This is a LaTeX source file (19 pages) of a paper due to appear in J. Phys. A: Math. Gen. BODY \documentstyle[12pt]{article} \if@twoside m \oddsidemargin 14truemm \evensidemargin 0mm \marginparwidth 85pt \else \oddsidemargin 7truemm \evensidemargin 7truemm \marginparwidth 68pt \fi \topmargin 5mm \headheight 0mm \headsep 0mm \textheight 225truemm \textwidth 150truemm \parindent=7mm \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \renewcommand{\baselinestretch}{1.} %\pagestyle{empty} %DEFINITION OF THE MACROS USED \newcommand{\ie}{{\em i.e.}} \newcommand{\eg}{{\em e.g.}} \newcommand{\cf}{{\em cf. }} \newcommand{\etc}{{\em etc. }} \newcommand{\QED}{\mbox{\rule[-1.5pt]{6pt}{10pt}}} \newcommand{\rhs}{{\em rhs }} \newcommand{\restr}{\vert\hskip -5.5pt \phantom{\vert}^{\scriptscriptstyle \backslash}} \newcommand{\R}{I\!\!R} \newcommand{\CC}{{\cal C}} \newcommand{\EE}{{\cal E}} \newcommand{\KK}{{\cal K}} \newcommand{\OO}{{\cal O}} \newcommand{\WW}{{\cal W}} \newcommand{\eps}{\varepsilon} \newtheorem{claim}{Claim}[section] \newtheorem{theorem}[claim]{Theorem} \newtheorem{lemma}[claim]{Lemma} \newtheorem{remark}[claim]{Remark} \newtheorem{conjecture}[claim]{Conjecture} %END OF THE DEFINITION \begin{document} \vspace*{20mm} \begin{flushleft} {\Large\bf Bound--state asymptotic estimates for \\ window--coupled Dirichlet strips and layers} \vspace{8mm} Pavel Exner$^{a,b}$ and Simeon A. Vugalter$^a$ \vspace{5mm} a) Nuclear Physics Institute, Academy of Sciences, 25068 \v{R}e\v{z} near Prague, \\ b) Doppler Institute, Czech Technical University, B\v rehov\'a 7, 11519 Prague, \\ {\em exner@ujf.cas.cz, vugalter@ujf.cas.cz} \end{flushleft} \vspace{8mm} \noindent We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel straight strips or planar layers coupled by a finite number $\,N\,$ of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lower asymptotic bounds on the gap between the eigenvalue and the essential spectrum in the planar case, as well as for $\,N=1\,$ in three dimensions. Based on these results, we formulate a conjecture on the weak--coupling asymptotic behaviour of such bound states. \vspace{8mm} \section{Introduction} There has been some interest recently to Laplacians on strips or layers. Such a system is trivial when the manifold is straight and the boundary conditions are translationally invariant, so there is a natural separation of variables. On the other hand, the spectral properties become nontrivial if the transverse modes are coupled, which can be achieved, \eg, if the manifold is bent, locally deformed, or coupled to another one \cite{ES,DE,BGRS,ESTV,EV1,EV2}. The interest stems from two sources. On the physical side, such operators with Dirichlet boundary conditions are used as models of various mesoscopic semiconductor structures. The corresponding solid--state literature is rather rich --- see \cite{DE,ESTV} for some references. On the other hand, bound states in systems with open geometries pose also mathematical questions such as the weak--coupling limit, validity of the semiclassical approximation, resonance scattering in such structures, \etc Some properties of them can be seen numerically \cite{ESTV} while analytical proofs are missing. Recall also that a closely related problem concerns Neumann Laplacians, namely the existence of trapped modes in acoustic waveguides \cite{ELV,DE}. In a recent paper \cite{EV1} we have studied a pair of parallel Dirichlet strips of widths $\,d_1, d_2\,$ coupled laterally through a window of a width $\,2a\,$ in the common boundary; we have shown that there are positive $\,c_1,\, c_2\,$ such that the gap between the ground state and the threshold of the continuous spectrum can be estimated as \begin{equation} \label{2-1} -c_1 a^4 \,\le\, \epsilon(a) -\left(\pi\over d\right)^2 \le\, -c_2 a^4 \end{equation} for any $\,a\,$ small enough. The numerical result of \cite{ESTV} suggests that the true asymptotics is of the same type, but proving this assertion and finding the coefficient in the leading term remains an open problem. The aim of the present paper is to generalize the above inequalities to the case of a finite number of connecting windows and to a higher dimension. In the following section we shall prove the bounds for a pair of strips with $\,N\,$ windows. In Section~3 we formulate the analogous problem for two layers and prove two--sided asymptotic bounds for a single window shrinking to a point. Proofs rely in both cases on variational estimates and follow the same basic strategy as in \cite{EV1}. On the other hand, the existence of multiple windows or the change in dimension require numerous modifications, which prompts us to present the argument with enough details. The upper and lower asymptotics bounds we are going to derive are in each case of the same type differing just by values of the constants. We are convinced that ground state has an asymptotic expansion and its lowest--order is given by functions analogous to our bounds. This conjecture is formulated in the concluding section. At the same time, our present method does not allow to squeeze the bounds, or even to come close to the true values as the Remark~\ref{optimality} below illustrates. \section{$N\,$ windows in dimension two} Consider a straight planar strip $\,\Sigma:= \R\times [-d_2,d_1]\,$. Given finite sequences $\,\CC\equiv\{x_k\}_{k=1}^N\,$ of mutually distinct points of the $\,x$--axis and $\,A=\{a_k\}_{k=1}^N\,$ with $\,a_k>0\,$, we denote $\,\WW_k:=[x_k\!-\!a_k,x_k\!+\!a_k]\,$ and set $\,\WW:= \bigcup_{k=1}^n \WW_k\,$. Then we define $\,H(d_1,d_2;\WW)\,$ as the Laplacian on $\,L^2(\Sigma)\,$ subject to the Dirichlet condition at $\,y=-d_2,d_1\,$ as well as at the $\,\R\setminus\WW\,$ part of the $\,x$--axis; this operator coincides with the Dirichlet Laplacian at the strip with the appropriate piecewise cut (see Fig.~1) defined in the standard way \cite[Sec.XIII.15]{RS4}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} \begin{picture}(120,80) \linethickness{2pt} \put(30,60){\line(1,0){350}} \put(30,35){\line(1,0){75}} \put(125,35){\line(1,0){40}} \put(195,35){\line(1,0){80}} \put(295,35){\line(1,0){85}} \put(30,20){\line(1,0){350}} \thinlines \put(205,1){\vector(0,1){70}} \put(108,8){$2a_1$} \put(105,3){\vector(1,0){20}} \put(125,3){\vector(-1,0){20}} \put(105,35){\line(0,-1){33}} \put(125,35){\line(0,-1){33}} \put(114,35){\circle{1.5}} \put(111,41){$x_1$} \put(173,8){$2a_2$} \put(165,3){\vector(1,0){30}} \put(195,3){\vector(-1,0){30}} \put(165,35){\line(0,-1){33}} \put(195,35){\line(0,-1){33}} \put(180,35){\circle{1.5}} \put(176,41){$x_2$} \put(277,8){$2a_3$} \put(275,3){\vector(1,0){20}} \put(295,3){\vector(-1,0){20}} \put(275,35){\line(0,-1){33}} \put(295,35){\line(0,-1){33}} \put(285,35){\circle{1.5}} \put(281,41){$x_3$} \put(380,35){\vector(1,0){20}} \put(35,35){\vector(0,1){25}} \put(35,60){\vector(0,-1){25}} \put(35,20){\vector(0,1){15}} \put(35,35){\vector(0,-1){15}} \put(18,45){$d_1$} \put(18,24){$d_2$} \put(210,71){y} \put(395,25){x} \end{picture} \vspace{5mm} \caption{Window--coupled planar waveguides} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Following the notation introduced in \cite{EV1} we put $\,d:=\max\{d_1,d_2\}\,$ and $\,D:=d_1\!+d_2\,$. If $\,d_1=d_2\,$, the operator decomposes into an orthogonal sum with respect to the $\,y$--parity; the nontrivial part is unitarily equivalent to the Laplacian on $\,L^2(\Sigma_+)\,$, where $\,\Sigma_+:= \R\times [0,d]\,$, with the Neumann condition at window part $\,\WW\,$ of the $\,x$--axis and Dirichlet at the remaining part of the boundary; we denote it by $\,H(d;\WW)\,$. If the specification is clear from the context, we will often denote the operator in question simply as $\,H\,$. We need a quantity to express the ``smallness" of the window set. We define \begin{equation} \label{IW2} I(\WW)\,:=\, \sum_{k=1}^N a_k|\WW_k|\,=\, 2\,\sum_{k=1}^N a_k^2\;; \end{equation} then the result of \cite{EV1} generalizes to the present situation as follows: \begin{theorem} \label{2-N thm} $\sigma_{\rm ess}(H(d_1,d_2;\WW))= [(\pi/d)^2\!,\infty)$. The discrete spectrum is contained in $\,((\pi/D)^2\!,(\pi/d)^2)\,$, finite, and non\-empty provided $\,\WW\ne\emptyset\,$. If $\,I(\WW)\,$ is sufficiently small, $\sigma_{\rm disc}(H(d_1,d_2;\WW))\,$ consists of just one simple eigenvalue $\,\epsilon(\WW) \le (\pi/d)^2$ and there are positive $\,c_1,\, c_2\,$ such that \begin{equation} \label{2-N} -c_1 I(\WW)^2 \,\le\, \epsilon(a) -\left(\pi\over d\right)^2 \le\, -c_2 I(\WW)^2 \end{equation} holds for any $\,I(\WW)\,$ small enough. \end{theorem} \noindent {\em Proof: (a) The upper bound.} In the symmetric case, $\,d_1=d_2\,$, the trial function will be chosen as $\,\psi=F+G\,$, where \begin{equation} \label{trial F} F(x,y)\,:=\, f_1(x)\chi_1(y)\,, \end{equation} with $$ f_1(x)\,:=\, \max\{\,\chi_{[x_1-a_1,x_N+a_N]}(x),\, e^{-\kappa|x-x_1+a_1|},\, e^{-\kappa|x-x_N-a_N|}\}\,, $$ and \begin{equation} \label{trial G} G(x,y)\,:=\,\sum_{k=1}^N G_k(x,y) \end{equation} with \begin{equation} \label{trial G_k} G_k(x,y)\,:=\,{2\eta_k a_k\over |\WW|}\,\chi_{[x_k-a_k,x_k+a_k]}(x)\, \cos\left(\pi (x\!-\!x_k)\over 2a_k\right)\, R_k(y)\,, \end{equation} where $\,|\WW|:= 2\sum_{k=1}^N a_k\,$, and \begin{equation} \label{R} R_k(y)\,:=\, \left\lbrace\; \begin{array}{lcc} e^{-\pi y/2a_k} & \quad \dots\quad & y\in\left\lbrack 0,{d\over2}\,\right\rbrack \\ \\ 2\left(1-{y\over d}\,\right) e^{-\pi d/4a_k} & \quad \dots\quad & y\in\left\lbrack {d\over2},d\,\right\rbrack \end{array} \right. \end{equation} for $\,k=1,2,\dots,N\,$. As before $\,\chi_n(y)= \sqrt{2\over d}\, \sin\left(\pi ny\over d\right)\,$, $\;n=1,2,\dots\,$, denote the ``transverse" eigenfunctions --- to be not confused with the indicator function $\,\chi_M\,$ of a set $\,M\,$. Notice that as long as we work with trial functions of $\,Q(H)\,$, the window smoothing employed in \cite{EV1} is in fact not needed --- \cf\cite{RS4}. The functional $\,L(\psi):= (H\psi, \psi)-\,\left(\pi\over d\right)^2\|\psi\|^2$ can be expressed as \begin{equation} \label{trial L} L(\psi)\,=\, \|\psi_x\|^2+ \|G_y\|^2 -\, \left(\pi\over d\right)^2 \|G\|^2- 2\, {\pi\over d}\, \sqrt{2\over d}\,\sum_{k=1}^N\, \int_{x_k-a_k}^{x_k+a_k} G_k(x,0)\, dx\,. \end{equation} Since $\,f_x,\, G_x\,$ have disjoint supports, we have $\,\|\psi_x\|^2= \|F_x\|^2\!+ \sum_{k=1}^N \|G_{k,x}\|^2$, where $\,G_{k,x}:= \partial_x G_k\,$. The $\,k$th term of the last sum equals $\,\eta_k^2\pi^2 a_k |\WW|^{-2}\, \|R_k\|^2_{L^2(0,d)}\,$, and $$ \|R_k\|^2_{L^2(0,d)}\,=\, {a_k\over\pi}\,+\, \left( {d\over 6}- {a_k\over\pi} \right)\, e^{-\pi d/2a_k}\,<\, {a_k\over\pi}\,(1+\eps_1) $$ for any $\,\eps_1>0\,$ and $\,a_k\,$ small enough. Obviously, $\,\int_{x_k-a_k}^{x_k+a_k} G_k(x,0)\,dx=\, {8\over\pi}\, \eta_k a_k^2|\WW|^{-1}\,$, and furthermore, a bound to $\,\|G_{k,y}\|^2$ follows from $$ \|R'_k\|^2_{L^2(0,d)}\,=\, {\pi\over 4a_k}\,+\, \left( {2\over d}- {\pi\over 4a_k} \right)\, e^{-\pi d/2a_k}\,<\, {\pi\over 4a_k} $$ for $\,a_k<\pi d/8\,$, which means that $\,\|G_{k,y}\|^2< \pi \sum_k \eta_k^2 a_k^2 |\WW|^{-2}\,$. Now we can put these estimates together using $\,\|F_x\|^2= \kappa\,$; neglecting the negative term $\,-\left(\pi\over d\right)^2 \|G\|^2\,$, we arrive at the inequality $$ L(\psi)\,<\, \kappa -\, {16\sqrt{2} \over d^{3/2}}\, \sum_{k=1}^N {\eta_k a_k^2\over |\WW|}\,+\, \pi(2+\eps_1)\sum_{k=1}^N {\eta_k^2 a_k^2\over |\WW|^2}\,. $$ The sum of the last two terms at the \rhs is minimized by $\,-\,{2^7 \over \pi d^3(2+\eps_1)}\, \sum_k a_k^2\,$. To conclude the argument, we have to estimate the trial function norm $\,\|\psi\|^2$ from below. The tail part is $\,\|\psi\|^2_{x\in\R\setminus\WW} =\, \kappa^{-1}$, while the window contributes by $$ \|\psi\|^2_{x\in\WW} \,\le \, 2 \|F\|^2_{x\in\WW}+ 2\|G\|^2_{x\in\WW} \,=\, |x_N\!-\!x_1\!+\!a_N\!+\!a_1|\, + 4\, \sum_{k=1}^N {\eta_k^2 a_k^3\over |\WW|^2}\, \|R_k\|^2_{L^2(0,d)}\,, $$ so $\,\|\psi\|^2> (1\!-\!\eps_2) \kappa^{-1}\,$ holds for any $\,\eps_2>0\,$ provided $\,|\WW|\,$ is small enough. Minimizing the obtained estimate of $\,L(\psi)/\|\psi\|^2$ over $\,\kappa\,$, we find \begin{equation} \label{trial bound} {L(\psi)\over\|\psi\|^2}\,<\,-\, (1-\eps_2)^{-1} \left(\,{2^6 \over \pi d^3(2+ \eps_1)}\, \sum_{k=1}^N a_k^2\, \right)^2 \end{equation} which yields the upper bound in (\ref{2-N}) for $\,d_1=d_2\,$. The extension to the nonsymmetric case proceeds as for $\,N=1\,$; the trial function is chosen in the above form for the wider channel, while in the narrower one it is given by (\ref{trial G}) transversally rescaled. \begin{remark} \label{optimality} {\rm The bound can be improved, for instance, by replacing the factorized form (\ref{trial G_k}) by a series, whose terms will be products of the trigonometric basis in the window with the functions $\,R_{k,n}(y)\,$ decaying as $\,\exp\{-{\pi ny\over 2a_k}\}\,$ around $\,y=0\;$ (in the above estimate we used just the first term of such a series). However, the gain is not large. To illustrate this fact, take $\,N=1\,$ and $\,d=\pi\,$. The use of the series leads then to the upper bound $\,\left(2a\over\pi\right)^4$ improving the coefficient by $\,(\pi^2/8)^2\approx 1.52\,$. A comparison to the numerically determined ground state \cite{ESTV} shows that the true asymptotic behaviour should be $\,\approx (2.23\, a)^4$, so the obtained $\,c_2\,$ is still two orders of magnitude off mark. The reason is obviously that the wavefunction is affected by the window outside the transverse ``window strip" as well. } \end{remark} Before proceeding to the lower bound, let us state some auxiliary results: \begin{lemma} \label{lemma 1} Let $\,J[\phi]:= \int_a^b \left(\phi'(t)^2\!+ m^2\phi(t)^2 \right)\, dt\,$ for $\,\phi\in C^2(a,b)\,$ with $\,\phi(a)=c_a\,$ (a fixed number). Given $\,m_0>0\,$, there is $\,\alpha_0>0\,$ such that \begin{equation} \label{interval bound} J[\phi]\,\ge\, \alpha_0 m c_a^2 \end{equation} holds for all $\,m\ge m_0\,$. \end{lemma} {\em Proof:} The mimimum is obviously reached with $\,\phi'(b)=0\,$. The corresponding Euler's equation is solved by $\,\phi_0(t)= d_1 e^{-mt}\!+d_2 e^{mt}\,$, where $\,d_1= c_a\left( e^{-ma}\!+ e^{m(a-2b)}\right)^{-1}\,$ and $\,d_2=d_1 e^{-2mb}\,$. Since $\,m^{-1}c_a^{-2} \inf J(\phi)>0\,$ for any $\,m\ge m_0\,$, it is sufficient to check that (\ref{interval bound}) remains valid as $\,m\to\infty\,$; evaluating the functional for $\,\phi_0\,$ we find $\,\lim_{m\to\infty} J(\phi)= mc_a^2\,$. \quad \QED \begin{lemma} \label{lemma 2} Suppose that $\,\phi\,$ minimizes $\,J[\phi]:= \int_a^{2a} \left(\phi'(t)^2\!+ p^2\phi(t)^2 \right)\, dt\,$ for positive $\,a,\,p\,$ within $\,C^2(a,2a)\,$ with the boundary condition $\,\phi(a)=c_a\,$; then \begin{equation} \label{exp bound} |\phi(2a)| \,\le\, 2|c_a|\, e^{-pa}\,. \end{equation} \end{lemma} {\em Proof:} Assume for definiteness that $\,c_a>0\,$. By the mentioned symmetry argument again, $\,\phi'(2a)=0\,$, and its explicit form is $\,\phi(t)= c_a\,\cosh p(2a\!-\!t)/ \cosh pa\,$, which yields $\,\phi(2a)\le 2c_a e^{-pa}$. \quad \QED \vspace{3mm} For the sake of completeness we reproduce also the following assertion the proof of which is given in \cite{EV1}: \begin{lemma} \label{lemma 3} Let $\,\phi\in C^2[0,d]\,$ with $\,\phi(0)=\beta\,$ and $\,\phi(d)=0\,$. If $\,(\phi,\chi_1)=0\,$, then for every $\,m>0\,$ there is $\,d_0>0\,$ such that \begin{equation} \label{combined bound} \int_0^d \phi'(t)^2 dt\,+\,\left(m\over a\right)^2 \int_0^a \phi(t)^2 dt\,-\, \left(\pi\over d\right)^2 \int_0^d \phi(t)^2 dt\,\ge\, {d_0\beta^2 \over a} \end{equation} holds for all $\,a\,$ small enough. \end{lemma} {\em (b) Proof of Theorem~\ref{2-N thm}, continued:} The lower bound is again the more difficult; however, we may restrict ourselves to the symmetric case only because inserting an additional Neumann boundary into the window we get a lower bound, and therefore we consider in the following the spectrum of $\,H\equiv H(d;\WW)\,$. We begin with a simple observation that it is sufficient to estimate $\,L(\psi):= (H\psi, \psi)-\, \left(\pi\over d\right)^2\|\psi\|^2$ from below for all {\em real} $\,\psi\,$ of a core of $\,H$, say, all $\,C^2$--smooth $\,\psi\in L^2(\Sigma_+)\,$ satisfying the boundary conditions, since $\,H\,$ commutes with complex conjugation. The main difficulty brought by the existence of multiple windows is that we are no longer allowed to restrict ourselves to trial functions symmetric with respect to the window centers. The strategy we employ is to split from the beginning a part of the kinetic--energy contribution to the functional, say, $\,{1\over 4}\, \|\psi_x\|^2$, which will be at the end used to mend the problems coming from the asymmetry, \ie, we begin with estimating $\,L_0(\psi):= L(\psi)-\,{1\over 4}\,\|\psi_x\|^2$. A trial function of the indicated set will be written in the form of a Fourier series, \begin{equation} \label{Fourier} \psi(x,y)\,=\, \sum_{n=1}^{\infty} c_n(x)\chi_n(y) \end{equation} with smooth coefficients $\,c_n(x)=(\psi(x,\cdot),\chi_n)\,$, which is uniformly convergent outside the windows, $\,x\not\in\WW\,$. We split further the lowest transverse--mode coefficient by putting \begin{equation} \label{f_1} f_1\,:=\, c_1\,-\, \sum_{k=1}^N \hat{f}_k\,, \end{equation} where \begin{equation} \label{f_1 hat} \hat f_k\,:=\, \left\lbrace \begin{array}{lll} c_k(x)\!-\!\alpha_k & \quad \dots \quad & x\in[x_k\!-\!2a_k,x_k\!+\!a_k] \\ 0 & \quad \dots \quad & {\rm otherwise} \end{array} \right. \end{equation} with $\,\alpha_k:=c_1(x_k\!-\!2a_k)\,$, \ie, each one of the functions $\,\hat f_k\,$ vanishes at the left endpoint of the appropriate extended window; in contrast to \cite{EV1} we double the left half of the window only. Writing the full trial function as \begin{equation} \psi(x,y) \,=\, F(x,y)+G(x,y)\,, \qquad F(x,y)\,:=\, f_1(x)\chi_1(y) \label{psi}\,, \end{equation} we can cast the reduced energy functional into the form \begin{equation} \label{L} L_0(\psi)\,=\, {3\over 4}\,\|\psi_x\|^2+ \|G_y\|^2 -\, \left(\pi\over d\right)^2 \|G\|^2- \sum_{k=1}^N\, 2\alpha_k\, {\pi\over d}\, \sqrt{2\over d}\, \int_{\WW_k} G(x,0)\, dx\,. \end{equation} Contributions to (\ref{L}) from different parts of the strip $\,\Sigma_+\,$ will be estimated separately. The out--of--window part consists of the sets \begin{eqnarray*} \omega_1 \!&=&\! \{\, (x,y)\,:\; x\le x_1\!-\!a_1\,\}\,, \\ \omega_k \!&=&\! \{\, (x,y)\,:\; x_{k-1}\!+\!a_{k-1}\le x \le x_k\!-\!a_k\,\}\,,\quad k=2,\dots,N\,, \\ \omega_{N+1} \!&=&\! \{\, (x,y)\,:\; x\ge x_N\!+\!a_N\,\}\,. \end{eqnarray*} The expansion (\ref{Fourier}) yields \begin{eqnarray*} \lefteqn{{1\over 4}\,\|\psi_x\|^2_{\omega_k}+ \|G_y\|^2_{\omega_k} -\,\left(\pi\over d\right)^2 \|G\|^2_{\omega_k}} \\ \\ && =\, {1\over 4}\,\sum_{n=1}^{\infty} \int_{\omega_k} c_n'(x)^2\, dx \,+\,\sum_{n=1}^{\infty} \left(\pi\over d\right)^2 \left(n^2\!-1\right) \int_{\omega_k} c_n(x)^2\, dx \,, \end{eqnarray*} and therefore $$ {1\over 4}\,\|\psi_x\|^2_{\omega_k}+ \|G_y\|^2_{\omega_k} -\,\left(\pi\over d\right)^2 \|G\|^2_{\omega_k} \,>\, \mu_0\, \sum_{n=2}^{\infty} nc_n(x_k\!-\!a_k)^2 $$ with some $\,\mu_0>0\,$ follows from Lemma~\ref{lemma 1} (applied to $\,c_n(-x)\,$) for $\,k=2,\dots,N\,$. The same inequality for $\,k=1\,$ is derived as in \cite{EV1}; for the right tail we use just the fact that the expression is positive so we can neglect it. Since $\,\psi_x=G_x\,$ inside the (left extended) windows, we arrive at the bound \begin{eqnarray} \label{L1} L_0(\psi) &\!>\!& {1\over 2}\,\|\psi_x\|^2_{x\not\in\WW}+\, \sum_{k=1}^N\, \bigg\lbrace\, {3\over 4}\,\|G_x\|^2_{x\in\WW_k} + \|G_y\|^2_{x\in\WW_k} -\,\left(\pi\over d\right)^2 \|G\|^2_{x\in\WW_k} \nonumber \\ \\ &\!+\!& \mu_0\, \sum_{n=2}^{\infty} nc_n(x_k\!-\!a_k)^2 -\, 2\alpha_k\, {\pi\over d}\, \sqrt{2\over d}\, \int_{\WW_k} G(x,0)\, dx\, \bigg\rbrace\,. \nonumber \end{eqnarray} Our next goal is to estimate the contribution to $\,\|G_x\|^2$ from the extended windows, $\,\EE_k:= [x_k\!-\!2a_k,x_k\!+\!a_k]\,$. In distinction to the case $\,N=1\,$, however, even the lowest--mode projection of $\,G\,$ may not vanish at the right endpoints of these intervals, so the inequality (5.6) of \cite{EV1} has to be modified. Fortunately, it is sufficient to change the coefficient: if a function $\,\tilde G:\, \Sigma_+\to C^2(\Sigma_+)\,$ vanishes for $\,x=x_k\!-\!2a_k\,$, the inequality (4.2) of \cite{EV1} in combination with a symmetry argument imply \begin{equation} \label{zero end bound} \|\tilde G_x\|^2_{x\in\EE_k}\,\ge\, \left( \pi\over 6a_k\right)^2 \|\tilde G\|^2_{x\in\EE_k}\,. \end{equation} To use this result we split the function by singling out the projection of $\,G\,$ onto the first transverse mode, \begin{equation} \label{G} G(x,y) \,=\, G_1(x,y)+G_2(x,y)\,, \qquad G_1(x,y)\,=\, \sum_{k=1}^N \,\hat f_k(x)\chi_1(y)\,. \end{equation} We have $$ {1\over 2}\,\|\psi_x\|^2_{x\in\EE_k\setminus\WW_k}+\,{3\over 4}\, \|G_x\|^2_{x\in\WW_k} \,\ge\, {1\over 2}\,\|G_x\|^2_{x\in\EE_k} \,=\, {1\over 2}\,\|G_{1,x}\|^2_{x\in\EE_k}+\, {1\over 2}\, \|G_{2,x}\|^2_{x\in\EE_k}\,, $$ and therefore \begin{eqnarray} \label{L2} \lefteqn{L_0(\psi) \,>\, {1\over 2}\,\|\psi_x\|^2_{x\not\in\EE}+\, \sum_{k=1}^N\, \bigg\lbrace\, {1\over 2}\, \|G_{2,x}\|^2_{x\in\EE_k}\,+\, \|G_y\|^2_{x\in\WW_k}\,-\, \left(\pi\over d\right)^2 \|G\|^2_{x\in\WW_k}} \nonumber \\ && \\ && +\, \mu_0\, \sum_{n=2}^{\infty} nc_n(x_k\!-\!a_k)^2 \,-\, 2\alpha_k\, {\pi\over d}\, \sqrt{2\over d}\, \int_{\WW_k} G(x,0)\, dx\,+\, {1\over 2}\, \left(\pi\over 6a_k\right)^2 \|G_1\|^2_{x\in\EE_k} \bigg\rbrace\,. \nonumber \end{eqnarray} with $\,\EE:= \bigcup_{k=1}^N\EE_k\,$. To proceed further we split the function $\,G_2\,$ in the $\,k$--th extended window as $\,G_2(x,y)= \hat G(x,y)+\Gamma(x,y)\,$, where $$ \Gamma(x,y)\,:=\, \sum_{n=2}^{\infty} c_n(x_k\!-\!2a_k)\chi_n(y)\,. $$ The second part is independent of $\,x\,$ while the first one vanishes at left endpoint, so $\,G_{2,x}= \hat G_x\,$ may be estimated by means of (\ref{zero end bound}) and the Schwarz inequality as \begin{eqnarray} \label{G_2 bound} \|G_{2,x}\|^2_{x\in\EE_k} &\!\ge\!& \left(\pi\over 6a_k\right)^2 \|\hat G\|^2_{x\in\EE_k} \,\ge\, \left(\pi\over 6a_k\right)^2 \|\hat G\|^2_{\Omega_k} \nonumber \\ \\ &\!\ge\!& {1\over 2}\left(\pi\over 6a_k\right)^2 \|G_2\|^2_{\Omega_k}\!- \left(\pi\over 6a_k\right)^2 \|\Gamma \|^2_{\Omega_k}\,, \nonumber \end{eqnarray} where we have denoted $\,\Omega_k:= \EE_k\times [0,a_k]\,$. To make use of the last estimate we have to find an upper bound to $\,\|\Gamma\|^2_{\Omega_k}\,$. To this end we notice that \begin{description} \item{\em (i)} instead of assuming $\,c_n\in C^2$, the lower bound can be looked for in a wider class of $\,\psi\,$ with piecewise continuous coefficients, \vspace{-1.8ex} \item{\em (ii)} on the other hand, we may restrict ourselves to those $\,\psi\,$ which satisfy for $\,x\in\EE_k\setminus \WW_k\,$ and $\,n\ge 2\,$ the inequality \begin{equation} \label{subexponential decay} |c_n(x)|\,\le\, c_n^{ex}(x)\,:=\, |c_n(a)|\, {\cosh\left( {\pi\over d}\,\sqrt{n^2\!-1}\, (x\!-\!x_k\!+\!2a_k)\right) \over \cosh\left( {\pi a_k\over d}\,\sqrt{n^2\!-1}\right)}\,. \end{equation} To see that we split the trial function in analogy with \cite{EV1}, $$ \tilde\psi(x,y)\,:=\, \left\lbrace \begin{array}{lll} \psi(x,y)-c_n(x)\chi_n(y) & \quad \dots \quad & x\in\EE_k \setminus\WW_k \\ \psi(x,y) & \quad \dots \quad & {\rm otherwise} \end{array} \right. $$ The basic expression $\,L(\psi)/\|\psi\|^2$ can be then rewritten as $$ \tilde L(\tilde\psi)-\left(\pi\over d\right)^2 \|\tilde\psi\|^2+ \sum_{k=1}^N \int_{\EE_k\setminus \WW_k} \left\lbrack\, c_n'(x)^2\, dx + \left({\pi\over d}\,\sqrt{n^2\!-1}\,\right)^2 c_n(x)^2 \right\rbrack\, dx \over \|\tilde\psi\|^2 + \sum_{k=1}^N \int_{\EE_k\setminus \WW_k} c_n(x)^2 dx \,, $$ where $\,\tilde L(\tilde\psi):= \int_{\Sigma_+} \left( |\tilde\psi_x|^2 +|\tilde\psi_y|^2 \right)(x,y)\, dx\,dy\,$. We may assume only those $\,\psi\,$ for which the numerator is negative; the part of its last term corresponding to the ``window neighborhoods" is minimized by the hyperbolic function $\,c_n^{ex}\,$ of (\ref{subexponential decay}) (see the proof of Lemma~\ref{lemma 2}). It follows that replacing $\,c_n(x)^2\,$ by $\,\min\{c_n(x)^2, c_n^{ex}(x)^2\}\,$ we can only get a larger negative number, while the positive denominator can be only diminished. \end{description} To estimate the norm of $\,\Gamma\,$ restricted to $\,\Omega_k\,$, we adapt again the argument of \cite{EV1} and divide the series into parts referring to small and large values of $\,y\,$, and employ, respectively, the smallness of $\,\|\chi_n\restr [0,a]\|\,$ and the bound (\ref{subexponential decay}). This yields \begin{eqnarray*} && \|\Gamma\|^2_{\Omega_k} \,=\, \int_{\EE_k} dx \int_0^{a_k} dy\, \left( \sum_{n=2}^{\infty} c_n[2a_k]\chi_n(y) \right)^2 \\ \\ &\!\le\!& 6a_k \int_0^{a_k} \left( \sum_{n=2}^{[a_k^{-1}]+1} c_n[2a_k]\chi_n(y) \right)^2\! dy \,+\, 6a_k \int_0^{a_k} \left( \sum^{\infty}_{2\le n=[a_k^{-1}]+2} c_n[2a_k]\chi_n(y) \right)^2 \!dy \\ \\ &\!\le\!& 24a_k \left( \sum_{n=2}^{[a_k^{-1}]+1} n^{-1} c_n[a_k]^2 \int_0^{a_k} \chi_n(y)^2 dy \right) \left( \sum_{n=2}^{[a_k^{-1}]+1} n \right) \\ \\ &\!+\!& 24a_k \left( \sum^{\infty}_{2\le n=[a_k^{-1}]+2} n c_n[a_k]^2 \int_0^{a_k} \chi_n(y)^2 dy \right) \left( \sum^{\infty}_{2\le n=[a_k^{-1}]+2} n^{-1} e^{-(2\pi a_k/d)\sqrt{n^2-1}} \right)\,, \end{eqnarray*} where $\,c_n[ja_k]:= c_n(x_k\!-\!ja_k)\,$ and $\,[\cdot]\,$ denotes the entire part; in the the last step we have used the bound $\,|c_n[2a_k]|<2|c_n[a_k]| \exp\left\lbrace -\,{\pi a_k\over d} \sqrt{n^2\!-1} \right\rbrace\,$ which follows from Lemma~\ref{lemma 2}. In analogy with \cite{EV1}, this implies the existence of a positive $\,C_k\,$ such that \begin{equation} \label{Gamma bound 2} \|\Gamma\|^2_{\Omega_k}\,\le\, C_k a_k^2\, \sum_{n=2}^{\infty} n c_n(x_k\!-\!a_k)^2\,. \end{equation} >From now on we consider continuous coefficient functions again. By (\ref{G_2 bound}) we have \begin{eqnarray*} \lefteqn{{1\over 2}\,\|G_{2,x}\|^2_{x\in\EE_k}+\, \mu_0\,\sum_{n=2}^{\infty} n c_n(x_k\!-\!a_k)^2} \\ && \\ && \ge\, \delta\left(\pi\over 12a_k\right)^2 \|G_2\|^2_{\Omega_k}\!- {\delta\over 2}\, \left(\pi\over 6a_k\right)^2 \|\Gamma \|^2_{\Omega_k}+\, \mu_0\,\sum_{n=2}^{\infty} n c_n(x_k\!-\!a_k)^2 \end{eqnarray*} for an arbitrary $\,\delta\in(0,1]\,$; if we choose the latter sufficiently small, the sum of the last two terms is nonnegative for each $\,k=1,\dots,N\,$ due to (\ref{Gamma bound 2}), so \begin{eqnarray} \label{L3} L_0(\psi) &\!>\!& {1\over 2}\,\|\psi_x\|^2_{x\not\in\EE}+\, \sum_{k=1}^N\, \bigg\lbrace\, \|G_y\|^2_{x\in\WW_k} \,-\,\left(\pi\over d\right)^2 \|G\|^2_{x\in\WW_k} \,+\, {m^2\over a_k^2}\, \|G_2\|^2_{\Omega_k} \nonumber \\ \\ &\!-\!& 2\alpha_k\, {\pi\over d}\, \sqrt{2\over d}\, \int_{\WW_k} G(x,0)\, dx\,+\, {1\over 2}\,\left(\pi\over 6a_k\right)^2 \|G_1\|^2_{x\in\EE_k} \bigg\rbrace\,, \nonumber \end{eqnarray} where we have denoted $\,m:=\, {\pi\over 12}\, \sqrt{\delta}\,$. Next we express the first term in the curly bracket using the decomposition (\ref{G}), properties of the transverse base, and an integration by parts, $$ \|G_y\|^2_{x\in\WW_k}=\, \|G_{1,y}\|^2_{x\in\WW_k}+\, \|G_{2,y}\|^2_{x\in\WW_k} -\,2\,{\pi\over d}\, \sqrt{2\over d}\, \int_{\WW_k} \hat f_k(x)G(x,0)\, dx\,. $$ As in \cite{EV1} we estimate the last term by the Schwarz inequality, substitute into (\ref{L3}), neglect $\,\|G_{1,y}\|^2_{x\in\WW_k}$ as well as $$ {\pi^2\over 72 a_k^2}\, \|G_1\|^2_{x\in\EE_k}-\, {\pi(\pi+\sqrt{2})\over d^2}\, \|G_{1,y}\|^2_{x\in\WW_k} $$ which is positive for $\,a_k\,$ small enough, obtaining \begin{eqnarray} \label{L4} L_0(\psi) &\!>\!& {1\over 2}\,\|\psi_x\|^2_{x\not\in\EE}+\, \sum_{k=1}^N\, \bigg\lbrace\, \|G_{2,y}\|^2_{x\in\WW_k} \,+\, {m^2\over a_k^2}\, \|G_2\|^2_{\Omega_k}\,-\,\left(\pi\over d\right)^2 \|G\|^2_{x\in\WW_k} \nonumber \\ \\ &\!-\!& {\pi\sqrt{2}\over d}\, \|G(\cdot,0)\|^2_{x\in\WW_k} -\,2\alpha_k\, {\pi\over d}\, \sqrt{2\over d}\, \int_{\WW_k} G(x,0)\, dx\, \bigg\rbrace\,. \nonumber \end{eqnarray} By Lemma~\ref{lemma 3}, the sum of the first three terms in the curly bracket is bounded from below by $\,{d_k\over a_k}\, \|G(\cdot,0)\|^2_{x\in\WW_k}\,$ for some $\,d_k>0\,$. Since $\,(d_k/2a_k) -(\pi\sqrt{2}/d)>0\,$ holds for $\,a_k\,$ small enough, we have $$ L_0(\psi) \,>\, {1\over 2}\,\|\psi_x\|^2_{x\not\in\EE}+\, \sum_{k=1}^N \biggl\lbrace {d_k\over 2a_k}\, \|G_2(\cdot,0)\|^2_{x\in\WW_k} \!-\,4\alpha_k\,{\pi\over d}\, \sqrt{a_k\over d}\,\|G_2(\cdot,0)\|_{x\in\WW_k}\, \biggr\rbrace\,, $$ where we have employed again the Schwarz inequality. The $\,k$--th term of the sum reaches its minimum w.r.t. the norm at $\,-\, {8\pi^2 \over d_kd^3}\, \alpha_k^2 a_k^2$. Returning to the original functional and neglecting in the first term of the last estimate all contributions except the one coming from the leftmost component of $\,\R\setminus\EE\,$, we see that there is a positive $\,\gamma\,$ such that \begin{equation} \label{L5} L(\psi) \,>\, {1\over 4}\,\|\psi_x\|^2+\, {1\over 2}\,\|\psi_x\|^2_{x\, {1\over 2}\,\|\psi_x\|^2_{x\, {1\over 2}\,\|\psi_x\|^2_{x\, {\int_{-\infty}^{x_1\!-\!2a_1} c'_1(x)^2 dx \,-\,\gamma\alpha_1^2 I(\WW) \over 2\int_{-\infty}^{x_1\!-\!2a_1} c_1(x)^2 dx}\;. $$ The \rhs is minimized by the function $\,c_1(x)= \alpha_1 e^{\kappa(x-x_1\!+2a_1)}$ which yields the value $\,(\kappa^2/2)-\gamma I(\WW)\kappa\,$; taking the minimum over $\,\kappa\,$ we find \begin{equation} \label{lower bound} {L(\psi)\over \|\psi\|^2} \,>\, -\,\gamma^2 I(\WW)^2\,. \quad \QED \end{equation} \section{Window--coupled layers} \setcounter{equation}{0} The setting of the three--dimensional problem is similar. We have a straight layer, $\,\Sigma:= \R^2\times [-d_2,d_1]\,$, and a set $\,\WW\subset \R^2\,$ which can be written as a finite union, $\,\WW:= \cup_{k=1}^N \WW_k\,$, whose components are open, connected sets of nonzero Lebesgue measure; without loss of generality we may suppose they are mutually disjoint. Then we define $\,H(d_1,d_2;\WW)\,$ as the Laplacian on $\,L^2(\Sigma)\,$ obeying the Dirichlet condition at the boundary of $\,\Sigma\,$, \ie, $\,y=-d_2,d_1\,$, as well as at $\,\R^2\setminus\WW\,$. This operator coincides again with the Dirichlet Laplacian \cite[Sec.XIII.15]{RS4} for the sliced layer the two parts of which are connected through the window set $\,\WW\,$. We use the same notation as above, $\,d:=\max\{d_1,d_2\}\,$ and $\,D:=d_1\!+d_2\,$. The nontrivial part of the symmetric case, $\,d_1=d_2\,$, reduces again to analysis of the Laplacian $\,L^2(\Sigma_+)\,$, where $\,\Sigma_+:= \R^2\times [0,d]\,$, with the Neumann condition at window part of the plane $\,y=0\,$ and Dirichlet at the remaining part of the boundary; this operator will be denoted as by $\,H(d;\WW)\,$. Our main aim here is to prove a weak--coupling asymptotic estimate for a pair of layers connected by a single window. \begin{theorem} \label{3-1 thm} $\sigma_{\rm ess}(H(d_1,d_2;\WW))= [(\pi/d)^2\!,\infty)$. The discrete spectrum is contained in $\,((\pi/D)^2\!,(\pi/d)^2)\,$, finite, and non\-empty provided $\,\WW\ne\emptyset\,$. Suppose further that $\,N=1\,$ and $\,\WW=aM\,$ for an nonempty open set $\,M\,$ contained in the unit ball $\,B_1\subset\R^2$. Then $\sigma_{\rm disc}(H(d_1,d_2;aM))\,$ consists of just one simple eigenvalue $\,\epsilon(aM) \le (\pi/d)^2$ for all $\,a\,$ small enough, and there are positive $\,c_1,\, c_2\,$ such that \begin{equation} \label{3-1} -\,\exp\left(-c_1 a^{-3}\right) \,\le\, \epsilon(a) -\left(\pi\over d\right)^2 \le\, -\,\exp\left(-c_2 a^{-3}\right)\,. \end{equation} \end{theorem} \noindent {\em Proof} is based again on variational estimates. {\em The upper bound} in the symmetric case, $\,d_1=d_2\,$, employs the trial function $\,\psi=F+\eta G\,$, where $\,F(x,y):= f_1(x)\chi_1(y)\,$ again with \begin{equation} \label{trial F_3} f_1(x)\,:=\,\min\left\lbrace\, 1,\; {K_0(\kappa|x|)\over K_0(\kappa a)} \right\rbrace\,, \end{equation} and \begin{equation} \label{trial G_3} G(x,y)\,:=\,\chi_{aM}(x)\phi_1(x)R(y)\,, \end{equation} where $\,\phi_1^{(a)}\,$ is the ground--state eigenfunction, $\,\|\phi_1^{(a)}\|=1\,$, of the operator $\,-\Delta_D^{aM}\,$ corresponding to the positive eigenvalue $\,\mu_1(a)= \mu_1(1)a^{-2}\,$, and \begin{equation} \label{R_3} R(y)\,:=\, \left\lbrace\; \begin{array}{lcc} e^{-\sqrt{\mu_1(a)}\,y} & \quad \dots\quad & y\in\left\lbrack 0,{d\over2}\,\right\rbrack \\ \\ 2\left(1-{y\over d}\,\right) \exp\left(-\,{d\over 2}\,\sqrt{\mu_1(a)}\right) & \quad \dots\quad & y\in\left\lbrack {d\over2},d\,\right\rbrack \end{array} \right. \end{equation} Using $\,-\chi''_1= (\pi/d)^2\chi_1\,$, a simple integration by parts, and the fact that the vector functions $\,\nabla f_1\,$ and $\,\nabla \phi_1^{(a)}\,$ have disjoint supports, we can express the reduced energy functional $\,L(\psi):= (H\psi, \psi)-\,\left(\pi\over d\right)^2\|\psi\|^2$ as \begin{eqnarray} \label{trial L_3} L(\psi) \!&=&\! \|\nabla f_1\|^2_{L^2(\R^2)} +\eta^2 \left( \mu_1(a) -\left(\pi\over d\right)^2 \right) \|R\|^2_{L^2(0,d)} \nonumber \\ \!&-&\! \eta^2 \|R'\|^2_{L^2(0,d)} -2\eta \chi'_1(0) \int_{aM} \phi_1^{(a)}(x)\, dx\,, \end{eqnarray} where the negative term in the bracket can be, of course, neglected. The second and the third term at the \rhs can be estimated in analogy with \cite{EV1}, $$ \mu_1(a) \|R\|^2_{L^2(0,d)} -\eta^2 \|R'\|^2_{L^2(0,d)}\,<\, {\sqrt{\mu_1(1)}\over 2a}\, (2\!+\eps_1) $$ for a fixed $\,\eps_1>0\,$ and any $\,a\,$ small enough. In a similar way, the last term equals $\,-2\eta\chi'_1(0)Ca\,$, where $\,C:= \int_M \phi_1^{(1)}(x)\,dx\,$. Finally, the first one can be evaluated by means of \cite[9.6.26]{AS},\cite[1.12.3.2]{PBM}, $$ K_0(\kappa a)^2 \|\nabla f_1\|^2_{L^2(\R^2)}\,=\, 2\pi\left\lbrack\, {1\over 2}\,\kappa^2a^2 K'_1(\kappa a)^2 -\,{1\over 2}\left(\kappa^2 a^2\!+1\right) K_1(\kappa a)^2 \,\right\rbrack\,. $$ Using $\,-K'_1(\xi)=K_0(\xi)+\xi^{-1}K_1(\xi)\,$ in combination with the asymptotic expressions $\,K_0(\xi)=-\ln\xi+\OO(1)\,,\; K_1(\xi)= \xi^{-1}\!+ \OO(\ln\xi)\,$, we find $$ \|\nabla f_1\|^2_{L^2(\R^2)}\,<\, -\: {2\pi(1+\eps_2)\over \ln\kappa a} $$ for a fixed $\,\eps_2\,$ and $\,a\,$ small enough. Substituting these estimates into (\ref{trial L_3}) and taking a minimum over $\,\eta\,$ we arrive at the bound \begin{equation} \label{upper L_3} L(\psi)\,<\, -\: {2\pi(1+\eps_2)\over \ln\kappa a} \,-\, {2\chi'_1(0)^2 C^2\over (2+\eps_1)\sqrt{\mu_1(1)}}\, a^3\,. \end{equation} It remains to find a lower bound to $$ \|\psi\|^2 \,\ge\, \|\psi\|^2_{|x|\ge a} -2\|F\|^2_{|x|\le a} -2\eta^2 \|F\|^2_{|x|\le a} \,=\, \|\psi\|^2_{|x|\ge a} -2\pi a^2 -2\eta^2\|R\|^2_{L^2(0,d)}\,. $$ The last term is $\,\OO(a)\,$, while the first one can be expressed as $$ K_0(\kappa a)^2 \|F\|^2_{|x|\ge a}\,=\, \pi a^2 \left\lbrack\, K_1(\kappa a)^2\! -K_0(\kappa a)^2\,\right\rbrack \,=\, {\pi\over \kappa^2}\,+\, \OO(a^2\ln\kappa a)\;; $$ using the asymptotic behaviour of $\,K_0\,$ we find $\,\|\psi\|^2 \ge \pi\kappa^{-2}(\ln\kappa a)^{-2} (1\!-\!\eps_3)\,$ for a fixed $\,\eps_3>0\,$ and $\,a\,$ small enough. Hence \begin{equation} \label{upper_3} {L(\psi)\over \|\psi\|^2}\,<\,-\, {\kappa^2\ln\kappa a\over \pi(1-\eps_3)}\, (Da^3\ln\kappa a +E)\,, \end{equation} where $\,E:= 2\pi(1+\eps_2)\,$ and $$ D\,:=\, {2\chi'_1(0)^2 C^2\over (2+\eps_1)\sqrt{\mu_1(1)}}\,. $$ Minimizing the \rhs of (\ref{upper_3}) with respect to $\,\kappa\,$, we conclude that to fixed positive $\,\eps_1,\,\eps_2\,$ and $\,\eps_3\in (0,1)\,$ there is a function $\,g\,$ such that \begin{equation} \label{upper bound_3} {L(\psi)\over \|\psi\|^2}\,<\,g(a) \qquad {\rm and} \qquad g(a)\,\approx\, -\, {1+\eps_2\over 1-\eps_3}\; {1\over a^2}\: e^{-2E/Da^3} \end{equation} as $\,a\to 0\,$. The upper bound in (\ref{3-1}) follows readily from (\ref{upper bound_3}); the extension to the nonsymmetric case is obtained as in \cite{EV1}. \begin{remark} {\rm In fact, one could suppose $\,M=B_1\,$ because the eigenvalue is pushed up if we reduce the window to a circle contained in $\,M\,$, and the obtained bound is all the same not optimal as in Remark~\ref{optimality}. In the rest of the proof we {\em embedd} $\,M\,$ into a circle leaving the question about relations between the constants and the geometry of $\,M\,$ to more sophisticated methods.} \end{remark} {\em The lower bound} can again be proven in the symmetric case only. We begin with auxiliary results. When constructing the trial function component (\ref{trial F_3}), we have used implicitly the fact that the functional $\,F\,:\; F(\phi)= \int_a^{\infty} \left( \phi'(t)^2\!+ m^2\phi(t)^2 \right)\, tdt\,$ on $\,C^2([a,\infty))\,$ with the condition $\,\phi(a)=\alpha\,$ and fixed positive $\,a,m\,$ and is minimized by \begin{equation} \label{Macdonald} \phi_0\,:\; \phi_0(t)\,=\, \alpha\, {K_0(mt)\over K_0(ma)}\,, \end{equation} as can be easily seen from solution of the appropriate Euler's equation. Furthermore, a two--dimensional analogy of the bound (4.2) in \cite{EV1} is given by the {\em Friedrichs inequality} \cite[Thm.~1.9]{Ne}: if $\,\Omega\subset\R^n,\; n\ge 2\,$, is a bounded domain with Lipschitz boundary, there is a positive $\,c\,$ such that \begin{equation} \label{Lipschitz} \|\nabla f\|^2 \ge\, c\|f\|^2 \end{equation} holds for every $\,f\in H_0^1(\Omega)\,$. The constant is, of course, easy to find for the circle $\,\Omega=B_a\,$ in terms of the appropriate Bessel zero, $\,c=j_{0,1}^2a^{-2}$. Repeating the argument of \cite{EV1} and the previous section, we infer that one has to find a lower bound to $\,L(\psi)/\|\psi\|^2$ over all real $\,\psi\in L^2(\Sigma)\,$, which are $\,C^2$, radially symmetric, and vanish at the boundary except in the window. We can express such a $\,\psi\,$ in the form of the series (\ref{Fourier}) again, where the convergence is uniform for $\,|x|\ge a\,$. The coefficients $\,c_n\,$ depend in fact only of $\,r:=|x|\,$. Moreover, in analogy with (\ref{subexponential decay}) we may restrict our attention to trial functions with \begin{equation} \label{dominated decay_3} |c_n(r)|\,\le\,|c_n(a)|\, {K_0\left( {\pi\over d}\, \sqrt{n^2\!-1}\, r \right) \over K_0\left( {\pi\over d}\, \sqrt{n^2\!-1}\, a \right)} \end{equation} for $\,n\ge 2\,$. As before we introduce \begin{equation} \label{F_3} F(x,y)\,:=\, \left\lbrace\; \begin{array}{lll} \alpha\chi_1(y) \quad & \dots & \quad 0\le r\le 2a \\ c_1(r)\chi_1(y) \quad & \dots & \quad r\ge 2a \end{array} \right. \end{equation} with $\,\alpha:=c_1(2a)\,$, and divide the rest $\,G(x,y)= \psi(x,y)-F(x,y)\,$ into $$ G_1(x,y)\,:=\, (c_1(r)-\alpha)\chi_1(y) $$ supported in the extended window region, $\,r\le 2a\,$, and $\,G_2(x,y)= \hat G(x,y)\!+\!\Gamma(x,y)\,$ with $$ \Gamma(x,y)\,:=\, \sum_{n=2}^{\infty} c_n(2a)\chi_n(y)\,. $$ We start estimating the reduced energy functional \begin{eqnarray} \label{L_3} L(\psi) \,=\, \|\nabla_x\psi\|^2 +\|G_y\|^2- \left(\pi\over d\right)^2\|G\|^2- 2\alpha\chi'_1(0) \int_{B_a} G(x,0)\,dx \end{eqnarray} from the ``external" contribution to the first ``two and a half" terms, \begin{eqnarray} \label{L_1} L_1\!&:=&\! {1\over 2}\,\|\nabla_x\psi\|^2_{r\ge a} +\|G_y\|^2_{r\ge a}- \left(\pi\over d\right)^2\|G\|^2_{r\ge a} \nonumber \\ \nonumber \\ \!&=&\! \pi\, \sum_{n=1}^{\infty}\, \int_a^{\infty} \left( c'_n(r)^2+\, 2\left(\pi\over d\right)^2(n^2\!-1) c_n(r)^2 \right)\, r\,dr \nonumber \\ \\ \!&\ge&\! \pi\, \sum_{n=2}^{\infty}\, \int_a^{\infty} \left( c'_n(r)^2+\, 2\left(\pi n\over d\right)^2 c_n(r)^2 \right)\, r\,dr \nonumber \\ \nonumber \\ \!&\ge&\! \pi\, \sum_{n=2}^{\infty} c_n(a)^2\, {\pi n\over d}\, a\, {K_1\left( {\pi n\over d}\, a \right) \over K_0\left( {\pi n\over d}\, a \right)} \,\ge\, {\pi^2 a\over d}\, \sum_{n=2}^{\infty} n\,c_n(a)^2\,, \nonumber \end{eqnarray} where in the last line we have used (\ref{Macdonald}), evaluated the integral as in the first part of the proof, and employed the inequality $\,K_1(\xi)\ge\,K_0(\xi)\,$ which follows from the well--known integral representation \cite[9.6.24]{AS}. Next we turn to \begin{equation} \label{L_2} L_2\,:= \,\|\nabla_x\psi\|^2_{r\le 2a} \,= \, \|\nabla_x G_1\|^2_{r\le 2a}+ \|\nabla_x G_2\|^2_{r\le 2a}\,. \end{equation} By assumption, $\,G_1\,$ vanishes at $\,r=2a\,$, so the first term can be estimated from (\ref{Lipschitz}) as \begin{equation} \label{G_1} \|\nabla_x G_1\|^2_{r\le 2a}\,\ge\,{C_1\over 4a^2}\, \|G_1\|^2_{r\le 2a}\,=\, {C_1\over a^2}\, \|G_1\|^2\,, \end{equation} where $\,4C_1:=j_{0,1}^2\,$. Furthermore, introducing the window neighbourhood $\,\Omega_a:= B_{2a}\times [0,a]\,$, we have \begin{eqnarray} \label{G_2} \|\nabla_x G_2\|^2_{r\le 2a}\!&=&\!\|\nabla_x \hat G\|^2_{r\le 2a}\,\ge\, {C_1\over a^2}\,\|\hat G\|^2_{r\le 2a} \nonumber \\ \\ \ge\,{C_1\over a^2}\,\|\hat G\|^2_{\Omega_{2a}} \!&\ge&\! {\delta C_1\over a^2}\,\|\hat G\|^2_{\Omega_{2a}} \,\ge\, {\delta C_1\over 2a^2}\,\|G_2\|^2_{\Omega_{2a}}-\, {\delta C_1\over a^2}\,\|\Gamma\|^2_{\Omega_{2a}} \nonumber \end{eqnarray} for all $\,a\le d\,$ and $\,\delta\in (0,1]\,$. The last norm can be estimated as in the previous cases by combining the smallness of the $\,\chi_n\,$ norm restricted to $\,[0,a]\,$ with the dominated decay (\ref{dominated decay_3}), \begin{eqnarray*} \|\Gamma\|^2_{\Omega_a} \!&=&\! 4\pi a^2 \int_0^a \left( \sum_{n=2}^{\infty} c_n(2a)\chi_n(y) \right)^2 dy \\ \\ &\!\le\!& 8\pi a^2 \left( \sum_{n=2}^{[a^{-1}]+1} n^{-1} c_n(a)^2 \int_0^a \chi_n(y)^2 dy \right) \sum_{n=2}^{[a^{-1}]+1} n \\ \\ &\!+\!& 8\pi a^2 \left( \sum^{\infty}_{2\le n=[a^{-1}]+2} n c_n(a)^2 \int_0^a \chi_n(y)^2 dy \right) \sum^{\infty}_{2\le n=[a^{-1}]+2} {K_0^2\left( {2\pi a\over d}\, \sqrt{n^2\!-1} \right) \over n K_0^2\left( {\pi a\over d}\, \sqrt{n^2\!-1} \right)} \\ \\ &\!\le\!& {16\pi a^3\over d}\, \left( {2\pi^2\over 3d^2}\,+\, \sum^{\infty}_{2\le n=[a^{-1}]+2} {K_0^2\left( {2\pi a\over d}\, \sqrt{n^2\!-1} \right) \over n K_0^2\left( {\pi a\over d}\, \sqrt{n^2\!-1} \right)} \right)\, \sum^{\infty}_{n=2} n c_n(a)^2\,. \end{eqnarray*} The sum in the bracket can be estimated as $$ \sum^{\infty}_{2\le n=[a^{-1}]+2} {K_0^2\left( {2\pi a\over d}\, \sqrt{n^2\!-1}\, \right) \over n K_0^2\left( {\pi a\over d}\, \sqrt{n^2\!-1}\, \right)}\,\le\, \int_{a^{-1}}^{\infty} {K_0^2\left( {2\pi a\over d}\, \sqrt{\xi^2\!-1}\, \right) \over \xi K_0^2\left( {\pi a\over d}\, \sqrt{\xi^2\!-1}\, \right)} \,d\xi \,\le\, \int_1^{\infty} {K_0^2\left( {\pi\xi\over d} \right) \over \xi K_0^2\left( {\pi\xi\over 2d} \right)} $$ for $\,a< {1\over 2}\sqrt{3}\,$, and the integral on the \rhs is convergent, because $\,K_0(\xi)\approx \sqrt{\pi\over 2\xi}\, e^{-\xi}\,$ as $\,\xi\to\infty\,$. Hence there is a positive $\,C_2\,$ independent of $\,\psi\,$ and $\,a\,$ such that \begin{equation} \label{Gamma_3 norm} {C_1\over a^2}\:\|\Gamma\|^2_{\Omega_a}\,<\, C_2 a \sum^{\infty}_{n=2} n c_n(a)^2\,. \end{equation} Combining the estimates (\ref{L_1})--(\ref{Gamma_3 norm}), we arrive at $$ L_1\!+L_2 \,\ge\, a\left({\pi^2\over d}\,-\delta C_2\right) \sum^{\infty}_{n=2} n c_n(a)^2+\, {C_1\over a^2}\, \|G_1\|^2 +\, {\delta C_1\over 2a^2}\, \|G_2\|^2_{\Omega_a}\,, $$ which gives \begin{equation} \label{L_12} L_1\!+L_2 \,\ge\, {C_1\over a^2}\, \|G_1\|^2 +\,{m^2\over a^2}\, \|G_2\|^2_{\Omega_a} \end{equation} for some $\,m>0\,$ and all sufficiently small $\,a\,$. The norm of $\,G_y\,$ is estimated as in the two--dimensional case \cite{EV1}, $$ \|G_y\|^2_{r\le a}\,\ge\, \|G_{2,y}\|^2_{r\le a} -\,{2\pi\over d^2}\, \left( 2\|G_1\|^2_{r\le a}\!+ d\|G_2(\cdot,0)\|^2_{r\le a}\right)\,, $$ which together with (\ref{L_12}) yields \begin{eqnarray*} L_1 \!\!&+&\!L_2\,+\:\|G_y\|^2_{r\le a}-\, \left(\pi\over d\right)^2 \|G\|^2_{r\le a} \\ \\ \!&\ge &\! \|G_{2,y}\|^2_{r\le a}-\, \left(\pi\over d\right)^2 \|G_2\|^2_{r\le a}+\, \left( {C_1\over a^2}\,-\,{\pi(\pi+4)\over d^2} \right) \|G_1\|^2_{r\le a} \\ \\ \!&+&\! {m^2\over a^2}\, \|G_2\|^2_{\Omega_a}-\, {2\pi\over d}\, \|G_2(\cdot,0)\|^2_{r\le a} \\ \\ \!&\ge &\! \left( {c_0\over a}\,-\,{2\pi\over d}\right) \|G_2(\cdot,0)\|^2_{r\le a} \,\ge\, {c_0\over 2a}\,\|G_2(\cdot,0)\|^2_{r\le a} \end{eqnarray*} for a positive $\,c_0\,$ and any $\,a\,$ small enough; in the second step we have neglected a positive term and employed Lemma~\ref{lemma 3}. Substituting from here to (\ref{L_3}) and using the Schwarz inequality, $$ \int_{B_a} G(x,0)\,dx \,\le\, \|G_2(\cdot,0)\|_{r\le a} \sqrt{\pi}\,a\, $$ we get \begin{eqnarray*} L(\psi) \!&\ge&\! {1\over 2}\, \|\nabla_x\psi\|^2_{r\ge 2a} -\,2\alpha a\chi'_1(0) \sqrt{\pi}\, \|G_2(\cdot,0)\|_{r\le a}+\, {c_0\over 2a}\, \|G_2(\cdot,0)\|_{r\le a}^2 \\ \\ \!&\ge&\! {1\over 2}\, \|\nabla_x\psi\|^2_{r\ge 2a} -\,{2\pi\alpha^2 \chi'_1(0)^2\over c_0}\, a^3 \,. \end{eqnarray*} The first term on the \rhs can be estimated from below by the first transverse--mode contribution. The same applies to $\,\|\psi\|^2$, so finally we find \begin{equation} \label{lbound_3} {L(\psi)\over \|\psi\|^2}\,\ge\, {\int_{2a}^{\infty} c'_1(r)^2 r\,dr -\,{\pi\chi'_1(0)^2\over c_0}\, a^3 c_1(2a)^2 \over 2 \int_{2a}^{\infty} c_1(r)^2 r\,dr}\,. \end{equation} In analogy with (\ref{Macdonald}) one has to solve the appropriate Euler's equation to check that the \rhs of (\ref{lbound_3}) is minimized by $\,c_1=\phi_{\kappa}$ for some $\,\kappa>0\,$, where $$ \phi_{\kappa}(r)\,:=\, c_1(2a)\, {K_0(\kappa r)\over K_0(2\kappa a)}\,. $$ Substituting into (\ref{lbound_3}), evaluating the integrals, and taking the asymptotics for small $\,a\,$, we infer that $$ {L(\psi)\over \|\psi\|^2}\,\ge\, -\,\kappa^2 \ln(2\kappa a)\, \left( {\pi\chi'_1(0)^2 \over c_0(1+\eps_2)}\, a^3\ln(2\kappa a)+ \,{1-\eps_1\over 1+\eps_2} \right) $$ holds for any fixed $\,\eps_1,\,\eps_2>0\,$ and all sufficiently small $\,a\,$. It remains to find the minimum of the \rhs with respect to $\,\kappa\,$. However, since it differs from (\ref{upper_3}) just by the values of the constants, the argument is concluded as in the first part of the proof. \quad \QED \section{Conclusions} \setcounter{equation}{0} To make sense of the derived bounds one has to take into account two aspects of the problem. First of all, we have mentioned already that the discrete spectrum can also be found numerically by means of the mode--matching method; a detailed description of the two--dimensional case is given in \cite{ESTV}. Although the method converges rather slowly if the window is narrow, the results obtained for a single window clearly suggest that the true asymptotics exists and is of the same type as our asymptotic bounds. Another insight can be obtained from comparing our result with the well-known weak--coupling asymptotics for Schr\"odinger operators in dimension one and two \cite{BGS,Kl,Si}. The ground state of the coupled strips in the narrow--window case is dominated the lowest transverse--mode component with long exponentially decaying tails and a local modification in the coupling region. In a similar way, a link can be made between window--connected layers and a two--dimensional Schr\"odinger operator. The comparison shows that the attractive interaction due to opening a narrow window (in particular, by changing the Dirichlet b.c. to Neumann at a short segment of the boundary in the symmetric case) acts effectively as a potential well of a depth proportional to the size of the window. \begin{conjecture} Let $\,H(d_1,d_2;\WW)\,$ be the operators described above. The ground--state eigenvalue behaves for small $\,|\WW|\,$ as \begin{eqnarray} \epsilon(a) \,\approx\, \left(\pi\over d\right)^2 -\,{1\over d^2}\, \left(\sum_{k=1}^N c_{2,k}(\nu) a_k^2 \right)^2 & \quad \dots \; & \dim\Sigma=2 \\ \nonumber \\ \epsilon(a) \,\approx\, \left(\pi\over d\right)^2 -\,{1\over d^2}\,\exp\left\lbrace -\left(\sum_{k=1}^N c_{3,k}(\nu) a_k^3 \right)^{-1}\right\rbrace & \quad \dots \; & \dim\Sigma=3 \end{eqnarray} where $\,\nu:=d^{-1}\min\{d_1,d_2\}\,$, and $\,a_k\,$ in the three-dimensional case is the scaling parameter of the $\,k$--th window. \end{conjecture} The conjecture is based on the described analogy only, and therefore it is difficult to say more about the coefficients. It is not excluded that they depend on the geometry of the window--center set for $\,N>1\,$; in the three-dimensional case the shapes of the scaled windows may also play role. We refrain from speculating about the nature of the error terms. On the other hand, we are convinced that the open ``constant cross--section" shape of our regions $\,\Sigma\,$ is crucial for the asymptotics. For instance, if $\,\Sigma\,$ is instead a bounded planar region with the Dirichlet boundary in which we open a window (to another bounded region the essential spectrum threshold of which is not lower) or a Neumann segment, we conjecture that leading term in the ground state shift is proportional to the {\em square} of the window width. Moreover, the same asymptotics is expected to be valid for higher eigenvalues provided the corresponding eigenfunctions are locally {\em symmetric} with respect to the window axis. In any case, proving of such asymptotic properties represents an intriguing mathematical problem. \section*{Acknowledgments} Thanks are due to P.~Lindovsk\'{y} for useful discussions. 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