These properties hold in fact for any subset Λ of the bonds containing at least a horizontal and a vertical bond. The string operators induce localized endomorphisms (in fact, automorphisms) of A. If γ is a path starting at a site x and extending to inﬁnity, write γn (n ∈ N) for the ﬁnite path consisting of the ﬁrst n bonds of the path γ. Proposition 3.1. Let Λ be a cone and let k = X, Y, Z. Choose a path γ k of type k in Λ extending to infinity. Consider the corresponding string operators Γγkn for

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Fig. 3. Example of a cone (bold bonds). The shaded region is the area bounded by two lines emanating from a point.

n ∈ N. For any A in A, define ρk (A) = lim Ad Γγkn (A), n→∞

(3.1)

where the limit is taken in norm. Then for each k, ρk defines an outer automorphism of the quasi-local algebra A. These automorphisms are localized in Λ. Proof. In the proof we will omit the symbol γ and write Γnk . Suppose A is an observable localized in a ﬁnite region Λ0 . Then one can ﬁnd n0 such that (γn \γn0 )∩ Λ0 = ∅ for all n > n0 . In other words, new parts of the path all lie outside Λ0 . But then it follows that Ad Γnk (A) = Ad Γnk 0 (A) for all n > n0 , hence the limit in Eq. (3.1) converges in norm for any local operator A. To deﬁne ρk on A, extend by continuity. Indeed, since each Γkn is a unitary operator, ρk (A) = A for each local observable. The local observables are normdense in A, so that ρk extends uniquely to A. By continuity of the ∗-operation and joint continuity of multiplication (in the norm topology), ρk is a ∗-endomorphism. The localization property immediately follows from locality: if B ∈ A(Λc ), then it commutes with Γnk for each n. The endomorphism ρk is in fact an automorphism. Indeed, because Pauli matrices square to the identity, ρk ◦ ρk is the identity. To see that the automorphisms are outer, it is enough to notice that the sequence Γnk is not a Cauchy sequence in A, hence it does not converge to an element in A. By [20, Theorem 6.3], it follows that the automorphisms are outer.e Note that the automorphism ρk depends on the choice of path γ k . If necessary, this path dependence will be emphasized by using the notation ρkγ . this follows because the GNS representation of ω0 ◦ ρk is disjoint from the GNS representation of ω0 , see Theorem 3.1.

e Alternatively,

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The automorphisms deﬁned in Proposition 3.1 induce states by composing with the ground state. Definition 3.4. Let x be a site and γ a path of type k = X, Y, Z starting at x and extending to inﬁnity. Deﬁne a state ωkx of A by ωkx (A) = ω0 (ρkγ (A)). At ﬁrst sight, this state appears to depend on the speciﬁc choice of path. However, this is not the case. Lemma 3.1. For each k = X, Y, Z and each site x of the same type, the state ωkx only depends on x, but not on the path γ. Proof. First consider the case k = Z, so that x is a point on the lattice. To prove independence of the path, consider another point y and let γ 1 and γ 2 be two paths from x to y. Denote the corresponding string operators by Γ1Z and Γ2Z . This allows to deﬁne two (a priori distinct) states ωix,y (A) = ω0 (ΓiZ AΓiZ ),

i = 1, 2.

Note that the string operators commute with plaquette operators, hence clearly ωix,y (Bp ) = 1 for each plaquette p. As for the star operators, note that each star has an even number (0, 2 or 4) of edges in common with the paths γ i , except at the endpoints x and y, where there are an odd number of edges in common. Let s be the star based at x. Suppose for the sake of example that it has one edge in common with the path γ 1 . Then, using the commutation relations for Pauli matrices, ω1x,y (As ) = ω0 (Γ1Z As Γ1Z ) = i2 ω0 (As ) = −1. A similar calculation holds in the case of three common edges, or for a star s containing the endpoint y. Summarizing, we ﬁnd that ω1x,y and ω2x,y coincide on the abelian algebra AXZ , taking the value 1 on all plaquette operators. On the star operators they take the value −1 if the star is based at either x or y, and 1 otherwise. A similar argument as given for the ground state now allows us to compute the value of the states on arbitrary elements of the local algebras, and it follows that both states coincide. There is in fact another way to see this. Let for example γ be a ﬁnite path of type Z. Let p be a plaquette such that p ∩ γ is non-empty. Then it is easy to see Z that ΓZ γ Bp = Γγ , where the path γ is obtained from γ by deleting the bonds of γ ∩ p and adding the bonds p\γ to the path γ. Hence once can use the plaquette operators to deform one path into another, provided the endpoints are the same. Since Z Z Z Z Z ω0 (ΓZ γ AΓγ ) = ω0 (Bp Γγ AΓγ Bp ) = ω0 (Γγ AΓγ )

it follows that the states coincide. A similar argument can be given for paths of type X. Now consider the case where γ 1 and γ 2 are two paths starting at x and extending to inﬁnity. Let A be a local observable, localized in some ﬁnite set Λ ⊂ B. Then

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there is an n0 such that the paths γn1 and γn2 do not return to Λ for n ≥ n0 . Consider a path γ ⊂ Λc from γn1 0 to γn2 0 . By locality and the result above, we then have γ1

γ1

γ2

γ2

γ1

γ1

n0 n0 n0 n0 γ γ ω0 (ρZ γ 1 (A)) = ω0 (ΓZ AΓZ ) = ω0 (ΓZ ΓZ AΓZ ΓZ )

= ω0 (ΓZn0 AΓZn0 ) = ω0 (ρZ γ 2 (A)). x By continuity this result extends to observables A ∈ A, hence the state ωZ is independent of the path. x and ωYx is essentially the same. The diﬀerence The argument for the states ωX is that one has to consider points x, y in the dual lattices, i.e. plaquettes of the lattice, together with paths on the dual lattice. For example, for k = X one ﬁnds −1 x, y ∈ p, x,y x,y (As ) = 1, ωX (Bp ) = ωX 1 otherwise. x The argument is now the same as for ωZ .

The state ωkx describes a single excitation. By the GNS construction, this leads to a corresponding representation πωkx of A. The GNS triple coming from the ground state ω0 will be denoted by (π0 , H0 , Ω). The remarkable feature is that representations corresponding to single excitations cannot be distinguished from the ground state representation when restricted to the complement of a cone. Theorem 3.1. Let Λ ⊂ B be any cone. Then π0 A(Λc ) ∼ = πωkx A(Λc ),

(3.2)

for k = X, Y, Z and any site x. In addition, πωkx ∼ = πωly if and only if k = l. This x holds for k = 0, X, Y, Z, where ω0 := ω0 . Proof. Let x be a site. Choose a path γ (of type k) in Λ, starting at x and going to inﬁnity. Consider ρk := ρkγ as above. Then π0 ◦ ρk is localized in Λ, in the sense that π0 ◦ ρk (A) = π0 (A) for all A ∈ A(Λc ). Moreover, it is a GNS representation for the state ωkx , essentially by deﬁnition of ωkx (the Hilbert space is H0 and Ω the cyclic vector).f Hence by uniqueness of the GNS representation, π0 ◦ ρk ∼ = πωkx . Together with localization this yields Eq. (3.2). Let y be another site. Consider a path γ from x to y, with corresponding string operator Γγk . Note that Ad Γγk ◦ ρk is precisely the automorphism induced by the path from y to inﬁnity, obtained by concatenating γ with γ. From unitarity of Γγk it is easy to see that πωkx ∼ = πωky , proving that the GNS representations of type k are equivalent, independent of the starting site. To complete the proof, we show that the representations are globally inequivalent. Note that ω0 is a pure state, hence the GNS representation is irreducible. The that ω0 and ωxk are automorphic states in the terminology of [21, Chap. 12]. The statement is then an example of Proposition 12.3.3 of the same reference.

f Note

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Fig. 4. Consider the state induced by thick path on the lattice. A path γ on the dual lattice (dashed) deﬁnes a string operator ΓγX . The state has value −1 on this operator.

GNS representations of the states ωkx can be obtained by composing π0 with an automorphism of A, hence they are also irreducible. But this implies that ω0 and ωk are factor states. Moreover, since the representations are irreducible, unitary equivalence is equivalent to quasi-equivalence of the states [21, Proposition 10.3.7]. Recall that in the situation at hand, two factor states ω1 and ω2 are quasi-equivalent such that for all ﬁnite if and only if for each ε > 0, there is a ﬁnite set of bonds Λ c sets Λ ⊂ Λ and B ∈ A(Λ), |ω1 (B) − ω2 (B)| < εB, by [6, Corollary 2.6.11]. We show that this inequality cannot hold. x , for some point x on the Consider for the sake of example the case ω0 and ωZ contains the star lattice. Set ε = 1. Without loss of generality, we can assume that Λ based at x. Since Λ is ﬁnite, it is possible to choose a closed non-self-intersecting is contained in the region bounded path γ in the dual lattice, such that the set Λ by the path (see Fig. 4). Consider the string operator ΓγX corresponding to this path. Then clearly this operator is localized in a ﬁnite region in the complement Recall that Γγ is the product of star operators enclosed by the path γ, in of Λ. X particular the star based at x. That is, ΓγX = Astar(x) As1 · · · Asn for certain stars s1 , . . . , sn . But this implies x x (ΓγX )| = |1 − ωZ (Astar(x) )| = 2 > ΓγX . |ω0 (ΓγX ) − ωZ

The other cases are similar, if necessary using plaquettes instead of stars. Remark 3.1. The fact that Λ is a cone is not essential at this point. What is important is that it should be possible to choose a path extending to inﬁnity contained in Λ. In particular, the proof implies that it is not possible to sharpen the result to unitary equivalence when restricted to the complement of a finite set. At one point in the analysis however, notably in the proof of Theorem 5.1, it is essential to be able to translate the support of any local observable to a region completely inside Λ. If Λ is a cone, this is always possible.

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In the language of algebraic quantum ﬁeld theory, the representations πωk are said to satisfy a selection criterion. Usually one imposes such a selection criterion to select physically relevant representations. Here, however, we start with physically reasonable constructions and arrive at the criterion. The criterion here can be interpreted as a lattice analogue of localization in spacelike cones, as considered in [9]. An example of a model admitting such representations, albeit a model mainly of mathematical interest, is constructed in [22]. The interpretation is that the excitations cannot be distinguished from the ground state outside a cone region. It would be interesting to know if there are other irreducible representations of A, not unitarily equivalent to the representations in Theorem 3.1, satisfying this criterion. One probably has to impose additional criteria to select physically relevant representations (cf. the condition on the existence of a mass gap in [9]). For the automorphisms considered here, a similar property can be derived. In particular, the automorphisms are covariant with respect to the time evolution. Moreover the generator has positive spectrum bounded away from zero. Note that the algebra A (being UHF) is simple, hence π0 is a faithful representation. To simplify notation, from now on we identify π0 (A) with A and often drop the symbol π0 , as was already done in the proof of Proposition 2.1. Proposition 3.2. Let γ be a path to infinity of type k. Then ργ is covariant for the action of αt . In fact, suppose γ is of type Z. Then, for all t ∈ R and A ∈ A, ργ (αt (A)) = eit(H0 +2As ) ργ (A)e−it(H0 +2As ) with Sp(H0 + 2As ) ⊂ [2, ∞). Here s is the starting point of γ. For the case k = X one has to replace As by Bp , where p is the plaquette where the path starts. The case k = Y has generator H0 + 2Bp + 2As , with spectrum contained in [4, ∞). Proof. We prove the result for paths of type X. The other cases are proved by making the obvious modiﬁcations. First note that for A ∈ Aloc , αt (A) = limΛ→Z2 eiHΛ t Ae−iHΛ t in norm. By the same reasoning as in the proof of Lemma 2.1, one sees that ργ (As ) = −As . Hence if Λ ⊃ star(s), we have ργ (HΛ ) = HΛ + 2As . By expanding the exponential into a power series, it is then clear that ργ (eitHΛ Ae−itHΛ ) = eit(HΛ +2As ) ργ (A)e−it(HΛ +2As ) . One then sees (remark in particular that As commutes with all local Hamiltonians) that for all A ∈ A we have ργ (αt (A)) = Ut ργ (A)Ut∗ , where Ut is the unitary Ut = exp(it(H0 + 2As )). It remains to show the spectrum condition. This can be done by similar methods as used in the proof of Proposition 2.1. The spectrum condition is equivalent to the inequality −iω(X ∗ δ(X)) + 2ω(X ∗ As X) − 2ω(X ∗ X) ≥ 0

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for all X ∈ Aloc . We then proceed as before: write X = XXZ + i Xi where / AXZ monomials in the Pauli matrices. After substituting XXZ ∈ AXZ and Xi ∈ this into the inequality, all terms containing XXZ vanish. By the same reasoning as in the proof of Proposition 2.1 one then sees that this inequality is indeed satisﬁed for all X ∈ Aloc . The following corollary is immediate. Corollary 3.1. The states ωγk are invariant with respect to αt . 4. Fusion, Statistics and Braiding The localized endomorphisms considered in the previous section can be endowed with a tensor product. In fact, it is possible to deﬁne a braiding in a canonical way. This braiding is related to the statistics of particles. In the DHR analysis, a crucial role in the construction is played by Haag duality in the vacuum sector. For dealing with cone localized endomorphisms, the appropriate formulation is the condition that for each cone Λ the following equality holds: π0 (A(Λ)) = π0 (A(Λc )) . Note that by locality, one always has π0 (A(Λ)) ⊂ π0 (A(Λc )) . Currently no general conditions from which Haag duality follows are known, but note that there are some results for quantum spin chains, e.g. [23, 24]. At the moment we do not have a proof of Haag duality, but since the ground state is known explicitly, one might hope that a direct proof is possible. Fortunately, in the present situation it is possible to do without Haag duality. To clarify this, ﬁrst note that Theorem 3.1 implies in particular that the localized automorphisms deﬁned by paths extending to inﬁnity are transportable. Definition 4.1. Let Λ be a cone and suppose that ρ is an endomorphism of A there is a unitary localized in Λ. Then ρ is called transportable, if for any cone Λ g equivalent endomorphism ρ localized in Λ. One of the applications of Haag duality is to get more control over the unitary setting up the equivalence. Speciﬁcally, one can show that the intertwiners are elements of the (weak closure) of cone algebras. Recall that an intertwiner V from an endomorphism ρ1 to ρ2 is an operator such that V ρ1 (A) = ρ2 (A)V for all A ∈ A. A unitary intertwiner is also called a charge transportation operator (or simply charge transporter). In our model we will be able to prove, without invoking Haag duality, that the charge transporters are elements of the weak closure of cone algebras. We again identify π0 (A) with A in the proof. g We

do not require that this unitary lives in A. More precisely, we demand that π0 ◦ ρ ∼ = π0 ◦ ρb.

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Lemma 4.1. Let γ 1 (respectively γ 2 ) be a path of type k starting at a site x (respectively y) and extending to infinity. Then there is a unitary intertwiner V from ρkγ1 to ρkγ2 such that Γkbγ V Ω = Ω (where Ω is the GNS vector for ω0 ) for any path γ from x to y. Moreover, if for each n a path γn from the nth site of γ1 to the nth site of γ2 is chosen such that limn→∞ dist( γn , x) = ∞, then for Vn = Γn1 Γkeγn Γn2 , where Γni is the string operator corresponding to the path γni , we have V = w-lim Vn . n→∞

(4.1)

In other words, Vn is a sequence of operators converging weakly to V. Proof. First note that As Ω = Bp Ω = Ω for all star and plaquette operators. Indeed, (As − I)Ω2 = ω0 ((As − I)∗ (As − I)) = 2 − 2 = 0, for all As . A similar calculation holds for the operators Bp . Note that this property can be interpreted as the ground state vector minimizing the value of each local Hamiltonian [1]. First note that a unitary V as in the statement is necessarily unique because any unitary intertwiner from ρkγ1 to ρkγ2 is a scalar multiple of V, by Schur’s lemma and irreducibility of π0 . To show existence, ﬁrst consider (for simplicity) the case where γ 1 and γ 2 start at the same site x. As remarked earlier in the proof of Theorem 3.1, Ω is a cyclic vector for ρk1 and for ρk2 (we will write ρk1 instead of ρkγ 1 in the proof). Moreover, the corresponding vector state is ωkx . By uniqueness of the GNS construction, there is a unitary V such that V ρk1 (A) = ρk2 (A)V for all A ∈ A, and V Ω = Ω. Choose paths γ n as in the statement of the lemma. The path obtained by concatenating γ n with the paths γn1 and γn2 can be seen as a loop based at x that gets larger and larger as n gets bigger. Now consider a sequence Vn of unitaries deﬁned by Vn = Γn1 Γn2 Γkeγn where Γni is deﬁned in the statement of the lemma. Note that Vn is a product of star and plaquette operators, since it is the path operator of a closed loop. Hence, Vn Ω = Ω by the observation above. Suppose B ∈ Aloc . Let N be such that γ n ∩ supp(B) = ∅ for all n ≥ N . Then from locality, one can easily verify that Vn ρk1 (B) = ρk2 (B)Vn for all n ≥ N , in other words, lim ρk1 (A)Ω, Vn ρk1 (B)Ω = lim ρk1 (A)Ω, ρk2 (B)Vn Ω = Ω, ρk1 (A)∗ ρk2 (B)Ω ,

n→∞

n→∞

for all A, B ∈ Aloc . On the other hand, for each A, B ∈ Aloc , ρk1 (A)Ω, V ρk1 (B)Ω = Ω, ρk1 (A)∗ ρk2 (B)Ω , since V Ω = Ω. The sequence Vn is uniformly bounded and because ρk1 (Aloc )Ω is dense in H0 , since ρk1 is an automorphism, it follows that Vn → V weakly. Seeing that any path γ from x to x is a loop, it is clear that Γkbγ V Ω = Ω.

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As for the general case, suppose γ 1 starts at the site x and γ 2 starts at the site y. Choose a path γ from x to y. Then ρ := Ad Γkeγ ◦ ρk1 is deﬁned by a path starting at y. By the argument above, there is a unitary V intertwining ρ and ρk2 such that V Ω = Ω. Set V = Γk V . It follows that V is an intertwiner from ρk to ρk that γ e

1

2

satisﬁes Γkγ V Ω = Ω for all paths γ from x to y, because Γkeγ Γkγ is the path operator of a loop. The claim on the converging net follows from the construction. A pleasant consequence of the above proof is that a speciﬁc sequence converging to the intertwiners is given, which makes it possible to do explicit calculations. A direct consequence of the lemma is that we have some control over the algebras containing the unitary intertwiners, a point where usually Haag duality is used. Theorem 4.1. Suppose Λ1 and Λ2 are two cones such that there is another cone Λ ⊃ Λ1 ∪ Λ2 . For k = X, Y, Z, consider ρki ∼ = πωk localized in Λi for i = 1, 2, defined by paths γ i extending to infinity. Let W be a unitary such that W ρk1 (A) = ρk2 (A)W for all A ∈ A. Then W ∈ A(Λ) . Proof. By Schur’s lemma, W is a multiple of the intertwiner V in the previous lemma. The geometric situation makes it clear that a net Vn as in the lemma can be chosen to be a net in A(Λ). This net converges weakly to V , by the previous lemma. Remark 4.1. Again it is not essential that Λ as in the theorem is a cone. It is enough to be able to chose paths γ n in as in Lemma 4.1 that lie inside Λ. But note that the smaller Λ is, the more control one has over the algebra where the intertwiners live in. Proposition 4.1. The representations ρk are covariant with respect to the action τx of translations. That is, for each x ∈ Z2 there is a unitary W (x) such that ρk (τx (A)) = W (x)ρk (A)W (x)∗ for all A ∈ A and the map x → W (x) is a group homomorphism. Proof. Let γ denote the string (starting at the site x0 ) deﬁning ρk . For x ∈ Z2 , consider the translated string γ = γ − x. This deﬁnes an automorphism ρ k . In fact, k k ρ = τ−x ◦ ρ ◦ τx . Then by Lemma 4.1, there is a unitary intertwiner Vx from ρk to ρ k . We choose Vx such that the condition in Lemma 4.1 is satisﬁed. Write U (x) for the unitaries that implement the translations in the GNS representation of ω0 . Deﬁne W (x) = U (x)Vx . It then follows that ρk (τx (A)) = W (x)ρk (A)W (x)∗ for all A ∈ Aloc , and hence by continuity for all A ∈ A. It remains to show that W (x) is a representation of Z2 . By irreducibility of ρk it follows that W (x + y) = λ(x, y)W (x)W (y) with λ a 2-cocycle of Z2 taking values in the unit circle. The claim is that λ is in fact trivial. This would follow from the equation U (y)∗ Vx U (y) = Vx+y Vy∗ for all x, y ∈ 2 Z . Note that the operator on the right-hand side is an intertwiner from ρkγ−y

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to ρkγ−(x+y) satisfying the condition in Lemma 4.1. This equation can be veriﬁed by noting that Vx+y and Vy commute with path operators (this should be clear from the construction of a converging net) and by the following observation: a path is a path from x0 − y to x0 − (x + y)) can be written as Γγ1 Γ∗γ2 operator Γbγ (where γ with γ1 a path from x0 to x0 − (x + y) and γ2 a path from x0 to x0 − y. Let Vxn be a sequence as in Lemma 4.1 converging weakly to Vx . Then for the translated sequence τ−y (Vxn ) w-lim τ−y (Vxn ) = Vx+y Vy∗ , n→∞

by the same lemma. The result follows since the map A → τ−y (A) = U (y)∗ AU (y) is weakly continuous, hence the left hand side is equal to U (y)∗ Vx U (y). It is possible to deﬁne a tensor product of localized endomorphisms. If ρ1 and ρ2 are localized in cones Λ1 and Λ2 , the basic idea is to deﬁne an endomorphism ρ1 ⊗ ρ2 by (ρ1 ⊗ ρ2 )(A) = ρ1 (ρ2 (A)). If Λ ⊃ Λ1 ∪ Λ2 is a cone, it follows that ρ1 ⊗ ρ2 is localized in Λ. In order to get a categorical tensor structure, one would then like to deﬁne a tensor product for intertwiners. If Ti , i = 1, 2 are intertwiners from ρi to σi (and Ti ∈ A), the reader will have no diﬃculty showing that T1 ⊗ T2 := T1 ρ1 (T2 ) is an intertwiner from ρ1 ⊗ ρ2 to σ1 ⊗ σ2 . In the terminology of category theory, this would turn the category of cone localized automorphisms with intertwiners as morphisms into a strict tensor category. The trivial endomorphism ι is the tensor unit. Note that the unit operator I of A can be regarded as an intertwiner from ρ to itself for any endomorphism ρ. To indicate this, we sometimes write Iρ . The distinction is important in the deﬁnition of the tensor product of intertwiners. There is, however, one problem with this deﬁnition: the intertwiners are elements of the algebra A(Λ) rather than of A(Λ) (recall that we identiﬁed π0 (A) with A). There is no reason why they should be contained in the quasi-local algebra A, because this algebra is not weakly closed in general. Since the localized endomorphisms are (a priori) only deﬁned on A, the above deﬁnition therefore does not make sense. A possible solution is to introduce an auxiliary algebra that contains the intertwiners [9]. Choose an arbitrary cone Λa , which will be ﬁxed from now on. The cone can be interpreted as a “forbidden” direction, not unlike the technique of puncturing the circle. Introduce a partial ordering on Z2 by deﬁning x ≤ y ⇔ (Λa + y) ⊂ (Λa + x) ⇔ (Λa + x)c ⊂ (Λa + y)c . Now (Z2 , ≤) is a directed set (each pair of points has an upper bound with respect to ≤), hence it is possible to take the (C ∗ )-inductive limit AΛa =

x∈Z2

A((Λa + x)c )

·

.

(4.2)

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Note that AΛa +x = AΛa for all x ∈ Z2 . Clearly, A ⊂ AΛa . Moreover, if Λ is a cone such that Λ ⊂ (Λa + x)c for some x, then A(Λ) ⊂ AΛa . An important pointh is that the automorphisms we consider can be extended to AΛa . Proposition 4.2. Let ρ be an automorphism defined by a path extending to infinity. Then ρ has a unique extension ρΛa to AΛa that is weakly continuous on A((Λa + x)c ) for any x ∈ Z2 . Moreover, ρΛa (AΛa ) ⊂ AΛa . In other words, it is an endomorphism of the auxiliary algebra. Proof. The proof is essentially the same as that of [9, Lemma 4.1], except at points where duality is used. First, let A ∈ A((Λa + x)c ). Since ρ is localizable, there is a unitary V such that ρ(A) = VAV ∗ (choose a unitary equivalent endomorphism localized in Λa + x). This implies that ρ is weakly continuous on A((Λa + x)c ) and the unique weakly continuous extension can be given by ρΛa (B) = VBV ∗ for B ∈ A((Λa + x)c ) . This procedure determines ρΛa on all of AΛa . To show that ρΛa maps AΛa into itself, ﬁrst note that ρ(A(Λ)) ⊂ A(Λ) for every ﬁnite set Λ ⊂ B. Hence, by weak continuity, ρΛa (A((Λa + x)c ) ) = ρ(A((Λa + x)c )) ⊂ A((Λa + x)c ) , which proves the claim. Remark 4.2. In the proof of Buchholz and Fredenhagen, Haag duality is used to show that the extensions map the auxiliary algebra into itself (see also Footnote h). The point is that using Haag duality it is possible to show that for representations localized in a cone Λ one has ρ(A(Λ)) ⊂ A(Λ) . Since we have an explicit description of the representations, we can directly prove the stronger statement ρ(A(Λ)) ⊂ A(Λ) for the automorphisms considered in our model. However, the intertwiners are typically not elements of A(Λ). a We now redeﬁne the tensor product as ρ1 ⊗ρ2 = ρΛ 1 ◦ρ2 . For the automorphisms that we have considered so far, this deﬁnition reduces to the old one. However, to deﬁne the tensor product of intertwiners, this deﬁnition is necessary. If S is an intertwiner from ρ1 to ρ1 and T an intertwiner from ρ2 to ρ2 such that T ∈ A(Λ) a for some cone Λ asymptotically disjoint from Λa , then S ⊗ T := SρΛ 1 (T ) is a well-deﬁned intertwiner from ρ1 ⊗ ρ2 to ρ1 ⊗ ρ2 . The tensor product gives rise to fusion rules. A fusion rule gives a decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations. In Kitaev’s model the rules are particularly simple. As remarked before, for each k = X, Y, Z, ρk ⊗ ρk = ι, where ι is the trivial endomorphism of A. Furthermore, essentially by deﬁnition, ρX ⊗ ρZ ∼ = ρY . This determines

h In

the case of algebraic quantum ﬁeld theory, the main point is to obtain endomorphisms of the auxiliary algebra from representations of the quasi-local algebra. In the present model, however, we already have automorphisms of A.

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the fusion rules for unitarily equivalent representations as well: unitaries setting up the equivalence can be deﬁned using the tensor product. Using the tensor product, in this case a braiding can then be deﬁned, similarly as in the DHR analysis [10]. This is a unitary operator ερ1 ,ρ2 intertwining ρ1 ⊗ ρ2 and ρ2 ⊗ ρ1 . First, consider two disjoint cones Λ1 and Λ2 that are both contained in (Λa + x)c for some x. We say that Λ1 < Λ2 if we can rotate Λ1 counterclockwise around the apex of the cone until it has non-empty intersection with Λa + x, such that at any intermediate angle it is disjoint from Λ2 . Note that for two disjoint cones either Λ1 < Λ2 or Λ2 < Λ1 . Now let ρ1 , ρ2 be two localized automorphisms, as considered above, such that ρ1 is localized in a cone Λ1 and ρ2 in Λ2 . Moreover, we demand that there is a cone 2 such that Λ 2 < Λ1 . Λ ⊃ Λ1 ∪Λ2 . Note that ρ1 ⊗ρ2 is localized in Λ. Choose a cone Λ ∗ 2 . This unitary can Then there is a unitary V such that V ρ2 (−)V is localized in Λ a be chosen in AΛa [25]. It then follows that ερ1 ,ρ2 := (V ⊗Iρ1 )∗ (Iρ1 ⊗V ) = V ∗ ρΛ 1 (V ) is an intertwiner from ρ1 ⊗ ρ2 to ρ2 ⊗ ρ1 . With this deﬁnition, one can prove the following result by adapting the proof in the DHR analysis (see e.g. [26]) in a suitable way. 2 < Λ1 , not on the Lemma 4.2. The braiding ερ,σ only depends on the condition Λ specific choices made. Moreover, it satisfies the braid equations ερ,σ⊗τ = (Iσ ⊗ ερ,τ )(ερ,σ ⊗ Iτ ) ερ⊗σ,τ = (ερ,τ ⊗ Iσ )(Iρ ⊗ εσ,τ ).

(4.3)

Furthermore, ερ,σ is natural in ρ and σ: if T is an intertwiner from ρ to ρ , then ερ ,σ (T ⊗ I) = (I ⊗ T )ερ,σ , and similarly for σ. In Lemma 4.1, a net converging to the charge transporters was explicitly constructed. This makes it possible to calculate the braiding operators exactly. In the subscript of the braiding, we will sometimes write X, Y or Z instead of ρX , ρY and ρZ . Theorem 4.2. Let ρ1 , ρ2 be automorphisms defined by strings extending to infinity in some cone Λ. Suppose that each automorphism is of type X or type Z. The braid operators in each of the possible cases are then given by εX,X = εZ,Z = I and εX,Z = ±I. If εX,Z = I, then εZ,X = −I and vice versa. disjoint from Λ, such that Λ < Λ and such that there is a Proof. Consider a cone Λ cone Λ ⊃ Λ ∪ Λ. There is a path γ 2 in Λ such that the corresponding automorphism The corresponding unitary charge ρ 2 is unitarily equivalent to ρ2 and localized in Λ. . By deﬁnition we then have ερ1 ,ρ2 = transporter V is then contained in A(Λ) a V ∗ ρΛ 1 (V ). a This can be calculated using weak continuity of ρΛ 1 and the explicit construction of Lemma 4.1 of a net converging to V . Indeed, let Vn → V be this net. Note that each Vn is a string operator of the same type as ρ2 . In particular, if ρ1 is of the

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Fig. 5. The path γn (dashed line) crosses the deﬁning path of ρ1 from the right. The dotted lines represent the deﬁning paths of ρ2 and ρb2 . a same type as ρ2 , then ρ1 (Vn ) = Vn for all n and hence ρΛ 1 (V ) = V . It follows that εX,X = εZ,Z = I. The situation where ρ1 is of type X and ρ2 is of type Z (or vice versa) is a bit more complicated. Recall that for the deﬁnition of the net Vn , for each n a path γ n is chosen, such that the distance to the starting points of the paths γ1 and γ2 goes to inﬁnity. The operator Vn is then the string operator corresponding to the string 2 , together with γ n . Note that, if n is big formed by the ﬁrst n bonds of γ2 and γ enough, this string crosses γ1 either an even number of times, or an odd number, independent of n. This property depends on whether the ﬁrst crossing is from the “left” or from the “right” (see Fig. 5), or if there is no crossing at all. By anti-commutation of the Pauli matrices, it follows that if the number of crossings is even, ρ1 (V ) = V , whereas if it is odd then ρ1 (V ) = −V . Hence, εX,Z = ±I. If the role of ρ1 and ρ2 is reversed, an odd number of crossings becomes an even number. This observation proves the last claim.

Since ρY = ρX ⊗ ρZ , the braid equations allow to compute the braiding with excitations of type Y . The braiding with the trivial automorphism is always trivial. This completely determines the braiding for all irreducible representations we consider. We note that the sign of, for example, εX,Z depends on the relative localization of both strings. Indeed, suppose we have two automorphisms ρ1 and ρ2 , deﬁned by strings γ1 of type X and γ2 of type Z, extending to inﬁnity and localized in Λ1 respectively Λ2 . Suppose moreover that Λ2 < Λ1 . It then follows that ερ1 ,ρ2 = I, 2 , do not cross γ1 . On the other since the paths in the proof, going from γ2 to γ hand, if Λ1 < Λ2 it follows that ερ1 ,ρ2 = −I. Note that this coincides with the situation in algebraic quantum ﬁeld theory in low dimensions [27, Sec. 2.2]. The ﬁnal piece of structure is that of conjugation. A conjugate can be interpreted as an anti-charge. Formally, a conjugate for an endomorphism ρ is a triple (ρ, R, R) such that R intertwines ι and ρ ⊗ ρ and R intertwines ι and ρ ⊗ ρ [28]. Here ι is the trivial endomorphism. The intertwiners R, R should satisfy ∗

R ρ(R) = I,

R∗ ρ(R) = I.

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A conjugate for an irreducible endomorphism ρ is called normalized if R∗ R = R R ∗ and standard if R∗ ρ(T )R = R T R for every intertwiner T from ρ to itself. If a conjugate exists, one can always ﬁnd a standard conjugate. Note that ρk ⊗ ρk = ι for k = X, Y, Z. It follows that in our model the automorphisms we consider have conjugates. These are particularly simple: ρk = ρk and one can choose the unit operators for the intertwiners R and R. This is trivially a standard conjugate. With the help of the braiding and conjugates one can deﬁne a twist. Let ρ be a cone localized endomorphism and (ρ, R, R) be a standard conjugate. The twist Θρ ∈ End(ρ) is then deﬁned by ∗

Θρ = (R ⊗ idρ ) ◦ (idρ ⊗ερ,ρ ) ◦ (R ⊗ idρ ). Note that if ρ is irreducible, Θρ = ωρ I for some phase factor. The (equivalence class of) ρ is called bosonic if ωρ = 1 and fermionic if ωρ = −1. Since the conjugates of ρk , k = X, Y, Z are particularly simple, the following corollary immediately follows from Theorem 4.2. Corollary 4.1. The excitations X and Z are bosonic and Y is fermionic. 5. Cone Algebras Let Λ be a cone. In this section we consider the von Neumann algebras associated to the observables localized in this cone. More precisely, deﬁne RΛ := π0 (A(Λ)) and RΛc := π0 (A(Λc )) . The main result in this section is that RΛ is an inﬁnite factor. Lemma 5.1. With the notation above, RΛ ∨ RΛc = B(H0 ). Proof. Note that for each set Λ ⊂ B one has RΛ = that B(H0 ) = π0 (A) = RΛ ∨ RΛc .

b∈Λ

π0 (A({b})). It follows

More can be said about the cone algebras. In fact, they are inﬁnite factors. In other words, RΛ is a factor of Type I∞ , Type II∞ or Type III. The basic idea of the proof, which is adapted from [23, Proposition 5.3], is to assume that RΛ admits a tracial state. It then follows that ω0 is tracial, which is a contradiction. In fact, Type I∞ can be ruled out as well. Theorem 5.1. RΛ is a factor of Type II∞ or Type III. Proof. To show that RΛ is a factor, we argue as in [23]. The center is Z(RΛ ) = RΛ ∩ RΛ . By taking commutants, Z(RΛ ) = RΛ ∨ RΛ . Note that RΛc ⊂ RΛ , hence by Lemma 5.1, Z(RΛ ) = B(H0 ). Assume that RΛ is a ﬁnite factor. Then there exists a unique tracial state ψ on RΛ . This induces a tracial state ψ = ψ ◦ π0 on A(Λ). By Propositions 10.3.12(i)

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and 10.3.14 of [21], it follows that the state ψ is factorial and quasi-equivalent to the restriction of ω0 to A(Λ). ⊂ Λ such that |ω0 (A) − Let ε > 0. By [6, Corollary 2.6.11], there is a ﬁnite set Λ ψ(A)| < εA for all A ∈ A(Λ\Λ). Now, let k > 0 be an integer. Consider local observables A, B with localization region contained in B(0, k) (that is, all bonds that can be connected to the origin of Z2 with a path of length at most k) and is ﬁnite, there is an x ∈ Z2 , such that τx (AB ) is norm 1. Since Λ is a cone and Λ localized in Λ\Λ. By translation invariance, x (AB ))| = |ω0 (τx (AB )) − ψ(τ x (AB ))| < ε, |ω0 (AB ) − ψ(τ and similarly for BA. Hence since ψ is a trace, x (AB )) − ω0 (BA) + ψ(τ x (BA))| < 2ε. |ω0 (AB ) − ω0 (BA)| = |ω0 (AB ) − ψ(τ Because k and ε were arbitrary, ω0 (AB ) = ω0 (BA) for all A, B ∈ Aloc , which is absurd. To see that the Type I case can be ruled out, note that RΛ is of Type I if and only if ω0 is quasi-equivalent to ω0,Λ ⊗ ω0,Λc . This can be seen by adapting the ⊂ B be any ﬁnite set. Then one can always proof of [29, Proposition 2.2]. Let Λ c ﬁnd a star s in Λ such that the intersection with both Λ and Λc is not empty. But for this star s, one has ω0 (As ) = 1. On the other hand, (ω0,Λ ⊗ ω0,Λc )(As ) = 0, essentially because Λ ∩ s is not a star any more. This implies that the states ω and ω0,Λ ⊗ ω0,Λc are not equal at inﬁnity. It follows that ω0 cannot be quasi-equivalent to ω0,Λ ⊗ ω0,Λc . We single out a useful consequence of this result. Corollary 5.1. Let Λ be a cone. Then RΛ contains isometries V1 , V2 such that Vi∗ Vj = δi,j I and V1 V1∗ + V2 V2∗ = I. Proof. By [30, Proposition V.1.36], there is a projection P such that P ∼ (I − P ) ∼ I, where ∼ denotes Murray–von Neumann equivalence with respect to RΛ . Hence, there are isometries V1 , V2 such that V1 V1∗ = P and V2 V2∗ = (I − P ). These isometries suﬃce. Although we have no proof for Haag duality for cones, we would like to point out an interesting consequence of this duality. For two cones Λ1 ⊂ Λ2 , write Λ1 Λ2 if any star or plaquette in Λ1 ∪ Λc2 is either contained in Λ1 or in Λc2 . Definition 5.1. We say that ω0 satisﬁes the distal split property for cones if for any pair of cones Λ1 Λ2 there is a Type I factor N such that RΛ1 ⊂ N ⊂ RΛ2 . With the assumption of Haag duality we can then prove the following theorem. Theorem 5.2. Suppose that π0 satisfies Haag duality for cones. Then ω0 has the distal split property for cones.

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Proof. Let Λ1 Λ2 be two cones. Note that it is enough to prove that RΛ1 ∨RΛ2 RΛ1 ⊗ RΛ2 , where denotes that the natural map A ⊗ B → AB (A ∈ RΛ1 , B ∈ RΛ2 ) extends to a normal isomorphism. Indeed, if this is the case, the result follows from [31, Theorem 1 and Corollary 1], since RΛ1 and RΛ2 are factors. Note that ω0 (AB ) = ω0 (A)ω0 (B) if A ∈ A(Λ1 ), B ∈ A(Λc2 ). Since ω0 is normal, this result is also valid for A ∈ RΛ1 and B ∈ RΛc2 . A result of Takesaki [32] then implies that RΛ1 ∪Λc2 = RΛ1 ∨ RΛc2 RΛ1 ⊗ RΛc2 . By Haag duality, RΛc2 = RΛ2 , which concludes the proof. Note that without Haag duality only the existence of a Type I factor RΛ1 ⊂ N ⊂ π0 (A(Λc2 )) can be concluded. The condition that Λ1 Λ2 is needed precisely to avoid the situation at the end of the proof of Theorem 5.1. 6. Equivalence with Repf D(Z2 ) If G is a ﬁnite group, one can form the quantum double D(G) of the group. The quantum double is a quasi-triangular Hopf algebra (see e.g. [33] for an introduction). It is well-known that Repf D(G), the category of ﬁnite dimensional D(G)-modules, is a modular tensor category [34]. In this section we will introduce the category ∆(Λ) of stringlike localized representations and show that it is equivalent to Repf D(Z2 ) (as braided tensor C ∗ -categories). This implies that for all practical purposes, the excitations are described by the representation theory of D(Z2 ). Lemma 6.1. Let ρ1 , ρ2 be two transportable endomorphisms of A, localized in a cone Λ. Then one can define a localized and transportable direct sum ρ1 ⊕ ρ2 . Proof. Let V1 , V2 ∈ RΛ be isometries as in Corollary 5.1. Deﬁne ρ(A) := V1 ρ1 (A)V1∗ + V2 ρ2 (A)V2∗ , for all A ∈ A. It follows that ρ is a ∗-representationi of A. Since Vi ∈ RΛ and RΛc ⊂ RΛ , it follows that ρ(A) = A for A ∈ A(Λc ), be another cone. Pick hence ρ is localized in Λ. To show transportability, let Λ isometries W1 , W2 ∈ RΛb as in Corollary 5.1. Since ρ1 and ρ2 are transportable, Deﬁne there are unitary operators Ui such that Ui ρi (−)Ui∗ is localized in Λ. ∗ ∗ ∗ ∗ ∗ W = W1 U1 V1 + W2 U2 V2 . Then W W = W W = I and W ρ(−)W is localized in hence ρ is transportable. This ρ, which is unique up to unitary equivalence, will Λ, be denoted by ρ1 ⊕ ρ2 . We will now introduce the category ∆(Λ). For technical reasons it is convenient to consider only representations localized in a ﬁxed cone Λ, since in that case clearly all intertwiners are in the algebra AΛa . Proceeding in this way, there is no problem in deﬁning the tensor product. It should be noted that the resulting category does that ρ is not necessarily an endomorphism of A any more, but rather of AΛa . This is, however, only a minor technicality and is not essential for what follows.

i Note

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not depend on the speciﬁc choice of cone Λ (see [25, Proposition 2.11] for a proof and for alternative approaches). The irreducible objects of the category ∆(Λ) are precisely the automorphisms localized in the cone Λ that are given by paths extending to inﬁnity. The morphisms are intertwiners from one endomorphism to another. By the lemma above, ﬁnite direct sums can be constructed, turning ∆(Λ) into a category with direct sums. In fact, by construction, each object can be decomposed into irreducibles. It is clear from the construction that the direct sums can be extended to endomorphisms of the auxiliary algebra. Hence the tensor product deﬁned in Sec. 4 can be deﬁned for all objects. Similarly, a braiding for direct sums can be constructed from Theorem 4.2. Conjugates for direct sums can be constructed from conjugates for the irreducible components. Summarizing, freely using terminology from [26, 35], we have the following result: Theorem 6.1. The category ∆(Λ) is a braided tensor C ∗ -category. The category obtained in this way is actually equivalent (as a braided tensor C ∗ -category) to the representation category of D(Z2 ) over the ﬁeld k = C. For the structure of Repf D(Z2 ) as a braided tensor C ∗ -category we refer to [13]. A highbrow way of seeing this is to appeal to the classiﬁcation results of modular tensor categories [36]. It is however possible to give an explicit construction of the equivalence. Note that equivalence as braided categories is in general stronger than equivalence as tensor categories. Indeed, there are non-isomorphic groups whose representation categories are equivalent as tensor categories but not as braided tensor categories [37]. On the other hand, every symmetric tensor category (satisfying certain additional properties) is the representation category of a compact group (determined up to isomorphism) [38]. Theorem 6.2. There is a braided equivalence of tensor C ∗ -categories ∆(Λ) → Repf D(Z2 ). Proof. Since Z2 is abelian, the irreducible representations of D(Z2 ) are labeled 2 [34, 39]. Here χe and by the elements e, f of Z2 and χe , χσ of the dual group Z χσ denote the trivial and the sign character of Z2 respectively. Write Vg,χ for the irreducible D(Z2 )-module induced by an element g and character χ. We obtain the following list of all irreducible modules of D(Z2 ): Π0 = Ve,χe ,

ΠX = Vf,χe ,

ΠY = Vf,χσ ,

ΠZ = Ve,χσ .

Recall that using the coproduct of D(Z2 ) the tensor product Πi ⊗ Πj can be made into a left D(Z2 )-module. The tensor product has the same fusion rules as ∆(Λ), e.g. ΠX ⊗ ΠY ∼ = ΠZ and Πk ⊗ Π0 ∼ = Π0 ⊗ Πk ∼ = Πk . On the side of ∆(Λ), choose paths of type X, Z such that the corresponding automorphisms ρX , ρZ satisfy εX,Z = −I. Deﬁne ρY = ρX ⊗ ρZ , and ρ0 = ι, the trivial endomorphism. Note that each irreducible representation in ∆(Λ) is unitarily

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equivalent to one of the ρk . This suggests to deﬁne a functor F : Repf D(Z2 ) → ∆(Λ) as follows: for irreducible modules, the most natural choice is to set F (Πk ) = ρk for k = 0, X, Y, Z. The irreducible modules have dimension one, hence the D(Z2 )linear maps between the irreducible modules are just the scalars. In order for F to be a linear functor, there is essentially only one choice of F (T ) for a morphism T . Note that F is full and faithful on the Hom-sets of irreducible objects. By construction every irreducible object of ∆(Λ) is isomorphic to an object in the image of F . In fact, F is a braided monoidal functor. By our particular choice of ρX , ρY and ρZ , one can choose the natural transformations F (V ⊗W ) → F (V )⊗F (W ), needed for the deﬁnition of a monoidal functor, to be identities. To see that F is indeed a braided functor, recall that for π1 , π2 ∈ Repf D(Z2 ), the braiding cπ1 ,π2 is the linear map intertwining π1 ⊗ π2 and π2 ⊗ π1 deﬁned by cπ1 ,π2 = σ ◦ (π1 ⊗ π2 )(R). Here σ is the canonical ﬂip and R is a universal R-matrix for D(Z2 ). It is then straightforward to verify that for irreducible modules, F sends the braiding of Repf D(Z2 ) to that of ∆(Λ). For example, cΠX ,ΠZ = −1 (where we omit the isomorphism of the underlying vector spaces). The extension of the functor to direct sums is left to the reader, as is the veriﬁcation that F preserves all the relevant structures of a braided tensor C ∗ -category. Since the irreducible objects of both categories are in 1-1 correspondence, and the functor F preserves direct sums and braidings, F sets up an equivalence of braided tensor C ∗ -categories. Note, for example, that F is full, faithful and essentially surjective. Indeed, it is tedious but relatively straightforward to deﬁne an inverse functor setting up the equivalence. Acknowledgments This research is funded by the Netherlands Organization for Scientiﬁc Research (NWO) Grant No. 613.000.608. I would like to thank M. Fannes for a discussion on the construction of the ground state, P. Fendley for the idea that single excitations can be obtained by moving one excitation of a pair to inﬁnity, and M. M¨ uger and N. P. Landsman for helpful discussions and a critical reading of the manuscript. Professors D. Buchholz and K. Fredenhagen gave useful references at the 27th LQP Workshop in Leipzig, where this work was presented. An anonymous referee pointed out a gap in the proof of the spectral gap in an earlier version, as well as a suggestion on how to ﬁx it. References [1] A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303 (2003) 2–30. [2] C. Nayak, S. H. Simon, A. Stern, M. Freedman and S. Das Sarma, Non-abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 1083–1159. [3] Z. Wang, Topological Quantum Computation, CBMS Regional Conference Series in Mathematics, Vol. 112, Published for the Conference Board of the Mathematical Sciences (Amer. Math. Soc., 2010).

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[4] R. Alicki, M. Fannes and M. Horodecki, A statistical mechanics view on Kitaev’s proposal for quantum memories, J. Phys. A 40 (2007) 6451–6467. [5] E. Dennis, A. Kitaev and J. Preskill, Topological quantum memory, J. Math. Phys. 43 (2002) 4452–4505. [6] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 1, Texts and Monographs in Physics, 2nd edn. (Springer-Verlag, New York, 1987). [7] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics. 2, Texts and Monographs in Physics, 2nd edn. (Springer-Verlag, Berlin, 1997). [8] R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Texts and Monographs in Physics, 2nd edn. (Springer-Verlag, Berlin, 1996). [9] D. Buchholz and K. Fredenhagen, Locality and the structure of particle states, Comm. Math. Phys. 84 (1982) 1–54. [10] S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. I, Comm. Math. Phys. 23 (1971) 199–230. [11] S. Doplicher, R. Haag and J. E. Roberts, Local observables and particle statistics. II, Comm. Math. Phys. 35 (1974) 49–85. [12] F. Nill and K. Szlach´ anyi, Quantum chains of Hopf algebras with quantum double cosymmetry, Comm. Math. Phys. 187 (1997) 159–200. [13] K. Szlach´ anyi and P. Vecserny´es, Quantum symmetry and braid group statistics in G-spin models, Comm. Math. Phys. 156 (1993) 127–168. [14] K. Fredenhagen and M. Marcu, Charged states in Z2 gauge theories, Comm. Math. Phys. 92 (1983) 81–119. [15] F. A. Bais, P. van Driel and M. de Wild Propitius, Quantum symmetries in discrete gauge theories, Phys. Lett. B 280 (1992) 63–70. [16] J. C. A. Barata and K. Fredenhagen, Charged particles in Z2 gauge theories, Comm. Math. Phys. 113 (1987) 403–417. [17] J. C. A. Barata and F. Nill, Electrically and magnetically charged states and particles in the (2 + 1)-dimensional ZN -Higgs gauge model, Comm. Math. Phys. 171 (1995) 27–86. [18] J. C. A. Barata and F. Nill, Dyonic sectors and intertwiner connections in (2 + 1)dimensional lattice ZN -Higgs models, Comm. Math. Phys. 191 (1998) 409–466. [19] B. Nachtergaele and R. Sims, Lieb–Robinson bounds and the exponential clustering theorem, Comm. Math. Phys. 265 (2006) 119–130. [20] D. E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford Mathematical Monographs (The Clarendon Press Oxford University Press, New York, 1998). [21] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II, Graduate Studies in Mathematics, Vol. 16 (American Mathematical Society, Providence, RI, 1997). [22] D. Buchholz and K. Fredenhagen, Locality and the structure of particle states in gauge field theories, in Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, Vol. 153 (Springer, 1982), pp. 368–371. [23] M. Keyl, T. Matsui, D. Schlingemann and R. F. Werner, Entanglement Haag-duality and type properties of infinite quantum spin chains, Rev. Math. Phys. 18 (2006) 935–970. [24] T. Matsui, Spectral gap, and split property in quantum spin chains, J. Math. Phys. 51 (2010) 015216, 8 pp. [25] P. Naaijkens, On the extension of stringlike localised sectors in 2+1 dimensions, Comm. Math. Phys. 303 (2011) 385–420.

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[26] H. Halvorson, Algebraic quantum field theory, in Philosophy of Physics, eds. J. Butterfield and J. Earman (Elsevier, 2006), pp. 731–922. [27] K. Fredenhagen, K.-H. Rehren and B. Schroer, Superselection sectors with braid group statistics and exchange algebras. II. Geometric aspects and conformal covariance, Rev. Math. Phys. 4 (1992) 113–157. [28] R. Longo and J. E. Roberts, A theory of dimension, K-Theory 11 (1997) 103–159. [29] T. Matsui, The split property and the symmetry breaking of the quantum spin chain, Comm. Math. Phys. 218 (2001) 393–416. [30] M. Takesaki, Theory of Operator Algebras. I, Encyclopaedia of Mathematical Sciences, Vol. 124 (Springer-Verlag, Berlin, 2002). [31] C. D’Antoni and R. Longo, Interpolation by type I factors and the flip automorphism, J. Funct. Anal. 51 (1983) 361–371. ohoku Math. J. (2) 10 (1958) [32] M. Takesaki, On the direct product of W ∗ -factors, Tˆ 116–119. [33] C. Kassel, Quantum Groups, Graduate Texts in Mathematics, Vol. 155 (SpringerVerlag, New York, 1995). [34] B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, University Lecture Series, Vol. 21 (American Mathematical Society, Providence, RI, 2001). [35] M. M¨ uger, Abstract duality for symmetric tensor ∗-categories, appendix to [26]. [36] E. Rowell, R. Stong and Z. Wang, On classification of modular tensor categories, Comm. Math. Phys. 292 (2009) 343–389. [37] P. Etingof and S. Gelaki, Isocategorical groups, Int. Math. Res. Notices 2001 (2001) 59–76. [38] S. Doplicher and J. E. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989) 157–218. [39] R. Dijkgraaf, V. Pasquier and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models. Recent advances in field theory (Annecy-le-Vieux, 1990), Nuclear Phys. B Proc. Suppl. 18B (1991) 60–72.

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Reviews in Mathematical Physics Vol. 23, No. 4 (2011) 375–407 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004321

EXISTENCE OF GROUND STATES OF HYDROGEN-LIKE ATOMS IN RELATIVISTIC QED I: THE SEMI-RELATIVISTIC PAULI–FIERZ OPERATOR

∗ ¨ MARTIN KONENBERG

Fakult¨ at f¨ ur Mathematik und Informatik, FernUniversit¨ at Hagen, L¨ utzowstraße 125, D-58084 Hagen, Germany [email protected] OLIVER MATTE† Institut f¨ ur Mathematik, TU Clausthal, Erzstraße 1, D-38678 Clausthal-Zellerfeld, Germany [email protected] EDGARDO STOCKMEYER Mathematisches Institut, Ludwig-Maximilians-Universit¨ at, Theresienstraße 39, D-80333 M¨ unchen, Germany [email protected]

Received 7 March 2011 We consider a hydrogen-like atom in a quantized electromagnetic ﬁeld which is modeled by means of the semi-relativistic Pauli–Fierz operator and prove that the inﬁmum of the spectrum of the latter operator is an eigenvalue. In particular, we verify that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true, for arbitrary values of the ﬁne-structure constant and the ultraviolet cut-oﬀ as long as the Coulomb coupling constant is less than 2/π. For Coulomb coupling constants larger than 2/π, we show that the quadratic form of the Hamiltonian is unbounded below. Keywords: Semi-relativistic Pauli–Fierz operator; quantum electrodynamics; existence of ground states. Mathematics Subject Classiﬁcation 2010: 81Q10, 81V10

1. Introduction The existence of atoms described in the framework of non-relativistic quantum electrodynamics (QED) is by now a well-established fact. The general picture is roughly ∗ Present

address: Fakult¨ at f¨ ur Physics, Universit¨ at Wien, Boltzmanngasse 5, 1090 Vienna, Austria. address: Mathematisches Institut, Ludwig-Maximilians-Universit¨ at, Theresienstraße 39, D-80333 M¨ unchen, Germany.

† Present

375

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that all exited bound states of an electronic Hamiltonian modeling an atom turn into resonances when the interaction with the quantized electromagnetic ﬁeld is taken into account. Only at the lower end of the spectrum there remains an eigenvalue corresponding to the ground states of the atomic system. Its analysis is particularly subtle as the whole spectrum is continuous up to its minimum in the presence of the quantized radiation ﬁeld. The existence of energy minimizing ground states for atoms and molecules in non-relativistic QED has been proven ﬁrst in [3, 5], for small values of the involved physical parameters. The latter are Sommerfeld’s ﬁne structure constant, e2 , and the ultraviolet cut-oﬀ, Λ. The existence of ground states for a molecular Pauli–Fierz–Hamiltonian has been shown in [11], for all values of e2 and Λ, assuming a certain binding condition which has been veriﬁed later on in [16]. In the last decade there appeared a large number of further mathematical contributions to non-relativistic QED. Here we only mention that ground state energies and projections have also been studied by means of infrared ﬁnite algorithms and renormalization group methods [1–6, 9]. In contrast to the situation in non-relativistic QED only a few mathematical works deal with models where the quantized radiation ﬁeld is coupled to relativistic particles. For instance, in [15, 16] the authors study a relativistic no-pair model of a molecule. They prove the stability of matter of the second kind and give an upper bound on the (positive) binding energy under certain restrictions on e2 , Λ, and the nuclear charges. In [19] two of the present authors consider a no-pair model of a hydrogenic atom and study the exponential localization of low-lying spectral subspaces. The same result is established in [19] also for the following operator which is investigated further in the present paper, (1.1) Hγ := (σ · (−i∇ + A))2 + 1 − γ/|x| + Hf . Here A is the quantized vector potential in the Coulomb gauge, Hf is the radiation ﬁeld energy, σ is a formal vector containing the Pauli spin matrices, and γ = e2 Z > 0 is the Coulomb coupling constant, Z > 0 denoting the atomic number. (The square-root of the ﬁne structure constant is included in the symbol A, which also depends on the choice of Λ.) Previous mathematical works dealing with this operator include [20] where the ﬁber decomposition of H0 with respect to different values of the total momentum is studied. We adopt the nomenclature of the latter paper and call Hγ the semi-relativistic Pauli–Fierz operator. Furthermore, the operator Hγ appears in the mathematical analysis of Rayleigh scattering [10] which is connected to the phenomenon of relaxation of an isolated atom to its ground state. (The electron spin has been neglected in [10] for notational simplicity.) An advantageous feature of semi-relativistic Hamiltonians in this situation is that the propagation speed of the electron is strictly less than the speed of light (which equals one in the units chosen in (1.1)). Moreover, it is shown in [27] that Hγ converges in norm resolvent sense to the non-relativistic Pauli–Fierz operator when the speed of light is re-introduced in (1.1) and goes to inﬁnity. A bound on the binding energy for Hγ is derived in [12]. (The bound obtained in [12] improves

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an earlier bound from a preprint version of the present article. Here we prove the same bound as in [12] by a variation of our original argument.) We remark that the existence of ground states in relativistic models of QED where all particles, including the electrons and positrons, are described by quantized ﬁelds is proven in [7]. To this end the authors employ infrared cut-oﬀs in the interaction part of the Hamiltonian which will not be necessary in our analysis below. Thanks to [19] we already know that Hγ is semi-bounded below on some natural dense domain, for all γ ∈ [0, 2/π], and, hence, has a physically distinguished self-adjoint realization. As already indicated above, it is also shown in [19] that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized with respect to the electron coordinates. Typically, localization estimates are important ingredients in the proofs of the existence of ground states. The bound on the binding energy derived in [12] and, by a diﬀerent method, in the present paper ensures that the bottom of the spectrum of Hγ is actually smaller than the ionization threshold. As a byproduct of our estimates we shall see that the quadratic form of Hγ is unbounded below, if γ > 2/π, so that the critical Coulomb coupling constant of the square-root operator does not change when the interaction with the quantized radiation ﬁeld is taken into account. The main theorem of this article asserts that the operator Hγ has an energy minimizing ground state eigenvector, for arbitrary values of e2 and Λ and for γ ∈ (0, 2/π). We remark that the ground state energy — in fact, every speculative eigenvalue — of Hγ is evenly degenerate since Hγ commutes with the time reversal operator [20]. In order to prove the existence of ground states we combine the strategies employed in [3, 5] and [11]. Roughly speaking we construct a sequence of approximating ground state eigenvectors — these are ground states of infrared cut-oﬀ Hamiltonians — along the lines of [3, 5], and apply a compactness argument similar to the one given in [11]. As in [3, 5], where the authors assumed e2 or Λ to be small, we prove the existence of ground states for the infrared cut-oﬀ Hamiltonians by means of a discretization procedure. A new observation based on the localization estimates actually permits to carry out the discretization argument, for all values of e2 and Λ. Another key ingredient in the proofs are infrared estimates on the approximating ground state eigenvectors, namely, a bound on the number of soft photons [3, 11] and a photon derivative bound [11]. In order to establish these bounds for the model treated here, the formal gauge invariance of Hγ is crucial. In fact, the no-pair models investigated in [15, 16, 19] are gauge invariant also and the present authors exploit this property to prove the existence of ground states for a no-pair model of a hydrogenic atom in the companion paper [14]. Although the general strategies to prove the existence of ground states in QED are fairly well known by now, their application to the model studied in the present paper and to no-pair models in QED is non-trivial, mainly due to the non-locality of the corresponding Hamiltonians. In fact, the electronic kinetic energy and the quantized vector potential in the Hamiltonian (1.1)

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are always linked together in a non-local way which leads to a variety of new mathematical problems in each of the steps in the existence proof mentioned above. To overcome these diﬃculties we employ various commutator estimates involving sign functions of the Dirac operator, multiplication operators, and the radiation ﬁeld energy. Some of them have already been derived in [19]. This article is organized as follows. In the next section, we introduce the semirelativistic Pauli–Fierz operator and state our main results more precisely. As a technical prerequisite we derive some estimates involving absolute values and sign functions of the Dirac operator in Sec. 3. In Sec. 4, we prove that binding occurs in our model. Section 5 is devoted to the proof of the existence of ground states and starts with a brief outline of the strategy. Finally, in Sec. 6 we prove the infrared bounds. 2. Definition of the Model and Main Results The semi-relativistic Pauli–Fierz operator acts in the Hilbert space H 2 , where ⊕ ν 2 3 ν ∼ Hm := L (Rx , C ) ⊗ Fb [Km ] = Cν ⊗ Fb [Km ]d3 x, H ν := H0ν , (2.1) R3x

for m ≥ 0. (In our proofs we employ spaces with m > 0 and it shall be convenient to choose ν = 4.) Here the bosonic Fock space, Fb [Km ] =

∞

(n)

Fb [Km ],

n=0

is modeled over the one photon Hilbert space Km , K := K0 , where Km := L2 (Am × Z2 , dk), Am := {|k| ≥ m}, dk := λ∈Z2

d3 k.

Am

The letter k = (k, λ) always denotes a tuple consisting of a photon wave vector, k ∈ R3 , and a polarization label, λ ∈ Z2 . The components of k are written as k = (k (1) , k (2) , k (3) ). The following subspace is dense in Hmν , ν := C0∞ (R3x , Cν ) ⊗ Cm , Dm

D ν := D0ν . (Algebraic tensor product.)

(2.2)

Here Cm ⊂ Fb [Km ] denotes the subspace of all elements (ψ (n) )∞ n=0 ∈ Fb [Km ] such that only ﬁnitely many components ψ (n) are non-zero and such that each ψ (n) is bounded and has a compact support. The quantized vector potential is deﬁned by means of the following coupling function with sharp ultraviolet cut-oﬀ at Λ > 0, (k) := e−ik·x g(k), Gphys x

g(k) ≡ ge,Λ (k) := −e

1{|k|≤Λ} ε(k), 2π |k|

(2.3)

for every x ∈ R3 and almost every k = (k, λ) ∈ R3 × Z2 . Here the square of the elementary charge, e > 0, is equal to Sommerfeld’s ﬁne-structure constant in our units where Planck’s constant, the speed of light, and the electron mass are equal to one. (Energies are measured in units of the rest energy of the electron and x is

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measured in units of one Compton wave length divided by 2π; we have e2 ≈ 1/137 in nature.) Writing k⊥ := (k (2) , −k (1) , 0),

k = (k (1) , k (2) , k (3) ) ∈ R3 ,

(2.4)

the polarization vectors are given by ε(k, 0) =

k⊥ , |k⊥ |

ε(k, 1) =

k ∧ ε(k, 0), |k|

(2.5)

for almost every k ∈ R3 . We let a† (f ) and a(f ) denote the standard bosonic creation and annihilation operators, for a photon state f ∈ Km . For a vector of functions f = (f (1) , f (2) , f (3) ) ∈ Km3 , we set a (f ) := (a (f (1) ), a (f (2) ), a (f (3) )), where a is a or a† . Then the quantized vector potential, A = (A(1) , A(2) , A(3) ), is the triplet of operators given by the direct integral

⊕

A := R3x

1Cν ⊗ A(x)d3 x,

A(x) := a† (Gphys ) + a(Gphys ). x x

(2.6)

(It will be clear from the context when ν = 2 or ν = 4.) We further set p := −i∇x ,

σ · (p + A) :=

3

σj (−i∂xj + A(j) ),

j=1

where σ1 , σ2 , σ3 are the Pauli spin matrices. An application of Nelson’s commutator theorem with test operator −∆ + Hf + 1 shows that σ · (p + A) is essentially selfadjoint on D 2 . We denote its closure again by the same symbol and deﬁne TA :=

(σ · (p + A))2 + 1

by means of the spectral calculus. Now the semi-relativistic Pauli–Fierz operator is a priori given as γ + Hf ϕ, ϕ ∈ D 2 . (2.7) Hγ ϕ := TA − |x| Here the radiation ﬁeld energy, Hf := dΓ(ω), is given as the second quantization of the dispersion relation ω(k) = |k|, k = (k, λ) ∈ R3 × Z2 . We recall that the second quantization of some real-valued Borel function, , on Am × Z2 , m ≥ 0, (n) is the direct sum dΓ( ) = ∞ ( ), where dΓ(0) ( ) := 0 and dΓ(n) ( ) is n=0 dΓ the maximal operator of multiplication with the symmetric function (k1 , . . . , kn ) → γ γ ≡ |x| ⊗ 1, Hf ≡ 1 ⊗ Hf ,

(k1 ) + · · · + (kn ). In (2.7) and henceforth we identify |x| etc. It has been shown in [19] that the quadratic form of Hγ is bounded from below on D 2 , for γ ∈ [0, 2π] and all values of e, Λ > 0; compare Inequality (3.5) below.

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In particular, Hγ has a self-adjoint Friedrichs extension which we again denote by the same symbol Hγ . In what follows we denote the ground state energy of Hγ by Eγ := inf σ[Hγ ],

γ ∈ (0, 2/π],

and its ionization threshold by Σ := inf σ[H0 ]. The following theorems are the main results of this paper. Theorem 2.1 (Binding). Let e, Λ > 0 and γ ∈ (0, 2/π]. Then Σ − Eγ ≥ |Eγel |,

√ where Eγel := inf σ[ 1 − ∆ − 1 − γ/|x|].

(2.8)

Proof. This theorem is a special case of Corollary 4.1 below. Theorem 2.2 (Critical Coupling Constant). For e, Λ >√0 and γ > 2/π, the quadratic form of TA − γ/|x| + Hf is unbounded below on Q( −∆) ∩ Q(Hf ). Proof. This theorem follows from Corollary 4.2. Theorem 2.3 (Existence of Ground States). Let e, Λ > 0 and γ ∈ (0, 2/π). Then Eγ is an evenly degenerated eigenvalue of Hγ . Proof. It is remarked in [20, §4] that every eigenvalue of Hγ is evenly degenerated. In fact, this follows from Kramers’ degeneracy theorem since Hγ commutes with the anti-unitary time reversal operator ϑ := σ2 CR, ϑ2 = −1, where C denotes complex conjugation and the electron parity R replaces x by −x. The fact that Eγ is an eigenvalue is proved in Sec. 5. Remark 2.1. (i) A binding condition for Hγ has been proven ﬁrst in a preprint version of the present article. There we gave, however, an estimate on the binding energy in terms of the lowest eigenvalue of the non-relativistic operator − 12 ∆ − γ/|x| which is smaller in absolute value than Eγel . The improved and more natural bound (2.8) has been obtained ﬁrst in [12] under the assumption that Hγ be essentially self-adjoint on some suitable dense domain. The latter property of Hγ can be veriﬁed, at least for all γ ∈ [0, 1/2) [14]. (ii) Every ground state eigenfunction of Hγ is exponentially localized with respect to the electron coordinates in the L2 -sense [19]; see Proposition 5.2 where we recall the precise statement. (iii) Theorems 2.1–2.3 actually hold true, for arbitrary choices of the polarization vectors ε(k, λ), λ ∈ Z2 , as long as ε(k, λ) is homogeneous of degree zero in k and {k/|k|, ε(k, 0), ε(k, 1)} is an orthonormal basis of R3 , for almost every k. For in this case the special form (2.5) of the polarization vectors can always be

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achieved by a suitable unitary transformation; see [26, Appendix] for details. Moreover, the sharp ultraviolet cut-oﬀ in (2.3) can be replaced by a smooth cutoﬀ implemented by some rapidly decaying function which is symmetric about the origin and Theorems 2.1–2.3 still remain valid. This follows by inspection of the proofs below. 3. The Dirac Operator 3.1. Operators acting on four-spinors It shall be convenient to work with a two-fold direct sum of the operator Hγ deﬁned in (2.7). For this permits to exploit earlier results on sign functions of the free Dirac operator minimally coupled to the quantized radiation ﬁeld and to have a familiar notation in the proofs. The full Hilbert spaces we shall work with in the rest of this paper, thus, are H 4 or Hm4 deﬁned in (2.1). In order to introduce the Dirac operator we ﬁrst recall that the Dirac matrices α1 , α2 , α3 , and β = α0 are hermitian (4 × 4)-matrices obeying the Cliﬀord algebra relations αi αj + αj αi = 2δij 1,

i, j ∈ {0, 1, 2, 3}.

In the standard representation they are given in terms of the Pauli matrices as αj = σ1 ⊗ σj , j ∈ {1, 2, 3}, and β = σ3 ⊗ 1. We shall also work with generalized coupling functions in what follows. For many of the technical results stated below are applied to truncated and discretized versions of the physical form factor (2.3). Recall that Am := {k ∈ R3 : |k| ≥ m}, for m ≥ 0. Hypothesis 3.1. Let : Am ×Z2 → [0, ∞) be a measurable function that depends on k ∈ Am only such that 0 < (k) ≤ |k|, for k = (k, λ) ∈ Am × Z2 with k = 0. For almost every k ∈ Am × Z2 , let G(k) be a bounded, twice continuously diﬀerentiable function, R3x x → Gx (k) ∈ R3 , such that the map (x, k) → Gx (k) is measurable and Gx (−k, λ) = Gx (k, λ), for almost every k and all x ∈ R3 , λ ∈ Z2 . Assume there exist d−1 , d0 , d1 , d2 ∈ (0, ∞) such that 2 (k) G(k)2∞ dk ≤ d2 , ∈ {−1, 0, 1, 2}, (3.1) where G(k)∞ := supx |Gx (k)|, and 2 (k)−1 ∇x ∧ G(k)2∞ dk ≤ d21 .

(3.2)

The free Dirac operator minimally coupled to A (deﬁned as in (2.6) with a general G and ν = 4) is now given as DA := α · (p + A) + β =

3

αj (−i∂xj + A(j) ) + β.

(3.3)

j=1 4 DA is essentially self-adjoint on Dm as a straightforward application of Nelson’s commutator theorem shows [16, 20]. We use the symbol DA again to denote its

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4 closure starting from Dm . Its spectrum is contained in the union of two half-lines, σ(DA ) ⊂ (−∞, −1] ∪ [1, ∞). The connection between the Dirac operator and the semi-relativistic Pauli–Fierz operator is given by the identities TA 0 (3.4) , TA := (σ · (p + A))2 + 1. |DA | = 0 TA

The assumption Gx (−k, λ) = Gx (k, λ) has been used in [19] to show that 2 1 ≤ |DA | + δdΓ( ) + (δ −1 + δk 2 )d21 , π |x|

(3.5)

4 for some k ∈ (0, ∞) and every δ > 0, in the sense of quadratic forms on Dm .

3.2. Comparison between operators with diﬀerent form factors x (k), k ∈ Am × Z2 , j ∈ {1, 2, 3}, is another form In the following we assume that G factor fulﬁlling Hypothesis 3.1 with new constants d−1 , . . . , d2 , that is, x (k)|2 dk ≤ d2 < ∞, ∈ {−1, 0, 1, 2}. 2 (k) sup |G (j)

x∈R3

as in (2.6) but with G in place of Gphys and put We deﬁne A 4 2 x (k)|2 }dk, (a) := 2+ sup {e−2a|x||Gx (k) − G

(k) x∈R3

a ≥ 0.

(3.6)

Well known estimates using α · z2 ≤ 2|z|2 , z ∈ C3 , show that + 1)−1/2 ≤ (a). e−a|x| α · (A − A)(dΓ( )

(3.7)

Next, we state some simple facts which are used repeatedly in the sequel: First, we may represent the sign function of DA in terms of its resolvent, RA (iy) := (DA − iy)−1 ,

y ∈ R,

as a strongly convergent principal value [13, Lemma VI.5.6], τ dy −1 RA (iy)ϕ , ϕ ∈ Hm4 . SA ϕ := DA |DA | ϕ = lim τ →∞ −τ π

(3.8)

Second, we recall that, for all y ∈ R, a ∈ [0, 1), and F ∈ C ∞ (R3x , R) with a ﬁxed sign and satisfying |∇F | ≤ a, we have iy ∈ (DA + iα · ∇F ), F (iy) := eF RA (iy)e−F = (DA + iα · ∇F − iy)−1 RA

on D(e−F ),

(3.9)

and F RA (iy) ≤ J(a)(1 + y 2 )−1/2 ,

(3.10)

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√ where J(0) := 1 and J(a) := 6/(1 − a2 ), for a ∈ (0, 1). The bound (3.10) is essentially well known; see, e.g., [8]. The proof given in [18] for classical vector potentials works for quantized ones as well. Finally, we abbreviate ˇ f := dΓ( ) + E H

(3.11)

in what follows, for some suﬃciently large E ≥ 1. Assuming E > (2d1 J(a))2 we constructed operators ΞF (iy) ∈ L (Hm4 ), y ∈ R, in [19] satisfying ˇ −1/2 RF (iy) = ΞF (iy)RF (iy)H ˇ −1/2 , H A A f f ΞF (iy) ≤ (1 − ca,E )−1 ,

(3.12)

ca,E := 2d1 J(a)/E 1/2 .

(3.13)

fulfill Hypothesis 3.1. Let a, κ ∈ [0, 1) and Lemma 3.1. Assume that , G, and G ∞ 3 assume that F ∈ C (Rx , R) satisfies |∇F (x)| ≤ a and F (x) ≥ a|x|, for all x ∈ R3 , and F (x) = a|x|, for large |x|. Then we find some C(κ) ∈ (0, ∞), depending only on κ, such that, for all E ≥ 1 with E > (2d1 J(a))2 , κ ˇ −1/2 e−F ≤ C(κ)(a)J(a)/(1 − ca,E ). |DA e | (SA e − SA )Hf

Proof. A short computation using (3.9) and (3.12) yields, for every ϕ ∈ −1/2

ˇ e−F H f

(3.14) 4 , Dm

(RA (−iy) − RA e (−iy))(DA e + iy)ϕ −1/2 −F

−F ˇ (−iy)H = Ξ−F (−iy)RA f

e

− A)R e (−iy)(D e + iy)ϕ. α · (A A A

4 4 4 Now, (DA e + iy)D is dense in Hm since DA e is essentially self-adjoint on Dm and −1/2 −F ˇ is bounded due to (3.7). Therefore, the previous computation H e α · (A − A) f

implies an operator identity in L (Hm4 ) whose adjoint reads

ˇ −1/2 e−F = R e (iy)Θ(y), (RA (iy) − RA e (iy))Hf A where ˇ − A)e−F H Θ(y) := α · (A f

−1/2

F RA (iy)[Ξ−F (−iy)]∗

satisﬁes Θ(y) ≤ (a)J(a)(1 + y 2 )−1/2 Ξ−F (iy) by (3.7) and (3.10). Combinκ 2 κ/2−1/2 we ﬁnd, for ing this with (3.8) and |DA e | RA e (iy) ≤ const(κ)(1 + y ) 4 ϕ, ψ ∈ Dm , −1/2

κ ˇ e−F ψ| ||DA e | ϕ | (SA − SA e )Hf dy κ (−iy)|D | ϕ|Θ(y)ψ = RA e e A π R

≤ const(κ)(a)J(a) sup Ξ−F (iy) y∈R

In view of (3.13) this implies (3.14).

R

dy ϕψ. (1 + y 2 )1−κ/2

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In the next lemma and the proofs of the infrared bounds we shall need the following bound which follows immediately from [19, Lemma 3.5]. (At least, for bounded F . Since its right-hand side depends only on a it can be generalized to all F satisfying the conditions given in the next lemma by some straightforward approximation ˇ 1/2 into itself and, for all argument.) We know that SA maps the domain of H f E > (2d1 J(a))2 , ˇ 1/2 SA H ˇ −1/2 e−F ≤ (1 + aJ(a))/(1 − ca,E ). eF H f f

(3.15)

fulfill Hypothesis 3.1. Let F ∈ C ∞ (R3x , R) Lemma 3.2. Assume that , G, and G satisfy |∇F (x)| ≤ a < 1 and F (x) ≥ a|x|, for all x ∈ R3 , and F (x) = a|x|, for large |x|. Then, for every E ≥ 1 with E > (2d1 J(a))2 , there is some C ≡ C(a, E, d1 ) ∈ 4 , (0, ∞) such that, for all , τ ∈ (0, 1] and ϕ ∈ Dm 4 1/2 ˇ 1/2 eF ϕ2 + C (a) ϕ2 . |ϕ | (|DA | − |DA ϕ2 + τ H e |)ϕ| ≤ |DA e| f 3 τ 2 1/2

4 Proof. Since SA maps Dm into the domains of D0 and Hf 4 , of Lemma 3.4(ii)]) we have the following identity on Dm

|DA | − |DA e | = DA e S∆ + α · (A − A)SA ,

(3.16)

(compare [19, Proof

S∆ := SA − SA e.

(3.17)

Thanks to (3.7) and (3.15) the second term on the right-hand side can be estimated as A ϕ| |ϕ | α · (A − A)S ˇ 1/2 SA H ˇ −1/2 e−F eF H ˇ 1/2 ϕ H ˇ −1/2 eF H ≤ ϕe−F α · (A − A) f f f f 2 ˇ 1/2 ϕ2 + (a) · const(a, E, d1 )ϕ2 , ≤ τ eF H f 4τ 4 for all ϕ ∈ Dm and τ > 0. To treat the ﬁrst term on the right-hand side of (3.17) we apply Lemma 3.1 with κ = 3/4 and ﬁnd some C∗ ∈ (0, ∞) such that 1/2 1/4 3/4 |DA S∆ ϕ2 ≤ |DA S∆ ϕ|DA S∆ ϕ e| e| e| 1/2 ˇ 1/2 ϕ S∆ ϕ1/2 C∗ (a)eF H ≤ S∆ ϕ | |DA e| f

≤

C 4 4 (a) 2 τ F ˇ 1/2 2 1 1/2 e Hf ϕ + |DA S∆ ϕ2 + ∗ 2 · 2 ϕ2 , e| 2 2 8τ (3.18)

4 for all ϕ ∈ Dm . In the last step we also used that S∆ ≤ 2. Solving (3.18) for 1/2 2 | S ϕ and replacing τ by 4ετ we arrive at |DA ∆ e 1/2 |ϕ | DA ϕ2 + e S∆ ϕ| ≤ ε|DA e|

1 1/2 |DA S∆ ϕ2 e| 4ε

4 4 1/2 ˇ 1/2 ϕ2 + C∗ (a) ϕ2 . ≤ ε|DA ϕ2 + τ eF H e| f 64ε3 τ 2

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fulfill Hypothesis 3.1. Then, for every Corollary 3.1. Assume that , G, and G γ ∈ [0, 2/π) and τ ∈ (0, 1], we find C ≡ C(γ, τ, d−1 , d0 , d1 , d˜−1 , d˜0 ) ∈ (0, ∞) and 4 , c ≡ c(γ, τ ) > 0 such that, in the sense of quadratic forms on Dm |DA | − γ/|x| + τ dΓ( ) ≥ c(|DA e | + 1/|x| + dΓ( )) − C.

(3.19)

Proof. We choose ∈ (0, 1) such that (γ + )/(1 − ) = 2/π. Then (3.5) with δ = τ /(2 − 2) implies |DA | −

τ γ + τ dΓ( ) ≥ |DA + dΓ( ) − C , e | + (|DA | − |DA e |) + |x| |x| 2

for some C ≡ C (τ, d1 ) ∈ (0, ∞). Applying (3.16) with F = 0 = a and τ replaced by τ /4 we obtain (3.19) with c := min{ − 2 , τ /4}. Corollary 3.2. Assume that and G fulfill Hypothesis 3.1. Then we find c ∈ (0, ∞), depending only on d1 , d0 , d−1 , such that inf σ[|DA | + dΓ( )] ≤ c. Proof. By virtue of Corollary 3.1 |DA | + dΓ( ) ≤ c(|D0 | + dΓ( )). Picking a minimizing sequence for the quadratic form on the right-hand side we conclude that inf σ[|DA | + dΓ( )] ≤ c. 4. Existence of Binding As a ﬁrst step towards the proof of the existence of ground states we have to verify that the inﬁmum of the spectrum of the semi-relativistic Pauli–Fierz operator is strictly less than its ionization threshold. This information will be exploited mathematically when we apply a bound on the spatial localization of low-lying spectral subspaces from [19]. The localization estimate in turn enters into the proof of the existence of ground states at various places, for instance, into the derivation of the infrared estimates. In our applications, it will actually be necessary to have a bound on the binding energy which is uniform in certain infrared cutoﬀ and discretization parameters introduced later on. In fact, we shall see that in all these cases the (positive) binding energy is always bounded from below by the binding energy of the purely electronic model. When the operator without electric potential is translation invariant such a bound has been derived recently in [12], improving an earlier bound from a preprint version of the present paper. Here we give an alternative proof for the same bound as in [12] which also applies to the discretized version of the semirelativistic Pauli–Fierz operator, which is not translation invariant anymore. In the ﬁrst two subsections below we introduce the infrared cut-oﬀ and discretized semi-relativistic Pauli–Fierz operators. After that we introduce a ﬁber integral representation of these operators for vanishing electric potentials. This representation is a key ingredient in our proof of the binding condition which is presented in the last of the four subsequent subsections.

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From now on we shall consider the following operator acting in H 4 , Hγ := |DA | − γ/|x| + Hf ,

γ ∈ [0, 2/π],

which, on account of (3.4), is just a two-fold direct sum of the semi-relativistic Pauli–Fierz operator Hγ . For then we can conveniently employ our results on the Dirac operator DA . By virtue of (3.5) we may deﬁne Hγ as a Friedrichs extension 0 , starting from D 4 . Again using (3.5), we may also deﬁne the operators Hγ,m , Hγ,m ε and Hγ,m introduced below with γ ∈ [0, 2π] by means of a Friedrichs extension and we shall always do so without further mentioning. 4.1. The infrared cut-oﬀ operator Hγ,m The infrared cut-oﬀ Hamiltonians, Hγ,m , m > 0, are given by Am (x) := a† (e−ik·x gm ) + a(e−ik·x gm ), Hγ,m := |DAm | −

gm := 1Am g,

γ + Hf . |x|

(4.1) (4.2)

Here Am = {|k| ≥ m}, g is deﬁned in (2.3), and Hf = dΓ(ω). We shall also encounter a version of Hγ,m acting in the truncated Hilbert space Hm4 : For every m > 0, we split the one-photon Hilbert space into two mutually orthogonal subspaces K = L2 (R3 × Z2 ) = Km ⊕ Km⊥ ,

Km = L2 (Am × Z2 ).

It is well known that Fb [K ] = Fb [Km ] ⊗ Fb [Km⊥ ] and we observe that all the operators a(1Am e−ik·x g (j) ), a† (1Am e−ik·x g (j) ), j ∈ {1, 2, 3}, x ∈ R3 , and Hf leave the Fock space factors associated to the subspaces Km and Km⊥ invariant. Hence, the same holds true also for DAm , SAm , and |DAm |. We designate operators acting in the Fock space factors Fb [Km ] or Fb [Km⊥ ] by a superscript 0 or ⊥, respectively. For instance, 0 Hf,m := dΓ(ωAm ×Z2 ),

⊥ Hf,m := dΓ(ω{|k|<m}×Z2 ).

Moreover, DAm ∼ = DA0m ⊗ 1 and SAm ∼ = SA0m ⊗ 1 under the isomorphism H4∼ = Hm4 ⊗ Fb [Km⊥ ],

Hm4 = L2 (R3x , C4 ) ⊗ Fb [Km ].

Under the same isomorphism Hγ,m decomposes as ⊥ , 0 Hγ,m = Hγ,m ⊗ 1 + 1 ⊗ Hf,m

0 Hγ,m := |DA0m| −

γ 0 + Hf,m . |x|

(4.3)

⊥ We let Ω⊥ = (1, 0, 0, . . .) denote the vacuum in Fb [Km⊥ ]. In view of Hf,m Ω⊥ = 0 we then observe that 0 Eγ,m := inf σ[Hγ,m ] = inf σ[Hγ,m ],

m > 0,

0 Σm := inf σ[H0,m ] = inf σ[H0,m ],

m > 0,

γ ∈ (0, 2/π],

(4.4) (4.5)

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0 by tensor-multiplying minimizing sequences for Hγ,m , γ ∈ [0, 2/π], with Ω⊥ . For 0 presentational reasons we consider Hγ,m to ﬁnd a lower bound on Σm − Eγ,m . ε 4.2. The discretized operator Hγ,m 0 Next, we deﬁne a discretized version of Hγ,m . To this end we decompose Am into a disjoint union of cubes with side length ε ∈ (0, m],

Qεm (ν), Qεm (ν) := (ν + [−ε/2, ε/2)3) ∩ Am , ν ∈ (εZ)3 . Am = ν∈(εZ)3

ε (k) ∈ (εZ)3 , such that Of course, for every k ∈ Am , we ﬁnd a unique vector, ν ε 3 ε ν ε (k)). To each ν ∈ (εZ) with Qm (ν) = ∅ we further associate some k ∈ Qm ( κ εm (ν) ∈ Qεm (ν) such that |κ εm (ν)| =

inf

k∈Qεm (ν)

|k|.

In this way we obtain a map ν ε : Am × Z2 → R3 , It is evident that the vectors

k = (k, λ) → ν ε (k) := κ εm ( ν ε (k)).

κ εm (ν)

(4.6)

can be chosen such that

ν ε (−k, λ) = −ν ε (k, λ),

for almost every k ∈ Am .

(4.7)

The set of Lebesgue measure zero where the identity (4.7) might not hold is contained in the union of all planes which are perpendicular to some coordinate axis and contain points of the lattice (εZ)3 . We deﬁne the ε-average of a locally integrable function, f , on Am × Z2 by 1 f (p, λ)d3 p, k = (k, λ) ∈ Am × Z2 . f ε (k) := ε |Qm ( ν ε (k))| Qεm (eν ε (k)) Alternatively, we may write, for every f ∈ Km , Qε (ν) | f 1 Qε (ν) , f ε = Pε f := 1 m m

(4.8)

ν∈(εZ)3 : Qεm (ν) =∅

Qε (ν) denotes the normalized characteristic function of the set Qεm (ν) so where 1 m that Pε is an orthogonal projection in Km . The discretized vector potential is now given as ε ε ) + a(e−iν ε ·x gm ), Aεm (x) := a† (e−iν ε ·x gm

ε gm = Pε [1Am g].

(4.9)

The dispersion relation is discretized in a slightly diﬀerent way, namely ωε (k) := inf{|p| : p ∈ Qεm ( ν ε (k))}, which entails max{m, (1 −

k = (k, λ) ∈ Am × Z2 ,

√ 3ε/m)ω} ≤ ωε ≤ ω

ε 0 Hf,m := dΓ(ωε ) ≤ Hf,m .

on Am × Z2 ,

(4.10) (4.11)

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Here the operators in the last line are acting in Fb [Km ]. For every m > 0, we deﬁne ε , acting in Hm4 , by a discretized Hamiltonian, Hγ,m ε ε Hγ,m := |DAεm | − γ/|x| + Hf,m .

(4.12)

4.3. Fiber decompositions of the free operators (γ = 0) Our bound on the binding energy for Hγ is based on a direct ﬁber decomposition of H 4 with respect to ﬁxed values of the total momentum p ⊗ 1 + 1 ⊗ pf , where pf := dΓ(k) := (dΓ(k (1) ), dΓ(k (2) ), dΓ(k (3) )),

(4.13)

is the photon momentum operator. A conjugation of the Dirac operator with the unitary operator eipf ·x — which is simply a multiplication with the phase (n) ei(k1 +···+kn )·x in each Fock space sector Fb [K ] — yields eipf ·x DA e−ipf ·x = α · (p − pf + A(0)) + β. ε 0 When we deal with Hγ,m , m ≥ ε ≥ 0, (Hγ,0 := Hγ ) then we replace pf by

pεf := dΓ(ν ε )

(all three components acting in Fb [Km ]),

where ν ε , ε > 0, is deﬁned in (4.6) and ν 0 := k on Am × Z2 . Then it is again easy to check that eipf ·x DAεm e−ipf ·x = α · (p − pεf + Aεm (0)) + β. ε

ε

A further conjugation with the Fourier transform, F : L2 (R3x ) → L2 (R3ξ ), with respect to the variable x turns the transformed Dirac operators into ⊕ ε ε ε (ξ)d3 ξ, A0 := A. (4.14) D (F ⊗ 1)eipf ·x DAεm e−ipf ·x (F −1 ⊗ 1) = m 0 R3

Here the operators ε (ξ) := α · (ξ − pε + Aε (0)) + β, D m f m

ξ ∈ R3 ,

acting in C4 ⊗ Fb [K ], for m = ε = 0, and in C4 ⊗ Fb [Km ], for m > 0, are ﬁber Hamiltonians of the transformed Dirac operator in (4.14) with respect to the isomorphisms ⊕ ⊕ C4 ⊗ Fb [K ]d3 ξ, Hm4 ∼ C4 ⊗ Fb [Km ]d3 ξ, (4.15) H4∼ = = R3

R3

respectively. Corresponding to (4.15) we then have the direct integral representation (compare, e.g., [24, Theorem XIII.85]) ⊕ ipεf ·x ε −ipεf ·x −1 ε H0,m e (F ⊗ 1) = H0,m (ξ)d3 ξ, (4.16) (F ⊗ 1)e R3

for m ≥ ε ≥ 0, where ε ε (ξ)| + H ε , H0,m (ξ) := |D m f,m

0 Hf,0 := Hf .

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4.4. Proof of the binding condition For m ≥ ε ≥ 0, we set ε ε Eγ,m := inf σ[Hγ,m ],

γ ∈ (0, 2/π],

ε Σεm := inf σ[H0,m ],

(4.17)

0 0 so that Eγ,m = Eγ,m , Eγ,0 = Eγ , Σ0m = Σm , Σ00 = Σ. We ﬁx some ρ > 0 in what follows. In view of the ﬁber decomposition (4.16) we know that the Lebesgue ε (ξ)] ∩ (Σεm − ρ, Σεm + ρ) = ∅ is measure of the set of all ξ ∈ R3 satisfying σ[H0,m strictly positive [24, Theorem XIII.85]. In particular, we ﬁnd some ξ ∈ R3 and ε (ξ )) such that some normalized ϕ ∈ Q(H0,m ε ϕ | H0,m (ξ )ϕ C4 ⊗Fb < Σεm + ρ.

(4.18)

ε := ei(pf −ξ )·x and observe as above We deﬁne the unitary transformation U ≡ Um that UDAεm U ∗ = α · (p + t ) + β, where ε

t := ξ − pεf + Aεm (0). It suﬃces to prove the binding condition for the unitarily equivalent operator ε ε UHγ,m U ∗ = (α · (p + t ))2 + 1 − γ/|x| + Hf,m . (4.19) ε Lemma 4.1. Let ϕ1 ∈ H 1/2 (R3 ) be real-valued, ϕ2 ∈ Q(Hf,m ) ⊂ C4 ⊗ Fb , ϕ := ϕ1 ⊗ ϕ2 , and µ ≥ 0. Then ϕ | (α · (p + t ))2 + µ2 ϕ ≤ ϕ | p2 + (α · t )2 + µ2 ϕ ≤ ϕ1 | ( p2 + µ2 − µ)ϕ1 ϕ2 2 + ϕ1 2 ϕ2 | (α · t )2 + µ2 ϕ2 .

Proof. For ξ ∈ R3 and η > 0, we abbreviate R0 (ξ, η) := (|ξ|2 + (α · t )2 + µ2 + η)−1 , RI (ξ, η) := ((α · (ξ + t ))2 + µ2 + η)−1 , and we observe that (α · (ξ + t ))2 = |ξ|2 + (α · t )2 + 2ξ · t on the domain of (α · t )2 . Applying the second resolvent identity twice we deduce that R0 − RI = 2R0 ξ · t R0 − {4R0 ξ · t RI ξ · t R0 }, where we dropped the ﬁxed arguments (ξ, η). The last operator {· · ·} is non-negative since RI ≥ 0. Furthermore, since ϕ1 is real-valued its Fourier transform satisﬁes 1 (−ξ)|. Since also R0 (ξ, η) = R0 (−ξ, η), the transformation ξ → −ξ |ϕ 1 (ξ)| = |ϕ applied to the integral below reveals that ϕ 1 ⊗ ϕ2 | R0 (·, η)ξ · t R0 (·, η)ϕ 1 ⊗ ϕ2 = ξ · R0 (ξ, η)ϕ2 | t R0 (ξ, η)ϕ2 C4 ⊗Fb |ϕ 1 (ξ)|2 d3 ξ = 0. R3

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We ﬁnally use the formula √ Tψ =

∞

0

1−

η T +η

dη ψ √ , π η

ψ ∈ D(T ),

valid for any self-adjoint operator T ≥ 0, to conclude that ϕ | ( (α · (p + t ))2 + µ2 − p2 + (α · t )2 + µ2 )ϕ ∞ √ ηdη ≤ 0. = ϕ 1 ⊗ ϕ2 | (R0 (·, η) − RI (·, η))ϕ 1 ⊗ ϕ2 π 0 Since p2 and α · t commute the second asserted inequality follows from the ele2 2 mentary bound µ + a + b + µ ≤ a + µ + b + µ2 , for a, b, µ ≥ 0. Corollary 4.1. Let e, Λ > 0, m ≥ ε ≥ 0, and γ ∈ (0, 2π]. Then √ ε Σεm − Eγ,m ≥ |Eγel |, Eγel := inf σ[ 1 − ∆ − 1 − γ/|x|].

(4.20)

√ Proof. First, assume γ < 2/π. Since the semi-group generated by 1 − ∆ is positivity improving we then ﬁnd some normalized, strictly positive eigenfunction, ϕ1 ∈ H 1/2 (R3 ), corresponding to Eγel . Then we apply Lemma 4.1 with ϕ2 = ϕ , ε ε ≤ ϕ | U Hγ,m U ∗ ϕ ≤ Eγel + Σεm + ρ, by the choice of ϕ = ϕ1 ⊗ ϕ2 , to get Eγ,m ϕ in (4.18). Since ρ > 0 is arbitrary, this proves the assertion if γ < 2/π. It is, el ε ε , as γ 2/π, and clear that Eγ,m ≥ E2/π,m , however, well known that Eγel E2/π for γ ∈ (0, 2/π). Corollary 4.2. Let e, Λ > 0 and γ > 2/π. Then we find normalized test functions 1/2 Φn , n ∈ N, in the algebraic tensor product H 1/2 (R3 ) ⊗ C4 ⊗ D(Hf ) such that U Φn | (|DA | − γ/|x| + Hf )U Φn → −∞,

n → ∞.

√ Proof. We set qγ [ϕ] := ϕ | ( −∆ − γ/|x|)ϕ, ϕ ∈ H 1/2 (R3 ), and assume that γ > 2/π. Then it is well known that qγ is unbounded below. For instance, the function ψ given in momentum space by ψ(ξ) := |ξ|−2 1{r≤|ξ|≤R} satisﬁes qγ [ψ] < 0, if 0 < r < R and the ratio R/r is suﬃciently large. ψ is real-valued since ψ is real and ψ(−ξ) = ψ(ξ). By scaling, the functions ϕn (x) := n3/2 ψ(nx) satisfy qγ [ϕn ] = nqγ [ψ], for n ∈ N. On account of Lemma 4.1 the functions Φn := ϕn ⊗ ϕ /ψ thus have the required properties.

5. Existence of Ground States 5.1. Outline of the proof In this section we prove our main Theorem 2.3. As in [3, 5] (see also [11] where a photon mass is introduced in a slightly diﬀerent way) we ﬁrst show that the infrared cut-oﬀ Hamiltonians, Hγ,m , m > 0, deﬁned in (4.1) and (4.2) possess

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ground state eigenfunctions, provided m > 0 is suﬃciently small. This is done in Sec. 5.3 by means of a discretization argument similar to the one in [5]. The implementation of the discretization procedure in [3, 5] requires a small coupling condition. By a modiﬁcation of the argument we observe, however, that this is actually not necessary. Before we turn to these issues we explain in Sec. 5.2 how to infer the existence of ground states for the limit operator Hγ from the fact that the Hγ,m have ground state eigenfunctions. Here we beneﬁt from a result from [11] saying that the spatial localization, a bound on the number of soft photons, and a photon derivative bound introduced in [11] allow to use standard imbedding theorems for Sobolev spaces to ensure the compactness of the set of approximating ground states. The ﬁrst of the latter key ingredients, the exponential localization estimate for low-lying spectral subspaces of the operators Hγ,m , has been proven in [19]. The proofs of the two infrared bounds are postponed to Sec. 6. We close this subsection by a general lemma which allows to prove the existence of imbedded eigenvalues by means of approximating sequences of operators and eigenvectors. It is a modiﬁed version of a result we learned from [3] and its assertion is actually stronger than necessary for our application. Lemma 5.1. Let T, T1 , T2 , . . . be self-adjoint operators acting in some separable Hilbert space, X, such that {Tj }j∈N converges to T in the strong resolvent sense. Assume that Ej is an eigenvalue of Tj with corresponding eigenvector φj ∈ D(Tj ). Assume further that {φj }j∈N converges weakly to some 0 = φ ∈ X . Then E := limj→∞ Ej exists and is an eigenvalue of T . If Ej = inf σ[Tj ], then T is semibounded below and E = inf σ[T ]. Proof. In what follows we abbreviate f := arctan. Then f (Tj )ψ → f (T )ψ, j → ∞, for every ψ ∈ X , since Tj → T in the strong resolvent sense. Let us assume for the moment that ψ ∈ X fulﬁlls ψ | φ = 0. Then we ﬁnd some j0 ∈ N such that ψ | φj = 0, for j ≥ j0 , and we may write f (Ej ) =

f (T )ψ | φj + f (Tj )ψ − f (T )ψ | φj , ψ | φj

j ≥ j0 .

The sequence {f (Ej )}j∈N thus has a limit f (E) := lim f (Ej ) = j→∞

ψ | f (T )φ . ψ | φ

In the case ψ | φ = 0 we may replace ψ by ψ = ψ + φ in the above argument to see that f (T )ψ | φ = 0 also. The equality ψ | f (T )φ = ψ | f (E)φ thus holds, for every ψ ∈ X , whence f (T )φ = f (E)φ. It follows that E := tan(f (E)) ≥ inf σ[T ] is an eigenvalue of T since u(σpp [f (T )]) ⊂ σpp [u(f (T ))] = σpp [(u ◦ f )(T ))], for every Borel measurable function u. Now assume that Ej = inf σ[Tj ]. Set λ := inf σ[T ], when T is semi-bounded below, and pick some λ ∈ σ[T ] ∩ (−∞, E − 1), when inf σ[T ] = −∞. Since Tj converges to T in the strong resolvent sense there is a sequence {Ej }j∈N with

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Ej ≤ Ej ∈ σ[Tj ] and Ej → λ. So E ≥ λ = limj→∞ Ej ≥ limj→∞ Ej = E. If inf σ[T ] = −∞, the ﬁrst inequality is strict and we get a contradiction. 5.2. Approximation by infrared cut-oﬀ electromagnetic ﬁelds In order to prove that Hγ has a ground state if this holds true for Hγ,m with suﬃciently small m > 0 we ﬁrst show that Hγ,m converges to Hγ in norm resolvent = 1Am Gphys , sense. If we choose a = 0, = ω, G = Gphys = e−ik·x g, and G m > 0, then the parameter deﬁned in (3.6) is equal to 4 2∗ (m) := (5.1) 2+ |g(k)|2 dk → 0, m 0. ω(k) {|k|≤m} Lemma 5.2. Let e, Λ, m > 0 and γ ∈ [0, 2/π). Then Hγ,m and Hγ have the same form domain, Q(Hγ,m ) = Q(Hγ ) = Q(|D0 |) ∩ Q(Hf ), and the form norms associated to Hγ,m and Hγ are equivalent. Moreover, Hγ,m converges to Hγ in the norm resolvent sense, as m 0. Proof. The ﬁrst assertion, which has already been observed in [27], follows from = 0). Moreover, we know that D 4 is a common Corollary 3.1 (with A = 0 or A form core of Hγ and Hγ,m , m > 0, and on D 4 we have Hγ − Hγ,m = (SA − SAm )DA + SAm α · (A − Am ). By virtue of Lemma 3.1 and (3.7) we thus ﬁnd some C ∈ (0, ∞) such that, for all m > 0 and ϕ ∈ D 4 , ˇ 1/2 ϕ|DA |1/2 ϕ + ϕH ˇ 1/2 ϕ) |ϕ | (Hγ − Hγ,m )ϕ| ≤ O(∗ (m))(H f f ≤ O(∗ (m))ϕ | (Hγ + C)ϕ.

(5.2)

ˇ f = Hf + E, for some suﬃciently large E > 0, and in the last step we used Here H Corollary 3.1. Since we may replace ϕ in (5.2) by any element of Q(Hγ,m ) = Q(Hγ ) the second assertion follows from [25, Theorem VIII.25]. In what follows we let [·]− : R → (−∞, 0] denote the negative part [t]− := min{t, 0},

t ∈ R.

Proposition 5.1 (Ground States with Mass). Let e, Λ > 0, γ ∈ (0, 2/π). Then there exists some m0 > 0 such that, for every m ∈ (0, m0 ), the operator 0 − Eγ,m − m [Hγ,m 4 ]− has finite rank. In particular, Eγ,m is an eigenvalue of both 0 Hγ,m and Hγ,m . 0 in Sec. 5.3. However, if φ0m is a ground Proof. We prove the assertion for Hγ,m 0 state eigenvector of Hγ,m then, by (4.3) and (4.4), φ0m ⊗ Ω⊥ is a ground state eigenvector of Hγ,m .

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To beneﬁt from this proposition we also need the following results. The ﬁrst one on the exponential localization of low-lying spectral subspaces is also applied to the ε 0 later on. (Recall our convention Hγ,0 = Hγ .) discretized operator Hγ,m Proposition 5.2 (Exponential Localization). There exist k0 , k1 ∈ (0, ∞) such that the following holds: Let e, Λ > 0, 0 ≤ ε ≤ m, γ ∈ (0, 2/π), and let I ⊂ (−∞, Σεm ) be some compact interval. Pick some a ∈ (0, 1) such that := Σεm − max I − 6a2 /(1 − a2 ) > 0 and ≤ 1. Then 2

ε ea|x|1I (Hγ,m ) ≤ (k1 /2 )(1 + |I|)(Σεm + k0 e2 Λ3 )ec(γ)a(Σm +k0 e ε

Λ3 )/

,

(5.3)

where |I| denotes the length of I and c : (0, 2/π) → (0, ∞) is some universal increasing function. In particular, we find m1 , δ1 > 0, and a1 ∈ (0, 1) such that ε sup{ea1 |x| 1J(m,ε) (Hγ,m ) : 0 ≤ ε ≤ m ≤ m1 } < ∞,

where J(m, ε) :=

ε ε , Eγ,m [Eγ,m

(5.4)

+ δ1 ]. The same estimates hold for Hγ,m , if ε = 0.

Proof. The bound (5.3) with k0 e2 Λ3 replaced by some constant times d21 is stated in [19, Theorem 2.5] for dispersion relations and form factors G fulﬁlling Hypothesis 3.1. In the cases = ω, G = e−ik·x g1Am , or = ωε , G = e−iν ε ·x gm,ε we can clearly choose d21 = const e2 Λ3 uniformly in 0 ≤ ε ≤ m ≤ m1 , for some m1 > 0. To prove (5.4) we pick some 0 < δ1 < |Eγel |, choose I = J(m, ε), and observe that, by Theorem 4.1, ≥ |Eγel | − δ1 − 6a21 /(1 − a21 ) ≥ const > 0, uniformly in 0 ≤ ε ≤ m ≤ m1 , provided that a1 ∈ (0, 1) is suﬃciently small. Finally, we know from Corollary 3.2 that all threshold energies Σεm are bounded from above by some constant depending only on the values of d−1 , d0 , d1 . Since these values can be chosen uniformly in 0 ≤ ε ≤ m ≤ m1 this concludes the proof. We recall the notation (a(k)ψ)(n) (k1 , . . . , kn ) = (n + 1)1/2 ψ (n+1) (k, k1 , . . . , kn ),

n ∈ N0 ,

(0) , 0, 0, . . .) = 0. The almost everywhere, for ψ = (ψ (n) )∞ n=0 ∈ Fb [K ], and a(k)(ψ following proposition is proved in Sec. 6.

Proposition 5.3 (Soft Photon Bound). Let e, Λ > 0 and γ ∈ (0, 2/π). Then there exist constants, m2 , C ∈ (0, ∞), such that, for all m ∈ [0, m2 ] and every normalized ground state eigenfunction, φm , of Hγ,m , we have a(k)φm 2 ≤ 1{m≤|k|≤Λ}

C , |k|

(5.5)

for almost every k = (k, λ) ∈ R3 × Z2 . The next proposition, which is also proved in Sec. 6, is the only place in the whole article where the special choice of the polarization vectors (2.5) enters into the analysis. Proposition 5.4 (Photon Derivative Bound). Let e, Λ > 0, γ ∈ (0, 2/π). Then there exist constants, m3 , C ∈ (0, ∞), such that, for all m ∈ [0, m3 ] and every

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normalized ground state eigenfunction, φm , of Hγ,m , we have 1 1 a(k, λ)φm − a(p, λ)φm ≤ C|k − p| + , |k|1/2 |k⊥ | |p|1/2 |p⊥ |

(5.6)

for almost every k, p ∈ R3 with m < |k| < Λ, m < |p| < Λ, and λ ∈ Z2 . (Here we use the notation introduced in (2.4).) We have now collected all prerequisites to show that inf σ[Hγ ] is an eigenvalue of Hγ . Proof of Theorem 2.3 by means of Propositions 5.1–5.4. Let φm denote a normalized ground state of Hγ,m , for m ∈ (0, m ], where m > 0 is the minimum of the constants m0 , m1 , m2 , m3 appearing in Propositions 5.1–5.4. Then the family {φm }m∈(0,m ] contains a weakly convergent sequence, {φmj }j∈N . We denote the weak limit of the latter by φ and verify that φ = 0 in the following. The assertion of Theorem 2.3 then follows from Lemmas 5.1 and 5.2. (In fact, we shall show that φmj → φ strongly in H 4 along a subsequence.) To verify that φ = 0 one can argue as in [11]. Essentially, we only have to replace the Rellich–Kondrashov theorem applied there by a suitable imbedding theorem for spaces of functions with fractional derivatives. (In the non-relativistic case the ground states φm possess weak derivatives with respect to the electron coordinates, whereas in our case we only have Inequality (5.11) below.) For the convenience of the ∞ (n) 2 3 reader we present the complete argument. Writing φm = (φm )∞ n=0 ∈ n=0 L (R × (n) Z4 ) ⊗ Fb [K ] we infer from the soft photon bound that ∞ ∞ 1 1 C 2 (n) 2 φ(n) ≤ nφ = , a(k)φm 2 dk ≤ m m n n n 0 0 0 n=n n=0 0

for m ∈ (0, m ] and some m-independent constant C ∈ (0, ∞). Given some ε > 0 we ﬁx n0 ∈ N so large that C/n0 < ε.

(5.7)

By virtue of (5.4) we further ﬁnd some R > 0 such that, for all m ∈ (0, m ], φm 2C4 ⊗Fb (x)d3 x < ε. (5.8) |x|≥R/2

(n)

In addition, the soft photon bound ensures that φm (x, ς, k1 , . . . , kn ) = 0, for almost every (x, ς, k1 , . . . , kn ) ∈ R3 × {1, 2, 3, 4} × (R3 × Z2 )n , kj = (kj , λj ), such that |kj | > Λ, for some j ∈ {1, . . . , n}. (Here and henceforth ς labels the four spinor components.) For 0 < n < n0 and some ﬁxed θ = (ς, λ1 , . . . , λn ) ∈ {1, 2, 3, 4} × Zn2 we set (n)

φm,θ (x, k1 , . . . , kn ) := φ(n) m (x, ς, k1 , λ1 , . . . , kn , λn ) and similarly for φ. Moreover, we set, for every δ ≥ 0, Qn,δ := {(x, k1 , . . . , kn ) : |x| < R − δ, δ < |kj | < Λ − δ, j = 1, . . . , n}.

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Fixing some small 0 < δ < min{m , R/2, Λ/4} we pick some cut-oﬀ function χ ∈ C0∞ (R3(n+1) , [0, 1]) such that χ ≡ 1 on Qn,2δ and supp(χ) ⊂ Qn,δ and deﬁne (n) (n) ψm,θ := χφm,θ . As a next step the photon derivative bound is used to show that (n)

{ψm,θ }m∈(0,δ] is a bounded family in the anisotropic Nikol’ski˘ı spacea Hqs (R3(n+1) ), where s = (1/2, 1/2, 1/2, 1, . . ., 1) and q = (2, 2, 2, p, . . . , p) with p ∈ [1, 2). In fact, employing the H¨older inequality (with respect to d3 x d3 k2 · · · d3 kn ) and the photon derivative bound (5.6), we obtain as in [11], for p ∈ [1, 2) and m ∈ (0, δ], (n) (n) |φm,θ (x, k1 + h, k2 , . . . , kn ) − φm,θ (x, k1 , . . . , kn )|p d3 x d3 k1 · · · d3 kn Q n,δ∩

{δ<|k1 +h|<Λ}

≤

([4π/3]n R3 Λ3(n−1) ) np/2

≤ C|h|

λ∈Z2

|(u,v)|

p 0

|(u,v)|<Λ

2−p 2

m<|k|<Λ, m<|k+h|<Λ

dk (3) + |(u, v)|p/2

a(k + h, λ)φm − a(k, λ)φm p d3 k Λ

|(u,v)|

dk (3) |k (3) |p/2

du dv |(u, v)|p

= C |h|p , (n)

where the constants C, C ∈ (0, ∞) do not depend on m ∈ (0, δ]. Since φm is symmetric in the photon variables the previous estimate implies [22, §4.8] that the (n) weak ﬁrst order partial derivatives of φm,θ with respect to its last 3n variables exist on Qn,δ and that (n) φm,θ pW r (Qn,δ ) p

=

(n) φm,θ pLp (Qn,δ )

+

n 3

(n)

∂k(i) φm,θ pLp (Qn,δ ) ≤ C ,

j=1 i=1

j

for m ∈ (0, δ] and some m-independent C ∈ (0, ∞), with r := (0, 0, 0, 1, . . . , 1). (n) The previous estimate implies ψm,θ Wpr (R3(n+1) ) ≤ C , for some C ∈ (0, ∞) which does not depend on m ∈ (0, δ]. Moreover, the anisotropic Sobolev space T (r ,...,r ) r1 , . . . , rd ∈ [0, 1], q1 , . . . , qd ≥ 1, we have Hq11,...,qdd (Rd ) := di=1 Hqriixi (Rd ). For ri ∈ [0, 1), ri d d a measurable function f : R → C belongs to the class Hqi xi (R ), if f ∈ Lqi (Rd ) and there is some M ∈ (0, ∞) such that a For

f (· + hei ) − f Lqi (Rd ) ≤ M |h|ri ,

h ∈ R,

(5.9)

where ei is the ith canonical unit vector in Rd . If ri = 1 then (5.9) is replaced by f (· + hei ) − 2f + f (· − hei )Lqi (Rd ) ≤ M |h|,

h ∈ R.

(5.10)

(r ,...,r )

Hq11,...,qdd (Rd ) is a Banach space with norm (r ,...,r )

f q11,...,qdd := max f Lqi (Rd ) + max Mi , 1≤i≤d

1≤i≤d

where Mi is the inﬁmum of all constants M > 0 satisfying (5.9) or (5.10), respectively. Finally, (r ,...,rd ) (r1 ,...,rd ) (Rd ) := Hq,...,q (Rd ). we abbreviate Hq 1

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Wpr (R3(n+1) ) is continuously imbedded into Hpr (R3(n+1) ); see, e.g., [22, §6.2]. Furthermore, since D 4 is a form core of Hγ,m , m > 0, Corollary 3.1 shows that (n) c−1 φ(n) m | |D0 |φm ≤ φm | Hγ,m φm + c = Eγ,m + c,

n ∈ N,

(5.11)

(n)

for some m-independent c ∈ (0, ∞). Therefore, {φm,θ }m∈(0,m] and, hence, (n)

{ψm,θ }m∈(0,m ] are bounded families in the Bessel potential, or, Liouville space

Lr2 (R3(n+1) ), r := (1/2, 1/2, 1/2, 0, . . ., 0), where the fractional derivatives are deﬁned by means of the Fourier transform. The imbedding Lr2 (R3(n+1) ) → (n) H2r (R3(n+1) ) is continuous, too [22, §9.3]. Altogether it follows that {ψm,θ }m∈(0,δ] is a bounded family in Hqs (R3(n+1) ). Now we may apply the compactness theo(n)

rem [21, Theorem 3.2]. The latter ensures that {ψm,θ }m∈(0,δ] contains a sequence which is strongly convergent in L2 (Qn,2δ ) provided 1 − 3n(p−1 − 2−1 ) > 0. Of course, we can choose p < 2 large enough such that the latter condition is fulﬁlled, for all n = 1, . . . , n0 − 1. By ﬁnitely many repeated selections of subsequences we (n) may hence assume without loss of generality that {φmj ,θ }j∈N converges strongly in (n)

L2 (Qn,2δ ) to φθ , for 0 ≤ n < n0 . In particular, by the choice of n0 and R in (5.7) and (5.8), 2

φ ≥ lim

j→∞

n 0 −1 n=0

(n)

φmj ,θ 2L2 (Qn,2δ )

θ

≥ lim φmj 2 − 2ε − c(δ) j→∞

= 1 − 2ε − c(δ), where we use the soft photon bound to estimate n 0 −1 n=1

≤

(n)

φmj ,θ 1{∃ i : |ki | ≤ 2δ ∨ |ki | ≥ Λ − 2δ}2

θ

λ∈Z2

{|k|≤2δ}∪ {|k|≥Λ−2δ}

2 3

a(k, λ)φmj d k ≤ C

2δ

Λ

+ 0

Λ−2δ

r2 dr =: c(δ) → 0, r

as δ 0. Since δ > 0 and ε > 0 are arbitrary we get φ = 1, whence φmj → φ strongly in H 4 . 5.3. Existence of ground states with infrared cut-oﬀ At the end of this subsection we prove Proposition 5.1. For this we ﬁrst have to extend our results on the spatial localization of low-lying spectral subspaces. This extension requires the following inequality, where J(a) is deﬁned below (3.10).

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Lemma 5.3. Let e, Λ > 0, m ≥ ε ≥ 0, γ ∈ [0, 2/π), and a ∈ [0, 1). Moreover, let F ∈ C ∞ (R3x , [0, ∞)) ∩ L∞ satisfy |∇F | ≤ a. Then ε |Re ϕ | [Hγ,m , eF ]e−F ϕ| ≤ 2a2 J(a)ϕ2 ,

4 ϕ ∈ Dm .

(5.12)

Proof. This lemma is a special case of [19, Lemma 5.7]. Lemma 5.4. Let e, Λ > 0, γ ∈ [0, 2/π), and m1 , a1 , δ1 , J(m, ε) be as in Proposition 5.2. Assume that F ∈ C ∞ (R3x , [0, ∞)) satisfies |∇F | ≤ a1 /2 and F (x) = a1 |x|, for large |x|. Then there is some C ∈ (0, ∞) such that, for all 0 ≤ ε ≤ m ≤ m1 , ε ε )), we have eF ψ ∈ Q(Hγ,m ) and and ψ ∈ Ran(1J(m,ε) (Hγ,m ε ε ˚γ,m ˚γ,m (H )1/2 eF ψ2 ≤ e2F ψH ψ + 2a2 J(a)eF ψ2 ,

(5.13)

˚ε := H ε − E ε , In particular, where H γ,m γ,m γ,m ˚ ε )1/2 eF 1J(m,ε) (H ε ) ≤ C , (H γ,m γ,m

(5.14)

where the constant C ∈ (0, ∞) neither depends on m nor on ε. Moreover, for ε ε } and ψ ∈ Ran(1J(m,ε) (Hγ,m )), we have eF ψ ∈ D(O1/2 ) O ∈ {|D0 |, |DAεm |, Hf,m and ε ) : 0 ≤ ε ≤ m ≤ m1 } < ∞. sup{O1/2 eF 1J(m,ε) (Hγ,m

(5.15)

4 and let F ∈ C ∞ (R3x , [0, ∞))∩L∞ such that |∇F | ≤ a1 /2. Proof. First, let ϕ ∈ Dm e Applying (5.12) with ϕ replaced by eF ϕ we obtain

˚ ε ϕ + eFe ϕ | [H ε , eFe ]e−Fe (eFe ϕ)] ˚ ε )1/2 eFe ϕ2 = Re[e2Fe ϕ | H (H γ,m γ,m γ,m e

e

˚ ε ϕ| + c(a)eF ϕ2 , ≤ |e2F ϕ | H γ,m

(5.16)

where c(a) = 2a2 J(a). In [19, Lemma 5.8] we proved the following inequality, ε ε eG ϕ ≤ c1 eG 2 ϕ | Hγ,m ϕ + c2 eG 2 ϕ2 , eG ϕ | Hγ,m

4 ϕ ∈ Dm ,

for every G ∈ C ∞ (R3 , [0, ∞)) ∩ L∞ with ∇G∞ < 1. In fact, we stated this γ is relatively form bounded with respect to inequality only for γ = 0. Since |x| ε H0,m with relative bound less than one it is clear, however, that it holds true for γ ∈ (0, 2/π) also, with new constants c1 , c2 ∈ (0, ∞) of course. In particular, if ε 4 ), ϕn ∈ Dm , n ∈ N, and ϕn → ψ with respect to the form norm ψ ∈ Q(Hγ,m e

e

e

e

e

e

ε ε of Hγ,m , then eF ψ, e2F ψ ∈ Q(Hγ,m ) and eF ϕn → eF ψ and e2F ϕn → e2F ψ with ε ε ) respect to the form norm of Hγ,m , too. We may thus replace ϕ by any ψ ∈ Q(Hγ,m ε ε )) in what follows. Then ψ ∈ D(Hγ,m ) in (5.16). We ﬁx some ψ ∈ Ran(1J(m,ε) (Hγ,m since J(m, ε) is bounded and we arrive at e

e

e

˚ ε ψ + c(a)eF ψ2 . ˚ ε )1/2 eF ψ2 ≤ e2F ψH (H γ,m γ,m

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Furthermore, we know from (5.3) that

e4F (x) ψ2F 4 (x)d3 x < ∞. We pick a b

sequence of bounded functions Fn ∈ C ∞ (R3 , [0, ∞)) such that |∇Fn | ≤ a1 /2, n ∈ N, and Fn F . Then eFn ψ → eF ψ and e2Fn → e2F ψ in Hm4 by dominated convergence. Inserting Fn for F in the previous estimate we conclude that the densely deﬁned linear functional ε ε ˚γ,m ˚γ,m )1/2 η = lim (H )1/2 eFn ψ | η, f (η) := eF ψ | (H n→∞

ε η ∈ Q(Hγ,m ),

is bounded, ε ˚γ,m |f (η)| ≤ (e2F ψH ψ + c(a)eF ψ2 )1/2 η,

ε η ∈ Q(Hγ,m ).

ε ε ε ˚γ,m )1/2 is self-adjoint with domain Q(Hγ,m ) it follows that eF ψ ∈ Q(Hγ,m ) Since (H ε 1/2 F ˚ and (Hγ,m ) e ψ = f . The inequality (5.14) follows from (5.13) and Proposition 5.2. The bound (5.15) follows from (5.14) and Corollary 3.1 where the constants can be chosen uniformly in 0 ≤ ε ≤ m ≤ m1 .

= e−iν ε ·x gm,ε , and = ωε , the parameter Choosing G = Gphys = e−ik·x g, G deﬁned in (3.6) equals 1 2ε := 1Am (k) 1 + sup {e−2a|x| |e−ik·x g(k) − e−iν ε (k)·x gm,ε (k)|2 }dk. ωε (k) x∈R3 It is well known and easy to see that, for ﬁxed e, Λ, a > 0, and m ∈ (0, 1], 2ε = o(ε0 ),

ε 0,

(5.17)

(Here and henceforth the little o-symbols depend on m which does, however, not do any harm.) In the next lemma we compare the ground state energies Eγ,m = 0 ε ε ] and Eγ,m = inf σ[Hγ,m ]. inf σ[Hγ,m Lemma 5.5. Let e, Λ > 0, m ∈ (0, m1 ], and γ ∈ (0, 2/π). Then ε + o(ε0 ). Eγ,m ≤ Eγ,m 0 ε Proof. By virtue of (4.10) and Corollary 3.1 we know that Q(Hγ,m ) = Q(Hγ,m ). In particular, we may pick some ρ ∈ (0, δ1 ] and try some normalized φρε ∈ ε 0 ε ε Ran(1[Eγ,m ,Eγ,m +ρ) (Hγ,m )) as a test function for Hγ,m . Here δ1 is the parameter appearing in Proposition 5.2 and we shall also employ the parameters a1 , m1 , and the interval J(m, ε) introduced there. We obtain 0 φρε Eγ,m ≤ φρε | Hγ,m ε 0 ε ≤ Eγ,m + ρ + φρε | (|DA0m | − |DAεm |)φρε + φρε | (Hf,m − Hf,m )φρε .

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Next, we choose F as in Lemma √ 5.4 and √ use (3.16) with = τ = ε 0 ε ε − Hf,m ≤ m3ε (1 − m3ε )−1 Hf,m , to get which implies Hf,m

399

and (4.10),

ε ρ ρ 1/2 ε 1/2 F ρ 2 Eγ,m ≤ Eγ,m + ρ + 1/2 e φε ε φε | |DAεm |φε + ε (Hf,m + E) ρ 2 0 ρ ε ρ + C3/2 ε φε + o(ε )φε | Hf,m φε , ε where E ≡ E(e, Λ) ∈ (0, ∞). By virtue of (5.15), (Hf,m + E)1/2 eF φρε ≤ C, where the constant C ∈ (0, ∞) neither depends on ρ nor 0 ≤ ε ≤ m ≤ m1 . Employing Corollary 3.1 once more we conclude that ε ε Eγ,m ≤ Eγ,m + ρ + C1/2 + o(ε0 )φρε | (Hγ,m + 1)φρε , ε

where the little o-symbol is uniform in ρ and ρ is arbitrarily small. Proof of Proposition 5.1. Let m1 , a1 , δ1 , and F be as in Proposition 5.2 and 0 χ := 1(−∞,Eγ,m +m/4] (Hγ,m ).

(5.18)

We assume that m ≤ m1 , m/4 ≤ δ1 , and ε ≤ m in the sequel so that (5.15) applies 1/2 to χ. On account of (3.16) with = τ = ε and (4.11) we have 0 ε χHγ,m χ ≥ χ{|DAεm | − γ/|x| + Hf,m }χ − o(ε0 )T1 , 0 where the norm of T1 := χ{|DA0m | + eF (Hf,m + E)eF }χ is bounded uniformly in m ∈ (0, m1 ] due to Lemma 5.4. (The constant E ≡ E(e, Λ) appears when we apply (3.16).) To proceed further we introduce the subspaces of discrete and ﬂuctuating photon states,

Kmd := Pε Km ,

Kmf := Km Kmd ,

where Pε is deﬁned in (4.8). The splitting Km = Kmd ⊕ Kmf gives rise to an isomorphism L2 (R3 , C4 ) ⊗ Fb (Km ) ∼ = (L2 (R3 , C4 ) ⊗ Fb [Kmd ]) ⊗ Fb [Kmf ] and we observe that the Dirac operator and the ﬁeld energy decompose under the above isomorphism as DAεm ∼ ⊗ 1f , = DAε,d m

ε,d ε,f ε Hf,m = Hf,m ⊗ 1f + 1d ⊗ Hf,m .

(5.19)

Here and in the following we designate operators acting in the Fock space factors Fb [Km ], ∈ {d, f }, by the corresponding superscript ∈ {d, f }. In fact, the discretized vector potential Aεm acts on the various n-particle sectors in Fb [Km ] by tensor-multiplying or taking scalar products with elements from Kmd (apart from symmetrization and a normalization constant). Denoting the projection onto the

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vacuum sector in Fb [Km ] by PΩ , writing PΩ⊥ := 1 − PΩ , ∈ {d, f }, and using ε,f PΩf = 0, we thus obtain Hf,m 0 χ{Hγ,m − Eγ,m − m/2}χ + o(ε0 )T1 ε,d ≥ χ{[|DAε,d | − γ/|x| + Hf,m − Eγ,m − m/2] ⊗ PΩf }χ m

(5.20)

ε,d ε + χ{[|DAε,d | − γ/|x| + Hf,m − Eγ,m ] ⊗ PΩ⊥f }χ m

(5.21)

ε,f ε + χ{1el ⊗ 1d ⊗ (Hf,m + Eγ,m − Eγ,m − m/2)PΩ⊥f }χ.

(5.22)

ε Here Eγ,m is deﬁned in (4.17). Setting ε,d | − γ/|x| + Hf,m Xεd := |DAε,d m ε we observe that Xεd − Eγ,m 1d ≥ 0, so that the term in (5.21) is non-negative. In fact, let ρ > 0 and pick some φd ∈ Q(Xεd ), φd = 1, satisfying φd | Xεd φd < inf σ(Xεd ) + ρ. Then ε )φd ⊗ Ωf = φd | Xεd φd ≤ inf σ(Xεd ) + ρ φd ⊗ Ωf | (|DAεm | − γ/|x| + Hf,m ε because of (5.19). Moreover, we know from Lemma 5.5 that Eγ,m − Eγ,m ≥ o(ε0 ), ε,f ⊥ ε 0. Since Hf,m PΩf ≥ mPΩ⊥f this implies that the term in (5.22) is non-negative also, provided that ε > 0 is suﬃciently small. In order to bound the remaining term in (5.20) from below we employ Corol = 0, = ωε ), (5.19), and H ε,f PΩf = 0 to get lary 3.1 (with A f,m ε,d | − γ/|x| + Hf,m ] ⊗ PΩf [|DAε,d m ε = (1 ⊗ PΩf ){|DAεm | − γ/|x| + Hf,m }(1 ⊗ PΩf ) ε,d ≥ ε[|D0 | + |x|2 + Hf,m ] ⊗ PΩf − (C(ε, γ, e, Λ) + ε|x|2 ) ⊗ PΩf ,

for all suﬃciently small values of ε > 0. Since χ is exponentially localized we further know that T2 := χ{|x|2 ⊗ PΩf }χ is a bounded operator. Therefore, we arrive at 0 χ{Hγ,m − Eγ,m − m/2}χ + o(ε0 )(T1 + T2 ) ε,d − C (ε, γ, e, Λ)] ⊗ PΩf }χ ≥ χ{[ε|D0 | + ε|x|2 + εHf,m ε,d − C (ε, γ, e, Λ)]− ⊗ PΩf }χ, ≥ χ{[ε|D0 | + ε|x|2 + εHf,m

(5.23)

ε,d have where [· · · ]− ≤ 0 denotes the negative part. Now, both |D0 | + |x|2 and Hf,m purely discrete spectrum as operators on the electron and photon Hilbert spaces and ε,d is the restriction of the discretized PΩf , of course, has rank one. (Recall that Hf,m ﬁeld energy to the Fock space modeled over the “2 -space” Kmd .) In particular, we observe that ε,d − := [ε|D0 | + ε|x|2 + εHf,m − C (ε, γ, e, Λ)]− ⊗ PΩf Wm,ε

is a ﬁnite rank operator, for every suﬃciently small ε > 0.

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We can now conclude the proof as follows: Given some suﬃciently small m > 0 we choose ε > 0 small enough such that, in particular, the terms in (5.21) and (5.22) are non-negative and o(ε0 )(T1 + T2) ≤ m/8. Since by deﬁnition (5.18) it 0 − Eγ,m − m/2}χ ≤ −(m/4)χ, we see that the left hand side of (5.23) holds χ{Hγ,m is bounded from above by −(m/8)χ, whence 0 − ) ≥ χWm,ε χ. −(m/8)1(−∞,Eγ,m+m/4] (Hγ,m 0 In particular, 1(−∞,Eγ,m +m/4] (Hγ,m ) is a ﬁnite rank projection.

6. Infrared Bounds 6.1. The gauge transformed operator In order to derive the infrared bounds (5.5) and (5.6) it is necessary to pass to a suitable gauge [5, 11]. For otherwise we would end up with a too singular infrared behavior of their right hand sides. To deﬁne an appropriate operator-valued gauge (i) transformation [11] we recall that, for i, j ∈ {1, 2, 3}, the components Am (x) and (j) 3 Am (y) of the magnetic vector potential at x, y ∈ R commute in the sense that all their spectral projections commute; see, e.g., [23, Theorem X.43]. Therefore, it makes sense to deﬁne ⊕ 3 (j) 1C4 ⊗ Ux d3 x, Ux := eixj Am (0) , x = (x1 , x2 , x3 ) ∈ R3 . U := R3

j=1

Then the gauge transformed vector potential is given by m := Am − 1 ⊗ Am (0) = a† (G m ) + a(G m ), A m ) = ⊕3 1C4 ⊗ a (G x,m )d3 x, and where a (G R x,m (k) := −e 1{m≤|k|≤Λ} G (e−ik·x − 1)ε(k) = (e−ik·x − 1)gm (k), 2π |k| for x ∈ R3 and almost every k = (k, λ) ∈ R3 × Z2 . In fact, using [U, α · Am ] = 0 ∗ ∗ we ﬁnd UD Am U ∗ = DA e m , thus, U SAm U = SA e m and U |DAm |U = |DA e m |. The crucial point observed in [5] is that the transformed vector potential has a better infrared behavior than Am in view of the estimate x,m (k)| ≤ |k| |x| |gm (k)|. |G

(6.1)

This avoids the appearance of infrared divergent integrals when the expectation value of the photon number operator is estimated by means of the soft photon bound. In this section we also include the case m = 0 where Hγ,0 := Hγ , A0 := A, etc. Assuming that φm is a ground state eigenvector of Hγ,m we further set f := UHf U ∗ , H

f , γ,m := UHγ,m U ∗ = |D e | − γ + H H Am |x|

φm := U φm ,

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γ,m φm = Eγ,m φm . We recall that, for f ∈ K with ωf ∈ K , so that H [a(f ), U ] = if | gm · xU,

f , a(f )] = −a(ωf ) + if | ωgm · x. [H

6.2. Soft photon and photon derivative bound In the following we set R(ωk ) := (Hγ,m − Eγ,m + ωk )−1 ,

ωk := ωk := |k|,

T (y, k) := R(ωk )DAm RAm (iy) = {|DAm |1/2 R(ωk )}∗ SAm |DAm |1/2 RAm (iy), where RAm (iy) = (DAm − iy)−1 , y ∈ R. For subcritical γ ∈ (0, 2/π), we ﬁnd T (y, k) ≤ const(γ, e, Λ)|k|−1 (1 + y 2 )−1/4 .

(6.2)

Lemma 6.1. For m ≥ 0, e, Λ > 0, γ ∈ (0, 2/π), and almost every k ∈ R3 × Z2 , x,m (k)SA φm − i(1 − ωk R(ωk ))gm (k) · xφm a(k)φm = −R(ωk )α · G m ∞ x,m (k)RAm (iy)φm dy . (6.3) + T (y, k)α · G π 0 γ,m − Eγ,m + ν)−1 , for every ν > 0, and ﬁx some Proof. We set R(ν) := (H 3 p = (p, µ) ∈ R × Z2 with p = 0. Let η ∈ U D 4 and f ∈ C0∞ ((R3 \{0}) × Z2 ). By a virial type argument we get γ,m − Eγ,m + ωp )η | a(f )φm (H † = [a† (f ), |DA e m |]η | φm + [a (f ), Hf ]η | φm + ωp η | a(f )φm 1/2 1/2 η | |DA [SA = η | [DA e m , a(f )]SA e m φm + SA e m |DA em| em| e m , a(f )]φm

+ iη | f | ωgm K · xφm + η | a((ωp − ω)f )φm .

(6.4)

Since |DA e m | ≤ Hγ,m +C and a(f )φm ∈ Q(Hγ,m ) by Lemma 6.2 below, the previous γ,m ). In particular, we may choose η = η (p) := identity extends to every η ∈ Q(H p )η, for some η ∈ H 4 . Next, we pick some h ∈ C ∞ (R3 , [0, ∞)) with supp h ⊂ R(ω 0 B1 (0) and R3 h(k)d3 k = 1, set h := −3 h(·/), and specify f := fp, , where fp, (k) := h (k − p)δµλ , for k = (k, λ) ∈ R3 × Z2 and > 0. Then (6.4) yields an identity between two locally bounded functions of p ∈ (R3 \{0}) × Z2 . After multiplying both sides of (6.4) with g ∈ C0∞ ((R3 \{0}) × Z2 , C) and integrating with respect to p we obtain (6.5) g(p)η | a(fp, )φm dp = C1 () + C2 () + C3 () + C4 (),

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where

C1 () := − C2 () := C3 () := i C4 () :=

403

p )η | α · fp, | G x,m S e φm dp, g(p)R(ω Am

1/2 1/2 R(ωp )η | |DA g(p)SA [SA e m |DA em| em| e m , a(fp, )]φm dp,

p )η | fp, | ωgm K · xφm dp, g(p)R(ω p )η | a((ωp − ω)fp, )φm dp. g(p)R(ω

Because of h ∗g → g in L2 and Fubini’s theorem the left-hand side of (6.5) tends to η | a(g)φm = U ∗ η | a(g)φm + iU ∗ η | g | gm · xφm , p )η | xφm , so that s3 is continuous on as 0. Next, we abbreviate s3 (p) := iR(ω 3 2 R \{0}. Then h ∗ (g s3 ) → g s3 in L and we further get C3 () = fp, | ω gm K · s3 (p)g(p)dp = h ∗ (g s3 ) | ω gm → g | ω s3 · gm , as 0. The term C1 () is discussed similarly. Furthermore, p )2 η2 dp + (4)−1 C5 (), |C4 ()| ≤ |g|(p)R(ω where, since |ωp − ω| ≤ on supp(fp, ), C5 () := |g|(p)a((ωp − ω)fp, )φm 2 dp 2 dp (ω = |g|(p) − ω )f (k)a(k) φ dk p k p, m fp, (k ) dk ≤ 2 |g|(p) fp, (k)ω(k)a(k)φm 2 dkdp. ω(k ) Here the integral in the curly brackets {· · ·} is bounded by some K ∈ (0, ∞) uniformly in p as long as ≤ dist(0, supp(g))/2, whence 2 C5 () ≤ K (|g| ∗ h )(k)ω(k)a(k)φm 2 dk. 1/2 Since |g| ∗ h → |g| in L∞ and φm ∈ D(Hf ) we conclude that C4 () → 0. (iy), a(f )] = R (iy)α · fp, (k) Finally, we treat C2 (). Using (3.8) and [RA p, em em A Gx,m (k)dkR e (iy) we obtain by means of Fubini’s theorem Am

C2 () =

λ∈Z2

R3

R3

g(p, λ)h (k − p)gm (k, λ)s2 (k, p)d3 k d3 p.

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Here we abbreviate

s2 (k, p) :=

dy , T(y, p)η | (e−ik·x − 1)αRA e m (iy)φm π R

1/2 1/2 where the norm of T(y, p) := {|DA R(ωp ) obeys the RA e m| e m (−iy)}SA e m |DA em| 2 −1/2 . Since s2 same bound as the one of T (y, p) in (6.2) and RA e m (iy) ≤ (1 + y ) 3 3 is continuous on (R \{0}) × R it is easy to see that C2 () converges to the second term on the right-hand side of ∗ g(k)U η | a(k)φm dk + i g(k)U ∗ η | gm (k) · xφm dk

=−

x,m (k)SAm φm dk g(k)U ∗ η | R(ωk )α · G

+ +i

g(k) 0

∞

x,m (k)RAm (iy)φm U ∗ η | T (y, k)α · G

dy dk π

g(k)U ∗ η|ωk R(ωk )gm (k) · xφm dk.

Next, we let g run over C0∞ ((R3 \{0})×Z2, C) and obtain an identity for all k outside some zero set which depends on η. Choosing η from a countable dense subset of H 4 we readily arrive at the assertion. Proof of Proposition 5.3. The soft photon bound (5.5) follows from (6.3) in combination with (6.1), (6.2), R(ωk ) ≤ |k|−1 , |x|φm ≤ C, |x|SAm φm ≤ C (by (3.15) and Proposition 5.2), and |x|RAm (iy)φm ≤ C(1 + y 2 )−1/2 (by (3.10)), uniformly in m ≥ 0. Proof of Proposition 5.4. Again, we use (6.3) to represent a(k)φm − a(p)φm and apply the bounds recalled in the proof of Proposition 5.3. Then it suﬃces to observe that, by the resolvent identity, R(ωk ) − R(ωp ) ≤ |k − p||k|−1 |p|−1 and to recall the following bound from [11]: For m < |k|, |p| < Λ and λ ∈ Z2 , 1 x,m (k, λ) − G x,m (p, λ)|} {|ωk gm (k, λ) − ωp gm (p, λ)| + |G |k| |k − p| x,m (p, λ)|} {|ωp gm (p, λ)| + |G |k||p| 1 1 + . ≤ C(1 + |x|)|k − p| |k|1/2 |k⊥ | |p|1/2 |p⊥ | +

(6.6)

For the sake of completeness we re-derive (6.6) in Sec. 6.3. Of course, here the special form of the polarization vectors (2.5) is exploited. Lemma 6.2. Let m ≥ 0, e, Λ > 0, γ ∈ (0, 2/π), and f ∈ K with ω −1/2 f ∈ K . γ,m ). Then a(f )φm ∈ Q(H

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˚ := Hγ,m − Eγ,m + 1. Thanks to [17, Theorem 5.2] we know that Proof. We put H ˚ −n/2 maps D(Hγ,m ) into itself, for every n ∈ N. ˚ n/2 ) ⊂ D(H n/2 ) and H n/2 H D(H f f n/2 n/2 ˚ −n/2 In particular, Hf φm = Hf H φm ∈ D(Hγ,m ) ⊂ Q(Hγ,m ) ⊂ D(|D0 |1/2 ). 1/2 Since φm ∈ D(Hf ) it is also clear that a(f )φm ∈ D(Hf ). Furthermore, a standard estimate based on the deﬁnitions of a(f ) and Hf shows that a(f )ψ ∈ D(|D0 |1/2 ), for 1/2 1/2 every ψ ∈ D(Hf ) with Hf ψ ∈ D(|D0 |1/2 ). In particular, a(f )φm ∈ D(|D0 |1/2 ), 1/2 thus a(f )φm ∈ Q(Hγ,m ) = D(|D0 |1/2 ) ∩ D(Hf ). γ,m ) we may equivalently show Now, to verify that a(f )φm = a(f )U φm ∈ Q(H that a(f )φm + if | gm · xφm ∈ Q(Hγ,m ). Therefore, it remains to prove that 1/2 xj φm ∈ Q(Hγ,m ) = D(|D0 |1/2 ) ∩ D(Hf ), for every component xj , j = 1, 2, 3, of x. We already know from Lemma 5.4, however, that eF φm ∈ Q(Hγ,m ), for every F ∈ C ∞ (R3 , [0, ∞)) such that F (x) = a|x|, for large |x|, and |∇F | ≤ a, provided that a > 0 is suﬃciently small. Since multiplication with xj e−F leaves the space 1/2 D(|D0 |1/2 ) ∩ D(Hf ) invariant we conclude that xj φm ∈ Q(Hγ,m ). 6.3. Elementary estimates on polarization vectors In this subsection we recall the proof of the bound (6.6). A similar bound has been used before in [11]. The elementary estimates below are the only place in our whole article where the special choice of the polarization vectors (2.5) is exploited. We set y⊥ := (y (2) , −y (1) , 0) and y◦ := y/|y|, for y = (y (1) , y (2) , y (3) ) ∈ R3 \{0}, and ∆h f (k, λ) := f (k + h, λ) − f (k, λ), for some function on R3 × Z2 . By means of (2.5) we ﬁnd (∆−h ε)(k, 0) = −|k⊥ |−1 h⊥ + (|(k − h)⊥ |−1 − |k⊥ |−1 )(k − h)⊥ , (∆−h ε)(k, 1) = ((k − h)◦ − k◦ ) ∧ ε(k − h, 0) + k◦ ∧ (∆−h ε)(k, 0), whence |∆−h ε(k, 0)| ≤ 2|h⊥ |/|k⊥ | ≤ 2|h|/|k⊥ |, |∆−h ε(k, 1)| ≤ 2|h|/|k| + |∆−h ε(k, 0)| ≤ 4|h|/|k⊥ |. Since gm (k) = |k|−1/2 ε(k)1m≤|k|≤Λ and |a−1/2 −b−1/2 | ≤ (|a−b|/2)(a−3/2 +b−3/2 ), a, b > 0, we further have, for m < |k|, |k − h| < Λ, 4|h| 1 |h| 1 + + , |∆−h gm (k)| ≤ 2 |k|3/2 |k|1/2 |k⊥ | |k − h|3/2 |h| 1 |∆−h (ωgm )| ≤ |∆−h gm (k)| + . |k| |k||k − h|1/2 x,m = (e−ik·x − 1)gm (k) and |eiy·x − eiz·x | ≤ |y − z||x|, we ﬁnd Moreover, since G 1 |h||x| x,m | ≤ |x||∆−h gm (k)| + |∆−h G , |k| |k||k − h|1/2

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again for m < |k|, |k − h| < Λ. Furthermore, it is clear that (∆h ω)(k − h)2 |h|2 ≤ . 2 2 |k| |k − h| |k| |k − h| Finally, by Young’s inequality, |h| |h| ≤ 1/2 3 |k||k − h|

2 |k|1/2 |k⊥ |

+

1 1/2 |k − h| |(k − h)⊥ |

.

Combining the above estimates with p = k − h we obtain (6.6). Acknowledgments This work has been partially supported by the German Research Foundation (DFG) (SFB/TR12). References [1] V. Bach, T. Chen, J. Fr¨ ohlich and I. M. Sigal, Smooth Feshbach map and operatortheoretic renormalization group methods, J. Funct. Anal. 203 (2003) 44–92. [2] V. Bach, J. Fr¨ ohlich and A. Pizzo, Infrared-ﬁnite algorithms in QED: The groundstate of an atom interacting with the quantized radiation ﬁeld, Comm. Math. Phys. 264 (2006) 145–165. [3] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Quantum electrodynamics of conﬁned nonrelativistic particles, Adv. Math. 137 (1998) 299–395. [4] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Renormalization group analysis of spectral problems in quantum ﬁeld theory, Adv. Math. 137 (1998) 205–298. [5] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Spectral analysis for systems of atoms and molecules coupled to the quantized radiation ﬁeld, Comm. Math. Phys. 207 (1999) 249–290. [6] V. Bach and M. K¨ onenberg, Construction of the ground state in nonrelativistic QED by continuous ﬂows, J. Diﬀer. Equ. 231 (2006) 693–713. [7] J.-M. Barbaroux, M. Dimassi and J. C. Guillot, Quantum electrodynamics of relativistic bound states with cutoﬀs, J. Hyperbol. Diﬀer. Eq. 1 (2004) 271–314. [8] A. Berthier and V. Georgescu, On the point spectrum of Dirac operators, J. Funct. Anal. 71 (1987) 309–338. [9] J. Fr¨ ohlich, M. Griesemer and I. M. Sigal, On spectral renormalization group, Rev. Math. Phys 21 (2009) 511–548. [10] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Asymptotic electromagnetic ﬁelds in models of quantum-mechanical matter interacting with the quantized radiation ﬁeld, Adv. Math. 164 (2001) 349–398. [11] M. Griesemer, E. H. Lieb and M. Loss, Ground states in non-relativistic quantum electrodynamics, Invent. Math. 145 (2001) 557–595. [12] F. Hiroshima and I. Sasaki, On the ionization energy of the semi-relativistic Pauli– Fierz model for a single particle, RIMS Kokyuroku Bessatsu 21 (2010) 25–34. [13] T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics (Springer, Berlin-Heidelberg, 1995). [14] M. K¨ onenberg, O. Matte and E. Stockmeyer, Existence of ground states of hydrogenlike atoms in relativistic quantum electrodynamics II: The no-pair operator, preprint (2010), 76 pp.; arXiv:1005.2109v1.

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[15] E. H. Lieb and M. Loss, A bound on binding energies and mass renormalization in models of quantum electrodynamics, J. Stat. Phys. 108 (2002) 1057–1069. [16] E. H. Lieb and M. Loss, Stability of a model of relativistic quantum electrodynamics, Comm. Math. Phys. 228 (2002) 561–588. [17] O. Matte, On higher order estimates in quantum electrodynamics, Doc. Math. 15 (2010) 207–234. [18] O. Matte and E. Stockmeyer, On the eigenfunctions of no-pair operators in classical magnetic ﬁelds, Integr. Equat. Oper. Th. 65 (2009) 255–283. [19] O. Matte and E. Stockmeyer, Exponential localization for a hydrogen-like atom in relativistic quantum electrodynamics, Comm. Math. Phys. 295 (2010) 551–583. [20] T. Miyao and H. Spohn, Spectral analysis of the semi-relativistic Pauli–Fierz Hamiltonian, J. Funct. Anal. 256 (2009) 2123–2156. [21] S. M. Nikol’ski˘ı, An imbedding theorem for functions with partial derivatives considered in diﬀerent metrics, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958) 321–336 (Russian); English translation in Amer. Math. Soc. Transl. (2 ) 90 (1970) 27–43. [22] S. M. Nikol’ski˘ı, Approximation of Functions of Several Variables and Imbedding Theorems, Die Grundlehren der Mathematischen Wissenschaften, Band 205 (Springer, New York, 1975). [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness (Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975). [24] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators (Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978). [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. (Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1980). [26] I. Sasaki, Ground state of a model in relativistic quantum electrodynamics with a ﬁxed total momentum, preprint (2006), 30 pp.; arXiv:math-ph/0606029v4. [27] E. Stockmeyer, On the non-relativistic limit of a model in quantum electrodynamics, preprint (2009), 13 pp.; arXiv:0905.1006v1.

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Reviews in Mathematical Physics Vol. 23, No. 4 (2011) 409–451 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004345

FAST SOLITONS ON STAR GRAPHS

RICCARDO ADAMI∗,§ , CLAUDIO CACCIAPUOTI†,¶ , DOMENICO FINCO‡, and DIEGO NOJA∗,∗∗ ∗Dipartimento

di Matematica e Applicazioni, Universit` a di Milano Bicocca, via R. Cozzi, 53, 20125 Milano, Italy

†Hausdorﬀ Center for Mathematics, Institut f¨ ur Angewandte Mathematik, Endenicher Allee, 60, 53115 Bonn, Germany ‡Facolt´ a di Ingegneria, Universit´ a Telematica Internazionale Uninettuno, Corso Vittorio Emanuele, II 00186 Roma, Italy §[email protected]

¶[email protected]

d.ﬁ[email protected] ∗∗[email protected]

Received 15 January 2011 Revised 30 March 2011 We deﬁne the Schr¨ odinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoﬀ boundary condition and the so-called δ and δ boundary conditions. Moreover, in the same setting, we study the collision of a fast solitary wave with the vertex and we show that it splits in reﬂected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale, the reﬂection and transmission coeﬃcients of the outgoing waves coincide with the corresponding coeﬃcients of the linear problem. In the analysis of the problem, we follow ideas borrowed from the seminal paper [17] about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski. The present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph. Keywords: Quantum graphs; nonlinear Schr¨ odinger equation; solitary waves. Mathematics Subject Classiﬁcation 2010: 35Q55, 81Q35, 37K40

1. Introduction In the present paper, we study the nonlinear wave propagation on graphs. As far as we know the subject of nonlinear Schr¨ odinger evolution on graphs is at its 409

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beginnings. An extensive literature on the behavior of linear wave and Schr¨ odinger equations on graphs exists ([22, 23, 21, 7, 8], and references therein) and a certain activity concerning the so called discrete nonlinear Schr¨ odinger equation (DNLSE) in chains with edges and graphs inserted (“decorations”, interpreted as defects in the chain), both from the physical and the numerical point of view (see, for example, [20, 9]). To our knowledge, however, there are only very few papers in which nonlinear Schr¨ odinger evolution on graphs has been introduced and studied (see [6, 26]). In the ﬁrst paper [6] (and similar ones quoted therein) NLS on graphs emerges in some models of quantum ﬁeld theory on “bulks”; the second recent paper [26] addresses from a physical point of view some general questions related to the ones here studied and is brieﬂy commented in the conclusions of the present paper. Besides several results on existence and stability of stationary states for the NLS on star graphs can be found in [2]. In any case, the study of nonlinear propagation in ramiﬁed structures could be of relevance in several branches of pure and applied science, from condensed matter physics to nonlinear ﬁber optics, hydrodynamics and ﬂuid transport (a nontraditional example is blood ﬂow in veins and arterias), and ﬁnally neural networks (see for example the study of reaction-diﬀusion type FitzHugh–Nagumo–Rall equations on networks in [12], and references therein). In all these examples, there is a strong dependence on the modelization. The nonlinear Schr¨ odinger equation with cubic nonlinearity is especially suitable to describe nonlinear electromagnetic pulse propagation in optical ﬁbers and, under the name of Gross–Pitaevskii equation, the dynamics of Bose–Einstein condensates. Better suited for other applications, for example the hydrodynamic ﬂow, is the KdV equation or its relatives, not treated here. Of course real networks are not strictly one dimensional, and an abstract graph, which is just a set of copies of R+ (“edges”, or “branches”) with functions living on it satisfying certain boundary conditions at 0 lacks some of the geometric meaningful characteristics of a real network, such as thickness and curvature of the branch or orientation between edges. On the other hand, problems related to wave propagation on networks are far from being well understood also for the linear propagation (see, for example, [4, 10]), and so we content ourselves with posing and analyzing the nonlinear problem in the idealized and simpliﬁed framework of an abstract graph. We study here the special case of a star graph with three edges. A generalization to a star graph with n edges would be straightforward, but here our interest is in clarifying the main features of the evolution and the techniques involved in its analysis. A preliminary point and not a trivial issue is the deﬁnition of the dynamics. Let us recall that for a star graph G, the linear Schr¨ odinger dynamics is deﬁned by giving a self-adjoint operator H on the product of n copies of L2 (R+ ) (brieﬂy L2 (G)), with a domain D(H) in which appears a linear condition involving the values at 0 of the functions of the domain and of their derivatives. The admissible boundary conditions characterize the interaction at the vertex of the star graph. For the nonlinear problem, we establish the well-posedness of the dynamics in the case of a star graph with some distinguished boundary condition at the vertex,

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namely the free (or Kirchhoﬀ), and the δ and δ boundary conditions (see Sec. 2 for the relevant deﬁnitions). To clarify the exact problem we are faced to, the diﬀerential equation to be studied is of the form d (1.1) i Ψt = HΨt − |Ψt |2 Ψt , t ≥ 0, dt where the function Ψ is, for a three-edge graph, a column vector   ψ1   Ψ = ψ2 , ψ3 that lies in the domain D(H) of the linear Hamiltonian H on the graph, so emboding the relevant boundary conditions. This is the abstract strong form of the equation, which is equivalent to a particular nonlinear coupled system of scalar equations. The coupling is not due to the nonlinearity, because of the deﬁnition   |ψ1 |2 ψ1   |Ψ|2 Ψ ≡ |ψ2 |2 ψ2 , |ψ3 |2 ψ3 but to the boundary condition at the vertex. For example, in the simple case of a Kirchhoﬀ boundary condition, the coupling between the edges is given by Ψ ∈ L2 (G)

s.t. ψi ∈ H 2 (R+ ),

ψ1 (0) = ψ2 (0) = ψ3 (0), ψ1 (0) + ψ2 (0) + ψ3 (0) = 0. The δ or δ boundary conditions allow a coupling between the values of the function Ψ and the values of their derivatives at the origin, but in principle the nature of the problem is unaltered. In the present paper, for several reasons, we prefer to write the dynamics in weak form, which is the following t e−iH(t−s) |Ψs |2 Ψs ds, t ≥ 0; (1.2) Ψt = e−iHt Ψ0 + i 0

here the Ψ belongs to the form domain D(Elin ), where Elin is the quadratic form of the linear Hamiltonian H. We interpret, according to the use, the form domain as the ﬁnite energy space. An adaptation of the methods in [3] gives the local wellposedness of Eq. (1.2) for every initial data in D(Elin ). Moreover, charge and energy conservation laws hold true for such weak solutions, and as a consequence, the NLS on graph admits global solutions. The generalization of the well-posedness result to the case of general self-adjoint boundary conditions will be treated elsewhere. Apart from well-posedness, the main goal of this paper is to provide information on the interaction between a solitary wave and the boundary condition at the vertex. As it is well known, the NLS on the line admits a family of solitary non dispersive

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solutions, rigidly translating with a ﬁxed velocity. A rich family of solitary solutions (“solitons”) is given by the action of the Galilei group on the function √ φ(x) = 2 cosh−1 x, x ∈ R, or explicitly, v

φx0 ,v (x, t) = ei 2 x e−it

v2 4

eit φ(x − x0 − vt)

x ∈ R,

t ∈ R,

v ∈ R.

We show that, after the collision of a solitary wave with the vertex there exists a timescale during which the dynamics can be described as the scattering of three split solitary waves, one reﬂected on the same branch where the originary soliton was running asymptotically in the past, and two transmitted solitary waves on the other branches. On the same timescale, the amplitudes of the reﬂected and transmitted solitary waves are given by the scattering matrix of the linear dynamics on the graph. The soliton-like character persists over time intervals that depend on the velocity of the impinging original soliton: the faster is the original soliton, the (logarithmically in the velocity v) longer is the survival time of the solitary wave behavior on every branch of the graph. The non-trivial point is that the persistence time of solitary behavior after collision with the vertex is much longer, for fast solitons, than the time over which it is reasonable to approximate the nonlinear dynamics with the linear one. The same timescale of the order ln v of persistence of solitary behavior appears in the paper [1] where the collision of two solitary waves with an underlying smooth potential is studied, and in the paper by Holmer, Marzuola and Zworski [17] on the fast NLS-soliton scattering by a delta potential on the line, which is the main source of inspiration for our result and for the techniques employed in the present paper. We give now an outline of our result and its proof. The initial data are of the following form   v χ(x)e−i 2 x φ(x − x0 ) , x0 ≥ v 1−δ , 0 < δ < 1, (1.3) Ψ0 (x) =  0 0 where χ is a cut oﬀ function, that is χ ∈ C ∞ (R+ ), χ = 1 in (2, +∞) and χ = 0 in (0, 1). Apart from a small tail term truncated by the cutoﬀ function, the ﬁrst component is the initial condition of a free (i.e. without external potentials) NLS which on the line yields a solitary wave running with velocity v; the center x0 of the initial soliton is chosen far from the vertex. We are interested in the evolution Ψt of this initial condition. The dynamics can be divided into three phases. The pre-interaction phase, where the evolved initial condition is far from the vertex, and the undisturbed NLS evolution dominates. At the end of this phase, the solution enters the vertex zone, and diﬀers (in L2 norm) from the evolved solitary wave by an exponentially small error in the velocity v. The second phase is the interaction phase, in which a substantial

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fraction of the mass of the initial soliton has reached the vertex, and the linear dynamics dominates due to the shortness of the interaction time, leaving the system at the end of this phase with three scattered waves, the amplitudes of which are given by the action of the scattering matrix of the associated linear graph on the incoming solitary wave. The size of the corresponding error is (again in L2 norm) a 1 suitable negative inverse power (v − 2 δ ) of the velocity. In the phase one, the main technical tool consists in the accurate use, as ﬁxed by [17], of the Strichartz’s estimates to control the deviations between the unperturbed NLS ﬂow and the NLS ﬂow on the graph. In the phase two, we need to compare the nonlinear evolution with the linear ﬂow on the graph in the relevant time interval. Finally there is the post-interaction phase, where the free NLS dynamics dominates again. However, now the initial data are not exact solitary waves, but waves with soliton-like proﬁles and “wrong” amplitudes (due to the scattering process in the interaction phase). The true evolution is compared with a reference dynamics given by the superposition of the nonlinear evolution of the outgoing scattered proﬁles, and it turns out that the error is, in L2 norm, of the order of an inverse power of velocity (depending on the size of the time interval of approximation). For a precise formulation, one has to tackle the problem of representing the reference soliton dynamics to be compared with the true dynamics. This problem arises because one would like to use crucial and known properties of NLS on the line (such as existence of an inﬁnite number of constants of motion), and various associated estimates, while on a star graph one has a NLS on halﬂines, jointly with boundary conditions. The problem occurs, of course, in each of the three phases in which the dynamics is decomposed. Our choice of reference dynamics is the following. We associate to every edge of the star graph a companion edge chosen between the other two, in such a way to have three ﬁctitious lines; then we glue the soliton on every single edge with the right tail on the companion edge, respecting the free nonlinear dynamics. One of the main technical points in the analysis of the true dynamics is to have a control in the errors brought by this schematization. More precisely, let us deﬁne   v2 v r˜e−i 4 t ei 2 x1 eit φ(x1 + x0 − vt)    ˜ 1t (x1 , x2 , x3 ) ≡  −i v2 t −i v x2 it Φ r˜e 4 e 2 e φ(x − x + vt), 2

0

0 

0



  v2 v ˜ 2t (x1 , x2 , x3 ) ≡  t˜e−i 4 t ei 2 x2 eit φ(x2 + x0 − vt) , Φ   v2 v t˜e−i 4 t e−i 2 x3 eit φ(x3 − x0 + vt)   v2 v t˜e−i 4 t e−i 2 x1 eit φ(x1 − x0 + vt)  ˜ 3t (x1 , x2 , x3 ) ≡  Φ 0  . v2 v −i t i x it 3 t˜e 4 e 2 e φ(x3 + x0 − vt)

(1.4)

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Each of these vectors represents a soliton on the ﬁctitious line given by an edge and its companion, multiplied by the scattering coeﬃcients of the linear dynamics considered, Kirchhoﬀ, δ or δ (and here left unspeciﬁed). Up to a small error, these functions represent outgoing waves at the end (t = t2 ) of the interaction phase, which is essentially a scattering process. Taking these as initial data for the free nonlinear dynamics on the pertinent ﬁctitious line, we deﬁne their time evolution Φjt as given by t j −iHj (t−t2 ) ˜ j Φt = e Φt2 + i dse−iHj (t−s) |Φjs |2 Φjs j = 1, 2, 3, t ≥ t2 , t2

where the Hj are the linear Hamiltonians that decouple the j + 2-branch from the others. With these premises, the main result of the paper is the following: Theorem 1.1. Let Ψt the unique, global solution to the Cauchy problem (1.1) with initial data (1.3). There exist τ∗ > 0 and T∗ > 0 such that for 0 < T < T∗ one has Ψt − Φ1t − Φ2t − Φ3t L2x (G) ≤ Cv −

T∗ −T τ∗

for every time t in the interval t2 < t < t2 + T ln v, where C is a positive constant independent of t and v. To be precise, the Hamiltonians in (1.1) to which the theorem refers have to be rescaled in order to give a nontrivial scattering matrix in the regime of high velocity (see Sec. 4, in particular Theorem 4.3). To get the previous result, Strichartz estimates do not suﬃce, and more direct properties coming from the integrable character of the cubic NLS are needed. In particular, thanks to the existence of an inﬁnite number of integrals of motion, in [17] a spatial localization property of the solution of NLS with smooth data is proven, with a polynomial (and not exponential) bound in time. This gives a control on the tails of the diﬀerence between the solution and the reference modiﬁed solitary dynamics. An analogous method applies in our case. Let us note that as a consequence of the previous result, we can give the scattered amplitudes in terms of the incoming amplitude and scattering coeﬃcients of the linear dynamics, in the time range of applicability of the main theorem (see Remark 4.4). We give a brief summary of the content of the various sections of the papers. In Sec. 2, we give some generalities on linear dynamics on graphs, including Hamiltonians, their quadratic forms, resolvents and propagators. Moreover the essential Strichartz estimates are recalled. In Sec. 3, local and global well posedness of nonlinear Schr¨ odinger equations on star graphs is proved. In Sec. 4 the main result (Theorem 4.3) is introduced and stated. Section 5 is devoted to the proof of the result. In Sec. 6, some ﬁnal remarks are given and further possible developments are discussed.

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1.1. Setting and notations We consider a graph G given by three inﬁnite half lines attached to a common vertex. In order to study a quantum mechanical problem on G, the natural Hilbert space is then L2 (G) ≡ L2 (R+ ) ⊕ L2 (R+ ) ⊕ L2 (R+ ). We denote the elements of L2 (G) by capital greek letters, while functions in L2 (R+ ) are denoted by lowercase greek letters. It is convenient to represent functions in L2 (G) as column vectors of functions in L2 (R+ ), namely   ψ1   Ψ = ψ2 . ψ3 The norm of L2 -functions on G is naturally deﬁned by  12  3 ΨL2 (G) :=  ψj 2L2 (R+ )  . j=1

Analogously, given 1 ≤ r ≤ ∞, we deﬁne the space Lr (G) as the set of functions on the graph whose components are elements of the space Lr (R+ ), and the norm is correspondingly deﬁned by  1r  3 ΨLr (G) =  ψj rLr (R+ )  , 1 ≤ r < ∞, j=1

ΨL∞ (G) = sup ψj L∞ (R+ ) . 1≤j≤3

When a functional norm refers to a function deﬁned on the graph, we omit the symbol G. Furthermore, from now on, when such a norm is L2 , we drop the subscript, and simply write · . Accordingly, we denote by (·, ·) the scalar product in L2 . As it is standard when dealing with Strichartz’s estimates, we make use of spaces of functions that are measurable as functions of both time (on the interval [T1 , T2 ]) and space (on the graph). We denote such spaces by Lp[T1 ,T2 ] Lr (G), with indices 1 ≤ r ≤ ∞, 1 ≤ p < ∞; we endow them with the norm

1/p ΨLp[T

T2

1 ,T2

ΨL∞ [T

Lr (G) ]

=

Lr (G)

=

1 ,T2 ]

T1

Ψs pLr ds

sup t∈[T1 ,T2 ]

,

1 ≤ p < ∞,

Ψs Lr .

The extension of the deﬁnitions given above to the case p = ∞ is straightforward. Besides, we need to introduce the spaces H 1 (G) ≡ H 1 (R+ ) ⊕ H 1 (R+ ) ⊕ H 1 (R+ ), H 2 (G) ≡ H 2 (R+ ) ⊕ H 2 (R+ ) ⊕ H 2 (R+ ),

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equipped with the norms Ψ2H 1 (G) =

3

ψi 2H 1 (R+ ) ,

Ψ2H 2 (G) =

i=1

3

ψi 2H 2 (R+ ) .

(1.5)

i=1

The product of functions is deﬁned componentwise,     |ψ1 |2 ψ1 ψ1 φ1     ΨΦ ≡ ψ2 φ2 , so that |Ψ|2 Ψ ≡ |ψ2 |2 ψ2 . |ψ3 |2 ψ3

ψ3 φ3

We denote by I the 3 × 3 identity matrix, while J is the 3 × 3 matrix whose elements are all equal to one. When an element of L2 (G) evolves in time, we use in notation the subscript t: for instance, Ψt . Sometimes we shall write Ψ(t) in order to highlight the dependence on time, or whenever such a notation is more understandable. 2. Summary on Linear Dynamics on Graphs 2.1. Hamiltonians and quadratic forms Standard references about the linear Schr¨ odinger equation on graphs are [7, 8, 22, 23, 21], to which we refer for more extensive treatments. Here we only give the deﬁnitions needed to have a self-contained exposition. We consider three Hamiltonian operators, denoted by HF , Hδα , Hδβ (with α, β ∈ R), and called, respectively, the Kirchhoﬀ, the Dirac’s delta, and the delta-prime Hamiltonian. These operators act as   −ψ1   (2.1) Ψ → −ψ2  −ψ3 on some subspace of H 2 (G), to be deﬁned by suitable boundary conditions at the vertex. Here and in the following subsection we collect some basic facts (see [21–23, 7]) on HF , Hδα , and Hδβ . The Kirchhoﬀ Hamiltonian HF acts on the domain D(HF ) := {Ψ ∈ H 2 (G) s.t. ψ1 (0) = ψ2 (0) = ψ3 (0), ψ1 (0) + ψ2 (0) + ψ3 (0) = 0}.

(2.2)

It is well known, see [21], that (2.2) and (2.1) deﬁne a self-adjoint Hamiltonian on L2 (G). Boundary conditions in (2.2) are usually called Kirchhoﬀ boundary conditions. We use the index F to remind that HF reduces to the free Hamiltonian on the line for a degenerate graph composed of two half lines. The quadratic form EF associated to HF is deﬁned on the subspace D(EF ) = {Ψ ∈ H 1 (G) s.t. ψ1 (0) = ψ2 (0) = ψ3 (0)}

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and reads EF [Ψ] =

3 i=1

+∞

0

417

|ψi (x)|2 dx.

The Dirac’s delta Hamiltonian is deﬁned on the domain D(Hδα ) := {Ψ ∈ H 2 (G) s.t. ψ1 (0) = ψ2 (0) = ψ3 (0), ψ1 (0) + ψ2 (0) + ψ3 (0) = αψ1 (0)}. 2

Hδα

(2.3) Hδα

Again, is a self-adjoint opeator on L (G) ([21]). It appears that generalizes the ordinary Dirac’s delta interaction with strength parameter α on the line, see, e.g. [5]. The quadratic form Eδα associated to Hδα is deﬁned on D(Eδα ) = {Ψ ∈ H 1 (G) s.t. ψ1 (0) = ψ2 (0) = ψ3 (0)} and is given by Eδα [Ψ] =

3 i=1

0

+∞

|ψi (x)|2 dx + α|ψ1 (0)|2 .

The delta-prime Hamiltonian is deﬁned on the domain D(Hδβ ) := {Ψ ∈ H 2 (G) s.t. ψ1 (0) = ψ2 (0) = ψ3 (0), ψ1 (0) + ψ2 (0) + ψ3 (0) = βψ1 (0)}.

(2.4)

2

Hδβ

Again, is a self-adjoint opeator on L (G) ([21]). The quadratic form Eδβ associated to Hδβ is deﬁned on D(Eδβ ) = H 1 (G) and is given by 2 3 +∞ 3 1 β 2 Eδ [Ψ] = |ψi (x)| dx + ψi (0) . β 0 i=1

i=1

Notice that does not reduce to the standard δ interaction on the line, see, e.g., [5], when it is restricted to a two-edge graph. Here we are following the notation in [22, 23]. The present δ vertex is called sometimes δs graph, where s is for symmetric. A discussion of the correct extension of the usual δ interaction is given in [15, 8]. For completeness, we give the operator domain (the action is the same as in the other cases). We use the denomination δ˜ to avoid confusion with the previously deﬁned interaction.  n  β ψj (0) = 0, ψj (0) − ψk (0) = (ψj (0) − ψk (0)), D(Hδ˜β ) := Ψ ∈ H 2 (G) s.t.  n j=1 Hδβ

j, k = 1, 2, . . . , n . Throughout the paper we restrict to the case of repulsive delta and delta-prime (δ ) interaction, i.e. α, β > 0. It is easily proved, for example by inspection of the

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operator resolvents in the following subsection, that such a condition prevents the corresponding Hamiltonian operator from possessing bound states. We also point out that the Hamiltonian Hδα is well deﬁned for α = 0 and that ˜ β˜ > 0, we shall consider the indeed Hδα |α=0 ≡ HF . We ﬁnally point out that, ﬁxed α, ˜ β/v

˜ and Hδ , namely, in the following we rescale α = α ˜v Hamiltonian operators Hδαv ˜ and β = β/v.

2.2. Resolvents, propagators and scattering For any complex number k with Im k > 0 we denote by RF (k), Rδα (k), Rδβ the resolvents (HF − k 2 )−1 , (Hδα − k 2 )−1 , (Hδβ − k 2 )−1 , respectively. We deﬁne the function Ut by x2

ei 4t , Ut (x) := √ 4πit

t = 0.

In the following we shall use the same symbol Ut to denote the operator in L2 (R) deﬁned by ∞ Ut (x − y)ψ(y)dy, t = 0. [Ut ψ](x) := −∞

Moreover, we deﬁne two integral operators Ut± acting on L2 (R+ ) +∞ Ut± : L2 (R+ ) → L2 (R+ ), [Ut± ψ](x) = Ut (x ± y)ψ(y)dy, 0

t = 0.

We stress that, according to our deﬁnitions, the operators Ut− and Ut have the same integral kernel, but act on diﬀerent Hilbert spaces. For the cases we consider, resolvents and propagators can be easily computed (see, e.g. [7, pp. 201–226] from resolvent formulas with generic boundary conditions in the vertex). The results are summarized in the following theorem. Theorem 2.1. For any complex number k with Im k > 0, the integral kernel of the resolvent operators RF (k), Rδα (k), Rδβ (k) are given by   −1 2 2 i ik|x−y| i ik(x+y) 1  e e RF (k; x, y) = I+ (2.5) 2 −1 2, 2k 2k 3 2 2 −1   α − ik 2ik 2ik ik(x+y) i ik|x−y| i e   e I− (2.6) Rδα (k; x, y) = α − ik 2ik ,  2ik 2k 2k α − 3ik 2ik 2ik α − ik   −1 + iβk 2 2 ik(x+y) i ik|x−y| i e   e 2 −1 + iβk 2 I− Rδβ (k; x, y) =  . 2k 2k 3 − iβk 2 2 −1 + iβk (2.7)

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Furthermore, the unitary group of the related time evolution operators reads   −1 2 2 1 2 e−iHF t = Ut− I + Ut+  2 −1 (2.8) 2 = (Ut− − Ut+ )I + Ut+ J, 3 3 2 2 −1 α Uδ,t (x, y) = [Ut (x − y) − Ut (x + y)]I 2 α +∞ −α u 3 + due Ut (x + y + u) J, Ut (x + y) − 3 3 0 3 2 +∞ β Uδ ,t (x, y) = [Ut (x − y) + Ut (x + y)]I − due− β u Ut (x + y + u)J, β 0

(2.9) (2.10)

where x, y ∈ R+ , α, β > 0. Proof. We start from the proof of (2.6). Let (Ψ)i = ψi for i = 1, 2, 3 and deﬁne RΨ by +∞ i i ikx e ci , i = 1, 2, 3, eik|x−y| ψi (y)dy + (2.11) (RΨ)i (x) = 2k 0 2k where ci = ci (Ψ) are constant to be speciﬁed. It is obvious that (RΨ)i satisﬁes d2 − 2 − k 2 (RΨ)i = 0, i = 1, 2, 3, (2.12) dx then it is suﬃcient to ﬁx ci such that RΨ belongs to D(Hαδ ) in order to compute δ . The boundary conditions in (2.3) translate to the following linear system for ci . Rα +∞  +∞  iky  e ψ (y)dy + c = eiky ψ2 (y)dy + c2 ,  1 1   0   0  +∞  +∞    eiky ψ2 (y)dy + c2 = eiky ψ3 (y)dy + c3 ,   0 0 (2.13) +∞   iky  e (ψ1 (y) + ψ2 (y) + ψ3 (y))dy − (c1 + c2 + c3 )    0    +∞    iα   = eiky ψ2 (y)dy + c2 , k 0 the solution is easily computed and it is given by  +∞ 2k 2k k + iα  iky  c1 = ψ1 (y) + ψ2 (y) + ψ3 (y) dy, e −   3k + iα 3k + iα 3k + iα  0    +∞  k + iα 2k 2k iky ψ1 (y) − ψ2 (y) + ψ3 (y) dy, c2 = e  3k + iα 3k + iα 3k + iα  0   +∞    2k 2k k + iα   c3 = ψ1 (y) + ψ2 (y) − ψ3 (y) dy, eiky 3k + iα 3k + iα 3k + iα 0

(2.14)

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which gives (2.6). In order to obtain (2.5) it is suﬃcient to put α = 0 in (2.6). Formula (2.7) can be proved by the same method. Now we prove (2.9). We start from the standard formula +∞ 1 α Rα (k; x, y)kdk, Uδ,t (x, y) = πi −∞ δ use (2.6) and recall the following identity 1 2π

+∞

−∞

eik(x+y) −ik2 t e dk = a − ik

2

+∞

e

0

−au e

i (x+y+u) 4t

√ du, 4πit

(2.15)

then we immediately arrive at (2.9). Formula (2.8) can be obtained by putting α = 0 into (2.9). Formula (2.10) can be proved in the same way. Corollary 2.2. From the expression of the resolvent one immediately has the reﬂection and transmission coeﬃcients: 1 r(k) = − , 3 k + iα rHδα (k) = − , 3k + iα βk + i , rH β (k) = δ βk + 3i

t(k) =

2 , 3

2k , 3k + iα 2i tH β (k) = − . δ βk + 3i tHδα (k) =

(2.16) (2.17) (2.18)

We refer to [21] for a comprehensive analysis of the scattering on star graphs. Indeed by using the results in [21] the reﬂection and transmission coeﬃcients can be obtained directly by the boundary conditions in the vertex. Throughout the paper we shall need some auxiliary dynamics to be compared with the dynamics described by (1.1), so, for later convenience, we introduce the two-edge Hamiltonians Hj and the corresponding two-edge propagators e−iHj t , j = 1, 2, 3. Let Hj be deﬁned by: D(Hj ) := {Ψ ∈ H 2 (G) s.t. ψj (0) = ψj+1 (0), ψj (0) + ψj+1 (0) = 0,

ψk (0) = 0, k = j, j + 1},   −ψ1   Hj Ψ := −ψ2 , −ψ3 where in Eq. (2.19) it is understood that j = {1, 2, 3} modulo 3. The Hamiltonian Hj couples the edges j and j + 1 with a Kirchhoﬀ boundary condition and sets a Dirichlet boundary condition for the remaining edge, so that there is free propagation between the edges j and j + 1 and no propagation between them and the edge j + 2.

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Fig. 1. The open graphs depict the Hamiltonians H1 , H2 and H3 . Under each graph we report the boundary conditions on vectors in the domain of the corresponding Hamiltonian.

With a straightforward computation, we have e−iHj t = Ut− I + Ut+ Tj , where Tj are the  0 T1 = 1 0

matrices  1 0 0 0; 0 −1

t ∈ R,



 −1 0 0 T2 =  0 0 1; 0 1 0

 0 T3 = 0 1

(2.19)  0 1 −1 0. 0 0

2.3. Strichartz’s estimates A key tool in our method is the extension of the standard Strichartz’s estimates α (see e.g. [13]) to the dynamics on G described by the propagators e−iHF t , e−iHδ t , β and e−iHδ t . In this subsection we use the symbol H to denote any of the three Hamiltonians of interest described in Sec. 2.1. As a preliminary step, we remark that from Eqs. (2.8)–(2.10), the standard dispersive estimate immediately follows: e−iHt ΨL∞ ≤

c t1/2

ΨL1 ,

t = 0.

(2.20)

Proposition 2.3 (Strichartz Estimates for e−iHt ). Let Ψ0 ∈ L2 (G), Γ ∈ LqR Lk , with 1 ≤ q, k ≤ 2, 2q + k1 = 52 , and deﬁne Ψ(t) = e

−iHt

Ψ0 ,

Φ(t) = 0

t

dse−iH(t−s) Γ(s).

The following estimates hold true: ΨLp Lr ≤ cΨ0

(2.21)

R

ΦLp Lr ≤ c ΓLq Lk R

R

(2.22)

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for any pair of indices (r, p) satisfying 2 1 1 + = . p r 2 The constants c in (2.21) and (2.22) are independent of T . 2 ≤ p, r ≤ ∞,

(2.23)

Proof. The proof is standard due to the dispersive estimate (2.20), see for instance [13, 19]. Remark 2.4. If H = Hδα (Hδβ ), the constants appearing in (2.20)–(2.22) are independent of α (β). Indeed, by the change of variable u → uα (u → u/β) the integral term in (2.9) ((2.10)) can be easily estimated independently of α (β), obtaining a dispersive estimate (2.20) independent of the parameters and therefore, by the standard Strichartz machinery, uniform inequalities (2.21) and (2.22). 3. Well-Posedness and Conservation Laws Here we treat the problem of the well-posedness (in the sense of, e.g., [13]), i.e., the existence and uniqueness of the solution to Eq. (1.2) in the energy domain of the system. Such a domain turns out to coincide with the form domain of the linear part of Eq. (1.1). Throughout this section, such a linear part is denoted by H, and, according to the particular case under consideration, it can be understood as the Hamiltonian operator HF , Hδα , or Hδβ . Correspondingly, we denote the associated energy domain simply by D(E). All of the following formulas can be specialized to the particular cases D(EF ), D(Eδα ), or D(Eδβ ). Let us stress that throughout the paper we do not approximate the dynamics in H 1 , but rather in L2 . Furthermore, local well-posedness in L2 is ensured by Strichartz estimates (Proposition 2.3), as is easily seen following the line exposed in [13, Chaps. 2 and 3]. Nonetheless, we prefer to deal with functions in the energy domain, since they are physically more meaningful. We follow the traditional line of proving, ﬁrst of all, local well-posedness, and then extending it to all times by means of a priori estimates provided by the conservation laws. For a more extended treatment of the analogous problem for a two-edge vertex (namely, the real line with a point interaction at the origin), see [3]. First, we endow the energy domain D(E) with the H 1 -norm deﬁned in (1.5). Second, we denote by D(E) the dual of D(E), i.e. the set of the continuous linear functionals on D(E). We denote the dual product of Γ ∈ D(E) and Ψ ∈ D(E) by Γ, Ψ . In such a bracket we sometimes exchange the place of the factor in D(E) with the place of the factor in D(E): indeed, the duality product follows the same algebraic rules of the standard scalar product. As usual, one can extend the action of H to the space D(E), with values in D(E) , by 1

1

HΨ1 , Ψ2 := (H 2 Ψ1 , H 2 Ψ2 ), where (·, ·) denotes the standard scalar product in L2 (G).

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Furthermore, for any Ψ ∈ D(E) the identity d −iHt e Ψ = −iHe−iHt Ψ dt holds in D(E) too. To prove it, one can ﬁrst test the functional element Ξ in the operator domain D(H), obtaining d −iHt eiH(t+h) Ξ − eiHt Ξ e Ψ, Ξ = lim Ψ, h→0 dt h

(3.1) d −iHt Ψ dt e

on an

= (Ψ, iHeiHt Ξ) = −iHe−iHt Ψ, Ξ . Then, the result can be extended to Ξ ∈ D(E) by a density argument. Besides, by (3.1), the diﬀerential version (1.1) of the Schr¨ odinger equation holds in D(E) . In order to prove a well-posedness result we need to generalize standard onedimensional Gagliardo–Nirenberg estimates to graphs, i.e. 1

−1

1

+1

ΨLp ≤ CΨ L2 2 p ΨL2 2 p ,

+∞ ≥ p ≥ 1,

(3.2)

where the C > 0 is a positive constant which depends on the index p only. The proof of (3.2) follows immediately from the analogous estimates for functions of the real line, considering that any function in H 1 (R+ ) can be extended to an even function in H 1 (R), and applying this reasoning to each component of Ψ. Proposition 3.1 (Local Well-Posedness in D(E)). For any Ψ0 ∈ D(E), there exists T > 0 such that Eq. (1.2) has a unique solution Ψ ∈ C 0 ([0, T ), D(E)) ∩ C 1 ([0, T ), D(E) ). Moreover, Eq. (1.2) has a maximal solution Ψmax deﬁned on an interval of the form [0, T ), and the following “blow-up alternative” holds: either T = ∞ or lim Ψmax D(E) = +∞, t

t→T

the function Ψmax evaluated at time t. where we denoted by Ψmax t Proof. We deﬁne the space X := L∞ ([0, T ), D(E)), endowed with the norm ΨX := supt∈[0,T ) Ψt D(E) . Given Ψ0 ∈ D(E), we deﬁne the map G : X → X as · e−iH(·−s) |Φs |2 Φs ds. GΦ := e−iH· Ψ0 + i 0

Notice that the nonlinearity preserves the space D(E). Indeed, since any component ψi of Ψ belongs to H 1 (R+ ), then |ψi |2 ψi belongs to L2 (R+ ) too, and so the energy space for the delta-prime case is preserved. Furthermore, the product preserves the continuity at zero required by the Kirchhoﬀ and the delta case.

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By estimates (3.2), one obtains |Φs |2 Φs D(E) ≤ CΦs 3D(E), so GΦX ≤ Ψ0 D(E) + C

T

0

Φs 3D(E) ds ≤ Ψ0 D(E) + CT Φ3X .

(3.3)

Analogously, given Φ, Ξ ∈ D(E), GΦ − GΞX ≤ CT (Φ2X + Ξ2X )Φ − ΞX .

(3.4)

We point out that the constant C appearing in (3.3) and (3.4) is independent of Ψ0 , Φ, and Ξ. Now let us restrict the map G to elements Φ such that ΦX ≤ 2Ψ0D(E) . From (3.3) and (3.4), if T is chosen to be strictly less than (8CΨ0 2D(E) )−1 , then G is a contraction of the ball in X of radius 2Ψ0 D(E) , and so, by the contraction lemma, there exists a unique solution to (1.2) in the time interval [0, T ). By a standard one-step boostrap argument one immediately has that the solution actually belongs to C 0 ([0, T ), D(E)), and due to the validity of (1.1) in the space D(E) we immediately have that the solution Ψ actually belongs to C 0 ([0, T ), D(E))) ∩ C 1 ([0, T ), D(E) ). The proof of the existence of a maximal solution is standard, while the blowup alternative is a consequence of the fact that, whenever the D(E)-norm of the solution is ﬁnite, it is possible to extend it for a further time by the same contraction argument. The next step consists in the proof of the conservation laws. Proposition 3.2. For any solution Ψ ∈ C 0 ([0, T ), D(E)) ∩ C 1 ([0, T ), D(E) ) to the problem (1.2), the following conservation laws hold at any time t: Ψt = Ψ0 ,

E(Ψt ) = E(Ψ0 ),

where the symbol E denotes the energy functional E(Ψt ) :=

1 1 Elin (Ψt ) − Ψt 4L4 . 2 4

Here the functional Elin coincides with EF , Eδα or Eδβ , according to the case one considers. Proof. The conservation of the L2 -norm can be immediately obtained by the validity of Eq. (1.1) in the space D(E): d d 2 Ψt = 2 Re Ψt , Ψt = 2 Im Ψt , HΨt = 0 dt dt

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by the self-adjointness of H. In order to prove the conservation of the energy, ﬁrst we notice that Ψt , HΨt is diﬀerentiable as a function of time. Indeed, 1 [ Ψt+h , HΨt+h − Ψt , HΨt ] h Ψt+h − Ψt Ψt+h − Ψt , HΨt+h + HΨt , = h h and then, passing to the limit h → 0, d d (Ψt , HΨt ) = 2 Re Ψt , HΨt = 2 Im |Ψt |2 Ψt , HΨt , dt dt

(3.5)

where we used the self-adjointness of H and (1.1). Furthermore, d d (Ψt , |Ψt |2 Ψt ) = (Ψ2t , Ψ2t ) = 4 Im |Ψt |2 Ψt , HΨt . dt dt From (3.5) and (3.6), one then obtains

(3.6)

d 1 d 1 d E(Ψt ) = Ψt , HΨt − (Ψt , |Ψt |2 Ψt )L2 = 0 dt 2 dt 4 dt and the proposition is proved. Corollary 3.3. The solutions are globally deﬁned in time. Proof. By estimate (3.2) with p = ∞ and conservation of the L2 -norm, there exists a constant M , that depends on Ψ0 only, such that 1 2 Ψ − M Ψt . 2 t Therefore a uniform (in t) bound on Ψt 2 is obtained. As a consequence, one has that no blow-up in ﬁnite time can occur, and therefore, by the blow-up alternative, the solution is global in time. E(Ψ0 ) = E(Ψt ) ≥

4. Main Result In this section, we describe the asymptotic dynamics of a particular initial state, which resembles a soliton for the standard cubic NLS on the line. According to Sec. 3, we use the symbol H to generically denote the linear part of the evolution, regardless of the fact that we are considering the Kirchhoﬀ, delta, or delta-prime boundary conditions. When necessary, we will distinguish between the three of them. We use the notation √ φ(x) = 2 cosh−1 x, x ∈ R, and for any x0 ∈ R and v ∈ R we deﬁne v

φx0 ,v (x, t) = ei 2 x e−it

v2 4

eit φ(x − x0 − vt),

x ∈ R,

t ∈ R.

(4.1)

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The function φx0 ,v represents a soliton for the cubic NLS on the line which at time t = 0 is centered in x = x0 and has velocity v. Therefore, φx0 ,v is the solution of the integral equation v

φx0 ,v (x, t) = [Ut ei 2 · φ(· − x0 )](x) t +i [Ut−s |φx0 ,v (·, s)|2 φx0 ,v (·, s)](x)ds, 0

t ∈ R.

(4.2)

Let χ be the cut oﬀ function χ ∈ C ∞ (R+ ), χ = 1 in (2, +∞) and χ = 0 in (0, 1). For later use we deﬁne also χ+ = χ[0,+∞) and χ− = χ(−∞,0] , where χ[a,b] denotes the characteristic function of the interval [a, b]. Moreover let x0 and v be two positive constants and 0 < δ < 1. We take as initial condition the following function   v χ(x)e−i 2 x φ(x − x0 ) , Ψ0 (x) =  0

x0 ≥ v 1−δ ,

(4.3)

0 and we denote by ΨH,t the solution of the equation ΨH,t = e

−iHt

Ψ0 + i

0

t

dse−iH(t−s) |ΨH,s |2 ΨH,s ,

t ≥ 0.

(4.4)

The choice of the vector Ψ0 is used to render the idea that the initial condition is a soliton centered away from the vertex and moving towards the vertex with velocity v. The cut oﬀ function χ in formula (4.3) is aimed at setting Ψ0 in the domain of the Hamiltonian H, see Sec. 2.1. Let us set t2 := x0 /v + v −δ and deﬁne the following functions: 

r˜H e−i

v2 4 t2

v

ei 2 x1 eit2 φ(x1 + x0 − vt2 )



  v2 v  Φ1H,t2 (x1 , x2 , x3 ) ≡  r˜H e−i 4 t2 e−i 2 x2 eit2 φ(x2 − x0 + vt2 ),

(4.5)

0 

0



2   ˜ −i v4 t2 ei v2 x2 eit2 φ(x2 + x0 − vt2 ) , Φ2H,t2 (x1 , x2 , x3 ) ≡    tH e v2 v t˜H e−i 4 t2 e−i 2 x3 eit2 φ(x3 − x0 + vt2 )   v2 v t˜H e−i 4 t2 e−i 2 x1 eit2 φ(x1 − x0 + vt2 )   . Φ3H,t2 (x1 , x2 , x3 ) ≡  0  

t˜H e

2 −i v4 t2

e

i v2 x3

(4.6)

(4.7)

eit2 φ(x3 + x0 − vt2 )

They represent solitons on the line multiplied by the scattering coeﬃcients of the linear dynamics r˜H and t˜H , that, in the particular regime we consider, are deﬁned

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as follows: r˜HF = −1/3, ˜ =− r˜Hδαv

r˜H β˜/v δ

t˜HF = 2/3

1 + 2iα ˜ , 3 + 2iα ˜

β˜ + 2i , = β˜ + 6i

˜ t˜Hδαv =

2 3 + 2iα ˜

˜ t˜H β/v =− δ

(4.8)

4i . β˜ + 6i

Remark 4.1. Notice that the coeﬃcients r˜H and t˜H can be obtained by the scattering coeﬃcients (2.17), (2.18), identifying k with v/2 and replacing α by αv ˜ and ˜ β by β/v. This is due to the fact that we implicitly considered a particle with mass equal to 1/2, therefore the momentum k is linked to the speed v by k = v/2. For any t > t2 we deﬁne the vectors ΦjH,t as the evolution of ΦjH,t2 with the nonlinear ﬂow generated by Hj , i.e. they are solutions of the equation t ΦjH,t = e−iHj (t−t2 ) ΦjH,t2 + i dse−iHj (t−s) |ΦjH,s |2 ΦjH,s , j = 1, 2, 3. (4.9) t2

Remark 4.2. The vectors ΦjH,t can be represented by   v2 (x ) e−i 4 t2 eit2 φref 1 t−t 2   , v2 Φ1H,t (x1 , x2 , x3 ) =  e−i 4 t2 eit2 φref (−x ) 2  t−t2 0   0  −i v2 t2 it2 tr  e 4 e φt−t2 (x2 ) , Φ2H,t (x1 , x2 , x3 ) =    2 −i v4 t2 it2 tr e φt−t2 (−x3 ) e  v2  e−i 4 t2 eit2 φtr t−t2 (−x1 )   Φ3H,t (x1 , x2 , x3 ) =  0 , e−i

v2 4 t2

(4.10)

(4.11)

eit2 φtr t−t2 (x3 )

and φtr where, for any t ≥ 0, the functions φref t t are the solutions to the following NLS on the line ∞ v (x) = r ˜ Ut (x − y)ei 2 y φ(y − v 1−δ )dy φref H t −∞

+i φtr t (x)

= t˜H

ds

0

∞

−∞

+i

t

0

∞

−∞

2 ref Ut−s (x − y)|φref s (y)| φs (y)dy,

(4.12)

v

Ut (x − y)ei 2 y φ(y − v 1−δ )dy

t

ds

∞

−∞

2 tr Ut−s (x − y)|φtr s (y)| φs (y)dy.

(4.13)

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Our main result is summarized in the following theorem: Theorem 4.3. Fixed α ˜ , β˜ > 0, let H be any of the self-adjoint operators HF , ˜ β/v ˜ , Hδ acting on L2 (G), where G is the three-edge star graph, and deﬁned by Hδαv (2.1)–(2.4). Call Ψt the unique, global solution to the Cauchy problem (4.4) with initial data (4.3). Then, there exist τ∗ > 0 and T∗ > 0 such that for 0 < T < T∗ one has Ψt − Φ1t − Φ2t − Φ3t L2x (G) ≤ Cv −

T∗ −T τ∗

(4.14)

for any time t in the interval x0 /v + v 1−δ < t < x0 /v + v 1−δ + T ln v. In (4.14), C is a positive constant independent of t and v, the functions Φjt are deﬁned by formulas (1.4), t˜ and r˜, given in (4.8), are the scattering coeﬃcients associated to H. The proof of the theorem will be broken into three steps or equivalently we break the time evolution of ΨH,t into three phases. Remark 4.4. A further consequence of Theorem 4.3, as in the case of [17], is the fact that fast solitons have reﬂection and transmission coeﬃcients which, up to negligible corrections, coincide with the corresponding coeﬃcients of the linear graph. For example, a deﬁnition of the transmission coeﬃcients along the branches j = 2, 3 could be given considering the ratio between the amount of mass on the j-edge and the total mass, in the limit t → ∞: Ψjt ≡ |tj (v)|, t→∞ Ψt lim

where Ψj denotes the restriction of the solution to the j-edge. In our case we do not have at our disposal the rigorous asymptotics for t → ∞; nevertheless, we can obtain a weaker result. We have the results of Theorem 1.1, which give, in the time interval t2 < t < t2 + T ln v, the estimate Ψjt = |t˜j | + O(v −σ ) Ψt for a certain σ > 0 and where t˜j is the scattering coeﬃcient of the linear Hamiltonian which describes the vertex. So, in the limit of fast solitons, i.e. v → ∞, one can assert that the ratio which deﬁnes the nonlinear scattering coeﬃcient converges to the corresponding linear scattering coeﬃcient. And analogously for the case of reﬂection coeﬃcient r(v), i.e. j = 1, one has (for t > t2 , as before) Ψ1t = |˜ r |. v→∞ Ψt lim

This is true for every coupling between the ones considered, i.e. Kirchhoﬀ, δ or δ .

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5. Proof of Theorem 4.3 In the proof, we drop the subscript H. When convenient, we specify the particular Hamiltonian operator we refer to. For any t ∈ R, we introduce the soliton  v2  v e−i 4 t e−i 2 x1 eit φ(x1 − x0 + vt)   v2 v  Φt (x1 , x2 , x3 ) =  (5.1)  e−i 4 t ei 2 x2 eit φ(x2 + x0 − vt) . 0 Then, the function Φt satisﬁes the equation t −iH1 t Φt = e Φ0 + i dse−iH1 (t−s) |Φs |2 Φs , 0

t ≥ 0.

(5.2)

5.1. Phase 1 We call “phase 1” the dynamics in the time interval (0, t1 ) with t1 = xv0 − v −δ . In this interval we approximate the solution by the soliton (5.1). The content of this subsection is the estimate of the error due to such an approximation, that is contained in Proposition 5.3. Before proving it, we need two lemmas. Lemma 5.1. Given 0 ≤ ta ≤ tb ≤ t1 , for the functions ∞ v K1 (t, x) := Ut−ta (x + y)e−i 2 y φ(y − x0 )dy 0

K2 (t, x) :=

t

+i

ds

∞

0

ta

v

Ut−s (x + y)e−i 2 y e−is

Ut−ta (x + y)e

t

+i

ds

∞

0

ta

eis φ3 (y − x0 + vs)dy, (5.3)

∞

0

v2 4

i v2 y

φ(y + x0 )dy v

Ut−s (x + y)ei 2 y e−is

v2 4

eis φ3 (y + x0 − vs)dy,

the following estimate holds: Ki Xta ,tb (R+ ) ≤ Ce−x0 +vtb ,

i = 1, 2,

2 + 6 6 + where Xta ,tb (R+ ) := L∞ [ta ,tb ] L (R ) ∩ L[ta ,tb ] L (R ).

Proof. Let us start with K1 . Adding and subtracting a contribution to negative values of y one can write ∞ v Ut−ta (x + y)e−i 2 y φ(y − x0 )dy K1 (t, x) = −∞

t

+i

ds ta

∞

−∞

v

Ut−s (x + y)e−i 2 y e−is

v2 4

eis φ3 (y − x0 + vs)dy

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−

0

v

−∞

−i

Ut−ta (x + y)e−i 2 y φ(y − x0 )dy

t

ds

−∞

ta v

= ei 2 x e −i

2 −it v4

ds ta

v

Ut−s (x + y)e−i 2 y e−is

eit φ(x + x0 − vt) −

t

0

0

−∞

v

0

v2 4

eis φ3 (y − x0 + vs)dy v

−∞

Ut−ta (x + y)e−i 2 y φ(y − x0 )dy

Ut−s (x + y)e−i 2 y e−is

v2 4

eis φ3 (y − x0 + vs)dy

(5.4)

where we used the integral equation (4.2). By a straightforward computation, the Xta ,tb (R+ )-norm of the ﬁrst term can be bounded by Ce−(x0 −vtb ) . To evaluate the size of the second term, let us write it as follows: ∞ v v − Ut−ta (x − y)ei 2 y φ(y + x0 )dy = [Ut−t ei 2 · φ(· + x0 )](x) a 0

v

= [Ut−ta χ+ ei 2 · φ(· + x0 )](x),

x > 0.

(5.5)

Using the one-dimensional homogeneous Strichartz’s estimates for Ut−ta , namely, the analogous of (2.21) for functions of the half line, we can estimate the Xta ,tb (R+ )norm of this term as v

v

Ut−ta χ+ ei 2 · φ(· + x0 )Xta ,tb (R) ≤ Cχ+ ei 2 · φ(· + x0 ) ≤ Ce−x0 , 2 6 6 where we used the notation Xta ,tb (R) := L∞ [ta ,tb ] L (R) ∩ L[ta ,tb ] L (R). The norm of the last term on the right-hand side of Eq. (5.4) can be estimated in a similar way by · [U·−s |φ−x0 ,v (s)|2 φ−x0 ,v (s)]ds ≤ Cφ3−x0 ,v L1[t ,t ] L2 (R+ ) a

Xta ,tb (R+ )

ta

b

1 ≤ C e−x0 +vtb . v

(5.6)

Therefore, from (5.4)–(5.6) we get K1 Xta ,tb (R+ ) ≤ Ce−x0 +vtb . To estimate K2 , the ﬁrst term in its deﬁnition (5.3) can be treated as in (5.5), while the second is estimated following the line of (5.6). Lemma 5.2. Given 0 ≤ ta ≤ tb ≤ t1 , let a and b two strictly positive numbers, with 1 . b≤ 2 8a + 4a Moreover, let y be a real, continuous function such that 0 ≤ y(ta ) ≤ a, and 0 ≤ y(t) ≤ a + by 2 (t) + by 3 (t),

for any t ∈ [ta , tb ].

(5.7)

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Then, max y(t) ≤ 2a.

t∈[ta ,tb ]

Proof. Consider the function fb (x) := bx3 + bx2 − x + a. Denoted ¯b := 8a21+4a , one has f¯b (2a) = 0. If b ≤ ¯b, then fb (x) ≤ f¯b (x) for any x > 0. Besides, notice that ˜ ∈ (0, 2a] s.t. fb (˜ x) = 0. Finally, since fb (0) = a > 0, then there must be a point x the function y is continuous, in order to satisfy the constraint (5.7) one must have y(t) ≤ x ˜ ≤ 2a,

for any t ∈ [ta , tb ].

Proposition 5.3. Let Ψt be the solution of Eq. (4.4), and Φt be the solution of Eq. (5.2). There exists C > 0, independent of t and v, such that Ψt − Φt ≤ Ce−v

1−δ

(5.8)

for any t ∈ [0, t1 ]. Proof. Let us deﬁne Ξt := Ψt − Φt , and ﬁx ta ∈ [0, t1 ]. Then, from Eqs. (4.4) and (5.2), we have Ξt = e−iH(t−ta ) Ξt−a + (e−iH(t−ta ) − e−iH1 (t−ta ) )Φt−a t t +i (e−iH(t−s) − e−iH1 (t−s) )|Φs |2 Φs ds + i e−iH(t−s) [|Ξs |2 Ξs + |Ξs |2 Φs ta

ta

2

+ |Φs | Ξs + 2 Re(Ξs Φs )Ξs + 2 Re(Ξs Φs )Φs ] = e−iH(t−ta ) Ξt−a + F (ta , t) t +i e−iH(t−s) [|Ξs |2 Ξs + |Ξs |2 Φs + |Φs |2 Ξs + 2 Re(Ξs Φs )Ξs ta

+ 2 Re(Ξs Φs )Φs ], where we deﬁned F (ta , t) := (e−iH(t−ta ) − e−iH1 (t−ta ) )Φta t +i (e−iH(t−s) − e−iH1 (t−s) )|Φs |2 Φs ds. ta 2 6 6 Let us ﬁx tb ∈ [ta , t1 ], and denote Xta ,tb = L∞ [ta ,tb ] L ∩ L[ta ,tb ] L . Then

ΞXta ,tb ≤ e−iH(·−ta ) Ξta Xta ,tb + F (ta , ·)Xta ,tb · + e−iH(·−s) [|Ξs |2 Ξs + |Ξs |2 Φs + |Φs |2 Ξs ta

+ 2 Re(Ξs Φs )Ξs + 2 Re(Ξs Φs )Φs ]

Xta ,tb

.

(5.9)

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Using (2.21) the ﬁrst term on the right-hand side can be estimated as e−iH(·−ta ) Ξta Xta ,tb ≤ CΞta . We estimate the integral term on the right-hand side of (5.9) also using Strichartz’s estimates. Let us analyze in detail the cubic term. Since both pairs of indices (∞, 2) and (6, 6) fulﬁll (2.23), in (2.22) we can choose q = k = 6/5 and obtain · −iH(·−s) 2 e |Ξ | Ξ ds ≤ C|Ξ· |2 Ξ· L6/5 L6/5 . (5.10) s s [ta ,tb ] ta

Xta ,tb

Moreover, by standard H¨ older estimates, |Ξ· |2 Ξ· L6/5

[ta ,tb ]

L6/5

≤ Ξ· 2L6 Ξ· L2 L6/5

[ta ,tb ]

≤ Ξ2L6

[ta ,tb ]

L6 ΞL2[ta ,t ] L2 b

1 2

≤ (tb − ta ) Ξ2L6

L6 ΞL[ta ,tb ] L2 .

(5.11)

≤ C(tb − ta )1/2 Ξ3Xta ,t .

(5.12)

[ta ,tb ]

Then, by (5.10) and (5.11), · −iH(·−s) 2 e |Ξs | Ξs , ds ta

Xta ,tb

∞

b

Notice that the constant C can be chosen independently of ta , tb , and of the boundary condition at the vertex. The other terms in the integral on the right-hand side of (5.9) can be estimated analogously. One ends up with 1

ΞXta ,tb ≤ CΞta + F (ta , ·)Xta ,tb + C(tb − ta ) 2 Ξ3Xta ,t

b

+ C(tb − ta )

2 3

Ξ2Xta ,t b

5 6

+ C(tb − ta ) ΞXta ,tb ,

where the arising norms of Φ were absorbed in the constant C. 5 If tb and ta are suﬃciently close, then C(tb − ta ) 6 < 1/2. Furthermore, since 2 1 the quantity tb − ta is upper bounded, one can estimate (tb − ta ) 3 by C(tb − ta ) 2 . >0 So, for some C 2 ta + CF (ta , ·)Xt ,t + C(t b − ta ) 2 [Ξ3 ΞXta ,tb ≤ CΞ Xta ,t + ΞXta ,t ]. a b 1

b

b

(5.13) Applying Lemma 5.2 to the function y(t) = ΞXta ,t , which is continuous and monotone, one has that, if 3 (Ξta + F (ta , ·)Xt ,t )2 + 4C 2 (Ξta + F (ta , ·)Xt ,t ))−2 , tb − ta ≤ (8C a b a b

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ta + 2CF (ta , ·)Xt ,t . From the immediate estimates then ΞXta ,tb ≤ 2CΞ a b Ξta ≤ 4,

F (ta , ·)Xta ,tb ≤ F (0, ·)X0,t1 ,

if one denotes 3 (4 + F (ta , ·)X0,t )2 + 4C 2 (4 + F (ta , ·)X0,t ))−2 , τ := (8C 1 1 t + 2CF (t, ·)Xt,t+τ . then for any t ∈ [0, t1 ) ΞXt,t+τ ≤ 2CΞ We divide the interval [0, t1 ] in N + 1 subintervals as follows   N −1 [0, t1 ] =  [jτ, (j + 1)τ ] ∪ [N τ, t1 ], j=0

where

N :=

x0 − v 1−δ , vτ

[·] = integer part.

Making use of Lemma 5.2, and noting that Ξ(j+1)τ ≤ ΞXjτ,(j+1)τ , one proves by induction that j+1 Ξ0 + ΞXjτ,(j+1)τ ≤ (2C)

j

j+1−k F (kτ, ·)X (2C) , kτ,(k+1)τ

k=0

j = 0, · · · , N − 1, N +1 Ξ0 + ΞXN τ,t1 ≤ (2C)

N −1

N +1−k F (kτ, ·)X (2C) kτ,(k+1)τ

k=0

(N τ, ·)XN τ,t , + 2CF 1

(5.14)

where the last inequality comes from the fact that t1 − N τ ≤ τ , so Lemma 5.2 applies to this last step too. The norm of Ξ as a function of the whole time interval [0, t1 ] can be estimated by ΞX0,t1 ≤

N −1

ΞXjτ,(j+1)τ + ΞXN τ,t1

j=0

≤

N

j+1 Ξ0 + (2C)

j=0

j N

j+1−k F (kτ, ·)X (2C) . (5.15) kτ,min{t1 ,(k+1)τ }

j=0 k=0

In order to prove the theorem using (5.15), we need more precise estimates for Ξ0 and F (jτ, ·)Xjτ,(j+1)τ . First, 2 ∞ φ2 (x − x0 )dx + φ2 (x + x0 ) Ξ0 2 ≤ 0

0

= 2(1 − tanh(x0 − 2)) ≤ Ce−2x0 .

(5.16)

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To estimate F (ta , ·)Xta ,tb we specialize F to the three cases under analysis. From the explicit propagators (2.8)–(2.10), we get     t −1 −1 2 −1 −1 2 1 +  i U + −1 −1 2 |Φs |2 Φs ds, FF (ta , t) = Ut−ta −1 −1 2 Φt−a + 3 3 ta t−s 2 2 2 2 2 2 +∞ α ˜ 2 + ˜ ˜v (ta , t) = FF (ta , t) − α due− 3 uv (Ut−t JΦt−a )(· + u) Fδαv a 9 0 t +∞ α ˜ 2 + −i α ˜v ds due− 3 uv (Ut−s J|Φs |2 Φs )(· + u), 9 ta 0     t 1 −1 0 1 −1 0 ˜ β/v + +  −1 Fδ (ta , t) = Ut−t Ut−s 1 0 Φt−a + i −1 1 0 |Φs |2 Φs ds a ta 0 0 2 0 0 2 2v +∞ − 3 vu + due β˜ (Ut−t JΦt−a )(· + u) − a β˜ 0 2v −i β˜

t

ds ta

+∞

0

due

3 −β ˜ vu

+ (Ut−s J|Φs |2 Φs )(· + u).

It is immediately seen that  FF (ta , t, x) =

−K1 (x, t) − K2 (x, t)



 1  −K1 (x, t) − K2 (x, t) ,   3 2K1 (x, t) + 2K2 (x, t),

where K1 and K2 were deﬁned in (5.3). Lemma 5.1 yields FF (ta , ·)Xta ,tb ≤ Ce−x0 +vtb .

(5.17)

Furthermore, since ˜ (ta , t, x) Fδαv

+∞ α ˜ 2 ˜v = FF (ta , t, x) − α due− 3 uv 9 0   K1 (x + u, t) + K2 (x + u, t)   × K1 (x + u, t) + K2 (x + u, t), K1 (x + u, t) + K2 (x + u, t)

after the change of variable u → uv we conclude ˜ (ta , ·)Xta ,tb ≤ Ce−x0 +vtb . Fδαv

(5.18)

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Finally, 

 K1 (x, t) − K2 (x, t) ˜ β/v Fδ (ta , t, x) = −K1 (x, t) + K2 (x, t) 0   K1 (x + u, t) + K2 (x + u, t) 3 2v +∞ − uv due β˜ K1 (x + u, t) + K2 (x + u, t) − β˜ 0 K1 (x + u, t) + K2 (x + u, t) yields, after the change of variable u → uv, ˜ β/v

Fδ (ta , ·)Xta ,tb ≤ Ce−x0 +vtb .

(5.19)

Now we go back to estimate (5.15). Due to (5.16)–(5.19), and estimating any geometric sum by the double of its largest term, which is justiﬁed if the rate of the sum is not less than two, we get ΞX0,t1

−x0 ≤ 2C Ce

N

j + 2C Ce −x0 +vτ (2C)

j=0

N +1 + Ce−x0 +vτ (2C)

N −1 vτ

e

k=0

k

2C

N +1 e−x0 + 4C Ce −x0 +vτ ≤ 2C(2C)

N −1

j vτ k e

j=0

k=0

j (2C)

2C

−x0 +vt1 + 2C Ce

N −1

ejvτ

j=0

2 e−x0 +N vτ + 2C Ce −x0 +vt1 + 8C C N +1 e−x0 + 8C Ce −x0 +N vτ + 8C C 2 e−x0 +N vτ ≤ 2C(2C) −x0 +vt1 . + 2C Ce

(5.20)

Concerning the ﬁrst term on the right-hand side of (5.20), we have x0 −v1−δ 1 vτ 1−δ 1−δ 2C  ≤ e−v ≤ Ce−v . e 

N e−x0 (2C)

From (5.20) and (5.21), we get ΞX0,t1 ≤ Ce−v is concluded.

1−δ

(5.21)

, so (5.8) follows and the proof

5.2. Phase 2 We call “phase 2” the evolution of the system in the time interval (t1 , t2 ) with t2 = xv0 + v −δ . Let us deﬁne the vector ΦSt := ΦS,in + ΦS,out t t

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Fig. 2. We represent the state ΦS t for t = t1 and t = t2 . The continuous lines stand for the real graph, the dashed lines represent the extension of each edge to negative values of x. At time t = t1 only the vector ΦS,in is signiﬁcantly supported on the real graph. At time t = t2 only the state ΦS,out has relevant support on the real graph, while the body of the soliton in ΦS,in has moved to the negative axis associated with the ﬁrst edge.

with





ΦS,in t

 φx0 ,−v (t) , :=  0 0

ΦS,out t

 r˜φ−x0 ,v (t)   :=  t˜φ−x0 ,v (t) , t˜φ−x0 ,v (t)

where the function φx0 ,v was deﬁned in Eq. (4.1) and the reﬂection and transmission coeﬃcients, r˜ and t˜, must be chosen accordingly to the Hamiltonian H taken in ˜ β/v ˜ , Hδ Eq. (4.4). The explicit expressions of r˜ and t˜ in all the cases H = HF , Hδαv can be read in formula (4.8). Proposition 5.4. Let t ∈ (t1 , t2 ) then there exists v0 > 0 such that for all v > v0 δ

Ψt − ΦSt ≤ C1 v − 2 ,

(5.22)

moreover δ

Ψt2 − ΦS,out ≤ C2 v − 2 , t2

(5.23)

where C1 and C2 are positive constants which do not depend on t and v. Proof. From the deﬁnition of Ψt , see Eq. (4.4), we have t dse−iH(t−s) |Ψs |2 Ψs . Ψt = e−iH(t−t1 ) Ψt1 + i t1

We start with the trivial estimate Ψt − ΦSt ≤ Ψt − e−iH(t−t1 ) Ψt1 + e−iH(t−t1 ) Ψt1 − e−iH(t−t1 ) ΦS,in t1 S + e−iH(t−t1 ) ΦS,in t1 − Φt

(5.24)

and estimate the right-hand side term by term. The estimates involved in the analysis of the ﬁrst term are similar to the ones used in the previous proposition,

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thus we omit the details. Similarly to what was done above we set Xt1 ,t2 = 2 6 6 L∞ [t1 ,t2 ] L ∩ L[t1 ,t2 ] L , then by Strichartz estimates (see Eq. (5.12) and Proposition 2.3) · −iH(·−t1 ) −iH(·−s) 2 Ψt1 Xt1 ,t2 ≤ dse |Ψs | Ψs Ψ − e t1

Xt1 ,t2

1/2

≤ C(t2 − t1 )

(5.25)

Ψ3Xt1 ,t2 ,

e−iH(·−t1 ) Ψt1 Xt1 ,t2 ≤ CΨt1 , which imply ΨXt1 ,t2 ≤ CΨt1 + C(t2 − t1 )1/2 Ψ3Xt1 ,t2 . By Lemma 5.2, one has that if (t2 − t1 ) ≤ [8C 3 Ψt1 2 + 4C 2 Ψt1 ]−2 , then ΨXt1 ,t2 ≤ 2CΨt1 ; using this estimate in the inequality (5.25) we get Ψt − e−iH(t−t1 ) Ψt1 ≤ Ψ − e−iH(·−t1 ) Ψt1 Xt1 ,t2 ≤ C(t2 − t1 )1/2 Ψt1 3 ≤ Cv −δ/2

(5.26)

where we used t2 − t1 = 2v −δ . We proceed now with the estimate of the second term on the right-hand side of inequality (5.24). Let us set   0 φ−x0 ,v (t1 ). Φtail t1 := 0 tail We notice that ΦS,in t1 + Φt1 = Φt1 where the vector Φt was deﬁned in Eq. (5.1) and rewrite Ψt1 by adding and subtracting Φt1 tail Ψt1 = Ψt1 − Φt1 + ΦS,in t1 + Φt1 .

The following trivial inequality holds true tail −v e−iH(t−t1 ) Ψt1 − e−iH(t−t1 ) ΦS,in t1 ≤ Ψt1 − Φt1 + Φt1 ≤ Ce

1−δ

(5.27)

where in the latter estimate we used Proposition 5.3 and the fact that Φtail t1 ≤ 1−δ 2e−v . Let us consider now the last term on the right-hand side of inequality (5.24). We are going to prove that for all t ∈ (t1 , t2 ) and for v big enough S −δ . e−iH(t−t1 ) ΦS,in t1 − Φt ≤ Cv

Let us introduce the functions (x) := φ− t

0

∞

Ut−t1 (x − y)φx0 ,−v (y, t1 )dy

(5.28)

(5.29)

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and φ+ t (x)

∞

:=

Ut−t1 (x + y)φx0 ,−v (y, t1 )dy.

0

(5.30)

(see Eqs. (5.35) First we prove a preliminary formula for the vector e−iH(t−t1 ) ΦS,in t1 and (5.36) below). For any constant a > 0, not dependent on v let us consider the term ∞ + e−uva [Ut−t φ (t1 )](u + x)du va 1 x0 ,−v 0

= va

∞

0

du

∞

0

v

dye−uva Ut−t1 (u + x + y)eiϕ(t1 ) e−i 2 y φ(y − x0 + vt1 )

2

where we set ϕ(t) := −t v4 + t. By integrating by parts we obtain the equality va

∞

0

+ e−uva [Ut−t φ (t1 )](u + x)du 1 x0 ,−v

= 2ia 0

∞

du

∞

0

dye

−uva

Ut−t1 (u + x + y)e

iϕ(t1 )

d −i v y 2 e φ(y − x0 + vt1 ) dy

= A1 (x, t) + A2 (x, t) + A3 (x, t)

(5.31)

with A1 (x, t) := −2ia

∞

0

A2 (x, t) := −2ia

due−uva Ut−t1 (u + x)eiϕ(t1 ) φ(−x0 + vt1 ),

∞

du

0

0

∞

dye

−uva

d Ut−t1 (u + x + y) dy

v

× eiϕ(t1 ) e−i 2 y φ(y − x0 + vt1 ), ∞ ∞ du dye−uva Ut−t1 (u + x + y)eiϕ(t1 ) A3 (x, t) := −2ia 0

v

× e−i 2 y

0

d φ(y − x0 + vt1 ) . dy

We notice that ∞ −uva due Ut−t1 (u + ·) 0

L2 (R+ )

≤

0

∞

due

−uva

Ut−t1 (u + ·)

= χ+ e−va· L2 (R) 1 . = 2va

L2 (R)

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Then the following estimate for the term A1 holds true ∞ −uva due Ut−t1 (u + ·) A1 (t)L2 (R+ ) ≤ 2a 2 0

L (R+ )

439

|φ(−x0 + vt1 )|

1−δ

e−v . (5.32) v 1/2 The term A3 is estimated by ∞ v A3 (t)L2 (R+ ) ≤ 2a due−uva [Ut−t1 χ+ e−i 2 · φ (· − x0 + vt1 )](−(u + ·))L2 (R+ ) ≤C

0

≤ 2φ L2 (R) a

∞

0

due−uva

C , v where we used the equality [Ut+ f ](x) = [Ut χ+ f ](−x). We compute ﬁnally the term A2 . By integration by parts ∞ ∞ d A2 (x, t) = −2ia Ut−t1 (u + x + y) du dye−uva du 0 0 ≤

(5.33)

v

× eiϕ(t1 ) e−i 2 y φ(y − x0 + vt1 ) ∞ + + 2 = 2iaφt (x) − 2iva e−uva [Ut−t φ (t1 )](u + x)du, 1 x0 ,−v 0

where the function we get

φ+ t

was deﬁned in Eq. (5.30). Using the last equality in Eq. (5.31)

va

∞

0

=

+ e−uva [Ut−t φ (t1 )](u + x)du 1 x0 ,−v

A1 (x, t) + A3 (x, t) 2ia φ+ . t (x) + 1 + 2ia 1 + 2ia

(5.34)

From the deﬁnition of ΦS,in and using the last equality with a = α ˜ /3 in the formula t αv ˜ for the integral kernel of e−iHδ t , see Eq. (2.9), it follows that   φx0 ,−v (t1 ) αv ˜ αv ˜  e−iHδ (t−t1 ) ΦS,in = e−iHδ (t−t1 )  0 t1 0 

φ−  t

  =   

 1 + 2iα ˜ + φ −   3 + 2iα ˜ t  Aα˜ (t)  2 2   φ+   − 3 Aα˜ (t) , 3 + 2iα ˜ t  Aα˜ (t)  2 + φ 3 + 2iα ˜ t

(5.35)

where the function φ− ˜ (x, t) := [(A1 (x, t) + t was deﬁned in Eq. (5.29) and we set Aα . Similarly using equality (5.34) with a = 3/β˜ in the A3 (x, t))/(1 + 2ia)]|a=α/3 ˜

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formula for the integral kernel of the propagator e−iHδ t , see Eq. (2.10), we get   β˜ + 2i + − φ + φ  t t    β˜ + 6i   (t) A ˜   β ˜ β/v 4i   2  = − (5.36) e−iHδ (t−t1 ) ΦS,in φ+  − Aβ˜ (t) t1 t ˜   3 β + 6i   Aβ˜ (t)   4i φ+ − t ˜ β + 6i where we introduced the notation Aβ˜ (x, t) := [(A1 (x, t) + A3 (x, t))/(1 + 2ia)]|a=3/β˜.

can be obtained from We notice that the analogous formula for e−iHF (t−t1 ) ΦS,in t1 Eq. (5.35) by setting α ˜ = 0 and Aα˜ = 0. To get the estimate (5.28) we show that, at the cost of an error of the order + (t2 − t1 ), for t ∈ (t1 , t2 ), the functions φ− t (x) and φt (x) can be approximated by the solitons φx0 ,−v (x, t) and φ−x0 ,v (x, t) respectively. We consider ﬁrst the function φ+ t , by adding and subtracting a suitable term to the right-hand side of Eq. (5.30) we get ∞ (x) = Ut−t1 (x + y)φx0 ,−v (y, t1 ) φ+ t −∞

t

+i

ds

−

−∞

t1

∞

Ut−s (x + y)|φx0 ,−v (y, s)|2 φx0 ,−v (y, s)dy

0

−∞

−i

Ut−t1 (x + y)φx0 ,−v (y, t1 )dy

t

ds

∞

−∞

t1

Ut−s (x + y)|φx0 ,−v (y, s)|2 φx0 ,−v (y, s)dy

= φ−x0 ,v (x, t) + I(x, t) + II(x, t),

(5.37)

where we used the fact that φx0 ,−v (−x, t) = φ−x0 ,v (x, t) and we set 0 Ut−t1 (x + y)φx0 ,−v (y, t1 )dy I(x, t) := − −∞

and

II (x, t) := −i

t

ds t1

∞

−∞

Ut−s (x + y)|φx0 ,−v (y, s)|2 φx0 ,−v (y, s)dy.

For the term I, we use the estimate IL2 (R+ ) ≤ φ(· − x0 + vt1 )L2 (R− ) ≤ 2e−v The term II is estimated by II L2 (R+ ) ≤ (t − t1 )φ3 L2 (R) ≤ Cv −δ .

1−δ

.

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Similarly, for the function φ− t2 , we get φ− t (x) = φx0 ,−v (x, t) + III (x, t) + IV (x, t) where we set

III (x, t) := −

and

IV (x, t) := −i

t

ds t1

0

−∞

∞

−∞

(5.38)

Ut−t1 (x − y)φx0 ,−v (t1 , y)dy

Ut−s (x − y)|φx0 ,−v (s, y)|2 φx0 ,−v (s, y)dy.

For t ∈ (t1 , t2 ) the estimates III L2 (R+ ) ≤ 2e−v

1−δ

;

IV L2 (R+ ) ≤ Cv −δ

are similar to the ones given above for the terms I and II . Then from Eqs. (5.37) and (5.38) we have −v φ+ t − φ−x0 ,v (t)L2 (R+ ) ≤ C[e

1−δ

+ v −δ ],

−v φ− t − φx0 ,−v (t)L2 (R+ ) ≤ C[e

1−δ

+ v −δ ]

for all t ∈ (t1 , t2 ). By using the last estimates and the estimates (5.32) and (5.33) in Eqs. (5.35) and (5.36) we get 1−δ e−v 1 −iH(t−t1 ) S,in S −v 1−δ −δ Φt1 − Φt ≤ C e + v + 1/2 + e v v which in turn implies that for v big enough the estimate (5.28) holds true. Using estimates (5.26)–(5.28) in the inequality (5.24) we get the estimate (5.22). The estimate (5.23) is a consequence of estimate (5.22) and of the fact that −v 1−δ = ΦS,in . ΦSt2 − ΦS,out t2 t2 ≤ Ce Remark 5.5. Notice that estimate (5.22), although not strictly necessary for the proof of Theorem 4.3, enforces the picture that in the phase 2 a scattering event is occurring. The true wavefunction can be approximated by the superposition of an incoming and an outgoing wavefunction. At the end of this phase, only the outgoing wavefunction is not negligibile. 5.3. Phase 3 Let us put t3 = t2 +T ln v. We call “phase 3” the evolution of the system in the time interval (t2 , t3 ). The approximation of Ψt during this time interval is the content of Theorem 4.3. We recall the following result (see [17]). ref be deﬁned as it was done in Eqs. (4.12) and Proposition 5.6. Let φtr t and φt (4.13) above. Then ∀ k ∈ N there exist two constants c(k) > 0 and σ(k) > 0 such

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that φγt L2 (R− ) + φγt L∞ (R− ) ≤

c(k)(ln v)σ(k) v k(1−δ)

(5.39)

for γ = {ref, tr}, uniformly in t ∈ [0, T ln v]. Notice that also the norms φγt Lp (R− ) , for 2 ≤ p ≤ ∞, are estimated by the right-hand side of (5.39). Now we can prove Theorem 4.3 Proof of Theorem 4.3. The strategy of the proof closely follows Proposition 5.3. We will just sketch the common part of the proof while proving in details the diﬀerent estimates. Let us deﬁne Ξt := Ψt − 3j=1 Φjt where the vectors Φjt were given in Eq. (4.9) and ﬁx ta ∈ [t2 , t3 ]. From Eqs. (4.4) and (4.9) it follows that the vector Ξt satisﬁes the following integral equation Ξt = e−iH(t−ta ) Ξta +

3

(e−iH(t−ta ) − e−iHj (t−ta ) )Φjta

j=1

t

+i

ds(e

−iH(t−s)

−e

−iHj (t−s)

ta t

+i

dse−iH(t−s)

ta

)|Φjs |2 Φjs

(1 − δj1 j2 δj2 j3 )Φjs1 Φjs2 Φjs3

j1 ,j2 ,j3

 2 3 3 −iH(t−s)  j 2 +i dse Φs Ξs + |Ξs | Φjs Ξs + ta j=1 j=1

t

 + 2 ReΞs

3

 Φjs 

j=1

 3

 Φjs 

j=1

= e−iH(t−ta ) Ξta + G(ta , t)  2 t 3 3  +i dse−iH(t−s) Ξs + Φjs Ξs + |Ξs |2 Φjs ta j=1 j=1  + 2 ReΞs

3 j=1

 Φjs 

 3

 Φjs  .

(5.40)

j=1

2 6 6 Let us ﬁx tb ∈ [ta , t3 ] and let Xta ,tb = L∞ [ta ,tb ] L ∩L[ta ,tb ] L . Using the Strichartz estimates as it was done in Proposition 5.3 (see Eqs. (5.9)–(5.13)), it is straightforward

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to prove that 3 ta + G(ta , ·)Xt ,t + (tb − ta )1/2 (Ξ2 ΞXta ,tb ≤ C[Ξ Xta ,t + ΞXta ,t )] a b b

b

depends only on the constants appearing in the Strichartz estimates. Using where C Lemma 5.2 as it was done in the proof of Proposition 5.3 it follows that there exists t + 2CG(t, ·)Xt,t+τ . τ > 0 such that, for any t ∈ [t2 , t3 ) one has ΞXt,t+τ ≤ 2CΞ We divide the interval [t2 , t3 ] in N + 1 subintervals: [t2 + jτ, t2 + (j + 1)τ ), with j = 0, . . . , N − 1; and [t2 + N τ, t3 ), and where N is the integer part of (t3 − t2 )/τ . Then proceeding by induction as we did in the proof of Proposition 5.3, see Eqs. (5.14) and (5.15), we get the inequality ΞXt2 ,t3 ≤

N

j+1 Ξt2 (2C)

j=0

+

j N

j+1−k G(t2 + kτ, ·)X (2C) . t2 +kτ,min{t3 ,t2 +(k+1)τ }

(5.41)

j=0 k=0

Now we estimate the initial data Ξt2 and the source term G(ta , ·)Xta ,tb with t2 ≤ ta ≤ tb ≤ t3 and tb − ta ≤ τ . By Proposition 5.4 (estimate (5.33)) and using the deﬁnitions (4.5)–(4.7) one has   t˜φx0 ,−v (t2 ) 3 j   Ξt2 = Ψt2 − Φt2 = Ψt2 − ΦS,out − r˜φx0 ,−v (t2 ), t2 j=1 t˜φx0 ,−v (t2 ) with Ψt2 − ΦS,out ≤ Cv −δ/2 . Since t2 ∞ ∞ 2 2 |φx0 ,−v (x, t2 )| dx = |φ(x − x0 + vt2 )| dx = 2 0

∞

v 1−δ

0

sech(x)2 dx

1−δ

=

4e−2v −2v 1−δ 1−δ ≤ 4e −2v 1+e

we have δ

Ξt2 ≤ C(v − 2 + e−2v

1−δ

δ

) ≤ Cv − 2 .

(5.42)

Let us now consider the source term G(ta , t). We use the estimate G(ta , ·)Xta ,tb ≤ G(t2 , ·)Xt2 ,t3 . To simplify the notation we set G(t) ≡ G(t2 , t) and 3 G1 (t) := (e−iH(t−t2 ) − e−iHj (t−t2 ) )Φjt2 j=1

t

+i

ds(e

−iH(t−s)

−e

t2 t

G2 (t) := i t2

dse−iH(t−s)

j1 ,j2 ,j3

−iHj (t−s)

)|Φjs |2 Φjs

,

(1 − δj1 j2 δj2 j3 )Φjs1 Φjs2 Φjs3 .

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By the deﬁnition of G(ta , t), see Eq. (5.40) it follows that G(t) = G1 (t) + G2 (t); we estimate G1 (t) and G2 (t) separately. We proceed ﬁrst with the estimate of the term G1 . From Eqs. (2.8)–(2.10) and (2.19) one can see that for any (column) vector F = (F1 , F2 , F3 ) ∈ L2  $ ∞ −uva +   +  va 0 e [Ut F1 ](u + ·)du Ut F1    2  $ ∞ −uva + [e−iHt − e−iHj t ]F = Mj Ut+ F2  − J  va 0 e [Ut F2 ](u + ·)du  , 3 $ ∞ −uva + + Ut F3 va 0 e [Ut F3 ](u + ·)du j = 1, 2, 3, where the constant a and the matrices Mj must be chosen accordingly to the Hamiltonian H: ˜ for H = Hδαv ,

α ˜ , 3

a=

2 Mj = −I + J − Tj ; 3

˜ β/v

for H = Hδ , a=

3 , β˜

Mj = I − Tj ;

and the formula for H = HF can be obtained by setting α ˜ = 0 in the formula for H = Hδα . Then, denoting by (G1 (x, t))l , l = 1, 2, 3, the lth component of the vector G1 one has 1 (x, t))l + (G %1 (x, t))l , (G1 (x, t))l = (G with 1 (x, t))l := (G

3

(Mj )lk

+ (Φjt2 )k ](x) [Ut−t 2

+i

j,k=1

%1 (x, t))l := − (G

t

t2

+ ds[Ut−s |(Φjs )k |2 (Φjs )k ](x)

,

3 2va ∞ −vua + e (Φjt2 )k ](u + x) [Ut−t 2 3 0 j,k=1

t

+i t2

+ ds[Ut−s |(Φjs )k |2 (Φjs )k ](u

+ x) du.

From the deﬁnition of the vectors Φjt , see Eqs. (4.10) and (4.11), we see that for 1 (x, t))l is a linear combination of four functions, each l = 1, 2, 3 the function (G

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ftγ,+ and ftγ,− , with γ being equal to ref and tr, given by ∞ 2 γ,+ −i v4 t2 it2 ft (x) := e e Ut−t2 (x + y)φγ0 (y)dy

ds

+i and

−i v4 t2 it2

e

ds

+i

y)|φγs−t2 (y)|2 φγs−t2 (y)dy

(5.43)

Ut−t2 (x + y)φγ0 (−y)dy

∞

0

t2

∞

0

t

Ut−s (x +

2

:= e

∞ 0

t2

ftγ,− (x)

0

t

Ut−s (x +

y)|φγs−t2 (−y)|2 φγs−t2 (−y)dy

,

and φref were deﬁned in Eqs. (4.12) and (4.13) respectively. where the functions φref t t %1 (x, t))l is a linear Similarly one can see that for each l = 1, 2, 3 the function (G combination of ∞ ∞ −vua ref ,+ va e ft (u + x)du, va e−vua fttr,+ (u + x)du, va

0

0

∞ 0

e−vua ftref ,− (u + x)du,

va

∞

0

e−vua fttr,− (u + x)du.

ftγ,+ .

We notice that, adding and subtracting a suitable First we study the function term in Eq. (5.43) and using the deﬁnitions (4.12) and (4.13), ftγ,+(x) can be written as ftγ,+(x) = I(x, t) + II (x, t) + III (x, t), with I(x, t) := e−i

v2 4 t2

eit2 φγt−t2 (−x), ∞ 2 −i v4 t2 it2 II (x, t) := −e e Ut−t2 (x − y)φα 0 (−y)dy, 0

III (x, t) := −ie

2 −i v4 t2

e

it2

t

ds t2

0

∞

Ut−s (x − y)|φγs−t2 (−y)|2 φγs−t2 (−y)dy.

2 ± Similarly to what was done above, we set Xt2 ,t3 (R± ) = L∞ [t2 ,t3 ] L (R ) ∩ 6 6 ± L[t2 ,t3 ] L (R ). By Proposition 5.6, we have

IXt2 ,t3 (R+ ) = φγ·−t2 Xt2 ,t3 (R− ) ≤

c (k)(ln v)σ (k) v k(1−δ)

(5.44)

where c (k) and σ (k) are constants, diﬀerent from the one appearing in Proposition 5.6. For our purposes we do not need to compute them.

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Using the one-dimensional Strichartz estimates for Ut , we have II Xt2 ,t3 (R+ ) ≤ Cχ− φγ0 L2 (R) ≤ Ce−2v

1−δ

.

(5.45)

Finally, the term III can be estimated using the inhomogeneous Strichartz estimate and Proposition 5.6 3 c(k)(ln v)σ(k) γ 3 III Xt2 ,t3 (R+ ) ≤ C(χ− φ·−t2 ) L1[t ,t ] L2 (R) ≤ C . (5.46) 2 3 v k(1−δ) Collecting the estimates (5.44)–(5.46), it follows that σ(k) c(k)(ln v) f·γ,+Xt2 ,t3 (R+ ) ≤ C . v k(1−δ) The estimate of ftγ,− is similar and we omit it. We have proved that for some c (k) and σ (k) possibly bigger than c(k) and σ(k) we have: σ (k) (k)(ln v) c 1 Xt ,t ≤ C G , 2 3 v k(1−δ) 1 (x, t))l , l = 1, 2, 3. 1 is the vector in L2 with components (G where G % 1 (t) = ((G %1 (t))1 , (G % 1 (t))2 , (G %1 (t))3 ) is a trivial consequence The estimate of G of the fact that ∞ −vua γ,± va e f (u + ·)du ≤ f·γ,±Xt2 ,t3 (R+ ) · + 0

Xt2 ,t3 (R )

c(k)(ln v)σ(k) ≤C v k(1−δ) from which it follows that

,

% 1 Xt ,t G 2 3 and

c (k)(ln v)σ (k) ≤C ; v k(1−δ)

G1 Xt2 ,t3

c (k)(ln v)σ (k) ≤C . v k(1−δ)

(5.47)

We analyze now the term G2 . Due to the presence of (1−δj1 j2 δj2 j3 ) the components of the vector (1 − δj1 j2 δj2 j3 )Φjs1 Φjs2 Φjs3 contains only terms (up to a phase) like 1 2 3 φγt−t (x)φγt−t (x)φγt−t (−x) 2 2 2

1 2 3 or φγt−t (x)φγt−t (−x)φγt−t (−x) 2 2 2

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where γ1 , γ2 and γ3 can be ref or tr. This can easily be seen by using Eqs. (4.10) and (4.11). By Strichartz methods, it is suﬃcient to estimate the L1[t2 ,t3 ] L2 (R+ ) norm of these terms. Then using H¨older’s inequality we have, for instance 1 1 2 3 φγ2 φγ3 (−·)L2 (R+ ) ≤ φγt−t 4 + φγt−t 4 + φγt−t ∞ − φγt−t 2 t−t2 t−t2 2 L (R ) 2 L (R ) 2 L (R )

c(k)(ln v)σ(k) . v k(1−δ) The second kind of terms can be estimated in the same way and we obtain c (k)(ln v)σ (k) G2 Xt2 ,t3 ≤ C , v k(1−δ) ≤C

which, together with the estimate (5.47), gives c (k)(ln v)σ (k) GXt2 ,t3 ≤ C . v k(1−δ)

(5.48)

Fix k such that k(1 − δ) > 2 then for v suﬃciently large (5.48) implies GXt2 ,t3 ≤ ΞXt2 ,t3 ≤ Cv −δ/2

N

1 . v

(5.49)

j+1 + v −1 (2C)

j=0

j N

j+1−k (2C)

j=0 k=0

N +1 + 2v −1 ≤ 2Cv −δ/2 (2C)

N

j+1 (2C)

j=0

N +1 + 2v −1 (2C) N +1 ≤ 2Cv −δ/2 (2C) % −δ/2 (2C) N. ≤ Cv Since N is the integer part of (t3 − t2 )/τ =

T τ

ln v we have

% −δ/2+ Tτ ΞXt2 ,t3 ≤ Cv

e ln(2C)

.

and T∗ = δτ∗ /2 and obtain We can ﬁnally set τ∗ ≡ τ / ln(2C) ΞL∞ [t ,t 2

3]

L2

% − ≤ ΞXt2 ,t3 ≤ Cv

T∗ −T τ∗

,

which concludes the proof of Theorem 4.3. 6. Conclusion and Perspectives In the present paper we have given a ﬁrst rigorous analysis of nonlinear Schr¨ odinger propagation on graphs. We have given a preliminary proof of local and global well posedness of the dynamics, and of energy and mass conservation laws for some distinguished vertex couplings, i.e Kirchhoﬀ, δ and δ couplings. Then we concentrated on the problem of collision of a fast solitary wave on the graph vertex (with

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couplings as before). It turns out that the solitary wave splits in reﬂected and transmitted components the form of which are again of solitary type, but with modiﬁed amplitudes controlled by scattering coeﬃcient given by the linear graph dynamics. This behavior holds true over times of the order ln v where v is the velocity of the impinging soliton. We add some other remarks on the result and further analysis and generalizations. To begin with, let us note that the real line with a point interaction at 0 can be interpreted as a degenerate graph with two edges. The cited paper [17] treats the special case of a δ interaction on the line, and our description shows how it could be possible to extend their results to other point interactions; among the examples treated in the present paper there is a version of the δ interaction, showing how to treat point interactions of a more singular character than the one given by a δ. Concerning more general issues, a sharper description of the post interaction phase can be achieved by an explicit characterization of the evolution of the modiﬁed solitary proﬁles, i.e. of the Φjt . This last part is somewhat delicate, and intersects with contemporary intense work on asymptotics for solitons in integrable and quasi integrable PDE, so we limit ourselves to the following remarks. In the case analyzed in [17], the asymptotic behavior of nonlinear Schr¨ odinger evolution of solitary waveforms with modiﬁed amplitudes is given, and making use of inverse scattering theory it is shown (Appendix B of the cited paper) that the evolution is close to a soliton up to times of order ln v and an error the order of which is an inverse power of v. Borrowing from these results, it is possible to get in our case too, but we omit details, the asymptotics of the Φjt , i.e. of the free NLS evolution of the modiﬁed solitary proﬁles outgoing from the phase two. It turns out that these outgoing wavefunctions can be approximated, on the same logarithmic timescale of Theorem 1.1, as new solitons with the same waveform of the unperturbed dynamics, modiﬁed amplitudes and phases, plus a dispersive (“radiation”) contribution. The meaning of this statement is that the L∞ norm of the diﬀerence between the evolved modiﬁed solitary proﬁles Φjt and such ﬁnal outgoing solitons has the usual 1 dispersive behavior, |t − t2 |− 2 . Let us note that, following this strategy, at the end of the phase three, there would be two types of errors: errors due to the approximation procedure in phase one and two (OL2 ); and errors arising from neglecting dispersion in the reconstruction of the outgoing solitons (OL∞ ). An important question concerns the possibility of extending the timescale of validity of approximation by the solitary outgoing waves. As a quite generic remark, this possibility could be related to the asymptotic stability of the system, or of systems immediately related to it. More concretely, in a diﬀerent type of model (scattering of two solitons on the line) in the already cited paper [1], some considerations are given on obtaining longer timescales of quasiparticle approximation in dependence of the initial data and external potential, but it is unclear whether similar considerations can be applied to the present case.

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Another issue is the nonlinearity. The fundamental asymptotics proved in [17] and used in the present paper relies on the integrable nature of cubic NLS, and it is not immediate to extend these results to more general nonlinearities. One can conjecture that for nonlinearities close to integrable ones which admit solitary waves, the outgoing waves are close to solitons over suitable timescales. Let us mention, however, the recent results of Perelman on the asymptotics of colliding solitons for nonlinearity close to integrable or L2 critical on the line ([25, 24]). A ﬁnal problem is the extension of results of the present work to more general graphs. We believe that results similar to the ones of the present papers are valid for more general boundary conditions at the vertex of a star graphs, with the same proof, under the condition of absence of eigenvalues for the linear Hamiltonian describing the graph. In presence of eigenvalues, some Strichartz estimates weaken, and a more reﬁned analysis is needed (see [14] for the analogous problem on the line with an attractive δ interaction). Of course, the extension of the present results to the case of star graphs with more than three edges has to be considered straightforward, while the extension to graphs having a less trivial topology is an open problem. Finally let us comment brieﬂy the recent paper [26]. In this partly heuristic paper the authors study a star graph (but also more general type of graphs are considered) with a NLS in which on every edge there is a diﬀerent strength βk in front of the the cubic term. The authors ﬁx a boundary condition which guarantees that mass and energy of the solution are (formally) conserved. Moreover according to the authors it is possible to derive a condition on the strengths βk which allow for complete transmission of an incoming solitary wave across the vertex. In these same situations the authors show that an inﬁnite chain of conserved quantities exists, deﬁned analogously to the case of the NLS on the line. The result, if formal, is interesting, and concerning the relation with ours we note the following. In the case of a three edge graph and more generally for a odd edge number, the complete transmission is made possible exactly by the ﬁne tuning of the coupling constants in front of the nonlinearities. For the case of a single medium with the same nonlinearity on every edge and Kirchhoﬀ boundary conditions, one can prove (see [2], where more generally the case of nonlinear bound states for δ boundary conditions is treated) that exact travelling solitons exist only in the case of a graph with an even number of edges, while in the case of an odd number of edges a stationary state is formed which is given by half a free soliton on every edge. Acknowledgements The present research was partially supported by INDAM-GNFM research project “Equazione di Schr¨ odinger non lineare interagente con difetti sulla retta e su graﬁ”. The Hausdorﬀ Research Institute for Mathematics is also acknowledged for the support. The authors are grateful to Sergio Albeverio for comments and discussions.

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References [1] W. K. Abu Salem, J. Fr¨ ohlich and I. M. Sigal, Colliding solitons for the nonlinear Schr¨ odinger equation, Comm. Math. Phys. 291 (2009) 151–176. [2] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Stationary states of NLS on star graphs (2011), 4 pp.; arXiv: 1104.3839[math-ph]. [3] R. Adami and D. Noja, Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect, J. Phys. A: Math. Theor. 42(49) (2009) 495302, 19pp. [4] S. Albeverio, C. Cacciapuoti and D. Finco, Coupling in the singular limit of thin quantum waveguides, J. Math. Phys. 48 (2007) 032103. [5] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, 2nd edn. (AMS Chelsea Publ., 2005); with an Appendix by P. Exner. [6] B. Bellazzini and M. Mintchev, Quantum ﬁelds on star graphs, J. Phys. A Math. Gen. 39(35) (2006) 1101–1117. [7] G. Berkolaiko, R. Carlson, S. Fulling and P. Kuchment, Quantum Graphs and Their Applications, Contemporary Math., Vol. 415 (American Math. Society, Providence, RI, 2006). [8] J. Blank, P. Exner and M. Havlicek, Hilbert Spaces Operators in Quantum Physics (Springer, New York, 2008). [9] R. Burioni, D. Cassi, P. Sodano, A. Trombettoni and A. Vezzani, Soliton propagation on chains with simple nonlocal defects, Phys. D 216 (2006) 71–76. [10] C. Cacciapuoti and P. Exner, Nontrivial edge coupling from a Dirichlet network squeezing: The case of a bent waveguide, J. Phys. A: Math. Theor. 40(26) (2007) F511–F523. [11] D. Cao Xiang and A. B. Malomed, Soliton defect collisions in the nonlinear Schr¨ odinger equation, Phys. Lett. A 206 (1995) 177–182. [12] S. Cardanobile and D. Mugnolo, Analysis of FitzHugh–Nagumo–Rall model of a neuronal network, Math. Methods Appl. Sci. 30 (2007) 2281–2308. [13] T. Cazenave, Semilinear Schr¨ odinger Equations, Courant Lecture Notes in Mathematics, Vol. 10 (Amer. Math. Soc., 2003). [14] K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities, Commun. Part. Diﬀ. Eqs. 34 (2009) 1074–1113. [15] P. Exner, Contact interactions on graph superlattices, J. Phys. A: Math. Gen. 29(1) (1996) 87–102. [16] R. H. Goodman, P. J. Holmes and M. I. Weinstein, Strong NLS soliton-defect interactions, Phys. D 192 (2004) 215–249. [17] J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Phys. 274 (2007) 187–216. [18] J. Holmer, J. Marzuola and M. Zworski, Soliton splitting by delta impurities, J. Nonlinear Sci. 7 (2007) 349–367. [19] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998) 955–980. [20] P. G. Kevrekidis, D. J. Frantzeskakis, G. Theocharis and I. G. Kevrekidis, Guidance of matter waves through Y -junctions, Phys. Lett. A 317 (2003) 513–522. [21] V. Kostrykin and R. Schrader, Kirchhoﬀ’s rule for quantum wires, J. Phys. A: Math. Gen. 32(4) (1999) 595–630. [22] P. Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media 14(1) (2004) S107–S128. [23] P. Kuchment, Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math. Gen. 38(22) (2005) 4887–4900.

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[24] G. Perelman, A remark on soliton-potential interaction for nonlinear Schr¨ odinger equations, Math. Res. Lett. 16(3) (2009) 477–486. [25] G. Perelman, Two soliton collision for nonlinear Schr¨ odinger equation in dimension 1, Ann. Inst. Henri Poincar´e, in print (2011). [26] Z. Sobirov, D. Matrasulov, K. Sabirov, S. Sawada and K. Nakamura, Integrable nonlinear Schr¨ odinger equation on simple networks: Connection formula at vertices, Phys. Rev. E 81 (2010) 066602.

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Reviews in Mathematical Physics Vol. 23, No. 5 (2011) 453–530 c World Scientific Publishing Company DOI: 10.1142/S0129055X11004333

REGULARITY OF BOUND STATES

JEREMY FAUPIN∗,‡ , JACOB SCHACH MØLLER†,§ and ERIK SKIBSTED†,¶ ∗Institut

de Math´ ematiques de Bordeaux, Universit´ e de Bordeaux 1, France

†Department

of Mathematical Sciences, Aarhus University, Denmark ‡[email protected] §[email protected] ¶[email protected] Received 1 July 2010 Revised 14 March 2011

We study regularity of bound states pertaining to embedded eigenvalues of a self-adjoint operator H, with respect to an auxiliary operator A that is conjugate to H in the sense of Mourre. We work within the framework of singular Mourre theory which enables us to deal with confined massless Pauli–Fierz models, our primary example, and many-body AC-Stark Hamiltonians. In the simpler context of regular Mourre theory, our results boil down to an improvement of results obtained recently in [8, 9]. Keywords: Mourre theory; non-relativistic QED; AC-Stark. Mathematics Subject Classification 2010: 81Q10, 81V10

Contents 1. Introduction 454 1.1. Singular Mourre theory . . . . . . . . . . . . . . . . . . . . . . . . . 457 1.2. The Nelson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 1.3. The AC-Stark model . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 2. Assumptions and Statement of Regularity Results

468

3. Preliminaries 472 3.1. Improved smoothness for operators of class C 1 (A) . . . . . . . . . . 472 3.2. Iterated commutators with N 1/2 . . . . . . . . . . . . . . . . . . . . 476 3.3. Approximating A by regular bounded operators . . . . . . . . . . . . 478 4. Proof of the Abstract Results 4.1. Proof of Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Proof of Theorem 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Theorem on more N -regularity . . . . . . . . . . . . . . . . . . . . . 453

481 481 490 491

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5. A Class of Massless Linearly Coupled Models 5.1. The model and the result . . . . . . . . 5.2. Application to the Nelson model . . . . 5.3. Expanded objects . . . . . . . . . . . . . 5.4. Mourre estimates . . . . . . . . . . . . . 5.5. Checking the abstract assumptions . . .

. . . . .

. . . . .

. . . . .

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. . . . .

. . . . .

. . . . .

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. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

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495 495 498 501 503 510

6. AC-Stark Type Models 517 6.1. The model and the result . . . . . . . . . . . . . . . . . . . . . . . . 517 6.2. Regularity of non-threshold bound states . . . . . . . . . . . . . . . 519 6.3. Regularity of non-threshold atomic type bound states . . . . . . . . 523

1. Introduction This paper is the ﬁrst in a series of two dealing with embedded eigenvalues and their bound states. Our arguments in both papers revolve around local positive commutator methods originating from Mourre’s seminal paper [30]. In fact, some of the central ideas employed in the present paper can be traced back to [14] by Froese and Herbst, where exponential decay of eigenfunctions for many-body Schr¨ odinger operators were ﬁrst extracted from a positive commutator estimate. See also [15] for a precursor pertaining to two-body operators. In contrast to the above mentioned works, we do not here study decay of bound states of a self-adjoint operator H in position space, but rather decay in the spectral representation for an auxiliary operator A conjugate to H in the sense of Mourre. More precisely, given a bound state ψ of H, we address the question Q(k): For a given k ∈ N, under what conditions on the pair of operators H and A does it hold true that ψ is in the domain of Ak ? It is a question that arises naturally in the context of second order perturbation theory for embedded eigenvalues because together with the Limiting Absorption Principle from [30], an aﬃrmative answer allows one to construct and analyze the so called Fermi Golden Rule operator describing level shifts to second order in perturbation theory. In [27] Fermi’s golden rule was formulated and veriﬁed in an abstract setup under the condition that ψ ∈ D(A2 ), following ideas from [1]. See also [5, 11, 13, 31]. For many-body Schr¨ odinger operators the conjugate operator is usually taken to be the generator of dilation and here the condition ψ ∈ D(A2 ) is fulﬁlled by the Froese–Herbst exponential bound. In other contexts however, it is a non-trivial question to answer. The ﬁrst results in an abstract setup are due to Cattaneo [8, 9], and the setting is regular Mourre theory. The adjective regular refers to setups where multiple commutators between H and A, in particular [H, A], are suitably controlled by resolvents of H. Results in this category range from Mourre’s original work [30] to the results relying on the C k (A) type conditions introduced by Amrein et al. [3]. See also [1, 5, 10, 18, 25, 27].

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In this paper, we address the question of regularity of bound states with respect to a conjugate operator A in the context of singular Mourre theory. In the second paper [13] the results obtained here are used to do second order perturbation theory of embedded eigenvalues, in particular we establish the validity of Fermi’s golden rule for an abstract class of Hamiltonians. By singular Mourre theory we refer to the situation where the ﬁrst commutator [H, A] is not controlled by the Hamiltonian itself, as in [11, 24, 21, 22, 31, 38]. Regular Mourre theory is a special case of the singular setup considered here, and our results thus extend those of [8, 9]. Roughly speaking, our answer to the question Q(k) is that control of k + 1 commutators suﬃces. We stress that even within regular Mourre theory we extend [8, 9] in that we reduce by one, from k + 2 to k + 1, the number of commutators one needs to control in order to answer the question in the aﬃrmative. Our result is optimal in terms of integer numbers of commutators, cf. Example 1.1 below. See also [32] where the regular Mourre theory analysis is extracted from this paper and conditions are established under which bound states become analytic vectors for A. Our main motivation is applications to massless models from quantum ﬁeld theory. In particular our results apply to the massless conﬁned Nelson model at arbitrary coupling strength. We can deal with infrared singularities that are slightly 1 weaker than the physical one, that is we can handle singularities of the form |k|− 2 + , for some > 0. As a by-product of our methods we also establish that all bound states are in the domain of the number operator. In Sec. 5, we in fact deal with a larger class of quantum ﬁeld theory models, sometimes called Pauli–Fierz models, which includes the Nelson model. For simplicity and concreteness we present our results in the introduction in the context of the Nelson model. This is done in Sec. 1.2 below. The reader can also consult [22, Sec. 2.3] for a discussion of the ﬁeld theory models considered in this paper and its sequel. In Sec. 6, we apply the abstract results of this paper to many-body AC-Stark Hamiltonians where we obtain a new regularity result. See Sec. 1.3 below for a formulation of the model and the result. The following example illustrates that if one desires bound states to be in the domain of the kth power of a conjugate operator, one needs at least control of k + 1 commutators. Example 1.1. Consider the one-dimensional Schr¨ odinger operator H = −∆+V on H = L2 (R), where V is a rank-one potential V = |φφ|. Here φ ∈ H is constructed as follows: Let k0 ∈ N and ∈ (0, 1/2). In momentum space we write φ as a sum of two functions φˆ = φˆ1 + φˆ2 , where we choose φ2 , or rather its Fourier transform, to be 0, |ξ| ≤ 1, φˆ2 (ξ) = 2 1 (ξ 2 − 1)k0 + 2 + e−ξ , |ξ| > 1.

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Having ﬁxed φ2 , we choose φ1 , such that ˆ 2 (ξ 2 − 1)−1 dξ = −1. |φ(ξ)| φˆ1 ∈ C0∞ − 21 , 12 , and

(1.1)

R

The key to the example is the singular behavior of φˆ near ξ 2 = 1. We have φ ∈ C ∞ (R) and d φ −k0 − 32 − , for all ≥ 0, dx (x) ≤ C (1 + |x|) (1.2) d φ xk ∈ L2 (R) ⇔ k ≤ k0 + 1. dx Furthermore, the normalization in (1.1) ensures that H has λ = 1 as an embedded eigenvalue with eigenfunction ψ = (−∆−1)−1 φ. Note that (ξ 2 −1)−1 φˆ decays faster than any polynomial. We have ψ ∈ C ∞ (R) and xk

d ψ ∈ L2 (R) ⇔ k ≤ k0 . dx

(1.3)

1 d d Let A denote the generator of dilations A = 2i (x dx + dx x). Introducing the k k−1 0 notation adA (H) = [adA (H), A] and adA (H) = H we formally compute

ik adkA (H) = −2k ∆ + ik adkA (V ). Due to (1.2), the iterated commutator adkA (V ) is compact if and only if k ≤ k0 + 1. Adding resolvents of H does not help. Furthermore i[H, A] obviously satisﬁes a Mourre estimate with compact error at positive energies: For any E > 0 1[H≥E] i[H, A]1[H≥E] ≥ E1[H≥E] − K, where K is compact and 1[H≥E] is the spectral projection for H associated with the Borel set [E, ∞). That is, we are within the scope of [8, 9]. We have the ﬁrst k0 + 1 commutators all H-bounded, with the (k0 + 2)nd commutator not controlled by any power of resolvents of H. Appealing to (1.3), we see that the bound state ψ is in the domain of Ak0 , but not in the domain of Ak0 +1 . That ψ ∈ D(Ak0 ) is a conclusion one cannot reach using [8, 9]. It is however attainable by the abstract result of the present paper. See Sec. 2. The additional information that ψ ∈ D(Ak0 +1 ) demonstrates that our result is optimal. As a last observation, more geared towards our second paper [13], let us perturb H by adding a small multiple of V to obtain Hσ = H + (1 + σ)V , with |σ| being small. First note that the operator Hσ can have at most one eigenvalue. Repeating some of the analysis from above, and appealing to the implicit function theorem, one can verify the following statements: There exist δ > 0 and a function λ : (−δ, 0] → R such that ∀ σ ∈ (0, δ) : σpp (Hσ ) = ∅, ∀ σ ∈ (−δ, 0] : σpp (Hσ ) = {λ(σ)}.

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The function λ satisﬁes that λ(0) = 1, λ is real analytic in (−δ, 0), λ ∈ C k0 +1 ((−δ, 0]) and λ ∈ C k0 +2 ((−δ, 0]). This establishes a natural limit on what one should expect from a perturbation theory for embedded eigenvalues. Indeed, it indicates that control of two commutators (corresponding to k0 = 1) may suﬃce for second order perturbation theory and this is in fact accomplished in [13, Sec. 5.1]. The strongest results, cf. [13, Sec. 5.2], require control of three commutators, since they rely on the condition ψ ∈ D(A2 ). 1.1. Singular Mourre theory Before we formulate our results more precisely, we pause to discuss on a more heuristic level the origin of conjugate operators, and how we are led naturally to singular Mourre theory. Consider the operator Mω of multiplication in momentum space L2 (Rd ) by a dispersion relation ω assumed to be locally Lipschitz. The connection between dynamics and structure of the spectrum of a self-adjoint operator is fairly well understood, starting from Kato-smoothness and the RAGE theorem [33–36]. When looking for a conjugate operator, one should study the dynamics of the operator Mω . It is natural to identify what states have (at least) ballistic motion, that is ﬁnd states ψ0 satisfying x2 ψt ≥ ct2 , for some c > 0. Here ψt = exp(−itMω )ψ0 . The position operator x is equal to i∇k . We can compute this quantity explicitly and we get t x · ∇ω + ∇ω · xψs ds x2 ψt = x2 ψ0 + 0

= x ψ0 + tx · ∇ω + ∇ω · xψ0 + t2 |∇ω|2 ψ0 . 2

We observe that if ψ0 has support away from zeroes of ∇ω, then the motion is at least ballistic. More precisely this is the case if essinf k∈supp ψ0 |∇ω(k)| ≥ c > 0. If ω = k 2 , the standard non-relativistic dispersion relation, we ﬁnd that ψ0 should be localized away from 0 in momentum space. Since |∇ω|2 = 4ω, the requirement on ψ0 can also be expressed as ψ0 ∈ EMω ([c/4, ∞))L2 (Rd ), where EMω denotes the spectral resolution associated with the self-adjoint operator Mω . We observe that the energy 0 has a special signiﬁcance for the case ω = k 2 and is called a threshold, in the sense that states localized in energy near a threshold may not have strict ballistic motion. A second example is ω = |k|. Here we observe that |∇ω| = 1, and hence all states ψ0 will exhibit ballistic motion. In other words this dispersion relation does not have thresholds. This of course reﬂects the constant (momentum independent) speed of light. See [21, Sec. 1.2] for a discussion of general dispersion relations.

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When picking a conjugate operator in Mourre theory, one is precisely looking for an observable a with at least ballistic growth. The choice often used is the Heisenberg derivative of x2 , where x is some suitably chosen position observable. That is, one would naturally be lead to consider 1 (x · ∇ω + ∇ω · x). 2 This is for example the case for the N -body problem, see e.g. [1, 8, 9, 27], and in the case of ﬁeld theory see [10, 11, 16, 17, 22, 38], where the position is the Newton– Wigner position dΓ(x). The free energy is dΓ(Mω ), and we get as conjugate operator A = dΓ(a), where a is as above. Here dΓ(b) denotes the second quantization of a one-body operator b, cf. the following subsection on Nelson’s model. It is often advantageous to modify the so obtained conjugate operator, to simplify proofs, or circumvent some technical issues. In this paper we need the modiﬁed generator of translations Aδ from [22] in order to deal with the conﬁned massless Nelson model, and more generally conﬁned massless Pauli–Fierz models. There are two issues that come up naturally when following the above guidelines for massless ﬁeld theory models, like the Nelson model. One is already apparent in the one-particle setup discussed above. If ω(k) = |k|, the resulting conjugate operator a, the generator of radial translations, does not have a self-adjoint realization. This appears to be a purely technical complication, that becomes a serious issue when one is in need of localizations in the operator a. The operator is not normal, so we do not have spectral calculus at hand, only resolvents. This has so far not been a serious issue when dealing with the limiting absorption principle [11, 22, 26, 27, 38], and perturbation theory around an uncoupled system [11, 24]. It does however become an obstacle when one tries to apply the conjugate operator a in the context of scattering theory [23]. In the present paper, non-self-adjointness of a is also a serious obstacle, which we overcome, as in [23], by passing to a so called expanded Hamiltonian. The idea is to write L2 (Rd ) ∼ L2 (R+ ) ⊗ L2 (S d−1 ) and double the Hilbert space to L2 (R) ⊗ L2 (S d−1 ). The dispersion relation in polar coordinates is just multiplication by r, which when extended linearly to negative r gives rise to the self-adjoint conjugate operator i∂/∂r ⊗ 1. We thus work with an expanded Hamiltonian, and in the end pull our results back to the physical Hamiltonian. The reader should keep this in mind when going through the abstract conditions in the following section. However passing to an expanded Hamiltonian is not a silver bullet, it comes with a price. The operator of multiplication by r is no longer bounded from below, making it hard to utilize energy localizations. For this reason we have to develop an abstract theory which does not demand that any naturally occurring object can be controlled by the (expanded) Hamiltonian. The second feature we want to discuss does not occur on the one-particle level, but only after second quantization. The free commutator becomes a=

i[dΓ(|k|), A] = N ,

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459

where N is the number operator. In the standard (regular) commutator based methods, one typically has the commutator bounded at least as a form on D(H). (This is for example a consequence of a C 1 (A) assumption.) This is not the case here and we call such a situation singular. One could of course avoid this issue by observing that the operators involved conserve particle number, and then rescale A by 1/n on the n-particle sector. However, perturbations are typically expressed in terms of ﬁeld operators, and straying from second quantized conjugate operators give rise to terms from the commutator with the perturbation, that have so far not been controllable. The dΓ(|k|)-unboundedness of the number operator has led authors to use a different conjugate operator instead, namely the second quantized generator of dilation given by dΓ((x · k + k · x)/2), normally associated with the dispersion relation k 2 . Here the commutator with dΓ(|k|) is dΓ(|k|) itself, so the issue disappears. However, this choice induces an artiﬁcial threshold at photon energy 0, which for a coupled system turns all eigenvalues of the atomic system into artiﬁcial thresholds. In order to circumvent this problem one can modify the generator of dilation by building the level shift from Fermi’s golden rule into the conjugate operator. This was done in [5] and gives rise to positive relatively bounded commutators, at weak coupling. There are however disadvantages to this approach. It does not cover situations where symmetries may cause embedded eigenvalues to persist to second order in perturbation theory. For the N -body problem in quantum mechanics, one can, for example, show that the underlying spectrum is absolutely continuous without a priori imposing Fermi’s golden rule, which can then subsequently be established [1, 27]. Works employing this choice of conjugate operator has, so far, not been able to address what happens outside the regime of weak coupling, which may be an issue since coupling constants typically are explicitly given numbers. In electron-photon models, the coupling constant involve the feinstructure constant 1/137 and in electron-phonon models from solid state physics, the coupling constants occurring may even be of the order 1. Eﬀective coupling constants may also depend on an ultraviolet cutoﬀ, thus imposing apparently artiﬁcial limitations on the size of the cutoﬀ. Finally the restriction on the size of the coupling constant is always locally uniform in energy. That is, all statements of this type holds only below a ﬁxed E0 . Papers employing the generator of dilation include [4, 5, 18]. We remark that in [24], the author modiﬁes the generator of radial translation, as it was done in [5] for the generator of dilations, in order to establish Fermi’s golden rule. We have no need for this construction since we follow the strategy of [1, 27, 31]. Instead of viewing the unboundedness of the ﬁrst commutator with respect to dΓ(|k|) as a technical problem, one can also adopt the point of view that it is a feature of the model which can be exploited. This is most obviously done for small coupling constants, where one gets a positive commutator globally in energy, modulo a compact error. This was done in [11, 17, 24, 38]. In [22] the extra positivity of the commutator is directly utilized to prove a Mourre estimate

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148-RMP

J. Faupin, J. S. Møller & E. Skibsted

at arbitrary coupling constant, the ﬁrst (and so far only) such result for massless models. Another piece of information one can extract is that the number operator has ﬁnite expectation in bound states. This was done in [38] for small coupling constants and generally in [22]. A more subtle property is that one can obtain a stronger limiting absorption principle, see [21, 31], which has so far not found an application. Here we prove in particular that bound states are in the domain of the number operator, not just in its form domain. We have not discussed positive temperature models, where one has a similar situation, except that so far no positive commutator estimates at arbitrary coupling has been proven, regardless of choice of conjugate operator. See e.g. [12, 19] and references therein. 1.2. The Nelson model The model describes a conﬁned atomic system coupled to a massless scalar quantum ﬁeld. The Hamiltonian K of the atomic system is K =−

P 1 ∆i + Vij (xi − xj ) + W (x1 , . . . , xP ) 2mi i=1 i<j

(1.4)

acting on K = L2 (R3P ). Here mi > 0 denotes the mass of the i’th particle located at xi ∈ R3 . We write x = (x1 , . . . , xP ) ∈ R3P . The external potential W is the conﬁnement and must satisfy (W0) W ∈ L2loc (R3P ) and there exist positive constants c0 , c1 and α > 2 such that W (x) ≥ c0 |x|2α − c1 . As for the pair potentials Vij , they should satisfy (V0) The Vij ’s are ∆-bounded with relative bound 0. The Hilbert space for the scalar bosons is the symmetric Fock-space F = Γ(L2 (R3 )) and the kinetic energy for the massless bosons is dΓ(|k|), the second quantization of the operator of multiplication with the massless dispersion relation |k|. The uncoupled Hamiltonian, describing the atomic system and the scalar ﬁeld is K ⊗ 1F + 1K ⊗ dΓ(|k|), as an operator on the full Hilbert space H = K ⊗ F. Our next task is to introduce a coupling of the form Iρ (x) =

P

φρ (xi ),

i=1

where φρ (y) is an ultraviolet and infrared regularized ﬁeld operator 1 (ρ(k)e−ik·y a∗ (k) + ρ(k)eik·y a(k))dk. φρ (y) = √ 2 R3

(1.5)

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We assume purely for simplicity that ρ only depends on k through its modulus. To conform with the notation used in [22], we introduce such that |k|ρ(k) = ρ˜(|k|).

ρ˜(r) = rρ(r, 0, 0),

For the interacting Hamiltonian, indexed by the coupling function ρ, HρN = K ⊗ 1F + 1K ⊗ dΓ(|k|) + Iρ (x)

(1.6)

to be essentially self-adjoint on D(K) ⊗ Γﬁn (C0∞ (R3 )), we need the following basic assumption on ρ. ∞ (ρ1) 0 (1 + r−1 )|˜ ρ(r)|2 dr < ∞. Here Γﬁn (V ) denotes the subspace of F consisting of elements η with only ﬁnitely many n-particle components η (n) nonzero, and those that are nonzero lie in the n-fold algebraic tensor product of the subspace V ⊆ L2 (R3 ). Note that Γﬁn (V ) is dense in F if V is dense in L2 (R3 ). In order to formulate the remaining assumption on ρ we introduce a function d ∈ C ∞ ((0, ∞)), which measures the amount of infrared regularization carried by ρ. It should, for some Cd > 0, satisfy d(r) = 1,

for r ≥ 1,

−Cd

d(r) ≤ d (r) < 0, r

lim d(r) = +∞.

r→0+

(1.7)

Note that the conditions above imply that 1 ≤ d(r) ≤ r−Cd , for r ∈ (0, 1]. In order to simplify some expressions below we make the additional assumption that 1

∀r ∈ (0, 1] : d(r) ≤ Cd r− 2 ,

(1.8)

for some Cd > 0. In practice we want to construct a d with as weak a singularity as possible, so this extra assumption is no restriction. We formulate the remaining conditions on ρ, of which the two ﬁrst also appeared in [22]. ∞ ρ(r)|2 + | ddrρ˜ (r)|2 ]dr < ∞. (ρ2) 0 (1 + r−1 )d(r)2 [r−2 |˜ ∞ 2 (ρ3) 0 | ddrρ2˜ (r)|2 dr < ∞. ∞ ρ(r)|2 dr < ∞. (ρ4) 0 r4 |˜ We remark that (ρ2) and (ρ4) imply (ρ1). A typical form of ρ, and hence ρ˜, would be |k|2

1

ρ(k) = e− 2Λ2 |k|− 2 + ,

r2

1

ρ˜(r) = e− 2Λ2 r 2 + .

(1.9)

One can construct a d by gluing together the functions 1 and r− , with 0 < < min{, 1/2}. The parameters Λ and are the ultraviolet and infrared regularization parameters. Ideally we would like to have Λ = ∞ and = 0. For the conditions (ρ1)–(ρ4) to be satisﬁed we must have 0 < Λ < ∞ and > 1. Observe that it is the condition (ρ3) on the second derivative of ρ˜ that causes the strongest restriction on .

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2011 13:31 WSPC/S0129-055X

148-RMP

J. Faupin, J. S. Møller & E. Skibsted

Observe that the set of ρ’s satisfying (ρ1)–(ρ4) is a complex vector space IN (d) ⊆ L (R3 ), which can be equipped with a norm matching the four conditions. That is 2 2 2 ∞ d˜ ρ d ρ ˜ ρ(r)|2 + (1 + r−1 )d(r)2 (r) + 2 (r) dr. (r4 + d(r)2 r−3 )|˜ ρ2N := dr dr 0 2

(1.10) In order to formulate our main theorem, we need to introduce an operator conjugate to HρN . We use the one constructed in [22], for which a Mourre estimate has been established under the assumptions above. Let χ ∈ C0∞ (R), with 0 ≤ χ ≤ 1, χ(r) = 1 for |r| < 1/2, and χ(r) = 0 for |r| > 1. For 0 < δ ≤ 1/2 we deﬁne a function on (0, ∞) by sδ (r) = χ(r/δ)d(δ)r−1 + (1 − χ)(r/δ)d(r)r−1 . Using this function we construct a vector-ﬁeld by sδ (k) = sδ (|k|)k, which equals k/|k| for |k| > 1 and d(δ)k/|k| for |k| < δ/2. The conjugate operator on the oneparticle sector is aδ =

1 (sδ · i∇k + i∇k · sδ ). 2

(1.11)

The operator is symmetric and closable on {f ∈ C0∞ (R3 )|f (0) = 0}. We denote again by aδ its closure which is a maximally symmetric operator, but not selfadjoint. It is a modiﬁcation, near k = 0, of the generator of radial translations k k ·i∇k + i∇k · |k| )/2. The conjugate operator is now the maximally symmetric a = ( |k| operator Aδ = 1K ⊗ dΓ(aδ ). The second quantization dΓ(a) of the generator of radial translations works as conjugate operator if one stays close to the uncoupled system. See [11, 24, 38]. It is not known if one really needs the modiﬁed generator of radial translations Aδ in order to get a Mourre estimate at arbitrary coupling. For an eigenvalue E ∈ σpp (HρN ) we write Pρ for the associated eigenprojection. It is known from [22] that Pρ has ﬁnite dimensional range. Finally we need the number operator N = 1K ⊗ dΓ(1L2 (R3 ) ). We will make use of the same notation for the (usual) number operator on F . Our main result of this paper, formulated in terms of the Nelson model, is Theorem 1.2. Suppose (W0) and (V0). Let E0 ∈ R and ρ0 ∈ IN (d) be given. There exist 0 < δ ≤ 1/2, r > 0 and C > 0 such that for any ρ ∈ IN (d), with ρ − ρ0 N ≤ r, and E ∈ σpp (HρN ) ∩ (−∞, E0 ] we have 1

1

Pρ : H → D(N 2 Aδ ) ∩ D(Aδ N 2 ) ∩ D(N )

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and 1

1

N 2 Aδ Pρ + Aδ N 2 Pρ + N Pρ ≤ C. We remark that for any δ > 0 small enough, one can ﬁnd r and C such that the conclusion of the theorem holds. See Theorem 5.2. The above suﬃces for our purpose and is a cleaner statement. We can implement a unitary transformation, the so-called Pauli–Fierz transform, which has the eﬀect of smoothening the infrared singularity. Let Uρ = exp(−iP φiρ/|k| (0)) be the unitary transformation with P ρ(k) Uρ a(k)Uρ∗ = a(k) − √ 2|k|

P ρ(k) and Uρ a∗ (k)Uρ∗ = a∗ (k) − √ . 2|k|

For

the transformation Uρ to be well-deﬁned we must require −2 |k| |ρ(k)|2 dk < ∞. To achieve this we strengthen (ρ1) to read R3 ∞ ρ(r)|2 dr < ∞. (ρ1 ) 0 (1 + r−2 )|˜

that

We then get

HρN = (1K ⊗ Uρ )HρN (1K ⊗ Uρ )∗ = Kρ ⊗ 1F + 1K ⊗ dΓ(|k|) + Iρ (x) − Iρ (0), (1.12) where P

Kρ = K −

vρ (xi ) +

i=1

and

vρ (y) = P

Observe that 1 φρ (y) − φρ (0) = √ 2

R3

R3

P2 2

∞

r−1 |˜ ρ(r)|2 dr1K

(1.13)

0

|ρ(k)|2 cos(k · y)dk. |k|

(1.14)

(ρ(k)(e−ik·y − 1)a∗ (k) + ρ(k)(eik·y − 1)a(k))dk.

The estimate |e±ik·y − 1| ≤ max{2, |k||y|} ≤ 2

|k| y, k

(1.15)

with η = (1 + |η|2 )1/2 , enables us to extract an extra infrared regularization using the decay in x supplied by the conﬁnement condition (W0). Keeping (1.8) and (ρ1 ) in mind, the remaining two assumptions on ρ now weaken to ∞ (ρ2 ) 0 | ddrρ˜ (r)|2 dr < ∞. ∞ 2 (ρ3 ) 0 r2 | ddrρ2˜ (r)|2 /(1 + r2 )dr < ∞.

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2011 13:31 WSPC/S0129-055X

148-RMP

J. Faupin, J. S. Møller & E. Skibsted

The condition (ρ4), being an ultraviolet condition, is unchanged. For the choice (1.9) to satisfy (ρ1 )–(ρ3 ) and (ρ4) we must have 0 < Λ < ∞ and > 0. Here the ﬁrst three conditions on ρ all require > 0. Observe again that the set of ρ satisfying (ρ1 )–(ρ3 ) and (ρ4) is a complex (d). We introduce the natural norm vector space IN 2 2 ∞ 2 d ρ˜ 2 d˜ ρ r 2 4 −2 2 ρ(r)| + (r) + (r) dr. (r + r )|˜ ρN := dr 1 + r2 dr2 0 Fix a ρ0 ∈ IN (d). There are now two avenues one can follow. Either one can continue as above, and for each ρ in a · N -ball around ρ0 we apply the transformation Uρ to arrive at the more regular Hamiltonian HρN that we can ﬁt into our class of Pauli–Fierz models. A second option would be to apply the same transformation Uρ0 regardless of ρ chosen near ρ0 . The advantage of this is two-fold: Firstly, we would be working in the same coordinate system for all ρ’s, which in the context of perturbation theory, cf. [13], is the most natural. Secondly, in this way the Hamiltonian will have a linear dependence on the “perturbation” ρ − ρ0 , which is a requirement in [13]. The drawback is that ρ − ρ0 has to be an element of IN (d), and for example cannot be a small multiple of ρ0 . To implement the latter approach, we now let ρ = ρ0 + ρ1 , with ρ1 ∈ IN (d), the space of regular interactions. We then employ the transformation Uρ0 which yields the transformed Hamiltonian HρN

= (1K ⊗

Uρ0 )HρN (1K

∗

⊗ Uρ0 ) =

HρN0

+ Iρ1 (x) −

P

vρ0 ,ρ1 (xi ),

(1.16)

i=1

where

ρ1 (k)ρ0 (k) −ik·y e Re vρ0 ,ρ1 (y) = P |k| R3

dk.

(1.17)

For an eigenvalue E ∈ σpp (HρN ) we write Pρ = (1K ⊗ Uρ )Pρ (1K ⊗ Uρ )∗ for the associated eigenprojection for HρN , and Pρ = (1K ⊗ Uρ0 )Pρ (1K ⊗ Uρ0 )∗ for the associated eigenprojection for HρN . Again Pρ and Pρ have ﬁnite dimensional ranges. Theorem 5.2 can be applied to the transformed Hamiltonian and we arrive at the following theorem. (d) be given. Theorem 1.3. Suppose (W0) and (V0). Let E0 ∈ R and ρ0 ∈ IN There exist 0 < δ ≤ 1/2, r > 0 and C > 0 such that (1) For any ρ ∈ IN (d) with ρ − ρ0 N ≤ r and E ∈ σpp (HρN ) ∩ (−∞, E0 ] we have 1

1

Pρ : H → D(N 2 Aδ ) ∩ D(Aδ N 2 ) ∩ D(N ) and 1

1

N 2 Aδ Pρ + Aδ N 2 Pρ + N Pρ ≤ C.

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(2) For any ρ1 ∈ IN (d) with ρ1 N ≤ r and E ∈ σpp (HρN ) ∩ (−∞, E0 ], where ρ = ρ0 + ρ1 , we have 1

1

Pρ : H → D(N 2 Aδ ) ∩ D(Aδ N 2 ) ∩ D(N ) and 1

1

N 2 Aδ Pρ + Aδ N 2 Pρ + N Pρ ≤ C. Unfortunately the transformation Uρ , with ρ ∈ IN (d), is too singular to allow for a recovery of the full set of regularity results for the original Hamiltonian HρN , as in Theorem 1.2. The only thing that remains after undoing the transformation is the following corollary to Theorem 1.3(1). The same argument using Theorem 1.3(2) would give a weaker result. Theorem 1.3(2) will however play a role in [13]. (d) be given. Corollary 1.4. Suppose (W0) and (V0). Let E0 ∈ R and ρ0 ∈ IN (d) with There exist 0 < δ ≤ 1/2, r > 0 and C > 0 such that for any ρ ∈ IN N ρ − ρ0 N ≤ r and E ∈ σpp (Hρ ) ∩ (−∞, E0 ] we have

Pρ : H → D(N )

and

N Pρ ≤ C.

We make a number of remarks concerning the results above. The domain of aδ is independent of δ, and in fact equals the domain of the generator of radial translations. The same is (presumably) false for the second quantized versions. This is the reason for the somewhat unpleasant formulation of the theorems in terms of Aδ . It should be read in the context of Mourre’s commutator method, and in [13] we need the regularity formulated in terms of Aδ . The statement that bound states are in the domain of the number operator is new. Previously it was only known that bound states are in the domain of N 1/2 . See [22]. The reader should ﬁrst and foremost read the results above with ρ = ρ0 . In the sequel [13] we need the locally uniform version to deduce a Fermi golden rule under minimal assumptions. In traditional approaches to Fermi’s golden rule, one typically require unperturbed bound states to be in the domain of the square of the conjugate operator. See [1, 27, 31]. In [13] we reduce the requirement to bound states ψ being in the domain of the conjugate operator itself, at the expense of a need for the norm Aδ ψ to be bounded uniformly in ρ in a ball around the unperturbed coupling function ρ0 and uniformly in E running over eigenvalues of Hρ in a ﬁxed compact interval. This motivates the somewhat unorthodox formulation in Theorem 1.2. The conditions (ρ3) and (ρ3 ) come from a need of handling the double commutator [[Hρ , Aδ ], Aδ ]. It is not a priori obvious that we should be able to place bound states in the domain of Aδ with control of just two commutators. In the context of regular Mourre theory the question is addressed in [8, 9] where the author(s) need three commutators to conclude a result of this type, which in view of Example 1.1 is not optimal. To deal eﬀectively with the infrared singularity, it is crucial to minimize the number of commutators needed.

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466

1.3. The AC-Stark model The model describes a system of N charged particles in a nonzero time-periodic Stark-ﬁeld with zero mean (AC-Stark ﬁeld). The particles are here taken threedimensional and we assume that the ﬁeld is 1-periodic and, for simplicity, that it is continuous i.e. that E˜ ∈ C([0, 1]; R3 ). The Hamiltonian is of the form

N 2 pi ˜ ˜ − qi E(t) · xi + V ; h(t) = (1.18) 2mi i=1 here xi , mi and qi are the position, the mass and the charge of the ith particle, respectively, and pi = −i∇xi is its momentum. The potential is of the form V = vij (xi − xj ), (1.19) 1≤i<j≤N

where the pair-potentials obey Conditions 1.5. Let k0 ∈ N be given. For each pair (i, j) the pair-potential R3 1 2 + vij where y → vij (y) ∈ R splits into a sum vij = vij (1) (2) (3) (4)

1 2 Diﬀerentiability: vij ∈ C k0 +1 (R3 ) and vij ∈ C k0 +1 (R3 \{0}). 1 (y)| ≤ C. Global bounds: For all α with |α| ≤ k0 +1 there are bounds |y||α| |∂yα vij 1 1 Decay at inﬁnity: |vij (y)| + |y · ∇y vij (y)| = o(1). 2 is compactly supported and for all α with |α| ≤ k0 + 1 Local singularity: vij |α|+1 2 |∂yα vij (y)| ≤ C; y = 0. there are bounds |y|

In the above conditions, the letter α denotes multiindices. Note that (1.18) and (1.19) with vij (y) = qi qj |y|−1 conform with Conditions 1.5 for any k0 . Introducing the inner product x · y = i 2mi xi · yi for x = (x1 , . . . , xN ), y = (y1 , . . . , yN ) ∈ R3N we can split R3N = XCM ⊕ X;

XCM = {x ∈ R3N | x1 = · · · = xN }.

There is a corresponding splitting ˜ = hCM (t) ⊗ I + I ⊗ h(t), h(t)

on L2 (XCM ) ⊗ L2 (X),

where hCM (t) = p2CM − ECM (t) · x Here ECM

Q ˜ ˜ and E = (E, . . . , E) = 2M

and h(t) = p2 − E(t) · x + V. q1 Q − 2m1 2M

˜ ..., E,

qN Q − 2mN 2M

E˜ ,

where Q = q1 + · · · + qN and M = m1 + · · · + mN are the total charge and mass of the system, respectively. In the special case where all the particles have identical charge to mass ratio, we see that the center of mass Hamiltonian is just an ordinary time-independent N -body Hamiltonian. Otherwise the Hamiltonian h(t) depends

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467

˜ (t, s), UCM (t, s) and U (t, s) the non-trivially on the time-variable t. We denote by U ˜ dynamics generated by h(t), hCM (t) and h(t), respectively, and observe that ˜ (t, s) = UCM (t, s) ⊗ U (t, s). U We shall address spectral properties of the monodromy operator U(1,0). Note that this is a unitary operator on L2 (X). Let A be the set of all cluster partitions a = {C1 , . . . , C#a }, 1 ≤ #a ≤ N , each given by splitting the set of particles {1, . . . , N } into non-empty disjoint clusters Ci . The spaces Xa , a ∈ A, are the spaces of conﬁgurations of the #a centers of mass of the clusters Ci (in the center of mass frame). The complement X a = X C1 ⊕ · · · ⊕ X C#a is the space of relative conﬁgurations within each of the clusters Ci . More precisely X Ci = {x ∈ X | xj = 0, j ∈ / Ci } and Xa = {x ∈ X | k, l ∈ Ci ⇒ xk = xl }. We will write xa and xa for the orthogonal projection of a vector x onto the subspace X a and its orthogonal complement respectively. Notice the natural ordering on A: a ⊂ b if and only if any cluster C ∈ a is contained in some cluster C ∈ b. Clearly the minimal and maximal elements are amin = {(1), . . . , (N )} and amax = {(1, . . . , N )}, respectively. Any pair (i, j) deﬁnes an N − 1 cluster decomposition (ij) ∈ A by letting C = {i, j} constitute a cluster and all others being one-particle clusters. For each a = amax the sub-Hamiltonian monodromy operator is U a (1, 0); it is deﬁned as the monodromy operator on Ha = L2 (X a ) constructed for a = amin from ha = (pa )2 − E(t)a · xa + V a , V a = (ij)⊂a vij (xi − xj ). If a = amin we deﬁne 1 U a (1, 0) = 1 (implying σpp (U amin (1, 0)) = {1}). The condition 0 E(t)dt = 0 leads to the existence of a unique 1-periodic function b such that 1 d b(t) = E(t) and b(t)dt = 0. dt 0 The set of thresholds is F (U (1, 0)) = e−iαa σpp (U a (1, 0)); a =amax

αa =

1

|b(t)a |2 dt.

(1.20)

0

We recall from [31] that the set of thresholds is closed and countable, and non-threshold eigenvalues, i.e. points in σpp (U (1, 0))\F (U (1, 0)), have ﬁnite multiplicity and can only accumulate at the set of thresholds. Moreover any corresponding bound state is exponentially decaying, the singular continuous spectrum σsc (U (1, 0)) = ∅ and there are integral propagation estimates for states localized away from the set of eigenvalues and away from F (U (1, 0)). These properties are known under Conditions 1.5 with k0 = 1. For completeness of presentation we mention that some of the results of [31] hold under more general conditions, in particular the exponential decay result does not require that the Coulomb singularity of each pair-potential (if present) is located at the origin (this applies to Born–Oppenheimer molecules in an AC-Stark ﬁeld).

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Letting A(t) =

1 (x · (p − b(t)) + (p − b(t)) · x), 2

(1.21)

and using a diﬀerent frame, we prove in Sec. 6: Theorem 1.6. Suppose Conditions 1.5, for some k0 ∈ N. Let φ be a bound state / F(U (1, 0)). Then for U (1, 0) pertaining to an eigenvalue e−iλ ∈ (1) φ ∈ D(A(1)k0 ) where A(t) is given by (1.21). 2 = 0 then φ ∈ D(|p|k0 +1 ). (2) If for all pairs (i, j) the term vij The result (1) is new for k0 > 1 while it is essentially contained in [31] for k0 = 1, see [31, Proposition 8.7(ii)]. We remark that the highest degree of smoothness known 2 = 0 is φ ∈ D(|p|), cf. [31, Theorem 1.8]. This holds without in general in the case vij the non-threshold condition. The result (2) overlaps with [29, Theorem 1.2], when N = 2 and “k0 = ∞”. 2. Assumptions and Statement of Regularity Results For a self-adjoint operator A on a Hilbert space H, we will make use of the C 1 (A) class of operators. This class consists a priori of bounded operators B with the property that [B, A] extends from a form on D(A) to a bounded form on H. The class is (consistently) extended to self-adjoint operators H, by requiring that (H − z)−1 is of class C 1 (A), for some (and hence all) z ∈ ρ(H), the resolvent set of H. We will use the notation H ∈ C 1 (A) to indicate that an operator H is of class C 1 (A). If H is of class C 1 (A) then D(H) ∩ D(A) is dense in D(H) and the form [H, A] extends by continuity from the form domain D(H) ∩ D(A) to a bounded form on D(H). The extension is denoted by [H, A]0 , and is also interpreted as an element of B(D(H), D(H)∗ ). If in addition [H, A]0 extends by continuity to an element of 1 1 (A). Note that being of class CMo (A) is B(D(H), H), then we say it is of class CMo equivalent to having the conditions of Mourre [30] satisﬁed for the ﬁrst commutator. See [20]. Conditions 2.1. Let H be a complex Hilbert space. Suppose there are given some self-adjoint operators H, A and N as well as a symmetric operator H with D(H ) = D(N ). Suppose N ≥ 1. Let R(η) = (A − η)−1 for η ∈ C\R. 1 (A). We abbreviate N = i[N, A]0 . (1) The operator N is of class CMo 1 (2) The operator N is of class C (H), and there exists 0 < κ ≤ 12 such that the commutator obeys 1

1

i[N, H]0 ∈ B(N − 2 +κ H, N 2 −κ H).

(2.1)

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(3) There exists a (large) σ > 0 such that for all η ∈ C with |Im η| ≥ σ we have as a form on D(H) ∩ D(N 1/2 ) i[H, R(η)] = −R(η)H R(η).

(2.2)

(Here it should be noticed that N −1/2 H N −1/2 and N ∓1/2 R(η)N ±1/2 are bounded if σ is large enough, cf. Remark 2.4(1).) (4) The commutator form i[H , A] deﬁned on D(A) ∩ D(N ) extends to a bounded operator 1

1

H := i[H , A]0 ∈ B(N − 2 H, N 2 H).

(2.3)

Condition 2.2. There are constants C1 , C2 , C3 ∈ R such that as a form on D(H)∩ D(N 1/2 ) N ≤ C1 H + C2 H + C3 1.

(2.4)

Condition 2.3. For a given λ ∈ R there exist c0 > 0, C4 ∈ R, fλ ∈ Cc∞ (R) with 0 ≤ fλ ≤ 1 and fλ = 1 in a neighborhood of λ, and a compact operator K0 on H such that as a form on D(H) ∩ D(N 1/2 ) H ≥ c0 1 − C4 fλ⊥ (H)2 H − K0 .

(2.5)

Here fλ⊥ := 1 − fλ . Remarks 2.4. (1) It follows from Condition 2.1(1) and an argument of Mourre [30, Proposition II.3], that there exists σ > 0 such that for |Im η| ≥ σ we have (A − η)−1 : D(N ) ⊆ D(N ) and (A − η)−1 D(N ) is dense in D(N ). By interpolation the same holds with N replaced by N α , 0 < α < 1, cf. Lemma 3.4 below. 1 (H). (2) From Condition 2.1(2) and Lemma 3.2 it follows that N 1/2 is of class CMo 1/2 1/2 In particular D(H) ∩ D(N ) is dense in D(N ). (3) Combining the above two remarks with Condition 2.1(3) and (3.14), we ﬁnd that given H, A and N , there can at most be one H such that Conditions 2.1(1)–2.1(3) are satisﬁed. (4) We remark that in practice we work with the weaker commutator estimate H ≥ c0 1 − Re{B(H − λ)} − K0 ,

(2.6)

where B = B(λ) is a bounded operator, with BD(N 1/2 ) ∪ B ∗ D(N 1/2 ) ⊆ D(N 1/2 ). The one in Condition 2.3 is however more standard. To see that Condition 2.3 implies the above bound choose B = C4 fλ⊥ (H)2 H(H − λ)−1 which under our Condition 2.1 satisﬁes the requirements on B by Lemma 3.3. We call H the first derivative of H. Similarly H is the second derivative of H. The estimate (2.4) is called the virial estimate, while (2.5) is the Mourre estimate at λ.

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Theorem 2.5. Suppose Conditions 2.1–2.3, and let ψ be a bound state, (H −λ)ψ = 0 (with λ as in Condition 2.3), obeying 1

ψ ∈ D(N 2 ).

(2.7)

Then ψ ∈ D(A) and Aψ ∈ D(N 1/2 ). By imposing assumptions on higher-order commutators between H and A we obtain a higher-order regularity result. For this we need the following condition, which coincides with Condition 2.1(4) if k0 = 1, but for k0 ≥ 2 it is stronger. Condition 2.6. There exists k0 ∈ N such that the commutator forms i adA (H ) deﬁned on D(A) ∩ D(N ), = 0, . . . , k0 , extend to bounded operators i adA (H ) ∈ B(N −1 H, H); ik0 adkA0 (H )

∈ B(N

− 12

= 0, . . . , k0 − 1.

(2.8)

1 2

H, N H).

(2.9)

We have the following extension of Theorem 2.5 to include higher orders: Theorem 2.7. Suppose Conditions 2.1–2.3 and Condition 2.6, and let ψ be a bound state, (H −λ)ψ = 0 (with λ as in Condition 2.3), obeying (2.7). Let k0 be given as in Condition 2.6. Then ψ ∈ D(Ak0 ), and for k = 1, . . . , k0 the states Ak ψ ∈ D(N 1/2 ). It should be noted that under the assumptions imposed in Theorems 2.5 and 2.7, it is crucial that N 1/2 is applied after the powers of A. The following result requires an additional assumption, and allows for arbitrary placement of N 1/2 amongst the at most k0 powers of A. The new condition (2.10) below is a generalization of Condition 2.1(1). Condition 2.8. Let N be given as in Condition 2.1(1). There exists k0 ∈ N such that the commutator forms i adA (N ) deﬁned on D(A) ∩ D(N ), = 0, . . . , k0 − 1, extend to bounded operators i adA (N ) ∈ B(N −1 H, H);

= 0, . . . , k0 − 1.

(2.10)

Moreover there exists κ1 > 0 such that the commutators (initially deﬁned as forms on D(N )) i adN (i adA (N )) ∈ B(N −1 H, N 1−κ1 H);

= 0, . . . , k0 − 1.

(2.11)

We have Corollary 2.9. Suppose Conditions 2.1–2.3, 2.6 and 2.8 (with the same k0 in Conditions 2.6 and 2.8). Let ψ ∈ D(N 1/2 ) be a bound state, (H − λ)ψ = 0 (with λ as in Condition 2.3). For any k, ≥ 0, with k + ≤ k0 , we have ψ ∈ D(Ak N 1/2 A ).

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We end with the following improvement of Theorem 2.5, which concludes in addition that bound states are in the domain of N . It requires the added assumption (2.11), with k0 = 1. Theorem 2.10. Suppose Conditions 2.1–2.3 and (2.11) for k0 = 1, and let ψ ∈ D(N 1/2 ) be a bound state (H − λ)ψ = 0 (with λ as in Condition 2.3). Then ψ ∈ D(N ), the states ψ, N 1/2 ψ ∈ D(A) and Aψ ∈ D(N 1/2 ). In Sec. 4.3, we in fact prove an extension of the above theorem, to include higher order estimates in N . These are applied in Sec. 6 to many-body AC-Stark Hamiltonians. Remarks 2.11. (1) The condition that N ≥ 1 is imposed partly for convenience of formulation. Obviously one can obtain a version of the above results upon imposing only that N is bounded from below (upon “translating” N → N + C ≥ 1 at various points in the above conditions). (2) The “standard” or “regular” Mourre theory, considered for example in [9], ﬁts in the semi-bounded case into the above scheme so that Theorem 2.7 holds. In fact (assuming here for simplicity that H is bounded from below) we have N := H + C ≥ 1 for a suﬃciently large constant C. Use this N and the same “conjugate operator” A in Conditions 2.1–2.3, 2.6 and 2.8. Note also that the standard Mourre estimate at energy λ reads fλ (H)i[H, A]0 fλ (H) ≥ c0 fλ2 (H) − K0 ;

c0 > 0,

K0 compact.

(2.12)

From (2.12) we readily conclude (2.5) with c0 = c0 /2, K0 = K0 an a suitable constant C4 ≥ 0. Although we shall not elaborate we also remark that the method of proof of Theorem 2.7 essentially can be adapted under the conditions of the standard Mourre theory, in fact only a simpliﬁed version is needed. Whence although we cannot literately conclude from Theorem 2.7 in the general non-semi-bounded case the result ψ ∈ D(Ak0 ) is still valid given standard conditions on repeated commutators ik adkA (H) for k ≤ k0 + 1. (3) Theorem 2.7 does not hold with one less commutator in Condition 2.6. Alternatively, under the conditions of Theorem 2.7 it is in general false that the bound state ψ ∈ D(Ak0 +1 ). Based on considerations for discrete eigenvalues this statement may at a ﬁrst thought appear surprising. See Example 1.1. Compared to [9] our method works with one less commutator, cf. (2), although the overall scheme of ours and the one of [9] are similar. (4) The proofs of Theorems 2.5 and 2.7, Corollary 2.9 and Theorem 2.10 are constructive in that they yield explicit bounds. Precisely, if we have a positive lower bound of the constant c0 in (2.5) that is uniform in λ belonging to some ﬁxed compact interval I as well as uniform bounds of the absolute value of the constants C1 , . . . , C4 of (2.4) and (2.5) (uniform in the same sense) and similarly for all possible operator norms related to Conditions 2.1, 2.6 and 2.8

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(and the B(λ) in Remark 2.4 if it is used) then there are bounds of the form, for example, 1

1

N 2 Ak ψ ≤ CN 2 ψ;

C = C(k, I, K0 );

here K0 = K0 (λ) is the compact operator of (2.5) and k ≤ k0 . Similar bounds are valid for the states Ak N 1/2 A ψ of Corollary 2.9 and for the state N ψ of Theorem 2.10. In the context of perturbation theory typically I will be a small interval centered at some (unperturbed) embedded eigenvalue λ0 and K0 = K0 (λ0 ). Whence the constant will depend only on the interval. For various models one can verify the condition (2.7) for all bound states ψ by a “virial argument”, cf. [22, 31, 38], along with a similar bound 1

N 2 ψ ≤ C(I)ψ. This virial argument is in a concrete situation related to the virial estimate (2.4). Clearly the above bounds can be used in combination, and this is precisely how we in Sec. 5 arrive at the Theorems 1.2 and 1.3. In [32] the case of regular Mourre theory is considered where the derivation of the bounds is simpler, and care is taken to derive good explicit bounds, which in particular are independent of any proof technical constructions. The bounds are good enough to formulate a reasonable condition on the growth of norms of multiple commutators which ensures that bound states are analytic vectors with respect to A. 3. Preliminaries In this section, we establish basic consequences of Conditions 2.1, and introduce a calculus of almost analytic extensions taylored to avoid issues with (A − η)−1 , when |Im η| is small. 3.1. Improved smoothness for operators of class C 1 (A) For an operator N of class C 1 (A) not much in the way of regularity can be expected, beyond the C 1 (A) property itself, and its equivalent formulations. See [3, 21]. Often one requires some additional smoothness properties to manipulate and estimate expressions in the two operators. The typical way of achieving improved smoothness is to impose conditions on i[N, A]0 stronger than what is implied by the C 1 (A) property itself. This is what is done in Conditions 2.1(1) and 2.1(2). This subsection is devoted primarily to the extraction of improved smoothness properties of the pair of operators N, H, aﬀorded to us by Conditions 2.1. Lemma 3.1. Let N ≥ 1 be of class C 1 (H) with [N, H]0 ∈ B(N −1/2 H, N 1/2 H). For any α ∈ ]0, 1[, the operator N α is of class C 1 (H).

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Proof. Let 0 < α < 1. It suﬃces to check for one η ∈ ρ(N α ) that (N α − η)−1 is of class C 1 (H). To this end we pick η = 0, and use the representation formula ∞ sin(απ) . (3.1) t−α (N + t)−1 dt, cα = N −α = cα π 0 Since N ∈ C 1 (H) we have for all t > 0 that the operator (N + t)−1 preserves D(H). In fact [H, (N + t)−1 ]φ = (N + t)−1 [N, H]0 (N + t)−1 φ;

φ ∈ D(H).

(3.2)

By combining (3.1) and (3.2), we can compute [N −α , H] considered as a form on D(H) as ∞ t−α (N + t)−1 [N, H]0 (N + t)−1 dt. (3.3) [N −α , H] = cα 0

Notice that the integral is absolutely convergent for any 0 < α < 1. This completes the proof. Lemma 3.2. Assume N ≥ 1 and H satisfy Condition 2.1(2) and let α ∈ ]0, 1[. Then N α ∈ C 1 (H) and for τ1 , τ2 ≥ 0, with

1 1 max 0, − κ − τ1 + max 0, − κ − τ2 < 1 − α, 2 2 1 (H). we have [N α , H]0 ∈ B(N −τ1 H, N τ2 H). In particular N 1/2 is of class CMo

Proof. That N α ∈ C 1 (H) follows from Lemma 3.1. We compute as a form on D(N α ) ∩ D(H) ∞ [N α , H] = cα tα (N + t)−1 [N, H]0 (N + t)−1 dt, (3.4) 0

where we have used the strongly convergent integral representation formula ∞ N α = cα tα (t−1 − (N + t)−1 )dt, (3.5) 0

which follows from (3.1). We thus get for τ1 , τ2 ≥ 0 ∞ 1 1 tα (N + t)−1 N 2 −κ−τ1 (N + t)−1 N 2 −κ−τ2 dt |ψ, [N α , H]ϕ| ≤ C 0

× N τ1 ψN τ2 ϕ. The integrand is of the order O(tα−2+θ ), with θ = max{0, 12 − κ − τ1 } + max{0, 12 − κ − τ2 }. It is integrable provided θ < 1 − α, which proves the lemma.

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We shall need a boundedness result: Lemma 3.3. Assume N ≥ 1 and H satisfy Condition 2.1(2) and let α ∈ ]0, 1/2+κ[. Suppose f ∈ C ∞ (R) is given such that dk f (t) = O(t−k ); dtk

k = 0, 1, . . .

Then N α f (H)N −α ∈ B(H).

(3.6)

Proof. Let ρ ∈ ]0, 1/2 + κ[, where 0 < κ ≤ 1/2 comes from Condition 2.1(2). From Lemma 3.2 applied with τ1 = max{0, ρ − κ} and τ2 = 0, we get [N ρ , H]0 ∈ B(N − max{0,ρ−κ} H, H).

(3.7)

is of class C 1 (H), We recall from [30, Proposition II.3] that if an operator N Mo then ∃σ > 0 : |Im η| ≥ σ ⇒ (H − η)−1

) and preserves D(N

(H − η)−1 ψ = (H − η)−1 Nψ N H]0 (H − η)−1 ψ + i(H − η)−1 i[N,

). for all ψ ∈ D(N

(3.8)

= N ρ , 0 < ρ < 1/2 + κ. The assumption is satisﬁed by (3.7). We apply this to N We shall show a representation formula for the special case f (x) = fη (x) = (x − η)−1 with v = Im η = 0. Now ﬁx α ∈ ]0, 1/2 + κ[. Using (3.7) and (3.8), multiple times with ρ = α − jκ, we obtain for |Im η| suﬃciently large and for all ψ ∈ D(N α ) N α (H − η)−1 ψ − (H − η)−1 N α ψ =

n

((H − η)−1 B1 ) · · · ((H − η)−1 Bj )(H − η)−1 N α−jκ ψ

j=1

+ ((H − η)−1 B1 ) · · · ((H − η)−1 Bn )((H − η)−1 Bn+1 )(H − η)−1 ψ,

(3.9)

where n is the biggest natural number for which α−nκ > 0 and the Bj ’s are bounded and independent of η. Next by analytic continuation we conclude that (3.9) is valid for all η ∈ C\R. Hence we have veriﬁed the adjoint version of (3.6) for f = fη ; v = 0. We shall now show (3.6) in general. Deﬁne a new function by h(t) = f (t)(t+i)−1 , ˜ denote an almost analytic extension of h such that (using the notation and let h η = u + iv) ˜ ∀n ∈ N : |∂¯h(η)| ≤ Cn η−n−2 |v|n .

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We shall use the representation 1 ˜ f (H) = (∂¯h)(η)(H − η)−1 (H + i)dudv π C 1 ˜ (∂¯h)(η)(1 + (η + i)(H − η)−1 )dudv, = π C

475

(3.10)

which should be read as a strong integral on D(H). We multiply by N α and N −α from the left and from the right, respectively. Inserting (3.9) we conclude the lemma. Observe that N −α being C 1 (H) preserves D(H). It will be important to work with the following “regularization” operators, ˜ cf. [30]: Let for any given self-adjoint operator A˜ and any positive operator N ˜ = −in(A˜ − in)−1 In (A)

˜ ) = n(N ˜ + n)−1 ; and Iin (N

n ∈ N.

(3.11)

In particular we shall use In (A) in conjunction with (2.2), In (H) in conjunction with (2.2), (2.4) and (2.5), while Iin (N ) will be used in conjunction with (2.1). Lemma 3.4. Assume the pairs N, A and N, H satisfy Conditions 2.1(1) and 2.1(2) respectively. Then 1

1

s-lim N 2 In (H)N − 2 = 1,

(3.12)

s-lim N In (A)N −1 = 1,

(3.13)

n→∞ n→∞

1

1

s-lim N 2 In (A)N − 2 = 1. n→∞

(3.14)

Proof. Observe ﬁrst that s-lim In (A) = 1 and s-lim A(A − in)−1 = 0, and similarly with A replaced by H. The statements (3.12) and (3.13) now follows from (3.8) and boundedness of the operators [N 1/2 , H]0 N −1/2 and [N, A]0 N −1 . This argument appears also in [30]. As for (3.14) we observe ﬁrst that N (In (A) − 1)N −1 is bounded uniformly in n. By interpolation the same holds true for N 1/2 (In (A) − 1)N −1/2 . The result now follows from observing that the result holds true strongly on the dense set D(N 1/2 ) by (3.13). We end with a small technical remark: Remark 3.5. Suppose N and H are as in Lemma 3.1 and 0 ≤ α < 1. Then D(H) ∩ D(N ) is dense in D(H) ∩ D(N α ) in the intersection topology. To see this let ψ ∈ D(H) ∩ D(N α ). Then ψn = Iin (N )ψ ∈ D(H) ∩ D(N ) since N is of class C 1 (H). We claim that ψn → ψ in D(H) ∩ D(N α ). Obviously N α ψn → N α ψ, so it remains to consider N N − 12 0 − 12 Hψn = Iin (N )Hψ + Iin (N )(N [N, H] N ) Iin (N )ψ. n n

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As in the proof above, the last term goes to zero and the ﬁrst term converges to Hψ proving the claim. 3.2. Iterated commutators with N 1/2 We address here the following question. Supposing Conditions 2.1(1) and (2.10) is satisﬁed for some k0 ≥ 1. One could reasonably assume that N 1/2 is also of class 1 (A) and admits k0 iterated N 1/2 -bounded commutators. We have however not CMo been able to establish this, but making the additional assumption (2.11) we answer the question in the aﬃrmative below. This permits us to deduce Corollary 2.9 from Theorem 2.7. The reader primarily interested in Theorem 2.7 may skip this subsection. We begin with a technical lemma. Let q ∈ N and ∈ (N∩{0})q , with 0 ≤ j < k0 for all j = 1, . . . , q. We abbreviate Nm = im adm A (N ), which is the iteratively deﬁned N -bounded operator from (2.10). Let for t ≥ 0 and q, as above   q 1 (3.15) 4Bq (t) = t 2  (N + t)−1 Nj  (N + t)−1 . j=1

Observe that Bq (t) is bounded for all t. Indeed it satisﬁes the bound Bq (t) = O(t−1/2 ) and is thus not norm integrable. However if ϕ ∈ D(N ) we have Bq (t)ϕ = O(t−3/2 ). The extra assumption (2.11) allows us to prove Lemma 3.6. Suppose Condition 2.1(1) and Condition 2.8. For any q ∈ N, ∈ (N ∪ {0})q (with 0 ≤ j < k0 as above) and ϕ ∈ D(N ) the map t → Bq (t)ϕ is integrable and there exist constants Cq such that ∞ ≤ Cq N 12 ϕ. B (t)ϕ dt q 0

Proof. We only have to prove the bound on the strong integral, since we already discussed strong integrability. We begin by analyzing the leftmost factors in Bq (t), −1 namely the N -bounded operator (N + t) N1 . We compute strongly on D(N ) (N + t)−1 N1 = (N1 N −1 )N (N + t)−1 − (N + t)−1 N (N −1 [N, N1 ]N −1+κ1 )N 1−κ1 (N + t)−1 = (N1 N −1 )N (N + t)−1 + O(t−κ1 ). (3.16) ∞ The contribution to the integral 0 Bq (t)N −1/2 dt coming from the last term is O(t−1−κ1 ) and hence norm-integrable. If q = 1 we can now ﬁnish the argument because the contribution to the integral coming from the ﬁrst term on the right-hand side of (3.16) is 1

(N1 N −1 )t 2 (N + t)−2 N,

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which on the domain of N integrates to the N 1/2 -bounded operator cN1 N −1/2 , for some c ∈ R. If q > 1 we write N (N + t)−1 = 1 − t(N + t)−1 . We can now bring out the next term N2 , and again the commutators with (N + t)−1 give norm-integrable contributions. Repeating this procedure successively until all the terms Nj are brought out to the left yields the formula   q 1 1 Nj N −1  t 2 (1 − t(N + t)−1 )q−1 (N + t)−2 N + O(t−1−κ1 )N 2 . Bq (t) =  j=1

We compute, by a change of variables, ∞ 1 1 t 2 (1 − t(N + t)−1 )q−1 (N + t)−2 dt = c N − 2 , 0

for some c ∈ R. This implies the lemma. Proposition 3.7. Assume Condition 2.1(1) and Condition 2.8. Then N 1/2 is of 1 (A) and the iterated commutators ip adpA (N 1/2 ), p ≤ k0 , extends from class CMo D(A) ∩ D(N 1/2 ) to N 1/2 -bounded operators. Proof. We already know from Lemma 3.1 that N 1/2 is of class C 1 (A). Hence we only need to establish that the iterated commutator forms extend to N 1/2 -bounded operators. Recall also that D(A)∩D(N ) is dense in D(A)∩D(N 1/2 ), cf. Remark 3.5, which implies that it suﬃces to show that the iterated commutator forms extend from D(A) ∩ D(N ) to N 1/2 -bounded operators. By Lemma 3.6 and the above remark, it suﬃces to prove, iteratively, the following representation formula p

i

adpA (N

1 2

)ϕ =

p

αp,q

q=1 1 +···+q =p−q

∞

Bq (t)ϕdt,

(3.17)

0

for ϕ ∈ D(N ). Note that the integrals are absolutely convergent. Here Bq (t) are deﬁned in (3.15). For p = 1, we compute using (3.5) ∞ 1 0 B1,n (t)ϕdt, i[An , N 2 ]ϕ = c 12 0

where the extra subscript n indicates that N0 = N has been replaced by In (A)N In (A). By (3.22) the integrand is O(t−3/2 ) uniformly in large n, and by (3.13) and Lebesgue’s theorem on dominated convergence we can thus compute ∞ 1 0 B1 (t)ϕdt. lim i[An , N 2 ]ϕ = c 12 n→∞

0

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Obviously this together with Lemma 3.6 implies that the form i adA (N 1/2 ) extends from D(A) ∩ D(N ) to an N 1/2 -bounded operator represented on D(N ) by the strongly convergent integral above. We can now proceed by induction, assuming that the iterated commutator p−1 p−1 adA (N 1/2 ) exists as an N 1/2 -bounded operator and is represented on D(N ) i p−1 (N 1/2 )] strongly on D(N ) by (3.17). Compute ﬁrst the commutator i[An , ip−1 adA using that In (A) i[An , N ] = −In (A)N+1

and i[An , (N + t)−1 ] = (N + t)−1 N (N + t)−1 .

Subsequently take the limit n → ∞ as above and appeal to Lemma 3.6 to conclude that the so computed limit in fact is an N 1/2 -bounded extension of the form p−1 (N 1/2 )] from D(A) ∩ D(N ) and represented on D(N ) as in (3.17). i[A, ip−1 adA Proof of Corollary 2.9. We can now argue that Corollary 2.9 is indeed a direct corollary of Theorem 2.7. Note that ψ ∈ D(N 1/2 Ak ) for all k ≤ k0 due to Theorem 2.7. We can now repeatedly use the fact that D(A) ∩ D(N 1/2 ) is dense in D(A) and Proposition 3.7 to compute for ϕ ∈ D(Ap ), with p + k ≤ k0 , 1

Ap ϕ, N 2 Ak ψ =

p

1

1

1

p−q βq ϕ, (adA (N 2 )N − 2 )N 2 Aq+k ψ,

q=0

with βq some real combinatorial factors. This completes the proof since the norm of the right-hand side is bounded by Cϕ. 3.3. Approximating A by regular bounded operators We recall now a construction from [31] (see [31, p. 203]). Consider an odd realvalued function g ∈ C ∞ (R) obeying g ≥ 0, that the function R t → tg (t)/g(t) has a smooth square root, that the function ]0, ∞[ t → g(t) is concave and the properties  for t > 3,  2 g(t) = t for |t| < 1,   −2 for t < −3. ˜ such Let h(t) = g(t)/t. We pick an almost analytic extension of h, denoted by h, that for some ρ > 0 (and using again the notation η = u + iv) ˜ ∀N : |∂¯h(η)| ≤ CN η−N −2 |v|N , 2/η for u > 6, |v| < ρ(u − 6), ˜ h(η) = −2/η for u < −6, |v| < ρ(6 − u). ˜ ˜ η ). ˜ such that h(η) = h(¯ We can choose h

(3.18)

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This gives the representation 1 ˜ g(t) = (∂¯h)(η)t(t − η)−1 du dv. π C

479

(3.19)

Let gm (t) = mg(t/m), for m ≥ 1. Using the properties of g one veriﬁes that for all t ∈ R the function m → gm (t)2 is increasing. We recall that there exists σ > 0 such that for |v| ≥ σ/m the operator

−1 A −η Rm (η) := m

(3.20)

(3.21)

preserves D(N ). See (3.8). Moreover we have uniformly in α ∈ [0, 1], m ∈ N and η that N α Rm (η)N −α ≤ C|v|−1 ;

η ∈ Vm> ,

(3.22)

where Vm> := {u + iv ∈ C : |v| ≥ σ/m}

and Vm< := {u + iv ∈ C : |v| < σ/m}.

This motivates the decomposition into smooth bounded real-valued functions gm = g1m + g2m , where

−1 t m ˜ g1m (t) = −η (∂¯h)(η) 1+η (3.23) dudv + Cm , π Vm> m m g2m (t) = π Cm

m = π

˜ (∂¯h)(η)η < Vm

t −η m

−1 dudv;

(3.24)

˜ ∂¯h(η)dudv. < Vm

Note that the integral in the expression for g2m is over a compact set (decreasing with m). This implies the property sup m∈N,t∈R

(k)

mn tk+1 |g2m (t)| ≤ Cn,k < ∞ for n, k ∈ N ∪ {0}.

(3.25)

Since gm and g2m are bounded functions, we conclude the same for g1m . At a key point in the proof we will need a smooth square root of the function tg g. We pick gˆ = pg ∈ C0∞ (R),

(3.26) where p(t) = tg (t)/g(t), which was assumed smooth. Clearly gˆ2 = tg g. Let p˜ ∈ C0∞ (C) be an almost analytic extension of p. It satisﬁes ∀N : |∂¯p˜(η)| ≤ CN |v|N .

(3.27)

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As above we put pm (t) = p(t/m) and make the splitting pm = p1m + p2m , where

−1 t 1 p1m (t) = −η (∂¯p˜)(η) dudv, (3.28) π Vm> m

−1 1 t ¯ −η p2m (t) = (∂ p˜)(η) dudv. (3.29) π Vm< m Let gˆm = pm gm and split gˆm = gˆ1m + gˆ2m by gˆ1m = p1m g1m

and gˆ2m = pm g2m + p2m g1m .

(3.30)

Clearly we can choose Cn,k in (3.25) possibly larger such that gˆ2m satisﬁes the same estimates. Since pm and p2m are uniformly bounded in m we get P := sup sup |p1m (t)| < ∞.

(3.31)

m∈N t∈R

We observe that the operators g2m (A) and p1m (A), p2m (A) are given by norm convergent integrals, whereas gm (A) and g1m (A) are given on the domain of As , for any s > 0, as strongly convergent integrals. From (3.20) and Lebesgue’s theorem on monotone convergence, we observe that ψ ∈ D(Ak ) is equivalent to supm gm (A)k ψ < ∞. Combining this with (3.25) we ﬁnd that for k ≥ 1 ψ ∈ D(Ak ) ⇔ ψ ∈ D(Ak−1 ) and

sup g1m (A)k ψ < ∞.

(3.32)

m

It will be convenient in the following when dealing with g1m to abbreviate 1 ¯˜ (∂ h)(η)dudv. π This is however not a complex measure, just a notation. Similarly we will on one occasion write dλp (η) = π1 (∂¯p˜)(η)dudv, which is in fact a complex measure. We have the following dλ(η) =

Lemma 3.8. As a result of the above constructions we have for any m ≥ 1 and (A), p1m (A) and Ag1m (A) 0 ≤ α ≤ 1 that the bounded operators g1m (A), g1m α preserve D(N ). Proof. Let ψ ∈ D(N ) and ϕ ∈ D(A). Observe that N −1 ϕ ∈ D(A), by the C 1 (A) property of N , cf. Condition 2.1(1). We can thus compute using the strongly convergent integral representation for g1m (A), and the notation introduced in (3.21), N ψ, g1m (A)N −1 ϕ =m N ψ, (1 + ηRm (η))N −1 ϕdλ(η) + Cm ψ, ϕ > Vm

= ψ, g1m (A)ϕ + i

> Vm

ηψ, Rm (η)N Rm (η)N −1 ϕdλ(η).

(3.33)

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By Condition 2.1(1), (3.18) and (3.22) we ﬁnd that for some constant Km we have |N ψ, g1m (A)N −1 ϕ| ≤ Km ψϕ.

(3.34)

This together with an interpolation argument concludes the proof. The cases g1m (A) and p1m (A) are done the same way. As for Ag1m (A) we write Aj = AIj (A) and compute (A)N −1 = Aj N g1m (A)N −1 − iIj (A)N N −1 N Ij (A)g1m (A)N −1 . N Aj g1m

To complete the proof by taking j → ∞ we need to argue that (A)D(N ) ⊆ D(A). N g1m

To achieve this we repeat the computation (3.33), with ψ replaced by Aψ, ψ ∈ D(A), . We get and g1m replaced by g1m (A)N −1 ϕ Aψ, N g1m (A)ϕ = ψ, Ag1m A 2 2 −1 ψ, {Rm (η)N Rm (η) + Rm (η) N Rm (η)}N ϕ dλ(η). η + > m Vm

The result now follows from writing and (3.22) as above.

A m Rm (η)

= 1 + ηRm (η)and appealing to (3.18)

4. Proof of the Abstract Results In this section we prove the abstract theorems formulated in Sec. 2 as well as an extended version of Theorem 2.10. The proofs are given in separate subsections. 4.1. Proof of Theorem 2.7 Let 1

Dk = {ϕ ∈ D(Ak )|∀0 ≤ j ≤ k : Aj ϕ ∈ D(N 2 )}. Using Conditions 2.1–2.3 and 2.6 we shall prove Theorem 2.7 by induction in k = 0, . . . , k0 that ψ ∈ Dk . We can assume without loss of generality that λ = 0. The proof relies on three estimates which we state ﬁrst in the form of three propositions. After giving the proof of Theorem 2.7, we then proceed to verify the propositions. We begin with some abbreviations and a deﬁnition. For a state ψ we introduce the notation ψm = g1m (A)k ψ

and ψˆm = gˆ1m (A)g1m (A)k−1 ψ = p1m (A)ψm .

Let σ > 0 be ﬁxed as in Remark 2.4(1), applied with N 1/2 in place of N .

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Definition 4.1. Let k ≥ 1. A family of forms {Rm }∞ m=1 on Dk−1 will be called a k-remainder if for all > 0 there exists C > 0 such that 1

1

|ψ, Rm ψ| ≤ N 2 ψm 2 + C N 2 (A − iσ)k−1 ψ2 ,

(4.1)

for any ψ ∈ Dk−1 and m ∈ N. Lemma 3.8 is repeatedly used below, mostly without comment, to justify manipulations. The ﬁrst proposition is a virial result, to be proved by a symmetrization of a commutator between H and a regularized version of A2k+1 . Proposition 4.2. Let 0 < k ≤ k0 and ψ ∈ Dk−1 be a bound state for H. There exists a k-remainder Rm , such that ψm , H ψm + 2kψˆm , H ψˆm = ψ, Rm ψ. The second result is an implementation of the virial bound (2.4) in Condition 2.2, which together with Proposition 4.2 makes it possible to deal with N 1/2 ψm . This is reminiscent of what was done in the proof of [31, Proposition 8.2]. The constant C2 appearing in the proposition comes from Condition 2.2. Proposition 4.3. Let ψ ∈ Dk−1 be a bound state. There exists C independent of m such that 1

1

N 2 ψm 2 ≤ 2C2 ψm , H ψm + C(ψm 2 + N 2 (A − iσ)k−1 ψ2 ) and 1 1 N 2 ψˆm 2 ≤ 2C2 ψˆm , H ψˆm + C(ψˆm 2 + N 2 (A − iσ)k−1 ψ2 ).

The third and ﬁnal input is an implementation of the positive commutator estimate in Condition 2.3. The constant c0 and the compact operator K0 appearing in the proposition come from Condition 2.3. >0 Proposition 4.4. Let ψ ∈ Dk−1 be a bound state. There exist constants C, C independent of m such that c0 m , K0 ψm − CN 12 (A − iσ)k−1 ψ2 ψm , H ψm ≥ ψm 2 − Cψ 2 and ψˆm , H ψˆm ≥

1 c0 ˆ 2 ˆ ψm − Cψm , K0 ψˆm − CN 2 (A − iσ)k−1 ψ2 . 2

Proof of Theorem 2.7. Let ψ be the bound state, which we take to be normalized. By assumption ψ ∈ D0 . Assume by induction that ψ ∈ Dk−1 , for some k ≤ k0 . We proceed to show that ψ ∈ Dk : From Proposition 4.2 we get the existence of a k-remainder Rm such that ψm H ψm + 2kψˆm , H ψˆm = ψ, Rm ψ.

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Estimating the right-hand side using (4.1) and Proposition 4.3 we ﬁnd a C > 0 such that 1 c0 ψm , H ψm + 2kψˆm , H ψˆm ≤ ψm 2 + CN 2 (A − iσ)k−1 ψ2 . 4 Finally, we appeal to Proposition 4.4 to derive the bound 1 c0 m , K0 ψm + 2k C ψˆm , K0 ψˆm . (4.2) ψm 2 ≤ CN 2 (A − iσ)k−1 ψ2 + Cψ 4

Pick Λ > 0 large enough such that 0 1[|A|>Λ] ≤ 2CK

c0 , 12(1 + 2kP 2 )

where P is given by (3.31). Write 1[|A|≤Λ] ψm = [1[|A|≤Λ] (gm (A) − g2m (A))]k ψ and estimate using (3.25) [|A|≤Λ] ψm , K0 ψm | ≤ 2C(Λ + C0,0 )k K0 ψψm 2C|1 ≤

2 (Λ + C0,0 )2k K0 2 12C c0 ψm 2 + ψ2 12 c0

and similarly 2k 2 4 2 [|A|≤Λ] ψˆm , K0 ψˆm | ≤ c0 ψm 2 + 24k C (Λ + C0,0 ) K0 P ψ2 . 2C|1 24k c0

Inserting 1 = 1[|A|≤λ] + 1[|A|>λ] ahead of the K0 ’s in (4.2) and appealing to the bounds above we get 1 c0 ψm 2 ≤ C(N 2 (A − iσ)k−1 ψ2 + ψ2 ), 8

for a suitable m-independent C. Recalling (3.32) we conclude that ψ ∈ D(Ak ). It remains to prove that Ak ψ ∈ D(N 1/2 ). Note that what we just established implies that ψm → Ak ψ in norm, cf. (3.20) and (3.25). We can now compute Ak ψ, N Iin (N )Ak ψ = lim ψm , N Iin (N )ψm . m→∞

But by Propositions 4.2 and 4.3 we have 1

ψm , N Iin (N )ψm ≤ N 2 ψm 2 1 1 ≤ N 2 ψm 2 + 2kN 2 ψˆm 2

≤ 2C2 (ψm , H ψm + 2kψˆm , H ψˆm ) + C = ψ, Rm ψ + C, where C > 0 is constant independent of m. The result now follows from (4.1) by ﬁrst taking the limit m → ∞, and subsequently n → ∞. Notice that Lebesgue’s

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theorem on monotone convergence applies, since Iin (N ) = n(N + n)−1 → 1 monotonously. The rest of the section is devoted to establishing Propositions 4.2–4.4. We begin with a deﬁnition and a series of lemmata. The σ in the deﬁnition below is the same σ that entered into Deﬁnition 4.1. l r and Em be families of forms on Dk−1 × D(N 1/2 ) and Definition 4.5. Let Em 1/2 l is a left-error if D(N ) × Dk−1 respectively. We say that Em 1

1

l |ψ, Em ϕ| ≤ CN 2 (A − iσ)k−1 ψN 2 ϕ. r We say that Em is a right-error if 1

1

r |ψ, Em ϕ| ≤ CN 2 ψN 2 (A − iσ)k−1 ϕ.

Remark 4.6. An example of a right-error that we will encounter below are forms N 1/2 Bm N 1/2 g1m (A) (A − iσ)−j , with − j ≤ k − 1 and supm Bm < ∞. To see that this is a right-error observe that it suﬃces to prove that N g1m (A) (A − iσ)−j−k+1 N −1 is uniformly bounded in m. The result then follows from interpolation. Since j + k − 1 ≥ , recalling that σ was chosen according to (3.8), we reduce the problem to showing that N g1m (A)(A − iσ)−1 N −1 is bounded uniformly in m. But this follows by a computation similar to (3.33), where the extra resolvent produces a bound which is uniform in m compared with the point wise bound (3.34). We introduce the notation Hn := HIn (H) = in(In (H) − 1),

(4.3)

which plays the role of a regularized Hamiltonian. See (3.11) for the deﬁnition of In (H). Lemma 4.7. We have the following limit in the sense of forms on D(N 1/2 ) lim i[Hn , g1m (A)] = − ηRm (η)H Rm (η)dλ(η). n→∞

> Vm

Proof. Observe ﬁrst that the integral on the right-hand side in the lemma is norm convergent. Compute as a form on D(A) using that the integral representation for g1m (A) is strongly convergent on D(A) i[Hn , g1m (A)] = ηi[Hn , Rm (η)]dλ(η). > Vm

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Recalling (4.3) we arrive at

485

i[Hn , g1m (A)] = in

> Vm

= > Vm

ηi[In (H), Rm (η)]dλ(η)

ηIn (H)i[H, Rm (η)]In (H)dλ(η).

Finally we employ Condition 2.1(3) to conclude that for each n, the following holds as a form identity on D(A) ∩ D(N 1/2 ) ηIn (H)Rm (η)H Rm (η)In (H)dλ(η). i[Hn , g1m (A)] = − > Vm

The integral on the right-hand side of the above identity is absolutely convergent in B(N −1/2 H; N 1/2 H). By density of D(A) ∩ D(N 1/2 ) in D(N 1/2 ), see Remark 2.4(2), the identity therefore extends to a form identity on D(N 1/2 ). The lemma now follows from (3.12). Lemma 4.8. Let 1 ≤ k ≤ k0 . r ˆ r such that, as forms on D(N 1/2 ) × Dk−1 , (1) There exist right-errors Em ,E m r lim i[Hn , g1m (A)k ] = Em ,

n→∞

ˆr . lim i[Hn , gˆ1m (A)g1m (A)k−1 ] = E m

n→∞

l r (2) There exist a left-error Em and a right-error Em such that, as forms on Dk−1 × 1/2 1/2 D(N ) and D(N ) × Dk−1 respectively, l (A)H + Em , lim lim i[Hn , g1m (A)k ]Aj = kg1m (A)k−1 Ag1m

j→∞ n→∞

r (A)g1m (A)k−1 + Em . lim lim Aj i[Hn , g1m (A)k ] = kH Ag1m

j→∞ n→∞

Proof. (1) also holds if we take the limit in the sense of forms on Dk−1 × D(N 1/2 ) and replace the right-error by a left-error. We will however not need that statement. One does however need its proof for the left-error part of (2). In the proof we will only work with right-errors. The other case is similar. We begin with (1) and prove only the ﬁrst statement leaving the second to the reader. We ﬁrst compute as a form on D(N 1/2 ). i[Hn , g1m (A)k ] = ki[Hn , g1m (A)]g1m (A)k−1

k +1 k + (−1) i adg1m (A) (Hn )g1m (A)k− .

(4.4)

=2

We now analyze the large n limit. The ﬁrst term on the right-hand side of (4.4) can be dealt with using Lemma 4.7 directly, observing that by Lemma 3.8 g1m (A)

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preserves the domain of N 1/2 . As for the terms involving higher order commutators, we again use Lemma 4.7 to compute lim i adg1m (A) (Hn ) = − ηRm (η) ad−1 g1m (A) (H )Rm (η)dλ(η) n→∞

> Vm

in the sense of forms on D(N 1/2 ). We can now employ Condition 2.6 to compute as forms on D(N 1/2 ) 1

1

() N2, lim i adg1m (A) (Hn ) = (−1) N 2 Bm

n→∞ ()

(4.5)

()

where Bm is a family of bounded operators with supm Bm < ∞, for all . They are given by 1 () Bm = η1 · · · η N − 2 Rm (η1 ) · · · Rm (η ) ad−1 A (H ) > (Vm )

1

× Rm (η ) · · · Rm (η1 )N − 2 dλ(η1 ) · · · dλ(η ).

(4.6)

From (4.4), (4.5) and Lemma 4.7, we thus obtain lim i[Hn , g1m (A)k ] = −k ηRm (η)H Rm (η)dλ(η)g1m (A)k−1 n→∞

> Vm

−

k

k =2

1

1

() N 2 g1m (A)k− . N 2 Bm

(4.7)

Combining this computation with Remark 4.6 yields (1). We now turn to part (2) of the lemma. In view of (4.7) we begin by computing as a form on D(N 1/2 ), using Condition 2.1(4) −k ηRm (η)H Rm (η)dλ(η) > Vm

=

kH g1m (A)

ik − m

> Vm

ηRm (η)H Rm (η)2 dλ(η).

(4.8)

We remark that the identity i[H , Rm (η)] = −m−1 Rm (η)H Rm (η) holds a priori as a form identity on D(N ). It extends by continuity to a form identity on D(N 1/2 ), which is what is used in the above computation. Note that the integral on the right-hand side is convergent as a form on D(N 1/2 ). From (4.7), (4.8) and Remark 4.6, we ﬁnd that r (A)Ag1m (A)k−1 + Em )Ij (A) lim i[Hn , g1m (A)k ]Aj = (kH g1m

n→∞

and hence by (3.14) we conclude the following identity as forms on D(N 1/2 ) r (A)Ag1m (A)k−1 + Em . lim lim i[Hn , g1m (A)k ]Aj = kH g1m

j→∞ n→∞

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To prove the second statement in (2) it remains to show that the commutator between Aj and i[Hn , g1m (A)k ] converges to a right-error. From (4.8) we get, as a form on D(N 1/2 ), ! ηRm (η)H Rm (η)dλ(η), Aj −k > Vm

= kIj (A)H

ik − m

Ij (A)g1m (A)

> Vm

ηRm (η)(H Aj − Aj H )Rm (η)2 dλ(η).

We can now take the limit j → ∞ and obtain ! 1 1 (1) lim −k ηRm (η)H Rm (η)dλ(η), Aj = N 2 Bm N 2, j→∞

(4.9)

> Vm

(1)

(1)

where Bm , is a family of bounded operators with supm Bm < ∞. It is given by

1 (1) − 12 η(Rm (η)H − H Rm (η))Rm (η)dλ(η) N − 2 . Bm = kN H g1m (A) − i > Vm

Here we used (3.14), that Aj Rm (η) = Rm (η)Aj = m(1 + ηRm (η))Ij (A), as well as Lebesgue’s theorem on dominated convergence. For the commutator between Aj and the second term on the right-hand side of (4.7) we compute 1

1

1

1

() ˜ () N 2 Ij (A), N 2 , Aj ] = Ij (A)N 2 B [N 2 Bm m () () ˜m are bounded operators with supm∈N Bm < ∞, for all . They are where B given by 1 ˜ () = B N − 2 Rm (η1 ) · · · Rm (η ) adA (H ) m > (Vm )

1

× Rm (η ) · · · Rm (η1 )N − 2 dλ(η1 ) · · · dλ(η ). We can now take the limit j → ∞ using (3.14), and the resulting expression together with (4.9), the formula (4.7) and Remark 4.6 yields that r . lim lim [i[Hn , g1m (A)k ], Aj ] = Em

j→∞ n→∞

Lemma 4.9. There exists a k-remainder Rm such that lim lim i[Hn , g1m (A)k Aj g1m (A)k ]

j→∞ n→∞

(A)H g1m (A)k } + Rm , = g1m (A)k H g1m (A)k + 2k Re{g1m (A)k−1 Ag1m

in the sense of forms on Dk−1 .

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Proof. We compute as a form on Dk−1 i[Hn , g1m (A)k Aj g1m (A)k ] = i[Hn , g1m (A)k ]Aj g1m (A)k + g1m (A)k i[Hn , Aj ]g1m (A)k + g1m (A)k Aj i[Hn , g1m (A)k ]. Using that limn→∞ i[Hn , Aj ] = Ij (A)H Ij (A), limj→∞ Ij (A)H Ij (A) = H (in the sense of forms on D(N 1/2 )), and Lemma 4.8(2), we conclude the result, with l r g1m (A)k + g1m (A)k Em . Rm = Em

Note that Rm is a k-remainder, in the sense of Deﬁnition 4.1. We now symmetrize the form g1m (A)k−1 Ag1m (A)H g1m (A)k , deﬁned on D(N 1/2 ).

Lemma 4.10. There exists a k-remainder Rm such that Re{g1m (A)k−1 Ag1m (A)H g1m (A)k } = g1m (A)k p1m (A)H p1m (A)g1m (A)k + Rm ,

in the sense of forms on Dk−1 . Proof. Step I. From the proof of Lemma 3.8, it follows that [N, Ag1m (A)]N −1 ,

[N, p21m (A)g1m (A)]N −1 ,

and N −1 p1m (A)N

(4.10) (4.11)

extend as forms from D(N ) to bounded operators with norm bounded uniformly in m. Step II. Boundedness of the forms in (4.10), together with the observation that − p21m g1m ∞ is bounded uniformly in m, implies after an interpolation tg1m argument that 1

1

N 2 (Ag1m (A) − p1m (A)2 g1m (A))N − 2

is bounded uniformly in m. Hence Re{g1m (A)k−1 Ag1m (A)H g1m (A)k } (1) , = g1m (A)k Re{p1m (A)2 H }g1m (A)k + Rm (1)

where Rm is a k-remainder. Step III. We compute as a form on D(N 1/2 ) A + iσ Rm (η)H Rm (η)dλp (η), (A + iσ)[p1m (A), H ] = −i > m Vm

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which is bounded uniformly in m as a form on D(N 1/2 ). This together with (4.11) and a interpolation argument as in step II, shows that (2) g1m (A)k Re{p1m (A)2 H }g1m (A)k = g1m (A)k p1m (A)H p1m (A)g1m (A)k + Rm , (2)

where Rm is a k-remainder. Here we used again Remark 4.6. This proves the lemma (1) (2) with Rm = Rm + Rm . Proof of Proposition 4.2. Combine Lemmas 4.9 and 4.10. Proof of Proposition 4.3. We only prove the ﬁrst estimate. The second is veriﬁed the same way. We can assume that λ = 0. We estimate using Condition 2.2 1

N 2 In (H)ψm 2 ≤ C1 In (H)ψm , HIn (H)ψm + C2 In (H)ψm , H In (H)ψm + C3 In (H)ψm 2 .

(4.12)

Note that HIn (H)ψm = Hn ψm = [Hn , g1m (A)k ]ψ. By Lemma 4.8(1) we ﬁnd that for any ϕ ∈ D(N 1/2 ) we have r ψ. lim ϕ, Hn ψm = ϕ, Em

n→∞

(4.13)

By this observation and the uniform boundedness principle there exists C = C(m) such that |ϕ, Hn ψm | ≤ CN 1/2 ϕ uniformly in n, for ϕ ∈ D(N 1/2 ). Applying this to ϕ = (In (H) − I)ψm , together with (4.13), now applied with ϕ = ψm , we get r ψ. lim In (H)ψm , HIn (H)ψm = ψm , Em

n→∞

(4.14)

r Here Em is a right-error. We can now take the limit n → ∞ in (4.12), and the result follows from Deﬁnition 4.5.

Proof of Proposition 4.4. As above we assume λ = 0 and prove only the ﬁrst bound. By Remark 2.4(4) it suﬃces to estimate using the bound (2.6) instead of the one in Condition 2.3. We get In (H)ψm , H In (H)ψm ≥ c0 In (H)ψm 2 + ReIn (H)ψm , BHIn (H)ψm − In (H)ψm , K0 In (H)ψm . (4.15) Arguing as in the part of the proof of Proposition 4.3 pertaining to (4.14), we ﬁnd that r ψ, lim ReIn (H)ψm , BHIn (H)ψm = Reψm , Em

n→∞

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r where Em is a right-error. Here (4.13) was used (twice) with ϕ replaced by Bϕ ∗ and B ϕ, where we used the assumption on B in Remark 2.4(4) to argue that Bϕ, B ∗ ϕ ∈ D(N 1/2 ) in (4.13). Inserting this limit into (4.15) yields

ψm , H ψm = lim In (H)ψm , H In (H)ψm n→∞

r ≥ c0 ψm 2 − ψm , K0 ψm + Reψm , Em ψ, r being a right-error. Using Deﬁnition 4.5 and Proposition 4.3, we conclude with Em the ﬁrst estimate.

4.2. Proof of Theorem 2.10 We shall show Theorem 2.10, which is an extension of Corollary 2.9 under the minimal condition k0 = 1. Proof of Theorem 2.10. We can without loss of generality take λ = 0. Due to Corollary 2.9, only the ﬁrst statement needs elaboration. The idea of the proof is to apply a virial argument for the commutator i[H, A] and the state N 1/2 ψ. We (1/2) = N 1/2 Iin (N ). divide the proof into three steps. Let Nn (1/2)

Step I. Due to Lemma 3.1, we have Nn

ψ ∈ D(H). We shall show that

sup HNn(1/2) ψ n∈N

< ∞.

(4.16)

We can use the representation formula (3.5) with α = 1/2 and commute H through N 1/2 , cf. (3.4). Whence it suﬃces to bound ∞ 1 1 t 2 (N + t)−1 [H, N ]0 (N + t)−1 Iin (N )N − 2 dt 0

independently of n. (Note that the contribution from commuting through the second factor Iin (N ) indeed is bounded independently of n.) By (2.1) we have 1

1

[H, N ]0 = N 2 −κ BN 2 −κ

for B bounded,

and we can estimate 1

3

(N + t)−1 i[H, N ]0 (N + t)−1 Iin (N )N − 2 ≤ Bt− 2 −κ

uniformly in n.

Hence the integrand is O(t−1−κ ) uniformly in n, and (4.16) follows. Step II. We shall show that sup ANn(1/2) ψ < ∞.

(4.17)

n∈N

Since φ := N 1/2 ψ ∈ D(A) due to Corollary 2.9 it suﬃces to bound the state [A, Iin (N )]φ independently of n. This is obvious from the representation [A, Iin (N )]φ = −i(N + n)−1 N Iin (N )φ, and whence (4.17) follows.

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Step III. We look at i[H, A]N (1/2) ψ = −2 ReiHNn(1/2) ψ, ANn(1/2) ψ. n

Due to (4.16) and (4.17), the right-hand side is bounded independently of n. We compute using Conditions 2.1(1) and 2.1(3) i[H, A]N (1/2) ψ = lim i[H, AIn˜ (A)]N (1/2) ψ = H N (1/2) ψ . n ˜ →∞

n

n

n

Whence using the virial estimate Condition 2.2 (and also Step I again) we conclude that N N (1/2) ψ ≤ C uniformly in n. n

Taking n → ∞ we obtain that indeed ψ ∈ D(N ). 4.3. Theorem on more N -regularity We formulate and prove an extended version of Theorem 2.10. Notice that under Conditions 2.1(1) and 2.1(2), and the additional condition (2.11) for k0 = 1, 1

1 1 (A) ∩ CMo (H), N 2 ∈ CMo

(4.18)

cf. Lemma 3.2 and Proposition 3.7. We impose the conditions of Corollary 2.9 and aim at an improvement of Corollary 2.9 and Theorem 2.10 in the case k0 ≥ 2. Let M0 = i[N 1/2 , A]0 . Then, cf. Proposition 3.7, 1

2 im adm A (M0 ) is N -bounded for m = 0, . . . , k0 − 1.

(4.19)

Here the commutators are deﬁned iteratively as extensions of forms on D(N 1/2 ) ∩ D(A) and they are considered as symmetric N 1/2 –bounded operators. We introduce the following N 1/2 –bounded operators: ∞ 1 1 t 2 (N + t)−1 i[N, H]0 (N + t)−1 dt, M1 = i[N 2 , H]0 = c 12 0

M2 = H N

− 12

1

and M3 = N − 2 H .

Notice that M3 ⊆ M2∗

and M2 ⊆ M3∗ .

(4.20)

We need to consider repeated commutation of Mj , j = 1, . . . , 3, with factors of T = A or T = N 1/2 . Condition 4.11. For all j = 1, . . . , 3, m = 1, . . . , k0 − 1 and all possible combinations of factors Tn ∈ {A, N 1/2 } where n = 1, . . . , m 1

im adTm · · · adT1 (Mj ) is N 2 -bounded.

(4.21)

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Notice that in (4.21) the commutators are deﬁned iteratively as extensions of forms on D(N 1/2 ) ∩ D(A) using (4.20) and the analogue properties for m ≥ 2 (−1)m−1 adTm−1 · · · adT1 (M3 ) ⊆ (adTm−1 · · · adT1 (M2 ))∗ , (−1)m−1 adTm−1 · · · adT1 (M2 ) ⊆ (adTm−1 · · · adT1 (M3 ))∗ . We shall prove the following extension of Corollary 2.9 and Theorem 2.10. Theorem 4.12. Suppose the conditions of Corollary 2.9 and for k0 ≥ 2 also Condition 4.11. Let ψ ∈ D(N 1/2 ) be a bound state (H − λ)ψ = 0 (with λ as in Condition 2.3). Then ψ ∈ D(Tk0 +1 · · · T1 ) where Tn ∈ {A, N 1/2 , 1} for n = 1, . . . , k0 + 1 and at least for one such n, Tn = A. Proof. We proceed by induction in k0 . The case k0 = 1 is the content of Theorem 2.10. So suppose k0 ≥ 2 and that the statement holds for k0 → k0 − 1. Consider any product S = Tk0 +1 · · · T1 not all factors being given by A. We shall show that ψ ∈ D(S). By Corollary 2.9 and the induction hypothesis we can assume that the factors Tn ∈ {A, N 1/2 } and that for at least two n’s Tn = N 1/2 . By using (4.19) and the induction hypothesis we can assume that Tk0 +1 = N 1/2 . Whence we can assume α with k = k0 introducing here the following notation for k = 1, . . . , k0 , S = N 1/2 Sk, = 0, . . . , k and α being a multiindex α ∈ {0, 1}k with j≤k αj = , α = Sαk · · · Sα1 =: Sk,

k

Sαj

1

where S0 = A and S1 = N 2 .

j=1

Partly motivated by the above considerations we introduce the following quantity for n ∈ N large and ∈ ]0, 1[ small f (n, ) =

k0 =0

2

−2 g(n, ); g(n, ) :=

1

N 2 Iin (N )Skα0 , ψ2 .

α∈{0,1}k0 α1 +···+αk0 =

We claim that for some constants K1 , K2 () > 0 independent of n f (n, ) ≤ 2 K1 f (n, ) + K2 ().

(4.22)

The theorem follows from (4.22) by ﬁrst choosing so small that 2 K1 ≤ 1/2, subtraction of the ﬁrst term on the right-hand side and then letting n → ∞. By Corollary 2.9 (or Theorem 2.7), supn g(n, = 0) < ∞, in agreement with (4.22). To see how the factor 2 comes about let us note that −22 = −( − 1)2 − ( + 1)2 + 2, whence (to be used later) we can for = 1, . . . , k0 − 1 bound the expression 2 (4.23) −2 g(n, − 1) g(n, + 1) ≤ 2 f (n, ). To show (4.22) we mimic the proof of Theorem 2.10. Again this is in three steps and we assume that λ = 0. We need to bound each term of g(n, ) for ≥ 1.

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Step I. Bounding HIin (N )Skα0 , ψ. We expand into terms; some can be bounded independently of n (using the induction hypothesis) while others will be estimated as C g(n, + 1) (assuming here that ≤ k0 − 1). We compute formally   k0 i[H, Iin (N )Skα0 , ] = i[H, Iin (N )]Skα0 , + Iin (N )i H, Sαj  , (4.24) j=1

where the second commutator is expanded as       k0 m=k k0 m−1 0  i H, Sαj  = Sαj  i[H, Sαm ]  Sαj  . j=1

m=1

j=m+1

(4.25)

j=1

In turn we have the expressions i[H, Iin (N )] = n−1 Iin (N )i[N, H]0 Iin (N ), i[H, Sαm ] = −M1 i[H, Sαm ] = M2 S1

if αm = 1,

(4.26a) (4.26b)

if αm = 0.

(4.26c)

We plug (4.26a)–(4.26c) into (4.24) and (4.25) and look at each term separately. Before embarking on a such examination we need to “ﬁx” the above formal computation. This is done in terms of multiple approximation somewhat similar to the one of the proof of Theorem 2.7. We replace H → Hp and the factors A → Aq and 1/2 N 1/2 → Niq = (N 1/2 )iq . More precisely it is convenient to introduce k0 diﬀerent q’s, say q1 , . . . , qk0 ; the q used for the jth factor Sαj is qj . For ﬁxed p and q’s the product rule applies for computing the commutator of the product and the analogues of (4.24) and (4.25) hold true. Now we can take the limit p → ∞. We can plug the modiﬁed expressions of (4.26a)–(4.26c) into (modiﬁed) (4.24) and (4.25). Actually (4.26a) is the same, but (4.26b) and (4.26c) are changed as 1

1

1

i[H, Niq2j ] = −Iiqj (N 2 )M1 Iiqj (N 2 ),

(4.27a)

i[H, Aqj ] = Iqj (A)M2 S1 Iqj (A).

(4.27b) 1/2

Of course we have a q-dependence of the various factors of either S1 → Niqj or S0 → Aqj . Eventually we take the limits in the q’s done in increasing order starting by taking q1 → ∞ and ending by taking qk0 → ∞. Before taking these limits we need to do some further commutation using Condition 4.11. For simplicity of presentation we ignore below in this process commutation with the regularizing factors of Iiqj (N 1/2 ) or Iqj (A) since in the limit they will disappear (a manifestation of this occurred also in the proof of Lemma 3.4). In other words we proceed now slightly formally using (4.24) and (4.25) with the plugged in expressions (4.26a)–(4.26c): From (4.26a), we obtain that i[H, Iin (N )] ≤ C so the contribution from the ﬁrst term of (4.24) can be estimated (uniformly in n) as i[H, Iin (N )]Skα0 , ψ ≤ CSkα0 , ψ ≤ C.

(4.28)

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As for the contribution from (4.26b), we compute  −Iin (N ) 

k0





m−1

Sαj  M1 

j=m+1

j=1







 1 2 Sαj  = T1  N

1≤j≤k0 j =m

 Sαj   + T2 ,

where 1 T1 = −Iin (N )M1 N − 2 .

Here T2 is given by repeated commutation using Condition 4.11. We apply this identity to the bound state ψ. Since T1 ≤ C the induction hypothesis gives similar bounds as (4.28) for the contribution from (4.26b). It remains to look at the contribution from (4.26c): We commute the factor M2 to the left and get similarly     k0 m−1 Iin (N )  Sαj  M2 S1  Sαj  j=m+1

j=1

 = T1 N Iin (N )  1 2



k0



Sαj  S1 

j=m+1

m−1

 Sαj  + T2 ,

j=1

where 1 T1 = Iin (N )M2 (N 2 Iin (N ))−1 .

As before T1 ≤ C (here we use that H is N -bounded) and the contribution from T2 is treated by using Condition 4.11 and the induction hypothesis. Consequently we get for ≤ k0 − 1 the total bound 1 g(n, + 1) + C 2 , HIin (N )Skα0 , ψ ≤ C (4.29) 1 and C 2 are independent of n, and for = k0 this bound without the ﬁrst where C term to the right. Step II. Bounding AIin (N )Skα0 , ψ. We claim that (recall ≥ 1) 3 AIin (N )Skα0 , ψ ≤ C

4 , g(n, − 1) + C

(4.30)

4 are independent of n. 3 and C where C To prove (4.30) we observe that it suﬃces by the induction hypothesis to bound Iin (N )ASkα0 , ψ. Since ≥ 1 there is a nearest factor of N 1/2 in the product Skα0 , that we move to the left in front of the factor A: 1

Iin (N )ASkα0 , = N 2 Iin (N )ASkβ0 −1,−1 + T.

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We apply this identity to the bound state ψ. The contribution from T is treated by using (4.19) and the induction hypothesis. This proves (4.30). Step III. We repeat Step III of the proof of Theorem 2.10 using now the proven estimates (4.29) and (4.30) to bound any term of g(n, ) for ≥ 1. In combination with (4.23) these bounds yield (4.22) with 1 C 3 (2k0 − 1) + 1; K1 = 2C2 C 1 and C 3 come from (4.29) and (4.30), here the constant C2 comes from (2.4) while C respectively. Notice that the cardinality of set {0, 1}k0 is 2k0 , so the factor 2k0 − 1 αj ≥ 1. arises by counting only those indices α ∈ {0, 1}k0 with Corollary 4.13. Suppose the conditions of Corollary 2.9 and for k0 ≥ 2 also Condition 4.11. Let ψ ∈ D(N 1/2 ) be a bound state (H − λ)ψ = 0 (with λ as in Condition 2.3). Then ψ ∈ D(N (k0 +1)/2 ). 5. A Class of Massless Linearly Coupled Models In this section we introduce a class of massless linearly coupled Hamiltonians, sometimes referred to as Pauli–Fierz Hamiltonians [7, 10, 11, 22]. The bulk of this section is spent on checking that an expanded version of the Hamiltonian does indeed satisfy the abstract assumptions of Sec. 2. In Sec. 5.2, we verify that the Nelson model described in Sec. 1.2 is indeed an example of the type of models discussed here. 5.1. The model and the result Consider the Hilbert space HPF = K ⊗ Γ(h), where K is the Hilbert space for a “small” quantum system, and Γ(h) is the symmetric Fock space over h = L2 (Rd , dk), describing a ﬁeld of massless scalar bosons. The Pauli–Fierz Hamiltonian HvPF acting on HPF is deﬁned by HvPF = K ⊗ 1Γ(h) + 1K ⊗ dΓ(|k|) + φ(v),

(5.1)

where K is a Hamiltonian on K describing the dynamics of the small system. We assume that K is bounded from below, and for convenience we require furthermore that K ≥ 0. The term dΓ(|k|) is the second √ quantization of the operator of multiplication by |k|, ∗ and φ(v) = (a (v) + a(v))/ 2. The form factor v is an operator from K to K ⊗ h, and a∗ (v), a(v) are the usual creation and annihilation operators associated to v. See [7, 22]. The hypotheses we make are slightly stronger than the ones considered in [22]. The ﬁrst one, Hypothesis (H0), expresses the assumption that the small system is conﬁned: (H0) (K + i)−1 is compact on K.

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Let 0 ≤ τ < 1/2 be ﬁxed. We will introduce a class of interactions which increase with τ . In order to formulate our assumption on the form factor v we introduce the subspace Oτ of B(D(K τ ); K ⊗ h) consisting of those operators which extend by continuity from D(K τ ) to an element of B(K; D(K τ )∗ ⊗ h). In other words Oτ := {v ∈ B(D(K τ ); K ⊗ h) | ∃ C > 0, ∀ ψ ∈ D(K τ ) : [(K + 1)−τ ⊗ 1h ]vψK⊗h ≤ CψK }. We also write v for the extension. It is natural to introduce a norm on Oτ by vτ = v(K + 1)−τ B(K;K⊗h) + [(K + 1)−τ ⊗ 1h ]vB(K;K⊗h) . Our ﬁrst assumption on the form factor interaction is the following: (I1) v, [1K ⊗ |k|−1/2 ]v ∈ Oτ . It is proved in [22] that if (I1) holds, HvPF is self-adjoint with domain D(HvPF ) = D(K ⊗ 1Γ(h) + 1K ⊗ dΓ(|k|)). ˜ deﬁned by The unitary operator T : L2 (Rd ) → L2 (R+ ) ⊗ L2 (S d−1 ) =: h (d−1)/2 u(ωθ) allows us to pass to polar coordinates. Lifting T to the (T u)(ω, θ) = ω ˜ PF := K ⊗ Γ(h). full Hilbert space as 1K ⊗ Γ(T ) gives a unitary map from HPF to H PF The Hamiltonian Hv is unitarily equivalent to vPF := K ⊗ 1 ˜ + 1K ⊗ dΓ(ω) + φ(˜ H v ), Γ(h)

(5.2)

˜ where v˜ = [1K ⊗ T ]v ∈ B(K; K ⊗ h). In polar coordinates the space of couplings consists of operators of the form ˜ where v ∈ Oτ . We write O τ = [1K ⊗ T ]Oτ and equip it with [1K ⊗ T ]v : K → K ⊗ h, ˜ ∗ v τ . Observe ˜ v ˜τ = vτ , when v˜ = [1K ⊗T ]v. the obvious norm ˜ v τ = [1K ⊗T ]˜ Let d be as in (1.7) and (1.8). We recall that d expresses the least amount of infrared regularization carried by a v satisfying (I2) below. The following further assumptions on the interaction are made: (I2) The following holds τ , v∈O [1K ⊗ (1 + ω −1/2 )ω −1 d(ω) ⊗ 1L2 (S d−1 ) ]˜ τ , [1K ⊗ (1 + ω −1/2 )d(ω)∂ω ⊗ 1L2 (S d−1 ) ]˜ v∈O ˜ v ∈ B(D(K τ ); K ⊗ h). (I3) [1K ⊗ ∂ω2 ⊗ 1L2 (S d−1 ) ]˜ In this paper we need an additional assumption compared to [22]. For bounded K, it is implied by (I1). Its presence is motivated by a desire to deal eﬀectively with infrared singularities. ˜ × D(K) to an element v − v˜K extends from [D(K) ⊗ h] (I4) The form [K ⊗ 1h˜ ]˜ of O 1 . 2

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1 is deﬁned as O τ . Supposing (I1), the statement above is meaningful. See Here O 2 also Remark 5.14 below. Remark 5.1. We remark that for separable Hilbert spaces K1 and K2 there are two natural subspaces of B(K1 ; K2 ⊗ h). Namely

2 d d 2 L (R ; B(K1 ; K2 )) = v : R → B(K1 ; K2 ) v(k)B(K1 ;K2 ) dk < ∞ L2w (Rd ; B(K1 ; K2 )) =

Rd

d 2 v : R → B(K1 ; K2 ) sup v(k)ψ2 dk < ∞ . ψ 1 ≤1 Rd

The functions v should be weakly measurable, to ensure that v(k)B(K1 ,K2 ) and v(k)ψ2 are measurable. Here · j denotes the norm on Kj . We have the obvious inclusions L2 (Rd : B(K1 ; K2 )) ⊆ L2w (Rd ; B(K1 ; K2 )) ⊆ B(K1 ; K2 ⊗ h). The ﬁrst inclusion is a contraction and the second an isometry. Both inclusions are strict as exempliﬁed by choosing K1 = K2 = L2 (R3x ), h = L2 (R3k ) and v1 (k) = e−|x−k| (read as a multiplication operator) for the ﬁrst inclusion and v2 (k, x) = |x − k|−1 e−|x−k| for the second. Here v2 induces the bounded operator v2 : L2 (R3x ) → L2 (R3k × R3x ) by the prescription (v2 ψ)(k, x) = v2 (k, x)ψ(x). (In [10, Sec. 2.16] and [22, Sect. 3.4] the second inclusion is claimed to be an equality.) We denote by IPF (d) the vector space of interactions v satisfying (I1)–(I4) and turn it into a normed vector space by equipping it (in polar coordinates) with the norm v ˜τ vPF := [1K ⊗ (1 + ω −3/2 d(ω)) ⊗ 1L2 (S d−1 ) ]˜ + 1K ⊗ (1 + ω −1 )d(ω)∂ω ) ⊗ 1L2 (S d−1 ) ]˜ v ˜τ + [(K + 1)−1/2 ⊗ ∂ω2 ⊗ 1L2 (S d−1 ) ]˜ v B(K;K⊗h) ˜ + [K ⊗ 1h ]˜ v − v˜K˜1 . 2

(5.3)

For any v0 ∈ IPF (d) and r > 0 write Br (v0 ) = {v ∈ IPF (d) | v − v0 PF ≤ r}

(5.4)

for the closed ball in IPF (d) with radius r around v0 . PF used in [22]. Let Let us recall the deﬁnition of the conjugate operator on H ∞ χ ∈ C0 ([0, ∞)) be such that χ(ω) = 0 if ω ≥ 1 and χ(ω) = 1 if ω ≤ 1/2. For 0 < δ ≤ 1/2, the function mδ ∈ C ∞ ([0, ∞)) is deﬁned by

ω ω mδ (ω) = χ d(δ) + (1 − χ) d(ω), δ δ

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˜ the operator a On h, ˜δ is deﬁned in the same way as in [22], that is a ˜δ := imδ (ω)

i dmδ ∂ + (ω), ∂ω 2 dω

D(˜ aδ ) = H01 (R+ ) ⊗ L2 (S d−1 ).

(5.5)

i dmδ ∂ − (ω), ∂ω 2 dω

D(˜ a∗δ ) = H 1 (R+ ) ⊗ L2 (S d−1 ).

(5.6)

Its adjoint is given by a ˜∗δ := imδ (ω)

We recall that H01 (R+ ) is the closure of C0∞ ((0, ∞)) in H 1 (R+ ). The conjugate PF is deﬁned by A δ := 1K ⊗ dΓ(˜ δ on H aδ ). Going back to HPF we get operator A −1 ˜δ T and aδ = T a δ [1K ⊗ Γ(T )]. Aδ = dΓ(aδ ) = [1K ⊗ Γ(T −1 )] A The operator aδ takes the form (1.11) when written in the original coordinates. We write N for the number operator 1K ⊗ dΓ(1h ) on HPF . For E ∈ σpp (HvPF ), we write Pv for the corresponding eigenprojection. Recall from [22, Theorem 2.4] that the range of Pv is ﬁnite dimensional under the assumptions (H0), (I1) and (I2). Theorem 5.2. Suppose (H0). Let v0 ∈ IPF (d) and J ⊆ R be a compact interval. There exists 0 < δ0 ≤ 1/2 such that for all 0 < δ ≤ δ0 the following holds: There exist γ > 0 and C > 0 such that for any v ∈ Bγ (v0 ) and E ∈ σpp (HvPF ) ∩ J we have 1

1

Pv : HPF → D(N 2 Aδ ) ∩ D(Aδ N 2 ) ∩ D(N ) and 1

1

N 2 Aδ Pv + Aδ N 2 Pv + N Pv ≤ C. Unfortunately we cannot employ our theory directly to conclude the above theorem, due to Aδ not being self-adjoint. Instead we use a trick of passing to an “expanded” model, for which we can use our abstract theory. The theorem above will then be a consequence of a corresponding theorem in the expanded picture. Remark 5.3. Under the hypotheses of Theorem 5.2, we also have that Pv : HPF → D(A∗δ N 1/2 )∩D(N 1/2 A∗δ ). This follows from Aδ ⊆ A∗δ . In particular this implies that Pv Aδ extends from D(Aδ ) to a bounded operator on HPF . Similar statements hold also for Pv Aδ N 1/2 and Pv N 1/2 Aδ . 5.2. Application to the Nelson model In this subsection we check the conditions (H0) and (I1)–(I4) for the Nelson model introduced in the introduction. After possibly adding a constant to W , we can assume that K ≥ 0. See (1.4) and (W0).

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We begin by remarking that it follows from (W0) and (V0) that 1

|x|α (K + 1)− 2 ∈ B(K) 1

|p|(K + 1)− 2 ∈ B(K).

(5.7) (5.8)

Here α > 2 is coming from (W0), |x| = |x1 | + · · · + |xP | and |p| = |p1 | + · · · + |pP |, where p = −i∇x . These bounds imply in particular (H0). Let ΨN : IN (d) → B(K; K ⊗ h) be deﬁned by ΨN (ρ) =

P

e−ik·x ρ.

=1

Clearly ΨN is a linear map and φ(ΨN (ρ)) = Iρ (x) such that HρN = K ⊗ 1F + 1K ⊗ dΓ(|k|) + φ(ΨN (ρ)), is a Pauli–Fierz Hamiltonian, cf. (1.5) and (1.6). Verifying the conditions (I1)–(I4) will be achieved if we can show that ΨN is a bounded operator from IN (d) to IPF (d). This also implies that results valid uniformly for v in a ball in IPF (d) will translate into results holding uniformly for ρ in a suﬃciently small ball in IN (d). See Remark 2.11(4). That the terms in the norm ΨN (ρ)PF , cf. (5.3), pertaining to the conditions (I1)–(I3) can be bounded by · N (or rather terms in · N pertaining to (ρ1)–(ρ3)), follows as in [22] after we have checked that |x|2 (K + 1)−τ is bounded for some positive τ < 1/2. To produce such a τ we invoke Hadamard’s three-line theorem. Consider the function z → |x|−iαz (K +1)iz/2 ∈ B(K). Observe that this function is bounded when Im z = 0 or Im z = 1, cf. (5.7). It now follows, cf. [33–36], that |x|sα (K + 1)−s/2 is bounded for 0 ≤ s ≤ 1. Choosing s = 2/α implies the desired bound with τ = α−1 < 1/2. This will be the τ used in the conditions (I1)–(I3). It remains to verify (I4). For this we compute ! P 1 −ik·xj −ik·xj −ik·xj −ik·xj 1 ρ−e ρK = − ∆ e ρ − ρe ∆ [K ⊗ 1h ]e 2m 2m =1 ! 1 1 =− ∆j e−ik·xj ρ − ρe−ik·xj ∆j 2mj 2mj =

e−ik·xj [−2k · pj + k 2 ]ρ 2m

= [−2k · pj − k 2 ]ρ

e−ik·xj . 2mj

(5.9)

From this computation and (5.8) we conclude that [K ⊗ 1]ΨN (ρ) − ΨN (ρ)K ∈ O1/2 as required by (I4) and the · 1/2 -norm of the diﬀerence is bounded by a constant times ρN . Here we need the term in · N coming from (ρ4).

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We can thus conclude Theorem 1.2 from Theorem 5.2. It remains to discuss the Nelson model after a Pauli–Fierz transformation. We recall that we have two transformations to consider, one giving rise to HρN and one to HρN . See (1.12) and (1.16). To identify these Hamiltonians as Pauli–Fierz (d) → B(K; K ⊗ h) by Hamiltonians, we introduce a linear map ΨN : IN ΨN (ρ) =

P

(e−ik·x − 1)ρ.

=1 With this notation we ﬁnd for ρ ∈ IN (d)

HρN = Kρ ⊗ 1Γ(h) + 1K ⊗ dΓ(|k|) + φ(ΨN (ρ)) and, specializing to ρ = ρ0 + ρ1 with ρ0 ∈ IN (d) and ρ1 ∈ IN (d), P HρN = Kρ0 − vρ0 ,ρ1 (x ) ⊗ 1Γ(h) + 1K ⊗ dΓ(|k|) + φ(ΨN (ρ0 ) + ΨN (ρ1 )). =1

See (1.13) for Kρ and (1.17) for vρ0 ,ρ1 . In order to apply Theorem 5.2 one should ﬁrst observe that ΨN is a bounded map from IN (d) to IPF (d). We leave it to the reader to establish this following the arguments in [22], using the key estimate (1.15). As for (I4), observe that the extra −ρ from (e−ik·xj − 1)ρ drops out when repeating (5.9) for ΨN (ρ). In particular we do not need (1.15) for (I4). Observe that for both the transformed Hamiltonians, the Hamiltonian for the conﬁned quantum system K is altered by the transformation, to obtain, e.g., Kρ in the case of HN . A priori the norm · PF is however deﬁned in terms of the operator K, and this deﬁnition we retain. However, when verifying the Mourre estimate in Sec. 5.4 and our abstract assumptions for Pauli–Fierz Hamiltonians in Sec. 5.5, we will naturally meet norms with the modiﬁed ρ-dependent K’s, and not the original K. We proceed to argue that the · PF norms arising in this way are equivalent, locally uniformly in ρ, (d) with respect to the appropriate normed space. Let for ρ ∈ IN Bρ

= Kρ − K = −

P =1

P2 vρ (x ) + 2

∞

r−1 |˜ ρ(r)|2 dr1K

0

and for ρ = ρ0 + ρ1 as above Bρ

=−

P

vρ0 ,ρ1 (x ) −

=1

P =1

P2 vρ0 (x ) + 2

∞

r−1 |ρ˜0 (r)|2 dr1K .

0

We observe the bounds 2 2 Bρ ≤ Cρ2 N and Bρ ≤ C(ρ0 N + ρ1 N ),

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for some ρ-independent constant C. In particular both Bρ and Bρ can be bounded locally uniformly in ρ, with respect to the appropriate norm. By yet another interpolation argument this implies that we can pass between · PF norms deﬁned with either K, Kρ , or Kρ0 + Bρ , and still retain bounds that are locally uniform in ρ. Finally we note that the above bounds also imply that by possibly adding to W a positive constant we still have Kρ ≥ 0 and Kρ0 + Bρ ≥ 0 locally in ρ. This ensures that (H0) is satisﬁed also for transformed Nelson Hamiltonians. In particular we still have e.g. |x|2 (Kρ + 1)−τ bounded. In conclusion, Theorem 1.3 also follows from Theorem 5.2. 5.3. Expanded objects ˜ by ( ve on H ( e := H PF ⊗ Γ(h) Let us now deﬁne the expanded operator H ˆ ( e := H PF ⊗ 1 ˜ − 1 e ⊗ dΓ(h), H v v Γ(h) HPF

(5.10)

ˆ is the operator of multiplication by where h ω2 ˆ . h(ω) = eω − 1 − 2 From the bound ω ≤

1 2

(5.11)

+ ω 2 /2 we ﬁnd that for ω ≥ 0 1 d ˆ ˆ h(ω) ≥ h(ω) + . dω 2

(5.12)

Since L2 (R+ ) ⊕ L2 (R+ ) L2 (R), it is known (see e.g. [11]) that there exists a unitary operator ˜ ⊗ Γ(h) ˜ → Γ(he ), U : Γ(h)

(5.13)

˜ ⊗ Γ(h), ˜ the unitary operator 1K ⊗ U is where he := L2 (R) ⊗ L2 (S d−1 ). On K ⊗ Γ(h) e e still denoted by U. It maps into H = K⊗Γ(h ). In this representation, the operator ( ve is unitary equivalent to the “expanded Pauli–Fierz Hamiltonian” Hve deﬁned as H an operator on He by ( e U −1 = K ⊗ 1Γ(he ) + 1K ⊗ dΓ(h) + φ(v e ), Hve := U H v where v e ∈ B(K, K ⊗ he ), and v e and h are deﬁned by ω if ω ≥ 0, v˜(ω) if ω ≥ 0, e h(ω) := v (ω) := ˆ 0 if ω ≤ 0. −h(−ω) if ω ≤ 0,

(5.14)

(5.15)

Note that h ∈ C 2 (R). The idea of expanding the Hilbert space in the above fashion has been used previously in [11, 12, 23, 28]. Our choice of expansion for the boson dispersion relation to the unphysical negative ω appears to be new. Previous implementations of the expansion all used the obvious linear expansion h(ω) = ω. ˜ is a core for dΓ(ω), then We remark that if CK ⊆ K is a core for K, C ⊆ Γ(h) PF , hence for H PF , and ﬁnally the algebraic tensor product CK ⊗ C is a core for H 0 v

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( e for any v ∈ IPF (d). The domain D(H e ) itself may CK ⊗ C ⊗ C is a core for H v v however be v dependent. (The argument for the contrary in [11, Sec. 5.2] seems wrong.) We have however set up our analysis such that knowledge of Hve ’s domain is not needed. See also Lemma 5.15 where an intersection domain is computed. Remark 5.4. We remark that if one is going for higher order results, i.e. ψ ∈ ˆ The choice D(Ak0 ) for k0 ≥ 2, one should use a diﬀerent h. ˆ k (ω) = eω − 1 − h 0

k 0 +1 =2

ω !

will work since the corresponding hk0 is in C k0 +1 (R) and the bound ˆ k (ω) 1 d ˆ h 0 + hk0 (ω) ≥ dω (k0 − 1)! 2 holds for ω ≥ 0 and k0 ≥ 1. For k0 = 1 this reduces to (5.12). Before introducing the conjugate operator on He that we shall use, let meδ ∈ C (R) be deﬁned by mδ (ω) if ω ≥ 0, meδ (ω) := d(δ) if ω ≤ 0. ∞

We set aeδ := imeδ (ω)

i dmeδ ∂ + (ω), ∂ω 2 dω

D(aeδ ) = H 1 (R) ⊗ L2 (S d−1 ),

(5.16)

and Aeδ := 1K ⊗ dΓ(aeδ ) as an operator on He . Note that both aeδ and Aeδ are self-adjoint. We can now formulated the expanded version of our regularity theorem Let −1 N e = 1K ⊗ dΓ(1he ) = U(N ⊗ 1Γ(h) ˜ ))U ˜ + 1H e PF ⊗ dΓ(1h

denote the expanded number operator. For E ∈ σpp (Hve ) we write Pve for the associated eigenprojection. Theorem 5.5. Suppose (H0). Let v0 ∈ IPF (d) and J ⊆ R be a compact interval. There exists a 0 < δ0 ≤ 1/2 such that for any 0 < δ ≤ δ0 the following holds: There exist γ > 0 and C > 0 such that for any v ∈ Bγ (v0 ) and E ∈ σpp (Hve ) ∩ J we have 1

1

Pve : He → D((N e ) 2 Aeδ ) ∩ D(Aeδ (N e ) 2 ) ∩ D(N e ) and 1

1

(N e ) 2 Aeδ Pve + Aeδ (N e ) 2 Pve + N e Pve ≤ C.

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In the next two subsections we verify that our abstract theory applies to the expanded model, but before doing so we pause to check that Theorem 5.2 does indeed follow from Theorem 5.5. For that we need a lemma. PF generated by A δ . Let Wδ,t , t ≥ 0, denote the contraction semigroup on H PF we have for t ≥ 0 Lemma 5.6. For any state ϕ ∈ H e

e−itAδ U(ϕ ⊗ Ω) = U(Wδ,t ϕ ⊗ Ω). k ) if and only if U(ϕ ⊗ Ω) ∈ D((Ae )k ). In particular, ϕ ∈ D(A δ δ Proof. It suﬃces to check the identity on a dense set of ϕ’s. Let ϕ ∈ K ⊗ Γﬁn (H01 (R+ )⊗L2 (S d−1 )) ⊆ D(A˜δ ). Then U(ϕ⊗Ω) ∈ K⊗Γﬁn (H 1 (R)⊗L2 (S d−1 )) ⊆ D(Aeδ ). The identify now follows by diﬀerentiating both sides of the equation and observing they satisfy the same diﬀerential equation, with the same initial conδ ϕ ⊗ Ω) valid for dition. Here we made use of the equality Aeδ U(ϕ ⊗ Ω) = U(A 1 + 2 d−1 )). ϕ ∈ K ⊗ Γﬁn (H0 (R ) ⊗ L (S Proof of Theorem 5.2. We only have to recall that bound states of Hve are PF , with precisely states on the form U(ϕ ⊗ Ω), where ϕ is a bound state for H v the same eigenvalue. This implies that eigenprojections for Hve are on the form PF . Theorem 5.5, together U[P ⊗ |ΩΩ|]U −1 where P is an eigenprojection for H v with Lemma 5.6, now implies Theorem 5.2. 5.4. Mourre estimates We begin by establishing a Mourre estimate for HvPF and Aδ in a form appropriate for use in this paper. At the end of the subsection we derive a Mourre estimate for Hve and Aeδ . Let Mδ := 1K ⊗ dΓ(mδ ) and Rδ = Rδ (v) := −φ(iaδ v) as operators on HPF . Let H be the closure of Mδ + Rδ with domain D(HvPF ) ∩ D(Mδ ). Recall from [22] that H = [HvPF , iAδ ]0 . Let f ∈ C0∞ (R) be such that 0 ≤ f ≤ 1, f (λ) = 1 if |λ| ≤ 1/2 and f (λ) = 0 if |λ| ≥ 1. In addition we choose f to be monotonously decreasing away from 0, i.e. λf (λ) ≤ 0. For E ∈ R and κ > 0 we set

λ−E fE,κ (λ) := f . κ The following “Mourre estimate” for HvPF is proved in [22]: Theorem 5.7 [22, Theorem 7.12]. Assume that Hypotheses (H0), (I1) and (I2) hold. Let E0 ∈ R. There exists δ0 ∈]0, 1/2] such that : For all E ≤ E0 , 0 < δ ≤ δ0

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and ε0 > 0, there exist C > 0, κ > 0, and a compact operator K0 on HPF such that the estimate ⊥ (HvPF )2 − K0 (5.17) Mδ + fE,κ (HvPF )Rδ fE,κ (HvPF ) ≥ (1 − ε0 )1HPF − CfE,κ

holds as a form on D(N 1/2 ). The following lemma is just a reformulation of [22, Proposition 4.1(i), Lemma 4.7 and Lemma 6.2(iv)]. We leave the proof to the reader. Lemma 5.8. Let v0 ∈ IPF (d). There exists c0 , c1 , c2 > 0, depending on v0 , such + c0 ≥ 0 and the following holds: for all w ∈ IPF (d) and 0 < δ ≤ 1/2 that HvPF 0 ±φ(w) ≤ c1 wPF (HvPF + c0 ) 0

and

± Rδ (w) ≤ c1 wPF (HvPF + c0 ). 0

+ i)−1 ≤ c2 wPF φ(w)(HvPF 0

and

Rδ (w)(HvPF + i)−1 ≤ c2 wPF . 0

The ﬁrst step we take is to translate the commutator estimate above into the form used in this paper, see Condition 2.3. In anticipation of the need for local uniformity of constants, we need to already at this step ensure that B = CB 1 can be chosen uniformly in E ∈ J, where J is compact interval. Corollary 5.9. Let J ⊆ R be a compact interval and v0 ∈ IPF (d). There exists δ0 ∈ ]0, 1/2] and CB > 0 such that for any E ∈ J, 0 > 0 and 0 < δ < δ0 the following holds. There exists κ > 0, C4 > 0 and a compact operator K0 such that the form inequality ⊥ Mδ + Rδ (v0 ) ≥ (1 − 0 )1HPF − C4 fE,κ (HvPF )2 − CB (HvPF − E) − K0 0 0

(5.18)

holds on D(N 1/2 ) ∩ D(HvPF ). 0 Proof. Let E0 be an upper bound for the interval J and take δ0 to be the one coming from Theorem 5.7, applied with v = v0 . Fix E ∈ J, 0 < δ < δ0 and 0 > 0. Apply Theorem 5.7 with 0 /2 in place of 0 . ) Compute as a form on D(HvPF 0 ⊥ ⊥ Rδ (v0 ) = fE,κ (HvPF )Rδ (v0 )fE,κ (HvPF ) + fE,κ (HvPF )Rδ (v0 )fE,κ (HvPF ) 0 0 0 0 ⊥ )Rδ (v0 )fE,κ (HvPF )}. + 2 Re{fE,κ (HvPF 0 0

Using Lemma 5.8 with w = v0 and abbreviating CB = c1 v0 PF , we estimate ⊥ ⊥ fE,κ (HvPF )Rδ (v0 )fE,κ (HvPF ) 0 0 ⊥ + c0 )fE,κ (HvPF )2 ≥ −c1 v0 PF (HvPF 0 0 ⊥ ⊥ = −CB (HvPF − E)fE,κ (HvPF )2 − CB (c0 + E)fE,κ (HvPF )2 0 0 0 ⊥ ≥ −CB (HvPF − E) − 3CB κ − CB (c0 + E)fE,κ (HvPF )2 . 0 0

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Using Lemma 5.8 again, we get ⊥ 2 Re{fE,κ (HvPF )Rδ (v0 )fE,κ (HvPF )} 0 0

≥−

4 0 ⊥ − Rδ (v0 )fE,κ (HvPF )2 fE,κ (HvPF )2 0 0 4 0

≥−

4c2 v0 2PF (|E| + κ + 1)2 ⊥ 0 − 2 fE,κ (HvPF )2 . 0 4 4ζ

Combining the equations above with Theorem 5.7 yields (5.18) with CB only depending on v0 . The above corollary suﬃces to prove Theorem 5.5 without local uniformity in v and E. The following lemma is designed to deal with uniformity of estimates in a small ball of interactions v around a ﬁxed (unperturbed) interaction v0 . Technically it replaces [22, Lemma 6.2(iv)]. > 0 and c0 , c1 , c2 > 0, Lemma 5.10. Let v0 ∈ IPF (d). There exists γ0 > 0, CB only depending on v0 , such that

(1) ∀ v ∈ Bγ0 (v0 ) : HvPF ≥ −c0 . (2) ∀ v ∈ Bγ0 (v0 ) : ±φ(v) ≤ c1 (HvPF + c0 ) and φ(v)(HvPF − i)−1 ≤ c2 . (HvPF + c0 ) and Rδ (v)(HvPF − (3) ∀ v ∈ Bγ0 (v0 ) and 0 < δ ≤ 1/2 : ±Rδ (v) ≤ CB −1 i) ≤ c2 . Proof. Let v0 ∈ IPF (d) be given. Let C1 (r, v) = [1K ⊗ ω −1/2 ]˜ v (K + r)−1/2 , for v ∈ IPF (d) and r > 0. √ We begin with (1). Fix r = r(v0 ) ≥ 1 such that 2C1 (r, v0 ) ≤ 1/3. This is possible due to (I1). Using [22, Proposition 4.1(i)] we get HvPF = H0PF + φ(v) = H0PF + φ(v0 ) + φ(v − v0 ) √ 1 ≥ H0PF − (H0PF + r) − 2C1 (1, v − v0 )(H0PF + 1) 3

r √ 1 √ = 1 − − 2C1 (1, v − v0 ) H0PF − − 2C1 (1, v − v0 ). 3 3 Using that ω −1/2 ≤ 2/3+ω −3/2/3 ≤ 2/3(1+ω −3/2d(ω)) we get C1 (r, v) ≤ 2vPF /3 for any v ∈ IPF (d) and r ≥ 1. This implies √ √ 2 2 2 2 2 r PF PF − v − v0 PF H0 − − v − v0 PF . Hv ≥ 3 3 3 3 √ Observe that the choice γ0 = 1/(2 2) ensures that we arrive at the bound r+1 . 3 Choose c0 = 1 + (r + 1)/3 such that HvPF + c0 ≥ 1. This proves (1). HvPF ≥ −

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As for (2) √ we observe ﬁrst that φ(v) = HvPF − H0PF ≤ HvPF . Next let r = r(v0 ) and γ0 = 1/(2 2) be as in the proof of (1) and estimate −φ(v) = −φ(v0 ) + φ(v0 − v) ≤

1 PF 1 2 r+1 (H + r) + (H0PF + 1) = H0PF + . 3 0 3 3 3

Writing H0PF = HvPF − Φ(v) we arrive at −φ(v) ≤ 2HvPF + r + 1. Combining with the choice of c0 in the proof of (1) now yields the ﬁrst estimate in (2), for a suﬃciently large c1 . As for the second part of (2) one can employ [22, Proposition 4.1(ii)] in place of [22, Proposition 4.1(i)] and argue as above. This gives a bound of the desired type for γ0 small enough. The choice γ0 = 1/8 works. Here one should observe that the constants Cj (r, v), j = 0, 1, 2, in [22] are all related to the norm · PF by Cj (1, v) ≤ 2vPF/3 as argued above for C1 . The statement in (3) now follows by appealing to [22, Proposition 4.1(i)] again √ ±Rδ (v) ≤ 2C1 (1, [1K ⊗ aδ ]v)(H0PF + 1) √ ≤ 2C1 (1, [1K ⊗ aδ ]v)((c1 + 1)HvPF + c1 c0 + 1). for which the ﬁrst estimate From (5.5) and (5.3), we conclude the existence of a CB in (3) is satisﬁed. Similarly for the second part of (3), where, as in the discussion of the second part of (2), one can make use of [22, Proposition 4.1(ii)].

We can now state and prove a commutator estimate that is uniform with respect to v from a small ball around v0 , and E in a compact interval. Given v0 , let γ0 denote the radius coming from Lemma 5.10. Corollary 5.11. Let J ⊆ R be a compact interval, v0 ∈ IPF (d), and 0 > 0. There exist a δ0 ∈ ]0, 1/2] such that for any 0 < δ < δ0 the following holds. There exists 0 < γ < γ0 , κ > 0, C4 > 0 and a compact operator K0 , with γ only depending on δ, 0 , J and v0 , such that the form inequality ⊥ Mδ + Rδ (v) ≥ (1 − 0 )1HPF − C4 fE,κ (HvPF )2 HvPF − K0

(5.19)

holds on D(N 1/2 ) ∩ D(HvPF ), for all E ∈ J and v ∈ Bγ (v0 ). Remark. We note that the constant C4 in Corollary 5.9 can, on inspection of the proof of [22, Theorem 7.12], be chosen uniformly in 0 < δ ≤ δ0 . Making use of this would allow us to choose γ independent of δ ≤ δ0 here, which would slightly simplify the exposition. We however choose not to test the readers patience on this issue. See Step II in the proof below.

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Proof. Given J, v0 and 0 , let γ0 be given by Lemma 5.10 and let CB > 0 δ0 > 0 be the constants coming from Corollary 5.9. For E ∈ J we apply Corollary 5.9, with 0 replaced by 0 /3, and get the form estimate ⊥ Mδ + Rδ (v0 ) ≥ (1 − 0 /3)1HPF − C4 (v0 , E)fE,κ(v (HvPF )2 0 0 ,E)

− CB (HvPF − E) − K0 (v0 , E). 0

(5.20)

The constants C4 , κ and the operator K0 also depend on δ, but this dependence does not concern us. We can assume that K0 ≥ 0. The key observation is that the constants C4 and κ, and the operator K0 above can be chosen independently of E ∈ J and v ∈ Bγ (v0 ), for some suﬃciently small γ which does not depend on δ ≤ δ0 . We divide the proof of the corollary into three steps, the two ﬁrst establish the observation mentioned in the previous paragraph. Step I. We begin by arguing that C4 , κ and K0 can be chosen independently of E ∈ J. By a covering argument it suﬃces to show that they can be chosen independently of E in a small neighborhood of E ∈ J. For the compact error, we remark that one should replace K0 by a ﬁnite sum K0 (v0 ) = K0 (v0 , E1 ) + · · · + K0 (v0 , Em ) of non-negative compact operators, which is again compact. Let E ∈ J be ﬁxed. Pick ζ1 = 0 /(6CB ) such that for |E − E | < ζ1 we have CB E ≥ CB E − 0 /6.

(5.21)

⊥ As for the term involving fE,κ we observe that for any self-adjoint operator S we have ⊥ fE,κ (S) − fE⊥ ,κ (S) = fE ,κ (S) − fE,κ (S) )

−1

−1 * S − E S − E 1 −z −z (∂¯f˜)(z) − dudv. = π C κ κ

Here z = u + iv. Estimating this we ﬁnd that |E − E | . κ ⊥ Writing a2 − b2 = (a − b)(a + b) we observe a similar bound for fE,κ (S)2 − fE⊥ ,κ (S)2 . Again we conclude that for ζ2 = κ(v0 , E)0 /(6CC4 (v0 , E)) we ﬁnd that for |E − E | < ζ2 : ⊥ fE,κ (S) − fE⊥ ,κ (S) ≤ C

⊥ (HvPF )2 ≥ −C4 (v0 , E)fE⊥ ,κ (HvPF )2 − 0 /6. −C4 fE,κ 0 0

(5.22)

The estimates (5.21) and (5.22) plus the aforementioned covering argument implies the form estimate ⊥ Mδ + Rδ (v0 ) ≥ (1 − 20 /3)1HPF − C4 (v0 )fE,κ(v (HvPF )2 0)

− CB (HvPF − E) − K0 (v0 ), for all E ∈ J.

(5.23)

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Step II. Secondly we argue that one can use the same constants C4 , κ, and compact operator K0 for v ∈ Bγ (v0 ), if γ is small enough. Using Lemma 5.10, we estimate Rδ (v0 ) = Rδ (v) + Rδ (v0 − v) ≤ Rδ (v) + C1 v − v0 PF (HvPF + C2 ). Writing C1 v − v0 PF (HvPF + C2 ) = C1 v − v0 PF (HvPF − E) + C1 v − v0 PF (C2 + E), We see that choosing γ1 = γ1 (0 , J, v0 ) small enough we arrive at the following bound 0 (5.24) Rδ (v0 ) ≤ Rδ (v) + C(HvPF − E) + 1HPF , 9 which holds for all v ∈ Bγ1 (v0 ) and E ∈ J. ⊥ For the fE,κ , contribution, we compute ) − fE,κ (HvPF ) fE,κ (HvPF 0 )

−1 PF

−1 * HvPF −E Hv − E 1 0 ¯ ˜ = −z −z (∂ f )(z) − dudv π C κ κ =

1 κπ

(∂¯f˜)(z) C

HvPF − E −z κ

−1

φ(v − v0 )

HvPF − E 0 −z κ

−1 dudv.

From Lemma 5.8 and the representation formula above, we ﬁnd that ⊥ ⊥ fE,κ (HvPF )2 − fE,κ (HvPF )2 ≤ Cv − v0 PF , 0

uniformly in E ∈ J. Arguing as above we thus ﬁnd a γ2 = γ2 (0 , J, v0 , δ) > 0 such that 0 ⊥ ⊥ (HvPF )2 ≥ −C4 (v0 )fE,κ (HvPF )2 − 1HPF (5.25) −C4 (v0 )fE,κ 0 9 for all v ∈ Bγ2 (v0 ). This is where the δ-dependence enters into the choice of γ through C4 . See the remark to the corollary. Using Lemma 5.10, we also get a γ3 = γ3 (0 , v0 ) > 0 such that −CB (HvPF − E) ≥ −CB (HvPF − E) − 0

0 1HPF , 9

(5.26)

for all v ∈ Bγ3 (v0 ). Combining (5.23) with (5.24)–(5.26) we conclude that the estimate (5.20) holds with the same C4 , κ and K0 , for all E ∈ J and v ∈ Bγ (v0 ), with γ = min{γ1 , γ2 , γ3 } only depending on 0 , J, v0 and δ. Step III. To conclude the proof we let γ, C4 , κ and K0 be ﬁxed by Steps I and II. Pick κ smaller than κ such that κ CB (1 + maxE∈J |E|)| ≤ 0 . The Corollary now

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follows from (5.20) and the estimate

⊥ (HvPF )2 HvPF . −CB (HvPF − E) ≥ −CB 1 + max |E| fE,κ E∈J

Observe that (5.20) holds with κ replaced by κ as well. The corresponding objects in the expanded Hilbert space are deﬁned as follows: We set Mδe := 1K ⊗ dΓ(meδ h ) and Rδe = Rδe (v) := −φ(iaeδ v e ). Note that + U −1 Mδe U = Mδ ⊗ 1Γ(h) ˜ + 1H ⊗ M δ ,

U −1 Rδe U = Rδ ⊗ 1Γ(h) ˜ ,

ˆ ) as an operator on Γ(h). ˜ From (5.12), we get +δ := dΓ(d(δ)h where M ! 1 ˆ + Mδ ≥ d(δ) dΓ(h) + N , 2

(5.27)

(5.28)

The Mourre estimate for Hve is stated in the following theorem. Theorem 5.12. Assume that Hypotheses (H0), (I1) and (I2) hold. Let v0 ∈ IPF (d), J a compact interval, and 0 > 0. There exists δ0 ∈ ]0, 1/2] such that for all 0 < δ ≤ δ0 , there exist 0 < γ < γ0 , C4 > 0, κ > 0, and a compact operator K0 on He such that ⊥ (Hve )2 Hve − K0 Mδe + Rδe ≥ (1 − 0 )1He − CfE,κ

(5.29)

for all E ∈ J and v ∈ Bγ (v0 ), as a form on D((Mδe )1/2 ) ∩ D(Hve ). Remark. As in Corollary 5.11, the constant γ can be chosen to only depend on 0 , J, v0 and δ, and as in the associated remark one can in fact choose it uniformly in 0 < δ ≤ δ0 . Proof. We ﬁx v0 , J and 0 as in the statement of the the theorem. We begin by taking δ0 to be the δ0 coming from Corollary 5.11. Secondly we ﬁx and c0 to be the two constants from Lemma 5.10(3). CB We can now choose 0 < δ0 ≤ δ0 such that

+ 2, max 2CB (E + c0 ) . (5.30) d(δ0 ) ≥ max CB E∈J

Here we used that limt→0+ d(t) = +∞. Fix now a 0 < δ ≤ δ0 and denote by γ the radius coming from Corollary 5.11. The above choices anticipates the proof below, but we make them here to make it evident that we pick the constants in the right order. We begin the veriﬁcation of the commutator estimate for v ∈ Bγ (v0 ) by computing as a form on D((Mδe )1/2 ) ∩ D(H e ) +δ + Rδ ⊗ 1]1 ⊗ P¯Ω . U −1 [Mδe + Rδe ]U = [Mδ + Rδ ] ⊗ PΩ + [Mδ ⊗ 1 + 1 ⊗ M (5.31)

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We apply Corollary 5.11 to the ﬁrst term in the right-hand side of (5.31), with the given δ (apart from v0 , J and 0 ). This yields a C4 , a κ > 0, and a compact operator K0 (apart from γ) such that the following bound holds ⊥ PF 2 ) HvPF − K0 ] ⊗ PΩ . [Mδ + Rδ ] ⊗ PΩ ≥ [(1 − 0 )1 − C4 fE,κ (Hv

(5.32)

Observe that the bound above also holds with κ replaced by any 0 < κ ≤ κ . To bound from below the second term on the right-hand side of (5.31), we use Lemma 5.10. Together with (5.28) and (5.30), this implies +δ + Rδ ⊗ 1]1 ⊗ P¯Ω [1 ⊗ M !

ˆ + 1 − C (H PF ⊗ 1 + c ) ⊗ 1g 1 ⊗ P¯Ω ≥ g 1 ⊗ d(δ) dΓ(h) B v 0 2 ! d(δ) e ˆ − C (H ( − C ≥ (d(δ) − CB )1 ⊗ dΓ(h) − E) + (E + c ) 1 ⊗ P¯Ω B v B 0 2 (e ≥ [2 − CB (Hv − E)]1 ⊗ P¯Ω .

(5.33)

ˆ ≥ 0. We now pick a 0 < κ ≤ κ such Here we also made use of (5.10) and that h 2 ⊥ ⊥ 2 + (fE,κ ) into (5.33) yields the that 3κCB ≤ 1. Inserting 1 = fE,κ + 2fE,κ fE,κ bound ⊥ ( ve )2 H ( ve ]1 ⊗ P¯Ω , +δ + Rδ ⊗ 1]1 ⊗ P¯Ω ≥ [1 − CB (1 + E )fE,κ (H [1 ⊗ M

where E = maxE∈J |E|. This estimate together with (5.31) and (5.32) lead (1 + E )} and K0 = to the statement of the theorem with C4 = min{C4 , CB −1 U[K0 ⊗ PΩ ]U . 5.5. Checking the abstract assumptions The purpose of this subsection is to complete the proof of Theorem 5.5. We do this by running through the abstract assumptions in Sec. 2 pertaining to Theorems 2.5 and 2.10, from which Theorem 5.5 then follows. In accordance with Remark 2.11(4), we ensure that all constants can be chosen locally uniformly in energy E and form factor v. This ensures local uniformity in Theorem 5.5. ). Observe that there exists e0 such that We ﬁx v0 ∈ IPF (d) and E0 ∈ σ(HvPF 0 e0 < inf σ(HvPF ) for all v ∈ Bγ0 (v0 ), where γ0 comes from Lemma 5.10. Put J = [e0 , E0 ]. Let γ and δ0 be ﬁxed by Theorem 5.12 and choose a δ < δ0 , which from now on is ﬁxed. We begin by postulating the objects for which the abstract assumptions in Conditions 2.1 should hold. We take H = He , H = Hve , A = Aeδ ,

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N = K ρ ⊗ 1Γ(he ) + 1K ⊗ dΓ(h ) + 1He ,

511

1 max 2τ, < ρ < 1, 2

H = [Mδe + Rδe ]|D(N ) . (5.34) The constant τ appearing above is the one from (I1). Observe that Rδe and Mδe are N -bounded. See Lemma 5.13 just below. We make use of the following dense subspace of H S = D(K) ⊗ Γﬁn (C0∞ (R) ⊗ L2 (S d−1 )) ⊆ He . The tensor product is algebraic. Observe that S is a core for H, N , and A. We recall that we can construct the group eitA explicitly. Let ψt denote the (global) ﬂow for the 1-dimensional ODE ψ˙ t (ω) = meδ (ψt (ω)). Then, for continuous compactly supported supported f , e

1

(eitaδ f )(ω) = e 2

Rt 0

(meδ ) (ψs (ω))ds

f (ψt (ω)).

This in particular implies that e

e

eitAδ = Γ(eitaδ ) : S → S.

(5.35)

We begin with the following lemma which implies that Rδe is N -bounded. Lemma 5.13. Let v ∈ Oτ and κ = 1/4 − τ /(2ρ). Then D(N 1−2κ ) ⊆ D(φ(v)), and for f ∈ D(N ) we have φ(v e )f ≤ Cvτ N 1−2κ f , where C does not depend on v nor on f . Proof. Adopting notation from [22] we put C0 (v) = v(K + 1)−τ 2 and C2 (v) = [(K + 1)−τ ⊗ 1h ]v2 . We estimate for f ∈ S, repeating the argument for [22, (3.14) and (3.16)], and get a∗ (v e )f 2 ≤ C0 (v)(K + 1)τ ⊗ 1Γ(he ) f 2 + C0 (v)f, (K + 1)2τ ⊗ N ef and a(v e )f 2 ≤ C2 (v)f, (K + 1)2τ ⊗ N ef . Observing the bound, with 2κ = 1/2 − τ /ρ and some C > 0, τ (K + 1)2τ ⊗ N e ≤ (K + 1)2ρ(1−2κ) ⊗ 1Γ(he ) ρ(1 − 2κ) +

1 (N e )2(1−2κ) ≤ C N 2(1−2κ) , 2(1 − 2κ)

yields Φ(v e )f ≤ Cvτ N 1−2κ f

(5.36)

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a priori as a bound for elements of S. The lemma now follows since S is a core for N . Condition 2.1(1). We make use of the fact (given the invariance of S mentioned in (5.35)) that our Condition 2.1(1) is equivalent to Mourre’s conditions, eitA D(N ) ⊆ D(N ) (i.e. D(N ) is invariant) and that i[N, A] extends from a form on S to an element of B(N −1 H; H). See [30, Proposition II.1]. From the computation i[h , aeδ ] = meδ h it follows that the following identity holds in the sense of forms on S N = i[N, Aeδ ] = 1K ⊗ dΓ(meδ h ).

(5.37)

Since meδ is bounded and supω∈R |h (ω)|/h (ω) < ∞, we ﬁnd that N extends from S to a bounded operator on D(N ), and the extension is in fact an element of B(N −1 H; H) as required. e It remains to check that D(N ) is invariant under eitAδ . For this we compute strongly on S e

e

N eitAδ = eitAδ (K ρ ⊗ 1Γ(he ) + 1K ⊗ dΓ(h ◦ ψ−t )). Since t → ψt (ω) is increasing and ω → h (ω) is decreasing (and positive) we ﬁnd for t ≤ 0 0 ≤ h ◦ ψ−t ≤ h . For positive t we estimate ω − Ct ≤ ψ−t (ω) ≤ ω, for some C > 0, where we used that meδ was a bounded function. This gives for t > 0 0 ≤ h ◦ ψ−t (ω) = max{1, e−ψ−t (ω) + ψ−t (ω)} ≤ max{1, e−ω+Ct + ω − Ct}. Using that e−ω+α + ω ≤ Cα (e−ω + ω), we get for any t a C = C (t) such that (h ◦ ψ−t )2 ≤ C (h )2 and hence by [22, Proposition 3.4] we arrive at dΓ(h ◦ ψ−t )2 ≤ C dΓ(h )2 . e

Since S was a core for N we now conclude that eitAδ D(N ) ⊆ D(N ). This completes the veriﬁcation of Condition 2.1(1). Condition 2.1(2). We begin by observing that N and H0e commute. In particular we can compute as a form on S i[N −1 , Hve ] = iN −1 φ(v e ) − iφ(v e )N −1 . This computation in conjunction with Lemma 5.13 implies that i[N −1 , Hve ] extends from a form on D(Hve ) to a bounded operator and hence N is of class C 1 (H).

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Since the commutator form i[N, H] extends from D(N ) ∩ D(H) to a bounded form on D(N ) it suﬃces to compute it on a core for N . Here we take again S and compute i[N, H] = [K ρ ⊗ 1Γ(he ) , φ(v e )] + φ(ih v e ) = φ([K ρ ⊗ 1he ]v e − v e K ρ ) + φ(iv e ).

(5.38) 1

That the second term extends by continuity to a bounded form on D(N 2 −κ ) follows from Lemma 5.13 (applied with iv e instead of v e ) and interpolation. In order to deal with the ﬁrst term in (5.38) we write −1 v − v˜K ρ ) ⊗ 1Γ(h) . φ([K ρ ⊗ 1he ]v e − v e K ρ ) = U(φ([K ρ ⊗ 1h˜ ]˜ ˜ )U

Here we need the new assumption (I4). We will immediately verify that the above expression extends to a bounded form on D(N 1/2−κ ) for some κ > 0. This implies the required property for i[H, N ]0 . We employ the representation formula (3.5) with K instead of N . Compute as a form on D(K ⊗ 1h˜ ) × D(K) ∞ (K ρ ⊗ 1h˜ )˜ v − v˜K ρ = −cρ tρ [((K + t)−1 ⊗ 1h˜ )˜ v − v˜(K + t)−1 ]dt 0

= B − cρ 1

∞

tρ ((K + t)−1 ⊗ 1h˜ )[˜ v K − (K ⊗ 1h˜ )˜ v ](K + t)−1 dt,

where B is the contribution from the integral between 0 and 1, which due to (I1) is a bounded operator. By (I4), we have 1

v K − (K ⊗ 1h˜ )˜ v )(K + 1)− 2 < ∞, c1 := (˜ 1

c2 := (K + 1)− 2 ⊗ 1h˜ (˜ v K − (K ⊗ 1h˜ )˜ v ) < ∞. Let τ < 1/2 be chosen such that ρ/2 > τ > ρ − 1/2. This is possible due to the choice of ρ. We estimate for ψ ∈ D(K ⊗ 1h˜ ) and ϕ ∈ D(K) ψ, ((K ρ ⊗ 1h˜ )˜ v − v˜K ρ )ϕ ≤ Bψϕ +

1 2

c1 cρ ψ(K + 1)τ ϕ. +τ −ρ

Similarly we get ψ, ((K ρ ⊗ 1h˜ )˜ v − v˜K ρ )ϕ ≤ Bψϕ +

1 2

c 2 cρ (K ⊗ 1h˜ + 1)τ ψϕ. + τ − ρ

We have thus established that the ﬁrst term in (5.38) is the (expanded) ﬁeld operator associated to an operator in Oτ . We can thus employ Lemma 5.13 again, this time with v e replaced by [K ρ ⊗ 1he ]v e − v e K ρ and κ replaced by 0 < κ = 1/4 − τ /(2ρ) < 1/4. Together with an interpolation argument this

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ensures that φ((K ρ ⊗ 1he )v e − v e K ρ ) extends by continuity to a bounded form on D(N 1/2−κ ). We have thus veriﬁed Condition 2.1(2) with the smallest of the two kappa’s. In addition we observe that the B(N −1/2+κ H; N 1/2−κ H)-norm of i[N, H]0 is bounded by a constant times vPF, cf. Remark 2.11(4). Remark 5.14. We observe from the discussion above that we could relax (I4) and v − v˜K ρ extends to an element of B(D(K η ); K ⊗ require instead that [K ρ ⊗ 1h˜ ]˜ ˜ ∩ B(K; D(K η )∗ ⊗ h), ˜ for some 1/2 ≤ η < 1 − τ , where τ is coming from (I1). h) This would still leave room to choose ρ and τ (in the argument above) such that 1 > ρ > 2τ and ρ/2 > τ > ρ + η − 1. While we do not know the domain of H, it turns out that we can indeed compute the intersection domain D(H) ∩ D(N ). This is done in the following lemma. Lemma 5.15. We have the identity D(H) ∩ D(N ) = D(K ⊗ 1Γ(he ) ) ∩ D(1K ⊗ dΓ(max{h , ω}))

(5.39)

and S is dense in D(H) ∩ D(N ) with respect to the intersection topology. Proof. Let for the purpose of this proof H0 = H0e , the unperturbed expanded Hamiltonian, and denote by D the right-hand side of (5.39). Since N controls the unphysical part of dΓ(h), due to the choice of extension of ω by an exponential, we observe that the identity (5.39) holds if H is replaced by H0 . Since H0 and N commute we ﬁnd that T0 = N + iH0 is a closed operator on D and it clearly generates a contraction semigroup. We now construct the formal operator sum N + iH in two diﬀerent ways. By Lemma 5.13, D(φ(v e )) ⊂ D(N 1−2κ ) and hence for u ∈ D φ(v e )u ≤ cN 1−2κ u + c u ≤

1 1 N u + c u ≤ T0 u + c u. 4 4

From this estimate, we deduce that T1 = T0 + iφ(v) =: N + iG is a closed operator on D and it generates a contraction semigroup. See [33–36, Lemma preceding Theorem X.50]. Here G is implicitly deﬁned as the operator sum G = H0 + φ(v e ) with domain D. On the other hand, since we have just established Condition 2.1(2), we conclude from [21, Theorem 2.25] that T2,± = N ± iH are closed operators on D(H) ∩ D(N ). ∗ = T2,∓ and since T2,± are both accretive we conclude that In addition we have T2,± T2,+ generates a contraction semigroup. See [33–36, Corollary to Theorem X.48]. We proceed to argue that T2 = T2,+ is an extension of T1 , i.e. T1 ⊂ T2 . Since S ⊆ D, G is a symmetric extension of H|S and S is a core for H we deduce that H is an extension of G. Hence indeed T1 ⊂ T2 .

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We now argue that in fact T1 = T2 , or more poignantly that their domains coincide. This will follow if the intersection of the resolvent sets is non-empty. Indeed, let ζ ∈ ρ(T1 ) ∩ ρ(T2 ). Then (T2 − ζ)(T1 − ζ)−1 = (T1 − ζ)(T1 − ζ)−1 = 1, and hence (T2 − ζ)−1 = (T1 − ζ)−1 and the domains must coincide. But by the Hille–Yosida theorem [33–36, Theorem X.47a] we have (−∞, 0) ⊂ ρ(T1 ) ∩ ρ(T2 ). Here we used that both T1 and T2 generate contraction semigroups. It remains to ascertain that S is dense in D with respect to the intersection topology of D(H)∩D(N ). We begin by verifying that S is dense in D with respect to the graph norm of T0 , which induces the intersection topology of D(H0 )∩D(N ) = D. Let ψ ∈ D. Observe ﬁrst that limn→∞ 1N e ≤n ψ → ψ in the graph norm of T0 , since N e and T0 commute. Similarly we ﬁnd that 1K ⊗Γ(1|ω|≤)ψ → ψ in the graph norm of T0 . Hence it suﬃces to approximate ψ ∈ D with Γ(1|ω|≤ )1N e ≤n ψ = ψ, for some and n, by elements from S in the graph norm of T0 . Fix now such a ψ, n and . Since S is a core for K ⊗ 1Γ(he ) we can ﬁnd a sequence {ψj } ⊂ S with ψj → ψ in D(K ⊗ 1Γ(he ) ). Put ψ˜j = 1Ne ≤n [1K ⊗ Γ(f )]ψj ∈ S, where f ∈ C0∞ (R), with 0 ≤ f ≤ 1 and f = 1 on [−, ]. Then ψ˜j → ψ in D(K ⊗ 1Γ(he ) ) as well. We now observe that T0 ψ˜j = (iK ⊗ 1Γ(he ) + Bn, )ψ˜j , for some bounded operator Bn, . This implies density of S in D in the graph norm of T0 . By the closed graph theorem H(T0 − ζ)−1 and N (T0 − ζ)−1 are bounded, and hence S is also dense in D(H)∩D(N ) = D with respect to the indicated intersection topology. Condition 2.1(3). Let σ be such that R(η) preserves D(N ) for η with |Im η| ≥ σ. It suﬃces to establish the identity R(η)H − HR(η) = −iR(η)H R(η), for η with |Im η| ≥ σ, as a form on D(H) ∩ D(N ), since this set is dense in D(H) ∩ D(N 1/2 ) by Remark 3.5. By Lemma 5.15, we can on the set D(H) ∩ D(N ) espress H and H as sums of operators H = H0e + φ(v e ) and H = dΓ(h ) − φ(iaeδ v e ). We are thus reduced to verifying the following two form identities on D(H) ∩ D(N ) R(η)H0e − H0e R(η) = −iR(η)1K ⊗ dΓ(meδ h )R(η), R(η)φ(v e ) − φ(v e )R(η) = iR(η)φ(iaeδ v e )R(η).

(5.40) (5.41)

Since all operators appearing in (5.40) commute with N e it suﬃces to verify this identity on each ﬁxed expanded particle sector with N e = n. Introduce for a positive integer the semibounded dispersion h (ω) = max{−, h(ω)} and a cutoﬀ

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expanded free Hamiltonian H0, = K ⊗ 1Γ(he ) + 1K ⊗ dΓ(h ). Then on a particle 1 (A) such that we can compute for |Im η| ≥ σn, sector H0, is of class CMo R(η)H0, − H0, R(η) = −iR(η)1K ⊗ dΓ(meδ h )R(η), as a form on 1[N e =n] D. Here σn, is some positive constant. Since both sides are analytic in η for |Im η| ≥ σ we conclude the above identity for all such η. Appealing to the explicit form of the domain D we ﬁnd that we can remove the cutoﬀ → ∞ by the dominated convergence theorem. This yields (5.40) for |Im η| ≥ σ. As for (5.41) we recall that we have already established that N is of class 1 (A). It is a consequence of the proof of [30, Proposition II.1], that i[φ(v e ), A] CMo read as a form on D(N ) ∩ D(A) can be represented by an extension from the form computed on S. Here we used (5.35). As a form on S we clearly have i[φ(v e ), A] = −φ(iaeδ v e ), which extends to an N -bounded operator by Lemma 5.13. The computation R(η)φ(v e ) − φ(v e )R(η) = R(η)[φ(v e ), A]R(η) as forms on D(N ) now concludes the veriﬁcation of (5.41), and hence of Condition 2.1(3). Condition 2.1(4). We compute ﬁrst as a form on S

dmeδ h + (meδ )2 h − φ((aeδ )2 v e ) i[H , A] = H = 1K ⊗ dΓ meδ dω and observe that the right-hand side extends by continuity to an N -bounded operator, cf. Lemma 5.13. Again, by the proof of [30, Proposition II.1], cf. (5.35), we conclude that the operator on the right-hand side of the formula also represents the commutator form i[H , A] on D(N ) ∩ D(A). Condition 2.2. By Lemma 5.15 and Remark 3.5, it suﬃces to check the form bound in the virial condition on S. In addition, since K ρ ≤ 1 + K, it suﬃces to check the estimate with ρ = 1. ˆ ≤ h ˆ , and hence h + h ≥ 0. Recalling (5.11) and (5.15), we observe that h Making use of this observation we ﬁnd that K ⊗ 1Γ(he ) + 1K ⊗ dΓ(h ) ≤ K ⊗ 1Γ(he ) + 1K ⊗ (dΓ(h) + 2dΓ(h )) ≤ K ⊗ 1Γ(he ) + 1K ⊗ dΓ(h) + 2Mδe . We now add and subtract Φ(v e ) + 2Rδe to obtain K ⊗ 1Γ(he ) + 1K ⊗ dΓ(h ) ≤ Hve + 2H − Φ(v e ) − 2Rδe . We now make use of the fact that 1

C = (Φ(v e ) + 2Rδe )(K ⊗ 1Γ(he ) + 1K ⊗ dΓ(h ) + 1)− 2 < ∞ to conclude the form estimate 1 1 K ⊗ 1Γ(he ) + 1K ⊗ dΓ(h ) ≤ Hve + 2H + (K ⊗ 1Γ(he ) + 1K ⊗ dΓ(h ) + 1) + C 2 . 2 2

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This completes the veriﬁcation of the virial bound. We again observe that the constants involved can be chosen independent of E in a bounded set and v ∈ Bγ (v0 ). Condition 2.3. This condition has already been essentially veriﬁed in the form of Theorem 5.12. We only need to observe that the form bound extends by continuity from D(H) ∩ D(N ) to D(H) ∩ D(N 1/2 ), cf. Remark 3.5. The condition (2.7). Let ψ e be a bound state for H = Hve . That is ψ e ∈ D(Hve ) vPF ) and Hve ψ e = Eψ e , for some E ∈ R. Recall that ψ e = U(ψ ⊗ Ω), where ψ ∈ D(H PF and Hv ψ = Eψ. From [22, Proposition 6.5] we conclude that ψ ∈ D(N 1/2 ). PF ⊗ 1 ˜ ). In particular we Hence we conclude that ψ e ∈ D(dΓ(h )1/2 ) ∩ UD(H v Γ(h) ﬁnd that ψ e ∈ D(H) ∩ D(N 1/2 ) and the result follows from the virial estimate in Condition 2.2. Observe again that N 1/2 ψ can be bounded uniformly in v ∈ Bγ0 (v0 ) and E ∈ [e0 , E0 ]. Condition 2.8 k0 = 1. This merely amounts to checking the statement in (2.11) with = 0. But this is trivially satisﬁed since [N, N ] = 0. See (5.37). This completes the veriﬁcation of the conditions needed to conclude Theorem 5.5 from Theorems 2.5 and 2.10. 6. AC-Stark Type Models 6.1. The model and the result We will work in the framework of generalized N -body systems, which we review brieﬂy. Let A be a ﬁnite index set and X a ﬁnite dimensional real vector-space with inner product. There is an injective map from A into the subspaces of X, A a → X a ⊆ X, and we write Xa = (X a )⊥ . We introduce a partial ordering on A: a ⊂ b ⇔ Xa ⊆ Xb and assume the following (1) There exist amin , amax ∈ A with X amin = {0} and X amax = X. (2) For each a, b ∈ A there exists c = a ∪ b ∈ A with Xa ∩ Xb = Xc . We will write xa and xa for the orthogonal projection of a vector x onto the subspaces X a and Xa respectively. We will work with a generalized potential Va (t, xa ), V = V (t, x) = a∈A\{amin }

where Va is a real-valued function on R × X a . In the conditions below α denotes multiindices.

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Conditions 6.1. Let k0 ∈ N be given. For each a = amin the following holds. The pair-potential R × X a (t, y) → Va (t, y) ∈ R is a continuous function satisfying (1) Periodicity: Va (t + 1, y) = Va (t, y), t ∈ R and y ∈ X a . (2) Diﬀerentiability in y: For all α with |α| ≤ k0 + 1 there exist ∂yα Va ∈ C(R × X a ). (3) Global bounds: For all α and k ∈ N ∪ {0} with |α| + k ≤ k0 + 1 there are global bounds |∂yα (y · ∇y )k Va (t, y)| ≤ C. (4) Decay at inﬁnity: |Va (t, y)| + |y · ∇y Va (t, y)| = o(1) uniformly in t. (5) Regularity in t: There exists ∂t Va ∈ C(R × X a ) and there is a global bound |∂t Va (t, y)| ≤ C. We consider under Condition 6.1 the Hamiltonian h = h(t) = p2 + V , p = −i∇, on the Hilbert space L2 (X). The corresponding propagator U satisﬁes: It is twoparameter strongly continuous family of unitary operators which solves the timedependent Schr¨ odinger equation d U (t, s)φ = h(t)U (t, s)φ for φ ∈ D(p2 ). dt The family satisﬁes the Chapman Kolmogorov equations i

U (s, r)U (r, t) = U (s, t),

r, s, t ∈ R,

the initial condition U (s, s) = 1 for any s ∈ R and the periodicity equation U (t + 1, s + 1) = U (t, s),

s, t ∈ R.

The operator U (1, 0) is called the monodromy operator. For each a = amax the sub-Hamiltonian monodromy operator is U a (1, 0); it is deﬁned as the monodromy operator on Ha = L2 (X a ) constructed for a = amin from ha = (pa )2 + V a , b a Va = amin =b⊂a Vb (t, x ). If a = amin we deﬁne U (1, 0) = 1 (implying amin (1, 0)) = {1}). The set of thresholds is then σpp (U σpp (U a (1, 0)). (6.1) F (U (1, 0)) = a =amax

We recall from [31] that the set of thresholds is closed and countable, and non-threshold eigenvalues, i.e. points in σpp (U (1, 0))\F (U (1, 0)), have ﬁnite multiplicity and can only accumulate at the set of thresholds. Moreover any corresponding bound state is exponentially decaying, the singular continuous spectrum σsc (U (1, 0)) = ∅ and there are integral propagation estimates for states localized away from the set of eigenvalues and away from F (U (1, 0)). It should be remarked that the weakest condition, Condition 6.1 with k0 = 1, corresponds to [31, Condition 1.1] (more precisely Condition 6.1 with k0 = 1 is slightly weaker than [31, Condition 1.1], and we also remark that [31] goes through with this modiﬁcation). All of the above properties are proven in [31] either under [31, Condition 1.1] or under weaker conditions allowing local singularities. In particular local singularities up to the Coulomb singularity are covered in [31]. See Sec. 6.3 for a new result for Coulomb systems.

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In the following subsection we establish the theorem below, which implies Theorem 1.6(2). Theorem 6.2. Suppose Conditions 6.1, for some k0 ∈ N. Let φ be an bound state / F(U (1, 0)). Then φ ∈ D(|p|k0 +1 ). for U (1, 0) pertaining to an eigenvalue e−iλ ∈ 6.2. Regularity of non-threshold bound states The principal tool in the proof of Theorem 6.2 will be Floquet theory (in common with [31] and other papers) which we brieﬂy review. The Floquet Hamiltonian associated with h(t) is H = τ + h(t) = H0 + V,

on H = L2 ([0, 1]; L2 (X)).

(6.2)

d , with periodic boundary conditions. Here τ is the self-adjoint realization of −i dt The spectral properties of the monodromy operator and the Floquet Hamiltonian are equivalent. We have the following relations

σpp (U (1, 0)) = e−iσpp (H) ,

σac (U (1, 0)) = e−iσac (H) ,

σsc (U (1, 0)) = e−iσsc (H) ,

and the multiplicity of an eigenvalue z = e−iλ of U (1, 0) is equal to the multiplicity of λ as an eigenvalue of H (regardless of the choice of λ). We also recall that the Floquet Hamiltonian is the self-adjoint generator of the strongly continuous unitary one-parameter group on H given by (e−isH ψ)(t) = U (t, t − s)ψ(t − s − [t − s]),

(6.3)

where [r] is the integer part of r. In particular any bound state of the monodromy operator, U (1, 0)φ = e−iλ φ, gives rise to a bound state of the Floquet Hamiltonian, Hψ = λψ, by the formula ψ(t) = eitλ U (t, 0)φ.

(6.4)

Proposition 6.3. Suppose Conditions 6.1 for some k0 ∈ N and suppose Hψ = λψ / F(U (1, 0)). Then ψ ∈ D(|p|k0 +1 ). for e−iλ ∈ Proof. We shall use Corollary 4.13 with H being the Floquet Hamiltonian and N = p2 + 1. This amounts to checking the assumptions given in terms of Conditions 2.1– 2.3, 2.6, 2.8 and (for k0 ≥ 2 only) 4.11 (same k0 ). We take A = 12 (x · p + p · x) and compute with direct reference to Conditions 2.1, 2.6 and 2.8 H = 2p2 − x · ∇V, i[N, H]0 = p · ∇V + ∇V · p =

(6.5a) dim X

((pj ∂j V + (∂j V )pj ),

(6.5b)

j=1

N = 2p2 ,

(6.5c)

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i adA (N ) = 2+1 p2 , i adN (i adA (N )) = 0;

≤ k0 − 1,

(6.5d)

il adlA (H ) = 2l+1 p2 + (−1)l+1 (x · ∇)l+1 V ;

l ≤ k0 .

(6.5e)

A comment on (6.5a) is due. We need to show Condition 2.1(3) using the expression (6.5a): First we remark that the operators τ , p2 and H0 are simultaneously diagonalizable. Therefore D(H)∩D(N ) = D(τ )∩D(N ) is dense in D(H)∩D(N 1/2 ). (See also Remark 3.5.) Moreover p2 , V and R(η) are obviously ﬁbered (i.e. they act on the ﬁber space L2 (X)) and R(η) preserves D(p2 ) and D(|p|) for |η| large enough. Whence as a form on D(τ ) ∩ D(N ) i[H, R(η)] = i[p2 + V, R(η)] = −R(η)i[p2 + V, A]R(η) = −R(η)H R(η). The last identity for ﬁber operators is well known in standard Mourre theory for Schr¨ odinger operators. Finally we extend the shown version of (2.2) by continuity to a form identity on D(H) ∩ D(N 1/2 ) yielding Condition 2.1(3). Clearly (2.4) holds with C1 = 0, C2 = 1/2 and C3 = 1 + sup x · ∇V (t, x)/2. As for (2.5) a stronger version follows from [31, Theorem 4.2] H ≥ c0 1 − C4 fλ⊥ (H)2 − K0 .

(6.6)

Finally it follows from [31, Proposition 4.1] that indeed the condition of Corollary 4.13, ψ ∈ D(N 1/2 ) = D(|p|), is fulﬁlled. This shows the proposition in the case k0 = 1. For k0 ≥ 2 it remains to verify Condition 4.11. For this purpose it is helpful to notice that i adA (pj ) = pj ,

(6.7a)

i adA ((N + tj )−1 ) = −2(N + tj )−1 (N − 1)(N + tj )−1 .

(6.7b)

Moreover all computations are in terms of ﬁber operators (in particular M1 , M2 and M3 are all ﬁbered operators), and recalling [30, Proposition II.1] and using the 1 (A) it suﬃces to do the computations in terms of forms on fact that N 1/2 ∈ CMo the Schwartz space S(X). Re M1 . We shall apply (6.7a) in combination with (6.5b) to verify the part of Condition 4.11 that involves M1 . Let us ﬁrst look at the particular choice in (4.21) for M1 given by taking all the T ’s equal N 1/2 . That is we will demonstrate that for m = 1, . . . , k0 − 1 im adm N 1/2 (M1 ) is |p|-bounded. We compute m

i

adm 12 N

(M1 ) = −(ic 21 )

m+1

∞

(N + tm+1 )

dtm+1 tm+1 · · ·

0 −1

1 2

0

adm+1 p2 (V

∞

(6.8)

1 2

dt2 t2 −1

)(N + tm+1 )

∞ 0

1

t12 (N + t1 )−1 · · ·

· · · (N + t1 )−1 dt1 ,

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and in turn,

adm+1 p2 (V ) =

521

cα,β pα (∂ α+β V )pβ = T1 + T2 + T3 ;

|α+β|=m+1

T1 =

cα,β pα (∂ α+β V )pβ ,

|α+β|=m+1, |β|≥1

T2 =

−i cα+β,0 pα (∂ α+2β V ),

|α+β|=m+1, |β|=1

T3 =

cα+β,0 pα (∂ α+β V )pβ .

|α+β|=m+1, |β|=1

Now in front of the bounded derivative of any of the terms of the expressions T1 , T2 and T3 we move the factor pα to the left in the integral representation and use the bound N s (N + t)−1 ≤ Cs (1 + t)s−1 ;

s ∈ [0, 1].

(6.9)

We obtain pα (N + tm+1 )−1 · · · (N + t1 )−1 ≤ Csm+1

m+1

(1 + tj )s−1 ;

s=

j=1

|α| . 2(m + 1)

β

Using (6.9) for the factors of p to the right (in case of T1 and T3 ) combined with the resolvents to the right and an additional factor N −1/2 we obtain 1

pβ (N + tm+1 )−1 · · · (N + t1 )−1 N − 2 ≤ Cσm+1

m+1

(1 + tj )σ−1 ;

σ=

j=1

|β| − 1 . 2(m + 1)

To treat T2 we notice that (N + tm+1 )−1 · · · (N + t1 )−1 ≤

m+1

(1 + tj )−1 .

(6.10)

j=1

Now the integrand with an additional factor N −1/2 to the right is a sum of terms either bounded (up to a constant) by m+1

|α|

1

tj2 (1 + tj ) 2(m+1)

−1

|β|−1

(1 + tj ) 2(m+1)

−1

j=1

=

m+1

1

3 1 − 2 − 2(m+1)

tj2 (1 + tj )

j=1

(these terms come from T1 and T3 ), or (for any term of T2 ) by m+1 j=1

1

|α|

tj2 (1 + tj ) 2(m+1) −1 (1 + tj )−1 =

m+1

1

tj2 (1 + tj ) 2(m+1) −2 . m

j=1

Whence in all cases the integral with an additional factor N −1/2 to the right is convergent in norm, which ﬁnishes the proof of the special case where all of the T ’s are equal to N 1/2 . The general case follows the same scheme. Some of the

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commutators with A “hit” the potential part introducing a change W (t, x) → −x · ∇W (t, x). Other commutators with A hit a factor pj in which case we apply (6.7a). Finally yet other commutators with A hit a factor (N + tj )−1 in which case we apply (6.7b) and (6.10). Re M2 and M3 . The contributions to (4.21) from the ﬁrst term of (6.5a), i.e. contributions from the expression 2p2 N −1/2 , vanish except for the case where all of the T ’s are equal to A. In this case we compute

m 1 d m m 2 − 12 i adA (2p N ) = 2t f (t)|t=p2 ; f (t) = 2t(t + 1)− 2 . (6.11) dt Obviously the right hand side of (6.11) is N 1/2 -bounded. The contributions to (4.21) from the expressions −x · ∇V N −1/2 and −N −1/2 x · ∇V are treated like the term M1 in fact slightly simpler. The iterated commutators are all bounded in this case. We leave out the details. Remark. Since H is not elliptic (more precisely |p|(H0 − i)−1 is unbounded) we do not see an “easy way” to get the conclusion of Proposition 6.3. For instance we need to use the assumption that e−iλ is non-threshold. See [29] for a related result for the one-body AC-Stark problem. Proof of Theorem 6.2. We mimic the proof of [31, Theorem 1.8]. Recall the notation Iin (N ) = n(N + n)−1 and Nin = N Iin (N ). Due to Proposition 6.3 and the representation (6.4) there exists t0 ∈ [0, 1[ such that U (t0 , 0)φ ∈ D(N (k0 +1)/2 ).

(6.12)

In particular ψ(t) = eitλ U (t, 0)φ ∈ D(p2 ) for all t. Next we compute d k0 +1 k0 +1 ψ(t), Nin ψ(t) = ψ(t), i[V, Nin ]ψ(t), dt p k0 −p k0 +1 i[V, Nin ]= Nin i[V, Nin ]Nin ,

(6.13a) (6.13b)

0≤p≤k0

i[V, Nin ] = −Iin (N )

dim X

((pj ∂j V + (∂j V )pj )Iin (N ).

(6.13c)

j=1

We plug (6.13c) into (6.13b) and then in turn (6.13b) into the right-hand side of (6.13a). We expand the sum and redistribute for each term at most k0 derivatives by pulling through the factor ∂j V obtaining terms on a more symmetric form, more precisely on the form N

k0 +1 2

ψ(t), Bn N

k0 +1 2

ψ(t)

where sup Bn < ∞.

(6.14)

n

Notice that for all terms the operator Bn involves at most k0 + 1 derivatives of V . Thanks to the Cauchy–Schwarz inequality and Proposition 6.3 any expression

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like (6.14) can be integrated on [t0 , 1] and the integral is bounded uniformly in n. In combination with (6.12) we conclude that k0 +1 ψ(1) < ∞, sup ψ(1), Nin n

whence φ = ψ(1) ∈ D(N

(k0 +1)/2

).

6.3. Regularity of non-threshold atomic type bound states The generator of the evolution of the a system of N particles in a time-periodic Stark-ﬁeld with zero mean (AC-Stark ﬁeld) is of the form hphy (t) = p2 − E(t) · x + Vphy

1 on L2 (X). Assuming that the ﬁeld is 1-periodic the condition 0 E(t)dt = 0 leads to the existence of unique 1-periodic functions b and c such that 1 d d b(t) = E(t), c(t) = 2b(t)) and c(t)dt = 0; dt dt 0 see [31] for details. For simplicity let us here assume that E ∈ C([0, 1]; X), see Remark 6.4 for an extension. The potential Vphy is a sum of time-independent real-valued “pair-potentials” Vphy = Vphy (x) = Va (xa ). a∈A\{amin }

In terms of these quantities we introduce Hamiltonians haux (t) = p2 + 2b(t) · p + Vphy , h(t) = p2 + Vphy (· + c(t)). The propagators Uphy , Uaux and U of hphy , haux and h, respectively, are linked by Galileo type transformations. Deﬁne t |b(s)|2 ds. S1 (t) = eic(t)·p and S2 (t) = ei(b(t)·x−α(t)) ; α(t) = 0

Then Uphy (t, 0) = S2 (t)Uaux (t, 0)S2 (0)−1 , −1

U (t, 0) = S1 (t)Uaux (t, 0)S1 (0) −1

Uphy (t, 0) = S2 (t)S1 (t)

(6.15a)

,

(6.15b) −1

U (t, 0)S1 (0)S2 (0)

.

(6.15c)

The bulk of [31] is a study of the Floquet Hamiltonian of h. Spectral information is consequently deduced for the monodromy operator U (1, 0). Finally the formula (6.15c) then gives spectral information for the physical monodromy operator Uphy (1, 0). The part of [31] concerning potentials with local singularities contains an incorrect reference in that it is referred to [39] for the existence of the propagator U (see [31, Remark 1.4]). However although the issue of Yajima’s paper is

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the existence of an appropriate dynamics for singular time-dependent potentials the paper as well as the method of proof is for the one-body problem only. This point is easily ﬁxed as follows, see Remark 6.4 for a more complicated procedure for E ∈ L1 ([0, 1]; X)\C([0, 1]; X): We use Yosida’s theorem which is in fact also alluded to in [31, Remark 1.4] (see [37, Theorem II.21] for a statement of the theorem). If Vphy is -bounded relatively to p2 (which is the case under the conditions considered in [31]) then indeed the propagator Uaux exists and we can use (6.15a) and (6.15b) to deﬁne Uphy and U . In particular we can use (6.15c) and obtain not only the existence of Uphy but various spectral information of the corresponding monodromy operator Uphy (1, 0) (see the introduction of [31] for details). We remark that the construction of the Floquet Hamiltonian of h is done independently of U although of course (6.3) may be taken as a deﬁnition. Let us for completeness note the following by-product of Yosida’s theorem (intimately related to its proof): Pick λ0 ∈ R such that haux (t) ≥ λ0 + 1 for all t. The crucial assumption in the theorem is the boundedness of the function −1 d −1 (6.16) t → (haux (t) − λ0 ) (haux (t) − λ0 ) . dt Since, by assumption E ∈ C([0, 1]; X), clearly the following constant is a bound of (6.16), C := 2 sup |E(t)| sup |p|(haux (t) − λ0 )−1 . t

t

We have the explicit bound of the dynamics restricted to D(p2 ). (haux (t) − λ0 )Uaux (t, 0)φ2 ≤ e2C|t| (haux (0) − λ0 )φ2

for φ ∈ D(p2 ).

Let us also note the following property of the dynamics restricted to D(|p|), cf. [37, Theorems II.23 and II.27], (haux (t) − λ0 )1/2 Uaux (t, 0)φ2 e

≤ eC|t| (haux (0) − λ0 )1/2 φ2

for φ ∈ D(|p|);

(6.17)

here := 2 sup |E(t)| sup |p|1/2 (haux (t) − λ0 )−1/2 2 . C t

t

Remark 6.4. If E ∈ L1 ([0, 1]; X) but possibly E ∈ C([0, 1]; X) we can still show that there exists an appropriate dynamics U under the conditions considered in [31], although possibly not one that preserves D(p2 ). We can use [37, Theorem II.27] directly on h. For the borderline case, the Coulomb singularity, Hardy’s inequality [31, (6.2)] is needed to verify the assumptions of this theorem; the details are not discussed here. This yields a dynamics U preserving D(|p|) which is good enough for getting the conclusions of [31] related to the condition E ∈ L1 ([0, 1]; X). The results presented below can similarly be extended to E ∈ L1 ([0, 1]; X).

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The following condition is an extension of [31, Condition 1.3] (which corresponds to k0 = 1 below). The Coulomb potential commonly used to describe atomic and molecular systems (here with moving nuclei) is included. Conditions 6.5. Let k0 ∈ N be given. For each a = amin the following holds. The pair-potential X a y → Va (y) ∈ R splits into a sum Va = Va1 + Va2 where (1) (2) (3) (4) (5)

Diﬀerentiability: Va1 ∈ C k0 +1 (X a ) and Va2 ∈ C k0 +1 (X a \{0}). Global bounds: For all α with |α| ≤ k0 +1 there are bounds |y||α| |∂yα Va1 (y)| ≤ C. Decay at inﬁnity: |Va1 (y)| + |y · ∇y Va1 (y)| = o(1). Dimensionality: Va2 = 0 if dim X a < 3. Local singularity: Va2 is compactly supported and for all α with |α| ≤ k0 + 1 there are bounds |y||α|+1 |∂yα Va2 (y)| ≤ C; y = 0.

We note that the part of time-dependent potential Vphy (·+c(t)) coming from the ﬁrst term Va1 of the splitting of Va in Condition 6.5 conforms with Condition 6.1. The part from Va2 does not, and we do not in general expect there to be an analogue of Theorem 6.2 in this case for k0 > 1. It is an open problem to determine whether there is an analogue statement of Theorem 6.2 for k0 = 1. Notice that the lowest degree of regularity, φ ∈ D(|p|), holds even without the non-threshold condition, cf. [31, Theorem 1.8]. On the other hand since the singularity is located at x = −c(t) we would expect and we will indeed prove regularity with respect to the observable 1 A = A(t) = ((x + c(t)) · p + p · (x + c(t))) 2 1 = S1 (t) (x · p + p · x)S1 (t)−1 . (6.18) 2 This regularity is the content of Theorem 6.6 stated below; see [31, Proposition 8.7(ii)] for a related result in the case k0 = 1 at the level of Floquet theory, cf. Proposition 6.7 stated below. The A-regularity statement of the theorem for k0 > 1 is new. The set of thresholds is deﬁned as before, see (6.1). Theorem 6.6. Suppose Conditions 6.5 for some k0 ∈ N. Let φ be a bound state for / F(U (1, 0)). Then φ ∈ D(A(1)k0 ) where U (1, 0) pertaining to an eigenvalue e−iλ ∈ A(1) is given by taking t = 1 in (6.18). The above theorem implies Theorem 1.6(1). We shall prove Theorem 6.6 along the same lines as that of the proof of Theorem 6.2. Whence we introduce the Floquet Hamiltonian by the expression (6.2) (with V = Vphy (·+ c(t))). By [31, Theorem 6.2] V is -bounded relatively to H0 whence H is self-adjoint. Proposition 6.7. Suppose Conditions 6.5 for some k0 ∈ N and suppose Hψ = λψ / F(U (1, 0)). Then for any k, ≥ 0, with k + ≤ k0 , we have ψ ∈ for e−iλ ∈ D(Ak pA ) where A is given by (6.18). Proof. It is tempting to try to apply Corollary 2.9 with H being the Floquet Hamiltonian, A being as stated and N = p2 + 1. In fact all of the conditions of

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Corollary 2.9 can be veriﬁed except for Condition 2.1(2) (notice that the formal analogue of (6.5b) might be too singular). This deﬁciency will be discussed at the end of the proof. All other conditions can be veriﬁed with H = 2p2 − (x + c) · ∇V + 2b · p,

(6.19a)

N = 2p2 ,

(6.19b)

i adA (N ) = 2+1 p2 ,

i adN (i adA (N )) = 0;

≤ k0 − 1,

i adA (H ) = 2+1 p2 + (−1)+1 ((x + c) · ∇)+1 V + 2b · p;

(6.19c) ≤ k0 .

(6.19d)

Comments are due. First, the second and the third terms of (6.19a) are bounded relatively to |p| uniformly in t, cf. the Hardy inequality [31, (6.2)], and whence indeed (6.19a) is N -bounded. We need to verify Condition 2.1(3) using the expression (6.19a): The operators p2 , V and R(η) are ﬁbered and R(η) preserves D(p2 ) and D(|p|) for |η| large enough (uniformly in t). Whence as a form on D(τ ) ∩ D(N ) i[h, R(η)] = −R(η)i[p2 + V, A]R(η) = −R(η)(2p2 − (x + c) · ∇V )R(η), i[τ, R(η)] = −R(η)2b · pR(η), and therefore i[H, R(η)] = −R(η)H R(η). Using again that D(H) ∩ D(N ) = D(τ ) ∩ D(N ) is dense in D(H) ∩ D(N 1/2 ), cf. Remark 3.5, the latter form identity can be extended by continuity to a form identity on D(H) ∩ D(N 1/2 ) yielding Condition 2.1(3). As for (6.19b)–(6.19d), Conditions 2.1(1) and 2.1(4), Conditions 2.6 and 2.8 the veriﬁcation is straightforward (omitted here). To show (2.4) we ﬁrst introduce the natural notation V = V 1 + V 2 reﬂecting the splitting of Conditions 6.5. Then we introduce 1 1 = (x + c) · ∇V 1 ; C = |p|− 2 ((x + c) · ∇V 2 − 2b · p)|p|− 2 and C

the norm is the operator norm on H. Then we note that N≤

1 1 1 H + C|p| + 1 + C, 2 2 2

understood as a form yielding (2.4) with C1 = 0, C2 = 1 and C3 = 1 + C 2 /4 + C 1/2 on D(N ). We have veriﬁed Condition 2.2. As for (2.5) a stronger version follows from [31, Proposition 6.4] H ≥ c0 1 − C4 fλ⊥ (H)2 − K0 .

(6.20)

/ F(U (1, 0)). The estimate (6.20) is valid Here we use the condition that e−iλ ∈ as a form on D(N 1/2 ). Finally it follows from [31, Theorem 6.3] that indeed the condition of Corollary 2.9, ψ ∈ D(N 1/2 ) = D(|p|), is fulﬁlled.

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Now to the deﬁciency given by the lack of Condition 2.1(2). Checking the proof of Corollary 2.9 it is realized that Condition 2.1(2) is used only to assure boundedness of N 1/2 BN −1/2 , where under the assumption (2.5) we have B = C4 fλ⊥ (H)2 H(H− λ)−1 . In our case we have a slightly stronger version of the Mourre estimate, (6.20), so what we really need is 1

1

N 2 BN − 2 ∈ B(H) where B = g(H); g(E) = fλ⊥ (E)2 (E − λ)−1 .

(6.21)

So let us show (6.21) without invoking a condition like Condition 2.1(2). Clearly it suﬃces to show that the commutator 1

[N 2 , g(H)] ∈ B(H). But

1 2

[N , g(H)] = c 12

∞

(6.22)

1

t 2 (N + t)−1 [N, g(H)](N + t)−1 dt,

0

[N, g(H)] = [H − V + I − τ, g(H)] = −[τ, g(H)] + T, 1 ¯g)(η)(H − η)−1 [τ, V ](H − η)−1 dudv, −[τ, g(H)] = (∂˜ π C −[τ, V ] = i2b · ∇V. Here the term T is bounded since V is bounded relatively to H; whence indeed T gives a bounded contribution to the commutator in (6.22). As for the contribution from the term −[τ, g(H)] only the part from V 2 is non-trivial. For that part we use [31, (6.6)] to obtain 1

(H − η)−1 2b · ∇V 2 (H − η)−1 ≤ Cmax(|Im η|−2 , |Im η|− 2 ). Whence we can bound the integral ¯g)(η)(H − η)−1 2b · ∇V 2 (H − η)−1 dudv (∂˜ C

≤C

C

1

¯g )(η)|max(|Im η|−2 , |Im η|− 2 )dudv < ∞. |(∂˜

This means that also the ﬁrst term −[τ, g(H)] is bounded and whence in turn its contribution to the commutator in (6.22) agrees with the statement of (6.22). We have proven (6.22). Proof of Theorem 6.6. We mimic the proof of Theorem 6.2. Recall the notation In (A) = −in(A − in)−1 and An = AIn (A). Due to Proposition 6.7 and the representation (6.4) there exists t0 ∈ [0, 1[ such that U (t0 , 0)φ ∈ D(|p|) ∩ D(A(t0 )k0 ).

(6.23)

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In particular ψ(t) = eitλ U (t, 0)φ ∈ D(|p|) for all t, cf. (6.15b) and (6.17). Moreover ψ(·) is diﬀerentiable as a D(|p|)∗ –valued function, and in this sense d ψ(t) = (h(t) − λ)ψ(t). dt

i Whence we can compute

d d Akn0 ψ(t)2 = 2 Re Akn0 ψ(t), i[h(t), Akn0 ] + Akn0 ψ(t) , dt dt

d d Apn i[h(t), An ] + An Ank0 −p−1 , i[h(t), Akn0 ] + Akn0 = dt dt

(6.24a) (6.24b)

0≤p≤k0 −1

d An = In (A)(2p2 + 2b · p − (x + c) · ∇V )In (A). (6.24c) dt We plug (6.24c) into (6.24b) and then in turn (6.24b) into the right-hand side of (6.24a). We expand the sum and redistribute for each term at most k0 − 1 factors of A obtaining terms on a more symmetric form, more precisely on the form i[h(t), An ] +

RepAk0 ψ(t), BpAk ψ(t)

where k ≤ k0 − 1

and

sup B < ∞.

(6.25)

n,t

Thanks to the Cauchy–Schwarz inequality and Proposition 6.7 any expression like (6.25) can be integrated on [t0 , 1] and the integral is bounded uniformly in n. In combination with (6.23) we conclude that supA(1)kn0 ψ(1)2 < ∞, n

whence φ = ψ(1) ∈ D(A(1) ). k0

Acknowledgment The ﬁrst author was partially supported by Center for Theory in Natural Sciences, Aarhus University. References [1] S. Agmon, I. Herbst and E. Skibsted, Perturbation of embedded eigenvalues in the generalized N -body problem, Comm. Math. Phys. 122 (1989) 411–438. [2] J. Aguilar and J.-M. Combes, A class of analytic perturbations for one-body Schr¨ odinger Hamiltonians, Comm. Math. Phys. 22 (1971) 269–279. [3] W. Amrein, A. Boutet de Monvel and V. Georgescu, C0 -Groups, Commutator Methods and Spectral Theory of N -Body Hamiltonians (Birkh¨ auser, 1996). [4] V. Bach, J. Fr¨ ohlich and I. M. Sigal, Quantum electrodynamics of conﬁned non-relativistic particles, Adv. Math. 137 (1998) 299–395. [5] V. Bach, J. Fr¨ ohlich, I. M. Sigal and A. Soﬀer, Positive commutators and the spectrum of Pauli–Fierz Hamiltonian of atoms and molecules, Comm. Math. Phys. 207 (1999) 557–587. [6] E. Balslev and J.-M. Combes, Spectral properties of many-body Schr¨ odinger operators with dilation analytic interactions, Comm. Math. Phys. 22 (1971) 280–294.

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[7] L. Bruneau and J. Derezi´ nski, Pauli–Fierz Hamiltonians deﬁned as quadratic forms, Rep. Math. Phys. 54 (2004) 169–199. [8] L. Cattaneo, Mourre’s inequality and embedded boundstates, Bull. Sci. Math. 129 (2005) 591–614. [9] L. Cattaneo, G. M. Graf and W. Hunziker, A general resonance theory based on Mourre’s inequality, Ann. Henri Poincar´e 7 (2006) 583–601. [10] J. Derezi´ nski and C. G´erard, Asymptotic completeness in quantum ﬁeld theory. Massive Pauli–Fierz Hamiltonians, Rev. Math. Phys. 11 (1999) 383–450 [11] J. Derezi´ nski and V. Jakˇsi´c, Spectral theory of Pauli–Fierz operators, J. Funct. Anal. 180 (2001) 243–327. [12] J. Derezi´ nski and V. Jakˇsi´c, Return to equilibrium for Pauli–Fierz systems, Ann. Henri. Poincar´e 4 (2003) 739–793. [13] J. Faupin, J. S. Møller and E. Skibsted, Second order perturbation theory for embedded eigenvalues, to appear in Comm. Math. Phys. [14] R. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for N -body Schr¨ odinger operators, Comm. Math. Phys. 87 (1982) 429–447. [15] R. Froese, I. Herbst, M. Hoﬀmann-Ostenhof and T. Hoﬀmann-Ostenhof, On the absence of positive eigenvalues for one-body Schr¨ odinger operators, J. Anal. Math. 41 (1982) 272–284. [16] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Asymptotic completeness for Rayleigh scattering, Ann. Henri Poincar´e 3 (2002) 107–170. [17] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Asymptotic completeness for Compton scattering, Comm. Math. Phys. 252 (2004) 415–476. [18] J. Fr¨ ohlich, M. Griesemer and I. M. Sigal, Spectral theory for the standard model of non-relativistic QED, Comm. Math. Phys. 283 (2008) 613–646. [19] J. Fr¨ ohlich and M. Merkli, Another return of “return to equilibrium”, Comm. Math. Phys. 251 (2004) 235–262. [20] V. Georgescu and C. G´erard, On the virial theorem in quantum mechanics, Comm. Math. Phys. 208 (1999) 275–281. [21] V. Georgescu, C. G´erard and J. S. Møller, Commutators, C0 -semigroups and resolvent estimates, J. Funct. Anal. 216 (2004) 303–361. [22] V. Georgescu, C. G´erard and J. S. Møller, Spectral theory of massless Pauli–Fierz models, Comm. Math. Phys. 249 (2004) 29–78. [23] C. G´erard, On the scattering theory of massless Nelson models, Rev. Math. Phys. 14 (2002) 1165–1280. [24] S. Gol´enia, Positive commutators, Fermi Golden Rule and the spectrum of zero temperature Pauli–Fierz Hamiltonians, J. Funct. Anal. 256 (2009) 2587–2620. [25] S. Gol´enia and T. Jecko, A new look at Mourre’s commutator theory, Complex Anal. Oper. Theory 1 (2007) 399–422. [26] M. H¨ ubner and H. Spohn, Spectral properties of the spin-boson Hamiltonian, Ann. Inst. Henri Poincar´e 62 (1995) 289–323. [27] W. Hunziker and I. M. Sigal, The quantum N -body problem, J. Math. Phys. 41 (2000) 3448–3510. [28] V. Jakˇsi´c and C.-A. Pillet, On a model for quantum friction II: Fermi’s golden rule and dynamics at positive temperature, Comm. Math. Phys. 176 (1996) 619–644. [29] Y. Kuwabara and K. Yajima, The limiting absorption principle for Schr¨ odinger operators with long-range time-periodic potentials, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) 833–851. ´ Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, [30] E. Comm. Math. Phys. 78 (1980/81) 391–408.

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[31] J. S. Møller and E. Skibsted, Spectral theory of time-periodic many-body systems, Adv. Math. 188 (2004) 137–221. [32] J. S. Møller and M. Westrich, Regularity of eigenstates in regular Mourre theory, J. Funct. Anal. 260 (2011) 852–878. [33] M. Reed and B. Simon, Methods of Modern Mathematical Physics I, Functional Analysis (Academic Press, 1972). , Methods of Modern Mathematical Physics II, Fourier Analysis, Self[34] Adjointness (Academic Press, 1975). , Methods of Modern Mathematical Physics III, Scattering Theory (Academic [35] Press, 1979). , Methods of Modern Mathematical Physics IV, Analysis of Operators [36] (Academic Press, 1978). [37] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton Series in Physics (Princeton University Press, 1971). [38] E. Skibsted, Spectral analysis of N -body systems coupled to a bosonic ﬁeld, Rev. Math. Phys. 10 (1998) 989–1026. [39] K. Yajima, Existence of solutions for Schr¨ odinger evolution equations, Comm. Math. Phys. 110 (1987) 415–426.

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Reviews in Mathematical Physics Vol. 23, No. 5 (2011) 531–551 c World Scientific Publishing Company DOI: 10.1142/S0129055X11004357

DYNAMICAL BACKREACTION IN ROBERTSON–WALKER SPACETIME

BENJAMIN ELTZNER∗,‡ and HANNO GOTTSCHALK†,§ ∗Max-Planck

Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany

†Bergische Universit¨ at Wuppertal, Fachgruppe Mathematik, Gaußstraße 20, D-42119 Wuppertal, Germany ‡[email protected] §[email protected]

Received 26 March 2010 Revised 16 March 2011 The treatment of a quantized field in a curved spacetime requires the study of backreaction of the field on the spacetime via the semiclassical Einstein equation. We consider a free scalar field in spatially flat Robertson–Walker spacetime. We require the state of the field to allow for a renormalized semiclassical stress tensor. We calculate the singularities of the stress tensor restricted to equal times in agreement with the usual renormalization prescription for Hadamard states to perform an explicit renormalization. The dynamical system for the Robertson–Walker scale parameter a(t) coupled to the scalar field is finally derived for the case of conformal and also general coupling. Keywords: Quantum field theory; cosmology. Mathematics Subject Classification 2010: 81T05

1. Introduction The studies of quantized ﬁelds in curved spacetimes usually assume a ﬁxed background spacetime on which a quantized ﬁeld is deﬁned [5]. Such a setting has also been used to investigate the simplest cosmological model of a homogeneous isotropic universe. In this context the mode spectrum of a scalar ﬁeld in a homogeneous isotropic universe undergoing an era of inﬂation has been found to be the famous and experimentally conﬁrmed scale free Harrison–Zeldovich spectrum [9, 10]. The work which is concerned with the backreaction of the ﬁeld on the evolution of the spacetime itself usually takes a mean ﬁeld approach in which quantum ﬂuctuations only contribute to the eﬀective potential of the (classical) expectation value of the ﬁeld [11, 12, 21]. The dynamics of the quantum degrees of freedom is not considered. In other approaches, the quantum degrees of freedom either decouple from the spacetime [17] due to conformal coupling for a massless ﬁeld, or a large 531

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mass of the ﬁeld is assumed [3], which implies another semiclassical approximation. This approximation in fact implies that the ﬁeld conﬁguration is dominated by renormalization ambiguities. Again the coupling of quantum degrees of freedom to the geometry of the space time is only approximate. In this work, which is partially based on [4], we determine the coupling of a free quantum ﬁeld to the scale parameter of the spatially ﬂat Robertson–Walker spacetime. The result is an equation for a dynamical system with inﬁnitely many degrees of freedom that can, at least in principle, be solved. The dynamical system is derived via an expansion of Riemann normal coordinates in the Robertson–Walker canonical coordinates up to ﬁfth order [1, 6]. This allows to calculate the singular terms of the Hadamard bidistribution and its ﬁrst and second time derivative restricted to equal time surfaces. Those terms are needed to renormalize the energy-momentum tensor of the free ﬁeld restricted to equal time surfaces. The latter establishes the dynamics of the Robertson–Walker scale factor (up to renormalization ambiguities). While ﬁnalizing this paper we got aware of the publication [15], which derives exactly the same equation of motion as we do for the case of conformal coupling using a completely independent argument. In fact, this work in the conformally coupled case also establishes the existence of solutions for small time intervals given that the initial state fulﬁlls certain conditions, which goes beyond the scope of this article.

2. General Description of Dynamics and Renormalization Approach Throughout the article we restrict to ﬂat Robertson–Walker (RW) spacetimes with spacetime dimension 4. This allows to use the standard Fourier transform on spatial sections of constant RW time in order to formulate the dynamics of the ﬁeld in terms of modes. As the stress tensor is formulated as a diﬀerential operator acting on the two-point function, we will formulate the dynamics of the ﬁeld in such a way that the two-point function of the ﬁeld and its associate momentum ﬁeld operator restricted to the given time is the dynamical variable. We will also restrict our attention to homogeneous, isotropic, quasifree and pure states that either are Hadamard states or are suﬃciently close to Hadamard in the sense that they allow the same renormalization prescription for their energy-momentum tensor. Hadamard states and adiabatic vacuum states on RW spacetime have been studied, e.g., by L¨ uders and Roberts [13], Juncker and Schrohe [7] and more recently by Olbermann [14]. The line element of a spatially ﬂat Robertson–Walker metric for a homogeneous, isotropic spacetime is ds2 = dt2 − a2 (t)dx 2

(1)

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where we call a(t) the scale parameter and deﬁne the Hubble parameter H(t) = aa˙ . We consider a scalar free ﬁeld of mass m coupled to the scalar curvature R(t) = ˙ −6(H(t) + 2H 2 (t)) with coupling ξ, ( − ξR + m2 )φ = 0,

(2)

where is the D’Alembertian. We use the abbreviation ωk2 = a−2 k 2 + m2 − ξR. The Klein–Gordon equation for the ﬁeld modes φk is given by φ¨k + 3H φ˙ k + ωk2 φk = 0.

(3)

Using the canonical momentum ﬁeld πk = a3 φ˙ k we consider the Hamiltonian form of this equation φ 0 a−3 φk ∂t k = . (4) πk −a3 ωk2 0 πk It is shown in [13] that the equal time two-point function, i.e. the two-point function on a Cauchy surface, of a state can be described by matrices Gφφ,k Gφπ,k , (5) Gπφ,k Gππ,k where Gφφ,k = Gφφ (t, k) is the Fourier transform in z = x −y of two equal time ﬁeld operators G(t, x, y ) = ω(φ(t, x)φ(t, y )), Gφπ,k the Fourier transform of one equal time ﬁeld operator and one canonically conjugated momentum operator etc., and k is the modulus of the momentum conjugated to z . Our normalization convention for the Fourier transform is F (f )(k) = R3 f (x)eik·x dx. The positivity of the state will enforce that the matrix is positive semideﬁnite with its determinant vanishing for a pure quasifree state. The symmetric part of the two-point function modes then fulﬁlls the linear system of equations      Gφφ,k 0 2a−3 0 Gφφ,k (6) ∂t G(φπ),k  = −a3 ωk2 0 a−3  G(φπ),k  . 3 2 Gππ,k 0 −2a ωk 0 Gππ,k Here and in the following ( ) stands for symmetrization and [ ] for antisymmetrization in the ﬁeld and momentum operator. This system of equations has one conserved quantity per mode Jk = Gφφ,k Gππ,k − G2(φπ),k

(7)

which reduces the number of degrees of freedom per mode to two, as required. The condition for a state to be pure and to induce a representation of the canonical commutation (CCR) algebra then is 1 (8) 4 which implies the vanishing of the above-mentioned determinant for all modes. ∀ k: Jk = −G2[φπ],k =

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Next we turn to the semi classical Einstein equation. This equation has the form Gµν = 8πG Tµν ω

(9)

where the expectation value of the stress tensor must be renormalized. In this equation Gµν is the Einstein tensor and G is the gravitational constant and both should not be confused with Gφφ,k etc. which are two-point functions. We will apply the point-splitting procedure as formulated in [16], see also [18]. In order to preserve general covariance of the renormalization prescription, we need to subtract the Hadamard bidistribution H, described in some detail in [5], from the two-point function before removing the point splitting. Actually, a suﬃciently precise approximation Hn of the Hadamard parametrix does the job as well (η) (η) lim D(v)µν (x, y)[φ(x)φ(y)ω − Hn (x, y)] + tµν (v) (10) Tµν (v) ω,λ,ξ = (x,y)→(v,v)

with, cf. [8, 18], tµν (v) =

δ δg µν (v)

(Am4 + Bm2 R + CR2 + DRαβ Rαβ )dg x

(11)

where A, B, C and D are real valued renormalization degrees of freedom and dg x stands for the canonical volume form |g|dx. (η) D(v)µν (x, y) is the symmetrization in x and y of the following second order partial diﬀerential operator (cf. [16, Eq. (10)] for the details, where we corrected some minor misprints)

1 ∂y,ν − gµν g γ,δ ∂x,γ ∂y,δ − m2 ∂x,µ 2

1 γδ + ξ Rµν − gµν R + 2gµν x + g ∂x,γ ∂y,δ − 2 Dx,µ ∂x,ν + ∂x,µ ∂y,ν 2 − ηgµν (x − ξR + m2 ).

(12)

Here ∂x,µ = δµν (v, x)∂x,ν , δ(v, x) being the geodesic transport and Dx,µ analogously denotes a covariant derivative. The quantities g, R and the Ricci tensor Rµν are evaluated at v. Note here we use sign convention (+ − −−) instead of (− + ++) in [16] which ﬂips the sign in the last expression in the brackets (and some more in the text below). The deﬁnition of Hn for d = 4 is n 1 σ(x, y) 1 1 u(x, y) − 2 log − vk (x, y)σ(x, y)k . (13) − 2 4π σ(x, y) 4π λ2 k! k=0

Here σ(x, y) is the squared geodesic distance, u(x, y) is the square root of the Van Vleck–Morette determinant (which for dimension 4 is U0 (x, y) as deﬁned in [16] up to normalization). u(x, y) and the vk (x, y) (corresponding to Uk+1 (x, y) of [16]) are

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functions that can be determined by a recursive system of diﬀerential equations that depends exclusively on invariant quantities and the operator − ξR + m2 : 2g µν (x)(∂µx σ)(∂νx u) + (x σ − 4)u = 0, 2g

µν

(x)(∂µx σ)(∂νx v0 )

(14) 2

+ ( σ − 2)v0 = −( + m − ξR(x))u, x

x

2g µν (x)(∂µx σ)(∂νx vk+1 ) + (x σ + 2k)vk+1 = −(x + m2 − ξR(x))vk .

(15) (16)

The coincidence limits vk (v, v) up to normalization coincide with the Hadamard– Minkashisundram–De Witt–Seely coeﬃcients [5, 16]. In particular, on RW space time, vk ((t, x), (t, y )) depends on x and y only through z 2 = (x − y ) 2 and the coincidence limit vk ((t, x), (t, x )) of the vk depends only on t. The same holds true for derivatives of vk wrt z 2 . The regularization at light-like separated x, y is done by adding an i(x0 −y 0 ) to σ(x, y), letting 0 and taking the real part. We do not need this in the following, as we will approach the coincidence limit from spatial directions exclusively. It is shown that for n ∈ N, the expression [φ(x)φ(y) ω − Hn (x, y)] can be extended to a function in C n (Cv × Cv ) where Cv is some convex normal neighborhood of v. The quantum averaged ﬁeld ﬂuctuations are deﬁned as φ2 ω,λ,ξ =

lim

[φ(x)φ(y)ω − Hn (x, y)]

(x,y)→(v,v)

(17)

where n = 0 suﬃces if no derivatives of this quantity are required. For η = 0, the corresponding term in (10) in the energy momentum tensor vanishes classically, but not quantum mechanically. For η = 1/3, the so-deﬁned energy-momentum tensor is conserved. For RW spacetimes and states of the form (5), the quantum averaged energy-momentum tensor depends only on t and is diagonal so that only energy conservation is non-trivial (1/3)

∇µ Tµ0

ω,λ,ξ = ρ˙ + H(ρ + p) = 0,

(18)

(1/3,λ) (1/3,λ) where ρ = T00 and p = a2 Tjj are energy density and pressure, ω,λ,ξ ω,λ,ξ respectively, see, e.g., [21]. Moretti also showed (1/3) 1 µν 2 − ξ + m φ2 ω,λ,ξ = ρ − 3p = −3 g Tµν ω,λ,ξ 6 +

1 v1 + cm2 + c m2 R + c R 4π 2

(19)

where v1 = v1 (t, ξ, m2 ) = v1 ((t, v ), (t, v )) and c, c , c can be calculated from A, B, C and D. The D’Alembert operator here does not require point splitting and hence on ﬂat RW spacetime can be replaced by d2 /dt2 + 3Hd/dt. Interestingly, this equation can be seen as the equation of state and the right-hand side gives the deviation of quantum matter from the state equation of hot (relativistic) matter p = ρ/3. For Minkowski space and ω the Minkowski vacuum, we expect p = ρ = 0 which

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can be achieved through λ2 = 4e 4 −2γ /m2 with γ the Euler constant. v1 has been calculated in [3] for the case of ﬂat RW spacetime as 1 ˙ 2 1 1 4 (HH + H ) + − ξ R v1 = 60 24 5 2 1 1 9 1 m4 2 2 ˙ 4 ˙ − ξ (H + 4H H + 4H ) − + − ξ m2 R. − (20) 2 6 8 4 6 In the case of a massless ﬁeld with conformal coupling m2 = 0, ξ = 1/6, the terms in the second line on the right-hand side of (19) is called the conformal anomaly. Combining (9) and (19) one arrives at the following equation of motion 2 d 1 d 2 −ξ + 3H + m φ2 ω,λ,ξ −R = 8πG −3 6 dt2 dt 1 + 2 v1 + cm4 + c m2 R + c R . (21) 4π 2

d d 2 We would like to ﬁnd an expression for ( dt 2 + 3H dt )φ ω,λ,ξ that does not ¨ ω,λ,ξ since only then second order time derivatives of the ﬁeld φ do include terms φφ d ˙ ω,λ,ξ + φ2 ω,λ,ξ equals φφ only occur on the left-hand side of (4). It is clear that dt 2 2 ¨ ˙ φφω,λ,ξ . Let h(t) = f (s, s )|s=s =t . Then, h(t) = (∂s + ∂s + 2∂s ∂s )f (s, s )|s=s =t . Applying this to our problem and using the fact [16, Lemma 2.1] that

φ( − ξR + m2 )φω,λ,ξ =

3 v1 , 2π 2

(22)

we conclude that 2 d d + 3H φ2 ω,λ,ξ dt2 dt 3 = 2φ˙ 2 ω,λ,ξ + 2φ(a−2 ∆ + ξR − m2 )φω,λ,ξ + 2 v1 . π

(23)

Here ∆ stands for the Laplacian on R3 . We summarize the discussion in the following theorem: Theorem 2.1. The equation of motion for semi-classical Einstein equation on flat Robertson–Walker spacetime can be written as follows: −R = 8πG (6ξ − 1)(φ˙ 2 ω,λ,ξ + a−2 φ∆φω,λ,ξ ) + [(2 − 6ξ)m2 − (1 − 6ξ)ξR]φ2 ω,λ,ξ +

36ξ − 5 v1 + cm4 + c m2 R + c R 4π 2 2

d d with R = −6(H˙ + 2H 2 ), R = ( dt 2 + 3H dt )R and v1 given by (20).

(24)

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From the above analysis it is clear that we need to compute φ2 ω,λ,ξ and its ˙ R, R, ˙ R, ¨ ... second time derivative in terms of the functions Gφφ , . . . and a, H, H, in order to combine (6) and (24) to a closed system of equations. Hence we need to calculate those terms in H(x, y)|x0 =y0 =t , ∆x H(x, y)|x0 =y0 =t ∂ ∂ and ∂x 0 ∂y 0 H(x, y)|x0 =y 0 =t that either are singular or contribute a (time dependent) quantity to the spatial coincidence limit z = (x − y ) → 0. 3. Leading Terms of the Hadamard Distribution We start the task described at the end of the preceding section via a perturbative u(x,y) of the calculation of the singularities in the leading term H (x, y) = −(4π 2 )−1 σ(x,y) Hadamard distribution. In normal coordinates X around y, σ takes the form 2σ = ηµν X µ X ν

(25)

with ηµν the Minkowski metric. Thus the computation is reduced to ﬁnding normal coordinates perturbatively in dependence of the canonical coordinates. We will need this expansion to ﬁfth order to ﬁx all singularities of limy→x H (x, y) and its ﬁrst and second derivatives as well as all homogeneous terms. The normal coordinates have been calculated perturbatively to ﬁfth order in [1] (using computer algebra support), formulas (11.12)–(11.16) for a general metric. We plug in

(n) α i j 2 α 0 i i 0 (26) ∂0n Γα βγ = δ0 δβ δγ δij Ln a + δi δβ δγ + δβ δγ H where H (n) denotes the nth derivative of H with respect to time and the Ln are L0 = H,

(27)

L1 = H˙ + 2H 2 ,

(28)

¨ + 6HH ˙ + 4H 3 , L2 = H

(29)

¨˙ + 8HH ¨ + 6H˙ 2 + 24HH ˙ 2 + 8H 4 . L3 = H

(30)

Using these formulae we get 1 1 2σ = z02 − a2 z 2 1 + Hz0 + (H˙ + H 2 )z02 + H 2 a2z 2 3 12 1 ¨ 1 ˙ 3 ˙ (H + 2HH)z (HH + 2H 3 )a2 z 2 z0 0 + 12 12 1 ¨˙ + 6HH ¨ + 2H˙ 2 − 8HH ˙ 2 − 4H 4 )z 4 (3H + 0 180 1 ¨ + 8H˙ 2 + 74HH ˙ 2 + 48H 4 )a2 z 2 z 2 (9 HH + 0 360 1 ˙ 2 + 4H 4 )a4 (z 2 )2 + O(z 7 ) (3HH + 360 +

(31)

where all time dependent terms are evaluated at y0 and we use the abbreviation z = x − y.

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Lemma 3.1. σ is symmetric under exchange of x and y. Proof. The proof is done by straightforward calculation, however on has to keep in mind that a, H and its derivative all have a suppressed argument y0 such that they have to be Taylor expanded. The terms of even orders in z do not change signs so it has to be shown, that they are not aﬀected by the terms from the Taylor expansion of the lower order coeﬃcients. Using abbreviations ay = a(y0 ) and the likes we get to the relevant orders a2x 1 ¨ 3 3 ˙ = 1+ 2Hy z0 +(H˙ y + 2Hy2 )z02 + (H y + 6Hy Hy + 4Hy )z0 a2y 3 1 ¨˙ ¨ y Hy + 6H˙ y2 + 24H˙ y Hy2 + 8Hy4 )z04 , (32) (H y + 8H 12 a2 1 ¨ 3 3 ˙ −Hx x2 z0 = −Hy z0 − (H˙ y + 2Hy2 )z02 − (H y + 6Hy Hy + 4Hy )z0 ay 2 +

1 ¨˙ 2 4 4 ¨ ˙2 ˙ − (H y + 8Hy Hy + 6Hy + 24Hy Hy + 8Hy )z0 , 6

(33)

1 ˙ a2 1 1 ¨ 3 3 ˙ (Hx + Hx2 ) x2 z02 = (H˙ y + Hy2 )z02 + (H y + 4Hy Hy + 2Hy )z0 3 ay 3 3 1 ¨˙ 2 4 4 ¨ ˙2 ˙ + (H y + 6Hy Hy + 4Hy + 14Hy Hy + 4Hy )z0 , 6 (34) −

1 ¨ (Hx + 12

a2 2H˙ x Hx ) x2 z03 ay

=− −

1 ¨ (Hy + 2H˙ y Hy )z03 12 1 ¨˙ ¨ y Hy + 2H˙ y2 + 4H˙ y Hy2 )z04 , (H y + 4H 12

(35)

1 2 a4x 1 2 1 ˙ H H + (Hy Hy + 2Hy3 )z0 = 12 x a4y 12 y 6 + −

1 ¨ (Hy Hy + H˙ y2 + 10H˙ y Hy2 + 8Hy4 )z02 , 12

(36)

a4 1 1 ˙ (Hx Hx + 2Hx3 ) x4 z0 = − (H˙ y Hy + 2Hy3 )z0 12 ay 12 −

1 ¨ (Hy Hy + H˙ y2 + 10H˙ y Hy2 + 8Hy4 )z02 . 12

(37)

Adding up the ﬁrst four and the last two terms one gets 1 1 ¨ 3 ˙ + 2HH)z 1 + Hy z0 + (H˙ + H 2 )z02 + (H 0, 3 12 1 2 1 Hy + (H˙ y Hy + 2Hy3 )z0 , 12 12 which proves the claim.

(38) (39)

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To calculate u we use the ﬁrst equation of well know Hadamard recursion which is deduced from the Klein–Gordon equation 2g µν (x)(∂µx σ)(∂νx u) + (x σ − 4)u = 0

(40)

where one has to take into account, that the derivatives are with respect to x and the metric is evaluated at x in the above formula, which leads to additional terms in the calculation. We make the ansatz u = 1 + µz02 + νz 2 + ρz03 + τz 2 z0 + φz04 + ψz 2 z02 + χ(z 2 )2

(41)

where all coeﬃcients are evaluated at y0 . The ﬁrst term being 1 and the absence of a ﬁrst order term follow from the requirements that u(x, x) = 1 and u(x, y) = u(y, x). The calculation of the coeﬃcients then yields 1 1 u = 1 − (H˙ + H 2 )z02 + (H˙ + 3H 2 )a2 z 2 4 12 1 ¨ 1 ¨ 3 ˙ ˙ + 6H 3 )a2 z 2 z0 (H + 8HH − (H + 2HH)z 0 + 8 24 1 ¨˙ − 36HH ¨ − 17H˙ 2 + 38HH ˙ 2 + 19H 4 )z 4 (−18H + 0 480 1 ¨˙ + 26HH ¨ + 17H˙ 2 + 52HH ˙ 2 + 1H 4 )a2 z 2 z 2 (3H + 0 240 1 ¨ + 3H˙ 2 + 36HH ˙ 2 + 29H 4 )a4 (z 2 )2 + O(z 5 ) (4HH + 480

(42)

which can be checked to be symmetric under interchange of x and y by a straight forward calculation. This equation has been derived using computer algebra support, but has been validated up to second order by hand calculations. From this we obtain Theorem 3.1. The most singular order of the Hadamard distribution is given by 1 1 1 − (H˙ + 2H 2 ) − a2z 2 12 1440 ˙ 3 + 51H 4 )a2 z 2 + O((z 2 )2 ), ¨ + 9H˙ 2 + 86HH × (12HH

−4π 2 H (z)|z0 =0 = −

(43) −4π 2 H˙ (z)|z0 =0 = −4π 2

H 1 ¨ ˙ − (H + 4HH) + O(z 2 ), a2 z 2 24

(44)

∂ ∂ 2 H2 1 H (z)|z0 =0 = 4 2 2 − 2 2 − 0 0 ∂x ∂y a (z ) a z 240 ˙ 2 + 17H 4 ) + O(z 2 ). ¨˙ + 16HH ¨ + 27H˙ 2 + 30HH × (4H (45)

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The proof is done by straightforward calculations where the occurring fractions in powers of Z = z 2 are expanded in a power series like L + MZ + NZ 2 MP − LQ L Z 2 = P + P2 P + QZ + SZ NP 2 − LPS − MPQ + LQ 2 2 Z . (46) + P3 Even if placeholders are inserted for the coeﬃcients of the powers of Z and z0 in σ and u the calculations remain very lengthy, as the coeﬃcient’s derivatives have to be taken into account. Therefore we will not show them here explicitly. We calculate the subleading terms of the Hadamard parametrix in position space to the order relevant to calculate the homogeneous term of the energy-momentum tensor. This means we use the Hadamard recursion

(47) 2g µν (x) ∂µx σ (∂νx v0 ) + (x σ − 2)v0 = −(x + m2 − ξR(x))u

2g µν (x) ∂µx σ (∂νx v1 ) + (x σ)v1 = −(x + m2 − ξR(x))v0 (48) to calculate 1 v0 = − 2

1 1 ¨ 2 ˙ − ξ R + m + (H + 4HH)z 0 6 4

1 ¨˙ + (87 − 480ξ)HH ¨ + (54 − 300ξ)H˙ 2 ((21 − 120ξ)H 240 ˙ 2 − (58 − 360ξ)H 4 + 30m2 (H˙ + H 2 ))z02 − (76 − 540ξ)HH +

1 ¨˙ + (3 − 60ξ)HH ¨ + (6 − 60ξ)H˙ 2 + (76 − 540ξ)HH ˙ 2 (−H 240 + (58 − 360ξ)H 4 − 10m2 (H˙ + 3H 2 ))a2 z 2 + O(z 3 ) +

and 1 v1 = 120

(49)

5 (1 − 6ξ)2 R2 − 2(H˙ + H 2 )H 2 12 + 5(1 − 6ξ)Rm2 + 15m4 + O(z). (1 − 5ξ)R +

(50)

4. Mode Expansion Two things remain to be done: The singular terms in (13) have to be related to the mode expansion Gφφ,k etc. of the two point functions and the homogeneous term of the energy momentum tensor has to be calculated. First, the expressions in Theorem 3.1 can easily be Fourier transformed with a method explained e.g. in [2]. Using the distributional Fourier transforms F (|z |−2 )(k) = 2π 2 k −1 , F (|z |−4 )(k) = −π 2 k with k = |k| and ignoring for now homogeneous terms that will only contribute terms proportional to the delta distribution in the zero mode and positive order terms in Z, we get the

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mode expressions 1 F (H |z0 =0 )(k) = a−2 , 2k H F (H˙ |z0 =0 )(k) = −a−2 , 2k 2 ∂ ∂ k −4 −2 H 0 + a . H | F ( k) = a z =0 ∂x0 ∂y 0 2 2k

(51) (52) (53)

Here it needs to be taken into account that all singular orders in z 2 that are encountered are locally integrable functions on R3 with the exception of the z14 ∂ ∂ ∂ ∂ term in ∂x 0 ∂y 0 H |z 0 =0 . Continuation of the ∂x0 ∂y 0 [φ(x)φ(y) − Hn (x, y)]|z 0 =0 for n ≥ 2 to z = 0 thus implicitly induces a rotational invariant regularization of z14 , as it can be written as a diﬀerence of already regularized terms and the newly found continuous function. The analytic regularization [2] of |z | ζ is well deﬁned except for poles at ζ = −3, −5, −7, . . . . It is clearly rotational invariant. Furthermore |z|ζ with ζ = −4 is the only invariant extension of z14 on R3 that preserves the scaling degree, as there are no rotation invariant linear combinations of ﬁrst order derivative of the delta distribution at zero. But the diﬀerence of two extensions with the given properties has to be such a linear combination, [2]. Hence we conclude that it has to ∂ ∂ coincide with the regularization induced by the continuation of ∂x 0 ∂y 0 [φ(x)φ(y)− Hn (x, y)]|z0 =0 to z = 0. The Fourier transform (53) can thus be calculated based on the analytic regularization prescription [2]. We now turn to the mode expansion of the subleading terms. Expanding the logarithmic factor with the help of (31) yields 2 n ∞ a 1 2 2 2 σ(x, y) 2 4 H = log ) − a z + O( z ) , + log( z − log − 0 λ2 λ2 12 z =0 n=1 (54) where the inﬁnite sum can be truncated after a few relevant terms. Inserting (31) to expand the powers σ(x, y)k in (13), we easily see that the singular contribution from the second term in (13) to [φ(x)φ(y) − Hn (x, y)]|z0 =0 is 2 a 1 2 v0 ((t, x), (t, y )) log 2 + log(z ) . (55) 4π 2 λ ∞ 1 2 2 2 1 Likewise, using σ(x,y) |z0 =0 = − z12 n=0 ( 12 H a z + O(z 4 ))n , it is easily shown ∂ that the expansion of derivatives of the second term in (13) with respect to ∂x 0 2 and ∂x∂0 ∂y0 at equal time into singular orders in z 2 which is truncated after some suﬃciently high order contains only terms ∼log(z 2 ) and ∼(z 2 )k , k ≥ 0, where the coeﬃcients are made of log(λ), log(a) and Uk ((t, x), (t, y ))-terms and their derivatives w.r.t. z 2 along with their time derivatives; for the case with the two time derivatives an additional term with the singularity structure z12 needs to be taken into account as well.

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One could calculate the coeﬃcients of the singular terms described above by calculating the derivatives of (49) and (50) and Fourier transforming the result by hand. However these calculations are very tedious, therefore we proceed diﬀerently here. As we have analyzed the general shape of the terms to account for, we can Fourier transform these terms and include them into the mode expansion with up to now undetermined coeﬃcients. Doing so, we use that F (log(|z|))(k ) = −4π 3/2 Γ( 32 )k −3 . Note that this distribution at 0 does not extend to a locally integrable function and hence requires a regularization prescription. The details can be found in the Appendix. The remaining singular orders that occur have mode expansion F (z 2 log(|z|))(k) = 24π 3/2 Γ( 32 )k −5 and F ((z 2 )n )(k) = (2π)3 (−∆)n δ(k), where we again neglect the latter for the time being. Next we multiply the leading terms in the singular order expansion with the powers of a to get expressions similar to the two-point function Hφφ =

a−2 , 2k

H(φπ) = − Hππ =

aH , 2k

a2 k a4 H 2 + . 2 2k

(56) (57) (58)

So, taking now into account the discussion above, we consider the ansatz Hφφ =

α3 a−2 α5 + 3 + 5 + O(k −7 ), 2k 2k 2k

H(φπ) = − Hππ =

β3 aH β5 + 3 + 5 + O(k −7 ), 2k 2k 2k

γ1 γ3 a2 k a4 H 2 + + + 3 + O(k −5 ), 2 2k 2k 2k

(59) (60) (61)

that captures all possible singular orders except for those concentrated in the zero mode. Note that Hn fulﬁlls the Klein–Gordon equation up to terms that vanish in the coincidence limit and a zero mode term. Plugging our ansatz of the singular order expansion into the system of equations (6) we obtain a couple of equations that help us to determine the unspeciﬁed coeﬃcients α˙ 5 2a−3 β3 2a−3 β5 α˙ 3 + 5 = + , 3 3 2k 2k 2k 2k 5 −

(62)

a(H˙ + 2H 2 ) (−ξR + m2 )a3 α3 + aα5 β˙ 3 (−ξR + m2 )a + aα3 + 3 =− − 2k 2k 2k 2k 3 +

a−3 γ3 a−3 γ1 + , 2k 2k 3

(63)

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γ˙ 1 γ˙ 3 2a4 H(H˙ + 2H 2 ) −2(−ξR + m2 )a4 H + 2aβ3 + + 3 =− 2k 2k 2k 2k −

2(−ξR + m2 )a3 β3 + 2aβ5 , 2k 3

(64)

where we suppress undetermined orders. Assuming these equations to hold order by order in k we get two sets of equations α˙ 3 = 2a−3 β3 , γ˙ 1 = −2aβ3 + 2a4 H α3 = a−4 γ1 −

1 − ξ R + m2 , 6

1 − ξ R − m2 , 6

(65) (66) (67)

and analogously α˙ 5 = 2a−3 β5 ,

(68) 3

2

γ˙ 3 = −2aβ5 − 2a β3 (−ξR + m ),

(69)

α5 = a−4 γ1 − a−1 β˙ 3 − a2 α3 (−ξR + m2 ).

(70)

We now want to solve these diﬀerential equations. The equation (65) + (66) −∂t (67) leads to a4 1 − ξ (R˙ + 2HR) + a4 Hm2 (71) γ˙ 1 − 2Hγ = 2 6 which can be solved using standard methods. The solutions for α3 and β3 can then be calculated straightforwardly without solving diﬀerential equations. Lemma 4.1. The solution to the system (65)–(67) is 1 1 − ξ R + m2 + Aa−2 , α3 = − 2 6 a3 1 β3 = − − ξ R˙ − AHa, 4 6 4 1 a γ1 = − ξ R + m2 + Aa2 , 2 6

(72) (73) (74)

where A is some constant. Proof. A diﬀerential equation of the form x(t) ˙ − 2f (t)x(t) = g(t)

(75)

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has the solution t t

t x(t) = g(t ) exp 2 f (t )dt dt + A exp 2 f (t )dt t

0

(76)

0

t with A being constant. In our case f (t) = H, such that exp (2 t12 H(t )dt ) = We get the solution

t 1 1 2 2 ˙ − ξ (R + 2HR) + Hm a2 dt + Aa2 γ1 = a 2 6 0 2 t 1 a = − ξ R + m2 a2 dt + Aa2 ∂t 2 0 6 a4 1 = − ξ R + m2 + Aa2 2 6

a2 (t1 ) a2 (t2 ) .

(77) (78) (79)

where the constant A absorbs all constant terms and thus changes from line to line. 1 4 2 2aβ3 = −γ˙ 1 + 2a H −ξ R+m (80) 6 1 a4 1 4 2 −ξ R+m − − ξ R˙ − 2Aa2 H = −2a H (81) 6 2 6 1 − ξ R + m2 + 2a4 H (82) 6 a4 1 − ξ R˙ − 2Aa2 H =− (83) 2 6 and

1 − ξ R − m2 6 1 1 1 − ξ R + m2 + Aa−2 − − ξ R − m2 = 2 6 6 1 1 − ξ R + m2 + Aa−2 =− 2 6

α3 = a−4 γ1 −

(84) (85) (86)

yield the other two coeﬃcients without solving diﬀerential equations. Applying the same strategy as before we get the solutions for the second system of equations 1 1 a2 ¨ + 5H R˙ − HR) ˙ − ξ (R −3 − ξ ξR2 α5 = (87) 8 6 6 − (4H˙ + 6H 2 + 6ξR)m2 − 3m4 , (88)

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a3 α˙ 5 , 2 1 1 a6 ¨ + H R˙ + HR) ˙ − ξ (R − − ξ ξR2 γ3 = − 8 6 6 − 2(H 2 + ξR)m2 − m4 ,

β5 =

545

(89) (90) (91)

where we suppress the integration constant as it vanishes for a pure state due an argument given below. Theorem 4.1. The singular part of the Hadamard modes is equal to the singularities of the two-point function of a pure state if the integration constant A in Lemma 4.1 vanishes. That is, (i) ∀ k: Hφφ,k Hππ,k − H2(φπ),k = 14 + O(k −6 ) (ii) ∀ k: Hφφ,k + Hππ,k > 0 + O(k −3 ) holds if and only if A = 0. Proof. The ﬁrst claim is equivalent to a2 H 2 + a−2 γ1 + a2 α3 − a2 H 2 = 0,

(92)

a−2 γ3 + a2 α5 + a4 H 2 α3 + α3 γ1 + 2aHβ3 = 0,

(93)

where Eq. (92) consists of the terms of order k −2 and Eq. (93) represents the order k −4 . Equation (92) gives 2A = 0. Analogously Eq. (93) forces the other integration constant to vanish. To show that the second claim is satisﬁed we use the fact that H is a solution of the Klein–Gordon equation (6) up to O(k −5 ). Then we use a form of the well-known deformation argument of Wald, which is explained in [18]. We deform the spacetime for early times such that a is constant at early times. Thus the claim holds exactly at early times, as we just get the Minkowski two-point function. Then we propagate H with Eq. (6) to some later time, where the spacetime is undeformed. Thus the claim is still satisﬁed at that time because the ﬁrst claim shown above ensures that Hφφ,k Hππ,k > 0 + O(k −6 ). Finally, we undeform the spacetime at early times and can thus show the claim to hold for all times using Eq. (6). From Theorem 4.1 we obtain a (tentative) parametrization of initial conditions of the ﬁeld degrees of freedom for the equation of motion for pure states. For k > 1 we set Gφφ,k = Hφφ,k +a(k), Gφπ,k = H(φπ),k +b(k) and Gππ,k = ( 14 +G2(φπ),k )/Gφφ,k where a(k)k 7 and b(k)k 7 are Cb∞ ((1, ∞)). Furthermore, a(k) needs to be chosen such that Gφφ,k > 0 for all such k. This can always be achieved as the leading term a−2 /k is positive. For k ≤ 1, these functions need to be continued to Cb∞ ([0, ∞)) such that the relations Gφφ,k + Gππ,k > 0 and (8) are preserved. Obviously with

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such initial conditions the trace of the initial energy momentum tensor is ﬁnite. In particular, Hadamard states fall into this class. From the construction of Hφφ,k , H(φπ),k and Hππ,k it follows that these quantities fulﬁll (6) up to order k −7 and k −5 , respectively. Thus, if the diﬀerece of Gφφ,k and Hφφ,k (multiplied by k 2 ) is integrable for large k, this property will prevail after an inﬁnitisimal time step. The argument for the diﬀerence of Gππ,k and Hππ,k is analoguous. For non pure states, a positive c(k) can be added to Gππ,k such that k 5 c(k) ∈ b C ([0, ∞)]. 5. Equation of Motion We begin with the easier case where the ﬁeld is conformally coupled to the mean curvature, i.e. ξ = 1/6. In this case, (24) simpliﬁes considerably, as it only contains the ﬁeld dependent term φφω,λ,ξ . This term, up to zero mode contributions, is

1 m2 a−2 + 3 dk, Gφφ,k − (94) 8π 3 R3 2k 4k 2 since α3 = − m2 . The renormalization prescription of R3 k −3 dk at 0 is given in the Appendix. The zero mode terms in the conformally coupled case are 2 a 1 1 R − 2 v0 log , (95) 72 4π λ2 where the ﬁrst term stems from the ﬁrst equation in Theorem 3.1 and the second one from (55). In the conformally coupled case, v0 (x, x) takes the simple form v0 = −

m2 . 2

(96)

Wrapping up, we obtain Theorem 5.1. For the case of conformal coupling, the equations of motion (24) is 2 m 1 m2 a−2 2 − R = 8πG + 3 dk + m +c R Gφφ,k − 8π 3 R3 2k 4k 72 2 1 a 1 m4 2 4 ˙ + (HH + H ) + + c R + 2 log 240 π 2 2880 π 2 4π λ2 1 − m4 − c (97) 32 π 2 where the k −3 integral is regularized at zero as explained in the Appendix.

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Wald’s ﬁfth axiom [19, 20] states that the equation of motion should be a second order equation. (97) is second order if and only if the renormalization constants A, B, C and D in (11) are chosen such that c = −1/(2880π 2). Note however, that for this value of c the joint system diﬀerential equations (6) and (97) is an implicit (inﬁnite dimensional) system of diﬀerential equations and for c = −1/(2880π 2) it is explicit, which is of advantage for proving existence and numerical solutions. The log(a) term that seems to be missing in [15] is due to the fact that the infra red prescription we employ for R3 fk(k) 3 dk (cf. Appendix) is not a dependent, whereas the prescription used in [15] is, leading to the absence of a standalone log(a) term in the latter case. For ξ = 1/6 the homogeneous terms of the Hadamard parametrix take a much more complicated form, as second derivatives of the parametrix must be taken into account. Ignoring a prefactor 8πG 4π 2 we have to calculate zero mode terms of u 1 − v0 log (2σ) − v1 2σ log (2σ) Cξ,m = (6ξ − 1)(∂x0 ∂y0 + a−2 ) −2 2σ 2 1 [(2 − 6ξ)m2 + (6ξ − 1)ξR](H˙ + 2H 2 ) 12 2

a 1 −2 + log (6ξ − 1) ∂x0 ∂y0 + a −v0 − v1 2σ λ2 2 − [(2 − 6ξ)m2 + (6ξ − 1)ξR]v0 −

(98)

in the coincidence limit x → x . This can be done using the expansions of u, v0 , v1 and σ 2 given in Sec. 3, applying the diﬀerential operator, setting z 0 = 0 and extracting terms of z -order zero. With a computer aided calculation this yields the result 1 ¨˙ + 53HH ¨ + 11H˙ 2 + 141HH ˙ 2 + 3H 4 ) Cξ,m = − (4H 30 1 ¨˙ + 113HH ¨ − 44H˙ 2 + 11HH ˙ 2 − 277H 4) + ξ(9H 5 ¨˙ + 12HH ¨ − 14H˙ 2 − 38HH ˙ 2 − 68H 4 ) − 108ξ 3 (H˙ + 2H 2 )2 − 6ξ 2 (H m2 ((21H˙ + 44H 2 ) − 6ξ(3H˙ + 8H 2 ) − 36ξ 2 (H˙ + 2H 2 )) 6 2 a m4 (1 − 6ξ) + log + 2 λ2 1 ¨˙ + 21HH ¨ + 42H˙ 2 + 152HH ˙ 2 + 116H 4) × − (3H 60

+

1 ¨˙ + 28HH ¨ + 31H˙ 2 + 106HH ˙ 2 + 58H 4 ) + ξ(4H 5

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¨˙ + 21HH ¨ − 6H˙ 2 − 36HH ˙ 2 − 72H 4 ) − 108ξ 3 (H˙ + 2H 2 )2 − ξ 2 (3H 3 2 2 4 ˙ + m (1 − 6ξ)(H + 2H ) − m (1 − 3ξ) . (99) 2 Although this result looks very tedious, one can simplify the result subsuming terms proportional to R, and m2 R into the renormalization degrees of freedom and the H independent term into a redeﬁnition of the scale parameter λ. The simpliﬁed form is 25 ˙ 5 2 1 2 2 4 ˙ Cξ,m = − − R − R − R + 13HH + 23H 30 3 6 36 1 3 25 ˙ 20 2 2 4 ˙ + ξ − R − R − R + 23HH + 43H 5 2 3 9 5 1 1 ˙ 2 + 4H 4 − 3ξ 3 R2 − 6ξ 2 − R − R˙ − R2 + 2HH 6 6 2 m2 m4 1 7 (1 − 6ξ) + − R + 2H 2 − 6ξ − R + 2H 2 + 6ξ 2 R + 6 2 2 2 2 a 5 2 1 1 2 4 ˙ + log − − R + R − 4HH − 4H λ2 60 2 6 1 2 5 2 2 4 ˙ + ξ − R + R − 2HH − 2H 5 3 12 1 1 1 − ξ 2 − R − R2 − 3ξ 3 R2 + m2 (1 − 6ξ) R − m4 (1 − 3ξ) . 2 2 4 (100) Plugging in the calculated homogeneous term we obtain the equation of motion for the general case. Theorem 5.2. The equation of motion in the general case is given by the following expression −6 a γ3 a2 k a4 H 2 + γ1 − − − −R = 8πG (6ξ − 1) G dk ππ,k 8π 3 R3 2 2k 2k 3

α5 a−2 k α3 a−2 − − 3 dk + 3 k 2 Gφφ,k − 8π R3 2 2k 2k

α3 ξ a−2 − 3 dk + 3 6m2 + (6ξ − 1)R Gφφ,k − 8π 2k 2k R3 36ξ − 5 1 4 2 v1 + cm + c m R + c R . (101) − 2 Cξ,m + 4π 4π 2 The regularization of the integrals over k −3 at zero is given in the Appendix. We would like to point out that due to the 4th time derivatives in γ3 and α5 , (101) again leads to an implicit system of diﬀerential equations, as the leading

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order time derivative can not be isolated, regardless of the renormalization degrees of freedom and the speciﬁc form of Cξ,m (t). Also, Wald’s ﬁfth axiom can never be fulﬁlled in this case. Appendix. Regularization Prescription on k−3 Here we give the details of the regularization of the seemingly infrared divergent integrals

m2 a−2 + 3 dk Gφφ,k − (102) 2k 4k R3 and related. For any Schwartz test function χ with χ(0) = 1. This integral can be re-written as

m2 m2 a−2 a−2 + 3 (1 − χ(k)) + Gφφ,k − χ(k)dk. Gφφ,k − χ(k) dk + 3 2k 4k 2k R3 R3 4k (103) Here the ﬁrst integral is regular. The second — seemingly infrared divergent — integral needs to be properly traced back to its deﬁnition as distributional Fourier transform of log(r) smeared with the test function χ, which gives a regular integral after shifting the Fourier transform to the test function. Let us quickly calculate the Fourier transform in the sense of tempered distributions of the locally integrable function log(r), r = |x| for the convenience of the reader. Let ϕ( ˇ x) = ϕ(−x), then d ζ r , F (ϕ) ˇ F (log(r))(k), ϕ = log(r), F (ϕ) ˇ = lim ζ0 dζ   ζ +3 Γ  d  2ζ+3 π 3/2 2 k −ζ−3 , ϕ . = lim (104)  ζ ζ0 dζ  Γ − 2 Since ζ approaches 0 from below, k −ζ−3 is locally integrable. Furthermore, k −ζ−3 , ϕ can be analytically continued in ζ to C\N0 ([2]). The Laurent series of k −ζ−3 in ζ = 0 has a pole proportional to δ0 . Hence, for ϕ(0) = 0, k −ζ−3 , ϕ is analytic at ζ = 0 and in particular has a ﬁnite ζ derivative. So let us from now on assume ϕ(0) = 0 and perform the ζ derivative at the right-hand side of (104). Due to the pole of Γ(− 2ζ ) at ζ = 0, the only term that contributes in the limit ζ 0 is

ζ +3 ζ Γ − 2 2 lim 2ζ+3−1 π 3/2 k −ζ−3 , ϕ 2 ζ0 ζ Γ − 2 3 = −4π 3/2 Γ k −3 , ϕ, 2 Γ

(105)

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where we used the Laurent series expansion of Γ at zero. Since, by dominated convergence, k −3 , ϕ = R3 k −3 ϕ(x)dx, we see that F (log(r))(k) is a regularization

2 of −4π 3/2 Γ 32 k −3 . Let now ϕ(0) = 0, and χ(k) = e−k . Since ϕ(x) = [ϕ(x) − 2 2 2 ϕ(0)e−k /2 ] + ϕ(0)e−k /2 = ϕ1 (x) + ϕ(0)e−k /2 . We have already shown how to evaluate F (log(r))(k) on ϕ1 . It remains to calculate ϕ(0) ϕ(0) 3 −k2 /2 −k2 /2 F (log(r)), ϕ(0)e = log(r), e = Γ . (106) 2 (2π)3/2 (2π)1/2 References [1] L. Brewin, Riemann normal coordinate expansions using Cadabra, Class. Quantum. Grav. 26(17) (2009) 175017. [2] F. Constantinescu, Distributionen und ihre Anwendungen in der Physik (Teubner, 1974). [3] C. Dappiaggi, K. Fredenhagen and N. Pinamonti, Stable cosmological models driven by a free quantum scalar ﬁeld, Phys. Rev. D 77 (2008) 1129–1163. [4] B. Eltzner, The semiclassical Einstein equation on Robertson–Walker spacetimes with backreaction, Diploma thesis, Bonn (2008). [5] S. A. Fulling, Aspects of Quantum Field Theory in Curved Spacetime (Cambridge University Press, 1989). ¨ [6] F. Gackstatter, Uber Volumendefekte und Kr¨ ummung bei der Robertson–Walker– Metrik und Konstruktion kosmologischer Modelle, Ann. Phys. 44(6) (1987) 423–439. [7] W. Junker and E. Schrohe, Adiabatic vacuum states on general spacetime manifolds: Deﬁnition, construction, and physical properties, Ann. Henri Poincar´e 3(6) (2002) 1113–1181. [8] T.-P. Hack, On the backreaction of scalar and spinor quantum ﬁelds on curved space times, Dissertation, Hamburg (2010). [9] E. R. Harrison, Fluctuations at the threshold of classical cosmology, Phys. Rev. D 1 (1970) 2726–2730. [10] Ya. B. Zel’dovich, A hypothesis unifying the structure and the entropy of the universe, Mon. Not. R. Astron. Soc. 160 (1972) 1P–3P. [11] A. D. Linde, Particle Physics and Inflationary Cosmology (Harwood, Chur, Switzerland, 1990). [12] A. D. Linde, Inflation and Quantum Cosmology (Academic Press, Boston, 1990). [13] C. L¨ uders and J. E. Roberts, Local quasiequivalence and adiabatic vacuum states, Comm. Math. Phys. 134 (1990) 29–63. [14] H. Olbermann, States of low energy on Robertson–Walker spacetimes, Class. Quantum Grav. 24 (2007) 5011–5030. [15] N. Pinamonti, On the initial conditions and solutions of the semi-classical Einstein equations in a cosmological scenario (January, 2010); to appear in Comm. Math. Phys.; gr/qc1001.0864v1. [16] V. Moretti, Comments on the stress-energy tensor operator in curved spacetime, Comm. Math. Phys. 232 (2003) 189–221; grqc/0109048. [17] A. A. Starobinski, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 (1980) 99–101. [18] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (The University of Chicago Press, 1994).

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[19] R. M. Wald, The back reaction eﬀect in particle creation in curved spacetime, Comm. Math. Phys. 54 (1977) 1–19. [20] R. M. Wald, Trace anomaly of a conformally invariant quantum ﬁeld in curved spacetime, Phys. Rev. D 17 (1978) 1477–1484. [21] S. Weinberg, Cosmology (Oxford University Press, 2008).

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Reviews in Mathematical Physics Vol. 23, No. 5 (2011) 553–574 c World Scientiﬁc Publishing Company DOI: 10.1142/S0129055X11004369

HYPERFINE SPLITTING OF THE DRESSED HYDROGEN ATOM GROUND STATE IN NON-RELATIVISTIC QED

L. AMOUR∗,‡ and J. FAUPIN†,§ ∗Laboratoire

de Math´ ematiques EDPPM, EA-4535, Universit´ e de Reims, Moulin de la Housse — BP 1039, 51687 REIMS Cedex 2, France †Institut

de Math´ ematiques de Bordeaux, UMR-CNRS 5251, Universit´ e de Bordeaux 1, 351 cours de la lib´ eration, 33405 Talence Cedex, France ‡[email protected] §[email protected] Received 24 June 2010 Revised 31 March 2011

We consider a spin- 21 electron and a spin- 21 nucleus interacting with the quantized electromagnetic ﬁeld in the standard model of non-relativistic QED. For a ﬁxed total momentum suﬃciently small, we study the multiplicity of the ground state of the reduced Hamiltonian. We prove that the coupling between the spins of the charged particles and the electromagnetic ﬁeld splits the degeneracy of the ground state. Keywords: Quantum electrodynamics; hyperﬁne splitting; multiplicity of the ground state; Feshbach–Schur map. Mathmatics Subject Classﬁcation 2010: 81V10, 81V45, 81Q10, 81Q15

1. Introduction This paper is concerned with the spectral analysis of the quantum Hamiltonian associated with a free hydrogen atom, in the context of non-relativistic QED. Before describing our result more precisely, we begin with recalling a few well-known facts about the spectrum of Hydrogen in the case where the corrections due to quantum electrodynamics are not taken into account. For more details, we refer the reader to classical textbooks on Quantum Mechanics (see, e.g., [29, 15]). See also [11, 26, 6]. We consider a neutral hydrogenoid system composed of one electron with spin 12 and one nucleus with spin 12 . The Pauli–Hamiltonian in L2 (R6 ; C4 ) associated with this system can be written in the following way: 1

H Pa :=

1 α 2 el 1 (pel − α 2 An (xel ))2 − σ · Bn (xel ) 2mel 2mel 1

+

1 1 α2 n α . (pn + α 2 Ael (xn ))2 + σ · Bel (xn ) − 2mn 2mn |xel − xn |

553

(1.1)

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Here the units are chosen such that = c = 1, where = h/2π, h is the Planck constant, and c is the velocity of light. The notations mel , xel and pel = −i∇xel (respectively, mn , xn and pn = −i∇xn ) stand for the mass, the position and the momentum of the electron (respectively of the nucleus), and α = e2 is the ﬁnestructure constant (with e the charge of the electron). Moreover, σ el = (σ1el , σ2el , σ3el ) (respectively, σn ) are the Pauli matrices accounting for the spin of the electron (respectively, of the nucleus), and An (xel ) is the vector potential of the electromagnetic ﬁeld generated by the nucleus at the position of the electron, that is An (xel ) = Cα1/2 (σ n ∧ (xel − xn ))/(mn |xel − xn |3 ) where C is a positive constant (and similarly for Ael (xn )). Finally, Bn (xel ) = ipel ∧ An (xel ) and Bel (xn ) = ipn ∧ Ael (xn ). The Hamiltonian H Pa can be derived from the Dirac equation in the nonrelativistic regime. It allows one to justify the so-called hyperﬁne structure of the ground state of the Hydrogen atom. More precisely, let H Pa (0) be the Hamiltonian obtained when the total momentum vanishes. Then H Pa (0) in L2 (R3 ; C4 ) can be decomposed into a sum of four terms, H Pa (0) = H0 + H1 + H2 + H3 , where H0 = p2r /(2µ) − α/|r| (here µ denotes the reduced mass of the atom and pr = −i∇r ), H1 is the orbital interaction, H2 is the spin-orbit interaction, and H3 is the spin-spin interaction (see, e.g., [6, Chap. 4] and [2] for details). It is seen that H0 has a 4-fold degenerate ground state. The correction terms, H1 , H2 , and H3 , produce an energy shift. Moreover, under the inﬂuence of the spin-spin interaction, the unperturbed ground state eigenvalue splits into two parts: a simple eigenvalue associated with a unique ground state, and a 3-fold degenerate eigenvalue. This phenomenon is referred to as the hyperﬁne splitting of the hydrogen atom ground state. Let us mention that this splitting explains the famous observed 21-cm hydrogen line. In this paper, we investigate the hyperﬁne structure of the hydrogen atom in the standard model of non-relativistic QED. We aim at establishing that a hyperﬁne splitting does occur in the framework of non-relativstic QED. The Hamiltonian is still given by the expression (1.1), except that An (xel ) and Ael (xn ) are replaced by the vector potentials of the quantized electromagnetic ﬁeld in the Coulomb gauge (and likewise for Bn (xel ) and Bel (xn ), precise deﬁnitions will be given in Sec. 2.1 below). Moreover the energy of the free photon ﬁeld is added. Since both the electron and the nucleus are treated as moving particles, the total Hamiltonian, Hg , is translation invariant. Here g denotes a coupling parameter depending on the ﬁne-structure constant α. The translation invariance implies that Hg admits a direct integral decomposition, Hg ∼ R3 Hg (P )dP , with respect to the total momentum P of the system. We set Eg (P ) := inf σ(Hg (P )). In [5], it is established that, for g and P suﬃciently small, Eg (P ) is an eigenvalue of Hg (P ), that is Hg (P ) has a ground state. We also mention [27] where the existence of a ground state for Hg (P ) is obtained for any value of g, under the assumption that Eg (0) ≤ Eg (P ). Using a method due to [23], it is proven in [5] that the multiplicity of Eg (P ) cannot exceed the multiplicity of E0 (P ) := inf σ(H0 (P )),

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where H0 (P ) := Hg=0 (P ) denotes the non-interacting Hamiltonian. In other words, (0 <) dim Ker(Hg (P ) − Eg (P )) ≤ dim Ker(H0 (P ) − E0 (P )).

(1.2)

Our purpose is to determine whether the inequality in (1.2) is strict, or, on the contrary, is an equality. Of course, the multiplicity of Eg (P ) depends on the value of the spins of the charged particles. If the spin of the electron is neglected and the spin of the nucleus is equal to 0, then E0 (P ) is simple, and hence, according to (1.2), Eg (P ) is also a simple eigenvalue. In particular, (1.2) is an equality. If the spin of the electron is taken into account, and the spin of the nucleus is equal to 0, then E0 (P ) is twice-degenerate. Using Kramer’s degeneracy theorem (see [28]), one can prove that the multiplicity of Eg (P ) is even. Therefore, by (1.2), Eg (P ) is also twice-degenerate, and hence (1.2) is again an equality. We refer the reader to [25, 32, 30, 22, 28] for results on the twice-degeneracy of the ground state of various QED models. Consider now a hydrogen atom composed of a spin- 12 electron and a spin- 12 nucleus (e.g. a proton). In this case, the multiplicity of E0 (P ) is equal to 4. Our main result states that dim Ker(Hg (P ) − Eg (P )) < dim Ker(H0 (P ) − E0 (P )) = 4,

(1.3)

for g = 0 small enough. Equation (1.3) can be interpreted as a hyperﬁne splitting of the ground state of Hg (P ). In other words, the Hamiltonian of a freely moving hydrogen atom at a ﬁxed total momentum in non-relativistic QED contains hyperﬁne interaction terms which split the degeneracy of the ground state, in the same way as for the Pauli–Hamiltonian of Quantum Mechanics mentioned above. Pursuing the analogy with the Pauli–Hamiltonian (1.1), one can conjecture that Eg (P ) is simple. Proving this is however beyond the scope of the present paper. We also mention that non-relativistic QED provides a suitable framework to rigorously justify radiative decay and Bohr’s frequency condition (see [9, 10, 1, 31] for the case of atomic systems with an inﬁnitely heavy nucleus). In particular, save for the ground state, all stationary states are expected to turn into metastable states with a ﬁnite lifetime. Hence in relation with the 21-cm hydrogen line mentioned above, one can expect that a resonance appears near the ground state energy Eg (P ), with a very small imaginary part. Showing this would presumably require the use of complex dilatations together with renormalization techniques as in [9]. The case of a nucleus of spin ≥ 1 is not considered here (for instance, the nucleus of deuterium, composed of one proton and one neutron, can be treated as a spin-1 particle), but we expect that a similar hyperﬁne splitting of the ground state occurs in this case also. As for positively charged hydrogenoid ions, the question of the existence of a ground state is more subtle than for the hydrogen atom. Indeed, it is proven in [21] that the Hamiltonian of a positive ion at a ﬁxed total momentum in non-relativistic QED does not have a ground state in Fock space. This result should be compared with the corresponding one for the model of a freely moving, dressed

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non-relativistic electron in non-relativistic QED, which has been studied recently by several authors (see, among other papers, [12, 13, 8, 21, 14, 28, 19]; see also [4]). Let us ﬁnally mention that the ground state degeneracy of the non-relativistic hydrogen atom conﬁned by its center of mass (see [3, 16]) could also be analyzed by the techniques developed here, provided that both the electron and the nucleus have a spin equal to 12 . 2. Definition of the Model and Statement of the Main Result 2.1. Definition of the model In the standard model of non-relativistic QED, the Hamiltonian associated with the system we consider acts on the Hilbert space H := Hat ⊗ Hph where Hat := L2 (R3 ; C2 ) ⊗ L2 (R3 ; C2 ) ∼ L2 (R6 ; C4 )

(2.1)

is the Hilbert space for the charged particles (the electron and the nucleus), and ∞ n (2.2) Hph := C ⊕ Sn L2 (R3 × {1, 2}) ⊗ n=1

is the symmetric Fock space for the photons. Here Sn denotes the symmetrization operator. The Hamiltonian of the system, H SM , is formally given by the expression 1 1 1 1 H SM := (pel − α 2 A(xel ))2 + (pn + α 2 A(xn ))2 + V (xel , xn ) + Hph 2mel 2mn 1

1

α 2 el α2 n − σ · B(xel ) + σ · B(xn ), 2mel 2mn

(2.3)

where xel , xn , pel , pn and α are deﬁned as in (1.1). For x ∈ R3 , A(x) is deﬁned by χΛ (k) λ −ik·x ∗ 1 aλ (k) + eik·x aλ (k) dk, (2.4) A(x) := 1 ε (k) e 2π 3 2 |k| λ=1,2 R and B(x) is given by

1 k i ∧ ελ (k) [e−ik·x a∗λ (k) − eik·x aλ (k)]dk, B(x) := − |k| 2 χΛ (k) 2π |k| R3 λ=1,2

(2.5) where the polarization vectors ε1 (k) and ε2 (k) are chosen in the following way: (k2 , −k1 , 0) ε1 (k) := 2 , k1 + k22

ε2 (k) :=

k (−k1 k3 , −k2 k3 , k12 + k22 ) ∧ ε1 (k) = 2 . |k| k1 + k22 k12 + k22 + k32

(2.6)

In (2.4) and (2.5), χΛ (k) denotes an ultraviolet cutoﬀ function which, for the sake of concreteness, we choose as χΛ (k) := 1|k|≤Λα2 (k).

(2.7)

Here, Λ is supposed to be a given arbitrary (large and) positive parameter. As explained in [10, 31], the model is physically relevant if we assume that 1 Λ

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α−2 . The reason for introducing α2 into the deﬁnition (2.7) will appear below (see (2.18)). As usual, for any h ∈ L2 (R3 × {1, 2}), we set ∗ ∗ ¯ λ)aλ (k)dk, h(k, λ)aλ (k)dk, a(h) := (2.8) h(k, a (h) := λ=1,2

R3

λ=1,2

R3

and Φ(h) := a∗ (h) + a(h), where the creation and annihilation operators, a∗λ (k) and aλ (k), obey the canonical commutation relations [aλ (k), aλ (k )] = [a∗λ (k), a∗λ (k )] = 0,

[aλ (k), a∗λ (k )] = δλλ δ(k − k ).

(2.9)

Hence, in particular, for j ∈ {1, 2, 3}, we have Aj (x) = Φ(hA j (x)) and Bj (x) = (x)), with Φ(hB j 1 χΛ (k) λ ε (k)e−ik·x , 2π |k| 12 j

k 1 i B λ 2 ∧ ε (k) e−ik·x . hj (x, k, λ) := − |k| χΛ (k) 2π |k| j hA j (x, k, λ) :=

(2.10) (2.11)

The Coulomb potential V (xel , xn ) is given by V (xel , xn ) ≡ V (xel − xn ) := −

α , |xel − xn |

(2.12)

and Hph is the Hamiltonian of the free photon ﬁeld, deﬁned by Hph := |k|a∗λ (k)aλ (k)dk. λ=1,2

(2.13)

R3

The 3-uples σ el = (σ1el , σ2el , σ3el ) and σ n = (σ1n , σ2n , σ3n ) are the Pauli matrices associated with the spins of the electron and the nucleus, respectively. They can be written as 4 × 4 matrices in the following way: 

0  0 σ1el =  1 0 

0  1 σ1n =  0 0

0 0 0 1

1 0 0 0

1 0 0 0

0 0 0 1

 0 1 , 0 0  0 0 , 1 0



0  0 σ2el =  i 0 

 0 −i 0 0 0 −i  , 0 0 0  i 0 0

 0 −i 0 0 i 0 0 0   σ2n =   0 0 0 −i  , 0 0 i 0



 0 0 0 1 0 0 , 0 −1 0 0 0 −1 (2.14)   1 0 0 0  0 −1 0 0 . σ3n =  0 0 1 0 0 0 0 −1 (2.15) 1  0 σ3el =  0 0

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In order to exhibit the perturbative behavior of the interaction between the char