Spectral shift functions and Dirichlet-to-Neumann maps

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh–Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these applications the spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps.


Introduction
Let A and B be self-adjoint operators in a separable Hilbert space H and assume that the m-th powers of their resolvents differ by a trace class operator, for some odd integer m ∈ N. It is known that in this case there exists a real-valued function ξ ∈ L 1 loc (R) such that R |ξ(λ)|(1+|λ|) −(m+1) dλ < ∞ and the trace formula Historically the trace formula (1.2) was first proposed and verified on a formal level by Lifshitz for the case that [B − A] is a finite-rank operator in [51] (see also [52]), and shortly afterwards in [44] Krein proved (1.2) rigorously in the more general case [B − A] ∈ S 1 (H) for all C 1 -functions ϕ with derivatives in the Wiener class. Furthermore, in [44] it was shown how the spectral shift function ξ can be computed with the help of the perturbation determinant corresponding to the pair {A, B}. For pairs of unitary operators and thus via Cayley transforms for the case m = 1 in (1.1) the spectral shift function and the trace formula were obtained later by Krein in [45]. Afterwards in [43] the more general case m > 1 in (1.1) for self-adjoint operators A and B with ρ(A) ∩ ρ(B) ∩ R = ∅ was discussed by Koplienko, and for odd integers m in (1.1) and arbitrary self-adjoint operators A and B see [74] by Yafaev or [73,Chapter 8,§11] and [76, Chapter 0, Theorem 9.4]. We also mention that the spectral shift function is closely connected with the scattering matrix via the famous Birman-Krein formula from [11,12]. For more details on the history, development and multifaceted applications of the spectral shift function in mathematical analysis we refer the reader to the survey papers [13,16,17], the standard monographs [73,76], and, for instance, to [14,19,24,26,27,31,46,47,67,70] and the more recent contributions [1,25,30,39,40,42,48,55,56,[64][65][66]68,75].
The main objective of the present paper is to prove a representation formula for the spectral shift function in terms of an abstract Titchmarsh-Weyl m-function of two self-adjoint operators satisfying the condition (1.1), and to apply this result to different self-adjoint realizations of second-order elliptic PDEs and Schrödinger operators with compactly supported potentials. In these applications the abstract Titchmarsh-Weyl m-function will turn out to be the energy dependent Neumann-to-Dirichlet map or Dirichlet-to-Neumann map associated to the elliptic differential expression and the Schrödinger operators on an interior and exterior domain, respectively.
More precisely, assume that A and B are self-adjoint operators in a separable Hilbert space H and consider the underlying closed symmetric operator which for convenience we assume is densely defined. We emphasize that neither A nor B needs to be semibounded in our approach. However, we first impose an implicit sign condition on the perturbation by assuming for some μ 0 ∈ ρ(A) ∩ ρ(B) ∩ R; in the semibounded case the condition (1.3) is equivalent to A ≤ B interpreted in the sense of the corresponding quadratic forms. We then make use of the concept of quasi boundary triples in extension theory of symmetric operators from [2,3] and construct an operator T such that T = S * and two boundary mappings 0 , 1  Since z → log M(z) is a Nevanlinna function it follows that the values of the spectral shift function ξ in (1.5) and (1.6) are nonnegative for a.e. λ ∈ R; this is rooted in the sign condition (1.3). In a second step we weaken the sign condition (1.3) and extend our representation of the spectral shift function to more general perturbations in the end of Sect. 4. We point out that the key difficulty in the proof of (1.5) and (1.6) is to ensure the existence of the limits on the right hand side of (1.5) and the trace class property of the function Im log M in the case k = 0, respectively, which are indispensable for (1.5) and (1.6). These problems are investigated separately in Sect. 3 on the logarithm of operator-valued Nevanlinna functions, where special attention is paid to the analytic continuation by reflection with respect to open subsets of the real line. We also mention that for the special case where (1.1) is a rank one or finite-rank operator and m = 1, our representation for the spectral shift function coincides with the one in [7,49]. Furthermore, for m = 1 in (1.1) a formula for the spectral shift function via a perturbation determinant involving boundary parameters and the Weyl function in the context of ordinary boundary triples was shown recently in [56] (see also [55]). We remark that our abstract result can also be formulated and remains valid in the special situation that the quasi boundary triple {G, 0 , 1 } is a generalized or ordinary boundary triple in the sense of [18,[21][22][23]32].
Our main reason to provide the general result in Sect. 4 for the spectral shift function in terms of the abstract notion of quasi boundary triples and their Weyl functions is its convenient applicability to various PDE situations, see also [2][3][4][5][6]8] for other related applications of quasi boundary triples in PDE problems. In Sect. 5 we consider a formally symmetric uniformly elliptic second-order partial differential expression L with smooth coefficients on a bounded or unbounded domain in R n , n ≥ 2, with compact boundary, and two self-adjoint realizations A β 0 and A β 1 of L subject to Robin boundary conditions β p γ D f = γ N f , where γ D and γ N denote the Dirichlet and Neumann trace operators, and β p ∈ C 1 (∂ ), p = 0, 1, are real-valued functions. It then turns out that the Robin realizations A β 0 and A β 1 satisfy for all k ∈ N 0 , k ≥ (n − 3)/4, and z ∈ ρ(A β 0 ) ∩ ρ(A β 1 ), and for any orthonormal basis (ϕ j ) j∈J in L 2 (∂ ), the function is a spectral shift function for the pair β ∈ R is such that β p (x) < β for all x ∈ ∂ , and N (z) denotes the (z-dependent) Neumann-to-Dirichlet map that assigns Neumann boundary values of solutions f z ∈ H 2 ( ) of L f z = z f z , z ∈ C\R, onto their Dirichlet boundary values. We note that the trace class property (1.7) was shown in [4,34] for the case k = 0 and in [6] for k ≥ 1. Moreover, in the case k = 0, that is, n = 2 or n = 3, it follows from (1.6) that the spectral shift function in (1.8) has the form In our second example, presented in Sect. 6, we consider a Schrödinger operator B = − + V with a compactly supported potential V ∈ L ∞ (R n ). Here we split the Euclidean space R n and the Schrödinger operator via a multi-dimensional Glazman decomposition and consider the orthogonal sum B D = B + ⊕ C of the Dirichlet realizations of − + V in L 2 (B + ) and L 2 (B − ), where B + is a sufficiently large ball which contains supp (V ) and B − := R n \B + . Similarly, the unperturbed operator A = − is decoupled and compared with the orthogonal sum A D = A + ⊕ C of the Dirichlet realizations of − in L 2 (B + ) and L 2 (B − ). Our abstract result applies to the pairs {B, B D } and {A, A D }, whenever k > (n − 2)/4, n ∈ N, n ≥ 2, and yields an explicit formula for their spectral shift functions ξ B and ξ A in terms of the (z-dependent) Dirichlet-to-Neumann maps associated to − and − + V on B + and B − . Since the spectra of the Dirichlet realizations A + = − and B + = − +V on the bounded domain B + are both discrete and bounded from below, the difference of their eigenvalue counting functions is a spectral shift function ξ + for the pair {A + , B + }, and hence also for the pair {A D , B D }. Then it follows that the function is a spectral shift function for the original pair {A, B} (cf. Theorem 6.1). We also mention that the trace class property of the resolvent differences of A and A D , and B and B D goes back to Birman [9] and Grubb [33], and that similar decoupling methods are often used in scattering theory, see, for instance, [20] or [71] for a slighty more abstract and general framework.
The applications in Sects. 5 and 6 serve as typical examples for the abstract formalism and results in Sect. 4. In this context we mention that one may compare in a similar form as in Sect. 5 the Dirichlet realization with the Neumann, or other selfadjoint Robin realizations of an elliptic partial differential expression, and that in principle also higher-order differential expressions with smooth coefficients could be considered. We refer the reader to [28,29,[35][36][37]54,57,58,63] for some recent related contributions in this area.
Finally, we briefly summarize the basic notation used in this paper: Let G, H, H, etc., be separable complex Hilbert spaces, (·, ·) H the scalar product in H (linear in the first factor), and I H the identity operator in H. If T is a linear operator mapping (a subspace of ) a Hilbert space into another, dom(T ) denotes the domain and ran(T ) is the range of T . The closure of a closable operator S is denoted by S. The spectrum and resolvent set of a closed linear operator in H will be denoted by σ (·) and ρ(·), respectively. The Banach space of bounded linear operators in H is denoted by L(H); in the context of two Hilbert spaces, H j , j = 1, 2, we use the analogous abbreviation L(H 1 , H 2 ). The p-th Schatten-von Neumann ideal consists of compact operators with singular values in l p , p > 0, and is denoted by S p (H) and S p (H 1 , H 2 ). For ⊆ R n nonempty, n ∈ N, we suppress the n-dimensional Lebesgue measure d n x and use the shorthand notation L 2 ( ) := L 2 ( ; d n x); similarly, if ∂ is sufficiently regular we write L 2 (∂ ) := L 2 (∂ ; d n−1 σ ), with d n−1 σ the surface measure on ∂ . We also abbreviate C ± := {z ∈ C | Im(z) ≷ 0} and N 0 = N ∪ {0}.

Quasi boundary triples and their Weyl functions
In this section we recall the concept of quasi boundary triples and their Weyl functions from extension theory of symmetric operators. We shall make use of these notions in Sect. 4 and formulate our main abstract result Theorem 4.1 in terms of the Weyl function of a quasi boundary triple. In Sects. 5 and 6 quasi boundary triples and their Weyl functions are used to parametrize self-adjoint Schrödinger operators and selfadjoint elliptic differential operators with suitable boundary conditions. We refer to [2,3] for more details on quasi boundary triples and to [4][5][6]8] for some applications; for the related notions of generalized and ordinary boundary triples see [18,[21][22][23]32,69].
Throughout this section let H be a separable Hilbert space and let S be a densely defined closed symmetric operator in H.

Definition 2.1
Let T ⊂ S * be a linear operator in H such that T = S * . A triple {G, 0 , 1 } is said to be a quasi boundary triple for T ⊂ S * if G is a Hilbert space and 0 , 1 : dom(T ) → G are linear mappings such that the following conditions (i)-(iii) are satisfied: (i) The abstract Green's identity holds for all f, g ∈ dom(T ). The next theorem from [2,3] is useful in the applications in Sects. 5 and 6; it contains a sufficient condition for a triple {G, 0 , 1 } to be a quasi boundary triple.

Theorem 2.2 Let H and G be separable Hilbert spaces and let T be a linear operator in H.
Assume that 0 , 1 : dom(T ) → G are linear mappings such that the following conditions (i)-(iii) hold: (i) The abstract Green's identity is a densely defined closed symmetric operator in H such that T = S * holds and the triple {G, 0 , 1 } is a quasi boundary triple for S * with A 0 = T ker( 0 ).
Next, we recall the definition of the γ -field γ and Weyl function M associated to a quasi boundary triple, which is formally the same as in [22,23] for the case of ordinary or generalized boundary triples. Let {G, 0 , 1 } be a quasi boundary triple for T ⊂ S * with A 0 = T ker( 0 ) and note that the direct sum decomposition Various properties of the γ -field and Weyl function were provided in [2,3], see also [18,[21][22][23]69] for the special cases of ordinary and generalized boundary triples. We briefly review some items which are important for our purposes. Note first that the values γ (z), z ∈ ρ(A 0 ), of the γ -field are operators defined on the dense subspace ran( 0 ) ⊂ G which map onto ker(T − z I H ) ⊂ H. The operators γ (z), z ∈ ρ(A 0 ), are bounded and admit continuous extensions γ (z) ∈ L(G, H). For the adjoint operators γ (z) * ∈ L(H, G), z ∈ ρ(A 0 ), it follows that and, in particular, ran(γ (z) * ) = ran( 1 dom(A 0 )) does not depend on z ∈ ρ(A 0 ). It is also important to note that (ran(γ (z) * )) ⊥ = ker(γ (z)) = {0} and hence In the same way as for ordinary boundary triples one verifies (2.4) and therefore z → γ (z)ϕ is holomorphic on ρ(A 0 ) for all ϕ ∈ ran( 0 ). The relation (2.4) extends by continuity to (2.5) and it follows that z → γ (z) is a holomorphic L(G, H)-valued operator function. According to [6,Lemma 2.4] the identities hold for all k ∈ N 0 and z ∈ ρ(A 0 ). The values M(z), z ∈ ρ(A 0 ), of the Weyl function M associated to a quasi boundary triple are operators in G and it follows from Definition 2.3 that dom(M(z)) = ran( 0 ) and ran(M(z)) ⊂ ran ( 1 ) hold for all z ∈ ρ(A 0 ). In particular, the operators M(z), z ∈ ρ(A 0 ), are densely defined in G. With the help of the abstract Green's identity one concludes that for z, z 0 ∈ ρ(A 0 ) and ϕ, ψ ∈ ran( 0 ) the Weyl function and the γ -field satisfy and hence M(z) ⊂ M(z) * and the operators M(z) are closable for all z ∈ ρ(A 0 ). From (2.7) it also follows that the Weyl function and the γ -field are connected via and M(z)ϕ = Re(M(z 0 ))ϕ for all z, z 0 ∈ ρ(A 0 ) and ϕ ∈ ran( 0 ). One observes that z → M(z)ϕ is holomorphic on ρ(A 0 ) for all ϕ ∈ ran( 0 ) and by (2.9) the imaginary part of M(z) is a bounded operator in G which admits a bounded continuation to Im(M(z)) = Im(z) γ (z) * γ (z) ∈ L(G). (2.11) Furthermore, the derivatives d k dz k M(z), k ∈ N, of the Weyl function are densely defined bounded operators in G and according to [6,Lemma 2.4] one has If the values M(z) are densely defined bounded operators for some, and hence for all (2.12) The next result will be used in the formulation and proof of our abstract representation formula for the spectral shift function in Sect. 4. The existence of a quasi boundary triple follows from [8, Proposition 2.9(i)] and the Krein-type resolvent formula in (2.14) is a special case of [3,Corollary 6.17] or [5,Corollary 3.9].

Proposition 2.4 Let A and B be self-adjoint operators in
(2.14)

Logarithms of operator-valued Nevanlinna functions
In this section we study the logarithm of operator-valued Nevanlinna (or Nevanlinna-Herglotz, resp., Riesz-Herglotz) functions. Here we shall recall some of the results formulated in [26,Section 2] which go back to [10,[60][61][62], and slightly extend and reformulate these in a form convenient for our subsequent purposes. We first recall the integral representation of the logarithm that corresponds to the cut along the negative imaginary axis, Next, let G be a separable Hilbert space and let K ∈ L(G) be a bounded operator such that Im(K ) ≥ 0 and 0 ⊂ ρ(K ). We use as the definition of the logarithm of the operator K . Then log(K ) ∈ L(G) by [26,Lemma 2.6] and in the special case that K ∈ L(G) is self-adjoint and 0 ∈ ρ(K ), it follows from [26,Lemma 2.7] that In the next lemma we show that besides log(K ) also log(K * ) is well-defined via (3.2) when K is a dissipative operator with spectrum off the imaginary axis (cf. [26, Lemmas 2.6, 2.7]).

L(G)
−1 it follows that the first integral in (3.5) is bounded. In order to show that the second integral in (3.5) is also bounded it suffices to show that and for f = 0 this yields This implies (3.6), and hence the second integral in the estimate (3.5) is finite. Thus, We recall that a function N : Nevanlinna function which admits an analytic continuation by reflection with respect to (−∞, 0), but it does not admit an analytic continuation by reflection with respect to any open subinterval of [0, ∞).
An operator-valued Nevanlinna function admits a minimal operator representation via the resolvent of a self-adjoint operator or relation in an auxiliary or larger Hilbert space (see, e.g., [10,38,50,60]). More precisely, if N : C + → L(G) is a Nevanlinna function and z 0 ∈ C + is fixed then there exists a Hilbert space K, a self-adjoint operator or self-adjoint relation L in K and an operator R ∈ L(G, K) (depending on the choice of z 0 ) such that holds for z ∈ C + . If N satisfies the condition (3.11) then L in (3.10) is a self-adjoint operator in K; cf. [50, Corollary 2.5]. The representation (3.10) also holds for z ∈ C − when N is extended onto C − via (3.9). Note that the model can be chosen minimal, that is, the minimality condition , and the open subset ρ(L) ∩ R is maximal with this property. Next, assume that N is an L(G)-valued Nevanlinna function and suppose that N (z) −1 ∈ L(G) for some, and hence (by [26,Lemma 2.3]) for all z ∈ C\R. Then we define for z ∈ C + the logarithm log(N (z)) in accordance with (3.2) by (3.12) and extend the function log(N ) onto C − by reflection, (3.14) The following theorem is a variant and slight extension of [26, Theorem 2.10], the new and important feature here is that we provide a sufficient condition in terms of the function N such that log(N ) admits an analytic continuation by reflection with respect to some real interval and a corresponding integral representation there. Theorem 3.3 Let N : C\R → L(G) be a Nevanlinna function and assume that N (z) −1 ∈ L(G) for some, and hence for all z ∈ C\R. Then there exists a weakly Lebesgue measurable operator-valued function λ → (λ) ∈ L(G) on R such that where C = Re(log(N (i))) ∈ L(G) is a self-adjoint operator and the integral is understood in the weak sense.
If, in addition, N admits an analytic continuation by reflection with respect to an open interval I ⊂ R such that σ (N (z)) ⊂ (ε, ∞) for some ε > 0 and all z ∈ I , then also log(N ) admits an analytic continuation by reflection with respect to I , (λ) = 0 for a.e. λ ∈ I , and (3.16) remains valid for z ∈ I .
Proof We make use of the representation (3.10) applied to the Nevanlinna function log(N ) with z 0 = i. Then there exists a Hilbert space K and R ∈ L(G, K) such that where C = Re(log(N (i))) ∈ L(G) is a self-adjoint operator. For h ∈ G it follows from (3.17) and (3.14) that for h ∈ G, z ∈ C\R, and (3.14) and the Stieltjes inversion formula imply that the measures are absolutely continuous with respect to the Lebesgue measure dλ and there exist measurable proving (3.15) and (3.16). Next, assume that N admits an analytic continuation by reflection with respect to an open interval I ⊂ R such that σ (N (z)) ⊂ (ε, ∞) for some ε > 0 and all z ∈ I . Fix some z 0 ∈ I and an open ball and hence the operators N (z), z ∈ B z 0 ∩ C + , satisfy the assumptions in Lemma 3.1. Therefore, the operators are well-defined, bounded operators in G. Thus for all z ∈ B z 0 , the operators log(N (z)) are well-defined via (3.12). It then follows from (3.12) We shall now also make use of the logarithm (3.19) which corresponds to the cut along the negative real axis. Since are well-defined operators and the function z → ln(N (z)) is analytic on B z 0 . In addition, (3.20) yields As log(z) = ln(z) (see (3.1)) for all z > 0 and N (z) is self-adjoint for z ∈ I it follows from the spectral theorem that log(N (z)) = ln(N (z)), z ∈ I, and hence log(N (z)) = ln(N (z)), z ∈ B z 0 , by analyticity. Therefore, (3.21) and It follows that z → log(N (z)) is analytic on B z 0 and the continuation of log(N ) onto (3.9)). This reasoning applies to all ν ∈ I and hence we have shown that log(N ) admits an analytic continuation by reflection with respect to I . Since the operator model for log(N ) is minimal the interval I belongs to ρ(L) and the representation (3.17) remains valid for z ∈ ρ(L). It follows that the measures dω h (·), h ∈ G, in (3.18) have no support in I and hence their Radon-Nikodym deriatives satisfy ξ h (λ) = 0 for a.e. λ ∈ I . It follows that ( (λ)h, h) G = 0 for a.e. λ ∈ I and all h ∈ G. Since (λ) ≥ 0 we conclude (λ) = 0 for a.e. λ ∈ I .
In the next proposition we provide a sufficient condition such that the values of the function are trace class operators and we express the traces of (λ) in terms of certain weak limits of the imaginary part of log(N ).
for some, and hence for all z ∈ C\R, and assume that N admits an analytic continuation by reflection with respect to an open interval I ⊂ R such that σ (N (ζ )) ⊂ (ε, ∞) for some ε > 0 and all ζ ∈ I . Consider (3.15), and assume, in addition, that for some k ∈ N 0 and some ζ ∈ I , Then 0 ≤ (λ) ∈ S 1 (G) for a.e. λ ∈ R, and holds for any orthonormal basis (ϕ j ) j∈J in G (J ⊆ N an appropriate index set ) and for a.e. λ ∈ R. Furthermore, if (3.23) holds for some ζ ∈ I and k = 0, that is,

25)
then Im(log(N (z))) ∈ S 1 (G) for all z ∈ C\R, the limit exists for a.e. λ ∈ R in the norm of S 1 (G), and The assumption (3.23) together with the integral representation (3.22) yields holds for all h ∈ G and all λ ∈ R, and therefore the Stietljes inversion formula yields holds for all λ ∈ R\A j , where A j ⊂ R, j ∈ J , is a set of Lebesgue measure zero. The countable union A := ∪ j∈J A j is also a set of Lebesgue measure zero and for all λ ∈ R\A and all ϕ j one has (3.30). Taking into acount that 0 ≤ (λ) ∈ S 1 (G) for a.e. λ ∈ R this implies for a.e. λ ∈ R, that is, (3.24) holds.
In the special case that (3.23) holds with k = 0 the formula (3.27) has the form for all z ∈ C\R. The last assertion on the existence of the limit Im(log(N (λ+i0))) for a.e. λ ∈ R in S 1 (G) is an immediate consequence of (3.31) and well-known results in [10,60,61] (cf. [26, Theorem 2.2(iii)]).
The following lemma will be useful in the proof of our main result, Theorem 4.1, in the next section; it also provides a sufficient condition for the assumption (3.23) in Proposition 3.4.
for some, and hence for all z ∈ C\R. Let ∈ N and assume that

32)
holds for all z ∈ C\R. Then and hold for all z ∈ C\R. Furthermore, if N admits an analytic continuation by reflection with respect to an open interval I ⊂ R such that σ (N (z)) ⊂ (ε, ∞) for some ε > 0 and all z ∈ I , and (3.32) is satisfied for z ∈ I , then also the assertions (3.33) and (3.34) are valid for all z ∈ I .
Proof We prove Lemma 3.5 for the case = 1 and leave the induction step to the reader. Assume that d dz N (z) ∈ S 1 (G) (3.35) holds for z ∈ C + (the proof works also for z ∈ I if N admits an analytic continuation by reflection with respect to I and σ (N (z)) ⊂ (ε, ∞) holds for some ε > 0 and all z ∈ I ). One notes that N (z) −1 ∈ L(G) implies the second assertion in (3.33) for = 1. In addition, one observes that log(N (z)) is well-defined and analytic for z ∈ C + according to (3.12) and Theorem 3.3. Since 0 ∈ ρ(N (z)) and  N (z) holds for all ϕ, ψ ∈ G and all z ∈ C + , and hence The assumption (3.35) yields From (3.36) and the properties of the trace class norm · S 1 (G) one gets , λ > 0, and hence the integral in (3.37) exists in trace class norm, that is, the first assertion in (3.33) holds for = 1. In order to prove (3.34) for = 1 we use (3.37) and cyclicity of the trace (i.e., tr G (C D) = tr G (DC) whenever C, D ∈ L(G) such that C D, DC ∈ S 1 (G)) and obtain Here we have used lim λ→+∞ tr G (N (z) + iλI G ) −1 d dz N (z) = 0 in the last step, which follows from , λ > 0.

A representation of the spectral shift function in terms of the Weyl function
Let A and B be self-adjoint operators in a separable Hilbert space H and assume that the closed symmetric operator S = A ∩ B, that is, is densely defined. According to Proposition 2.4 we can choose a quasi boundary triple {G, 0 , 1 } with γ -field γ and Weyl function M such that and In the next theorem we find an explicit expression for a spectral shift function of the pair {A, B} in terms of the Weyl function M, see [49,Theorem 1] for the case that the difference of (the first powers of) the resolvents A and B is a rank one operator, [7,Theorem 4.1] for the finite-rank case, and [56, Theorem 3.4 and Remark 3.5] for a different representation via a perturbation determinant involving the Weyl function and boundary parameters of an ordinary boundary triple. In the present situation of infinite dimensional perturbations and differences of higher powers of resolvents a much more careful analysis is necessary, in particular, the properties of the logarithm of operator-valued Nevanlinna functions discussed in Sect. 3 will play an essential role. In Theorem 4.1 an implicit sign condition on the perturbation is imposed via the resolvents which leads to a nonnegative spectral shift function; this condition will be weakend afterwards (cf. (4.25) and (4.29)). In the special case that A and B are semibounded operators the sign condition (4.4) is equivalent to the inequality t A ≤ t B of the semibounded closed quadratic forms t A and t B corresponding to A and B. In order to ensure that for some k ∈ N 0 the difference of the 2k + 1th-powers of the resolvents of A and B is a trace class operator a set of S p -conditions on the γ -field and the Weyl function are imposed. In the applications in Sects. 5 and 6 these conditions are satisfied. that for some k ∈ N 0 , all p, q ∈ N 0 , and all z ∈ ρ(A) ∩ ρ(B), and Then the following assertions (i) and (ii) hold: (i) The difference of the 2k + 1th-powers of the resolvents of A and B is a trace class operator, that is, valid for all z ∈ (C\R) ∪ I ζ 0 , and (λ) = 0 for a.e. λ ∈ I ζ 0 .
First, it follows from (2.8) and the assumption that M(z 1 ) is bounded for some z 1 ∈ ρ(A) that M(z) is bounded for all z ∈ ρ(A) and hence the closures are bounded operators defined on G, that is, M(z) ∈ L(G), z ∈ ρ(A), and Im M(z) ≥ 0, z ∈ C + , (4.12) by (2.11). Since −M −1 is the Weyl function corresponding to the quasi boundary triple {G, 1 , − 0 }, where B = T ker( 1 ) is self-adjoint according to (4.2), it follows from the assumption that M(z 2 ) −1 is bounded for some and since ran(γ (ζ 0 ) * ) is dense in G (see (2.3)), it follows that the bounded operator M(ζ 0 ) −1 is nonnegative. The same is true for M(ζ 0 ) and the closure M(ζ 0 ), and from (4.13) one concludes σ M(ζ 0 ) ⊂ (ε, ∞) for some ε > 0. Since ζ 0 ∈ ρ(A) ∩ ρ(B) the Nevanlinna function M admits an analytic continuation by reflection with respect to a real neighborhood of ζ 0 , and it follows that (4.16) holds for all λ in a sufficiently small interval Step 2 In this step we show that for z ∈ (C\R) ∪ I ζ 0 , the trace class property (4.8) holds, and that In fact, for z ∈ (C\R) ∪ I ζ 0 one computes and by assumption (4.5) each summand is a trace class operator; in the last step the product rule for holomorphic operator functions was applied, see, e.g. [6, (2.6)]. This proves (4.8). Furthermore, making use of both assumptions (4.5) and (4.6), the cyclicity of the trace (see, e.g., [72, Theorem 7.11(b)]), and one obtains Noting that assumption (4.7) and Lemma 3.5 with = 2k + 1 imply and that one concludes the trace formula (4.17).
In the special case k = 0 Theorem 4.1 can be slightly improved. Here the essential feature is that Proposition 3.4 can be applied under the assumption (3.25), so that the limit Im(log(M(λ + i0))) exists in S 1 (G) for a.e. λ ∈ R.

Corollary 4.2 Let A and B be self-adjoint operators in a separable Hilbert space H and assume that for some ζ 0 ∈ ρ(A) ∩ ρ(B) ∩ R the sign condition
holds. Assume that the closed symmetric operator S = A∩ B in (4.1) is densely defined and let {G, 0 , 1 } be a quasi boundary triple with γ -field γ and Weyl function M such that (4.2), and hence also (4.3), hold. Assume that M(z 1 ), M(z 2 ) −1 are bounded (not necessarily everywhere defined ) operators in G for some z 1 , z 2 ∈ ρ(A) and that γ (z 0 ) ∈ S 2 (G, H) for some z 0 ∈ ρ(A). Then the following assertions (i)-(iii) hold: (i) The difference of the resolvents of A and B is a trace class operator, that is, (ii) Im log M(z) ∈ S 1 (G) for all z ∈ C\R and the limit for a.e. λ ∈ R, In the next step we replace the sign condition (4.4) in the assumptions in Theorem 4.1 by some weaker comparability condition. Again, let A and B be self-adjoint operators in a separable Hilbert space H and assume that there exists a self-adjoint operator C in H such that is a spectral shift function for the pair {A, B}, and in the special case where G A = G B := G and (ϕ j ) j∈J is an orthonormal basis in G, one infers that for a.e. λ ∈ R.
(4. 30) We emphasize that in contrast to the spectral shift function in Theorem 4.1, the spectral shift function ξ in (4.29) and (4.30) is not necessarily nonnegative.

Elliptic differential operators with Robin boundary conditions
In this section we consider a uniformly elliptic formally symmetric second-order differential expression L on a bounded or unbounded domain in R n with compact boundary, and we determine a spectral shift function for a pair {A β 0 , A β 1 } consisting of two self-adjoint Robin-realizations of L. We shall assume throughout this section that the following hypothesis holds.
Hypothesis 5.1 Let n ∈ N, n ≥ 2, and ⊆ R n be nonempty and open such that its boundary ∂ is nonempty, C ∞ -smooth, and compact. Consider the differential expression on , where the real-valued coefficients a jk ∈ C ∞ ( ) satisfy a jk (x) = a k j (x) for all x ∈ and j, k = 1, . . . , n, their first partial derivatives are bounded in , and a ∈ C ∞ ( ) is a real-valued, bounded, measurable function. Furthermore, it is assumed that L is uniformly elliptic on , that is, for some C > 0, We briefly recall the definition and some mapping properties of the Dirichlet and (oblique) Neumann trace maps associated with the differential expression L. For a function f ∈ C ∞ ( ) we denote its trace by γ D f = f | ∂ and we set where n(x) = (n 1 (x), . . . , n n (x)) is the unit normal vector at x ∈ ∂ pointing out of the domain . Let C ∞ 0 ( ) := {h| | h ∈ C ∞ 0 (R n )} and recall that the mapping , γ ν f } can be extended to a continuous surjective mapping and that Green's second identity is valid for all f, g ∈ H 2 ( ); cf. [53]. We will also use the fact that The following lemma is a variant of [5,Lemma 4.7]; it will be useful for the S pestimates in this and the next section.

Lemma 5.2
Let ⊆ R n be as in Hypothesis 5.1, let X ∈ L L 2 ( ), H t (∂ ) , and assume that ran(X ) ⊆ H s (∂ ) for some s > t ≥ 0. Then X is compact and Assume that β 0 ∈ C 1 (∂ ) and β 1 ∈ C 1 (∂ ) are real-valued functions. For p = 0, 1 we consider the elliptic differential operators in L 2 ( ), which correspond to the densely defined, closed, semibounded quadratic forms defined on H 1 ( ) × H 1 ( ). Both operators A β 0 and A β 1 are self-adjoint in L 2 ( ) and semibounded from below. For β ∈ R we shall also make use of the self-adjoint Robin realization which corresponds to the densely defined, closed, semibounded quadratic form Next, we define the Neumann-to-Dirichlet map associated to L as a densely defined operator in L 2 (∂ ). First one notes that for β 0 = 0 in (5.6) (or β = 0 in (5.8)) one obtains 10) where A N denotes the self-adjoint Neumann realization of L in L 2 ( ). One recalls that for ϕ ∈ H 1/2 (∂ ) and z ∈ ρ(A N ), the boundary value problem admits a unique solution f z ∈ H 2 ( ); this follows, for instance, from (5.3) and z ∈ ρ(A N ). The corresponding solution operator is denoted by (5.12) and it is clear that dom(P ν (z)) = H 1/2 (∂ ) and ran(P ν (z)) ⊆ H 2 ( ). For z ∈ ρ(A N ) the Neumann-to-Dirichlet map associated to L is defined as it maps the (oblique) Neumann boundary values γ ν f z of solutions f z ∈ H 2 ( ) of (5.11) onto the Dirichlet boundary values γ D f z . It follows from the properties of the trace maps that dom(N (z)) = H 1/2 (∂ ) and ran(N (z)) ⊆ H 3/2 (∂ ). (5.14) In the next theorem a spectral shift function for the pair {A β 0 , A β 1 } is expressed in terms of the limits of the Neumann-to-Dirichlet map N (z) and the functions β 0 and β 1 in the boundary conditions of the Robin realizations A β 0 and A β 1 . We mention that the trace class condition for the difference of the 2k + 1-th powers of the resolvents was shown for k = 0 in [4,34] and for k ∈ N in [6]. Theorem 5.3 Assume Hypothesis 5.1, let A β 0 and A β 1 be the self-adjoint Robin realizations of L in L 2 ( ) in (5.6), let β ∈ R such that β p (x) < β for all x ∈ ∂ and p = 0, 1 and let A β be the self-adjoint Robin realizations of L in (5.8). Furthermore, let where N (z) denotes the closure in L 2 (∂ ) of the Neumann-to-Dirichlet map associated with L in (5.13). Then the following assertions (i) and (ii) hold for k ∈ N 0 such that k ≥ (n − 3)/4: (i) The difference of the 2k + 1th-powers of the resolvents of A β 0 and A β 1 is a trace class operator, that is, for a.e. λ ∈ R, is a spectral shift function for the pair {A β 0 , A β 1 } such that ξ(λ) = 0 for λ < min(σ (A β )) and the trace formula Proof The proof of Theorem 5.3 consists of three steps. In the first step we construct a suitable quasi boundary triple such that the self-adjoint operators A β and A β 1 correspond to the kernels of the boundary mappings 0 and 1 , and in the second and third step we show that the pair {A β , A β 1 } and the γ -field and Weyl function satisfy the assumptions in Theorem 4.1. The same reasoning applies to the pair {A β , A β 0 }, and hence Theorem 4.1 can be applied to both pairs {A β , A β 1 } and {A β , A β 0 }, which together with the considerations at the end of Sect. 4 yield the assertions in Theorem 5.3.
Step 1 The basic techniques in this step have been used in a similar framework, for instance, in [2,3,5,8]. We consider the closed symmetric operator S = A β ∩ A β 1 , which is given by where we have used that β − β 1 (x) = 0 for all x ∈ ∂ . In this step we check that the operator satisfies T = S * and that L 2 (∂ ), 0 , 1 , where is a quasi boundary triple for T ⊂ S * such that and for all z ∈ ρ(A β ) ∩ ρ(A N ), where A N is the self-adjoint Neumann realization in (5.10), the corresponding γ -field γ and Weyl function M in L 2 (∂ ) are given by and dom(M(z)) = H 1/2 (∂ ). (5.20) We will use Theorem 2.2 for this purpose. For f, g ∈ dom(T ) = H 2 ( ) one obtains with the help of Green's identity (5.4), and hence condition (i) in Theorem 2.2 holds. Since One notes that for z ∈ ρ(A N ) and f z ∈ ker(T − z I L 2 ( ) ) one has N (z)γ ν f z = γ D f z according to (5.13), and hence and From this and (5.12) it follows that the γ -field corresponding to {L 2 (∂ ), 0 , 1 } has the form (5.19). One also concludes from (5.24) and (5.23) that holds for all f z ∈ ker(T − z I L 2 ( ) ) and z ∈ ρ(A β ) ∩ ρ(A N ). Thus the Weyl function corresponding to the quasi boundary triple {L 2 (∂ ), 0 , 1 } has the form (5.20).
Step 2 In this step we verify that the pair {A β , A β 1 } satisfies the sign condition (4.4) and that the values of Weyl function and its inverse are bounded operators; see the assumptions of Theorem 4.1.
The assumption β > β 1 (x) shows that the semibounded quadratic forms a β and a β 1 in (5.7) and (5.9) corresponding to A β and A β 1 satisfy the inequality a β ≤ a β 1 . Hence min(σ (A β )) ≤ min(σ (A β 1 )) and for ζ < min(σ (A β )) the forms a β − ζ and a β 1 − ζ are both nonnegative, satisfy the inequality a β − ζ ≤ a β 1 − ζ , and hence the resolvents of the corresponding nonnegative self-adjoint operators A β − ζ I L 2 ( ) and A β 1 − ζ I L 2 ( ) satisfy the inequality are bounded operators for some z 1 , z 2 ∈ C\R. According to [5,Lemma 4.4] the closure N (z), z ∈ C\R, of the Neumann-to-Dirichlet map in (5.13) in L 2 (∂ ) is compact, and hence βN (z) − I L 2 (∂ ) and β 1 N (z) − I L 2 (∂ ) are densely defined bounded operators in L 2 (∂ ), and for z ∈ C\R their closures are First, one notes that N (z) ⊆ N (z) * , z ∈ C\R, holds by (5.4), and this yields that also Q(z) ⊆ Q(z) * , z ∈ C\R. Hence the operator Q(z) is closable in L 2 (∂ ). Moreover, as Q(z) is defined on H 1/2 (∂ ) and maps into H 1/2 (∂ ), it follows that Q(z) is a closed operator in H 1/2 (∂ ), and hence is bounded. Therefore, the dual operator , is also bounded. One verifies that Q(z) is an extension of Q(z) and hence by interpolation and (5.27) and (5.28), the restriction is a bounded operator in L 2 (∂ ) and an extension of Q(z).
Hence for all z ∈ C\R the operator Q(z) is bounded in L 2 (∂ ) and its closure is The same reasoning with Q(z) replaced by Q 1 (z) shows that for all z ∈ C\R the operator Q 1 (z) is bounded in L 2 (∂ ) and Step 3 In this step we verify that the γ -field and Weyl function corresponding to the quasi boundary triple L 2 (∂ ), 0 , 1 in Step 1 satisfy the S p -conditions in the assumptions of Theorem 4.1 for dimensions n ∈ N, n ≥ 2, and k ≥ (n − 3)/4, that is, we verify for all p, q ∈ N 0 and all z ∈ ρ(A β ) ∩ ρ(A β 1 ) the conditions In the following we shall often use the smoothing property Here we have used in the last step that satisfies the boundary condition βγ D g − γ ν g = 0. It follows from (2.6) and (5.36) that and hence, ran((γ (z) * ) (q) ) ⊂ H 2q+3/2 (∂ ) by (5.35) and (5.5). From Lemma 5.2 with s = 2q + (3/2) and t = 0 one concludes that for all z ∈ ρ(A β ), q ∈ N 0 , and hence by (2.6) also for all z ∈ ρ(A β ), p ∈ N 0 . Furthermore, by (2.12) and with the help of (5.36) it follows in the same way as in (5.37) that Moreover, γ (z) ∈ S y (L 2 (∂ ), L 2 ( )) for y > 2(n − 1)/3 by (5.38) and hence it follows from (5.39) and the well-known property P Q ∈ S w for P ∈ S x , Q ∈ S y , and x −1 + y −1 = w −1 , that d j dz j M(z) ∈ S w L 2 (∂ , w > (n − 1)/(2 j + 1), z ∈ ρ(A β ), j ∈ N. (5.40) One observes that that M(z) −1 is bounded, and by (5.40) that also we leave the formal induction step to the reader. Therefore, and one has M(z) by (5.37) and each summand (and hence also the finite sum) on the right-hand side is in Hence the assumptions in Theorem 4.1 are satisfied with S in (5.15), the quasi boundary triple in (5.17) and the corresponding γ -field and Weyl function in (5.19) and (5.20), respectively. Now Theorem 4.1 yields assertion (i) in Theorem 5.3 with A β 0 replaced by A β and for any orthonormal basis (ϕ j ) j∈J in L 2 (∂ ) the function ξ 1 (λ) = j∈J lim ε↓0 π −1 Im log(M 1 (λ + iε)) ϕ j , ϕ j L 2 (∂ ) for a.e. λ ∈ R, is a spectral shift function for the pair {A β , A β 1 } such that ξ 1 (λ) = 0 for λ < min(σ (A β )) ≤ min(σ (A β 1 )) and the trace formula The same construction as above with β 1 replaced by β 0 yields an analogous representation for a spectral shift function ξ 0 of the pair {A β , A β 0 }. Finally it follows from the considerations in the end of Sect. 4 (see (4.29)) that for a.e. λ ∈ R is a spectral shift function for the pair {A β 0 , A β 1 } such that ξ(λ) = 0 for λ < min(σ (A β )) ≤ min(σ (A β p )), p = 0, 1. This completes the proof of Theorem 5.3.
In space dimensions n = 2 and n = 3 one can choose k = 0 in Theorem 5.3, and hence the resolvent difference of A β 1 and A β 0 is a trace class operator. In this situation Corollary 4.2 leads to the following slightly stronger statement.

Schrödinger operators with compactly supported potentials
In this section we determine a spectral shift function for the self-adjoint operators is a compactly supported real-valued function. Thus we consider the self-adjoint operators in L 2 (R n ), and we fix an open ball B + ⊂ R n such that supp (V ) ⊂ B + . The n − 1 dimensional sphere ∂B + is denoted by S. We shall also make use of the self-adjoint Dirichlet realizations of − and − + V in L 2 (B + ). Their spectra are discrete and bounded from below. The eigenvalue counting functions are denoted by N ( · , A + ) and N ( · , B + ), respectively; recall that N (λ, A + ) and N (λ, B + ) stand for the total number of eigenvalues (multiplicities counted) of A + and B + in (−∞, λ), λ ∈ R.
The main ingredient in the proof of Theorem 6.1 below is a decoupling technique for the operators A and B, where artificial Dirichlet boundary conditions on the sphere S will be imposed. We shall use the extension of the L 2 (S) scalar product onto the dual pair H 1/2 (S) × H −1/2 (S) via where ı is a uniformly positive self-adjoint operator in L 2 (S) defined on the dense subspace H 1/2 (S) (and in the following ι is regarded as an isomorphism from H 1/2 (S) onto L 2 (S)), and ı −1 is the extension of ı −1 to an isomorphism from H −1/2 (S) onto L 2 (S). A typical and convenient choice for ı is (− S + I L 2 (S) ) 1/4 , where − S is the Laplace-Beltrami operator on the sphere S; for other choices see also [8,Remark 5.3].
Since ·, · in (6.3) is an extension of the L 2 (S) scalar product, Green's identity can also be written in the form for f + , g + ∈ H 2 (B + ). Here γ + D and γ + N denote the Dirichlet and Neumann trace operators in (5.3) (with and ∂ replaced by B + and S, respectively). Let B − := R n \B + and let γ − D and γ − N be the Dirichlet and Neumann trace operators on B − ; the normal vector in the definition of γ − N is pointing in the outward direction of B − and hence opposite to the normal of B + . Besides (6.4) we also have the corresponding Green's identity on B − , that is, holds for all f − , g − ∈ H 2 (B − ).

Hence, the operators
are everywhere defined and bounded in L 2 (S).
In the next theorem we obtain a representation for a spectral shift function for {A, B} in (6.1) via a decoupling technique and Theorem 4.1. The considerations in the beginning of Step 1 of the proof of Theorem 6.1 are similar as in [8,Section 5.2] and hence some details are omitted. Theorem 6.1 Let n ∈ N, n ≥ 2, and k ∈ N, k > (n − 2)/4, and suppose that V ∈ L ∞ (R n ) is real-valued with support in the open ball B + . In addition, let N(z) and N V (z) be as in (6.6), and denote the eigenvalue counting functions of the Dirichlet operators A + and B + in L 2 (B + ) by N ( · , A + ) and N ( · , B + ), respectively. Then the following assertions (i) and (ii) hold: (i) The difference of the 2k + 1th-powers of the resolvents of A and B is a trace class operator, that is, is a spectral shift function for the pair {A, B} such that ξ(λ) = 0 for λ < min(σ (B)) ≤ 0 and the trace formula Proof Besides the self-adjoint operators A = − and B = − + V in (6.1), and the Dirichlet realizations A + = − and B + = − + V in L 2 (B + ) in (6.2) we shall also make use of the Dirichlet realization A − of − in L 2 (B − ) given by as well as the orthogonal sums in L 2 (R n ) = L 2 (B + ) ⊕ L 2 (B − ), (6.8) For any orthonormal basis (ϕ j ) j∈J in L 2 (S) we shall first prove the representation ξ A (λ) = j∈J lim ε↓0 π −1 Im log(N(λ + iε)) ϕ j , ϕ j L 2 (S) (6.9) for a spectral shift function ξ A of the pair {A, A D } and the representation for a spectral shift function ξ B of the pair {B, B D }.
Step 1 In this step we consider the operators B and B D as self-adjoint extensions of the closed symmetric S = B ∩ B D , which is given by Furthermore, consider the operator Since H 1 (R n ) ⊂ (H 1 0 (B + ) × H 1 0 (B − )) this implies b ≤ b D and yields the sign condition (B − ζ I L 2 (R n ) ) −1 ≥ (B D − ζ I L 2 (R n ) ) −1 in the assumptions of Theorem 4.1 for all ζ < min(σ (B)) ≤ min(σ (B D )); see the beginning of Step 2 in the proof of Theorem 5.3.
Hence, the assumptions in Theorem 4.1 are satisfied with S in (6.11), the quasi boundary triple in (6.12), and the corresponding Weyl function in (6.13). Thus, The-orem 4.1 yields assertion (i) with A replaced by B D and for any orthonormal basis (ϕ j ) j∈J in L 2 (S) the function ξ B (λ) = j∈J lim ε↓0 π −1 Im log(N V (λ + iε)) ϕ j , ϕ j L 2 (S) for a.e. λ ∈ R in (6.10) is a spectral shift function for the pair {B, B D } and the trace formula is valid for all z ∈ ρ(B) ∩ ρ(B D ).

Remark 6.2
We note that the spectral shift function ξ in Theorem 6.1 is continuous for λ > 0 since V ∈ L ∞ (R n ) is compactly supported (see, e.g., [76,Theorem 9.1.20]). On the other hand the spectral shift function ξ + of {A + , B + } is a step function and hence the difference of the spectral shift functions ξ A and ξ B of the pairs {A, A D } and {B, B D } cancel the discontinuities of ξ + for λ > 0.