Spectral analysis of selfadjoint elliptic differential operators, Dirichlet-to-Neumann maps, and abstract Weyl functions

The spectrum of a selfadjoint second order elliptic differential operator in $L^2(\mathbb{R}^n)$ is described in terms of the limiting behavior of Dirichlet-to-Neumann maps, which arise in a multi-dimensional Glazman decomposition and correspond to an interior and an exterior boundary value problem. This leads to PDE analogs of renowned facts in spectral theory of ODEs. The main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding Weyl functions, and are applied to the PDE setting afterwards.


Introduction
The Titchmarsh-Weyl function is an indispensable tool in direct and inverse spectral theory of ordinary differential operators and more general systems of ordinary differential equations; see the classical monographs [17,55] and [11,18,27,28,29,34,38,44,51,52] for a small selection of more recent contributions. For a singular second order Sturm-Liouville differential operator of the form L + = − d 2 dx 2 + q + on R + with a real-valued, bounded potential q + the Titchmarsh-Weyl function m + can be defined as where f λ is a square-integrable solution of L + f = λf on R + ; cf. [55,56]. The function m + : C \ R → C belongs to the class of Nevanlinna (or Riesz-Herglotz) functions and it is a celebrated fact that it reflects the complete spectral properties of the selfadjoint realizations of L + in L 2 (R + ). E.g. the eigenvalues of the Dirichlet realization A D are precisely those λ ∈ R, where lim ηց0 iηm + (λ + iη) = 0, the isolated eigenvalues among them coincide with the poles of m + , and the absolutely continuous spectrum of A D (roughly speaking) consists of all λ with the property 0 < Im m + (λ + i0) < +∞. If L = − d 2 dx 2 + q is a singular Sturm-Liouville expression on R with q real-valued and bounded, it is most natural to use decomposition methods of Glazman type for the analysis of the corresponding selfadjoint operator in L 2 (R); cf. [30]. More precisely, the restriction of L to R + gives rise to the Titchmarsh-Weyl function m + in (1.1), and similarly a Titchmarsh-Weyl function m − associated to the restriction of L to R − is defined. In that case usually the functions are employed for the description of the spectrum. Whereas the scalar function m seems to be more convenient it will in general not contain the complete spectral data, a drawback that is overcome when using the 2 × 2-matrix function m. Some of these observations were already made in [36,55], similar ideas can also be found in [33,35,39] for Hamiltonian systems, and more recently in an abstract operator theoretical framework in [19,21], see also [7,8].
One of the main objectives of this paper is to extend the classical spectral analysis of ordinary differential operators via the Titchmarsh-Weyl functions in (1.2) to the multidimensional setting. For this consider the second order partial differential expression with smooth, bounded coefficients a jk , a j : R n → C and a : R n → R bounded, and assume that L is formally symmetric and uniformly elliptic on R n . Let A be the selfadjoint operator associated to (1.3) in L 2 (R n ). Our main goal is to describe the spectral data of A, that is, isolated and embedded eigenvalues, continuous, absolutely continuous and singular continuous spectral points, in terms of the limiting behaviour of appropriate multidimensional counterparts of the functions in (1.2). Note first that the multidimensional analogue of the Titchmarsh-Weyl function (1.1) is the Dirichlet-to-Neumann map, and in order to define suitable analogues of the functions in (1.2) we proceed as follows: Split R n into a bounded domain Ω i with smooth boundary Σ and let Ω e = R n \ Ω i be the exterior of Ω i . For λ ∈ C \ R the Dirichlet-to-Neumann maps for L in Ω i and Ω e , respectively, on the compact interface Σ are given by Λ i (λ)u λ,i | Σ := ∂u λ,i ∂ν Li Σ and Λ e (λ)u λ,e | Σ := ∂u λ,e ∂ν Le Σ , λ ∈ C \ R, where u λ,j ∈ H 2 (Ω j ) solve Lu λ,j = λu λ,j , j = i, e, and u λ,j | Σ and We mention that in connection with Schrödinger operators in R 3 the function M in (1.4) was already used in [2] for the extension of a classical convergence property of the Titchmarsh-Weyl function to the three-dimensional case, see also [5,6,50]. We also remark that for Schrödinger operators on exterior domains with C 2 -boundaries the connection of the spectrum to the limits of the Dirichlet-to-Neumann map was already investigated by the authors in [10].
In this paper our approach to Titchmarsh-Weyl functions and their connection to spectral properties of corresponding selfadjoint differential operators is more abstract and of general nature, based on the concepts of (quasi) boundary triplets and their Weyl functions. Recall first that for a symmetric operator S in a Hilbert space H a boundary triple {G, Γ 0 , Γ 1 } consists of a "boundary space" G and two linear mappings Γ 0 , Γ 1 : dom S * → G, which satisfy an abstract Green identity and a maximality condition. The corresponding Weyl function M is defined as where f λ ∈ H solves the equation S * f = λf ; the values M (λ) of the Weyl function M are bounded operators in the Hilbert space G. The example of the Sturm-Liouville expression L + in the beginning of the introduction fits into this scheme: There H = L 2 (R + ), S is the minimal operator associated with the differential expression L + in L 2 (R + ), G = C, and the mappings Γ 0 , Γ 1 are given by where S * is the maximal operator associated with L + in L 2 (R + ). Then the corresponding Weyl function is m + in (1.1), the selfadjoint Dirichlet operator A D coincides with S * ↾ ker Γ 0 , and the spectrum can be described with the help of the limits of the Weyl function. The correspondence between the spectrum of the particular selfadjoint extension A 0 := S * ↾ ker Γ 0 and the limits of the Weyl function is not a special feature of the boundary triple for the above Sturm-Liouville equation. In fact, it holds as soon as the symmetric restriction S (and, thus, the boundary mappings Γ 0 and Γ 1 ) is chosen properly. More abstract considerations from [22,41,42,43] yield that the operator A 0 (and hence its spectrum) is determined up to unitary equivalence by the Weyl function if and only if the symmetric operator S is simple or completely non-selfadjoint, that is, there exists no nontrivial subspace of H which reduces S to a selfadjoint operator. This condition can be reformulated equivalently as where γ(ν) = (Γ 0 ↾ ker(S * − ν)) −1 is the so-called γ-field and clsp denotes the closed linear span; cf. [40]. Under the assumption that S is simple a description of the absolutely continuous and singular continuous spectrum in the framework of boundary triples and their Weyl functions was given in [12]; for more recent related work see also [13,14,15,16,32,45,46,48,49,53]. The concept of boundary triples and their Weyl functions was extended in [3] in such a way that it is conveniently applicable to PDE problems. For that one defines boundary mappings Γ 0 , Γ 1 on a suitable, smaller subset of the domain of the maximal operator and requires Green's identity (1.5) only to hold on this subset; the definition of the Weyl function associated to such a quasi boundary triple {G, Γ 0 , Γ 1 } is as in (1.6), except that only solutions in the domain of the boundary maps are used; cf. Section 2.1. For the second order elliptic operator L in (1.3) restricted to the smooth domain Ω i ⊂ R n one may choose G = L 2 (Σ), in which case the corresponding Weyl function is (minus) the Dirichlet-to-Neumann map −Λ i . Based on orthogonal couplings of symmetric operators and extending abstract ideas in [19] also the functions M and M in (1.4) can be interpreted as Weyl functions associated to properly chosen quasi boundary triples; e.g., M corresponds to the pair of boundary mappings where u j ∈ H 2 (Ω j ), j = i, e. Moreover, ker Γ 0 is the domain of the unique selfadjoint operator A associated with L in L 2 (R n ). When trying to link the spectral properties of A to the limiting behavior of the function M it is necessary to extend the known results for boundary triples to the more general notion of quasi boundary triples. Moreover, a subtle difficulty arises: The symmetric operator S corresponding to the boundary mappings in (1.8) may possess eigenvalues and thus in general is not simple.
In the abstract part of the present paper we show how this difficulty can be overcome. In the general setting of quasi boundary triples and their Weyl functions we show that a local simplicity condition on an open interval (or, more generally, a Borel set) ∆ ⊂ R suffices to characterize the spectrum of A 0 in ∆. To be more specific, we assume that where E(∆) denotes the spectral projection of A 0 = S * ↾ ker Γ 0 on ∆; this is a local version of the condition (1.7). Under this assumption we provide characterizations of the isolated and embedded eigenvalues and the corresponding eigenspaces, as well as the continuous, absolutely continuous and singular continuous spectrum of A 0 in ∆ in terms of the limits of M (λ) when λ approaches the real axis. For instance, we prove that the eigenvalues of A 0 in ∆ are those λ, where lim ηց0 iηM (λ+iη)g = 0 for some g ∈ ran Γ 0 , and that the absolutely continuous spectrum of A 0 can be characterized by means of the points λ where 0 < Im(M (λ + i0)g, g) G < ∞. Moreover, we prove inclusions and provide conditions for the absence of singular continuous spectrum. Afterwards we apply the obtained results to the selfadjoint elliptic differential operator associated to L in (1.3) in L 2 (R n ). We prove that, despite the fact that the underlying symmetric operator fails to be simple in general, the whole absolutely continuous spectrum of A 0 can be recovered from the mapping M in (1.4). Moreover, we prove that the eigenvalues of A 0 and the corresponding eigenfunctions can be characterized by limiting properties of M as far as the eigenfunctions do not vanish on the interface Σ. A complete picture of the spectrum of A 0 is obtained when using the function M in (1.4). This paper is organized in the following way. In Section 2 we recall the basic facts on quasi boundary triples and corresponding Weyl functions and discuss the local simplicity property (1.9) in detail. In Section 3 the connection between the spectra of selfadjoint operators and corresponding abstract Weyl functions is investigated. Section 4 contains the application of the abstract results to the mentioned PDE problems.
Finally, let us fix some notation. For a selfadjoint operator A in a Hilbert space H we denote by σ(A) (σ p (A), σ c (A), σ ac (A), σ sc (A), σ s (A), respectively) the spectrum (set of eigenvalues, continuous, absolutely continuous, singular continuous, singular spectrum, respectively) of A and by ρ(A) = C \ σ(A) its resolvent set.

Quasi boundary triples, associated Weyl functions, and a local simplicity condition
In this preliminary section we first recall the concepts of quasi boundary triples, their γ-fields and their Weyl functions. Afterwards we discuss a local simplicity property of symmetric operators, which will be assumed to hold in most of the results of Section 3.
2.1. Quasi boundary triples. The notion of quasi boundary triples was introduced in [3] as a generalization of the notions of boundary triples and generalized boundary triples, see [20,22,23,31,37]. The basic definition is the following. Definition 2.1. Let S be a closed, densely defined, symmetric operator in a separable Hilbert space H and let T ⊂ S * be an operator whose closure coincides with S * , i.e., T = S * . A triple {G, Γ 0 , Γ 1 } consisting of a Hilbert space G and two linear mappings Γ 0 , Γ 1 : dom T → G is called a quasi boundary triple for S * if the following conditions are satisfied.
(i) The range of the mapping Γ : In the following we suppress the indices in the scalar products and simply write (·, ·), when no confusion can arise.
We recall some facts on quasi boundary triples, which can be found in [3,4]. Let S be a closed, densely defined, symmetric operator in H. A quasi boundary triple {G, Γ 0 , Γ 1 } for S * exists if and only if the defect numbers of S are equal. What we will use frequently is that if {G, Γ 0 , Γ 1 } is a quasi boundary triple for S * then dom S = ker Γ 0 ∩ker Γ 1 . Recall also that a quasi boundary triple with the additional property ran (Γ 0 , Γ 1 ) ⊤ = G × G becomes an (ordinary) boundary triple and that, in particular, in this case the boundary mappings Γ 0 , Γ 1 are defined on dom S * and (2.1) holds with T replaced by S * . In particular, in the case of finite defect numbers the notions of quasi boundary triples and (ordinary) boundary triples coincide. For more details on quasi boundary triples we refer to [3,4].
In order to prove that a triple {G, Γ 0 , Γ 1 } is a quasi boundary triple for the adjoint S * of a given symmetric operator S it is not necessary to know S * explicitly, as the following useful proposition shows; cf. [3, Theorem 2.3] for a proof.
Proposition 2.2. Let T be a linear operator in a separable Hilbert space H, let G be a further Hilbert space, and let Γ 0 , Γ 1 : dom T → G be linear mappings which satisfy the following conditions.
2.2. γ-fields and Weyl functions. Let S be a closed, densely defined, symmetric operator in H and let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T = S * with A 0 = T ↾ ker Γ 0 . In order to define the γ-field and the Weyl function corresponding to {G, Γ 0 , Γ 1 } note that the direct sum decomposition holds for each λ ∈ ρ(A 0 ) and that, in particular, the restriction of Γ 0 to ker(T − λ) is injective. The following definition is formally the same as for ordinary and generalized boundary triples. It follows immediately from the definition that for each λ ∈ ρ(A 0 ) the operator M (λ) satisfies the equality and that ran γ(λ) = ker(T − λ) holds. We summarize some properties of the γ-field and the Weyl function. For the proofs of items (i)-(iv) in the next lemma we refer to [3, Proposition 2.6], item (v) is a simple consequence of (ii) and (iii).
(i) γ(λ) is a bounded operator from G to H defined on the dense subspace ran Γ 0 . The adjoint γ(λ) * : H → G is defined on H and is bounded. It is given by (ii) The identity holds for all g ∈ ran Γ 0 . (iii) The γ-field and the Weyl function are connected via is an operator in G defined on the dense subspace ran Γ 0 and satisfies for all g ∈ ran Γ 0 . In particular, for every g ∈ ran Γ 0 the function λ → M (λ)g is holomorphic on ρ(A 0 ) and each isolated singularity of λ → M (λ)g is a pole of first order. Moreover, lim ηց0 iηM (ζ + iη)g exists for all g ∈ ran Γ 0 and all ζ ∈ R.

2.3.
Simple symmetric operators and local simplicity. Let S be a closed, densely defined, symmetric operator in the separable Hilbert space H. Recall that S is said to be simple or completely non-selfadjoint if there is no nontrivial Sinvariant subspace H 0 of H which reduces S to a selfadjoint operator in H 0 , see [1,]. According to [40] the simplicity of S is equivalent to the density of the span of the defect spaces of S in H, i.e., S is simple if and only if holds; here clsp stands for the closed linear span. Assume that {G, Γ 0 , Γ 1 } is a quasi boundary triple for T = S * with A 0 = T ↾ ker Γ 0 . Then it follows that S is simple if and only if (2.3) holds with ker(S * − ν) replaced by ker(T − ν). Moreover, if γ is the γ-field corresponding to the quasi boundary triple {G, Γ 0 , Γ 1 } we conclude that S is simple if and only if holds. We also mention that the set C \ R in (2.4) can be replaced by any set G ⊂ ρ(A 0 ) which has an accumulation point in each connected component of ρ(A 0 ); cf. Lemma 2.5 (v) below.
Our aim is to generalize the notion of simplicity and to replace it by some weaker, local condition, which is satisfied in, e.g., the applications in Section 4. Instead of (2.4) we will assume that holds on a Borel set (later on usually an open interval) ∆; here E(·) denotes the spectral measure of A 0 . This condition will be imposed in many of the general results in Section 3. In the next lemma we discuss this condition and some consequences of it.
Lemma 2.5. Let S be a closed, densely defined, symmetric operator in H and let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T = S * with A 0 = T ↾ ker Γ 0 . Then the following holds.
Proof. Assertion (i) is a consequence of item (ii) since (2.5) holds with ∆ = R when S is simple. For (ii) note that the inclusion ⊃ in (2.6) clearly holds. For the converse inclusion let u ∈ E(∆ ′ )H. As ∆ ′ ⊂ ∆ we have u ∈ E(∆)H and hence there exists a sequence (v n ), n = 1, 2, . . . , in the linear span of {E(∆)γ(ν)g : ν ∈ C \ R, g ∈ ran Γ 0 } which converges to u. Then (E(∆ ′ )v n ), n = 1, 2, . . . , is a sequence in the linear span of In order to prove (iii) let δ j be as in the assumptions and let ∆ = j δ j . The inclusion ⊃ in (2.5) again is obvious. For the converse inclusion let u ∈ E(∆)H and define h ∈ ran Γ 0 and hence the assertion follows if we verify for all µ ∈ C \ R, h ∈ ran Γ 0 , and all j. For this purpose consider some fixed E(δ j )γ(µ)h. According to Lemma 2.4 (ii) we have for all ν ∈ C \ R and all g ∈ ran Γ 0 , and hence H in (2.9) can be rewritten in the form It follows that for η, ε > 0 the element belongs to H, where we have written δ j = (α j , β j ). From this and Stone's formula it follows which proves (2.10) and, hence, yields the inclusion ⊂ in (2.5). Item (iii) is proved. In order to verify (iv), assume that Su = λu for some u ∈ dom S and λ ∈ ∆. Then A 0 u = λu and hence u ∈ E(∆)H. On the other hand, for g ∈ ran Γ 0 and ν ∈ C \ R it follows together with Lemma 2.4 (i) that This implies u = 0 and thus S does not possess eigenvalues in ∆.

Spectral properties of selfadjoint operators and corresponding Weyl functions
This section contains the main abstract results of this paper. We describe the spectral properties of a given selfadjoint operator by means of a corresponding Weyl function. For this we fix the following setting.
Assumption 3.1. Let S be a closed, densely defined, symmetric operator in the separable Hilbert space H and let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T = S * with corresponding γ-field γ and Weyl function M . Moreover, let A 0 = T ↾ ker Γ 0 and denote by E(·) the spectral measure of A 0 .

3.1.
Eigenvalues and corresponding eigenspaces. Let us start with a characterization of the isolated and embedded eigenvalues as well as the corresponding eigenspaces of a selfadjoint operator by means of an associated Weyl function. We write s-lim for the strong limit of an operator function.
is bijective, where cl τ denotes the closure in the normed space ran τ .
Remark 3.3. Recall that the limit (R λ M )g = lim ηց0 iηM (λ + iη)g exists for all λ ∈ R and all g ∈ ran Γ 0 by Lemma 2.4 (iv). Moreover, if λ is an isolated singularity of M , that is, there exists an open neighborhood O of λ such that M is strongly holomorphic on O \ {λ}, then R λ M = 0 if and only if for some g ∈ ran Γ 0 the G-valued function ζ → M (ζ)g has a pole at λ. In this case R λ M coincides with the residue Res λ M of M at λ in the strong sense, i.e., where C denotes the boundary of an open ball B such that M is strongly holomorphic in a neighborhood of B except the point λ. We also remark that without additional assumptions the Weyl function is not able to distinguish between isolated and embedded eigenvalues of A 0 ; cf. Proposition 3.6 below.
Proof of Theorem 3.2. Let λ ∈ R be fixed. Note first that the mapping From this it follows immediately that the mapping τ in (3.1) and (3.2) is well-defined and bijective. In order to verify (3.3) let g ∈ ran Γ 0 and denote by E(·) the spectral measure of A 0 . Then holds for all ν ∈ C \ R. Since by Lemma 2.4 (i) the operator γ(ν) * is bounded, it follows from (3.4) for all ν ∈ C \ R, and together with Lemma 2.4 (v) we conclude that the limit on the left hand side of (3.5) coincides with .
With the help of Lemma 2.4 (i), (3.5) and (3.6) we obtain where we have used Γ 1 (ker(S − λ)) = {0}. From this the first inclusion in (3.3) follows immediately. For the second inclusion in (3.3) note that the mapping Γ 1 ↾ K is continuous as In fact, fix ν ∈ C \ R and let u ∈ K be orthogonal to P γ(ν)g for all g ∈ ran Γ 0 . Then by Lemma 2.4 (i), which implies Γ 1 u = 0 as ran Γ 0 is dense. Hence we have u ∈ ker Γ 0 ∩ ker Γ 1 = dom S and this implies u ∈ K ∩ ker(S − λ), so that u = 0. Now the second inclusion in (3.3) follows together with (3.7) and the fact that Γ 1 ↾ K is continuous. Hence the mapping τ in (3.2) is well-defined and bijective. If K is finite-dimensional then clearly the closure in (3.2) can be omitted and we end up with the bijectivity of (3.1).
As an immediate consequence of Theorem 3.2 all eigenvalues of A 0 which are not eigenvalues of S can be characterized as "generalized poles" of the Weyl function.
Corollary 3.4. Let Assumption 3.1 be satisfied, and assume that λ ∈ R is not an eigenvalue of S. Then λ is an eigenvalue of A 0 if and only if R λ M := slim ηց0 iηM (λ + iη) = 0. If the multiplicity of the eigenvalue λ is finite then the mapping is bijective; if the multiplicity of the eigenvalue λ is infinite then the mapping is bijective, where cl τ denotes the closure in the normed space ran τ .

3.2.
Continuous, absolutely continuous, and singular continuous spectra. In this subsection we describe the continuous, absolutely continuous, and singular continuous spectrum of a selfadjoint operator A 0 by means of the limits of an associated Weyl function M . Again we fix the setting in Assumption 3.1. It is clear that an additional minimality or simplicity condition must be imposed. Usually one assumes that the underlying symmetric operator S is simple; cf. [12]. However, for our purposes the weaker assumption of local simplicity in Section 2.3 is more appropriate: in order to characterize the spectrum of A 0 in an open interval ∆ ⊂ R we assume that For instance, in Theorem 3.2 it turned out that an eigenvalue λ of A 0 with its full multiplicity can only be detected by the Weyl function if λ / ∈ σ p (S). This condition corresponds to the identity (3.8) with ∆ replaced by {λ}; cf. Lemma 2.5 (iv).
In the next theorem we agree to say that the Weyl function M can be continued analytically to some point λ ∈ R if there exists an open neighborhood O of λ in C such that ζ → M (ζ)g can be continued analytically to O for all g ∈ ran Γ 0 . We mention that the proof of (i) is similar to the proof of [25, Theorem 1.1]. If S is simple then (i) and (ii) hold for all λ ∈ R.
Proof. (i) Recall first that by Lemma 2.4 (iv) the function λ → M (λ)g is analytic on ρ(A 0 ) for each g ∈ ran Γ 0 , which proves the implication (⇒). In order to verify the implication (⇐) in (i), let us assume that M can be continued analytically to some λ ∈ ∆, that is, there exists an open neighborhood O of λ in C with O ∩ R ⊂ ∆ such that ζ → M (ζ)g can be continued analytically to O for each g ∈ ran Γ 0 . Choose a, b ∈ R with λ ∈ (a, b), [a, b] ⊂ O, and a, b / ∈ σ p (A 0 ). The spectral projection E ((a, b)) of A 0 corresponding to the interval (a, b) is given by Stone's formula where the integral on the right-hand side is understood in the strong sense. Using the identity in Lemma 2.4 (v) and (3.9) a straight forward calculation leads to for all g ∈ ran Γ 0 and all ν ∈ C \ R, since ζ → (M (ζ)g, g) admits an analytic continuation into O for all g ∈ ran Γ 0 . Thus the assumption (3.8) and [a, b] ⊂ ∆ together with Lemma 2.5 (ii) imply E((a, b)) = 0. In particular, λ ∈ ρ(A 0 ).
(ii) According to Lemma 2.5 (iv) the condition (3.8) implies that S does not have eigenvalues in ∆. Hence item (ii) follows immediately from item (i) and Corollary 3.4.
If S is simple then by Lemma 2.5 (i) the assumption (3.8) is satisfied for ∆ = R. Hence (i) and (ii) hold for all λ ∈ R.
Now we return to the characterization of eigenvalues. We formulate a sufficient condition under which the Weyl function is able to distinguish between isolated and embedded eigenvalues. Proof. Let λ ∈ R and let ∆ ⊂ R be an open interval with λ ∈ ∆ such that (3.8) holds. Then λ ∈ σ p (S) by Lemma 2.5 (iv) and hence the assertions in Corollary 3.4 hold for λ. Moreover, if λ is an isolated eigenvalue of A 0 then by Lemma 2.4 (iv) there exists an open neighborhood O of λ such that ζ → M (ζ)g is holomorphic on O \ {λ} for all g ∈ ran Γ 0 . From Corollary 3.4 we conclude that there exists g ∈ ran Γ 0 such that lim ηց0 iηM (λ + iη)g = 0. (3.10) Hence Lemma 2.4 (iv) implies that M has a pole of first order in the strong sense at λ. Conversely, if M has a pole (of first order) in the strong sense at λ then there exists g ∈ ran Γ 0 such that (3.10) holds. According to Lemma 2.4 (iv) the order of the pole is one and, hence, for all g ∈ ran Γ 0 . It follows with the help of Corollary 3.4 that λ is an eigenvalue of A 0 . Moreover, Theorem 3.5 (i) implies that there exists an open neighborhood . Hence λ is isolated in the spectrum of A 0 . This completes the proof.
Next we discuss the relation of the function M to the absolutely continuous and singular continuous spectrum of A 0 . In the special case of ordinary boundary triples and ∆ = R the following results reduce to those in [12]. For our purposes a localized version and an extension to quasi boundary triples is necessary. The proofs presented here are somewhat more direct than those in [12]; in particular, the integral representation of Nevanlinna functions and the corresponding measures are avoided.
In the following for a finite Borel measure µ on R we denote the set of all growth points of µ by supp µ, that is, Note that supp µ is closed with µ(R \ supp µ) = 0 and that supp µ is minimal with this property, that is, each closed set S ⊂ R with µ(R \ S) = 0 satisfies supp µ ⊂ S. Moreover, for a Borel set χ ⊂ R we define the absolutely continuous closure (also called essential closure) by where | · | denotes the Lebesgue measure, and the continuous closure by Observe that cl ac (χ) and cl c (χ) both are closed and that cl ac (χ) ⊂ cl c (χ) ⊂ χ holds, but in general the converse inclusions are not true. In fact, cl ac (χ) = ∅ if and only if |χ| = 0, and cl c (χ) = ∅ if and only if χ is countable.
The following lemma can partly be found in, e.g., the monographs [26] or [54].
Lemma 3.7. Let µ be a finite Borel measure on R and denote by F its Borel transform, Then the limit Im F (x + i0) = lim yց0 Im F (x + iy) exists and is finite for Lebesgue almost all x ∈ R. Let µ ac and µ s be the absolutely continuous and singular part, respectively, of µ in the Lebesgue decomposition µ = µ ac + µ s , and decompose µ s into the singular continuous part µ sc and the pure point part. Then the following assertions hold.
Proof. From [54, Lemma 3.14 and Theorem 3.23] it follows immediately that assertion (i) is true, that the limit Im F (x + i0) exists and is finite for Lebesgue almost all x ∈ R, and that µ s R \ {x ∈ R : Im F (x + i0) = +∞} = 0, (3.12) which implies (ii). In order to verify (iii) note first that lim yց0 yF (x+iy) = iµ({x}) holds for all x ∈ R since For x ∈ R \ cl c (M sc ) by definition there exists ε > 0 such that (x − ε, x + ε) ∩ M sc is countable; thus µ sc ((x − ε, x + ε) ∩ M sc ) = 0. With the help of (3.13) it follows The absolutely continuous spectrum of a selfadjoint operator in some interval ∆ can be characterized in the following way.
is satisfied for each open interval δ ⊂ ∆ with δ ∩ σ p (S) = ∅. Then the absolutely continuous spectrum of A 0 in ∆ is given by cl ac x ∈ ∆ : 0 < Im(M (x + i0)g, g) < +∞ . Proof. The proof of Theorem 3.8 consists of two separate steps in which the assertions (3.17) and (3.19) below will be shown. The identity (3.15) is then an immediate consequence of (3.17) and (3.19) (note that the right hand side in (3.19) does not depend on ζ ∈ C \ R). We fix some notation first. Let us set D ∆ := E(∆)γ(ζ)g : ζ ∈ C \ R, g ∈ ran Γ 0 (3. 16) and define the measures µ u := (E(·)u, u) for u ∈ H. Denote by P ac the orthogonal projection in H onto the absolutely continuous subspace H ac of A 0 . Observe that the spectral measure of the absolutely continuous part of A 0 is E(·)P ac and that the absolutely continuous measures µ u,ac are given by µ u,ac = (E(·)P ac u, P ac u) = µ Pacu .
Step 1. In this step the identity by assumption. With the help of Lemma 2.5 (iii) we conclude and therefore P ac E(∆)H = clsp P ac D ∆ .
(ii) If the condition (3.14) is satisfied for each open interval δ ⊂ ∆ with δ ∩ σ p (S) = ∅ then the singular continuous spectrum of cl c x ∈ ∆ : Im(M (x + i0)g, g) = +∞, lim Proof. We show the statements (i) and (ii) at once. Let us define Note first that the same arguments as in Step 1 of the proof of Theorem 3.8 imply In order to apply Lemma 3.7 (ii) and (iii), respectively, we calculate the limits that appear there. In fact, it follows from (2.2) that for each g ∈ ran Γ 0 and each and lim yց0 y(M (x + iy)g, g) = |x − ζ| 2 lim yց0 y (A 0 − (x + iy)) −1 γ(ζ)g, γ(ζ)g (3.28) hold; cf. (3.21) for the first identity and the text below (3.21) for its interpretation as a possible improper limit. Let u = E(∆)γ(ζ)g ∈ D ∆ and let be the Borel transform of µ u = (E(·)u, u). Then for all x ∈ R. From this we conclude with the help of (3.27) that Similarly, from (3.28) we obtain for u = E(∆)γ(ζ)g ∈ D ∆ . Thus the assertions of the theorem follow from (3.26).
As a further immediate corollary of Theorem 3.12 we formulate a sufficient criterion for the absence of singular continuous spectrum in terms of the limiting behaviour of the function M . The corresponding result for ordinary boundary triples (in the special case ∆ = R) can be found in [12]. As a further corollary of the theorems of this section we provide sufficient criteria for the spectrum of the operator A 0 to be purely absolutely continuous or purely singular continuous, respectively, in some set. for all g ∈ ran Γ 0 and all x ∈ ∆. Then the following assertions hold.

Second order elliptic differential operators on R n
In this section we show how the spectrum of a selfadjoint second order elliptic differential operator on R n , n ≥ 2, can be described with the help of a Titchmarsh-Weyl function acting on an n − 1-dimensional compact interface Σ which splits R n into a bounded domain Ω i and an unbounded domain Ω e with common boundary Σ.
We consider the differential expression where a jk , a j ∈ C ∞ (R n ) together with their derivatives are bounded and satisfy a jk (x) = a kj (x) for all x ∈ R n , 1 ≤ j, k ≤ n, and a ∈ L ∞ (R n ) is real valued. Moreover, we assume that L is uniformly elliptic on R n , that is, there exists E > 0 with n j,k=1 The selfadjoint operator associated with L in L 2 (R n ) is given by where H 2 (R n ) is the usual L 2 -based Sobolev space of order 2 on R n . In Sections 4.1 and 4.2 two different choices of Titchmarsh-Weyl functions for the differential expression L, both acting on the interface Σ, are studied.

A Weyl function corresponding to a transmission problem.
We first consider a Weyl function for the operator A 0 which appears in transmission problems in connection with single layer potentials (see, e.g. [47,Chapter 6] and [50]) and which was also used in [2] to generalize the classical limit point/limit circle analysis from singular Sturm-Liouville theory to Schrödinger operators in R 3 . Let Σ be the boundary of a bounded C ∞ -domain Ω i ⊂ R n and denote by Ω e the exterior of Σ, that is, Ω e = R n \ Ω i . In the following we make use of operators induced by L in L 2 (Ω i ) and L 2 (Ω e ), respectively. For j = i, e we write L j for the restriction of the differential expression L to functions on Ω j . For functions in L 2 (Ω j ) we use the index j and we write u = u i ⊕ u e for u ∈ L 2 (R n ). As Σ is smooth, the selfadjoint Dirichlet operator associated with L j in L 2 (Ω j ) is given by where u j | Σ denotes the trace of u j at Σ = ∂Ω j . Let H s (Σ) be the Sobolev spaces of orders s ≥ 0 on Σ. We recall that for each λ ∈ ρ(A D,j ) and each g ∈ H 3/2 (Σ) there exists a unique solution u λ,j ∈ H 2 (Ω j ) of the boundary value problem L j u j = λu j , u j | Σ = g. This implies that for each λ ∈ ρ(A D,j ) the Dirichlet-to-Neumann map is well-defined; here the conormal derivative with respect to L j in the direction of the outer unit normal ν j = (ν j,1 , . . . , ν j,n ) ⊤ at Σ = ∂Ω j is defined by Note that the outer unit normals at ∂Ω i and ∂Ω e coincide up to a minus sign. The operator Λ i (λ) + Λ e (λ) is invertible for all λ ∈ ρ(A 0 ) ∩ ρ(A D,i ) ∩ ρ(A D,e ) and, hence, the operator function . We remark that the values M (λ) are bounded operators in L 2 (Σ) with domain H 1/2 (Σ); cf. Lemma 4.2 below for the details.
The following theorem is the main result of this section. It states that the absolutely continuous spectrum of A 0 can be recovered completely from the knowledge of the function M in (4.4), while the eigenvalues and corresponding eigenspaces may be only partially visible for the function M . This depends on the choice of the interface Σ and the fact that the symmetric operator may have eigenvalues. In particular, in general S is not simple; cf. Example 4.5 and Example 4.6 below.
Theorem 4.1. Let A 0 , Σ, S, and M be as above, let λ, µ ∈ R such that λ ∈ σ p (S), µ ∈ σ p (S), and let ∆ ⊂ R be an open interval. Then the following assertions hold.
The proof of Theorem 4.1 makes use of the following two lemmas and is given at the end of this subsection.
In the next lemma it is shown that S satisfies the local simplicity in the assumptions of the results in Section 3. Lemma 4.3. Let A 0 be the selfadjoint elliptic operator in (4.2) with spectral measure E(·) and let S be the symmetric operator in (4.5). Let {L 2 (Σ), Γ 0 , Γ 1 } be the quasi boundary triple in Lemma 4.2 and let γ be the corresponding γ-field. Then clsp E(δ)γ(ν)g : g ∈ H 1/2 (Σ), ν ∈ C \ R = E(δ)L 2 (R n ) holds for every open interval δ ⊂ R such that δ ∩ σ p (S) = ∅.
Proof. For j = i, e we consider the densely defined, closed, symmetric operators in L 2 (Ω j ) and the operators is a quasi boundary triple for S * j , j = i, e; cf. [3,Proposition 4.1]. For λ ∈ ρ(A D,j ), j = i, e, the corresponding γ-fields are given by where u λ,j is the unique solution in H 2 (Ω j ) of L j u j = λu j , u j | Σ = ϕ. It follows in the same way as in [10,Proposition 2.2] that S e is simple; the simplicity of S i follows from a unique continuation argument, see, e.g. [9, Proposition 2.5]. Therefore we have and hence Here and in the following ⊕ denotes the orthogonality of the closed subspaces L 2 (Ω i ) and L 2 (Ω e ) in L 2 (R n ). Let now δ ⊂ R be an open interval such that δ ∩ σ p (S) = ∅ and let T be as in (4.8). Since γ i (ν)g ⊕ γ e (ν)g : g ∈ H 3/2 (Σ) = ker(T − ν) = ran γ(ν), ν ∈ C \ R, (4.11) we have to verify that We note first that the inclusion H δ ⊂ E(δ)L 2 (R n ) is obviously true. For the opposite inclusion we conclude from (4.10) that it suffices to verify Let us show the statements in (4.12). We start with the second one. Let us fix µ ∈ C \ R. By Lemma 2.4 (ii) we have Since A D,i and A D,e are both semibounded from below we may choose λ 0 ∈ R such that σ(A D,j ) ⊂ (λ 0 , ∞), j = i, e. Recall that the spectrum of A D,i is purely discrete and let λ 1 < λ 2 < . . . be the distinct eigenvalues of A D,i . Then for all η, ε > 0 and k = 0, 1, 2, . . . the function belongs to H δ , and as (λ k , λ k+1 ) ⊂ ρ(A D,i ), Stone's formula implies where E e (·) is the spectral measure of A D,e . Next we show that for the eigenvalues λ k , k = 1, 2, . . . , of A D,i the property holds. For this consider the element for some fixed h ∈ H 3/2 (Σ). Clearly, as u ∈ ker((A D,i ⊕A D,e )−λ k ) and as A D,i ⊕A D,e is a selfadjoint extension of the symmetric operator S in (4.5) we may write u in the form u = u D ⊕u S with u S ∈ ker(S − λ k ) and where ⊕ and ⊖ indicate the orthogonality of subspaces in ker((A D,i ⊕ A D,e ) − λ k ). Then for each v ∈ ν∈C\R ran (S − ν) and each ν ∈ C \ R one has (4.16) Since the limit holds for all w ∈ dom S * we conclude that In particular, (4.15) implies (y, u D ) = 0. Therefore we obtain from the identity (4.16) with ν = λ k + iη in the limit This shows that u D is orthogonal to ν∈C\R ran (S − ν) and hence Therefore (4.11) implies Note that if the eigenvalue λ k of A D,i is contained in the interval δ then by assumption λ k ∈ σ p (S) and hence u = u D in this case. If λ k ∈ δ then u S ∈ ker(S − λ k ) ⊂ ker(A 0 − λ k ) implies that u S is orthogonal to ran E(δ), so that E(δ)u S = 0. Summing up we have for any eigenvalue λ k , k = 1, 2, . . . , of A D,i that (4.17). We have shown (4.14). Let m ∈ N. Then we have E e ((λ k , λ k+1 ))γ e (µ)h and from (4.13) and (4.14) we conclude Taking the limit m ր +∞ we obtain E(δ)(0 ⊕ γ e (µ)h) ∈ H δ . We have proved the second statement in (4.12).
can be verified in the same way as (4.14), where E i (·) is the spectral measure of A D,i . Hence for m ∈ N we conclude and in the limit m ր +∞ we obtain the first statement in (4.12). Now (4.12) together with (4.10) imply the inclusion E(δ)L 2 (R n ) ⊂ H δ . This completes the proof of Lemma 4.3.
As a consequence of Lemma 4.3 we obtain the following corollary. Proof of Theorem 4.1. Let {L 2 (Σ), Γ 0 , Γ 1 } be the quasi boundary triple for T = S * in Lemma 4.2. Then T ↾ ker Γ 0 corresponds to the selfadjoint elliptic differential operator A 0 in (4.2) and the associated Weyl function coincides with the operator function M in (4.4). Taking Lemma 4.3 into account, item (i) follows from Corollary 3.4 and items (ii)-(iv) are consequences of Theorem 3.5 and Proposition 3.6 when choosing an open interval δ ∋ λ with δ ∩ σ p (S) = ∅. Moreover, item (v) follows from Theorem 3.8 and Corollary 3.11, and item (vi) is due to Theorem 3.12 and Corollary 3.15.
We point out that in the case that the symmetric operator S is simple the assertions in Theorem 4.1 hold for all λ, µ ∈ R. On the other hand, without further assumptions, it may happen that S possesses eigenvalues. In this case at least the parts of the eigenspaces of A which do not belong to S can be characterized in terms of the function M ; cf. Theorem 3.2. The next examples illustrate that a proper choice of the interface Σ may avoid eigenvalues of S. Example 4.5. Assume that L equals the Laplacian outside some compact set K ⊂ R n and choose Σ to be the boundary of any smooth, bounded domain Ω i ⊃ K. Then S does not have any eigenvalues. Indeed, if u ∈ H 2 (R n ) satisfies Lu = λu on R n and u| Σ = 0 then u| Ωe belongs to ker(A D,e − λ) and must vanish. Then a unique continuation argument implies u = 0. Hence S is simple by Corollary 4.4 and the assertions in Theorem 4.1 hold for all λ, µ ∈ R. Example 4.6. Let the coefficients of L be chosen in a way such that for some bounded, smooth domain Ω i ⊂ R n the operator A D,i in L 2 (Ω i ) is strictly positive; for instance this happens if − 2 E n j=0 a j 2 ∞ + inf a ≥ 0 on Ω i , where E is an ellipticity constant for L, see (4.1). If we choose Σ = ∂Ω i then S has no nonpositive eigenvalues, otherwise Su = λu for some λ ≤ 0 and u ∈ dom S with u = 0, and a unique continuation argument yields that u i is nontrivial, thus u i is an eigenfunction of A D,i corresponding to the eigenvalue λ ≤ 0, a contradiction. Hence in this situation all non-positive eigenvalues of A 0 and the corresponding eigenspaces can be described completely in terms of the function M .

4.2.
A block operator matrix Weyl function associated with a decoupled system. In this section we consider a different Weyl function for the operator A 0 , which corresponds to a symmetric operator which is always simple, independently of the choice of the interface Σ. This symmetric operator is the orthogonal sum of the minimal symmetric realizations S i and S e of L in L 2 (Ω i ) and L 2 (Ω e ), respectively, in the proof of Lemma 4.3, and hence an infinite dimensional restriction of the symmetric operator in (4.5); it can be viewed as a decoupled symmetric operator. Let Λ i and Λ e be the Dirichlet-to-Neumann maps for the interior and exterior elliptic boundary value problem, respectively, defined in (4.3), and let A N,e u e = L e u e , dom A N,e = u e ∈ H 2 (Ω e ) : ∂u e ∂ν Le Σ = 0 , be the selfadjoint realization of L e in L 2 (Ω e ) with Neumann boundary conditions. In Lemma 4.8 below it will turn out that the function is well defined on ρ(A 0 ) ∩ ρ(A D,i ) ∩ ρ(A N,e ) and can be viewed as the Weyl function of a quasi boundary triple for (S i ⊕S e ) * , where A 0 in (4.2) corresponds to the kernel of the first boundary mapping. We mention that a scalar analog of the function M in (4.18) appears in connection with λ-dependent Sturm-Liouville boundary value problems in [24] and in more general abstract form in [19], see also [8] for more details and references. In the present setting Lemma 4.8 and Lemma 4.9 below combined with the results in Section 3 lead to an improvement of items (i)-(iv) in Theorem 4.1. The assertions (v) and (vi) in Theorem 4.1 remain valid with M and H 1/2 (Σ) replaced by M and H 1/2 (Σ) × H 3/2 (Σ), respectively, but will not be formulated again. is bijective, where cl τ denotes the closure in the normed space ran τ .  We provide a quasi boundary triple such that M in (4.18) is the corresponding Weyl function. As indicated above we make use of the densely defined, closed, symmetric operators S j u j = L j u j , dom S j = u j ∈ H 2 (Ω j ) : u j | Σ = ∂u j ∂ν Lj Σ = 0 , in L 2 (Ω j ) for j = i, e, which appeared already the proof of Lemma 4.3 and which are both simple. Besides the operators S j also the operators appear in the formulation of the next lemma. which is dense in (L 2 (Σ) × L 2 (Σ)) 2 . Moreover, C ∞ 0 (R n \ Σ) is a dense subspace of L 2 (R n ) which is contained in ker Γ 0 ∩ ker Γ 1 . Green's identity implies that (2.1) holds, and as H 2 (R n ) is contained in ker Γ 0 the selfadjoint operator A 0 is contained in (T i ⊕ T e ) ↾ ker Γ 0 . Hence the assumptions (i)-(iii) in Proposition 2.2 are satisfied and it follows that {L 2 (Σ) × L 2 (Σ), Γ 0 , Γ 1 } is a quasi boundary triple for S * i ⊕ S * e such that A 0 = (T i ⊕ T e ) ↾ ker Γ 0 . Let us verify that the corresponding Weyl function is given by M in (4.18). For this let λ ∈ ρ(A 0 ) ∩ ρ(A D,i ) ∩ ρ(A N,e ) and let u λ = u λ,i ⊕ u λ,e ∈ dom (T i ⊕ T e ) be such that L j u λ,j = λu λ,j , j = i, e. Then we have Lemma 4.9. The symmetric operator S i ⊕ S e is simple.
Proof of Theorem 4.7. Let {L 2 (Σ)×L 2 (Σ), Γ 0 , Γ 1 } be the quasi boundary triple in Lemma 4.8. Then (T i ⊕ T e ) ↾ ker Γ 0 corresponds to the selfadjoint elliptic differential operator A 0 in (4.2) and the associated Weyl function coincides with the operator function M in (4.18). Taking Lemma 4.9 into account, item (i) follows from Corollary 3.4 and items (ii)-(iv) are consequences of Theorem 3.5 and Proposition 3.6.