A geometric approximation of $\delta$-interactions by Neumann Laplacians

We demonstrate how to approximate one-dimensional Schr\"odinger operators with $\delta$-interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small protuberance with"room-and-passage"geometry. We show that in the limit when the perpendicular size of the strip tends to zero, and the room and the passage are appropriately scaled, the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense to the above singular Schr\"odinger operator. Also we prove Hausdorff convergence of the spectra. In both cases estimates on the rate of convergence are derived.


Introduction
Schrödinger operators with potentials supported on a discrete set of points have attracted considerable attention over several past decades due to numerous applications in different fields of science and engineering. In particular, such operators serve as solvable models in quantum mechanics. The term "solvable" reflects the fact that their mathematical and physical quantities, like spectrum, eigenfunctions and resonances, can be calculated in many cases explicitly. We refer to the monograph [1] for a comprehensive introduction to this topic. Note that in the literature such models are also called Schrödinger operators with point interactions.
Investigation of these operators was originated by the famous Kronig-Penney model [18], concerning a non-relativistic electron moving in a fixed crystal lattice. Its mathematical representation is a one-dimensional Schrödinger operator with a singular potential supported at Z = {0, ±1, ±2, . . . }: One says that the conditions (1.2) correspond to a δ-interaction with strength γ supported at the point z.
In the present paper we wish to contribute to the understanding of ways how Schrödinger operators with δ-interactions can be approximated by differential operators with regular coefficients. In what follows, we restrict ourselves to a Schrödinger operator on an open interval Ω 0 (bounded or not) with a single δ-interaction with strength γ supported at 0; we denote this operator by A 0 . The obtained results can easily be extended to Schrödinger operators with finitely and even countably many δ-interactions, see Section 5.
One way of approximation is given by a sequence of Schrödinger operators with smooth δ-like potentials, see e.g. [1,Sec. 1.3.2]. In the present work we discuss another approach, in which the desired δ-interaction is generated by geometry. Namely, we construct a waveguide-like domain Ω ε such that its Neumann Laplacian A ε converges (in a kind of norm-resolvent topology) to the desired operator A 0 as the perpendicular size of the guide tends to zero. Since the approximating operator is non-negative (minus the Laplacian), we can only expect γ ≥ 0 in the limit (in our convention, γ ≥ 0 leads to a non-negative operator with δ-interaction, see (2.7) for the corresponding form). The approximating domain will be a thickened version of Ω 0 with a decoration near 0 ∈ Ω 0 given by a small passage P ε and a larger (but shrinking) room R ε (see Figure 1). In particular, the room (a square with side length b ε = ε β ) can be chosen to shrink arbitrarily slow (β ∈ (0, 1/2)), and the passage of height h ε = ε α and width d ε = γε α+1 can shrink arbitrarily fast (α > 0). Nevertheless, the area ε 2β of the room R ε is still shrinking compared to the transversal shrinking rate ε of the strip S ε . The strength of the δ-interaction is given by γ = d ε /(h ε ε). Note that we can interpret the quotient of the width d ε and the height h ε of the passage as the (vertical) conductance of P ε .
It is interesting to compare our results with the the case, when the room is joined directly to the strip near 0 as on Figure 2. Such a geometrical configuration was considered in [11,19] (see also [23] for a version with generalised norm resolvent convergence). The critical value 1/2 (in dimension 2) of the parameter β appears here as well. Namely, if β ∈ (0, 1/2), then the limit operator is the direct sum of the Laplacian with Dirichlet boundary condition at 0 (hence decouples) and the 0 operator on a one-dimensional space as in our case. In both cases ( Figure 1 and Figure 2) the room area decays slower than the transversal shrinking rate, which attracts particles and leads to an own state (at energy 0). If the room is directly attached to the strip (Figure 2), then it prevents transport along the strip, while in the present situation ( Figure 1) it leads to a repulsive (i.e. with γ ≥ 0) interaction at 0. Note that for the waveguide as on Figure 2 in the critical case b ε = ε 1/2 the limiting operator is the Laplacian with another peculiar conditions at 0 (see the so-called "borderline case" in [11] or [23]); these conditions resemble (1.2) with the coupling constant γ being replaced by a quantity dependent on the spectral parameter. Domains with attached protuberances with "room-and-passage" geometry are widely used in spectral theory in order to demonstrate various peculiar effects. For example, R. Courant and D. Hilbert [9] used such a domain as an example of a small perturbation breaking the continuity of eigenvalues of the Neumann Laplacian; see [3] for more details. In [14] such "rooms-and-passages" were used to construct a domain such that its Neumann Laplacian has prescribed essential spectrum (see also the overview [4] for more details). Homogenization problems in domains with corrugated "room-and-passage"-like boundary were studied in [6,7]. Finally, various peculiar examples in the theory of Sobolev spaces are based on domains with such a geometry, see [2,10,13] and the monograph [20].
As the spaces change while passing to the limit ε → 0, we use the framework of generalised norm resolvent convergence developed by the second author in [22] and [23]. We provide a self-contained presentation including a new proof of spectral convergence (cf. Theorem 3.5) in Section 3. As usual the generalised norm resolvent convergence is not much harder to show than other concepts such as versions of strong resolvent convergence used e.g. in homogenization theory [21,Chap. III], [27,Chap. XI].
Actually, our limit operator is not the Laplacian with δ-interaction itself (as already mentioned above), but its direct sum with the null operator on a one-dimensional space. In Remark 2.3 we give some light on the appearance of this extra component.
The work is organized as follows. In Section 2 we set the problem and formulate the main results, Theorem 2.1 concerning the norm resolvent convergence and Theorem 2.2 concerning the spectral convergence. Note that we treat even more general operators − d 2 where V 0 is a regular potential. In Section 3 we give two abstract results designed for studying convergence of operators in varying Hilbert spaces. Using them we prove the main results in Section 4. Finally, in Section 5 we discuss the case of countably many point interactions.
2. Setting of the problem and the main result 2.1. The waveguide Ω ε and the operator A ε . Throughout the paper we denote points in Let ε ∈ (0, ε 0 ] be a small parameter. We set Moreover, we claim ε 0 to be sufficiently small, namely 3) The first assumption in (2.3) implies the second one leads to 5) and the last one yields | ln γ| ≤ | ln ε| (it will be used in the proof of Lemma 4.3). Note that, since either γ ≤ 1 or γ −1 ≤ 1, one has ε 0 < 1. Finally, we introduce the domains and the resulting domain Ω ε given by (here int(·) stands for the interior of a subset of R 2 ). Due to (2.4)-(2.5) the geometry of Ω ε is exactly as shown in Figure 1, i.e. the bottom part (respectively, the top part) of ∂P ε is contained in the top part of ∂S ε (respectively, the bottom part of ∂R ε ). In the Hilbert space H ε := L 2 (Ω ε ) we introduce the sesquilinear form with a real-valued potential V ε ∈ L ∞ (Ω ε ). This form is densely defined in L 2 (Ω ε ), nonnegative and closed, consequently [17, Theorem VI.2.1] there is a unique non-negative selfadjoint operator A ε acting in L 2 (Ω ε ) such that the domain inclusion dom(A ε ) ⊂ dom(a ε ) and the equality The main goal of this work is to describe the behavior of the resolvent and the spectrum of A ε as ε → 0. In the next subsection we introduce the anticipated limiting operator.
In the space H 0 we introduce the sesquilinear form a 0 defined by where V 0 ∈ L ∞ (Ω 0 ). It is easy to see that the above form is densely defined in H 0 , nonnegative and closed. We denote by A 0 the self-adjoint operator associated with a 0 . It is easy to show that its domain is given by where 0 C is the null-operator in C, and A 0 is defined by the operation

Resolvent convergence.
Our first goal is to prove (a kind of) norm resolvent convergence of the operator A ε to the operator A 0 . Since these operators act in different Hilbert spaces H ε = L 2 (Ω ε ) and H 0 = L 2 (Ω 0 ) ⊕ C, respectively, the standard definition of norm resolvent convergence cannot be applied here and should be modified in an appropriate way. The modified definition should be adjusted in such a way that it still implies the convergence of spectra as it takes place in the classical situation. The standard approach (see, e.g. the abstract scheme in [16] and its applications to homogenization in perforated spaces [ where R ε and R 0 are the resolvents of A ε and A 0 , respectively, and J ε : H 0 → H ε is a suitable bounded linear operator being "almost isometric" in a sense that For the problem we deal in this paper it is natural to define the operator J ε as follows: The operator J ε is natural because it is an isometry, namely one has Along with J ε we also introduce the operator J ε : H ε → H 0 by It is easy to see that J u H 0 ≤ u H ε . Moreover, J ε is the adjoint of J ε , as To guarantee the closeness of the resolvents of the operators A ε and A 0 , the potentials V ε and V 0 have to be close in a suitable sense. Namely, we choose the family {V ε } ε>0 as follows In order to simplify the presentation we assume further that and hence both A ε and A 0 are non-negative operators; the general case needs only slight modifications. We denote by R ε and R 0 the resolvents of A ε and A 0 , respectively: We are now in position to formulate the first result of this work. Below, · X→Y stands for the norm of an operator acting between normed spaces X and Y. where C 1 > 0 is a constant independent of ε (see Remark 4.6 for more details).
Obviously, solely the estimate (2.13) provide no information on the closeness of spectra -simply because (2.13) holds for arbitrary operators R ε : H ε → H ε and R 0 : H 0 → H 0 if we choose J ε = 0 and J ε = 0. Therefore, in order to get some information on the closeness of spectra, we need additional conditions on the operators J ε and J ε . Such conditions are formulated in the abstract Theorem 3.5 below, and also in the original concept of quasiunitary equivalence in [22] and [23], see Section 3.3.

Spectral convergence.
Our second result concerns Hausdorff convergence of spectra. Recall (see, e.g. [26]), that for closed sets X, Y ⊂ R the Hausdorff distance between X and Y is given by (2.14) The notion of convergence provided by this metric is too restrictive for our purposes. Indeed, the closeness of σ(A ε ) and σ(A 0 ) in the metric d H (·, ·) would mean that these spectra look nearly the same uniformly on all parts of [0, ∞) -a situation, which is not guaranteed by norm resolvent convergence. To overcome this difficulty, we introduced the new metric d H (·, ·), which is given by With respect to this metric two spectra can be close even if they differ significantly at high energies. Note Note that by the spectral mapping theorem Note that the best convergence rate in (2.16) is provided when α = 1/3 and β = 1/6.

Remark 2.3. We denote
acting as −∆ + V ε and with Neumann boundary conditions on ∂F ε \ D + ε and Dirichlet conditions on D + ε . Using similar methods as in the proof of Theorem 2.2, one can show that the δ-coupling at 0 is caused by the Dirichlet conditions on D + ε (these boundary conditions can be regarded as an infinite potential). Also, let A B ε be the Neumann Laplacian on B ε . The first eigenvalue of A B ε is zero for each ε > 0, while the next eigenvalues escape to infinity as ε → 0. Hence, we get

Abstract toolbox
In this section we present two abstract results serving to compare the resolvents (Theorem 3.1) and the spectra (Theorem 3.5) of two self-adjoint non-negative operators acting in two different Hilbert spaces. The first result was established by the second author in [22], and the second result (in a slightly weaker form) was proven by the first author and G. Cardone in [8]. For convenience of the reader we will present complete proofs here.
Throughout this section H and H are two Hilbert spaces, a and a are closed, densely defined, non-negative sesquilinear forms in H and H, respectively. We denote by A and A the non-negative, self-adjoint operators associated with a and a, by R and R we denote the resolvents of A and A, respectively: Along with H and H we also introduce spaces H 1 and H 1 consisting of functions from dom(a) = dom(A 1/2 ) and dom( a) = dom( A 1/2 ), respectively, and equipped with the norms and spaces H 2 and H 2 consisting of functions from dom(A) and dom( A), respectively, and equipped with the norms Note that Moreover, due to the non-negativity of A and A one has the estimates Proof. Let g ∈ H and v ∈ H be arbitrary. We set f : Using (3.5)-(3.8) one can estimate all terms in the right-hand-side of the above equality, Evidently, (3.10) implies (3.9). The theorem is proven.

Remark 3.2.
It is well-known [17,Theorem VI.3.6], that convergence of sesquilinear forms with common domain implies norm resolvent convergence of the associated operators (see the recent paper [5, Theorem 2] for a quantitative version of this result); in these theorems the convergence of the forms a ε to the form a means that the inequality Nevertheless, in most of the applications one is able to establish stronger estimate (3.8) (cf. Lemma 4.5). Moreover, sometimes (for example, when studying convergence of graph-like manifolds [22,23]), one even can prove the stronger inequality which can be regarded as a counterpart to (3.11).

Remark 3.4.
Usually in applications the operators J and J appear in a natural way (as, for example, J defined in (2.8) and J defined in (2.10) in our case), while the other two operators J 1 and J 1 should be constructed as "almost" restrictions of J and J to H 1 and H 1 , respectively, modified in such a way that they respect the form domains (see conditions (3.5)-(3.6) above). J R − R J H→H ≤ η, (3.15) and, moreover,

Quasi-unitary operators.
Let us here finally comment on the concept originally introduced in [22] and [23]: Definition 3.7. We say that J and J are δ-quasi-unitary for some δ ≥ 0 if and also (3.7) holds. We say that A and A are δ-quasi-unitarily equivalent, if (additionally to (3.29)-(3.30)) J and J also fulfil RJ − J R H→ H ≤ δ, J R − R J H→H ≤ δ.
The above concept allows to generalise norm resolvent convergence in the sense that A ε converges to A 0 in generalised norm resolvent sense (with convergence speed δ ε ) if A ε and A 0 are δ ε -quasi-unitarily equivalent with δ ε → 0 as ε → 0 (cf. (4.1)). This concept generalises the classical norm resolvent convergence in the sense that if H ε = H 0 = H and choosing J = J = I the identity operator on H, then the generalised norm resolvent convergence is just the classical norm resolvent convergence R ε − R 0 H→H ≤ δ ε → 0 as ε → 0.
As for the classical norm resolvent convergence we have [22,23] (see also [25] for a brief up-to-date version and more details): . If A ε converges to A 0 in generalised norm resolvent convergence with convergence speed δ ε , then we have as ε → 0 for suitable functions Ψ (e.g. measurable and continuous in a neighbourhood U of σ(A 0 ) with Ψ(λ)(λ + 1) 1/2 → 0 as λ → ∞. If Ψ is holomorphic on U, then the above norm of the resolvent difference is of order δ ε .
The above proposition applies in particular to the heat operator with Ψ = Ψ t and Ψ t (λ) = e −λt or spectral projections Ψ = 1 I and ∂I ∩ σ(A 0 ) = ∅. We also showed spectral convergence, see [22,23] for details. Nevertheless, the result in Theorem 3.5 is more explicit as it imitates the proof of (3.13) of Herbst and Nakamura [15] and gives better error estimates.

Preliminaries.
For the proof of Theorems 2.1 and 2.2 we will use abstract results given in Section 3 (Theorems 3.1 and 3.5, respectively). Recall, that these abstract results serve to compare the resolvents and spectra of self-adjoint non-negative operators A and A acting in different Hilbert spaces H and H, respectively. We will apply these abstract theorems for (recall that H ε = L 2 (Ω ε ) and H 0 = L 2 (Ω 0 ) ⊕ C, and A ε and A 0 are the self-adjoint operators acting in these spaces associated with the sesquilinear forms a ε (2.6) and a 0 (2.7)). Similarly to (3.1)-(3.2) we introduce Hilbert spaces H k ε and H k 0 , k = 1, 2, consisting of functions u ∈ dom(A k/2 ε ) and f ∈ dom(A k/2 0 ), respectively, equipped with the norms Note that for Sobolev spaces we use a sans serif font, e.g.
Recall, that the sets D ± ε are given in (2.17). We introduce several other subsets of Ω ε : Note that Y ε ⊂ S ε due to (2.3). We also denote By u D we denote the mean value of the function u(x) in the domain D, i.e.
where |D| denotes the area of D. Also we keep the same notation if D is a segment (for example, D 0 ε ); in this case we integrate with respect to the natural coordinate on this segment, and |D| denotes its length.
In the following, we need the standard Sobolev inequality.
where the constant I > 1 depends only on the length |I| of I.
where H 1 0 and H 1 ε are the energy spaces associated with the forms a 0 and a ε , i.e.

Proof of the quasi-unitary equivalence.
In this subsection, we additionally show that the identification operators are also quasi-unitarily equivalent, see Section 3.3). From this concept, also used in [22,23], the convergence of operator functions as in Proposition 3.8 follows. Note that we have given a more explicit (and better) estimate on the spectral convergence in Theorem 3.5 here, although it was already shown in [22,23]. We have commented on the differences in Remark 3.10. holds (in fact, (4.39) is equivalent to (4.44)). Using (4.44) with D = {x 1 } × (−ε, 0) and D = R ε , and the estimate (4.6), we obtain with C 18 := (C 6 + π −2 ) 1/2 (for the last step note that β < 1 and C 6 > 1/π 2 ). The lemma is proven.

(5.2)
In L 2 (Ω 0 ) we consider the operator A Z defined by the operation − d 2 dx 2 + V 0 on Ω 0 \ Z, Neumann conditions at − (provided − > −∞) and + (provided + < ∞), and δ-coupling with strength γ z at z ∈ Z. The case Z = Z, γ z = γ, V 0 = 0 (5.3) corresponds to the Kronig-Penney model. To approximate A Z , we consider the domain Ω ε = int(S ε ∪ z∈Z (P z,ε ∪ R z,ε ) , (5.4) consisting of the straight strip S ε = Ω 0 × (−ε, 0), and the family of "rooms" R z,ε and "passages" P z,ε given by Here d ε = γ z ε α+1 , h ε = ε α , b ε = ε β with α > 0 and 0 < β < 1 2 . We choose ε to be small enough, such that the bottom part (respectively, the top part) of ∂P z,ε is contained in the top part of ∂S ε (respectively, the bottom part of ∂R z,ε ), and moreover the neighbouring rooms are disjoint; this can be achieved due to (5.1)-(5.2). As before, where ∆ Ω ε is the Neumann Laplacian in Ω ε , and the potential V ε is defined as in (2.12). As in the case of a single δ-interaction one has the estimate where C > 0 is a constant independent of ε. The proof of (5.6) is similar to the proof of Theorem 2.2. It relies on the abstract Theorems 3.1 and 3.5 being applied to H ε := L 2 (Ω ε ), A ε as in (5.5), H 0 := L 2 (Ω 0 ) ⊕ 2 (Z), A 0 := A Z ⊕ 0 2 (Z) and appropriately modified operators J ε , J ε , J 1 ε , J 1 ε . One of the byproducts of this result is the tool for constructing periodic Neumann waveguides with spectral gaps. Namely, it is known [1, Sec. III.2.3] that the spectrum of the Kronig-Penney operator (1.1) has infinitely many gaps provided γ = 0. Then, using the estimate (5.6), we conclude that for any m ∈ N the Neumann Laplacian on the periodic domain Ω ε (5.4) (cf. (5.3)) has at least m gaps provided ε sufficiently small enough.
As another byproduct of our careful calculations of the constants, we may also allow that Z = Z ε depends on ε in such a way that inf z,z ∈Z ε ,z =z |z − z | = 2 ε > 0 is still positive as in (5.1), but with ε → 0. Moreover, we may allow that γ z = γ z,ε depends on ε such that γ z,ε → ∞ as ε → 0. If, for example, ε = ε τ and γ z,ε = γ z,1 ε −ω , then we can show a spectral estimate as in (5.6) remains valid, namely we have d H (σ(A ε ), σ( A Z ε ) ∪ {0}) → 0 as ε → 0 provided τ > 0 and ω > 0 are sufficiently small. Such results are useful when approximating other point interactions by δ-interactions of the form A Z ε , as e.g. done for certain self-adjoint vertex conditions on a metric graph, see [12] and the references therein.