Spectral analysis of photonic crystals made of thin rods

Abstract

In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of given frequencies. In such materials electromagnetic waves with these frequencies can not propagate; this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging to a predefined range there exists a transverse electric mode that can propagate in the medium. These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in $L^{2} (R^{2})$ . The proofs rely on perturbation theory of linear operators, Floquet–Bloch analysis, and properties of Schrödinger operators with point interactions.

Keywords

Photonic crystals spectral gaps inverse problem thin rods electromagnetic waves periodic differential operators perturbation theory Floquet–Bloch analysis point interactions

1. Introduction

1.1. Motivation

Photonic crystals gained a lot of attention in the recent decades both from the physical and the mathematical side. An electromagnetic wave can propagate in the crystal if and only if its frequency does not belong to a photonic band gap. Therefore, photonic crystals can be seen as an optical analogue of semiconductors giving a physical motivation to study them. The idea of designing periodic dielectric materials with photonic band gaps was proposed in [30,44]. In the recent years a great advance in fabrication of such crystals was achieved.

Despite a substantial progress in the physical and mathematical investigation of photonic crystals with a given geometry (see [16,28] and the references therein), the important task of designing photonic crystals having band gaps of a certain predefined structure still remains challenging. In this paper we are interested in the following inverse problem: $\begin{matrix} How to design a photonic crystal such that finitely many \\ predefined frequencies ω_{1}, \dots, ω_{N} belong to photonic band gaps? \end{matrix}$

In order to tackle this problem we employ a special class of photonic crystals made of very thin infinite rods with high dielectric permittivity embedded into vacuum. To be more precise, let $\hat{Γ}$ be a parallelogram in $R^{2}$ which is spanned by two linearly independent vectors and let the points $y^{(1)}, \dots, y^{(N)} \in \hat{Γ}$ be pairwise distinct. The basis cell of the crystal consists of N infinite rods with large relative dielectric permittivities $ε^{(n)} ≫ 1$ , $n = 1, \dots, N$ , whose cross sections are small bounded domains in $R^{2}$ localized near the points $y^{(n)}$ , $n = 1, \dots, N$ , and surrounded by vacuum with relative dielectric permittivity $ε = 1$ . Then the crystal is built by repeating the basis cell in a periodic way such that the whole Euclidean space $R^{3}$ is filled; cf. Fig. 1 and Section 1.3 for a more precise definition.

Fig. 1.

Cross section of the crystal. The basis period cell is colored in gray and contains three rods ( $N = 3$ ).

Special crystals of the above type, which are known in the applied literature also as arrow fibres, have already been investigated in [28,29,37,39,40] by physical and numerical experiments. They were one of the first photonic crystals treated in the literature because they are comparably simple to produce. The results in the above mentioned papers indicate that these crystals may have band gaps for electromagnetic waves polarized in a special way. Our goal is to provide an analytic proof of this and related results.

1.2. Maxwell’s equations and propagation of electromagnetic waves

Under conventional physical assumptions for our model (absence of currents and electric charges, linear constitutive relations and no magnetic properties, i.e. the relative magnetic permeability satisfies $μ \equiv 1$ ) and a suitable choice of the units, Maxwell’s equations for time harmonic fields take the following simple form (see [22]): $\begin{matrix} (1.1) & div (ε E) = 0, curl E = \frac{i ω}{c} H, div H = 0, curl H = - \frac{i ω}{c} ε E . \end{matrix}$ Here, the three-dimensional vector fields E and H are the electric and the magnetic field, respectively, $ω > 0$ is the frequency of the wave, ε is the relative dielectric permittivity and $c > 0$ stands for the speed of light. Choose a system of coordinates such that the $x_{3}$ -axis is parallel to the rods building our photonic crystal. In this paper, we are interested in $x_{3}$ -independent waves, i.e. $E = E (x_{1}, x_{2})$ and $H = H (x_{1}, x_{2})$ . This assumption is reasonable, as the physical parameters also depend only on $x_{1}$ and $x_{2}$ . In the physical literature such waves are often called standing waves, as they propagate strictly parallel to the $x_{1}$ - $x_{2}$ -plane perpendicular to the rods and not in the $x_{3}$ -direction. The frequency $ω > 0$ belongs to a photonic band gap if the Maxwell equations (1.1) possess no bounded solutions.

Maxwell’s equations can be regarded as a generalized eigenvalue problem for the Maxwell operator $\begin{matrix} M (\begin{matrix} E \\ H \end{matrix}) = (\begin{matrix} 0 & i c ε^{- 1} curl \\ - i c curl & 0 \end{matrix}) (\begin{matrix} E \\ H \end{matrix}), \end{matrix}$ which is defined on an appropriate subspace of $L^{2} (R^{2}, C^{3}, ε d x) \times L^{2} (R^{2}, C^{3}, d x)$ that takes the constraints $div (ε E) = 0$ and $div H = 0$ into account; cf. [22] for more details. According to [22, Sec. 7.1] the operator $M$ is self-adjoint in $L^{2} (R^{2}, C^{3}, ε d x) \times L^{2} (R^{2}, C^{3}, d x)$ . Using periodicity and the results of [33] it can be shown that ω belongs to a photonic band gap of the crystal if and only if $ω \notin σ (M)$ . Therefore, the existence and location of photonic band gaps can be analyzed by means of the spectral analysis of $M$ .

Since ε is periodic in the variables $x_{1}$ and $x_{2}$ and independent of $x_{3}$ , the operator $M$ can be decomposed as $M = M_{1} \oplus M_{2}$ , where $M_{1}$ acts on so-called transverse magnetic $TM$ -modes having the form $\begin{matrix} E = {(0, 0, E_{3})}^{⊤}, H = {(H_{1}, H_{2}, 0)}^{⊤}, \end{matrix}$ and $M_{2}$ acts on transverse electric $TE$ -modes given by $\begin{matrix} E = {(E_{1}, E_{2}, 0)}^{⊤}, H = {(0, 0, H_{3})}^{⊤}; \end{matrix}$ see [22, Sec. 7.1]. Therefore, it holds $σ (M) = σ (M_{1}) \cup σ (M_{2})$ and it suffices to perform the spectral analysis of $M_{1}$ and $M_{2}$ separately to characterize $σ (M)$ . The spectra of $M_{j}$ , $j = 1, 2$ , have their own independent physical meaning when taking polarization of electromagnetic waves to $TE$ - and $TM$ -modes into account. Moreover, $σ (M_{j})$ , $j = 1, 2$ , are in simple direct correspondence with the spectra of certain scalar differential operators on $L^{2} (R^{2})$ ; cf. Section 1.3.

Several mathematical approaches to treat the spectral problems for the operators $M_{j}$ , $j = 1, 2$ , have been developed. Purely numerical methods are elaborated in e.g. [37,39,40]. A combination of numerical and analytical methods is suggested in [26]. For a wide class of geometries a method based on boundary integral equations is efficient [5,6]. An analytic approach for high contrast media is proposed by A. Figotin and P. Kuchment for crystals of a different geometry from ours which are composed of periodically ordered vacuum bubbles surrounded by an optically dense material with very large dielectric permittivity and of small width. In a series of papers [20–24] these authors showed that such crystals have an arbitrarily large number of photonic band gaps. Their approach largely inspired the methods used in the present paper.

Finally, we point out that topics of recent interest in this active research field include the analysis of guided perturbations of photonic crystals [7,10,12,35], of materials with non-linear constitutive relations [17,19], and of photonic crystals made of metamaterials [13,18].

1.3. Notations and statement of the main results

In order to formulate our main results, we fix some notations. We set $0 : = (0, 0) \in R^{2}$ . For $x \in R^{2}$ and $r > 0$ we define $B_{r} (x) : = {y \in R^{2} : | y - x | < r}$ . For $α, β ⩾ 0$ and $A, B \subset R^{2}$ we use the notation $\begin{matrix} α A + β B : = {α x + β y \in R^{2} : x \in A, y \in B} . \end{matrix}$ For a measurable set $Ω \subset R^{2}$ we denote its Lebesgue measure by $| Ω |$ and its characteristic function by $1_{Ω}$ . As usual, $L^{1} (Ω)$ stands for the space of integrable functions over Ω. For $f \in L^{1} (Ω)$ we introduce the notation ${⟨ f ⟩}_{Ω} = \int_{Ω} f (x) d x$ . The $L^{2}$ -space over $Ω \subset R^{2}$ with the usual inner product is denoted by $(L^{2} (Ω), {(\cdot, \cdot)}_{L^{2} (Ω)})$ and the $L^{2}$ -based Sobolev spaces by $H^{k} (Ω)$ , $k = 1, 2$ , respectively. For a self-adjoint operator $T$ in a Hilbert space we denote its spectrum by $σ (T)$ and its resolvent set by $ρ (T) : = C ∖ σ (T)$ .

Let $N \in N$ be fixed and let $λ_{1}, \dots, λ_{N} \in (0, \infty)$ be pairwise distinct; without loss of generality we assume that $0 < λ_{1} < λ_{2} < \dots < λ_{N}$ . These numbers are associated to the frequencies $ω_{1}, \dots, ω_{N}$ that are desired to be contained in photonic band gaps of the crystal via the relations $λ_{n} = {(\frac{ω_{n}}{c})}^{2}$ . Moreover, let $a_{1}, a_{2} \in R^{2}$ be linearly independent. We set $\begin{matrix} (1.2) & Λ : = {n_{1} a_{1} + n_{2} a_{2} \in R^{2} : (n_{1}, n_{2}) \in Z^{2}} and \hat{Γ} : = {s_{1} a_{1} + s_{2} a_{2} \in R^{2} : s_{1}, s_{2} \in [0, 1)} . \end{matrix}$ For the points in Λ we often use the notation $y_{n} = n_{1} a_{1} + n_{2} a_{2}$ , $n = (n_{1}, n_{2}) \in Z^{2}$ . We choose pairwise distinct points $y^{(1)}, \dots, y^{(N)} \in \hat{Γ}$ and define $\begin{matrix} (1.3) & Y : = {y^{(1)}, \dots, y^{(N)}} . \end{matrix}$ Let $Ω \subset R^{2}$ be a bounded domain with $0 \in Ω$ and let $r > 0$ be sufficiently small such that $\begin{matrix} (y^{(n)} + r Ω) \cap (y^{(m)} + r Ω) = \emptyset, n \neq m . \end{matrix}$ For $n = 1, \dots, N$ we define $\begin{matrix} (1.4) & μ_{n} (x) : = \frac{2 π}{λ_{n} | Ω |} x + c_{n} x^{2} + {\tilde{μ}}_{n} (x), \end{matrix}$ where $c_{n} \in R$ , $n = 1, \dots, N$ , are some constant parameters and ${\tilde{μ}}_{n} (x) = o (x^{2})$ , as $x \to 0 +$ . Finally, we introduce the function $w_{r} : R^{2} \to R$ by $\begin{matrix} (1.5) & w_{r} : = 1 + \frac{1}{r^{2}} \sum_{n = 1}^{N} μ_{n} (\frac{1}{| ln r |}) 1_{Λ + y^{(n)} + r Ω} . \end{matrix}$ The relative dielectric permittivity $ε_{r} : R^{3} \to R$ , which describes the physical properties of the crystal, is expressed through $w_{r}$ by $ε_{r} (x_{1}, x_{2}, x_{3}) : = w_{r} (x_{1}, x_{2})$ . We would like to emphasize that the particular choice of the parameters in the definition of $μ_{n}$ in (1.4) is of great importance to ensure that the photonic crystal with this dielectric permittivity $ε_{r}$ has band gaps in neighborhoods of the initially given numbers $λ_{1}, \dots, λ_{N}$ . In particular, if the coefficient at the linear term is chosen differently, then the auxiliary operators $H_{r, λ}$ introduced below in (1.7) do not converge to a non-trivial limit operator, as $r \to 0 +$ . The extra parameters $c_{1}, \dots, c_{N}$ will allow us to control the properties of these limit operators such that they have desired spectra. On the other hand, the terms ${\tilde{μ}}_{1}, \dots, {\tilde{μ}}_{N}$ do not affect the convergence behavior. This is of interest in applications as small errors, that might obviously occur in the fabrication of such photonic crystals, do not change its properties.

In order to treat the spectral problem for the associated Maxwell operator $M = M_{1} \oplus M_{2}$ described in Section 1.2, we introduce two partial differential operators in $L^{2} (R^{2})$ by $\begin{array}{l} (1.6a) & Θ_{r} f : = - w_{r}^{- 1} Δ f, & dom Θ_{r} & : = H^{2} (R^{2}), \\ (1.6b) & Υ_{r} f : = - div (w_{r}^{- 1} grad f), & dom Υ_{r} & : = {f \in H^{1} (R^{2}) : div (w_{r}^{- 1} grad f) \in L^{2} (R^{2})} . \end{array}$ According to [22, Sec. 7.1] we have $\begin{array}{l} ω \in σ (M_{1}) ⟺ {(\frac{ω}{c})}^{2} \in σ (Θ_{r}), \\ ω \in σ (M_{2}) ⟺ {(\frac{ω}{c})}^{2} \in σ (Υ_{r}) . \end{array}$

Following the strategy of [22,24] in order to investigate the spectral properties of $Θ_{r}$ we introduce a family of auxiliary Schrödinger operators $\begin{matrix} (1.7) & H_{r, λ} f : = - Δ f - λ (w_{r} - 1) f, dom H_{r, λ} : = H^{2} (R^{2}), λ ⩾ 0 . \end{matrix}$ It is not difficult to check that $\begin{matrix} λ \in σ (Θ_{r}) ⟺ λ \in σ (H_{r, λ}) . \end{matrix}$ We show that the Schrödinger operators $H_{r, λ}$ converge (as $r \to 0 +$ ) in the norm resolvent sense to Hamiltonians with point interactions supported on $Y + Λ$ . This convergence result is already demonstrated in a more general setting in [9], but for our special form of $w_{r}$ we provide a refined analysis of the approximation including an estimate for the order of convergence. For similar results in the case of a single point interaction in $R^{2}$ and in other space dimensions see [2,4,27] or the monograph [3] and the references therein. Using the known spectral properties of these limit operators with point interactions and continuity arguments one can prove that the initially given number $λ_{n}$ belongs to a gap of $σ (H_{r, λ_{n}})$ , $n = 1, \dots, N$ , if the geometry of the crystal and the parameters in the definition of $w_{r}$ are chosen appropriately. This leads to the existence of gaps in $σ (Θ_{r})$ in the vicinities of $λ_{n}$ which is the first main result of this paper and whose proof is provided in Section 3.

Theorem 1.1.
There exist linearly independent vectors $a_{1}, a_{2} \in R^{2}$ and coefficients $c_{1}, \dots, c_{N}$ such that $\begin{matrix} ⋃_{n = 1}^{N} (λ_{n} - λ_{1} r^{2} | ln r |, λ_{n} + λ_{1} r^{2} | ln r |) \subset ρ (Θ_{r}) \end{matrix}$ for all sufficiently small $r > 0$ .

Concerning the analysis of $Υ_{r}$ , there are several works on similar divergence type operators with high contrast coefficients, see e.g. [25,46]. They have in common that the parameter becomes large on a domain whose diameter divided by the size of the period cell is constant and thus, these results do not apply in our setting. Other closely related results on the spectral analysis of divergence type operators with high contrast coefficients can be found e.g. in [14,21,32,45].

In our setting we use the Floquet–Bloch decomposition to show that any compact subinterval of $[0, \infty)$ is contained in the spectrum of $Υ_{r}$ for sufficiently small $r > 0$ . The proof of Theorem 1.2 is based on a standard technique in the analysis of differential operators with periodic coefficients. Nevertheless, to the best of our knowledge this result is not contained in the existing literature; hence, we provide its complete proof in Section 4.
Theorem 1.2.
For any $L > 0$ there exists $r_{0} = r_{0} (L) > 0$ such that $[0, L] \subset σ (Υ_{r})$ for all $0 < r < r_{0}$ .

We conclude this section by a discussion and interpretation of our results. According to Theorem 1.1, for a given set of pairwise distinct frequencies $ω_{1}, \dots, ω_{N} \in (0, + \infty)$ there exists a geometry of the crystal (a suitable period cell and coefficients $c_{1}, \dots, c_{N}$ ) such that $\begin{matrix} {{(\frac{ω_{1}}{c})}^{2}, {(\frac{ω_{2}}{c})}^{2}, \dots, {(\frac{ω_{N}}{c})}^{2}} \subset ρ (Θ_{r}), \end{matrix}$ if the diameter of the rods (related to $r > 0$ ) is sufficiently small. Moreover, we have an estimate for the size of the gap around each $ω_{n}$ . In particular, our results demonstrate a way how to construct photonic crystals such that $TM$ -modes with frequencies in the vicinities of $ω_{n}$ , $n = 1, \dots, N$ , can not propagate through it. At the same time, in view of Theorem 1.2 there are no gaps in compact subintervals of the spectrum of $Υ_{r}$ for small $r > 0$ . Restricting the frequencies to certain ranges, as it is typically the case in applications, there exists an $r_{0} > 0$ such that for any frequency in this range there is a $TE$ -mode with this frequency which can propagate through the crystal for any $r \in (0, r_{0})$ . These results perfectly match the experimental data and numerical tests in [28,29,39] performed for the special case of the square lattice and $N = 1$ .
1.4. Organization of the paper

In Section 2 we introduce Schrödinger operators with point interactions supported on a lattice and collect some results about their spectra. These results are employed in the spectral analysis of $Θ_{r}$ in Section 3. Next, the operator $Υ_{r}$ is investigated in Section 4. Finally, the Appendix contains the technical analysis of the convergence of $H_{r, λ}$ in the norm resolvent sense to a Schrödinger operator with point interactions supported on a lattice.

2. Schrödinger operators with point interactions supported on lattices

In this preliminary section we fix some notations that are associated to lattices of points. Furthermore, we introduce Schrödinger operators with point interactions supported on a lattice and discuss their spectral properties. These preparations will be useful in the spectral analysis of $Θ_{r}$ .

Let two linearly independent vectors $a_{1}, a_{2} \in R^{2}$ be given and let the lattice Λ and the period cell $\hat{Γ}$ be defined by (1.2). Next, we introduce the associated dual lattice Γ by $\begin{matrix} (2.1) & Γ : = {n_{1} b_{1} + n_{2} b_{2} \in R^{2} : n_{1}, n_{2} \in Z}, \end{matrix}$ where $b_{1}, b_{2} \in R^{2}$ are defined via $a_{m} b_{l} = 2 π δ_{m l}$ for $m, l = 1, 2$ . The Brillouin zone $\hat{Λ} \subset R^{2}$ corresponding to the lattice Λ is defined by $\begin{matrix} (2.2) & \hat{Λ} : = {s_{1} b_{1} + s_{2} b_{2} \in R^{2} : s_{1}, s_{2} \in [- \frac{1}{2}, \frac{1}{2}]} . \end{matrix}$ In what follows we are going to discuss Hamiltonians with point interactions supported on Λ following the lines of [3, Sec. III.4]. Let $- Δ$ be the self-adjoint free Laplacian in $L^{2} (R^{2})$ with the domain $dom (- Δ) = H^{2} (R^{2})$ . Its resolvent is denoted by $R_{0} (ν) : = {(- Δ - ν)}^{- 1}$ . For $ν \in ρ (- Δ) = C ∖ [0, \infty)$ the integral kernel $G_{ν}$ of $R_{0} (ν)$ is given by $\begin{matrix} (2.3) & G_{ν} (x - y) = \frac{i}{4} H_{0}^{(1)} (\sqrt{ν} | x - y |), \end{matrix}$ where $Im \sqrt{ν} > 0$ and $H_{0}^{(1)}$ is the Hankel function of the first kind and order zero; cf. [1, Chap. 9] for details on Hankel functions. Next, we set $\begin{matrix} (2.4) & {\tilde{G}}_{ν} (x) = \{\begin{matrix} G_{ν} (x), & x \neq 0, \\ 0, & x = 0 . \end{matrix} \end{matrix}$ For $α \in R$ and $m, l \in Z^{2}$ we define $\begin{matrix} q_{α, Λ}^{m l} (ν) : = (α - \frac{1}{2 π} (γ - ln \frac{\sqrt{ν}}{2 i})) δ_{m l} - {\tilde{G}}_{ν} (y_{m} - y_{l}), \end{matrix}$ where $γ = 0.5772 \dots$ is the Euler–Mascheroni constant and $y_{p} = p_{1} a_{1} + p_{2} a_{2}$ for $p = (p_{1}, p_{2}) \in Z^{2}$ . Eventually, we introduce for $ν \in C ∖ R$ the matrix $\begin{matrix} (2.5) & Q_{α, Λ} (ν) : = {q_{α, Λ}^{m l} (ν)}_{m, l \in Z^{2}}, \end{matrix}$ which induces a closed operator in $ℓ^{2} (Z^{2})$ that admits a bounded and everywhere defined inverse, if $Im \sqrt{ν}$ is sufficiently large; cf. [3, Thm. III.4.1]. We denote this operator again by $Q_{α, Λ} (ν)$ as no confusion will arise. The matrix elements of the inverse $Q_{α, Λ} {(ν)}^{- 1}$ in $ℓ^{2} (Z^{2})$ are denoted by $r_{α, Λ}^{m l} (ν)$ .

Definition 2.1.
The Schrödinger operator $- Δ_{α, Λ}$ with point interactions supported on Λ with coupling constant $α \in R$ is defined as the self-adjoint operator in $L^{2} (R^{2})$ with the resolvent $\begin{matrix} (2.6) & R_{α, Λ} (ν) : = {(- Δ_{α, Λ} - ν)}^{- 1} = R_{0} (ν) + \sum_{m, l \in Z^{2}} r_{α, Λ}^{m l} (ν) {(\cdot, \overline{G_{ν} (\cdot - y_{l})})}_{L^{2} (R^{2})} G_{ν} (\cdot - y_{m}), \end{matrix}$ where $ν \in C ∖ R$ and $y_{p} = p_{1} a_{1} + p_{2} a_{2}$ for $p = (p_{1}, p_{2}) \in Z^{2}$ .
Remark 2.2.
The operator $- Δ_{α, Λ}$ formally corresponds to a Hamiltonian with δ-potentials of strength α supported on Λ as it is often used in the physical literature, i.e. $\begin{matrix} (2.7) & - Δ_{α, Λ} = - Δ + λ (α) \sum_{y \in Λ} δ_{y} \end{matrix}$ with a coupling constant $λ (α)$ which is a re-normalization of α; cf. [3]. We would like to point out that one has to be careful with such an interpretation. In fact, we will see in Theorem 3.1 below that $- Δ_{α, Λ}$ can be approximated by Schrödinger operators with squeezed potentials, but these potentials do not converge to $α \sum_{y \in Λ} δ_{y}$ in the sense of distributions.

Next, we are going to investigate the spectrum of $- Δ_{α, Λ}$ . For this purpose, we introduce for $α \in R$ the numbers $E_{j} = E_{j} (α, Λ)$ , $j \in {0, 1, 2}$ , as follows: $E_{0}$ is the smallest zero of the function1
¹
Equation (2.8) differs from the condition in [3, Eq. (4.42) in Sec. III.4], as the term $\frac{1}{2 π} ln 2$ is missing there (it disappeared in the convergence analysis in [3, Eq. (4.29) in Sec. III.4]).

$\begin{matrix} (2.8) & E \mapsto g (E, 0) + \frac{1}{2 π} (γ + ln 2) - α, \end{matrix}$ where $g (E, θ)$ is defined for $θ \in \hat{Λ}$ and $E \notin {| x + θ |^{2} : x \in Γ}$ by $\begin{matrix} g (E, θ) : = \frac{1}{4 π^{2}} lim_{R \to \infty} [\sum_{x \in Γ : | x + θ | ⩽ R} \frac{| \hat{Λ} |}{| x + θ |^{2} - E} - 2 π ln R] . \end{matrix}$ Similarly, the number $E_{1}$ is given by the smallest zero of the function $\begin{matrix} (2.9) & E \mapsto g (E, θ_{0}) + \frac{1}{2 π} (γ + ln 2) - α, \end{matrix}$ where $θ_{0} : = - \frac{1}{2} (b_{1} + b_{2})$ . Eventually, let $b_{-} \in {b_{1}, b_{2}}$ be a vector satisfying $| b_{-} | = min {| b_{1} |, | b_{2} |}$ . Then, we set $E_{2} : = min {\tilde{E}, \frac{1}{4} | b_{-} |^{2}}$ , where $\tilde{E}$ is the smallest positive solution of equation (2.8).2
²
Note that $\tilde{E}$ is equal to $E_{b_{-}}^{α, Λ} (0)$ in the notation of [3, Sec. III.4]. The fact that $E_{b_{-}}^{α, Λ} (0)$ is the smallest positive solution of equation (2.8) can be shown in the same way as in the proof of [3, Thm. III.1.4.4]. Observe that (2.9) is modified similarly as (2.8) compared to [3].

All the numbers $E_{0}$ , $E_{1}$ , and $E_{2}$ are well defined; cf. [3, Sec. III.4]. In the next proposition we summarize some fundamental spectral properties of $- Δ_{α, Λ}$ that can be found e.g. in [3, Thm. III.4.7].
Proposition 2.3.
Let $α \in R$ and Λ be as in ( 1.2 ). Let the Schrödinger operator $- Δ_{α, Λ}$ be as in Definition 2.1 and let $θ_{0}$ , $g (\cdot, \cdot)$ and $E_{j} = E_{j} (α, Λ)$ , $j = 0, 1, 2$ , be as above. Then the following claims hold.
$σ (- Δ_{α, Λ}) = [E_{0}, E_{1}] \cup [E_{2}, \infty)$ .

$E_{0} < 0$ and $E_{2} > 0$ for all $α \in R$ .

$E_{1} < 0$ if and only if $α < g (0, θ_{0}) + \frac{1}{2 π} (γ + ln 2)$ . 3
³
This condition differs from Eq. (4.51) in [3, Thm. III.4.7], the term $\frac{1}{2 π} (γ + ln 2)$ is missing; but it should be there; cf. [3, Eqs (4.29) and (4.42) in Sec. III.4].

There exists an $α_{1} = α_{1} (Λ) \in R$ such that $E_{2} ⩽ E_{1}$ for any $α ⩾ α_{1}$ . In particular, $σ (- Δ_{α, Λ}) = [E_{0}, + \infty)$ holds for all $α ⩾ α_{1}$ .

By Proposition 2.3 the operator $- Δ_{α, Λ}$ has a gap in its spectrum, if the interaction strength is chosen in a proper way. In the rest of this section, we are going to investigate this gap in more detail. In particular, we will show that for a given compact interval $[a, b] \subset R$ there exist a lattice Λ and an interaction strength α such that $[a, b]$ is contained in the spectral gap of $- Δ_{α, Λ}$ . To this aim we introduce for $k > 0$ the unitary scaling operator $\begin{matrix} U_{k} : L^{2} (R^{2}) \to L^{2} (R^{2}), (U_{k} f) (x) : = k^{- 1} f (k^{- 1} x) . \end{matrix}$ Its inverse $U_{k}^{- 1} : L^{2} (R^{2}) \to L^{2} (R^{2})$ clearly acts as $(U_{k}^{- 1} f) (x) = k f (k x)$ . In the next proposition we show that this rescaling yields, up to multiplication with a constant, a unitary equivalence between point interaction operators with suitably modified geometries of lattices and strengths of interactions.
Proposition 2.4.
Let $α \in R$ and Λ be as in ( 1.2 ). For $k > 0$ set $Λ_{k} : = k^{- 1} Λ$ and $α_{k} : = α - \frac{ln k}{2 π}$ . Let the Schrödinger operators $- Δ_{α, Λ}$ and $- Δ_{α_{k}, Λ_{k}}$ be as in Definition 2.1 . Then it holds $\begin{matrix} U_{k}^{- 1} (- Δ_{α, Λ}) U_{k} = k^{- 2} (- Δ_{α_{k}, Λ_{k}}) . \end{matrix}$
Proof.
Let $ν \in C ∖ R$ . We show $\begin{matrix} U_{k}^{- 1} R_{α, Λ} (ν) U_{k} = k^{2} R_{α_{k}, Λ_{k}} (k^{2} ν), \end{matrix}$ which yields then the claim. By (2.6) it holds $\begin{matrix} U_{k}^{- 1} R_{α, Λ} (ν) U_{k} = U_{k}^{- 1} R_{0} (ν) U_{k} + \sum_{m, l \in Z^{2}} r_{α, Λ}^{m l} (ν) {(\cdot, U_{k}^{- 1} \overline{G_{ν} (\cdot - y_{l})})}_{L^{2} (R^{2})} U_{k}^{- 1} G_{ν} (\cdot - y_{m}) . \end{matrix}$ Since $U_{k}^{- 1} (- Δ - ν) U_{k} = k^{- 2} (- Δ - k^{2} ν)$ , we get $\begin{matrix} (2.10) & U_{k}^{- 1} R_{0} (ν) U_{k} = k^{2} R_{0} (k^{2} ν) . \end{matrix}$ Using the definition of $U_{k}^{- 1}$ we obtain for any $y \in Λ$ the relation $\begin{matrix} (2.11) & U_{k}^{- 1} G_{ν} (\cdot - y) = k G_{k^{2} ν} (\cdot - k^{- 1} y) \end{matrix}$ almost everywhere in $R^{2}$ . This implies $\begin{matrix} {(\cdot, U_{k}^{- 1} \overline{G_{ν} (\cdot - y)})}_{L^{2} (R^{2})} = {(\cdot, k \overline{G_{k^{2} ν} (\cdot - k^{- 1} y)})}_{L^{2} (R^{2})} . \end{matrix}$ Eventually, a straightforward calculation yields $\begin{array}{l} q_{α, Λ}^{m l} (ν) & = (α - \frac{1}{2 π} (γ - ln \frac{\sqrt{ν}}{2 i})) δ_{m l} - {\tilde{G}}_{ν} (y_{m} - y_{l}) \\ (2.12) & = (α_{k} - \frac{1}{2 π} (γ - ln \frac{k \sqrt{ν}}{2 i})) δ_{m l} - {\tilde{G}}_{k^{2} ν} (k^{- 1} (y_{m} - y_{l})) = q_{α_{k}, Λ_{k}}^{m l} (k^{2} ν) . \end{array}$ Hence, the identity $r_{α, Λ}^{m l} (ν) = r_{α_{k}, Λ_{k}}^{m l} (k^{2} ν)$ follows. Finally, employing (2.10), (2.11) and (2.12) we get $\begin{array}{l} U_{k}^{- 1} R_{α, Λ} (ν) U_{k} & = U_{k}^{- 1} R_{0} (ν) U_{k} + \sum_{m, l \in Z^{2}} r_{α, Λ}^{m l} (ν) {(\cdot, U_{k}^{- 1} \overline{G_{ν} (\cdot - y_{l})})}_{L^{2} (R^{2})} U_{k}^{- 1} G_{ν} (\cdot - y_{m}) \\ = k^{2} R_{0} (k^{2} ν) + k^{2} \sum_{m, l \in Z^{2}} r_{α_{k}, Λ_{k}}^{m l} (k^{2} ν) {(\cdot, \overline{G_{k^{2} ν} (\cdot - k^{- 1} y_{l})})}_{L^{2} (R^{2})} G_{k^{2} ν} (\cdot - k^{- 1} y_{m}) \\ = k^{2} R_{α_{k}, Λ_{k}} (k^{2} ν) . \end{array}$ □

The following useful statement follows immediately from Propositions 2.3 and 2.4.
Proposition 2.5.
Let $a, b \in R$ with $a < b$ be given. Then there exists a lattice Λ and a coupling $α \in R$ such that the interval $[a, b]$ is contained in a gap of the spectrum of the Schrödinger operator $- Δ_{α, Λ}$ in Definition 2.1 , i.e. $\begin{matrix} [a, b] \subset ρ (- Δ_{α, Λ}) . \end{matrix}$
Proof.
According to Proposition 2.3 one can find a lattice $Λ_{0} = {n_{1} a_{1} + n_{2} a_{2} \in R^{2} : n_{1}, n_{2} \in Z}$ and a coupling constant $α_{0} \in R$ such that $\begin{matrix} 0 \notin σ (- Δ_{α_{0}, Λ_{0}}) = [E_{0}, E_{1}] \cup [E_{2}, \infty) . \end{matrix}$ Furthermore, by Proposition 2.4 it holds for any $k > 0$ $\begin{matrix} σ (- Δ_{α_{k}, Λ_{k}}) = k^{2} σ (- Δ_{α_{0}, Λ_{0}}) = [k^{2} E_{0}, k^{2} E_{1}] \cup [k^{2} E_{2}, \infty), \end{matrix}$ where $α_{k} = α_{0} - \frac{1}{2 π} ln k$ and $Λ_{k} : = k^{- 1} Λ_{0}$ . It remains to choose the parameter $k > 0$ so large that $k^{2} E_{1} < a < b < k^{2} E_{2}$ . Then the lattice $Λ = Λ_{k}$ and the coupling coefficient $α = α_{k}$ fulfill all the requirements. □

Finally, we define Schrödinger operators with point interactions supported on a shifted lattice. For this purpose we introduce for $y \in R^{2}$ the unitary translation operator $T_{y} : L^{2} (R^{2}) \to L^{2} (R^{2})$ acting as $(T_{y} f) (x) : = f (x - y)$ . Then $\begin{matrix} (2.13) & - Δ_{α, y + Λ} : = T_{y}^{- 1} (- Δ_{α, Λ}) T_{y} \end{matrix}$ is the Schrödinger operator with point interactions supported on $y + Λ$ . Since $T_{y}$ is a unitary operator, we have $σ (- Δ_{α, y + Λ}) = σ (- Δ_{α, Λ})$ .
3. Spectral analysis of the operator $Θ_{r}$

This section is devoted to the proof of Theorem 1.1 on the operator $Θ_{r}$ defined in (1.6a). Since the spectrum of $Θ_{r}$ is still difficult to investigate, we consider instead the spectral problem for the auxiliary family of Schrödinger operators $H_{r, λ}$ in (1.7). Since $w_{r}, w_{r}^{- 1} \in L^{\infty} (R^{2}; R)$ , the operator $H_{r, λ}$ is well defined and self-adjoint in $L^{2} (R^{2})$ and it holds that $λ \in σ (Θ_{r})$ if and only if $λ \in σ (H_{r, λ})$ for all $λ ⩾ 0$ .

Let the numbers $0 < λ_{1} < λ_{2} < \dots < λ_{N}$ be given. First, we prove that $H_{r, λ_{n}}$ , $n = 1, \dots, N$ , converges in the norm resolvent sense to a Schrödinger operator with point interactions supported on $y^{(n)} + Λ$ . In view of the spectral properties of these Hamiltonians with point interactions (summarized in Section 2), it turns out that there exists a lattice Λ and constants $c_{1}, \dots, c_{N}$ (that appear in the definition of $w_{r}$ ) such that $λ_{n}$ belongs to a gap of $σ (H_{r, λ_{n}})$ . Finally, employing a perturbation argument, we deduce the claim of Theorem 1.1.

The following theorem treats the convergence of $H_{r, λ}$ to a Schrödinger operator with point interactions. Since the proof of this statement is rather long and technical, it is postponed to the Appendix.

Theorem 3.1.
Let $H_{r, λ}$ , $λ ⩾ 0$ , and $- Δ_{α, y + Λ}$ , $α \in R$ , $y \in R^{2}$ , be defined as in ( 1.7 ) and in ( 2.13 ), respectively, and let $ν \in C ∖ R$ . Then the following claims hold.
There exists a constant $κ = κ (Λ, λ_{1}, \dots, λ_{N}, Ω, ν) > 0$ such that for any $n \in {1, \dots, N}$ and all sufficiently small $r > 0$ $\begin{matrix} ‖ {(H_{r, λ_{n}} - ν)}^{- 1} - {(- Δ_{α_{n}, y^{(n)} + Λ} - ν)}^{- 1} ‖ ⩽ κ | ln r |^{- 1}, \end{matrix}$ where the coefficient $α_{n}$ is given by $\begin{matrix} (3.1) & α_{n} = - c_{n} \frac{λ_{n} | Ω |}{4 π^{2}} + \frac{C}{2 π | Ω |^{2}} with C = \int_{Ω} \int_{Ω} ln | x - z | d x d z . \end{matrix}$

For $λ \notin {λ_{1}, \dots, λ_{N}}$ there exists a constant $κ^{'} = κ^{'} (Λ, λ_{1}, \dots, λ_{N}, λ, Ω, ν) > 0$ such that for all sufficiently small $r > 0$ $\begin{matrix} ‖ {(H_{r, λ} - ν)}^{- 1} - {(- Δ - ν)}^{- 1} ‖ ⩽ κ^{'} | ln r |^{- 1} . \end{matrix}$

Remark 3.2.
The assumption $λ_{n} \neq λ_{m}$ for $n \neq m$ is motivated by our application, but it is only technical. If we drop this assumption, then one can still prove convergence of $H_{r, λ_{n}}$ to a Schrödinger operator with point interactions supported on a more complicated lattice with (in general) non-constant interaction strength; cf. [9]. However, in this case the spectral analysis of the limit operator is a rather difficult problem. For special interesting geometries there are results available in the literature [36].

Combining the statements of Theorem 3.1 and of Proposition 2.5 with the perturbation result [43, Satz 9.24 b)], we obtain the following claim on the spectrum of $H_{r, λ_{n}}$ .
Proposition 3.3.
Let $0 < λ_{1} < λ_{2} < \dots < λ_{N}$ , let $a > 0$ be fixed and define $η : = \frac{2 π}{| Ω |} + 1$ . Let the operator $H_{r, λ_{n}}$ be as in ( 1.7 ). Then there exist a lattice Λ and constants $c_{1}, \dots, c_{N}$ (that appear in ( 1.5 )) such that $\begin{matrix} (λ_{n} - η - a, λ_{n} + η + a) \subset ρ (H_{r, λ_{n}}) \end{matrix}$ for all sufficiently small $r > 0$ and all $n \in {1, \dots, N}$ .
Proof.
Set $I_{n} : = (λ_{n} - η - a, λ_{n} + η + a)$ and $J_{n} : = (λ_{n} - η - 2 a, λ_{n} + η + 2 a)$ . Then, by Proposition 2.5 there exist a lattice Λ and a coupling constant $α \in R$ such that $\begin{matrix} (λ_{1} - η - 2 a, λ_{N} + η + 2 a) \subset ρ (- Δ_{α, Λ}) . \end{matrix}$ This implies, in particular, that for any $n \in {1, \dots, N}$ $\begin{matrix} {\overline{I}}_{n} \subset J_{n} \subset ρ (- Δ_{α, y^{(n)} + Λ}) = ρ (- Δ_{α, Λ}), \end{matrix}$ where the last equation holds due to translational invariance. Next, choose the constants $c_{n}$ in (1.5) as $\begin{matrix} (3.2) & c_{n} = \frac{4 π^{2}}{λ_{n} | Ω |} (\frac{C}{2 π | Ω |^{2}} - α), \end{matrix}$ where C is given as in (3.1). Theorem 3.1(i) implies that $H_{r, λ_{n}}$ converges in the norm resolvent sense to $- Δ_{α, y^{(n)} + Λ}$ . Finally, let $E : = E (I_{n})$ and $E_{r} : = E_{r} (I_{n})$ be the spectral projections corresponding to the interval $I_{n}$ and the operators $- Δ_{α, y^{(n)} + Λ}$ and $H_{r, λ_{n}}$ , respectively. Since $H_{r, λ_{n}}$ converges in the norm resolvent sense to $- Δ_{α, y^{(n)} + Λ}$ , it follows from [43, Satz 9.24 b)] that $‖ E - E_{r} ‖ < 1$ for all sufficiently small $r > 0$ . Hence, employing [43, Satz 2.58 a)] we conclude $\begin{matrix} dim ran E_{r} = dim ran E = 0 \end{matrix}$ for all sufficiently small $r > 0$ . This implies $I_{n} \subset ρ (H_{r, λ_{n}})$ . □

Now, we are prepared to prove the main result about the spectrum of $Θ_{r}$ .
Proof of Theorem 1.1.
Let $a > 0$ be given. Set $η : = 2 π | Ω |^{- 1} + 1$ and $I_{n} : = (λ_{n} - η - a, λ_{n} + η + a)$ . Choose a lattice Λ and the constants $c_{1}, \dots, c_{N}$ (that appear in (1.5)) such that $I_{n} \subset ρ (H_{r, λ_{n}})$ for all sufficiently small $r > 0$ , which is possible by Proposition 3.3. Recall that $λ \in ρ (Θ_{r})$ if and only if $λ \in ρ (H_{r, λ})$ ; we are going to verify this property for λ belonging to a small neighborhood of $λ_{n}$ . Since $I_{n} \subset ρ (H_{r, λ_{n}})$ , it follows from the spectral theorem that $\begin{matrix} (3.3) & {‖ (H_{r, λ_{n}} - ν) f ‖}_{L^{2} (R^{2})} ⩾ η ‖ f ‖_{L^{2} (R^{2})} \end{matrix}$ for all $ν \in (λ_{n} - a, λ_{n} + a)$ and all $f \in H^{2} (R^{2})$ . Note that the definition of $w_{r}$ in (1.5) implies $\begin{matrix} ‖ w_{r} - 1 ‖_{L^{\infty}} ⩽ \frac{η}{λ_{1}} \frac{1}{r^{2} | ln r |} \end{matrix}$ for $r > 0$ small enough. Therefore, it holds for $ζ \in R$ with $| ζ | < λ_{1} r^{2} | ln r |$ that $\begin{matrix} (3.4) & | ζ | ‖ w_{r} - 1 ‖_{L^{\infty}} < λ_{1} r^{2} | ln r | \cdot \frac{η}{λ_{1}} \frac{1}{r^{2} | ln r |} = η . \end{matrix}$ For small enough $r > 0$ we have $| ζ | < λ_{1} r^{2} | ln r | < a$ and the estimate (3.3) implies for $f \in H^{2} (R^{2})$ $\begin{array}{l} {‖ H_{r, λ_{n} + ζ} f - (λ_{n} + ζ) f ‖}_{L^{2} (R^{2})} & = {‖ (- Δ - (λ_{n} + ζ) (w_{r} - 1) - (λ_{n} + ζ)) f ‖}_{L^{2} (R^{2})} \\ ⩾ {‖ (H_{r, λ_{n}} - (λ_{n} + ζ)) f ‖}_{L^{2} (R^{2})} - {‖ ζ (w_{r} - 1) f ‖}_{L^{2} (R^{2})} \\ ⩾ (η - | ζ | ‖ w_{r} - 1 ‖_{L^{\infty}}) ‖ f ‖_{L^{2} (R^{2})} . \end{array}$ This and (3.4) imply $λ_{n} + ζ \in ρ (H_{r, λ_{n} + ζ})$ , which yields $λ_{n} + ζ \in ρ (Θ_{r})$ . □

We conclude this section with an explanation how to construct a crystal such that given numbers $0 < λ_{1} < λ_{2} < \dots < λ_{N}$ belong to gap(s) of $σ (Θ_{r})$ . First, for a given lattice $Λ_{0}$ we choose $α_{0} \in R$ such that $α_{0} < g (0, θ_{0}) + \frac{1}{2 π} (γ + ln 2)$ , where $θ_{0} = - \frac{1}{2} (b_{1} + b_{2})$ and $b_{1}$ and $b_{2}$ are the basis vectors of the dual lattice $Γ_{0}$ . By Proposition 2.3 it holds that $0 \in ρ (- Δ_{α_{0}, Λ_{0}})$ . Finding (or estimating) the smallest zeros of the function $\begin{matrix} E \mapsto g (E, θ) + \frac{1}{2 π} (γ + ln 2) - α_{0} \end{matrix}$ yields an approximation for the upper and the lower endpoints of the bands of the spectrum of $- Δ_{α_{0}, Λ_{0}}$ ; cf. Proposition 2.3. Next, choose $k > 0$ as in the proof of Proposition 2.5 such that $\begin{matrix} (λ_{1} - η - 2 a, λ_{N} + η + 2 a) \subset ρ (- Δ_{α_{k}, Λ_{k}}), \end{matrix}$ where $η = 2 π | Ω |^{- 1} + 1$ , $Λ_{k} : = k^{- 1} Λ_{0}$ , $α_{k} = α_{0} - \frac{ln k}{2 π}$ , and a is a small positive constant. Finally, we define the constants $c_{n}$ via the formula (3.2) in the proof of Proposition 3.3 (with α replaced by $α_{k}$ ). Then the crystal that is specified via the lattice $Λ_{k}$ and $w_{r}$ as in (1.5) satisfies ${λ_{1}, \dots, λ_{N}} \subset ρ (Θ_{r})$ for all sufficiently small $r > 0$ .
4. Spectral analysis of the operator $Υ_{r}$

In this section we prove that there are no gaps in the spectrum of the operator $Υ_{r} = - div (w_{r}^{- 1} grad f)$ in bounded subsets of $[0, \infty)$ , if $r > 0$ is sufficiently small. The methods employed in this section are completely different from the methods in Section 3, partly because the aim is to prove the statement of an opposite type. Using the Floquet–Bloch theory for differential operators with periodic coefficients we will see that $σ (Υ_{r})$ consists of bands and that the ‘lowest bands’ overlap for small $r > 0$ . The proof of this result is inspired by ideas coming from [21] and makes additionally use of a result in [41] on the convergence of eigenvalues of the Laplace operator on domains with small holes.

First, we set up some notations. For a fixed $r ⩾ 0$ we define the sesquilinear form $\begin{matrix} h_{r} [f, g] : = {(w_{r}^{- 1} \nabla f, \nabla g)}_{L^{2} (R^{2}; C^{2})}, dom h_{r} : = H^{1} (R^{2}), \end{matrix}$ with $w_{r}$ given by (1.5) for $r > 0$ and $w_{r} \equiv 1$ for $r = 0$ . It is clear that $h_{r}$ is well defined and symmetric. Moreover, by the definition of $w_{r}$ , there exists for any sufficiently small fixed $r ⩾ 0$ a constant $κ_{r} \in (0, 1]$ such that $\begin{matrix} 0 ⩽ κ_{r} ‖ \nabla f ‖_{L^{2} (R^{2}; C^{2})}^{2} ⩽ h_{r} [f] ⩽ ‖ \nabla f ‖_{L^{2} (R^{2}; C^{2})}^{2} \end{matrix}$ for all $f \in H^{1} (R^{2})$ . This implies that $h_{r}$ is closed. Thus, by the first representation theorem [31, Thm. VI 2.1] there exists a uniquely determined self-adjoint operator associated to the form $h_{r}$ , which is $Υ_{r}$ as in (1.6b) for $r > 0$ and the free Laplacian $- Δ$ for $r = 0$ .

In order to describe the spectrum of $Υ_{r}$ , $r ⩾ 0$ , we use that its coefficients are periodic with respect to the lattice Λ given by (1.2). Let the period cell $\hat{Γ}$ and the Brillouin zone $\hat{Λ}$ associated to Λ be given by (1.2) and (2.2), respectively, and define for $θ \in \hat{Λ}$ the subspace $H (θ)$ of $L^{2} (\hat{Γ})$ as the set of all $f \in H^{1} (\hat{Γ})$ that satisfy the so-called semi-periodic boundary conditions, i.e. $\begin{matrix} H (θ) : = {f \in H^{1} (\hat{Γ}) : f (t a_{2}) = e^{- i θ a_{1}} f (t a_{2} + a_{1}), f (t a_{1}) = e^{- i θ a_{2}} f (t a_{1} + a_{2}), t \in [0, 1)} . \end{matrix}$ Defining now for $r ⩾ 0$ and $θ \in \hat{Λ}$ the form $\begin{matrix} h_{r, θ} [f, g] : = {(w_{r}^{- 1} \nabla f, \nabla g)}_{L^{2} (\hat{Γ}; C^{2})}, dom h_{r, θ} : = H (θ), \end{matrix}$ we see, similarly as above, that it satisfies the assumptions of the first representation theorem. Hence, there exists a uniquely determined self-adjoint operator $Υ_{r, θ}$ in $L^{2} (\hat{Γ})$ associated to $h_{r, θ}$ .

It is not difficult to see that for all $θ \in \hat{Λ}$ the operator $Υ_{r, θ}$ , $r ⩾ 0$ , has a compact resolvent. Hence, its spectrum is purely discrete and we denote its eigenvalues (counted with multiplicity) by $\begin{matrix} 0 ⩽ λ_{r, 1} (θ) ⩽ λ_{r, 2} (θ) ⩽ λ_{r, 3} (θ) ⩽ \dots . \end{matrix}$ Since $w_{r}^{- 1}$ is periodic, we can apply the results from [11, Sec. 4] (cf. also the footnote on p. 3 of [11]) and get, combined with [15] or [34, Thm. 5.9], the following characterization for the spectrum of $Υ_{r}$ .

Proposition 4.1.

For any $r ⩾ 0$ it holds $\begin{matrix} σ (Υ_{r}) = ⋃_{n = 1}^{\infty} [a_{r, n}, b_{r, n}], a_{r, n} : = min_{θ \in \hat{Λ}} λ_{r, n} (θ), b_{r, n} : = max_{θ \in \hat{Λ}} λ_{r, n} (θ) . \end{matrix}$ Moreover, for all $n_{0} \in N$ there exists $β = β (n_{0}) \in (0, 1)$ such that $b_{0, n} > (1 + β) a_{0, n + 1}$ holds for $n = 1, 2, \dots, n_{0}$ .

Our goal is to show that for sufficiently small $r > 0$ the relation $b_{r, n} > a_{r, n + 1}$ persists. For this purpose, we need the following auxiliary lemma, which provides a useful estimate for the $L^{2}$ -norm of a function in a finite union of disks with radius r in terms of the $H^{1}$ -norm over the whole domain. In what follows it will be convenient to use the notation $\begin{matrix} (4.1) & B_{s} : = Y + B_{s} (0), s > 0, \end{matrix}$ where Y is as in (1.3).

Lemma 4.2.

Let $r > 0$ be sufficiently small and let $B_{r}$ be as in ( 4.1 ). Then, there exists a constant $κ > 0$ such that $\begin{matrix} ‖ f ‖_{L^{2} (B_{r})}^{2} ⩽ κ r ‖ f ‖_{H^{1} (\hat{Γ} ∖ B_{r})}^{2} + κ r^{2} ‖ f ‖_{H^{1} (\hat{Γ})}^{2} \end{matrix}$ holds for all $f \in H^{1} (\hat{Γ})$ .

Proof.

Throughout the proof $κ > 0$ denotes a generic constant. Choose $R > 0$ so small that $\begin{matrix} B_{R}^{(n)} : = \overline{B_{R} (y^{(n)})} \subset \hat{Γ} for all n \in {1, \dots, N} and B_{R}^{(n)} \cap B_{R}^{(m)} = \emptyset for n \neq m . \end{matrix}$ Moreover, assume also that $r \in (0, R)$ . For fixed $n \in {1, \dots, N}$ we are going to prove the inequality $\begin{matrix} (4.2) & ‖ f ‖_{L^{2} (B_{r}^{(n)})}^{2} ⩽ κ r ‖ f ‖_{H^{1} (\hat{Γ} ∖ B_{r})}^{2} + κ r^{2} ‖ f ‖_{H^{1} (\hat{Γ})}^{2} . \end{matrix}$ By summing up over n the claimed result follows.

We denote by $cl (\hat{Γ})$ the closure of $\hat{Γ}$ . Since $C^{\infty} (cl (\hat{Γ}))$ is dense in $H^{1} (\hat{Γ})$ , it suffices to prove (4.2) for smooth functions. Let $f \in C^{\infty} (cl (\hat{Γ}))$ be fixed. We use for its equivalent in polar coordinates $(ρ, ϕ)$ centered at $y^{(n)}$ the symbol $f (ρ, ϕ) : = f (y_{1}^{(n)} + ρ cos ϕ, y_{2}^{(n)} + ρ sin ϕ)$ , where $y_{1}^{(n)}$ and $y_{2}^{(n)}$ denote the coordinates of $y^{(n)}$ . Let $ρ > 0$ be such that $ρ < r$ . Employing the main theorem of calculus we conclude $\begin{matrix} f (ρ, ϕ) = f (r, ϕ) - \int_{ρ}^{r} (\partial_{t} f) (t, ϕ) d t . \end{matrix}$ This implies $\begin{matrix} (4.3) & ρ \int_{0}^{2 π} {| f (ρ, ϕ) |}^{2} d ϕ ⩽ 2 ρ \int_{0}^{2 π} {| f (r, ϕ) |}^{2} d ϕ + 2 ρ \int_{0}^{2 π} {| \int_{ρ}^{r} (\partial_{t} f) (t, ϕ) d t |}^{2} d ϕ . \end{matrix}$ Similarly as above, one finds $\begin{matrix} (4.4) & ρ \int_{0}^{2 π} {| f (r, ϕ) |}^{2} d ϕ ⩽ 2 ρ \int_{0}^{2 π} {| f (R, ϕ) |}^{2} d ϕ + 2 ρ \int_{0}^{2 π} {| \int_{r}^{R} (\partial_{t} f) (t, ϕ) d t |}^{2} d ϕ . \end{matrix}$ By the trace theorem [38, Thm. 3.37] applied for the domain $\hat{Γ} ∖ B_{R}$ and using that $\hat{Γ} ∖ B_{R} \subset \hat{Γ} ∖ B_{r}$ we get $\begin{array}{l} \int_{0}^{2 π} {| f (R, ϕ) |}^{2} ρ d ϕ & ⩽ \int_{0}^{2 π} {| f (R, ϕ) |}^{2} R d ϕ = ‖ f |_{\partial B_{R}^{(n)}} ‖_{L^{2} (\partial B_{R}^{(n)})}^{2} \\ (4.5) & ⩽ κ ‖ f ‖_{H^{1} (\hat{Γ} ∖ B_{R})}^{2} ⩽ κ ‖ f ‖_{H^{1} (\hat{Γ} ∖ B_{r})}^{2}, \end{array}$ where the constant κ depends only on R and is independent of r. Using the expression for the gradient in polar coordinates and the Cauchy–Schwarz inequality, we obtain $\begin{array}{l} ρ \int_{0}^{2 π} {| \int_{r}^{R} (\partial_{t} f) (t, ϕ) d t |}^{2} d ϕ & ⩽ r R \int_{0}^{2 π} \int_{r}^{R} {| (\partial_{t} f) (t, ϕ) |}^{2} d t d ϕ \\ ⩽ R \int_{0}^{2 π} \int_{r}^{R} \frac{r}{t} {| (\nabla f) (y_{1}^{(n)} + t cos ϕ, y_{2}^{(n)} + t sin ϕ) |}^{2} t d t d ϕ \\ (4.6) & ⩽ κ ‖ f ‖_{H^{1} (\hat{Γ} ∖ B_{r})}^{2} . \end{array}$ Equations (4.4), (4.5) and (4.6) imply $\begin{matrix} (4.7) & ρ \int_{0}^{2 π} {| f (r, ϕ) |}^{2} d ϕ ⩽ κ ‖ f ‖_{H^{1} (\hat{Γ} ∖ B_{r})}^{2} \end{matrix}$ with κ depending just on R. In a similar way as in (4.6) one shows $\begin{array}{l} ρ \int_{0}^{2 π} {| \int_{ρ}^{r} (\partial_{t} f) (t, ϕ) d t |}^{2} d ϕ & ⩽ r ρ \int_{0}^{2 π} \int_{ρ}^{r} {| (\partial_{t} f) (t, ϕ) |}^{2} d t d ϕ \\ (4.8) & = r \int_{0}^{2 π} \int_{ρ}^{r} \frac{ρ}{t} {| (\partial_{t} f) (t, ϕ) |}^{2} t d t d ϕ ⩽ r ‖ f ‖_{H^{1} (\hat{Γ})}^{2} . \end{array}$ Hence, integrating (4.3) from 0 to r with respect to ρ and using (4.7) and (4.8) we obtain $\begin{array}{l} ‖ f ‖_{L^{2} (B_{r}^{(n)})}^{2} & = \int_{0}^{r} \int_{0}^{2 π} {| f (ρ, ϕ) |}^{2} ρ d ϕ d ρ ⩽ κ \int_{0}^{r} (‖ f ‖_{H^{1} (\hat{Γ} ∖ B_{r})}^{2} + r ‖ f ‖_{H^{1} (\hat{Γ})}^{2}) d ρ \\ = κ r ‖ f ‖_{H^{1} (\hat{Γ} ∖ B_{r})}^{2} + κ r^{2} ‖ f ‖_{H^{1} (\hat{Γ})}^{2} . \end{array}$ □

After these preliminary considerations, we are prepared to show that $Υ_{r}$ has no gaps in the spectrum in any fixed compact subinterval of $[0, + \infty)$ , if $r > 0$ is sufficiently small.

Proof of Theorem 1.2.

Throughout this proof $κ, κ^{'}, κ^{″}, κ^{‴} > 0$ denote generic constants. As no confusion can arise, we will use the abbreviation $(\cdot, \cdot)$ for both scalar products ${(\cdot, \cdot)}_{L^{2} (\hat{Γ})}$ and ${(\cdot, \cdot)}_{L^{2} (\hat{Γ}; C^{2})}$ , and the shorthand $‖ \cdot ‖$ for the respective norms $‖ \cdot ‖_{L^{2} (\hat{Γ})}$ and $‖ \cdot ‖_{L^{2} (\hat{Γ}; C^{2})}$ .

Recall the definitions of the numbers $a_{0, n}$ and $b_{0, n}$ from Proposition 4.1. Fix $L > 0$ and let $n_{0} : = min {n \in N_{0} : b_{0, n} > L}$ . Furthermore, we choose $β \in (0, 1)$ such that $b_{0, n} > (1 + β) a_{0, n + 1}$ for all $n ⩽ n_{0}$ (note that such a β exists by Proposition 4.1). Using the min-max principle [42, Sec. XIII.1] we obtain $\begin{matrix} λ_{r, n} (θ) = min_{\begin{array}{c} V \subset H (θ) \\ dim V = n \end{array}} max_{\begin{array}{c} f \in V \\ ‖ f ‖ = 1 \end{array}} (w_{r}^{- 1} \nabla f, \nabla f) ⩽ min_{\begin{array}{c} V \subset H (θ) \\ dim V = n \end{array}} max_{\begin{array}{c} f \in V \\ ‖ f ‖ = 1 \end{array}} ‖ \nabla f ‖^{2} = λ_{0, n} (θ), for all θ \in \hat{Λ} . \end{matrix}$ Moreover, let the vectors $θ_{n}^{\pm} \in \hat{Λ}$ be such that $a_{0, n} = λ_{0, n} (θ_{n}^{-})$ and $b_{0, n} = λ_{0, n} (θ_{n}^{+})$ for all $n \in N$ . We aim to prove that $λ_{r, n + 1} (θ_{n + 1}^{-}) < λ_{r, n} (θ_{n}^{+})$ holds for all sufficiently small $r > 0$ and for any $n < n_{0}$ . In addition, we will show that $λ_{r, n_{0}} (θ_{n}^{+}) > L$ , which yields then the claim. Fix $n \in N$ such that $n < n_{0}$ . Choose an n-dimensional subspace $W^{+} = W^{+} (n, r) \subset H (θ_{n}^{+})$ such that $\begin{matrix} λ_{r, n} (θ_{n}^{+}) = min_{\begin{array}{c} V \subset H (θ_{n}^{+}) \\ dim V = n \end{array}} max_{\begin{array}{c} f \in V \\ ‖ f ‖ = 1 \end{array}} (w_{r}^{- 1} \nabla f, \nabla f) = max_{\begin{array}{c} f \in W^{+} \\ ‖ f ‖ = 1 \end{array}} (w_{r}^{- 1} \nabla f, \nabla f) . \end{matrix}$ Let $f \in W^{+}$ with $‖ f ‖ = 1$ and fix $R > 0$ such that $Ω \subset B_{R} (0)$ . Furthermore, we define $B : = B_{r R}$ as in (4.1); in particular, we have $Y + r Ω \subset B$ . Since $w_{r} \equiv 1$ on $\hat{Γ} ∖ B$ , we get $\begin{matrix} (4.9) & (w_{r}^{- 1} \nabla f, \nabla f) ⩾ ‖ \nabla f ‖_{L^{2} (\hat{Γ} ∖ B; C^{2})}^{2} . \end{matrix}$ Combining the inequalities $b_{0, n_{0}} ⩾ λ_{0, n} (θ_{n}^{+}) ⩾ λ_{r, n} (θ_{n}^{+})$ , the estimate (4.9), and Lemma 4.2 we obtain $\begin{array}{l} ‖ f ‖_{L^{2} (\hat{Γ} ∖ B)}^{2} & = ‖ f ‖^{2} - ‖ f ‖_{L^{2} (B)}^{2} \\ ⩾ 1 - κ r ‖ f ‖_{H^{1} (\hat{Γ} ∖ B)}^{2} - κ r^{2} ‖ f ‖_{H^{1} (\hat{Γ})}^{2} \\ ⩾ 1 - κ r (1 + r) - κ r ‖ \nabla f ‖_{L^{2} (\hat{Γ} ∖ B; C^{2})}^{2} - κ r^{2} ‖ \nabla f ‖^{2} \\ ⩾ 1 - κ^{'} r - κ^{″} (r + | ln r |^{- 1}) \cdot (w_{r}^{- 1} \nabla f, \nabla f) \\ ⩾ 1 - κ^{'} r - κ^{″} (r + | ln r |^{- 1}) b_{0, n_{0}} \\ ⩾ 1 - κ^{‴} | ln r |^{- 1} \end{array}$ for all sufficiently small $r > 0$ . Thus, we conclude $\begin{matrix} (w_{r}^{- 1} \nabla f, \nabla f) ⩾ ‖ \nabla f ‖_{L^{2} (\hat{Γ} ∖ B; C^{2})}^{2} ⩾ \frac{‖ \nabla f ‖_{L^{2} (\hat{Γ} ∖ B; C^{2})}^{2}}{‖ f ‖_{L^{2} (\hat{Γ} ∖ B)}^{2}} (1 - κ | ln r |^{- 1}) . \end{matrix}$ Taking now the maximum over all normalized functions $f \in W^{+}$ we deduce $\begin{matrix} λ_{r, n} (θ_{n}^{+}) = max_{\begin{array}{c} f \in W^{+} \\ ‖ f ‖ = 1 \end{array}} (w_{r}^{- 1} \nabla f, \nabla f) ⩾ (1 - κ | ln r |^{- 1}) max_{\begin{array}{c} f \in W^{+} \\ ‖ f ‖ = 1 \end{array}} \frac{‖ \nabla f ‖_{L^{2} (\hat{Γ} ∖ B; C^{2})}^{2}}{‖ f ‖_{L^{2} (\hat{Γ} ∖ B)}^{2}} . \end{matrix}$ Note that $dim span {f |_{\hat{Γ} ∖ B} : f \in W^{+}} = n$ holds for all sufficiently small $r > 0$ . Indeed, suppose that this is not the case. Then, there exists $f \in W^{+}$ , $‖ f ‖ = 1$ , such that $‖ f ‖_{L^{2} (\hat{Γ} ∖ B)} = 0$ . Thus, in view of $supp f \subset B$ , Lemma 4.2 implies $\begin{matrix} 1 = ‖ f ‖_{L^{2} (B)}^{2} ⩽ κ r^{2} ‖ f ‖_{H^{1} (\hat{Γ})}^{2} ⩽ κ r^{2} + κ | ln r |^{- 1} h_{r, θ_{n}^{+}} [f] ⩽ κ b_{0, n_{0}} | ln r |^{- 1}, \end{matrix}$ which is a contradiction. Hence, we obtain $\begin{matrix} \frac{λ_{r, n} (θ_{n}^{+})}{1 - κ | ln r |^{- 1}} ⩾ max_{\begin{array}{c} f \in W^{+} \\ ‖ f ‖ = 1 \end{array}} \frac{‖ \nabla f ‖_{L^{2} (\hat{Γ} ∖ B; C^{2})}^{2}}{‖ f ‖_{L^{2} (\hat{Γ} ∖ B)}^{2}} ⩾ min_{\begin{array}{c} V \subset \tilde{H} (θ_{n}^{+}) \\ dim V = n \end{array}} max_{\begin{array}{c} f \in V \\ f \neq 0 \end{array}} \frac{‖ \nabla f ‖_{L^{2} (\hat{Γ} ∖ B; C^{2})}^{2}}{‖ f ‖_{L^{2} (\hat{Γ} ∖ B)}^{2}} = μ_{n} (θ_{n}^{+}), \end{matrix}$ where $\tilde{H} (θ_{n}^{+}) : = {f |_{\hat{Γ} ∖ B} : f \in H (θ_{n}^{+})} \subset L^{2} (\hat{Γ} ∖ B)$ and $μ_{n} (θ_{n}^{+})$ is the n-th eigenvalue of the self-adjoint operator in $L^{2} (\hat{Γ} ∖ B)$ associated to the closed, symmetric and densely defined form $\begin{matrix} \tilde{H} (θ_{n}^{+}) ∋ f \mapsto ‖ \nabla f ‖_{L^{2} (\hat{Γ} ∖ B; C^{2})}^{2} . \end{matrix}$ The above form corresponds to the Laplace operator in $L^{2} (\hat{Γ} ∖ B)$ with semi-periodic boundary conditions on $\partial \hat{Γ}$ and Neumann boundary conditions on $\partial B$ . Finally, it is known from [41, Sec. 3], that $μ_{n} (θ_{n}^{+})$ converges to $λ_{0, n} (θ_{n}^{+}) = b_{0, n}$ , as $r \to 0 +$ . Thus, it follows that for sufficiently small $r > 0$ $\begin{matrix} (4.10) & b_{r, n} = λ_{r, n} (θ_{n}^{+}) ⩾ (1 - κ | ln r |^{- 1}) μ_{n} (θ_{n}^{+}) ⩾ {(1 + β)}^{- 1} b_{0, n} > a_{0, n + 1} ⩾ λ_{r, n + 1} (θ_{n + 1}^{-}) = a_{r, n + 1} . \end{matrix}$ Therefore, the first $n_{0}$ bands in $σ (Υ_{r})$ overlap. It follows by a similar argument that $\begin{matrix} (4.11) & b_{r, n_{0}} = λ_{r, n_{0}} (θ_{n}^{+}) ⩾ (1 - κ | ln r |^{- 1}) μ_{n_{0}} (θ_{n}^{+}) > L \end{matrix}$ for sufficiently small $r > 0$ . From (4.10) and (4.11) the claimed inclusion $[0, L] \subset σ (Υ_{r})$ can be deduced. □

Footnotes

Acknowledgements

The authors thank S. Albeverio, J. Behrndt, P. Exner, F. Gesztesy, and D. Krejčiřík for useful hints how to solve the approximation problems. Moreover, M. Holzmann acknowledges financial support under a scholarship of the program “Aktion Austria – Czech Republic” during a research stay in Prague by the Czech Centre for International Cooperation in Education (DZS) and the Austrian Agency for International Cooperation in Education and Research (OeAD). V. Lotoreichik was supported by the grant No. 17-01706S of the Czech Science Foundation (GAČR).

Approximation of Schrödinger operators with infinitely many point interactions in R 2

This appendix is devoted to the proof of Theorem 3.1. Let $N \in N$ , $λ_{1}, \dots, λ_{N} \in (0, \infty)$ , Y, Λ, $w_{r}$ be as in Section 1.3 and let the self-adjoint operator $H_{r, λ}$ be as in (1.7).

Let $λ > 0$ and let a sufficiently small $r > 0$ be fixed. First, we derive a resolvent formula for $H_{r, λ}$ . To this aim, we define the set $\begin{matrix} Ω_{r} : = ⋃_{y \in Y + Λ} (y + r Ω) \subset R^{2}, \end{matrix}$ and introduce the operators $\begin{matrix} (A.1) & u_{r} : L^{2} (R^{2}) \to L^{2} (Ω_{r}), (u_{r} f) (x) : = (w_{r} (x) - 1) f (x), \end{matrix}$ and $\begin{matrix} (A.2) & v_{r} : L^{2} (Ω_{r}) \to L^{2} (R^{2}), (v_{r} f) (x) : = \{\begin{matrix} f (x), & x \in Ω_{r}, \\ 0, & else . \end{matrix} \end{matrix}$ Note that $‖ u_{r} ‖ = ‖ w_{r} - 1 ‖_{L^{\infty}} = {max}_{n \in {1, \dots, N}} μ_{n} (| ln r |^{- 1}) r^{- 2}$ and $‖ v_{r} ‖ = 1$ . Moreover, the multiplication operator in $L^{2} (R^{2})$ associated to $(w_{r} - 1)$ can be factorized as $(w_{r} - 1) = v_{r} u_{r}$ . Recall that we denote ${(- Δ - ν)}^{- 1}$ , $ν \in ρ (- Δ) = C ∖ [0, \infty)$ , by $R_{0} (ν)$ . With these notations in hands we can derive an auxiliary resolvent formula for $H_{r, λ}$ .

In order to rewrite the resolvent formula (A.3) in a way which is convenient to study its convergence, we set $H : = ⨁_{y \in Y + Λ} L^{2} (Ω)$ and define the function $\begin{matrix} μ : R_{+} \times (Y + Λ) \to R_{+}, μ (r, y) = μ_{n} (| ln r |^{- 1}) for y \in y^{(n)} + Λ . \end{matrix}$ Furthermore, for $ν \in C ∖ R$ we define the operators $A_{r} (ν) : H \to L^{2} (R^{2})$ , $E_{r} (ν) : L^{2} (R^{2}) \to H$ by $\begin{array}{l} (A.4a) & A_{r} (ν) Ξ : = \sum_{y \in Y + Λ} \int_{Ω} G_{ν} (\cdot - y - r z) {[Ξ]}_{y} (z) d z, \\ (A.4b) & {[E_{r} (ν) f]}_{y} : = \int_{R^{2}} G_{ν} (r \cdot - z + y) f (z) d z, \end{array}$ and $B_{r} (ν), C_{r} (ν), D_{r} (ν) : H \to H$ by $\begin{array}{l} (A.5a) & {[B_{r} (ν) Ξ]}_{y} : = μ (r, y) \int_{Ω} G_{ν} (r (\cdot - z)) {[Ξ]}_{y} (z) d z, \\ (A.5b) & {[C_{r} (ν) Ξ]}_{y} : = \sum_{y_{1} \in (Y + Λ) ∖ {y}} \int_{Ω} G_{ν} (r (\cdot - z) + y - y_{1}) {[Ξ]}_{y_{1}} (z) d z, \\ (A.5c) & {[D_{r} (ν) Ξ]}_{y} : = μ (r, y) {[Ξ]}_{y} . \end{array}$ In the above formulae ${[Ξ]}_{y}$ denotes the component of $Ξ \in ⨁_{y \in Y + Λ} H_{y}$ belonging to $H_{y}$ , where $H_{y}$ are separable Hilbert spaces. To analyse the operators in (A.4a)–(A.4b) and (A.5a)–(A.5c) we require several auxiliary unitary mappings: the identification mapping $I_{r} : ⨁_{y \in Y + Λ} L^{2} (y + r Ω) \to L^{2} (Ω_{r})$ , the translation operator $T_{r} : ⨁_{y \in Y + Λ} L^{2} (r Ω) \to ⨁_{y \in Y + Λ} L^{2} (y + r Ω)$ , and the scaling transformation $S_{r} : H \to ⨁_{y \in Y + Λ} L^{2} (r Ω)$ defined by $\begin{array}{l} (I_{r} Ξ) (x) : = {[Ξ]}_{y} (x) (x \in y + r Ω, y \in Y + Λ), \\ {[T_{r} Ξ]}_{y} (x) : = {[Ξ]}_{y} (x - y) (x \in y + r Ω), \\ {[S_{r} Ξ]}_{y} (x) : = \frac{1}{r} {[Ξ]}_{y} (\frac{x}{r}) (x \in r Ω) . \end{array}$ Note that the inverses of these mappings act as $\begin{array}{l} {[I_{r}^{- 1} f]}_{y} (x) = f (x) (x \in y + r Ω), \\ {[T_{r}^{- 1} Ξ]}_{y} (x) = {[Ξ]}_{y} (x + y) (x \in r Ω), \\ {[S_{r}^{- 1} Ξ]}_{y} (x) = r {[Ξ]}_{y} (r x) (x \in Ω) . \end{array}$ It will also be convenient to define the product $\begin{matrix} J_{r} : = r^{- 1} I_{r} T_{r} S_{r} . \end{matrix}$ In the following lemma we state some of the basic properties of the operators $A_{r} (ν)$ , $B_{r} (ν)$ , $C_{r} (ν)$ , $D_{r} (ν)$ and $E_{r} (ν)$ .

After all these preparations it is not difficult to transform the resolvent formula for $H_{r, λ}$ from Proposition A.1 into another one which is more convenient for the investigation of its convergence. For this purpose, we define for $λ ⩾ 0$ and $ν \in C ∖ R$ the operator $\begin{matrix} (A.8) & F_{r} (ν, λ) : = λ {(1 - λ B_{r} (ν))}^{- 1} D_{r} (ν) . \end{matrix}$ Note that $F_{r} (ν, λ)$ is well defined, as it is known from the one-center case that each component of the diagonal operator $1 - λ B_{r} (ν)$ is boundedly invertible; see [3, Eq. (5.49) in Chap. I.5]. Hence, thanks to its diagonal structure it is clear that also ${(1 - λ B_{r} (ν))}^{- 1}$ exists as a bounded and everywhere defined operator.

Now we have all the tools to analyse the convergence of $H_{r, λ}$ in the norm resolvent sense. For this purpose, it is sufficient to compute the limits of the operators $A_{r} (ν)$ , $C_{r} (ν)$ , $E_{r} (ν)$ and $F_{r} (ν, λ)$ separately. The obvious candidates for the limits of $A_{r} (ν)$ , $C_{r} (ν)$ and $E_{r} (ν)$ , as $r \to 0 +$ , are given by $A_{0} (ν), C_{0} (ν)$ , and $E_{0} (ν)$ that are defined as in (A.4a)–(A.4b) and (A.5a)–(A.5c) with $r = 0$ . The convergence of $F_{r} (ν, λ)$ is more subtle, as $G_{ν} (0)$ is not defined. The known analysis of the convergence in the one-center case [3, Chap. I.5] suggests the following limit operator: $\begin{matrix} (A.9) & F (ν, λ) : H \to H, {[F (ν, λ) Ξ]}_{y} (x) : = \frac{q (y, ν, λ)}{| Ω |^{2}} {⟨ {[Ξ]}_{y} ⟩}_{Ω}, \end{matrix}$ where ${⟨ f ⟩}_{Ω} = \int_{Ω} f d x$ and $q (y, ν, λ)$ is given by $\begin{matrix} q (y, ν, λ) = \{\begin{matrix} 2 π {ln \frac{\sqrt{ν}}{2 i} - γ + 2 π α_{n}}^{- 1}, & λ = λ_{n}, y \in y^{(n)} + Y, n \in {1, \dots, N}, \\ 0, & else, \end{matrix} \end{matrix}$ with $α_{n}$ as in (3.1). Before going further with the proof of the convergence of ${(H_{r, λ} - ν)}^{- 1}$ , we recall the asymptotics of the integral kernel $G_{ν} (x - y)$ of $R_{0} (ν)$ . In a way similar to [8, Prop. A.1], one can prove the following claim.

Now, we are prepared to investigate the convergence of $A_{r} (ν)$ , $C_{r} (ν)$ , $E_{r} (ν)$ , and $F_{r} (ν, λ)$ , as $r \to 0 +$ .

Since we know now the convergence properties of all the involved operators in the resolvent formula of $H_{r, λ}$ , we are ready to prove Theorem 3.1.

References

Abramowitz and

I.A.

Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington, D.C., 1964, for sale by the Superintendent of Documents.

Albeverio,

Gesztesy,

Høegh-Krohn and

Holden, Point interactions in two dimensions: Basic properties, approximations and applications to solid state physics, J. Reine Angew. Math. 380 (1987), 87–107.

Albeverio,

Gesztesy,

Høegh-Krohn and

Holden, Solvable Models in Quantum Mechanics, 2nd edn, AMS Chelsea Publishing, Providence, RI, 2005, with an Appendix by Pavel Exner.

Albeverio,

Gesztesy,

Høegh-Krohn and

Kirsch, On point interactions in one dimension, J. Operator Theory 12 (1984), 101–126.

Ammari,

Kang and

Lee, Layer Potential Techniques in Spectral Analysis, Mathematical Surveys and Monographs, Vol. 153, American Mathematical Society, Providence, RI, 2009.

Ammari,

Kang and

Lee, Asymptotic analysis of high-contrast phononic crystals and a criterion for the band-gap opening, Arch. Ration. Mech. Anal. 193(3) (2009), 679–714. doi:10.1007/s00205-008-0179-4.

Ammari and

Santosa, Guided waves in a photonic bandgap structure with a line defect, SIAM J. Appl. Math. 64(6) (2004), 2018–2033. doi:10.1137/S0036139902404025.

Behrndt,

Exner,

Holzmann and

Lotoreichik, Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces, Math. Nachr. 290(8–9) (2017), 1215–1248. doi:10.1002/mana.201500498.

Behrndt,

Holzmann and

Lotoreichik, Convergence of 2d-Schrödinger operators with local scaled short-range interactions to a Hamiltonian with infinitely many δ-point interactions, PAMM 14 (2014), 1005–1006. doi:10.1002/pamm.201410482.

10.

B.M.

Brown,

Hoang,

Plum,

Radosz and

Wood, Gap localization of TE-modes by arbitrarily weak defects, J. Lond. Math. Soc. (2) 95(3) (2017), 942–962. doi:10.1112/jlms.12046.

11.

B.M.

Brown,

Hoang,

Plum and

I.G.

Wood, Floquet–Bloch theory for elliptic problems with discontinuous coefficients, in: Spectral Theory and Analysis,

Janas,

Kurasov,

Laptev,

Naboko and

Stolz, eds, Operator Theory: Advances and Applications, Vol. 214, Springer, Basel, 2011, pp. 1–20.

12.

B.M.

Brown,

Hoang,

Plum and

I.G.

Wood, Spectrum created by line defects in periodic structures, Math. Nachr. 287(17–18) (2014), 1972–1985. doi:10.1002/mana.201300165.

13.

Chen and

Lipton, Resonance and double negative behavior in metamaterials, Arch. Ration. Mech. Anal. 209(3) (2013), 835–868. doi:10.1007/s00205-013-0634-8.

14.

K.D.

Cherednichenko and

Cooper, Resolvent estimates for high-contrast elliptic problems with periodic coefficients, Arch. Ration. Mech. Anal. 219(3) (2016), 1061–1086. doi:10.1007/s00205-015-0916-4.

15.

B.E.J.

Dahlberg and

Trubowitz, A remark on two-dimensional periodic potentials, Comment. Math. Helv. 57(1) (1982), 130–134. doi:10.1007/BF02565850.

16.

Dörfler,

Lechleiter,

Plum,

Schneider and

Wieners, Photonic Crystals. Mathematical Analysis and Numerical Approximation, Oberwolfach Seminars, Vol. 42, Springer, Berlin, 2011.

17.

Effenberger,

Kressner and

Engström, Linearization techniques for band structure calculations in absorbing photonic crystals, Internat. J. Numer. Methods Engrg. 89(2) (2012), 180–191. doi:10.1002/nme.3235.

18.

Engström, On the spectrum of a holomorphic operator-valued function with applications to absorptive photonic crystals, Math. Models Methods Appl. Sci. 20(8) (2010), 1319–1341. doi:10.1142/S0218202510004611.

19.

Engström,

Langer and

Tretter, Rational eigenvalue problems and applications to photonic crystals, J. Math. Anal. Appl. 445(1) (2017), 240–279. doi:10.1016/j.jmaa.2016.07.048.

20.

Figotin and

Kuchment, Band-gap structure of the spectrum of periodic Maxwell operators, J. Statist. Phys. 74(1–2) (1994), 447–455. doi:10.1007/BF02186820.

21.

Figotin and

Kuchment, Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model, SIAM J. Appl. Math. 56(1) (1996), 68–88. doi:10.1137/S0036139994263859.

22.

Figotin and

Kuchment, Band-gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals, SIAM J. Appl. Math. 56(6) (1996), 1561–1620. doi:10.1137/S0036139995285236.

23.

Figotin and

Kuchment, 2D photonic crystals with cubic structure: Asymptotic analysis, in: Wave Propagation in Complex Media, Minneapolis, MN, 1994, IMA Vol. Math. Appl., Vol. 96, Springer, New York, 1998, pp. 23–30. doi:10.1007/978-1-4612-1678-0_2.

24.

Figotin and

Kuchment, Spectral properties of classical waves in high-contrast periodic media, SIAM J. Appl. Math. 58(2) (1998), 683–702 (electronic). doi:10.1137/S0036139996297249.

25.

Hempel and

Lienau, Spectral properties of periodic media in the large coupling limit, Comm. Partial Differential Equations 25(7–8) (2000), 1445–1470. doi:10.1080/03605300008821555.

26.

Hoang,

Plum and

Wieners, A computer-assisted proof for photonic band gaps, Z. Angew. Math. Phys. 60(6) (2009), 1035–1052. doi:10.1007/s00033-008-8021-2.

27.

Holden,

Høegh-Krohn and

Johannesen, The short-range expansion in solid state physics, Ann. Inst. H. Poincaré Phys. Théor. 41(4) (1984), 335–362.

28.

J.D.

Joannopoulos,

S.G.

Johnson,

J.N.

Winn and

R.D.

Meade, Photonic Crystals: Molding the Flow of Light, 2nd edn, Princeton University Press, 2008.

29.

J.D.

Joannopoulos,

P.R.

Villeneuve and

Fan, Photonic crystals: Putting a new twist on light, Nature 386 (1997), 143–149, Erratum: ibid. vol. 387, p. 830, 1997.

30.

John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (1987), 2486–2489. doi:10.1103/PhysRevLett.58.2486.

31.

Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995, reprint of the 1980 edition.

32.

Khrabustovskyi, Periodic elliptic operators with asymptotically preassigned spectrum, Asymptot. Anal. 82(1–2) (2013), 1–37.

33.

Klein,

Koines and

Seifert, Generalized eigenfunctions for waves in inhomogeneous media, J. Funct. Anal. 190(1) (2002), 255–291. doi:10.1006/jfan.2001.3887.

34.

Kuchment, An overview of periodic elliptic operators, Bull. Amer. Math. Soc. (N. S.) 53(3) (2016), 343–414. doi:10.1090/bull/1528.

35.

Kuchment and

B.-S.

Ong, On guided electromagnetic waves in photonic crystal waveguides, Amer. Math. Soc. Transl. Ser. 2(231) (2010), 99–108.

36.

Lee, Dirac cones for point scatterers on a honeycomb lattice, SIAM J. Math. Anal. 48(2) (2016), 1459–1488. doi:10.1137/14095827X.

37.

A.A.

Maradudin and

A.R.

McGurn, Photonic band structure of a truncated, two-dimensional, periodic dielectric medium, J. Opt. Soc. Am. B 10(2) (1993), 307–313. doi:10.1364/JOSAB.10.000307.

38.

McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

39.

R.D.

Meade,

A.M.

Rappe,

K.D.

Brommer and

J.D.

Joannopoulos, Nature of the photonic band gap: Some insights from a field analysis, J. Opt. Soc. Am. B 10(2) (1993), 328–332. doi:10.1364/JOSAB.10.000328.

40.

Plihal and

A.A.

Maradudin, Photonic band structure of two-dimensional systems: The triangular lattice, Phys. Rev. B 44 (1991), 8565–8571. doi:10.1103/PhysRevB.44.8565.

41.

Rauch and

Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal. 18 (1975), 27–59. doi:10.1016/0022-1236(75)90028-2.

42.

Reed and

Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.

43.

Weidmann, Lineare Operatoren in Hilberträumen. Teil 1. Mathematische Leitfäden, B. G. Teubner, Stuttgart, 2000, Grundlagen.

44.

Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett. 58 (1987), 2059–2062. doi:10.1103/PhysRevLett.58.2059.

45.

V.V.

Zhikov, On an extension and an application of the two-scale convergence method, Mat. Sb. 191(7) (2000), 31–72. doi:10.4213/sm491.

46.

V.V.

Zhikov, On spectrum gaps of some divergent elliptic operators with periodic coefficients, St. Petersburg Math. J. 16 (2005), 773–790. doi:10.1090/S1061-0022-05-00878-2.