Resonant rigidity for Schrödinger operators in even dimensions
aa r X i v : . [ m a t h . SP ] D ec RESONANT RIGIDITY FOR SCHR ¨ODINGER OPERATORS IN EVENDIMENSIONS
T.J. CHRISTIANSEN
Abstract.
This paper studies the resonances of Schr¨odinger operators with bounded, com-pactly supported, real-valued potentials on R d , where the dimension d is even. If the poten-tial V is non-trivial and d = 4, then the meromorphic continuation of the resolvent of theSchr¨odinger operator has infinitely many poles, with a quantitative lower bound on theirdensity. A somewhat weaker statement holds if d = 4. We prove several inverse-type results.If the meromorphic continuations of the resolvents of two Schr¨odinger operators − ∆ + V and − ∆ + V have the same poles, V , V ∈ L ∞ c ( R d ; R ), k ∈ N and if V ∈ H k ( R d ; R ),then V ∈ H k as well. Moreover, we prove that certain sets of isoresonant potentials arecompact. We also show that the poles of the resolvent for a smooth potential determinethe heat coefficients and that the (resolvent) resonance sets of two potentials in L ∞ c ( R d ; R )cannot differ by a nonzero finite number of elements away from 0. Introduction
This paper proves some results about resonances of the Schr¨odinger operator − ∆ + V on R d when d is even and the potential V ∈ L ∞ c ( R d ; R ). For example, we show that if V ∈ L ∞ c ( R d ; R ) is nontrivial and d = 4 then − ∆ + V has infinitely many (resolvent) resonances.If d = 4, then either 0 is a resonance of − ∆ + V , or there are infinitely many resonances.Hence, one could say that this demonstrates the resonant rigidity of the 0 potential amongall potentials in L ∞ c ( R d ; R ). If V , V ∈ L ∞ c ( R d ; R ) have the same resolvent resonance set,including multiplicities, and if V ∈ H k ( R d ) for some k ∈ N , then V ∈ H k ( R d ) as well.These results are inspired by analogous results of Smith and Zworski [39] in odd dimension d ≥
3. In addition, we show that if V , V ∈ C ∞ c ( R d ; R ) have the same resonances, includingmultiplicities, then they have the same heat coefficients. The compactness of the set ofpotentials in L ∞ c ( R ; R ) with support in a fixed compact set and having the same poles as afixed potential V ∈ C ∞ c ( R ; R ) then follows rather directly by results of [7, 15]. There is aweaker result in higher dimensions. See [21] for analogous results in dimension d = 1 ,
3. Asa whole, these results can be interpreted as saying something about the rigidity of the set ofpotentials V ∈ L ∞ c ( R d ; R ) having the same resonances.For Schr¨odinger operators on R d , the manifold on which the resonances lie is determinedby the parity of the dimension d . Set R V ( λ ) = ( − ∆ + V − λ ) − when 0 < arg λ < π . Then if d is odd R V has a meromorphic continuation, as an operator from L c ( R d ) to L ( R d ), to thecomplex plane. If d is even, the continuation is to Λ, the logarithmic cover of C \ { } . The Key words and phrases.
Schr¨odinger operator, resonance, scattering theory, inverse problem. poles of this continuation to Λ are the nonzero (resolvent) resonances. Here we explicitlyinclude as resonances poles corresponding to eigenvalues and lying in the physical space { λ : 0 < arg λ < π } , although conventions differ on this.The following theorem provides a quantitative lower bound on the number of resonancesfor a Schr¨odinger operator in even dimensions. We describe a point λ ∈ Λ by specifying itsnorm | λ | and argument arg λ , where we do not identify points whose arguments differ by anonzero integral multiple of 2 π . For Hypothesis 2.1, which is a hypothesis about the natureof the singularity of the resolvent at the origin (if it is unbounded there), see Section 2. Weremark that Hypothesis 2.1 holds generically. The definition of the multiplicity of a nonzeropole of the resolvent is given in (3.1). Theorem 1.1.
Let d be even. Let V ∈ L ∞ c ( R d ; R ) and suppose V . If d = 4 , suppose inaddition that Hypothesis 2.1 holds. Then − ∆ + V has infinitely many (resolvent) resonances.In fact, set N ( r ) to be the number of poles of the resolvent on Λ , counted with multiplicity,that have /r < | λ | < r and | arg λ | < log r . Then, for any ǫ > , lim sup r →∞ N ( r )(log r ) − ǫ = ∞ . S´a Barreto [34] ( d ≥
4) and Chen [8] ( d = 2) proved a related, stronger, bound for V ∈ C ∞ c ( R d ; R ), V
0. They showed lim sup r →∞ N ( r )(log r )(log log r ) − p = ∞ for any p >
1, and didnot require Hypothesis 2.1 when d = 4. Each of these lower bounds is much smaller than theupper bounds known to hold in even dimensions [23, 40, 41], and the lower bounds whichare known to hold generically, see [12].Interestingly, in odd dimensions d >
3, the result of [39] analogous to Theorem 1.1 isthat any non-trivial, real-valued potential V must have at least one resonance, although inodd dimensions it is known that any non-trivial, smooth, real-valued potential must haveinfinitely many, and there is a quantitative lower bound (e.g. [35] and references therein).In dimension 3, any nontrivial V ∈ L ∞ c ( R ; R ) must have infinitely many ([39]), and indimension d = 1, asymptotics of the resonance-counting function are known [17, 36, 45]. Bothhere and in [39], it is important that we require V to be real-valued, since in dimension atleast 3 there are examples of complex-valued potentials with no resonances, and in dimension d = 2 no resonances away from 0 [1, 9, 10]. See [39] and references therein for further resultsin odd dimensions.It is important to emphasize that in the assumptions of Theorems 1.2 and 1.3 and elsewe-here we use the notions of multiplicity of a resonance defined in this paper. The multiplicityof points of Λ as a resonance is rather standard and is recalled in Section 3. However, thenotion of the multiplicity of 0 as a resonance is a rather subtle point. The one we use heremight more properly be called a normalized or weighted multiplicity, and can be found inSection 2. For other purposes a different notion of multiplicity of 0 as a resonance than thatof this paper may be preferable. ESONANT RIGIDITY IN EVEN DIMENSIONS 3
The preliminary steps in the proof of Theorem 1.1 give rather directly some results aboutthe heat coefficients for smooth potentials in even dimensions. Recall that here and else-where we include poles of the resolvent corresponding to eigenvalues of − ∆ + V among theresonances. Theorem 1.2.
Let d be even, and let V , V ∈ C ∞ c ( R d ; R ) . Suppose V and V have thesame resolvent resonances, including multiplicities. Then V and V have the same heatcoefficients. A similar result holds in odd dimension d . Two potentials in C ∞ c ( R ; R ) with the sameresonance set which does not include 0 have the same heat coefficients, except, possibly, forthe first–that is, the integral of V . If d ≥
5, the first two heat coefficients may differ if 0 isan eigenvalue of both Schr¨odinger operators. This follows from bounds on the determinantof the scattering matrix and the number of resonances, Hadamard’s factorization theorem,a trace formula, and the behavior of the determinant of the scattering matrix near 0. For adifferent proof in dimensions d = 1 and d = 3, see [21].For R >
0, set B (0 , R ) = { x ∈ R d : | x | < R } . Fix R >
0, and let V ∈ C ∞ c ( R d ; R ) satisfysupp V ⊂ B (0 , R ). Set(1.1) Iso ( V , R ) = { V ∈ C ∞ c ( R d ; R ) : supp V ⊂ B (0 , R ) and − ∆ + V and − ∆ + V have the same resolvent resonances, including multiplicities } and, for c > s ≥ Iso ( V , R , s, c ) = { V ∈ Iso ( V , R ) : k V k H s ≤ c } . Theorem 1.3.
Let V ∈ C ∞ c ( R d ; R ) , with supp V ⊂ B (0 , R ) . If d = 2 , then Iso ( V , R ) is compact in the topology of C ∞ ( B (0 , R )) . If d ≥ is even and s > d/ − , then Iso ( V , R , s, c ) is compact in C ∞ ( B (0 , R )) . Hislop and Wolf have proved analogs of Theorems 1.2 and 1.3 in dimensions d = 1 and d = 3, [21]. In dimension 1, some stronger results are due to Zworski [47] and Korotyaev[28, 29, 30]. Our proof of Theorem 1.3 uses results of Br¨uning [7] and Donnelly [15] forisospectral Schr¨odinger operators on compact Riemannian manifolds, together with Theorem1.2. We remark that again it is necessary to assume the potentials are real-valued, asexamples of [1, 10] give large families of isoresonant complex-valued potentials which are noteven bounded in L ∞ .Our proof of Theorem 1.1 uses an adaptation of techniques from [39] and [34]. The centralnovel technical results are Theorems 1.4 and 4.1. Theorem 1.4 gives a relationship betweenthe determinants of two scattering matrices if the difference of the sets of their poles is nottoo big. When combined with techniques of [39] or [34], we shall see that Theorem 1.4 hasa number of corollaries, among them Theorems 1.1, 1.2, and 1.5.In the statement of Theorem 1.4, for z ∈ C , S j ( e z ) means we evaluate S j at the point inΛ having argument Im z and norm e Re z . T.J. CHRISTIANSEN
Theorem 1.4.
Let d be even, and V j ∈ L ∞ c ( R d ; R ) for j = 1 , . Set P j = − ∆ + V j and let S j ( λ ) be the associated scattering matrix, unitary for λ > . Set F ( z ) = det S ( e z )det S ( e z ) . Let { z l } denote the distinct poles of F ( z ) , and M ( z l ) their multiplicities. Suppose that forsome ǫ > , (1.2) X | z l |
Theorem 1.4 has several corollaries, including Theorems 1.1, 1.2, and 1.5. Some of theseare inspired by analogous results of Smith and Zworski [39] in the case of d ≥ Theorem 1.5.
Let d be even. Suppose V , V ∈ L ∞ c ( R d ; R ) are as in the statement ofTheorem 1.4, including the conditions on the poles of F . If d = 4 , assume that Hypothesis2.1 holds for P and P . If for some k ∈ N , V ∈ H k ( R d ) , then V ∈ H k ( R d ) as well. Again, this result is not true if we omit the hypothesis that the potentials are real-valued.The paper [39] proves a similar result in odd dimensions: if V , V ∈ L ∞ c ( R d ; R ) have thesame resonances, including multiplicities, and if V ∈ H k ( R d ), then V ∈ H k ( R d ).We note that a consequence of our Theorem 1.5 is that for V ∈ C ∞ c ( R d ; R ) supported in B (0 , R ), we could replace the definition (1.1) by the equivalent Iso ( V , R ) = { V ∈ L ∞ c ( R d ; R ) : supp V ⊂ B (0 , R ) and − ∆ + V and − ∆ + V have the same resolvent resonances, including multiplicities } . Another corollary of Theorem 1.4 is the following.
Corollary 1.6.
Let d be even, and let V , V ∈ L ∞ c ( R d ; R ) . Then, if d = 4 , the resolventresonance sets of V and V cannot differ by a nonzero finite number of nonzero elements.If d = 4 , the same is true, provided that Hypothesis 2.1 holds for both potentials, or bothpotentials are smooth. We emphasize here that we cannot at this point exclude the possibility that the resonancesets of V and V are the same except that the point 0 has different multiplicities as anelement of the resonance sets for the two potentials. We remark that Korotyaev has studiedthe rigidity of the resonance set for a larger class of potentials in one dimension, i.e., d = 1. Heshowed that within this larger class of potentials, finitely many resonances can be “moved”within certain restrictions. For the full line case see [30, Theorem 1.3] and for the half-line[28, 29].1.1. Notational conventions.
Throughout this paper we shall use the convention that C stands for a positive constant, the value of which may change from line to line withoutcomment. By the physical space in Λ we mean the copy of the upper half plane { Im λ > } in Λ on which the resolvent R V ( λ ) is bounded from L ( R d ) to L ( R d ). Our convention isthat this corresponds to { λ ∈ Λ : 0 < arg λ < π } , and we identify this with the upper halfplane when convenient. When (0 , ∞ ) = R + is considered as a subset of Λ, it is identifiedwith the set of points with argument 0. Likewise, if λ ∈ Λ, by λ > λ ∈ (0 , ∞ ) wemean that the point λ ∈ Λ has argument 0.The set of resonances of − ∆ + V includes all poles of R V on Λ, including those corre-sponding to eigenvalues, and should be repeated with multiplicity. Moreover, 0 may be aresonance of − ∆ + V , with multiplicity as defined in Section 2.The dimension d is assumed to be even in subsequent sections, except in Section 4, where d can be even or odd, but d ≥ T.J. CHRISTIANSEN
The set L ∞ c ( R d ; R ) = { f ∈ L ∞ ( R d ; R ) : f has compact support } , and similarly for L c ( R d ), C ∞ c ( R d ).1.2. Organization of the paper.
We briefly outline the organization of this paper. InSection 2 we discuss the behavior of the resolvent near 0 and fix the notion of the multiplicityof 0 as a resonance which we shall use in this paper. This section also includes Hypothesis2.1. Section 3 recalls some results about the relationship between the poles and zeros ofthe determinant of the scattering matrix and the poles of the (meromorphically continued)resolvent, and recalls the definition of the multiplicity of a nonzero resonance.An important technical step in the proof of Theorem 1.4 is Theorem 4.1, which is proved inSection 4. Theorem 4.1 is a result on the high-energy behavior of the logarithmic derivativeof the determinant of the scattering matrix of − ∆ + V under the assumption only that V ∈ L ∞ c ( R d ; R )–that is, without any additional regularity assumed on V . Theorem 4.1 isvalid in both even and odd dimensions. See Section 4 for the statement of the theorem andreferences to earlier results.In Section 5 we turn to the behavior of the determinant of the scattering matrix near 0.In Section 6 we write the function F from Theorem 1.4 using a canonical product, and thenprove Theorem 1.4 in Section 7. The proof uses results about both the high and low energybehavior of the logarithmic derivative of the scattering matrix. Section 8 includes proofs ofTheorems 1.5, 1.1, and 1.2 and Corollary 1.6.In Section 9 we consider some questions related to linear independence of elements of theimage of the singular part of the resolvent at different points on Λ.2. Resonances at and Hypothesis 2.1 Let V ∈ L ∞ c ( R d ; R ) with d even. All but at most one of the resonances of − ∆ + V lie onΛ, the logarithmic cover of C \ { } . However, 0 may be a resonance as well. Since this is adelicate question and is related to Hypothesis 2.1, we address this here. For further detailsin even dimensions we refer the reader to [6] for dimension 2, [25] for dimension 4, and [24]for dimension d ≥
6. We recall the results which are most important to us here, clarifyingthe language we shall use to describe the possible scenarios.If for each χ ∈ C ∞ c ( R d ), lim ǫ ↓ χ ( − ∆ + V + ǫ ) − χ exists as a bounded operator on L ( R d ),then 0 is not a resonance of − ∆ + V . If this limit does not exist for some χ ∈ C ∞ c ( R d ),then our convention is that 0 is a resonance of − ∆ + V , even if this singularity is caused by − ∆ + V having 0 as an eigenvalue.Let P denote projection onto the L null space of − ∆ + V , with P = 0 if 0 is not aneigenvalue of − ∆ + V . Then if P = 0 the leading singularity of ( − ∆ + V + ǫ ) − is givenby P ǫ − .If d ≥
6, then for any χ ∈ C ∞ c ( R d )(2.1) lim ǫ ↓ χ (( − ∆ + V + ǫ ) − P /ǫ ) χ ESONANT RIGIDITY IN EVEN DIMENSIONS 7 exists, [24]. However, the behavior of the resolvent near 0 is more complicated in lowerdimensions.If for some χ ∈ C ∞ c ( R d ) the limit (2.1) fails to exist, we will say that 0 is a “non-eigenvalueresonance” of − ∆ + V . We shall use this notation even if 0 is simultaneously an eigenvalue of − ∆ + V , so that it is possible for 0 to be both an eigenvalue resonance and a non-eigenvalueresonance. (We note that here is an awkwardness that arises from our convention that thesquare roots of eigenvalues in the closure of the physical space are resonances. Had we notadopted that convention, we could just say that 0 is a resonance in this case. Many writersdo choose this other convention. However, our convention is more convenient for some otherpurposes.) The non-eigenvalue resonances at 0 correspond to elements of the null space of − ∆ + V which are bounded (if d = 2) or decaying at infinity (if d = 4), but which are notin L ( R d ).In dimension d = 4, some of our techniques do not work if 0 is a non-eigenvalue resonanceof − ∆ + V . Hence for some results in dimension 4 we shall need to assume the followinghypothesis. Hypothesis 2.1.
Let P denote projection onto the L null space of − ∆ + V , if any. Thenfor any χ ∈ C ∞ c ( R ) , lim ǫ ↓ χ (cid:0) ( − ∆ + V + ǫ ) − − ǫ − P (cid:1) χ exists. We note that if this limit exists for one nontrivial χ ∈ C ∞ c ( R ) with χ V = V , then itexists for any χ ∈ C ∞ c ( R ). Moreover, this hypothesis is true generically.Finally, we shall need a notion of the multiplicity of 0 as a resonance of − ∆ + V . If 0 isnot a non-eigenvalue resonance, this is straightforward, and is the dimension of the L nullspace of − ∆ + V . However, if 0 is a non-eigenvalue resonance, there are several possiblenotions. We choose the one which is most convenient for our purposes, but which may notbe the most natural in terms of the dimension of the space of bounded/decaying elementsof the null space of − ∆ + V .Let S ( λ ) denote the scattering matrix of − ∆ + V and let − µ ≤ − µ ≤ ... ≤ − µ K ≤ − ∆ + V , repeated according to multiplicity. Our assumptions on V ensure that there are at most finitely many eigenvalues and they are real and non-positive.We shall use the heat trace repeatedly in our proofs, and we recall one expression for it here.The Birman-Krein formula tells us, for t > e t (∆ − V ) − e t ∆ ) = 12 πi Z ∞ tr (cid:18) S ( λ ) − ddλ S ( λ ) (cid:19) e − tλ dλ + K X k =1 e tµ k + β ( V, d )see [14, 20], [16, Section 3.8], and references therein. Here β ( V, d ) is 0 whenever 0 is nota non-eigenvalue resonance of − ∆ + V . However, the converse is not true and the exactbehavior is rather subtle; see [5, 6]. Our notion of multiplicity of 0 as a resonance wouldbetter be called a normalized or weighted multiplicity, but in the interest of brevity we shall T.J. CHRISTIANSEN just refer to it as multiplicity. We define the multiplicity of 0 as a resonance of − ∆ + V tobe β ( V, d ) + dim { f ∈ L ( R d ) : ( − ∆ + V ) f = 0 } . Note that if 0 is an eigenvalue of − ∆ + V then it makes a contribution in (2.2) to the sumover the eigenvalues.3. Multiplicities of poles of the resolvent and scattering determinant
In this section we clarify the relationship between the poles of the resolvent and the polesof the determinant of the scattering matrix. This is well-known in odd dimensions, but is abit more subtle in even dimensions. The discussion here is taken from [13], though modifiedto reflect the fact that we need only a somewhat simplified version for our specific case ofthe Schr¨odinger operator on R d .Let d be even. We consider here the case of poles of the resolvent which lie on Λ–thatis, all of the poles except the (possible) pole at 0. We define the notion of the multiplicity µ R V of the pole of the resolvent as follows. Given λ ∈ Λ, define γ λ to be a small positivelyoriented circle centered at λ that contains no poles of the resolvent except, possibly, a poleat λ . Here we locally identify a subset of Λ with a subset of the complex plane. Define, for λ ∈ Λ,(3.1) µ R V ( λ ) def = rank Z γ λ R V ( λ )2 λdλ = rank Z γ λ R V ( λ ) dλ. We shall also consider poles of the determinant of the scattering matrix, a scalar functionon Λ. Following [13], let f be a (scalar) function meromorphic on Λ. If f ( λ ) = 0, definem sc ( f, λ ) to be the multiplicity of λ as a zero of f . If f has a pole at λ , define m sc ( f, λ )to be minus the order of the pole of f at λ . If λ is neither a pole nor a zero of f , setm sc ( f, λ ) = 0. Thus m sc ( f, · ) is positive at zeros and negative at poles. It should bethought of as measuring the order of vanishing of the function f at λ .From [13, Theorem 4.5], if V ∈ L ∞ c ( R d ; R ), then for λ ∈ Λ,(3.2) µ R V ( λ ) − µ R V ( λ ) = − m sc (det S ( λ ) , λ ) , where λ = | λ | e − i arg λ . Thus the determinant of the scattering matrix does not necessarilyhave a pole at each pole of the resolvent. However, if the determinant of the scatteringmatrix has a pole at λ , then the resolvent must have a pole at λ .The following lemma and its corollary will be used in the proofs of Theorem 1.1 andCorollary 1.6. Lemma 3.1.
Let V , V ∈ L ∞ c ( R d ; R ) , and for j = 1 , let S j denote the scattering matrixfor the operator − ∆ + V j . Suppose det S ( λ ) / det S ( λ ) is analytic on all of Λ . Let µ R j ( λ ) denote the quantity (3.1) for the operator − ∆ + V j . Then, for τ > , θ ∈ R , µ R ( τ e i ( θ + kπ ) ) − µ R ( τ e i ( θ + kπ ) ) = µ R ( τ e iθ ) − µ R ( τ e iθ ) , if k ∈ Z ESONANT RIGIDITY IN EVEN DIMENSIONS 9 and µ R ( τ e i ( − θ + kπ ) ) − µ R ( τ e i ( − θ + kπ ) ) = µ R ( τ e iθ ) − µ R ( τ e iθ ) , if k ∈ Z . Proof.
Note that our assumption on det S ( λ ) / det S ( λ ) implies that this ratio is non-vanishing on all of Λ. Hence, for all λ ∈ Λ,(3.3) m sc (det S ( λ ) , λ ) = m sc (det S ( λ ) , λ ) . By (3.2) and (3.3),(3.4) µ R ( τ e iθ ) − µ R ( τ e iθ ) = µ R ( τ e − iθ ) − µ R ( τ e − iθ )for any τ > θ ∈ R . Since R j ( λ ) = R ∗ j ( e iπ λ ), µ R j ( λ ) = µ R j ( e iπ λ ) and(3.5) µ R j ( τ e ± iθ ) = µ R j ( τ e i ( π ∓ θ ) ) . Combining this with (3.4) gives µ R ( τ e i ( π ± θ ) ) − µ R ( τ e i ( π ± θ ) ) = µ R ( τ e iθ ) − µ R ( τ e iθ ) . Applying (3.4) again, this gives µ R ( τ e − i ( π ± θ ) ) − µ R ( τ e − i ( π ± θ ) ) = µ R ( τ e i ( π ± θ ) ) − µ R ( τ e i ( π ± θ ) )= µ R ( τ e iθ ) − µ R ( τ e iθ ) . (3.6)Using (3.4), (3.5), and (3.6) inductively proves the lemma. (cid:3) Lemma 3.1 has the following corollary as a special case. This corollary will be used in theproof of Theorem 1.1.
Corollary 3.2.
Let V ∈ L ∞ c ( R d ; R ) and denote the associated scattering matrix S ( λ ) . If det S ( λ ) is analytic on all of Λ but − ∆ + V has a negative eigenvalue, then the meromorphiccontinuation of the resolvent of − ∆ + V has infinitely many poles. In particular, in this caseif − ρ is an eigenvalue of − ∆ + V with multiplicity m > and if ρ > , then e iπ ( k +1 / ρ ∈ Λ is a pole of χR V ( λ ) χ of multiplicity m for every k ∈ Z .Proof. In Lemma 3.1, take V = V , V ≡ τ = ρ and θ = π/ (cid:3) High energy behavior of the determinant of the scattering matrix onthe real line
The proof of Theorem 1.4 which we shall give uses Theorem 4.1, a result about the large λ behavior of the determinant of the scattering matrix on the positive real axis. A strongerresult is well known for smooth potentials V [14, 20, 33] and [44, Theorem 9.2.12], or even,if d = 3 for some potentials with less regularity [26], but still with more regularity than L ∞ . A related but slightly different result for dimensions 2 and 3 and V ∈ L ∞ c ( R d ; R ) is [43,Theorem 3.12] or [44, Theorem 9.1.14]. However, we are unaware of a result which is validin all dimensions d ≥ d is allowed to be evenor odd, but we always assume d ≥ Theorem 4.1.
Let d ≥ be even or odd. Let V ∈ L ∞ c ( R d ; R ) , and let S ( λ ) be the associatedscattering matrix. Then for λ ∈ R + (i.e., arg λ = 0 ), i ddλ det S ( λ )det S ( λ ) = − ( d − c d Z V ( x ) dxλ d − + O ( λ d − / ) as λ → ∞ . Here c d = π (2 π ) − d vol( S d − ) . Note that when d = 2, the coefficient of λ d − is 0.4.1. Reduction of the proof of Theorem 4.1 to Lemma 4.5.
The proof uses an explicitexpression for the scattering matrix (e.g. [42, (8.1)])(4.1) S ( λ ) = I − πi Γ ( λ )( V − V R V ( λ ) V )Γ ( λ ) ∗ , if λ ∈ (0 , ∞ )where (Γ ( λ ) f ) ( ω ) = 2 − / (2 π ) − d/ λ ( d − / ˆ f ( λω ) , ω ∈ S d − and ˆ f ( ξ ) = R e − ix · ξ f ( x ) dx . Consistent with our notation elsewhere, for λ > R V ( λ ) =( − ∆ + V − ( λ + i ) − .Let χ ∈ C ∞ c ( R d ). By [27, Theorem 16.1], with R ( λ ) = ( − ∆ − λ ) − when Im λ > (cid:13)(cid:13)(cid:13)(cid:13) d j dλ j χR ( λ ) χ (cid:13)(cid:13)(cid:13)(cid:13) ≤ C j λ − when λ ∈ (0 , ∞ ), j ∈ N . Here the value of C j depends on χ as well as on j .Since k Γ ( λ ) χf k = π − Im h χR ( λ ) χf, f i [42, (10.1)], we have(4.3) k Γ ( λ ) χ k ≤ Cλ − / , (cid:13)(cid:13)(cid:13)(cid:13) ddλ Γ ( λ ) χ (cid:13)(cid:13)(cid:13)(cid:13) ≤ Cλ − / when λ ∈ (0 , ∞ ) . With k · k HS denoting the Hilbert-Schmidt norm(4.4) k Γ ( λ ) χ k HS = 12(2 π ) d Z x ∈ supp χ Z ω ∈ S d − (cid:12)(cid:12) λ ( d − / e − iλx · ω χ ( x ) (cid:12)(cid:12) dσ ω dx ≤ Cλ d − , λ ∈ (0 , ∞ )where dσ denotes the density on S d − . Likewise, k ddλ Γ ( λ ) χ k HS ≤ Cλ d − for λ ∈ (1 , ∞ ).For an operator A depending on λ , denote by ˙ A ( λ ) the derivative of A ( λ ) with respect to λ . Recall that for λ ∈ (0 , ∞ ), S ∗ ( λ ) S ( λ ) = I , so that(4.5) tr[ S ∗ ( λ ) ˙ S ( λ )] = tr[ S − ( λ ) ˙ S ( λ )] = ddλ det S ( λ )det S ( λ ) , λ ∈ (0 , ∞ ) . Set(4.6) B ( λ ) = − πi Γ ( λ ) V Γ ∗ ( λ ) and B ( λ ) = 2 πi Γ ( λ ) V R ( λ ) V Γ ∗ ( λ ) . ESONANT RIGIDITY IN EVEN DIMENSIONS 11
Lemma 4.2.
Let B , B be as defined in (4.6). Then for λ ∈ (0 , ∞ ) , tr( S ∗ ( λ ) ˙ S ( λ )) = tr (cid:16) ˙ B ( λ ) + ˙ B ( λ ) + B ∗ ( λ ) ˙ B ( λ ) (cid:17) + O ( λ d − ) when λ → ∞ . Proof.
We use the expression (4.1) for the scattering matrix. Let χ ∈ C ∞ c ( R d ) satisfy χV = V . Since, for large λ ∈ R + , χR V ( λ ) χ = χR ( λ ) χ P ∞ j =0 ( − V R ( λ ) χ ) j , using the bounds(4.2-4.4) we see that (cid:13)(cid:13)(cid:13)(cid:13) d j dλ j [ S ( λ ) − ( I + B ( λ ) + B ( λ ))] (cid:13)(cid:13)(cid:13)(cid:13) tr ≤ Cλ d − for λ ∈ R , λ ≥ , j = 0 , (cid:13)(cid:13)(cid:13)(cid:13) d j dλ j [ S ( λ ) − ( I + B ( λ ) + B ( λ ))] (cid:13)(cid:13)(cid:13)(cid:13) L → L ≤ Cλ − for λ ∈ R , λ ≥ , j = 0 , . Moreover,(4.7) k B ( λ ) k L → L ≤ Cλ − , k B ( λ ) k tr ≤ Cλ d − . Hence, by (4.1) tr( S ∗ ( λ ) ˙ S ( λ )) = tr[( I + B ∗ ( λ ))( ˙ B ( λ ) + ˙ B ( λ ))] + O ( λ d − ) . Similarly, (cid:12)(cid:12)(cid:12) tr[ B ∗ ( λ ) ˙ B ( λ )] (cid:12)(cid:12)(cid:12) ≤ k B ∗ ( λ ) k tr k ˙ B ( λ ) k L → L ≤ Cλ d − λ − for some constant C , since k ˙ B k L → L = O ( λ − ). (cid:3) Lemma 4.3.
Let B be as defined in (4.6). Then tr[ ˙ B ( λ )] = − ic d ( d − (cid:18)Z V ( x ) dx (cid:19) λ d − where c d = π (2 π ) − d vol( S d − ) .Proof. Here we can evaluate the trace of ˙ B as the integral of its Schwartz kernel over thediagonal. Hence, with dσ denoting the usual measure on S d − ,tr[ ˙ B ( λ )]= − πi (2 π ) d Z ω ∈ S d − Z x ∈ R d e iλx · ( ω − θ ) ↾ θ = ω V ( x ) (cid:0) ( d − λ d − + λ d − ix · ( ω − θ ) (cid:1) ↾ θ = ω dx dσ ω = − πi ( d − π ) d λ d − Z ω ∈ S d − Z x ∈ R d V ( x ) dx dσ ω = − πi ( d − π ) d λ d − vol( S d − ) Z V ( x ) dx. (cid:3) Lemma 4.4.
Let B be as defined in (4.6) and λ ∈ (0 , ∞ ) . Then Im tr[ B ∗ ( λ ) ˙ B ( λ )] = 0 . Proof.
We havetr[ B ∗ ( λ ) ˙ B ( λ )] = 4 π tr { Γ ( λ ) V Γ ∗ ( λ )[ ˙Γ ( λ ) V Γ ∗ ( λ ) + Γ ( λ ) V ˙Γ ∗ ( λ )] } . We use the cyclicity of the trace to write this astr[ B ∗ ( λ ) ˙ B ( λ )] = 4 π tr { Γ ( λ ) V [Γ ∗ ( λ ) ˙Γ ( λ ) + ˙Γ ∗ ( λ )Γ ( λ )] V Γ ∗ ( λ ) } . Since this is the trace of a self-adjoint operator, its imaginary part is 0. (cid:3)
Note that for λ ∈ (0 , ∞ ), S ( λ ) is unitary so that [ ddλ det S ( λ )] / det S ( λ ) is pure imaginary.Thus, by Lemmas 4.2, 4.3, and 4.4, to prove Theorem 4.1 it remains only to estimate theimaginary part of tr[ ˙ B ( λ )]. Hence the proof of Theorem 4.1 is completed by the followinglemma. Lemma 4.5.
For λ ∈ (0 , ∞ ) , Im[tr( ˙ B ( λ ))] = O ( λ d − / ) as λ → ∞ . The proof of this lemma is the content of the next subsection.4.2.
Proof of Lemma 4.5.
In this section we study Im tr B ( λ ), and prove Lemma 4.5.Writing the integral as a trace of the Schwartz kernel over the diagonal,tr B ( λ ) = πi (2 π ) − d λ d − Z ω ∈ S d − Z x ∈ R d e − iλx · ω V ( x ) W ( x, ω, λ ) dxdσ ω where W ( x, ω, λ ) = ( R ( λ )( V ( • ) e iλω ·• )( x ). Using Parseval’s formula to evaluate the x inte-gral, we get tr B ( λ ) = i π ) − d +1 λ d − lim ǫ ↓ Z ω ∈ S d − Z η ′ ∈ R d | ˆ V ( η ′ − λω ) | | η ′ | − ( λ + iǫ ) dη ′ dσ ω . Making the substitution η ′ = λη + λω gives(4.8) tr B ( λ ) = i π ) − d +1 λ d − lim ǫ ↓ Z ω ∈ S d − Z η ∈ R d | ˆ V ( λη ) | | η + ω | − − iǫ dη dσ ω . We shall use the following lemma to justify some manipulations in our estimate of thederivative of Im tr B ( λ ). Since we shall want to estimate the size of the derivative of theimaginary part of (4.8), we take (twice) the average of the distributions lim ǫ ↓ / ( | η + ω | − ± iǫ ) in the following lemma and in Lemma 4.9. Lemma 4.6.
Let ψ ∈ C ∞ ( R ) be supported in a neighborhood of , and let φ ∈ C ∞ ( R d ) .Then (4.9) lim δ ↓ lim ǫ ↓ Z ω ∈ S d − Z η ∈ R d X ± | η + ω | − ± iǫ ψ ( | η | /δ ) φ ( η ) dηdσ ω = 0 and (4.10) lim δ ↓ lim ǫ ↓ Z ω ∈ S d − Z η ∈ R d X ± | η + ω | − ± iǫ ψ (( | η | − /δ ) φ ( η ) dηdσ ω = 0 . ESONANT RIGIDITY IN EVEN DIMENSIONS 13
Proof.
We give the proof of (4.10), as the proof of (4.9) is similar.Using polar coordinates η = ρθ and [22, (3.2.13)],(4.11) lim ǫ ↓ Z ω ∈ S d − Z η ∈ R d X ± | η + ω | − ± iǫ ψ (( | η | − /δ ) φ ( η ) dηdσ ω = − Z ω ∈ S d − Z θ ∈ S d − Z ∞ log | ρ + 2 θ · ω | ∂∂ρ [ ρ d − ψ (( ρ − /δ ) φ ( ρθ )] dρdσ θ dσ ω . The integrand is absolutely integrable, and we may change the order of integration as con-venient. We would like to understand the behavior of(4.12) Z ω ∈ S d − log | ρ + 2 θ · ω | dσ ω near ρ = 2. If ρ is near 2, the integrand in (4.12) is smooth away from a neighborhood of ω = − θ . Therefore, the integral over { ω ∈ S d − : | ω + θ | ≥ ǫ ′ > } results in a functiondepending smoothly on ρ near ρ = 2. Thus we concentrate on a neighborhood of ω = − θ in S d − . The function θ · ω has a nondegenerate critical point at θ = − ω . By first integratingover level sets in ω ∈ S d − of θ · ω and then using the Morse lemma, we may then introducea variable s so that Z { ω ∈ S d − : | θ · ω +1 | <ǫ ′ } log | ρ + 2 θ · ω | dσ ω = Z | s | <ǫ ′′ log | ρ − s | h ( s ) ds for some smooth function h . But then Z | s | <ǫ ′′ log | ρ − s | h ( s ) ds − Z | s | <ǫ ′′ log | − s | h ( s ) ds = Z | s | <ǫ ′′ log (cid:12)(cid:12)(cid:12)(cid:12) ρ − s s (cid:12)(cid:12)(cid:12)(cid:12) h ( s ) ds. By a change of variables, for ρ = 2, ρ near 2,(4.13) Z | s | <ǫ ′′ log (cid:12)(cid:12)(cid:12)(cid:12) ρ − s s (cid:12)(cid:12)(cid:12)(cid:12) h ( s ) ds = p | ρ − | Z | s | <ǫ ′′ / √ | ρ − | log (cid:12)(cid:12)(cid:12)(cid:12) sgn( ρ −
2) + s s (cid:12)(cid:12)(cid:12)(cid:12) h ( s p | ρ − | ) ds. Since log [ s − | sgn( ρ −
2) + s | ] = O ( s − ) when | s | → ∞ , the integral on the right hand sideof (4.13) is bounded independently of ρ near 2, hence(4.14) Z ω ∈ S d − log | ρ + 2 θ · ω | dσ ω − Z ω ∈ S d − log | θ · ω | dσ ω = O ( p | ρ − | ) . But then
Z Z ω,θ ∈ S d − Z ∞ log | ρ + 2 θ · ω | ∂∂ρ [ ρ d − ψ (( ρ − /δ ) φ ( ρθ )] dρdσ θ dσ ω = Z Z ω,θ ∈ S d − Z ∞ [log | ρ + 2 θ · ω | − log | θ · ω | + log | θ · ω | ] × ∂∂ρ [ ρ d − ψ (( ρ − /δ ) φ ( ρθ )] dρdσ θ dσ ω = Z Z ω,θ ∈ S d − Z ∞ [log | ρ + 2 θ · ω | − log | θ · ω | ] ∂∂ρ [ ρ d − ψ (( ρ − /δ ) φ ( ρθ )] dρdσ θ dσ ω (4.15)since Z Z ω,θ ∈ S d − Z ∞ log | θ · ω | ∂∂ρ [ ρ d − ψ (( ρ + 2) /δ ) φ ( ρθ )] dρdσ θ dσ ω = 0by the support properties of ψ and using that log | θ · ω | is independent of ρ . Finally,using (4.14) shows that the final integrand in (4.15) is O ( δ − / ) in L ∞ , O ( δ / ) in L . Hencethe limit as δ ↓ (cid:3) In practice to understand (4.8) we shall want to evaluate the ω integrals first, and inter-change the order of the limit and the integral over η ∈ R d . We shall use the following twolemmas to rigorously justify this and to understand the limit as ǫ ↓ ω integral. Lemma 4.7.
Let h ∈ C ∞ c (( − , R ) . Then there is a C > so that for < ǫ < , t > , (4.16) (cid:12)(cid:12)(cid:12)(cid:12) Re Z t + 2 ts − iǫ h ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C t ( t + 1) (1 + | log | ǫ/ ( t − t ) || ) . Moreover, if t > , then setting g ( t ) = lim ǫ ↓ Re Z t + 2 ts − iǫ h ( s ) ds the function g is continuous function on (0 , ∞ ) and satisfies | g ( t ) | ≤ C/ ( t ( t + 1)) .Proof. If t ≥ δ > < t < z to denote the principal branch of the logarithm. Then for ǫ > Z t + 2 ts − iǫ h ( s ) ds = Re 1 t Z t + 2 s − iǫ/t h ( s ) ds = Re 1 t Z u − iǫ/t h ( u − t/ du = − Re 12 t Z log(2 u − iǫ/t ) h ′ ( u − t/ du (4.17)with the change of variable u = s + t/
2. The estimate (4.16) follows from this and thesupport and smoothness properties of h and integrability of log(2 u − iǫ/t ) and log | u | . From ESONANT RIGIDITY IN EVEN DIMENSIONS 15 (4.17) we can see that the limit as ǫ ↓ g ( t ) = − t Z log | u | h ′ ( u − t/ du, and is continuous for positive t . Moreover, for 0 < t < , | g ( t ) | ≤ C/t . (cid:3) This next lemma proves a similar result. Notice a difference between this lemma andthe previous one comes from the denominator in the integrand having a stationary point at s = 0. Lemma 4.8.
Let ˜ h ∈ C ∞ c (( − / , / R ) . Then there is a C > so that for < ǫ < and t > (cid:12)(cid:12)(cid:12)(cid:12) Re Z t − t √ − s − iǫ ˜ h ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct | t − | / (cid:0) (cid:12)(cid:12) log( ǫ/ | t − t | ) (cid:12)(cid:12)(cid:1) . Moreover, if t > , t = 2 , then setting g ( t ) = lim ǫ ↓ Re Z t − t √ − s − iǫ ˜ h ( s ) ds the function g is continuous function on (0 , ∪ (2 , ∞ ) and satisfies | g ( t ) | ≤ C/ ( t | t − | / ) .Proof. The lemma is immediate if t ≥ δ >
2, so we shall assume 0 < t < t = 2.By the Morse Lemma, there is a smooth function u = u ( s ) defined for | s | ≤ / √ − s = 1 − u , and s can be written as a smooth function of u . Then(4.19) Z t − t √ − s − iǫ ˜ h ( s ) ds = Z t − t (1 − u ) − iǫ h ( u ) du where h ( u ) = ˜ h ( s ( u )) dsdu . The integral in (4.19) can be rewritten Z t − t (1 − u ) − iǫ h ( u ) du = 12 t | t − | Z sgn( t −
2) + u | t − | − i ǫ t | t − | h ( u ) du = 12 t | t − | / Z sgn( t −
2) + u − i ǫ t | t − | h ( u | t − | / ) du. Let α = α ( ǫ, t ) be such that α = − sgn( t −
2) + i ǫ t | t − | . Then, with log z denoting theprincipal branch of the logarithm defined on C \ ( −∞ , t | t − | / Z sgn( t −
2) + u − i ǫ t | t − | h ( u | t − | / ) du = Re 14 tα | t − | / Z (cid:18) u − α − u + α (cid:19) h ( u | t − | / ) du = Re − tα Z (log( u − α ) − log( u + α )) h ′ ( u | t − | / ) du. Now we use that | log( u ± α ) | ≤ ( | log | u ± α || + π/
2) and the support properties of h toobtain (4.18). This also shows the limit as ǫ ↓ < | t − | < < t <
2, then when ǫ = 0, α = 1 / √
2, and g ( t ) = −√ t Z log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ u − √ u + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ′ ( u | t − | / ) du. This integral yields a continuous function of t for t ∈ (0 ,
2) which is bounded as claimed. If2 < t <
4, then when ǫ = 0, α = i/ √
2, and g is given by g ( t ) = √ t Im Z (log( u − i/ √ − log( u + i/ √ h ′ ( u | t − | / ) du. Since | Im(log( u − i/ √ − log( u + i/ √ | ≤ π , this integral gives a continuous function of t ∈ (2 ,
4) which is bounded as claimed. (cid:3)
The previous two lemmas help to prove the next result.
Lemma 4.9.
There is a
C > so that for | η | 6 = 0 , , < ǫ < , (4.20) (cid:12)(cid:12)(cid:12)(cid:12) Re Z ω ∈ S d − | η + ω | − − iǫ dσ ω (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + | log( ǫ/ ( | η | − | η | ) | ) | η ||| η | − | / . Moreover, lim ǫ ↓ Re Z ω ∈ S d − | η + ω | − − iǫ dσ ω = g ( | η | ) where g ( t ) is continuous for t ∈ (0 , ∪ (2 , ∞ ) . Moreover, t p | t − | g ( t ) is bounded for t ≤ ,and there is a constant C so that | g ( t ) | ≤ C/t when t ≥ .Proof. When ǫ >
0, by a change of variable of integration (a rotation) one can see that Z ω ∈ S d − | η + ω | − − iǫ dσ ω = Z ω ∈ S d − | η | + 2 ω · η − iǫ dσ ω depends on η only through | η | . Hence, without loss of generality, to evaluate the integralaway from η = 0 we may assume η/ | η | = (1 , , ..., t in place of | η | , we wishto bound Re Z ω ∈ S d − t + 2 ω t − iǫ dσ ω , where ω = ( ω , ..., ω d ). It immediate that the integral is smooth when t > t ) as claimed, even when ǫ = 0. Hence below we may assume 0 < t ≤ S d − , depending only on ω , so that 1 = χ + ( ω ) + χ − ( ω ) + χ m ( ω ), χ ± are supported near ω = ± χ m is 0 in a neighborhood of ω = ±
1, and allthree functions are smooth and real-valued. We shall want χ − supported close to ω = − δ , with δ > χ m ( ω ) = 0 if | ω ± | < δ . Since t > (cid:12)(cid:12)(cid:12)(cid:12) t + 2 ω t − iǫ χ + ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct .
ESONANT RIGIDITY IN EVEN DIMENSIONS 17
Hence the integral over the support of χ + gives a contribution to (4.20) and to g whichbehaves as claimed.On the support of χ m we can use ω as a coordinate and can write, with t > δ > Z ω ∈ S d − χ m ( ω ) t + 2 ω t − iǫ dσ ω = c ′ d Re Z − δ − δ χ m ( ω ) t + 2 tω − iǫ (1 − ω ) ( d − / dω (4.21)where c ′ d is a positive constant. Here we have integrated over a space of dimension d −
2, thelevel surfaces of ω in S d − . Now applying Lemma 4.7 we see the integral over the supportof χ m gives a contribution to (4.20) and to g which behaves as claimed.Now consider the contribution from ω near −
1. Writing ω = ( ω , ω ′ ) and taking s = | ω ′ | ,Re Z ω ∈ S d − χ − ( ω ) t + 2 ω t − iǫ χ − ( ω ) dσ ω = ˜ c d Re Z δ − δ χ − ( −√ − s ) t − t √ − s − iǫ s d − √ − s ds for some constant ˜ c d . Here again we have integrated over the level surfaces of ω , a space ofdimension d −
2. Applying Lemma 4.8 completes the proof. (cid:3)
Lemma 4.10.
Let V ∈ L ∞ c ( R d ; R ) , and B be as defined in (4.6). Then for λ ∈ (0 , ∞ ) , ddλ Im tr B ( λ ) = 12 (2 π ) − d +1 λ d − ddλ Z | ˆ V ( λη ) | g ( | η | ) dη + O ( λ d − ) as λ → ∞ . Here g ( | η | ) is the function defined in Lemma 4.9.Proof. Using (4.8) ddλ
Im tr B ( λ ) = 2( d − λ − Im tr B ( λ )+ λ d − π ) d − ddλ lim ǫ ↓ Re Z ω ∈ S d − Z | ˆ V ( λη ) | | η + ω | − − iǫ dηdσ ω Using the estimate (4.7), the first term on the right above is O ( λ d − ).Since V ∈ L ∞ ( R d ), then ˆ V , ∂∂η j ˆ V ( η ) ∈ C ∞ ( R d ) ∩ L ( R d ), and we can change the order ofthe limit and the integral, getting, by Lemmas 4.6 and 4.9 ddλ lim ǫ ↓ Re Z ω ∈ S d − Z | ˆ V ( λη ) | | η + ω | − − iǫ dηdσ ω = ddλ Z | ˆ V ( λη ) | g ( | η | ) dη where g ( | η | ) is the function defined in Lemma 4.9. (cid:3) We may now prove Lemma 4.5.
Proof.
Recall that ˆ V ∈ L ( R d ) is smooth. Using Lemma 4.10 we must estimate(4.22) ddλ Z | ˆ V ( λη ) | g ( | η | ) dη = d X j =1 Z η j h ˆ V j ( λη ) ˆ V ( λη ) + ˆ V ( λη ) ˆ V j ( λη ) i g ( | η | ) dη where ˆ V j ( η ) = ∂∂η j ˆ V ( η ) . We have used the properties of ˆ V , ddη j ˆ V , and g to justify the interchange the order ofdifferentiation and integration.We take just one term on the right in (4.22), as the others are handled in exactly the sameway. Introducing polar coordinates(4.23) Z η j ˆ V j ( λη ) ˆ V ( λη ) g ( | η | ) dη = Z ∞ Z θ ∈ S d − ρθ j ˆ V j ( λρθ ) ˆ V ( λρθ ) g ( ρ ) ρ d − dσ θ dρ. We will write this integral as the sum of three integrals, depending on the size of ρ : 0 ≤ ρ ≤ − λ − , ρ ≥ λ − , and | − ρ | ≤ λ − . Recalling that ρ | ρ − | / g ( ρ ) is bounded when ρ ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z − λ − Z θ ∈ S d − θ j ρ ˆ V j ( λρθ ) ˆ V ( λρθ ) g ( | ρ | ) ρ d − dσ θ dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cλ / Z − λ − Z θ ∈ S d − | ˆ V j ( λρθ ) ˆ V ( λρθ ) | ρ d − dσ θ dρ. Integrating over all of ρ ∈ (0 , ∞ ) and doing a change of variables gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z − λ − Z θ ∈ S d − ˆ V j ( λρθ ) ˆ V ( λρθ ) θ j ρg ( | ρ | ) ρ d − dσ θ dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cλ / − d Z ∞ Z θ ∈ S d − (cid:12)(cid:12)(cid:12) ˆ V j ( ρθ ) ˆ V ( ρθ ) (cid:12)(cid:12)(cid:12) ρ d − dσ θ dρ ≤ Cλ / − d k V kk x j V k . (4.24)Similarly, when ρ > | g ( ρ ) | ≤ C | ρ − | − / ρ − . Thus (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ λ − Z θ ∈ S d − ˆ V j ( λρθ ) ˆ V ( λρθ ) θ j ρg ( | ρ | ) ρ d − dσ θ dρ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cλ / Z ∞ λ − Z θ ∈ S d − (cid:12)(cid:12)(cid:12) ˆ V j ( λρθ ) ˆ V ( λρθ ) (cid:12)(cid:12)(cid:12) ρ d − dσ θ dρ ≤ Cλ / Z ∞ Z θ ∈ S d − (cid:12)(cid:12)(cid:12) ˆ V j ( λρθ ) ˆ V ( λρθ ) (cid:12)(cid:12)(cid:12) ρ d − dσ θ dρ ≤ λ / − d Z R d (cid:12)(cid:12)(cid:12) ˆ V j ( η ) ˆ V ( η ) (cid:12)(cid:12)(cid:12) dη ≤ Cλ / − d k V kk x j V k . (4.25)Now we consider the region with 2 − λ − ≤ ρ ≤ λ − . Here we use that for compactlysupported V Z θ ∈ S d − (cid:12)(cid:12)(cid:12) ˆ V ( λθ ) (cid:12)(cid:12)(cid:12) dS θ ≤ Cλ − ( d − − k V k ESONANT RIGIDITY IN EVEN DIMENSIONS 19 where the constant C depends on the support of V ; this is essentially our bound on λ − ( d − / Γ ( λ ) χ applied to V , see (4.3). Hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z λ − − λ − Z θ ∈ S d − ˆ V j ( λρθ ) ˆ V ( λρθ ) θ j ρg ( | ρ | ) ρ d − dσ θ dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z λ − − λ − Z θ ∈ S d − (cid:12)(cid:12)(cid:12) ˆ V j ( λρθ ) ˆ V ( λρθ ) (cid:12)(cid:12)(cid:12) | ρ − | − / ρ d − dσ θ dρ ≤ C k V k L k x j V k L λ − ( d − − Z λ − − λ − | ρ − | − / dρ ≤ C k V k| L k x j V k| L λ − d +1 − / = O ( λ − d +1 / ) . (4.26)Using (4.24), (4.25), and (4.26) shows that (4.22) is of order O ( λ − d +1 / ) as λ → ∞ . Hence,by Lemma 4.10, Im tr B ( λ ) = O ( λ d − / ), finishing the proof of the lemma. (cid:3) The determinant of the scattering matrix near λ → ∞ , λ ∈ (0 , ∞ ). We next consider thebehavior of the same quantity, but for λ near 0. Lemma 5.1.
Let d ≥ be even. Let V ∈ L ∞ c ( R d ; R ) , with S denoting the scattering matrixof − ∆ + V . Let P denote projection onto the L null space of − ∆ + V , with P = 0 if the L null space is empty. If d = 4 , assume Hypothesis 2.1 holds. Then ddλ det S ( λ )det S ( λ ) = − πi ( d − π ) d vol( S d − ) λ d − kP V k + O ( λ d − | log λ | ) as λ → , arg λ = 0 . If d = 4 but without assuming Hypothesis 2.1, (5.1) ddλ det S ( λ )det S ( λ ) = O ( λ − | log λ | − ) , as λ → , arg λ = 0 . Before proving the lemma, we make several comments.We note that (5.1) is proved in [34]. We include an outline of the proof of (5.1) here forcompleteness and for the convenience of the reader.Note that the term hP V, V i = kP V k may be nonzero (this nonzero contribution hasgone unnoticed in some places). Suppose φ ∈ L ( R d ), ( − ∆+ V ) φ = 0. Then for R sufficientlylarge, h V, φ i = Z V ( x ) φ ( x ) dx = Z | x | First we suppose that either d = 4 or, if d = 4, Hypothesis 2.1 holds. By results of[24, 25], near 0, with 0 ≤ arg λ ≤ π , the resolvent R V ( λ ) satisfies(5.2) R V ( λ ) = − λ − P + log λ B , + B , + O ( | λ | | log λ | ) , for some operators B j,k which are bounded from L c ( R d ) to L ( R d ). Moreover, the operator ddλ R V ( λ ) also has an expansion near λ = 0, and the expansion can be found by formallydifferentiating (5.2), with error term for the derivative O (1). We note that by results of[32], for any M > R V ( λ ) has an expansion in powers of λ and log λ that is valid for | arg λ | < M .We shall use the expression for the scattering matrix (4.1). Writing Γ ( λ, ω, x ) for theSchwartz kernel of Γ ( λ ), near λ = 0(5.3) Γ ( λ, ω, x ) V ( x ) = 2 − / (2 π ) − d/ λ ( d − / (1 − iλx · ω + O ( | λ | )) V ( x )with O ( | λ | ) error uniform in x and ω , since V has compact support. Thus, using in additionthat S d − is compact, the same error holds for the corresponding operators using the L → L norm or the Hilbert-Schmidt norm. Then(5.4)(Γ ( λ ) V R V ( λ ) V Γ ∗ ( λ )) ( ω, θ ) = 12(2 π ) d λ d − (cid:20) − hP V, V i λ + i f ( ω ) λ − i f ( θ ) λ + O ( | log λ | ) (cid:21) near λ = 0. Here f ( θ ) = Z V ( x ) x · θ ( P V )( x ) dx. The error O ( | log λ || λ | d − ) here holds for the corresponding operators in the L → L or thetrace norm. Moreover, formally differentiating (5.4) results in an expansion for the derivativeof the left hand side near 0, with the resulting error bounded by O ( | λ | d − ).We use ddλ det S ( λ )det S ( λ ) = tr( S ∗ ( λ ) ˙ S ( λ )) . (5.5)Using the expression (4.1) and the fact that P V = 0 if d = 4, we see thattr( S ∗ ( λ ) ˙ S ( λ )) = tr (cid:20) πi ddλ (Γ ( λ ) V R V ( λ ) V Γ ∗ ( λ )) (cid:21) + O ( | λ | d − ) ESONANT RIGIDITY IN EVEN DIMENSIONS 21 near λ = 0 with arg λ near 0. Using (5.4) and the fact that here the trace is given by theintegral of the Schwartz kernel over the diagonal ω = θ , this meanstr( S ∗ ( λ ) ˙ S ( λ ))= πi (2 π ) d Z S d − (cid:2) − ( d − λ d − hP V, V i + i ( d − λ d − ( f ( θ ) − f ( θ )) + O ( | λ | d − | log λ | ) (cid:3) dσ θ = − πi (2 π ) d ( d − 4) Vol( S d − ) λ d − kP V k + O ( | λ | d − | log λ | ) . We outline how to modify the proof if d = 4 without the assumption of Hypothesis 2.1.In this case, by [25] near λ = 0, 0 ≤ arg λ < π , R V ( λ ) = − λ − P + λ − ( a − λ ) − B − , − + log λB , + B , + O (1 / | log λ | ) , where a is a constant depending on V and the operators B j,k are bounded from L c ( R d )to L ( R d ). This expansion can also be differentiated, with resulting error O (1 / | λ log λ | ).Using again that P V = 0 in dimension d = 4, we get(Γ ( λ ) V R V ( λ ) V Γ ∗ ( λ )) ( ω, θ ) = b ( a − λ ) + O ( | λ | )for some constant b . Continuing as in the previous case, we prove (5.1). (cid:3) Writing F in terms of canonical products We turn now more directly to the proof of Theorem 1.4. As a next step, we write F , andthen F ′ /F , using canonical products. The proof uses many of the same components as theproofs of Propositions 2.1 and 2.2 of [34]. In particular, the proof of the next proposition isvery similar to that of [34, Proposition 2.1], which itself uses techniques as in, for example,[46]. We use some of the notation of [34] to highlight the similarities.For z ∈ C set E ( z ) = (1 − z ) and E p ( z ) = (1 − z ) exp p X k =1 z k k ! for p ∈ N . Central to the proof is the observation that while the scattering matrix S j ( λ ) for − ∆ + V j is a meromorphic function of λ ∈ Λ, S j ( e z ) is a meromorphic function of z ∈ C .For the next proposition, we use much of the notation of Theorem 1.4. Note, however,that we do not need assumption (1.2), nor do we need any additional hypotheses for the d = 4 case. Proposition 6.1. Let d be even, and V j ∈ L ∞ c ( R d ; R ) for j = 1 , . Set P j = − ∆ + V j andlet S j ( λ ) be the associated scattering matrix, unitary for λ > . Set F ( z ) = det S ( e z )det S ( e z ) . Let { z l } denote the distinct poles of F ( z ) , and M ( z l ) their multiplicities. Suppose that thereis an m ∈ (0 , ∞ ) such that (6.1) X M ( z l ) | z l | m < ∞ . Let m ∈ N ∪ { } be the smallest nonnegative integer satisfying m + 1 ≥ m , and set P ( z, m ) = Y ( E m ( z/z l )) M ( z l ) , Q ( z, m ) = Y ( E m ( z/z l )) M ( z l ) . Then F ( z ) = e g ( z ) P ( z, m ) Q ( z, m ) where g is an entire function satisfying | g ( z ) | ≤ C exp( C | z | ) , some C > .Proof. Since det S j ( e z ) det S ∗ j ( e z ) = 1, the set { z l } is the set of zeros of F ( z ). Hence F ( z ) Q ( z, m ) /P ( z, m ) is an entire nowhere zero function, so the only thing to prove is thebound on | g ( z ) | .An intermediate result of the proof of [34, Proposition 2.1] is that for every R > ρ j = ρ j ( R ) ∈ ( R/ , R ) so that(6.2) | det S j ( e z ) | ≤ C exp(exp( C | z | )) , when | z | = ρ j with constant C independent of R . For our application we will need to know that we canchoose ρ = ρ ∈ ( R/ , R ) so that (6.2) holds, and to understand this we explain the originof the ρ j . The need to choose ρ = ρ is the main point of divergence from the proof of [34,Propostion 2.1], and we outline enough of the proof to show how to make the modificationsnecessary.Let χ ∈ C ∞ c ( R d ) satisfy χV j = V j for j = 1 , 2. Then we can write S j ( λ ) = I + A j ( λ )where for a nonzero constant c ′ d A j ( λ ) = c ′ d E χ ( e iπ λ )( I + V j R ( λ ) χ ) − V j ( E χ ( λ )) t and E χ ( e iπ λ ) = Γ ( λ ) χ. For a bounded linear operator B , let µ ( B ) ≥ µ ( B ) ≥ ... denote the characteristic values of B . Then | det S j ( λ ) | ≤ ∞ Y l =1 (1 + µ l ( A j ( λ )))and(6.3) µ l ( A j ( λ )) ≤ | c ′ d | µ l ( E χ ( e iπλ λ )) k ( I + V j R ( λ ) χ ) − kk V j k L ∞ k (( E χ ( λ )) t k . Only two terms involve V j , and for j = 1 , k V j k L ∞ ≤ C for some C . Thus to prove (6.2)we need to find regions where we can bound k ( I + V j R ( λ ) χ ) − k independently of j . ESONANT RIGIDITY IN EVEN DIMENSIONS 23 As in [46] (and just as in [34]), we use [19, Theorem V.5.1],(6.4) k ( I + V j R ( e z ) χ ) − k ≤ (1 + k V j R ( e z ) χ k d/ ) det( I + | V j R ( e z ) χ | d/ ) | det( I + ( V j R ( e z ) χ ) d/ ) | . We have, by choosing the constant C to be the larger of the corresponding constants for j = 1 , (cid:12)(cid:12) det( I + ( V j R ( e z ) χ ) d/ ) (cid:12)(cid:12) ≤ det( I + | V j R ( e z ) χ | d/ ) ≤ exp( C exp[( d + 1) | z | ])by [23, Proposition 2.1].Let f be an analytic function in the disc { z ∈ C : | z | ≤ eR } , with f (0) = 1. Then byCartan’s estimate (e.g. [31, Theorem I.11]), if 0 < η < e/ 2, then outside a family of discsthe sum of whose radii does not exceed 4 ηR ,log | f ( z ) | > − (2 + log 3 e η ) log M f (2 eR ) , where M f ( s ) = max {| f ( z ) | : | z | ≤ s } . Now we apply Cartan’s estimate to f j ( z ) = det( I + ( V j R ( e z ) χ ) d/ ) , choosing η = 1 / ρ = ρ ∈ ( R/ , R ) so that for a constant C independent of R , we have(6.6) (cid:12)(cid:12) det( I + ( V j R ( e z ) χ ) d/ ) (cid:12)(cid:12) ≥ exp( − C exp( C | z | )) , for j = 1 , , if | z | = ρ = ρ . Otherwise, the sum of the sums of the radii of the exceptional circles for f and for f wouldexceed ( R − R/ / R/ 4. But if η = 1 / 40, then 2(4 ηR ) = R/ ρ = ρ ∈ ( R/ , R ). It follows from the identity S j ( λ ) S ∗ j ( λ ) = I thatdet( S j ( e z ))det( S j ( e z )) = 1 . Hence (6.2) gives the same bound on the reciprocal of det( S j ( e z )) on the circle | z | = ρ ( R ),and(6.7) (cid:12)(cid:12)(cid:12)(cid:12) det S ( e z )det S ( e z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp(exp C | z | )) , if | z | = ρ = ρ . The bounds on canonical products (e.g. [31, Section I.4]) mean that for δ > | Q ( z, m ) | ≤ C exp( C | z | m + δ ) , | P ( z, m ) | ≤ C exp( C | z | m + δ )with constants depending on δ . Applying this together with (6.7) and using that we havechosen ρ = ρ ∈ ( R/ , R ) gives | F ( z ) Q ( z, m ) | ≤ C exp(exp( C | z | ))) , if | z | = ρ . But since F ( z ) Q ( z, m ) is entire and we can find such a ρ = ρ ∈ ( R/ , R ) for each R > 1, bythe maximum principle we get | F ( z ) Q ( z, m ) | ≤ C exp(exp( C | z | )). Now Cartan’s estimateapplied to the function P ( z, m ) shows that if δ > R > ρ ′ = ρ ′ ( R ) ∈ ( R/ , R ) so that | P ( z, m ) | ≥ C exp( − C ′ | z | m + δ ) when | z | = ρ ′ , with constants C, C ′ independent of R . Thus | exp( g ( z )) | ≤ C exp( C exp | z | )) , if | z | = ρ ′ . Again using the maximum principle, the fact that g is entire, and our ability to find such a ρ ′ for each R > | exp( g ( z )) | ≤ C exp(exp C | z | )) , z ∈ C . This implies that Re g ( z ) ≤ C exp( C | z | ) . By Carath´eodory’s theorem (e.g. [31, Theorem I.8]), | g ( z ) | ≤ C exp( C | z | ) . (cid:3) Note that the hypotheses of Theorem 1.4 ensure that P M ( z k ) > | z k | − (1 − ǫ / < ∞ . Lemma 6.2. Under the hypotheses of Theorem 1.4, set P ( z ) = Q M ( z k ) > E ( z/z k ) and Q ( z ) = Q M ( z k ) > E ( z/z k ) . Then there is a constant C so that (cid:12)(cid:12)(cid:12)(cid:12) F ′ ( z ) F ( z ) P ( z ) Q ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp( C | z | ) . Proof. We use the functions P , Q , and g from Proposition 6.1, noting our assumption onthe poles of F in Theorem 1.4 includes that the sum (6.1) converges for some finite value of m .By Proposition 6.1,(6.9) F ′ ( z ) F ( z ) = g ′ ( z ) + P ′ ( z, m ) P ( z, m ) − Q ′ ( z, m ) Q ( z, m ) . Hence(6.10) F ′ ( z ) F ( z ) P ( z ) Q ( z ) = g ′ ( z ) P ( z ) Q ( z ) + P ′ ( z, m ) P ( z, m ) P ( z ) Q ( z ) − Q ′ ( z, m ) Q ( z, m ) P ( z ) Q ( z ) . Note that P ′ /P and Q ′ /Q have simple poles which coincide with the zeros of P and Q ,respectively. Hence F ′ P Q /F is entire. Since | g ( z ) | ≤ C exp( C | z | ), by Cauchy’s estimatethe same inequality holds for g ′ ( z ), with perhaps a new constant C . Moreover, P ( z ), Q ( z )satisfy (6.8) with m = 1 − ǫ / 2. Hence it is easy to see that the first term in (6.10) is boundedas claimed.Consider P ′ ( z, m ) P ( z, m ) P ( z ) = − X k M ( z k ) z k ( z/z k ) m − z/z k ! Y l (1 − z/z l ) ! = − X k M ( z k ) z m z km +1 Y l : l = k (1 − z/z l ) ! . ESONANT RIGIDITY IN EVEN DIMENSIONS 25 Using the bounds on canonical products and the assumption on { z l } we can bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y l : l = k (1 − z/z l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp( C | z | − ǫ / )with a constant C chosen independent of k . But X k (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) M ( z k ) z m z km +1 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = X k M ( z k ) | z k | m +1 ! | z | m ≤ C | z | m for some constant C using our assumptions on the convergence of P M ( z k ) / | z k | m . Thus thesecond term is bounded as desired. The third term in (6.10) is bounded in exactly the sameway as the second. (cid:3) Proof of Theorem 1.4 The next proposition proves Theorem 1.4 when d = 2, and is an important step in the proofof Theorem 1.4 for d ≥ 4. Here we use the notation δ j,k for the Kronecker delta function. Inthe statement of the proposition, we understand λ d − , λ d − ∈ C . More carefully this mightbe denoted ( p ( λ )) d − , ( p ( λ )) d − , where p : Λ → C is the natural projection, where points ofΛ with argument differing by an integral multiple of 2 π are identified. Proposition 7.1. Under the assumptions and using the notation of Theorem 1.4, F ( z ) isan entire function. Moreover, if d ≥ S ( λ ) = exp (cid:0) − ic d (1 − δ d, ) α λ d − − ic d α λ d − (cid:1) det S ( λ ) where c d = π (2 π ) − d vol( S d − ) . Here α = kP , V k − kP , V k α = Z ( V ( x ) − V ( x )) dx, with P ,j denoting projection onto the L null space of P j . If d = 2 , det S ( z ) = det S ( z ) .Proof. We first prove the proposition assuming either that d = 4 or d = 4 and Hypothesis2.1 holds.Consider the function defined by(7.2) G ( z ) = e − ( d − / z (cid:18) F ′ ( z ) F ( z ) + i ( d − − δ d, ) c d α e ( d − z + i ( d − c d α e ( d − z (cid:19) × Y M ( z l ) > (1 − z/z l )(1 − z/z l ) . To motivate our definition of G , notice that by Lemma 5.1 the first term after F ′ ( z ) /F ( z ),when evaluated at z = x ∈ R , is (up to sign) the leading term of F ′ ( x ) /F ( x ) when x → −∞ .By Theorem 4.1 the next term corresponds to the leading term when x → ∞ . Note that when d = 2, the coefficients of both α and α are 0, and when d = 4 the coefficient of α is0. The multiplication by Q (1 − z/z l )(1 − z/z l ) ensures the function G is analytic.By our assumptions on { z l } and estimates for canonical products (e.g. [31, Theorem 1.7]),there is a constant C so that(7.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y M ( z l ) > (1 − z/z l )(1 − z/z l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C exp( C | z | − ǫ / ) . Combining this with the result of Lemma 6.2, we find there is a constant C so that(7.4) | G ( z ) | ≤ Ce C | z | . Since G is analytic, G is a function of exponential type (see e.g. [4]).By Theorem 4.1 and (7.3), for x ∈ R , | G ( x ) | = O ( e − x/ C | x | − ǫ / ) as x → ∞ . Next we consider the behavior of G ( x ) when x ∈ R , x → −∞ . If d ≥ 4, by Lemma 5.1then G ( x ) = O ( | x | e x/ C | x | − ǫ / ) as x → −∞ . If d = 2, results of [6] (when R V j ( x ) = 0)or [11] (for R V j = 0), imply that F ′ ( x ) /F ( x ) = O ( x − ) when x → −∞ . Combiningthese asymptotics with (7.3) implies that for x ∈ R , there is a constant C > G ( x ) = O (exp( x/ C | x | − ǫ / )) when x → −∞ .Now noting the bound (7.4) on | G ( z ) | and the fact that G ( z ) decays exponentially on thereal axis, [4, Corollary 5.1.14] shows that G ≡ 0. Hence F ′ ( z ) /F ( z ) ≡ d = 2, and F ′ ( z ) /F ( z ) = − i ( d − c d α e ( d − z − i ( d − c d α e ( d − z if d ≥ . Returning to the λ variable we find, for d ≥ ddλ det S ( λ )det S ( λ ) = ddλ det S ( λ )det S ( λ ) − i ( d − c d α λ d − − ic d ( d − α λ d − . Recalling that lim λ ∈ R + ,λ ↓ det S j ( λ ) = 1 ([6] or [11, Proposition 6.1]) finishes the proof if d = 4 or if d = 4 and Hypothesis 2.1 holds.To complete the proof, we consider the remaining case, which is d = 4 and V , V ∈ C ∞ c ( R ; R ). In this case we define, in analogy with (7.2), G ( z ) = e z (cid:18) F ′ ( z ) F ( z ) + i c α e z (cid:19) Y M ( z l ) > (1 − z/z l )(1 − z/z l ) . Using that V , V ∈ C ∞ c ( R ; R ), by [20, 33],(det S j ( λ )) − ddλ det S j ( λ ) = − ic Z V j ( x ′ ) dx ′ λ + O ( λ − N ) , λ > , λ → ∞ for any N ∈ N . Hence for x > G ( x ) = O ( e (2 − N ) x ) as x → ∞ for any N ∈ N . On theother hand, by Lemma 5.1, if x ∈ R , G ( x ) = O ( x − e x + x − x ) = O ( x − e x ) as x → −∞ . Nowusing that G is an entire function of exponential type decaying exponentially on the realaxis, as before [4, Corollary 5.1.14] shows that G ≡ 0. The remainder of the proof followsas in the first case. (cid:3) ESONANT RIGIDITY IN EVEN DIMENSIONS 27 The function f defined below will appear in the proof of Lemma 7.3. Lemma 7.2. Set f ( t, r ) = 2 t − rt + 2 r t for t ∈ [0 , , r ≥ .Then for fixed r ≥ , f ( t, r ) is an increasing function of t ∈ [0 , , and ≤ f ( t, r ) ≤ / (1 − r + 4 r ) = 4 / (1 − r ) . For fixed t ∈ [0 , , f ( t , r ) is a decreasing function of r ≥ , lim r →∞ f ( t , r ) = 0 and f (0 , r ) = 0 .Proof. These properties are immediate from inspection and elementary calculus. (cid:3) Although we state the following lemma for the determinant of the scattering matrix fora Schr¨odinger operator, it is valid for a much larger class of operators. In fact, if P is anappropriate self-adjoint “black-box” perturbation of − ∆ on R d , d even, then the followinglemma is valid for the scattering matrix of P . See [37] for the definition of the black-boxperturbation. (For the following to hold, though, we need in addition the existence of aninvolution on the underlying Hilbert space which commutes with P and which agrees withcomplex conjugation “at infinity.”) Note that in the following lemma, since ρ > k ∈ N ,the point ρe iπk ∈ Λ projects in the complex plane to the real axis, so that the square of theprojection lies in the continuous spectrum of P . Lemma 7.3. Let d be even, V ∈ L ∞ c ( R d ; R ) and let S ( λ ) denote the scattering matrixassociated to − ∆ + V . Let k ∈ N and fix ρ > . Then lim k →∞ det S ( ρe ikπ ) = 1 .Proof. Since det S ( e ikπ λ ) = det S ( λ ) det S ( λe iπ ) · · · det S ( λe i ( k − π ) det S ( λe ikπ )det S ( λ ) det S ( λe iπ ) · · · det S ( λe i ( k − π )we have using [13, Lemma 3.4] (compare the proof of [13, Proposition 3.5]) thatdet S ( e ikπ λ ) = det(( k + 1) S ( λ ) − kI )det( kS ( λ ) − ( k − I )= det (cid:0) [( kS ( λ ) − ( k − I + S ( λ ) − I ][ kS ( λ ) − ( k − I ] − (cid:1) = det( I + [ S ( λ ) − I ][ I + k ( S ( λ ) − I )] − ) . Now we specialize to λ = ρ ∈ (0 , ∞ ), and recall that S ( ρ ), being unitary, has eigenvalues { e iθ j } , θ j ∈ R . The θ j depend on ρ , but since ρ is fixed, we do not denote this dependence.Then det S ( e ikπ ρ ) = Y j (cid:18) e iθ j − 11 + k ( e iθ j − (cid:19) . For k ≥ | k ( e iθ − | ≥ so that | ( e iθ j − k ( e iθ j − − | ≤ | e iθ j − | . Hence given ǫ > 0, using the trace classproperties of S ( τ ) − I we can find J ∈ N independent of k ∈ N so that(7.6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Y j>J (cid:18) e iθ j − 11 + k ( e iθ j − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ/ . By a straightforward computation (cid:12)(cid:12)(cid:12)(cid:12) e iθ j − 11 + k ( e iθ j − (cid:12)(cid:12)(cid:12)(cid:12) = f (1 − cos θ j , k )where f is the function defined in Lemma 7.2. Thenlim k →∞ (cid:12)(cid:12)(cid:12)(cid:12) e iθ j − 11 + k ( e iθ j − (cid:12)(cid:12)(cid:12)(cid:12) = 0 for all j ≤ J . Since the product over j ≤ J is a finite product, this means lim k →∞ Q j ≤ J (cid:16) e iθj − k ( e iθj − (cid:17) =1, and we can find K ∈ N so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Y j ≤ J (cid:18) e iθ j − 11 + k ( e iθ j − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ǫ/ k ≥ K , k ∈ N . Together with (7.6) this shows that for 1 > ǫ > (cid:12)(cid:12) − det S ( e ikπ ρ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Y j (cid:18) e iθ j − 11 + k ( e iθ j − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ for k ≥ K, k ∈ N . Since ǫ > (cid:3) There does not seem to be anything analogous to Lemma 7.3 in odd dimensions. Thislemma allows us to improve Proposition 7.1 to Theorem 1.4. Proof of Theorem 1.4. Proposition 7.1 has proved most of Theorem 1.4, including the d = 2case. Thus, suppose d ≥ 4. From Proposition 7.1,det S ( λ ) = e iq ( λ ) det S ( λ )where q : Λ → C is given by(7.7) q ( λ ) = − c d (1 − δ d, ) α λ d − − c d α λ d − . It remains to show that e iq ( λ ) ≡ ρ > k ∈ N , q ( ρe iπk ) = q ( ρ ) since d − d − ρ > k ∈ N , det S ( ρe iπk ) = e iq ( ρ ) det S ( ρe iπk ) . Fixing ρ > 0, by Lemma 7.3 lim k →∞ det S ( ρe iπk ) =1 = lim k →∞ det S ( ρe iπk ), so that e iq ( ρ ) = 1. Since this is true for all ρ > 0, we must have e iq ≡ (cid:3) ESONANT RIGIDITY IN EVEN DIMENSIONS 29 Proofs of Theorems 1.5, 1.1, and 1.2 and Corollary 1.6 With Theorem 1.4 in hand, the remainder of the proof of Theorem 1.5 is similar to theproof of [39, Theorem 1.2] and the proof of Theorem 1.1 is similar to the proof of [34,Theorem 1.1]. We include the proofs for the convenience of the reader. Proof of Theorem 1.5. Recall that P j = − ∆ + V j for j = 1 , 2, and set P = − ∆. Let − µ l,j ,l = 1 , ..., L j denote the non-positive eigenvalues of P j , repeated according to multiplicity.For t > 0, we recall (2.2),(8.1) tr( e − tP j − e − tP ) = 12 πi Z ∞ tr (cid:18) S − j ( λ ) ddλ S j ( λ ) (cid:19) e − tλ dλ + L j X l =1 e tµ l,j + β ( V j , d ) . Using that det S ( λ ) = det S ( λ ) by Theorem 1.4, and (8.1)(8.2) tr( e − tP − e − tP ) = L X l =1 e tµ l, − L X l =1 e tµ l, + β ( V , d ) − β ( V , d ) . In particular, by (8.2)(8.3) tr( e − tP − e − tP ) = t − ( d − / f ( t ) with f ( t ) ∈ C ∞ ([0 , ∞ )).Note that the parity of d is important here.By [39, Theorem 4] and our assumption that V ∈ H k , there are constants c , c , ..., c m +1 so thattr( e − tP − e − tP ) = (4 πt ) − d/ ( c t + c t + ... + c k +1 t k +1 + r k +2 ( t ) t k +2 ) when t ↓ | r k +2 ( t ) | ≤ C for 0 ≤ t ≤ 1. But combining this with (8.3) implies that that there areconstants ˜ c , ˜ c , ..., ˜ c k +1 and a function ˜ r k +2 ( t ) so thattr( e − tP − e − tP ) = (4 πt ) − d/ (˜ c t + ˜ c t + ... + ˜ c k +1 t k +1 + ˜ r k +2 ( t ) t k +2 ) when t ↓ | ˜ r l +2 ( t ) | ≤ C for 0 ≤ t ≤ 1. Hence, again by [39, Theorem 4], V ∈ H k ( R d ). (cid:3) A natural question to ask is the following: suppose P and P satisfy the conditions ofTheorem 1.4. Theorem 1.4 implies that det S and det S have exactly the same zeros andpoles on Λ, including multiplicity. It is natural, then, to ask if P and P have the sameresolvent resonances away from 0. While it seems likely that they do, it is possible todescribe a scenario in which the symmetric difference of their (resolvent) resonance sets,where elements are repeated with multiplicity, is infinite. This sort of scenario is the settingof Lemma 3.1, which is used in the proof of Theorem 1.1. Proof of Theorem 1.1. The proof is by contradiction. Suppose that for some nontrivialpotential V and some ǫ > r →∞ N ( r )(log r ) − ǫ < ∞ . If d = 4, suppose in addition that Hypothesis 2.1 holds. Let S denote the scattering matrixof − ∆ + V . Since the poles of det S ( λ ) are a subset of the poles of the resolvent, by Theorem1.4 with V = V and V ≡ S ( λ ) has no poles. Then we can apply Theorem 1.5 with V = V and V ≡ V ∈ H m ( R d ) for all m ∈ N . Hence V ∈ C ∞ c ( R d ; R ).Now we essentially follow [34] to show that there must be infinitely many poles of thecut-off resolvent of − ∆ + V . We fill in a few details omitted in [34]. By our Theorem 1.4(see also the proof of [34, Theorem 1.1]),det S ( λ ) = 1 . As in the proof of Theorem 1.5 we consider the heat trace H ( t ) for t > 0. Using det S ( λ ) ≡ H ( t ) = tr (cid:0) e − t ( − ∆+ V ) − e t ∆ (cid:1) = K X k =1 e tµ k + β ( V, d ) , t > − µ , ..., − µ K are the non-positive eigenvalues of − ∆ + V .It is well-known that(8.5) H ( t ) ∼ ∞ X j =1 C j ( V ) t j − d/ , t ↓ . For us it is important to note that(8.6) C ( V ) = α ,d Z V ( x ) dx with nonzero constant α ,d ; see e.g. [2, 14] and references therein. If d > 4, using that C ( V ) = 0 we have immediately a contradiction between (8.4) and (8.5), showing thatdet S ( λ ) must have, by Theorem 1.4, infinitely many poles. In fact, we must havelim sup N ( r ) / (log r ) − ǫ = ∞ for all ǫ > S ( λ ) are a(perhaps proper) subset of the poles of the resolvent by [13], we prove the theorem when d > 4. Note that in this case we have actually proved a potentially stronger result thanclaimed in the statement of the theorem, as we have proved a lower bound on the countingfunction for the poles of the determinant of the scattering matrix.Next we shall show that if d = 2 or d = 4 and Hypothesis 2.1 holds, det S has no poles,and V 0, then − ∆ + V must have at least one strictly negative eigenvalue. For d = 2,this follows immediately again by comparing the expansions (8.4) and (8.5), and noting that C ( V ), in the d = 2 case the coefficient of t in the expansion of H ( t ) at t = 0, is nonzero.Hence there must be at least one negative eigenvalue.Now we turn to the case of d = 4. In [3], Benguria and Yarur showed that in dimension d = 3, if W ∈ L ∞ ( R ; R ) goes to 0 sufficiently rapidly at infinity, then 0 cannot be aneigenvalue of − ∆ + W if − ∆ + W does not have a negative eigenvalue. With a modificationto their proof, in particular, a change to [3, Theorem 2] using the function g ( r ) = r − , one ESONANT RIGIDITY IN EVEN DIMENSIONS 31 can show the same is true in dimension d = 4, at least if the potential W has compactsupport. Using this and returning to the case at hand, if V ∈ C ∞ c ( R ; R ), V 0, has det S analytic, since C ( V ) = 0 by comparing (8.4) and (8.5) (recalling β ( V, 4) = 0 by assumption) − ∆ + V must have at least one eigenvalue, and hence at least one negative eigenvalue.Now in the d = 2 and d = 4 cases, we have shown that − ∆ + V must have at least onenegative eigenvalue if det S ( λ ) has no poles and V 0. Thus by Corollary 3.2 the cut-offresolvent of − ∆ + V has infinitely many poles. Moreover, the explicit location of the polesshows that lim sup r →∞ N ( r )(log r ) − ǫ = ∞ . (cid:3) Proof of Theorem 1.2. Under the hypotheses of Theorem 1.2, (8.2) holds. However, we havealso assumed that the negative eigenvalues of P and P agree, as do their multiplicities.Hence for t > e − tP − e − tP ) = n ( P ) + β ( V , d ) − n ( P ) − β ( V , d )where n ( P j ) is the dimension of the L null space of P j . However, we have defined n ( P j ) + β ( V j , d ) to be the multiplicity of 0 as a resonance of P j , so that tr( e − tP − e − tP ) = 0.Therefore 0 = tr( e − tP − e − tP ) ∼ ∞ X l =1 ( C l ( V ) − C l ( V )) t l − d/ , as t ↓ . This proves the result immediately. (cid:3) To prove Theorem 1.3, we shall use an intermediary step of considering Schr¨odinger opera-tors on a flat torus M . Given R > 0, we shall define a corresponding flat torus M = M ( R )by identifying opposite sides of { x ∈ R d : max | x j | ≤ R + 1 } . Henceforth we omit the R dependence of M , as we shall hold R fixed. If V ∈ C ∞ c ( R d ; R ) has its support in B (0 , R ), then we can consider V as an element of C ∞ c ( M ; R ) in a natural way. Thinkingof V ∈ C ∞ ( M ; R ) gives a corresponding Schr¨odinger operator P V,M = − ∆ M + V acting on(a domain in) L ( M ), where ∆ M ≤ M . Then it is wellknown that(8.7) tr M e − tP V,M ∼ t − d/ ∞ X l =0 C l,M ( V ) t − l as t ↓ M denotes the trace on L ( M ).Recall we denote the heat coefficients of V on R d by C l ( V ); see (8.5). Lemma 8.1. Suppose V , V ∈ C ∞ c ( R d ; R ) are supported in B (0 , R ) . Then C l ( V ) = C l ( V ) for all l ∈ N if and only if C l,M ( V ) = C l,M ( V ) for all l ∈ N .Proof. Let us denote by P = − ∆ + V and P = − ∆ + V the Schr¨odinger operators on R d ,and by P ,M = − ∆ M + V and P ,M = − ∆ M + V the Schr¨odinger operators on M .Note that C l ( V ) = C l ( V ) for all l ∈ N if and only if tr( e − tP − e − tP ) = O ( t N ) for all N as t ↓ ∞ . Let χ ∈ C ∞ c ( R d ) have its support in B (0 , R + 1 / 2) and be equal to 1 on B (0 , R + 1 / e − tP − e − tP ) − tr( χe − tP χ − χe − tP χ ) = O ( t N ) as t ↓ N ∈ N . Note thattr( χe − tP χ − χe − tP χ ) = tr M ( χe − tP χ − χe − tP χ )since χ is supported in B (0 , R + 1 / M is equipped with the flat metric. But thenusing [38, Lemma 1.5], we see thattr M ( χe − tP χ − χe − tP χ ) − tr M ( χe − tP ,M χ − χe − tP ,M χ ) = O ( t N ) as t ↓ N ∈ N , and by a second application of [38, Lemma 1.5] thattr M ( χe − tP ,M χ − χe − tP ,M χ ) − tr M ( e − tP ,M − e − tP ,M ) = O ( t N ) as t ↓ N ∈ N . But tr M ( e − tP ,M − e − tP ,M ) = O ( t N ) as t ↓ N ∈ N if and only if C l,M ( V ) = C l,M ( V ) for all l ∈ N . (cid:3) We shall need another lemma for our proof of Theorem 1.3. Lemma 8.2. For n ∈ N , let V n ∈ L ∞ c ( R d ; R ) , with supp V n ⊂ B ( R , , and suppose µ R Vn ( λ ) = µ R Vm ( λ ) for all λ ∈ Λ and all n, m ∈ N . Then if V n → V ∗ in L ∞ ( R ) ,then µ R Vn ( λ ) = µ R V ∗ ( λ ) for all λ ∈ Λ .Proof. This proof is the same as in the odd-dimensional case. We recall a proof as in [21]for the convenience of the reader.Since V n → V ∗ , supp V ⊂ B ( R , χ ∈ C ∞ c ( R d ) be 1 on B ( R , λ ∈ Λ, it iswell known (cf. [16, Section 3.4]) that(8.8) µ R V ∗ ( λ ) = m sc (det p ( I + V ∗ R ( λ ) χ ); λ ) , where p ∈ N , p > d/ p is the regularized determinant defined for operatorsof the type I + B , with B in the p -Schatten class.Fix p > d/ p ∈ N , and set h n ( λ ) = det p ( I + V n R ( λ ) χ ) and h ∗ ( λ ) = det p ( I + V ∗ R ( λ ) χ ).Then h n → h ∗ , uniformly on compact sets of Λ. Now we locally identify a neighborhoodof λ ∈ Λ with an open set in the complex plane. Given λ in Λ, choose a small circle γ λ in Λ so that no zeros of h ∗ or of h n lie on γ λ . We may in addition ensure that there areno zeros of h n inside γ λ , except, possibly, at λ . This is possible since the zeros of both h ∗ and h n are isolated, and the zeros of h n are independent of n . Using Hurwitz’s Theoremthere is an N ∈ N so that if n > N , h n and h ∗ have the same number of zeros, counted withmultiplicity, inside γ λ . By choosing γ λ appropriately, this shows that if m sc ( h n ; λ ) = 0 , then m sc ( h ∗ ; λ ) = 0 . Applying this again for other values of λ shows that in generalm sc ( h n ; λ ) = m sc ( h ∗ ; λ ). (cid:3) ESONANT RIGIDITY IN EVEN DIMENSIONS 33 Proof of Theorem 1.3. By Theorem 1.2, if V ∈ Iso ( V , R ), then C l ( V ) = C l ( V ) for all l ∈ N . By Lemma 8.1, C l,M ( V ) = C l,M ( V ) for all l ∈ N .We use results of Br¨uning [7] and Donnelly [15]. Although the results of [7, 15] are stated asresults for isospectral Schr¨odinger operators on compact manifolds, a careful reading showsthat the proofs of the results therein use the isospectrality of the Schr¨odinger operators onlyto show that the operators have the same heat coefficients, and not any additional propertiesof isospectral Schr¨odinger operators. Suppose { V n } ⊂ Iso ( V , R ) if d = 2 (respectively, { V n } ⊂ Iso ( V , R , s, c ) if d ≥ V n have the same heat coefficients on M , { V n } has a subsequence which converges in C ∞ ( M ) to a function V ∗ , necessarily in C ∞ ( M ). By selecting a subsequence and relabelingif necessary, we can assume V n → V ∗ . Because V n has its support in B ( R , 0) for each n , sodoes V ∗ . Since the heat coefficients C l ( V ) are continuous functions of V and C l ( V n ) = C l ( V )for each l, n ∈ N , C l ( V ∗ ) = C l ( V ) for each l ∈ N . If d ≥ k V ∗ k H s ≤ c , since this holds foreach V n . It remains only to show that − ∆ + V ∗ is isoresonant with − ∆ + V .By Lemma 8.2, V ∗ and V have the same nonzero resonances with the same multiplicities.Recall that this means that they have the same negative eigenvalues (if any) with the samemultiplicities. Using (2.2) and Theorem 1.4, for t > e t (∆ − V ∗ ) − e t (∆ − V ) ) = n ( − ∆ + V ∗ ) + β ( V ∗ , d ) − n ( − ∆ + V ) − β ( V , d )where n ( − ∆ + V ∗ ) is the dimension of the L null space of − ∆ + V ∗ , and similarly for V ∗ replaced by V . However, since − ∆ + V and − ∆ + V ∗ have the same heat coefficients, forany N ∈ N there is a C > | tr( e t (∆ − V ∗ − e t (∆ − V ) ) | ≤ Ct N for 0 < t < 1. Hence n ( − ∆ + V ∗ ) + β ( V ∗ , d ) = n ( − ∆ + V ) + β ( V , d ) . But we have chosen the left (respectively right) hand side to be the definition of the multi-plicity of 0 as a resonance of − ∆ + V ∗ (resp. − ∆ + V ), completing the proof. (cid:3) Proof of Corollary 1.6. Suppose to the the contrary that there are potentials V , V ∈ L ∞ c ( R d ; R ) so that the nonzero poles of the meromorphically continued resolvents R V j ( λ ),including multiplicities, differ by a nonzero finite number of elements. Then since the polesof determinants of the associated scattering matrices S j ( λ ) are a subset of the poles of R V j ,the hypotheses of Theorem 1.4 are fulfilled. Hence by Theorem 1.4, det S ( λ ) / det S ( λ ) ≡ τ > θ ∈ R , µ R ( τ e iθ ) − µ R ( τ e iθ ) = m = 0 . This must hold for some τ > θ ∈ R if the set of (nonzero) poles of the resolvents of P and P are not identical. But then by Lemma 3.1, µ R ( τ e i ( θ + kπ ) ) − µ R ( τ e i ( θ + kπ ) ) = m = 0 for all k ∈ Z . But this contradicts the assumption that the difference of the number of poles of the resol-vents, counted with multiplicity, is a finite number. (cid:3) The singular part of R V ( λ ) at λ e iπk and linear independence In this section we denote by p the natural projection, p : Λ → C , which identifies pointswhose arguments differ by an integral multiple of 2 π .Suppose R V ( λ ) has poles at λ , λ ∈ Λ, with, for j = 1 , f j in the range of thesingular part of R V ( λ ) at λ j , with ( − ∆ + V − ( p ( λ j )) ) f j = 0. It is easy to see from thecondition that f j is in the null space of − ∆ + V − ( p ( λ j )) that if ( p ( λ )) = ( p ( λ )) and f and f are both nontrivial, then they are linearly independent. This argument can easilybe extended to m distinct points { λ , ..., λ m } in Λ, as long as the set { ( p ( λ )) , ..., ( p ( λ m )) } consists of m distinct points in the complex plane. This argument does not work, however,if some elements of the set { ( p ( λ )) , ..., ( p ( λ m )) } coincide. Since on Λ there are infinitelymany points which project to any given point in the complex plane, the question of linearindependence of the ranges of the singular parts of the resolvent may be complicated. Wedo not attempt to fully answer this question here. However, we do show that in some sense,made precise in Proposition 9.2 below, the ranges of the singular part of the resolvent for aset of distinct points in Λ are “usually” linearly independent.To state our results, we introduce a little notation. Suppose λ ∈ Λ is a pole of R V ( λ ) oforder l . Then we will say f is in the range of the most singular part of R V ( λ ) at λ if f is inthe range of (( λ − λ ) l R V ( λ )) ↾ λ = λ . Here we locally identify a neighborhood of λ in Λ withan open set in the complex plane so that ( λ − λ ) l makes sense.In even dimension d , for any l ∈ Z (9.1) R ( e iπl λ ) = R ( λ ) − ilT ( λ )where ( T ( λ ) f ) ( x ) = 12 (2 π ) − d λ d − Z Z e iλω · ( x − y ) f ( y ) dydσ ω where dσ ω is the usual measure on S d − . Lemma 9.1. Let d be even, V ∈ L ∞ c ( R d ; R ) , and m ∈ Z , m ≥ . Suppose for some λ = ρe iϕ , ≤ ϕ < π , ρ > , there are distinct integers k , k , ..., k m and not identically functions f k l in the range of the most singular part of R V at λ e iπk l , so that { f k , f k , ..., f k l } are linearly dependent. Then for each integer p , R V ( λ ) has a pole at λ = e ipπ λ .Proof. Since f k l is in the image of the most singular part of R V at e iπk l λ , we have ( − ∆+ V − ρ e iϕ ) f k l = 0. Set φ k l = ( − ∆ − ρ e iϕ ) f k l = − V f k l . Since { f k l } forms a linearly dependentset, so do { φ k l } . Moreover, by unique continuation, none of the functions φ k l are the zerofunction. Since R V ( λ ) = R ( λ )( I + V R ( λ )) − so that f k l is in the image of R ( λ e iπk l ), thefunction φ k l is in the null space of I + V R ( λ e iπk l ) = I + V ( R ( λ ) − ik l T ( λ )).By relabeling and decreasing m if necessary, we can assume that no proper subset of the { φ k l } is linearly dependent. Then there are nonzero constants c , ..., c m so that φ k = m X l =2 c l φ k l . ESONANT RIGIDITY IN EVEN DIMENSIONS 35 But 0 = ( I + V R ( λ ) − ik V T ( λ )) φ k = ( I + V R ( λ ) − ik V T ( λ )) m X l =2 c l φ k l = − iV T ( λ ) m X l =2 c l ( k − k l ) φ k l ;that is, P ml =2 c l ( k − k l ) φ k l is in the null space of V T ( λ ). If p ∈ { k , ..., k m } there is nothingto prove. So we assume p 6∈ { k , ..., k m } . Then( I + V R ( λ ) − ipV T ( λ )) m X l =2 c l ( k − k l ) k l − p φ k l = i m X l =2 c l ( k − k l ) V T ( λ ) φ k l = iV T ( λ ) m X l =2 c l ( k − k l ) φ k l = 0 . By our assumption that no proper subset of { φ k , ..., φ k m } is linearly dependent, the function g = P ml =2 c l ( k − k l ) k l − p φ k l is nontrivial.Hence I + V R ( e ipπ λ ) has a nontrivial null space. But it is well known that R V ( λ ) has apole whenever I + V R ( λ ) has nontrivial null space. (cid:3) Proposition 9.2. If the hypotheses of Lemma 9.1 hold, then ϕ = π/ and − ρ is an eigen-value of − ∆ + V .Proof. Suppose the hypotheses of Lemma 9.1 hold. Then by Lemma 9.1 R V ( λ ) has a poleat λ , and λ lies in the closure of the physical region, more particularly in the regionwith 0 ≤ arg λ < π . But the only poles of R V ( λ ) with 0 ≤ arg λ < π correspond to thesquare roots of eigenvalues. Since V is real-valued, the eigenvalues of − ∆ + V are real, and ϕ = π/ (cid:3) Acknowledgments. It is a pleasure to thank Rafael Benguria, Peter Hislop, Antˆonio S´aBarreto, and Maciej Zworski for helpful conversations. 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