Geometric Approximation of Point Interactions in Two-Dimensional Domains for Non-Self-Adjoint Operators

Abstract

We define the notion of a point interaction for general non-self-adjoint elliptic operators in planar domains. We show that such operators can be approximated in a geometric way by cutting out a small cavity around the point, at which the interaction is concentrated. On the boundary of the cavity, we impose a special Robin-type boundary condition with a nonlocal term. As the cavity shrinks to a point, the perturbed operator converges in the norm resolvent sense to a limiting one with a point interaction containing an arbitrary prescribed complex-valued coupling constant. The mentioned convergence holds in a few operator norms, and for each of these norms we establish an estimate for the convergence rate. As a corollary of the norm resolvent convergence, we prove the convergence of the spectrum.

1. Introduction

Operators with point interactions are among the interesting objects in the spectral theory and have been studied in many works. They come from quantum mechanics as an idealized model of Hamiltonians of systems, in which the interaction is concentrated in a small area [1]. The first rigorous description of such operators is due to Berezin and Faddeev [2], and later these operators were intensively studied. To avoid mentioning all papers and monographs devoted to such operators, we will just cite two well-known books [3,4], which provide quite an extensive presentation of the theory of the operators with point interactions.

One of the important aspects in studying the operators with point interactions is how to approximate them by the operators with regular coefficients. The most natural way is through an appropriately large potential concentrated on a small set. Such an approximation was studied in detail, and the corresponding results can be found in [3] (Secs. I.1.2, I.2.2, I.3.2, I.5), see also some selected papers [5,6,7,8,9,10,11,12,13,14] and the references therein. Recently, in [15], an alternative way of approximation was proposed for two-dimensional operators. The idea was to cut out a small cavity around the point, at which the interaction is concentrated, and to introduce a Robin condition on its boundary. The coefficient in this condition was singular in the small parameter $\epsilon$ characterizing the size of the cavity; namely, it behaved as $O\left({\epsilon }^{-1}{ln}^{-1}\epsilon \right)$. This approximation was introduced for a general self-adjoint second order differential operator with varying coefficients, and it was shown that the approximating operator converged to the limiting one with a point interaction in a norm resolvent sense. Apart from establishing the convergence, the estimates for the convergence rates were obtained. The limiting operator with the point interaction involved a coupling constant $\beta$, which was real and was calculated in terms of certain integrals of the coefficients in the aforementioned Robin condition. It turned out that the proposed way of approximating had a serious disadvantage; namely, the coupling constant could not exceed a certain upper bound. The presence of such an upper bound was not just a technical issue, but it was an unavoidable feature of the approximation.

It should also be mentioned that perturbation by small cavities is a classical one, and it was studied in many contexts and works. Beginning from very classical works [16,17], there have been plenty of papers and books in which the issues on convergence and asymptotic expansions were studied, especially on problems from the homogenization theory, see, for instance, [18,19,20,21] and many references therein. The norm resolvent convergence for a domain with a single cavity was established in [22,23] in the case when, on the boundary of the cavity, one of the classical boundary conditions was established that was independent of the size of the cavity. Similar results for domains perforated by many closely spaced cavities, that is, for the problems from the homogenization theory, were established in [24,25,26,27,28,29,30,31,32]. In particular, the results obtained in the cited papers showed that, in the limit, delta interactions can appear on certain manifolds due to the shapes and distributions of the cavities.

In the present work, we again consider a model proposed in [15], but now we show how to modify it to omit all restrictions for the coupling constant in the limiting operator with the point interaction. Namely, we first consider a general non-self-adjoint operator in an arbitrary two-dimensional domain with varying coefficients and define the notion of the point interaction for such an operator. Then, we again cut out a small cavity around a point, at which the interaction is concentrated, and we impose a Robin-type condition on its boundary, keeping the differential expression unchanged. The main difference with the situation studied in [15] is that now this boundary condition involves a non-local integral term. This term is the main feature of our model. We show that the obtained perturbed operator condition converges to the operator with the point interaction in the norm resolvent sense, and now there are no restrictions for the coupling constant in the limiting operator. In other words, by appropriately choosing the functions describing the Robin-type condition on the boundary of the cavity in the limit, we can obtain an arbitrary prescribed complex-valued coupling constant for the point interaction, and this approximation works for general non-self-adjoint operators. This is our main result and the main advantage in comparison with [15]. It should also be stressed that even if the limiting operator is self-adjoint but contains a coupling constant exceeding the aforementioned upper bound from [15], we nevertheless have to deal with the nonlocal Robin-type condition on the boundary of the cavity, and this condition makes the perturbed operator non-self-adjointness. This non-self-adjointness is an important ingredient, which allows us to approximate an arbitrary value of the coupling constant.

We establish the norm resolvent convergence in a few natural operator norms, and in each case we estimate the convergence rate. On the base of the convergence result, we study the convergence of the spectrum. We show that the essential spectrum is invariant with respect to the perturbation by the cavity. We also explicitly find a narrow neighborhood of the limiting spectrum containing the spectrum of the perturbed operator.

2. Problem and Main Results

Let $x=(x_1,x_2)$ be Cartesian coordinates in ${\mathbb{R}}^2$ and $\mathrm{\Omega }\subseteq {\mathbb{R}}^2$ be a domain. The domain $\mathrm{\Omega }$ can be bounded or unbounded, and the case $\mathrm{\Omega }={\mathbb{R}}^2$ is admitted. The smoothness of the boundary of $\mathrm{\Omega }$ is supposed to be $C^2$ once this boundary is non-empty.

Let $A_{ij}=A_{ij}\left(x\right)$, $A_j=A_j\left(x\right)$, and $A_0=A_0\left(x\right)$ be given functions defined on $\overline{\mathrm{\Omega }}$ such that $A_{ij},A_j\in C^3\left(\overline{\mathrm{\Omega }}\right)$, $A_0\in C^2\left(\overline{\mathrm{\Omega }}\right)$. The functions $A_{ij}$ are real-valued, while the functions $A_j$ and $A_0$ are complex-valued. The standard ellipticity condition is supposed to be satisfied:

$$A_{ij}=A_{ji},\sum _{i,j=1}^2{A}_{ij}\left(x\right){\xi }_i{\xi }_j⩾c_0({\xi }_1^2+{\xi }_2^2),{\xi }_i\in \mathbb{R},x\in \overline{\mathrm{\Omega }},$$

where $c_0$ is a fixed positive constant independent of x and $\xi$.

By ${\mathcal{H}}_{\mathrm{\Omega }}$, we denote an unbounded operator in $L_2\left(\mathrm{\Omega }\right)$ with a differential expression

$$\widehat{\mathcal{H}}:=-\sum _{i,j=1}^2\frac{\partial }{\partial x_i}{A}_{ij}\frac{\partial }{\partial x_j}+\sum _{j=1}^2{A}_j\frac{\partial }{\partial x_j}+A_0$$

subject to a boundary condition

$$\mathcal{B}u=0\mathrm{on}\partial \mathrm{\Omega }$$

(1)

with some boundary operator $\mathcal{B}$ of at most first order. The operator $\mathcal{B}$ in (1) can be chosen arbitrarily. For instance, the identities $\mathcal{B}u=u$ and $\mathcal{B}u=\frac{\partial u}{\partial \mathrm{n}}+b_{0}u$ describe, respectively, the Dirichlet condition and the Robin condition with the parameter $b_0$; here $\frac{\partial u}{\partial \mathrm{n}}$ is the corresponding conormal derivative. It is also possible to choose quasi-periodic boundary conditions as well as the mixing of different types of boundary conditions on various subsets of $\partial \mathrm{\Omega }$. In the case $\partial \mathrm{\Omega }=\varnothing$, condition (1) is omitted.

To rigorously define the operator ${\mathcal{H}}_{\mathrm{\Omega }}$, we first introduce a sesquilinear form:

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\mathrm{\Omega }}(u,v):=& \sum _{i,j=1}^2{\left(A_{ij}\frac{\partial u}{\partial x_j},\frac{\partial v}{\partial x_i}\right)}_{L_2\left(\mathrm{\Omega }\right)}+\sum _{j=1}^2{\left(A_j\frac{\partial u}{\partial x_j},v\right)}_{L_2\left(\mathrm{\Omega }\right)}+{(A_{0}u,v)}_{L_2\left(\mathrm{\Omega }\right)}+{(bu,v)}_{L_2(\partial \mathrm{\Omega })}\hfill \end{array}$$

in $L_2\left(\mathrm{\Omega }\right)$ on a domain $\mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right)$ being an appropriate subspace in $W_2^1\left(\mathrm{\Omega }\right)$ and b is some function defined on $\partial \mathrm{\Omega }$; both this subspace and the function b are determined by a specific choice of the boundary operator $\mathcal{B}$. The form ${\mathfrak{h}}_{\mathrm{\Omega }}$ is supposed to be sectorial, and by the first representation theorem [33] (Ch. VI, Sect. 2.1) there exists a unique associated m-sectorial operator in $L_2\left(\mathrm{\Omega }\right)$. This operator is exactly ${\mathcal{H}}_{\mathrm{\Omega }}$.

We suppose that there exists a bounded subdomain ${\mathrm{\Omega }}_0\subset \mathrm{\Omega }$ separated from $\partial \mathrm{\Omega }$ by a positive distance such that the restriction of each function from the domain $\mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$ to ${\mathrm{\Omega }}_0$ is an element of $W_2^2\left({\mathrm{\Omega }}_0\right)$. For an arbitrary subdomain $\tilde{\mathrm{\Omega }}\subset \mathrm{\Omega }$, we denote

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\tilde{\mathrm{\Omega }}}(u,v):=& \sum _{i,j=1}^2{\left(A_{ij}\frac{\partial u}{\partial x_j},\frac{\partial v}{\partial x_i}\right)}_{L_2\left(\tilde{\mathrm{\Omega }}\right)}+\sum _{j=1}^2{\left(A_j\frac{\partial u}{\partial x_j},v\right)}_{L_2\left(\tilde{\mathrm{\Omega }}\right)}\hfill \\ \hfill & +{(A_{0}u,v)}_{L_2\left(\tilde{\mathrm{\Omega }}\right)}+{(bu,v)}_{L_2(\partial \mathrm{\Omega }\cap \partial \tilde{\mathrm{\Omega }})},\hfill \end{array}$$

where u and v are the restrictions of the functions from $\mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right)$ to $\tilde{\mathrm{\Omega }}$. We assume that the inequalities

$$\begin{array}{cc}\hfill & Re\left({\mathfrak{h}}_{\mathrm{\Omega }}(u,u)-{\mathfrak{h}}_{\tilde{\mathrm{\Omega }}}(u,u)+c_1{\parallel u\parallel }_{L_2(\mathrm{\Omega }\backslash \tilde{\mathrm{\Omega }})}^2\right)⩾c_2{\parallel u\parallel }_{W_2^1(\mathrm{\Omega }\backslash \tilde{\mathrm{\Omega }})}^2,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & Re{\mathfrak{h}}_{\tilde{\mathrm{\Omega }}}(u,u)+c_1{\parallel u\parallel }_{L_2\left(\tilde{\mathrm{\Omega }}\right)}^2⩾c_2{\parallel u\parallel }_{W_2^1\left(\tilde{\mathrm{\Omega }}\right)}^2,\hfill \end{array}$$

(3)

are satisfied for $\tilde{\mathrm{\Omega }}={\mathrm{\Omega }}_0$ and $\tilde{\mathrm{\Omega }}={\omega }_{\epsilon }$ and for all $u\in \mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right)$ with positive constants $c_1$, $c_2$ independent of u and $\epsilon$.

We fix a point $x_0\in {\mathrm{\Omega }}_0$ and consider a boundary value problem:

$$\begin{array}{cc}\hfill & (\widehat{\mathcal{H}}+c_1)G=0\mathrm{in}\mathrm{\Omega }\backslash \left\{x_0\right\},\mathcal{B}G=0\mathrm{on}\partial \mathrm{\Omega },\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & G\left(x\right)=ln|{\mathrm{A}}^{-\frac{1}{2}}(x-x_0)|+a+O\left(|x-x_0|ln|x-x_0|\right),x\to x_0,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & (1-\chi )G\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right),\chi G\in W_2^2(\mathrm{\Omega }\backslash B_{\delta }\left(x_0\right))\cap C^2(\overline{{\mathrm{\Omega }}_0}\backslash \left\{x_0\right\}),\delta >0,\hfill \end{array}$$

(6)

where $a\in \mathbb{C}$ is fixed, $c_1$ is the constant from (2) and (3) and

$$\mathrm{A}:=\left(\begin{array}{cc}{A}_{11}\left(x_0\right)& A_{12}\left(x_0\right)\\ A_{21}\left(x_0\right)& A_{22}\left(x_0\right)\end{array}\right).$$

We shall show below in Lemma 5 that problem (4)–(6) is uniquely solvable in $W_2^2(\mathrm{\Omega }\backslash B_{\delta }\left(x_0\right))\cap C^2(\overline{{\mathrm{\Omega }}_0}\backslash \left\{x_0\right\})$ for some $\delta >0$; hereinafter, the symbol $B_r\left(a\right)$ stands for a circle of radius r centered at the point a.

The first main object of our study is an operator with a point interaction at the point $x_0$, which is denoted by ${\mathcal{H}}_{0,\beta }$ with an arbitrary $\beta \in \mathbb{C}\backslash \left\{a\right\}$. This is the operator with the differential expression $\widehat{\mathcal{H}}$ in $\mathrm{\Omega }$ in $L_2\left(\mathrm{\Omega }\right)$ on the domain

$$\mathfrak{D}\left({\mathcal{H}}_{0,\beta }\right):=\left\{u=u\left(x\right):u\left(x\right)=v\left(x\right)+{(\beta -a)}^{-1}v\left(x_0\right)G\left(x\right),v\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)\right\}$$

(7)

acting by rule

$${\mathcal{H}}_{0,\beta }u={\mathcal{H}}_{\mathrm{\Omega }}v-c_1{(\beta -a)}^{-1}v\left(x_0\right)G,$$

(8)

where $c_1$ is the above constant from (2) and (3).

The main aim of the paper is to show that the introduced operator ${\mathcal{H}}_{0,\beta }$ can be approximated in the sense of the norm resolvent convergence by cutting out a small cavity in $\mathrm{\Omega }$ and imposing an appropriate Robin condition on its boundary. We introduce the cavity as ${\omega }_{\epsilon }:=\{x:(x-x_0){\epsilon }^{-1}\in \omega \}$, where $\epsilon$ is a small positive parameter and $\omega$ is a bounded simply connected domain $\omega \subset {\mathbb{R}}^2$ with a $C^3$-boundary. This cavity shrinks to the point $x_0$ as $\epsilon$ goes to zero. By $s\in [0,|\partial \omega \left|\right]$, we denote the arc length of $\partial \omega$; then $s_{\epsilon }:=\epsilon s$ naturally serves as the arc length on $\partial {\omega }_{\epsilon }$. We denote ${\mathrm{\Omega }}_{\epsilon }:=\mathrm{\Omega }\backslash \overline{{\omega }_{\epsilon }}$. We approximate ${\mathcal{H}}_{0,\beta }$ by an unbounded operator ${\mathcal{H}}_{\epsilon }$ in $L_2\left(\mathrm{\Omega }\right)$ with the differential expression $\widehat{\mathcal{H}}$ subject to condition (1) and to a nonlocal Robin condition on $\partial {\omega }_{\epsilon }$:

$$\begin{array}{cc}\hfill & \frac{\partial u}{\partial \mathrm{n}}={\ell }_{\epsilon }u\mathrm{on}\partial {\omega }_{\epsilon },\hfill \end{array}$$

(9)

$$\begin{array}{c}{\ell }_{\epsilon }u:={\epsilon }^{-1}\alpha \left({\epsilon }^{-1}·,{ln}^{-1}\epsilon \right)u+{\epsilon }^{-1}{ln}^{-1}\epsilon {\alpha }_2({\epsilon }^{-1}·)\left(u-{\langle {\alpha }_3({\epsilon }^{-1}·)u\rangle }_{\partial {\omega }_{\epsilon }}ln\epsilon \right),\hfill \\ \begin{array}{c}\frac{\partial }{\partial \mathrm{n}}:=\sum _{i,j=1}^2{A}_{ij}{\nu }_i\frac{\partial }{\partial x_i},\alpha (s,\mu ):={\alpha }_0\left(s\right)+\mu {\alpha }_1\left(s\right),{\langle u\rangle }_{\partial {\omega }_{\epsilon }}:=\frac{1}{|\partial {\omega }_{\epsilon }|}\underset{\partial {\omega }_{\epsilon }}{\int }uds,\hfill \end{array}\hfill \end{array}$$

(10)

where $\nu =({\nu }_1,{\nu }_2)$ is the normal unit on $\partial {\omega }_{\epsilon }$ directed inside ${\omega }_{\epsilon }$. We let

$${\alpha }_0\left(s\right)=\frac{\nu ·{\mathrm{A}}^{\frac{1}{2}}\mathrm{x}\left(s\right)}{|{\mathrm{A}}^{-\frac{1}{2}}{\mathrm{x}\left(s\right)|}^2},$$

where $\mathrm{x}=\mathrm{x}\left(s\right)$ is a parametric equation of the boundary $\partial \omega$. The functions ${\alpha }_i={\alpha }_i\left(s\right)$, $i=1,2,3$ are defined on $\partial \omega$, complex-valued and continuous with respect to the arc length $s\in [0,|\partial \omega \left|\right]$. We also assume that the function ${\alpha }_3$ satisfies the conditions

$$\underset{\partial \omega }{\int }{\alpha }_3\left(s\right)ds=0,\underset{\partial \omega }{\int }{\alpha }_3\left(s\right)ln\left|{\mathrm{A}}^{-\frac{1}{2}}\mathrm{x}\left(s\right)\right|ds=1,$$

(11)

while the function ${\alpha }_2$ reads as

$${\alpha }_2={\alpha }_2^0+\overline{{\alpha }_3},$$

(12)

where ${\alpha }_2^0$ is some complex constant and $\overline{{\alpha }_3}$ is the complex-conjugation of the function ${\alpha }_3$.

We define the operator ${\mathcal{H}}_{\epsilon }$ rigorously following the scheme proposed in [15]. Namely, let $\chi$ be an infinitely differentiable cut-off function with values in $[0,1]$ equaling to one in $B_{2R_1}\left(x_0\right)$ and vanishing outside ${\mathrm{\Omega }}_0$. We introduce ${\mathcal{H}}_{\epsilon }$ as the operator in $L_2\left({\mathrm{\Omega }}_{\epsilon }\right)$ with the differential expression $\widehat{\mathcal{H}}$ on the domain $\mathfrak{D}\left({\mathcal{H}}_{\epsilon }\right)$ consisting of the functions $u\in W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)$ satisfying condition (9) such that

$$(1-\chi )u\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right),\chi u\in W_2^2({\mathrm{\Omega }}_0\backslash {\omega }_{\epsilon })$$

and acting as

$${\mathcal{H}}_{\epsilon }u:={\mathcal{H}}_{\mathrm{\Omega }}(1-\chi )u+\widehat{\mathcal{H}}\chi u.$$

(13)

We let

$$K_0:=\underset{\partial \omega }{\int }\left(\left({\alpha }_0\left(s\right)+{\alpha }_2\left(s\right)\right)ln\left|{\mathrm{A}}^{-\frac{1}{2}}\mathrm{x}\left(s\right)\right|+{\alpha }_1\left(s\right)\right)ds,K_1:=-\pi trA+{\alpha }_2^0|\partial \omega |,$$

(14)

and we suppose that

$$Re(K_0+K_{1}a)<0,K_1\ne 0.$$

(15)

We also let

$$\beta :=-\frac{K_0}{K_1}$$

(16)

assuming that $\beta \ne a$.

By ${\parallel ·\parallel }_{X\to Y}$, we denote the norm of a bounded operator acting from a Hilbert space X into a Hilbert space Y. For an arbitrary sesquilinear form $\mathfrak{h}$, we shall often use a shorthand notation $\mathfrak{h}\left[u\right]:=\mathfrak{h}(u,u)$.

Our main results are as follows.

Theorem 1. 

The operator ${\mathcal{H}}_{\epsilon }$ is m-sectorial, while the operator ${\mathcal{H}}_{0,\beta }$ is closed. There exists a real number ${\lambda }_0$ independent of ε such that the half-plane $Re\lambda ⩽{\lambda }_0$ is inside the resolvent sets of both operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_{0,\beta }$. The operator ${\mathcal{H}}_{\epsilon }$ converges to ${\mathcal{H}}_{0,\beta }$ in the norm resolvent sense as $\epsilon \to +0$, and the following estimates hold:

$$\begin{array}{cc}\hfill & \parallel {({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽C{|ln\epsilon |}^{-1},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \parallel {({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽C{|ln\epsilon |}^{-\frac{1}{2}},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \parallel {({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to W_2^1(\mathrm{\Omega }\backslash B_r\left(x_0\right))}⩽C{|ln\epsilon |}^{-1},\hfill \end{array}$$

(19)

and in the latter estimate, the radius r of the ball is chosen so that $B_r\left(x_0\right)\subset {\mathrm{\Omega }}_0$. The symbol C stands for positive constants independent of ε. For each $f\in L_2\left(\mathrm{\Omega }\right)$, the estimate

$$|{\mathfrak{h}}_{\mathrm{\Omega }\backslash {\mathrm{\Omega }}_0}\left[{({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}f-{({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}f\right]{|}^{\frac{1}{2}}⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}$$

(20)

holds, where C is a constant independent of ε and f.

Our next result describes the spectral convergence of the operator ${\mathcal{H}}_{\epsilon }$. By $\sigma (·)$ and ${\sigma }_{ess}(·)$ we denote the spectrum and the essential spectrum of a given operator. The essential spectrum is introduced in terms of characteristic sequences.

Theorem 2. 

The essential spectra of the operators ${\mathcal{H}}_{\mathrm{\Omega }}$, ${\mathcal{H}}_{0,\beta }$ and ${\mathcal{H}}_{\epsilon }$ coincide. For each compact set $Q\subset \mathbb{C}$, the inclusion

$$\sigma \left({\mathcal{H}}_{\epsilon }\right)\cap Q\subseteq \left\{\lambda \in Q:\parallel {({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left(\mathrm{\Omega }\right)}⩾C{\mathrm{\Lambda }}^{-2}|ln\epsilon |\right\}$$

(21)

holds, where C is some fixed constant independent of ε, λ and Q, while

$$\mathrm{\Lambda }:=1+\underset{\lambda \in Q}{sup}|\lambda +c_1|.$$

(22)

Let us briefly discuss the problem and the main results. The first main point is that our operators ${\mathcal{H}}_{0,\beta }$ and ${\mathcal{H}}_{\epsilon }$ are non-self-adjoint. The coupling constant $\beta$ in the definition of the operator ${\mathcal{H}}_{0,\beta }$ is allowed to be complex, and the differential expression $\widehat{\mathcal{H}}$ involves complex-valued coefficients at lower derivatives. To the best of our knowledge, non-self-adjoint operators with complex-valued point interactions were not introduced and considered before, and from this point of view, our work is the first that deals with such operators. The definition of the operator ${\mathcal{H}}_{\epsilon }$ has two main features. The first is the non-local Robin boundary conditions (9) and (10) defining the domain of this operator. The nonlocality is due to the presence of the mean value ${\langle ·\rangle }_{\partial {\omega }_{\epsilon }}$, and it plays a crucial role in establishing our main results; we shall discuss this aspect a bit later. The second main feature of the operator ${\mathcal{H}}_{\epsilon }$ is the definition of its action in (13). Such a definition allows us to consider general second order differential operators with coefficients behaving arbitrarily at infinity once the domain $\mathrm{\Omega }$ is unbounded. Formula (13) says that the behavior of the coefficients $A_{ij}$, $A_j$, $A_0$ at infinity as well as the boundary conditions on $\partial \mathrm{\Omega }$ are inherited from the operator ${\mathcal{H}}_{\mathrm{\Omega }}$, which is defined via its sesquilinear form ${\mathfrak{h}}_{\mathrm{\Omega }}$ on an appropriate subspace of $W_2^1\left(\mathrm{\Omega }\right)$. Assumptions (2) and (3) are naturally interpreted as an appropriate coercivity of the considered forms.

Our main result is formulated in Theorem 1, and it states that the operator ${\mathcal{H}}_{0,\beta }$ can be approximated in the norm resolvent sense by the operator ${\mathcal{H}}_{\epsilon }$. The constant $\beta$ is given by Formula (16) and despite conditions (15), this constant can take arbitrary values. Indeed, the constants $K_0$ and $K_1$ can take arbitrary values due to the arbitrary choice of the constant ${\alpha }_2^0$ and the function ${\alpha }_1$, see (14). This is why, for a given $\beta$, we can always choose $K_0$ and $K_1$ to achieve Formula (16); the constants $K_0$ and $K_1$ can be both multiplied by an arbitrary complex non-zero constant k still keeping the latter formula for $\beta$. The first inequality in (16) can be rewritten as $Re{K}_1(\beta -a)>0$, and to satisfy it, we just need to appropriately choose the mentioned constant k for a given $\beta$.

The fact that we can approximate the operator ${\mathcal{H}}_{0,\beta }$ with an arbitrary $\beta$ is the main advantage in comparison with a similar result in [15]. In the cited work, the constant $\beta$ was assumed to be real and could not exceed a certain constant. Such a restriction was unavoidable and came from a similar classical Robin condition on the boundary of the cavity ${\omega }_{\epsilon }$, which contained no nonlocal terms. In the present work, we introduce a nonlocality in the boundary condition on $\partial {\omega }_{\epsilon }$, and this allows us to approximate the operator ${\mathcal{H}}_{0,\beta }$ with an arbitrary $\beta \ne a$. In view of the above discussion, this is achieved by choosing an appropriate constant ${\alpha }_2^0$, and it is easy to see that for ${\alpha }_2^0\ne 0$ the operator ${\mathcal{H}}_{\epsilon }$ is non-self-adjoint. In other words, even if the operator ${\mathcal{H}}_{0,\beta }$ is self-adjoint, we can approximate it for the values of $\beta$ excluded in work [15] by using the non-self-adjoint operator ${\mathcal{H}}_{\epsilon }$, and precisely this non-self-adjointness plays a crucial role.

One more important feature to be stressed is that the shape of the cavity $\omega$ should be different from a ball. Indeed, if the cavity coincides with a ball, then the function $\mathrm{x}\left(s\right)$ is constant, and conditions (11) cannot be obeyed.

The norm resolvent convergence of the operator ${\mathcal{H}}_{\epsilon }$ to ${\mathcal{H}}_{0,\beta }$ is characterized by estimates (17)–(20). The first three estimates correspond to three different operator norms when we treat the difference of the resolvents as acting from $L_2\left(\mathrm{\Omega }\right)$ into $L_2\left({\mathrm{\Omega }}_{\epsilon }\right)$, into $W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)$ and into $W_2^1(\mathrm{\Omega }\backslash B_r\left(x_0\right))$. If the difference of the resolvents is treated as an operator acting into $W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)$, the convergence rate is ${|ln\epsilon |}^{-\frac{1}{2}}$, cf. (18). In a weaker norm of the operators acting from $L_2\left(\mathrm{\Omega }\right)$ into $L_2\left({\mathrm{\Omega }}_{\epsilon }\right)$, the convergence rate is twice improved, cf. (17). A similar situation holds, when the difference of the resolvents is treated as acting into $W_2^1(\mathrm{\Omega }\backslash B_r\left(x_0\right))$, cf. (19). The explanation of such a situation is obviously due to the logarithmic singularity at the point $x_0$ of the functions from the domain of the operator ${\mathcal{H}}_{0,\beta }$, and this singularity is in fact eliminated either by considering a weaker norm or by dealing with the domain $\mathrm{\Omega }\backslash B_r\left(x_0\right)$ separated from $x_0$ by a positive distance. If the domain $\mathrm{\Omega }$ is unbounded and the coefficients $A_{ij}$, $A_j$, $A_0$ are not uniformly bounded, then we also have a convergence in the sense of the form ${\mathfrak{h}}_{\epsilon }$, see estimate (20). In the latter estimate, the convergence rate is naturally the same as in (19).

As a corollary of the above discussed results, the convergence of the spectrum holds as well. It is described in Theorem 2. Namely, the essential spectrum of the operators ${\mathcal{H}}_{\epsilon }$, ${\mathcal{H}}_{0,\beta }$ and ${\mathcal{H}}_{\mathrm{\Omega }}$ coincide, and the perturbation by boundary condition (9) can change only the point spectrum and the residual spectrum. The spectrum of the operators ${\mathcal{H}}_{\epsilon }$ is located in a narrow neighborhood of the spectrum of the operator ${\mathcal{H}}_{0,\beta }$ described in (21). Indeed, the inequality for the norm of the resolvent of ${\mathcal{H}}_{0,\beta }$ in this embedding can hold only because the norm is sufficiently large, and this is a narrow neighborhood of the spectrum of ${\mathcal{H}}_{0,\beta }$.

Summarizing what was said above, we see that cutting out a small hole and imposing nonlocal Robin condition (9) on its boundary is an alternative way of approximating two-dimensional operators with point interactions including general non-self-adjoint operators. It is important that this procedure works for arbitrary coupling constants. This is achieved owing to the nonlocality, which makes the approximating operator necessarily non-self-adjoint, even if the operator with the point interaction is self-adjoint. From this point of view, our results describe the situation in which non-self-adjoint operators necessarily appear as a tool for approximating standard models of point interactions from the classical theory of self-adjoint operators.

3. Auxiliary Statements

In this section, we formulate several auxiliary lemmata, which will then be employed in proving our main results. First, we reproduce three lemmata proved in [15] (Sect. 3).

Lemma 1. 

The identity

$$\underset{\partial \omega }{\int }{\alpha }_0\left(s\right)\mathrm{d}s=\underset{\partial \omega }{\int }\frac{\nu ·{\mathrm{A}}^{\frac{1}{2}}\mathrm{x}\left(s\right)}{|{\mathrm{A}}^{-\frac{1}{2}}{\mathrm{x}\left(s\right)|}^2}\mathrm{d}s=-\pi tr\mathrm{A}$$

holds true.

The definition of ${\omega }_{\epsilon }$ yields the existence of a fixed positive constant $R_1$ independent of $\epsilon$ such that

$${\omega }_{\epsilon }\subset B_{R_1\epsilon }\left(x_0\right)\subset B_{2R_1\epsilon }\left(x_0\right)\subset B_{R_1}\left(x_0\right)\subset B_{2R_1}\left(x_0\right)\subset {\mathrm{\Omega }}_0\subset \mathrm{\Omega }.$$

Lemma 2. 

For all $v\in W_2^1(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })$ satisfying the condition

$$\underset{\partial {\omega }_{\epsilon }}{\int }vds=0$$

(23)

the inequality

$${\parallel v\parallel }_{L_2(\partial {\omega }_{\epsilon })}^2⩽C\epsilon {\parallel \nabla v\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })}^2$$

(24)

holds, where C is a constant independent of ε and v.

Lemma 3. 

For all $\phi \in C(\partial \omega )$ and all $v\in W_2^2\left(B_{2R_1}\left(x_0\right)\right)$, the inequality

$$|{\epsilon }^{-1}\underset{\partial {\omega }_{\epsilon }}{\int }\phi \left({\epsilon }^{-1}{s}_{\epsilon }\right)v\left(x\right)d{s}_{\epsilon }-c\left(\phi \right)v\left(x_0\right)|⩽{C\epsilon |ln\epsilon |}^{\frac{1}{2}}{\parallel v\parallel }_{W_2^2\left(B_{2R_1}\left(x_0\right)\right)},c\left(\phi \right):=\underset{\partial \omega }{\int }\phi \left(s\right)\mathrm{d}s,$$

holds true, where C is a constant independent of ε and v.

Lemma 4. 

For all $v\in W_2^1(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })$ the estimate

$${\parallel v\parallel }_{L_2(\partial {\omega }_{\epsilon })}^2⩽C\epsilon \left({|ln\epsilon |\parallel \nabla v\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })}^2+{\parallel v\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })}^2\right)$$

(25)

is valid, where C is a fixed constant independent of ε and v. If, in addition, the function v is defined on the entire ball $B_{2R_1}\left(x_0\right)$ and belongs to $W_2^2\left(B_{2R_1}\left(x_0\right)\right)$, then the estimate

$${\parallel v\parallel }_{L_2(\partial {\omega }_{\epsilon })}^2⩽C\epsilon {\parallel v\parallel }_{W_2^2\left(B_{2R_1}\left(x_0\right)\right)}^2$$

(26)

holds, where C is a fixed constant independent of ε and v.

Proof. 

Estimate (25) was proved in Lemma 3.3 in [15]. Due to the standard embedding $W_2^2\left(B_{2R_1}\left(x_0\right)\right)\subset C\left(\overline{B_{2R_1}\left(x_0\right)}\right)$, we have the estimate

$${\left|v\left(x\right)\right|}^2⩽C{\parallel v\parallel }_{W_2^2\left(B_{2R_1}\left(x_0\right)\right)}^2,x\in \overline{B_{2R_1\left(x_0\right)}},$$

with some constant C independent of v and x. Integrating this estimate over $\partial {\omega }_{\epsilon }$, we obtain (26). The proof is complete. □

Summing inequalities (2) and (3) with $\tilde{\mathrm{\Omega }}={\mathrm{\Omega }}_0$, we immediately obtain

$$Re{\mathfrak{h}}_{\mathrm{\Omega }}(u,u)+c_1{\parallel u\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2⩾c_2{\parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}^2$$

(27)

For all $u\in \mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right)$. In view of the m-sectoriality of the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ and the positivity of $c_2$, this inequality implies that the point $-c_1$ is in the resolvent set of the operator ${\mathcal{H}}_{\mathrm{\Omega }}$, and the inverse operator ${({\mathcal{H}}_{\mathrm{\Omega }}+c_1)}^{-1}:L_2\left(\mathrm{\Omega }\right)\to W_2^1\left(\mathrm{\Omega }\right)$ is well-defined and bounded.

Lemma 5. 

Boundary value problems (4)–(6) have a unique solution that belongs to $W_2^2(\mathrm{\Omega }\backslash B_{\delta }\left(x_0\right))\cap C^2(\overline{{\mathrm{\Omega }}_0}\backslash \left\{x_0\right\})$ for all sufficiently small $\delta >0$. This solution has the following asymptotics at $x_0$:

$$\begin{array}{cc}\hfill G\left(x\right)=& lnr+a+r\left(\right(a_{1}sin\theta +a_{2}cos\theta )lnr\hfill \\ \hfill & +P(sin\theta ,cos\theta ))+O\left(r^2{ln}^{2}r\right),x\to x_0,\hfill \end{array}$$

(28)

where $a,a_1,a_2\in \mathbb{C}$ are some numbers, P is a polynomial and $(r,\theta )$ are polar coordinates corresponding to the variables $y={\mathrm{A}}^{-\frac{1}{2}}(x-x_0)$. The belongings $\chi G\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$ and $(\widehat{\mathcal{H}}+c_1)\chi G\in L_2\left(\mathrm{\Omega }\right)$ hold and the identity

$${\mathfrak{h}}_{\mathrm{\Omega }}\left((1-\chi )G,u\right)+c_1{\left((1-\chi )G,u\right)}_{L_2\left(\mathrm{\Omega }\right)}=-{\left((\widehat{\mathcal{H}}+c_1)\chi G,u\right)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}$$

(29)

is satisfied for each $u\in \mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right)$.

Proof. 

We establish the solvability of problem (4)–(6) following the lines of the proof of Lemma 3.2 in [15]. We pass to the variables y in the vicinity of the point $x_0$ and introduce a function

$$\begin{array}{cc}\hfill G_0\left(x\right)=& ln\left|y\right|+\left|y\right|\left((a_{1}sin\theta +a_{2}cos\theta )lnr+P(sin\theta ,cos\theta )\right)\hfill \\ \hfill & +{\left|y\right|}^2\left(P_1(sin\theta ,cos\theta ){ln}^2\left|y\right|+P_2(sin\theta ,cos\theta )ln\left|y\right|+P_1(sin\theta ,cos\theta )\right),\hfill \end{array}$$

where P and $P_i$ are some polynomials, and $a_1$ and $a_2$ are some constants determined by the condition that the function $F_0\left(x\right):=(\widehat{\mathcal{H}}+c_1)G\left(x\right)$ behaves at $x_0$ as

$$F_0\left(x\right)=O\left(\right|y|ln|y\left|\right),x\to x_0;$$

it is clear that this function is also continuous in a punctured vicinity of the point $x_0$. Then we seek the solution to problems (4)–(6) as

$$G\left(x\right)=G_0\left(x\right)\chi \left(x\right)+G_1\left(x\right),$$

(30)

and for a new unknown function $G_1$, we obtain the operator equation

$$({\mathcal{H}}_{\mathrm{\Omega }}+c_1)G_1=-(\widehat{\mathcal{H}}+c_1)\chi G_0=:F,F=\chi F_0+F_1,$$

(31)

where $F_1\in C\left(\overline{{\mathrm{\Omega }}_0}\right)$ is compactly supported in a ${\mathrm{\Omega }}_0$ function vanishing in the vicinity of the point $x_0$. The m-sectoriality of the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ and estimate (27) imply that the resolvent ${(\mathcal{H}-\mathrm{\Omega }+c_1)}^{-1}$ is well-defined and, hence, $G_1={({\mathcal{H}}_{\mathrm{\Omega }}+c_1)}^{-1}F$. By the standard Schauder estimate, it also belongs to $C^{2+\gamma }\left(\overline{B_{R_1}\left(x_0\right)}\right)$ with some $\gamma \in (0,1)$. By the Taylor formula, this implies the asymptotics

$$G_1\left(x\right)=a+a_3{y}_1+a_4{y}_2+O\left(\right|x-x_0{|}^2),x\to x_0,a_3,a_4=const,$$

which completes the proof of the solvability of problems (4)–(6).

It follows from (30) and the definition of the function $G_0$ that the function $(1-\chi )G$ is an element of $\mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$. The integral identity corresponding to Equation (31) reads as

$${\mathfrak{h}}_{\mathrm{\Omega }}(G_1,u)+c_1{(G_1,u)}_{L_2\left(\mathrm{\Omega }\right)}=-{((\widehat{\mathcal{H}}+c_1)\chi G_0,u)}_{L_2\left(\mathrm{\Omega }\right)},u\in \mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right).$$

(32)

It is clear that $\chi G_1,(1-\chi )\chi G_0\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$, and due to the smoothness of the coefficients $A_{ij}$, $A_j$, and $A_0$, we can integrate by parts and obtain the identities

$$\begin{array}{cc}\hfill & {\mathfrak{h}}_{\mathrm{\Omega }}(\chi G_1,u)+c_1{(\chi G_1,u)}_{L_2\left({\mathrm{\Omega }}_0\right)}={\left((\widehat{\mathcal{H}}+c_1)\chi G_1,u\right)}_{L_2\left({\mathrm{\Omega }}_0\right)},\hfill \\ \hfill & {\mathfrak{h}}_{\mathrm{\Omega }}\left((1-\chi )\chi G_0,u\right)+c_1{\left((1-\chi )\chi G_0,u\right)}_{L_2\left({\mathrm{\Omega }}_0\right)}={\left((\widehat{\mathcal{H}}+c_1)(1-\chi )\chi G_0,u\right)}_{L_2\left({\mathrm{\Omega }}_0\right)}.\hfill \end{array}$$

We deduct the former identity from one in (32) and then add the latter identity. In view of (30) and the fact that the function $(\widehat{\mathcal{H}}+c_1)\chi G_0$ vanishes in $B_{R_1}\left(x_0\right)$ due to the equation in (4), we then arrive at (29). The proof is complete. □

We introduce an adjoint differential expression and a corresponding conormal derivative:

$$\begin{array}{cc}\hfill & {\widehat{\mathcal{H}}}^{\mathrm{*}}:=-\sum _{i,j=1}^2\frac{\partial }{\partial x_i}{A}_{ij}\frac{\partial }{\partial x_j}-\sum _{j=1}^2\frac{\partial }{\partial x_j}{A}_j+A_0\mathrm{in}{\mathrm{\Omega }}_0,\hfill \\ \hfill & \frac{\partial }{\partial {\mathrm{n}}^{\mathrm{*}}}:=\sum _{i,j=1}^2{A}_{ij}{\nu }_i\frac{\partial }{\partial x_i}+\sum _{j=1}^2{A}_j{\nu }_j\mathrm{on}\partial {\omega }_{\epsilon }.\hfill \end{array}$$

(33)

Lemma 6. 

The identities

$$\begin{array}{cc}\hfill & {\mathfrak{h}}_{\epsilon }(\zeta G,u)+c_1{(\zeta G,u)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}={\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,u\right)}_{L_2(\partial {\omega }_{\epsilon })}+{((\widehat{\mathcal{H}}+c_1)\zeta G,u)}_{L_2({\mathrm{\Omega }}_{\epsilon })},\hfill \\ \hfill & {\mathfrak{h}}_{\epsilon }(u,\zeta G)+c_1{(u,\zeta G)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}={\left(u,\frac{\partial G}{\partial {\mathrm{n}}^{\mathrm{*}}}-{\ell }_{\epsilon }G\right)}_{L_2(\partial {\omega }_{\epsilon })}+{(u,({\widehat{\mathcal{H}}}^{\mathrm{*}}+c_1)\zeta G)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}\hfill \end{array}$$

(34)

are valid for all $u\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$, where $\zeta =\zeta \left(x\right)$ is an arbitrary cut-off function equaling to one in a fixed neighborhood of the point $x_0$ and vanishing outside a bigger neighborhood.

Proof. 

It follows from Lemma 5 that the function $\zeta G$ is an element of $C^2(\overline{{\mathrm{\Omega }}_0\backslash {\omega }_{\epsilon }})\cap \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$ for each $\epsilon$. Integrating then once by parts in the integrals

$$\underset{{\mathrm{\Omega }}_0\backslash {\omega }_{\epsilon }}{\int }\overline{u}(\widehat{\mathcal{H}}+c_1)\zeta Gdx\mathrm{and}\underset{{\mathrm{\Omega }}_0\backslash {\omega }_{\epsilon }}{\int }u\overline{({\widehat{\mathcal{H}}}^{\mathrm{*}}+c_1)\zeta G}dx,$$

we easily prove identity (34). The proof is complete. □

4. Closedness and $\mathit{m}$-Sectoriality

In this section, we prove that the operator ${\mathcal{H}}_{0,\beta }$ is closed for all $\beta$, while the operator ${\mathcal{H}}_{\epsilon }$ is m-sectorial.

Let $u_n\in \mathfrak{D}\left({\mathcal{H}}_{0,\beta }\right)$ be an arbitrary sequence such that $u_n\to u$ and ${\mathcal{H}}_{0,\beta }{u}_n\to f$ in $L_2\left(\mathrm{\Omega }\right)$ as $n\to +\infty$. By $v_n\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$, we denote a sequence of functions associated with $u_n$ in the sense of the identity

$$u_n\left(x\right)=v_n\left(x\right)+{(\beta -a)}^{-1}v\left(x_0\right)G\left(x\right).$$

(35)

The definition of the action of the operator ${\mathcal{H}}_{0,\beta }$ in (8) implies that

$$({\mathcal{H}}_{0,\beta }+c_1)u_n=({\mathcal{H}}_{\mathrm{\Omega }}+c_1)v_n\to f+c_{1}u\mathrm{in}{L}_2\left(\mathrm{\Omega }\right)\mathrm{as}n\to \infty .$$

Estimate (27) ensures that the constant $-c_1$ is in the resolvent set of the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ and hence, the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ is invertible and the estimate holds:

$$\parallel {({\mathcal{H}}_{\mathrm{\Omega }}+c_1)}^{-1}{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to W_2^1\left(\mathrm{\Omega }\right)}⩽C,$$

(36)

where C is some fixed constant. The assumed smoothness of the functions $A_{ij}$, $A_j$ and $A_0$ and the standard smoothness improving theorems then imply

$$\parallel {({\mathcal{H}}_{\mathrm{\Omega }}+c_1)}^{-1}{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to W_2^2\left({\mathrm{\Omega }}_0\right)}⩽C,$$

where C is another fixed constant. This estimate and (36), (36) yield that the functions $v_n$ converge in $W_2^1\left(\mathrm{\Omega }\right)\cap W_2^2\left({\mathrm{\Omega }}_0\right)$ to function v as $n\to +\infty$. Since the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ is m-sectorial, it is closed and the established convergence implies that $v\in \mathfrak{D}\left({\mathrm{\Omega }}_{\mathrm{\Omega }}\right)$ and

$$({\mathcal{H}}_{\mathrm{\Omega }}+c_1)v=f+c_{1}u.$$

(37)

The well-known embedding of $W_2^2\left({\mathrm{\Omega }}_0\right)$ into $C\left(\overline{{\mathrm{\Omega }}_0}\right)$ yields that $v_n\left(x_0\right)\to v\left(x_0\right)$. Hence, it follows from identity (35) that

$$u_n\to v+{(\beta -a)}^{-1}v\left(x_0\right)G\mathrm{in}{L}_2\left(\mathrm{\Omega }\right)\mathrm{as}n\to \infty$$

and $u=v+{(\beta -a)}^{-1}v\left(x_0\right)G$. This means that $u\in \mathfrak{D}\left({\mathcal{H}}_{0,\beta }\right)$ and in accordance with (8) and (37), we see that ${\mathcal{H}}_{0,\beta }u=f$ and conclude that operator ${\mathcal{H}}_{0,\beta }$ is closed.

We proceed in proving the m-sectoriality of the operator ${\mathcal{H}}_{\epsilon }$. We first define a sesquilinear form

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\epsilon }(u,v):=& {\mathfrak{h}}_{\mathrm{\Omega }}\left((1-\chi )u,(1-\chi )v\right)+{\mathfrak{h}}_{{\mathrm{\Omega }}_0\backslash {\omega }_{\epsilon }}\left(\chi u,(1-\chi )v\right)\hfill \\ \hfill & +{\mathfrak{h}}_{{\mathrm{\Omega }}_0\backslash {\omega }_{\epsilon }}\left((1-\chi )u,\chi v\right)+{\mathfrak{h}}_{{\mathrm{\Omega }}_0\backslash {\omega }_{\epsilon }}(\chi u,\chi v)-{({\ell }_{\epsilon }u,v)}_{L_2(\partial {\omega }_{\epsilon })}\hfill \end{array}$$

(38)

on the domain

$$\mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right):=\left\{u:(1-\chi )u\in \mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right),\chi u\in W_2^1({\mathrm{\Omega }}_0\backslash {\omega }_{\epsilon })\right\}.$$

(39)

Reproducing literal calculations in formulae (36)–(39) in [15] (Section 4.1), we easily confirm that this is the form associated with the operator ${\mathcal{H}}_{\epsilon }$, that is,

$${\mathfrak{h}}_{\epsilon }(u,v)={({\mathcal{H}}_{\epsilon }u,v)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}\mathrm{for}\mathrm{all}u\in \mathfrak{D}\left({\mathcal{H}}_{\epsilon }\right),v\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right).$$

For each $u\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$, we define

$${\langle u\rangle }_{B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon }}:=\frac{1}{mes{B}_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon }}\underset{B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon }}{\int }udx,u_{\perp }:=u-{\langle u\rangle }_{B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon }}.$$

The function $\tilde{u}\left(\xi \right):=u_{\perp }(x_0+\epsilon \xi )$ belongs to $W_2^1(B_{R_1}\left(0\right)\backslash \omega )$ and obviously satisfies the identity

$$\underset{B_{R_1}\left(0\right)\backslash \omega }{\int }\tilde{u}dx=0.$$

By the Poincaré inequality, we then have

$$\parallel \tilde{u}{\parallel }_{L_2(B_{R_1}\left(0\right)\backslash \omega )}⩽C{\parallel {\nabla }_{\xi }\tilde{u}\parallel }_{L_2(B_{R_1}\left(0\right)\backslash \omega )},$$

where C is some constant independent of $\tilde{u}$. Returning back to the function $u_{\perp }$ in the above inequality, we obtain

$$\parallel u_{\perp }{\parallel }_{L_2(B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon })}^2⩽C{\epsilon }^2{\parallel \nabla u\parallel }_{L_2(B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon })}^2$$

(40)

with the same constant C.

In the vicinity of the boundary $\partial {\omega }_{\epsilon }$, we introduce local coordinates $(\tau ,s_{\epsilon })$, where $s_{\epsilon }$ is the arc length of $\partial {\omega }_{\epsilon }$, while $\tau$ is the distance measured along the unit normal $\nu$ directed inside ${\omega }_{\epsilon }$. Since the boundary $\partial \omega$ is $C^3$-smooth, the introduced variables are well-defined as $dist(x,\partial {\omega }_{\epsilon })⩽\epsilon {\tau }_0$, where ${\tau }_0>0$ is some fixed number independent of $\epsilon$. By ${\chi }_1={\chi }_1\left(t\right)$, we denote an infinitely differentiable cut-off function equaling to one as $\left|t\right|<\frac{{\tau }_0}{3}$ and vanishing as $\left|t\right|>\frac{2{\tau }_0}{3}$. Then, we continue each function $u\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$ inside ${\omega }_{\epsilon }$ as follows:

$$u(s_{\epsilon },\tau ):={\langle u\rangle }_{B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon }}+{\chi }_1\left({\epsilon }^{-1}\tau \right)u_{\perp }(s_{\epsilon },-\tau )\mathrm{for}\tau >0.$$

(41)

Using estimate (40), it is straightforward to confirm that the continued function satisfies the estimates

$${\parallel u\parallel }_{L_2\left({\omega }_{\epsilon }\right)}⩽{C\parallel u\parallel }_{L_2(B_{2R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon })}{,\parallel \nabla u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽C{\parallel u\parallel }_{W_2^1(B_{2R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon })},$$

(42)

where C is some constant independent of $\epsilon$ and u. Until the end of this section, we suppose that each $u\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$ is continued in the aforementioned way; we keep the same notation for the continuation.

It follows from the definition of the forms ${\mathfrak{h}}_{\epsilon }$ and ${\mathfrak{h}}_{{\mathrm{\Omega }}_0}$ that

$${\mathfrak{h}}_{\epsilon }((1-\chi )u,v)={\mathfrak{h}}_{{\mathrm{\Omega }}_0}((1-\chi )u,v),{\mathfrak{h}}_{\epsilon }(u,(1-\chi )v)={\mathfrak{h}}_{{\mathrm{\Omega }}_0}(u,(1-\chi )v)$$

for all $u,v\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$ continued in accordance with (41). This identity allows us to rewrite formula (38) as follows:

$${\mathfrak{h}}_{\epsilon }(u,v):={\mathfrak{h}}_{\mathrm{\Omega }}(u,v)-{\mathfrak{h}}_{{\omega }_{\epsilon }}(u,v)-{({\ell }_{\epsilon }u,v)}_{L_2(\partial {\omega }_{\epsilon })}.$$

(43)

Inequalities (42), (27) and (2) with $\tilde{\mathrm{\Omega }}={\omega }_{\epsilon }$ and the definition of the form ${\mathfrak{h}}_{{\omega }_{\epsilon }}$ yield the estimates

$$|{\mathfrak{h}}_{{\omega }_{\epsilon }}{\left[u\right]|⩽C\parallel u\parallel }_{W_2^1\left({\omega }_{\epsilon }\right)}^2⩽C{\parallel u\parallel }_{W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)}^2⩽CRe({\mathfrak{h}}_{\mathrm{\Omega }}\left[u\right]-{\mathfrak{h}}_{{\omega }_{\epsilon }}\left[u\right]+c_1{\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2),$$

(44)

where C are some constants independent of $\epsilon$ and u. Employing this estimate and the sectoriality of the form ${\mathfrak{h}}_{\mathrm{\Omega }}$, we find:

$$|Im({\mathfrak{h}}_{\mathrm{\Omega }}\left[u\right]-{\mathfrak{h}}_{{\omega }_{\epsilon }}\left[u\right])|⩽CRe\left({\mathfrak{h}}_{\mathrm{\Omega }}\left[u\right]-{\mathfrak{h}}_{{\omega }_{\epsilon }}\left[u\right]+c_1{\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2\right),$$

(45)

where C is some constant independent of $\epsilon$ and u.

We consider an auxiliary boundary value problem:

$$\begin{array}{c}{∆}_{\xi }X=\frac{|\partial \omega |}{mes{B}_{R_1}\left(0\right)\backslash \omega }\mathrm{in}{B}_{R_1}\left(0\right)\backslash \omega ,\\ \frac{\partial X}{\partial \nu }=1\mathrm{on}\partial \omega ,\frac{\partial X}{\partial \nu }=0\mathrm{on}{B}_{R_1}\left(0\right),\end{array}$$

where $\nu$ is the unit normal directed outside $B_{R_1}\left(0\right)\backslash \omega$. Such a problem is obviously solvable, and due to the standard Schauder estimates, it belongs to $C^1\left(\overline{B_{R_1}\left(0\right)\backslash \omega }\right)$. We denote $X_{\epsilon }\left(x\right):=X\left({\epsilon }^{-1}(x-x_0)\right)$ and, using the above problem, we integrate by parts as follows:

$$\epsilon \underset{B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon }}{\int }{\left|u\right|}^2∆X_{\epsilon }dx={\parallel u\parallel }_{L_2(\partial {\omega }_{\epsilon })}^2-\epsilon \underset{B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon }}{\int }\nabla {\left|u\right|}^2·\nabla X_{\epsilon }dx.$$

Using then the Cauchy–Schwarz inequality, we easily obtain:

$${\parallel u\parallel }_{L_2(\partial {\omega }_{\epsilon })}^2⩽{\delta \parallel \nabla u\parallel }_{L_2(B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon })}^2+C({\epsilon }^{-1}+{\delta }^{-1}){\parallel u\parallel }_{L_2(B_{R_1\epsilon }\left(x_0\right)\backslash {\omega }_{\epsilon })}^2$$

For each $\delta >0$, where C is a positive constant independent of $\epsilon$, u and $\delta$. This estimate with $\delta$ replaced by $\delta \epsilon |ln\epsilon |$ and the definition of the operator ${\ell }_{\epsilon }$ yields

$$\begin{array}{cc}\hfill \left|{({\ell }_{\epsilon }u,u)}_{L_2(\partial {\omega }_{\epsilon })}\right|⩽& C{\epsilon }^{-1}{|ln\epsilon |}^{-1}{\parallel u\parallel }_{L_2(\partial {\omega }_{\epsilon })}^2\hfill \\ \hfill ⩽& {\delta \parallel \nabla u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2+C{\epsilon }^{-1}(1+{\delta }^{-1}{|ln\epsilon |}^{-1}{)\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2\hfill \end{array}$$

(46)

For each $\delta >0$, where C is a positive constant independent of $\epsilon$, u and $\delta$. This estimate with a sufficiently small but fixed $\delta$, (45) and (2) with $\tilde{\mathrm{\Omega }}={\omega }_{\epsilon }$ gives

$$|Im{\mathfrak{h}}_{\epsilon }\left[u\right]|⩽CRe{\mathfrak{h}}_{\epsilon }\left[u\right]+C{\epsilon }^{-2}{\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2,$$

where C are positive constants independent of $\epsilon$ and u. This proves the sectoriality of the form ${\mathfrak{h}}_{\epsilon }$.

Let us prove that the form ${\mathfrak{h}}_{\epsilon }$ is closed. Estimates (44) and (46) yield

$$Re{\mathfrak{h}}_{\epsilon }\left[u\right]+C{\epsilon }^{-2}{\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2⩾C{\parallel u\parallel }_{W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)}^2,$$

(47)

where C are some fixed constants independent of $\epsilon$ and u. Let $u_n\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$ be an arbitrary sequence such that $u_n\to u$ in $L_2\left({\mathrm{\Omega }}_{\epsilon }\right)$ and ${\mathfrak{h}}_{\epsilon }[u_n-u_m]\to 0$ as $n,m\to \infty$. Estimate (47) then immediately implies that the sequence $u_n$ is fundamental in $W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)$ and $u_n\to u$ in $W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)$ as $n\to \infty$ and $u\in W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)$. In view of inequality (42), the continuations of $u_n$ defined in accordance with (41) also converge to the continuation of the function u in $W_2^1\left(\mathrm{\Omega }\right)$ as $n\to \infty$. Therefore, due to formula (43) and inequalities (44) and (46), we obtain the convergence ${\mathfrak{h}}_{\mathrm{\Omega }}[u_n-u_m]\to 0$ as $n,m\to \infty$ and, hence, due to the closedness of the form ${\mathfrak{h}}_{\mathrm{\Omega }}$, the function u is an element of $\mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right)$ and ${\mathfrak{h}}_{\mathrm{\Omega }}[u_n-u]\to 0$ as $n\to \infty$. According to (39), this implies that u belongs to $\mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$, while formula (43) and the convergence of $u_n$ to u in $W_2^1\left(\mathrm{\Omega }\right)$ yield that ${\mathfrak{h}}_{\epsilon }[u_n-u]\to 0$ as $n\to \infty$. This proves that the form ${\mathfrak{h}}_{\epsilon }$ is closed. Hence, the operator ${\mathcal{H}}_{\epsilon }$ is m-sectorial.

5. Resolvent Sets

In this section, we prove the existence of the number ${\lambda }_0$ mentioned in the formulation of Theorem 1 for the operator ${\mathcal{H}}_{\epsilon }$, and we also provide a preliminary description of the resolvent set of the operator ${\mathcal{H}}_{0,\beta }$.

We begin with the latter operator. It follows immediately from inequality (27) that the half-plane

$$\mathrm{\Xi }:=\{\lambda \in \mathbb{C}:Re\lambda <{\lambda }_0\}$$

(48)

is in the resolvent set of the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ for each ${\lambda }_0<c_2-c_1$. Hence, for each $\lambda \in \mathrm{\Xi }$, the function $w=w(·,\lambda ):={({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )}^{-1}G$ is well-defined and belongs to $\mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$ and to $W_2^2\left({\mathrm{\Omega }}_0\right)$. Due to standard properties, the resolvent ${({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )}^{-1}$ is holomorphic in $\lambda \in \mathrm{\Xi }$ as a bounded operator from $L_2\left(\mathrm{\Omega }\right)$ to $W_2^1\left(\mathrm{\Omega }\right)$ and $W_2^2\left({\mathrm{\Omega }}_0\right)$. Due to the embedding $W_2^2\left({\mathrm{\Omega }}_0\right)\subset C\left(\overline{{\mathrm{\Omega }}_0}\right)$, the function $w(x_0,\lambda )$ is holomorphic in $\lambda \in \mathrm{\Xi }$ as a scalar function.

We choose $\lambda \in \mathrm{\Xi }$, and by straightforward calculations, we confirm that the resolvent equation $({\mathcal{H}}_{0,\beta }-\lambda )u=f$ possesses a solution

$$\begin{array}{c}u=v+\frac{v\left(x_0\right)}{\beta -a}G,v=v_0+\frac{c_1-\lambda }{\beta -a+(\lambda -c_1)w(x_0,\lambda )}w(x,\lambda ),\\ w(·,\lambda ):={({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )}^{-1}G,\end{array}$$

(49)

which is well-defined provided

$$\beta -a+(\lambda -c_1)w(x_0,\lambda )\ne 0.$$

(50)

The function in the left-hand side of the above inequality is holomorphic in the half-plane $\mathrm{\Xi }$ and it is non-zero as $\lambda =c_1\in \mathrm{\Xi }$. By the uniqueness theorem for the holomorphic functions we conclude that condition (50) is satisfied everywhere in $\mathrm{\Xi }$ except at most a countable set of points, which can accumulate at infinity only; we denote the latter set by $\theta$. Then, the first formula in (49) is that for the resolvent ${({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}$ for $\lambda \in \mathrm{\Xi }\backslash \theta$. Formula (49) and the standard smoothness improving estimate also imply the estimate

$$\parallel v_0{\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}+\parallel v_0{\parallel }_{W_2^2\left(B_{2R_1}\left(x_0\right)\right)}+|v_0\left(x_0\right){|⩽C\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)},$$

(51)

with some constant $C=C\left(\lambda \right)$ independent of f.

We proceed to the operator ${\mathcal{H}}_{\epsilon }$. For each $u\in W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)$, we let

$$u^{\perp }:=u-{\langle u\rangle }_G{\chi }_{2}G,{\langle u\rangle }_G:=\frac{{\langle u\rangle }_{\partial {\omega }_{\epsilon }}}{{\langle G\rangle }_{\partial {\omega }_{\epsilon }}},u=u^{\perp }+{\langle u\rangle }_G{\chi }_{2}G,$$

(52)

where ${\chi }_2={\chi }_2\left(x\right)$ is an infinitely differentiable cut-off function equaling to one on $B_{R_2}\left(x_0\right)$ and vanishing outside $B_{2R_2}\left(x_0\right)$ for some fixed positive $R_2<R_1$. Using then identities (34), we obtain:

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\epsilon }\left[u\right]+c_1{\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2=& {\mathfrak{h}}_{\epsilon }\left(u^{\perp }+{\langle u\rangle }_G{\chi }_{2}G,u\right)+c_1{\left(u^{\perp }+{\langle u\rangle }_G{\chi }_{2}G,u\right)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}\hfill \\ \hfill =& {\mathfrak{h}}_{\epsilon }(u^{\perp },u)+c_1{(u^{\perp },u)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}\hfill \\ \hfill & +{\langle u\rangle }_G{\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,u\right)}_{L_2(\partial {\omega }_{\epsilon })}\hfill \\ \hfill & +{\langle u\rangle }_G{((\widehat{\mathcal{H}}+c_1){\chi }_{2}G,u)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}\hfill \\ \hfill =& {\mathfrak{h}}_{\epsilon }[u^{\perp }]+c_1\parallel u^{\perp }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2+{\left|{\langle u\rangle }_G\right|}^2{\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,G\right)}_{L_2(\partial {\omega }_{\epsilon })}\hfill \\ \hfill & +\overline{{\langle u\rangle }_G}{\left(u^{\perp },\frac{\partial G}{\partial {\mathrm{n}}^{\mathrm{*}}}-{\ell }_{\epsilon }G\right)}_{L_2(\partial {\omega }_{\epsilon })}\hfill \\ \hfill & +{\langle u\rangle }_G{\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,u^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}\hfill \\ \hfill & +\overline{{\langle u\rangle }_G}{(u^{\perp },({\widehat{\mathcal{H}}}^{\mathrm{*}}+c_1){\chi }_{2}G)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}\hfill \\ \hfill & +{\langle u\rangle }_G{((\widehat{\mathcal{H}}+c_1){\chi }_{2}G,u)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}.\hfill \end{array}$$

(53)

It follows from asymptotics (28) and Equation (4) for the function G that

$$\parallel (\widehat{\mathcal{H}}+c_1){\chi }_2{G\parallel }_{L_2\left({\mathrm{\Omega }}_0\right)}⩽C;$$

Hereinafter, by C we denote various inessential constants independent of u, $\epsilon$ and spatial variables. Hence, in view of definition (33) of the differential expression ${\widehat{\mathcal{H}}}^{\mathrm{*}}$, asymptotics (28) and the assumed smoothness of the coefficients $A_j$, we have

$$({\widehat{\mathcal{H}}}^{\mathrm{*}}+c_1){\chi }_{2}G=({\widehat{\mathcal{H}}}^{\mathrm{*}}-\widehat{\mathcal{H}}){\chi }_{2}G+(\widehat{\mathcal{H}}+c_1){\chi }_{2}G=-2\sum _{j=1}^{2}Re{A}_j(x_0)\frac{\partial }{\partial x_j}{\chi }_{2}G+F_2,$$

where $F_2$ is some compactly supported function from $L_2\left(\mathrm{\Omega }\right)$. Integrating by parts, we easily find that

$$\begin{array}{cc}\hfill -2\sum _{j=1}^{2}Re{A}_j\left(x_0\right){\left(u^{\perp },\frac{\partial }{\partial x_j}{\chi }_{2}G\right)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}=& -2\sum _{j=1}^{2}Re{A}_j\left(x_0\right){(u^{\perp },{\nu }_{j}G)}_{L_2(\partial {\omega }_{\epsilon })}\hfill \\ \hfill & +2\sum _{j=1}^{2}Re{A}_j\left(x_0\right){\left(\frac{\partial u^{\perp }}{\partial x_j},{\chi }_{2}G\right)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}.\hfill \end{array}$$

By estimate (24) we then obtain:

$$\left|2\sum _{j=1}^{2}Re{A}_j\left(x_0\right){\left(u^{\perp },\frac{\partial }{\partial x_j}{\chi }_{2}G\right)}_{L_2\left(\mathrm{\Omega }\right)}\right|⩽C\left(\epsilon |ln\epsilon |+\parallel {\chi }_2{G\parallel }_{L_2\left(B_{2R_2}\left(x_0\right)\right)}\right){\parallel \nabla u^{\perp }\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })}.$$

Hence,

$$\begin{array}{cc}\hfill |& \overline{{\langle u\rangle }_G}{(u^{\perp },({\widehat{\mathcal{H}}}^{\mathrm{*}}+c_1)\chi G)}_{L_2({\mathrm{\Omega }}_{\epsilon })}+{\langle u\rangle }_G{((\widehat{\mathcal{H}}+c_1)\chi G,u)}_{L_2({\mathrm{\Omega }}_{\epsilon })}|\hfill \\ \hfill & ⩽C\left({\parallel u\parallel }_{L_2({\mathrm{\Omega }}_{\epsilon })}+(\epsilon |ln\epsilon |+\parallel {\chi }_2{G\parallel }_{L_2(B_{2R_2}(x_0))})\parallel \nabla u^{\perp }{\parallel }_{L_2(B_{2R_1}(x_0)\backslash {\omega }_{\epsilon })}\right)\left|{\langle u\rangle }_G\right|.\hfill \end{array}$$

Choosing $R_2$ as small but fixed, we can make the norm $\parallel {\chi }_2{G\parallel }_{L_2\left(B_{2R_2}\left(x_0\right)\right)}$ arbitrarily small. Having this fact and the above estimate in mind, we choose a fixed $R_2$ so that

$$\begin{array}{cc}\hfill |\overline{{\langle u\rangle }_G}{(u^{\perp },({\widehat{\mathcal{H}}}^{\mathrm{*}}+c_1)\chi G)}_{L_2({\mathrm{\Omega }}_{\epsilon })}& +{\langle u\rangle }_G{\left((\widehat{\mathcal{H}}+c_1)\chi G,u\right)}_{L_2({\mathrm{\Omega }}_{\epsilon })}|\hfill \\ \hfill & ⩽C({\parallel u\parallel }_{L_2({\mathrm{\Omega }}_{\epsilon })}^2+{\left|{\langle u\rangle }_G\right|}^2)+\frac{c_2}{3}{\parallel \nabla u^{\perp }\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })}^2.\hfill \end{array}$$

(54)

Using the definition of the operator ${\ell }_{\epsilon }$ in (10), asymptotic (5) and condition (11), by straightforward calculations we find:

$$\begin{array}{cc}\hfill \frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G=& {\epsilon }^{-1}{ln}^{-1}\epsilon {\Phi }_1({\epsilon }^{-1}·)+{\epsilon }^{-1}{ln}^{-2}\epsilon {\Phi }_2(·,\epsilon ),\hfill \\ \hfill \frac{\partial G}{\partial {\mathrm{n}}^{\mathrm{*}}}-{\ell }_{\epsilon }G=& {\epsilon }^{-1}{ln}^{-1}\epsilon {\Phi }_1({\epsilon }^{-1}·)+{\epsilon }^{-1}{ln}^{-2}\epsilon {\Phi }_3(·,\epsilon ),\hfill \end{array}$$

(55)

where ${\Phi }_2$ and ${\Phi }_3$ are continuous on $\partial {\omega }_{\epsilon }$ functions bounded uniformly in $\epsilon$, while the function ${\Phi }_1$ is given by the identity

$${\Phi }_1\left(s\right):=-\left({\alpha }_0\left(s\right)+{\alpha }_2\left(s\right)\right)\left(ln|{\mathrm{A}}^{-\frac{1}{2}}\mathrm{x}\left(s\right)|+a\right)-{\alpha }_1\left(s\right).$$

Using these identities and (14), we see that

$${\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,G\right)}_{L_2(\partial {\omega }_{\epsilon })}=-K_0-K_{1}a+{O\left(\right|ln\epsilon |}^{-1}).$$

(56)

Estimate (24) and identity (55) also give:

$$\left|{\left(u^{\perp },\frac{\partial G}{\partial {\mathrm{n}}^{\mathrm{*}}}-{\ell }_{\epsilon }G\right)}_{L_2(\partial {\omega }_{\epsilon })}\right|+\left|{\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,u^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}\right|⩽{C|ln\epsilon |}^{-1}{\parallel \nabla u^{\perp }\parallel }_{L_2\left({\mathrm{\Omega }}_0\right)}.$$

(57)

We continue the function $u^{\perp }$ in accordance with (41) and use representation (43) with $u=v=u^{\perp }$. Then, we apply inequality (27) and we obtain:

$$Re{\mathfrak{h}}_{\epsilon }\left[u^{\perp }\right]+c_1\parallel u^{\perp }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2⩾c_2{\parallel u^{\perp }\parallel }_{W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)}^2-Re{({\ell }_{\epsilon }{u}^{\perp },u^{\perp })}_{L_2(\partial {\omega }_{\epsilon })}.$$

(58)

Since the function $u^{\perp }$ satisfies condition (23), we find that

$${\ell }_{\epsilon }{u}^{\perp }={\epsilon }^{-1}\left(\alpha ({\epsilon }^{-1}·,{ln}^{-1}\epsilon )+{ln}^{-1}\epsilon {\alpha }_2({\epsilon }^{-1}·)\right)u^{\perp }-{\epsilon }^{-1}{\alpha }_2({\epsilon }^{-1}·){\langle {\alpha }_3({\epsilon }^{-1}·)u^{\perp }\rangle }_{\partial {\omega }_{\epsilon }}$$

(59)

and by estimate (24), we obtain

$$\left|{\left({\epsilon }^{-1}\left(\alpha ({\epsilon }^{-1}·,{ln}^{-1}\epsilon )+{ln}^{-1}\epsilon {\alpha }_2({\epsilon }^{-1}·)\right)u^{\perp },u^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}\right|⩽{C|ln\epsilon |}^{-1}{\parallel \nabla u^{\perp }\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2,$$

(60)

while (12) implies

$$\begin{array}{cc}\hfill Re{\left({\epsilon }^{-1}{\alpha }_2({\epsilon }^{-1}·){\langle {\alpha }_3({\epsilon }^{-1}·)u^{\perp }\rangle }_{\partial {\omega }_{\epsilon }},u^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}=& |\partial \omega |Re{\langle {\alpha }_3({\epsilon }^{-1}·)u^{\perp }\rangle }_{\partial {\omega }_{\epsilon }}{\langle {\alpha }_2({\epsilon }^{-1}·)\overline{u^{\perp }}\rangle }_{\partial {\omega }_{\epsilon }}\hfill \\ \hfill =& |\partial \omega ||{\langle {\alpha }_3({\epsilon }^{-1}·)u^{\perp }\rangle }_{\partial {\omega }_{\epsilon }}{|}^2>0.\hfill \end{array}$$

Hence, due to (58)–(60),

$$Re{\mathfrak{h}}_{\epsilon }\left[u^{\perp }\right]+c_1\parallel u^{\perp }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2⩾(c_2-{C|ln\epsilon |}^{-1})\parallel \nabla u^{\perp }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2.$$

This inequality and (56), (54) and (57) allow us to estimate from below the form in (53):

$$\begin{array}{cc}\hfill Re{\mathfrak{h}}_{\epsilon }\left[u\right]+c_1{\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2⩾& \left(\frac{2c_2}{3}-C{|ln\epsilon |}^{-1}\right)\parallel u^{\perp }{\parallel }_{W_2^1\left({\mathrm{\Omega }}_{\epsilon }\right)}^2+c_1{\parallel u^{\perp }\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2\hfill \\ \hfill & -C{\delta }^{-1}{\parallel u\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2+Re(-K_0-K_{1}a-\delta -{C|ln\epsilon |}^{-1}{\left)\right|\langle u\rangle }_G{|}^2,\hfill \end{array}$$

(61)

where $\delta >0$ can be chosen arbitrary and C are some constants independent of u, $\epsilon$ and $\delta$. Hence, in view of condition (15), we can choose a sufficiently small but fixed $\delta$ and conclude that there exists a fixed ${\lambda }_0\in \mathbb{R}$ independent of $\epsilon$ such that for all $\lambda$ in the corresponding half-plane $\mathrm{\Xi }$ defined in (48) we have the estimate

$$Re\left({\mathfrak{h}}_{\epsilon }\left[u\right]+\lambda {\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2\right)⩾c_3\left(\parallel \nabla u^{\perp }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2{+|\langle u\rangle }_G{|}^2+{\parallel u\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2\right),$$

(62)

where $c_3$ is some fixed positive constant independent of $\epsilon$ and u. This estimate and the m-sectoriality of the operator ${\mathcal{H}}_{\epsilon }$ then ensures that the half-plane $\mathrm{\Xi }$ is in the resolvent set of this operator.

6. Convergence

In this section, we prove estimates (17), (18) and (20). Since the half-plane $\mathrm{\Xi }$ is in the resolvent set of the operator ${\mathcal{H}}_{0,\beta }$ except for many points accumulating at infinity only, and this half-plane is also a part of the resolvent set of the operator ${\mathcal{H}}_{\epsilon }$, we can choose a fixed $\lambda \in \mathrm{\Xi }$, which is in the resolvent sets of both operators. In what follows, the number $\lambda$ is supposed to be chosen exactly in this way.

We take an arbitrary $f\in L_2\left(\mathrm{\Omega }\right)$ and we let $u_0:={({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}f$, $u_{\epsilon }:={({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}f$; in the latter definition by f, we naturally mean the restriction of the function f to ${\mathrm{\Omega }}_{\epsilon }$. Then, the function $w_{\epsilon }=u_{\epsilon }-u_0$ is an element of $\mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$. We continue this function inside ${\omega }_{\epsilon }$ in accordance with (41) and the continuation, denoted by the same symbol, belongs to $\mathfrak{D}\left({\mathfrak{h}}_{\mathrm{\Omega }}\right)$.

The integral identity corresponding to the equation for $u_{\epsilon }$ with a test function $w\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$ reads as

$${\mathfrak{h}}_{\epsilon }(u_{\epsilon },w)-\lambda {(u_{\epsilon },w)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}={(f,w)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}.$$

(63)

According to (7), we represent the function $u_0\in \mathfrak{D}\left({\mathcal{H}}_{0,\beta }\right)$ as

$$u_0=v_0+{(\beta -a)}^{-1}{v}_0\left(x_0\right)G,v_0\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right),$$

and by (8), we then have

$$({\mathcal{H}}_{0,\beta }-\lambda )u_0=({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )v_0-{(\beta -a)}^{-1}(\lambda +c_1)v_0\left(x_0\right)G=f.$$

(64)

We continue the function w from (63) in accordance with (41) and then write the associated integral identity for the operator $({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )$ with the continued test function $w_{\epsilon }$:

$${\mathfrak{h}}_{\mathrm{\Omega }}(v_0,w)-\lambda {(v_0,w)}_{L_2\left(\mathrm{\Omega }\right)}={(f,w)}_{L_2\left(\mathrm{\Omega }\right)}+{(\beta -a)}^{-1}(\lambda +c_1)v_0\left(x_0\right){(G,w)}_{L_2\left(\mathrm{\Omega }\right)}$$

and hence, in view of (43),

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\epsilon }(v_0,w)-\lambda {(v_0,w)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}=& {(f,w)}_{L_2\left(\mathrm{\Omega }\right)}+{(\beta -a)}^{-1}(\lambda +c_1)v_0\left(x_0\right){(G,w)}_{L_2\left(\mathrm{\Omega }\right)}\hfill \\ \hfill & -{({\ell }_{\epsilon }{v}_0,w)}_{L_2(\partial {\omega }_{\epsilon })}-{\mathfrak{h}}_{{\omega }_{\epsilon }}(v_0,w)+\lambda {(v_0,w)}_{L_2\left({\omega }_{\epsilon }\right)}.\hfill \end{array}$$

(65)

Integrating by parts, we easily find that

$${\mathfrak{h}}_{{\omega }_{\epsilon }}(v_0,w)-\lambda {(v_0,w)}_{L_2\left({\omega }_{\epsilon }\right)}={((\widehat{\mathcal{H}}-\lambda )v_0,w)}_{L_2\left({\omega }_{\epsilon }\right)}+{\left(\frac{\partial v_0}{\partial \mathrm{n}},w\right)}_{L_2(\partial {\omega }_{\epsilon })},$$

and therefore, in view of (64), we can rewrite identity (65) as

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\epsilon }(v_0,w)-\lambda {(v_0,w)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}=& {(f,w)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}+{(\beta -a)}^{-1}(\lambda +c_1)v_0\left(x_0\right){(G,w)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}\hfill \\ \hfill & +{\left(\frac{\partial v_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{v}_0,w\right)}_{L_2(\partial {\omega }_{\epsilon })}.\hfill \end{array}$$

(66)

It follows from the definition of the cut-off function $\chi$ and formulae (29) and (43) that

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\epsilon }\left((1-\chi )G,w\right)& -\lambda {\left((1-\chi )G,w\right)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}={\mathfrak{h}}_{\mathrm{\Omega }}\left((1-\chi )G,w\right)-\lambda {\left((1-\chi )G,w\right)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}\hfill \\ \hfill =& -{((\widehat{\mathcal{H}}+c_1)\chi G,w)}_{L_2({\mathrm{\Omega }}_{\epsilon }}-(\lambda +c_1){\left((1-\chi )G,w\right)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}.\hfill \end{array}$$

Using then the first identity in (34), we find:

$${\mathfrak{h}}_{\epsilon }(G,w)-\lambda {(G,w)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}={\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,w\right)}_{L_2(\partial {\omega }_{\epsilon })}-(\lambda +c_1){\left(G,w\right)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}.$$

We multiply this identity by ${(\beta -a)}^{-1}{v}_0\left(x_0\right)$ and deduct it and (66) from (63). This gives:

$${\mathfrak{h}}_{\epsilon }(w_{\epsilon },w)-\lambda {(w_{\epsilon },w)}_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}=-{\left(\frac{\partial u_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{u}_0,w\right)}_{L_2(\partial {\omega }_{\epsilon })}.$$

(67)

For $w=w_{\epsilon }$ the above identity becomes:

$${\mathfrak{h}}_{\epsilon }\left[w_{\epsilon }\right]-\lambda {\parallel w_{\epsilon }\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}^2=-{\left(\frac{\partial u_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{u}_0,w_{\epsilon }\right)}_{L_2(\partial {\omega }_{\epsilon })}.$$

(68)

Our next step is to estimate the right hand side of the obtained identity.

We represent the function $w_{\epsilon }$ in accordance with (52)

$$w_{\epsilon }^{\perp }:=w_{\epsilon }-{\langle w_{\epsilon }\rangle }_G\chi G,w_{\epsilon }=w_{\epsilon }^{\perp }+{\langle w_{\epsilon }\rangle }_G\chi G$$

(69)

and we rewrite the right hand side in (68) as follows:

$$\begin{array}{cc}\hfill {\left(\frac{\partial u_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{u}_0,w_{\epsilon }\right)}_{L_2(\partial {\omega }_{\epsilon })}=& {\left(\frac{\partial v_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{v}_0,w_{\epsilon }^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}+{\langle w_{\epsilon }\rangle }_G{\left(\frac{\partial v_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{v}_0,G\right)}_{L_2(\partial {\omega }_{\epsilon })}\hfill \\ \hfill & +{(\beta -a)}^{-1}{v}_0\left(x_0\right){\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,w_{\epsilon }^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}\hfill \\ \hfill & +{(\beta -a)}^{-1}{v}_0\left(x_0\right){\langle w_{\epsilon }\rangle }_G{\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,G\right)}_{L_2(\partial {\omega }_{\epsilon })}.\hfill \end{array}$$

(70)

Conditions (11), Lemma 3 with $\phi ={\alpha }_3$ and inequality (51) yield the estimate

$$|{\langle {\alpha }_3({\epsilon }^{-1}·)v_0\rangle }_{\partial {\omega }_{\epsilon }}|⩽C{\epsilon }^{\frac{1}{2}}{|ln\epsilon |\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)};$$

(71)

Hereinafter in this section, by C we denote various inessential constants independent of $v_0$, $u_0$, f, $w_{\epsilon }$, $\epsilon$ and spatial variables. Since the function $w_{\epsilon }^{\perp }$ satisfies condition (23), it follows the definition of the operator ${\ell }_{\epsilon }$ in (10), the first identity in (55) and estimates (24)–(26), (51) and (71) that

$$\begin{array}{cc}\hfill & \left|{\left(\frac{\partial v_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{v}_0,w_{\epsilon }^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}\right|⩽{C|ln\epsilon |}^{-1}\parallel v_0{\parallel }_{W_2^2\left(B_{2R_1\left(x_0\right)}\right)}{\parallel \nabla w_{\epsilon }^{\perp }\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })}\hfill \\ \hfill & \phantom{\left|{\left(\frac{\partial v_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{v}_0,w_{\epsilon }^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}\right|}⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}{\parallel \nabla w_{\epsilon }^{\perp }\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })},\hfill \\ \hfill & \left|{(\beta -a)}^{-1}{v}_0\left(x_0\right){\left(\frac{\partial G}{\partial \mathrm{n}}-{\ell }_{\epsilon }G,w_{\epsilon }^{\perp }\right)}_{L_2(\partial {\omega }_{\epsilon })}\right|⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}{\parallel \nabla w_{\epsilon }^{\perp }\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })}.\hfill \end{array}$$

(72)

In the same way, taking into consideration asymptotic (5), we obtain:

$$\begin{array}{cc}\hfill & \left|{\langle w_{\epsilon }\rangle }_G{\left(\frac{\partial v_0}{\partial \mathrm{n}},G\right)}_{L_2(\partial {\omega }_{\epsilon })}\right|⩽{C\epsilon |ln\epsilon |}^{\frac{3}{2}}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}\left|{\langle w_{\epsilon }\rangle }_G\right|,\hfill \\ \hfill & \left|{\langle w_{\epsilon }\rangle }_G{\left({\ell }_{\epsilon }{v}_0,G-ln\epsilon \right)}_{L_2(\partial {\omega }_{\epsilon })}\right|⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}\left|{\langle w_{\epsilon }\rangle }_G\right|.\hfill \end{array}$$

(73)

It follows from Lemma 3 with $\phi \equiv 1$ and estimate (51) that

$$|{\langle v_0\rangle }_{\partial {\omega }_{\epsilon }}-v_0\left(x_0\right)|⩽{C\epsilon |ln\epsilon |}^{\frac{1}{2}}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}.$$

(74)

The definition of the operator ${\ell }_{\epsilon }$ yields

$$\begin{array}{cc}\hfill -{({\ell }_{\epsilon }{v}_0,ln\epsilon )}_{L_2(\partial {\omega }_{\epsilon })}=& -\underset{\partial {\omega }_{\epsilon }}{\int }{\epsilon }^{-1}\left(\alpha ({\epsilon }^{-1}{s}_{\epsilon },{ln}^{-1}\epsilon )ln\epsilon +{\alpha }_2\left({\epsilon }^{-1}{s}_{\epsilon }\right)\right)v_0\left(x\right)d{s}_{\epsilon }\hfill \\ \hfill & +{\epsilon }^{-1}{\langle {\alpha }_3({\epsilon }^{-1}·)v_0\rangle }_{\partial {\omega }_{\epsilon }}ln\epsilon \underset{\partial {\omega }_{\epsilon }}{\int }{\alpha }_2\left({\epsilon }^{-1}{s}_{\epsilon }\right)d{s}_{\epsilon }.\hfill \end{array}$$

Using Lemma 3 with $\phi =\alpha (·,{ln}^{-1}\epsilon )$, Lemma 1 and (5), (74) and (71), we obtain:

$$|{\langle w_{\epsilon }\rangle }_G{({\ell }_{\epsilon }{v}_0,ln\epsilon )}_{L_2(\partial {\omega }_{\epsilon })}-v_0\left(x_0\right)K_1{\langle w_{\epsilon }\rangle }_G|⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}\left|{\langle w_{\epsilon }\rangle }_G\right|.$$

(75)

This inequality, (56) and (16) imply

$$\begin{array}{cc}\hfill \left|{(\beta -a)}^{-1}{v}_0\left(x_0\right){\langle w_{\epsilon }\rangle }_G\right(\frac{\partial G}{\partial \mathrm{n}}& -{\ell }_{\epsilon }G,G{)}_{L_2(\partial {\omega }_{\epsilon })}-v_0\left(x_0\right)K_1{\langle w_{\epsilon }\rangle }_G|\hfill \\ \hfill & ⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}\left|{\langle w_{\epsilon }\rangle }_G\right|.\hfill \end{array}$$

This estimate, (75), (73), (72) and (70) give the desired estimate for the right hand side in (68):

$$\left|{\left(\frac{\partial u_0}{\partial \mathrm{n}}-{\ell }_{\epsilon }{u}_0,w_{\epsilon }\right)}_{L_2(\partial {\omega }_{\epsilon })}\right|⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}(\parallel \nabla w_{\epsilon }^{\perp }{\parallel }_{L_2(B_{2R_1}\left(x_0\right)\backslash {\omega }_{\epsilon })}+|{\langle w_{\epsilon }\rangle }_G|).$$

We take the real part of identity (68) and use the above estimate and (62) with $u=w_{\epsilon }$. This yields:

$$\parallel \nabla w_{\epsilon }^{\perp }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}+|{\langle w_{\epsilon }\rangle }_G|+\parallel w_{\epsilon }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}.$$

(76)

It follows from asymptotic (5) that

$${\parallel \chi G\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽{C,\parallel \nabla \chi G\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽{C|ln\epsilon |}^{\frac{1}{2}},{\parallel \chi G\parallel }_{W_2^1\left(\tilde{\mathrm{\Omega }}\right)}⩽C,$$

(77)

where a domain $\tilde{\mathrm{\Omega }}$ was described in the formulation of Theorem 1. These estimates, (76) and (69) imply

$$\begin{array}{cc}\hfill & \parallel w_{\epsilon }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)},\parallel \nabla w_{\epsilon }{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽{C|ln\epsilon |}^{-\frac{1}{2}}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)},\hfill \\ \hfill & \parallel \nabla w_{\epsilon }{\parallel }_{L_2\left(\tilde{\mathrm{\Omega }}\right)}⩽{C|ln\epsilon |}^{-1}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)},\hfill \end{array}$$

(78)

and these inequalities prove (17)–(19) for the considered values of $\lambda$.

We choose an arbitrary subdomain $\tilde{\mathrm{\Omega }}\subset \mathrm{\Omega }$ separated from $x_0$ by a positive distance and such that the domain $\mathrm{\Omega }\backslash \tilde{\mathrm{\Omega }}$ is bounded. Let ${\chi }_{\tilde{\mathrm{\Omega }}}$ be an infinitely differentiable cut-off function equaling to one in $\mathrm{\Omega }\backslash B_{2R_3\left(x_0\right)}$ and vanishing on $B_{R_3}\left(x_0\right)$, where $R_3$ is some fixed number such that $\tilde{\mathrm{\Omega }}\subset \mathrm{\Omega }\backslash B_{2R_3\left(x_0\right)}$. Assuming that the function $w_{\epsilon }$ is continued inside ${\omega }_{\epsilon }$ in accordance with (41), we write identity (67) with $w={\chi }_{\tilde{\mathrm{\Omega }}}^2{w}_{\epsilon }$ and use Formula (43) to obtain:

$${\mathfrak{h}}_{\mathrm{\Omega }}(w_{\epsilon },{\chi }_{\tilde{\mathrm{\Omega }}}^2{w}_{\epsilon })-\lambda {\parallel {\chi }_{\tilde{\mathrm{\Omega }}}{w}_{\epsilon }\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2=0.$$

Using an obvious identity

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\mathrm{\Omega }}(w_{\epsilon },{\chi }_{\tilde{\mathrm{\Omega }}}^2{w}_{\epsilon })=& {\mathfrak{h}}_{\mathrm{\Omega }}\left[{\chi }_{\tilde{\mathrm{\Omega }}}{w}_{\epsilon }\right]+\sum _{i,j=1}^2{\left(A_{ij}\frac{\partial {\omega }_{\epsilon }}{\partial x_j},{\chi }_{\tilde{\mathrm{\Omega }}}{w}_{\epsilon }\frac{\partial {\chi }_{\tilde{\mathrm{\Omega }}}}{\partial x_i}\right)}_{L_2\left(\mathrm{\Omega }\right)}\hfill \\ \hfill & -\sum _{i,j=1}^2{\left(A_{ij}\frac{\partial {\omega }_{\epsilon }}{\partial x_j},{\chi }_{\tilde{\mathrm{\Omega }}}{w}_{\epsilon }\frac{\partial {\chi }_{\tilde{\mathrm{\Omega }}}}{\partial x_i}\right)}_{L_2\left(\mathrm{\Omega }\right)}-\sum _{j=1}^2{\left(A_j{w}_{\epsilon }\frac{\partial {\chi }_{\tilde{\mathrm{\Omega }}}}{\partial x_j},{\chi }_{\tilde{\mathrm{\Omega }}}{w}_{\epsilon }\right)}_{L_2\left(\mathrm{\Omega }\right)}\hfill \end{array}$$

and estimates (78), we immediately obtain:

$$\left|{\mathfrak{h}}_{\tilde{\mathrm{\Omega }}}\left[{\chi }_{\tilde{\mathrm{\Omega }}}{w}_{\epsilon }\right]\right|⩽{C|ln\epsilon |}^{-2}{\parallel f\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2,$$

and employing estimates (78) and the definition of the cut-off function ${\chi }_{\tilde{\mathrm{\Omega }}}$, we obtain estimate (20).

Now we are going to show that the half-plane $\mathrm{\Xi }$ is a subset of the resolvent set of the operator ${\mathcal{H}}_{0,\beta }$. In other words, we are going to prove that the set $\theta$ introduced in Section 5 is empty. We argue by contradiction. Let ${\lambda }_{\mathrm{*}}$ be a point in $\theta \subset \mathrm{\Xi }$. Then, the resolvent ${({\mathcal{H}}_{\epsilon }-{\lambda }_{\mathrm{*}})}^{-1}$ is well-defined and by (62), (52) and (77) we see that

$$\parallel {({\mathcal{H}}_{\epsilon }-{\lambda }_{\mathrm{*}})}^{-1}{\parallel }_{L_2\left({\mathrm{\Omega }}_{\epsilon }\right)\to L_2\left({\mathrm{\Omega }}_{\epsilon }\right)}⩽C.$$

By ${\mathcal{A}}_{\epsilon }$ we denote the operator of multiplication by ${\epsilon }^{-2}$ in $L_2\left({\omega }_{\epsilon }\right)$. Then, it follows from the above inequality that

$$\parallel {({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-{\lambda }_{\mathrm{*}})}^{-1}{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left(\mathrm{\Omega }\right)}⩽C.$$

(79)

Then, we choose and fix some $\lambda \in \mathrm{\Xi }\backslash \theta$ and due to the embedding $C\left(\overline{B_{R_1}\left(x_0\right)}\right)\subset W_2^2\left(B_{R_1}\left(x_0\right)\right)$ and (51), (5) and (49), we have the estimate

$$\parallel {({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left({\omega }_{\epsilon }\right)}⩽C\epsilon |ln\epsilon |.$$

This estimate, the definition of the operator ${\mathcal{A}}_{\epsilon }$ and (17) yield

$$\parallel {\mathcal{L}}_{\epsilon }{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left(\mathrm{\Omega }\right)}⩽C{|ln\epsilon |}^{-1},{\mathcal{L}}_{\epsilon }:={({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}.$$

(80)

We consider the resolvent equation

$$({\mathcal{H}}_{0,\beta }-{\lambda }_{\mathrm{*}})u=f,$$

(81)

which we immediately rewrite as

$$u+(\lambda -{\lambda }_{\mathrm{*}}){({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}u={({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}f$$

and then, in view of the definition of the operator ${\mathcal{L}}_{\epsilon }$ in (80),

$$(\mathcal{I}+(\lambda -{\lambda }_{\mathrm{*}})({({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-\lambda )}^{-1}-{\mathcal{L}}_{\epsilon }))u={({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}f,$$

(82)

where $\mathcal{I}$ stands for the identity mapping. The obvious identities

$$\begin{array}{cc}\hfill & \mathcal{I}+(\lambda -{\lambda }_{\mathrm{*}}){({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-\lambda )}^{-1}={({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-\lambda )}^{-1}({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-{\lambda }_{\mathrm{*}}),\hfill \\ \hfill & {\mathcal{Q}}_{\epsilon }:={\left(\mathcal{I}+(\lambda -{\lambda }_{\mathrm{*}}){({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-\lambda )}^{-1}\right)}^{-1}=\mathcal{I}+({\lambda }_{\mathrm{*}}-\lambda ){({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-{\lambda }_{\mathrm{*}})}^{-1}\hfill \end{array}$$

allow us to rewrite Equation (82) as

$$\left(\mathcal{I}-(\lambda -{\lambda }_{\mathrm{*}}){\mathcal{Q}}_{\epsilon }{\mathcal{L}}_{\epsilon }\right)u={\mathcal{Q}}_{\epsilon }{({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}f.$$

(83)

Estimates (79) and (80) yield that

$$\parallel (\lambda -{\lambda }_{\mathrm{*}}){\mathcal{Q}}_{\epsilon }{\mathcal{L}}_{\epsilon }{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left(\mathrm{\Omega }\right)}⩽C{|ln\epsilon |}^{-1}$$

and this allows us to solve Equation (83):

$$u={\left(\mathcal{I}-(\lambda -{\lambda }_{\mathrm{*}}){\mathcal{Q}}_{\epsilon }{\mathcal{L}}_{\epsilon }\right)}^{-1}{\mathcal{Q}}_{\epsilon }{({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}f.$$

(84)

This is the solution for Equation (81) and hence, the resolvent ${({\mathcal{H}}_{0,\beta }-{\lambda }_{\mathrm{*}})}^{-1}$ is well-defined. Hence, $\theta =\varnothing$ and estimates (17)–(20) are valid for all $\lambda \in \mathrm{\Xi }$.

7. Spectrum

In this section, we study the spectrum of the operator ${\mathcal{H}}_{\epsilon }$ and prove Theorem 2. First, we are going to show that the essential spectra of the operators ${\mathcal{H}}_{\mathrm{\Omega }}$, ${\mathcal{H}}_{0,\beta }$ and ${\mathcal{H}}_{\epsilon }$ coincide.

Given $\lambda \in {\sigma }_{ess}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$, let $u_n\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$ be a corresponding characteristic sequence, that is,

$$({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )u_n=:f_n\to 0\mathrm{in}{L}_2\left(\mathrm{\Omega }\right)\mathrm{as}n\to \infty ,$$

and the sequence $u_n$ is bounded and non-compact in $L_2\left(\mathrm{\Omega }\right)$. It follows from the identity

$${\mathfrak{h}}_{\mathrm{\Omega }}\left[u_n\right]-\lambda {\parallel u_n\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2={(f_n,u_n)}_{L_2\left(\mathrm{\Omega }\right)}$$

and inequality (27) that the sequence $u_n$ is also bounded in $W_2^1\left(\mathrm{\Omega }\right)$. Applying then standard smoothness improving theorems, we also find that this sequence is also bounded in $W_2^2\left({\mathrm{\Omega }}_0\right)$. Employing the compact embedding of $W_2^2\left({\mathrm{\Omega }}_0\right)$ into $W_2^1\left({\mathrm{\Omega }}_0\right)$ and the boundedness of the sequence $u_n$ in $L_2\left(\mathrm{\Omega }\right)$, we can select its subsequence, also denoted by $u_n$, which converges weakly in $W_2^1\left(\mathrm{\Omega }\right)$ and $W_2^2\left({\mathrm{\Omega }}_0\right)$ and strongly in $W_2^1\left({\mathrm{\Omega }}_0\right)$ as $n\to \infty$ to the function $u_{†}\in W_2^1\left(\mathrm{\Omega }\right)$. The operator ${\mathcal{H}}_{\mathrm{\Omega }}$ is densely defined and, hence, the adjoint operator ${\mathcal{H}}_{\mathrm{\Omega }}^{\mathrm{*}}$ is well-defined. Then, for each function $v\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}^{\mathrm{*}}\right)$, we have

$${(f_n,v)}_{L_2\left(\mathrm{\Omega }\right)}={\left(({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )u_n,v\right)}_{L_2\left(\mathrm{\Omega }\right)}={\left(u_n,({\mathcal{H}}_{\mathrm{\Omega }}^{\mathrm{*}}-\overline{\lambda })v\right)}_{L_2\left(\mathrm{\Omega }\right)}$$

Passing to the limit as $n\to \infty$ in the above identity, due to the weak convergence of $u_n$, we immediately obtain

$${\left(u_{†},({\mathcal{H}}_{\mathrm{\Omega }}^{\mathrm{*}}-\overline{\lambda })v\right)}_{L_2\left(\mathrm{\Omega }\right)}=0\mathrm{for}\mathrm{all}v\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}^{\mathrm{*}}\right).$$

Since the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ is closed, we have ${\mathcal{H}}_{\mathrm{\Omega }}^{\mathrm{*}\mathrm{*}}={\mathcal{H}}_{\mathrm{\Omega }}$, and by the above identity, we conclude that

$$u_{†}\in \mathfrak{D}\left({\mathcal{H}}_{\mathrm{\Omega }}^{\mathrm{*}}\right),({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )u_{†}=0.$$

We let ${\tilde{u}}_n:=u_n-u_{†}$, and we see immediately that ${\tilde{u}}_n$ is also a characteristic sequence of the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ corresponding to $\lambda$. Moreover,

$$({\mathcal{H}}_{\mathrm{\Omega }}-\lambda ){\tilde{u}}_n=f_n$$

(85)

and ${\tilde{u}}_n\to 0$ weakly in $W_2^1\left(\mathrm{\Omega }\right)$ and strongly in $W_2^1\left({\mathrm{\Omega }}_0\right)$. Then, the sequence $\chi {\tilde{u}}_n$ converges to zero strongly in $W_2^1\left(\mathrm{\Omega }\right)$. We then introduce one more sequence ${\widehat{u}}_n:=(1-\chi )u_n$ and in view of (85), we find that it is also a characteristic sequence of the operator ${\mathcal{H}}_{\mathrm{\Omega }}$ corresponding to $\lambda$. A specific feature of the latter sequence is that each function ${\widehat{u}}_n$ vanishes on $B_{R_1}\left(x_0\right)$. Using this fact, we easily see that the sequence ${\widehat{u}}_n$ is in the domains of both operators ${\mathcal{H}}_{0,\beta }$ and ${\mathcal{H}}_{\epsilon }$ and

$$({\mathcal{H}}_{0,\beta }-\lambda ){\widehat{u}}_n=({\mathcal{H}}_{\epsilon }-\lambda ){\widehat{u}}_n=f_n\to 0\mathrm{as}n\to \infty .$$

Hence, $\lambda \in {\sigma }_{ess}\left({\mathcal{H}}_{0,\beta }\right)$ and $\lambda \in {\sigma }_{ess}\left({\mathcal{H}}_{\epsilon }\right)$ and therefore,

$${\sigma }_{ess}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)\subseteq {\sigma }_{ess}\left({\mathcal{H}}_{0,\beta }\right),{\sigma }_{ess}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)\subseteq {\sigma }_{ess}\left({\mathcal{H}}_{\epsilon }\right).$$

(86)

In the same way as above, but employing estimate (61) instead of (27), we also establish that

$${\sigma }_{ess}\left({\mathcal{H}}_{\epsilon }\right)\subseteq {\sigma }_{ess}\left({\mathcal{H}}_{0,\beta }\right),{\sigma }_{ess}\left({\mathcal{H}}_{\epsilon }\right)\subseteq {\sigma }_{ess}\left({\mathcal{H}}_{\mathrm{\Omega }}\right).$$

It remains to show that ${\sigma }_{ess}\left({\mathcal{H}}_{0,\beta }\right)\subseteq {\sigma }_{ess}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$. Let $\lambda \in {\sigma }_{ess}\left({\mathcal{H}}_{0,\beta }\right)$ and $u_n$ be a corresponding characteristic sequence. Then, each function $u_n$ satisfies representation (35) and

$$({\mathcal{H}}_{0,\beta }-\lambda )u_n=({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )v_n-\frac{\lambda +c_1}{\beta -a}{v}_n\left(0\right)G=f_n\to 0\mathrm{in}{L}_2\left(\mathrm{\Omega }\right).$$

(87)

Then, we integrate by parts as follows:

$$\begin{array}{cc}\hfill {\left(({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )u_n,\chi G\right)}_{L_2\left({\mathrm{\Omega }}_{\delta }\right)}=& {\left((\widehat{\mathcal{H}}-\lambda )u_n,\chi G\right)}_{L_2\left({\mathrm{\Omega }}_{\delta }\right)}\hfill \\ \hfill =& -{\left(\frac{\partial u_n}{\partial \mathrm{n}},G\right)}_{L_2(\partial {\omega }_{\delta })}+{\left(u_n,\frac{\partial G}{\partial {\mathrm{n}}^{\mathrm{*}}}\right)}_{L_2(\partial {\omega }_{\delta })}\hfill \\ \hfill & +{\left(u_n,({\widehat{\mathcal{H}}}^{\mathrm{*}}-\overline{\lambda })\chi G\right)}_{L_2\left({\mathrm{\Omega }}_{\delta }\right)}\hfill \\ \hfill =& -{\left(\frac{\partial v_n}{\partial \mathrm{n}},G\right)}_{L_2(\partial {\omega }_{\delta })}+{\left(v_n,\frac{\partial G}{\partial {\mathrm{n}}^{\mathrm{*}}}\right)}_{L_2(\partial {\omega }_{\delta })}\hfill \\ \hfill & +\sum _{j=1}^2{(A_j{\nu }_j{v}_n,G)}_{L_2(\partial {\omega }_{\delta })}+\frac{v_n\left(0\right)}{\beta -a}\sum _{j=1}^2{(A_j{\nu }_{j}G,G)}_{L_2(\partial {\omega }_{\delta })}\hfill \\ \hfill & +{(u_n,({\widehat{\mathcal{H}}}^{\mathrm{*}}-\overline{\lambda })\chi G)}_{L_2({\mathrm{\Omega }}_{\delta }{)}^{’}}\hfill \end{array}$$

(88)

where $\delta$ is small enough. In the same way as the first inequality in (73) was established, we find that

$$\left|{\left(\frac{\partial v_n}{\partial \mathrm{n}},G\right)}_{L_2(\partial {\omega }_{\delta })}\right|⩽{C\delta |ln\delta |}^{\frac{3}{2}}{\parallel v_n\parallel }_{W_2^2\left(\mathrm{\Omega }\right)}$$

(89)

with a constant C independent of $\delta$ and $v_n$. Employing asymptotic (5), it is straightforward to confirm that

$$\frac{\partial G}{\partial {\mathrm{n}}^{\mathrm{*}}}={\delta }^{-1}{\alpha }_0({\delta }^{-1}·)+{\Phi }_4(·,\delta )\mathrm{on}\partial {\omega }_{\delta },$$

(90)

where ${\Phi }_4$ is a continuous $\partial {\omega }_{\delta }$ function bounded uniformly in $\delta$. Estimate (26) implies immediately that

$$\begin{array}{cc}\hfill & \left|{(v_n,{\Phi }_4)}_{L_2(\partial {\omega }_{\delta })}\right|⩽C\delta {\parallel v_n\parallel }_{W_2^2\left(B_{2R_1}\left(x_0\right)\right)},\hfill \\ \hfill & \left|\sum _{j=1}^2{(A_j{\nu }_j{v}_n,G)}_{L_2(\partial {\omega }_{\delta })}\right|⩽C\delta |ln\delta |\parallel v_n{\parallel }_{W_2^2\left(B_{2R_1}\left(x_0\right)\right)},\hfill \\ \hfill & \left|\frac{v_n\left(0\right)}{\beta -a}\sum _{j=1}^2{(A_j{\nu }_{j}G,G)}_{L_2(\partial {\omega }_{\delta })}\right|⩽{C\delta |ln\delta |}^2\left|v_n\left(0\right)\right|,\hfill \end{array}$$

(91)

where C are constants independent of $\delta$ and $v_n$. Lemma 3 with $\varphi ={\alpha }_0$, $\epsilon =\delta$ and Lemma 1 yield:

$$\left|{\delta }^{-1}{\left(v_n,{\alpha }_0({\delta }^{-1}·)\right)}_{L_2(\partial {\omega }_{\delta })}+\pi v_n\left(x_0\right)trA\right|⩽{C\delta |ln\delta |}^{\frac{1}{2}}{\parallel v_n\parallel }_{W_2^2\left(B_{2R_1}\left(x_0\right)\right)},$$

where C is a constant independent of $\delta$ and $v_n$. Using this estimate and (89)–(91), we pass to the limit in (88) as $\delta \to 0$ and use the identity in (87):

$${(f_n,\chi G)}_{L_2\left(\mathrm{\Omega }\right)}={\left(({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )u_n,\chi G\right)}_{L_2\left(\mathrm{\Omega }\right)}=-\pi v_n\left(x_0\right)trA+{(u_n,({\widehat{\mathcal{H}}}^{\mathrm{*}}-\overline{\lambda })\chi G)}_{L_2\left(\mathrm{\Omega }\right)}.$$

(92)

Expressing $v_n\left(x_0\right)$ from this formula, in view of the properties of the characteristic sequence $u_n$, we see that the scalar sequence $v_n\left(x_0\right)$ is bounded uniformly in n. Hence, selecting a subsequence from $v_n$, we can suppose that $v_n\left(x_0\right)$ is a converging sequence. Hence, the sequence $v_n=u_n-v_n\left(x_0\right)G$ is bounded and non-compact in $L_2\left(\mathrm{\Omega }\right)$. The identity in (87) implies that the sequence $({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )v_n$ is bounded in $L_2\left(\mathrm{\Omega }\right)$ and, as proceeding as above, we find a subsequence in $v_n$, still denoted by $v_n$, which converges weakly in $W_2^1\left(\mathrm{\Omega }\right)$ and $W_2^2\left({\mathrm{\Omega }}_0\right)$ and strongly in $W_2^1\left({\mathrm{\Omega }}_0\right)$ to some function $v_{†}$ as $n\to \infty$. In the same way as identity (92) was established, we confirm that

$$\pi v_n\left(x_0\right)trA={\left(({\mathcal{H}}_{\mathrm{\Omega }}-\lambda )v_n,\chi G\right)}_{L_2\left(\mathrm{\Omega }\right)}-{(v_n,({\widehat{\mathcal{H}}}^{\mathrm{*}}-\overline{\lambda })\chi G)}_{L_2\left(\mathrm{\Omega }\right)}.$$

Passing then to the limit as $n\to \infty$, we see that $v_n\left(x_0\right)\to v_{†}\left(x_0\right)$ as $n\to \infty$. Proceeding as above, we also see easily that the function $u_{†}:=v_{†}+{(\beta -a)}^{-1}{v}_{†}\left(x_0\right)G$ is in the domain of the operator ${\mathcal{H}}_{0,\beta }$ and is in the kernel of the operator ${\mathcal{H}}_{0,\beta }-\lambda$. This allows us to reproduce the arguing in the proof of (86) and to confirm that the sequence $(1-\chi )(u_n-u_{†})$ is the characteristic one of ${\mathcal{H}}_{0,\beta }$ corresponding to $\lambda$. This sequence is also characteristic one for ${\mathcal{H}}_{\mathrm{\Omega }}$ corresponding to $\lambda$ and this proves the inclusion ${\sigma }_{ess}\left({\mathcal{H}}_{0,\beta }\right)\subseteq {\sigma }_{ess}\left({\mathcal{H}}_{\mathrm{\Omega }}\right)$.

We proceed to proving inclusion (21). We choose a compact set $Q\subset \mathbb{C}$ and let $\lambda \in Q$. In the same way as solution (84) of Equation (81) was found, we easily obtain a similar formula for the resolvent of the operator ${\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }$:

$$\begin{array}{cc}\hfill & {({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }-\lambda )}^{-1}={\left(\mathcal{I}-(\lambda +c_1)\mathcal{M}\left(\lambda \right){\mathcal{T}}_{\epsilon }\right)}^{-1}\mathcal{M}\left(\lambda \right){({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }+c_1)}^{-1},\hfill \\ \hfill & \mathcal{M}\left(\lambda \right):=\mathcal{I}+(\lambda +c_1){({\mathcal{H}}_{0,\beta }-\lambda )}^{-1},{\mathcal{T}}_{\epsilon }:={({\mathcal{H}}_{\epsilon }\oplus {\mathcal{A}}_{\epsilon }+c_1)}^{-1}-{({\mathcal{H}}_{0,\beta }+c_1)}^{-1}.\hfill \end{array}$$

The above formula is well-defined provided

$$|\lambda +c_1|\mathcal{M}\left(\lambda \right){\mathcal{T}}_{\epsilon }<1.$$

(93)

It follows from estimate (17) and the definition of the operator ${\mathcal{A}}_{\epsilon }$ that

$$\parallel {\mathcal{T}}_{\epsilon }{\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left(\mathrm{\Omega }\right)}⩽C{|ln\epsilon |}^{-1},$$

where C is a constant independent of $\epsilon$ and $\lambda \in Q$. The norm of the operator $\mathcal{M}\left(\lambda \right)$ is estimated as follows:

$${\parallel \mathcal{M}\left(\lambda \right)\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left(\mathrm{\Omega }\right)}⩽\mathrm{\Lambda }{\parallel {({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left(\mathrm{\Omega }\right)}$$

with constant $\mathrm{\Lambda }$ defined in (22). Hence, inequality (93) is satisfied provided

$$C{\mathrm{\Lambda }}^2{|ln\epsilon |}^{-1}{\parallel {({\mathcal{H}}_{0,\beta }-\lambda )}^{-1}\parallel }_{L_2\left(\mathrm{\Omega }\right)\to L_2\left(\mathrm{\Omega }\right)}<1$$

and the numbers $\lambda \in Q$ determined by this inequality are in the resolvent set of the operator ${\mathcal{H}}_{\epsilon }$. This proves inclusion (22).