Content-Type: multipart/mixed; boundary="-------------0312101028453" This is a multi-part message in MIME format. ---------------0312101028453 Content-Type: text/plain; name="03-533.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-533.comments" 39 pages ---------------0312101028453 Content-Type: text/plain; name="03-533.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-533.keywords" Branched quantum wave guides; convergence of eigenvalues; singular limit; Laplacian on a manifold ---------------0312101028453 Content-Type: application/x-tex; name="graph.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="graph.tex" %-------------------------------------------------------------------- % graph.tex Convergence of spectra of graph-like thin mfds % Olaf Post, Pavel Exner % 2002-09-23 Institut f"ur Reine und Angewandte Mathematik % RWTH Aachen, Germany % last change email: post@iram.rwth-aachen.de % 2003-12-10, 2003-10-06, 09-01, 2003-03-27, 2003-02-19, 2002-10-18 %-------------------------------------------------------------------- \documentclass[12pt,reqno]{amsart} % leqno: eq-numbers on right side \listfiles % file list in .log file \usepackage{a4} \usepackage{amsmath} \usepackage{amssymb} \usepackage{graphicx} % for graphics %\usepackage[notref,notcite]{showkeys} % show labels (testing phase) %------------------------------------------------------------ % Theorem environments %------------------------------------------------------------ %\swapnumbers % 1.1 Theorem instead of Theorem 1.1 \theoremstyle{plain} % body italics \newtheorem{thm}{Theorem}[section] \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \theoremstyle{definition} % body roman \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \newtheorem{exmp}[thm]{Example} %------------------------------------------------------------ % Equation numbering %------------------------------------------------------------ \numberwithin{equation}{section} \renewcommand{\thesubsection}{\arabic{section}.\Alph{subsection}} %------------------------------------------------------------ % Own definitions %------------------------------------------------------------ \DeclareMathOperator{\dom} {dom} \DeclareMathOperator{\spec} {spec} \DeclareMathOperator{\vol} {vol} \DeclareMathOperator{\tr} {tr} %------------------------------------------------------------ % circle over symbol %------------------------------------------------------------ \newlength{\maxbreite}% \newlength{\maxhoehe}% \newlength{\maxtiefe}% \newcommand{\stelldrueber}[3][0pt]{% Vorbereitung f"ur Kreis "uber Symbol \settowidth{\maxbreite}{#3}% \settoheight{\maxhoehe}{#3}% \settodepth{\maxtiefe}{#2}% \addtolength{\maxhoehe}{\maxtiefe}% {\makebox[\maxbreite]{\raisebox{\maxhoehe}{\hspace{#1}#2}}% \makebox[0pt][r]{#3}}% } \newcommand{\overcirc}[1] % Kreis "uber Symbol 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L_2(#1)-spaces \newcommand{\Lsqrloc}[1]{L_{2,\mathrm{loc}}({#1})} % L_{2,loc}(#1)-spaces \newcommand{\Sob}[2][1]{\HS^{#1}({#2})} % Sobolev spaces \newcommand{\Sobn}[2][1]{{\overcirc {\mathcal H}{}^{#1}({#2})}}% Sob-Raum H_0 \newcommand{\norm}[2][{}]{\|{#2}\|_{#1}} % norm \newcommand{\normsqr}[2][{}]{\|{#2}\|^2_{#1}} % norm squared \newcommand{\bignorm}[2][{}]{\bigl\|{#2}\bigr\|_{#1}} % norm \newcommand{\bignormsqr}[2][{}]{\bigl\|{#2}\bigr\|^2_{#1}}% norm squared \newcommand{\Bignorm}[2][{}]{\Bigl\|{#2}\Bigr\|_{#1}} % norm \newcommand{\Bignormsqr}[2][{}]{\Bigl\|{#2}\Bigr\|^2_{#1}}% norm squared \newcommand{\iprod}[3][{}]{\langle{#2},{#3}\rangle_{#1}} % inner product \newcommand{\bd} {\partial} % symbol for boundary of a set \newcommand{\dcup}{\dot \cup} % symbol for disjoint union \newcommand{\bigdcup}{\stelldrueber[.45ex]% {$\scriptscriptstyle \! \bullet\mspace{1mu}$}{$\bigcup$}} %\newcommand{\bigdcup}{\bigcup^{\bullet}} % symbol for disjoint union \newcommand{\restr}[1]{{\restriction}_{#1}} % symbol for map restriction \newcommand{\orth}{\bot} % symbol for orthogonality \newcommand{\normder}{\partial_\mathrm{n}} % symbol for normal derivative \newcommand{\map}[3]{{#1}\colon{#2}\longrightarrow{#3}} % maps \newcommand{\set}[2]{\{ \, #1 \, | \, #2 \, \} } % set \usepackage{bbm} % blackboard (see: /usr/share/texmf/mytex/bbm.dvi) \newcommand{\one}{\mathbbm 1} % blackboard 1 %\renewcommand{\one}{\mathbf 1} % if bbm does'nt work % uncomment this! % \reversemarginpar % set marginpar on inner side % \newcommand{\look}[1]{\rule{3ex}{2.5mm} % \marginpar{\rule{1ex}{2.5mm}\hspace*{-2ex} % \textbf {\textit{\footnotesize #1}}}} \newcommand{\Neu}{{\mathrm N}} % symbol for Neumann bd cond \newcommand{\Dir}{{\mathrm D}} % symbol for Dirichlet bd cond \newcommand{\laplacian}[1]{\Delta_{{#1}}} % symbol for Laplacian on mfd \newcommand{\laplacianD}[1]{\Delta^\Dir_{{#1}}}% symb f Dir-Laplacian \newcommand{\laplacianND}[1]{\Delta^{\Neu \Dir} _{{#1}}}% symb f ND-Laplacian \newcommand{\laplacianN}[1]{\Delta^\Neu_{{#1}}}% symb f Neu-Laplacian \newcommand{\laplacianT}[1]{\Delta^{\theta}_{{#1}}}% symb f theta-Laplacian \newcommand{\EW}[2]{\lambda_{#1}({#2})} % Eigenvalue of Laplacian on #2 \newcommand{\EWD}[2]{\lambda^\Dir_{#1}({#2})}% EV of Dir Laplacian \newcommand{\EWN}[2]{\lambda^\Neu_{#1}({#2})}% EV of Neu Laplacian \newcommand{\EWT}[2]{\lambda^{\theta}_{#1}({#2})}% EV of theta Laplacian % Some general symbol names in this article (useful for global % replacement), no brackets necessary as arguments (e.g. $\Sob \Veps$) % % I hope this is not to pedantic and cryptic ... % Furthermore, one can globally change subscripts (e.g. $I_{j,k}$ instead of % $I_{jk}$ and so on. % \newcommand{\Mnull}{{M_0}} % symbol for graph \newcommand{\Meps}{{M_\eps}} % symbol for thickened graph % Symbols for edge \newcommand{\Ij}{{I_j}} % simple edge \newcommand{\Ijk}{{I_{jk}}} % half simple edge \newcommand{\pI}{{I^+}} % extended interval of edge nbhd \newcommand{\pIjk}{{I_{jk}^+}} % extended interval of edge nbhd \newcommand{\nI} {{I^0}} % added interval \newcommand{\nIjk}{{I_{jk}^0}} % eps-indep % Symbols for thickened edge \newcommand{\Ueps}{{U_\eps}} % with metric \newcommand{\Uepsj}{{U_{\eps, j}}} % more precise: with index \newcommand{\Uepsjk}{{U_{\eps, jk}}} % half of the edge \newcommand{\Uj}{{U_j}} % only as mfd, with index \newcommand{\Ujk}{{U_{jk}}} % half of the edge only as mfd \newcommand{\tUeps}{{\tilde U_\eps}} % with cylindrical metric \newcommand{\tUepsj}{{\tilde U_{\eps, j}}} % more precise: with index \newcommand{\tUepsjk}{{\tilde U_{\eps, jk}}} % more precise: with index % Symbols for enlarged thickened edge (part of vertex added) \newcommand{\pUeps}{{U_\eps^+}} % with metric \newcommand{\tpUeps}{{\tilde U_\eps^+}} % with cylindrical metric \newcommand{\pUepsjk}{{U_{\eps, jk}^+}} % more precise: with index \newcommand{\pU}{{U^+}} % only as mfd \newcommand{\pUjk}{{U_{jk}^+}} % only as mfd, with index % Symbols for thickened vertex region \newcommand{\Veps}{{V_\eps}} % with metric \newcommand{\Vepsk}{{V_{\eps, k}}} % more precise: with index \newcommand{\Vk}{{V_k}} % only as manifold % Symbols for enlarged vertex region \newcommand{\pVeps}{{V_\eps^+}} % with metric \newcommand{\pVepsk}{{V_{\eps, k}^+}} % more precise: with index \newcommand{\pV}{{V^+}} % only as manifold \newcommand{\pVk}{{V_k^+}} % only as manifold % Symbols for limit vertex region (alpha=0) \newcommand{\Vnullk}{{V_{0, k}}} % more precise: with index % Symbols for $\alpha$-scaled thickened vertex region (subset of $\Vk$) \newcommand{\mVeps}{{V_\eps^-}} % with metric \newcommand{\mVepsk}{{V_{\eps, k}^-}} % more precise: with index \newcommand{\mV}{{V^-}} % only as manifold \newcommand{\mVk}{{V_k^-}} % only as manifold % Symbol for "bottle neck" \newcommand{\Aeps}{A_\eps} % with metric \newcommand{\tAeps}{\tilde A_\eps} % with product metric \newcommand{\Aepsjk}{A_{\eps, jk}} % more precise: with index \newcommand{\Ajk}{A_{jk}} % only as manifold %------------------------------------------------------------ % begin of the document %------------------------------------------------------------ \begin{document} \title{Convergence of spectra of graph-like thin manifolds} \author{Pavel Exner} \address{Department of Theoretical Physics, NPI, Academy of Sciences, 25068 \v{R}e\v{z} near Prague, and Doppler Institute, Czech Technical University, B\v{r}ehov\'{a}~7, 11519 Prague, Czechia} %\email{exner@ujf.cas.cz} \date{\today} \author{Olaf Post} \address{Institut f\"ur Reine und Angewandte Mathematik, Rheinisch-Westf\"alische Technische Hochschule Aachen, Templergraben 55, 52062 Aachen, Germany} %\email{post@iram.rwth-aachen.de} \date{\today} %------------------------------------------------------------ % Subject classifications %------------------------------------------------------------ %Subjclass contains the Classification of the paper following the 1991 %Mathematics Subject Classifications %\subjclass{35P20, 58G18, 47F05} %\keywords{Eigenvalues, spectral gap, perturbation of periodic structures} %------------------------------------------------------------ % Abstract. %------------------------------------------------------------ \begin{abstract} We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices. \end{abstract} \maketitle %------------------------------------------------------------ % The main part %------------------------------------------------------------ \section{Introduction} Graph models of quantum systems have a long history. Already half a century ago Ruedenberg and Scherr \cite{ruedenberg-scherr:53} used this idea to calculate spectra of aromatic carbohydrate molecules, however, a real boom started from the late eighties when semiconductor graph-type structures became small and clean enough so that coherent effects in the corresponding quantum transport can play the dominating role. From the mathematical point of view these models were analyzed first thoroughly in \cite{exner-seba:89}, for recent developments and bibliography see \cite{kostrykin-schrader:99}. The free Hamiltonian of a graph model is the (differential) Laplacian on the graph. To define it properly one has to specify the boundary conditions which couple the wave functions at the vertices. They have to define a self-adjoint operator, however, this requirement itself does not specify the conditions uniquely: in a vertex joining $n$ graph edges we have $n^2$ free parameters. A natural idea to remove this non-uniqueness is to regard the graph model as a limit case of a more realistic one with a unique Hamiltonian; an appropriate choice is a ``thickened graph'' composed of thin tubes which gives the original graph in the limit of a vanishing tube radius. Unfortunately, it is not easy to see what happens with spectral and/or scattering properties in such a limit. The spectral convergence when the ``thick graph'' is planar with Neumann boundary conditions has been solved recently by Kuchment--Zeng \cite{kuchment-zeng:01}, and Rubinstein--Schatzman \cite{rubinstein-schatzman:01}; Saito~\cite{saito:00} showed the convergence of the resolvent. Note that Colin de Verdi\`ere~\cite{colin:86b} already established a similar result to prove that the first non-zero eigenvalue of a compact manifold of dimension greater than $2$ can have arbitrary high (finite) multiplicity. The physically more interesting situation with Dirichlet boundary represents a longstanding open problem. These two cases do not exhaust all possible ways in which a family of manifolds can approach a graph. One more choice are manifolds without a boundary of codimension one in $\R^\nu,\: \nu\ge 3$, which encloses the graph like a system of ``sleeves''\footnote{Kuchment and Zeng \cite{kuchment-zeng:01} speak also about sleeves having meaning graph edges thickened into strips; what we have in mind here is rather a cylindrical surface with the graph edge as its axis.}, with the limit consisting of the sleeve diameter shrinking. It is particularly interesting from the viewpoint of recent efforts to build circuits based on carbon nanotubes. Recall that recently discovered techniques --- see, e.g., \cite{andriotis:01, papadopoulos:00, terrones:02} --- allow to fabricate branched nanotubes and thus in principle objects very similar to the mentioned ``sleeved graphs''. In this paper we consider a more abstract setting of the problem which covers the ``strip graphs'' of \cite{kuchment-zeng:01, rubinstein-schatzman:01} and their generalizations to higher dimensions, as well as the ``sleeved graphs'' described above. \sloppy Let us briefly describe the structure of the paper. In Section~\ref{sec:prelim} we define the Laplacian on a graph and give an abstract eigenvalue comparision tool (Lemma~\ref{lem:main}). In Section~\ref{sec:graph.mfd} we define the graph like manifolds associated to a graph. In Subsection~\ref{ssec:motivation} we motivate the four different limiting procedures on the vertex neighbourhoods discussed in Sections~\ref{sec:fast.decay}~--~\ref{sec:alpha.null}. In Section~\ref{sec:edge.nbh} we define the limit procedure of the edge neighbourhoods which remain the same in all cases. In the last section (Sec.~\ref{sec:applications} we give an application on the spectral convergence result in the case of periodic graphs. %------------------------------------------------------------ \section{Preliminaries} \label{sec:prelim} %------------------------------------------------------------ %------------------------------------------------------------ \subsection{Laplacian on a graph} %\label{ssec:laplacian.graph} Suppose $\Mnull$ is a finite connected graph with vertices $v_k$, $k \in K$ and edges $e_j$, $j \in J$. Suppose furthermore that $e_j$ has length $\ell_j>0$, i.e., $e_j \cong \Ij:=[0,\ell_j]$. We clearly can make $\Mnull$ into a metric measure space with measure given by $p_j(x) dx$ on the edge $e_j$ where $\map {p_j} {\Ij} {(0,\infty)}$ is a smooth density function for each $j \in J$. We then have % ------------- % \begin{align*} \Lsqr \Mnull &= \bigoplus_{j \in J} \Lsqr {\Ij, p_j(x)dx}\\ \normsqr[\Mnull] u &= \sum_{j \in J} \normsqr[\Ij] {u_j} = \sum_{j \in J} \int_{\Ij} |u_j(x)|^2 p_j(x)dx. \end{align*} % ------------- % We let $\Sob \Mnull$ be the completion of % ------------- % \begin{displaymath} \set{u \in \Cont \Mnull} {u_j := u \restr {e_j} \in \Cont[1]{\Ij}} \end{displaymath} % ------------- % where the closure is taken with respect to the norm % ------------- % \begin{displaymath} \normsqr[1,\Mnull] u := \sum_{j \in J} (\normsqr[\Ij] {u_j} + \normsqr[\Ij]{u_j'}). \end{displaymath} % ------------- % Note that the weakly differentiable functions $\Sob {\Ij}$ on an interval are continuous, therefore $\Sob \Ij \subset \Cont \Ij$. Next we associate with the graph a positive quadratic form, % ------------- % \begin{displaymath} \normsqr[\Mnull] {u'} := \sum_{j \in J} \normsqr[\Ij]{u_j'} \end{displaymath} for all $u \in \Sob \Mnull$. It allows us to define the \emph{(differential) Laplacian on the (weighted) graph $\Mnull$} as the unique self-adjoint and non-negative operator $\laplacian \Mnull$ associated with the closed form $u \mapsto \normsqr[\Mnull] {u'}$ (see \cite[Chapter~VI]{kato:66}, \cite{reed-simon-1} or \cite{davies:96} for details on quadratic forms). In other words, the operator and the quadratic form are related by % ------------- % \begin{equation} \label{eq:graphform} \normsqr[\Mnull] {u'} = \iprod {\laplacian \Mnull u} u \end{equation} % ------------- % for $u \in \Cont[1] \Mnull$ belonging to the domain of $\laplacian \Mnull$. On the edge $e_j$, the operator $\laplacian \Mnull$ is given formally by % ------------- % \begin{equation} \label{eq:formaledge} \laplacian \Mnull u = % - \frac d {dx_j} \bigl( p_j \frac d {dx_j} u_j \bigr). - \frac 1 {p_j(x)}( p_j(x) u_j')'. \end{equation} % ------------- % Note that the domain of $\laplacian \Mnull$ consists of all functions $u \in \Cont \Mnull$ which are twice weakly differentiable on each edge. Furthermore, each function $u$ satisfies (weighted) \emph{Kirchhoff boundary conditions}\footnote{This is the usual terminology, not quite a fortunate one. The name suggests that the probability current at the vertex obeys the conservation law analogous to Kirchhoff's law in an electric circuit. While this claim is valid, the current conservation requirement is equivalent to selfadjointness and thus also satisfied for the other operators mentioned below.} at each vertex $v_k$, i.e., % ------------- % \begin{equation} \label{eq:kirchhoff} \sum_{\text{$j$, $e_j$ meets $v_k$}} p_j(v_k) u_j'(v_k) = 0 \end{equation} % ------------- % for all $k \in K$ where the derivative is taken on each edge in the direction away from the vertex. In particular, we assume Neumann boundary conditions at a vertex with only one edge emanating.\footnote{This hypothesis is made for convenience only and our result will not change if it is replaced by any other boundary condition at the ``loose ends'', in particular, by Dirichlet or $\theta$-periodic ones (cf.~Subsection~\ref{ssec:per.graph}).} If we assume that $p$ is continuous on $\Mnull$, we can omit the factors $p_j(v_k)$ in~\eqref{eq:kirchhoff}. Note that different values of $p_j(v_k)$ for $j$ can correspond in our limiting result to different radii of the thickened edges which are attached to a vertex neighbourhood (see (\ref{eq:def.radius}) below). As we have mentioned in the introduction there are other self-adjoint operators which act according to (\ref{eq:formaledge}) on the graph edges but satisfy different boundary conditions at the vertices --- see \cite{exner-seba:89, kostrykin-schrader:99} for details. The corresponding quadratic forms differ from (\ref{eq:graphform}) by an extra term. In general there are many admissible boundary conditions; a graph vertex joining $n$ edges gives rise to a family with $n^2$ real parameters. An example is represented by the so-called $\delta$ coupling for which the corresponding domain consists of all functions $u \in \Cont \Mnull$ which are twice weakly differentiable on each edge, and (\ref{eq:kirchhoff}) is replaced by % ------------- % \begin{equation} \label{eq:delta} \sum_{\text{$j$, $e_j$ meets $v_k$}} p_j(v_k) u_j'(v_k) = \kappa u(v_k) \end{equation} % ------------- % with a fixed $\kappa\in\mathbb{R}$, where $u(v_k)$ is the common value of all the $u_j(v_k)$ at the vertex. One can ask naturally whether such graph Hamiltonians can be obtained from a family of graph-shaped manifolds. In Section~\ref{sec:borderline} we will discuss a particular case of the limiting procedure leading to the spectrum which --- although it does \emph{not} correspond to a graph operator with the generalized boundary condition described above --- is at least \emph{similar} to that with a $\delta$ coupling. The difference is that in the boundary conditions (\ref{eq:delta}) the coupling constant $\kappa$ is replaced by a quantity dependent on the spectral parameter, the corresponding operator being defined not on $L^2(M_0)$ but on a slightly enlarged Hilbert space --- cf.~(\ref{def:lim.border})--(\ref{def:delta.spectral}). In Section~\ref{sec:slow.decay} we obtain another limit operator due to a different limiting procedure. This operator is again no graph operator with boundary conditions as above, but decouples and the graph part corresponds to a fully decoupled operator with Dirichlet boundary conditons at each vertex. The spectrum of $\laplacian \Mnull$ is purely discrete. We denote the corresponding eigenvalues by $\EW k {\laplacian \Mnull} = \EW k \Mnull$, $k \in \N$, written in the ascending order and repeated according to multiplicity. With this eigenvalue ordering, we can employ the \emph{min-max principle} (in the present form it can be found, e.g., in~\cite{davies:96}): the $k$-th eigenvalue of $\laplacian \Mnull$ is expressed as % ------------- % \begin{equation} \label{eq:max.min} \EW k \Mnull = \inf_{L_k} \sup_{u \in L_k \setminus \{0\} } \frac {\normsqr{q_0(u)}}{\normsqr u} \end{equation} % ------------- % where the infimum is taken over all $k$-dimensional subspaces $L_k$ of $\Sob \Mnull$. %------------------------------------------------------------ \subsection{Comparison of eigenvalues} %\label{ssec:eigenvalues} Let us now formulate a simple consequence of the min-max principle which will be crucial for the proof of our main results. Suppose that $\HS$, $\HS'$ are two separable Hilbert spaces with the norms $\norm \cdot$ and $\norm \cdot '$. We need to compare eigenvalues $\lambda_k$ and $\lambda'_k$ of non-negative operators $Q$ and $Q'$ with purely discrete spectra defined via quadratic forms $q$ and $q'$ on $\mathcal D \subset \HS$ and $\mathcal D' \subset \HS$. We set $\normsqr[Q,n] u := \normsqr u + \normsqr {Q^{n/2}u}$. % ------------- % \begin{lem} \label{lem:main} Suppose that $\map \Phi {\mathcal D}{\mathcal D'}$ is a linear map such that there exist constants $n_1, n_2 \ge 0$ and $\delta_1, \delta_2 \ge 0$ such that % ------------- % \begin{align} \label{eq:est.norm} \normsqr u & \le {\norm{\Phi u}'}^2 + \delta_1 \normsqr[Q,n_1] u\\ \label{eq:est.quad.form} q(u) & \ge \, q'(\Phi u) - \delta_2 \normsqr[Q,n_2] u \end{align} % ------------- % for all $u \in \mathcal D$ and that $\mathcal D \subset \dom Q^{\max\{n_1,n_2\}/2}$. Then to each $k$ there is a positive function $\eta_k$ given by~\eqref{eq:eta.k} satisfying $\eta_k:=\eta(\lambda_k, \delta_1, \delta_2) \to 0$ as $\delta_1, \delta_2 \to 0$, such that % ------------- % \begin{displaymath} \lambda_k \ge \lambda_k' - \eta_k. \end{displaymath} % ------------- % \end{lem} % ------------- % \begin{proof} Let $\phi_1, \dots, \phi_k$ be an orthonormal system of eigenvectors corresponding to the eigenvalues $\lambda_1, \dots, \lambda_k$. For $u$ in the linear span $E_k$ of $\phi_1, \dots, \phi_k$, we have % ------------- % \begin{equation} \label{eq:est.ev} \normsqr[Q,n] u \le (1+\lambda_k^n) \normsqr u \end{equation} % ------------- % and % ------------- % \begin{multline} \label{eq:est.rayleigh} \frac{q'(\Phi u)}{{\norm {\Phi u}'}^2} - \frac{q(u)}{\normsqr u} = \frac{q(u)}{\normsqr u} \frac{\normsqr u - {\norm {\Phi u}'}^2} {{\norm {\Phi u}'}^2} + \frac{q'(\Phi u) - q(u)} {{\norm {\Phi u}'}^2}\\ \le \left( \frac{q(u)}{\normsqr u} \delta_1 \normsqr[Q,n_1] u + \delta_2 \normsqr[Q,n_2] u \right) \frac 1 {{\norm {\Phi u}'}^2}\\ \le (\lambda_k(1+\lambda_k^{n_1}) \delta_1 + (1+\lambda_k^{n_2})\delta_2) \frac {\normsqr u}{{\norm {\Phi u}'}^2} \end{multline} % ------------- % where we have used \eqref{eq:est.norm} and \eqref{eq:est.quad.form} to get the first inequality and \eqref{eq:est.ev} to get the second one. From relation \eqref{eq:est.norm} we follow % ------------- % \begin{equation} \label{eq:est.norm2} (1-(1+\lambda_k^{n_1})\delta_1) \normsqr u \le {\norm {\Phi u}'}^2 \end{equation} % ------------- % and thus we can estimate the r.h.s.\ of \eqref{eq:est.rayleigh} by % ------------- % \begin{equation} \label{eq:eta.k} \eta_k:= \eta(\lambda_k, \delta_1, \delta_2) := \frac {\lambda_k(1+\lambda_k^{n_1}) \delta_1 + (1+\lambda_k^{n_2})\delta_2} {1-(1+\lambda_k^{n_1}) \delta_1} \end{equation} % ------------- % provided $0 \le \delta_1 < 1/(1+\lambda_k^{n_1})$. From \eqref{eq:est.norm2} we also conclude that $\norm u = 0$ holds if $\norm {\Phi u}' = 0$, i.e., that $\Phi(E_k)$ is $k$-dimensional. From the min-max principle applied to the quadratic form $q'$ we obtain % ------------- % \begin{displaymath} \lambda_k' \le \sup_{u \in E_k \setminus \{0\}}\frac{q'(\Phi u)}{{\norm{\Phi u}'}^2} \le \sup_{u \in E_k \setminus \{0\}}\frac{q(u)}{\normsqr u} + \eta_k = \lambda_k + \eta_k \end{displaymath} % ------------- % which is the desired result. \end{proof} %------------------------------------------------------------ \section{Graph-like manifolds} \label{sec:graph.mfd} %------------------------------------------------------------ %------------------------------------------------------------ \subsection{Laplacian on a manifold} %\label{ssec:laplacian.mfd} Throughout this paper we study manifolds of dimension $d \ge 2$. For a Riemannian manifold $X$ (compact or not) without boundary we denote by $\Lsqr X$ the usual $L_2$-space of square integrable functions on $X$ with respect to the volume measure $dX$ on $X$. In a chart, the volume measure has the density $(\det G)^{1/2}$ with respect to the Lebesgue measure, where $\det G$ is the determinant of the metric tensor $G:=(g_{ij})$ in this chart. The norm of $\Lsqr X$ will be denoted by $\norm[X] \cdot$. For $u \in \Cci X$, the space of compactly supported smooth functions, we set % ------------- % \begin{displaymath} \check q_X(u):=\normsqr[X] {d u} = \int_{X} |d u|^2 dX. \end{displaymath} % ------------- % Here the $1$-form $d u$ denotes the exterior derivative of $u$ whose squared norm in coordinates is given by % ------------- % \begin{displaymath} |d u |^2= \sum_{i,j} g^{ij} \partial_i u \, \partial_j \overline u = G^{-1} \nabla u \cdot \nabla \overline u \end{displaymath} % ------------- % where $(g^{ij})$ is the component representation of the inverse matrix $G^{-1}$. We denote the closure of the non-negative quadratic form $\check q_X$ by $q_X$. Note that the domain $\dom q_X$ of the closed quadratic form $q_X$ consists of functions in $L_2(X)$ such that the weak derivative $d u$ is also square integrable, i.e., $q_X(u) < \infty$. We define the \emph{Laplacian} $\laplacian X$ (for a manifold without boundary) as the unique self-adjoint and non-negative operator associated with the closed quadratic form $q_X$, i.e., the operator and the quadratic form are related by % ------------- % \begin{displaymath} q_X(u)=\iprod {\laplacian X u} u \end{displaymath} % ------------- % for $u \in \Cci X$ (see again \ \cite[Chapter~VI]{kato:66}, \cite{reed-simon-1} or \cite{davies:96}). Thus the Laplacian action on smooth functions $u$ is given in a fixed chart by % ------------- % \begin{equation} \label{eq:lapl.local} \laplacian X u = -(\det G)^{-1/2} \sum_{i,j} \partial_i \bigl( (\det G)^{1/2} g^{ij} \, \partial_j u \bigr). \end{equation} % ------------- % If $X$ is a compact manifold with piecewise smooth boundary $\bd X \ne \emptyset$ we can define the Laplacian with Neumann boundary condition via the closure $q_X$ of the quadratic form $\check q_X$ defined on $\Ci X$, the space of smooth functions with derivatives continuous up to the boundary of $X$. Note that the usual conditions on the normal derivative occurs only in the \emph{operator} domain via the Gauss-Green formula. In a similar way other boundary conditions at $\bd X$ may be introduced. The spectrum of $\laplacian X$ (with any boundary condition if $\bd X \ne \emptyset$) is purely discrete as long as $X$ is compact and the boundary conditions are local. We denote the corresponding eigenvalues by $\EW k {\laplacian X} = \EW k X$, $k \in \N$, written in increasing order and repeated according to multiplicity. %------------------------------------------------------------ \subsection{General estimates on manifolds} %\label{ssec:gen.est} We will employ (partial) averaging processes on edge and vertex neighbourhoods which correspond to projection onto the lowest (transverse) mode. We start with such a general Poincar\'{e}-type estimate: % ------------- % \begin{lem} \label{lem:minmax.2nd.neu} Let $X$ be a connected, compact manifold with smooth boundary $\bd X$. For $u \in \Sob X$ define the constant function $u_0(x) := \frac 1 {\vol X} \int_X u\, dX$. Then we have $\normsqr[X]{u_0} \le \normsqr[X] u$, % ------------- % \begin{displaymath} \normsqr[X] {u - u_0} \le \frac 1 {\EWN 2 X} \normsqr[X] {du} \qquad \text{and} \qquad \normsqr[X] u - \normsqr[X] {u_0} \le \frac 1 {\delta \EWN 2 X} + \delta \normsqr[X] u \end{displaymath} % ------------- % for $\delta>0$. \end{lem} % ------------- % \begin{proof} The first inequality follows directly from Cauchy-Schwarz. For the second one, note that $u - u_0$ is orthogonal to the first eigenfunction of the Neumann Laplacian. By the min-max principle we obtain % ------------- % \begin{displaymath} \EWN 2 X \normsqr[X] {u - u_0} \le \normsqr[X]{d(u-u_0)} = \normsqr[X]{du}, \end{displaymath} % ------------- % because $u_0$ is constant on $X$. Since $X$ is connected, we have $\EWN 2 X>0$. The last inequality follows from \begin{equation} \label{ineq:norm} | \normsqr u -\normsqr {u_0}| \le 2 \norm {u - u_0} \norm u \le \frac 1 \delta \normsqr{u - u_0} + \delta \normsqr u \end{equation} for all $\delta>0$. \end{proof} Next, we often need the following continuity of the map which restricts a function on $X$ to the boundary $\bd X$. To this aim we use standard Sobolev embedding theorems: % ------------- % \begin{lem} \label{lem:rest.est} There exists a constant $c_1>0$ depending only on $X$ and the metric $g$ such that % ------------- % \begin{displaymath} \normsqr[\bd X] {u \restr {\bd X}} \le c_1(\normsqr[X] u + \normsqr[X] {d u}) \end{displaymath} % ------------- % for all $u \in \Sob U$. \end{lem} % ------------- % \begin{proof} See e.g. \cite[Ch.~4, Prop.~4.5]{taylor:96}. An alternative proof similar to the proof of Lemma~\ref{lem:poincare} exists, and follows easily from~\eqref{eq:rest.est} together with a cut-off function. \end{proof} %------------------------------------------------------------ \subsection{Definition of the graph-like manifold} %\label{ssec:def.mfd} For each $0 < \eps \le 1$ we associate with the graph $\Mnull$ a compact and % ------------- % \begin{figure}[h] \begin{center} %------------------------------------------------------------ % \input{edge.vertex.pstex_t} \begin{picture}(0,0)% \includegraphics{graph1.pstex}% \end{picture}% \setlength{\unitlength}{4144sp}% \begin{picture}(5282,2500)(374,-1970) \put(1591,-586){$\Uepsj$}% \put(1561,-1674){$e_j$}% \put(5641,-331){$\Vepsk$}% \put(5416,-1771){$v_k$}% \put(4579,374){$\eps$}% \put(3169,-436){$\eps$}% \end{picture} %------------------------------------------------------------ \caption{The associated edge and vertex neighbourhoods with $F=\Sphere^1$, i.e., $\Uepsj$ and $\Vepsk$ are $2$-dimensional manifolds with boundary.} \label{fig:edge.vertex} \end{center} \end{figure} % ------------- % connected Riemannian manifold $\Meps$ of dimension $d \ge 2$ equipped with a metric $g_\eps$ to be specified below. We suppose that $\Meps$ is the union of compact subsets $\Uepsj$ and $\Vepsk$ such that the interiors of $\Uepsj$ and $\Vepsk$ are mutually disjoint for all possible combinations of $j \in J$ and $k \in K$. We think of $\Uepsj$ as the thickened edge $e_j$ and of $\Vepsk$ as the thickened vertex $v_k$ (see Figures~\ref{fig:edge.vertex} and~\ref{fig:mfd}). Note that the second picture describes the situation only rougly, since it assumes that $\Meps$ is embedded in $\R^\nu$. More correctly, we should think of $\Meps$ as an abstract manifold obtained by identifying the appropriate boundary parts of $\Uepsj$ and $\Vepsk$ via the connection rules of the graph $M_0$. % ------------- % \begin{figure}[h] \begin{center} %------------------------------------------------------------ % \input{mfd.pstex_t} \begin{picture}(0,0)% \includegraphics{graph2.pstex}% \end{picture}% \setlength{\unitlength}{4144sp}% \begin{picture}(6263,2015)(308,-1370) \put(6571,-601){$\Vepsk$}% \put(5581,-1186){$\Uepsj$}% \put(3601,-1231){$\Meps$}% \put(2926,-736){$v_k$}% \put(316,-1231){$\Mnull$}% \put(2206,-1006){$e_j$}% \end{picture} %------------------------------------------------------------ \caption{On the left, we have the graph $\Mnull$, on the right, the associated graph-like manifold (in this case, $F=\Sphere^1$ and $\Meps$ is a $2$-dimensional manifold).} \label{fig:mfd} \end{center} \end{figure} % ------------- % This manifold need not to be embedded, but the situation when $\Meps$ is a submanifold of $\R^\nu$ ($\nu \ge d$) can be viewed also in this abstract context (see Remark~\ref{rem:why.2.metrics}). As a matter of convenience we assume that $\Uepsj$ and $\Vepsk$ are independent of $\eps$ as manifolds, i.e., only their metric $g_\eps$ depend on $\eps$. This can be achieved in the following way: for the edge regions we assume that $\Uepsj$ is diffeomorphic to $\Ij \times F$ for all $0 < \eps \le 1$ where $F$ denotes a compact and connected manifold (with or without a boundary) of dimension $m:=d-1$. For the vertex regions we assume that the manifold $\Vepsk$ is diffeomorphic to a $\eps$-independent manifold $\Vk$ for $0 < \eps \le 1$. Pulling back the metrics to the diffeomorphic manifold we may assume that the underlying differentiable manifold is independent of $\eps$. Therefore, $\Uepsj = \Uj = \Ij \times F$ and $\Vepsk = \Vk$ with an $\eps$-depending metric $g_\eps$. For further purposes, we need a decomposition of $e_j \cong \Ij$ into two halves. We reverse the orientation of one such half so that each half is directed away from its adjacent vertex and collect all halves $\Ijk$ ending at the vertex $v_k$, i.e., $j \in J_k$, where % ------------- % \begin{equation} \label{eq:def.Jk} J_k := \set{j \in J}{\text{$e_j$ meets $v_k$}}\footnote{For each loop $e_j$ at $v_k$, i.e., each edge beginning \emph{and} ending at $v_k$, we need to replace the label $j$ by two distinct labels $j_1,j_2$ belonging to $J_k$ in order to collect \emph{both} halfs of the edge.} \end{equation} % ------------- % We denote $\Ujk := \Ijk \times F$ (and similar notation with subscript $\eps$). For further references, we denote the midpoint of the edge $e_j \cong \Ij$ by $x_j^*$ and the endpoint of $\Ij$ corresponding to the edge $v_k$ by $x_{jk}^0$, e.g., $\Ijk = [x_j^*, x_{jk}^0]$. %------------------------------------------------------------ \subsection{Notation} In the sequel, we are going to suppress the edge and vertex subscripts $j$ and $k$ unless a misunderstanding may occur. Similarly we set, e.g., $U:= U_1$, in other words we omit the subscript $\eps$ if we only mean the underlying $\eps$-independent manifold with metric $g_1$, i.e., if we fix $\eps = 1$. %------------------------------------------------------------ \subsection{Motivation for the different limit operators} \label{ssec:motivation} Let us shortly motivate why the limit operator of $\laplacian \Meps$ as $\eps \to 0$ should depend on the volume decay of the vertex neighbourhoods $\Vepsk$ in comparison with $\vol_{d-1} \bd \Vepsk$ (or $\vol_d \Uepsj$, which is of the same order when $\eps \to 0$ as we will see in Section~\ref{sec:edge.nbh}). For simplicity, we assume that the radius of the transversal direction on the edge $\Uepsj$ is $\eps$ (i.e., $p_j \equiv 1$). The assumptions on the edge neighbourhoods will be specified in the next section. We stress that our aim in this subsection is to present a heuristic idea, not a proof (for a suitable reasoning cf.\ \cite{ruedenberg-scherr:53} or \cite{kuchment:02}). Suppose $\phi=\phi_\eps$ is an eigenfunction of $\laplacian \Meps$ w.r.t the eigenvalue $\lambda=\lambda_\eps$. By the Gauss-Green formula, we have at the vertex $\Veps=\Vepsk$ \begin{equation} \label{eq:gauss-green} \lambda \int_\Veps \phi \, \overline u \, d\Veps = \int_\Veps \iprod {d\phi}{du} d\Veps + \int_{\bd \Veps} \partial_\mathrm{n} \phi \, \overline u \, d\partial \Veps \end{equation} for all $u \in \Sob \Meps$. Assume that $\lambda_\eps \to \lambda_0$ and $\phi_\eps \to \phi_0=(\phi_{0,j})_j$. If the vertex volume $\vol_d \Veps$ decays faster then the boundary area $\vol_{d-1} \bd \Veps$ only the boundary integral over $\bd \Veps$ survives in the limit $\eps \to 0$ and leads to \begin{displaymath} 0=\sum_{j \in J_k} \phi_{0,j}'(v_k) \end{displaymath} which is exactly the Kirchhoff boundary condition mentioned above in~\eqref{eq:kirchhoff}. This \emph{fast decaying} vertex volume case will be treated in Section~\ref{sec:fast.decay}. If the vertex volume decays slower than $\vol_{d-1} \bd \Veps$, the integrals over $\Veps$ are dominant. In this case, $\vol \Vepsk \gg \vol \Uepsj$ and only slowly varying eigenfunctions on $\Vepsk$ lead to bounded eigenvalues $\lambda=\lambda_\eps$. Since $\vol \Vepsk \gg \vol \Uepsj$, normalized eigenfunctions are nearly vanishing on $\Vepsk$ viewed from the scale on $\Uepsj$. This roughly explains, why we end up in a decoupled operator with Dirichlet boundary conditions on $\Mnull$ plus extra zero eigenmodes at the vertices (the zero eigenmodes also survive the limit $\eps \to 0$). This \emph{slow decaying} vertex volume case will be shortly be discussed in Section~\ref{sec:slow.decay}. In the borderline case when $\vol_d \Veps \approx \vol_{d-1} \bd \Veps$, we also expect the eigenfunctions to vary slowly on $\Vepsk$ (since $\vol_d \Vepsk \to 0$), so the integral over $\iprod {d\phi}{du}$ should tend to $0$, and in the limit \begin{displaymath} \lambda_0 \phi_0(v_k) = \sum_{j \in J_k} \phi_{0,j}'(v_k). \end{displaymath} This \emph{borderline case} will be treated in Section~\ref{sec:borderline}. If $\vol \Vepsk$ does not tend to $0$, i.e., when $\Vepsk$ tends to a compact $d$-dimensional manifold $\Vnullk$ without boundary (and \emph{not} to a point as in the cases above), we still expect a decoupled operator with Dirichlet boundary conditions on the edges by the same arguments as in the slow decaying case. In addition, not only the lowest eigenmode of $\Vepsk$ but all eigenmodes survive, i.e., the limit operator should consist of the direct sum of all Dirichlet Laplacians on the edges plus the Laplacians on $\Vnullk$, $k \in K$. This \emph{non-decaying} vertex volume case will be treated in Section~\ref{sec:alpha.null}. It needs an extra effort to prove rigorously the conclusions of the above reasoning; recall that we have assumed e.g.\ that $\lambda_\eps \to \lambda_0$ (which we want to show in this paper) and $\norm[\infty]{\phi_\eps},\norm[\infty]{d \phi_\eps} \le c$ which is in general not true for normalized ($L_2$-)eigenfunctions since $\vol \Meps \to 0$ as $\eps \to 0$. %------------------------------------------------------------ \section{Edge neighbourhoods} \label{sec:edge.nbh} %------------------------------------------------------------ %------------------------------------------------------------ \subsection{Definition of the thickened edges} %\label{ssec:def.edge} Suppose that $U=I \times F$ with metric $g_\eps$, where $I$ corresponds to some (part of an) edge and $F$ denotes (as before) a compact and connected Riemannian manifold of dimension $m=d-1$ with metric $h$. Without loss of generality we may assume that $\vol F = 1$. We define another metric $\tilde g_\eps$ on $\Ueps$ by % ------------- % \begin{equation} \label{eq:def.met.edge} \tilde g_\eps := dx^2 + \eps^2 r_j^2(x) \, h(y), \qquad (x,y) \in \Uj = \Ij \times F \end{equation} % ------------- % where \begin{equation} \label{eq:def.radius} r_j(x):=(p_j(x))^{1/m} \end{equation} defines a smooth function (specifying the radius of the fibre $\{x\} \times F$ at the point $x$), where $p_j$ is the density function on the edge $e_j$ introduced in Section~\ref{sec:prelim}. We denote by $G_\eps$ and $\tilde G_\eps$ the $d \times d$-matrices associated to the metrics $g_\eps$ and $\tilde g_\eps$ with respect to the coordinates $(x,y)$ (here $y$ stands for suitable coordinates on $F$) and assume that the two metrics coincide up to an error term as $\eps\to 0$, more specifically % ------------- % \begin{equation} \label{eq:asym.met.edge} G_\eps = \tilde G_\eps + \begin{pmatrix} o(1) & o(\eps) \\ o(\eps) & o(\eps^2) \end{pmatrix} = \begin{pmatrix} 1 + o(1) & o(\eps) \\ o(\eps) & \eps^2 r_j \, H + o(\eps^2) \end{pmatrix}, \end{equation} % ------------- % i.e., % ------------- % \begin{displaymath} g_{\eps,xx} = 1 + o(1), \quad g_{\eps,xy_\alpha} = o(\eps), \quad g_{\eps,y_\alpha y_\beta} = \eps^2 r_j^2(x) \, h_{\alpha\beta}(y) + o(\eps^2). \end{displaymath} uniformly on $U$. To summarize, we assume that the metric $g_\eps$ is equal to the product metric $\tilde g_\eps$ up to error terms. %------------------------------------------------------------ %\subsection{rem} \begin{rem} \label{rem:why.2.metrics} This is a central assumption in our construction which describes how in fact the family of manifolds shrinks to the graph $M_0$. One of the reasons why we introduce a pair of metrics is the following. While our construction uses intrinsic metric properties of the manifolds only, we want it to be applicable to manifolds embedded into some Euclidean space (e.g.\ in $\R^{d+1}$ if $F=\Sphere^1$). This is impossible if the cylindrical sleeves have the same length as the underlying graph edges, but it can be achieved with the length modified by a factor of $o(1)$. It will be one of our aims to show that within the prescribed error margin such a ``practically important'' metric yields the same result as the product metric which is easier to handle. \end{rem} In the next two examples we illustrate that a suitable smooth $\eps$-neighbourhood of an embedded graph in $\R^\nu$ ($\nu \ge d$) as well as an embedded \emph{curved} edge is covered by the above abtract setting of a pure product metric and an ``almost'' product metric. %------------------------------------------------------------ \begin{exmp} \label{ex:embedded} \textbf{Embedded graphs.} If we set $\nu=d=2$ and $F=[-1,1]$ (or more generally $\nu=d \ge 2$, $F:=\set{x \in \R^m}{|x| \le 1}$) and if $\Meps$ is a suitable $\eps$-neighbourhood of $\Mnull$, we recover the situation treated in \cite{kuchment-zeng:01}. Note that the error term $o(1)$ in the first component in~\eqref{eq:asym.met.edge} allows us to start with a graph $\Mnull$ embedded in some Euclidean space $\R^\nu$ as we have already mentioned in Remark~\ref{rem:why.2.metrics}. The edge neighbourhoods $\Uepsj$ of the embedded manifold $\Meps$ have a length \emph{smaller} than the length of the corresponding edge, with the error term $o(1)$. By a simple transformation of the variable $x$ we return to our setting with a fixed length of $\Uj = \Ij \times F$. \end{exmp} \begin{exmp} \label{ex:curved} \textbf{Curved edges and variable transversal radius.} Suppose $\Ueps$ is the $\eps$-neighbourhood of a smooth curve $\map{\vec \gamma=\vec \gamma_j}{\Ij}{\R^d}$ parametrized by arc-length. If, e.g., $\nu=d=2$ and $F=[-1,1]$ a chart is given by % ------------- % \begin{displaymath} \map \Psi {\Ij \times [-1,1]} \Uepsj, \qquad (x,y) \mapsto \vec \gamma(x) + \eps r_j(x) y \, \vec n(x), \end{displaymath} % ------------- % i.e., we thicken the curve $\vec \gamma$ in its normal direction $\vec n(x)$ at the point $\vec \gamma(x)$ by the factor $\eps r(x) = \eps r_j(x)$. The corresponding metric in $(x,y)$-coordinates is given by % ------------- % \begin{displaymath} G_\eps = \begin{pmatrix} (1+ \eps \kappa y r)^2 + \eps^2 y^2\dot r^2 & \eps^2 r \dot r y\\ \eps^2 r \dot r y & \eps^2 r^2 \end{pmatrix} = \begin{pmatrix} 1+O(\eps) & O(\eps^2) \\ O(\eps^2) & \eps^2 r^2 \end{pmatrix} \end{displaymath} % ------------- % where $\kappa:=\dot \gamma_1 \ddot \gamma_2 - \dot \gamma_2 \ddot \gamma_2$ is the curvature of the generating curve $\vec \gamma$. Therefore, the error term $o(1)$ comes from the curvature of the embedded curve $\vec \gamma$ whereas the off-diagonal error terms come from the variable radius of the transversal direction (note that $\dot r=0$ if $r(x)$ is constant). %% only needed, when explicit error estimates are wanted: % Furthermore, the metric density satisfies \begin{displaymath} \det G_\eps^{\frac12} = |1+\eps \kappa y r| \eps r = (1 + O(\eps)) \det \tilde G_\eps^{\frac12}. \end{displaymath} We give a general treatment in the next lemma. \end{exmp} % ------------- % end of the examples %------------------------------------------------------------ \subsection{Estimates on the thickened edges} %\label{ssec:est.edges} Following the philosophy explained in Remark~\ref{rem:why.2.metrics}, we start with pointwise estimates where we compare the product metric $\tilde g_\eps$ with the original metric $g_\eps$. Note that the assumption~\eqref{eq:asym.met.edge}, while fully sufficient for our purposes, is optimal in a sense, i.e., that the following lemma ceases to be valid if we weaken its hypothesis even slightly. % ------------- % \begin{lem} \label{lem:metric} Suppose that $g_\eps$, $\tilde g_\eps$ are given as in~\eqref{eq:def.met.edge} and~\eqref{eq:asym.met.edge}, then % ------------- % \begin{align} \label{eq:met.vol} (\det G_\eps)^\frac12 &= (1 + o(1)) (\det \tilde G_\eps)^\frac12 \\ \label{eq:met.1st.comp} g_\eps^{xx} & \!:= (G_\eps^{-1})_{xx} = 1 + o(1) \\ \label{eq:met.1st.der} |d_x u|^2 &\le (1+ o(1)) |d u|_{g_\eps}^2\\ \label{eq:met.2nd.der} |d_F u|_h^2 &\le o(\eps) |d u|_{g_\eps}^2 \end{align} % ------------- % where $d_x$ and \ $d_F$ are the (exterior) derivative with respect to $x \in I$ and \ $y \in F$, respectively. All the estimates are uniform in $(x,y)$ as $\eps \to 0$. \end{lem} \begin{proof} The first equation follows from % ------------- % \begin{displaymath} \begin{split} \det(G_\eps \tilde G_\eps^{-1}) &= \det \begin{pmatrix} 1 + o(1) & o(\eps)\\ o(\eps) & \eps^2 H + o(\eps^2) \end{pmatrix} \begin{pmatrix} 1 & 0\\ 0 & \eps^{-2} H^{-1} \end{pmatrix}\\ &= \det \begin{pmatrix} 1 + o(1) & o(\eps^{-1})\\ o(\eps) & \one + o(1) \end{pmatrix} = 1 + o(1). \end{split} \end{displaymath} % ------------- % For the second one, we consider the upper left component of % ------------- % \begin{displaymath} \begin{split} G_\eps^{-1} - \tilde G_\eps^{-1} &= -\tilde G_\eps^{-1} (G_\eps - \tilde G_\eps) \tilde G_\eps^{-1} + o(G_\eps - \tilde G_\eps)\\ &= \begin{pmatrix} 1 & 0 \\ 0 & O(\eps^{-2}) \end{pmatrix} \begin{pmatrix} o(1) & o(\eps) \\ o(\eps) & o(\eps^2) \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & O(\eps^{-2}) \end{pmatrix} + o(1) \\ &= \begin{pmatrix} o(1) & o(\eps^{-1}) \\ o(\eps^{-1}) & o(\eps^{-2}) \end{pmatrix}. \end{split} \end{displaymath} % ------------- % Inequality~\eqref{eq:met.1st.der} is equivalent to % ------------- % \begin{displaymath} \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix} \le (1+ o(1)) G_\eps^{-1} \end{displaymath} % ------------- % in the sense of quadratic forms. This will be true if we show that % ------------- % \begin{displaymath} \begin{pmatrix} 1 & 0 \\ 0 & \delta \one \end{pmatrix} \le (1 + o(1)) G_\eps^{-1} \end{displaymath} % ------------- % for some $\delta>0$, where $\one$ is the $m\times m$ unit matrix, which in turn means % ------------- % \begin{displaymath} (1+ o(1)) \begin{pmatrix} 1 & 0 \\ 0 & \delta^{-1} \one \end{pmatrix} \ge G_\eps. \end{displaymath} % ------------- % However, % ------------- % \begin{displaymath} G_\eps = \tilde G_\eps + \begin{pmatrix} o(1) & o(\eps) \\ o(\eps) & o(\eps^2) \end{pmatrix} = \begin{pmatrix} 1 + o(1) & 0 \\ 0 & O(\eps^2) \end{pmatrix} + \begin{pmatrix} 0 & o(\eps) \\ o(\eps) & 0 \end{pmatrix} \end{displaymath} % ------------- % and the eigenvalues of the last matrix are of order $o(\eps)$, so % ------------- % \begin{displaymath} G_\eps \le \begin{pmatrix} 1 + o(1) & 0 \\ 0 & O(\eps^2) \end{pmatrix} + o(\eps) \one = \begin{pmatrix} 1 + o(1) & 0 \\ 0 & o(\eps) \end{pmatrix} \le (1 + o(1)) \begin{pmatrix} 1 & 0 \\ 0 & c \one \end{pmatrix} \end{displaymath} % ------------- % for some constant $c>0$, and therefore it is sufficient to choose $\delta0$ such that % ------------- % \begin{displaymath} c_- \eps^2 g(x)(v,v) \le g_\eps(x)(v,v) \le c_+ \eps^{2\alpha} g(x)(v,v) \end{displaymath} % ------------- % for all $v \in T_x \Vk$ and all $x \in \Vk$. The number $\alpha$ in the exponent is assumed to satisfy the inequalities % ------------- % \begin{equation} \label{eq:est.alpha.fast} \frac {d-1} d < \alpha \le 1\,; \end{equation} % ------------- % notice that $\alpha \le 1$ is needed for \eqref{eq:met.vertex} to make sense with $0<\eps\le 1$. Thus the edge and vertex parts of the manifold need not shrink at the same rate but the vertex shrinking should not be too slower than that of the edges. This hypothesis expressed by (\ref{eq:met.vertex}) plays a central role here; other shrinking regimes will be discussed in the following sections. Note that the manifold $\Vepsk$ shrinks at most as $\eps$ (in each direction) by the lower bound in~\eqref{eq:met.vertex}. This ensures that a global \emph{smooth} metric $g_\eps$ exists on $\Meps$ with the requirements on $\Uepsj$ and $\Vepsk$. Therefore, we do not need an intermediate part (called \emph{bottle neck}) between the edge and vertex neighbourhoods interpolating between the different scalings as in Sections~\ref{sec:slow.decay} and~\ref{sec:borderline} (see also Remark~\ref{rem:alpha.bigger.1}). We easily obtain the following global estimates: % ------------- % \begin{lem} \label{lem:est.norm.quad} There are $c_1^\pm, c_2^\pm > 0$ such that % ------------- % \begin{align} \label{eq:est.vert.norm} c_1^- \eps^d \normsqr[V] u & \le \normsqr[\Veps] u \le c_1^+ \eps^{\alpha d} \normsqr[V] u \\ \label{eq:est.vert.quad} c_2^- \eps^{d-2\alpha} \normsqr[V] {du} & \le \normsqr[\Veps] {du} \le c_2^+ \eps^{\alpha d-2} \normsqr[V] {du} \end{align} % ------------- % for all $u \in \Sob \Veps = \Sob V$. \end{lem} % ------------- % \begin{proof} Using assumption~\eqref{eq:met.vertex} we obtain % ------------- % \begin{displaymath} c_-^{d/2} \eps^d (\det G)^{1/2} \le (\det G_\eps)^{1/2} \le c_+^{d/2} \eps^{\alpha d} (\det G)^{1/2} \end{displaymath} % ------------- % and % ------------- % \begin{displaymath} c_+^{-1} \eps^{-2 \alpha} G^{-1} \le G_\eps^{-1} \le c_-^{-1} \eps^{-2} G^{-1} \end{displaymath} % ------------- % which implies the result with $c_1^\pm:= c_\pm^{d/2}$ and $c_2^\pm:= c_\pm^{d/2} c_\mp^{-1}$. \end{proof} %------------------------------------------------------------ \subsection{Convergence of the spectra} The limit operator will concentrate only on the edge part in this case, therefore we define % ------------- % \begin{equation} \label{eq:lim.slow} \HS_0 := \Lsqr \Mnull, \qquad \mathcal D_0 := \Sob \Mnull, \qquad q_0(u):= \normsqr[\Mnull] {u'} = \sum_j \normsqr[\Ij] {u_j'}, \end{equation} % ------------- % i.e., the limit operator $Q_0$ is $\laplacian \Mnull$ (see Def.~\eqref{eq:formaledge}). With the above preliminaries we can finally formulate the main result of this section: % ------------- % \begin{thm} \label{thm:ev.conv} Under the stated assumptions $\EW k \Meps \to \EW k \Mnull$ as $\eps\to 0$. \end{thm} % ------------- % \noindent Recall that the eigenvalues $\EW k \Meps$ are by assumption ordered in the ascending order, multiplicity taken into account, so the label of a particular eigenvalue curve may change as $\eps$ moves. The spectrum of the manifold is in general richer than that of the graph and a part of the eigenvalues escapes to $+\infty$ as $\eps\to 0$; the proof presented below shows that this happens, roughly speaking, for all states with the transverse part of the eigenfunction orthogonal to the ground state. Our aim is to find a two sided estimate on each eigenvalue $\EW k \Meps$ by means of $\EW k \Mnull$ with an error which is $o(1)$ w.r.t.\ the parameter $\eps$. %------------------------------------------------------------ \subsection{An upper bound} %\label{ssec:up.est} The mentioned upper eigenvalue estimate now reads as follows: % ------------- % \begin{thm} \label{thm:ev.above} $\EW k \Meps \le \EW k \Mnull + o(1)$ holds as $\eps\to 0$. \end{thm} % ------------- % \noindent To prove it, we define the transition operator by % ------------- % \begin{equation} \label{eq:trans.op.above} \Phi_\eps u (z):= \begin{cases} \eps^{-m/2} u(v_k) & \text{if $z \in \Vk$},\\ \eps^{-m/2} u_j(x) & \text{if $z=(x,y) \in \Uj$} \end{cases} \end{equation} % ------------- % for any $u \in \Sob \Mnull$. Theorem~\ref{thm:ev.above} is then implied by Lemma~\ref{lem:main} in combination with the following result. % ------------- % \begin{lem} \label{lem:above} We have $\Phi_\eps u \in \Sob \Meps$, i.e., $\Phi_\eps$ maps the quadratic form domain of the Laplacian on the graph into the quadratic form domain of the Laplacian on the manifold. Furthermore, for $u \in \Sob \Mnull$ we have % ------------- % \begin{align} \normsqr[\Mnull] u - \normsqr[\Meps] {\Phi_\eps u} &\le o(1) \, \normsqr[\Mnull] u\\ \normsqr[\Meps] {d \, \Phi_\eps u} - q_0(u) & = o(1) \, q_0(u). \end{align} % ------------- % \end{lem} % ------------- % \begin{proof} The first assertion is true since $\Phi_\eps u$ is constant on each thickened vertex $\Vepsk$ and continuous on $\bd \Vepsk$. Clearly, $\Phi_\eps u$ is weakly differentiable on each thickened edge $\Uepsj$. Moreover, we have % ------------- % \begin{multline*} \normsqr[\Mnull] u - \normsqr[\Meps] {\Phi_\eps u} \le \sum_{j \in J} (\normsqr[\Ij] u - \normsqr[\Uepsj] {\Phi_\eps u}) \\ = \sum_{j \in J} (\normsqr[\Ij] u - (1+o(1)) \normsqr[\tUepsj] {\Phi_\eps u})= o(1) \sum_{j \in J} \normsqr[\Ij] u = o(1) \normsqr[\Mnull] u \end{multline*} % ------------- % where we have neglected the contribution to the norm of $\Phi_\eps u$ from the vertex parts of $\Meps$ and employed eqs.~\eqref{eq:met.vol} and~\eqref{eq:ind.of.2nd}. The second relation follows from % ------------- % \begin{multline*} \normsqr[\Meps] {d \, \Phi_\eps u} - q_0(u) = \sum_{j \in J} ((1+o(1))\normsqr[\tUepsj]{g_\eps^{xx} d_x \Phi_\eps u} - \normsqr[\Ij]{u'})\\ = \sum_{j \in J} ((1+o(1))\normsqr[\Ij] {u'} - \normsqr[\Ij]{u'}) = o(1) \, q_0(u) \end{multline*} % ------------- % in the same way as above and with \eqref{eq:met.1st.comp}; recall that $\Phi_\eps u$ is constant on $\Vepsk$ and independent of $y \in F$ on $\Uepsj$. \end{proof} % ------------- % Note that to get the upper bound we did not use any estimate for the metric on the vertex neighbourhoods $\Vepsk$. %------------------------------------------------------------ \subsection{A lower bound} %\label{ssec:low.est} The opposite estimate is more difficult. Here, we will also employ averaging processes on the vertex neigbourhoods $\Vepsk$ which correspond to projection onto the lowest (constant) mode: % ------------- % \begin{equation} \label{def:vert.av} C u = C_k u := \frac 1 {\vol {\Vk}} \int_{\Vk} u \, d \Vk. \end{equation} % ------------- % Recall that $V=\Vk$ denotes the manifold $\Vk$ with the metric $g=g_1$ (see Remark~\ref{rem:no.eps} for the reason why we use $\Vk$ instead of $\Vepsk$). % ------------- % \begin{lem} \label{lem:diff.av} The inequality % ------------- % \begin{displaymath} |C_k u - N_j u(x^0)|^2 \le O(\eps^{2\alpha-d}) \normsqr[\Vepsk] {du} \end{displaymath} % ------------- % holds for all $u \in \Sob \Vepsk$ where the point $x^0=x^0_{jk} \in \bd \Ij$ corresponds to the vertex $v_k$. \end{lem} % ------------- % \begin{proof} % ------------- % \begin{align*} |C_k u - N_j u(x^0)|^2 & \le \int_F |C_k u - u(x^0,y)|^2 \, dF(y) \\ & \le c_1 \left(\normsqr[\Vk] {C_k u - u} + \normsqr[\Vk] {du} \right)\\ & \le c_1 \left(\frac 1 {\EWN 2 {\Vk}}+1 \right) \normsqr[\Vk] {du} \\ & \le O(\eps^{2\alpha-d}) \normsqr[\Vepsk] {du} \end{align*} % ------------- % holds by Lemma~\ref{lem:rest.est} and Lemma~\ref{lem:minmax.2nd.neu} with $X=\Vk$ and metric $g=g_1$, and Lemma~\ref{lem:est.norm.quad}. \end{proof} % ------------- % \begin{lem} \label{lem:diff.vol.av} We have % ------------- % \begin{displaymath} \normsqr[\Veps] {u - C u} \le O(\eps^\beta) \normsqr[\Veps] {du} \end{displaymath} % ------------- % for all $u \in \Sob \Veps$, where $\beta :=(2+d)\alpha - d$. \end{lem} % ------------- % \begin{proof} Using again Lemmas~\ref{lem:minmax.2nd.neu} and~\ref{lem:est.norm.quad} we infer % ------------- % \begin{displaymath} \normsqr[\Veps] {u - C u} \le c_1^+ \eps^{\alpha d} \normsqr[V] {u - C u} \le c_1^+ \eps^{\alpha d} \, \frac 1 {\EWN 2 V} \normsqr[V] {du} \le O(\eps^{\alpha d - d + 2\alpha}) \normsqr[\Veps] {du}. % ------------- % \end{displaymath} \end{proof} % ------------- % \noindent Notice that $\beta>0$ is equivalent to $\alpha > d/(d+2)$ and the last inequality is satisfied due to~\eqref{eq:est.alpha.fast} and the fact that $d \ge 2$ holds by assumption. % ------------- % \begin{rem} \label{rem:no.eps} For Lemma~\ref{lem:diff.vol.av}, the ``natural'' averaging $C_\eps u := \int_\Veps u \,d\Veps$ would yield the same result whereas Lemma~\ref{lem:diff.av} leads to the estimate $O(\eps^{\beta-d})$ which is worse since $2\alpha>\beta$. \end{rem} % ------------- % We conclude that in the fast decaying case the edge neighbourhoods lead to no spectral contribution in the limit $\eps \to 0$: % ------------- % \begin{cor} \label{cor:vertex.small} The inequality % ------------- % \begin{displaymath} \normsqr[\Veps] u \le O(\eps^{\alpha d - m})(\normsqr[\Ueps \cup \Veps] u + \normsqr[\Ueps \cup \Veps]{du}) \end{displaymath} % ------------- % holds true for all $u \in \Sob {\Ueps \cup \Veps}$. \end{cor} % ------------- % \begin{proof} We start from the telescopic estimate % ------------- % \begin{multline*} \norm[\Veps] u \le \norm[\Veps] {u - C u} + \norm[\Veps] {C u - N u(x^0)} + \norm[\Veps] {N u(x^0)} \\ \le O(\eps^{\beta/2}) \norm[\Veps]{du} +(\vol \Veps)^{1/2} \left( O(\eps^{(2\alpha-d)/2}) \normsqr[\Veps]{du} + O(\eps^{-m} ) (\normsqr[\Ueps] u + \normsqr[\Ueps] {du}\right)^{1/2} \\= O(\eps^{(\alpha d - m)/2})(\normsqr[\Ueps \cup \Veps] u + \normsqr[\Ueps \cup \Veps] {du})^{1/2} \end{multline*} % ------------- % where we have used Lemmas~\ref{lem:diff.vol.av},~\ref{lem:diff.av}, and~\ref{lem:av.edge}, and furthermore the inequality~\eqref{eq:est.vert.norm} to obtain $\vol \Veps = O(\eps^{\alpha d})$. \sloppy Finally, note that $\beta = (d+2)\alpha - d > \alpha d - m >0$ and that $\alpha d - m > 0$ is equivalent to assumption~\eqref{eq:est.alpha.fast}. \end{proof} % ------------- % Now we define the transition operator by % ------------- % \begin{equation} \label{eq:trans.op.below} (\Psi_\eps u)_j (x):= \eps^{m/2} (N_j u (x) + \rho(x)(C_k u - N_j u(x^0)) \quad \text{for $x \in \Ijk$} \end{equation} % ------------- % where $\map \rho \R {[0,1]}$ is a smooth function such that % ------------- % \begin{equation} \label{eq:smooth.fct} \rho(x^0)=1 \quad \text{and} \quad \rho(x)=0 \quad \text{for all $|x-x^0| \ge \frac12 \min_{j \in J} \ell(\Ij)$} \end{equation} % ------------- % where $\ell(\Ij)$ denotes the length of the edge $e_j \cong \Ij$. Furthermore, $x^0=x^0_{jk} \in \bd \Ij$ is the edge point which can be identified with the vertex $v_k$. Recall that $\Ijk$ denotes the (closed) half of the interval $\Ij \cong e_j$ adjacent with the vertex $v_k$ and directed away from $v_k$. % ------------- % \begin{lem} \label{lem:below} We have $\Psi_\eps u \in \Sob \Mnull$ if $u \in \Sob \Meps$. Furthermore, % ------------- % \begin{align} \normsqr[\Meps] u - \normsqr[\Mnull] {\Psi_\eps u} &\le o(1) (\normsqr[\Meps] u + \normsqr[\Meps] {du}) \\ q_0(\Psi_\eps u) - \normsqr[\Meps] {du} &\le o(1) (\normsqr[\Meps] u + \normsqr[\Meps] {du}) \end{align} % ------------- % for all $u \in \Sob \Meps$. \end{lem} % ------------- % \begin{proof} The first assertion follows from $(\Psi_\eps u)_j(x^0_{jk}) = C_k u$. Furthermore, we have % ------------- % \begin{multline*} \normsqr[\Meps] u - \normsqr[\Mnull] {\Psi_\eps u} \\ \le \sum_{k \in K} \Bigl( \normsqr[\Vepsk] u + \sum_{j \in J_k} \bigl( \normsqr[\Uepsjk] u - \eps^m \normsqr[\Ijk] {Nu + \rho \cdot (Cu - Nu(x^0))} \bigr) \Bigr) \\ \le \sum_{k \in K} \Bigl( \normsqr[\Vepsk] u + \sum_{j \in J_k} \bigl( \normsqr[\Uepsjk] u - \normsqr[\Ijk] {\eps^{m/2} Nu} \bigr) \\ + \sum_{j \in J_k} \bigl( \delta \normsqr[\Ijk] {\eps^{m/2} Nu} + \eps^m \delta^{-1} \normsqr[\Ijk] \rho |Cu - Nu(x^0)|^2 \bigr) \Bigr) \end{multline*} % ------------- % where we have used the inequality % ------------- % \begin{equation} \label{eq:quad.lower} (a+b)^2 \ge (1-\delta)a^2 - \frac 1 \delta b^2, \qquad \delta>0. \end{equation} % ------------- % The last term in the sum can be estimated by $O(\eps^m)\delta^{-1}|C u - N u(x^0)|^2$. Applying Lemma~\ref{lem:diff.av} we arrive at the bound by $O(\eps^{m+2\alpha-d})\delta^{-1}(\normsqr[\Meps] u + \normsqr[\Meps] {du})$. Note that $m+2\alpha-d=2\alpha-1>0$ since $\alpha>1/2$. Set $\delta := \eps^{(2\alpha-1)/2}$. The remaining terms can be estimated by Corollary~\ref{cor:vertex.small}, Lemma~\ref{lem:diff.norm.av}, and estimate~\eqref{eq:edge.av.cont}. The second inequality can be proven in the same way, namely % ------------- % \begin{multline*} q_0(\Psi_\eps u) - \normsqr[\Meps] {du} \le \sum_{\substack{k \in K\\ j \in J_k}} \Bigl( \eps^m \normsqr[\Ijk] {(Nu)' + \rho' \cdot (Cu - Nu(x^0))} - \normsqr[\Uepsjk] {du} \Bigr) \\ \le \sum_{\substack{k \in K\\ j \in J_k}} \bigl( \normsqr[\Ijk] {\eps^{m/2} (Nu)'} - \normsqr[\Uepsjk] {du} + \delta \normsqr[\Ijk] {\eps^{m/2} (Nu)'} + \frac{2\eps^m} \delta \normsqr[\Ijk] {\rho'} |Cu - Nu(x^0)|^2 \bigr) \end{multline*} % ------------- % where we have used % ------------- % \begin{equation} \label{eq:quad.upper} (a+b)^2 \le (1+\delta)a^2 + \frac 2 \delta b^2, \qquad 0 < \delta \le 1, \end{equation} % ------------- % with $\delta := \eps^{(2\alpha-1)/2}$. Since the norm involving $\rho'$ is a fixed constant, the result follows from Lemma~\ref{lem:diff.quad.av} and Lemma~\ref{lem:diff.av}. \end{proof} Using Lemma~\ref{lem:below} we arrive at the sought lower bound. Note that the error term $\eta_k$ in~\eqref{eq:eta.k} can be estimated by some $\eps$-independent quantity because $\lambda_k=\EW k \Meps \le c_k$ by the upper bound given in Theorem~\ref{thm:ev.above}. % ------------- % \begin{thm} \label{thm:ev.below} We have $\EW k \Mnull \le \EW k \Meps + o(1)$. \end{thm} % ------------- % \noindent Theorem~\ref{thm:ev.conv} now follows easily by combining the last result with Theorem~\ref{thm:ev.above}. \begin{rem} \label{rem:alpha.bigger.1} If we allow a maximal shrinking factor $\alpha''>1$ in~\eqref{eq:met.vertex}, i.e., \begin{displaymath} c_- \eps^{2\alpha''} g \le g_\eps \le c_+ \eps^{2\alpha} g \end{displaymath} we would have to introduce the bottle necks already in this section and we would need \begin{displaymath} \alpha \le \alpha'' < \frac2d \alpha + \frac m d; \end{displaymath} the second inequality is due to Lemma~\ref{lem:diff.av} since we would need $m + (2\alpha - \alpha''d) > 0$ in the proof of Lemma~\ref{lem:below}. We have omitted this general setting to keep the previous section simple. \end{rem} %------------------------------------------------------------ \section{Slowly decaying vertex volume} \label{sec:slow.decay} %------------------------------------------------------------ If the volume of the vertex region decays significantly slower than the volume of the edge neighbourhoods, the limit operator is different. At the ends of the edges we have Dirichlet boundary conditions, whereas for each vertex $v_k$, $k \in K$, we obtain an additional eigenmode. In other words, we add a point measure at each vertex to the given measure on the graph $\Mnull$; the corresponding Hilbert space and quadratic form (domain) is therefore given by % ------------- % \begin{equation} \label{def:lim.slow} \HS_0 := \Lsqr \Mnull \oplus \C^K, \qquad \mathcal D_0 := \bigoplus_j \Sobn \Ij \oplus \C^K, \qquad q_0(u):= \sum_j \normsqr[\Ij] {u_j'}. \end{equation} % ------------- % For elements of $\HS_0$ we write $u=((u_j)_{j \in J},(u_k)_{k \in K})$ where $u_j \in \Lsqr {\Ij, p_j(x)dx}$ and $u_k \in \C$. We sometimes omit the indices and simply write $u$ instead of $u_j$. Note that the point contributions $u_k$ do not occur in the quadratic form, i.e., the additional eigenmodes have zero energy. Furthermore, the associated operator \begin{displaymath} Q_0 := \bigoplus_{j \in J} \laplacianD \Ij \oplus \boldsymbol 0 \end{displaymath} corresponds to a fully decoupled graph, i.e., a collection of independent edges, and its spectrum consists of all Dirichlet eigenvalues of the intervals $\Ij$ and $0$. Here, $\boldsymbol 0$ corresponds to the zero operator on $\C^K$. In order to define assumptions such that a smooth metric $g_\eps$ exists globally with different length scalings on the vertex and edge neighbourhoods, we need to introduce some additional notation (see Figure~\ref{fig:add.mfd}): % ------------- % \begin{figure}[h] \begin{center} %------------------------------------------------------------ % \input{add.mfd.pstex_t} \begin{picture}(0,0)% \includegraphics{graph3.pstex}% \end{picture}% \setlength{\unitlength}{4144sp}% \begin{picture}(6155,3698)(55,-3063) \put(4861,-61){$\Aepsjk$}% \put(3601,479){$\Vepsk$}% \put(4636,-1681){$\Uepsjk$ (transversal scaling $\eps$)}% \put(2341,-421){$\mVepsk$}% \put(1576,-691){(scaling inbetween $\eps^\alpha$ and $\eps^{\alpha'}$)}% \end{picture} %------------------------------------------------------------ \caption{The decomposition with the different scaling areas.} \label{fig:add.mfd} \end{center} \end{figure} % ------------- % let $\mVk$ be a closed submanifold of $\Vk$ of the same dimension with a positive distance from all adjacent edge neighbourhoods $\Ujk$, $j \in J_k$. Furthermore, we assume that the cylindrical structure of the half vertex neighbourhood $\Ujk$ extends to the component of $\Vk \setminus \mVk$ where $\Ujk$ meets $\Vk$, i.e., the closure of $\Vk \setminus \mVk$ is diffeomorphic to the disjoint union of cylinders $[0,1] \times F$. We denote the extended cylinder containing $\Ujk$ together with the corresponding cylindrical end (the \emph{bottle neck}) of $\Vk$ by $\pUjk=\pIjk \times F$ and the bottle neck alone by $\Ajk = \nIjk \times F$. Note that $\Ajk = \pUjk \cap \Vk$ and that $\pIjk = \Ij \cup \nIjk$. Again, we use the subscript $\eps$ to indicate the corresponding Riemannian manifold with metric $g_\eps$. %------------------------------------------------------------ \subsection{Assumption on the smaller vertex neighbourhood} We first fix the scaling behaviour on the smaller vertex neighbourhood $\mVk$. Here, we assume that % ------------- % \begin{equation} \label{eq:met.vertex.slow} c_- \eps^{2\alpha} g \le g_\eps \le c_+ \eps^{2\alpha'} g \qquad \text{on $\mVk$} \end{equation} (for the notation see~\eqref{eq:met.vertex}) where \begin{equation} \label{eq:est.alpha.slow} 0 < \alpha < \frac{d-1} d, \end{equation} % ------------- % i.e., $\mVepsk$ scales at most as $\eps^\alpha$ in each direction and at least as $\eps^{\alpha'}$ where % ------------- % \begin{equation} \label{eq:est.alpha'.slow} \frac d {d+2} \alpha < \alpha' \le \alpha, \end{equation} e.g., a homogenious scaling ($\alpha'=\alpha$) would do. Note that $\alpha' \le \alpha$ is necessary in order that \eqref{eq:met.vertex.slow} makes sense whereas $\alpha d/(d+2) < \alpha'$ ensures that the second Neumann eigenvalue of $\mVeps$ tends to $\infty$ as we will need in Lemma~\ref{lem:diff.vol.av.slow}. %------------------------------------------------------------ \subsection{Assumptions on the bottle neck} Roughly speaking, we have to avoid that the bottle neck has more than a single neck separating $\Vepsk$ in more than one part as $\eps \to 0$. In that case more than one zero eigenmode occur in the limit. Such a counterexample will be given in Remark~\ref{rem:no.assumpt}. We use the same notation as in Section~\ref{sec:edge.nbh} for the metric $g_\eps$ on the bottle neck $A=\Ajk$ and set \begin{equation} \label{eq:def.met.edge.slow} \tilde g_\eps := a_\eps^2(x) dx^2 + r_\eps^2(x) h(y), \qquad (x,y) \in A=\nI \times F \end{equation} for the (pure) product metric on $A$. Here, $a_\eps=a_{\eps,jk}$ and $r_\eps=r_{\eps,jk}$ are strictly positive smooth functions. Note that $r_\eps$ defines the radius of the fibre $\{x\} \times F$ at the point $x$. Again, we denote by $G_\eps$ and $\tilde G_\eps$ the $d \times d$-matrices associated to the metrics $g_\eps$ and $\tilde g_\eps$ with respect to the coordinates $(x,y) \in \nI \times F$) and assume that the two metrics coincide up to an error term as $\eps\to 0$, more specifically % ------------- % \begin{equation} \label{eq:asym.met.edge.slow} G_\eps = \tilde G_\eps + \begin{pmatrix} o(a_\eps^2) & o(a_\eps r_\eps) \\ o(a_\eps r_\eps) & o(r_\eps^2) \end{pmatrix} = \begin{pmatrix} (1+o(1)) a_\eps^2 & o(a_\eps r_\eps) \\ o(a_\eps r_\eps) & (H + o(1)) r_\eps^2 \end{pmatrix}, \end{equation} % ------------- % uniformly on $A$. We prove the following lemma in the same way as Lemma~\ref{lem:metric}: \begin{lem} \label{lem:metric.slow} Suppose that $g_\eps$, $\tilde g_\eps$ are given as above then % ------------- % \begin{align} \label{eq:met.vol.slow} (\det G_\eps)^\frac12 &= (1 + o(1)) \, (\det \tilde G_\eps)^\frac12 \\ \label{eq:met.1st.comp.slow} g_\eps^{xx} & \!:= (G_\eps^{-1})_{xx} = a_\eps^{-2}(1 + o(1)) \\ \label{eq:met.1st.der.slow} a_\eps^{-2} |d_x u|^2 &\le O(1) \, |d u|_{g_\eps}^2 \end{align} % ------------- % where $d_x$ denotes the partial derivative with respect to $x$. \end{lem} To make a smooth junction between the metrics on $\Uj$ and $\mVk$ possible, we assume that \begin{align*} a_\eps(x)&=\eps^\alpha, & r_\eps(x)&=\eps^\alpha &&\text{near $x^+$}\\ a_\eps(x)&=1, & r_\eps(x)&=\eps r_- &&\text{near $x^0$} \end{align*} where $x \in \nI=[x^+,x^0]$ and $r_-:=r_j(x^0)$ (the radius of the fibre at $x^0$, see also equation~\eqref{eq:def.met.edge}). Furthermore, we assume that \begin{equation} \label{eq:est.a.r} a_\eps(x) \le \begin{cases} \eps^\alpha & \text{on $[x^+, x^0-\delta_0,]$}\\ 1 & \text{on $[x^0-\delta_0, x^0]$} \end{cases} \quad \eps r_- \le r_\eps(x) \le \begin{cases} \eps^\alpha & \text{on $[ x^+, x^+ + \delta_+]$}\\ \eps r_+ & \text{on $[x^+ + \delta_+, x^0]$} \end{cases} \end{equation} for some constant $r_+ \ge r_-$, where $\delta_0=\eps^\alpha$ and \begin{figure}[h] \begin{center} %------------------------------------------------------------ % \input{a.r.pstex_t} \begin{picture}(0,0)% \includegraphics{graph4.pstex}% \end{picture}% \setlength{\unitlength}{4144sp}% \begin{picture}(5730,1926)(58,-1276) \put(530,494){$a_\eps(x)$}% \put(2773,-1073){$x$}% \put( 58,209){$1$}% \put( 58,-781){$\eps^\alpha$}% \put(3545,494){$r_\eps(x)$}% \put(5788,-1073){$x$}% \put(3073,209){$\eps^\alpha$}% \put(3073,-781){$\eps r_-$}% \put(3061,-601){$\eps r_+$}% \put(1466,-1276){$x^0-\delta_0$}%1666 \put(2161,-1276){$x^0$}% \put(316,-1276){$x^+$}% \put(3691,-1276){$x^++\delta_+$}% \put(5176,-1276){$x^0$}% \put(3331,-1276){$x^+$}% \end{picture} %------------------------------------------------------------ \caption{The functions $a_\eps$ and $r_\eps$ in its allowed area (in grey).} \label{fig:a.r} \end{center} \end{figure} $\delta_+=\eps^{(1-\alpha)m}=\eps^\alpha \eps^{m - \alpha d}$. These assumptions are needed in Lemma~\ref{lem:poincare}, e.g.\ to assure that the eigenfunctions of $\Meps$ do not concentrate on $\Aepsjk$ (i.e., \eqref{eq:rest.small} holds). %------------------------------------------------------------ \subsection{Convergence of the spectra} With the above prerequisites we can finally formulate the main result of this section: % ------------- % \begin{thm} \label{thm:ev.conv.slow} Under the stated assumptions $\EW k \Meps \to \EW k {Q_0}$ as $\eps\to 0$. More precisely, the first $|K|$ eigenvalues tend to $0$, while the remaining bounded eigenvalue branches tend to Dirichlet eigenvalues of the intervals $I_j$, i.e., \begin{align} \label{eq:lim.slow.0} \EW k \Meps &\to 0 & \text{if $1 \le k \le |K|$}\\ \label{eq:lim.slow.bigger.0} \EW k \Meps &\to \EWD {k-|K|} {\bigdcup_{j \in J} I_j} & \text{if $k > |K|$}, \intertext{where $\EWD n {\bigdcup_{j \in J} I_j}$ denotes the Dirichlet eigenvalues $\EWD l {\laplacian \Ij}$ of the operators on $\Ij$ ($j \in J$) defined as in~\eqref{eq:formaledge}, reordered with respect to multiplicity. In particular, if the length of all the edges $I_j$ is $\ell$ and $p_j(x)=1$ for all $j$, we have} \EW k \Meps &\to \EWD m {[0,\ell]} = \pi^2 m^2/\ell^2 & \text{if $k=(m-1)|J|+1,\dots, m|J|$.} \end{align} \end{thm} % ------------- % Again, our aim is to find a two sided estimate on each eigenvalue $\EW k \Meps$ by means of $\EW k {Q_0}$ with an error which is $o(1)$ w.r.t.\ the parameter $\eps$. %------------------------------------------------------------ \subsection{An upper bound} %\label{ssec:up.est} The following upper eigenvalue estimate is slightly more difficult to show than in the previous section: % ------------- % \begin{thm} \label{thm:ev.above.slow} $\EW k \Meps \le \EW k {Q_0} + o(1)$ holds as $\eps\to 0$. \end{thm} % ------------- % \noindent To prove it, we define the transition operator by % ------------- % \begin{displaymath} \label{eq:trans.op.above.slow} \Phi_\eps u (z):= \begin{cases} (\vol \mVepsk)^{-1/2} u_k & \text{if $z \in \Vk$},\\ \eps^{-m/2} u_j(x) + (\vol \mVepsk)^{-1/2} \rho(x) u_k & \text{if $z=(x,y) \in \Uj$} \end{cases} \end{displaymath} % ------------- % for any $u \in \mathcal D_0$, where $\rho$ is a smooth function as in~\eqref{eq:smooth.fct} and $x^0=x^0_{jk}$ denotes the endpoint of the half-edge $\Ijk$ corresponding to the vertex $v_k$. Theorem~\ref{thm:ev.above.slow} is then implied by Lemma~\ref{lem:main} in combination with the following result. % ------------- % \begin{lem} \label{lem:above.slow} We have $\Phi_\eps u \in \Sob \Meps$, i.e., $\Phi_\eps$ maps the quadratic form domain $\mathcal D_0$ into the quadratic form domain of the Laplacian on the manifold. Furthermore, % ------------- % \begin{align} \normsqr[\HS_0] u - \normsqr[\Meps] {\Phi_\eps u} &\le o(1) \, \normsqr[\HS_0] u\\ \normsqr[\Meps] {d \, \Phi_\eps u} - q_0(u) & \le o(1) \, (\normsqr[\HS_0] u + q_0(u)) \end{align} % ------------- % \end{lem} % ------------- % \begin{proof} Since $u_j \restr {\bd \Ij} = 0$, the function $\Phi_\eps u$ agrees on $\bd \Vepsk$ for both definitions. Clearly, $\Phi_\eps u$ is weakly differentiable on each thickened edge $\Uepsj$. Moreover, we have % ------------- % \begin{multline*} \normsqr[\HS_0] u - \normsqr[\Meps] {\Phi_\eps u} = \sum_{k \in K} \biggl( \sum_{j \in J_k} \Bigl( \normsqr[\Ijk] u - \normsqr[\Uepsjk] {\Phi_\eps u} \Bigr) + \Bigl( |u_k|^2 - \normsqr[\Vepsk] {\Phi_\eps u} \Bigr) \biggr) \\ \le \sum_{\substack{k \in K\\ j \in J_k}} \left( \bigl( \normsqr[\Ijk] u - (1+o(1)) \Bignormsqr[\tUepsjk] {\eps^{-m/2} u + (\vol \mVepsk)^{-1/2} \rho \, u_k} \bigr) \right) \\ + \sum_{k \in K} \left( |u_k|^2 - \normsqr[\mVepsk] {\Phi_\eps u} \right) \end{multline*} % ------------- % where we have used equation~\eqref{eq:met.vol} and that $\mVepsk \subset \Vepsk$. Note that the latter sum in the last line is equal to $0$. To estimate the remaining sum, remember that $\Phi_\eps u$ is independent of $y$ on $\Uepsjk$. Therefore we can apply equation~\eqref{eq:ind.of.2nd}, and inequality~\eqref{eq:quad.lower} with $\delta:=\eps^{(m-\alpha d)/2}$ yields the upper estimate % ------------- % \begin{multline*} \sum_{\substack{k \in K\\ j \in J_k}} \biggl( \normsqr[\Ijk] u - (1+o(1)) \Bigl( (1-\delta) \normsqr[\Ijk] u - \frac{\eps^m}{\delta \vol \mVepsk} \normsqr[\Ijk]\rho |u_k|^2 \Bigr) \biggr) \\ \le \sum_{\substack{k \in K\\ j \in J_k}} \Bigl( (\delta + o(1)) \normsqr[\Ijk] u + \frac{O(\eps^{m-\alpha d})}{\delta} |u_k|^2 \Bigr) = o(1) \normsqr[\HS_0] u. \end{multline*} % ------------- % In the last inequality, we have used the estimate $(\vol \mVepsk)^{-1} \le O(\eps^{-\alpha d})$ which follows from the lower bound of~\eqref{eq:met.vertex.slow}. Note that $\delta=o(1)$ by assumption~\eqref{eq:est.alpha.slow}. The second relation follows from % ------------- % \begin{multline*} \normsqr[\Meps] {d \, \Phi_\eps u} - q_0(u) = \sum_{\substack{k \in K\\ j \in J_k}} \biggl( \bigl(1+o(1)\bigr) \normsqr[\tUepsjk]{g_\eps^{xx} d_x \Phi_\eps u} - \normsqr[\Uepsjk]{u'} \biggr)\\ = \sum_{\substack{k \in K\\ j \in J_k}} \biggl( \bigl(1+o(1)\bigr) \eps^m \normsqr[\Ijk]{\eps^{-m/2}u' + (\vol \mVepsk)^{-1/2} \rho'\, u_k} - \normsqr[\Ijk]{u'} \biggr) \\ \le \sum_{\substack{k \in K\\ j \in J_k}} \bigl(1+o(1)\bigr) \biggl( \delta \normsqr[\Ijk]{u'} + \frac{2 \eps^m}{\delta \vol \mVepsk} \normsqr[\Ijk] {\rho'} |u_k|^2 \biggr) \\ \le o(1) \bigl(\normsqr[\HS_0] u + q_0(u)\bigr) \end{multline*} % ------------- % in the same way as above together with \eqref{eq:met.1st.comp} for the second equality and~\eqref{eq:quad.upper} in the last line; recall that $\Phi_\eps u$ is constant on $\Vepsk$. \end{proof} Note that we need a counterpart to $\normsqr[\Vepsk] u$ on the limit problem Hilbert space $\HS_0$. In the case of a fast decaying vertex volume in the previous section, the correponding norm vanished (see Corollary~\ref{cor:vertex.small}), but here we need the additional subspace $\C^K$ in $\HS_0$ coming from extra point measures at the vertices. Furthermore, note that the upper bound estimate on $\EW k \Meps$ already proven in Lemma~\ref{lem:above} remains valid in this setting, but it is too rough for the present purpose. \subsection{A lower bound} %\label{ssec:low.est} Again, the opposite estimate is more difficult. We will employ averaging processes also on the vertex neigbourhoods; this time with the $\eps$-scaled manifold $\mVepsk$: % ------------- % \begin{equation} \label{def:vert.av.eps} C_\eps^- u = C_{\eps,k}^- u := \frac 1 {\vol {\mVepsk}} \int_{\mVepsk} u \, d \mVepsk. \end{equation} % ------------- % In the first lemma, we prove an estimate similar to the one in Lemma~\ref{lem:diff.vol.av}. Note that $\norm {C_\eps^- u} \le \norm u$ by Cauchy-Schwarz, but we need the opposite inequality. % ------------- % \begin{lem} \label{lem:diff.vol.av.slow} For any $u \in \Sob \mVeps$ we have % ------------- % \begin{displaymath} \normsqr[\mVeps] u - \normsqr[\mVeps] {C_\eps^- u} \le o(1) % O(\eps^{((d+2)\alpha' - d \alpha)/2}) \bigl(\normsqr[\mVeps] u + \normsqr[\mVeps] {du} \bigr). \end{displaymath} % ------------- % \end{lem} % ------------- % \begin{proof} Apply Lemma~\ref{lem:minmax.2nd.neu} with $X=\mVeps$ and $\delta =\eps^{((d+2)\alpha' - d \alpha)/2}$. From the min-max principle we obtain $\EWN 2 \mVeps \ge O(\eps^{d(\alpha-\alpha')-2\alpha'})$. Note that $\delta=o(1)$ since $(d+2)\alpha' - d \alpha>0$ by~\eqref{eq:est.alpha'.slow}. \end{proof} % ------------- % The next three results are valid independently of the assumptions on $\alpha$ given in~\eqref{eq:est.alpha.slow} and \eqref{eq:est.alpha'.slow}. We will need these results also for the borderline case $\alpha=(d-1)/d$ in the next section. We need an estimate on the average $|Nu(x^0)|^2$. Since on the bottle neck $\Aepsjk$, the estimates are quite delicate, we first prove the result for $|Nu(x^+)|^2$ (i.e., on $\bd \mVepsk$ where the scaling of the metric is of the right order. The error is controlled by~\eqref{eq:diff.av.slow}. Note that this estimate is a counterpart to the estimate in Lemma~\ref{lem:av.edge} where we extended the function to the \emph{edge} neighbourhood $\Uepsj$ (useful in the case of fast decaying vertex volume, $\alpha d - m > 0$). This is not possible here, since $\alpha d - m < 0$. Therefore, we extend the function to the \emph{vertex} neighbourhood $\mVepsk$. % ------------- % \begin{lem} \label{lem:av.vert.slow} The inequality % ------------- % \begin{displaymath} |Nu(x^+)|^2 \le \normsqr[F] {u(x^+,\cdot)} \le O(\eps^{-\alpha d}) \, \bigl( \normsqr[\mVeps] u + \normsqr[\mVeps]{du} \bigr) \end{displaymath} % ------------- % holds for any $u \in \Sob \mVeps$. \end{lem} % ------------- % \begin{proof} We have % ------------- % \begin{multline*} |Nu(x^+)|^2 \le \int_F |u(x^+,y)|^2 dF(y) \\ \le c_1 \bigl( \normsqr[\mV] u + \normsqr[\mV]{du} \bigr) \le O(\eps^{-\alpha d}) \bigl( \normsqr[\mVeps] u + \normsqr[\mVeps]{du} \bigr) \end{multline*} % ------------- % by Lemma~\ref{lem:rest.est} with $X=\mV$ and the lower bound in assumption~\eqref{eq:est.alpha.slow}. \end{proof} % ------------- % The next lemma is the key ingredient in dealing with the bottle neck. Here, we prove two Poincar\'e-like estimates. Since we want to avoid a cut-off function (leading to divergent terms when being differentiated) we only prove an estimate on the \emph{difference} and not on $Nu(x^0)$ itself in~\eqref{eq:diff.av.slow}. For the same reason, an integral over $F$ remains in~\eqref{eq:rest.small}. Note that $I_\eps^+(x^0) = \vol \Aeps$. % ------------- % \begin{lem} \label{lem:poincare} There is a constant $C>0$ such that % ------------- % \begin{gather} \label{eq:diff.av.slow} |Nu(x^0) - Nu(x^+)|^2 \le C I_\eps^-(x^0) \, \normsqr[\Aeps] {du}, \\ \label{eq:rest.small} \normsqr[\Aeps] u \le 4 I_\eps^+(x^0) \, \normsqr[F] {u(x^+,\cdot)} + 4 C I_\eps^{+-} \, \normsqr[\Aeps]{du} \end{gather} % ------------- % for all $u \in \Sob \Aeps$ where \begin{displaymath} I_\eps^\pm(x) := \int_{x+}^x a_\eps(x') r_\eps^{\pm m}(x') \, dx' \quad \text{and} \quad I_\eps^{+-} := \int_{x^+}^{x^0} a_\eps(x) r_\eps^m(x) I_\eps^-(x) \, dx. \end{displaymath} Furthermore, under the assumption~\eqref{eq:est.a.r}, we have \begin{displaymath} I_\eps^-(x^0) = o(\eps^{-m}), \qquad I_\eps^+(x^0) = o(\eps^{\alpha d}) \qquad \text{and} \qquad I_\eps^{+-} = o(1). \end{displaymath} % ------------- % \end{lem} % ------------- % \begin{proof} For a smooth function $u$ we have % ------------- % \begin{equation} \label{eq:diff.int} u(x,y) - u(x^+,y) = \int_{x^+}^x \partial_x u(x',y) \, dx'. % ------------- % \end{equation} For the first assertion, we set $x=x^0$, foremost integrate over $y \in F$ and then apply Cauchy-Schwarz % ------------- % \begin{multline*} |Nu(x^0) - Nu(x^+)|^2 \le \int_F \int_{x^+}^{x^0} a_\eps^2(x') \det G_\eps(x',y)^{-\frac12} \, dx' \times{} \\ {} \times \int_{x^+}^{x^0} a_\eps^{-2}(x') |\partial_x u(x',y)|^2 \det G_\eps(x',y)^\frac12 \, dx'\, dF(y). \end{multline*} % ------------- % The first integrand over $x'$ can be estimated by $C a_\eps(x') r_\eps^{-m}$ applying~\eqref{eq:met.vol.slow}. Therefore, the first integral is smaller than $C I_\eps^-(x^0)$. The second integral together with the integral over $F$ can be estimated by $O(1) \, \normsqr[\Aeps]{du}$ applying~\eqref{eq:met.1st.der.slow}. For the second assertion, we first apply Cauchy-Schwarz (and~\eqref{eq:quad.upper} with $\delta=1$) to~\eqref{eq:diff.int} and than integrate over $y \in F$ to obtain % ------------- % \begin{multline} \label{eq:rest.est} \int_F |u(x,y)|^2 \, dF(y) \le 2 \int_F |u(x^+,y)|^2 \, dF(y) + 2 \int_F \int_{x^+}^x a_\eps^2(x') \det G(x',y)^{-\frac12} \, dx' \times {}\\ {} \times \int_{x^+}^x a_\eps^{-2} (x') \, |\partial_x u (x',y)|^2 \det G(x',y)^\frac12 \,dx' \,dF(y). \end{multline} % ------------- % The first integral over $x'$ can be estimated as before by $C I_\eps^-(x)$. Finally, multiplying with $a_\eps(x) r_\eps^m(x)$ and integrating over $x \in \nI$ yields \begin{displaymath} \normsqr[\tAeps] u \le 2 I^+_\eps(x_0) \, \normsqr[F]{u(x^+,\cdot)} + 2 C I_\eps^{+-} \,\normsqr[\Aeps]{du}. \end{displaymath} Applying~\eqref{eq:met.vol.slow} once more we obtain the desired estimate over $\Aeps$ instead of $\tAeps$ (note that $2/(1+o(1)) \le 4$ provided $\eps$ is small enough). The general case of non-smooth functions can easily shown with approximation arguments. The integral estimates follow from \begin{displaymath} I_\eps^-(x^0) \le \int_{x^+}^{x^0 - \delta_0} \eps^\alpha (\eps r_-)^{-m} \, dx + \int_{x^0-\delta_0}^{x^0} (\eps r_-)^{-m} \, dx = (\eps^\alpha + \delta_0) O(\eps^{-m}). \end{displaymath} Since $\delta_0=\eps^\alpha$, we have $I_\eps^-(x^0) \le O(\eps^{\alpha-m})$. Next, we have \begin{multline*} I_\eps^+(x^0) \le \int_{x^+}^{x^+ + \delta_+} \eps^\alpha \eps^{\alpha m} \, dx + \int_{x^+ + \delta_+}^{x^0 - \delta_0} \eps^\alpha (\eps r_+)^m \, dx \\ + \int_{x^0-\delta_0}^{x^0} (\eps r_+)^m \, dx= \delta_+ O(\eps^{\alpha d}) + ( \eps^\alpha + \delta_0) O(\eps^m) \end{multline*} and therefore $I_\eps^+(x^0) \le O(\eps^{\alpha + m})= O(\eps^{\alpha + m - \alpha d}) O(\eps^{\alpha d})$ since $\delta_+=\eps^\alpha \eps^{m-\alpha d}$. The last assertion follows from $I_\eps^{+-} \le I_\eps^-(x_0) \, I_\eps^+(x_0) \le O(\eps^{2\alpha})$. \end{proof} The following corollary is again independent of the assumption we made about $\alpha$ in~\eqref{eq:est.alpha.slow} and \eqref{eq:est.alpha'.slow}, in particular, it is also valid in the setting of the borderline case of Section~\ref{sec:borderline}. % ------------- % \begin{cor} \label{cor:rest.small} For all $u \in \Sob \Veps$ we have % ------------- % \begin{displaymath} \normsqr[\Aeps] u \le o(1) \, \bigl( \normsqr[\Veps] u + \normsqr[\Veps]{du} \bigr). \end{displaymath} % ------------- % \end{cor} % ------------- % \begin{proof} We only have to put together~\eqref{eq:rest.small} and Lemma~\ref{lem:av.vert.slow}. \end{proof} % ------------- % We now formulate a consequence of the preceding lemmas under the assumption~\eqref{eq:est.alpha.slow}. \begin{cor} \label{cor:av.shifted} Suppose $0<\alpha< m/d=(d-1)/d$. Then we have % ------------- % \begin{displaymath} |Nu(x^0)|^2 \le o(\eps^{-m}) \, \bigl( \normsqr[\Veps] u + \normsqr[\Veps] {du} \bigr) \end{displaymath} % ------------- % for all $u \in \Sob \Veps$. \end{cor} % ------------- % \begin{proof} Applying \eqref{eq:quad.lower} with $\delta=1/2$ to \eqref{eq:diff.av.slow} we obtain % ------------- % \begin{displaymath} |Nu(x^0)|^2 \le o(\eps^{-m}) \normsqr[\Aeps]{du} + 4|Nu(x^+)|^2. \end{displaymath} % ------------- % The second term is of order $O(\eps^{-\alpha d})$ by Lemma~\ref{lem:av.vert.slow} and therefore also of order $o(\eps^{-m})$ by the assumption on $\alpha$. \end{proof} % ------------- % In this section, we define the transition operator by % ------------- % \begin{equation} \label{eq:trans.op.below.slow} \begin{split} (\Psi_\eps u)_j (x) &:= \eps^{m/2} N_j u (x) - \rho(x) N_j u (x^0) \quad \text{for $x \in I_{jk}$}\\ (\Psi_\eps u)_k &:= (\vol \mVepsk)^{1/2} C_{\eps,k}^- u \end{split} \end{equation} % ------------- % where $\rho$ is a smooth function as in~\eqref{eq:smooth.fct} and $x^0 = x^0_{jk}$ denotes the endpoint of the half-edge $\Ijk$ corresponding to the vertex $v_k$. % ------------- % \begin{lem} \label{lem:below.slow} We have $\Psi_\eps u \in \mathcal D_0$ if $u \in \Sob \Meps$. Furthermore, % ------------- % \begin{align} \normsqr[\Meps] u - \normsqr[\HS_0] {\Psi_\eps u} &\le o(1) \bigl( \normsqr[\Meps] u + \normsqr[\Meps] {du} \bigr) \\ q_0(\Psi_\eps u) - \normsqr[\Meps] {du} &\le o(1) \bigl( \normsqr[\Meps] u + \normsqr[\Meps] {du} \bigr) \end{align} % ------------- % for all $u \in \Sob \Meps$. \end{lem} % ------------- % \begin{proof} The first assertion follows from the fact that $(\Psi_\eps u)_j(x^0) = 0$. Furthermore, we have % ------------- % \begin{multline*} \normsqr[\Meps] u - \normsqr[\HS_0] {\Psi_\eps u} \le % \sum_{k \in K} % \Bigr( % (\normsqr[\mVepsk] u - |(\Psi_\eps u)_k|^2) + % \sum_{j \in J_k} % \bigr( % \normsqr[\Aepsjk] u + \normsqr[\Uepsjk] u - % \normsqr[\Ijk] {(\Psi_\eps u)_j} % \bigr) % \Bigl) \\ \le \sum_{k \in K} \biggr( \bigl( \normsqr[\mVepsk] u - \normsqr[\mVepsk]{C_\eps^- u} \bigr) \\ + \sum_{j \in J_k} \bigr( \normsqr[\Aepsjk] u + \normsqr[\Uepsjk] u - \eps^m \bignormsqr[\Ijk] {N u - \rho \cdot N u (x^0)} \bigr) \biggl). \end{multline*} % ------------- % The first difference is of the desired form by Lemma~\ref{lem:diff.vol.av.slow}. Furthermore, the integral over the ``bottle necks'' $\Aepsjk$ can be estimated in the needed way by Corollary~\ref{cor:rest.small}. Applying~\eqref{eq:quad.lower} to the remaining difference in the last sum we obtain the upper estimate by % ------------- % \begin{equation} \label{eq:below.slow} \bigl( \normsqr[\Uepsjk] u - \eps^m \normsqr[\Ijk] {N u} \bigr) + \delta \, \eps^m \normsqr[\Ijk] {N u} + \frac {\eps^m} \delta \normsqr[\Ijk] \rho |N u (x^0)|^2 \end{equation} % ------------- % For the first two terms we obtain the sought bound by virtue of Lemma~\ref{lem:diff.norm.av} and \eqref{eq:edge.av.cont}; for the remaining term one has to apply Corollary~\ref{cor:av.shifted}. The second inequality can be proven in the same way, namely % ------------- % \begin{multline*} q_0(\Psi_\eps u) - \normsqr[\Meps] {du} \\ = \sum_{k \in K} \Bigr( -\normsqr[\mVepsk] {du} + \sum_{j \in J_k} \bigr( \eps^m \bignormsqr[\Ijk] {(N u)' - \rho' \, N u (x^0)} - \normsqr[\Uepsjk] {du} \bigr) \Bigl) \end{multline*} % ------------- % We omit the norm contribution from $\mVepsk$ and estimate the remaining difference with~\eqref{eq:quad.upper} and obtain (up to the summation) % ------------- % \begin{displaymath} \bigl(\eps^m \normsqr[\Ijk] {(N u)'} - \normsqr[\Uepsjk] {du} \bigr) + \delta \, \eps^m \normsqr[\Ijk] {(N u)'} + 2 \frac{\eps^m} \delta \normsqr[\Ijk] {\rho'} |N u (x^0)|^2. \end{displaymath} % ------------- % For the first difference we obtain the needed estimate by virtue of Lemma~\ref{lem:diff.quad.av}. An upper bound for the remaining term is of the same form as before. \end{proof} % ------------- % Using Lemma~\ref{lem:below.slow} we arrive at the sought lower bound. Note that the error term $\eta_k$ in~\eqref{eq:eta.k} can be estimated by some $\eps$-independent quantity because $\lambda_k=\EW k \Meps \le c_k$ by Theorem~\ref{thm:ev.above.slow}. % ------------- % \begin{thm} \label{thm:ev.below.slow} We have $\EW k {Q_0} \le \EW k \Meps + o(1)$. \end{thm} % ------------- % \noindent Theorem~\ref{thm:ev.conv.slow} now follows easily by combining the last result with Theorem~\ref{thm:ev.above.slow}. %------------------------------------------------------------ \begin{rem} \label{rem:no.assumpt} Without assumption~\eqref{eq:est.a.r} on the metric on the bottle neck $\Ajk$, the second Neumann eigenvalue of $\Vepsk$ could tend to $0$ (and not to $\infty$, as required in the proof of Lemma~\ref{lem:diff.vol.av.slow}), for example, if $\Vepsk$ separates into more than one part as $\eps \to 0$ (i.e., $\Vepsk$ has an additional ``throat''). \end{rem} %------------------------------------------------------------ \section{The borderline case} \label{sec:borderline} % ------------- % \subsection{Definition of the thickened vertices} %\label{ssec:def.vert} If the volume of the vertex region decays at the same rate as the volume of the edge neighbourhoods, the limit operator acts again in the extended Hilbert space introduced in the previous section but it is not decoupled anymore. Thus it is not supported by the graph alone, in particular, it is not the Hamiltonian with the boundary conditions \eqref{eq:delta}. We start with the definition of the limit operator. The corresponding Hilbert space and quadratic form are given by % ------------- % \begin{equation} \label{def:lim.border} \HS_0 := \Lsqr \Mnull \oplus \C^K, \qquad q_0(u):= \sum_j \normsqr[\Ij] {u_j'}\,, \end{equation} % ------------- % where the form domain $\mathcal D_0$ of $q_0$ is given by those functions $u=((u_j)_{j\in J},(u_k)_{k\in K})$ such that % ------------- % \begin{equation} \label{def:lim.dom.border} u \in \Sob \Mnull \oplus \C^K \qquad \text{and} \qquad (\vol \mVk)^{1/2} u_j(v_k) = u_k \end{equation} % ------------- % for all $j \in J_k$ and $k \in K$, i.e., values of the functions at the edge endpoints $v_k\equiv x^0_{jk}$ are now coupled with the additional wave function components; recall that $\mVk$ denotes the manifold $\mVepsk$ with $\eps=1$. The corresponding operator $Q_0$ is given by % ------------- % \begin{equation} \label{def:lim.op.border} Q_0 u = \biggl( \Bigl( -\frac1{p_j} (p_j u_j')' \Bigr)_{\!j}, \Bigl( - (\vol \mVk)^{-\frac12} \sum_{j \in J_k} p_j(v_k) u_j'(v_k) \Bigr)_{\!k} \biggr) \,; \end{equation} % ------------- % it depends parametrically on $\vol(\mVk)$ but we refrain from marking this fact explicitly. Again, this operator has a purely discrete spectrum provided the graph $\Mnull$ is finite. As we have said, $Q_0$ is not a graph operator with the conditions \eqref{eq:delta}. Nevertheless, there is a similarity between the two noticed by Kuchment and Zeng in \cite{kuchment-zeng:03}. To solve the spectral problem $Q_0 u=\lambda u$ one has to find $(u_j)_{j\in J}$ such that $-(p_j u_j')'/p_j = \lambda u_j$ and at the vertices the functions satisfy the conditions % ------------- % \begin{equation} \label{def:delta.spectral} \sum_{j \in J_k} p_j(v_k) u_j'(v_k) = - \lambda (\vol \mVk)^{\frac12} u_k \,. \end{equation} % ------------- % This looks like \eqref{eq:delta}, the difference is that the coefficient at the right-hand side is not a constant but a multiple of the spectral parameter; in physical terms one may say that the coupling strength at a vertex is proportional to the energy. After this digression let us return to the limiting properties. We adopt again the assumption~\eqref{eq:est.a.r} in this section. Instead of (\ref{eq:met.vertex.slow}) we suppose now that on the vertex neighbourhood the metric satisfies the relation % ------------- % \begin{equation} \label{eq:met.vertex.border} g_\eps = \eps^{2\alpha} g + o(\eps^{2\alpha}) \qquad \text{on $\mVk$} \end{equation} % ------------- % with % ------------- % \begin{equation} \label{eq:est.alpha.border} \alpha = \frac {d-1} d\,, \end{equation} which corresponds to the above mentioned equal decay rate for the volume of the edge and vertex neighbourhoods. In particular, we have \begin{equation} \label{eq:norm.quad.border} \normsqr[\mVeps] u = \eps^{\alpha d} (1+o(1)) \normsqr[\mV] u \quad \text{and} \quad \normsqr[\mVeps] {du} = \eps^{\alpha (d-2)} (1+o(1)) \normsqr[\mV] {du} \end{equation} and \begin{equation} \label{eq:vol.border} \vol (\mVeps) = \eps^{\alpha d} (1+o(1)) \vol (\mV) \end{equation} for each $\mV=\mVk$ as in Lemmas~\ref{lem:metric} and~\ref{lem:est.norm.quad}. %------------------------------------------------------------ \subsection{Convergence of the spectra} %------------------------------------------------------------ With the above prerequisites we can finally formulate the main result of this section: % ------------- % \begin{thm} \label{thm:ev.conv.border} Under the stated assumptions $\EW k \Meps \to \EW k {Q_0}$ as $\eps\to 0$. \end{thm} % ------------- % \noindent To prove it, our aim is again to find a two sided estimate on each eigenvalue $\EW k \Meps$ by means of $\EW k {Q_0}$ with an error which is $o(1)$ w.r.t.\ the parameter $\eps$. %------------------------------------------------------------ \subsection{An upper bound} %------------------------------------------------------------ Again, we first show the easier upper eigenvalue estimate: % ------------- % \begin{thm} \label{thm:ev.above.border} $\EW k \Meps \le \EW k {Q_0} + o(1)$ holds as $\eps\to 0$. \end{thm} % ------------- % \noindent We define the transition operator by % ------------- % \begin{equation} \label{eq:trans.op.above.border} \Phi_\eps u (z):= \begin{cases} \vol(\mVepsk)^{-1/2} u_k & \text{if $z \in \Vk$},\\ \begin{aligned} \eps^{-m/2} & u_j(x) + \rho(x) \times {} \\ & {} \times \bigl( \vol(\mVepsk)^{-1/2} u_k - \eps^{-m/2} u_j(x^0) \bigr) \end{aligned} & \text{if $z=(x,y) \in \Uj$} \end{cases} \end{equation} % ------------- % for any $u \in \mathcal D_0$, where $\rho$ is a smooth function as in~\eqref{eq:smooth.fct} and $x^0=x^0_{jk}$ denotes the endpoint of the half-edge $\Ijk$ away from the vertex $v_k$. Theorem~\ref{thm:ev.above.border} is then implied by Lemma~\ref{lem:main} in combination with the following result. % ------------- % \begin{lem} \label{lem:above.border} We have $\Phi_\eps u \in \Sob \Meps$, i.e., $\Phi_\eps$ maps the quadratic form domain $\mathcal D_0$ into the quadratic form domain of the Laplacian on the manifold. Furthermore, % ------------- % \begin{align} \normsqr[\HS_0] u - \normsqr[\Meps] {\Phi_\eps u} &\le o(1) \, \normsqr[\HS_0] u\\ \normsqr[\Meps] {d \, \Phi_\eps u} - q_0(u) & \le o(1) \, (\normsqr[\HS_0] u + q_0(u)) \end{align} % ------------- % \end{lem} % ------------- % \begin{proof} The argument is analogous to the proof of Lemma~\ref{lem:above.slow}. The only difference is that we need the following estimate % ------------- % \begin{displaymath} \eps^m \bigl| \vol(\mVepsk)^{-1/2} u_k - \eps^{-m/2} u_j(x^0) \bigr|^2 = \bigl| \eps^{m/2} (\vol \mVepsk)^{-1/2} - (\vol \mVk)^{-1/2} \bigr|^2 \, | u_k |^2 %\le o(1) \normsqr[\HS_0] u \end{displaymath} % ------------- % since $u \in \mathcal D_0$. The last difference is of order $o(1)$ by~\eqref{eq:vol.border}. \end{proof} % ------------- % %------------------------------------------------------------ \subsection{A lower bound} %------------------------------------------------------------ The estimate on $\EW k \Meps$ from below can be found in analogy with the slow-decay case in Section~\ref{sec:slow.decay}. Furthermore, we need the following averaging operator % ------------- % \begin{displaymath} C_k^- u := \frac 1 {\vol \mVk} \int_\mVk u \, d\mVk. \end{displaymath} % ------------- % Since we have an exact scaling of the metric of order $\eps^\alpha$ by~\eqref{eq:est.alpha.border}, we also could use the $\eps$-depending manifold $\mVepsk$ here (cf.\ also Remark~\ref{rem:no.eps}). % ------------- % \begin{lem} \label{lem:diff.av.border} For all $u \in \Sob \mVepsk$ we have % ------------- % \begin{displaymath} \bigl| C^-_k u - N_j u(x^0) \bigr|^2 \le o(\eps^{-m}) \, \normsqr[\Vepsk] {du} \end{displaymath} % ------------- % \end{lem} % ------------- % \begin{proof} We have % ------------- % \begin{displaymath} \bigl| C^-_k u - N_j u(x^0) \bigr| \le \bigl| C^-_k u - N_j u(x^+) \bigr| + \bigl| N_j u(x^+) - N_j u(x^0) \bigr|. \end{displaymath} % ------------- % The first difference can be estimated in the same way as Lemma~\ref{lem:diff.av} (replacing $\Vk$ by $\mVk$ and using estimate~\eqref{eq:norm.quad.border}, i.e., we arrive at % ------------- % \begin{displaymath} \bigl| C^-_k u - N_j u(x^+) \bigr|^2 \le O(\eps^{-(d-2)\alpha}) \, \normsqr[\mVeps]{du}; \end{displaymath} % ------------- % recall that now we have $\alpha d = m$. For the second difference, use~\eqref{eq:diff.av.slow}. \end{proof} % ------------- % % ------------- % \begin{lem} \label{lem:diff.vol.av.border} For all $u \in \Sob \mVeps$, we have \begin{displaymath} \normsqr[\mVeps] u - \normsqr[\mVeps]{C^- u} \le O(\eps^\alpha) (\normsqr[\mVeps] u + \normsqr[\mVeps] {du}). \end{displaymath} \end{lem} % ------------- % \begin{proof} The proof is similar to the proof of Lemma~\ref{lem:diff.vol.av.slow}: inequality~\eqref{ineq:norm} together with~\eqref{eq:norm.quad.border} and Lemma~\ref{lem:minmax.2nd.neu} for $X=\mV$ implies % ------------- % \begin{multline*} \normsqr[\mVeps] u - \normsqr[\mVeps] {C^- u} \le \frac 1 \delta \eps^{\alpha d} (1+o(1)) \normsqr[\mV] {u - C^-u} + \delta ( \normsqr[\mVeps] u + \vol \mVeps |C^- u|^2) \\ \le \frac {\eps^{2\alpha}(1+o(1))} {\delta \EWN 2 \mV} \normsqr[\mVeps] {du} + \delta ( \normsqr[\mVeps] u + \vol \mVeps \normsqr[\mV] u) \end{multline*} % ------------- % Applying~\eqref{eq:vol.border} and \eqref{eq:norm.quad.border} once more, the result follows setting $\delta=\eps^\alpha$. \end{proof} Now we define the transition operator by % ------------- % \begin{equation} \label{eq:trans.op.below.border} \begin{split} (\Psi_\eps u)_j (x) &:= \eps^{m/2} \Bigl(N_j u (x) + \rho(x) \bigr(C^-_k u - N_j u (x^0) \bigr)\Bigr) \quad \text{for $x \in I_{jk}$} \\ (\Psi_\eps u)_k &:= \eps^{m/2} (\vol \mVk)^{1/2} C^-_k u \end{split} \end{equation} % ------------- % where $\rho$ is a smooth function as in~\eqref{eq:smooth.fct} and $x^0=x^0_{jk}$ denotes the endpoint of the half-edge $\Ijk$ corresponding to the vertex $v_k$. % ------------- % \begin{lem} \label{lem:below.border} We have $\Psi_\eps u \in \mathcal D_0$ if $u \in \Sob \Meps$. Furthermore, % ------------- % \begin{align} \normsqr[\Meps] u - \normsqr[\HS_0] {\Psi_\eps u} &\le o(1) \bigl( \normsqr[\Meps] u + \normsqr[\Meps] {du} \bigr) \\ q_0(\Psi_\eps u) - \normsqr[\Meps] {du} &\le o(1) \bigl( \normsqr[\Meps] u + \normsqr[\Meps] {du} \bigr) \end{align} % ------------- % for all $u \in \Sob \Meps$. \end{lem} % ------------- % \begin{proof} The arguments are the same as in the proof of Lemma~\ref{lem:below.slow}. For the vertex contribution, we need the estimate % ------------- % \begin{displaymath} \normsqr[\mVepsk] u - \eps^m (\vol \mVk) |C^-_k u|^2 = (\normsqr[\mVepsk] u - \normsqr[\mVepsk] {C^-_k u}) + \Bigl( \frac{\vol \mVepsk}{\eps^m \vol \mVk} - 1 \Bigr) \eps^m \normsqr[\mVk] {C^-_k u}. \end{displaymath} % ------------- % The first difference can be treated with Lemma~\ref{lem:diff.vol.av.border} and leads to an error term $O(\eps^\alpha)$. The second term is of order $o(1) \normsqr[\mVepsk] u$ by~\eqref{eq:norm.quad.border}, \eqref{eq:vol.border} and Cauchy-Schwarz. Furthermore, Corollary~\ref{cor:rest.small} is also true in this setting (independent on the particular $\alpha$). We also need Lemma~\ref{lem:diff.av.border}. \end{proof} Using Lemma~\ref{lem:below.border} we arrive at the sought lower bound. Again, the error term $\eta_k$ in~\eqref{eq:eta.k} can be estimated by some $\eps$-independent quantity because $\lambda_k=\EW k \Meps \le c_k$ by Theorem~\ref{thm:ev.above.border}. % ------------- % \begin{thm} \label{thm:ev.below.border} We have $\EW k {Q_0} \le \EW k \Meps + o(1)$. \end{thm} % ------------- % \noindent Theorem~\ref{thm:ev.conv.border} now follows easily by combining the last result with Theorem~\ref{thm:ev.above.border}. %------------------------------------------------------------ \section{Non-decaying vertex volume} \label{sec:alpha.null} %------------------------------------------------------------ In this section, we treat the case when the vertex volume does not tend to $0$. In some sense, this case corresponds to $\alpha = 0$ in the previous notation but we need more assumptions to precise the convergence of the manifold $\Vepsk$ to a manifold $\Vnullk$ as $\eps \to 0$. We cite only the result here since it has already been presented in~\cite{post:03a} or with a more detailed proof in~\cite{post:00}. A related result corresponding to the embedded case (see Example~\ref{ex:embedded}) as in \cite{kuchment-zeng:01} was proven by Jimbo and Morita in \cite{jimbo-morita:92} or for manifolds (with non-smooth junctions between edge and vertex neighbourhoods) by Ann\'e and Colbois in \cite{anne-colbois:95}. In this section, we assume that the transversal direction is a sphere, i.e., $F=\Sphere^m$. Let $\Vnullk$ be a compact $d$-dimensional manifold without boundary for $k \in K$. To each edge $j \in J_k$ eminating from the vertex $v_k$, we associate a point $x_{jk}^0 \in \Vnullk$ such that $x_{jk}^0$ ($j \in J_k$) are mutually distinct points with lower bound $2 \eps_0>0$ on their distance to each other. We assume for simplicity that the metric at $x^0=x_{jk}^0$ is locally flat within a distance $\eps_0$ from $x^0$ (the general case can be found in~\cite{post:03a}). Then the metric in polar coordinates $(x,y) \in (0,\eps_0) \times \Sphere^m$ looks locally like \begin{displaymath} g = dx^2 + x^2 \, h_y \end{displaymath} %(cf.~\cite[Prop.~E.III.7]{bgm:71}) where $h_y$ is the standard metric on $\Sphere^m$. Modifying the factor before $h_y$, we define a new metric by \begin{displaymath} g_\eps = dx^2 + r_\eps^2(x) \, h_y \end{displaymath} with a smooth monotone function $\map {r_\eps} {(0,\eps_0)}{(0,\infty)}$ such that \begin{displaymath} r_\eps(x) = \begin{cases} \eps & \text{for $0 < x < \eps/2$}\\ x & \text{for $2\eps < x < \eps_0$}. \end{cases} \end{displaymath} We denote the (completion of the) manifold $(\Vnullk \setminus \bigcup_{j \in J_k} x_{jk}^0, g_\eps)$ by $\Vepsk$. Note that this manifold has $|J_k|$ attached cylindrical ends of order $\eps$ at each point $x_{jk}^0$. Now we can construct the graph-like manifold $\Meps$ as in Section~\ref{sec:graph.mfd}. As in the slow decaying case of Section~\ref{sec:slow.decay} the limit operator \begin{displaymath} Q_0 := \bigoplus_{j \in J} \laplacianD \Ij \oplus \bigoplus_{k \in K} \laplacian \Vnullk \end{displaymath} decouples and the next result follows (cf.~\cite[Theorem~1.2]{post:03a} or~\cite{post:00}): \begin{thm} \label{thm:ev.conf.null} We have $\EW k \Meps \to \EW k {Q_0}$ as $\eps \to 0$. \end{thm} %------------------------------------------------------------ \section{Applications} \label{sec:applications} %------------------------------------------------------------ Finally we comment on consequences of the spectral convergence. We begin with a general remark stating that we only have uniform control over a \emph{compact} spectral interval: \begin{rem} \label{rem:not.uniform} Note that the convergence $\EW k \Meps \to \EW k \Mnull$ cannot be uniform in $k \in \N$: if this were the case, the theta-function \begin{displaymath} \Theta_\eps(t):= \tr \eu^{-t\laplacian \Meps} = \sum_k \eu^{-t \EW k \Meps} \end{displaymath} would converge to $\Theta_0(t)$. But Weyl asymtotics are different in the two cases, \begin{displaymath} \Theta_\eps(t) \sim \frac{\vol_d \Meps}{(4\pi t)^{d/2}}, \qquad \text{whereas} \qquad \Theta_0(t) \sim \frac{\vol_1 \Mnull}{(4 \pi t)^{1/2}} \end{displaymath} as $t \to 0$ (cf.~\cite[Sec.~VI.4]{chavel:84} and~\cite[Thm.~1]{roth:84}). Recall that $d \ge 2$ and $\vol_1 \Mnull := \sum_j \ell (\Ij)$, i.e. the sum over the length of each edge. \end{rem} %------------------------------------------------------------ \subsection{Periodic graphs} \label{ssec:per.graph} %------------------------------------------------------------ Suppose we have an infinite graph $X_0$ on which a discrete, finitely generated group $\Gamma$ operates such that the quotient $M_0 := X_0 / \Gamma$ is a finite graph. In the same way as in the previous sections, we can associate a family of graph-like compact manifolds $\Meps$ to the graph $M_0$. By a lifting procedure we obtain a (non-compact) covering manifold $X_\eps$ of $\Meps$ with deck transformation group $\Gamma$, i.e., $\Meps$ is isometric to $X_\eps / \Gamma$. Furthermore, $X_\eps$ is a graph-like manifold collapsing to the infinite graph $X_0$. We are interested in spectral properties of the non-compact manifolds $X_\eps$. Assuming that $\Gamma$ is abelian, we can apply Floquet theory (a non-commutative version is work in progress, cf.~\cite{lledo-post:pre03}). Instead of investigating $\laplacian {X_\eps}$ we analyze a family of operators $\laplacianT \Meps$, $\theta \in \hat \Gamma$, where $\hat \Gamma$ is the dual group, i.e., the group of homomorphisms from $\Gamma$ into the unit circle $\Torus^1$. The operator $\laplacianT \Meps$ acts on a complex line bundle over the compact manifold $\Meps$, or equivalently, over the closure of a fundamental domain $D_\eps \subset X_\eps$ with $\theta$-periodic boundary conditions. We call the closure $\overline D_\eps$ a \emph{period cell} and denote it also by $\Meps$ (for details see e.g.\ \cite{reed-simon-4} or~\cite{post:03a}). The direct integral decomposition implies \begin{displaymath} \spec \laplacian {X_\eps} = \bigcup_{k \in \N} B_k(\eps), \qquad B_k(\eps) := \set{\EWT k \Meps} {\theta \in \hat \Gamma} \end{displaymath} where $B_k(\eps)$ is a compact subset of $[0,\infty)$, called the \emph{$k$-th band}.\footnote{Note that $\hat \Gamma$ is connected iff $\Gamma$ is torsion free, e.g., if $\Gamma=\Z \times \Z_2$ then $\hat \Gamma \cong \Torus^1 \times \Z_2$ which is homeomorphic to two disjoint copies of the unit circle $\Torus^1$. Therefore, the bands $B_k(\eps)$ being the continuous image of $\hat \Gamma$ under the map $\theta \mapsto \EWT k \Meps$ need not to be intervals. } A similar assertion holds for the limit operator on $X_\eps$. %------------------------------------------------------------ \subsection{Spectral gaps} %------------------------------------------------------------ We are interested in the existence of \emph{spectral gaps} of the operator $\laplacian {X_\eps}$, i.e., the existence of an interval $[a,b]$, $00$ since $X_0$ is non-compact and $X_0/\Gamma$ is compact. Denote $r:=r_0 + r_1 + \dots + r_a$. We assume that $X_0$ is the (metric) Cayley graph accociated to $\Gamma$ w.r.t.\ the canonical generators $\eps_1, \dots, \eps_r$ ($\eps_j$ equals $1$ at the $j$-th component and $0$ otherwise), i.e., the set of vertices is $\Gamma$ and two vertices $\gamma_1,\gamma_2$ are connected iff $\gamma_2=\eps_j \gamma_1$ for some $1 \le j \le r$ (see Figure~\ref{fig:cayley}). For simplicity, we assume that each edge has length $1$. Note that $X_0$ is $2r$-regular, i.e., each vertex meets $2r$ edges. \begin{figure}[h] \begin{center} %------------------------------------------------------------ % \input{cayley.pstex_t} \begin{picture}(0,0)% \includegraphics{graph5.pstex}% \end{picture}% \setlength{\unitlength}{4144sp}% \begin{picture}(4422,728)(214,-286) \put(2971,-286){$X_0$}% \put(4636,-286){$M_0$}% \end{picture} %------------------------------------------------------------ \caption{The Cayley graph associated to the group $\Gamma = \Z \times \Z_2$ and the corresponding period cell. Note that $\laplacian {X_0}$ has no spectral gaps} \label{fig:cayley} \end{center} \end{figure} We want to calculate the eigenfunctions and eigenvalues of the $\theta$-periodic operator $\laplacianT \Mnull$, i.e., functions $u_j$ on $\Ij \cong [0,1]$ satisfying $-u_j''=\lambda u_j$ with the boundary conditions \begin{equation} \label{eq:bd.cond} u_j(0) = u(0), \quad \eu^{-\im\theta_j} u_j(1) = u(0) \quad \text{and} \quad \sum_{k=1}^r \bigl( \eu^{-\im \theta_k} u_k'(1) - u_k'(0) \bigr) = 0 \end{equation} for all $j=1, \dots, r$. Here, $\theta \in \Torus^{r_0} \times \Torus_{p_1}^{r_1} \times \dots \times \Torus_{p_a}^{r_a}$ where $\Torus_p := \set{\xi \in \R/\Z}{\eu^{\im \xi p} = 1}$ is the group of $p$-th unit roots (isomorphic to $\Z_p$). Note that we have identified $\theta \in \Torus^r$ with $\gamma \mapsto \eu^{\im \theta \cdot \gamma} \in \hat \Gamma$. If $\lambda=\omega^2 > 0$ (and $\omega>0$) we make the Ansatz \begin{displaymath} u_j(x):=Z \cos(\omega x) + A_j \sin(\omega x) \end{displaymath} and arrive at the coefficient matrix $M(\omega)$ given by \begin{displaymath} {\scriptsize \begin{pmatrix} \eu^{-\im \theta_1} \sin \omega & 0 & \cdots & 0 & (\eu^{-\im \theta_1} \cos \omega - 1) \\ 0 & \eu^{-\im \theta_2} \sin \omega & \cdots & 0 & (\eu^{-\im \theta_2} \cos \omega - 1) \\ \vdots &&& 0 & \vdots \\ 0 & 0 & \cdots & \eu^{-\im \theta_r} \sin \omega & (\eu^{-\im \theta_r} \cos \omega - 1) \\ (\eu^{-\im \theta_1} \cos \omega - 1) & (\eu^{-\im \theta_2} \cos \omega - 1) & \cdots & (\eu^{-\im \theta_r} \cos \omega - 1) & (- \sin \omega \sum_{k=1}^r \eu^{-\im \theta_k}) \end{pmatrix} } \end{displaymath} for the variables $A_1, \dots, A_r, Z$. A direct calculation shows that \begin{displaymath} \det M(\omega) = 2 (\sin^{r-1}\omega) \eu^{-\im \sum_k \theta_k} \sum_k (\cos \omega - \cos \theta_k). \end{displaymath} Non-trivial solutions of the eigenvalue problem exist iff $\det M(\omega)=0$, i.e. if $\omega = \ell \pi$, $\ell \in \N$, or \begin{equation} \label{eq:omega.theta} \cos \omega = \frac 1 r \sum_{k=1}^r \cos \theta_k. \end{equation} The solutions $\omega = \ell \pi$ correspond to Dirichlet eigenfunctions on each edge and produce therefore bands degenerated to a point $\{ (\ell\pi)^2 \}$. The multiplicity is $r-1$ provided $\theta \ne 0$ (if $\ell$ is even) resp.\ $\theta \ne \pi$ (if $\ell$ is odd) and $r+1$ if $\theta = 0$ resp.\ $\theta = \pi$ (modolo $2\pi$). If $\omega \ne \ell \pi$, the eigenvalues are simple. Note that the bands at $\omega^2 = (\ell \pi)^2$ do not overlap, but \emph{touch} each other. For $\omega=0$, we need a special Ansatz. The only possibility is the case of periodic boundary conditions ($\theta=0$); the eigenvalue is simple. \begin{thm} \label{thm:gaps.ex} If one of the orders $p_1, \dots, p_a$ is odd, the operator $\laplacian {X_0}$ has infinitely many spectral gaps. In particular, Theorem~\ref{thm:gaps} applies. If all orders $p_1, \dots, p_a$ are even, we have $\spec \laplacian {X_0} = [0, \infty)$. \end{thm} \begin{proof} We analyze the behaviour of $\omega$ in dependence of the continuous parameters $\theta_1, \dots, \theta_{r_0} \in \Torus^{r_0}$ given by the relation~\eqref{eq:omega.theta}. We have gaps iff $\frac 1 r \sum_{k=1}^r \cos \theta_k$ in~\eqref{eq:omega.theta} does not cover the whole interval $[-1,1]$. We reach the maximal value $1$ iff all $\theta_j=0$ ($j=1, \dots, r$) and the minimal value $-1$ iff all $\theta_j=\pi$ ($j=1, \dots, r$). The latter can only occur if all group orders are even. Note that in this case, the whole interval $[-1,1]$ can be covered by an appropriate choice of the $\theta_j$'s, $j=r_0+1, \dots, r$. \end{proof} We cannot say anything about the (non-)existence of gaps in the case when all orders $p_1, \dots, p_a$ are even. If e.g.\ $\Gamma = \Z \times \Z_2$, the bands do not overlap, but touch each other and fill the whole half line $[0,\infty)$ (cf.\ Remark~\ref{rem:overlap}). %------------------------------------------------------------ \subsection{Cayley graphs with loops} %------------------------------------------------------------ If we set one (or more) of the group orders $p_j$ equal to $1$ we formally attach a loop (or more) at each vertex (see Figure~\ref{fig:cayley.loop}). \begin{figure}[h] \begin{center} %------------------------------------------------------------ % \input{cayley.loop.pstex_t} \begin{picture}(0,0)% \includegraphics{graph6.pstex}% \end{picture}% \setlength{\unitlength}{4144sp}% \begin{picture}(4422,1074)(214,-373) \put(2971,-286){$X_0$}% \put(4636,-286){$M_0$}% \end{picture} %------------------------------------------------------------ \caption{The Cayley graph associated to the group $\Gamma = \Z \times \Z_2 \times \Z_1$, where the trivial group $\Z_1$ leads to the attachment of a loop at each vertex. On the right, the corresponding period cell is shown. Note that $\laplacian {X_0}$ has spectral gaps in contrast to the example without loops.} \label{fig:cayley.loop} \end{center} \end{figure} \begin{thm} \label{thm:gaps.ex.loop} The Laplacian of a Cayley graph associated to an arbitrary finitely generated abelian discrete group $\Gamma$ has spectral gaps provided we attach at each vertex a fixed number of loops. \end{thm} \begin{proof} Formally, the assertion follows from Theorem~\ref{thm:gaps.ex}. Note that $\hat \Z_1 = \{0\}$, i.e., the corresponding component of $\theta$ cannot be $\pi$ and therefore, the minimum $-1$ cannot be achieved in~\eqref{eq:omega.theta}. \end{proof} This is an analogue of gap generation by decoration of the graph as discussed by Aizenman and Schenker in \cite{aizenman-schenker:00}. %------------------------------------------------------------ \subsection{Cayley graphs and the borderline case} %------------------------------------------------------------ In the borderline case, the eigenvalue problem of the limit operator is more complicated. Here, functions $u_j$ on $\Ij \cong [0,1]$ satisfy $-u_j''=\lambda u_j$ with the boundary conditions \begin{equation} \label{eq:bd.cond.2} u_j(0) = u(0), \quad \eu^{-\im\theta_j} u_j(1) = u(0) \quad \text{and} \quad \sum_{k=1}^r \bigl( \eu^{-\im \theta_k} u_k'(1) - u_k'(0) \bigr) = c \lambda u(0) \end{equation} for all $j=1, \dots, r$ where $c^2$ is the volume of the (unscaled) vertex neighbourhood (cf.~\eqref{def:lim.op.border}). Again, we make the Ansatz \begin{displaymath} u_j(x):=Z \cos(\omega x) + A_j \sin(\omega x) \end{displaymath} with $\lambda=\omega^2 > 0$ and arrive at the coefficient matrix $M_c(\omega)$ given by \begin{displaymath} {\scriptsize \begin{pmatrix} \eu^{-\im \theta_1} \sin \omega & 0 & \cdots & 0 & (\eu^{-\im \theta_1} \cos \omega - 1) \\ 0 & \eu^{-\im \theta_2} \sin \omega & \cdots & 0 & (\eu^{-\im \theta_2} \cos \omega - 1) \\ \vdots &&& 0 & \vdots \\ 0 & 0 & \cdots & \eu^{-\im \theta_r} \sin \omega & (\eu^{-\im \theta_r} \cos \omega - 1) \\ (\eu^{-\im \theta_1} \cos \omega - 1) & (\eu^{-\im \theta_2} \cos \omega - 1) & \cdots & (\eu^{-\im \theta_r} \cos \omega - 1) & -(\sin \omega \sum_{k=1}^r \eu^{-\im \theta_k} + c \omega) \end{pmatrix} } \end{displaymath} for the variables $A_1, \dots, A_r, Z$. Note that formally the case $c=0$ corresponds to the Kirchhoff boundary condition case and $M_0(\omega) = M(\omega)$. In a similar way as before we have \begin{displaymath} \det M(\omega) = (\sin^{r-1}\omega) \eu^{-\im\sum_k \theta_k} \bigl( 2 \sum_k (\cos \omega - \cos \theta_k) - c \omega \sin \omega \bigr). \end{displaymath} Non-trivial solutions of the eigenvalue problem exist iff $\det M(\omega)=0$, i.e. if $\omega = \ell \pi$, $\ell \in \N$, or \begin{equation} \label{eq:omega.theta.2} \cos \omega - \frac1{2rc} = \frac 1 r \sum_{k=1}^r \cos \theta_k. \end{equation} Again, the solutions $\omega = \ell \pi$ correspond to Dirichlet eigenfunctions on each edge and produce therefore bands degenerated to a point $\{ (\ell\pi)^2 \}$. The situation in the case $c>0$ is more complicated since the relation~\eqref{eq:omega.theta.2} is no longer $2\pi$-periodic in $\omega$. In such a case the spectrum could be more complicated; recall the example of a lattice graph discussed in~\cite{exner-gawlista:96} shows where number-theoretic properties of parameters play role. This interesting question will be considered separately. %------------------------------------------------------------ \section*{Acknowledgements} %------------------------------------------------------------ The authors appreciate P.~Kuchment who made available to them the paper \cite{kuchment-zeng:03} prior to publication. O.P. is grateful for the hospitality extended to him at the Nuclear Physics Institute of Czech Academy of Sciences where a part of this work was done. 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Lett. \textbf{89} (2002), 075505. \end{thebibliography} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% TeX-master: t %%% End: ---------------0312101028453 Content-Type: application/postscript; name="graph1.pstex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="graph1.pstex" %!PS-Adobe-2.0 EPSF-2.0 %%Title: edge.vertex.pstex %%Creator: fig2dev Version 3.2 Patchlevel 3c %%CreationDate: Tue Dec 9 19:44:44 2003 %%For: post@post (Olaf Post) %%BoundingBox: 0 0 334 158 %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} 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1347 446 l 1340 439 l 1328 426 l 1314 410 l 1298 391 l 1282 372 l 1268 353 l 1255 336 l 1246 321 l 1239 307 l 1234 294 l 1231 282 l 1230 270 l 1231 260 l 1232 249 l 1236 239 l 1240 228 l 1245 218 l 1252 208 l 1260 198 l 1268 188 l 1278 180 l 1288 172 l 1298 165 l 1308 160 l 1319 156 l 1329 152 l 1340 151 l 1350 150 l 1360 151 l 1371 152 l 1381 156 l 1392 160 l 1402 165 l 1412 172 l 1422 179 l 1432 187 l 1440 196 l 1448 206 l 1455 215 l 1460 225 l 1464 235 l 1468 244 l 1469 253 l 1470 263 l 1469 274 l 1465 286 l 1458 299 l 1448 313 l 1435 328 l 1418 345 l 1400 362 l 1381 379 l 1366 392 l 1355 401 l 1351 404 l 1350 405 l gs col0 s gr % Polyline n 450 900 m 450 901 l 447 904 l 440 911 l 428 924 l 414 940 l 398 959 l 382 978 l 368 997 l 355 1014 l 346 1029 l 339 1043 l 334 1056 l 331 1068 l 330 1080 l 331 1090 l 332 1101 l 336 1111 l 340 1122 l 345 1132 l 352 1142 l 360 1152 l 368 1162 l 378 1170 l 388 1178 l 398 1185 l 408 1190 l 419 1194 l 429 1198 l 440 1199 l 450 1200 l 460 1199 l 471 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901 l 1347 904 l 1340 911 l 1328 924 l 1314 940 l 1298 959 l 1282 978 l 1268 997 l 1255 1014 l 1246 1029 l 1239 1043 l 1234 1056 l 1231 1068 l 1230 1080 l 1231 1090 l 1232 1101 l 1236 1111 l 1240 1122 l 1245 1132 l 1252 1142 l 1260 1152 l 1268 1162 l 1278 1170 l 1288 1178 l 1298 1185 l 1308 1190 l 1319 1194 l 1329 1198 l 1340 1199 l 1350 1200 l 1360 1199 l 1371 1198 l 1381 1194 l 1392 1190 l 1402 1185 l 1412 1178 l 1422 1171 l 1432 1163 l 1440 1154 l 1448 1144 l 1455 1135 l 1460 1125 l 1464 1115 l 1468 1106 l 1469 1097 l 1470 1088 l 1469 1076 l 1465 1064 l 1458 1051 l 1448 1037 l 1435 1022 l 1418 1005 l 1400 988 l 1381 971 l 1366 958 l 1355 949 l 1351 946 l 1350 945 l gs col0 s gr % Polyline n 1800 900 m 1800 901 l 1797 904 l 1790 911 l 1778 924 l 1764 940 l 1748 959 l 1732 978 l 1718 997 l 1705 1014 l 1696 1029 l 1689 1043 l 1684 1056 l 1681 1068 l 1680 1080 l 1681 1090 l 1682 1101 l 1686 1111 l 1690 1122 l 1695 1132 l 1702 1142 l 1710 1152 l 1718 1162 l 1728 1170 l 1738 1178 l 1748 1185 l 1758 1190 l 1769 1194 l 1779 1198 l 1790 1199 l 1800 1200 l 1810 1199 l 1821 1198 l 1831 1194 l 1842 1190 l 1852 1185 l 1862 1178 l 1872 1171 l 1882 1163 l 1890 1154 l 1898 1144 l 1905 1135 l 1910 1125 l 1914 1115 l 1918 1106 l 1919 1097 l 1920 1088 l 1919 1076 l 1915 1064 l 1908 1051 l 1898 1037 l 1885 1022 l 1868 1005 l 1850 988 l 1831 971 l 1816 958 l 1805 949 l 1801 946 l 1800 945 l gs col0 s gr % Polyline n 2250 900 m 2250 901 l 2247 904 l 2240 911 l 2228 924 l 2214 940 l 2198 959 l 2182 978 l 2168 997 l 2155 1014 l 2146 1029 l 2139 1043 l 2134 1056 l 2131 1068 l 2130 1080 l 2131 1090 l 2132 1101 l 2136 1111 l 2140 1122 l 2145 1132 l 2152 1142 l 2160 1152 l 2168 1162 l 2178 1170 l 2188 1178 l 2198 1185 l 2208 1190 l 2219 1194 l 2229 1198 l 2240 1199 l 2250 1200 l 2260 1199 l 2271 1198 l 2281 1194 l 2292 1190 l 2302 1185 l 2312 1178 l 2322 1171 l 2332 1163 l 2340 1154 l 2348 1144 l 2355 1135 l 2360 1125 l 2364 1115 l 2368 1106 l 2369 1097 l 2370 1088 l 2369 1076 l 2365 1064 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