Approximation of Schrödinger operators with δ-interactions supported

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1 Mathematische Nachrichten, 29 December 215 Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces Jussi Behrndt 1,, Pavel Exner 2,3, Markus Holzmann 1,, and Vladimir Lotoreichik 3, 1 Institut für Numerische Mathematik Graz University of Technology, Steyrergasse 3, 81 Graz, Austria 2 Doppler Institute for Mathematical Physics and Applied Mathematics Czech Technical University in Prague, Břehová 7, Prague, Czech Republic 3 Department of Theoretical Physics Nuclear Physics Institute, Czech Academy of Sciences, 2568 Řež, Czech Republic Received XXXX, revised XXXX, accepted XXXX Published online XXXX Key words Schrödinger operators, δ-interactions supported on hypersurfaces, approximation by scaled regular potentials, norm resolvent convergence, spectral convergence MSC 21 81Q1, 35J1 We show that a Schrödinger operator A δ,α with a δ-interaction of strength α supported on a bounded or unbounded C 2 -hypersurface R d, d 2, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator A δ,α with a singular interaction is regarded as a self-adjoint realization of the formal differential expression α δ, δ, where α: R is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result. 1 Introduction Singular Schrödinger operators with potentials supported by subsets of the configuration space of a lower dimension are often used as models of physical systems because they are easier to solve, the original differential equation being reduced to the analysis of an algebraic or functional problem. The best known about them are solvable models with point interactions used in physics since the 193s see [25], the rigorous analysis of which started from the seminal paper [8]; for a survey see the monograph [2]. In the last two decades the attention focused on interactions supported on curves, surfaces, and more complicated sets composed of them, which are used to model leaky quantum systems in which the particle is confined to such manifolds or complexes, but the tunnelling between different parts of the interaction support is not neglected; for a review see [15] or [19, Chapter 1]. While these models are useful and mathematically accessible, one has to keep in mind that the singular interaction represents an idealized form of the actual, more realistic description. This naturally inspires the question about approximations of such singular potentials by regular ones. In the simplest case of a point interaction this problem was already addressed in the 193s in [36]. Starting from the 197s the approximation of Hamiltonians with point interactions supported on a finite or an infinite set of points in R d, d {1, 2, 3}, was treated systematically; cf. the monograph [2] and the references therein. Corresponding author behrndt@tugraz.at Phone: , Fax: exner@ujf.cas.cz holzmann@math.tugraz.at lotoreichik@ujf.cas.cz

2 2 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces Apart from that, the literature on the approximation of Schrödinger operators with δ-potentials supported on curves in R 2 and surfaces in R 3 is less complete; there are results available for the special cases that is a sphere in R 3 [5,33], that is the boundary of a star-shaped domain in the plane [31], and that is a smooth planar curve or surface and the interaction strength is constant [16,18]. In all of the above mentioned works convergence in the norm resolvent sense is shown. Abstract approaches developed in [3,34] cover more cases but imply only strong resolvent convergence in this context. We point out that the usage of scaled regular potentials is not the unique way of an approximation of δ-interactions supported on hypersurfaces, other mechanisms of approximation are discussed in e.g. [11, 12, 2, 3]. It is also worth mentioning that the approximation of δ-interactions supported on special periodic structures in R 2 has important applications in the mathematical theory of photonic crystals, see [21] and the references therein. The aim of the present paper is to analyze the general case where the interaction support is a C 2 -smooth hypersurface R d, d 2, which is not necessarily bounded or closed, and the interaction strength is an arbitrary real valued bounded measurable function α on. Following the approach of [16, 18] we show that the corresponding singular Schrödinger operator can be approximated in the norm resolvent sense by a family of regular ones with potentials suitably scaled in the direction perpendicular to. We pay particular attention to the order of convergence and provide all preparatory technical integral estimates in a complete and self-contained form. We shall also mention some spectral consequences of the general approximation result. In the following we describe our main result. Let d 2 and let R d be a bounded or unbounded orientable C 2 -hypersurface as in Definition 2.1, and consider the symmetric sesquilinear form a δ,α [f, g] = f, g α f L 2 R d ;C d g dσ, dom a δ,α = H 1 R d, where α L is a real valued function and f, g denote the traces of functions f, g H 1 R d on. Standard arguments yield that a δ,α is a densely defined, closed, and semibounded form in L 2 R d, and hence there exists a unique self-adjoint operator A δ,α in L 2 R d such that A δ,α f, g = a δ,α [f, g], f dom A δ,α, g dom a δ,α, 1.1 see Lemma 2.6 for more details. The operator A δ,α is regarded as a Schrödinger operator with a δ-interaction of strength α supported on which corresponds to the formal singular differential expression α δ, δ ; cf. [1] and [7, Theorem 3.3]. The choice of the negative potential sign is motivated by the fact that interesting spectral features are in this context usually associated with attractive interactions. Let ν be the continuous unit normal vector field on, choose β > sufficiently small as in Hypothesis 2.1 and consider layer neighborhoods Ω ε of of the form Ω ε := { x + tνx : x, t ε, ε }, < ε β. Fix a real valued potential V L R d with support in Ω β, define the scaled potentials V ε L R d with support in Ω ε by V ε x := { β ε V x + β ε tνx, if x = x + tνx Ω ε,, else, 1.2 and consider the corresponding self-adjoint Schrödinger operators H ε f = f V ε f, dom H ε = H 2 R d. With these preparatory considerations we can formulate the main result of the present paper. Theorem 1.1 Let R d, d 2, be an orientable C 2 -hypersurface as in Definition 2.1 which satisfies Hypothesis 2.1, let Q L R d be real valued, and let V L R d be real valued with support in Ω β. Define α L as the transversally averaged value of t V x + tνx by αx := β β V x + sνx ds

3 mn header will be provided by the publisher 3 for a.e. x and let A δ,α be the corresponding Schrödinger operator with a δ-interaction of strength α supported on. Then there exists a λ < such that, λ ρa δ,α + Q ρh ε + Q for all ε > sufficiently small and for every λ, λ there is a constant c = cd, λ,, V, Q > such that Hε + Q λ A δ,α + Q λ c ε 1 + ln ε holds for all ε > sufficiently small. In particular, H ε + Q converges to A δ,α + Q in the norm resolvent sense, as ε +. Let us briefly describe the structure of the proof of Theorem 1.1 and the contents of this paper. Section 2 contains preliminary material, definitions and properties of the hypersurfaces and their layer neighborhoods, as well as a representation of the resolvent of the Schrödinger operator A δ,α which goes back to [1]. The heart of the proof of Theorem 1.1 is in Section 3. The main part of this section deals with the special case Q =. For this purpose, the potentials V ε in 1.2 are factorized with the standard Birman-Schwinger method and useful representations of the resolvent of H ε are provided. The convergence analysis is then done by comparing the different resolvent representations from Theorem 2.7 and Proposition 3.3, and essentially reduces to convergence properties of certain integral operators discussed in Lemma 3.4. However, the proof of Lemma 3.4 requires various refined technical estimates for integrals containing the Green s function for the free Laplacian which are outsourced to Appendix A. We wish to mention that in Appendix A particular attention is paid to keep the present paper self-contained. Therefore all necessary estimates are presented in full detail and complete rigorous form; as a result Appendix A is of mainly technical nature. The statement of Theorem 1.1 in the general case with Q follows then from the previous considerations by a simple perturbation argument. Eventually, there is a short Appendix B in which it is shown that boundaries of bounded C 2 -domains satisfy the assumptions imposed on the hypersurfaces in this paper. Finally, we agree that throughout the paper c, C, C k, C k, k N, denote constants that do not depend on space variables and on ε. In the formulation of the results we usually write C = C... to emphasize on which parameters these constants depend, but in the proofs we will mostly omit this. Acknowledgements. Jussi Behrndt and Vladimir Lotoreichik gratefully acknowledge financial support by the Austrian Science Fund FWF: Project P N26. Pavel Exner and Vladimir Lotoreichik also acknowledge financial support by the Czech Science Foundation: Project S. Markus Holzmann thanks the Czech Centre for International Cooperation in Education DZS and the Austrian Agency for International Cooperation in Education and Research OeAD for financial support under a scholarship of the program Aktion Austria - Czech Republic during a research stay in Prague. 2 Preliminaries This section contains some preliminary material that will be useful in the main part of the paper. In Section 2.1 we recall certain basic facts from differential geometry of hypersurfaces in Euclidean spaces and of layers built around these hypersurfaces. Then, in Section 2.2 we define Schrödinger operators with δ-interactions supported on hypersurfaces in a mathematically rigorous way. 2.1 Hypersurfaces and their layer neighborhoods In this section we introduce several notions associated to hypersurfaces and layers around these hypersurfaces. We follow the presentation from [26], which we adopt for our applications. We start with a suitable definition of a class of hypersurfaces in the Euclidean space R d. We wish to emphasize that the hypersurfaces considered here are in general unbounded and not necessarily closed; note also that the index set I in the parametrization below is assumed to be finite. Definition 2.1 We call R d, d 2, a C 2 -hypersurface and {ϕ i, U i, V i } i I a parametrization of, if I is a finite index set and the following holds: a U i R d and V i R d are open sets and ϕ i : U i V i is a C 2 -mapping for all i I;

4 4 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces b rank Dϕ i u = d 1 for all u U i and i I; c ϕ i U i = V i and ϕ i : U i V i is a homeomorphism; d i I V i; e there exists a constant C > such that ϕ i u ϕ i v C u v for all u, v U i and i I. Let R d be a C 2 -hypersurface with parametrization {ϕ i, U i, V i } i I. Then the inverse mappings ϕ i : V i U i are often called charts and the family {ϕ i, V i, U i } i I atlas of. For x = ϕ i u u U i, i I we denote the tangent hyperplane by T x := span { 1 ϕ i u,..., d ϕ i u }. The tangent hyperplane T x is independent of the parametrization of and dim T x = d 1 holds by Definition 2.1 b. Subsequently, it is assumed that is orientable, i.e. there exists a globally continuous unit normal vector field on. From now on we fix a continuous unit normal vector field which is unique up to multiplication with and denote it by νx for x. Then the mapping U i u νϕ i u is continuously differentiable for all i I and j νϕ i u T ϕiu for all u U i and j {1,..., d 1}, see, e.g., [26, Lemma 3.9 and Section 3F]. The first fundamental form I x associated to is the bilinear form on the tangent hyperplane T x defined by I x [a, b] := a, b, a, b T x, where, denotes the standard scalar product in R d. For x = ϕ i u u U i, i I the matrix representing I x in the canonical basis { j ϕ i u} d j=1 of T x is given by G i u = k ϕ i u, l ϕ i u d k,l=1 2.1 and also known as the metric tensor of. Observe that G i u = Dϕ i u Dϕ i u. Together with condition b in Definition 2.1 this implies that G i u is positive definite. Finally, we introduce the notion of the Weingarten map or shape operator. Definition 2.2 Let R d, d 2, be an orientable C 2 -hypersurface with parametrization {ϕ i, U i, V i } i I and let νx, x, be a continuous unit normal vector field on. For x = ϕ i u u U i, i I the Weingarten map W x: T x T x is the linear operator acting on the basis vectors { j ϕ i u} d j=1 of T x as W x j ϕ i u := j νϕ i u. The Weingarten map W x is well-defined but its sign depends on the choice of the continuous unit normal vector field, independent of the parametrization and symmetric with respect to the inner product induced by the first fundamental form, see e.g. [26, Lemma 3.9] for the case d = 3. For x = ϕ i u u U i, i I the matrix associated to the linear mapping W x corresponding to the canonical basis { j ϕ i u} d j=1 of T x will be denoted by L i u. The eigenvalues {µ j u} d j=1 of L iu are the principal curvatures of and do not depend on the choice of the parametrization, see [26, Definition 3.46]. In particular, the quantity det1 tl i u for t R, which will appear later frequently, is independent of the parametrization and will also be denoted by det1 tw x. Furthermore, the eigenvalues of W x depend continuously on x, as the entries of L i depend continuously on u U i see the text after Definition 3.1 in [26] and ϕ i : U i V i is a homeomorphism. Next, we discuss a convenient definition of an integral for functions defined on the C 2 -hypersurface. For this fix a parametrization {ϕ i, U i, V i } i I of as in Definition 2.1 with a finite index set I and choose a partition of unity subordinate to {V i } i I, that is a family of functions χ i : R d [, 1], i I, with the following properties:

5 mn header will be provided by the publisher 5 i χ i C R d for all i I; ii supp χ i V i for all i I; iii i I χ ix = 1 for any x. Note that some of the functions χ i, i I, are not compactly supported, if is unbounded. A function f : C is said to be measurable integrable, if U i u χ i ϕ i ufϕ i u is measurable integrable, respectively for all i I. If f : C is integrable, we define the integral of f over as fxdσx := χ i ϕ i ufϕ i u det G i udu, 2.2 i I U i where du := dλ d u denotes the usual d 1-dimensional Lebesgue measure on U i and G i u is the matrix of the first fundamental form given in 2.1. The measure σ in 2.2 coincides with the canonical Hausdorff measure on which is independent of the parametrization of ; cf. [27, Appendix C.8]. Therefore, the above definition of the integral does not depend on the parametrization of and the choice of the partition of unity. We denote the space of equivalence classes of square integrable functions f : C with respect to σ by L 2. Next, we introduce layer neighborhoods of a C 2 -hypersurface and we impose some additional conditions on in Hypothesis 2.1 below. For this it is useful to define the functions ι ϕi : U i R R d, ι ϕi u, t := ϕ i u + tνϕ i u, i I. 2.3 The Jacobian matrix of ι ϕi, i I, is given by the d d matrix Dι ϕi u, t = Dϕ i u1 tl i u νϕ i u and the absolute value of the determinant of this matrix can be expressed as det Dι ϕi u, t = det1 tl i u det G i u; 2.4 cf. [28, Section 2] and [14, Section 3]. We will also make use of the mapping ι : R R d, ι x, t := x + tνx, 2.5 and layer neighborhoods Ω β of of the form Ω β := ι β, β, β >. 2.6 We employ the following hypothesis for the hypersurface. Hypothesis 2.1 Let R d be an orientable C 2 -hypersurface with parametrization {ϕ i, U i, V i } i I. Assume that there exists β > such that a the restriction of the mapping ι on β, β is injective; b there is a constant η, 1 such that det1 tw x 1 η, 1+η for all x and all t β, β; c there exists a constant c > such that the mappings ι ϕi in 2.3 satisfy ι ϕi u, t ι ϕi v, s 2 c 2 u v 2 + s t 2 for all u, v U i, s, t β, β and all i I.

6 6 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces All the assumptions of Hypothesis 2.1 are satisfied for the boundary of a compact and simply connected C 2 - domain; see Appendix B. We also mention that a similar set of assumptions was imposed in [1, Section 4]. In the next proposition it will be shown that item c in Hypothesis 2.1 implies that the eigenvalues of W are uniformly bounded on. In particular, this shows that b in Hypothesis 2.1 is automatically satisfied if β > is small enough. Proposition 2.3 Let R d be an orientable C 2 -hypersurface and assume that item c in Hypothesis 2.1 holds. Then the eigenvalues of the matrix of the Weingarten map are uniformly bounded on. P r o o f. Let β > be as in Hypothesis 2.1 c and suppose that the eigenvalues of the Weingarten map are not uniformly bounded. Then for some i I there exists u U i and an eigenvalue µ of L i u such that µ > β. Choose a sequence s n β, β such that s n are not eigenvalues of L i u and s n µ. Then det1 s n L i u and det1 s n L i u and as G i u is positive definite and has uniformly bounded values by Definition 2.1 e, the same holds for det1 s n L i u det G i u, that is, det Dι ϕi u, s n and det Dι ϕi u, s n ; cf From Dι ϕ i ι ϕi u, s n = Dι ϕi u, s n we conclude det Dι 1 ϕ i ι ϕi u, s n =. 2.7 det Dι ϕi u, s n On the other hand, by Hypothesis 2.1 c the mapping ι ϕ i hence Dι ϕ i is bounded; this contradicts 2.7. is Lipschitz continuous on ι ϕi U i β, β and In the next example we provide a C 2 -hypersurface which does not satisfy Hypothesis 2.1; here a curve in R 2 with unbounded curvature at infinity is discussed. Example 2.4 Consider the curve ϕ: R R 2 u, u u, sint2 dt and observe that ϕr is an orientable C 2 -hypersurface in R 2 with parametrization {ϕ, R, R 2 }. If we fix the unit normal vector field by 1 sinu νu = sin 2, u 2 1/2 1 then the corresponding 1 1-matrix of the Weingarten map is given by Lu = 2u cosu sin 2 u 2 3/2. Clearly, L is unbounded and hence, item b in Hypothesis 2.1 is not satisfied. Under Hypothesis 2.1, the mapping ι in 2.5 is bijective from β, β onto Ω β. This allows us to identify functions f supported on Ω β with functions f defined on β, β via the natural identification fx = fι x, t = fx, t, x = ι x, t, x, t β, β. Subsequently, L 1 Ω β is equipped with the d-dimensional Lebesgue measure Λ d and L 1 β, β is equipped with the measure σ Λ 1. In the next proposition it is shown that L 1 Ω β and L 1 β, β can be identified and a useful change of variables formula is provided.

7 mn header will be provided by the publisher 7 Proposition 2.5 Let R d be an orientable C 2 -hypersurface, assume that Hypothesis 2.1 is satisfied and let Ω β be as in 2.6. Then the following assertions are true. i Let ι β := ι β,β. Then, there exist constants < c 1 c 2 < + such that c 1 f L1 Ω β f ι β L1 β,β c 2 f L1 Ω β, f L 1 Ω β. In particular, f L 1 Ω β if and only if f ι β L 1 β, β. ii For f L 1 Ω β the identity fxdx = Ω β β β fx + tνx det1 tw x dtdσx holds, where W is the Weingarten map associated to. P r o o f. Let {ϕ i, U i, V i } i I be a parametrization of with a finite index set I, let {χ i } i I be a partition of unity subordinate to the covering {V i } i I and set χ i x := χ i x, i I, for x = ι β x, t = x + tνx Ω β with x and t β, β. The family { χ i } i I satisfies χ i x = 1 for all x Ω β 2.8 i I due to the properties of the partition of unity {χ i } i I. Let f L 1 Ω β, let ι ϕi be as in 2.3 and let Ω i,β := ι ϕi U i β, β. Using 2.8 we get fxdx = Ω β Ω β i I χ i xfxdx = i I Ω i,β χ i xfxdx. Making the substitution x = ι ϕi u, t, i I, in each summand of the last formula, we get with the aid of 2.4 Ω β fxdx = i I U i β β χ i ι ϕi u, tfι ϕi u, t det1 tl i u det G i udtdu. Using χ i ι ϕi u, t = χ i ϕ i u, 2.2 and ι ϕi u, t = ι β ϕ i u, t, we end up with Ω β fxdx = β β fι β x, t det1 tw x dσx dt. Together with Hypothesis 2.1 b this formula implies assertions i and ii. 2.2 Schrödinger operators with δ-interactions on hypersurfaces In this section we recall a representation for the resolvent of the self-adjoint Schrödinger operator A δ,α in L 2 R d with a δ-interaction supported on the C 2 -hypersurface. As in [1] the operator A δ,α is defined via the corresponding quadratic form with the help of the first representation theorem [24, Theorem VI 2.1]; the functions in the domain of A δ,α then satisfy the typical δ-type boundary conditions on, see, e.g., [7, Theorem 3.3]. In the following A δ,α is the unique self-adjoint operator in L 2 R d associated to the form a δ,α in the next lemma; cf. 1.1.

8 8 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces Lemma 2.6 Let R d be a C 2 -hypersurface, assume that Hypothesis 2.1 is satisfied and let α L be a real valued function. Then the symmetric sesquilinear form a δ,α [f, g] := f, g L 2 R d ;C d α f g dσ, dom a δ,α := H 1 R d, 2.9 is densely defined, closed and bounded from below in L 2 R d ; here f, g denote the traces of functions f, g H 1 R d on. P r o o f. First, we note that Lemma A.3 i together with [23, Theorem VII 2, Remark VI 1] see also [23, Section VIII 1.1] imply that for 1 2 < s 1 there is a bounded trace operator from Hs R d to L 2. In particular, the form a δ,α in 2.9 is well-defined. Moreover, since H 1 R d is dense in L 2 R d, the form a δ,α is densely defined in L 2 R d. Subsequently, fix some s 1 2, 1 and c s > such that f L 2 c s f Hs R d for all f H 1 R d. Let ε > and use [22, Theorem 3.3] or [37, Satz e] to see that there exists a Cε > such that αf, f L2 α f 2 L 2 c s α f 2 H s R d c s α ε f 2 H 1 R d + Cε f 2 L 2 R d. 2.1 Thus, for sufficiently small ε > the form f αf, f on H 1 R d is relatively bounded with respect to the closed and nonnegative form f f, f L 2 R d ;C d on H 1 R d with bound smaller than one. Then by [24, Theorem VI 1.33] the form a δ,α in 2.9 is closed and bounded from below. Next, we provide a formula for the resolvent of the Schrödinger operator A δ,α. For this purpose some notations are required. The free Laplace operator in L 2 R d with domain H 2 R d is denoted by ; it is clear that coincides with A δ, α in the above lemma. The spectrum of is given by σ = [,. For λ, ρ we define the function G λ x := 1 d/2 1 x λ x K 2π d/2 d/2, x R d \ {}, 2.11 λ where K d/2 denotes a modified Bessel function of the second kind and order d 2 1, see [1] for the definition and the properties of these functions. Then, Rλfx := λ f x = G λ x yfydy; R d cf. [35, Section 7.4]. Next we define for λ, integral operators γλ, Mλ and provide the integral representation for the adjoint of γλ γλ: L 2 L 2 R d, γλξx:= G λ x y ξy dσy ; 2.12a Mλ: L 2 L 2, Mλξx := G λ x y ξy dσy ; 2.12b γλ : L 2 R d L 2, γλ fx = G λ x yfydy. R d 2.12c For our later considerations the resolvent formula in the next theorem is particularly useful. In the proof of item a and later in Section 3 the Schur test for integral operators will be used frequently, see, e.g., [24, Example III 2.4] or [37, Satz 6.9]. Theorem 2.7 Let R d be a C 2 -hypersurface which satisfies Hypothesis 2.1 and let α L be a real valued function. Then the following statements are true. a For λ, the operators γλ, Mλ and γλ in 2.12 are bounded and everywhere defined.

9 mn header will be provided by the publisher 9 b There exists a λ < such that 1 αmλ admits a bounded and everywhere defined inverse for all λ, λ. These λ belong to ρa δ,α and it holds A δ,α λ = Rλ + γλ 1 αmλ αγλ. P r o o f. a Let λ,. In order to prove that γλ is well-defined and bounded we use the Schur test. In fact, from Proposition A.4 i and Proposition A.2 i we obtain γλ 2 sup Gλ x y dσy sup Gλ x y dx <. x R d y R d In a similar way one can show that Mλ and γλ are bounded. Item b is essentially a variant of [1, Lemma 2.3]. In fact, let us define for Borel sets B R d the measure m by mb := σb, 2.13 where σ is the measure in 2.2. Then mr d \ =, the spaces L 2 R d ; m and L 2 can be identified and for f L 1 one has fx dσx = fx dmx, R d where f is some extension of f onto the m-null set R d \. Moreover, the estimate 2.1 shows that the measure m in 2.13 satisfies [1, eq. 2.1] with γ = α. Now it is easy to see that the integral operators γλ, Mλ, and γλ in 2.12 can be identified with the operators R m dx i λ, R mm i λ and R dx m i λ in [1], respectively. The assertion in item b follows from [1, Lemma 2.3 ii and iii]. 3 Approximation of A δ,α by Schrödinger operators with regular potentials In this section we prove Theorem 1.1, the main result of this paper. First, in Section 3.1 we recall briefly the definitions of the layer neigborhoods, the scaled potentials and the associated Hamiltonians H ε from the Introduction, and we derive a resolvent formula for H ε which is convenient in the convergence analysis. Section 3.2 contains the main part of the proof of Theorem 1.1. It is efficient to prove Theorem 1.1 for the special case Q = first; all technical estimates in Appendix A and all preparatory steps in Sections 3.1 and 3.2 are tailormade for this case. The general case Q is treated with a simple perturbation argument in the last step of the proof of Theorem 1.1. Finally, in Section 3.3 we discuss some connections between the spectral properties of H ε and of A δ,α that follow from Theorem Preliminary considerations on H ε Let d 2 and R d be a C 2 -hypersurface which satisfies Hypothesis 2.1. Then the mapping ι in 2.5 is injective on β, β for some in the following fixed β > as in Hypothesis 2.1. Recall the definition of the layer Ω ε from 2.6 for ε, β], fix a real valued potential V L R d with supp V Ω β and consider the scaled potentials { β ε V ε x = V x + β ε tνx, if x = x + tνx Ω ε,, else, where ν is the continuous unit normal vector field on. Observe that V β = V and supp V ε Ω ε. The associated self-adjoint Schrödinger operators are given by H ε f = f V ε f, dom H ε = H 2 R d. 3.1

10 1 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces Our main objective in this section is to derive the resolvent formula for H ε in Proposition 3.3 which turns out to be particularly convenient for our convergence analysis. We start with the standard factorization of the potentials V ε = v ε u ε, where and u ε : L 2 R d L 2 Ω ε, u ε fx := V ε x 1/2 fx, x Ω ε, 3.2 v ε : L 2 Ω ε L 2 R d, v ε hx := { sign V ε x V ε x 1/2 hx, x Ω ε,, else. 3.3 Recall that for λ ρ = C \ [, the resolvent of is denoted by Rλ = λ. The following proposition contains a first auxiliary resolvent formula for H ε. Proposition 3.1 Let H ε be defined as in 3.1 and let u ε, v ε and Rλ be given as above. Then the following assertions are true. i For all λ C \ [, with 1 ρu ε Rλv ε one has λ ρh ε and H ε λ = Rλ + Rλv ε 1 u ε Rλv ε u ε Rλ. ii For all M, 1 there exists λ M < such that u ε Rλv ε M holds for all ε, β] and λ < λ M. In particular, for these λ the results from i apply and hence, λ M ρh ε for all ε, β]. P r o o f. i Let λ C \ [, be such that 1 ρu ε Rλv ε. Note that λ is not an eigenvalue of H ε, as otherwise 1 σ p u ε Rλv ε ; cf. [9, Lemma 1]. Next, we define the operator T λ := Rλ + Rλv ε 1 u ε Rλv ε u ε Rλ. This operator is well defined and bounded, as 1 ρu ε Rλv ε. Using V ε = v ε u ε we conclude that H ε λt λf = λ v ε u ε T λf = f + v ε 1 u ε Rλv ε u ε Rλf v ε u ε Rλf v ε u ε Rλv ε 1 u ε Rλv ε u ε Rλf = f + v ε 1 u ε Rλv ε u ε Rλf v ε u ε Rλf v ε 1 u ε Rλv ε u ε Rλf + v ε u ε Rλf = f holds for any f L 2 R d. Hence, H ε λ is bijective, which implies that λ ρh ε, and This proves assertion i. H ε λ = T λ = Rλ + Rλv ε 1 u ε Rλv ε u ε Rλ. ii Let λ, and recall that Rλ can be expressed by Rλf = G R d λ yfydy with G λ as in Let M, 1 be fixed. Using the Schur test and that the absolute value of the integral kernel of u ε Rλv ε is symmetric, we find that uε Rλv ε sup x Ω ε Ω ε V ε x 1/2 G λ x y V ε y 1/2 dy β ε V L sup G λ x y dy. x Ω ε Ω ε Hence, the claimed result follows from Proposition A.4 ii, as this shows the existence of a number λ M < such that β ε V L G λ x y dy M Ω ε for any x Ω ε, all λ < λ M and any ε, β].

11 mn header will be provided by the publisher 11 Next, we transform the resolvent formula from Proposition 3.1 into another one, which is more convenient for the convergence analysis. This requires several preparatory steps. Recall that for an interval I R the space L 2 I is equipped with the product measure σ Λ 1 of the Hausdorff measure on and the one-dimensional Lebesgue measure. Define the functions u, v L, 1 by ux, t := βv x + βtνx 1/2 and vx, t := sign V x + βtνx ux, t. 3.4 The following operators are essential to state a convenient resolvent formula for H ε. We define for ε [, β] and λ, the integral operators A ε λ: L 2, 1 L 2 R d, B ε λ: L 2, 1 L 2, 1 and C ε λ: L 2 R d L 2, 1 as 1 A ε λξx := G λ x y εsνy vy, s 3.5a det1 εsw y Ξy, sdsdσy ; 1 B ε λξx, t := ux, t G λ x + εtνx y εsνy vy, s det1 εsw y Ξy, sdsdσy ; C ε λfx, t := ux, t G λ x + εtνx yfydy. R d 3.5b 3.5c In order to investigate the properties of A ε λ, B ε λ and C ε λ, we introduce several auxiliary operators. For ε, β] define the embedding operator I ε : L 2 ε, ε L 2 Ω ε, I ε Φx + tνx := Φx, t. 3.6 It follows from Hypothesis 2.1 b and Proposition 2.5 that the operator I ε is bounded, everywhere defined and bijective. Its inverse is given by I ε : L 2 Ω ε L 2 ε, ε, I ε hx, t = hx + tνx. Furthermore, we consider the scaling operator S ε : L 2, 1 L 2 ε, ε, S ε Ξx, t := 1 ε Ξ The operator S ε is unitary and its inverse is S ε : L 2 ε, ε L 2, 1, Sε Φx, t = εφx, εt. In the next lemma some properties of A ε λ, B ε λ and C ε λ are provided. x, t. 3.7 ε Lemma 3.2 Let λ, and let ε, β]. Moreover, let the operators u ε, v ε, I ε and S ε be defined by 3.2, 3.3, 3.6 and 3.7, respectively, and let the integral operators A ε λ, B ε λ and C ε λ be as in 3.5. Then the following assertions are true. i It holds A ε λ = Rλv ε I ε S ε and C ε λ = S ε Iε u ε Rλ. In particular, the operators A ε λ and C ε λ are bounded and everywhere defined. ii It holds B ε λ = Sε Iε u ε Rλv ε I ε S ε. Moreover, for any M, 1 there exists λ M < such that B ε λ M for all λ, λ M and all ε, β]. In particular, B ε λ is bounded and everywhere defined and for λ < λ M the operator 1 B ε λ has a bounded and everywhere defined inverse.

12 12 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces P r o o f. Let ε, β] and λ, be fixed. i We show the two formulae A ε λ = Rλv ε I ε S ε and C ε λ = Sε Iε u ε Rλ. The operators A ε λ and C ε λ are then automatically bounded and everywhere defined, as the operators S ε, I ε as well as their inverses, u ε, v ε and Rλ have these properties. Let Ξ L 2, 1. By the definitions of I ε and S ε it holds that I ε S ε Ξy + sνy = 1 Ξ y, s. ε ε Furthermore, the definitions of v ε and v see 3.4 imply for any h L 2 Ω ε that v ε h y + sνy = 1 ε sign V y + β ε sνy βv y + β 1/2 ε sνy h y + sνy = 1 v y, s h y + sνy ε ε for a.e. y, s ε, ε. Using the transformation Ω ε y = y + sνy y, s ε, ε and Proposition 2.5 we find that Rλvε I ε S ε Ξ x = G λ x yv ε I ε S ε Ξydy R d ε = G λ x y sνy 1 ε vξ y, s det1 sw y dsdσy ε = ε 1 = A ε λξx. Since this is true for a.e. x R d, the first formula of item i is shown. Next, we show the assertion on C ε λ. A simple calculation yields that G λ x y εrνy v y, r det1 εrw y Ξ y, r drdσy 3.8 S ε Iε u ε gx, t = εi u ε gx, εt = ε ε 1 ε βv x + β ε εtνx 1/2 gx + εtνx 3.9 = ux, tgx + εtνx for any g L 2 R d and a.e. x, t, 1. Using 3.9, we find that Sε Iε u ε Rλfx, t = ux, t G λ x + εtνx yfydy = C ε λfx, t R d for all f L 2 R d and a.e. x, t, 1. Thus, the second formula is shown as well. ii Using 3.9 and 3.8 we get that S ε Iε u ε Rλv ε I ε S ε Ξ x, t = ux, t Rλv ε I ε S ε Ξ x + εtνx 1 = ux, t G λ x + εtνx y εrνy = B ε λξx, t. v y, r det1 εrw y Ξ y, r drdσy Therefore, we obtain the desired formula in item ii. In particular, this formula implies that B ε λ is bounded and everywhere defined.

13 mn header will be provided by the publisher 13 Let M, 1 be fixed. Note that I ε and Iε are uniformly bounded for all ε, β] by Proposition 2.5 and Hypothesis 2.1 b. Furthermore, recall that S ε is unitary. Hence, using Proposition 3.1 ii we find that B ε λ = S ε Iε for all λ < with λ sufficiently large and all ε, β]. u ε Rλv ε I ε S ε S I uε Rλv ε Iε S ε M After all these preparatory steps it is simple to transform the resolvent formula for H ε from Proposition 3.1 into another one which is more convenient for the convergence analysis. Proposition 3.3 Let H ε be defined as in 3.1 and let A ε λ, B ε λ and C ε λ be as in 3.5. Then there exists a λ < such that, λ ρh ε and H ε λ = Rλ + A ε λ 1 B ε λ Cε λ for all ε, β] and λ < λ. P r o o f. Let the operators u ε, v ε, I ε, S ε be defined as in 3.2, 3.3, 3.6 and 3.7, respectively. Choose λ < such that 1 u ε Rλv ε and 1 B ε λ are boundedly invertible for any λ < λ and all ε, β] such a λ exists by Proposition 3.1 and Lemma 3.2. Then, it holds by Proposition 3.1 and Lemma 3.2 that H ε λ = Rλ + Rλv ε 1 u ε Rλv ε u ε Rλ = Rλ + A ε λsε Iε 1 Iε S ε B ε λsε = Rλ + A ε λ 1 B ε λ C ε λ, which proves the statement of this proposition. ε ε I ε Iε S ε C ε λ 3.2 Proof of Theorem 1.1 In this section we prove Theorem 1.1. The argument essentially reduces to special the case Q =, which will be treated first. In this situation we have to show that the family of operators H ε converges in the norm resolvent sense to A δ,α, as ε +. Because of Theorem 2.7 and Proposition 3.3 it is sufficient to investigate the convergence of A ε λ, B ε λ and C ε λ separately. This is done in the following lemma. In the proof we make use of the integral estimates in Appendix A and we frequently use the Schur test for integral operators; cf. [24, Example III 2.4] or [37, Satz 6.9]. Lemma 3.4 Let λ, and let A ε λ, B ε λ and C ε λ be defined as in 3.5. Then there exists a constant c = cd, λ,, V > such that for all sufficiently small ε > the following estimates hold: A ε λ A λ, C ε λ C λ cε 1 + ln ε 1/2, Bε λ B λ cε 1 + ln ε. P r o o f. First, we provide an estimate related to the Weingarten map W y. Let µ 1 y,..., µ d y be the eigenvalues of the matrix of W y, which are independent of the parametrization of and uniformly bounded on ; cf. Proposition 2.3. This implies for s, 1 and ε, 1 the existence of a constant c 1 > such that d D ε y, s := 1 det1 εsw y = 1 1 εsµ k y c 1ε, y. 3.1 Fix λ,. In order to find an estimate for A ε λ A λ, we introduce for ε > the auxiliary integral operator Âελ: L 2, 1 L 2 R d by Âε λξ x := 1 k=1 G λ x y εsνy vy, sξy, sdsdσy.

14 14 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces The quantities A ε λ Âελ and Âελ A λ are estimated separately for sufficiently small ε >. To estimate Âελ A ε λ we introduce the functions F A 1,ε : R d [, ] and F A 1,ε :, 1 [, ] by 1 F 1,εx A := Gλ x y εsνy vy, s Dε y, sdsdσy, F 1,εy A, s := Gλ x y εsνy vy, s Dε y, sdx. R d With the aid of 3.1, Proposition A.5 i and Proposition A.2 ii, we find that sup F A 1,ε c A,1 ε and sup F A 1,ε c A,1 ε with a constant c A,1 = c A,1 d, λ,, V >. Using these bounds and the Schur test we obtain  ε λ A ε λ 2 sup F A 1,ε sup F A 1,ε c 2 A,1 ε To estimate Âελ A λ, we introduce the functions F A 2,ε : R d [, ] and F A 2,ε :, 1 [, ] by 1 F 2,εx A := G λ x y εsνy G λ x y vy, s dsdσy, F 2,εy A, s := Gλ x y εsνy G λ x y vy, s dx. R d We find with the help of Proposition A.9 i and Proposition A.6 that sup F A 2,ε c A,2 ε1 + ln ε and sup F A 2,ε c A,2 ε with a constant c A,2 = c A,2 d, λ,, V >. Using these bounds and the Schur test we get Âε λ A λ 2 sup F A 2,ε sup F A 2,ε c 2 A,2 ε ln ε Combining the estimates 3.11, 3.12 and applying the triangle inequality for the operator norm, we conclude that there exists a constant c A = c A d, λ,, V > such that Aε λ A λ Aε λ Âελ + Âε λ A λ c A,1 ε + c A,2 ε1 + ln ε 1/2 c A ε 1 + ln ε 1/2. Next, we analyze the convergence of B ε λ. For this purpose, we introduce for ε > the auxiliary operator B ε λ: L 2, 1 L 2, 1 as Bε λξ 1 x, t := ux, t G λ x + εtνx y εsνy vy, sξy, sdsdσy. As in the analysis of convergence of A ε λ, we separately prove the estimates for B ε λ B ε λ and B ε λ B λ, which yield then the claimed convergence result. To estimate B ε λ B ε λ, we introduce the functions F B 1,ε, F B 1,ε :, 1 [, ] by F 1,εx B, t := F 1,εy B, s := 1 1 ux, tg λ x + εtνx y εsνy vy, s D ε y, sdsdσy, ux, tg λ x + εtνx y εsνy vy, s Dε y, sdtdσx.

15 mn header will be provided by the publisher 15 Using 3.1 and Proposition A.5 ii, we get sup F B 1,ε c B,1 ε and sup F B 1,ε c B,1 ε with a constant c B,1 = c B,1 d, λ,, V >. Employing these bounds and again applying the Schur test we obtain that Bε λ B ε λ 2 sup F B 1,ε sup F B 1,ε c 2 B,1 ε To estimate B ε λ B λ we introduce the auxiliary function F B 2,ε :, 1 [, ] as F2,εx B, t := 1 ux, t G λ x +εtνx y εsνy G λ x y vy, y dsdσy. Using that the absolute value of the integral kernel of B ε λ B λ is symmetric and applying Proposition A.9 ii, we obtain with the help of the Schur test that Bε λ B λ sup F B 2,ε c B,2 ε 1 + ln ε 3.14 with a constant c B,2 = c B,2 d, λ,, V >. Putting together the estimates in 3.13, 3.14 and employing the triangle inequality, we eventually deduce that there is a constant c B = c B d, λ,, V > such that B ε λ B λ B ε λ B ε λ + Bε λ B λ c B,1 ε + c B,2 ε 1 + ln ε c B ε 1 + ln ε. Finally, we analyze the convergence of C ε λ. Using again the Schur test, Proposition A.9 i and Proposition A.6 we find Cε λ C λ 1 2 sup ux, t G λ x + εtνx y G λ x y dtdσx y R d sup ux, t G λ x + εtνx y G λ x y dy x,t,1 R d c C ε ln ε with a constant c C follows. = c C d, λ,, V >. Setting c := max{c A, c B, c C }, the claimed result of this lemma Theorem 2.7, Proposition 3.3 and Lemma 3.4 contain the essential ingredients to prove Theorem 1.1. In the first two steps of the proof the special case Q = is discussed, the general situation is treated in the last step. Proof of Theorem 1.1. For λ < and ε [, β] the operators A ε λ, B ε λ and C ε λ are defined as in 3.5. Step 1: First, we prove that 1 B λ exists and is bounded and everywhere defined for λ < with λ sufficiently large. Let M, 1 be fixed and choose λ M < such that B ε λ M for any λ < λ M and all ε, β]; recall that such a λ M exists by Lemma 3.2 ii. Hence, the operators 1 B ε λ are bounded and everywhere defined for λ < λ M and ε, β], and it holds 1 Bε λ 1 1 M Because of 3.15 and Lemma 3.4, we can apply [24, Theorem IV 1.16], which yields that 1 B λ is boundedly invertible. Moreover, for λ < λ M we conclude from [24, Theorem IV 1.16] and 3.15 that 1 Bε λ 1 B λ 1 B ε λ 2 1 B ε λ B λ 1 B ε λ B ελ B λ c 1 ε 1 + ln ε

16 16 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces holds for all ε > sufficiently small and a constant c 1 = c 1 d, λ,, V >. Step 2: From Proposition 3.3, Lemma 3.4 and the estimates in Step 1 we obtain H ε λ Rλ + A λ1 B λ C λ = Aε λ1 B ε λ C ε λ A λ1 B λ C λ c2 ε 1 + ln ε with a constant c 2 = c 2 d, λ,, V >. It remains to verify that Rλ + A λ1 B λ C λ = A δ,α λ, 3.16 where α L is defined as in the theorem, αx = β β V x + sνx ds. In order to show 3.16 let u and v be given by 3.4, introduce the bounded operators Û : L2 L 2, 1 and V : L 2, 1 L 2 by Ûξx, t := ux, tξx and 1 V Ξ x := vx, sξx, sds defined almost everywhere, and note that V Û is the multiplication operator with α in L2. Furthermore, recall the definition of the bounded operators γλ, Mλ and the formula for γλ from 2.12; cf. Theorem 2.7. Then, we observe that 1 A λξx = G λ x y vy, sξy, sdsdσy = 1 G λ x y vy, sξy, sds dσy = γλ V Ξ x for any Ξ L 2, 1 and a.e. x R d. Thus, we conclude A λ = γλ V. In a similar way, one finds that 1 B λξx, t = ux, t G λ x y vy, sξy, sdsdσy = ux, t 1 G λ x y vy, sξy, sds dσy = ÛMλ V Ξ x, t for any Ξ L 2, 1 and a.e. x, t, 1, which implies B λ = ÛMλ V. Finally, one sees that C λfx, t = ux, t G λ x yfydy = Ûγλ x, t R d for all f L 2 R d and a.e. x, t, 1, which implies C λ = Ûγλ. Thus, we get lim H ε λ = Rλ + A λ 1 B λ C λ ε = Rλ + γλ V 1 ÛMλ V Ûγλ Since α = V Û we conclude from V 1 ÛMλ V Û = 1 V ÛMλ V Û = 1 αmλ α,

17 mn header will be provided by the publisher 17 and 3.17 together with Theorem 2.7 that lim H ε λ = Rλ + γλ 1 αmλ αγλ = A δ,α λ. ε + This completes the proof of Theorem 1.1 in the case Q =. Step 3: Let Q L R d be real valued and let λ R be such that λ < λ M Q L. Then λ is smaller than the lower bound of the operators A δ,α + Q and H ε + Q for all ε, β]. Using the formula we compute H ε + Q λ = 1 H ε + Q λ Q H ε λ, H ε + Q λ A δ,α + Q λ [ ] = H ε + Q λ A δ,α + Q λ 1 A δ,α + Q λ [ = H ε + Q λ 1 H ε + Q λ Q A δ,α λ ] A δ,α λa δ,α + Q λ = 1 H ε + Q λ Q [ H ε λ A δ,α λ ] 1 QA δ,α + Q λ. This implies Hε + Q λ A δ,α + Q λ 1 H ε + Q λ Q H ε λ A δ,α λ 1 QA δ,α + Q λ. Since the norm 1 Hε + Q λ Q is uniformly bounded in ε, the result of Step 2 yields the desired claim. 3.3 Consequences for the spectra of H ε and A δ,α In this section we discuss how Theorem 1.1 can be used to deduce certain spectral properties of H ε from those of A δ,α and vice versa. First, the approximation result in Theorem 1.1 combined with the results in [7, 16, 18] on the existence of geometrically induced bound states for Schrödinger operators with δ-interactions supported on curves and surfaces can be used to show the existence of such bound states also for the operators with regular potentials in the approximating sequence provided that the potential well is sufficiently narrow and deep. This application is motivated by Open Problem 7.1 in the review paper [15]. In order to formulate the respective claims we introduce several geometric notions. We say that the hypersurface R d, d 2, is a local deformation of the hypersurface R d, if and if there exists a compact set K R d such that \ K = \ K. Furthermore, we introduce for θ, π/2 the broken line L R 2 and the infinite circular conical surface C R 3 by L := { x, y R 2 : y = cotθ x }, C := { x, y, z R 3 : z = cotθ x 2 + y 2}. Proposition 3.5 Let R d be as in Definition 2.1 such that Hypothesis 2.1 holds and assume that satisfies one of the following assumptions. a d = 2 and is a local deformation of a straight line; b d = 2 and is a local deformation of a broken line; c d = 3 and is a C 4 -smooth local deformation of a plane, which admits a global natural parametrization in the sense of [13, Section 2-3, Definition 2]; d d = 3 and is a local deformation of an infinite circular conical surface.

18 18 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces Let the layer Ω β R d be as in 2.6 and let V L R d be real valued with supp V Ω β. Assume that α L associated to V as in Theorem 1.1 is a positive constant, which in case c is assumed to be sufficiently large. Then for all sufficiently small ε > the self-adjoint operator H ε in 3.1 has a non-empty discrete spectrum below the threshold of its essential spectrum. P r o o f. As there is no danger of confusion, we denote the value of the constant positive function α L by α as well. First, recall that by Theorem 1.1 the self-adjoint operators H ε converge in the norm resolvent sense to the self-adjoint lower-semibounded operator A δ,α as ε +. Next, in all the cases it is known that σ ess A δ,α = [ α 2 /4, ; cf. [6, Theorems 4.2 and 4.1] and [7, Theorem 3.3]. Moreover, Proposition 3.1 implies that the operators H ε are bounded from below uniformly in ε, β]. Hence, the result [37, Satz 9.24 a] yields inf σ ess H ε α 2 /4, ε Moreover, in all the cases it is known that σ d A δ,α, α 2 /4 ; cf. [16, Theorem 5.2] for a, b, [18, Theorem 4.3] for c and [7, Theorem 3.3] for d. Choose now a finite interval a, b, α 2 /4 with a, b ρa δ,α such that σ d A δ,α a, b. By 3.18 we get for all sufficiently small ε > that inf σ ess H ε > b. Eventually, it follows from [37, Satz 2.58 a and Satz 9.24 b] that for all sufficiently small ε > the dimensions of the spectral subspaces of H ε corresponding to a, b are equal to the dimension of the spectral subspace of A δ,α corresponding to the same interval; i.e. dim ran E Hε a, b = dim ran E Aδ,α a, b > Since b < inf σ ess H ε, this implies the claimed result. In the next proposition we show that Schrödinger operators with potential wells supported in curved periodic strips have gaps in their spectra under reasonable assumptions. This proposition can be proven in a similar way as Proposition 3.5. It suffices to combine a result [17, Corollary 2.2] on the existence of gaps in the negative spectrum for the Schrödinger operator with a strong δ-interaction supported on a periodic curve with the main result of this paper and with standard statements on spectral convergence. Note that a similar idea was earlier used in a different, albeit related context in [38]. Proposition 3.6 Let R 2 be as in Definition 2.1 such that Hypothesis 2.1 holds. Suppose that is parametrized via the tuple {ϕ, R, R 2 } with ϕ 1. For ϕs = ϕ 1 s, ϕ 2 s define the signed curvature of by κ := ϕ 1ϕ 2 ϕ 1ϕ 2. Assume that is not a straight line and that κ satisfies the following conditions. a κ C 2 R; b κs + L = κs for some L > and all s R; c L κsds = ; d T κsds < π/2 for all T [, L]. Let Ω β R 2 be as in 2.6 and let V L R d be real valued with supp V Ω β. Assume that α L associated to V as in Theorem 1.1 is a sufficiently large positive constant. Then for all sufficiently small ε > the self-adjoint operator H ε in 3.1 has at least one spectral gap in the interval,. One can also use Theorem 1.1 to obtain spectral results for A δ,α from those of H ε. To illustrate this idea, we show in the following example that for d 3 the operator A δ,α does not have any negative bound states if is a sphere with radius R > and if α, d 2 R is a constant. The same result is also contained in [5, Theorem 2.3] for d = 3 and in [4, Theorem 4.1] for arbitrary d 3. The proofs there are of a different nature than ours. Example 3.7 Let d 3, R > and = B, R. Let α, d 2 R be fixed. Let q C R + be non-negative with supp q R/2, 3R/2 such that 3R/2 R/2 qrdr = α.

19 mn header will be provided by the publisher 19 Define the radially symmetric potential V L R d by V x := q x, x R d. Furthermore, define the scaled potentials V ε and the operators H ε as in the Introduction with β = R/2 and ε, R/2]. By Theorem 1.1 the operators H ε converge in the norm resolvent sense to A δ,α as ε +. For any ε, R/2] the potential V ε is also radially symmetric and we get with the help of Lebesgue s dominated convergence theorem that for q ε x := V ε x lim ε + ε rq ε rdr = lim ε + = lim ε + ε 3R/2 R/2 R + r R 2ε q R + Rr 2ε 2εz R + R 2ε dr qzdz = αr < d 2. Hence, using Bargmann s estimate [32, Theorem 3.2], we obtain that H ε has no negative eigenvalues for all sufficiently small ε >. Because of Theorem 1.1 and [37, Satz 9.24 a] it follows that A δ,α has no negative eigenvalues as well. A Estimates related to Green s function In this appendix we provide estimates for integrals that contain Green s function G λ x = 1 2π d/2 x λ 1 d/2 K d/2 λ x, x R d \ {}, d 2, A.1 from The estimates are formulated in a way such that they can be applied directly in the main part of the paper. We note that some of the estimates below are known, but exact references are difficult to find in the mathematical literature. Therefore, in order to keep this paper self-contained we also provide complete proofs of standard estimates as, e.g., in Proposition A.1. Throughout this appendix we assume that is an orientable C 2 -hypersurface which satisfies Hypothesis 2.1, and we denote by ν the continuous unit normal vector field of. In the first preliminary proposition we discuss the asymptotics of G λ and G λ. Proposition A.1 Let λ, and let G λ be as in A.1. Then there exists a constant R = Rd > such that the following assertions hold. i There is a constant c = cd such that for all x R λ Gλ x { c 1 + ln λ x, if d = 2, c x 2 d, if d 3, holds. Moreover, there exists a constant C = Cd such that for all x > G λ x 1 d/2 x C e λ x. λ In particular, G λ L 1 R d. R λ it holds ii The function R d \ {} x G λ x is continuously differentiable. Furthermore, there exist constants c = cd, λ and C = Cd, λ such that for all x Gλ x c x 1 d is true and for all x > R λ it holds G λ x Ce λ x. In particular, G λ L 1 R d ; C d. R λ

20 2 J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik: Approximation of δ-interactions supported on hypersurfaces P r o o f. i Due to the asymptotic behavior of K d/2, see [1, Section 9.6 and 9.7], there exist an R = Rd > and constants C 1, C 2, C 3 > such that for any p, R] { C ln p, if d = 2, K d/2 p C 2 p 1 d/2 A.2, if d 3, is satisfied and for any p > R K d/2 p C 3 e p A.3 holds. Hence, the claimed asymptotics follow from the definition of G λ. It is not difficult to check that these asymptotics imply G λ L 1 R d. ii Recall first that the mapping C \, ] z K ν z is holomorphic by [1, Section 9.6]. Therefore, R d \ {} x G λ x is continuously differentiable and we obtain G λ x = 1 2π d/2 x 1 d x d/2 λ x x 2 K 1 d/2 d/2 λ + 1 d/2 x λ x A.4 λ K d/2 λ for d 3 and G λ x = 1 x λ K λ x 2π x A.5 in the case d = 2. Since K d/2 z = K d/2z + d 2 2z K d/2z A.6 by [1, eq ] we conclude from A.2 that for d 3 and x R λ K d/2 λ x Kd/2 λ x d Kd/2 λ x λ x C 4 x d/2 d 2 + C 5 2 λ x x 1 d/2 C 6 x d/2 holds with some constant C 6 = C 6 d, λ >. It is easy to see that the same estimate is also true in the case d = 2. Hence, A.2, A.4 and A.5 yield G λ x C 7 x d/2 Kd/2 λ x + x 1 d/2 λ x c x 1 d. In the same way, using A.3, A.4, A.5, and A.6 one finds for all x G λ x Ce λ x K d/2 R λ that holds with some constant C = Cd, λ >. Finally, it is not difficult to check that the asymptotics for G λ imply G λ L 1 R d ; C d. We start now with the estimates which are needed to show Lemma 3.4. The first proposition in this context is only based on the integrability of G λ ; cf. Proposition A.1 i. Proposition A.2 Let be a C 2 -hypersurface which satisfies Hypothesis 2.1 and let λ,. Then the following statements are true.