Content-Type: multipart/mixed; boundary="-------------0606070702570" This is a multi-part message in MIME format. ---------------0606070702570 Content-Type: text/plain; name="06-177.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-177.keywords" waveguide, distant perturbation, asymptotics, spectrum ---------------0606070702570 Content-Type: application/x-tex; name="run_wav_short5.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="run_wav_short5.tex" \documentclass[a4paper,12pt]{article} \addtolength{\topmargin}{-1.5 true cm} \addtolength{\textheight}{1.5 true cm} \addtolength{\textwidth}{1 true cm} \usepackage{amsmath, amsthm, amsfonts, amssymb} \usepackage{graphicx} %\usepackage{srcltx} %\usepackage{color} %\usepackage{showkeys} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}[lemma]{Proposition} \theoremstyle{remark} \newtheorem{remark}{Remark}[section] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\th{\theta} \def\Om{\Omega} \def\om{\omega} \def\e{\varepsilon} \def\g{\gamma} \def\G{\Gamma} \def\l{\lambda} \def\p{\partial} \def\D{\Delta} \def\bs{\backslash} \def\k{\varkappa} \def\E{\mbox{\rm e}} \def\a{\alpha} \def\b{\beta} %\def\t{\widetilde} \def\si{\sigma} \def\Si{\Sigma} \def\d{\delta} \def\L{\Lambda} \def\z{\zeta} \def\vs{\varsigma} \def\r{\rho} \def\vp{\varphi} \def\vw{\varpi} \def\vr{\varrho} \def\vt{\vartheta} \def\h{\widehat} \def\Odr{\mathcal{O}} \def\H{W_2} \def\Hloc{W_{2,loc}} %\def\essspec{\sigma_{ess}} %\def\discspec{\sigma_{disc}} \def\di{\,\mathrm{d}} \def\Th{\Theta} \def\I{\mathrm{I}} \def\iu{\mathrm{i}} \def\Ups{\Upsilon} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\card}{card} \DeclareMathOperator{\RE}{Re} \DeclareMathOperator{\IM}{Im} \DeclareMathOperator{\spec}{\sigma} \DeclareMathOperator{\discspec}{\sigma_{disc}} \DeclareMathOperator{\essspec}{\sigma_{ess}} \DeclareMathOperator{\conspec}{\sigma_{c}} \DeclareMathOperator{\pointspec}{\sigma_{p}} \DeclareMathOperator{\resspec}{\sigma_{r}} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\tr}{tr} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \allowdisplaybreaks \title{\textbf{ Distant perturbation asymptotics in window-coupled waveguides. \\ I.~The non-threshold case}} \author{D.~Borisov$^{a,b}$ and P.~Exner$^{a,c}$} \date{} \maketitle \begin{quote} {\small {\em a) Nuclear Physics Institute, Academy of Sciences, 25068 \v Re\v z \\ \phantom{a) }near Prague, Czechia \\ b) Bashkir State Pedagogical University, October Revolution St.~3a, \\ \phantom{a) }450000 Ufa, Russia \\ c) Doppler Institute for Mathematical Physics and Applied Mathematics, \\ \phantom{a) }Czech Technical University, B\v rehov{\'a} 7, 11519 Prague, Czechia } \\ \phantom{a) }\texttt{borisovdi@yandex.ru}, \texttt{exner@ujf.cas.cz}} \end{quote} \begin{quote} {\small We consider a pair of adjacent quantum waveguides, in general of different widths, coupled laterally by a pair of windows in the common boundary, not necessarily of the same length, at a fixed distance. The Hamiltonian is the respective Dirichlet Laplacian. We analyze the asymptotic behavior of the discrete spectrum as the window distance tends to infinity for the generic case, i.e. for eigenvalues of the corresponding one-window problems separated from the threshold. } \end{quote} %%%%%%%%%%%%%%%%%%%%%% \section{Introduction} %%%%%%%%%%%%%%%%%%%%%% Quantum mechanics exhibits various effects which defy our intuition based on ``classical'' experience. A nice class of examples are bound states in hard-wall tubes induced solely by their geometric properties such as bends, protrusions, or ``windows''. Such systems are interesting not only \emph{per se} but also from the practical point of view as models of various nanophysical devices, and in a reasonable approximation also of flat electromagnetic waveguides. Among numerous questions such models pose an important one concerns behavior of the spectra in case of two distant perturbations. One can think of it as of an analogue of the exponential spectral shift for a pair of distant potential wells, despite the fact that the usual methods of the Schr\"odinger operator theory do not work here. The aim of the present paper is to study this problem in a model example of a pair of laterally coupled waveguides, or adjacent straight hard-wall strip in the plane, coupled by a pair of ``windows'' in the common boundary -- we refer to \cite{ESTV}, \cite{BE}, \cite{B1} for a bibliography concerning such models. In our recent paper \cite{BE} we dealt with the symmetric situation where the widths $d_1$, $d_2$ of the two channels were the same and so were the window widths $a_1, a_2$. The technique used in these papers employed substantially the fact that the problem can be decomposed into parts with a definite parity, which allows one to study a single-window problem with a perturbation which consists of an additional Dirichlet or Neumann boundary condition at a segment far from the window. The approach based on symmetry works no longer if $a_1\ne a_2$. The man aim of the present work is to demonstrate a different technique, suitable for the general case, which reduces the question to analysis of a boundary perturbation at the distant window. This technique follows the main ideas of \cite{B2}, where the Dirichlet Laplacian in an $n$-dimensional tube with a pair of distant perturbations described by two arbitrary operators was studied. It was assumed in \cite{B2} that these operators are defined on functions from $\H^2$ vanishing at the boundary, and this assumption was employed substantially. This is obviously not true in the problem we study, since the windows enlarge the domain of the Laplacian beyond the Sobolev space $\H^2$. At the same time, the general approach of \cite{B2} works in our case with the appropriate modifications. Moreover, since we restrict ourselves to the two-dimensional case and specify the nature of the distant perturbations, we are able to obtain a more detailed result in comparison with the general case in \cite{B2}. In order not to make this study too technical we concentrate in this paper at the generic case when the ``unperturbed'' energy is an \emph{isolated} eigenvalue of the one-window problem, leaving the computationally involved discussion of threshold resonances to a sequel. The problem will be properly formulated and the results stated in the next section; the rest of the paper is devoted to the proofs. \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Statement of the problem and the results} %%%%%%%%%%%%% Let $x=(x_1,x_2)$ be Cartesian coordinates in the plane, $\Pi^+:=\{x: 00$ the essential spectrum of $H(a)$ equals $[1,+\infty)$ while $\discspec({H}(a))$ is non-empty consisting of finitely many simple eigenvalues. The eigenfunction associated with an eigenvalue $\l_m(a)$ has a definite parity: it is even or odd with respect to $x_1$ if $m$ is odd or even, respectively. In the particular case $d=\pi$ the eigenfunctions are even in the variable $x_2$. \end{proposition} % ------------- % The eigenfunctions associated with the eigenvalues $\l_n(a)$ will be denoted as $\psi_n(\cdot,a)$ and assumed to be normalized, i.e. to be unit vectors in $L_2(\Pi_a)$. It is easy to check that $\psi_n(\cdot,a)\in C^\infty(\Pi_a)$. We put $\sigma_*:= \discspec{H(a_+)}\cup\discspec{H(a_-)}$; an element $\l_*\in \sigma_*$ of this set will be called \emph{simple} if $\l_*$ belongs to one of the sets $\discspec(H(a_\pm))$ only and \emph{double} otherwise. Furthermore, we set $\boldsymbol{a}:= (a_+,a_-)$. \smallskip With these preliminaries we can formulate the first main result of this paper. % ------------- % \begin{theorem}\label{th1.0} For any $l>0$, $a_\pm>0$ the operator $H$ has the essential spectrum equal to $[1,+\infty)$ and finitely many isolated eigenvalues. The number of the isolated eigenvalues of $H$ is independent of the window distance provided $l\geqslant\max\{a_-,a_+\}$. In the limit $l\to+\infty$ each isolated eigenvalue of the operator $H$ converges to one of the numbers from the set $\sigma_*$ or to the threshold of $\essspec(H)$. \end{theorem} % ------------- % By $\Xi_a$ we indicate the set of all bounded domains $S\subset\Pi_a$ having smooth boundary and separated from the edges of $\g_a$ by a positive distance; we stress that the case $\p S\cap \p\Pi_a\not=\emptyset$ is not excluded. For any $\l$ such that $\RE\l\leqslant 1$ we denote $\kappa_1^+=\kappa_1^+(\l):= \sqrt{1-\l}$ and $\kappa_1^-=\kappa_1^-(\l):= \sqrt{\frac{\pi^2}{d^2}-\l}$, where the branch of the root is specified by the requirement that the functions are analytic in $\mathbb{S}_\d$ and $\sqrt{1}=1$. The following statements will be proven in Section~\ref{s: onewin}. % ------------- % \begin{proposition}\label{lm1.2} In the limit $x_1\to\pm\infty$ the eigenfunction $\psi_n$ of $H(a)$ behaves as % ------------- % \begin{equation}\label{1.4} \begin{aligned} &\psi_n(x,a)=(\pm 1)^{n+1}c(\l_n,a)\E^{-\kappa_1^+(\l_n)|x_1|}\sin x_2+\Odr(\E^{-\RE\sqrt{4-\l}|x_1|}),&& x_2\in[0,\pi], \\ &\psi_n(x,a)=\Odr(\E^{-\RE \kappa_1^-(\l)|x_1|}), && x_2\in[-d,0], \end{aligned} \end{equation} % ------------- % if $d<\pi$, and % ------------- % \begin{equation}\label{1.4a} \begin{aligned} &\psi_n(x,a)=(\pm 1)^{n+1}c(\l_n,a)\E^{-\kappa_1^+(\l_n)|x_1|}\sin |x_2|+\Odr(\E^{-\RE\sqrt{4-\l}|x_1|}),&& x_2\in[-\pi,\pi], \end{aligned} \end{equation} in the case of equal-width channels, $d=\pi$. In these relations \begin{equation}\label{1.5} c(\l_n,a)=\frac{1}{\pi \kappa_1^+(\l_n)}\int\limits_{\g_a}\psi_n(x,a)\E^{\kappa_1^+(\l_n)x_1}\di x_1=\frac{(-1)^{n+1}}{\pi \kappa_1^+(\l_n)}\int\limits_{\g_a}\psi_n(x,a)\E^{-\kappa_1^+(\l_n)x_1}\di x_1, \end{equation} % ------------- % and $c(\l_i,a)\not=0$, $i=1,2$. The asymptotic relations (\ref{1.4}), (\ref{1.4a}) give rise to valid formul{\ae} when both their sides are differentiated. \end{proposition} % ------------- % % ------------- % \begin{proposition}\label{lm1.3} For any $\l\in(-\infty,1)\setminus\discspec(H(a))$ there exists a unique solution of the boundary value problem % ------------- % \begin{equation}\label{1.3a} \begin{gathered} (\D+\l)U=0,\quad x\in\Pi_a\setminus\g_a,\qquad U=0,\quad x\in\p\Pi_a, \\ \frac{\p U}{\p x_2}\Big|_{x_2=+0}- \frac{\p U}{\p x_2}\Big|_{x_2=-0}=\E^{\kappa_1^+(\l)x_1}, \quad x\in\g_a, \end{gathered} \end{equation} % ------------- % belonging to $\H^1(\Pi_a)$. For large values of $|x_1|$ this function is infinitely differentiable and in the limit $x_1\to+\infty$ it behaves as % ------------- % \begin{equation}\label{1.3b} \begin{aligned} &U(x,\l,a)=c(\l,a)\E^{-\kappa_1^+(\l_n)x_1}\sin x_2+\Odr(\E^{-\RE\sqrt{4-\l}x_1}),&& x_2\in[0,\pi], \\ &U(x,\l,a)=\Odr(\E^{-\RE \kappa_1^-(\l)x_1}), && x_2\in[-d,0], \end{aligned} \end{equation} % ------------- % if $d<\pi$, and \begin{equation}\label{1.3c} \begin{aligned} &U(x,\l,a)=c(\l,a)\E^{-\kappa_1^+(\l_n)x_1}\sin |x_2|+\Odr(\E^{-\RE\sqrt{4-\l}x_1}),&& x_2\in[-\pi,\pi], \end{aligned} \end{equation} % ------------- % in the case $d=\pi$, where the coefficient is given by % ------------- % \begin{equation}\label{1.10} c(\l,a)=\frac{1}{\pi \kappa_1^+(\l)}\int\limits_{\g_a}U(x,\l,a)\E^{\kappa_1^+(\l)x_1}\di x_1. \end{equation} This coefficient is negative for $\l<\l_1(a)$. The asymptotic relations (\ref{1.3b}), (\ref{1.3c}) give rise to valid formul{\ae} when both their sides are differentiated. \end{proposition} % ------------- % For the double window $\g_+\cup\g_-$ we indicate by $\Xi$ the set of all bounded domains $S\subset\Pi$ having smooth boundary and separated from the edges of $\g_\pm$ by a positive distance; the case $\p S\cap \p\Pi\not=\emptyset$ is again not excluded. For brevity we will introduce a two-valued symbol, $\tau:=1$ if $d<\pi$ and $\tau:=2$ if $d=\pi$. Continuing the list of the main results we make the following claims. % ------------- % \begin{theorem}\label{th1.2} Suppose that $\l_*\in\sigma_*$ is simple being an eigenvalue $\l_n(a_\pm)$ of the operator $H(a_\pm)$. Then there is a unique eigenvalue of the operator $H$ converging to $\l_*$ as $l\to+\infty$. This eigenvalue is simple and behaves asymptotically as follows, % ------------- % \begin{align} &\l^\pm(l,\boldsymbol{a})=\l_*+\mu^\pm(l,\boldsymbol{a}) \E^{-4\kappa_1^+(\l_*)l}+\Odr(l^2\E^{-8\kappa_1^+(\l_*)l}+ \E^{-2l(\kappa_1^+(\l_*)+\rho)}),\label{1.11} \\ & \begin{aligned} &\mu^\pm(l,\boldsymbol{a}):=\tau\pi c(\l_*,a_\pm)c^2(\l_*,a_\mp) \kappa_1^+(\l_*), \end{aligned}\label{1.12} \end{align} % ------------- % where $\rho=\rho(\l):=\min\{\kappa_1^-(\l),\sqrt{4-\l}\}$ if $d<\pi$, and $\rho=\rho(\l):=\sqrt{4-\l}$ if $d=\pi$. The associated eigenfunction $\psi^\pm(x,l,\boldsymbol{a})$ satisfies the relation % ------------- % \begin{equation}\label{1.13} \psi^\pm(x,l,\boldsymbol{a})=\psi_n(x_1\mp l,x_2,a_\pm)+\Odr(\E^{-2\kappa_1^+(\l_*)l}) \end{equation} % ------------- % in the norms of both the $\H^1(\Pi)$ and $\H^2(S)$ for each $S\in\Xi$. \end{theorem} % ------------- % % ------------- % \begin{theorem}\label{th1.3} Suppose that $\l_*\in\sigma_*$ is double and $\l_*=\l_n(a_-)= \l_m(a_+)$. Then there exist either two simple eigenvalues $\l^\pm(l,\boldsymbol{a})$ or one double eigenvalue $\l^-(l,\boldsymbol{a})=\l^+(l,\boldsymbol{a})$ of the operator $H$ converging to $\l_*$ as $l\to+\infty$. The asymptotic expansions of these eigenvalues read as follows, % ------------- % \begin{align} &\l^\pm(l,\boldsymbol{a})=\l_*\pm|\mu(l,\boldsymbol{a})| \E^{-2\kappa_1^+(\l_*)l}+ \Odr(l\E^{-4\kappa_1^+(\l_*)l}+\E^{-2\rho(\l_*)l}),\label{1.14} \\ & \begin{aligned} &\mu(l,\boldsymbol{a})=(-1)^{m+1}\tau\pi \kappa_1^+(\l_*)c(\l_*,a_-)c(\l_*,a_+), \end{aligned}\label{1.15} \end{align} % ------------- % \end{theorem} % ------------- % % ------------- % \begin{theorem}\label{th1.4} Suppose the hypothesis of Theorem~\ref{th1.3} holds true. If $\mu(l,\boldsymbol{a})\not=0$, the eigenvalues $\l^+(l,\boldsymbol{a})$ $\l^-(l,\boldsymbol{a})$ do not coincide and are simple. The associated eigenfunctions $\psi^\pm(x,l,\boldsymbol{a})$ satisfy the relations % ------------- % \begin{equation}\label{1.16} \psi^\pm(x,l,\boldsymbol{a})=\psi_n(x_1+l,x_2,a_-)\mp \psi_m(x_1-l,x_2,a_+) \sgn\mu(l,\boldsymbol{a})+\Odr(\E^{-2\kappa_1^+(\l_*)l}) \end{equation} % ------------- % in the norms of $\H^1(\Pi)$ and $\H^2(S)$ for each $S\in\Xi$. If $\l^-(l,\boldsymbol{a})=\l^+(l,\boldsymbol{a})$ is a double eigenvalue, the associated eigenfunctions $\psi^\pm(x,l, \boldsymbol{a})$ satisfy the relations % ------------- % \begin{equation}\label{1.17} \begin{aligned} &\psi^+(x,l,\boldsymbol{a})=\psi_n(x_1+l,x_2,a_-) +\Odr(\E^{-2\kappa_1^+(\l_*)l}), \\ &\psi^-(x,l,\boldsymbol{a})=\psi_m(x_1-l,x_2,a_-) +\Odr(\E^{-2\kappa_1^+(\l_*)l}), \end{aligned} \end{equation} % ------------- % in the norm of $\H^1(\Pi)$ and $\H^2(S)$ for each $S\in\Xi$. Finally, if $\mu(l,\boldsymbol{a})=0$ and $\l^-(l,\boldsymbol{a}) \not=\l^+ (l,\boldsymbol{a})$, the eigenvalues $\l^\pm(l, \boldsymbol{a})$ are simple and the associated eigenfunctions satisfy the relations % ------------- % \begin{equation}\label{1.18} \psi^\pm(x,l,\boldsymbol{a})=c_+^\pm\psi_n(x_1+l,x_2,a_-)+c_-^\pm \psi_m(x_1-l,x_2,a_+)+\Odr(\E^{-2\kappa_1^+(\l_*)l}), \end{equation} % ------------- % where the vectors $\boldsymbol{c}^\pm:=\left( \begin{smallmatrix} c_+^\pm \\ c_-^\pm \end{smallmatrix} \right)$ are nontrivial solutions to the system (\ref{4.16}) with $\l=\l^\pm$ such that $\|\boldsymbol{c}^\pm\|_{\mathbb{R}^2}=1$. \end{theorem} % ------------- % The leading terms of the asymptotics (\ref{1.11}), (\ref{1.14}) are non-zero provided the corresponding coefficients $c(\l_*,a_\pm)$ are non-zero. We know from Propositions~\ref{lm1.2}, \ref{lm1.3} that this is true at least for $c(\l,a)$ as $\l\leqslant\l_1(a)$ or $\l=\l_2(a)$. For instance, if $\l_1(a_-)<\l_1(a_+)$, the eigenvalue of the operator $H$ converging to $\l_1(a_-)$ has the asymptotic expansion (\ref{1.11}), and the coefficient (\ref{1.12}) of leading term is non-zero. Moreover, due to Proposition~\ref{lm1.3} this coefficient is negative. If $a_\pm$ are such that $\l_1(a_-)=\l_2(a_+)$, the eigenvalues of the operator $H$ converging to $\l_*=\l_1(a_-)=\l_2(a_+)$ have the asymptotics expansions (\ref{1.14}), and the coefficients of the leading terms are non-zero. By Theorem~\ref{th1.4} the ``perturbed'' eigenvalues are simple and the associated eigenfunctions satisfy the identities (\ref{1.16}) in this case. We also stress that in this case the leading terms of the asymptotic expansions (\ref{1.14}) have the same modulus but different signs. This phenomenon is known in double-well problems with symmetric wells. It also occurs in the symmetric case, $a_-=a_+$ and $d_-=d_+$, as we have shown in \cite{BE}. We conjecture that the coefficient $c(\l,a)$ is non-zero for all values $a$ and $\l<1$. If it is true, this fact would imply that the leading terms in the asymptotics (\ref{1.11}), (\ref{1.14}) are non-zero. In turn, this fact together with Theorem~\ref{th1.4} would imply that a double $\l_*\in\si_*$ splits into two simple ''perturbed'' eigenvalues and the formul{\ae} (\ref{1.16}) are valid for the associated eigenfunctions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Analysis of the one-window problem} \label{s: onewin} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we shall study the following boundary value problem, % ------------- % \begin{equation}\label{3.11} \begin{gathered} (\D+\l)u=0,\quad x\in\Pi_a\setminus\g_a, \qquad u=0,\quad x\in\G_a, \\ \frac{\p u}{\p x_2}\Big|_{x_2=+0}- \frac{\p u}{\p x_2}\Big|_{x_2=-0}=f, \quad x\in\g_a. \end{gathered} \end{equation} % ------------- % The function $f$ is assumed to be an element of $L_2(\g_a)$. A solution to this problem is understood in a generalized sense, more specifically, as a function belonging to $\H^1(\Pi_a^b,\G_a^b)$ for each $b>0$ and satisfying the equation % ------------- % \begin{equation}\label{3.12} -(\nabla u,\nabla \z)_{L_2(\Pi_a)}+\l(u,\z)_{L_2(\Pi_a)}- (f,\z)_{L_2(\g_a)}=0 \end{equation} % ------------- % for any $\z\in C_0^\infty(\Pi_a)$. By standard smoothness-improving results about solutions to elliptic boundary value problems, cf.~\cite[Ch. 4, \S 2]{Mh}, the said solution belongs to $C^\infty(\overline{\Pi^+}\cup\overline{\Pi^-}\setminus \overline{\g}_a)$. As we have said in the introduction we will deal in this paper with the non-threshold case only. Thus the parameter $\l$ is supposed to belong to $\mathbb{S}_\d$ for a fixed $\d>0$, where $\mathbb{S}_\d$ is a set of all $\l$ separated from the halfline $[1,+\infty)$ by a distance not less than $\d$. We seek a solution to the problem (\ref{3.11}) belonging to $L_2(\Pi_a)$. We fix $\b>a$ and put $P:=\{x: |x_1|a+\b/3$ or $|x_2|>d_0/3$. We extend the function $\widetilde{u}$ in an even way for $x_2<0$ setting $\widetilde{u}(x):=\widetilde{u}(x_1,-x_2)$ as $x_2<0$ and denote $u_f(x):=\chi_1(x)\widetilde{u}(x)$. % ------------- % \begin{lemma}\label{lm3.5} The function $u_f$ belongs to $\H^1(\Pi_a,\G_a)\cap C^\infty(\Pi_a)$ and satisfies the equation % ------------- % \begin{equation}\label{3.15} -(\nabla u_f,\nabla \z)_{L_2(\Pi_a)}+\l(u_f,\z)_{L_2(\Pi_a)} -(f,\z)_{L_2(\g_a)}=(F,\z)_{L_2(\Pi_a)} \end{equation} % ------------- % for any $\z\in C_0^\infty(\Pi_a)$, where % ------------- % \begin{equation*} F=T_{1}(\l,a)f:=2\nabla\widetilde{u}\cdot\nabla\chi_1+ \widetilde{u}\D\chi_1\,. \end{equation*} % ------------- % The operator $T_{1}: L_2(\g_a)\to L_2(\{x: |x_1|0$. By Theorem~4.6.8 of \cite[Ch. 4, \S 4.6]{BEH} any solution of this problem belonging to $L_2(\Pi_a)$ is an element of the operator domain of $H(a)$. In this way the problem (\ref{3.25}) can be cast into the form $(H(a)-\l)\widehat{u}=F$, which in turn gives $\widehat{u}=\big(H(a)-\l\big)^{-1}F$. Let us next denote $T_{3}(\l,a):=T_{2}(\l,a) +(H(a)-\l)^{-1} T_{1}(\l,a)$. In order to analyze properties of this operator we need an additional notation and a lemma. For any numbers $b_1,b_2,b_3\in \mathbb{R}$ we set $\Om_\pm:=\{x: \pm x_1>b_1,\, b_2b_1\}\cap\p\Om_\pm$. % ------------- % \begin{lemma}\label{lm3.12} Let $v\in\H^1(\Om_\pm)$ be a solution to the problem % ------------- % \begin{equation*} (\D+\l)v=0,\quad x\in\Om^\pm,\qquad v=0,\quad x\in\om^\pm, \end{equation*} % ------------- % and $\,0b_4,b_2b_1$. The coefficients $\a_j$ satisfy the condition % ------------- % \begin{equation}\label{3.42} \frac{\pi}{2}\sum\limits_{j=1}^{\infty} |\a_j|^2=\|v(\pm b_1,\cdot,\l)\|_{L_2(b_2,b_3)}. \end{equation} % ------------- % \end{lemma} This lemma is a particular case of Lemma~3.3 of \cite{B2} so we skip the proof. % ------------- % \begin{lemma}\label{lm3.13} The operator $T_{3}(\l,a)$ is bounded and meromorphic in $\l\in \mathbb{S}_\d$ as a map from $L_2(\g_a)$ into $\H^1(\Pi_a,\G_a)$ and into $\H^2(S)$ for each $S\in\Xi_a$. Its poles coincide with the eigenvalues of the operator $H(a)$. For any $\l$ close to an eigenvalue $\l_n$ of $H(a)$ the representation % ------------- % \begin{equation} T_{3}(\l,a)= \frac{\psi_n}{\l-\l_n}T_{4}(a)+T_{5}(\l,a)\label{3.9} \end{equation} % ------------- % holds true. Here $T_{4}(a)f:=(f,\psi_n)_{L_2(\g_a)}$ and the operator $T_{5}$ is bounded and holomorphic in $\l\in \mathbb{S}_\d$ as a map from $L_2(\g_a)$ into $\H^1(\Pi_a,\G_a)$. The operator $T_{5}$ is also bounded and holomorphic as a map into $\H^2(S)$ for each $S\in\Xi_a$. \end{lemma} % ------------- % \begin{proof} In accordance with \cite[Ch. 5, \S 3.5]{K} the operator $(H(a)-\l)^{-1}$ is bounded and meromorphic in $L_2(\Pi_a)$, its poles coincide with the eigenvalues of $H(a)$ and for $\l$ close to $\l_n$ the representation % ------------- % \begin{equation}\label{3.27} \big(H(a)-\l\big)^{-1}=-\frac{\psi_n}{\l-\l_n} (\cdot,\psi_n)_{L_2(\Pi_a)}+T_{6}(\l,a) \end{equation} % ------------- % is valid, where the operator $T_{6}(\l,a)$ is bounded and holomorphic in $\l$ in the vicinity of $\l_n$. The function $\check{u}:=T_{6}(\l,a)F$ is a solution to the boundary value problem (\ref{3.25}) with $F$ replaced by $\check{F}:=F-(F,\psi_n)_{L_2(\Pi_a)}\psi_n$; it means that % ------------- % \begin{equation*} \|\nabla\check{u}\|_{L_2(\Pi_a)}^2-\l\|\check{u}\|_{L_2(\Pi_a)}^2= (\check{F},\check{u})_{L_2(\Pi_a)}. \end{equation*} % ------------- % This relation together with (\ref{3.27}) imply that the operator $T_{6}$ is bounded and holomorphic as a map into $\H^1(\Pi_a)$ as well. Using again the smoothness-improving theorems mentioned above we conclude that the operator $T_{6}$ is also bounded and holomorphic in $\lambda$ as a map into $\H^2(S)$ for each $S\in\Xi_a$. Since the function $\psi_n$ is an element of $\H^1(\Pi_a,\G_a)$, the relation (\ref{3.15}) is valid for $\z=\psi_n$. For any $f\in L_2(\g_a)$ the function $T_{1}(\l,a)f$ is compactly supported, hence we have % ------------- % \begin{equation*} (T_{1}(\l_n,a)f,\psi_n)_{L_2(\Pi_a)}=-(\nabla u_f,\nabla \psi_n)_{L_2(\Pi_a)}+\l(u_f,\psi_n)_{L_2(\Pi_a)}- (f,\psi_n)_{L_2(\g_a)}. \end{equation*} % ------------- % According to Lemma~\ref{lm3.5}, the function $u_f$ belongs to $\H^1(\Pi_a,\G_a)$, which allows us to proceed with the calculations, % ------------- % \begin{gather*} -(\nabla u_f,\nabla \psi_n)_{L_2(\Pi_a)}+\l(u_f,\psi_n)_{L_2(\Pi_a)}=0, \\ (T_{1}(\l_n,a)f,\psi_n)_{L_2(\Pi_a)}=-(f,\psi_n)_{L_2(\g_a)}=-T_{4}f. \end{gather*} % ------------- % Substituting the relation thus obtained together with (\ref{3.27}) into the definition of the operator $T_{3}$ and taking into account Lemma~\ref{lm3.5}, we arrive finally at the statement of the lemma. \end{proof} % ------------- % Let us next fix a number $\widetilde{a}>0$. For any $l\geqslant(a+\widetilde{a})$ we define operators $T_{7}^\pm(\l,l,a,\widetilde{a})$ which map an arbitrary $v\in \H^1(\Pi_a^a)$ into the function % ------------- % \begin{align} &(T_{7}^\pm v)(x_1,\l,l):=\sum\limits_{j=1}^{\infty} j\a_j^\pm \E^{\mp \kappa_j^+(\l)(x_1\mp a)}\E^{-2\kappa_j^+(\l)l}- \sum\limits_{j=1}^{\infty} \frac{\pi j}{d}\b_j^\pm \E^{\mp \kappa_j^-(\l)(x_1\mp a)}\E^{-2\kappa_j^-(\l)l},\nonumber \\ &\a_j^\pm=\frac{2}{\pi}\int\limits_0^\pi v(a,x_2)\sin jx_2\di x_2,\quad \b_j^\pm=\frac{2}{d}\int\limits_{-d}^0 v(a,x_2)\sin \frac{\pi j}{d}x_2\di x_2,\nonumber%\l%abel{3.35} \\ &\kappa_j^+(\l):=\sqrt{j^2-\l},\quad \kappa_j^-(\l):=\sqrt{\frac{\pi^2 j^2}{d^2}-\l},\quad j\geqslant 2.\label{3.35a} \end{align} The branch of the root in the definition of the functions $\kappa_j$ is specified by the requirement that the functions are analytic in $\mathbb{S}_\d$ and $\sqrt{1}=1$. % ------------- % \begin{lemma}\label{lm3.7} The operators $T_{7}^\pm: \H^1(\Pi_a^a)\to L_2(\g_{\widetilde{a}})$ are well defined, bounded and holomorphic in $\l\in \mathbb{S}_\d$. The estimates % ------------- % \begin{equation*} \left\|\frac{\p^i T_{7}^\pm}{\p\l^i}\right\|\leqslant Cl^i\E^{-(2l-a-\widetilde{a})\RE \kappa_1^+(\l)},\quad i=0,1,2, \end{equation*} % ------------- % hold true uniformly w.r.t. $\l\in \mathbb{S}_\d$ and $l\geqslant (a+\widetilde{a})$. \end{lemma} % ------------- % \begin{proof} We will prove the lemma for $T_{7}^+$ only, the argument for $T_{7}^-$ is similar. The function $u$ belongs to $\H^1(\Pi_a^a)$, hence we have the estimate % ------------- % \begin{equation*} \sum\limits_{j=1}^{\infty}\big(|\a_j^\pm|^2+|\b_j^\pm|^2\big) \leqslant C\|u\|_{\H^1(\Pi_a)}^2, \end{equation*} % ------------- % where the constant $C$ is independent of $\l\in \mathbb{S}_\d$ and $l\geqslant (a+\widetilde{a})$. Employing this inequality we infer that % ------------- % \begin{align*} &\left\|\sum\limits_{j=1}^{\infty} j\a_j^+ \E^{- \kappa_j^+(\l)(\cdot- a)}\E^{-2\kappa_j^+(\l)l}\right\|_{L_2(\g_{\widetilde{a}})} \leqslant \sum\limits_{j=1}^{\infty} j|\a_j| \E^{-2l\RE \kappa_j^+(\l) } \|\E^{-\kappa_j^+(\l)(\cdot-a)}\|_{L_2(-\widetilde{a},\widetilde{a})} \\ &\leqslant C\sum\limits_{j=1}^{\infty} \frac{j|\a_j^+|}{\sqrt{\RE \kappa_j^+(\l)}}\E^{-(2l-a-\widetilde{a})\RE \kappa_j^+(\l)} \\ &\leqslant C\left(\sum\limits_{j=1}^{\infty} |\a_j|^2\right)^{1/2}\left(\sum\limits_{j=1}^{\infty} \frac{j^2\E^{-2(2l-a-\widetilde{a})\RE \kappa_j^+(\l)}}{|\kappa_j^+(\l)|}\right)^{1/2} \\ &\leqslant C\|v\|_{\H^1(\Pi_a^a)}\E^{-(2l-a-\widetilde{a}) \RE \kappa_1^+(\l)} \left(\sum\limits_{j=1}^{\infty} \frac{j^2\exp\left(-2(2l-a-\widetilde{a})\RE (\kappa_j^+(\l)-\kappa_1^+(\l))\right)}{\sqrt{|\IM\l|}}\right)^{1/2} \\ &\leqslant C\E^{-(2l-a-\widetilde{a})}\|v\|_{\H^1(\Pi_a^a)}, \end{align*} % ------------- % where $C$ is independent of $\l\in \mathbb{S}_\d$. In the same way one can prove that % ------------- % \begin{align*} &\left\|\sum\limits_{j=1}^{\infty} \frac{\pi j}{d}\b_j^+ \E^{- \kappa_j^-(\l)(\cdot- a)}\E^{-2\kappa_j^-(\l)l}\right\|_{L_2(\g_{\widetilde{a}})} \leqslant C\E^{-(2l-a-\widetilde{a})\RE \kappa_1^-(\l)}\|v\|_{\H^1(\Pi_a^a)}, \end{align*} % ------------- % The last two estimates imply that the operator $T_{7}^+:\H^1(\Pi_a^a)\to L_2(\g_{\widetilde{a}})$ is well defined and bounded. One can check easily that % ------------- % \begin{align*} \left(\frac{\p T_{7}^+ v}{\p\l}\right)(x_1,\l,l):=&\sum\limits_{j=1}^{\infty} \frac{j\a_j^+}{2\kappa_j^+(\l)}(x_1-a+2l)\E^{-\kappa_j^+(\l)(x_1- a)}\E^{-2\kappa_j^+(\l)l} \\ &-\sum\limits_{j=1}^{\infty} \frac{\pi j \b_j^+(x_1-a+2l)}{2\kappa_j^-(\l)d}\E^{-\kappa_j^-(\l)(x_1- a)}\E^{-2\kappa_j^-(\l)l}. \end{align*} % ------------- % Repeating the argument which yielded the estimate for $T_{7}^+v$ we can establish that % ------------- % \begin{align*} &\left\|\frac{\p T_{7}^\pm v}{\p\l}\right\|_{L_2(\g_{\widetilde{a}})} \leqslant Cl\E^{-(2l-a-\widetilde{a})\RE \kappa_1^-(\l)}\|v\|_{\H^1(\Pi_a^a)} \end{align*} % ------------- % with the constant $C$ independent of $\l\in \mathbb{S}_\d$ and $l\geqslant (a+\widetilde{a})$. Consequently, the operator $\frac{\p T_{7}^+}{\p\l}$ exists, it is bounded and the stated estimate for its norm holds true. The norm estimate for $\frac{\p^2 T_{7}^+}{\p\l^2}$ is obtained in a similar way. \end{proof} % ------------- % For any $l\geqslant (a+\widetilde{a})$ we define operators $T_{8}^\pm(\l,l,a,\widetilde{a})$ which map any $f\in L_2(\g_a)$ into the function % ------------- % \begin{equation*} \frac{\p u}{\p x_2}(x_1\pm 2l,+0,\l)-\frac{\p u}{\p x_2}(x_1\pm 2l,-0,\l),\quad x_1\in(-\widetilde{a},\widetilde{a}). \end{equation*} % ------------- % Here $u$ is a solution to the boundary value problem (\ref{3.11}) belonging to $\H^1(\Pi)$. Taking into account Lemma~\ref{lm3.13} together with the boundedness of the embedding $\H^1(\Pi_a)$ into $L_2(\{x: |x_1\pm 2l|<\widetilde{a}, x_2=0\})$, we conclude that the operators $T_{8}^\pm: L_2(\g_a)\to L_2(\g_{\widetilde{a}})$ are bounded and holomorphic in $\l\in\mathbb{S}_\d$. % ------------- % \begin{lemma}\label{lm3.6a} The poles of the operators $T_{8}^\pm$ coincide with the eigenvalues of the operator $H(a)$. For any compact set $K\subset \mathbb{S}_\d$ separated from $\discspec(H(a))$ by a positive distance the estimates % ------------- % \begin{equation}\label{3.21a} \left\|\frac{\p^i T_{8}^\pm}{\p\l^i}\right\|\leqslant Cl^i\E^{-2 l \RE \kappa_1^+(\l)},\quad i=0,1, \end{equation} % ------------- % hold true with $C$ which is independent of $\l\in K$ and $l$. For any $\l$ close to an eigenvalue $\l_n$ of the operator $H(a)$ the representation % ------------- % \begin{equation} T_{8}^\pm(\l,l,a,\widetilde{a})=\frac{\phi_n^\pm}{\l-\l_n} T_{4}(a) + T_{9}^\pm(\l,l,a,\widetilde{a})\label{3.17} \end{equation} % ------------- % is valid, where % ------------- % \begin{equation}\label{3.17a} \phi_n^\pm(x_1,l,a):=\frac{\p\psi_n}{\p x_2}(x_1\pm 2l,+0,a)-\frac{\p\psi_n}{\p x_2}(x_1\pm 2l,-0,a),\quad x_1\in(-\widetilde{a},\widetilde{a}). \end{equation} % ------------- % The operators $T_{9}^\pm: L_2(\g_a)\to L_2(\g_{\widetilde{a}})$ are bounded and holomorphic w.r.t. $\l$ in the vicinity of $\l_n$ and satisfy the estimates % ------------- % \begin{equation}\label{3.16} \left\|\frac{\p^i T_{9}^\pm}{\p\l^i}\right\|\leqslant Cl^{i+1}\E^{-2l\RE \kappa_1^+(\l)},\quad i=0,1, \end{equation} % ------------- % where the constant $C$ is independent of $\l$ and $l$. \end{lemma} % ------------- % \begin{proof} Due to Lemma~\ref{lm3.12} we have % ------------- % \begin{equation}\label{3.27a} T_{8}^\pm(\l,l,a,\widetilde{a})f=T_{7}^\pm(\l,l,a,\widetilde{a})u. \end{equation} % ------------- % Here $u$ is a solution to the boundary value problem (\ref{3.11}). Using this identity and the representation (\ref{3.9}), we arrive at (\ref{3.17}), where $T_{9}^\pm$ is bounded operator holomorphic in $\lambda$. Moreover, % ------------- % \begin{align*} T_{9}^\pm(\l,l,a,\widetilde{a})=&\frac{ T_{7}^\pm(\l,l,a,\widetilde{a})- T_{7}^\pm(\l_n,l,a,\widetilde{a})}{\l-\l_n} T_{4}(a)+T_{7}^\pm(\l,l,a,\widetilde{a})T_{5}^\pm(\l,a) \\ =&\left(\frac{1}{\l-\l_n}\int\limits_{\l_n}^{\l}\frac{\p T_{7}^\pm}{\p\l}(z,l,a,\widetilde{a})\di z\right) T_{4}(a)+T_{7}^\pm(\l,l,a,\widetilde{a})T_{5}^\pm(\l,a), \\ \frac{T_{9}^\pm}{\p\l}(\l,l,a,\widetilde{a})=& \left(\frac{1}{(\l-\l_n)^2}\int\limits_{\l_n}^{\l} \int\limits_{z_1}^{\l}\frac{\p^2 T_{7}^\pm}{\p\l^2}(z_2,l,a,\widetilde{a})\di z_2\di z_1\right) T_{4}(a) \\ &\hphantom{\Bigg(}+\frac{\p}{\p\l}\left( T_{7}^\pm(\l,l,a,\widetilde{a})T_{5}^\pm(\l,a)\right). \end{align*} % ------------- % Applying now Lemma~\ref{lm3.7} we obtain the estimates (\ref{3.16}). The operators $\frac{\p^i T_{3}}{\p\l^i}$, $i=0,1$, are bounded uniformly in $\l\in K$, thus in view of the relation (\ref{3.27a}) and Lemma~\ref{lm3.7} we arrive readily at the estimates (\ref{3.21a}). \end{proof} % ------------- % Concluding this section we shall prove Propositions~\ref{lm1.2} and \ref{lm1.3}. % ------------- % \begin{proof}[Proof of Proposition~\ref{lm1.2}] Applying Lemma~\ref{lm3.12} to $\psi_n$ with $b_1=\pm a$, $b_2=0$, $b_3=\pi$ and $b_2=-d$, $b_3=0$, we obtain the formul{\ae} (\ref{1.4}), (\ref{1.4a}). The factor $(\pm 1)^{n+1}$ in these formul{\ae} is due to the definite parity of $\psi_n$ w.r.t. $x_1$. The formula (\ref{1.5}) for $c_1^{(n)}$ follows from the chain of relations obtained by integration by parts, % ------------- % \begin{align*} 0&=\int\limits_{\Pi^+}\E^{\pm \kappa_1^+(\l_n)x_1}\sin x_2(\D+\l_n)\psi_n(x,a)\di x \\ &=\int\limits_{\g_a}\E^{\pm \kappa_1^+(\l_n)x_1}\psi_n(x,a)\di x-(\pm 1)^{n+1}\pi \kappa_1^+(\l_n)c(\l_n,a). \end{align*} % ------------- % It remains to check the inequalities $c(\l_i,a)\not=0$, $i=1,2$. The eigenfunction $\psi_1$ associated with the ground state can be chosen non-negative. Moreover, $\psi_1$ is not identically zero at $\g_a$, since otherwise it would be an eigenfunction of the negative Dirichlet Laplacian in $\Pi^+$ and would correspond to the eigenvalue $\l_1<1$. At the same time, the spectrum of the mentioned operator is the halfline $[1,+\infty)$. The described properties of $\psi_1$ and the formula (\ref{1.15}) imply that $c(\l_1,a)\not=0$. According to Proposition~\ref{lm1.1} the eigenfunction $\psi_2$ is odd w.r.t. $x_1$. It allows us to modify the formula (\ref{1.5}), \begin{equation}\label{3.20} \begin{aligned} c(\l_2,a)=&\frac{1}{\pi \kappa_1^+(\l_2)}\int\limits_{\g_a} \psi_2(x,a)\sinh \kappa_1^+(\l_2)x_1\di x_1 \\ =&\frac{2}{\pi \kappa_1^+(\l_2)} \int\limits_0^a\psi_2(x_1,0,a)\sinh \kappa_1^+(\l_2)x_1\di x_1. \end{aligned} \end{equation} The eigenvalue $\l_2$ is the ground state of the negative Laplacian in $\Pi_a\cap\{x: x_1>0\}$ subject to Dirichlet boundary condition on $\p\Pi_a\cap\{x: x_1>0\}$ and to Neumann one on $\{x: x_1=0, -d0$ such that % ------------- % \begin{equation}\label{4.0} (\nabla\psi,\nabla\z)_{L_2(\Pi)}=\l(\psi,\z)_{L_2(\Pi)}, \end{equation} % ------------- % holds for each $\z\in C^\infty_0(\Pi)$; it follows from the smoothness-improving theorem mentioned above that such a $\psi$ belongs to $C^\infty(\Pi)$. We assume that $\l\in \mathbb{S}_\d$, with $\d>0$ is chosen in such a way that $\sigma_*\subset \mathbb{S}_\d$. Let $f_\pm=f_\pm(\cdot,l)\in L_2(\g_{a_\pm})$ be an arbitrary pair of functions. Denote by $u_\pm$ the solutions of the problem (\ref{3.11}) with $a=a_\pm$ and $f=f_\pm\in L_2(\g_{a_\pm})$ and assume that $u_\pm\in L_2(\Pi_{a_\pm})$. We will seek a solution to the problem (\ref{4.1}) in the form % ------------- % \begin{equation}\label{4.2} \psi(x,\l,l)=u_+(x_1-l,x_2,\l,l)+u_-(x_1+l,x_2,\l,l). \end{equation} % ------------- % Suppose for a moment that the function $\psi$ defined in this way solves the problem (\ref{4.1}). In such a case the function $\psi$ is infinitely differentiable at the points of the segments $\g_{a_\pm}$, and therefore % ------------- % \begin{equation*} \frac{\p\psi}{\p x_2}(x_1,+0,\l,l)-\frac{\p\psi}{\p x_2}(x_1,-0,\l,l)=0,\quad x\in\g_\pm. \end{equation*} % ------------- % Substituting from (\ref{4.2}) into this identity, we obtain a pair of equations, % ------------- % \begin{equation}\label{3.4a} f_\pm(x_1) + \frac{\p u_\mp}{\p x_2}(x_1\pm 2l,+0,\l,l)-\frac{\p u_\mp}{\p x_2}(x_1\pm 2l,-0,\l,l)=0,\quad x\in\g_{a_\pm}. \end{equation} % ------------- % Denote $\boldsymbol{f}=(f_+,f_-)\in L_2(\g_{a_+})\oplus L_2(\g_{a_-})$. The following lemma states that the last equation is equivalent to the original problem (\ref{4.1}). % ------------- % \begin{lemma}\label{lm4.1} To any solution $\boldsymbol{f}\in L_2(\g_{a_+})\oplus L_2(\g_{a_-})$ of (\ref{3.4a}) and functions $u_\pm$ solving (\ref{3.11}) for $a=a_\pm$, $f=f_\pm$ there exists a unique $L_2(\Pi)$-solution of (\ref{4.1}) given by (\ref{4.2}). Reversely, for any solution $\psi$ of (\ref{4.1}) there are unique $\boldsymbol{f}\in L_2(\g_{a_+})\oplus L_2(\g_{a_-})$ solving (\ref{3.4a}) and unique functions $u_\pm\in L_2(\Pi_{a_{\pm}})$ satisfying (\ref{3.11}) with $a=a_\pm$, $f=f_\pm$ such that $\psi$ is given by (\ref{4.2}). This equivalence holds for any $\l\in \mathbb{S}_\d$ and $l\geqslant\max\{a_-,a_+\}+1$. \end{lemma} % ------------- % \begin{proof} Suppose that $\boldsymbol{f}\in L_2(\g_{a_+})\oplus L_2(\g_{a_-})$ is a solution to the equations (\ref{3.4a}), where the functions $u_\pm\in L_2(\Pi_{a_\pm})$ solve the problem (\ref{3.11}) $a=a_\pm$ and $f=f_\pm$. We define $\psi$ in accordance with (\ref{4.2}). The functions $u_\pm$ are elements of $L_2(\Pi)$, hence the same is true for $\psi$. Moreover, the function $\psi$ belongs obviously to $\H^1(\Pi^b,\G^b)$ for each $b>0$ and vanishes on $\G$. Let us check that the function $\psi$ satisfies the equation (\ref{4.0}). To this purpose, we indicate by $\chi_2=\chi_2(x_1)$ an infinitely differentiable cut-off function being equal to one if $|x_1+l|<\max\{a_+,a_-\}+1/2$ and vanishing if $|x_1+l|>\max\{a_+,a_-\}+1$. For any $\z\in C_0^\infty(\Pi)$ we have % ------------- % \begin{equation}\label{4.4} \begin{aligned} &\big(\nabla u_+(x_1-l,x_2,\l,l),\nabla\z\big)_{L_2(\Pi)}-\l\big(u_+(x_1-l,x_2,\l,l), \z\big)_{L_2(\Pi)} \\ &=\big(\nabla u_+(x_1-l,x_2,\l,l),\nabla(\z\chi_2)\big)_{L_2(\Pi)}- \l\big(u_+(x_1-l,x_2,\l,l), \z\chi_2)\big)_{L_2(\Pi)} \\ &+\big(\nabla u_+(x_1-l,x_2,\l,l),\nabla(\z(1-\chi_2))\big)_{L_2(\Pi)}- \l\big(u_+(x_1-l,x_2,\l,l), \z(1-\chi_2)\big)_{L_2(\Pi)}. \end{aligned} \end{equation} % ------------- % Since $u_+(\cdot)$ is an element of $C^\infty(\Pi_{a_+}\setminus \overline{Q}^{a_+})$, we can integrate by parts, % ------------- % \begin{align*} &\big(\nabla u_+(x_1-l,x_2,\l,l),\nabla\z\chi_2)\big)_{L_2(\Pi)}- \l\big(u_+(x_1-l,x_2,\l,l), \z\chi_2\big)_{L_2(\Pi)} \\ &=-\int\limits_{\g_-}\left(\frac{\p u_+}{\p x_2}(x_1-l,+0,\l,l)-\frac{\p u_+}{\p x_2}(x_1-l,-0,\l,l)\right)\z\di x_1- \\ &-\int\limits_\Pi\z\chi_2(\D+\l) u_+(x_1-l,x_2,\l,l)\di x= \int\limits_{\g_-} f_-(x_1+l)\z\di x_1. \end{align*} % ------------- % We have employed here the equation satisfied by $u_+$ as well as the equation (\ref{3.4a}) for $f_+$. Since $\z(x_1+l,x_2)(1-\chi_2(x_1+l))\in C^\infty_0(\Pi_{a_+})$, we can use the identity (\ref{3.12}) to infer that \begin{align*} \big(\nabla u_+(x_1-l,x_2,\l,l),\nabla\z(1-\chi_2)\big)_{L_2(\Pi)}-& \big(u_+(x_1-l,x_2,\l,l),\z(1-\chi_2)\big)_{L_2(\Pi)} \\ &=-\int\limits_{\g_+}f_+(x_1-l)\z\di x_1. \end{align*} We substitute now the last two relations into (\ref{4.4}) and arrive at the identity \begin{align*} &\big(\nabla u_+(x_1-l,x_2,\l,l),\nabla\z\big)_{L_2(\Pi)}- \l\big(u_+(x_1-l,x_2,\l,l), \z\big)_{L_2(\Pi)} \\ &=(f_-(x_1+l),\z)_{L_2(\g_-)}-(f_+(x_1-l),\z)_{L_2(\g_+)}. \end{align*} % ------------- % In the same way one can check that % ------------- % \begin{align*} &\big(\nabla u_-(x_1+l,x_2,\l,l),\nabla\z\big)_{L_2(\Pi)}- \l\big(u_-(x_1+l,x_2,\l,l), \z\big)_{L_2(\Pi)} \\ &=(f_+(x_1-l),\z)_{L_2(\g_+)}-(f_-(x_1+l),\z)_{L_2(\g_-)}; \end{align*} % ------------- % summing the last two relations we arrive at the relation (\ref{4.0}) for the function $\psi$. Let $\psi$ be a solution to the problem (\ref{4.1}) belonging to $L_2(\Pi)$. By smoothness-improving theorems the function $\psi$ belongs to $C^\infty(\{x: -1\leqslant x_1\leqslant 1, 0\leqslant x_2\leqslant \pi\})$ and to $C^\infty(\{x: -1\leqslant x_1\leqslant 1, -d\leqslant x_2\leqslant 0\})$. This allows us to define the numbers % ------------- % \begin{align*} &\a_j^\pm=\a_j^\pm(\l,l):= \frac{2}{\pi}\int\limits_0^\pi\left(\psi(0,x_2,\l,l)\pm \frac{1}{\kappa_j^+}\frac{\p\psi}{\p x_1}(0,x_2,\l,l)\right)\sin j x_2\di x_2, \\ &\b_j^\pm=\b_j^\pm(\l,l):= \frac{2}{d}\int\limits_0^\pi\left(\psi(0,x_2,\l,l)\pm \frac{1}{\kappa_j^-}\frac{\p\psi}{\p x_1}(0,x_2,\l,l)\right)\sin \frac{\pi j}{d}x_2\di x_2. \end{align*} % ------------- % Using these numbers, we introduce the functions $u_\pm$ in the following way: % ------------- % \begin{align*} &u_\pm(x_1\mp l,x_2,\l,l):=\sum\limits_{j=1}^{\infty} \a_j^\pm(\l,l)\E^{\pm \kappa_j^+ x_1}\sin j x_2,&& \pm x_1\leqslant 0,\quad x_2\in(0,\pi), \\ &u_\pm(x_1\mp l,x_2,\l,l):=\sum\limits_{j=1}^{\infty} \b_j^+(\l,l)\E^{\pm \kappa_j^- x_1}\sin \frac{\pi j}{d} x_2,&& \pm x_1\leqslant 0,\quad x_2\in(-d,0), \\ &u_\pm(x_1\mp l,x_2,\l,l):=\psi(x,\l,l)-u_\pm(x_1\pm l,x_2,\l),&& \pm x_1>0,\quad x_2\in(-d,\pi). \end{align*} % ------------- % Proceeding in the same way as in the proof of Lemma~4.1 in \cite{B2}, we check that the functions $u_\pm$ are well defined and % ------------- % \begin{align}\label{4.6} % ------------- % &u_\pm\in \H^1(\Pi_{a_\pm},\G_{a_\pm})\cap\H^2(S), \quad S\in\Xi_a, \\ &(\D+\l)u_\pm(x_1\mp l,x_2,\l,l)=0,\quad x\in\Pi\setminus\{x: x_1=0\}.\label{4.8} \end{align} The relation (\ref{4.2}) follows from the definition of the functions $u_\pm$. Now we set % ------------- % \begin{equation}\label{4.8a} f_\pm(x_1,l):=-\frac{\p u_\mp}{\p x_2}(x_1\pm 2l,+0,\l,l)+ \frac{\p u_\mp}{\p x_2}(x_1\pm 2l,-0,\l,l),\quad x_1\in(-a_\pm,a_\pm); \end{equation} % ------------- % in view of (\ref{4.6}) we can conclude that $f_\pm\in L_2(\g_{a_\pm})$. We also note that the definition of $u_\pm$ and the smoothness of $\psi$ at $\g_\pm$ imply % ------------- % \begin{equation}\label{4.7} f_\pm(x_1,l)=\frac{\p u_\pm}{\p x_2}(x_1,+0,\l,l)-\frac{\p u_\pm}{\p x_2}(x_1,-0,\l,l),\quad x_1\in(-a_\pm,a_\pm). \end{equation} % ------------- % % ------------- % Let us check the integral equation (\ref{3.12}) for the function $u_+=u_+(x,\l)$. Taking into account (\ref{4.8}), (\ref{4.7}), and integrating by parts, we get % ------------- % \begin{align*} &-(\nabla u_+,\nabla\z)_{L_2(\Pi_a)}+\l(u_+,\z)_{L_2(\Pi_a)} \\ &= \left(\frac{\p u_+}{\p x_2}(x_1,+0,\l)- \frac{\p u_+}{\p x_2}(x_1,-0,\l),\z\right)_{L_2(\g_{a_+})}=(f_+,\z)_{L_2(\g_{a_+})}. \end{align*} % ------------- % for any $\z\in C_0^\infty(\Pi_{a_+})$. In the same way one can check that % ------------- % \begin{equation*} -(\nabla u_-,\nabla\z)_{L_2(\Pi_a)}+\l(u_-,\z)_{L_2(\Pi_a)} =(f_-,\z)_{L_2(\g_{a_-})} \end{equation*} % ------------- % for any $\z\in C_0^\infty(\Pi_{a_-})$, thus $u_\pm$ are solutions to the problem (\ref{3.11}) with $a=a_\pm$ and $f=f_\pm$. The equations (\ref{3.4a}) follow from (\ref{4.8a}). \end{proof} % ------------- % Suppose that $\l\in \mathbb{S}_\d\setminus\sigma_*$. In that case the functions $u_\pm$ introduced above can be represented as $u_\pm=T_{3}(\l,a_\pm)f_\pm$, thus the equations (\ref{3.4a}) become % ------------- % \begin{equation}\label{4.3a} \boldsymbol{f}+T_{8}(\l,l,\boldsymbol{a})\boldsymbol{f}=0, \end{equation} % ------------- % where the operator $T_{8}: L_2(\g_{a_+})\oplus L_2(\g_{a_-})\to L_2(\g_{a_+})\oplus L_2(\g_{a_-})$ is defined by % ------------- % \begin{equation*} T_{8}(\l,l,\boldsymbol{a})\boldsymbol{f}:=\big( T_{8}^+(\l,l,a_-,a_+)f_-,T_{8}^-(\l,l,a_+,a_-)f_+\big). \end{equation*} % ------------- % Now we are ready to demonstrate the first one of our main results. \begin{proof}[Proof of Theorem~\ref{th1.0}] If $a_\pm=0$ the essential spectrum of the operator $H$ is obviously $[1,+\infty)$, and an elementary argument using Dirichlet-Neumann bracketing \cite[Ch. X\!I\!I\!I, \S 15]{RS} and the minimax principle \cite[Ch. X\!I\!I\!I, \S 1]{RS} shows that the threshold of the essential spectrum of $H$ is one, i.e. $\essspec(H)\subseteq[1,+\infty)$. The opposite inclusion can be shown easily; one needs to employ Weyl's criterion (see, for instance, proof of Lemma~2.1 in \cite{B2}). The operator $H$ being self-adjoint, its isolated eigenvalues are real, and in view of the above observation they are smaller than one; we arrange them conventionally in the ascending order counting multiplicity. Next we use bracketing again in a way analogous to \cite{ESTV}: we add Neumann boundaries at segments corresponding to $x_1$ at the endpoints of $\g_\pm$ and $x_2\in (-d,\pi)$. In this way we get an operator estimating $H$ from below, and since only the window parts contribute to the spectrum below one we infer by minimax that $H$ has finitely many eigenvalues for any $l>0$ and their number has a bound independent of $l$. Let $K\subset \mathbb{S}_\d$ be any compact set separated from $\sigma_*$ by a positive distance. By the estimates (\ref{3.21a}) the operator $T_{8}$ has a norm being strictly less than one for $\l\in K$ and $l$ large enough. For such $\l$ and $l$ the equation (\ref{4.3a}) has thus a trivial solution only, and in view of Lemma~\ref{lm4.1} this implies that the operator $H$ has no eigenvalues in the set $K$ if $l$ is large enough. This means that each eigenvalue of the operator $H$ has to converge to one of the numbers from the set $\sigma_*$ or to the threshold of the essential spectrum. \end{proof} The eigenvalues $H$, i.e. those $\l$ for which the problem (\ref{4.1}) has a nontrivial $L_2(\Pi)$-solution, coincide in view of Lemma~\ref{lm4.1} with the values of $\l$ for which the equation (\ref{3.4a}) has a nontrivial solution. In the case considered here we deal only with the eigenvalues of $H$ which converge to a value $\l_*\in\sigma_*$ separated from the threshold, in other words, being smaller than one. Our aim is to solve the equation (\ref{3.4a}) and to obtain in this way an equation for the aforementioned values of $\l$. Consider a $\l_*\in\sigma_*$; if $\l_*=\l_n$ is an eigenvalue of the operator $H(a_+)$ we set % ------------- % \begin{equation*} \boldsymbol{\phi}_*^+(\cdot,l):=\big(0,\phi_n^-(\cdot,l,a_+)\big)\in L_2(\g_{a_+})\oplus L_2(\g_{a_-}),\quad T_{4}^+\boldsymbol{f}:=(f_+,\psi_n)_{L_2(\g_{a_+})}, \end{equation*} % ------------- % where $\phi_n^-$ is determined by $\psi_n$ in accordance with (\ref{3.17a}) and $\psi_n$ is an eigenfunction associated with $\l_n$, in the opposite case we put % ------------- % \begin{equation*} \boldsymbol{\phi}_*^+(\cdot,l):=(0,0)\in L_2(\g_{a_+})\oplus L_2(\g_{a_-}),\quad T_{4}^+\boldsymbol{f}:=0. \end{equation*} % ------------- % Analogously, if $\l_*=\l_n$ is an eigenvalue of $H(a_-)$ we set % ------------- % \begin{equation*} \boldsymbol{\phi}_*^-(\cdot,l):=\big(\phi_n^+(\cdot,l,a_-),0\big)\in L_2(\g_{a_+})\oplus L_2(\g_{a_-}),\quad T_{4}^-\boldsymbol{f}:=(f_-,\psi_n)_{L_2(\g_{a_-})}, \end{equation*} % ------------- % where $\phi_n^+$ corresponds to $\psi_n$ according to (\ref{3.17a}) and $\psi_n$ is an eigenfunction associated with $\l_n$, otherwise % ------------- % \begin{equation*} \boldsymbol{\phi}_*^-(\cdot,l):=(0,0)\in L_2(\g_{a_+})\oplus L_2(\g_{a_-}),\quad T_{4}^-\boldsymbol{f}:=0. \end{equation*} % ------------- % Given a number $\l_*\in\si_*$, we consider the equation (\ref{3.4a}) for $\l$ in the vicinity of $\l_*$. Assume first that $\l\ne\l_*$, in which case the equation (\ref{3.4a}) is equivalent to (\ref{4.3a}). In view of Lemma~\ref{lm3.6a} the operator $T_{8}$ is bounded and meromorphic as a function of $\l\in \mathbb{S}_\d$, and the numbers $\l_*\in\sigma_*$ are poles of $T_{8}$. For any $\l$ close to $\l_*$ the operator $T_{8}$ can be thus represented as % ------------- % \begin{equation}\label{4.12} T_{8}(\l,l,\boldsymbol{a}) =\boldsymbol{\phi}_*^+(\cdot,l)\frac{T_{4}^+ }{\l-\l_*} + \boldsymbol{\phi}_*^-(\cdot,l)\frac{T_{4}^- }{\l-\l_*} +T_{9}(\l,l,\boldsymbol{a}), \end{equation} % ------------- % where the operator $T_{9}$ acts as % ------------- % \begin{equation*} T_{9}(\l,l,\boldsymbol{a})\boldsymbol{f}:=\big( T_{8}^+(\l,l,a_-,a_+)f_-,T_{9}^-(\l,l,a_+,a_-)f_+\big) \end{equation*} % ------------- % if $\l_*\in\discspec(H(a_+))\setminus\discspec(H(a_-))$, % ------------- % \begin{equation*} T_{9}(\l,l,\boldsymbol{a})\boldsymbol{f}:=\big( T_{9}^+(\l,l,a_-,a_+)f_-,T_{8}^-(\l,l,a_+,a_-)f_+\big) \end{equation*} % ------------- % if $\l_*\in\discspec(H(a_-))\setminus\discspec(H(a_+))$, and finally, % ------------- % \begin{equation*} T_{9}(\l,l,\boldsymbol{a})\boldsymbol{f}:=\big( T_{9}^+(\l,l,a_-,a_+)f_-,T_{9}^-(\l,l,a_+,a_-)f_+\big) \end{equation*} % ------------- % if $\l_*\in\discspec(H(a_-))\cap\discspec(H(a_+))$. The operator $T_{9}$ on $L_2(\g_{a_+})\oplus L_2(\g_{a_-})$ is bounded and holomorphic w.r.t. $\l$ in the vicinity of $\l_*$, and the estimate % ------------- % \begin{equation}\label{4.13} \left\|\frac{\p^i T_{9}}{\p\l^i}\right\|\leqslant Cl^{i+1}\E^{-2l\RE \kappa_1^+(\l)},\quad i=0,1, \end{equation} % ------------- % holds true with a constant $C$ which is independent on $\l$ and $l$. We substitute the representation (\ref{4.12}) into (\ref{4.3a}) to obtain % ------------- % \begin{equation*}%\l%abel{4.14} \boldsymbol{f}+\frac{T_{4}^+\boldsymbol{f}}{\l-\l_*} \boldsymbol{\phi}_*^++ \frac{T_{4}^-\boldsymbol{f}}{\l-\l_*} \boldsymbol{\phi}_*^-+T_{9}\boldsymbol{f}=0. \end{equation*} % ------------- % Since the norm of $T_{9}$ is small for large $l$ due to (\ref{4.13}), the operator $(\I+T_{9})^{-1}$ is well defined being bounded in $L_2(\g_{a_+})\oplus L_2(\g_{a_-})$. We apply this operator to the last equation arriving at % ------------- % \begin{equation}\label{4.15} \boldsymbol{f}+\frac{T_{4}^+\boldsymbol{f}}{\l-\l_*}\boldsymbol{\Phi}_*^+ + \frac{T_{4}^-\boldsymbol{f}}{\l-\l_*}\boldsymbol{\Phi}_*^-=0, \end{equation} % ------------- % where $\boldsymbol{\Phi}_*^\pm(\cdot,\l,l)= (\I+T_{9}(\l,l, \boldsymbol{a}))^{-1} \boldsymbol{\phi}_*^\pm(\cdot,l)$. The last equation implies that % ------------- % \begin{equation}\label{4.17a} \boldsymbol{f}=c_+\boldsymbol{\Phi}^+_*+ c_-\boldsymbol{\Phi}^-_* \end{equation} % ------------- % for some numbers $c_\pm$. We substitute from here into (\ref{4.15}) obtaining % ------------- % \begin{equation}\label{4.18} \boldsymbol{\Phi}_*^+\left(c_+\left(1+\frac{A_{11}}{\l-\l_*}\right) +c_-\frac{A_{12}}{\l-\l_*}\right)+\boldsymbol{\Phi}_*^- \left(c_+\frac{A_{21}}{\l-\l_*}+ c_-\left(1+\frac{A_{22}}{\l-\l_*}\right)\right)=0, \end{equation} % ------------- % where the quantities $A_{ij}=A_{ij}(\l,l)$ are defined by % ------------- % \begin{gather*} A_{11}(\l,l):=T_{4}^+\Phi_*^+(\cdot,\l,l),\quad A_{12}(\l,l):=T_{4}^+\Phi_*^-(\cdot,\l,l), \\ A_{21}(\l,l):=T_{4}^-\Phi_*^+(\cdot,\l,l),\quad A_{22}(\l,l):=T_{4}^-\Phi_*^-(\cdot,\l,l). \end{gather*} % ------------- % The definition of $\boldsymbol{\Phi}_*^\pm$ together with the estimate (\ref{4.13}) imply for $l$ large enough % ------------- % \begin{equation}\label{4.16a} \boldsymbol{\Phi}_*^\pm=\boldsymbol{\phi}_*^\pm+ \Odr\big(\E^{-2l\RE \kappa_1^+(\l)}\|\boldsymbol{\phi}_*^\pm\|\big). \end{equation} % ------------- % If $\boldsymbol{\phi}_*^+\not=0$, and $\boldsymbol{\phi}_*^-=0$, in particular, we have % ------------- % \begin{equation}\label{4.19} \boldsymbol{\Phi}_*^+\not=0,\quad \boldsymbol{\Phi}_*^-=0,\quad A_{12}=A_{22}=0, \end{equation} % ------------- % and in this case the equation (\ref{4.18}) holds if and only if % ------------- % \begin{equation*} c_+\left(1+\frac{A_{11}}{\l-\l_*}\right)=0. \end{equation*} % ------------- % If $\boldsymbol{f}$ corresponds to an eigenfunction $\psi$ of the problem (\ref{4.1}) by (\ref{4.2}), the number $c_+$ is non-zero. Indeed, in the opposite case (\ref{4.17a}) and (\ref{4.19}) would imply that $\boldsymbol{f}=0$, which by Lemma~\ref{lm4.1} results in $\psi=0$. Consequently, the equation (\ref{4.3a}) has in this case a nontrivial solution if and only if % ------------- % \begin{equation}\label{4.20} \l-\l_*+A_{11}(\l,l)=0. \end{equation} % ------------- % If $\l$ is a root of this equation, the corresponding nontrivial solution of (\ref{4.3a}) can be expressed as (\ref{4.17a}) with $c_+\not=0$ and $c_-=0$. In the case $\boldsymbol{\phi}_*^+=0$ and $\boldsymbol{\phi}_*^-\not=0$ similar arguments lead us to the conclusion that the equation (\ref{4.3a}) has a nontrivial solution if and only if % ------------- % \begin{equation}\label{4.21} \l-\l_*+A_{22}(\l,l)=0, \end{equation} % ------------- % and the corresponding non-trivial solution can be written as (\ref{4.17a}) with the coefficients $c_+=0$ and $c_-\not=0$. Finally, if both the functions $\boldsymbol{\phi}_*^\pm$ are non-zero, they are linearly independent by definition and the same is true for the functions $\boldsymbol{\Phi}_*^\pm$. Hence the equation (\ref{4.3a}) holds if and only if % ------------- % \begin{equation}\label{4.16} \left((\l-\l_*)\mathrm{E}+ \mathrm{A}(\l,l)\right)\boldsymbol{c}=0, \end{equation} % ------------- % where $\mathrm{E}$ is the unit matrix, and % ------------- % \begin{equation*} \mathrm{A}(\l,l):= \begin{pmatrix} A_{11}(\l,l)& A_{12}(\l,l) \\ A_{21}(\l,l)& A_{22}(\l,l) \end{pmatrix}, \quad \boldsymbol{c}:= \begin{pmatrix} c_+ \\ c_- \end{pmatrix}. \end{equation*} % ------------- % The column $\boldsymbol{c}$ is non-zero, since otherwise (\ref{4.17a}) and (\ref{4.19}) would imply $\boldsymbol{f}=0$, thus the system (\ref{4.16}) of linear equations has a nontrivial solution if and only if % ------------- % \begin{equation}\label{4.20a} \det \left((\l-\l_*)\mathrm{E}+\mathrm{A}(\l,l)\right)=0, \end{equation} % ------------- % which can be rewritten as % ------------- % \begin{equation}\label{4.17} (\l-\l_*)^2+(\l-\l_*)\tr \mathrm{A}(\l,l)+\det \mathrm{A}(\l,l)=0\,; \end{equation} % ------------- % the corresponding non-trivial solution of the equation (\ref{4.3a}) is given by (\ref{4.17a}), where $\left(\begin{smallmatrix} c_+ \\ c_- \end{smallmatrix}\right)$ is a nontrivial solution of (\ref{4.16}). Assume now that $\l=\l_*$. Let $\l_*$ coincide with an eigenvalue $\l_n$ of the operator $H(a_+)$ being not at the same time an eigenvalue of $H(a_-)$. In this case we again can claim that $u_-=T_{3}(\l_*,a_-)f_-$, on the other hand, the boundary value problem for $u_+$ with $\l=\l_*$ is solvable if and only if % ------------- % \begin{equation}\label{3.50} 0=\int\limits_{\g_{a_+}} f_+\psi_n(x,a_+)\di x=T_{4}^+\boldsymbol{f}. \end{equation} % ------------- % This follows from Lemma~\ref{lm3.13}. The function $u_+$ is given by $u_+=T_{5}(\l_*,a_+)f_+ - c_+\psi_+$, where $c_+$ is a constant. We can substitute now the described $u_\pm$ into (\ref{3.4a}) and obtain % ------------- % \begin{gather} \boldsymbol{f}+T_{9}(\l,l,\boldsymbol{a})\boldsymbol{f}= c_+\boldsymbol{\phi}_*^+,\nonumber \\ \boldsymbol{f}=c_+\boldsymbol{\Phi}_*^+.\label{3.51} \end{gather} % ------------- % This function will generate a solution to the problem (\ref{4.1}) if and only if (\ref{4.20}) holds true. Substituting (\ref{3.51}) into (\ref{3.50}), we arrive at the equation (\ref{4.20}) with $\l=\l_*$. If $c_+\not=0$ holds in (\ref{3.51}) we see that the formula (\ref{3.50}) coincides with (\ref{4.17a}) with $c_-=0$. Consequently, in the case $\l_*\in\spec(H(a_-))\setminus\spec(H(a_+))$ the equation (\ref{4.20}) determines all the values of $\l$ in the vicinity of $\l_*$ for which the equation (\ref{3.4a}) has a nontrivial solution; these nontrivial solutions are given by (\ref{4.17a}) with $c_+\not=0$ and $c_-=0$. In the same way one can check that the equation (\ref{4.21}) determines the sought values of $\l$ in the case when $\l_*$ is an eigenvalue of the operator $H(a_-)$ and not of $H(a_+)$. The corresponding nontrivial solutions of (\ref{3.4a}) have $c_+=0$ and $c_-\not=0$. Finally, if $\l_*\in\sigma_*$ is double and $\l=\l_*$, the solvability conditions of the boundary value problems for $u$ are $T_{4}^\pm \boldsymbol{f}=0$. If this holds true, the functions $u_\pm$ are given by $u_\pm=T_{5}(\l,a_\pm) -c_\pm \psi_\pm(\cdot)$, where $c_\pm$ are constants and $\psi_\pm$ are the eigenfunctions of $H(a_\pm)$ associated with $\l_*$. The equation (\ref{3.4a}) becomes % ------------- % \begin{equation*} \boldsymbol{f}+T_{9}(\l_*,l,\boldsymbol{a})\boldsymbol{f}= c_+\boldsymbol{\phi}_*^++ c_-\boldsymbol{\phi}_*^-, \end{equation*} % ------------- % which yields the relation (\ref{4.17a}). The solvability conditions $T_{4}^\pm \boldsymbol{f}=0$ are nothing else than the system of linear equations (\ref{4.16}). In this way (\ref{4.17a}), (\ref{4.16}), and (\ref{4.20a}) describe the sought values of $\l$ in the vicinity of $\l_*$ and the corresponding nontrivial solutions of (\ref{3.4a}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proofs of Theorems~\ref{th1.2}-\ref{th1.4}} \label{s: remain_proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Now we are going to demonstrate the remaining part of our claims. \begin{proof}[Proof of Theorem~\ref{th1.2}] We will give the proof for the case $\l_*=\l_n(a_-)$, the argument for $\l_*=\l_n(a_+)$ is similar. In accordance with the results of the previous section the eigenvalue $\l^-(a,l)$, if it exists, it must be a root of the equation (\ref{4.21}). Let us prove first that there is a unique root which converges to $\l_*$ as $l\to+\infty$. Proposition~\ref{lm1.2} implies that the relation % ------------- % \begin{equation}\label{5.1} \phi_n^+(x_1,l,a_-)=\tau c(\l_*,a_-)\E^{-2\kappa_1^+(\l_*)l}\E^{-\kappa_1^+(\l_*)x_1}+ \Odr(\E^{-2 \rho(\l_*) l}), \end{equation} % ------------- % holds in the norm of $L_2(\g_{a_+})$, hence by the definition of $\boldsymbol{\phi}_*^-$ we have % ------------- % \begin{equation}\label{5.2} \boldsymbol{\phi}_*^-(x_1,l)=\tau c(\l_*,a_-)\E^{-2\kappa_1^+(\l_*)l} (\E^{-\kappa_1^+(\l_*)x_1},0)+ \Odr(\E^{-2 \rho(\l_*) l}). \end{equation} % ------------- % This formula in combination with the estimate (\ref{4.13}) lead to the relation % ------------- % \begin{equation}\label{5.1a} A_{22}(\l,l)=\Odr(\E^{-2\kappa_1^+(\l_*)l}). \end{equation} % ------------- % Since $T_{9}$ is holomorphic w.r.t. $\l$ and has a small norm for large $l$, we infer that the left-hand side of the last equation is holomorphic in $\l$. For a small $\d$ take the circle of those $\l$ such that $|\l-\l_*|=\d$. In view of (\ref{5.1a}) the function $A_{22}$ satisfies the estimate $|A_{22}|<\d$ if $l$ is large enough and $|\l-\l_*|=\d$; by Rouch\'{e} theorem it implies that the function $\l\mapsto \l-\l_*+A_{22}(\l,l)$ has the same number of zeros in the disk $\{\l: |\l-\l_*|<\d\}$ as the function $\l\mapsto \l-\l_*$ does. The number $\d$ is arbitrary, so we can conclude that there is a unique root of the equation (\ref{4.21}) converging to $\l_*$ as $l\to+\infty$. As a consequence, there exists a unique eigenvalue of the operator $H$ converging to $\l_*$ as $l\to+\infty$; we will denote this eigenvalue as $\l^-(l,\boldsymbol{a})$. The estimate (\ref{5.1a}) implies at the same time that % ------------- % \begin{equation}\label{5.1b} \l^-(l,\boldsymbol{a})-\l_*=\Odr(\E^{-2\kappa_1^+(\l_*)l}). \end{equation} % ------------- % Let us derive the asymptotic expansion (\ref{1.11}) for the eigenvalue $\l^-(l,\boldsymbol{a})$. In order to do it, we will need to know the asymptotic behavior for $A_{22}$ in a way more precise than (\ref{5.1a}). For the sake of brevity we will write shortly $\l$ instead of $\l^-(l,\boldsymbol{a})$. The relations (\ref{5.2}) together with the estimates (\ref{4.13}), (\ref{5.1b}) imply that % ------------- % \begin{equation} \begin{aligned} &A_{22}(\l,l)=T_{4}^-(\I+T_{9}(\l,l,\boldsymbol{a}))^{-1} \boldsymbol{\phi}_*^-(\cdot,l) \\ &=T_{4}^-\boldsymbol{\phi}_*^-(\cdot,l)- T_{4}^- T_{9}(\l,l,\boldsymbol{a})\boldsymbol{\phi}_*^-(\cdot,l)+ \Odr(\|T_{9}\|^2\|\boldsymbol{\phi}_*^-\|) \\ &=-T_{4}^- T_{9}(\l_*,l,\boldsymbol{a})\boldsymbol{\phi}_*^-(\cdot,l)+ \Odr\left(|\l-\l_*|\left\|\frac{\p T_{9}}{\p\l}\right\|\|\boldsymbol{\phi}_*^-\| +\|T_{9}\|^2\|\boldsymbol{\phi}_*^-\|\right) \\ &=-T_{4}^-T_{8}^-(\l,l, \boldsymbol{a})\phi_n^+(\cdot,l,a_-)+ \Odr\left(|\l-\l_*| l^2\E^{-4\kappa_1^+(\l_*)l}+l\E^{-6\kappa_1^+(\l_*)l}\right). \end{aligned}\label{5.4a} \end{equation} % ------------- % Taking into account the estimate (\ref{3.21a}) for $\|T_{8}^-\|$ and the relation (\ref{5.1}), we can proceed with the calculations obtaining % ------------- % \begin{equation}\label{5.3} \begin{aligned} A_{22}(\l,l)=&-\tau c(\l,a_-)\E^{-2\kappa_1^+(\l_*)l}\big( T_{8}^-(\l_*,l,a_+,a_-) \E^{-\kappa_1^+(\l_*)x_1},\psi_*\big)_{L_2(\g_{a_-})} \\ &+\Odr\left(|\l-\l_*|l^2\E^{-4\kappa_1^+(\l_*)l}+\E^{-2l( \kappa_1^+(\l_*)+\rho(\l_*))}\right), \end{aligned} \end{equation} % ------------- % where we have denoted $\psi_*(x)=\psi_n(x,a_-)$. In view of the relation (\ref{3.27a}) the function $T_{8}^-(\l_*,l,a_+,a_-)\E^{\kappa_1^+(\l_*)x_1}$ coincides with $T_{7}^-(\l,l,a_+,a_-)u$, where $u$ is the solution to the problem (\ref{3.11}) with $a=a_+$, $\l=\l_*$, and $f=\E^{-\kappa_1^+(\l_*)x_1}$. It is clear that $u(x)=U(-x_1,x_2,\l_*,a_+)$, and in view of (\ref{1.5}), (\ref{1.3b}), (\ref{1.3c}) we obtain % ------------- % \begin{align*} \big(T_{8}^-(\l_*,l,&a_+,a_-)\E^{-\kappa_1^+(\l_*)x_1}, \psi_*\big)_{L_2(\g_{a_-})} \\ &=c(\l_*,a_+)\E^{-2\kappa_1^+(\l_*)l} \big(\E^{\kappa_1^+(\l_*)x_1}, \psi_*\big)_{L_2(\g_{a_-})}+\Odr(\E^{-2\rho(\l_*)l}) \\ &=\pi c(\l_*,a_+)c(\l_*,a_-)\kappa_1^+(\l_*) \E^{-2\kappa_1^+(\l_*)l} +\Odr(\E^{-2\rho(\l_*)l}). \end{align*} % ------------- % Substituting these identities into (\ref{5.3}), we finally arrive at the following formula, % ------------- % \begin{equation*} A_{22}(\l,l)=\mu^-(l,\boldsymbol{a})\E^{-4\kappa_1^+(\l_*)l} +\Odr\left(|\l-\l_*| l^2\E^{-4\kappa_1^+(\l_*)l}+\E^{-2l( \kappa_1^+(\l_*)+\rho(\l_*))}\right), \end{equation*} % ------------- % where $\mu^-(l,\boldsymbol{a})$ is defined by (\ref{1.12}). It allows us to rewrite the equation (\ref{4.21}) as % ------------- % \begin{equation*} (\l-\l_*)\big(1+\Odr(l^2\E^{-4\kappa_1^+(\l_*)l})\big)= \mu^-(l,\boldsymbol{a})\E^{-4\kappa_1^+(\l_*)l} +\Odr\left(\E^{-2l( \kappa_1^+(\l_*)+\rho(\l_*))}\right)\,; \end{equation*} % ------------- % expressing $(\l-\l_*)$ from here we get the asymptotic expansion (\ref{1.11}) and the formula (\ref{1.12}). Next we have to prove the asymptotic expansion for the eigenfunction associated with $\l^-$. The nontrivial solution of the equation (\ref{3.4a}) is given by (\ref{4.17a}) with $c_+=0$ and $c_-=1$, i.e. as $\boldsymbol{f}=\boldsymbol{\Phi}_*^-$. We substitute it into the relation $u_+=T_{3}(\l^+,a_+)f_+$ and take into account (\ref{5.2}), (\ref{4.16a}); this yields % ------------- % \begin{equation*} u_+=T_{3}(\l^+,a_+)f_+=\Odr(\|\boldsymbol{\phi}_*^-\|)= \Odr(\E^{-2\kappa_1^+(\l_*)l}), \end{equation*} % ------------- % which holds true in $\H^1(\Pi_{a_+})$ and in $\H^2(S)$ for each $S\in\Xi_{a_+}$. If $\l^-\not=\l_*$, we obtain similarly with the help of Lemma~\ref{lm3.13} % ------------- % \begin{equation}\label{5.9a} u_-=T_{3}(\l^-,a_-)f_-=\frac{\psi_*}{\l^--\l_*} T_{4}^-\boldsymbol{\Phi}_*^-+T_{5}^-(\l^-,a_-)f_-= \frac{ A_{22}(\l^-,l)\psi_*}{\l^--\l_*}+ \Odr(\|\boldsymbol{\phi}_*^-\|). \end{equation} % ------------- % Due to the equation (\ref{4.21}) it follows that % ------------- % \begin{equation*} u_-=-\psi_*+\Odr(\E^{-2\kappa_1^+(\l_*)l}) \end{equation*} % ------------- % holds in $\H^1(\Pi_a)$ and $\H^2(S)$ for each $S\in\Xi_{a_-}$. If $\l^-=\l_*$, the last relation holds again; in order to prove it, it is sufficient to employ the identity \begin{equation*} u_-=T_{5}(\l_*,a_-)f_--c_-\psi_*=T_{5}(\l_*,a_-)f_--\psi_*. \end{equation*} The relations obtained in this way together with (\ref{4.2}) lead to (\ref{1.13}). \end{proof} % ------------- % \begin{proof}[Proof of Theorem~\ref{th1.3}] The general lines of the proof are similar to those of the previous one. According to the results of the previous section the eigenvalues of $H$ converging to $\l_*$ are roots of the equation (\ref{4.17}). First we will check that the function at the left-hand side of this equation has either two simple zeroes or one second-order zero converging to $\l_*$ as $l\to+\infty$. To this aim, we need to estimate the functions $A_{ij}$. Proposition~\ref{lm1.2} implies % ------------- % \begin{equation}\label{5.7} \phi_m^-(x_1,l,a_+)=(-1)^m\tau c(\l_*,a_+)\E^{-2\kappa_1^+(\l_*)l}\E^{-\kappa_1^+(\l_*)x_1}+ \Odr(\E^{-2 \rho(\l_*) l}), \end{equation} % ------------- % This formula together with (\ref{5.1}) allow us to conclude that % ------------- % \begin{equation}\label{5.8} A_{ij}(\l,l)=\Odr(\E^{-2\kappa_1^+(\l_*)l}), \end{equation} % ------------- % hence for any small $\d$ we have the inequality % ------------- % \begin{equation*} |(\l-\l_*)\tr A(\l,l)+\det A(\l,l)|<\d^2\quad \text{as}\quad |\l-\l_*|=\d, \end{equation*} % ------------- % if $l$ is large enough. Since the functions $A_{ij}$ are holomorphic, by Rouch\'{e} theorem this inequality implies that the function $\l\mapsto D(\l,l):=\det\big((\l-\l_*) \mathrm{E}+\mathrm{A}(\l,l)\big)$ has the same number of zeroes (with the order taken into account) as the function $\l\mapsto(\l-\l_*)^2$ does. The last function has $\l_*$ as a second-order zero, of course, so it follows that the function $D(\cdot,l)$ has either two simple zeroes or a second-order zero, converging to $\l_*$ as $l\to+\infty$. In what follows we denote these roots as $\l^\pm$, the case of the second-order zero corresponds to the equality $\l^+=\l^-$. As it was established in the previous section, the nontrivial solutions of the equation (\ref{3.4a}) associated with the roots of (\ref{4.17}) are given by (\ref{4.17a}) with the coefficients $c_\pm$ solving the system of linear equations (\ref{4.16}). If the numbers $\l^\pm$ solve (\ref{4.20a}), the system (\ref{4.16}) has at least one nontrivial solution corresponding to $\l^+$ and $\l^-$. Suppose that $\l^+\not=\l^-$. Then $\l^\pm$ are simple zeroes of the function $D(\cdot,l)$, and in view of the above discussion the system (\ref{4.16}) has exactly one non-trivial solution for $\l=\l^+$ and $\l=\l^-$. Hence in the case $\l^+\not=\l^-$ the operator $H$ has exactly two simple eigenvalues converging to $\l_*$ as $l\to+\infty$. Let us check that if the system (\ref{4.16}) has two linear independent solutions referring to $\l=\l^\pm$ it follows that $\l^\pm$ is a second-order zero of the function $D(\cdot,l)$. Indeed, two linear independent solutions exist if and only if % ------------- % \begin{equation}\label{5.19} A_{11}(\l^\pm,l)=A_{22}(\l^\pm,l)=\l_*-\l^\pm,\quad A_{12}(\l^\pm,l)=A_{21}(\l^\pm,l)=0. \end{equation} % ------------- % The derivative of $D(\l,l)$ with respect to $\l$ equals % ------------- % \begin{align*} \frac{\p D}{\p\l}(\l,l)=&2(\l-\l_*)+(A_{11}(\l,l)+A_{22}(\l,l)) \\ &+(\l-\l_*)\left(\frac{\p A_{11}}{\p\l}(\l,l)+\frac{\p A_{22}}{\p\l}(\l,l)\right) \\ &+A_{11}(\l,l)\frac{\p A_{22}}{\p\l}(\l,l)-A_{12}(\l,l)\frac{\p A_{21}}{\p\l}(\l,l) \\ &+A_{22}(\l,l)\frac{\p A_{11}}{\p\l}(\l,l)-A_{21}(\l,l)\frac{\p A_{12}}{\p\l}(\l,l). \end{align*} % ------------- % Substituting from (\ref{5.19}) into this expression, we see that % ------------- % \begin{equation*} \frac{\p D }{\p\l}(\l,l)=0 \quad\text{as}\quad \l=\l^\pm, \end{equation*} % ------------- % thus $\l^\pm$ is a second-order zero. It is more complicated to check existence of a double eigenvalue of the operator $H$ if $\l^+=\l^-=:\widetilde{\l}$. It is equivalent to the fact that for $\l=\widetilde{\l}$ the system (\ref{4.16}) has two linear independent solutions, and this in turn is equivalent to the relations (\ref{5.19}). Let us prove that they hold. Consider the boundary value problem % ------------- % \begin{equation}\label{3.55} \begin{gathered} (\D+\l)u=0,\quad x\in\Pi_a\setminus(\g_+\cup\g_-), \qquad u=0,\quad x\in\p\Pi, \\ \frac{\p u}{\p x_2}\Big|_{x_2=+0}- \frac{\p u}{\p x_2}\Big|_{x_2=-0}=-g_\pm, \quad x\in\g_\pm. \end{gathered} \end{equation} % ------------- % Here $g_\pm\in L_2(\g_\pm)$ are arbitrary functions, and the parameter $\l$ is supposed to range in a small neighborhood of $\l_*$ without coinciding with $\l_*$ and $\widetilde{\l}$. This problem is uniquely solvable provided we seek a $L_2(\Pi)$-solution to (\ref{3.55}). In a complete analogy with the proof of Lemma~\ref{lm4.1} one can check easily that the problem (\ref{3.55}) is equivalent to the equation % ------------- % \begin{equation}\label{3.56} \boldsymbol{f}+T_{8}(\l,l,\boldsymbol{a}) \boldsymbol{f}=\boldsymbol{g}, \end{equation} % ------------- % where $\boldsymbol{g}=(g_+,g_-)\in L_2(\g_{a_-})\oplus L_2(\g_{a_+})$, while the solution $u$ of (\ref{3.55}) is given by % ------------- % \begin{equation*}%\l%abel{3.57} u(x,\l,l)=u_+(x_1-l,x_2,\l,l)+u_-(x_1+l,x_2,\l,l),\quad u_\pm:=T_{3}(\l,a_\pm)f_\pm. \end{equation*} % ------------- % We can solve the equation (\ref{3.56}) in the same way as the equation (\ref{4.3a}), obtaining as a result that % ------------- % \begin{equation}\label{3.58} \boldsymbol{f}+\frac{T_{4}^+ \boldsymbol{f}}{\l-\l_*}\boldsymbol{\Phi}_*^+ + \frac{T_{4}^-\boldsymbol{f}}{\l-\l_*}\boldsymbol{\Phi}_*^-= \boldsymbol{G},\quad \boldsymbol{G}:= \big(\I+T_{9}(\l,l,\boldsymbol{a})\big)^{-1}\boldsymbol{g}. \end{equation} % ------------- % Hence the function $\boldsymbol{f}$ is of the form % ------------- % \begin{equation}\label{3.59} \boldsymbol{f}=C_+\boldsymbol{\Phi}^+_*+ C_-\boldsymbol{\Phi}^-_*+\boldsymbol{G}, \end{equation} % ------------- % where $C_\pm=C_\pm(\l,l)$ are constants to be found. Denoting $\boldsymbol{C}:=\left(\begin{smallmatrix} C_+ \\ C_- \end{smallmatrix}\right)$ and substituting (\ref{3.59}) into (\ref{3.58}), we obtain an equation for $\boldsymbol{C}$, % ------------- % \begin{equation}\label{3.60} \big((\l-\l_*)\mathrm{E}+\mathrm{A}(\l,l)\big) \boldsymbol{C} =\boldsymbol{h},\quad \boldsymbol{h}:= \begin{pmatrix} -T_{4}^+ \boldsymbol{G} \\ -T_{4}^- \boldsymbol{G} \end{pmatrix}. \end{equation} % ------------- % The solution of this system is given by Cramer's formula, % ------------- % \begin{equation}\label{3.61} \begin{aligned} &C_+(\l,l)=\frac{A_{12}(\l,l)T_{4}^-\boldsymbol{G}- \big(\l-\l_*+A_{22}(\l,l)\big) T_{4}^+\boldsymbol{G}} {D(\l,l)}\,, \\ &C_-(\l,l)=\frac{A_{21}(\l,l) T_{4}^+\boldsymbol{G}-\big(\l-\l_*+ A_{11}(\l,l)\big)T_{4}^-\boldsymbol{G}}{D(\l,l)}\,. \end{aligned} \end{equation} % ------------- % Using now (\ref{3.60}) and Lemma~\ref{lm3.13}, we infer that % ------------- % \begin{equation}\label{3.62} \begin{aligned} u_+(\cdot,\l,l)=&-C_+(\l,l)\psi_m(\cdot,a_+)+C_+(\l,l)T_{5}(\l,a_+) \Phi_{*,+}^+ \\ &+ C_-(\l,l)T_{5}(\l,a_+) \Phi_{*,+}^-+ T_{5}(\l,a_+)G_+, \\ u_-(\cdot,\l,l)=&-C_-(\l,l)\psi_n(\cdot,a_-)+C_-(\l,l)T_{5}(\l,a_-) \Phi_{*,-}^+ \\ &+ C_-(\l,l)T_{5}(\l,a_-)\Phi_{*,-}^-+ T_{5}(\l,a_-)G_-, \end{aligned} \end{equation} % ------------- % where $\Phi_{*,\pm}^\pm$ and $G_\pm$ are the components of the vectors $\boldsymbol{\Phi}_*^\pm$ and $\boldsymbol{G}$, % ------------- % \begin{equation*} \boldsymbol{\Phi}_*^\pm=(\Phi_{*,+}^\pm,\Phi_{*,-}^\pm),\quad \boldsymbol{G}=(G_+,G_-). \end{equation*} % ------------- % Since the number $\widetilde{\l}$ is a second-order zero of $D(\cdot,l)$, we conclude from (\ref{3.61}) that the coefficients $C_\pm$ have, in general, a second-order pole at $\widetilde{\l}$, and the same is true for $u_\pm$. Taking into account (\ref{3.62}), we conclude that the solution of (\ref{3.55}) can be represented as % ------------- % \begin{equation}\label{3.63} u(x,\l,l)=u_{-1}^+(x,\l,l)C_+(\l,l)+ u_{-1}^-(x,\l,l)C_-(\l,l) +\Odr(1),\quad \l\to\widetilde{\l}. \end{equation} % ------------- % In a complete analogy with the proof of Lemma~\ref{lm3.13} one can check easily that the solution of the problem (\ref{3.55}) has a simple pole at $\widetilde{\l}$. Hence the function $u_{-1}^+(x,\l,l)C_+(\l,l)+ u_{-1}^-(x,\l,l)C_-(\l,l)$ has a simple pole at $\widetilde{\l}$. For $x$ from a neighborhood of $\g_+$ this function satisfies due to (\ref{3.61}), (\ref{3.62}) the relation % ------------- % \begin{align*} &D(\l,l)\left(u_{-1}^+(x,\l,l)C_+(\l,l)+ u_{-1}^-(x,\l,l)C_-(\l,l)\right) \\ &= \Big(\big(\l-\l_*+A_{22}(\l,l)\big)T_{4}^+\boldsymbol{G}- A_{12}(\l,l)T_{4}^-\boldsymbol{G} \Big)\psi_m^+(x_1-l,x_2)+ \Odr(\E^{-2\kappa_1^+(\l_*)l}). \end{align*} % ------------- % Since $\widetilde{\l}$ is by assumption a second-order zero of $D(\cdot,l)$, the obtained identity yields that % ------------- % \begin{equation*} \widetilde{\l}-\l_*+A_{22}(\widetilde{\l},l)= A_{12}(\widetilde{\l},l)=0. \end{equation*} % ------------- % Observing the behavior of the function $u$ for $x$ in the vicinity of $\g_-$, one can prove in the same way that % ------------- % \begin{equation*} \widetilde{\l}-\l_*+A_{11}(\widetilde{\l},l)= A_{21}(\widetilde{\l},l)=0. \end{equation*} % ------------- % This completes the check of the relations (\ref{5.19}) for $\l^+=\l^-$ showing that in this case the operator $H$ has a double eigenvalue converging to $\l_*$ as $l\to+\infty$. We proceed to calculation of the asymptotic expansions for the root(s) of the equation (\ref{4.17}). Substituting the estimates (\ref{5.8}) into (\ref{4.17}) we obtain % ------------- % \begin{equation}\label{5.9} \l-\l_*=o(\E^{-\kappa_1^+(\l_*)l}). \end{equation} % ------------- % This relation in combination with (\ref{5.1}), (\ref{5.4a}) and the estimate (\ref{3.21a}) imply that % ------------- % \begin{equation}\label{5.10} A_{22}(\l,l)=\Odr(l\E^{-4\kappa_1^+(\l_*)l}). \end{equation} % ------------- % It is easy to establish an expression for $A_{11}$ similar to (\ref{5.4a}), which together with (\ref{5.7}) and (\ref{5.9}) yield % ------------- % \begin{equation}\label{5.11} A_{11}(\l,l)=\Odr(l\E^{-4\kappa_1^+(\l_*)l}). \end{equation} % ------------- % Proceeding in the same way as in (\ref{5.4a}) we obtain a chain of relations, % ------------- % \begin{align*} A_{12}(\l,l)&=T_{4}^+(\I+T_{9}(\l,l,\boldsymbol{a}))^{-1} \boldsymbol{\phi}_*^-(\cdot,l)= T_{4}^+\boldsymbol{\phi}_*^-(\cdot,l)+ \Odr(\|T_{9}\|\|\boldsymbol{\phi}_*^-\|) \\ &=(\phi_n^+(\cdot,l,a_-),\psi_m(\cdot,a_+))_{L_2(\g_{a_+})}+ \Odr(l\E^{-4\kappa_1^+(\l_*)l}). \end{align*} % ------------- % Due to (\ref{5.1}) and (\ref{1.5}) we have % ------------- % \begin{align*} (\phi_n^+(\cdot,l,a_-),&\psi_m(\cdot,a_+))_{L_2(\g_{a_+})} \\ &= c(\l_*,a_-)\E^{-2\kappa_1^+(\l_*)l} \big(\E^{-\kappa_1^+(\l_*)x_1},\psi_m(\cdot,a_+)\big)_{L_2(\g_{a_+})}+ \Odr(\E^{-2\rho(\l_*)l}) \\ &=\mu(l,\boldsymbol{a})\E^{-2\kappa_1^+(\l_*)l}+ \Odr(\E^{-2\rho(\l_*)l}), \end{align*} % ------------- % where $\mu(l,\boldsymbol{a})$ is given by (\ref{1.15}). Consequently, % ------------- % \begin{equation}\label{5.12} A_{12}(\l,l)=\mu(l,\boldsymbol{a}) \E^{-2\kappa_1^+(\l_*)l}+ \Odr(\E^{-2\rho(\l_*)l}+l\E^{-4\kappa_1^+(\l_*)l}), \end{equation} % ------------- % and in the same way one can show that % ------------- % \begin{equation}\label{5.13} A_{21}(\l,l)=\mu(l,\boldsymbol{a}) \E^{-2\kappa_1^+(\l_*)l}+ \Odr(\E^{-2\rho(\l_*)l}+l\E^{-4\kappa_1^+(\l_*)l}). \end{equation} % ------------- % The equation (\ref{4.17}) is equivalent to the following pair of the equations, % ------------- % \begin{equation}\label{5.14} \l-\l_* =\frac{-\tr A(\l,l)\pm\sqrt{(A_{11}(\l,l)-A_{22}(\l,l))^2+ 4A_{12}(\l,l)A_{21}(\l,l)}}{2}\,. \end{equation} % ------------- % If $c(\l_*,a_-)c(\l_*,a_+)=0$, these equations together with (\ref{5.10})--(\ref{5.13}) imply that % ------------- % \begin{equation*} \l-\l_*=\Odr(\E^{-2\rho(\l_*)l}+l\E^{-4\kappa_1^+(\l_*)l}). \end{equation*} % ------------- % which proves the asymptotic expansion (\ref{1.14}) in the case $\mu(l,\boldsymbol{a})=0$. Suppose on the contrary that $c(\l_*,a_-)c(\l_*,a_+)\not=0$. In this case the function $(A_{11}-A_{22})^2+4A_{12}A_{21}$ is non-zero as $\l=\l_*$, and therefore its square root is holomorphic w.r.t. $\l$. Using this fact and the relations (\ref{5.10})--(\ref{5.13}), one can show easily in analogy with the similar argument for the equation (\ref{4.17}) that each of the equations (\ref{5.14}) has a unique root converging to $\l_*$ as $l\to+\infty$. Hence one of the roots of (\ref{4.17}) satisfies the first of the equations (\ref{5.14}), while the other satisfies the other one. Substituting now from (\ref{5.10})--(\ref{5.13}) into (\ref{5.14}), we arrive immediately at the asymptotics (\ref{1.14}), (\ref{1.15}) in the case $\mu(l,\boldsymbol{a})\not=0$. \end{proof} \begin{proof}[Proof of Theorem~\ref{th1.4}] Let $\boldsymbol{c}$ be a nontrivial solution to the system (\ref{4.16}), where $\l$ is $\l^+$ or $\l^-$. Without loss of generality we may assume that $\|\boldsymbol{c} \|_{\mathbb{R}^2}=1$. Modifying (\ref{4.17a}), we choose the corresponding nontrivial solution of the equation (\ref{3.4a}) as $\boldsymbol{f}=-c_+\boldsymbol{\Phi}^+_*- c_- \boldsymbol{\Phi}^-_*$. In analogy with (\ref{5.9a}) we then obtain % ------------- % \begin{align*} u_-&=\frac{\psi_n(\cdot,a_-)}{\l-\l_*}T_{4}^-\boldsymbol{f} +\Odr(\E^{-2\kappa_1^+(\l_*)l}) \\ &=-\frac{c_+A_{21}(\l,l)+c_-A_{22}(\l,l)} {\l-\l_*}\psi_n(\cdot,a_-)+\Odr(\E^{-2\kappa_1^+(\l_*)l}), \end{align*} % ------------- % which holds true in $\H^1(\Pi_{a_-})$ and $\H^2(S)$ for each $S\in\Xi_{a_-}$. Employing now the system (\ref{4.16}) we can write % ------------- % \begin{equation*} c_+A_{21}(\l,l)+c_-A_{22}(\l,l)=-c_-(\l-\l_*), \end{equation*} % ------------- % hence % ------------- % \begin{equation*} u_-=c_-\psi_n(\cdot,a_-)+\Odr(\E^{-2\kappa_1^+(\l_*)l}), \end{equation*} % ------------- % and in the same way one can prove that % ------------- % \begin{equation*} u_+=c_+\psi_m(\cdot,a_+)+\Odr(\E^{-2\kappa_1^+(\l_*)l}). \end{equation*} % ------------- % in the norm of $\H^1(\Pi_{a_-})$ and $\H^2(S)$ for each $S\in \Xi_{a_-}$. The last two relations prove the sought formul{\ae} (\ref{1.18}). Suppose that $\l^+=\l^-$, then (\ref{4.16}) has two nontrivial solutions, which means that $(\l-\l_*)\mathrm{E} +\mathrm{A}(\l,l)=0$; we can choose these solutions as $(c_+,c_-)=(-1,0)$ and $(c_+,c_-)=(0,-1)$. Substituting these values into (\ref{1.18}), we arrive at (\ref{1.17}). Suppose that $\mu(l,\boldsymbol{a})\not=0$. In view of (\ref{1.14}) it implies that $\l^+(\l,a)\not=\l^-(\l,a)$, i.e. that $\l^\pm(\l,a)$ are simple eigenvalues. In this case the relations (\ref{1.14}) and (\ref{5.11}), (\ref{5.12}) yield % ------------- % \begin{equation}\label{5.21} \begin{aligned} &\l^\pm-\l_*+A_{11}(\l^\pm,l)=\pm|\mu(l,\boldsymbol{a})|\E^{-2\kappa_1^+(\l_*)l} (1+\Odr(\E^{-2\kappa_1^+(\l_*)l}))\not=0, \\ &A_{12}(\l^\pm,l)=\mu(l,\boldsymbol{a})\E^{-2\kappa_1^+(\l_*)l}(1+ \Odr(\E^{-2\kappa_1^+(\l_*)l}))\not=0. \end{aligned} \end{equation} % ------------- % Since the matrix $(\l^\pm-\l_*)\mathrm{E}+\mathrm{A}(\l^\pm,l)$ has rank one, we can choose nontrivial solutions of (\ref{4.16}) as % ------------- % \begin{align*} &c_+^\pm:=\pm\frac{\sqrt{2}(\l^\pm-\l_*+A_{11}(\l^\pm,l))} {\sqrt{(\l^\pm-\l_*+A_{11}(\l^\pm,l)^2+A_{12}^2(\l^\pm,l)}}\,, \\ &c_-^\pm:=\pm\frac{\sqrt{2}A_{12}(\l^\pm,l)} {\sqrt{(\l^\pm-\l_*+A_{11}(\l^\pm,l)^2+A_{12}^2(\l^\pm,l)}}\,. \end{align*} % ------------- % In view of to (\ref{5.21}) we then have % ------------- % \begin{equation*} c_+^\pm=1+\Odr(\E^{-2\kappa_1^+(\l_*)l}),\quad c_-^\pm=\mp\sgn\mu(l,\boldsymbol{a}) +\Odr(\E^{-2\kappa_1^+(\l_*)l}). \end{equation*} % ------------- % Substituting from here into (\ref{1.18}) we arrive immediately at (\ref{1.16}). \end{proof} \subsection*{Acknowledgment} D.B. was supported by \emph{Marie Curie International Fellowship} within the 6th European Community Framework Programm (MIF1-CT-2005-006254), and in part by the Russian Foundation for Basic Researches (No. 06-01-00138). 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I\!V: Analysis of Operators}, Academic Press, New York 1978. \end{thebibliography} \end{document} ---------------0606070702570 Content-Type: application/postscript; name="pic.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pic.eps" %!PS-Adobe-3.0 EPSF-3.0 %%Creator: jpeg2ps V1.8 by Thomas Merz %%Title: pic.jpg %%CreationDate: Wed Jun 07 10:41:26 2006 %%BoundingBox: 20 20 592 161 %%DocumentData: Clean7Bit %%LanguageLevel: 2 %%EndComments %%BeginProlog %%EndProlog %%Page: 1 1 /languagelevel where {pop languagelevel 2 lt}{true} ifelse { (JPEG file 'pic.jpg' needs PostScript Level 2!\n) dup print flush /Helvetica findfont 20 scalefont setfont 100 100 moveto show showpage stop } if save /RawData currentfile /ASCII85Decode filter def /Data RawData << >> /DCTDecode filter def 20 20 translate 572.00 140.92 scale /DeviceRGB setcolorspace { << /ImageType 1 /Width 755 /Height 186 /ImageMatrix [ 755 0 0 -186 0 186 ] /DataSource Data /BitsPerComponent 8 /Decode [0 1 0 1 0 1] >> image Data closefile RawData flushfile showpage restore } exec s4IA0!"_al8O`[\!-n$Jl-n$Jl-n$Jl-n$Jl-n$Jl-n$Jl-n$Jl-n$Jl-n$Jl-n$Jl-n$Jl-n $Jl-n,,2!"fJ:\cVI*!?qLF&HMtG!WU(<*rl9A"T\W)!10ZOeE%*6F"? 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