Spectral theory for Schr\"odinger operators with $\delta$-interactions supported on curves in $\mathbb R^3$

The main objective of this paper is to systematically develop a spectral and scattering theory for selfadjoint Schr\"odinger operators with $\delta$-interactions supported on closed curves in $\mathbb R^3$. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten--von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix.


Introduction
Schrödinger operators with singular interactions supported on sets of Lebesgue measure zero were suggested in the physics literature as solvable models in quantum mechanics in [12,38,46,49,61]. They appear, e.g., in the modeling of zerorange interactions of quantum particles [22,23,52,53], in the theory of photonic crystals [42], and in quantum few-body systems in strong magnetic fields [20]. The mathematical investigation of their spectral and scattering properties attracted a lot of attention during the last decades. First studies were mostly devoted to singular interactions supported on a discrete set of points, see the monograph [4] and [35,Chapter 5]. Later on, singular interactions supported on more general curves, surfaces, and manifolds gained much attention; there is an extensive literature on Schrödinger operators with δ-interactions supported on manifolds of codimension one, see, e.g, [5,9,16,18,27,30,35,36,37] and the references therein. Manifolds of higher codimension were first treated in [17] in the very special case of an interaction supported on a straight line in R 3 . More general curves were considered in [13,19,28,31,32,33,34,45,47,48,54,56,60].
In the present paper we systematically develop a spectral and scattering theory for Schrödinger operators with singular interactions supported on curves in the three-dimensional space. More specifically, for a compact, closed, regular C 2 -curve Σ ⊂ R 3 we consider the selfadjoint Schrödinger operator −∆ Σ,α in L 2 (R 3 ), which corresponds to the formal differential expression where α ∈ R\{0} is the inverse strength of interaction. The mathematically rigorous definition of −∆ Σ,α is more involved than in the case of, e.g., a curve in R 2 or a hypersurface in R 3 . For our purposes an explicit characterization of the domain and action of −∆ Σ,α is essential; here the key difficulty is to define an appropriate generalized trace map for functions which are not sufficiently regular; see Section 2 for the details. Our method is strongly inspired by [56] and the abstract concept of boundary triples [7,8,21,24,25], and can also be viewed as a special case of the more general approach in [54] (see Example 3.5 therein); cf. [19,31,34,60] for equivalent alternative definitions.
The main results of this paper deal with spectral and scattering properties of −∆ Σ,α and extend and complement results in [19,26,28,29,32,45,56]. First we verify that the operator −∆ Σ,α is in fact selfadjoint; along with this, in Theorem 3.1 we establish a Krein type formula for the resolvent difference of −∆ Σ,α and the free Laplacian −∆ free . Using this formula we show that the resolvent difference is compact; in particular, the essential spectrum of −∆ Σ,α equals [0, ∞). Moreover, we provide a Birman-Schwinger principle for the negative eigenvalues of −∆ Σ,α and employ this principle for a more detailed study of these eigenvalues. In fact, in Theorem 3.3 we show that the negative spectrum is always finite and we prove upper and lower estimates for the number of negative eigenvalues, depending on the (inverse) strength of interaction α and the geometry of the curve; these results complement the estimates in [19,32,44,45]. In the case that Σ is a circle our estimates lead to an explicit formula for the number of negative eigenvalues. As a further main result, in Theorem 3.6 we prove that amongst all curves of a fixed length the principle eigenvalue of −∆ Σ,α is maximized by the circle. With this result we give an affirmative answer to an open problem formulated in [27,Section 7.8].
Our proof is inspired by related considerations for δ-interactions supported on loops in the plane in [26,29]. Another group of results focuses on a more detailed comparison of −∆ Σ,α with the free Laplacian. From a careful analysis of the operators involved in the Krein type resolvent formula we obtain an asymptotic upper bound for the singular values s 1 (λ) ≥ s 2 (λ) ≥ . . . of the resolvent difference (1.2) in Theorem 3.2, (1. 3) s k (λ) = O 1 k 2 ln k as k → +∞.
In particular, the resolvent difference in (1.2) belongs to the Schatten-von Neumann class S p for any p > 1/2; this improves the trace class estimate in [19] and is in accordance with a previous observation for periodic curves in [28,Remark 4.1]. Note that, as a consequence of (1.3), the absolutely continuous spectrum of −∆ Σ,α equals [0, ∞) and the wave operators for the scattering pair {−∆ free , −∆ Σ,α } exist and are complete. In Theorem 3.8 a representation of the associated scattering matrix is given in terms of an explicit operator function which acts in L 2 (Σ); this complements earlier investigations in [19,Section 3]. Its proof relies on an abstract approach developed recently in [11]. The paper is organized as follows. In Section 2 we discuss in detail the mathematically rigorous definition of the operator −∆ Σ,α . Section 3 contains all main results of this paper. Their proofs are carried out in the remainder of this paper. In fact, Section 4 is preparatory and contains the analysis of the Birman-Schwinger operator. The actual proofs of Theorems 3.1-3.8 are contained in Section 5. In a short appendix the notions of quasi boundary triples and their Weyl functions from extension theory of symmetric operators are reviewed and it is shown how the operators −∆ free and −∆ Σ,α fit into this abstract scheme.

Acknowledgements
Jussi Behrndt, Christian Kühn, Vladimir Lotoreichik, and Jonathan Rohleder gratefully acknowledge financial support by the Austrian Science Fund (FWF), project P 25162-N26. Vladimir Lotoreichik also acknowledges financial support by the Czech Science Foundation, project 14-06818S. Rupert Frank acknowledges support through NSF grant DMS-1363432. The authors also wish to thank Johannes Brasche and Andrea Posilicano for helpful discussions and the anonymous referees for their helpful comments which led to various improvements.

Definition of the operator −∆ Σ,α
In this section we define the operator −∆ Σ,α associated with the differential expression (1.1) in L 2 (R 3 ). On a formal level we interpret the action of (1.1) as It will be shown that A α gives rise to a selfadjoint operator in L 2 (R 3 ). The key difficulty in the definition of this operator is to specify a suitable domain. Note that the Sobolev space H 2 (R 3 ) is not a suitable domain as u| Σ · δ Σ ∈ L 2 (R 3 ) for all those u ∈ H 2 (R 3 ) which do not vanish identically on Σ. On the other hand, any proper subspace of H 2 (R 3 ) will turn out to be too small for −∆ Σ,α to become selfadjoint in L 2 (R 3 ). Thus it is necessary to include suitable more singular elements in the domain of the operator. This requires the definition of a generalized trace u| Σ for functions u ∈ L 2 (R 3 ) which are not sufficiently regular.
The identity (2.4) indicates that in general the trace of γ λ h on Σ does not exist due to the singularity of the integral kernel. This motivates the following regularization. Here and in the following we denote by C 0,1 (Σ) the space of all complex-valued Lipschitz continuous functions on Σ. Moreover, for x = σ(s 0 ) ∈ Σ and δ > 0 let be the open interval in Σ with center x and length 2δ. In order to define the trace of γ λ h in a generalized sense, for λ ≤ 0, h ∈ C 0,1 (Σ) and x ∈ Σ we set due to technical reasons the case λ = 0 is included here although γ λ is defined for λ < 0 only. It will be shown in Proposition 4.5 that B λ is a well-defined, essentially selfadjoint operator in L 2 (Σ) for each λ ≤ 0 and that the domain of its closure B λ is independent of λ. Note that the basic idea in the definition of B λ is to remove the singularity of γ λ h on Σ. We remark that the limit in the definition of B λ can also be viewed as the finite part in the sense of Hadamard of the first summand as δ ց 0; cf. [51,Chapter 5]. A procedure of this type is frequently employed to define hypersingular integral operators.
With the help of B λ we can make the following definition.
Accordingly, for a function u = u c + γ λ h with u c ∈ H 2 (R 3 ) and h ∈ dom B λ we define its generalized trace u| Σ on Σ by Note that u| Σ is well-defined. Indeed, the representation of u as a sum is unique since γ λ h ∈ H 2 (R 3 ) implies h = 0. Moreover, the definition of u| Σ is independent of the choice of λ < 0; cf. Section 4.3.
Furthermore, note that the expression A α in (2.1) is no longer formal, but makes sense as we have defined the generalized trace u| Σ . Now we are able to define the Schrödinger operator −∆ Σ,α corresponding to the differential expression in (1.1) in a rigorous way.
where λ < 0 is arbitrary and the generalized trace u| Σ is defined in (2.8).
Observe that the operator −∆ Σ,α is well-defined since dom B λ and the trace u| Σ do not depend on the choice of λ. Note also that for α = +∞ we formally have so that the Schrödinger operator with δ-interaction of strength 0 on Σ coincides with the free Laplacian −∆ free ; this will be made precise in Theorem 3.1 (ii) below.
Remark 2.4. The definition of −∆ Σ,α relies on the generalized trace in Definition 2.2 and, thus, on the operator B λ . As mentioned above, the operator B λ is designed in such a way that the singularity of γ λ h on Σ is removed; this is done here by the term ln δ 2π . However, an alternative choice ln δ 2π + c with an arbitrary δindependent constant c ∈ R can be made. This leads to a different operator −∆ Σ,α , which can be transformed into the operator in Definition 2.3 by adding the same constant c to α. For instance, for c = − ln 2 2π one obtains the family of operators considered in [60].

Main results
In this section we present all main results of this paper. It will be shown that −∆ Σ,α is selfadjoint and its spectral and scattering properties will be analyzed. This section is focused on the main statements and does not contain their proofs; these are postponed to Section 5 below. In the following we denote by σ p (−∆ Σ,α ), σ ess (−∆ Σ,α ), and ρ(−∆ Σ,α ) the point spectrum, essential spectrum, and resolvent set of −∆ Σ,α , respectively.
In the first theorem we check that −∆ Σ,α is a selfadjoint operator in L 2 (R 3 ), prove a Birman-Schwinger principle for its negative eigenvalues and compare its resolvent to the resolvent of the free Laplacian −∆ free in a Krein type formula, which also implies that the difference of the resolvents is compact.
Next we investigate the resolvent difference of −∆ Σ,α and the free Laplacian in more detail.
The logarithmic factor in the estimate for the singular values in the above theorem is related to the fact that the eigenvalues of B λ behave asymptotically as − ln k 2π , see Proposition 4.5 (iii).
In the following theorem we show that the discrete spectrum of −∆ Σ,α is always finite and give estimates for the number N α of negative eigenvalues, counted with multiplicities. Let R = L 2π and define the intervals which are disjoint and satisfy R = ∞ r=−1 I r . Moreover, set where σ is the parametrization of Σ fixed in the beginning of Section 2 and τ denotes an arc length parametrization of a circle of radius R. then N α = 0. Otherwise, where r ≥ −1 and l ≥ 0 are such that α + d Σ ∈ I r and α − d Σ ∈ I l . In particular, N α is finite and the operator −∆ Σ,α is bounded from below.
In the next corollary the upper and lower bounds on the number N α of negative eigenvalues in Theorem 3.3 are made more explicit. This also leads to an asymptotic bound N α = e −2πα+O(1) as α → −∞. We mention that a slightly better asymptotic bound was obtained in [32]. For convenience we make a very small technical restriction and consider the case α + d Σ < ln(4R) 2π − 1 π only.
In the case where Σ is a circle we have d Σ = 0 and hence from Theorem 3.3 and Corollary 3.4 we immediately obtain the following explicit expressions for the number of negative eigenvalues. For a similar formula in a related context see [45] (cf. also [19]).
Next, we investigate the behavior of the smallest eigenvalue of −∆ Σ,α when varying Σ among all curves of a given length L. It turns out that circles are the unique maximizers of the minimum of the spectrum σ(−∆ Σ,α ) in the case that negative eigenvalues exist. The analog of the following theorem for curves in the two-dimensional space was shown in [26,29].
Theorem 3.6. Let T be a circle in R 3 of radius R = L 2π and assume that Σ is not a circle. Let α < ln(4R) 2π . Then where −∆ T ,α denotes the Schrödinger operator with δ-interaction of strength 1 α supported on the circle T .
Finally, we regard the pair {−∆ free , −∆ Σ,α } as a scattering system consisting of the unperturbed Laplacian −∆ free and the singularly perturbed operator −∆ Σ,α . The following corollary is an immediate consequence of Theorem 3.2 and the Birman-Krein theorem [15].
Corollary 3.7. The absolutely continuous spectrum of −∆ Σ,α is given by Moreover, the wave operators for the scattering pair {−∆ free , −∆ Σ,α } exist and are complete.
In the next theorem we express the scattering matrix of the scattering system {−∆ free , −∆ Σ,α } in terms of the limits of a certain explicit operator function, using a result in [11]; we refer to [6,43,58,62] and Appendix A for more details on scattering theory. For our purposes it is convenient to consider the symmetric operator S in L 2 (R 3 ) defined as which turns out to be the intersection of the selfadjoint operators −∆ free and −∆ Σ,α . Then S is a densely defined, closed, symmetric operator with infinite defect numbers. Furthermore, in general S contains a selfadjoint part which can be split off. More precisely, consider the closed subspace where the closed symmetric operator S 1 is completely non-selfadjoint or simple (cf. [3, Chapter VII]) in H 1 and S 2 is a selfadjoint operator in H 2 with purely absolutely continuous spectrum. In the following let L 2 (R, dλ, H λ ) be a spectral representation of the selfadjoint operator S 2 in H 2 ; cf. [6,Chapter 4].
where h ∈ L 2 (Σ) and x ∈ Σ. Then the following assertions hold.

The operator B λ and the generalized trace
In this section we discuss properties of the operator B λ in (2.7) and of the generalized trace defined in (2.8). We verify that the latter is well-defined and independent of λ. Our investigation of the operator B λ is split into two parts: first the special case of a circle Σ is treated, and afterwards the results are extended by perturbation arguments to the general case.

4.1.
Properties of B λ for a circle. Throughout this subsection we assume that Σ is a circle of radius R = L 2π . Without loss of generality we assume that Σ lies in the xy-plane and is centered at the origin. We will make use of its arc length parametrization σ : [0, L] → R 3 , σ(t) = R cos(2πt/L), R sin(2πt/L), 0 and occasionally use the formula which holds for elementary geometric reasons. Furthermore, for x = σ(t) ∈ Σ and δ > 0 let I Σ δ (x) be the open interval in Σ with center x and length 2δ as in (2.6). Let us first prove the following preliminary lemma. Its proof is partly inspired by [60, Lemma 1].
Lemma 4.1. Let λ ≤ 0 and x ∈ Σ. Then the limit exists in R, is independent of x and equals In particular, k λ → −∞ as λ → −∞.
Proof. First of all, it follows from the symmetry of the circle Σ that k λ is indeed independent of x (if it exists). Hence, without loss of generality, we can choose x = σ(0). Using (4.1) and the substitution s = π L t we obtain where we have used π L = 1 2R in the last equality. As sin( π Hence in the limit δ ց 0 the equation (4.2) becomes In particular, k λ exists and is finite. By monotone convergence we have As a first step towards the study of the operator B λ on the circle we show properties of B 0 in the following lemma.
Then the following assertions hold.
(ii) B 0 is bounded from above, has a compact resolvent, and its eigenvalues ν k (0), k = 1, 2, . . . , ordered nonincreasingly and counted with multiplicities, are given by For every x ∈ Σ we can write Note that the first integral exists due to the fact that h is Lipschitz continuous. According to Lemma 4.1 (for λ = 0) we can write the above equation as To show the symmetry of B 0 let g, h ∈ C 0,1 (Σ) be arbitrary. Using (4.3) we get where the last equality follows from the fact that the integrand is skew-symmetric with respect to x, y. Thus B 0 is symmetric. Next we calculate the eigenvalues of B 0 ; this will also lead us to the essential selfadjointness of B 0 . Consider the functions h k defined by h k (x) = sin(kt/R) with x = σ(t) and k ∈ N. Then by (4.3) and (4.1) we have ds.
Due to the identity sin(ks/R) − sin(kt/R) = 2 sin( ks−kt 2R ) cos( ks+kt 2R ) this leads to We split the interval of integration into two parts and obtain with the substitution z = s − t + L for the first integral where we have used in the last step that sin is an odd function and that the formulas sin(x+π) = − sin(x) and cos(x+π) = − cos(x) hold for all x ∈ R. For the remaining second integral the substitution z = s − t yields With the help of (4.5) and (4.6) and the substitution s = z/(2R) the identity (4.4) implies where π 0 sin(ks) cos(ks) 2π sin(s) ds = 0 was used in the last step. Furthermore, using the identity 2 sin 2 (ks) = 1 − cos(2ks) and the indefinite integrals given in [41, 2.526 1. and 2.539 4.] we get Hence (4.7) yields By an analogous computation we see that also where h k (x) = cos(kt/R) with x = σ(t). Moreover, for the constant function h(x) = 1 on Σ we clearly have Since the functions h, h k , h k are eigenfunctions of B 0 by (4.8), (4.9) and (4.10) and span a dense subspace of L 2 (Σ), it follows that the symmetric operator B 0 is actually essentially selfadjoint in L 2 (Σ). Furthermore, by (4.8), (4.9) and (4.10), the selfadjoint closure B 0 has a pure point spectrum and its eigenvalues, counted with multiplicities, are given by ν k (0), k = 1, 2, . . . , in item (ii). Since these eigenvalues are bounded from above and converge to −∞ as k → +∞, it follows that B 0 is bounded from above and has a compact resolvent.
Let us now turn to the operator B λ on the circle for general λ < 0. Lemma 4.3. Let λ ≤ 0, let Σ be a circle of radius R and let B λ be defined in (2.7). Then the following assertions hold.
(i) B λ is a well-defined, essentially selfadjoint operator in L 2 (Σ) and the identity dom B λ = dom B 0 holds. (ii) B λ is bounded from above and has a compact resolvent. (iii) The eigenvalues ν k (λ) of B λ , k = 1, 2, . . . , ordered nonincreasingly and counted with multiplicities, satisfy The eigenspace corresponding to ν 1 (λ) is given by the constant functions on Σ.
Proof. Note first that the operator B λ can be written as The integral operator M λ has a real, symmetric kernel, which is square integrable since for all Thus M λ is a compact, selfadjoint operator in L 2 (Σ). Hence, due to Lemma 4.2 and (4.11) B λ is well-defined and essentially selfadjoint in L 2 (Σ) with In particular, B λ has a compact resolvent and dom B λ = dom B 0 , which shows (i).
Next we show that B λ is bounded from above by the number k λ defined in Lemma 4.1. For every h ∈ C 0,1 (Σ) and x ∈ Σ we can write where again the integral exists due to the Lipschitz continuity of h. Hence where in the last step we first changed the roles of x and y and then the order of integration. Addition of the last two lines yields with equality if and only if h is constant, that is, B λ (and, thus, B λ ) is bounded from above by k λ , which shows (ii). Moreover it follows ν 1 (λ) = k λ . By Lemma 4.1 this implies ν 1 (λ) → −∞ as λ → −∞ and thus ν k (λ) → −∞ as λ → −∞ for all k. This finishes the proof of (iv). It remains to verify the asymptotic behaviour of the eigenvalue ν k (λ) for k → +∞ as claimed in (iii). According to [1,Equation 4.
where γ ≈ 0.577216 denotes the Euler-Mascheroni constant. Hence By Lemma 4.2 (ii) for the eigenvalues of B 0 this implies and consequently (4.14) From (4.12) we conclude with the help of the min-max principle The latter together with (4.13) and (4.14) implies which completes the proof of the lemma.

4.2.
Properties of B λ in the general case. In this subsection Σ is an arbitrary compact, closed, regular C 2 -curve in R 3 of length L without self-intersections. In the following we explore properties of B λ by using the results of the previous subsection for the case of a circle. This will be done by a perturbation argument. Let T be a circle in R 3 with radius R = L 2π which is parametrized with respect to the arc length by a function τ : [0, L] → R 3 . In order to distinguish the operators B λ on Σ from those on the circle T we denote the latter by B T λ . Moreover, recall that σ : [0, L] → R 3 is an arc length parametrization of Σ. We define an operator D λ by for h ∈ L 2 (Σ). Furthermore, let J : L 2 (Σ) → L 2 (T ) be the unitary operator defined by Our studies of B λ will rely on the following properties of D λ . Lemma 4.4. For each λ ≤ 0 the operator D λ in (4.15) is well-defined, compact and selfadjoint in L 2 (Σ), and D λ ≤ C holds for all λ ≤ 0 and some C > 0 which is independent of λ. In the special case λ = 0 the estimate holds with d Σ given in (3.3). Moreover, the relation is satisfied for all λ ≤ 0.
Proof. In order to study the integral in the definition (4.15) of D λ we identify the parametrizations σ, τ of Σ and T , respectively, with their L-periodic continuations to all of R. Let s, t ∈ R with |s − t| for z > 0. Then from which it follows that f ′ is monotonously nondecreasing on (0, ∞) and, thus, |f ′ | is monotonously nonincreasing on (0, ∞). Hence with (4.20) Note, that there exist ε σ > 0 and ε τ > 0 such that for all Recall that Σ is a C 2 -curve. Hence we get with Taylor's theorem (for each component) for some suitable ζ 1 , ζ 2 and ζ 3 ∞ and |σ ′ (s)| = 1 it follows Analogously we get with C τ := τ ′′ for some suitable ξ 1 , ξ 2 and ξ 3 . Hence By changing the roles of σ and τ we observe Note that e −x (x + 1) ≤ 1 for x ≥ 0. Together with (4.19), (4.22) and for all s, t ∈ R with |s − t| ≤ L 2 . For arbitrary s, t ∈ R there exists k ∈ such that |(s + kL) − t| ≤ L 2 . As σ and τ are L-periodic it follows that (4.23) holds for all s, t ∈ R. From (4.23) we conclude that the integral kernel of the operator D λ is bounded with a bound C independent of λ. Thus with C = CL, the definition of D λ in (4.15) and estimate (4.23) it follows for all h ∈ L 2 (Σ) and C does not depend on λ. In particular, D λ is a well-defined, compact operator in L 2 (Σ) whose operator norm can be estimated by a constant independent of λ. Since the integral kernel of D λ is real and symmetric it follows that D λ is selfadjoint. For λ = 0 the estimate (4.17) follows immediately from the definition of D λ . In order to verify the relation (4.18) note that h ∈ C 0,1 (Σ) if and only if h := Jh ∈ C 0,1 (T ) and in this case for every h ∈ C 0,1 (Σ) and x = σ(t) ∈ Σ. This identity and the definitions of B λ and D λ lead to the relation (4.18). Now we are in the position to prove all properties of B λ which are required for the proofs of the main results of this paper.
where k λ is given in Lemma 4.1. Since k λ → −∞ as λ → −∞ by Lemma 4.1 we conclude from (4.24) that ν k (λ) → −∞ as λ → −∞ for each k. From (4.18) and the min-max principle it follows where ν T k (λ) denotes the k-th eigenvalue of B T λ . We obtain with the help of Lemma 4.3 (iii) that This proves the assertion (iii). In order to show the remaining assertions in (iv) let λ, µ ≤ 0 and define the operator D λ,µ : As in the proof of Lemma 4.3 one shows that D λ,µ is a compact, selfadjoint operator with L.
In particular, D λ,µ → 0 as λ → µ. From this and (4.25) it follows with the min-max principle that ν k (λ) → ν k (µ) for all k, that is, all the functions λ → ν k (λ) are continuous. For the strict monotonicity let λ, µ < 0. If h ∈ dom B λ = dom B µ it follows from the definition of γ λ and γ µ in (2.3) that (4.26) in particular, γ λ h− γ µ h ∈ H 2 (R 3 ). Note also that γ λ − γ µ is continuous from L 2 (Σ) to H 2 (R 3 ) since γ λ − γ µ is defined on L 2 (Σ) and is closed as a mapping from L 2 (Σ) to H 2 (R 3 ). According to Lemma 2.1 we have for almost all x ∈ R 3 \ Σ. As the integral in (4.27) is continuous with respect to x we obtain (4.27) for all x ∈ R 3 . In particular, for all x ∈ Σ and h ∈ C 0, Due to (4.28) and (4.26) we observe Since the mapping h → hδ Σ is continuous from L 2 (Σ) to H −2 (R 3 ) (see (2.2)), −∆ − λ is an isomorphism between H s (R 3 ) and H s−2 (R 3 ) for all s ∈ R, and the trace map is continuous from H 2 (R 3 ) to L 2 (Σ) we conclude and hence the limit lim n→∞ B µ h n exists and equals B µ h. Using the continuity of γ λ − γ µ as a mapping from L 2 (Σ) into H 2 (R 3 ), the continuity of the trace and (4.28) we observe for all h ∈ dom B λ = dom B µ . From (4.29), (4.26) and (2.2) we obtain Since γ λ is an injective operator it follows that the function λ → B λ h, h L 2 (Σ) is strictly increasing on (−∞, 0), as its derivative is positive, i.e., whenever λ < µ < 0. From this and the min-max-principle for λ < µ < 0 we obtain where we have used that the operators −B λ and −B µ are bounded from below; cf. (ii). Thus ν k (λ) < ν k (µ) for λ < µ < 0 and by continuity the same holds in the case λ < µ = 0. This proves the remaining assertion in (iv).

4.3.
Well-definedness of the generalized trace. In this subsection we verify that the definition of the generalized trace u| Σ in (2.8) is independent of the choice of λ < 0. Observe first that if for some λ < 0 then h ∈ dom B µ for any µ < 0 by Proposition 4.5 (i) and It follows as in (4.26) that γ λ h−γ µ h belongs to H 2 (R 3 ), and hence also v c ∈ H 2 (R 3 ). Thus if u admits the decomposition (4.30) with respect to some λ < 0 then u admits the decomposition (4.31) with respect to any µ < 0. Note also that for fixed λ < 0 both elements u c and h in the decomposition (4.30) are unique. Let now λ, µ < 0 and assume that with u c , v c ∈ H 2 (R 3 ) and h, k ∈ dom B λ = dom B µ . Then it follows from the above considerations and the uniqueness of the decompositions in (4.32) that (4.33) v c = u c + γ λ h − γ µ h and h = k.

Using (4.29) it follows from (4.33) that
This shows that the definition of the generalized trace in (2.8) is independent of the choice of λ.

Proofs of the main results
In this section we provide the complete proofs of the results in section 3.

Proof of Theorem 3.3 and Corollary 3.4. Let us first prove Theorem 3.3.
For λ ≤ 0 let us denote by ν j (λ) the eigenvalues of the operator B λ , ordered nonincreasingly and counted with multiplicities; cf. Proposition 4.5 (iii). We remark that by Theorem 3.1 (i) and Proposition 4.5 (iv) the number N α of negative eigenvalues of −∆ Σ,α counted with multiplicities coincides with the number of eigenvalues of B 0 larger than α, counted with multiplicities. Moreover, let T be a circle of radius R = L 2π , where L is the length of Σ. We denote by B T λ the analog of B λ where Σ is replaced by the circle T , and by ν T j (λ) the eigenvalues of its closure. From (4.18) with λ = 0 it follows with the min-max principle that
In the remaining case α + d Σ ∈ I r with r = −1 it is clear that and the upper estimate for N α follows as above. This completes the proof of the theorem.
Let us now turn to the proof of the corollary. As in Theorem 3.3 let r and l such that α + d Σ ∈ I r and α − d Σ ∈ I l . The condition α + d Σ < ln(4R) 2π − 1 π ensures 1 ≤ r ≤ l. The proof is based on the estimates for the harmonic sum H k = k j=1 1 j , k ≥ 1, see e.g. [40, (9.89)]. Since k j=1 1 and therefore ln l < −2π(α − d Σ ) + ln R − γ. (5.10) Using N α ≤ 2l + 1 from Theorem 3.3 and the estimate (5.10) we get which yields the upper estimate for N α in (3.4).
In fact the identities in (5.20) hold by construction and Proposition A.5. In order to verify the abstract Green identity for the boundary maps in (5.19) recall from (A.17) in the proof of Proposition A.5 that for u, v ∈ dom T such that u = u c + γ η h and v = v c + γ η k the identity and hence the Green identity is valid. The same argument as in the proof of Proposition A.5 shows that the range of the mapping u → (Γ 0 u, Γ 1 u) ⊤ is dense in L 2 (Σ) × L 2 (Σ). Hence {L 2 (Σ), Γ 0 , Γ 1 } is a quasi boundary triple for S * . Since Γ 0 is the same map as in Proposition A.5 the corresponding γ-field has the same form as in Proposition A.5. The form of the Weyl function in (5.21) follows from (3.5) in the same way as in the proof of Proposition A.5; cf. (4.26), (4.28), and Remark A.6. Now we complete the proof of Theorem 3.8. Consider the quasi boundary triple {L 2 (Σ), Γ 0 , Γ 1 } in (5.19). It follows from (5.21), (2.3) and the proof of Theorem 3.2 that Moreover, since η < 0 was chosen such that 0 ∈ ρ(B η − α) it is clear that the operator M (η) −1 = (B η − α) −1 is bounded in L 2 (Σ). Note also that holds by (5.21). Hence the assumptions in Theorem A.4 are satisfied and the assertions (i), (iii), and (iv) in Theorem 3.8 follow. Observe that by (5.21) and (A.4) holds for λ ∈ C \ [0, ∞). Therefore (5.22) and [6, Proposition 3.14] yield that the limit N (λ + i0) exists in the Hilbert-Schmidt norm for a.e. λ ∈ [0, ∞), that is, assertion (ii) in Theorem 3.8 holds. This completes the proof.

Appendix A. Quasi boundary triples and their Weyl functions
In this appendix we briefly review the abstract notions of quasi boundary triples and their Weyl functions from extension theory of symmetric operators in Hilbert spaces, and relate them to the Schrödinger operators −∆ free and −∆ Σ,α . Furthermore, we recall a representation formula for the scattering matrix in terms of the Weyl function of a quasi boundary triple from [11], which is the main ingredient in the proof of Theorem 3.8. For more details on quasi boundary triples and their Weyl functions we refer the reader to [7,8], and for generalized and ordinary boundary triples to [21,24,25].
Definition A.1. Let S be a densely defined, closed, symmetric operator in a Hilbert space (H, ·, · H ) and assume that T is a linear operator in H such that T = S * . A triple {G, Γ 0 , Γ 1 } is a quasi boundary triple for S * if (G, ·, · G ) is a Hilbert space and Γ 0 , Γ 1 : dom T → G are linear mappings such that the following holds.
(i) For all u, v ∈ dom T one has (ii) The range of the mapping (Γ 0 , Γ 1 ) ⊤ : dom T → G × G is dense.
If {G, Γ 0 , Γ 1 } is a quasi boundary triple for T = S * then Moreover, if ran Γ 0 = G then {G, Γ 0 , Γ 1 } is a generalized boundary triple in the sense of [25,Section 6], and if ran(Γ 0 , Γ 1 ) ⊤ = G × G then {G, Γ 0 , Γ 1 } is an ordinary boundary triple; cf. [21,24]. In the latter case it follows that T = S * and hence the abstract Green identity in Definition A.1 (i) holds for all u, v ∈ dom S * . We remark that for an ordinary boundary triple condition (iii) in Definition A.1 is automatically satisfied. A quasi boundary triple {G, Γ 0 , Γ 1 } for T = S * is a useful tool to describe the extensions of S which are contained in T via abstract boundary conditions in the auxiliary Hilbert space G. However, in this context it is important to note that not all selfadjoint extensions of S in H are covered, but only those which are also restrictions of T . Furthermore, a selfadjoint parameter Θ in G does not automatically lead to a selfadjoint extension via as one is used to from the theory of ordinary boundary triples. In general A Θ in (A.1) is only symmetric in H, not necessarily closed, and one has to impose additional conditions on Θ or on other involved objects to ensure selfadjointness of the extension A Θ , see, e.g. [7,8].
Next we recall [8,Theorem 6.11] which is very useful for the construction of quasi boundary triples and provides a method to determine the adjoint of a symmetric operator.
Theorem A.2. Let T be a linear operator in a Hilbert space (H, ·, · H ), let (G, ·, · G ) be a Hilbert space, and assume that Γ 0 , Γ 1 : dom T → G are linear mappings such that the following holds.
Next we recall the notion of the γ-field and Weyl function associated to a quasi boundary triple {G, Γ 0 , Γ 1 } for T = S * . First of all it follows from the direct sum decomposition dom T = dom A 0+ ker(T − λ), λ ∈ ρ(A 0 ), and dom A 0 = ker Γ 0 that the restriction of the boundary map Γ 0 onto ker(T − λ) is invertible. The inverse is a densely defined operator from G into H. The function λ → γ(λ) is called the γ-field associated to {G, Γ 0 , Γ 1 }. The Weyl function M associated to {G, Γ 0 , Γ 1 } is defined by The values M (λ) of the Weyl function are densely defined operators in G, which may be unbounded and not closed in general. If one views the boundary maps Γ 0 and Γ 1 as abstract Dirichlet and Neumann trace maps then the values of the Weyl function can be interpreted as abstract analogues of the Dirichlet-to-Neumann map in the theory of elliptic PDEs. For λ, µ ∈ ρ(A 0 ) and h ∈ ran Γ 0 we note the useful identities as well as for the γ-field and Weyl function, and refer the reader for more details and proofs of the above identities to [7,8].
The following theorem from [7,8] contains a Krein type resolvent formula and provides a criterion to show selfadjointness of the extension A Θ in (A.1). Theorem A.3. Let S be a densely defined, closed, symmetric operator in a Hilbert space (H, ·, · H ) and let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T = S * with A 0 = T ↾ ker Γ 0 and γ-field γ and Weyl function M . Let Θ be an operator in G and let Assume, in addition, that λ ∈ ρ(A 0 ) is not an eigenvalue of A Θ or, equivalently, ker(Θ − M (λ)) = {0}. Then the following assertions hold.
In particular, if Θ is a symmetric operator in G and ran γ(λ) * is contained in dom(Θ − M (λ)) −1 for some λ ∈ C + and some λ ∈ C − then A Θ is selfadjoint in H and the resolvent formula (A.5) holds for all λ ∈ ρ(A Θ ) ∩ ρ(A 0 ) and all u ∈ H.
Next we provide a slightly generalized variant of the representation formula for the scattering matrix from [11]. Let again S be a densely defined, closed, symmetric operator in a Hilbert space (H, ·, · H ) and let {G, Γ 0 , Γ 1 } be a quasi boundary triple for T = S * with A 0 = T ↾ ker Γ 0 and γ-field γ and Weyl function M . Assume, in addition, that the extension is selfadjoint in H; in general A 1 is only symmetric in H and not necessarily closed. Denote the absolutely continuous subspaces of A 0 and A 1 by H ac (A 0 ) and H ac (A 1 ), respectively, let P ac (A 0 ) be the orthogonal projection onto H ac (A 0 ) and let be the absolutely continuous part of A 0 . If the difference of the resolvents of A 0 and A 1 is a trace class operator, that is, for some, and hence for all, λ ∈ ρ(A 0 ) ∩ ρ(A 1 ) then the wave operators exist and satisfy ran W ± (A 0 , A 1 ) = H ac (A 1 ) according to the Birman-Krein theorem [15]. It follows that the scattering operator is unitary in the absolutely continuous subspace H ac (A 0 ) of A 0 , and that S(A 0 , A 1 ) is unitarily equivalent to a multiplication operator {S(λ)} λ∈R in a spectral representation of the absolutely continuous part A ac 0 of A 0 . The family {S(λ)} λ∈R is called the scattering matrix of the pair {A 0 , A 1 }; cf. [6,43,58,62].
In general the underlying closed symmetric operator S is not simple (or completely non-selfadjoint) and hence its selfadjoint part is reflected in the scattering matrix of {A 0 , A 1 }. More precisely, if S is not simple then there is a nontrivial orthogonal decomposition of the Hilbert space H = H 1 ⊕ H 2 such that where S 1 is a simple symmetric operator in H 1 and S 2 is a selfadjoint operator in H 2 .
Since A 0 and A 1 are selfadjoint extensions of S there exist selfadjoint extensions B 0 and B 1 of S 1 in H 1 such that In the following let L 2 (R, dλ, H λ ) be a spectral representation of the absolutely continuous part S ac 2 of the selfadjoint operator S 2 in H 2 . Now we can formulate a variant of [11, Theorem 3.1 and Corollary 3.3] which is suitable for our purposes. Instead of generalized boundary triples the result is stated for quasi boundary triples here.
In the following we show how the objects of this manuscript fit in the abstract scheme of quasi boundary triples. Let −∆ free be the selfadjoint Laplacian in L 2 (R 3 ) with domain H 2 (R 3 ) and let −∆ Σ,α be the Schrödinger operator with a δ-interaction of strength 1 α supported on Σ from Definition 2.3. Consider the symmetric operator (A.9) Su = −∆u, dom S = u ∈ H 2 (R 3 ) : u| Σ = 0 , and define the operator T in L 2 (R 3 ) by where η < 0 is chosen such that 0 ∈ ρ(B η − α) (see Proposition 4.5 (ii) and (iv)) and γ η h = (−∆ − η) −1 (hδ Σ ) is as in (2.3). It follows from the remark below Definition 2.2 that the sum in the definition of dom T is direct. Furthermore, T is a well-defined operator in L 2 (R 3 ) since for an element u = u c + γ η h ∈ dom T with u c ∈ H 2 (R 3 ) and h ∈ dom B η one has Note also that (A. 12) ker(T − η) = γ η h : h ∈ dom B η .
In the next proposition we specify a quasi boundary triple {L 2 (Σ), Γ 0 , Γ 1 } for the adjoint of the symmetric operator S such that −∆ free = T ↾ ker Γ 0 . Proof. In order to show that the mappings in (A.13) yield a quasi boundary triple for S * we make use of Theorem A.2. Note first that the identities S = T ↾ ker Γ 0 ∩ ker Γ 1 and − ∆ free = T ↾ ker Γ 0 hold. Hence it remains to check that the Green identity holds for all u, v ∈ dom T and that the range of the mapping u → (Γ 0 u, Γ 1 u) ⊤ is dense in L 2 (Σ) × L 2 (Σ). In order to verify (A.17) decompose u, v ∈ dom T in the form u = u c + γ η h and v = v c + γ η k, where u c , v c ∈ H 2 (R 3 ) and h, k ∈ dom B η .
Next it will be shown that the γ-field and Weyl function corresponding to Note, however, that Γ 1 is not surjective and {L 2 (Σ), Γ 0 , Γ 1 } is not an ordinary boundary triple.