Written for the jubilee volume dedicated to the 60th birthday of Sergio Albeverio BODY \documentclass{amsproc} %------- % if required %\usepackage{amssymb} %-------- % for figures % \usepackage{graphicx} %------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% ==================================================================== %%% @LaTeX-file{ %%% filename = "cmsams-l.cls", %%% version = "1.2b", %%% date = "1996/10/25", %%% time = "16:48:24 EDT", %%% checksum = "45397 37 138 1396", %%% author = "Michael Downes", %%% address = "American Mathematical Society, PO Box 6248, %%% Providence, RI 02940, USA", %%% email = "tech-support@ams.org (Internet)", %%% supported = "yes", %%% abstract = "This is a LaTeX documentclass for the Conference %%% Proceedings, Canadian Mathematical Society.", %%% docstring = "The checksum field above contains a CRC-16 checksum %%% as the first value, followed by the equivalent of %%% the standard UNIX wc (word count) utility output %%% of lines, words, and characters. This is produced %%% by Robert Solovay's checksum utility.", %%% } %%% ==================================================================== % \NeedsTeXFormat{LaTeX2e} % \ProvidesClass{cmsams-l} % [1996/10/25 v1.2b CMSAMS Author Class] % \DeclareOption*{\PassOptionsToClass{\CurrentOption}{amsproc}} % \ProcessOptions % \LoadClass{amsproc}[1996/10/24] % \def\publname{Canadian Mathematical Society\newline % Conference Proceedings\hfil} % \def\ISSN{0731-1036} % \endinput %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\restr}{\vert\hskip -5.5pt \phantom{\vert}^{\scriptscriptstyle \backslash}} \newcommand{\QED}{\mbox{\rule[-1.0pt]{5pt}{8pt}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Point interactions in a tube} % author one information \author{Pavel Exner} \address{Nuclear Physics Institute, Academy of Sciences, 25068 \v Re\v z near Prague; Doppler Institute, Czech Technical University, B\v rehov{\'a} 7, 11519 Prague, Czech Republic} \curraddr{} \email{exner@ujf.cas.cz} \thanks{The research has been partially supported by GAAS under the contract A1048801.} % author two information %\author{} %\address{} %\curraddr{} %\email{} %\thanks{} \subjclass{} \begin{abstract} We discuss discuss spectral and scattering properties of a particle confined to a straight Dirichlet tube in $\mathbb{R}^3$ with a family of point interactions. \end{abstract} \maketitle \noindent Point interactions belong to the list of problems to which Sergio Albeverio made a significant contribution. This topic combines a practical importance as a source of numerous solvable models with an aesthetic appeal as the monograph \cite{AGHH} witnesses. At the same time it is far to be closed; despite the extensive and thorough character of the mentioned treatise new questions still arise. One of them concerns point interactions in tubular regions which represent a natural model for a ``quantum wire" with impurities. The simplest situation when the tube is a straight planar strip was investigated in \cite{EGST}; we refer to this paper for a detailed motivation and bibliography. In the present paper we present a brief discussion of a straight tube in $\mathbb{R}^3$ with a family of point interactions. \section{The one-center case} \noindent Let $\Omega:= \mathbb{R}\times M$ where $M\subset\mathbb{R}^2$ is a closed compact set; we suppose that it is pathwise connected and $\partial M$ has the segment property \cite{RS}. The free Hamiltonian is the corresponding Dirichlet Laplacian, $H_0= -\Delta_D^{\Omega}$ with the domain $W^{2,2}_0(\Omega)$. It can be expressed by means of the one-dimensional Laplacian and $-\Delta_D^M$. The last named operator has a purely discrete spectrum; we denote by $\chi_n,\: \nu_n$ its eigenfuctions and eigenvalues, respectively. For any $z\in\mathbb{C}\setminus [\nu_0,\infty)$ the free resolvent is an integral operator with the kernel % \begin{equation} \label{free kernel} G_0(\vec x_1,\vec x_2;z)\,\equiv\, (H_0\!-z)^{-1}(\vec x_1,\vec x_2)\,=\, {i\over 2}\, \sum_{n=0}^{\infty}\, \frac{e^{ik_n(z)|x_1-x_2|}}{k_n(z)}\, \chi_n(\vec y_1) \chi_n(\vec y_2)\,, \end{equation} % where $\vec x_j=(x_j,\vec y_j)$ and $k_n(z):=\, \sqrt{z\!-\!\nu_n}$, which is defined and smooth except at $\vec x_1= \vec x_2$. It is a multivalued function of $z$ with cuts $[\nu_n,\infty), \: n=0,1,\dots$. Suppose now that a point interaction is situated at $\vec a= (a,\vec b) \in M^o$. We define it as in \cite{AGHH}, i.e. as a self-adjoint extension of the operator $-\Delta_D^\Omega \restr C_0^{\infty}(\Omega\setminus\{\vec a\})$. We employ generalized boundary values % \begin{equation} \label{generalized bv} L_0(\psi,\vec a)\,:=\, \lim_{\vec x\to\vec a}\, \psi(\vec x) |\vec x\!-\!\vec a|\,, \quad\; L_1(\psi,\vec a)\,:=\, \lim_{\vec x\to\vec a} \left\lbrack \psi(\vec x)- {L_0(\psi,\vec a)\over |\vec x\!-\!\vec a|} \right\rbrack\,; \end{equation} % then the extension in question is specified by the boundary condition % \begin{equation} \label{bc} L_1(\psi,\vec a)+4\pi\alpha L_0(\psi,\vec a)\,=\,0 \end{equation} % for a given $\alpha\in\mathbb{R}$. We shall denote it $H(\alpha,\vec a)$; the case $\alpha=\infty$, i.e. $L_0(\psi,\vec a)=0$, corresponds to the free Hamiltonian $H_0$. The resolvent of $H(\alpha,\vec a)$ is obtained by Krein's formula. Mimicking the argument of \cite {EGST} we get % \begin{equation} \label{Krein} (H(\alpha,\vec a)\!-\!z)^{-1}(\vec x_1,\vec x_2)\,=\, G_0(\vec x_1,\vec x_2;z) +\, {G_0(\vec x_1,\vec a;z) G_0(\vec a,\vec x_2;z)\over \alpha-\xi(\vec a;z)}\,, \end{equation} % where $\xi(\vec a;z)$ is the regularized Green's function at $\vec a$, % \begin{equation} \label{xi} \xi(\vec a;z)\,=\, \lim_{u\to 0}\, \left\lbrack {i\over 2}\, \sum_{n=0}^{\infty}\, {e^{ik_n u}\over k_n} |\chi_n(\vec b)|^2 -\, {1\over 4\pi u}\, \right\rbrack\,. \end{equation} % The existence of the limit follows from the kernel behaviour at the singularity \cite{Ti}. However, we also need a prescription how to compute it and this differs from the two-dimensional case. We use semiclassical properties of the above series terms \cite{RS}. The transverse eigenvalues behave as $\nu_n\approx 4\pi|M|^{-1}n$ for $n\to\infty$ so $k_n= 2i\sqrt{\pi}|M|^{-1/2} \sqrt{n}+ \mathcal{O}(1)$. On the other hand, the probability densities $|\chi_n|^2$ are rapidly oscillating functions. Since $M$ supports no potential the mean value of these oscillations equals the constant $|M|^{-1}$; assuming that $|\chi_n(\vec b)|^2$ oscillates around this value as $n\to\infty$ we can assess the divergence rate of the first series. Next we use the identity % \begin{equation*} %\label{reg integral} {1\over 4\pi u}\,=\, \beta \int_0^{\infty} {e^{-\gamma u\sqrt{s}} \over \sqrt{s}}\, ds \end{equation*} % with $\gamma:=2\sqrt{\pi}|M|^{-1/2}$ and $\beta^{-1}:= 4\sqrt{\pi|M|}$ and write the r.h.s. as a sum of the integrals over $(n,n\!+\!1)$ obtaining % \begin{equation*} %\label{xi2} \xi(\vec a;z)\,=\, \lim_{u\to 0}\, \left\lbrack \sum_{n=0}^{\infty}\, {e^{-\kappa_n u}\over 2\kappa_n} |\chi_n(\vec b)|^2 +\, {e^{-\gamma u\sqrt{n+1}} -e^{-\gamma u\sqrt{n}}\over 4\pi u}\, \right\rbrack\,, \end{equation*} % where $\kappa_n(z):= -ik_n(z)=\sqrt{\nu_n\!-\!z}$. In combination with the preceding argument, it is easy to check that the summand has a uniform bound of order $o(n^{-3/2})$. Hence the limit can be interchanged with the sum and % \begin{equation} \label{xi3} \xi(\vec a;z)\,=\, \sum_{n=0}^{\infty}\, \left\lbrack {|\chi_n(\vec b)|^2\over 2\kappa_n(z)} +\, {\sqrt{n} -\sqrt{n+1}\over 2\sqrt{\pi|M|}}\, \right\rbrack\,. \end{equation} % \begin{remarks} (i) We do not give here details of the oscillation argument. Notice that the conclusions made below can be obtained even without it, up to an additive renormalization of the coupling constant, since by the Weyl formula and the uniform boundedness of the $\chi_n$'s the difference $\xi(\vec a;z)-\xi(\vec a;z_0)$ is given by a convergent series. \\ (ii) The scaling behaviour for $\Omega^{\sigma}=\mathbb{R} \times M^{\sigma}$ with $M^{\sigma}:= \sigma M, \; \sigma>0$ is more complicated than in the two-dimensional case. We have $\xi(\vec{a^{\sigma}};z\sigma^{-2})= \sigma^{-1} \xi(\vec a;z)$, so the singularities of the resolvent kernel are related by % \begin{equation} \label{scale} \epsilon^{\sigma}(\alpha^{\sigma},\vec{a^{\sigma}}) \,=\, \sigma^{-2} \epsilon(\alpha,\vec{a})\,, \quad \alpha^{\sigma}:= \sigma^{-1} \alpha\,. \end{equation} % \end{remarks} \begin{proposition} \label{spectr1} The operator $H(\alpha,\vec a)$ has for any $\alpha\in\mathbb{R}$ a single eigenvalue $\epsilon(\alpha,\vec a) \in(-\infty,\nu_0)$. The corresponding eigenfunction is % \begin{equation} \label{ef1} \psi(\vec x;\alpha,\vec a) \,=\, \sum_{n=0}^{\infty} {e^{-\kappa_n(\epsilon)|x-a|}\over 2\kappa_n(\epsilon)} \chi_n(\vec y) \chi_n(\vec b)\,. \end{equation} % The function $\epsilon(\cdot,\vec a)$ is strictly increasing and behaves as % \begin{equation} \label{weak1} \epsilon(\alpha,\vec a) \,=\, \nu_0- \left(|\chi_0(\vec b)|^2 \over 2\alpha \right)^2 + \mathcal{O}(\alpha^{-3})\,, \end{equation} % in the limit of weak coupling, $\alpha\to +\infty$. Moreover, there are no eigenvalues embedded in $\sigma_c(H(\alpha,\vec a))= [\nu_0,\infty)$. \end{proposition} % {\sc Proof:} Due to (\ref{xi3}), $\xi(\vec a;\cdot)$ is strictly increasing with ${\rm Ran\,}\xi= \mathbb{R}$ and $\xi(\vec a;z)= {1\over 2} |\chi_0(\vec b)|^2 (\nu_0\!-z)^{-1/2} +\mathcal{O}(1)$ as $z\to \nu_0-$. The non-normalized eigenfunction (\ref{ef1}) is given by the residue term in (\ref{Krein}). To check the absence of embedded eigenvalues we have to show that $\xi(\vec a;z)= \alpha$ has no solutions on $[\nu_0,\infty)$. Away of the thresholds, this follows from % \begin{equation} \label{im1} {\rm Im\,}\xi(\vec a;z)\,=\, \sum_{\{n:\, \nu_n0\,. \end{equation} % If $|\chi_n(\vec b)|^2\ne 0$, the resolvent kernel has a finite limit as $z$ approaches $\nu_n$, otherwise it has the same singularity as $G_0(\vec x_1,\vec a;\cdot)$ there, so in neither case it has a pole. \QED \\ [2mm] % Mimicking the argument of \cite{EGST} we also get % \begin{proposition} \label{scat1} The on-shell S-matrix at energy $z=k^2$ is a $2N_{\rm open}\times 2N_{\rm open}$ unitary matrix with elementary blocks % \begin{equation} \label{Sblock} S_{nm}\,=\, \sqrt{k_m\over k_n} \left( \begin{array}{cc} t_{nm} & r_{nm} \\ \tilde r_{nm} & \tilde t_{nm} \end{array} \right)\,, \quad n,m= 1,\dots,N_{\rm open}\,, \end{equation} % where $N_{\rm open} := {\rm card}\{ \nu_n:\: \nu_n\nu_0$ cannot be an eigenvalue corresponding to an eigenvector from the subspace $\bigoplus_{\{n:\,\nu_n0\,$, which follows from the identity % \begin{equation*} %\label{Bochner} (1+\kappa|x|)\, e^{-\kappa|x|} \,=\, {2\kappa^3\over \pi}\, \int_{\mathbb{R}} {e^{ipx}\over (p^2+\kappa^2)^2}\, dp \end{equation*} % and Bochner's theorem \cite[Sec.IX.2]{RS}. In fact, since the measure in the last integral is pointwise positive, $\,{d\over dz}\, \Lambda(z)\,$ is even strictly positive; it means that all the eigenvalues of $\,\Lambda(\alpha,\vec a;z)\,$ are decreasing functions of $\,z\,$ and $H(\alpha,\vec a)$ has at most $N$ eigenvalues. To check that the ground state is non-degenerate, we have to demonstrate that the lowest eigenvalue of $\Lambda(z)$ is simple for any $z\in (-\infty,\nu_0)$, which is equivalent to the claim that the matrix semigroup $ \lbrace\, e^{-t\Lambda(z)} :\, t\ge 0\,\rbrace$ is positivity preserving \cite[Sec.XIII.12]{RS}. The last property is ensured if all the non-diagonal elements of $\Lambda(z)$ are negative; we have $\Lambda(z)_{jm}= -G_0(\vec a_j,\vec a_m;z)$ by (\ref{Lambda}) so the desired result follows from the positivity of the free-resolvent kernel. The coefficients may be therefore chosen of the same sign for the ground state; in fact, as strictly positive because $d_{j_0}=0$ would mean that the eigenfunction is smooth at $\vec x=\vec a_{j_0}$ so the corresponding interaction is absent, $\alpha_{j_0}= \infty$. (b) Suppose now that $H\varphi= z\varphi$ for some $z>\nu_0$. We adapt again the argument from \cite[Sec.II.1]{AGHH} and pick an arbitrary $z'\in\rho(H)$; then there is a vector $\psi_0\in D(H_0)$ which allows us to write % \begin{equation} \label{eigenvector decomposition} \varphi\,=\, \psi_0+\, \sum_{j=1}^N d_j G_0(\cdot,\vec a_j;z')\,. \end{equation} % Furthermore, we expand $\psi_0$ as a series, $\psi_0(\vec x)= \sum_{n=0}^{\infty} g_n(x) \chi_n(\vec y)\,$ with the coefficients $g_n\in L^2(\mathbb{R})$. Using the identity $(H_0-z)\psi_0= (z-z') \sum_{j=1}^N d_j G_0(\cdot,\vec a_j;z')$ and the fact that $\{ \chi_n\}$ is an orthonormal basis in $L^2(M)$, we obtain a system of equations; by the Fourier-Plancherel operator it is transformed into % \begin{equation} \label{g hat} (p^2-z+\nu_n)\hat g_n(p)\,=\, {z-z'\over 2\pi}\, \sum_{j=1}^N d_j \chi_n(\vec b_j)\, {e^{-ipa_j}\over p^2-z'+\nu_n} \,. \end{equation} % If $g_n\in L^2$ the same has to be true for $\hat g_n$; this is impossible if $z>\nu_n$ and the r.h.s. of (\ref{g hat}) is nonzero at $\pm p_n$, where $p_n:= \sqrt{\nu_n\!-z}$, since $\hat g_n^2$ would have then a non-integrable singularity. If $N>1$, it might happen that the r.h.s. of (\ref{g hat}) is not zero identically. However, if the $a_j$ are mutually different, $\sum_{j=1}^N d_j\chi_n(\vec b_j)\, e^{\mp ipa_j}=0$ implies $d_j=0$ by linear independence. On the other hand, if some of them coincide we find $\sum_j d_j\chi_n(\vec b_j)=0$ where the index runs through the values with the same longitudinal coordinate $a_j$, and therefore $\hat g_n=0$ again. The condition $\nu_n1$. Examples can be constructed as in \cite{EGST} (or other similar systems -- cf.\cite{ELV}) using $M$ with a symmetry: one has to choose a family of weak enough point interactions with the same symmetry. (c) We denote $\,A:= {\rm diag}(\alpha_1,\dots,\alpha_N)$, and use the decomposition % \begin{equation*} %\label{weak decomp} \Lambda(z)\,=\, \left( A-\tilde\Gamma(z) \right) \left\lbrack\, I- \left( A-\tilde\Gamma(z) \right)^{-1} {M_1\over 2\sqrt{\nu_0-z}}\, \right\rbrack\,, \end{equation*} % where $\tilde\Gamma(z)$ is a remainder independent of $\,\alpha\,$, whose norm is bounded as $z\to \nu_0-$, and $M_1$ is the matrix defined above. The first factor is regular for $|A|$ large enough. Since $M_1$ is rank one we have to solve the equation % \begin{equation*} %\label{implicit} \eta \sum_{j,k=1}^N \chi_0(\vec b_j) \alpha_j \left(I- \tilde\Gamma(z)A^{-1} \right)_{jk} \chi_0(\vec b_k) - \sum_{j=1}^N \chi_0(\vec b_j)^2\,=\,0 \end{equation*} % with $\eta:= 2\sqrt{\nu_0\!-z}$; then (\ref{weak ev}) follows by the implicit-function theorem. \QED % \begin{remarks} (i) We get also the weak-coupling asymptotics for the eigenfunction: % \begin{eqnarray*} \psi(x;\alpha,\vec a) &\!\approx\!& \chi_0(\vec y) \: {2\sum_{j=1}^N \alpha_j\chi_0(\vec b_j)^2 \over \left( \sum_{j=1}^N \chi_0(\vec b_j)^2 \right)^2}\: \sum_{j=1}^N e^{-\sqrt{\nu_0-\epsilon}|x-a_j|} \chi_0(\vec b_j)^2 \\ \\ && +\, \sum_{n=1}^{\infty}\, \chi_n(\vec b_j)\: \sum_{j=1}^N {e^{-\sqrt{\nu_n-\nu_0}|x-a_j|} \over \sqrt{\nu_n-\nu_0}}\, \chi_n(\vec b_j) \chi_0(\vec b_j)\,. \end{eqnarray*} % The leading term is a product of $\,\chi_1(\vec y)\,$ with a linear combination of the eigenfunctions of one-dimensional point interactions placed at $a_j,\; j=1,\dots,N\,$. \\ (ii) The scattering problem can be treated as in the one-center case. Existence and completeness of wave operators follow from the Kato--Birman theory \cite{RS}. The reflection and transmission amplitudes from the $n$-th to the $m$-th channel are % \begin{eqnarray} \label{rt_N} r_{nm}(z) &\!=\!& {i\over 2}\, \sum_{j,k=1}^N (\Lambda(z)^{-1})_{jk} {\chi_m(\vec b_j) \chi_n(\vec b_k)\over k_m(z)}\, e^{i(k_ma_j+k_na_k)}\;, \nonumber \\ t_{nm}(z) &\!=\!& \delta_{nm}\,+\, {i\over 2}\, \sum_{j,k=1}^N (\Lambda(z)^{-1})_{jk} {\chi_m(\vec b_j) \chi_n(\vec b_k)\over k_m(z)}\, e^{-i(k_ma_j-k_na_k)} \end{eqnarray} % and the unitarity condition now reads % \begin{eqnarray} \label{unitarity_N} \sum_{\{m:\: 0\le \nu_m< z\}} k_m (t_{nm}\overline t_{sm}+r_{nm}\overline r_{sm}) &\!=\!& \delta_{ns} k_n\,, \nonumber \\ \sum_{\{m:\: 0\le \nu_m< z\}} k_m \left(\tilde t_{nm}\overline r_{sm} +\tilde r_{nm}\overline t_{sm}\right) &\!=\!& 0 \,, \end{eqnarray} % because $\,S_{nm}\,$ is given again by (\ref{Sblock}), where the tilded quantities are obtained by mirror transformation, $\,a_j\to -a_j\,$. \end{remarks} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The periodic case} \noindent In the infinite-center case we restrict ourselves to the periodic situation, i.e. we suppose that the set $\{\alpha,\vec a\}_{per}=\{[\alpha_j,\vec a_j]:\; j=1,2,\dots\,\}$ in $\Omega$ is countably infinite and has a periodic pattern with a period $\ell>0$ and $N$ perturbations in each cell, which we denote again as $\{\alpha,\vec a\}$. Following the Floquet-Bloch decomposition, we find the unitary operator $U:\, L^2(\Omega)\to L^2(\mathcal{B}, (\ell/2\pi)d\theta; L^2(\hat\Omega))$, where % \begin{equation} \label{WS cell} \hat\Omega\,:=\, [0,\ell)\times M\,, \qquad \mathcal{B}\,:=\, \left\lbrack\, -{\pi\over\ell},\,{\pi\over\ell}\right) \times M \;; \end{equation} % the $x$-projections of these sets are the Wigner-Seitz cell of the underlying one-dimensional lattice and the corresponding Brillouin zone, respectively. By means of $U$, the operator $H(\alpha,\vec a)$ is unitarily equivalent to % \begin{equation} \label{FB decomposition} U\,H(\{\alpha,\vec a\}_{per})\,U^{-1}\,=\, {\ell\over 2\pi}\, \int_{|\theta\ell|\le\pi}^{\oplus} H(\alpha,\vec a;\theta)\,d\theta\,, \end{equation} % where $H(\alpha,\vec a;\theta)$ is the point-interaction Hamiltonian on $L^2(\hat\Omega)$, i.e. the Laplacian satisfying (\ref{bc N}) at the points $\vec a_j$, Dirichlet b.c. for $x\in [0,\ell)\,, \: \vec y\in\partial M$, and % \begin{equation} \label{Bloch bc} \psi(\ell-,\vec y)\,=\,e^{i\theta\ell} \psi(0+,\vec y)\,, \qquad {\partial\psi\over\partial x}(\ell-,\vec y)\,=\,e^{i\theta\ell}\; {\partial\psi\over\partial x}(0+,\vec y) \end{equation} % for $\vec y\in M$. The resolvent of $H(\alpha,\vec a;\theta)$ can be derived by modifying the argument of \cite[Sec.5]{EGST}. The ``free'' eigenvalues % \begin{equation} \label{FB free eigenvalues} \epsilon_{mn}(\theta)\,:=\, \left({2\pi m\over\ell}\,+\theta\right)^2 +\nu_n\,, \quad m\in\mathcal{Z}\,,\; n=0,2,\dots\,, \end{equation} % correspond to the eigenfunctions $\eta^{\theta}_m\otimes\chi_n$, where $\chi_n$ are as above and $\eta^{\theta}_m(x):= \ell^{-1/2}\, e^{i(2\pi m+\theta\ell)x/\ell },\, m\in\mathcal{Z}$. Moreover, the free resolvent kernel is in analogy with \cite{EGST} obtained by a partial summation of the appropriate double series and equals % \begin{eqnarray} \label{FB free resolvent} G_0(\vec x_1,\vec x_2;\theta;z) &\!=\!& \sum_{n=0}^{\infty}\, {\sinh((\ell\!-\!|x_1\!-\!x_2|) \sqrt{\nu_n\!-\!z}) +e^{2i\eta\theta\ell} \sinh(|x_1\!-\!x_2| \sqrt{\nu_n\!-\!z}) \over \cosh(\ell\sqrt{\nu_n\!-\!z}) -\cos(\theta\ell)} \nonumber \\ && \times\:{\chi_n(\vec y_1) \chi_n(\vec y_2) \over 2\sqrt{\nu_n\!-\!z}}\,, \end{eqnarray} % where $\eta:={\rm sgn}(x_1\!-x_2)$. The full kernel is then expressed by a formula analogous to (\ref{KreinN}) with % \begin{equation} \label{FB lambda} \lambda(\alpha,\vec a,\theta;z)\,=\, \Lambda(\alpha,\vec a,\theta;z)^{-1}\,, \end{equation} % where $\Lambda_{jr}= -G_0(\vec a_j,\vec a_r)$ for $j\ne r$, while the diagonal elements are given by % \begin{equation} \label{FB Lambda} \Lambda_{jj} \,=\, \alpha_j\,-\,{1\over 2}\, \sum_{n=0}^{\infty}\, \left( {\sinh(\ell\sqrt{\nu_n\!-\!z}) \over \cosh(\ell\sqrt{\nu_n\!-\!z}) -\cos\theta\ell}\: {\chi_n(\vec y)^2\over \sqrt{\nu_n\!-\!z}}\,+\, {\sqrt{n}- \sqrt{n\!+\!1}\over \sqrt{\pi|M|}} \right)\,. \end{equation} % We may also write the last formula as $\Lambda_{jj}(\alpha,\vec a,\theta;z) =\alpha_j- \xi(\vec a_j,\theta;z)$, where the function $\xi$ is for $z\in\mathbb{R}$ more explicitly given by % \begin{eqnarray} \label{FB xi} \xi(\vec a_j,\theta;z) &\!=\!& {1\over 2}\, \sum_{\{n:\: \nu_n\le z\}}\, \left( {\sin(\ell\sqrt{z\!-\!\nu_n}) \over \cos(\ell\sqrt{z\!-\!\nu_n}) -\cos\theta\ell}\: {\chi_n(\vec b_j)^2\over \sqrt{z\!-\!\nu_n}}\,+\, {\sqrt{n}- \sqrt{n\!+\!1}\over \sqrt{\pi|M|}} \right) \nonumber \\ &\!+\!& {1\over 2}\, \sum_{\{n:\: \nu_n>z\}}\, \left( {\sinh(\ell\sqrt{\nu_n\!-\!z}) \over \cosh(\ell\sqrt{\nu_n\!-\!z}) -\cos\theta\ell}\: {\chi_n(\vec b_j)^2\over \sqrt{\nu_n\!-\!z}}\,+\, {\sqrt{n}- \sqrt{n\!+\!1}\over \sqrt{\pi|M|}} \right)\,. \end{eqnarray} % It is defined everywhere except at % \begin{equation} \label{FB singularities} \mathcal{E}(\vec a,\theta)\,:=\, \{\, \epsilon_{mn}(\theta)\in\mathcal{E}(\theta)\,:\; \chi_n(\vec b_j)\ne 0\;\}\,, \end{equation} where $\mathcal{E}(\theta)$ is the eigenvalue set (\ref{FB free eigenvalues}). Of course, the r.h.s. of (\ref{FB singularities}) makes no sense if $z=\nu_n$ and $\chi_n(\vec b_j)\ne 0$, but applying general results on self-adjoint extensions to the one-center case, we can establish {\em a posteriori} that $\xi$ can be defined there by continuity. Moreover, we find that the function $\xi(\vec a_j,\theta;\cdot)$ is monotonously increasing between any pair of neighboring singularities. Then we have the following result. % \begin{proposition} \label{spec-per} For a given $\alpha\in\mathbb{R}^N$ the operator $H(\alpha,\vec a;\theta)$ has $N$ eigenvalues $\epsilon_{mnj}(\alpha,\vec a;\theta),\: j=1,\dots,N$, in any gap of the set (\ref{FB singularities}) determined by % \begin{equation} \label{FB pole} \det \Lambda(\alpha,\vec a,\theta;z)\,=\,0\,. \end{equation} % The corresponding eigenfunctions are $\psi(\vec x) = \sum_{j=1}^N d_j G_0(\vec x,\vec a_j,\theta;z)$ where the $d_j$'s are determined by $\Lambda(\alpha,\vec a,\theta;z)$ as in Theorem~\ref{spectrN}. \end{proposition} % The spectrum of the original Hamiltonian $H(\{\alpha,\vec a\}_{per})$ consist then of bands, % \begin{equation} \label{spec-band} \sigma\left(H(\{\alpha,\vec a\}_{per})\right)\,=\, \bigcup_{mnj} \left\{\, \epsilon_{mnj}(\alpha,\vec a;\theta):\: \theta\in \mathcal{B}\, \right\}\,. \end{equation} % One is interested, of course, in its absolute continuity and existence of gaps. We restrict ourselves to the simplest nontrivial situation. % \begin{example} Let $N=1$, i.e. let each cell contain a single point interaction. The condition (\ref{FB pole}) then simplifies to % \begin{equation} \label{FB pole_1} \xi(\vec a,\theta;z)\,=\,\alpha\,. \end{equation} % The left hand side is monotonously increasing between its singularities, i.e. the points of $\mathcal{E}(\vec a,\theta)$. This means that for fixed $\alpha,\,\theta$ there is a sequence $\{\epsilon_r(\alpha,\vec a,\theta)\}_{r=0} ^{\infty}$ arranged in the ascending order; each of them depends, in fact, only on the transverse component $\vec b$ of the vector $\vec a$. The lowest one satisfies % \begin{equation*} %\label{ground band} \epsilon_0(\alpha,\vec a,\theta)\,<\,\nu_0+\theta^2 \end{equation*} % and between each two neighboring points of $\mathcal{E}(\vec a,\theta)$ there is just one of the other eigenvalues. It is also clear that any of $\epsilon_r(\alpha,\vec a,\theta)$ is continuous with respect to the parameters and $\epsilon_r(\cdot,\vec a,\theta)$ is increasing for fixed $\vec b$ and $\theta$. Concerning the $\theta$-dependence, the implicit-function theorem tells us that % \begin{equation*} %\label{theta dep} {\partial\epsilon_r(\alpha,\vec a,\theta)\over \partial\theta}\,=\, -\, \left.{\partial\xi(\vec a,\theta;z)\over \partial\theta}\, \left( {\partial\xi(\vec a,\theta;z)\over \partial z} \right)^{-1}\, \right|_{(\epsilon_r,\theta)} \end{equation*} % whenever the denominator is nonzero. Away of the thresholds, $z=\nu_n$, and the points of $\mathcal{E}(\theta)$, a straightforward differentiation shows that $\xi(\vec a,\cdot;\cdot)$ is analytic in both variables. By (\ref{FB xi}) the numerator is not identically zero; hence the derivative $\partial\epsilon_r(\alpha,\vec a,\theta)/ \partial\theta$ may be zero at some points but never in an interval and {\em the spectrum of $H(\{\alpha,\vec a\}_{per})$ is absolutely continuous} \cite[Sec.XIII.16]{RS}. Let us turn now to the question about the number of gaps. Below $z=\nu_0$ the spectrum may be estimated by means of extrema of the function $\xi$ which yield $\theta$-independent bounds: we have $\xi(\vec a,\theta;z) \le \xi_+(\vec a,z)$ where % \begin{equation*} %\label{xi plus} \xi_+(\vec a,z):= \max_{|\theta\ell|\le\pi} \xi(\vec a,\theta;z) \,=\, {1\over 2} \sum_{n=0}^{\infty}\,\left( {\chi_0(\vec b)^2 \over \sqrt{\nu_n\!-\!z}}\, \coth\left({\ell\over 2}\sqrt{\nu_n\!-\!z} \right) \,+\, {\sqrt{n}-\sqrt{n\!+\!1}\over \sqrt{\pi|M|}} \right) \end{equation*} % and a similar formula for the minimum, $\xi_-(\vec a,z)$, with $\,\coth\,$ replaced by $\,\tanh$. Both functions are continuously increasing and tend to $-\infty$ as $z\to -\infty$. On the other hand, $\xi_+(\vec a,\cdot)$ diverges as $z\to \nu_0-$ while the lower bound $\xi_-(\vec a,\cdot)$ has a finite limit. This shows, in particular, that the spectral condition (\ref{FB pole_1}) has no solution for any $\theta$ in a left neighborhood of $\nu_0$ provided % \begin{eqnarray} \label{one gap} \alpha\!&<&\! \xi_-(\vec a,\nu_0-) \,=\, {\ell\over 4}\, \chi_0(\vec b)^2 -\,{1\over 2\sqrt{\pi|M|}} \nonumber\\ &\!+\!& {1\over 2}\,\sum_{n=1}^{\infty}\,\left( {\chi_0(\vec b)^2 \over \sqrt{\nu_n\!-\!\nu_0}}\, \tanh\left({\ell\over 2}\sqrt{\nu_n\!-\!\nu_0} \right) \,+\, {\sqrt{n}-\sqrt{n\!+\!1}\over \sqrt{\pi|M|}} \right)\,; \end{eqnarray} % in other words, that a gap exists. The condition (\ref{one gap}) is satisfied for a strong enough coupling if the point-interaction spacing is kept fixed. On the other hand, inspecting the right-hand-side we see that the gap exists also for any fixed $\alpha$ and the spacing $\ell$ large enough. In this respect the spectrum is similar to that of a straight polymer in $\mathbb{R}^3$ described in \cite[Sec.III.1]{AGHH}. However, for our ``coated polymer'' a much stronger result is valid: we shall show that under a suitable choice of parameters it can have {\em any finite number} of gaps. To this end, we consider $z\in(\nu_0+\varepsilon, \nu_1-\varepsilon)$ for a fixed $\varepsilon>0$ and $\ell\gg \sqrt{|M|}$, and rewrite the right hand side of the relation (\ref{FB xi}) as % \begin{equation*} %\label{decomp} \xi(\vec a,\theta;z)\,=\, \xi_0(\vec a,\theta;z)+ \eta(\vec a,\theta;z)\,, \end{equation*} % where % \begin{equation*} %\label{leading} \xi_0(\vec a,\theta;z)\,:=\, {\sin(\ell\sqrt{z\!-\!\nu_0}) \over \cos(\ell\sqrt{z\!-\!\nu_0}) -\cos\theta\ell}\; {\chi_0(\vec b)^2\over 2\sqrt{z\!-\!\nu_0}} \end{equation*} % and $\eta(\vec a,\theta;z)$ is the rest. The latter is monotonously increasing with respect to z and its derivative is bounded everywhere below the second threshold, in particular, in the chosen interval of energies. Moreover, $\eta(\vec a,\theta;z)$ is bounded from above by % \begin{equation*} %\label{upper} \eta_+(\vec a,z)\,:=\, -\,{1\over 2\sqrt{\pi|M|}} \,+\, {1\over 2} \sum_{n=1}^{\infty}\,\left( {\chi_0(\vec b)^2 \over \sqrt{\nu_n\!-\!z}}\, \coth\left({\ell\over 2}\sqrt{\nu_n\!-\!z} \right) \,+\, {\sqrt{n}-\sqrt{n\!+\!1}\over \sqrt{\pi|M|}} \right) \end{equation*} % and the corresponding minimum, $\eta_-(\vec a,z)$, is obtained when $\,\coth\,$ in the last expression is replaced by $\,\tanh$. These estimates shrink as $\ell$ becomes large: using the inequality $\coth u-\tanh u< 5\, e^{-2u}$ for $2u\ge 1$, we find % \begin{equation*} %\label{corridor} \eta_+(\vec a,z)-\eta_-(\vec a,z)\,<\, {5\over 2}\, {\chi_1(\vec b)^2 \over \sqrt{\nu_1\!-\!z}}\, e^{-\ell\sqrt{\nu_1\!-\!z}} \,+\, {5\over 2}\, \sum_{n=2}^{\infty}\, {\chi_n(\vec b)^2 \over \sqrt{\nu_n\!-\!z}}\, e^{-\ell\sqrt{\nu_n\!-\!z}}\,. \end{equation*} % The series can be estimated by an integral, which yields for a fixed $z\in(\nu_0+\varepsilon, \nu_1-\varepsilon)$ the behavior % \begin{equation} \label{corridor2} \eta_+(\vec a,z)-\eta_-(\vec a,z)\,=\, \mathcal{O}\left(\sqrt{|M|}\, \ell^{-1} \right)\,. \end{equation} % On the other hand, the function $g_{\theta}(u):= \sin u (\cos u-\cos\theta\ell)^{-1}$ is increasing between any two zeros of its denominator. In the intervals, where it is positive, it can be estimated from below by the appropriate branch of $\tan\left( {u\over 2}+\pi m\right)$; when it is negative, we have a similar estimate from above with $\,\tan\,$ replaced by $\,-\cot$. Hence independently of $\theta$ we have either % \begin{equation*} %\label{xi 0 up} \xi_0(\vec a,\theta;z)\,\ge\, {\chi_n(\vec b)^2\over 2\sqrt{z\!-\!\nu_0}}\; \tan\left( {\pi\over 2}\,\left\lbrace {\ell\over\pi} \sqrt{z\!-\!\nu_0} \right\rbrace \right) \end{equation*} % or % \begin{equation*} %\label{xi 0 down} \xi_0(\vec a,\theta;z)\,\le\, -\,{\chi_n(\vec b)^2\over 2\sqrt{z\!-\!\nu_0}}\; \cot\left( {\pi\over 2}\,\left\lbrace {\ell\over\pi} \sqrt{z\!-\!\nu_0} \right\rbrace \right)\,, \end{equation*} % where $\{\cdot\}$ denotes the fractional part. Putting the estimates together we see that the oscillating part dominates, so for sufficiently large $|\alpha|$ there are gaps having $\nu_0+\,\left( \pi m\over\ell\right)^2$ as one endpoint provided it belongs to $(\nu_0+\varepsilon, \nu_1-\varepsilon)$. In addition, $\tan u+\cot u\ge 2$ which means that the gap between the lower and the upper bound to $\xi(\vec a,\theta;z)$ never closes within the chosen interval if $\ell/\sqrt{|M|}$ is large enough; we infer that {\em for any} $\alpha\in\mathbb{R}$ the operator $H(\{\alpha,\vec a\}_{per})$ can have {\em an arbitrary finite number of gaps} in its spectrum provided the spacing of the point-interaction array is large enough. \end{example} % Conclusions of the example can be extended to a finite number of point perturbations per cell. In a similar way one can treat a toroidal tube supporting point interactions and threaded by a magnetic flux, etc. A more difficult question concerns the finiteness of the number of open gaps. Recall that the gap number depends strongly on the dimension of a periodic system: for one-dimensional systems it is generically infinite \cite[Sec.XIII.16]{RS}, \cite[Sec.III.2]{AGHH}, while for higher dimensions it is finite by the {\em Bethe-Sommerfeld conjecture}. The latter is known to be true, in particular, for periodic potentials or lattices of point interactions in $\mathbb{R}^3$ -- see \cite{AG,Sk}. The tube boundary makes things more complicated, but one still expects that at high energies gaps will close due to overlapping of contributions from different transverse modes. Notice in this connection that in the example we have been looking for gaps in the energy interval where transport is possible in the lowest transverse mode only. Nevertheless, it is not easy to demonstrate that no gaps remain open above a certain energy. In a similar way, the ``mixed dimensionality" of waveguide systems inspires other questions such as existence of a mobility edge in tubes with random point interactions, etc. Generally speaking, solvable models whose genealogy can be traced back to the treatise \cite{AGHH} will represent for long a useful laboratory for the spectral theory. \begin{thebibliography}{article} % \bibitem{AGHH} S.~Albeverio, F.~Gesztesy, R.~H\o egh-Krohn, H.~Holden: {\em Solvable Models in Quantum Mechanics}, Springer, Heidelberg 1988. % \bibitem{AG} S.~Albeverio, V.A.~Geyler: The band structure of the general periodic Schr\"odinger operator with point interactions, {\em Commun. Math. Phys.} (1999), to appear % \bibitem{ELV} D.V.~Evans, M.~Levitin, D.~Vassiliev: Existence theorems for trapped modes, {\em J. Fluid. Mech.} {\bf 261} (1994), 21--31. % \bibitem{EGST} P.~Exner, R.~Gawlista, P.~\v Seba, M.~Tater: Point interactions in a strip, {\em Ann. Phys.} {\bf 252} (1996), 133--179. % \bibitem{RS} M.~Reed, B.~Simon: {\em Methods of Modern Mathematical Physics, II.~Fourier Analysis. Self-Adjointness, III.~Scattering Theory, IV.~Analysis of Operators}, Academic Press, New York 1975--1979. % \bibitem{Sk} M.M.~Skriganov: The spectrum band structure of the three--dimensional Schr\"odinger operator with a periodic potential, {\em Invent. Math.} {\bf 80} (1985), 107--121. % \bibitem{Ti} E.C.~Titchmarch: {\em Eigenfunction Expansions Associated with Second--Order Differential Equations}, vol.II, Clarendon Press, Oxford 1958. % \end{thebibliography} \end{document} ENDBODY