A relative trace formula for obstacle scattering
AA RELATIVE TRACE FORMULA FOR OBSTACLE SCATTERING
FLORIAN HANISCH, ALEXANDER STROHMAIER, AND ALDEN WATERS
Abstract.
We consider the case of scattering of several obstacles in R d for d ≥ are unitarily equivalent. For suitablefunctions that decay sufficiently fast we have that the difference g (∆) − g (∆ ) is atrace-class operator and its trace is described by the Krein spectral shift function. Inthis paper we study the contribution to the trace (and hence the Krein spectral shiftfunction) that arises from assembling several obstacles relative to a setting where theobstacles are completely separated. In the case of two obstacles we consider the Laplaceoperators ∆ and ∆ obtained by imposing Dirichlet boundary conditions only on oneof the objects. Our main result in this case states that then g (∆) − g (∆ ) − g (∆ ) + g (∆ ) is a trace class operator for a much larger class of functions (including functionsof polynomial growth) and that this trace may still be computed by a modificationof the Birman-Krein formula. In case g ( x ) = x the relative trace has a physicalmeaning as the vacuum energy of the massless scalar field and is expressible as anintegral involving boundary layer operators. Such integrals have been derived in thephysics literature using non-rigorous path integral derivations and our formula providesboth a rigorous justification as well as a generalisation. Introduction
Let d ≥ O be an open subset in R d with compact closure and smooth boundary ∂ O . The (finitely many) connected components will be denoted by O j with some index j . We will think of these as obstacles placed in R d . Removing these obstacles from R d results in a non-compact open domain M = R d \O with smooth boundary ∂ O , which isthe disjoint union of connected components ∂ O j . We will assume throughout that M isconnected.The positive Laplace operator ∆ on R d with Dirichlet boundary conditions at ∂ O isby definition the self-adjoint operator constructed from the energy quadratic form withDirichlet boundary conditions q D ( φ, φ ) = (cid:104)∇ φ, ∇ φ (cid:105) L ( R d ) , dom( q D ) = { φ ∈ H ( R d ) | φ | ∂ O = 0 } . (1)The Hilbert space L ( R d ) splits into an orthogonal sum L ( R d ) = L ( O ) ⊕ L ( M ) and theLaplace operator leaves each subspace invariant. In fact, the spectrum of the Laplacianon L ( O ) is discrete, consisting of eigenvalues of finite multiplicity, whereas the spectrumon L ( M ) is purely absolutely continuous. The above decomposition is therefore alsothe decomposition into absolutely continuous and pure point spectral subspaces. In thispaper we are interested in the fine spectral properties of the Laplace operator on L ( M )but it will be convenient for notational purposes to consider instead the Laplace operator Supported by Leverhulme grant RPG-2017-329. a r X i v : . [ m a t h . SP ] F e b F. HANISCH, A. STROHMAIER, AND A. WATERS ∆ on L ( R d ) with Dirichlet boundary conditions imposed on ∂ O as defined above. Letus denote by ∆ be the Laplace operator on L ( R d ) without boundary conditions.Scattering theory relates the continuous spectrum of the operator ∆ to that of the op-erator ∆ . A full spectral decomposition of ∆ analogous to the Fourier transform in R d can be achieved for ∆. The discrete spectrum of ∆ consists of eigenvalues of theinterior Dirichlet problem on O and the continuous spectrum is described by generalisedeigenfunctions E λ (Φ). We now explain the well known spectral decompositions in moredetail. A similar description as below is true in the more general black-box formalismin scattering theory as introduced by Sj¨ostrand and Zworski [24], and follows from themeromorphic continuation of the resolvent and its consequences. The exposition be-low follows [25] and we refer the reader to this article for the details of the spectraldecomposition.There exists an orthonormal basis ( φ j ) in L ( O ) consisting of eigenfunctions of ∆ anda family of generalised eigenfunctions E λ (Φ) on M indexed by functions Φ ∈ C ∞ ( S d − )such that ∆ φ j = λ j φ j , φ j | ∂ O = 0 , φ j ∈ C ∞ ( O ) , ∆ E λ (Φ) = λ E λ (Φ) , E λ (Φ) | ∂ O = 0 , E λ (Φ) ∈ C ∞ ( M ) , with E λ (Φ) = e − i λr r d − Φ + e i λr r d − e − i π ( d − τ ◦ S λ (Φ) + O ( r − d +12 ) (2)as r → ∞ for any λ >
0. Here τ : C ∞ ( S d − ) → C ∞ ( S d − ) , τ Φ( θ ) = Φ( − θ ) is theantipodal map and the scattering operator S λ : C ∞ ( S d − ) → C ∞ ( S d − ) is implicitlydetermined be the above asymptotic. The generalised eigenfunctions E λ (Φ) togetherwith the eigenfunctions φ j provide the full spectral resolution of the operator ∆. Wedefine the eigenvalue counting function N O ( λ ) of O by N O ( λ ) = { λ j | λ j ≤ λ } . The scattering matrix is a holomorphic function in λ on the upper half space and it isof the form S λ = id + A λ , where A λ is a holomorphic family of smoothing operators on S d − . It can be shown that A λ extends to a continuous family of trace-class operatorson the real line and one has the following estimate on the trace norm (cid:107) A λ (cid:107) = (cid:40) O ( λ d − ) for d > ,O ( λ − log( λ ) ) for d = 2 (3)for all | λ | < in a fixed sector in the logarithmic cover of the complex plane, c.f. [25,Theorem 1.11] or [6, Lemma 2.5] in case d ≥
3. In fact A λ extends to a meromorphicfamily on the entire complex plane in case d is odd. It extends to a meromorphic familyon the logarithmic cover of the complex plane in case d is even, and in this case theremay be logarithmic terms in the expansion about the point λ = 0. The behaviour near λ = 0 is conveniently described by the fact that A λ is Hahn-holomorphic near zero inany fixed sector of the logarithmic cover of the complex plane (see Appendix B).It follows that the Fredholm determinant det ( S λ ) is well defined, holomorphic in λ inthis sector, and for any choice of branch of the logarithm, we have that ddλ log det( S λ ) = Tr( S − λ ddλ S λ ) , (4) RELATIVE TRACE FORMULA 3 for λ at which det( S λ ) is non-zero. Moreover, det( S λ ) = 1 + O ( λ ) for | λ | < λ ) >
0. Since S λ is unitary on the positive real axis this allows one to fix a uniquebranch for the logarithm on the positive real line. These statements are well known forholomorphic families of matrices but can easily be shown to extend to the Fredholmdeterminant, for example by using Theorem 3.3 and Theorem 6.5 in [23].The general Birman-Krein formula [4] relates the spectral functions of two operatorsunder the assumption that the difference of certain powers of the resolvent is trace-class.It has been observed by Kato and Jensen [12] that this formalism applies to obstaclescattering. In this context the Birman-Krein formula states that if f is an even Schwartzfunction then Tr (cid:18) f (∆ ) − f (∆ ) (cid:19) = − (cid:90) ∞ f (cid:48) ( λ ) ξ O ( λ ) dλ, (5)where the Krein spectral shift function ξ O ( λ ) is given by ξ O ( λ ) = 12 π i log det ( S λ ) + N O ( λ ) . (6)This formula is well known. It is stated for even and compactly supported functions in[12] (see also the textbook [26, Ch. 8] for in case d = 3), but has been generalised to thecase of conical manifolds without boundary in [6]. A direct proof of the stated formulacan be inferred from the Birman-Krein formula as stated and proved in [25].There has been significant interest in the asymptotics of the scattering phase and thecorresponding Weyl law and its error term starting with the work by Majda and Ralston[15] and subsequent papers proving Weyl laws in increasing generality. We mention herethe paper by Melrose [18] establishing the Weyl law for the spectral shift function forsmooth obstacles in R d in odd dimensions, and Parnovski who establishes the result formanifolds that are conic at infinity [21] and possibly have a compact boundary.The Krein-spectral shift function is related to the ζ -regularised determinant of the Dirich-let to Neumann operator N λ which is the sum of the interior and the exterior Dirichletto Neumann operator of the obstacle O . We have ξ O ( λ ) = lim ε → + arg det ζ N λ +i ε . (7)This was proved in a quite general context by Carron in [5]. A similar formula involvingthe double layer operator instead of the single layer operator is proved for planar exteriordomains by Zelditch in [27], inspired by the work of Balian and Bloch [1] in dimension3, and also of Eckmann and Pillet [7] in the case of planar domains in dimension 2. Itis worth noting that a representation of the scattering matrix that allows to reduce thecomputation of the spectral shift function to the boundary appears implicitly in theirproof of inside-outside duality for planar domains [8]. In a more general framework ofboundary triples formulae that somewhat resemble this one were proved more recentlyin [11], although this paper does not use the ζ -regularised determinant.1.1. Setting.
In the present paper we investigate the contribution to the spectral shiftfrom assembling the objects O from individual objects O j . If ∂ O j are the N connectedcomponents of the boundary we define the following self-adjoint operators on L ( R d ). F. HANISCH, A. STROHMAIER, AND A. WATERS ∆ = the Laplace operator with Dirichlet boundary conditions on ∂ O as defined before.∆ j = the Laplace operator with Dirichlet boundary conditions on ∂ O j (1 ≤ j ≤ N ) . ∆ = the ”free” Laplace operator on R d with domain H ( R d ) . We are now interested in the following relative traceTr f (∆ ) − f (∆ ) − N (cid:88) j =1 f (∆ j ) − f (∆ ) = Tr f (∆ ) − N (cid:88) j =1 f (∆ j ) + ( N − f (∆ ) , which is the trace of the operator D f = f (∆ ) − f (∆ ) − N (cid:88) j =1 f (∆ j ) − f (∆ ) . If f is an even Schwartz function the Birman-Krein formula applies and we simply haveTr ( D f ) = − (cid:90) ∞ ξ O ( λ ) − N (cid:88) j =1 ξ O j ( λ ) f (cid:48) ( λ ) dλ. We therefore define the relative spectral shift function ξ rel ( λ ) = ξ O ( λ ) − N (cid:88) j =1 ξ O j ( λ ) . The contributions of N O and N O j in the relative spectral shift function cancel and ξ rel ( λ ) = 12 π i log (cid:18) det S λ det( S ,λ ) · · · det( S N,λ ) (cid:19) , where S j,λ are the scattering matrices of ∆ j , i.e. associated to the objects O j . Thisshows that ξ rel is a holomorphic function near the positive real axis and that ξ (cid:48) rel has ameromorphic continuation to the logarithmic cover of the complex plane. In particularthe restriction of ξ rel ( λ ) to R is continuous.Our main results are concerned with the properties of the operators f (∆ ) − f (∆ ) and D f for a class of functions f that is much larger than the class usually admissible in theBirman-Krein formula. In order to state the main theorems let us first introduce thisclass of functions.Assume 0 < (cid:15) ≤ π and let S (cid:15) be the open sector S (cid:15) = { z ∈ C | z (cid:54) = 0 , | arg( z ) | < (cid:15) } . RELATIVE TRACE FORMULA 5
We define the following two spaces of functions. The space E (cid:15) will be defined by E (cid:15) = { f : S (cid:15) → C | f is holomorphic in S (cid:15) , ∃ α > , ∀ (cid:15) > , | f ( z ) | = O ( | z | α e (cid:15) | z | ) } . Definition 1.1.
We define the space P (cid:15) as the set of functions in E (cid:15) whose restrictionto [0 , ∞ ) is polynomially bounded and that extend continuously to the boundary of S (cid:15) inthe logarithmic cover of the complex plane. Remark 1.2.
Reference to the logarithmic cover of the complex plane is only neededin case (cid:15) = π . In this case functions in P π are required to have continuous limits fromabove and below on the negative real axis. We do not however require that these limitscoincide. The space P (cid:15) contains in particular f ( z ) = z a , a > < (cid:15) ≤ π .When working with the Laplace operator it is often convenient to change variables anduse λ as a spectral parameter. For notational brevity we therefore introduce anotherclass of functions as follows. Definition 1.3.
The space (cid:101) P (cid:15) is defined to be the space of functions f such that f ( λ ) = g ( λ ) for some g ∈ P (cid:15) . For 0 < (cid:15) ≤ π we also define the contours Γ (cid:15) in the complex plane as the boundary curvesof the sectors S (cid:15) . In case (cid:15) = π the contour is defined as a contour in the logarithmiccover of the complex plane. We also let (cid:101) Γ (cid:15) be the corresponding contour after the changeof variables, i.e. the pre-image in the upper half space under the map z → z of Γ (cid:15) .These sectors and contours are illustrated below. Figure 1.
Sectors and contours in the complex plane1.2.
Main results.
Our first result is about the behaviour of the integral kernel of f (∆ ) − f (∆ ) away from the object O . Theorem 1.4.
Suppose that Ω ⊂ M is an open subset M such that dist(Ω , O ) > .Assume that f ∈ (cid:101) P (cid:15) for some < (cid:15) ≤ π and that near λ = 0 we have | f ( λ ) | = O ( | λ | a ) for some a > . Let p Ω be the multiplication operator with the indicator function of F. HANISCH, A. STROHMAIER, AND A. WATERS Ω . Then, the operator p Ω (cid:18) f (∆ ) − f (∆ ) (cid:19) p Ω extends to a trace-class operator T f : L (Ω) → L (Ω) with smooth integral kernel k f ∈ C ∞ (Ω × Ω) . Moreover, Tr ( T f ) = (cid:90) Ω k f ( x, x ) dx. For large dist( x, ∂ O ) we have | k f ( x, x ) | ≤ C Ω (dist( x, ∂ O )) d − a (8) where the constant C Ω depends on Ω and f . It is not hard to see that in general the operator f (∆ ) − f (∆ ) is not trace-class inthe case O (cid:54) = ∅ and the above trace is dependent on the cut-off p Ω . The main resultof this paper however is that D f is trace-class and a modification of the Birman-Kreinformula applies to a rather large class of functions. We prove that D f is densely definedand bounded and therefore extends uniquely to the entire space by continuity. We willnot distinguish this unique extension notationally from D f . Theorem 1.5.
Suppose that f ∈ (cid:101) P (cid:15) for some < (cid:15) ≤ π . Then the operator D f is trace-class in L ( R d ) and has integral kernel k f ∈ C ∞ ( O × O ) ⊕ C ∞ ( M × M ) . Moreover, Tr( D f ) = (cid:90) R d k f ( x, x ) dx. Theorem 1.6.
Let δ = min j (cid:54) = k dist( O j , O k ) be the minimal distance between distinctobjects and let < δ (cid:48) < δ . Then there exists a unique function Ξ , holomorphic in theupper half space, such that(1) Ξ (cid:48) has a meromorphic extension to the logarithmic cover of the complex plane, andto the complex plane in case d is odd.(2) for any (cid:15) > there exists C δ (cid:48) ,(cid:15) > with | Ξ (cid:48) ( λ ) | ≤ C δ (cid:48) ,(cid:15) e − δ (cid:48) Im( λ ) , if Im( λ ) ≥ (cid:15) | λ | , | Ξ( λ ) | ≤ C δ (cid:48) ,(cid:15) e − δ (cid:48) Im( λ ) , if Im( λ ) ≥ (cid:15) | λ | . (3) for λ > we have π Im Ξ( λ ) = − i2 π (Ξ( λ ) − Ξ( − λ )) = ξ rel ( λ ) . (4) if f ∈ (cid:101) P (cid:15) for some < (cid:15) ≤ π , then Tr ( D f ) = − i2 π (cid:90) (cid:101) Γ (cid:15) Ξ( λ ) f (cid:48) ( λ ) dλ. We note here that the right hand side of the Birman-Krein formula is not well definedfor these functions since neither ξ rel ( λ ) nor ξ (cid:48) rel ( λ ) are in L ( R + ) in general: it followsfrom the wave trace expansion of [2] that the cosine transform of ξ rel ( λ ) may have adiscontinuity at lengths of non-degenerate simple bouncing ball orbits. This happens forexample for two spheres. RELATIVE TRACE FORMULA 7
The function Ξ can be explicitly given in terms of boundary layer operators. Let Q λ bethe usual single layer operator on the boundary ∂ O (see Section 2). One can define the”diagonal part” ˜ Q λ of this operator by restricting the integral kernel to the subset N (cid:91) j =1 ∂ O j × ∂ O j ⊂ ∂ O × ∂ O , thus excluding pairs of points on different connected components of ∂ O (see Section 3for more details). We then have Theorem 1.7.
The operator Q λ (cid:101) Q − λ − is trace-class and Ξ( λ ) = log det (cid:16) Q λ (cid:101) Q − λ (cid:17) . Since π Im Ξ( λ ) = ξ rel ( λ ) this theorem also yields a gluing formula for the spectral shiftfunction. In particular, choosing f ( x ) = x and (cid:15) = π this gives the formulaTr (cid:18) ∆ − N (cid:88) j =1 ∆ j + ( N − (cid:19) = − π (cid:90) ∞ Ξ(i λ ) dλ. (9)Since the operators ∆ are unbounded it may come as a surprise that the linear combi-nation in the above formula is trace-class.Our method of proof is to reduce the computations to the boundary using single anddouble layer operators. Some of the ingredients, namely a polynomial bound of theDirichlet-to-Neumann operator (Corollary 2.7) and on the inverse of the single layeroperator (Corollary 2.8), both in a sector of the complex plane, may be interesting intheir own right. Similar estimates for the Dirichlet-to-Neumann map in the complexplane have been proved in [14] (see also [3]). Recently there has been interest in boundsof the Dirichlet-to-Neumann operator and layer potentials and their inverses on the realline (see e.g. [10]).1.3. Applications in Physics.
The left hand side of (9) has an interpretation in thephysics of quantum fields. It is the vacuum energy of the free massless scalar field of theassembled objects relative to the objects being separated. This energy is used to computeCasimir forces between objects and the above formula provides a formula for the forces interms of the function Ξ which can be constructed out of the scattering matrix. We notethat the right hand side of Eq.(9) was used to compute Casimir energies between objectsand derived using non-rigorous path integral methods (see for example [9]). This wasan important development that lead to more efficient numerical algorithms. We wouldlike to refer to [13] and references therein. Our work therefore make these statementsmathematically rigorous and also generalises them.
F. HANISCH, A. STROHMAIER, AND A. WATERS Layer potentials
The Green’s function for the Helmholtz equation, i.e. the integral kernel of the resolvent(∆ − λ ) − will be denoted by G λ, . We have explicitly, G λ, ( x, y ) = i4 (cid:18) λ π | x − y | (cid:19) d − H (1) d − ( λ | x − y | ) . (10)In particular, in dimension three: G λ, ( x, y ) = 14 π e i λ | x − y | | x − y | . As usual we identify integral kernels and operators, so that G λ, coincides with theresolvent (∆ − λ ) − for Im( λ ) >
0. Note that in case the dimension d is even theabove Hankel function fails to be analytic at zero. In that case one can however write G λ, = ˜ G λ, + F λ λ d − log( λ ) , where ˜ G λ, is an entire family of operators H s ( R d ) → H s +2loc ( R d ) and F λ is an entirefamily of operators with smooth integral kernel F λ ( x, y ) = 12i (2 π ) − ( d − (cid:90) S d − e i λθ ( x − y ) dθ, that is even in λ , c.f. [17] Ch. 2. Moreover we have that G − λ, ( x, y ) = G λ, ( x, y ) λ > . (11)The single and double layer potential operators are continuous maps S λ : C ∞ ( ∂ O ) → C ∞ ( O ) ⊕ C ∞ ( M ) ⊂ C ∞ ( R d \ ∂ O )and D λ : C ∞ ( ∂ O ) → C ∞ ( O ) ⊕ C ∞ ( M ) ⊂ C ∞ ( R d \ ∂ O ) , given by S λ f ( x ) = (cid:90) ∂ O G λ, ( x, y ) f ( y ) dy, D λ f ( x ) = (cid:90) ∂ O ( ∂ ν,y G λ, ( x, y )) f ( y ) dy, where ∂ ν,y denotes the outward normal derivative. We also define Q λ : C ∞ ( ∂ O ) → C ∞ ( ∂ O ) and K λ : C ∞ ( ∂ O ) → C ∞ ( ∂ O ) by Q λ f ( x ) = (cid:90) ∂ O G λ, ( x, y ) f ( y ) dy,K λ f ( x ) = (cid:90) ∂ O ( ∂ ν,y G λ, ( x, y )) f ( y ) dy. Let as usual γ + denote the exterior restriction map C ∞ ( M ) → C ∞ ( ∂ O ) and ∂ + ν : C ∞ ( M ) → C ∞ ( ∂ O ) the restriction of the inward pointing normal derivative. Similarly, γ − denotes the interior restriction map C ∞ ( O ) → C ∞ ( ∂ O ) and ∂ − ν : C ∞ ( O ) → C ∞ ( ∂ O )the restriction of the outward pointing normal derivative. If w is a function in C ∞ ( M )with (∆ − λ ) w = 0 such that either RELATIVE TRACE FORMULA 9 (a) Im λ > w ∈ L ( M ), or(b) λ ∈ R and w satisfies the Sommerfeld radiation condition.Then we have for x ∈ M (for example [26]): w ( x ) = S λ ( ∂ + ν w )( x ) + D λ ( γ + w )( x ) . If w ∈ C ∞ ( O ) satisfies (∆ − λ ) w = 0 then w ( x ) = S λ ( ∂ − ν w )( x ) − D λ ( γ − w )( x ) . We have the following well known relations (for example [26]): γ ± S λ = Q λ ,γ ± D λ = ±
12 Id + K λ . The following summarises the mapping properties of the layer potential operators.
Proposition 2.1.
The maps Q λ , S λ , and K λ extend by continuity to larger spaces asfollows:(1) If s < and Im λ > then S λ extends to a holomorphic family of maps S λ : H s ( ∂ O ) → H s +3 / ( R d ) . (2) If Im λ > and Re λ (cid:54) = 0 then (cid:107)S λ (cid:107) H − ( ∂ O ) → L ( R d ) ≤ C √ | λ | | Re( λ ) Im( λ ) | .(3) If s < then S λ extends to a family of maps S λ : H s ( ∂ O ) → H s +3 / ( R d ) of theform S λ = s λ + m λ λ d − log( λ ) , where s λ = F λ γ ∗ is an entire family of operators s λ : H s ( ∂ O ) → H s +3 / ( R d ) and m λ : H s ( ∂ O ) → C ∞ ( R d ) is an entire family ofsmoothing operators. Moreover, m λ = 0 in case the dimension d is odd.(4) The operator Q λ can be written as Q λ = q λ + r λ λ d − log( λ ) , where q λ is an entirefamily of pseudodifferential operators of order − , and r λ = γF λ γ ∗ is an entirefamily of smoothing operators. In case d is odd we have r λ = 0 .(5) The operator K λ can be written as K λ = k λ + ˜ r λ λ d − log( λ ) , where k λ is an en-tire family of pseudodifferential operators of order , and ˜ r λ is an entire family ofsmoothing operators. In case d is odd we have ˜ r λ = 0 .Proof. First note that S λ = G λ, ◦ γ ∗ , where γ : H s + ( R d ) → H s ( ∂ O ) is the restrictionmap for s > γ ∗ : H − s ( ∂ O ) → H − s − ( R d ) is its dual. The first statement (1) thenfollows from the mapping property of G λ, since G λ, : H s ( R d ) → H s +2 ( R d ) continuously.Statement (3) follows in the same way from the continuity of ˜ G λ, : H sc ( R d ) → H s +2loc ( R d )which can easily be obtained from the explicit representation of the integral kernel. Itis well known that Q λ and K λ are pseudodifferential operators and their full symboldepends holomorphically on λ . The representations in (4) and (5) then follow from theexplicit form of the integral kernel. To show (2) recall that the map γ ∗ : H − ( ∂ O ) → H − ( R d ) is continuous. Thus, the operator norm of S λ : H − ( ∂ O ) → L ( R d ) is boundedby a constant times the norm of (1 + ∆ ) (∆ − λ ) − . Using the spectral representationof ∆ one sees the this norm equals sup x ∈ R | √ x x − λ | ≤ √ | λ | | Re( λ ) Im( λ ) | . (cid:3) For 0 < (cid:15) < π , let the sector D (cid:15) in the upper half plane be given by D (cid:15) := { z ∈ C | (cid:15) ≤ arg( z ) ≤ π − (cid:15) } . Let the function ρ be defined as ρ ( t ) := t d − if d = 2 , | log t | + 1 if d = 4 and 0 ≤ t ≤ t = 1 if d ≥
51 for d > S λ . Proposition 2.2.
For (cid:15) ∈ (0 , π ) , for all λ ∈ D (cid:15) we have the following bounds:(1) Let Ω ⊂ R d be an open set with smooth boundary. Set δ = dist(Ω , ∂ O ) > and let < δ (cid:48) < dist(Ω , ∂ O ) . Assume that χ ∈ L ∞ ( R d ) has support in Ω . Let s ∈ R , thenthere exists C δ (cid:48) ,(cid:15) > such that for λ ∈ D (cid:15) we have that χ S λ : H s ( ∂ O ) → L ( R d ) isa Hilbert-Schmidt operator whose Hilbert-Schmidt norm is bounded by (cid:107) χ S λ (cid:107) HS( H s → L ) ≤ C δ (cid:48) ,(cid:15) ρ (Im λ ) e − δ (cid:48) Im λ (13) (2) For λ ∈ D (cid:15) we have (cid:107)S λ (cid:107) H − ( ∂ O ) → L ( R d ) ≤ C δ (cid:48) ,(cid:15) (cid:16) ρ (Im λ ) + 1 (cid:17) . Proof.
Let k ∈ N and P be any an invertible, formally self-adjoint, elliptic differentialoperator of order 2 k on ∂ O with smooth coefficients, for example P = ∆ k∂ O + 1. Since ∂ O is compact this implies that P − : H s ( ∂ O ) → H s +2 k ( ∂ O ). The integral kernel of χ S λ P is given by χ ( x ) P y G λ, ( x, y ). Here, ( x, y ) ∈ Ω × ∂ O and the distance betweensuch x and y is bounded below by δ . By Lemma A.1, we have (cid:107) χP y G λ, (cid:107) L (Ω × ∂ O ) ≤ C δ (cid:48) ,(cid:15) ρ (Im λ ) / e − δ (cid:48) Im λ . In particular, we conclude from [22], Proposition A.3.2 that χ S λ P is a Hilbert Schmidtoperator L ( ∂ O ) → L ( R d ) and its Hilbert Schmidt norm is bounded by (cid:107) χ S λ P (cid:107) HS ( L → L ) ≤ C δ (cid:48) ,(cid:15) ρ (Im λ ) / e − δ (cid:48) Im λ . Since P − : H − k ( ∂ O ) → L ( ∂ O ) is bounded we conclude that χ S λ : H − k ( ∂ O ) → L ( R n ) is Hilbert Schmidt and its corresponding norm is estimated by (cid:107) χ S λ (cid:107) HS ( H − k → L ) ≤ C δ (cid:48) ,(cid:15) ρ (Im λ ) / e − δ (cid:48) Im λ . This yields (13) for all s > − k . This proves part (1) and it remains to show (2). By(2) of Prop. 2.1 we only need to check this estimate near zero. We choose a compactlysupported positive smooth cutoff function χ which equals one for all points of distanceless than 1 from O . We show that the operators χ S λ and (1 − χ ) S λ satisfy thedesired bound for λ near zero. It follows from (3) of Prop. 2.1 that in case d ≥ χ S λ : H − ( ∂ O ) → H ( R d ) is bounded near zero. In case d = 2 we have (cid:107) χ S λ (cid:107) H − ( ∂ O ) → H ( R d ) ≤ C δ (cid:48) ,(cid:15) log( λ ) near zero. Either way (cid:107) χ S λ (cid:107) L ( ∂ O ) → L ( R d ) ≤ RELATIVE TRACE FORMULA 11 C δ (cid:48) ,(cid:15) ρ (Im λ ) near λ = 0. The operator (1 − χ ) S λ is Hilbert-Schmidt with Hilbert-Schmidt norm bounded by C δ (cid:48) ,(cid:15) ρ (Im λ ) as a map from H − ( ∂ O ) to L ( R d ) from part(1). Thus, (cid:107) (1 − χ ) S λ (cid:107) H − ( ∂ O ) → L ( R d ) ≤ C δ (cid:48) ,(cid:15) ρ (Im λ ) . (cid:3) The layer potential operators can be used to solve the exterior boundary value problem.Let Im λ ≥ f ∈ C ∞ ( ∂ O ). Let B + λ : f (cid:55)→ w be the solution operator of the exteriorDirichlet problem (∆ − λ ) w = 0 , γ + w = f, where w ∈ L ( M ) in case Im λ > w satisfies the Sommerfeld radiation condition incase λ ∈ R . Similarly, if λ is not an interior Dirichlet eigenvalue, let B − λ : f (cid:55)→ w be thesolution operator of the interior problem. Together they constitute the solution operatorof the Dirichlet problem B λ = B − λ ⊕ B + λ mapping to L ( R d ) = L ( O ) ⊕ L ( M ). Then B λ is holomorphic in λ on the upper half space. The operator Q λ is a pseudodifferentialoperator of order − H s ( ∂ O ) to H s +1 ( ∂ O ). It is invertible away from the Dirichleteigenvalues of the interior problem. Hence, for all Im λ > B λ = S λ ( Q λ ) − . In case the dimension d is odd we conclude from holomorphic Fredholm theory that( Q λ ) − is a meromorphic family of Fredholm operators of finite type on the complexplane. This provides a meromorphic continuation of B λ to the entire complex plane inthis case. In case the dimension d is even we may not apply meromorphic Fredholmtheory directly to Q λ at the point λ = 0 since the family fails to be holomorphic atthat point due to the presence of log-terms. One can however apply the theory of Hahnholomorphic functions instead to arrive at the following conclusion. A summary of Hahnholomorphic functions and their properties can be found in Appendix B. Lemma 2.3.
For any < (cid:15) < π/ and s ∈ R the operator family Q − λ is holomorphic inthe sector D (cid:15) and continuous on its closure as a family of bounded operators H s ( ∂ O ) → H s − ( ∂ O ) . In particular the family is bounded near zero in the sector. If d is odd then Q − λ : H s ( ∂ O ) → H s − ( ∂ O ) is a meromorphic family of operators of finite type in thecomplex plane. If d is even then Q − λ : H s ( ∂ O ) → H s − ( ∂ O ) is Hahn-meromorphicof finite type on any sector of the logarithmic cover of the complex plane that is Hahn-holomorphic at zero.Proof. First note that for Im( λ ) > Q λ is a holomorphic family of el-liptic pseudodifferential operators of order − λ . It is therefore a holomorphic family of Fredholm operators of index zero from H s ( ∂ O ) → H s +1 ( ∂ O ). Now note that if u ∈ ker Q λ , then u is smooth and S λ u is aLaplace eigenfunction with eigenvalue λ that vanishes at the boundary and has Neu-mann data u . If λ is in the upper half space then λ is not a Dirichlet eigenvalue. There-fore u cannot be a Dirichlet eigenfunction and we conclude that u = 0. By analyticFredholm theory Q λ is invertible in the upper half plane as a map H s ( ∂ O ) to H s +1 ( ∂ O )and the inverse Q − λ is holomorphic. This argument also extents to zero in odd di-mensions. However in even dimensions the family fails to be holomorphic at zero. Wetherefore use Fredholm theory for Hahn meromorphic operator valued functions. First note by 2.1 that S λ and Q λ are analytic in a larger sector D (cid:15) and Hahn-meromorphicnear zero. For d ≥ S λ and Q λ are of the form S λ = s λ + m λ log λ and Q λ = q λ + r λ log λ where r hasconstant integral kernel, γ ◦ F (see (4) of Prop. 2.1). If follows from Fredholm the-ory that Q − λ is Hahn-meromorphic of finite type on this sector as an operator family H s +1 ( ∂ O ) → H s ( ∂ O ). We will show that Q − λ is Hahn-holomorphic at zero. Let A bethe most singular term in the Hahn-expansion of ( Q λ ) − , i.e. A = lim λ → r ( λ )( Q λ ) − for r ( λ ) = λ α ( − log( λ )) − β such that lim λ → r ( λ ) = 0. Then A has finite rank andlim λ → Q λ A = 0. Let u = Aw . If d ≥ Q u = 0 and therefore S u solves theDirichlet problem with zero eigenvalue and Neumann data u . Since 0 is not an interioreigenvalue the Neumann data must vanish and therefore u = 0. In case d = 2 the argu-ment is slightly more subtle since Q λ is not regular at λ = 0. In this case Q λ u is regularat zero and therefore Hahn-holomorphic. Comparing expansion coefficients this implies r u = 0 and thus m u = 0. Consequently, S λ u = s λ u is Hahn-holomorphic near zeroand lim λ → S λ u exists and solves the interior Dirichlet problem with Neumann data u .By the same argument as before this implies u = 0. Hence in all dimensions Aw = 0 forall w ∈ H s +1 ( ∂ O ) and thus A = 0. This means Q − λ has no singular terms in its Hahnexpansion and is therefore Hahn-holomorphic near zero and hence bounded. (cid:3) The exterior and interior Dirichlet to Neumann maps N ± λ : C ∞ ( ∂ O ) → C ∞ ( ∂ O ) aredefined as N ± λ = ∂ ν B ± λ and related to the above operators by N ± λ = Q − λ ( 12 Id ∓ K λ )(see [26], Ch. 7 Prop. 11.2) and therefore N + λ + N − λ = Q − λ . (14)Since the operator family K λ : H s ( ∂ O ) → H s ( ∂ O ) is continuous near λ = 0 we imme-diately obtain the following. Corollary 2.4.
For any < (cid:15) < π/ and s ∈ R the operator family N ± λ is holomor-phic in the sector D (cid:15) and continuous on its closure as a family of bounded operators H s ( ∂ O ) → H s − ( ∂ O ) . In particular the family is bounded near zero in the sector. Lemma 2.5.
For λ ∈ C satisfying Re( λ ) < , there exists a constant C > s.t. (cid:107)N ± λ (cid:107) H / ( ∂ O ) → H − / ( ∂ O ) ≤ C ( | λ | + 1) ( −
2) Re( λ ) + C. The same estimate holds when
Re( λ ) is replaced by Im( λ ) under the assumption that Im( λ ) < Proof.
Let u ∈ C ∞ ( M, C ) ∩ H ( M, C ) be a solution of (∆ − λ ) u = 0. Since ν + is theoutward normal for M , the real part of Green’s first identity reads (cid:90) M |∇ u | − Re( λ ) | u | dx = Re (cid:90) ∂M ¯ u∂ + ν udS. RELATIVE TRACE FORMULA 13
Using the dual pairing between H / ( ∂ O ) and H − / ( ∂ O ) to estimate the right handside and dropping the first derivatives, we obtain (cid:107) u (cid:107) L ( M ) ≤ ( − Re( λ )) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂M ¯ u∂ + ν udS (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( − Re( λ )) − (cid:107) γ + u (cid:107) H / (cid:107) ∂ + ν u (cid:107) H − / (15)Choose R > O is contained in B R (0). Choose a cutoff function χ ∈ C ∞ ( M ) such that χ ( M ) ⊂ [0 , χ | B R (0) and supp( χ ) ⊂ B R (0). Setting ˜ u := χ · u , thisfunction satisfies the equation (∆+1)˜ u = χ (1+ λ ) u +[∆ , χ ] u , where [∆ , χ ] is a differentialoperator of order one with compact support. We now apply elliptic estimates on boundeddomains from [16, Lemma 4.3 and Theorem 4.10 (i)]. We are in case (i) of theorem 4.10since the homogeneous problem (∆ + 1) u = 0 with Dirichlet boundary conditions hasonly the trivial solution, as − (cid:107) ∂ + ν ˜ u (cid:107) H − / ( ∂M ) ≤ C (cid:107) ˜ u (cid:107) H ( M ) ≤ C ( (cid:107) (∆ + 1)˜ u (cid:107) H − ( M ) + (cid:107) ˜ u (cid:107) H / ( ∂ O ) ) ≤ C ( (cid:107) χ (1 + λ ) u (cid:107) H − ( M ) + (cid:107) [∆ , χ ] u (cid:107) H − ( M ) + (cid:107) ˜ u (cid:107) H / ( ∂ O ) ) ≤ C ( | λ | · (cid:107) χu (cid:107) L ( M ) + (cid:107) u (cid:107) L ( M ) + (cid:107) ˜ u (cid:107) H / ( ∂ O ) ) (16)Since ∂ + ν (˜ u ) | ∂M = ∂ + ν ( u ) | ∂M and γ + ˜ u = γ + u , combining (15) and (16) implies (cid:107) ∂ + ν u (cid:107) H − / ( ∂M ) ≤ C | λ | + 1( − Re( λ )) / (cid:107) γ + u (cid:107) / H / (cid:107) ∂ + ν u (cid:107) / H − / + C (cid:107) u (cid:107) H / ( ∂ O ) ≤ C ( | λ | + 1) ( −
2) Re( λ ) (cid:107) γ + u (cid:107) H / + 12 (cid:107) ∂ + ν u (cid:107) H − / + C (cid:107) u (cid:107) H / ( ∂ O ) Rearranging this estimate yields the first estimate on (cid:107)N + λ (cid:107) . Assuming Im( λ ) < λ ) replaced by Im( λ ) and we arrive at a corresponding estimatefor (cid:107)N + λ (cid:107) . (cid:3) Remark 2.6.
The estimate (15) can be improved in the following way. Choosing (cid:15) ∈ (0 , and Re( λ ) < − (cid:15) , then we may use Green’s identity to get an estimate for (cid:107) u (cid:107) H ( M ) : (cid:15) (cid:107) u (cid:107) H ( M ) ≤ (cid:90) |∇ u | − Re( λ ) | u | dx ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂M ¯ u∂ + ν udS (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) γ + u (cid:107) H / (cid:107) ∂ + ν u (cid:107) H − / For 0 < (cid:15) < π , recall the sector D (cid:15) in the upper half plane is given by D (cid:15) := { z ∈ C | (cid:15) ≤ arg( z ) ≤ π − (cid:15) } . Note that for λ ∈ D (cid:15) , we have the estimateIm( λ ) = | Im( λ ) | ≤ | λ | ≤ C (cid:15) Im( λ ) , where C (cid:15) := sin( (cid:15) ) − is independent of λ ∈ D (cid:15) .The lemma above yields the following uniform estimate: Corollary 2.7.
For any (cid:15) > , there exists C = C ( (cid:15) ) > such that (cid:107)N ± λ (cid:107) H / ( ∂ O ) → H − / ( ∂ O ) ≤ C (1 + | λ | ) for all λ in the sector D (cid:15) . Proof.
It is enough to prove the statement for | λ | ≥ r >
0, since N ± λ is continuous at λ = 0. Lemma 2.5 now implies the estimate for any (cid:15) > r > (cid:15) ≤ arg( λ ) ≤ π/ − (cid:15) . Indeed, the estimate involving Re( λ ) covers the region where π/ (cid:15) ≤ arg( λ ) ≤ π/ − (cid:15) whereas the one based on Im( λ ) is valid for (cid:15) ≤ arg( λ ) ≤ π/ − (cid:15) . Inorder to extend the estimate to all of D (cid:15) , we observe that N ± λ ( f ) = N ± ¯ λ ( ¯ f ) = N ±− ¯ λ ( ¯ f )by construction. Moreover, the map λ (cid:55)→ − ¯ λ maps the sector 3 π/ − (cid:15) ≤ arg( λ ) ≤ π − (cid:15) to (cid:15) ≤ arg( λ ) ≤ π/ (cid:15) . Since we already established the estimate in the latter region,this proves the statement on all of D (cid:15) . (cid:3) Taking into account (14), we also obtain
Corollary 2.8.
For any (cid:15) > we have there exists a constant C = C ( (cid:15) ) such that (cid:107) Q − λ (cid:107) H / ( ∂ O ) → H − / ( ∂ O ) ≤ C (1 + | λ | ) . for all λ in the sector D (cid:15) . For f ∈ C ∞ ( R d ) we have that g = (∆ − λ ) − f −S λ Q − λ γ (∆ − λ ) − f satisfies Dirichletboundary conditions and (∆ − λ ) g = f . This shows that we have the following formulafor the difference of resolvents(∆ − λ ) − − (∆ − λ ) − = −S λ Q − λ γ (∆ − λ ) − = −S λ Q − λ S t λ (17)as maps from C ∞ ( R d ) to C ∞ ( O ) ⊕ C ∞ ( M ). Here S t λ is the transpose operator to S λ obtained from the real inner product, i.e. S t λ f = S t λ f . We will now use the propertiesof Q λ to establish trace class properties of suitable differences of resolvents. Theorem 2.9.
Let (cid:15) > and also suppose that Ω is a smooth open set in R d whosecomplement contains O . Let δ = dist( ∂ O , Ω) . If p is the projection onto L (Ω) in L ( R d ) then the operator p (∆ − λ ) − p − p (∆ − λ ) − p is trace class for all λ ∈ D (cid:15) . Moreover for any δ (cid:48) ∈ (0 , δ ) , its trace norm is bounded by (cid:107) p (∆ − λ ) − p − p (∆ − λ ) − p (cid:107) ≤ C δ (cid:48) ,(cid:15) ρ (Im λ ) e − δ (cid:48) Im( λ ) , (18) for all λ ∈ D (cid:15) , where ρ is defined in proposition 2.2. The operator (∆ − λ ) − − (∆ − λ ) − has integral kernel k λ in C ∞ (Ω × Ω) for all λ ∈ D (cid:15) . On the diagonal in Ω × Ω , we havethere exists C , C depending on Ω and (cid:15) such that | k λ ( x, x ) | ≤ C e − C dist( x,∂ O ) Im λ (dist( x, ∂ O )) d − , (19) for large dist( x, ∂ O ) for d ≥ and | k λ ( x, x ) | ≤ C (1 + | log( λ dist( x, ∂ O )) | ) e − C (dist( x,∂ O )) Im λ (20) for d = 2 . RELATIVE TRACE FORMULA 15
Proof of Theorem 2.9.
From (17), we have p (∆ − λ ) − p − p (∆ − λ ) − p = − p S λ Q − λ S t λ p = − p S λ Q − λ ( p S λ ) t . Choosing χ to be the characteristic function of Ω, p S λ coincides with χ S λ . By proposition2.2, p S λ is a Hilbert Schmidt operator H s ( O ) → L ( R d ) for all s ∈ R and consequently,( p S λ ) t : L ( R d ) → H s ( ∂ O ) is Hilbert Schmidt as well. Since Q − λ is bounded by Corol-lary 2.8, we have factorised p (∆ − λ ) − p − p (∆ − λ ) − p into a product two HilbertSchmidt and a bounded operator, which is trace-class by [22], (A.3.4) and (A.3.2). Thenorm estimates in Proposition 2.2 and Corollary 2.8 then imply the bound (18) on thetrace norm.The kernel of − p S λ Q − λ S t λ p is smooth since Q − λ is a pseudodifferential operator of order1 and p S is smoothing for dist(Ω , ∂ O ) >
0. For x ∈ Ω fixed, G λ, ( x, · ) ∈ C ∞ ( ∂ O ) andhence in H s ( ∂ O ) for all s ∈ R . Using the dual pairing (cid:104) , (cid:105) between H / and H − / , wehave for x, x (cid:48) ∈ Ω in dimensions d ≥ | k λ ( x, x (cid:48) ) | = |(cid:104) G λ, ( x (cid:48) , · ) , Q − λ ( G λ, ( x, · )) (cid:105)|≤ (cid:107) G λ, ( x (cid:48) , · ) (cid:107) H / · (cid:107) Q − λ (cid:107) H / ( ∂ O ) → H − / ( ∂ O ) · (cid:107) G λ, ( x, · ) (cid:107) H / (21) ≤ C (dist( x, ∂ O )dist( x (cid:48) , ∂ O )) − d · e − C Im λ (dist( x,∂ O )+dist( x (cid:48) ,∂ O )) , where we have used Corollary A.4 with 0 < < min(dist( x, ∂ O ) , dist( x (cid:48) , ∂ O )) as well asCorollary 2.8. The constants C , C depend on (cid:15) and Ω. Similarly, for d = 2 we havethe estimate | k λ ( x, x (cid:48) ) | ≤ C (1 + | log( λ dist( x, ∂ O )) | )(1 + | log( λ dist( x (cid:48) , ∂ O )) | ) × (22) e − C Im λ (dist( x,∂ O )+dist( x (cid:48) ,∂ O )) (cid:3) The structure of Q λ in a multi-component setting The boundary ∂ O consists of N connected components ∂ O j . We therefore have anorthogonal decomposition L ( ∂ O ) = ⊕ Nj =1 L ( ∂ O j ). Let p j : L ( ∂ O ) → L ( ∂ O j ) be thecorresponding orthogonal projection and Q j,λ := p j Q λ p j . We can then write Q λ as Q λ = N (cid:88) j =1 Q j,λ + (cid:88) j (cid:54) = k p j Q λ p k =: (cid:101) Q λ + T λ . (23)Note that Q j,λ , regarded as a map from L ( ∂ O j ) → L ( ∂ O j ), does not depend on theother components and equals Q λ for O = O j . (cid:101) Q λ describes the diagonal part of theoperator Q λ with respect to the decomposition L ( ∂ O ) = ⊕ Nj =1 L ( ∂ O j ) whereas T λ isthe off diagonal remainder. Let δ := min j (cid:54) = k dist( ∂ O j , ∂ O k ) > . We have the following:
Proposition 3.1.
For (cid:15) > we have that T λ is a holomorphic family of smoothingoperators on the sector D (cid:15) , such that for λ ∈ D (cid:15) with | λ | > we have (cid:107)T λ (cid:107) H − s → H s = C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ , (cid:107) ddλ T λ (cid:107) H − s → H s = C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ for any s ∈ R and any δ (cid:48) with < δ (cid:48) < δ . If d is odd, we also have that T λ : H − s ( ∂ O ) → H s ( ∂ O ) is a holomorphic family on the complex plane. If d is even then for any s ∈ R the family T λ : H − s ( ∂ O ) → H s ( ∂ O ) is holomorphic in any sector of the logarithmiccover of the complex plane and Hahn-meromorphic at zero. If d > then T λ is Hahnholomorphic at zero.Proof. The first estimate is a consequence of Lemma A.2 and Corollary A.3, where theprecise behaviour of the kernel in various regimes is given. Indeed, as dist( ∂ O j , ∂ O k ) > j (cid:54) = k , these statements yield an estimate for (cid:107) P x P y G (cid:107) L ( ∂ O j × ∂ O k ) which implies theestimate for (cid:107)T λ (cid:107) H − s → H s . Taking into account (32), the kernel estimates can be extendedto dG λ, /dλ and this gives the second estimate. The holomorphic and meromorphicproperties of T λ follow immediately from the corresponding properties of the kernelsestablished in Prop. 2.1, (4) bearing in mind that T λ consists of off-diagonal contributionsof Q λ . (cid:3) As observed before, Q λ and (cid:101) Q λ are invertible for Im λ >
0. For these λ , it then followsfrom (23) that Q − λ − (cid:101) Q − λ = − Q − λ T λ (cid:101) Q − λ = − (cid:101) Q − λ T λ Q − λ . (24)Note that the right hand side of this equation is a smoothing operator because T λ issmoothing by Proposition 3.1 and Q − λ , (cid:101) Q − λ are pseudodifferential operators of order 1. Proposition 3.2.
For (cid:15) > , we have that Q λ (cid:101) Q − λ − is a holomorphic family ofsmoothing operators in the sector D (cid:15) that is continuous on the closure D (cid:15) . If δ (cid:48) > isany positive real number smaller than δ , then, for any s ∈ R we have (cid:107) Q λ (cid:101) Q − λ − (cid:107) H → H s ≤ C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ . If d is odd then for any s ∈ R the family Q λ (cid:101) Q − λ − H − s ( ∂ O ) → H s ( ∂ O ) is meromor-phic of finite type in the complex plane and regular at zero. If d is even then for any s ∈ R the family Q λ (cid:101) Q − λ − H − s ( ∂ O ) → H s ( ∂ O ) is meromorphic of finite type in anysector of the logarithmic cover of the complex plane and Hahn-holomorphic at zero.Proof. First note that ˜ Q − λ is Hahn-holomorphic as a family of operators H s +1 ( ∂ O ) → H s ( ∂ O ). This was shown in the proof of Lemma 2.3 using Fredholm theory. If d ≥ T λ is a Hahn-holomorphic family of operators H s ( ∂ O ) → H r ( ∂ O ) for any r >
0. Incase d = 2 the family T λ is Hahn-meromorphic with singular term having integral kernel( γF ) log( λ ), where γF has constant integral kernel. From the singularity expansion of˜ Q λ and ˜ Q λ ˜ Q − λ = 1 we see that F γ ∗ ˜ Q − = 0. This shows that also in dimension two T λ is a Hahn-holomorphic family of operators H s ( ∂ O ) → H r ( ∂ O ) for any r > T λ (cid:101) Q − λ is Hahn holomorphic as a family of operators H s ( ∂ O ) → H − s ( ∂ O ). It istherefore continuous at zero. From (24), we conclude Q λ (cid:101) Q − λ − T λ (cid:101) Q − λ . RELATIVE TRACE FORMULA 17
This shows now that Q λ (cid:101) Q − λ − D (cid:15) . This implies the claimed bounds for | λ | ≤
1. The bounds for | λ | > (cid:3) Recall that the Fredholm determinant is defined for operators of the form 1 + T where T is trace class, see e.g. [23]. The previous proposition now implies Corollary 3.3.
For (cid:15) > the Fredholm determinant det (cid:16) Q λ (cid:101) Q − λ (cid:17) is well defined, holo-morphic in the sector D (cid:15) , and continuous on the closure D (cid:15) . Moreover, the function Ξ( λ ) = log det (cid:16) Q λ (cid:101) Q − λ (cid:17) satisfies on D (cid:15) the following bounds | Ξ( λ ) | ≤ C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ , | Ξ (cid:48) ( λ ) | ≤ C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ for any δ (cid:48) with < δ (cid:48) < δ .Proof. First note that the Proposition 3.2 implies that Q λ (cid:101) Q − λ − H ( ∂ O ) and its trace-norm is bounded by C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ . Since it is also a smoothing operator it is trace-class as an operator on L ( ∂ O ) and all the eigenvectors for non-zero eigenvalues are smooth. It follows that thetrace of Q λ (cid:101) Q − λ − L ( ∂ O ) and the Fredholm determinant of Q λ (cid:101) Q − λ can also be computed in the Hilbert space H ( ∂ O ). This implies | det (cid:16) Q λ (cid:101) Q − λ (cid:17) − | = C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ for sufficiently large | λ | and therefore also | Ξ( λ ) | = C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ for sufficiently large | λ | . Now note that boundedness for λ in a compact set is implied bythe estimate for the | Ξ (cid:48) ( λ ) | for all λ by integration. Hence, we get the claimed estimatefor Ξ( λ ) once we have shown the bound for | Ξ (cid:48) ( λ ) | .Since for every λ in the sector Q λ is an isomorphism H s ( ∂ O ) → H s +1 ( ∂ O ) and thedeterminant can be computed in H s for any s ∈ R we have det Q λ (cid:101) Q − λ = det (cid:101) Q − λ Q λ .We will compute this determinant in H − ( ∂ O ) . Since (cid:101) Q − λ Q λ − (cid:101) Q − λ T λ is a holo-morphic family of smoothing operators for Im( λ ) > (cid:48) ( λ ) = Tr (cid:18) ( ddλ Q λ ) Q − λ − ( ddλ ˜ Q λ ) (cid:101) Q − λ (cid:19) . We have used ddλ (cid:16) (cid:101) Q − λ Q λ (cid:17) = − (cid:101) Q − λ ( ddλ (cid:101) Q λ ) (cid:101) Q − λ Q λ + (cid:101) Q − λ ddλ Q λ = (cid:101) Q − λ (cid:18) ( ddλ Q λ ) Q − λ − ( ddλ (cid:101) Q λ ) (cid:101) Q − λ (cid:19) Q λ , and thereforeTr (cid:18)(cid:16) (cid:101) Q − λ Q λ (cid:17) − ddλ (cid:16) (cid:101) Q − λ Q λ (cid:17)(cid:19) = Tr (cid:18) ( ddλ Q λ ) Q − λ − ( ddλ (cid:101) Q λ ) (cid:101) Q − λ (cid:19) . In the last step we have again used that the trace may be computed in any Sobolevspace. We obtain( ddλ Q λ ) Q − λ − ( ddλ ˜ Q λ ) (cid:101) Q − λ = (cid:18) ddλ (cid:16) Q λ − (cid:101) Q λ (cid:17)(cid:19) Q − λ + (cid:18) ddλ ˜ Q λ (cid:19) (cid:16) Q − λ − (cid:101) Q − λ (cid:17) (25)= (cid:18) ddλ T λ (cid:19) Q − λ − (cid:18) ddλ ˜ Q λ (cid:19) (cid:101) Q − λ T λ Q − λ . This is a smoothing operator and we will compute its trace in the Hilbert space H .Now let P be an elliptic invertible pseudodifferential operator of order one on ∂ O . Let f ( λ ) be the trace-norm of P ddλ ( T λ ) as a trace-class operator from H − → H − , andlet f ( λ ) be the trace-norm of P T λ as a trace-class operator from H − → H − . Then, | Tr (cid:16)(cid:16) ddλ T λ (cid:17) Q − λ (cid:17) | ≤ (cid:107) P − (cid:107) H − → H f ( λ ) (cid:107) Q − λ (cid:107) H → H − , | Tr (cid:16)(cid:16) ddλ ˜ Q λ (cid:17) (cid:101) Q − λ T λ Q − λ (cid:17) | ≤ (cid:107) P − (cid:107) H − → H (cid:107) P (cid:107) H → H − f ( λ ) × (cid:107) Q − λ (cid:107) H → H − (cid:107) (cid:101) Q − λ (cid:107) H → H − (cid:107) ddλ (cid:101) Q λ (cid:107) H − → H . Therefore, for some
C >
0, we have | Ξ (cid:48) ( λ ) | ≤ C (cid:16) f ( λ ) (cid:107) Q − λ (cid:107) H → H − + f ( λ ) (cid:107) Q − λ (cid:107) H → H − (cid:107) (cid:101) Q − λ (cid:107) H → H − (cid:107) ddλ (cid:101) Q λ (cid:107) H − → H (cid:17) . For | λ | > G λ, to establish the bounds (cid:107) G λ, (cid:107) H − ( R d ) → H ( R d ) ≤ C, (cid:107) ddλ G λ, (cid:107) H − ( R d ) → H ( R d ) ≤ C | λ | , for all λ in D (cid:15) . Therefore, using Q λ = γ ◦ G λ, ◦ γ ∗ , we obtain (cid:107) Q λ (cid:107) H − ( R d ) → H ( R d ) ≤ C (cid:48) , (cid:107) ddλ Q λ (cid:107) H − ( R d ) → H ( R d ) ≤ C (cid:48) | λ | , for | λ | > λ in D (cid:15) , and the same bounds hold for (cid:101) Q λ and its derivative. Now,using Prop. 3.1 we obtain | f ( λ ) | ≤ C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ and | f ( λ ) | ≤ C δ (cid:48) ,(cid:15) e − δ (cid:48) Im λ for | λ | > | λ | >
1. It now remains to establish that | Ξ (cid:48) ( λ ) | is boundedfor | λ | ≤ (cid:16) ddλ T λ (cid:17) Q − λ , and (cid:16) ddλ ˜ Q λ (cid:17) (cid:101) Q − λ T λ Q − λ (26)are Hahn-holomorphic families of trace-class operators on H ( ∂ O ). We know that Q − λ , (cid:101) Q − λ are Hahn holomorphic as maps H s ( ∂ O ) → H s − ( ∂ O ). If d ≥ P T λ and P ddλ T λ are Hahn holomorphic as family of trace-class operators on H − ( ∂ O ),and ddλ ˜ Q λ is Hahn holomorphic as a family of maps H s ( ∂ O ) → H s +1 ( ∂ O ). It followsthen that the above are Hahn holomorphic families of trace-class operators and thereforethe trace is continuous at zero. The two dimensional case is slightly more complicatedsince P T λ , P ddλ T λ and ddλ ˜ Q λ are Hahn-meromorphic. In this case the Hahn-expansion RELATIVE TRACE FORMULA 19 can be used to compute the singular terms of these families. If we split the correspond-ing Hahn-meromorphic, possibly not holomorphic integral kernel term, γF into diagonaland off diagonal kernel terms γF = (cid:103) γF + W then the integral kernels (cid:103) γF and W areconstant on every connected component of ∂ O . This splitting implies (cid:103) γF γ ∗ (cid:101) Q − = 0, (cid:103) γF γ ∗ Q − = 0, W γ ∗ (cid:101) Q − = 0, and W γ ∗ Q − = 0. We have modulo Hahn-holomorphicterms P T λ ∼ P W γ ∗ log λ, P ddλ T λ ∼ P W γ ∗ λ , ddλ ˜ Q λ ∼ (cid:103) γF γ ∗ λ . Therefore, in the expressions (26) all singular terms cancel out and the operator families(26) are Hahn holomorophic. (cid:3)
Corollary 3.4. If d is odd then the function Ξ (cid:48) is meromorphic on the complex plane.If d is even then the function Ξ (cid:48) is meromorphic on the logarithmic cover of the complexplane and Hahn-holomorphic at zero on any sector of the logarithmic cover of the complexplane. The relative setting
We consider the resolvent difference R rel ,λ = (cid:0) (∆ − λ ) − − (∆ − λ ) − (cid:1) − N (cid:88) j =1 (cid:0) (∆ j − λ ) − − (∆ − λ ) − (cid:1) . Using (17), we conclude (∆ j − λ ) − − (∆ − λ ) − = −S λ Q − j,λ S t λ and hence R rel ,λ = −S λ Q − λ S t λ + S λ (cid:101) Q − λ S t λ . We have the following improvement of Theorem 2.9 in the relative setting:
Theorem 4.1.
Let δ (cid:48) be any positive constant smaller than δ . Then the operator R rel ,λ is trace-class and its trace norm can be estimated by (cid:107) R rel ,λ (cid:107) ≤ C δ (cid:48) ,(cid:15) ρ (Im λ ) e − δ (cid:48) Im( λ ) . Proof.
As before, let P be an elliptic invertible pseudodifferential operator of order oneon ∂ O and f ( λ ) be the trace-norm of P T λ as a trace-class operator from H − → H − .Computing the trace norm of R rel ,λ we see that (cid:107) R rel ,λ (cid:107) = (cid:107)S λ ( Q − λ − ˜ Q − λ ) S t λ (cid:107) = (cid:107)S λ ( Q − λ T λ ˜ Q − λ ) S t λ (cid:107) ≤ ||S λ Q − λ || H → L || P − || H − → H f ( λ ) || ˜ Q − λ S t λ || L → H − ≤ C δ (cid:48) ,(cid:15) (1 + | λ | ) ρ (Im λ ) e − δ (cid:48) Im( λ ) by Propositions 2.2 and 3.1 and Corollary 2.8. (cid:3) Theorem 4.2.
Let f ∈ ˜ P (cid:15) . Then, D f extends to a trace-class operator and Tr( D f ) = − i π (cid:90) ˜Γ (cid:15) f (cid:48) ( λ )Ξ( λ ) dλ. Proof.
Let f ( λ ) = g ( λ ) with g ∈ P (cid:15) . Assume without loss of generality that 0 < (cid:15) < π/ g n defined by g n ( z ) = g ( z ) exp( − n z ) is an admissible function for the Riesz-Dunford functional calculus for every n ∈ N . Let f n be the corresponding function suchthat f n ( λ ) = g n ( λ ). We then have that f n (∆ ) = g n (∆) is a bounded operator and f n (∆ ) = g n (∆) = − i π (cid:90) ˜Γ (cid:15) λf n ( λ )(∆ − λ ) − dλ. By standard functional calculus for self-adjoint operators we have for every φ in thedomain of f (∆ ) that lim n →∞ f n (∆ ) φ = f (∆ ) φ. Since g is polynomially bounded on the real line the domain of smoothness of the operator∆ is contained in the domain of f (∆ ). In particular the domain of f (∆ ) the space ofsmooth functions compactly supported in R d \ ∂ O and the operator f (∆ ) has a uniquedistributional kernel. Therefore, D f n = − i π (cid:90) ˜Γ (cid:15) λf n ( λ ) R rel ,λ dλ where ˜Γ (cid:15) is the boundary of D (cid:15)/ (see paragraph after Definition 1.3). If φ ∈ C ∞ ( R d \ ∂ O )we have D f n φ → D f φ in L . Using the bound (cid:107) R rel ,λ (cid:107) ≤ C δ (cid:48) ,(cid:15) ρ (Im λ ) e − δ (cid:48) Im λ and since | f ( λ ) | = O ( | λ | a ) for some a > D f n is a Cauchy sequenceconverging in the Banach space of trace-class operators to D f = − i π (cid:90) ˜Γ (cid:15) λf ( λ ) R rel ,λ dλ. Therefore D f is trace-class and the trace commutes with the integral. Using S t λ S λ = γG λ, γ ∗ = 12 λ γ ddλ G λ, γ ∗ = 12 λ ddλ Q λ , and Tr (cid:18) ( ddλ T λ ) (cid:101) Q − λ (cid:19) = 0we find Tr ( R rel ,λ ) = Tr (cid:16) −S t λ S λ Q − λ + S t λ S λ (cid:101) Q − λ (cid:17) = − λ Tr (cid:18) ( ddλ Q λ ) Q − λ − ( ddλ Q λ ) (cid:101) Q − λ (cid:19) = − λ Tr (cid:18) ( ddλ Q λ ) Q − λ − ( ddλ (cid:101) Q λ ) (cid:101) Q − λ − ( ddλ T λ ) (cid:101) Q − λ (cid:19) (27)= − λ Tr (cid:18) ( ddλ Q λ ) Q − λ − ( ddλ (cid:101) Q λ ) (cid:101) Q − λ (cid:19) = − λ Ξ (cid:48) ( λ ) . We have used the fact that R rel ,λ is a trace class operator to perform the algebraicmanipulations. We then have that:Tr( D f ) = − i π (cid:90) ˜Γ (cid:15) λf ( λ )Tr ( R rel ,λ ) dλ = i2 π (cid:90) ˜Γ (cid:15) f ( λ )Ξ (cid:48) ( λ ) dλ = − i2 π (cid:90) ˜Γ (cid:15) f (cid:48) ( λ )Ξ( λ ) dλ. RELATIVE TRACE FORMULA 21 (cid:3)
Theorem 4.3.
For λ > we have π Im Ξ( λ ) = − i2 π (Ξ( λ ) − Ξ( − λ )) = ξ rel ( λ ) . Proof.
Recall that Ξ (cid:48) has a meromorphic extension to the logarithmic cover of the com-plex plane. In particular we can choose a non-empty interval ( a, b ) ⊂ R + such that Ξ (cid:48) is holomorphic near ( a, b ) and − ( a, b ). Now assume that f is an arbitrary even functionthat is compactly supported in − ( a, b ) ∪ ( a, b ). Let d m ( z ) = d x d y be the Lebesguemeasure on C . By the Helffer-Sj¨ostrand formula we have f (∆ / ) = 2 π (cid:90) Im( z ) > z ∂ ˜ f∂z (∆ − z ) − d m ( z ) , where ˜ f is a compactly supported almost analytical extension of f (see [28] Prop 7.2and [29] p.169/170). This implies that D f = 2 π (cid:90) Im( z ) > z ∂ ˜ f∂z R rel ,z d m ( z ) , and hence Tr( D f ) = − π (cid:90) Im( z ) > ∂ ˜ f∂z Ξ (cid:48) ( z )d m ( z ) . by (27). Using Stokes’ theorem in the form of [30] (p.62/63), we therefore obtainTr( D f ) = i2 π (cid:90) ba (cid:0) Ξ (cid:48) ( x ) + Ξ (cid:48) ( − x ) (cid:1) f ( x )d x. Comparing this with the Birman-Krein formula gives − i2 π (Ξ (cid:48) ( x ) + Ξ (cid:48) ( − x )) = ξ (cid:48) rel ( x ) forall x ∈ ( a, b ). Since both functions are meromorphic this shows that this identity holdseverywhere. We conclude that i2 π (Ξ( λ ) − Ξ( − λ )) + ξ rel ( λ ) is constant in the upper halfspace. Since Ξ( λ ) is Hahn-meromorphic and bounded, it is Hahn holomorphic in anysector of the complex plane. This implies that (Ξ (cid:48) ( λ ) − Ξ (cid:48) ( − λ )) → λ → ξ rel ( λ ) → λ →
0. Hence, i2 π (Ξ( λ ) − Ξ( − λ )) + ξ rel ( λ ) vanishes everywhere. Wehave by definition for λ > − Ξ (cid:48) ( − λ ) = Tr (cid:18)(cid:18) ddλ (cid:16) Q − λ − (cid:101) Q − λ (cid:17)(cid:19) Q − − λ + (cid:18) ddλ ˜ Q − λ (cid:19) (cid:16) Q − − λ − (cid:101) Q − − λ (cid:17)(cid:19) = Tr (cid:18) ( Q − λ ) ∗ (cid:18) ddλ (cid:16) Q λ − (cid:101) Q λ (cid:17)(cid:19) ∗ + (cid:16) Q − λ − (cid:101) Q − λ (cid:17) ∗ (cid:18) ddλ ˜ Q λ (cid:19) ∗ (cid:19) = Tr (cid:18)(cid:18) ddλ (cid:16) Q λ − (cid:101) Q λ (cid:17)(cid:19) Q − λ + (cid:18) ddλ ˜ Q λ (cid:19) (cid:16) Q − λ − (cid:101) Q − λ (cid:17)(cid:19) = Ξ (cid:48) ( λ )where Q − − λ = ( Q − λ ) ∗ for λ >
0. Here we have used the fact that we can switch theposition of the individual operators under the trace if they are trace class. This impliesIm(Ξ (cid:48) ( λ )) = − i2 (Ξ (cid:48) ( λ ) + Ξ (cid:48) ( − λ )). This again shows that Im(Ξ( λ )) + i2 (Ξ( λ ) − Ξ( − λ )) is constant on the positive real line. The right hand side vanish at zero by the sameargument as before. To finally establish that Im(Ξ( λ )) = − i2 (Ξ( λ ) − Ξ( − λ )) it onlyremains to show that Ξ( λ ) is real at λ = 0. This follows immediately from the fact that Q λ and (cid:101) Q λ both have purely real kernels on the positive imaginary axis, which impliesthat Ξ( λ ) is real valued for all λ on the positive imaginary line. (cid:3) Proofs of main theorems
Proof of Theorem 1.4.
The proof of this theorem is similar to the proof of Theorem 4.2.Assume that f is in ˜ P (cid:15) and f ( z ) = g ( z ) so that g ∈ P (cid:15) . As in the proof of Theorem 4.2we conclude that the domain of g (∆) contains C ∞ ( M ). The operator g (∆) therefore hasunique distributional integral kernel k ∈ D (cid:48) ( M × M ). There exists a sector S (cid:15) such thatthe sequence ( g n ) of functions g n ( z ) = g ( z ) e − n z is in E (cid:15) . Since all g n are admissible forthe Riesz-Dunford holomorphic functional calculus, we obtain pg n (∆) p − pg n (∆ ) p = (cid:90) Γ (cid:15) g n ( z ) (cid:0) p (∆ − z ) − p − p (∆ − z ) − p (cid:1) dz. By the estimate on the resolvent the integrand consists of trace-class operators and theintegral converges in the Banach space of trace-class operators. For g ∈ E (cid:15) we obtainthat pg (∆) p − pg (∆ ) p is trace-class and we also get the following estimate for its tracenorm (cid:107) pg (∆) p − pg (∆ ) p (cid:107) ≤ C (cid:90) Γ (cid:15) | g ( λ ) | | λ | e − C (cid:48) | λ | dλ. We apply this estimate to g n as well as to g n − g m to conclude that pg n (∆) p − pg n (∆ ) p is a sequence of trace-class operators that is Cauchy in the Banach space of trace-classoperators. By Borel functional calculus ( pg n (∆) p − pg n (∆ ) p ) h converges to ( pg (∆) p − pg (∆ ) p ) h in L (Ω) for any h ∈ C ∞ (Ω). This establishes the first part of the Theorem.The rest follows immediately from the bounds on the kernel in Theorem 2.9. Indeed, indimensions d ≥ x, ∂ Ω), we have | pf (∆ ) p − pf (∆ ) p | ≤ C (cid:90) ˜Γ (cid:15) | f ( λ ) | e − C Im λ dist( x,∂ Ω) dist( x, ∂ Ω) d − | λ | dλ ≤ C Ω dist( x, ∂ Ω) d − a . Similarly, for large dist( x, ∂
Ω) and d = 2, we find | pf (∆ ) p − pf (∆ ) p | ≤ C (cid:90) ˜Γ (cid:15) | f ( λ ) | e − C Im λ dist( x,∂ Ω) (1 + | log( λd ( x, ∂ Ω)) | ) | λ | dλ ≤ C Ω dist( x, ∂ Ω) a (cid:3) Proof of Theorem 1.6.
The first two points of Theorem 1.6 follow directly from Corollary3.3. The last third point follows from Theorem 4.3, and the last is a consequence ofTheorem 4.2. The only point that remains to be shown is uniqueness of the functionΞ. Suppose there is another function ˜Ξ with the same properties. Then the difference
RELATIVE TRACE FORMULA 23
Θ = Ξ − ˜Ξ is holomorphic in the upper half plane, decays rapidly along the positiveimaginary line, and we have (cid:90) ∞ λp ( λ ) exp( − λ (cid:48) (i λ ) dλ = 0for any even polynomial p . Since functions of the form p ( λ ) exp( − λ ) are dense in L ( R + )and Θ (cid:48) (i λ ) is in L ( R + ) we conclude that Θ (cid:48) vanishes on the positive imaginary line.Since Θ (cid:48) is holomorphic in the upper half space and meromorphic on the logarithmiccover of the complex plane it must vanish. Thus, Θ is constant. The decay now impliesΘ = 0 and hence Ξ = ˜Ξ. (cid:3) Proof of Theorem 1.7.
It was shown in the main text that Ξ has all the claimed proper-ties of Theorem 1.6. (cid:3)
Acknowledgement
The authors would like to thank James Ralston and Steve Zelditch for useful comments.
Appendix A. Estimates for the resolvent kernel
From the Nist digital library [20] eqs. 10.17.13,14, and 15, we have that H (1) ν ( z ) = (cid:18) πz (cid:19) e iω (cid:32) (cid:96) − (cid:88) k =0 ( ± i ) k a k ( ν ) z k + R ± (cid:96) ( ν, z ) (cid:33) (28)where (cid:96) ∈ N , ω = z − νπ − π and (cid:12)(cid:12) R ± (cid:96) ( ν, z ) (cid:12)(cid:12) ≤ | a (cid:96) ( ν ) |V z, ± i ∞ (cid:16) t − (cid:96) (cid:17) exp (cid:0) | ν − |V z, ± i ∞ (cid:0) t − (cid:1)(cid:1) , where V z,i ∞ (cid:0) t − (cid:96) (cid:1) may be estimated in various sectors as follows V z,i ∞ (cid:16) t − (cid:96) (cid:17) ≤ | z | − (cid:96) , ≤ ph z ≤ π,χ ( (cid:96) ) | z | − (cid:96) , − π ≤ ph z ≤ π ≤ ph z ≤ π ,2 χ ( (cid:96) ) | Im z | − (cid:96) , − π < ph z ≤ − π or π ≤ ph z < π .Here, χ ( (cid:96) ) is defined by χ ( x ) := π / Γ (cid:0) x + 1 (cid:1) / Γ (cid:0) x + (cid:1) . For z →
0, we have the estimate [20] equations 10.7.7 and 2. H (1) ν ( z ) ∼ (cid:40) − ( i/π )Γ ( ν ) ( z ) − ν for ν > i/π ) log z for ν = 0 (29)Here ∼ means that the quotient of left- and right hand side converges to 1 as z → Combining (28) and (29), we conclude that for ν ≥
0, there exist positive constants C and r such thatfor | z | ≤ r : | H (1) ν ( z ) | ≤ C (cid:40) | z | − ν for ν > | log( z ) | for ν = 0 (30)for | z | ≥ r : | H (1) ν ( z ) | ≤ C | z | − / e − Im z (31)Here, r > C depends on ν and the choice of r .(31) can be obtained from (28) for any choice of (cid:96) ∈ N since negative powers of | z | arebounded above for | z | ≥ r . We will apply these estimates with ν = d − , in that case,the logarithm in (30) corresponds precisely to d = 2.Finally, we recall (see [20], (10.6.7)) that derivatives of Hankel functions can be expressedas d jz H (1) κ ( z ) = 2 − j · (cid:80) jl =0 ( − l (cid:0) jl (cid:1) H (1) κ − j +2 l . (32)In the following we assume that O and M are as in the main body of the text. Recallthat the integral kernel of the free resolvent is given by G λ, ( x, y ) = i4 (cid:18) λ π | x − y | (cid:19) ν d H (1) ν d ( λ | x − y | ) where ν d = d − . (33)We will subsequently prove norm and pointwise estimates for G λ, and its derivatives,which are used in the main body of the text. Lemma A.1.
Let Ω ⊂ M be an open set with dist(Ω , O ) = δ > and λ ∈ D (cid:15) . Then,for any < δ (cid:48) < δ and any m ∈ R there exists C δ (cid:48) ,(cid:15),m > such that we have (cid:107) G λ, (cid:107) H m (Ω × ∂ O ) ≤ C δ (cid:48) ,(cid:15),m ρ (Im λ ) e − δ (cid:48) Im λ . (34) Proof.
Let us set λ = θ | λ | and note that Im( θ ) ≥ sin( (cid:15) ) >
0. We consider a smoothtubular neighbourhood of the boundary, say U which is such that dist(Ω , U ) = δ > δ (cid:48) < δ < δ and ∂ O ⊂ U . Since the kernel G λ, satisfies the Helmholtz equation in bothvariables away from the diagonal we have (∆ x + ∆ y ) k G λ, ( x, y ) = (2 λ ) k G λ, ( x, y ). Wethen change variables so that r := | x − y | ≥ δ . By homogeneity, all of the integration willbe carried out in this variable, with the angular variables only contributing a constant.Substituting s := Im λ r , equations (31) and (30) imply for all k ∈ N (cid:107) ∆ k G λ, (cid:107) L (Ω × U ) ≤ C k (Im λ ) k ∞ (cid:90) δ | G λ, ( r ) | r d − dr ≤ C k (Im λ ) d +4 k − (cid:18)(cid:90) s Im λδ h ( s ) ds + (cid:90) ∞ s | e − s · Im θ | ds (cid:19) , (35)where h ( s ) := s − d +3 for d ≥ h ( s ) := s log( s | λ | / Im λ ) + s in case d = 2. We haveused that | λ | can be bounded by a multiple of Im λ ) in the sector. Moreover, we mayassume without loss of generality that s >
1. Note that the second summand in (35)can be bounded by C k | Im λ | d +4 k − e − δ (cid:48) Im( λ ) , (36) RELATIVE TRACE FORMULA 25 the constant being independent of λ . The first term can be computed and estimatedexplicitly. A short computation shows that for d > (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s Im λδ h ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:40) C · | Im λδ | − d + C (cid:48) for d (cid:54) = 4 C · | log(Im λδ ) | + C (cid:48) for d = 4 , (37)whereas for d = 2, we may estimate the integral by C · ( | Im λ | δ ) · (cid:0) log( | Im λ | δ ) − log( | Im λ | δ ) + (cid:1) + C · | Im λδ | + C (cid:48) . (38)Here, the constants C and C (cid:48) depend on the dimension. Combining (35) and (38) whileadjusting constants proves the lemma. Let C δ (cid:48) ,(cid:15),k denote a generic constant dependingon δ (cid:48) , (cid:15), k . As a result for all k ∈ R we can conclude (cid:107) G λ, (cid:107) H k (Ω × U ) ≤ C δ (cid:48) ,(cid:15),k ρ (Im λ ) e − δ (cid:48) Im λ . (39)Furthermore, the trace theorem then implies (cid:107) G λ, (cid:107) H m (Ω × ∂ O ) ≤ C δ (cid:48) ,(cid:15),m ρ (Im λ ) e − δ (cid:48) Im λ . (40)for all m ∈ R , whence the Lemma is proved. (cid:3) Lemma A.2.
Let x ∈ R d \ O . Let dist( x , ∂ O ) be abbreviated as δ ( x ) . For λ ∈ D (cid:15) ,we have whenever δ ( x ) | λ | ≤ for any multi-indices α, β ∈ N d the estimates sup y ∈ ∂ O | ∂ αx ∂ βy G λ, ( x , y ) | ≤ C · (cid:40) δ ( x ) − ( d − | α | + | β | ) · ( d ≥ | log( δ ( x ) λ ) | + δ ( x ) − ( | α | + | β | ) ( d = 2) . (41) The constant depends on α and β . If | α | + | β | > , there is indeed no log -contributionfor d = 2 . In the case that δ ( x ) | λ | > , we conclude sup y ∈ ∂ O | ∂ αx ∂ βy G λ, ( x , y ) | ≤ C | λ | | α | + | β | + d − e − Im λδ ( x ) (42) for all d . Assuming that < δ ( x ) , combining the estimates gives for x ∈ Ω , Ω ⊂ R d \O : sup y ∈ ∂ O | ∂ αx ∂ βy G λ, ( x , y ) | ≤ C e − C δ ( x ) Im λ δ ( x ) − ( d − (43) where C , C depend on Ω , α, β and (cid:15) .Proof. We make the change of variables r := | x − y | >
0. For d ≥
3, we can split theintegral kernel into two parts, | rλ | > | rλ | <
1. We do the ∂ βy estimates only withthe ∂ αx estimates following by symmetry. From (30) and (31), we have that | ∂ βy G λ, ( x , y ) | ≤ sup δ ( x ) 1, we have thatsup δ ( x ) Let Ω and λ be chosen as above. Let P y , P x be differential operatorson ∂ O and Ω of order k and (cid:96) , respectively. Assume that P x has bounded coefficients.For | λδ ( x ) | ≤ , we have sup y ∈ ∂ O | P x P y G λ, ( x , y ) | ≤ C · (cid:40) δ ( x ) − ( d − + δ ( x ) − ( d − k + (cid:96) ) ( d ≥ | log( δ ( x ) λ ) | + δ ( x ) − k − (cid:96) ( d = 2) . If P x (1) = 0 = P y (1) (no constant terms), we can improve the estimate by replacing theright hand side by C ( δ ( x ) − ( d − + δ ( x ) − ( d + k + (cid:96) − ) for all d ≥ . If | λδ ( x ) | ≥ , wehave sup y ∈ ∂ O | P x P y G λ, ( x , y ) | ≤ C | λ | k + (cid:96) + d − e − (Im λ ) · δ ( x ) for all d ≥ . Assuming δ ( x ) > , we find for all d that sup y ∈ ∂ O | P x P y G λ, ( x , y ) | ≤ C e − C Im λδ ( x ) · δ ( x ) d − . Here, all constants depend on P x , P y , Ω and (cid:15) .Proof. Any coordinate derivative ∂ αy on ∂ O can be expressed in terms of the Cartesianderivatives ∂ i on R d , with local coefficients in c αi ∈ C ∞ ( ∂ O ): ∂ αy = (cid:80) i c αi ∂ i . Hence, wehave locally in y ∈ ∂ O that( P y G λ, ( x , · ))( y ) = (cid:88) | I |≤ k c I ( y )( ∂ I G λ, ( x , · ))( y )for a I (locally defined) smooth functions on ∂ O . Note that we have contributions with | I | < k since c αi are in general not constant. P x acts in an analogous fashion. Using theestimates from Lemma A.2 and the fact that ∂ O is compact, we obtain the corollary.For | λδ ( x ) | ≤ 1, we have only kept the smallest and largest power of δ ( x ). (cid:3) Since integrals over the compact space ∂ O can easily be estimated in terms of L ∞ -norms,we also obtain estimates for Sobolev norms. For simplicity, we only state it for the caseof a finite distance between ∂ O and Ω: RELATIVE TRACE FORMULA 27 Corollary A.4. Let (cid:15) > , λ ∈ D (cid:15) and Ω as above such that dist( ∂ O , Ω) > . For any s ∈ R and x ∈ Ω , we have (cid:107) G λ, ( x , · ) (cid:107) H s ( ∂ O ) ≤ C e − C (Im λ ) δ ( x ) · δ ( x ) d − . Here, C and C depend on s, (cid:15) and the choice of Ω ; the estimate is valid for all d ≥ . Appendix B. Hahn holomorphic and Hahn meromorphic functions The theory of Hahn analytic functions was developed in [19] in a very general setting.This appendix is taken directly from [25]. For the purposes of this paper we restrictour considerations to so-called z -log( z )-Hahn holomorphic functions and refer to theseas Hahn holomorphic. To be more precise, suppose that Γ ⊂ R is a subgroup of R .We endow Γ and R with the lexicographical order. Recall that a subset A ⊂ Γ is calledwell-ordered if any subset of A has a smallest element. A formal series (cid:88) ( α,β ) ∈ Γ a α,β z α ( − log z ) − β will be called a Hahn-series if the set of all ( α, β ) ∈ Γ with a α,β (cid:54) = 0 is a well orderedsubset of Γ.In the following let Z be the logarithmic covering surface of the complex plane withoutthe origin. We will use polar coordinates ( r, ϕ ) as global coordinates to identify Z as aset with R + × R . Adding a single point { } to Z we obtain a set Z and a projectionmap π : Z → C by extending the covering map Z → C \{ } by sending 0 ∈ Z to0 ∈ C . We endow Z with the covering topology and Z with the topology generatedby the open sets in Z together with the open discs D (cid:15) := { } ∪ { ( r, ϕ ) | ≤ r < (cid:15) } .This means a sequence (( r n , ϕ n )) n converges to zero if and only if r n → 0. The coveringmap is continuous with respect to this topology. For a point z ∈ Z we denote by | z | its r -coordinate and by arg( z ) its ϕ coordinate. We will think of the positive real axisembedded in Z as the subset { z | arg( z ) = 0 } . Define the following sectors D [ σ ] δ = { z ∈ Z | ≤ | z | < δ, | ϕ | < σ } .In the following fix σ > V . We say a function f : D [ σ ] δ → V is Hahn holomorphic near 0 in D [ σ ] δ if there exists a Hahn series with coefficients in V that converges normally to f , i.e. such that f ( z ) = (cid:88) ( α,β ) ∈ Γ a α,β z α ( − log z ) − β and (cid:88) ( α,β ) ∈ Γ (cid:107) a α,β (cid:107)(cid:107) z α ( − log z ) − β (cid:107) L ∞ ( D [ σ ] δ ) < ∞ and there exists a constant C > a α,β = 0 if − β > Cα . This implies also that a α,β = 0 in case ( α, β ) < (0 , V is a Banach algebra the setof Hahn holomorphic functions with values in V is an algebra. A meromorphic functionon D [ σ ] δ \{ } is called Hahn meromorphic with values in a Banach space V if near zeroit can be written as a quotient of a Hahn holomorphic function with values in V and aHahn holomorphic function with values C . Note that the algebra of Hahn holomorphicfunctions with values in C is an integral domain and Hahn meromorphic functions with values in C form a field. There exists a well defined injective ring homomorphism fromthe field of Hahn meromorphic functions into the field of Hahn series. This ring homo-morphism associates to each Hahn meromorphic function its Hahn series. The theoryis in large parts very similar to the theory of meromorphic functions. In particular thefollowing very useful statement holds: if V is a Banach space and f : D [ σ ] δ → V Hahnmeromorphic and bounded, then f is Hahn holomorphic. The main result of [19] statesthat the analytic Fredholm theorem holds for this class of functions. References [1] R. Balian and C. Bloch. Distribution of eigenfrequencies for the wave equation in a finite domain.I. Three-dimensional problem with smooth boundary surface. Ann. Physics , 60:401–447, 1970.[2] C. Bardos, J.-C. Guillot, and J. Ralston. 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