Content-Type: multipart/mixed; boundary="-------------0601111605739"

This is a multi-part message in MIME format.

---------------0601111605739
Content-Type: text/plain; name="06-10.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="06-10.comments"

36 pages, 4 figures
---------------0601111605739
Content-Type: text/plain; name="06-10.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="06-10.keywords"

convergence of spectrum, metric graph, non-compact manifold, Laplacian
---------------0601111605739
Content-Type: application/x-tex; name="conv-res-graph.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="conv-res-graph.tex"

%-----------------------------------------------------------------------------
% conv-res.tex                     Singular limits and spectral convergence
%                                  Graph-like manifolds and other examples
%
% Olaf Post, 2004-08-31
% "Anderung: 2005-12-21, 2005-11-10, 2005-09-17, 2005-08-16
%            2005-04-04, 2005-03-02, 2005-02-01, 2004-09-07, 2004-09-01
%-----------------------------------------------------------------------------
\documentclass[12pt,reqno]{amsart}

\usepackage{a4}
\addtolength{\textheight}{2.2cm}   % to save a little bit paper in the
                                  % draft ...
\usepackage[german,english]{babel}



%------------------------------------------------------------
% begin of general math definitions
%------------------------------------------------------------

%\input{/home/post/Aktuell/mymath.tex}
%-----------------------------------------------------------------------------
% mymath.tex                       General math abbreviations and commands
%
%
% Olaf Post, 2005-11-30
% "Anderung: 2005-11-30
%-----------------------------------------------------------------------------
%\documentclass[12pt,reqno,letterpaper]{amsart}

%\usepackage{a4}
%\addtolength{\textheight}{2cm}   % to save a little bit paper in the
                                 % draft ...
%\usepackage[german,english]{babel}
\usepackage{amsmath, amssymb}
%\usepackage{amscd}                  % use commutative diagrams
%\usepackage{diagrams}               % use commutative diagrams
\usepackage{graphicx}                      % for graphics
%\usepackage{epsfig}                 % for xfig still using epsfig
\newcommand{\color}[2][{}]{}        % figure-x.pstex_t file contains this
%                                    % command which is not implemented???

\usepackage{bbm}         % blackboard (see: /usr/share/texmf/mytex/bbm.dvi)

%\usepackage[notref,notcite]{showkeys}      % show labels (testing phase)

%------------------------------------------------------------
% theorem environments
%------------------------------------------------------------

%\swapnumbers                    % 1.1 Theorem instead of Theorem 1.1
\theoremstyle{plain}            % body italics
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{theorem*}{Theorem}
\newtheorem{maintheorem}{Main Theorem}[section]
\newtheorem*{maintheorem*}{Main Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}       % body roman
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\theoremstyle{remark}           
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem*{notation*}{Notation}


\newcommand{\Sec}[1]{Section~\ref{sec:#1}}
\newcommand{\Secs}[2]{Sections~\ref{sec:#1},~\ref{sec:#2}}
\newcommand{\App}[1]{Appendix~\ref{sec:#1}}
\newcommand{\Subsec}[1]{Subsection~\ref{sec:#1}}
\newcommand{\Ex}[1]{Example~\ref{ex:#1}}
\newcommand{\Eq}[1]{Equation~\eqref{eq:#1}}
\newcommand{\Thm}[1]{Theorem~\ref{thm:#1}}
\newcommand{\Thms}[2]{Thms.~\ref{thm:#1},~\ref{thm:#2}}
\newcommand{\Lem}[1]{Lemma~\ref{lem:#1}}
\newcommand{\Cor}[1]{Corollary~\ref{cor:#1}}
\newcommand{\Rem}[1]{Remark~\ref{rem:#1}}
\newcommand{\Def}[1]{Definition~\ref{def:#1}}
\newcommand{\Not}[1]{Notation~\ref{not:#1}}
\newcommand{\Fig}[1]{Figure~\ref{fig:#1}}

%------------------------------------------------------------
% Equation numbering
%------------------------------------------------------------
\numberwithin{equation}{section}

%------------------------------------------------------------
% Math names 
%------------------------------------------------------------
\DeclareMathOperator{\card}   {card}
\DeclareMathOperator{\dist}   {dist}
\DeclareMathOperator{\dom}    {dom}
\DeclareMathOperator{\ran}    {ran}
\DeclareMathOperator{\Hom}    {Hom}
\DeclareMathOperator{\supp}   {supp}
\DeclareMathOperator{\vol}    {vol}
\DeclareMathOperator{\length} {length}
\DeclareMathOperator{\const}  {const}
\DeclareMathOperator{\Ric}    {Ric}
\DeclareMathOperator{\Scal}   {Scal}
\DeclareMathOperator{\injrad} {inj\,rad}
\DeclareMathOperator{\intr}   {int}  % topological interiour

\newcommand{\specsymb} {\sigma} % symbol for spectrum

\renewcommand{\Re}     {\mathrm {Re}\,}
\renewcommand{\Im}     {\mathrm {Im}\,}

\newcommand{\spec}[2][{}]   {\sigma_{\mathrm{#1}}(#2)}
\newcommand{\essspec}[1]{\spec[ess] {#1}}
\newcommand{\disspec}[1]{\spec[disc]{#1}}

\newcommand{\GL}{\mathrm{GL}}
\newcommand{\SL}{\mathrm{SL}}

%------------------------------------------------------------
% 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
{\stelldrueber[.45ex]{$\scriptscriptstyle\circ$}{${#1}$}}


%------------------------------------------------------------
% general math shortcuts
%------------------------------------------------------------
\newcommand{\R}{\mathbb{R}} % symbol for real numbers
\newcommand{\C}{\mathbb{C}} % symbol for complex numbers
\newcommand{\N}{\mathbb{N}} % symbol for natural numbers
\newcommand{\Z}{\mathbb{Z}} % symbol for integers
\newcommand{\Sphere}{\mathbb{S}} % symbol for sphere
\newcommand{\Torus}{\mathbb{T}} % symbol for torus


\newcommand{\eps}{\varepsilon} % shortcut
\renewcommand{\phi}{\varphi}   % shortcut
\newcommand{\e}{\mathrm e}  %Euler number
\newcommand{\im}{\mathrm i} % complex unit
\newcommand{\dd}{\,\mathrm d} % symbol for integration d

\newcommand{\wt}{\widetilde}           % shortcut
\newcommand {\qf}[1]{\mathfrak{#1}}    % font for quadratic forms
\newcommand {\Exp}[2][{}]{\mathbb{E}_{{#1}}{(#2)}} % Expectation value

%------------------------------------------------------------
% Banach spaces
%------------------------------------------------------------
\newcommand{\HS}{\mathcal H}           % symbol for Hilbert/Sobolev space
\newcommand{\Lin}[1]{\mathcal B({#1})} % symbol for bounded linear operators

\newcommand{\Sobsymb} {\mathsf H}      % symbol for Sobolev space
\newcommand{\Contsymb} {\mathsf C}     % symbol for cont. space
\newcommand{\Lsymb}    {\mathsf L}     % symbol for int L-spaces
\newcommand{\lsymb}    {\ell}          % symbol for int l-spaces

%------------------------------------------------------------
% spaces of continuous functions
%------------------------------------------------------------
\newcommand{\Ci} [1]{\Contsymb^\infty ({#1})}
                                       % C^\infty(#1)-spaces
\newcommand{\Cci}[1]{\Contsymb_{\mathrm c}^\infty ({#1})} 
                                       % C_c^\infty(#1)-sp
\newcommand{\Cont}[2][{}]{\Contsymb^{#1}({#2})}
                                       % space of cont (C^k) functions
\newcommand{\Contb}[2][{}]{\Contsymb_{\mathrm b}^{#1}({#2})}
                                       % space of C^k-bounded functions 
\newcommand{\Contc}[2][{}]{\Contsymb_{\mathrm c}^{#1}({#2})}   
                                       % space of C^k fct. wit compact supp
%------------------------------------------------------------
% L_p-spaces
%------------------------------------------------------------
\newcommand{\Lp}[2][p]{\Lsymb_{#1}({#2})} % L_#1(#2)-spaces
\newcommand{\Lploc}[2][p]{\Lsymb_{#1,\mathrm{loc}}({#2})} 
                                       % same, but loc
\newcommand{\lp}[2][p]{\lsymb_{#1}({#2})}% l_#1(#2)-spaces

\newcommand{\Lsqr}[2][{}]{\Lsymb_2^{#1}({#2})} % L_2(#1)-spaces 
                                       % (with optional subscript
\newcommand{\Lsqrloc}[2]{\Lsymb_{2,\mathrm{loc}}({#1})} 
                                       % L_{2,loc}(#1)-spaces
\newcommand{\lsqr}[1]{\lsymb_2({#1})}           % l_2(#1)-spaces

\newcommand{\Linfty}[2][{}]{\Lsymb_\infty^{#1}({#2})} % L_2(#1)-spaces 
                                       % (with optional subscript
\newcommand{\Linftyloc}[2]{\Lsymb_{\infty,\mathrm{loc}}({#1})} 
                                       % L_{2,loc}(#1)-spaces
\newcommand{\linfty}[1]{\lsymb_\infty({#1})}           % l_2(#1)-spaces

%------------------------------------------------------------
% H-Sobolev spaces
%------------------------------------------------------------
\newcommand{\Sob}[2][1]{\Sobsymb^{#1}({#2})} % Sobolev spaces
\newcommand{\Sobn}[2][1]{{\overcirc {\Sobsymb}{}^{#1}({#2})}}
                                       % Sob-Raum H_0

%------------------------------------------------------------
% norms and squared norms
%------------------------------------------------------------
\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



%------------------------------------------------------------
% inner products
%------------------------------------------------------------
\newcommand{\iprod}[3][{}]{\langle{#2},{#3}\rangle_{#1}}  % inner product
\newcommand{\bigiprod}[3][{}]{\bigl\langle{#2},{#3}\bigr\rangle_{#1}}
\newcommand{\Bigiprod}[3][{}]{\Bigl\langle{#2},{#3}\Bigr\rangle_{#1}}

%------------------------------------------------------------
% topological/set symbols
%------------------------------------------------------------
\newcommand{\set}[2]{\{ \, #1 \, | \, #2 \, \} } % set { #1 | #2 }
\newcommand{\bigset}[2]{\bigl\{ \, #1 \, \bigl|\bigr. \, #2 \, \bigr\} }
\newcommand{\Bigset}[2]{\Bigl\{ \, #1 \, \Bigl|\Bigr. \, #2 \, \Bigr\} }

\newcommand{\map}[3]{{#1}\colon{#2}\longrightarrow{#3}} % maps

\newcommand{\bd}  {\partial}                % symbol for boundary of a set
\newcommand{\clo}[1]{\overline{{#1}}} %            symbol for closure

\newcommand{\compl}[1]{#1^{\mathrm c}}       % complement of a set
\newcommand{\dcup}{\mathrel{\uplus}}               % symbol for disjoint union
\newcommand{\bigdcup}{\operatorname*{\biguplus}}
%\newcommand{\bigdcup}{\stelldrueber[.45ex]%
%   {$\scriptscriptstyle \bullet\mspace{1mu}$}{$\bigcup$}}
\newcommand{\disjcup}{\mathrel{\overline{\dcup}}} % symbol for disjoint union
\newcommand{\bigdisjcup}{\operatorname*{\overline{\biguplus}}}
%\newcommand{\bigdcup}{\bigcup^{\bullet}}        % symbol for disjoint union


\newcommand{\restr}[1]{{\restriction}_{#1}} % symbol for map restriction

%------------------------------------------------------------
% other symbols
%------------------------------------------------------------
\newcommand{\conj}[1]{\overline {{#1}}}       % symbol for complex conjugation
\newcommand{\orth}{\bot}                    % symbol for orthogonality
\newcommand{\normder}{\partial_\mathrm{n}}  % symbol for normal derivative

\newcommand{\1}{\mathbbm 1}                    % blackboard 1
%\renewcommand{\1}{\mathbf 1}                  % if bbm does'nt work
                                                 % uncomment this!
%------------------------------------------------------------
% symbol for Laplacians
%------------------------------------------------------------
\newcommand{\Neu}{{\mathrm N}}              % symbol for Neumann bd cond
\newcommand{\Dir}{{\mathrm D}}              % symbol for Dirichlet bd cond
\newcommand{\NeuDir}{{\Neu \Dir}}             % symbol for mixed bd cond
\newcommand{\laplacian}[2][{}]{\Delta_{{#2}}^{{#1}}} 
                                            % symbol for Laplacian on mfd
\newcommand{\laplacianD}[1]{\laplacian[\Dir]{#1}} % symb f Dir-Laplacian
\newcommand{\laplacianN}[1]{\laplacian[\Neu]{#1}} % symb f Neu-Laplacian
\newcommand{\laplacianND}[1]{\laplacian[\NeuDir]{#1}} % symb f ND-Laplacian

%------------------------------------------------------------
% symbols for eigenvalues (EigenWerte)
%------------------------------------------------------------
\newcommand{\EW}[3][{}]{\lambda^{{#1}}_{#2}({#3})}
                                             % Eigenvalue of Laplacian on #2
\newcommand{\EWD}[2]{\EW[\Dir]{#1}{#2}}      % EV of Dir Laplacian
\newcommand{\EWN}[2]{\EW[\Neu]{#1}{#2}}      % EV of Neu Laplacian
\newcommand{\EWND}[2]{\EW[\NeuDir]{#1}{#2}} % EV of NeuDir Laplacian
%------------------------------------------------------------



%------------------------------------------------------------
% remarks in draft
%------------------------------------------------------------
\newcommand{\look}[1]{\textbf{*}%\rule{3ex}{2.5mm}
    \footnote{ #1 }}
%\reversemarginpar     % set marginpar on inner side 
%\newcommand{\look}[1]{\textbf{*}%\rule{3ex}{2.5mm}
%    \marginpar{\hspace*{-1ex}\fbox{\parbox{2cm}{
%       \textbf {\selectlanguage{german} \textit{\tiny \noindent #1}}}}}}



%------------------------------------------------------------
% end of general math definitions
%------------------------------------------------------------

%------------------------------------------------------------
% begin of specific math definitions
%------------------------------------------------------------


% Symbols for vertex/edge indices on M_\eps
\newcommand{\vxeps}{{\eps,v}}
\newcommand{\edeps}{{\eps,e}}
\newcommand{\vxedeps}{{\eps,v,e}}




%------------------------------------------------------------
% begin of the document
%------------------------------------------------------------

\begin{document}
\title[Spectral convergence of quasi-one-dimensional
  spaces]{Spectral convergence of non-compact quasi-one-dimensional
  spaces}

\author{Olaf Post} 
\address{Department of Mathematics, 
  University of Kentucky,
  751~Patterson~Office~Tower, 
  Lexington, Kentucky 40506-0027, USA}
\email{post@ms.uky.edu}
%\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 strructures}

%------------------------------------------------------------
% Abstract.
%------------------------------------------------------------

\begin{abstract}
  We consider a family of non-compact manifolds $X_\eps$ (``graph-like
  manifolds'') approaching a metric graph $X_0$ and establish
  convergence results of the related natural operators, namely the
  (Neumann) Laplacian $\laplacian {X_\eps}$ and the generalised
  Neumann (Kirchhoff) Laplacian $\laplacian {X_0}$ on the metric
  graph.  In particular, we show the norm convergence of the
  resolvents, spectral projections and eigenfunctions. As a
  consequence, the essential and the discrete spectrum converge as
  well. Neither the manifolds nor the metric graph need to be compact,
  we only need some natural uniformity assumptions.  We provide
  examples of manifolds having spectral gaps in the essential
  spectrum, discrete eigenvalues in the gaps or even manifolds
  approaching a fractal spectrum. The convergence results will be
  given in a completely abstract setting dealing with operators acting
  in different spaces, applicable also in other geometric situations.
\end{abstract}

\maketitle


%\enlargethispage*{2cm}

\section{Introduction}
\label{sec:intro}

The aim of this article is to show that non-compact
quasi-one-dim\-ensional spaces can be approximated by the underlying
metric graph.  A \emph{metric} or \emph{quantum graph} is a graph
considered as one-dimensional space where each edge is assigned a
length. A \emph{quasi-one-dimensional space} consists of a family of
\emph{graph-like} manifolds, i.e., a family of manifolds $X_\eps$
shrinking to the underlying metric graph $X_0$. The family of
graph-like manifolds is constructed of building blocks $U_\vxeps$ and
$U_\edeps$ for each vertex $v \in V$ and edge $e \in E$ of the graph,
respectively (cf.~\Fig{edge.vertex}).
%------------------------------------------------------------
\begin{figure}[h]
  \begin{center}
%    \input{edge.vertex.pstex_t}
\begin{picture}(0,0)%
 \includegraphics{graph1.eps}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
\begin{picture}(5282,2500)(374,-1970)
  \put(1591,-586){$U_\edeps$}%
  \put(1561,-1674){$e$}%
  \put(5641,-331){$U_\vxeps$}%
  \put(5416,-1771){$v$}%
  \put(4579,374){$\eps$}%
  \put(3169,-436){$\eps$}%
\end{picture}
    \caption{The associated edge and vertex neighbourhoods with
      $F_\eps=\Sphere^1_\eps$, i.e., $U_\edeps$ and $U_\vxeps$ are
      $2$-dimensional manifolds with boundary.}
    \label{fig:edge.vertex}
  \end{center}
\end{figure}
%------------------------------------------------------------
The cross section of the edge neighbourhood $U_\edeps$ as well as the
boundary component of $U_\vxeps$, where $U_\edeps$ meet, consists of a
manifold $F_\eps$ with radius of order $\eps$.  The cross section
could have a boundary resulting in a manifold $X_\eps$ \emph{with}
boundary. In addition, the vertex neighbourhoods $U_\vxeps$ are
assumed to be small.  The simplest example is the $\eps$-neighbourhood
$X_\eps$ of a quantum graph $X_0$ embedded in $\R^2$. In this case,
the cross section is $F_\eps=(-\eps,\eps)$.  A simple boundaryless
example is given by the \emph{surface} of a pipeline network according
to the underlying graph $X_0$ (cf.~\Fig{mfd}). Here, the cross section
consists of a circle of radius $\eps$.
%------------------------------------------------------------
\begin{figure}[htb]
  \begin{center}
%   \input{graph2.pstex_t}
    \begin{picture}(0,0)%
      \includegraphics{graph2.eps}%
    \end{picture}%
    \setlength{\unitlength}{4144sp}%
    \begin{picture}(6263,3204)(308,-2913)
      \put(3601,-2176){$X_\eps$}
      \put(6571,-1546){$U_{\eps,v}$} 
      \put(4611,-1581){$U_{\eps,e}$}
      \put( 316,-2176){$X_0$}
      \put(2926,-1681){$v$}
      \put(1476,-1451){$e$}
    \end{picture}
    \caption{On the left, we have the graph $X_0$, on the right, the
      associated family of graph-like manifolds. Here,
      $F_\eps=\Sphere^1_\eps$ is the transversal section of radius
      $\eps$ and $X_\eps$ is a $2$-dimensional manifold.}
    \label{fig:mfd}
  \end{center}
\end{figure}
%------------------------------------------------------------


On the graph-like manifold $X_\eps$ we consider the Laplacian $\wt
H:=\laplacian {X_\eps} \ge 0$ acting in the Hilbert space $\wt \HS :=
\Lsqr {X_\eps}$.  If $X_\eps$ has a boundary, we impose \emph{Neumann}
boundary conditions.  On the graph, we choose the natural Laplacian
$H:=\laplacian {X_0} \ge 0$, namely, the generalised Neumann
(Kirchhoff) Laplacian acting on each edge as a one-dimensional
weighted Laplacian (cf.\ Eq.~\eqref{eq:formaledge}). On each vertex,
we assume continuity and current conservation (cf.\ 
Eq.~\eqref{eq:kirchhoff}).  Note that $\laplacian {X_0}$ acts on
$\HS:=\oplus_e \Lsqr e$ where each edge $e$ is identified with the
interval $(0,\ell_e)$ ($0<\ell_e \le \infty$) --- in contrast to the
\emph{discrete} graph Laplacian acting as difference operator on the
space of vertices, $\lsqr V$. For a relation between these two
operators see~\Sec{disc.graph}.

In this article, we concentrate on the spectrum of such systems. Our
main result is the following:
%------------------------------------------------------------
\begin{maintheorem*}[\Thm{graph}]
  Suppose $X_\eps$ is a family of (non-compact) graph-like manifolds
  associated to a metric graph $X_0$. If $X_\eps$ and $X_0$ satisfy
  some natural uniformity conditions, then the resolvent of
  $\laplacian {X_\eps}$ converges in norm to the resolvent of
  $\laplacian {X_0}$ (with suitable identification operators) as $\eps
  \to 0$. In particular, the corresponding essential and discrete
  spectra converge uniformly in any bounded interval. Furthermore, the
  eigenfunctions converge as well.
\end{maintheorem*}
%------------------------------------------------------------

The \emph{uniformity conditions} are precisely stated in
\Sec{ass.graph}.  For example we need a global lower bound on the edge
length $\ell_e\ge \ell_0$ and a global upper bound on the vertex
degree $\deg v \le d_0$.  In the case when the graph $X_0$ is embedded
in $\R^2$ (cf.~\Sec{emb.graph}) the uniformity conditions mean in
particular, that we need a global bound on the curvature of an edge
and a global lower bound on the angle between two different edges at a
vertex \emph{although} both quantities do not enter into the limit
operator and space.


In contrast to previous articles (cf.~\cite{rubinstein-schatzman:01,
  kuchment-zeng:01, kuchment-zeng:03, exner-post:05}) we allow here
\emph{infinite} structures, i.e., we drop the condition of
\emph{compactness} of $X_\eps$ and $X_0$. Therefore, we cannot use the
variational principle in order to characterise the discrete spectrum.
The appropriate substitute is an abstract convergence criterion
provided in \App{abstr.mod}. The basic idea is to define a
``distance'' between the operators $\laplacian {X_0}$ and $\laplacian
{X_\eps}$ with suitable identification operators
(cf.~\Def{closeness}). We have formulated the abstract results fully
in terms of this ``distance'' in order to trace the parameter
dependence on $\eps$ of the operator $\laplacian {X_\eps}$, the
Hilbert space $\Lsqr {X_\eps}$ \emph{and} the identification operators
between the graph and the manifold. The ``distance'' can be calculated
in terms of the associated sesquilinear forms which makes the
verification quite simple in our main model. As a consequence, we show
norm resolvent convergence which implies all other convergence results
like convergence of the spectral projections, convergence of the
eigenfunctions and convergence of the spectra. Note that
in~\cite{rubinstein-schatzman:01, kuchment-zeng:01, kuchment-zeng:03,
  exner-post:05}, only convergence of eigenvalues has been
established. Our results here show, that the eigenvectors converge as
well.  We will show in a forthcoming paper that this abstract
convergence criterium has applications in other geometrical
situations.

A related result on \emph{non-compact} spaces has been established
in~\cite{saito:00}. Saito considered metric trees (allowing also
arbitrary small edges, i.e., no lower bound on $\ell_e$) together with
a suitable $\eps$-neighbourhood, but showed only \emph{weak}
convergence of the resolvents.  In~\cite{evans-saito:00} the authors
prove \emph{exact} relations (equality, inclusion) of the essential
spectrum of the Neumann Laplacian on a thickened tree (for fixed
$\eps$ in our notation) and the corresponding metric graph.


Our spectral convergence result has many applications in different
situations: First, we can consider graph-like manifolds as a kind of
toolbox in order to construct manifolds with prescribed spectrum, at
least approximately. For example, we are able to construct manifolds
with gaps in the essential spectrum (cf.~\Thms{gaps}{ew.graph}~) also
in the non-periodic case: Using the recent result on \emph{graph
  decoration} one can construct metric graphs with spectral gaps
(cf.~\cite{aizenman-schenker:00, beg:03, bgl:05, kuchment:05} and
\Sec{examples}). Our spectral convergence result then immediately
states that an associated graph-like manifold also has gaps. In the
periodic case (i.e., on covering spaces with compact quotient), we
have of course the same result once we ensure the existence of gaps on
the quantum graph. For the existence of spectral gaps on periodic
manifolds (not necessarily graph-like in our sense) we refer
to~\cite{post:03a, lledo-post:pre04, lledo-post:pre05} and the
references therein. The periodic case can often be reduced to the
spectral convergence on a compact space.

An example with arbitrary many gaps in a \emph{compact} spectral
interval is given by a fractal-like manifold in~\Thm{sierpinski}.  The
graph-like manifold is constructed according to a Sierpi\'nski graph.
It was shown in~\cite{teplyaev:98} that the discrete Laplacian on a
Sierpi\'nski graph has pure point spectrum which is purely essential
and of fractal nature. Using a nice relation between the spectrum of
the discrete graph Laplacian and the metric graph Laplacian with
constant edge length $\ell_e=\ell$ developed in~\cite{cattaneo:97}
(cf.~\Thm{disc.met}, see also~\cite{exner:97b}) we are able to
construct a family of graph-like manifolds $X_\eps$ such that the
spectrum of $\laplacian {X_\eps}$ approaches a fractal set. In
particular, the spectrum of $\laplacian {X_\eps}$ has an arbitrary (a
priori finite) number of spectral gaps in the \emph{compact} interval
$[0, \Lambda]$ provided $\eps$ is small enough. Such fractal manifolds
have been constructed in~\cite{bcg:01} in order to provide examples of
smooth spaces sharing properties of fractal spaces in large scales
(e.g.~heat kernel estimates).

  
Finally, our result shows rigorously, that the physically intuition of
modeling quasi-one-dimensional spaces by its singular limit is
correct, also on \emph{infinite} structures. Graph models have a long
history in modeling properties of networks, complicated organic
molecules~\cite{ruedenberg-scherr:53} or quite recently,
nanostructures, i.e., structures, which are too small to be considered
classically, but still too large to be described on a conventional
quantum level, see e.g.~\cite{aghh:05, kostrykin-schrader:99,
  kuchment:02, kuchment:04, kuchment:05}. On the one side quantum
graphs provide a solvable model in quantum mechanics in the sense that
many quantities can be calculated explicitely essentially by solving
systems of ODEs.  On the other side, the structure of a metric graph
is still rich enough in order to provide a good model for branched
structures.  For example, a spectral gap corresponds to ``forbidden
modes'', i.e., a particle with an energy in the gap cannot propagate
through the system. In this sense, transport properties on $X_\eps$
are approximately described by the quantum graph $X_0$.  Furthermore,
a bound state (of finite degeneracy) on $X_\eps$ (i.e., an
eigenfunction corresponding to a discrete eigenvalue) can be
approximated by its (mostly explicitly known) eigenvector on $X_0$
(cf.~\Thm{ew.graph}). In a forthcoming paper~\cite{exner-post:pre06}
we will deal with the convergence of resonances, i.e., eigenvalues of
a suitable dilated Hamiltonian. The methods needed there differ from
the ones given in~\App{abstr.mod} since the dilated operators are no
longer self-adjoint (even not normal).


With our methods here, we consider the discrete and the essential
spectrum only since they can be characterised by the dimension of
spectral projections. A finer analysis of the spectrum needs more
elaborated methods, such as scattering theory. Our results presented
here are considered as a first step in dealing with the
above-mentioned structures. We will concentrate on the relation
between scattering and transport properties on the two systems in a
forthcoming paper.

The paper is organised as follows.  In \Sec{graph} we define properly
graph-like manifolds and metric graphs and show that the abstract
convergence result can be used in this situation for
$(H,\HS)=(\laplacian{X_0}, \Lsqr{X_0})$ and $(\wt H,\wt \HS) =
(\laplacian{X_\eps},\Lsqr{X_\eps})$.  In \Sec{ex.app} we discuss
various examples of graph-like manifolds to which our result applies.
We also derive several consequences of the spectral convergence.  In
\App{abstr.mod} we develop the abstract framework in order to show the
spectral convergence for arbitrary pairs $(H,\HS)$ and $(\wt H, \wt
\HS)$ being at a ``distance'' $\delta$ to each other.


%------------------------------------------------------------
\section{Graph-like manifolds}
\label{sec:graph}
%------------------------------------------------------------

In this section we apply the abstract setting developed in
\App{abstr.mod} to the example of a family of manifolds $X_\eps$
converging to a (metric) graph $X_0$.  This situation has already been
treated in a quite general way in~\cite{exner-post:05} based
on~\cite{kuchment-zeng:01, rubinstein-schatzman:01} with the only
restriction that the graph is \emph{compact} (i.e., finite and each
edge has finite length) and each manifold $X_\eps$ is \emph{compact}.
Under these assumptions, the spectra of the operators considered are
purely discrete (for a precise definition see below). The main result
in~\cite{exner-post:05} states that the $k$-th eigenvalue of the
Laplacian $\laplacian {X_\eps}$ converges to the $k$-th eigenvalue of
the limit operator. The proof uses the min-max principle and
comparison of the appropriate Rayleigh quotients.

If the manifold and the metric graph are non-compact, more elaborated
methods are needed. Namely, we establish in \App{abstr.mod} norm
resolvent convergence from which all other convergence results follow.
The norm resolvent convergence is reduced to the verification of
several natural conditions provided in \Def{closeness}. In order that
these conditions are satisfied we need the uniformity
assumptions~\eqref{eq:deg.bd}--\eqref{eq:vol.vertex} in our model. The
eigenvalue convergence already proven in~\cite{exner-post:05} appears
as a special case (cf.\ \Cor{ev.conv}).


%------------------------------------------------------------
\subsection{Metric graphs}
\label{sec:met.graph}
%------------------------------------------------------------

Let us first describe the metric graph $X_0$ and the family of
graph-like manifolds $X_\eps$; the necessary assumptions in order that
the convergence results hold will be given later.  Suppose
$X_0=(V,E,\bd, \ell)$ is a countable, connected metric graph, i.e.,
$V$ denotes the set of vertices, $E$ the set of edges and $\map \bd E
{V \times V}$, $\bd e = (\bd_+e,\bd_-e)$ denotes the pair of the end
point and the starting point of the edge $e$.  For each vertex $v \in
V$ we denote by
\begin{equation*}
  E_v^\pm := \set {e \in E} {\bd_\pm e = v}
\end{equation*}
the edges starting ($-$)/ending ($+$) at $v$. Let $E_v := E_v^+ \dcup
E_v^-$ be the \emph{disjoint} union of all edges emanating at $v$.
The \emph{degree} of a vertex $v$ is the number of vertices emanating
from $v$, i.e.,
\begin{equation}
  \label{eq:degree}
  \deg v := |E_v| = |E_v^+| + |E_v^-|.
\end{equation}
We assume that $X_0$ is \emph{locally finite}, i.e., $\deg v \in \N$.
Note that we allow loops, i.e. edges $e$ with $\bd_+ e = \bd_- e = v$.
A loop $e$ will be counted twice in $\deg v$ and occurs twice in $E_v$
due to the disjoint union. In addition, we assume that $\bd e$ always
consists of two elements, even if $\bd_-e = \bd_+e = v$ for a loop
$e$.  We also allow multiple edges, i.e., edges $e_1 \ne e_2$ having
the same starting and end points.

Finally, $\map \ell E {(0,\infty]}$ assigns a length $\ell_e$ to each
edge $e \in E$, making the graph $(V,E,\bd)$ a \emph{metric} or
\emph{quantum} graph. Clearly, $X_0$ becomes a metric space.  We
identify each edge $e$ with the interval $(0,\ell_e)$.  In the case of
an infinite edge, a ``lead'', (i.e., $\ell_e=\infty$) we assume that
there is only one vertex $\bd e = \bd_-e$ at the end corresponding to
$0$, i.e., there is no vertex at $\infty$.  For a general survey on
quantum graphs consult e.g.~\cite{kuchment:04, kuchment:05}. We stress
that our graphs need by no way to be embedded in some Euclidean space.
\begin{remark}
  Note that for a metric graph, the notion ``compact'' and ``finite''
  have a different meaning: A \emph{finite} metric graph is a graph
  with finitely many vertices and edges, whereas a \emph{compact}
  graph must in addition have finite edge length for each edge.
  Therefore, a compact metric graph is finite but not vice versa
  (think e.g.\ of a star-shaped metric graph with one vertex and a
  finite number of leads attached to the vertex).
\end{remark}
We also assign a \emph{density} $p_e$ to each edge $e \in E$, i.e., a
measurable function $\map {p_e} e {(0,\infty)}$.  For simplicity, we
assume that $p_e$ is smooth in order to obtain a smooth metric in the
graph-like manifold. The data $(V,E,\bd,\ell,p)$, $p=(p_e)_e$ describe
a \emph{weighted} metric graph.

The Hilbert space associated to such a graph is
\begin{equation*}
  \HS := \Lsqr {X_0} = 
  \bigoplus_{e \in E} \Lsqr e
\end{equation*}
which consists of all functions $f$ with finite norm
\begin{equation*}
  \normsqr f = \normsqr[X_0] f = 
  \sum_{e \in E} \normsqr[e] {f_e} =
  \sum_{e \in E} \int_e |f_e(x)|^2 p_e(x) \dd x.
\end{equation*}
We define the limit operator $H$ via the quadratic form
\begin{displaymath}
  \qf h(f) := 
  \sum_{e \in E} \normsqr[e] {f'_e} =
  \sum_{e \in E} \int_e |f'_e(x)|^2 p_e(x) \dd x
\end{displaymath}
for functions $f$ in
\begin{equation*}
  \HS_1 := \Sob {X_0} :=
  \Cont {X_0} \cap \bigoplus_{e \in E} \Sob e.
\end{equation*}
Note that a weakly differentiable function on an interval $e$, i.e.,
$f_e \in \Sob e$, is automatically continuous. Therefore, the
continuity is only a condition at each vertex. Furthermore, $\qf h$ is
a closed form, i.e., $\HS_1$ together with the norm
\begin{equation*}
  \normsqr[1] f = \normsqr[1, X_0] f := \normsqr[X_0] f + \qf h(f)
\end{equation*}
is complete.

The associated self-adjoint, non-negative operator $H=\laplacian
{X_0}$ is given by
\begin{equation}
\label{eq:formaledge}
  (\laplacian {X_0} f)_e = -\frac 1{p_e}\big(p_e \, f_e'\big)'
\end{equation}
on each edge $e$.  If we assume the global lower
bound~\eqref{eq:length} (cf.\ page~\pageref{eq:length}) on the length
$\ell_e$ of the edge $e$ then the domain $\HS_2$ of $H=\laplacian
{X_0}$ consists of all functions $f \in \Lsqr {X_0}$ such that
$\laplacian {X_0} f \in \Lsqr {X_0}$
(cf.~e.g.~\cite[Thm.~17]{kuchment:04}). Furthermore, each function $f$
satisfies the so-called \emph{(generalised) Neumann boundary
  condition} (sometimes also named \emph{Kirchhoff}) at each vertex
$v$, i.e., $f$ is continuous at $v$ and
\begin{equation}
  \label{eq:kirchhoff}
  \sum_{e \in E_v} p_e(v) f'_e(v) = 0
\end{equation}
for all vertices $v \in V$ where the derivative is taken \emph{away}
from the vertex, i.e.\ we set $f'_e(v) := f'_e(0)$ if $v=\bd_-e$ and
$f'_e(v) := -f'_e(\ell_e)$ if $v=\bd_+e$ (considering $f_e$ as
function on the interval $(0,\ell_e)$).  We will call $\laplacian
{X_0}$ the \emph{(generalised) weighted Neumann Laplacian} on $X_0$.
For details on operators on non-compact or infinite metric graphs we
refer e.g.  to~\cite{kuchment:04, kuchment:05}.



%------------------------------------------------------------
\subsection{Graph-like manifolds}
\label{sec:graph.mfd}
%------------------------------------------------------------

Let $X_0$ be a weighted metric graph as defined in the previous
section.  The corresponding family of graph-like manifolds $X_\eps$ is
given as follows: For each $0 < \eps \le \eps_0$ we associate with the
graph $X_0$ a connected Riemannian manifold $X_\eps$ of dimension $d
\ge 2$ with or without boundary equipped with a metric $g_\eps$ to be
specified below\footnote{The boundary of $X_\eps$ (if there is any)
  need not to be smooth; we allow singularities on the boundary of the
  vertex neighbourhood $U_\vxeps$, see e.g.\ \Sec{emb.graph}
  and~\Fig{graph.emb}}. We suppose that $X_\eps$ is the union of the
closure of open subsets $U_\edeps$ and $U_\vxeps$ such that the
$U_\edeps$ and $U_\vxeps$ are mutually disjoint for all possible
combinations of $e \in E$ and $v \in V$, i.e.,
\begin{equation}
  \label{eq:disjcup}
  X_\eps = \bigcup_{e \in E} \clo U_\edeps \cup 
   \bigcup_{v \in V} \clo U_\vxeps.
\end{equation}
We think of $U_\edeps$ as the thickened edge $e$ and of $U_\vxeps$ as
the thickened vertex $v$ (see Figures~\ref{fig:edge.vertex}
and~\ref{fig:mfd}).  Note that Figure~\ref{fig:mfd} describes the
situation only roughly, since it assumes that $X_\eps$ is embedded in
$\R^\nu$. More correctly, we should think of $X_\eps$ as an abstract
manifold obtained by identifying the appropriate boundary parts of
$U_\edeps$ and $U_\vxeps$ via the connection rules of the graph $X_0$.
This manifold need not to be embedded, but the situation when $X_\eps$
is a submanifold of $\R^\nu$ ($\nu \ge d$) can be viewed also in this
abstract context; note that the $\eps$-neighbourhood of an embedded
metric graph in $\R^d$ is also included as example
(cf.~\cite{exner-post:05} and \Sec{emb.graph}).

As a matter of convenience we assume that $U_\edeps$ and
$U_\vxeps$ are independent of $\eps$ as manifolds, i.e., only their
metrics $g_\eps$ depend on $\eps$.  This can be achieved in the
following way: for the edge regions we assume that $U_\edeps$ is
diffeomorphic to $U_e := e \times F$ for all $0 < \eps \le \eps_0$
where $F$ denotes a compact and connected manifold (with or without a
boundary) of dimension $m:=d-1$.  We fix a metric $h$ on $F$ and
assume for simplicity that $\vol F = 1$.

For the vertex regions we assume that the manifold $U_\vxeps$ is
diffeomorphic to an $\eps$-independent manifold $U_v$ for $0 < \eps
\le \eps_0$.  Pulling back the metric to the diffeomorphic manifold
$U_e$ resp.\ $U_v$ we may assume that the underlying differentiable
manifold is independent of $\eps$.  Therefore, $U_\edeps \cong (U_e,
g_\edeps)$ and $U_\vxeps = (U_v, g_\vxeps)$.

We use the obvious notation for functions $u$ on $X_\eps$ like
$u_e$ and $u_v$ as restrictions on $U_\edeps$ and
$U_\vxeps$, respectively.  The corresponding Hilbert space is then
\begin{equation*}
  \wt \HS := \Lsqr {X_\eps} = 
  \bigoplus_{e \in E} \Lsqr {U_\edeps} \oplus
      \bigoplus_{v \in V} \Lsqr {U_\vxeps}
\end{equation*}
which consists of all functions $u$ with finite norm
\begin{multline*}
  \normsqr u =
  \normsqr[X_\eps] u = 
  \sum_{e \in E} \normsqr[U_\edeps] {u_e} +
     \sum_{v \in V} \normsqr[U_\vxeps] {u_v} \\ =
  \sum_{e \in E} 
     \int_{e \times F} |u_e|^2 \det g_\edeps^{1/2} \dd x \dd y
   + \sum_{v \in V} 
        \int_{U_v} |u_v|^2 \det g_\vxeps^{1/2} \dd z
\end{multline*}
where $y$ and $z$ represent coordinates of $F$ and $U_v$,
respectively.

The operator $\wt H$ we are considering will be the Laplacian on
$X_\eps$, i.e., $\wt H = \laplacian {X_\eps}$.  If $F$ has non-trivial
boundary, we assume Neumann boundary conditions on the boundary part
coming from $\bd F$. We define $\laplacian {X_\eps}$ via its quadratic
form $\wt{\qf h}$ given by
\begin{equation}
  \label{eq:def.quad.mfd}
  \wt{\qf h} (u) = \int_{X_\eps} |\dd u|_{g_\eps}^2 \dd X_\eps
\end{equation}
for functions $u \in \wt \HS_1 = \Sob {X_\eps}$ where the latter space
denotes the completion of the space of smooth functions with
\emph{bounded} support w.r.t.\ the norm
\begin{equation}
  \label{eq:def.quad.norm}
  \normsqr[1] u = 
  \normsqr[1, X_\eps] u :=
  \normsqr[X_\eps] u + \wt{\qf h}(u).
\end{equation}



%------------------------------------------------------------
\subsection{Quasi-unitary operators}
\label{sec:quasi-unitary}
%------------------------------------------------------------

Let us fix the identification operators $\map J \HS {\wt \HS}$ and
$\map {J'} {\wt \HS} \HS$ and their analogues on the quadratic form
domains. Roughly speaking, $J$ extends the function $f$ at $x \in e$
constantly onto the cross section $F_\edeps(x):=(\{x\} \times
F,h_\edeps) \subset (U_e, g_\edeps)$, where $h_\edeps$ is the induced
metric of the restriction, and $J'$ is the transversal average of $u$
at $x$, i.e., the Fourier coefficient of $u(x,\cdot)$ w.r.t.\ the
first (constant) eigenfunction on $F_\edeps(x)$. We will first show
what estimates are necessary in order that $J$, $J'$ and their
quadratic form analogues become \emph{quasi-unitary} in the sense of
\Def{closeness}. In a second step we provide the necessary assumptions
on the graph (\Sec{ass.graph}) and on the manifold (\Sec{ass.mfd}).
Finally, we provide some necessary estimates (\Sec{est.graph}) and
finish the proof of quasi-unitarity.


We define the operator $\map J \HS {\wt \HS}$ by
\begin{equation}
  \label{eq:gr.trans.op}
  J f (z):=
  \begin{cases}
    \eps^{-m/2} f_e(x) & \text{if $z=(x,y) \in U_e$},\\
    0 & \text{if $z \in U_v$}
  \end{cases}
\end{equation}
and the operator $\map {J_1} {\HS_1} {\wt \HS_1}$ by
\begin{equation}
  \label{eq:gr.trans.op1}
  J_1 f (z):=
  \begin{cases}
    \eps^{-m/2} f_e(x) & \text{if $z=(x,y) \in U_e$},\\
    \eps^{-m/2} f(v)   & \text{if $z \in U_v$}
  \end{cases}
\end{equation}
Note that the latter operator is well-defined since functions in
$\HS_1$ are continuous (cf.\ \Lem{point.vx}).  For the operators in
the opposite direction, we first introduce the following averaging
operators
\begin{gather*}
  (N_e u)(x) :=
   \iprod[F]{\phi_{F,1}} {u_e(x,\cdot)} =
   \int_F u_e(x,y) \dd F(y),\\
   C_v u := 
   \iprod[U_v] {\phi_{U_v,1}} {u_v} =
   \frac 1 {\vol U_v} \int_{U_v} u \dd U_v
\end{gather*}
for $u \in \wt \HS = \Lsqr {X_\eps}$ giving the coefficient
corresponding to the first (transversal) eigenfunction $\phi_1$ on
$U_e$ resp.\ $U_v$. Note that these eigenfunctions are constant and
that $\vol F = 1$.

We define $\map {J'} {\wt \HS} \HS$ by
\begin{equation}
  \label{eq:gr.trans.op.}
  (J' u)_e (x):=
    \eps^{m/2} (N_e u)(x),
      \qquad x \in e
 \end{equation}
and the operator $\map {J_1'} {\wt \HS_1} {\HS_1}$ by
\begin{multline}
  \label{eq:gr.trans.op1.}
  (J_1' u)_e (x):=
    \eps^{m/2} \Bigl[
        N_e u (x) \\ + 
        \rho^+_e(x) \bigl[ C_{\bd_+ e} u - N_e u (\bd_+ e) \bigr] +
        \rho^-_e(x) \bigl[ C_{\bd_- e} u - N_e u (\bd_- e) \bigr]
    \Bigr]
\end{multline}
for $x \in e$.  Here, $\map {\rho^\pm_e} \R {[0,1]}$ are the
continuous, piecewise affine functions given by
\begin{equation}
  \label{eq:def.cut.off}
  \rho^+_e(\bd_+ e)=1           \quad \text{and} \quad
  \rho^+_e(x)=0
      \quad \text{for all $\dist(x, \bd_+ e) \ge \min \{1, \ell_e/2\}$}
\end{equation}
and similarly for $\rho^-_e$ and $\bd_- e$.  Note that $(J_1' u)_e(v)
= C_v u$ for $v = \bd_\pm e$. In particular, $J_1' u$ is a continuous
function on $X_0$.  Again, the operator $J_1'$ is only defined on $\wt
\HS_1 = \Sob {X_\eps}$.


The closeness assumptions of \Sec{ass} now reads as follows:
\begin{gather}
  \label{eq:j.scale.gr}
  \normsqr{Jf - J_1f} = 
  \sum_{v \in V} \eps^{-m} \vol U_\vxeps |f(v)|^2\\
  \label{eq:j.scale..gr}
  \tag{\ref{eq:j.scale.gr}'}
  \normsqr{J' u - J_1' u} =
  \sum_{e \in E} \sum_ {v \in \bd e} 
     \eps^m \normsqr[e]{\rho^\pm_e} \, | C_v u - N_e u(v)|^2 \\
  \label{eq:j.adj.gr}
  \begin{split}
     |\iprod {Jf} u &- \iprod f {J' u} | \\ {}=
      &\Bigl| \sum_{e \in E} \int\limits_{e \times F} 
              \conj f(x) u(x,y) \, \eps^{-m/2}
      \bigl[ \dd U_\edeps(x,y) - 
              \eps^m \dd F(y) \, p_e(x) \dd x \bigr] \Bigr|
     \end{split}
\end{gather}
\begin{gather}
  \label{eq:j.comm.gr}
  \begin{split}
     |\wt{\qf h} (J_1 f, u) - \qf h(f, J_1' & u)| \\ {}=
      \Bigl| \sum_{e \in E} \int\limits_{e \times F} 
             \conj f'(x) \, \partial_x &u(x,y) \, \eps^{-m/2}
      \bigl[ g_\edeps^{xx} \dd U_\edeps(x,y) - 
              \eps^m \dd F(y) \, p_e(x) \dd x \bigr] \\
      &\qquad\qquad{}-\sum_{e \in E} \sum_{v \in \bd e} 
             \eps^{-m/2} (C_v u - N_e u(v)) 
                 \iprod[e] {f_e'}{(\rho^\pm_e)'} \Bigr|
     \end{split}\\
  \label{eq:j.inv..gr}
  \normsqr{J J' u - u} =
  \sum_{e \in E} \normsqr[U_\edeps] {N_e u - u} +
    \sum_{v \in V} \normsqr[U_\vxeps] u\\
  \label{eq:j.bdd.gr}
  \normsqr{J f} = 
  \sum_{e \in E} \int_{e \times F} |f(x)|^2 
         \eps^{-m} \dd U_\edeps(x,y)\\
  \label{eq:j.bdd..gr}
  \tag{\ref{eq:j.bdd.gr}'}
  \normsqr{J' u} \le
%  \int_e \Bigl| \int_F u(x,y) \dd F(y) \Bigr|^2 \eps^m p_e(x) \dd x \le
  \int_{e \times F} |u(x,y)|^2 \, \eps^m \dd F(y) \, p_e(x) \dd x 
\end{gather}
Here, the sign in $\rho^\pm_e$ is used according to $v=\bd_\pm e$.
Note that $J' J f = f$. From the RHS we can deduce the necessary
assumptions given precisely in the next section. For example,
from~\eqref{eq:j.scale.gr} it follows that we must have $\vol
U_\vxeps=o(\eps^m)$, and from~\eqref{eq:j.adj.gr}
and~\eqref{eq:j.comm.gr} we see that $g_\edeps$ must be close to a
product metric on $U_e = e \times F$.

\subsection{Assumptions on the graph}
\label{sec:ass.graph}
We precise here the necessary assumptions in order to estimate the RHS
of~\eqref{eq:j.scale.gr}--\eqref{eq:j.bdd.gr}
and~\eqref{eq:j.bdd..gr}. For the graph data we require that the
degree is uniformly bounded, i.e., that there exists $d_0 \in \N$ such
that
\begin{equation}
  \label{eq:deg.bd}
  \tag{G1}
  \deg v \le d_0, \qquad v \in V.
\end{equation}
We assume in addition that there is a uniform lower bound on the set
of length, i.e., there exists $\ell_0>0$ (without loss of generality
$\ell_0 \le 1$) such that
\begin{equation}
  \label{eq:length}
  \tag{G2}
  \ell_e \ge \ell_0 \qquad \text{for all $e \in E$.}
\end{equation}
We assume that the density function $p_e$ is uniformly bounded, i.e.,
there exist constants $p_\pm > 0$ such that
\begin{equation}
  \label{eq:density}
  \tag{G3}
  \begin{split}
    p_- \le p_e(x), \qquad &
       \dist(x, \bd_\pm e) \le \min \{1,\ell_e/2\}, \qquad
        e \in E,\\
   p_e(x) \le p_+, \qquad &x \in e, \qquad e \in E.
  \end{split}
\end{equation}
Since $r_e(x):=p_e(x)^{1/m}$ will correspond to the radius of the
cross section of $U_\edeps$ at $x \in e$, we also denote $r_\pm :=
p_\pm^{1/m}$ the maximal/minimal radius. We want to stress that we
allow a sequence of edges $e_n$ such that $\ell_{e_n} \to \infty$ or
even external edges (i.e., edges with infinite length). In both cases,
we do not impose a global \emph{lower} bound on the density function
$p_e=r_e^m$. E.g., a horn-like shape of radius $r_e(x)=r_{e,0}
x^{-\beta}$ for the associated edge neighbourhood $U_\edeps$ is
allowed for an external edge $e$ (cf.\ also \Rem{horn}).

\begin{definition}
  \label{def:unif.graph}
  A \emph{uniform} weighted metric graph is a weighted metric graph
  $X_0=(V,E,\bd,\ell,p)$ satisfying
  \eqref{eq:deg.bd}--\eqref{eq:density}.
\end{definition}
From these assumptions we conclude the following estimates:

\begin{lemma}
  \label{lem:ed.vx}
  Suppose that $(a(v))_{v \in V}$ is a family of non-negative
  numbers.  Then
  \begin{equation}
  \label{eq:sum.ed.vx}
    \sum_{e \in E} \sum_{v \in \bd e} a(v) =
    \sum_{e \in E} \bigl(a(\bd_+ e) + a(\bd_- e)\bigr) =
    \sum_{v \in V} (\deg v) a(v) \le
    d_0 \sum_{v \in V} a(v)
  \end{equation}
  due to~\eqref{eq:deg.bd}. Furthermore,
  \begin{equation}
  \label{eq:sum.vx.ed}
    \sum_{v \in V} \sum_{e \in E_v} b_e = 2 \sum_{e \in E} b_e
  \end{equation}
  for a family $(b_e)_{e \in E}$.
\end{lemma}
\begin{proof}
  Inequality~\eqref{eq:sum.ed.vx} is clear, and from the disjoint
  union
  \begin{equation*}
    E = \bigdcup_{v \in V} E_v^+ = \bigdcup_{v \in V} E_v^-
  \end{equation*}
  the second equality follows immediately.
\end{proof}
The next lemma is needed in order to estimate $f(v)$:
\begin{lemma}
  \label{lem:point.vx}
  We have
  \begin{equation*}
    \sum_{v \in V} |f(v)|^2 \le \frac 4 {\ell_0 p_-} \normsqr[1] f
  \end{equation*}
  for all $f \in \HS_1 = \Sob {X_0}$.
\end{lemma}
\begin{proof}
  The estimate follows easily from
  \begin{equation}
    \label{eq:point.vx}
    |f(0)|^2 \le
    \frac 2 {p_-} \max \Bigl\{ \frac 1 \ell, \ell \Bigr\}
       \int_0^\ell \bigl( |f(x)|^2 + |f'(x)|^2 \bigr) p(x) \,
         \dd x
  \end{equation}
  where $f \in \Sob{0,\ell}$ and $p_- := \inf_{0 \le x \le \ell} p(x)$
  (cf.~\cite[Lemma~8]{kuchment:04}) applied to $\ell := \min \{ 1,
  \ell_e/2 \}$ together with~\eqref{eq:length} and~\eqref{eq:density}.
\end{proof}

Finally, we can estimate the cut-off function $\rho^\pm_e$
using~\eqref{eq:length} and~\eqref{eq:density}:
\begin{lemma}
  \label{lem:cut.off}
  The estimate
  \begin{equation}
  \label{eq:cut.off}
    \normsqr[e] {\rho^\pm_e} \le p_+ \qquad \text{and} \qquad
    \normsqr[e] {(\rho^\pm_e)'} \le \frac {2 p_+} {\ell_0}
  \end{equation}
  holds for all $e \in E$.
\end{lemma}


\subsection{Assumptions on the manifold}
\label{sec:ass.mfd}
Guided by the classical example of an embedded graph
(cf.~\Sec{emb.graph} and~\eqref{eq:met.emb}) we assume that the metric
$g_\edeps$ on the edge neighbourhood $U_e=e \times F$ is given as a
perturbation of the product metric
\begin{equation}
  \label{eq:def.met.edge}
  \overline g_\edeps := \dd x^2 + \eps^2 \, r_e^2(x) \, h(y), \qquad 
  (x,y) \in U_e = e \times F
\end{equation}
with
\begin{equation*}
  \label{eq:def.rad}
  r_e(x) := (p_e(x))^{1/m}
\end{equation*}
where $h$ is the fixed metric on $F$, $m=\dim F = d-1$ and $p_e$ is
the density function of the metric graph on the edge $e$.

We denote by $G_\edeps$ and $\overline G_\edeps$ the $d \times d$-matrices
associated to the metrics $g_\edeps$ and $\overline g_\edeps$ with respect
to the coordinates $(x,y)$ and assume that the two metrics coincide up
to an error term as $\eps\to 0$, more specifically
\begin{equation}
  \label{eq:asym.met.ed}
  \tag{G4}
  G_\edeps = \overline G_\edeps +
    \begin{pmatrix}
      o(1) & o(\eps)r_e \\ o(\eps)r_e &
      o(\eps^2)r_e^2
    \end{pmatrix} =
    \begin{pmatrix}
      1 + o(1) & o(\eps)r_e \\ o(\eps)r_e &
      (\eps^2   H + o(\eps^2))r_e^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_e^2(x) \, h_{\alpha\beta}(y) + o(\eps^2).
%\end{displaymath}
\emph{uniformly} on $U_e$. We also assume that these error estimates
are uniform in $e$, i.e., that $o(\eps^i)$ does not depend on the edge $e
\in E$. As in \cite[Lemma~4.3]{exner-post:05} (replacing $\eps$ by
$\eps r_e\le \eps r_+$) we can show the following estimates
\begin{gather}
  \label{eq:met.vol}
    \dd U_\edeps(x,y) = (1 + o_1(1)) \eps^m \dd F(y) \, p_e(x) \dd x \\
%    (\det g_\edeps)^\frac12 = (1 + o_1(1)) \eps^m \det h^\frac12 p_e(x) \\
  \label{eq:met.1st.comp}
    g_\edeps^{xx}  \!:= (G_\edeps^{-1})_{xx} = 1 + o_2(1) \\
  \label{eq:met.1st.der}
    |\dd_x u|^2 \le O_3(1) |\dd u|_{g_\edeps}^2\\
  \label{eq:met.2nd.der}
    |\dd_F u|_h^2 \le o_4(\eps) |\dd u|_{g_\edeps}^2
\end{gather}
where $\dd_x$ and $\dd_F$ are the (exterior) derivatives with respect
to $x \in e$ and $y \in F$, respectively. Here, $o_1(1)$ and $o_2(1)$
depend only on $o(\eps^j)$ in~\eqref{eq:asym.met.ed} whereas $O_3(1)$
and $o_4(1)$ depend also on $r_+$.  The index $i$ in $o_i(\cdot)$ is
added in order to trace the error estimates in the formulas below. All
the estimates are uniform on $U_e$ and uniform in $e \in E$ as $\eps
\to 0$.

On the vertex neighbourhood $U_v$ we assume that the metric $g_\vxeps$
satisfies
% ------------- %
\begin{equation}
  \label{eq:met.vx}
  \tag{G5}
  c_- \eps^2 g_v \le g_\vxeps \le c_+ \eps^{2\alpha} g_v
\end{equation}
% ------------- %
in the sense that there are constants $c_-, c_+>0$ \emph{independent}
of $v$ such that
% ------------- %
\begin{displaymath}
  c_- \eps^2 g_v(z)(w,w) \le 
  g_\vxeps(z)(w,w) \le 
  c_+ \eps^{2\alpha} g_v(z)(w,w)
\end{displaymath}
% ------------- %
for all $w \in T_z U_v$ and all $z \in U_v$ where $g_v$ is the metric
$g_\vxeps$ with $\eps=1$. The number $\alpha$ in the exponent is
assumed to satisfy the inequalities
% ------------- %
 \begin{equation}
  \label{eq:est.alpha}
  \tag{G6}
  \frac {d-1} d < \alpha \le 1.
\end{equation}
In addition, we assume that
\begin{equation}
  \label{eq:vol.vertex}
  \tag{G7}
  c_{\vol} := \sup_{v \in V} \vol U_v < \infty 
    \quad \text{and} \quad
  \lambda_2 := \inf \EWN 2 {U_v} > 0
\end{equation}
where $\EWN 2 {U_v}$ denotes the second (i.e., the first non-zero)
Neumann eigenvalue of $\laplacian {U_v}$.

\begin{definition}
  \label{def:unif.mfd}
  A family of graph-like manifolds $X_\eps$ w.r.t.\ the uniform metric
  graph $X_0$ will be called \emph{uniform}
  if~\eqref{eq:asym.met.ed}--\eqref{eq:vol.vertex} are satisfied.
\end{definition}
We will discuss several examples of uniform metric graphs and
graph-like manifolds in \Sec{ex.app}. Let us just finish this
subsection with a few comments on the assumptions.

\begin{remark}
  \label{rem:unif.mfd}
  \begin{enumerate}
  \item The condition~\eqref{eq:asym.met.ed} is motivated by the
    classical example of a curved edge embedded in $\R^2$,
    cf.~\Sec{emb.graph} and~\eqref{eq:met.emb}.
  \item We have assumed an upper bound of the density $p_e$ (i.e., the
    radius function $r_e$) on the \emph{whole} edge
    in~\eqref{eq:density}. A careful analysis of the proof
    of~\eqref{eq:met.vol}--\eqref{eq:met.2nd.der} shows that the first
    two estimates remain globally true even if $p_e$ has no global
    bound $p_+$ on $e$. But the last two estimates need the global
    bound $p_e(x) \le p_+$ for all $x \in e$ and $e \in E$. Therefore,
    an infinite edge cannot have a neighbourhood $U_\edeps$ with
    growing radius $r_e(x) \to \infty$ such as a conical end. It is
    also forbidden that a sequence of edges $e_n$ has neighbourhoods
    with radius functions $r_{e_n}$ unbounded in $n$.
  \item 
    \label{rem:bottle.neck}
    Note that the upper estimate in assumption~\eqref{eq:met.vx} does
    not apply to points \emph{at the border} of $\bd U_v$, since we
    still assume that $g_\vxeps$ is the restriction of a \emph{global}
    metric $g_\eps$ on $X_\eps$; and the cross-section on $U_\edeps$
    has a metric of order $O(\eps^2) h$.  Nevertheless it is not
    excluded that $g_\vxeps$ scales differently \emph{away} from $\bd
    U_v$ (for a detailed discussion of such scalings
    cf.~\cite[Sec.~6]{exner-post:05}).
  \item
    \label{rem:critical}
    The reason for the critical exponent $(d-1)/d$
    in~\eqref{eq:est.alpha} is roughly the following: If $\alpha$
    satisfies~\eqref{eq:est.alpha} then $\vol U_\vxeps \le
    O(\eps^{\alpha d})$ decays faster than $\vol U_\edeps =
    O(\eps^{d-1})$. Other decay rates are discussed
    in~\cite{kuchment-zeng:03, exner-post:05}.
  \item Assumption~\eqref{eq:vol.vertex} roughly assures that $U_v$
    remains small (as family in $v \in V$): Suppose that there is an
    infinite sequence $(v_n) \subset V$ such that $U_{v_n}= U_{v_0}$
    as sets and that $g_{v_n} = \rho_n^2 g$ for a sequence $\rho_n \to
    \infty$ of positive numbers. This behaviour does not
    contradict~\eqref{eq:met.vx} since~\eqref{eq:met.vx} is only a
    \emph{relative} bound w.r.t.\ a fixed metric $g_v$.
    But~\eqref{eq:vol.vertex} is no longer satisfied.
    
    In addition, the eigenvalue estimate assures that $U_\vxeps$ as
    well as $(U_{v_n})_n$ for a sequence $(v_n)_n \subset V$ do not
    separate into two (or more) parts as $\eps \to 0$ or $n \to
    \infty$, respectively. This could happen e.g.\ if $U_\vxeps$ or
    $(U_{v_n})_n$ consists of two (or more) large parts joined by
    small cylinders which shrink as $\eps \to 0$ resp.\ $n \to
    \infty$.  This could lead to a decoupling of the edges emanating
    from such a vertex or such a sequence of vertices.
  \item We want to stress that $U_v$ cannot either become too small as
    $v \to \infty$ for some sequence of vertices. This is at first
    sight not obvious, since a priori,~\eqref{eq:met.vx} is not a
    restriction of $g_v$ as family in $v \in V$
    and~\eqref{eq:vol.vertex} roughly says that $U_v$ remains small as
    family in $v$. But one has to take into account that $g_\edeps$
    and $g_\vxeps$ are restrictions of a global \emph{smooth} metric
    $g_\eps$. In particular, the metrics on the common boundary $\bd_e
    U_v$ of $U_e$ and $U_v$ must be the same; and therefore, the
    uniform estimates of $g_\edeps$ in~\eqref{eq:asym.met.ed} become
    uniform estimates of $g_\vxeps$ (take e.g.\ a sequence $\rho_n \to
    0$ and argue as in the previous remark).
  \end{enumerate}
\end{remark}
\begin{remark}
  \label{rem:collar.vx}
  Without loss of generality we can assume that $\bd_e U_v$ has a
  collar neighbourhood $U_{e,v}=(0,\ell_-) \times F$ in which the
  metric $g_v$ on $U_v$ has the form
  \begin{equation*}
    g_{e,v}=\dd \overline x^2 + h_{\overline x}
  \end{equation*}
  for $(\overline x,y) \in U_{e,v}$, i.e., the collar neighbourhood
  has length $\ell_-$ for some global constant $0<\ell_-<1$ (e.g.,
  $\ell_-=\ell_0/2$, cf.~also \Fig{graph.emb} where
  $\ell_-<\ell_0/2$).  Here, $\bd_e U_v$ is the part of the boundary
  (diffeomorphic to $F$) where the edge neighbourhood of $e$ meets. In
  addition, we assume that estimates similar to \eqref{eq:met.vol}
  and~\eqref{eq:met.1st.der} are fulfilled (with $\eps=1$, $x$
  replaced by $\overline x$ and $g_\edeps$ replaced by $g_{e,v}$).
  
  If this condition is not satisfied, we just have to modify the
  decomposition of the manifold into edge and vertex neighbourhoods in
  order that $U_\vxeps$ has at least a cylindrical part of length
  $\eps \ell_-$ (taken from the edge neighbourhood). The desired
  estimates \eqref{eq:met.vol} and~\eqref{eq:met.1st.der} follow now
  from~\eqref{eq:asym.met.ed} with the new variable $\overline
  x=x/\eps$.
\end{remark}


\subsection{Main result}
\label{sec:est.graph}
We are now able to prove the following lemmas needed to complete the
proof of the closeness assumptions. Mainly, the estimates are already
given in~\cite{exner-post:05} but since there, we only considered
compact graphs and compact manifolds, a precise control of the
constants was not necessary. We do not repeat the proofs given there
but we state the necessary results together with their dependence on
the constants given in the uniformity assumptions.

\begin{lemma}
  \label{lem:rest0}
  We have
  \begin{equation*}
    \normsqr[\bd_e U_v] u \le 
    c_{\mathrm{tr}} \big( \normsqr[U_v] u + \normsqr[U_v] {du} \big)
  \end{equation*}
  for all $u \in \Sob {U_v}$ where
  \begin{equation*}
    c_{\mathrm{tr}} :=
    \frac{2p_+(1+o_1(1))(1+O_3(1))}{p_-(1-o_1(1)) \ell_-}.
  \end{equation*}
\end{lemma}
\begin{proof}
  The estimate is an immediate consequence of~\eqref{eq:point.vx} and
  \Rem{collar.vx}.
\end{proof}

The following lemma roughly states that the transversal average on the
boundary $\bd_e U_v$ is close to the total average on $U_v$. The proof
is based on the fact that the second Neumann eigenvalue of $U_\vxeps$
tends to $\infty$ as $\eps \to 0$
(cf.~\cite[Lemma~5.5]{exner-post:05}):
\begin{lemma}
  \label{lem:rest}
  The inequality
  \begin{equation*}
    \eps^m |C_v u - N_e u(v)|^2 \le 
    \widetilde c_{\mathrm{tr}} \eps^{2\alpha - 1} \normsqr[U_\vxeps] {d u}
  \end{equation*}
  holds for all functions $u \in \Sob {U_\vxeps}$ and $v =
  \bd_\pm e$ where
  \begin{equation*}
    \widetilde c_{\mathrm{tr}} :=
        \frac{c_-^2}{p_- (1-o_1(1)) c_+^d} 
             \Big( \frac 1 {\lambda_2} + 1 \Big)
              c_{\mathrm{tr}}.
  \end{equation*}
\end{lemma}
Note that $\eps^{2\alpha - 1} \to 0$ as $\eps \to 0$ since $\alpha >
(d-1)/d \ge 1/2$.

The next lemma assures that higher transversal modes does not
contribute too much (cf.~\cite[Lemmas~3.1 and~4.4]{exner-post:05}).
Essentially, it is the observation, that $N_e u - u$ is the projection
onto the orthogonal complement of the first (constant) eigenfunction
on $F$:
\begin{lemma}
  \label{lem:ed.av}
  We have
  \begin{equation*}
    \normsqr[U_\edeps] {N_e u - u} \le 
    c_{\mathrm {ed}} o_4(\eps) \normsqr[U_\edeps] {d u}
  \end{equation*}
  for all $u \in \Sob {U_\edeps}$ where
  \begin{equation*}
    c_{\mathrm {ed}} := \frac{(1+o_1(1))}{(1-o_1(1))\EW 2 F}.
  \end{equation*}
\end{lemma}

Finally, we need to assure that there is no concentration at the
vertex neighbourhoods in any bounded spectral interval
(cf.~\cite[Corollary~5.8]{exner-post:05}):
\begin{lemma}
\label{lem:vx}
  The estimate
  \begin{equation*}
    \normsqr[U_\vxeps] u \le 
    c_{\mathrm{vx}} \eps^{\alpha d - m} \normsqr[1, \widehat U_\vxeps] u
  \end{equation*}
  holds for all $u$ in $\Sob {\widehat U_\vxeps}$ where $\widehat
  U_\vxeps := U_\vxeps \cup \bigcup_{e \in E_v} U_\edeps$ and
  $c_{\mathrm {vx}}$ depends only on $\ell_0$, $p_\pm$, $o_1(1)$,
  $O_3(1)$, $c_\pm$, $c_{\vol}$, $\lambda_2$ and $\widetilde
  c_{\mathrm{tr}}$.
%   \begin{equation*}
%     c_{\mathrm{vx}} := 
%     c_+^{d/2} \Bigl(
%       \frac {c_+^{1/2}}{\lambda_2^{1/2} c_-^{d/4}} +
%     c_{\vol}^{1/2}(\widetilde c_{\mathrm{tr}}^{1/2} + c_
%     \Bigr)^2
%   \end{equation*}
\end{lemma}

We are now able to estimate the RHS of the closeness
assumptions~\eqref{eq:j.scale.gr}--\eqref{eq:j.bdd.gr}
and~\eqref{eq:j.bdd..gr}. For the first one, we have
\begin{equation*}
  \normsqr{Jf - J_1f} \le 
%  \sum_{v \in V} \eps^{-m} \vol U_\vxeps |f(v)|^2 \le
      \frac{4 c_+^{d/2} c_{\vol}}{\ell_0 p_-}
         \eps^{\alpha d - m} \normsqr[1] f
\end{equation*}
for $f \in \HS_1$ using \Lem{point.vx} and~\eqref{eq:met.vx}. Note
that $\eps^{\alpha d - m} \to 0$ as $\eps \to 0$ due
to~\eqref{eq:est.alpha}. Next, we have
\begin{equation*}
  \normsqr{J' u - J_1' u} \le
%  \sum_{e \in E} \sum_ {v \in \bd e} 
%     \eps^m \normsqr[e]{\rho^\pm_e} \, | C_v u - N_e u(v)|^2 \\ \le
     p_+ \tilde c_{\mathrm{tr}} \eps^{2\alpha - 1} 
         \sum_{e \in E} \sum_{v \in \bd e} \normsqr[U_\vxeps] {du} \le
     d_0 \, p_+ \tilde c_{\mathrm{tr}} \eps^{2\alpha - 1} \, \wt{\qf h} (u)
\end{equation*}
using \Lem{rest}, \Lem{cut.off} and \Eq{sum.ed.vx}.
%Here, the
%sign in $\rho^\pm_e$ is used according to $v=\bd_\pm e$.
The estimation of~\eqref{eq:j.adj.gr}, i.e.,
\begin{equation*}
  |\iprod {Jf} u - \iprod f {J' u} | \le 
%   o_1(1) \sum_{e \in E} \int_e \int_F 
%     |f(x) u(x,y)| \, \eps^{m/2} \dd F(y)  p_e(x) \dd x \\ \le
  o_1(1) \norm f \norm u 
\end{equation*}
follows from~\eqref{eq:met.vol}. Similarly,
the estimate~\eqref{eq:j.comm.gr} can be proven by
\begin{equation*}
  |\wt{\qf h}(J_1 f, u) - \qf h(f, J_1' u)| \le
%  \Big|\sum_{e \in E}  
%       \int_{U_\edeps}
%            g_\eps^{xx} f' \,\partial_x u \, \eps^{-m/2} \dd U_\edeps -
%       \int_e  f'(x) (N_e u)'(x) \eps^{m/2} p_e(x) \dd x
%  \Big| \\ +
%  \sum_{e \in E} \sum_{v \in \bd e} 
%        \norm[e]{f'} \, \norm[e] {(\rho^\pm_e)'}\, 
%             \eps^{m/2} |C_v u - N_e u(v)| p_e(x) \,\dd x \\ \le 
  \Bigl( o(1)  + 
     \Bigl[\frac {2d_0 p_+ \widetilde c_{\mathrm{tr}}} {\ell_0} \Bigr]^{1/2} 
      \eps^{\alpha - 1/2}
  \Bigr) \qf h(f)^{1/2} \wt{\qf h} (u)^{1/2}
\end{equation*}
where $o(1)$ depends on the errors in~\eqref{eq:asym.met.ed}. Again, we
have used \Lem{rest}, \Lem{cut.off} and \Eq{sum.ed.vx}.
Estimate~\eqref{eq:j.inv..gr} follows from
\begin{multline*}
  \normsqr{J J' u - u} =
  \sum_{e \in E} \normsqr[U_\edeps] {N_e u - u} +
    \sum_{v \in V} \normsqr[U_\vxeps] u \\ \le
  c_{\mathrm {ed}} o_4(\eps) \sum_{e \in E} \normsqr[U_\edeps] {d u} +
    c_{\mathrm {vx}} \eps^{\alpha d - m} \sum_{v \in V} 
                          \normsqr[1, \widehat U_\vxeps] {u} \\ \le
  3 (c_{\mathrm {ed}} o_4(\eps) + c_{\mathrm {vx}} \eps^{\alpha d - m})
        \normsqr[1] u
\end{multline*}
where we also used~\eqref{eq:sum.vx.ed} in the second estimate.
Finally, Assumption~\eqref{eq:j.bdd} follows from
\begin{equation*}
  \normsqr{J f} \le (1+ o_1(1) \normsqr f \qquad \text{and} \qquad
  \normsqr{J' u} \le \frac 1 {1 - o_1(1)} \normsqr u
\end{equation*}
and~\eqref{eq:met.vol}. We therefore have proven
\begin{theorem}
  \label{thm:graph}
  Suppose that the metric graph $X_0$ and the family of graph-like
  manifolds $X_\eps$ is given as below and satisfy the uniformity
  conditions~\eqref{eq:deg.bd}--\eqref{eq:vol.vertex}. Then the
  generalised weighted Neumann Laplacian on the graph $(\laplacian
  {X_0}, \Lsqr {X_0})$ and the (Neumann) Laplacian on the manifold
  $(\laplacian {X_\eps}, \Lsqr {X_\eps})$ are $\delta$-close of order
  $1$ where $\delta=o(1)$ as $\eps \to 0$.  In particular, all the
  results of \App{abstr.mod} are true, e.g., the convergence of
  eigenfunctions stated in \Thm{eigenvectors} or the spectral
  convergence in \Thm{spectrum}.
\end{theorem}

In particular, we have the following consequence of the convergence
results of \App{abstr.mod}:
\begin{remark}
  Due to \Thm{other.est} and \Thms{cont}{meas}, we can approximate the
  complicated operator on the manifold $\phi(\laplacian{X_\eps})$ by
  the simpler operator $J\phi(\laplacian{X_0})J'$ up to an error,
  where e.g.\ $\phi(\lambda)=(\lambda+1)^{-1}$ (resolvent),
  $\phi_t(\lambda)=\e^{-t\lambda}$ (heat operator) or $\phi=\1_I$
  (spectral projection).  Saito~\cite{saito:00} obtained a similar but
  weaker assertion in the resolvent case.
\end{remark}



\section{Examples and applications of the spectral convergence}
\label{sec:ex.app}
In this section we give several classes of examples for uniform graphs
and manifolds. We also provide consequences of the above-mentioned
spectral convergence.

\subsection{Embedded graphs and graph-like manifolds}
\label{sec:emb.graph}

Let us start with an \emph{embedded} metric graph as an explicite
example in order to illustrate the geometric meaning of the uniformity
assumptions~\eqref{eq:deg.bd}--\eqref{eq:vol.vertex}.  This situation
has originally been treated~in~\cite{kuchment-zeng:01,
  rubinstein-schatzman:01} on compact metric graphs (cf.\ 
also~\cite[Ex.~4.2]{exner-post:05})), and on trees in~\cite{saito:00}:

Suppose that the weighted metric graph $X_0=(V,E,\bd,\ell,p)$ is
isometrically embedded in $\R^2$ via the maps
$\map{\psi_e}{(0,\ell_e)}{\R^2}$. Then
  \begin{equation*}
    \Psi_\edeps(x,y) := \psi_e(x) + \eps r_e(x) y n_e(x), \qquad
    (x,y) \in (0,\ell) \times [-1/2,1/2] \cong e \times F
  \end{equation*}
  defines a weighted tubular neighbourhood of the edge $e$ considered
  as subset of $\R^2$. Here, $n_e$ is a unit vector field normal to
  the tangent vector field $\dot \psi_e$ and $r_e(x)$ defines the
  radius of the neighbourhood. Let $X_\eps$ be the union of the
  closures of the range of all $\Psi_\edeps$, $e \in E$. Denote by
  \begin{equation*}
    \kappa_e := \dot \psi_1 \ddot \psi_2 - \ddot \psi_1 \dot \psi_2
  \end{equation*}
  the signed curvature of the embedded edge $e$. We assume that there
  are global constants $\ell_0, \beta_0, \kappa_0, r_\pm, \dot r_0>0$
  such that
  \begin{align}
    \label{eq:angle} \tag{E1}
    \measuredangle (e,e') &\ge \beta_0, &%\quad e \ne e' \in E_v, &
    \tan (\beta_0/2)        &> r_+/\ell_0\\
    \label{eq:curve} \tag{E2}
    && |\kappa_e(x)|         &\le \kappa_0, \quad x \in e\\
    \label{eq:len.rad} \tag{E3}
    \ell_e                &\ge \ell_0, &
    r_e (x)               &\ge r_-, \quad d(x,\bd e) \le \min\{1,\ell_e/2\}\\
    \label{eq:rad.upp} \tag{E4}
    |\dot r_e(x)|         &\le \dot r_0, &
    r_e (x)               &\le r_+, \quad x \in e.
  \end{align}
  Here, $\measuredangle(e,e')$ denotes the angle between the tangent
  vectors $\dot \psi_e$ and $\dot \psi_{e'}$ of the embedded edges at
  the vertex $v$ for $e, e' \in E_v$, $e \ne e'$.
  
  We claim that under these assumptions, $X_\eps$ is a \emph{uniform}
  family of graph-like manifolds\footnote{Of course, there are other
    possibilities how to define $X_\eps$ close to the vertices, which
    still lead to a \emph{uniform} graph-like manifold. One can e.g.\ 
    smoothen the singularities at the boundary $\bd U_\vxeps \cap \bd
    X_\eps$ (cf.~\Fig{graph.emb}). In our case, $\bd X_\eps$ has
    singularities but this does not matter in the proof of~\Thm{graph}
    and \Thm{emb.graph}.} embedded in $\R^2$ associated to the
  \emph{uniform} metric graph $X_0$ with weights $p_e := r_e$:
  \begin{theorem}
    \label{thm:emb.graph}
    Under the assumptions~\eqref{eq:angle}--\eqref{eq:rad.upp} \sloppy
    the metric graph $X_0$ and the associated weighted neighbourhood
    $X_\eps$ satisfy~\eqref{eq:deg.bd}--\eqref{eq:vol.vertex} for $\eps$
    small enough. In particular, the generalised weighted Neumann
    Laplacian on the graph $(\laplacian {X_0}, \Lsqr {X_0})$ and the
    Neumann Laplacian $(\laplacian[\Neu] {X_\eps}, \Lsqr {X_\eps})$
    are $\delta$-close of order $1$ where $\delta=O(\eps^{1/2})$ as
    $\eps \to 0$, and therefore, the convergence results of
    \App{abstr.mod} are true.
  \end{theorem}
  \begin{proof}
    The bounded degree assumption~\eqref{eq:deg.bd} follows
    from~\eqref{eq:angle} with $d_0 = \lceil 2\pi/\beta_0 \rceil$.
    \eqref{eq:length} and \eqref{eq:density} are clear. We decompose
    $X_\eps$ into the edge and vertex neighbourhoods $U_\edeps$ and
    $U_\vxeps$, resp., in the way that
    \begin{equation*}
      U_\edeps=
      \Psi_\edeps\bigl( 
             (\eps \ell_0/2, \ell_e - \eps \ell_0/2) \times F \bigr)
    \end{equation*}
    denotes the edge neighbourhood of $e$ shortened by the amount
    $\eps \ell_0$ belonging to the vertex neighbourhoods of $\bd_-e$
    and $\bd_+e$. A straightforward calculation now shows that the
    metric of $U_\edeps \cong (e \times F, g_\eps)$ is represented by
    \begin{equation}
      \label{eq:met.emb}
      G_\eps =
      \begin{pmatrix}
        [(1+ \eps \kappa_e r_e y)^2 + \eps^2 y^2\dot r_e^2]
                           (1-\eps \ell_0/\ell_e)^2 &
        \eps^2 r_e \dot r_e y (1-\eps \ell_0/\ell_e)\\ 
        \eps^2 r_e \dot r_e y (1-\eps \ell_0/\ell_e) &  \eps^2 r_e^2
      \end{pmatrix}.
%       \\=
%      \begin{pmatrix}
%        1+O(\eps) & O(\eps^2) \\ O(\eps^2) &
%        \eps^2 r^2
%      \end{pmatrix}.
    \end{equation}
    The uniformity condition~\eqref{eq:asym.met.ed} is fulfilled due
    to the curvature assumption~\eqref{eq:curve} and due
    to~\eqref{eq:rad.upp} and we obtain
    \begin{align*}
      o_1(1) &= 2r_+ \eps (1+\eps \kappa_0 r_+)^{1/2}, &
      1+o_2(1) &%= (1-\eps \kappa_0 r_+)^{-2}
                =1+\eps C(\kappa_0,r_+),\\
      O_3(1) &= C(\kappa_0, r_+, \dot r_0), &
      o_4(\eps) &= 2 r_+^2 \eps^2
    \end{align*}
    provided $0 < \eps < \eps_0 = \eps_0(\kappa_0,r_+,\dot r_0)$,
    where $C(\cdot)$ depend only on the indicated constants, since,
    e.g.,
    \begin{equation}
      \label{eq:met.dens.emb}
      \det G_\eps^{1/2} = \eps r_e (1+\eps
      \ell_0/\ell_e)|1+\eps \kappa_e r_e y|.
  \end{equation}
    
    A similar calculation on $U_\vxeps \cap \Psi_{\eps,e(e \times
      F)}$, $v \in \bd e$, now using $\overline x = x/\eps$ and $y \in
    F$ as coordinates (as far as possible on $U_\vxeps$) shows
    that~\eqref{eq:met.vx} and~\eqref{eq:est.alpha} are satisfied
    with $\alpha=1$ and $c_\pm$ depending only on $\kappa_0$, $r_\pm$
    and $\dot r_0$. As unscaled set $U_v$ we use the ``straightened''
    version of $\frac 1 \eps U_\vxeps$, i.e., we replace the edge
    parts in $\frac 1 \eps U_{v, \eps}$ by their tangentials at $v$;
    the tubular neighbourhood has width $r_e(0)$ (cf.\ 
    \Fig{graph.emb}).
    \begin{figure}[h]
      \begin{center}
%        \input{graph.emb.pstex_t}
        \begin{picture}(0,0)%
           \includegraphics{graph.emb.eps}%
        \end{picture}%
        \setlength{\unitlength}{4144sp}%
        \begin{picture}(7103,3171)(98,-2390)
           \put(3849,-709){$e$}%
           \put(4697,-1396){$U_v$}%
           \put(6571,-511){$r_e(0)$}%
           \put(5671,-1096){$\ell_0/2$}%
           \put(4816,344){$\ell_0/2$}%
           \put(6121, 74){$\measuredangle(e,e')$}%
           \put(3015,-536){$r_e(x)$}%
           \put(2757,-776){$x$}%
           \put(2476,614){$e'$}%
           \put(2103,-1396){$U_{\eps,v}$}%
           \put(3256,-909){$U_{\eps,e}$}%
         \end{picture}%
        \caption{Decomposition of the weighted neighbourhood $X_\eps$
          and the unscaled vertex neighbourhood $U_v$. In dark grey we
          denoted the collar neighbourhoods mentioned in
          \Rem{collar.vx}. Note that these rectangles exists due
          to~\eqref{eq:angle}.}
      \label{fig:graph.emb}
      \end{center}
    \end{figure}
    
    Next, the volume estimate of~\eqref{eq:vol.vertex} is satisfied
    due to the angle assumption~\eqref{eq:angle}:
    \begin{equation*}
      \vol U_v \le \sum_{e \in E_v} r_e(v) \ell_0/2 \le
      d_0 r_+ \ell_0/2.
    \end{equation*}
    Due to~\eqref{eq:angle} we can place at each end of $U_v$ a
    rectangle (denoted in dark grey in \Fig{graph.emb}) of length
    $\ell_-=(\ell_0 - r_+ \cot (\beta_0/2))/2>0$ and width $r_e(0) \in
    [r_-,r_+]$. These rectangles are the collar neighbourhoods
    mentioned in \Rem{collar.vx}. The Neumann eigenvalues depend
    continuously on the angles $\measuredangle(e,e') \ge \beta_0$ and
    on the widths $r_e(0) \in [r_-,r_+]$, so~\eqref{eq:vol.vertex}
    follows.
\end{proof}

\begin{remark}
  Although the curvature $\kappa_e$ of the edge and the angle
  $\measuredangle(e,e')$ between two adjacent edges are not detectable
  in the limit (at least not in our first order approximation of an
  eigenvalue $\lambda(\eps)=\lambda(0)+o(1)$), we nevertheless need
  the uniform assumptions~\eqref{eq:angle} and~\eqref{eq:curve}. Using
  the direct eigenvalue estimates of \Rem{ew.better} on compact spaces
  one can show that $\lambda_k(\eps)=\lambda_k(0)+O(\eps^{1/2})$. It
  would be interesting whether one can detect information on the
  curvature or the angles between the edges via an asymptotic
  expansion of $\lambda_1(\eps)$. For a curved tubular neighbourhood
  with Dirichlet boundary conditions around a closed curve of length
  $\ell$ with positive curvature in $\R^3$, the first eigenvalue
  expands as
  \begin{equation*}
    \lambda_1^\Dir(\eps) = 
           \frac{\lambda_1}{\eps^2} - 
           \frac 3{4L} \int_0^\ell \kappa_1(s)^2 \dd s + O(\eps)
  \end{equation*}
  (cf.~\cite[Thm.~4.1]{karp-pinsky:88}) where $\lambda_1$ is the first
  Dirichlet eigenvalue of the unit disc and $\kappa_1$ is the
  curvature of the curve.
\end{remark}
  
\begin{remark}
  \label{rem:horn}
  Note that we do not need a \emph{global} lower bound on the radius
  $r_e$, i.e., infinite edges with a shrinking neighbourhood are
  allowed (e.g.\ horn-like shapes as in~\cite{davies-simon:92}. If the
  spectrum of the Laplacian on the corresponding edge neighbourhood
  $U_\edeps$ is $[0,\infty)$, our analysis does not give new
  information.  More interesting cases are provided if the spectrum on
  $U_\edeps$ is purely discrete, e.g. for radial functions decaying
  fast enough like $r_e(x)=\e^{-x^\beta}$, $\beta>1$
  (cf.~\cite{evans-harris:89, davies-simon:92}). The weighted graph
  Laplacian on $e$ now has the form $(Hf)_e=-f_e''+m \beta x^{\beta-1}
  f_e'$. An example of a horn-like end with infinitely many spectral
  gaps in the essential spectrum was constructed
  in~\cite[Thm.~3]{lott:01}. In principle, these results can be
  recovered with our analysis.
  % siehe Rechnung 2005-09-26
\end{remark}  


\subsection{Examples constructed from a finite number of building blocks}
\label{sec:examples}

\subsubsection*{Covering manifolds}
An important class of examples satisfying the uniformity conditions
are coverings with a compact quotient: Clearly, a covering metric
graph $X_0 \to M_0$ with compact quotient $M_0$ is \emph{uniform}.
Similarly, an associated family of graph-like covering manifold
$X_\eps \to M_\eps$ with compact quotient $M_\eps$ is \emph{uniform}
once~\eqref{eq:asym.met.ed}--\eqref{eq:est.alpha} are fulfilled.

For abelian covering groups (and for some classes of non-abelian
\sloppy groups, cf.~\cite{lledo-post:pre04}) the spectral convergence
on \emph{compact} graphs and manifolds would be enough: The Floquet
theory allows to describe the spectrum on the covering via a family of
Laplacians on the compact quotient. Nevertheless our result here is
more general since we can treat an \emph{arbitrary} covering.

More generally, the assumptions
\eqref{eq:asym.met.ed}--\eqref{eq:vol.vertex} are fulfilled if the number
of isomorphism classes of $U_v$ and $U_e$ are finite, i.e., if we
construct the graph-like manifold out of a finite number of building
blocks as in a plumber's shop. An example is given in \Sec{disc.graph}
where we construct a graph-like manifold according to the
Sierpi\'nski graph (cf.~\Fig{sierpinski}).  Here, only two different
vertex neighbourhood building blocks are necessary. Note that this
manifold has a \emph{fractal} structure, not locally, but
\emph{globally}.


  

\subsubsection*{Spectral gaps and eigenvalues in gaps}
%\label{sec:coverings}

Typically, operators on coverings have a \emph{band-gap} type
spectrum, i.e., the spectrum is the locally finite union of compact
intervals (maybe reduced to a point). The spectral convergence ensures
e.g.\ the existence of spectral gaps as $\eps \to 0$ once $\spec
{\laplacian {X_0}}$ has spectral gaps and $\eps$ is small
enough:
\begin{theorem}
  \label{thm:gaps} Suppose that $M_0$ is a compact graph with
  associated uniform graph-like manifold $M_\eps$. Denote by $X_0$
  resp.\ $X_\eps$ a covering of $M_0$ resp.\ $X_\eps$ such that
  $X_\eps$ is a graph-like manifold associated to $X_0$. If the
  generalised Neumann Laplacian on $X_0$ has a spectral gap $(a,b)$,
  i.e., $\spec {\laplacian{X_0}} \cap (a,b) = \emptyset$, then the
  Laplacian $\laplacian{X_\eps}$ has a spectral gap close to $(a,b)$
  provided $\eps$ is small enough.
\end{theorem}
Covering metric graphs $X_0 \to M_0$ with spectral gaps are given
e.g.~in~\cite[Sec.~9.4--9.6]{exner-post:05}; the simplest example is
maybe the Cayley graph of the group $\Gamma = \Z \times \Z_p$ where
$\Z_p$ denotes the cyclic group of order $p$ has a spectral gap iff
$p$ is odd. A similar example consists of a regular rooted tree
(cf.~\Thm{tree.mfd}). A different procedure of creating gaps in a
metric graph is provided by the so-called \emph{graph-decoration}.
Roughly speaking, the new graph $\hat X_0$ is obtained from a given
infinite graph $X_0$ by attaching a fixed (compact) graph $M_0$ to
each vertex $v$ of $X_0$. The Laplacian on $\hat X_0$ now has spectral
gaps. This result has been established for a discrete graph
in~\cite{aizenman-schenker:00}. The general case for quantum graphs is
announced in~\cite{kuchment:05} and proved in the case when $X_0$ is a
compact graph (in the sense that there is no spectrum of the decorated
graph near (certain parts) of the spectrum of the decorating graph).
For quantum graphs, there are related examples leading to gaps
(cf.~\cite{ael:94}, \cite{exner:95}). A similar effect by attaching a
single loop to each vertex of a periodic graph has been used
in~\cite[Sec.~9]{exner-post:05}. The case of periodically arranged
manifolds connected by line sements or through points has been
analysed in~\cite{beg:03, bgl:05}.

Finally, another class of examples is given by \emph{fractal} metric
graphs, i.e., graphs arranged in a self-similar manner
(cf.~\Thm{sierpinski}). The main point here is that the metric graph
spectrum has a fractal structure, so once, $\eps$ is small enough, the
corresponding Laplacian on a graph-like manifold has an arbitrary (but
a priori finite) number of spectral gaps \emph{in the compact}
spectral interval $[0,\Lambda]$.
  

\subsubsection*{Eigenvalues in gaps}
%  \label{ex:graph.ew}
Suppose that $X_0$ is a uniform metric graph such that its generalised
Neumann Laplacian has a spectral gap, namely, $\spec {\laplacian
  {X_0}} \cap (a,b) = \emptyset$. By the previous example, a
corresponding \emph{uniform} graph-like manifold $X_\eps$ has also a
spectral gap close to $(a,b)$ provided $\eps$ is small enough.
  
Now if we change the metric graph locally, e.g.\ by attaching a loop
of length $\ell_1$ at a fixed vertex $v_1 \in V$ (call the perturbed
graph $\hat X_0$) then the generalised Neumann Laplacian on $\hat X_0$
has additional eigenvalues $\lambda_k=(2\pi k/\ell_1)^2$ with
eigenfunctions located on the loop and vanishing at $v_1$ and on the
rest of the graph (maybe there are more additional eigenvalues). For
example, if $2\pi/\sqrt b < \ell_1 < 2\pi/\sqrt a$ then the ground
state of the loop lies in $(a,b)$.  Note that the essential spectrum
of $X_0$ and the perturbed metric graph $\hat X_0$ are the same since
the perturbation is compact; in particular, only \emph{discrete}
eigenvalues occur in the spectral gap. Due to the spectral convergence
\Thm{graph}, a corresponding (uniform) graph-like manifold now must
have (at least) an eigenvalue in $(a,b)$.  Furthermore, if the
corresponding eigenvalue of the quantum graph is simple (i.e., there
are no other eigenvalues of the looped graph $\hat X_0$ at
$\lambda_1=(2\pi/\ell_0)^2$), then there is a unique eigenvalue close
to $\lambda_1$ of multiplicity $1$, and the corresponding
eigenfunction of the graph-like manifold is close to the eigenfunction
of the metric graph in the sense of \Thm{eigenvectors}. More
generally, we have:
\begin{theorem}
  \label{thm:ew.graph}
  Suppose that $\laplacian {X_0}$ has a discrete eigenvalue $\lambda$
  of multiplicity $m$ and $\spec {\laplacian {X_0}} \cap I =
  \{\lambda\}$. Let $X_\eps$ be a graph-like manifold associated to
  $X_0$. Then $\laplacian {X_\eps}$ has $m$ eigenvalues (not all
  necessarily distinct) converging to $\lambda$ as $\eps \to 0$ and
  the eigenprojections converge in norm, i.e., $\norm{J'P_\eps J -
    P_0} \to 0$ as $\eps \to 0$ where $P_\eps$ is the eigenprojection
  of $\laplacian {X_\eps}$ onto the interval $I$, $\eps \ge 0$. 
  
  If in particular, $\lambda$ is a simple eigenvalue with
  eigenfunction $\phi$, then there exists an eigenfunction $\phi_\eps$
  of $\laplacian {X_\eps}$ such that $\norm {J\phi - \phi_\eps} \to
  0$.
\end{theorem}
Roughly speaking, the theorem says, that an eigenfunction on the
graph-like manifold is approximately given by the corresponding
eigenfunction on the graph, extended constantly in the transversal
direction (and set to $0$ in the vertex neighbourhood).

 
Eigenvalues in gaps have been discussed e.g.~in~\cite{aadh:94,
  post:03b} (see also the references therein). One can interprete such
a local perturbation as an \emph{impurity} of a periodic structure,
say, a crystal or a semi-conductor. The additional eigenvalue in the
gap now corresponds to an additional energy level and a bound state.
Note that in~\cite{post:03b} it was \emph{assumed} that $X_\eps$ is a
\emph{covering} manifold and that it fulfilled a \emph{gap} condition
in the sense that there is a fundamental domain $D$ such that $\EWD k
D < \EWN {k+1} D$ for some $k$ (Dirichlet and Neumann eigenvalues on
$D$). This is in general a stronger condition than just the assumption
that $\laplacian {X_\eps}$ has a spectral gap.  Here, we only need to
know that the generalised Neumann Laplacian $\laplacian {X_0}$ on the
graph has a spectral gap which is often easy to show.



\subsection{Equilateral metric graphs and discrete graphs}
\label{sec:disc.graph}
In this subsection we will analyse the special case when all length of
the metric graph $X_0=(V,E,\bd, \ell)$ are the same, say
$\ell_e=\ell$. Under these assumptions, there is a nice relation
between the spectrum of the metric graph Laplacian $\laplacian {X_0}$
(with weights $p_e=1$) acting on $\Lsqr{X_0} = \oplus_e \Lsqr{0,\ell}$
and the discrete Laplacian $\laplacian G$ of the graph $G=(V,E,\bd)$.
Roughly speaking, the spectrum of $\laplacian {X_0}$ consists of
infinitely many (distorted) copies of the spectrum of $\laplacian G$
arranged in a row.  This relation allows us to profit from the vast
literature on $\laplacian G$ and to carry over calculations of spectra
of the discrete to the continuous graph Laplacian.

The \emph{discrete} Laplacian is given by
\begin{equation}
  \label{eq:def.disc.lap}
  (\laplacian G a)(v) := -\frac 1 {\deg v} \sum_{w \sim v}
  (a(w)-a(v)),
  \qquad a \in \lsqr V
\end{equation}
where $w \sim v$ means that $v$ and $w$ are joined by an edge. The
discrete Laplacian acts in the weighted space $\lsqr V$ consisting of
all sequences $a=(a_v)_v$ with finite weighted norm
\begin{equation*}
  \norm[\lsqr V] a := \sum_{v \in V}{\deg v}|a(v)|^2.
\end{equation*}
This operator is bounded and has spectrum in $[0,2]$.

\enlargethispage*{3ex}
In~\cite{cattaneo:97} (cf.~\cite{exner:97b} for more general boundary
conditions on the metric graph) one can find the following nice
relation between the spectrum of the generalised Neumann Laplacian
$\laplacian {X_0}$ and the discrete Laplacian $\laplacian G$. The
Dirichlet spectrum $\Sigma^\Dir = \set{ (k\pi/\ell)^2}{k \in \N}$ of
an individual edge $e \cong (0,\ell)$ always plays a special role.
Since we are only interested in qualitative results (e.g.\ the fractal
example in \Thm{sierpinski}) we exclude the Dirichlet spectrum here. A
more detailed discussion on the Dirichlet spectrum can be found
in~\cite{kuchment:05} and~\cite{cattaneo:97}. We set
$g(\lambda)=1-\cos(\ell \sqrt \lambda)$. Note that $\Sigma^\Dir =
g^{-1}\{0,2\}$.
\begin{theorem}
  \label{thm:disc.met}
  Assume that $G=(V,E,\bd)$ is a countable, connected graph with $\deg
  v \in \{2, 3, \dots, d_0\}$, $v \in V$, and without self-loops then
  \begin{equation*}
    \spec[\bullet]{\laplacian {X_0}} \setminus \Sigma^\Dir =
    g^{-1}\bigl(\spec[\bullet]{\laplacian G} \setminus \{0, 2\}\bigr),
  \end{equation*}
  i.e., for $\lambda \ne \Sigma^\Dir$, we have $\lambda \in
  \spec[\bullet]{\laplacian {X_0}}$ iff $g(\lambda) \in
  \spec[\bullet]{\laplacian G}$. Here $\bullet \in \{\mathrm p,
  \mathrm c, \emptyset, \mathrm {pp}, \mathrm{disc}, \mathrm{ess}\}$
  denotes either the point spectrum (the set of eigenvalues
  $\sigma_{\mathrm p}$), the complement of the
  eigenvalues\footnote{Note that $\sigma_{\mathrm c}$ does in general
    not coincide with the continuous spectrum,
    cf.~\cite{reed-simon-1}.} ($\sigma_{\mathrm c}= \sigma \setminus
  \sigma_{\mathrm p}$), the entire spectrum ($\sigma$), the pure point
  spectrum (the closure of the set of eigenvalues, i.e.,
  $\sigma_{\mathrm{pp}}=\clo{\sigma_{\mathrm p}}$), the discrete or
  the essential spectrum.  Furthermore, the eigenvalue $\lambda$ of
  $\laplacian {X_0}$ has multiplicity $m$ iff the eigenvalue
  $g(\lambda)$ of $\laplacian G$ has multiplicity $m$.
\end{theorem}
\begin{proof}
  The assertion has been proved for the point spectrum and its
  complement in~\cite{cattaneo:97} and therefore also for the entire
  spectrum. Since $g$ is a local homeomorphism on the complement of
  $\Sigma^\Dir$, the statement on $\sigma_{\mathrm{pp}}$ also
  follows. In addition, note that
  \begin{gather*}
    \map {U_\lambda} {E_{g(\lambda)}(\laplacian G)} 
                     {E_\lambda(\laplacian{X_0})},\\
    (U_\lambda a)_e(x) := 
        \frac {\sqrt \ell} {\sqrt 2 \sin (\ell \sqrt \lambda)} \bigl( a(\bd_-e)
    \sin ((\ell - x) \sqrt \lambda) + a(\bd_+e) \sin (x \sqrt \lambda) \bigr)
  \end{gather*}
  is an isometry from the eigenspace of $\laplacian G$ w.r.t.\ the
  eigenvalue $g(\lambda)$ onto the eigenspace of $\laplacian {X_0}$
  w.r.t.\ the eigenvalue $\lambda \notin \Sigma^\Dir$.  In particular,
  multiplicities of eigenvalues are preserved. Furthermore, $\lambda$
  is isolated in $\spec{\laplacian {X_0}}$ iff $g(\lambda)$ is
  isolated in $\spec{\laplacian G}$, so the statement on the discrete
  spectrum follows. For the essential spectrum note that we already
  have the statement for the entire spectrum and for the discrete
  spectrum, and that $\sigma_{\mathrm{ess}}=\sigma \setminus
  \sigma_{\mathrm{disc}}$.
\end{proof}

Since the spectrum of the discrete Laplacian has been explored in many
cases, the previous theorem allows us to determine the corresponding
spectrum of the equilateral metric graph leading to interesting
examples of graph-like manifolds. For simplicity, we fix the edge
length to $\ell=1$.


\subsubsection*{Homogeneous trees}
Let $G$ be a homogeneous routed tree of degree $d_0 \ge 3$, then
  \begin{equation}
    \label{eq:sp.tree}
    \spec[\mathrm p]{\laplacian G} = \emptyset \qquad\text{and}\qquad
    \spec[\mathrm c]{\laplacian G} =
      \left[ 1 - \frac{2\sqrt{d_0-1}} {d_0}, 
             1+\frac{2\sqrt{d_0-1}} {d_0} \right].
  \end{equation}
  \Thm{disc.met} now describes $\spec[\mathrm c]{\laplacian{X_0}}$.
  Since $\spec{\laplacian G} = \spec[\mathrm c]{\laplacian G}
  \subsetneq [0,2]$, the metric graph Laplacian has spectral gaps
  (i.e., no spectrum at) $I_0:=(0, \omega_0^2)$ and
\begin{equation*}
    I_k:=\bigl((k\pi - \omega_0)^2, (k\pi)^2 \bigr) \cup
         \bigl((k\pi)^2, (k\pi + \omega_0)^2\bigr), \qquad k \in \N,
\end{equation*}
where $\omega_0=\arccos (2\sqrt{d_0-1}/d_0)$. A more detailed analysis
done in~\cite{cattaneo:97} shows that $\spec[\mathrm p]{\laplacian
  {X_0}}=\Sigma^\Dir$. All eigenvalues have infinite multiplicity, so
$\spec {\laplacian {X_0}}$ is purely essential and has band-gap
structure with gaps exactly at $I_k$, $k \in \N_0$. In particular,
$\inf \spec {\laplacian {X_0}} = \omega_0^2>0$ and we have:
\begin{theorem}
  \label{thm:tree.mfd}
  Suppose $X_\eps$ is a family of uniform graph-like manifolds
  associated to the regular tree of degree $d_0 \ge 3$ with equal edge
  lengths. Then the Laplacian on $X_\eps$ has spectral gaps and its
  essential spectrum is non-empty near any $\lambda \in \essspec
  {\laplacian {X_0}}$ provided $\eps$ is small enough
  (cf.~\Cor{ess.sp}). Furthermore, $\Sigma_0(X_\eps):=\inf \spec
  {\laplacian{X_\eps}} \to \omega_0^2$ as $\eps \to 0$ and in
  particular, $\Sigma_0(X_\eps) \ge c>0$ for $\eps$ small enough.
\end{theorem}
\begin{remark}
  Note that if $X_\eps$ would be a Riemannian covering of a compact
  manifold $M_\eps \cong X_\eps /\Gamma$, then the non-amenability of
  $\Gamma$ (if e.g.~$\Gamma$ contains the non-abelian free group
  $\Z^{*2}$ as subgroup) implies $\Sigma_0(X_\eps)>0$
  (cf.~\cite{brooks:81}). An \emph{unrooted} tree of degree $d_0$ can
  be considered as \emph{unoriented} Cayley graph of the group
  $\Z_2^{*d_0}$ (free product of $d_0$ groups of order $2$) with
  respect to the generators $\gamma_i$, $i=1, \dots, d_0$. But the
  elements of order $2$ act on a corresponding graph-like manifold as
  reflections. In particular, these elements have fixed points and the
  quotient is no longer a smooth manifold so we cannot use the
  covering argument here in order to show $\Sigma_0(X_\eps)>0$.
\end{remark}

\begin{remark}
  \label{rem:brooks}  
  There is a nice upper estimate for $\Sigma_{\mathrm{ess}}(X):=\inf
  \essspec {\laplacian X}$: Denote by
  \begin{equation*}
    \mu(X) := \lim_{r \to \infty} \frac 1 r \log \vol_d B(x_0,r)
  \end{equation*}
  the growth rate of a ball of radius $r$ in $X$. It can be easily
  seen that $\mu(X)$ is independent of $x_0$. Brooks~\cite{brooks:81b}
  showed that if $\vol_d(X)=\infty$ then
  \begin{equation*}
    \Sigma_{\mathrm{ess}}(X) \le \frac 14 \mu(X)^2.
  \end{equation*}
  A priori, this estimate is not sharp as there are amenable groups
  $\pi$ of exponential growth (cf.~\cite{milnor:68}): For the
  universal cover $\wt M$ of a compact manifold $M$ with amenable
  fundamental group $\pi_1(M)=\pi$ we have $\Sigma_{\mathrm{ess}}(\wt
  M)=0$ but $\mu(\wt M)>0$.
  
  In our case, we can easily calculate $\mu(X_\eps)$ approximately: If
  $x_0$ is a point in the root vertex neighbourhood then an increase
  of $r$ by $\ell_0=1$ encounters another generation in the tree.
  Therefore $\vol_d B(x_0,r) \approx \vol_d \hat U_\vxeps
  ((d_0-1)^r-1)$ for large $r$ where $\hat U_\vxeps$ is a (sample)
  vertex neighbourhood together with the $d_0$ adjacent half edge
  neighbourhoods. We obtain $\mu(X_\eps) \approx \log(d_0-1)$ and in
  particular, our estimate shows
  \begin{equation*}
    \Sigma_0(X_\eps)=
    \Sigma_{\mathrm{ess}}(X_\eps) \approx 
      \arccos^2 \Bigl( \frac{2\sqrt{d_0-1}} {d_0} \Bigr) < 
      \frac 1 4 \log^2(d_0-1) \approx \frac 14 \mu(X_\eps)^2
  \end{equation*}
  if $d_0 \ge 3$. The inequality provides an (approximative)
  \emph{lower} bound on $\mu(X_\eps)$ and the difference of the LHS
  and RHS is e.g.\ smaller than 1\% if $d_0=3$). For further estimates
  of this type using isoperimetric constants we refer
  to~\cite{brooks:81b}. We only want to mention here that
  isoperimetric constants on graph-like manifolds can quite easily be
  calculated.
\end{remark}


\subsubsection*{Sierpi\'nski graphs}
Let us give another interesting example admitting a fractal spectrum.
Let $G_1$ be the complete graph with three vertices. Suppose
$G_n=(V_n,E_n)$ has already been constructed. Then $G_{n+1}$ is
obtained from three disjoint copies $G_n^{(i)}$ ($i=1,2,3$) of $G_n$
and the equivalent relation $\sim$, i.e., $G_{n+1} := G_n^{(1)} \dcup
G_n^{(2)} \dcup G_n^{(3)}/{\sim}$ where $\sim$ identifies the vertex
$v_i(G_n^{(j)})$ with $v_j(G_n^{(i)})$, $i\ne j$, $i,j=1,2,3$. Here,
$v_1(G_n)$ denotes the lower left vertex of degree $2$, $v_2(G_n)$ the
lower right vertex of degree $2$ and $v_3(G_n)$ the upper vertex of
degree $2$ (cf.~\Fig{sierpinski}). Note that each $G_n$ has exactly
three vertices of degree $2$, the other have degree $4$. Furthermore,
$|V_n|=(3^n+1)/2$ and $|E_n|=3^n$. Now, $G_n$ embeds into $G_{n+1}$
via $G_n = G_n^{(1)} \hookrightarrow G_{n+1}$. Finally,
the Sierpi\'nksi graph is given by $G := \bigcup_{n \in \N} G_n$
(cf.~\Fig{sierpinski}).
\begin{figure}[h]
  \begin{center}
%     \input{sierpinski.pstex_t}
    \begin{picture}(0,0)%
      \includegraphics{sierpinski.eps}
    \end{picture}%
    \setlength{\unitlength}{4144sp}%
    \begin{picture}(5857,4184)(496,-3481)
      \put(496,-2941){$v_1(G_3)$}%
      \put(1576,-1456){$v_3(G_3)$}%
      \put(5446,-3031){$v_2(G_4)$}%
      \put(3106,-3436){$v_2(G_3)$}%
      \put(541,-3166){$=v_1(G_4)$}%
      \put(2566,254){$v_3(G_4)$}%
    \end{picture}%
  \caption{The first four generations $G_4$ of the infinite Sierpi\'nski
    graph, each edge having unit length. The graph $G_3$ is denoted
    with thick edges and is naturally embedded into $G_4$.}
  \label{fig:sierpinski}
\end{center}
\end{figure}
%\enlargethispage*{2cm}
Note that $G$ has one vertex of degree $2$, all other vertices have
degree $4$. The nature of the spectrum of the discrete Laplacian
$\laplacian G$ was calculated by~\cite[Thm.~2]{teplyaev:98}: We have
\begin{equation}
  \label{eq:spec.sier}
  \spec {\laplacian G} = J \cup D, \qquad
  D := \Bigl\{\frac 32\Bigr\} \cup 
       \bigcup_{n=0}^\infty p^{-n} \Bigl\{\frac 34 \Bigr\},
  \qquad
  J := \clo D,
\end{equation}
where $p(z):=z(5-4z)$. and $p^{-n}$ is the $n$-th pre-image, i.e.,
$p^{-n}\{3/4\} = \set{z \in \R}{p^{\circ n}(z)=3/4}$. The spectrum of
$\laplacian G$ is pure point, each eigenvalue has multiplicity
$\infty$, so the spectrum is also purely essential. The set $D$
consists of the isolated eigenvalues and the set $J$ is a Cantor set
of Lebesgue measure $0$ (the Julia set of the polynomial $p$). Due to
\Thm{disc.met} the spectrum of the metric graph Laplacian
$\laplacian{X_0}$ (with $\ell=1$) is pure point and purely essential.
It is given by
\begin{equation*}
  \spec {\laplacian {X_0}} \setminus \Sigma^\Dir =
  g^{-1}(D) \cup g^{-1}(J \setminus \{0\}).
\end{equation*}
Since $g(\lambda)=1-\cos \sqrt \lambda$ is a local homeomorphism
mapping measure $0$ set into measure $0$ sets and vice versa,
$g^{-1}(D)$ consists of isolated eigenvalues of $\laplacian {X_0}$ of
infinite multiplicity and $g^{-1}(J \setminus \{ 0\})$ is a Cantor set
of measure $0$ (away from $\Sigma^\Dir$).

Now our spectral convergence result on graph-like manifolds leads to
an example of a family of smooth manifolds approaching a fractal
spectrum:
\begin{theorem}
  \label{thm:sierpinski}
  Suppose $X_\eps$ is a family of uniform graph-like manifolds
  associated to the Sierpi\'nski graph with equal edge lengths. Then
  the essential spectrum of the Laplacian on $X_\eps$ approaches the
  fractal spectrum of $\laplacian {X_0}$ in any fixed interval
  $[0,\Lambda]$. The discrete spectrum of $\laplacian{X_\eps}$ is
  either empty or merges into the essential spectrum as $\eps \to 0$
  (cf.~\Cor{ess.sp}). In particular, $\spec {\laplacian {X_\eps}}$ has
  an arbitrary large number of spectral gaps in the \emph{compact}
  interval $[0,\Lambda]$ provided $\eps$ is small enough.
\end{theorem}


\subsection{The decoupling case}
\label{sec:decoupling}

We obtain similar convergence results on \emph{non-compact} graph-like
manifolds $X_\eps$ with slower decay of $\vol U_\vxeps$ (i.e., large
vertex neighbourhoods $U_\vxeps$) as discussed in the compact case
in~\cite{kuchment-zeng:03} and~\cite[Sec.5--8]{exner-post:05}.  If for
example the vertex volume $\vol U_\vxeps$ decays slower than the edge
volume $\vol U_\edeps$ then the Laplacian on $X_\eps$ converges also
in the non-compact case to the \emph{decoupled} limit operator $H :=
\bigoplus_e H_e \oplus 0$ acting in the enlarged Hilbert space $\HS :=
\Lsqr X_0 \oplus \C^V$.  Here, $H_e$ is the Dirichlet operator on the
interval $(0,\ell_e) \cong e$ and $0$ is the null operator on $\C^V$.
Therefore, the limit operator has pure point spectrum
\begin{equation}
  \label{eq:spec.dec}
    \spec H = \Bigset{\lambda_k(e):=\frac {\pi^2 k^2} {\ell_e^2}} 
                  {e \in E, k \in \N_0}
\end{equation}
and the multiplicity of $\lambda$ is $|V|$ if $\lambda=0$ and $|\set{e
  \in E}{\lambda=\lambda_k(e)}|$ if $\lambda>0$. We omit the details
here since the proof is rather similar to the non-decoupling case
treated below. Note that we need here some uniformity assumptions on
the vertex neighbourhood $U_\vxeps$ ensuring that $U_\vxeps$ does not
become too \emph{small}, e.g., we need a \emph{lower} bound on $\vol
U^-_\vxeps$.  Here $U^-_\vxeps$ is the subset of $U_\vxeps$ where the
metric satisfies $g_\vxeps \cong \eps^{2\alpha} g_v$ and $0 < \alpha <
(d-1)/d$ (cf.~\Rem{unif.mfd}.\ref{rem:bottle.neck}
and~\ref{rem:unif.mfd}.\ref{rem:critical}).

\begin{theorem}
  \label{thm:decoupling}
  Suppose that $X_\eps$ is a graph-like manifold with large vertex
  neighbourhoods associated to a metric graph $X_0$ with edge length
  $(\ell_e)_e$. Then the Laplacian on $X_\eps$ (with Neumann boundary
  conditions, if $\bd X_\eps \ne \emptyset$) approaches the decoupled
  operator $H$. In particular, its spectrum converges to the set
  $\spec H$ given in~\eqref{eq:spec.dec} in any compact spectral
  interval.
\end{theorem}
  
In a similar way, the case of Dirichlet boundary conditions and small
junctions as in~\cite{post:05a} can be extended to the non-compact
case. The Laplacian on $X_\eps$ with Dirichlet boundary conditions has
to be shifted by $\lambda_1^\Dir(F)/\eps^2$ where $\lambda_1^\Dir(F)$
is the first Dirichlet eigenvalue of the cross section. The limit
operator also decouples and consists of the Dirichlet eigenvalues
only.

Since the graph structure is no more visible in the limit due to the
decoupling, we just give a simple example of a graph consisting of a
half-line with one vertex of degree $1$ at $0$ and the others of
degree $2$.  We label the edges by $n \in \N$. By an appropriate
choice of the length $\ell_e$ we can construct a manifold with certain
spectral properties. If we consider e.g.\ $\ell_n:=\sqrt n/\pi$ then
$\lambda_k(n)=k^2/n$. Since every rational number $r$ can be written
in the form $r=k^2/n$ ($r=p/q=p^2/(pq)$), the operator $H$ has dense
point spectrum consisting of all non-negative rational numbers.
Therefore the spectrum of $H$ is $[0,\infty)$ and purely essential.
Our analysis here is too weak to say anything more on the nature of
the Laplacian on the graph-like manifold $X_\eps$ with large vertex
neighbourhoods than it is purely essential in any bounded spectral
interval. We expect that $\laplacian {X_\eps}$ also has pure point
spectrum, but one needs arguments from scattering theory to prove
this.

Other choices of the length are possible, e.g., a set of rationally
independent length $\ell_e$, $e \in E$. Once one is able to determine
$\spec H$ by number theoretical arguments one has a Laplacian
$\laplacian {X_\eps}$ with a spectrum close to this set.




%------------------------------------------------------------
% We want the Appendix numbers A.1 etc.
\setcounter{section}{0}
\renewcommand{\thesection}{\Alph{section}}
\section{Appendix}
%\label{sec:appendix}
\label{sec:abstr.mod}
%------------------------------------------------------------
In the appendix we prove our main technical tool, the convergence
results for arbitrary pairs of self-adjoint non-negative operators and
Hilbert spaces $(H,\HS)$ and $(\wt H, \wt \HS)$ being close to each
other (cf.~\Def{closeness}). Although most of the techniques are
standard, we repeat the arguments here since usually, one has a
\emph{fixed} Hilbert space and a careful analysis of the dependence on
some parameter entering in the operator \emph{and} Hilbert space is
not necessary. We do not mention the parameter explicitely but express
all convergence results in terms of a ``distance'' $\delta$ of
$(H,\HS)$ and $(\wt H,\wt \HS)$.

%------------------------------------------------------------
\subsection{Scale of Hilbert spaces associated with a non-negative
  operator}
\label{sec:scale}
%------------------------------------------------------------

To a Hilbert space $\HS$ with inner product $\iprod \cdot \cdot$ and
norm $\norm \cdot$ together with a non-negative, unbounded, operator
$H$, we associate the scale of Hilbert spaces
\begin{equation}
  \label{eq:scale.hs}
  \HS_k := \dom (H+1)^{k/2}, \qquad 
    \norm[k] u := \norm{(H+1)^{k/2}u}, \quad k \ge 0.
\end{equation}
For negative exponents, we define
\begin{equation}
  \label{eq:scale.hs.neg}
  \HS_{-k} := \HS_k^*.
\end{equation}
Note that $\HS = \HS_0$ embeds naturally into $\HS_{-k}$ via $u \mapsto
  \iprod u \cdot$ since
\begin{displaymath}
  \norm[-k] {\iprod u \cdot} =
  \sup_{v \in \HS_k} \frac{|\iprod u v|}{\norm[k] v} =
  \sup_{w \in \HS_0} \frac{|\iprod {R^{k/2}u} w|}{\norm[0] w} =
  \norm[0]{R^{k/2} u},
\end{displaymath}
where
\begin{equation}
  \label{def:res}
  R := (H+1)^{-1}
\end{equation}
denotes the resolvent of $H \ge 0$. The last equality used the natural
identification $\HS \cong \HS^*$ via $u \mapsto \iprod u \cdot$.
Therefore, we can interprete $\HS_{-k}$ as the completion of $\HS$ in
the norm $\norm[-k] \cdot$. With this identification, we
have
\begin{equation}
  \label{eq:norm.iprod}
  \norm[-k] u =  \sup_{v \in \HS_k} \frac{|\iprod u v|}{\norm[k] v}, \qquad
  \text{for all $k \in \R$.}
\end{equation}
For a second Hilbert space $\wt \HS$ with inner product $\iprod \cdot
\cdot$ and norm $\norm \cdot$ together with a non-negative, unbounded,
operator $\wt H$, we define in the same way a scale of Hilbert spaces
$\wt \HS_k$ with norms $\norm[k] \cdot$.

Guided by the classical application $H=\laplacian X$ in $\HS=\Lsqr X$
for a complete manifold $X$, we call $k$ the \emph{regularity order}.
In this case, $\HS_k$ corresponds to the $k$-th Sobolev space
$\Sob[k]X$.

%------------------------------------------------------------
\subsection{Operators on scales}
\label{sec:op}
%------------------------------------------------------------

Suppose we have two scales of Hilbert spaces $\HS_k$, $\wt \HS_k$
associated to the non-negative operators $H$, $\wt H$ with resolvents
$R := (H+1)^{-1}$, $\wt R := (\wt H+1)^{-1}$, respectively. The norm
of an operator $\map A {\HS_k} {\wt \HS_{-\wt k}}$ is
\begin{equation}
  \label{eq:norm.op}
  \norm[k \to -\wt k] A :=
  \sup_{u \in \HS_k} \frac{\norm[-\wt k]{Au}}{\norm[k] u} =
  \norm[0 \to 0] {\wt R^{\wt k/2} A R^{k/2}}.
\end{equation}
The norm of the adjoint $\map {A^*} {\wt \HS_{\wt k}} {\HS_{-k}}$ is given by
\begin{equation}
  \label{eq:norm.adj}
  \norm[\wt k \to -k] {A^*} =
  \norm[k \to -\wt k] A.
\end{equation}
Furthermore,
\begin{equation}
  \label{eq:norm.op.ineq}
  \norm[k \to -\wt k] A \le
  \norm[m \to -\wt m] A \qquad \text{provided} \qquad k \ge m,  \wt k \ge \wt m
\end{equation}
since
\begin{displaymath}
  \norm[k \to -\wt k] A =
  \norm[0 \to 0] {\wt R^{\wt k/2} A R^{k/2}} =
  \norm[0 \to 0] {\wt R^{(\wt k - \wt m)/2} \wt R^{\wt m/2} A 
                    R^{m/2} R^{(k-m)/2}} \le
  \norm[m \to -\wt m] A
\end{displaymath}
and $\norm R, \norm {\wt R} \le 1$.


\subsection{Closeness assumption}
\label{sec:ass}

In this section we state our main assumptions on the two operators $H$
and $\wt H$ acting in the Hilbert spaces $\HS$ and $\wt \HS$.  We
think of $(\wt H,\wt \HS)$ being a perturbation of $(H,\HS)$, or that
$(H,\HS)$ describes a simplified model (say, on a metric graph $X_0$)
which is close to a more complicated model given by $(\wt H, \wt \HS)$
(say, on a graph-like manifold $X_\eps$).  We want to state
assumptions under which $(H,\HS)$ and $(\wt H,\wt \HS)$ are close in a
sense to be specified in \Def{closeness}.

Let us explain the following concept of \emph{quasi-unitary} operators
in the case of unitary operators (cf.\ also \Ex{nrconv}): Suppose we
have a unitary operator $\map J \HS {\wt \HS}$ with inverse $\map {J'
  = J^*} {\wt \HS} \HS$ respecting the quadratic form domains, i.e.
$\map {J_1:=J \restr {\HS_1}} {\HS_1} {\wt \HS_1}$ and $\map {J_1'=J^*
  \restr {\wt \HS_1}} {\wt \HS_1}{\HS_1}$. If
\begin{displaymath}
  J_1'^* H = \wt H J_1
\end{displaymath}
then $H$ and $\wt H$ are unitarily equivalent and have therefore the
same spectral properties. The main point here is that $J$ respects the
quadratic form domain and therefore, $\map {{J'_1}^*} {\HS_{-1}}
{\wt \HS_{-1}}$ is an extension of $\map J \HS {\wt \HS}$. In this sense, the
above equality says that $J$ is an intertwining operator.

We want to lessen the assumption such that the spectral properties are
not the same but still at close quarters.  We now start with the
definition of $\delta$-closeness:

\begin{definition}
  \label{def:closeness}
  Suppose we have linear operators
  \begin{equation}
  \label{eq:trans.op}
  \begin{array}{ccc}
    \map J      \HS     {\wt \HS},    & \qquad & \map  {J'} {\wt \HS} \HS\\
    \map {J_1} {\HS_1} {\wt \HS_1}, & \qquad & \map {J_1'} {\wt \HS_1} {\HS_1}.
  \end{array}
  \end{equation}
  Let $\delta>0$ and $k \ge 1$. We say that $(H,\HS)$ and $(\wt H,\wt
  \HS)$ are \emph{$\delta$-close} with respect to the
  \emph{quasi-unitary maps} $(J,J_1)$ and $(J',J_1')$ of \emph{order
    $k$} iff the following conditions are fulfilled:
\begin{align}
  \label{eq:j.scale}
  \norm[1 \to 0] {J - J_1} &\le \delta, & 
  \norm[1 \to 0] {J'-J_1'} &\le \delta\\
  \label{eq:j.adj}
  \norm[0 \to 0] {J-{J'}^*} &\le \delta, &\\
  \label{eq:j.comm}
  \norm[k \to -1] {\wt H J_1 - J_1'^* H} & \le \delta, &\\ 
  \label{eq:j.inv}
  \norm[1 \to 0] {\1 - J' J} & \le \delta, & 
  \norm[1 \to 0] {\1 - J J'} & \le \delta\\ 
  \label{eq:j.bdd}
  \norm[0 \to 0] J   & \le 2, & 
  \norm[0 \to 0] {J'} & \le 2.
\end{align}
\end{definition}

Note that all operators make sense on the given domains, e.g.,
\begin{displaymath}
   \map{(\wt H J_1 - J_1'^* H) \restr {\HS_k} = 
        (\wt H J_1 - J_1'^* H) \1_{k \to 1}} 
           {\HS_k} {\wt \HS_{-1}} 
\end{displaymath}
where $\map {\1_{k \to 1}} {\HS_k} {\HS_1}$ is the natural inclusion
map. Strictly speaking, we should also write $J \1_{1 \to 0} - \wt \1_{1
  \to 0} J_1$ in \eqref{eq:j.scale} and $(\1_{0 \to 0} - J' J)\1_{1
  \to 0}$ in \eqref{eq:j.inv}, e.g., but we refrain from it in order
to keep the notation readable.

We can interprete~\eqref{eq:j.comm} in the sense that $J_1$ and $J'_1$
are \emph{quasi-intertwining} operators.  Since~\eqref{eq:j.scale}
assures the closeness of $J$ and $J_1$ resp.\ $J'$ and $J_1'$ we call
$J$ resp.\ $J'$ also \emph{quasi-intertwining}.
Furthermore,~\eqref{eq:j.inv} together with~\eqref{eq:j.adj} says that
$J$ and $J'$ are \emph{quasi-unitary}. In our application of a
graph-like manifold converging to a metric graph, cf.\ \Sec{graph}),
the regularity order $k$ equals $1$. In this case, the assumptions are
symmetric in $H$ and $\wt H$, but we will also meet situations in a
forthcoming paper where $k>1$ is needed.

For concrete applications, the following equivalent characterisation
of \eqref{eq:j.scale}--\eqref{eq:j.bdd} will be useful:
\begin{align}
  \label{eq:j.scale.}
  \tag{\ref{eq:j.scale}'}
  \norm[0]{Jf-J_1f} &\le \delta \norm[1] f, &
  \norm[0]{J' u-J_1' u} &\le \delta \norm[1] u\\
  \label{eq:j.adj.}
  \tag{\ref{eq:j.adj}'}
  |\iprod {Jf}  u - \iprod f {J' u}| & \le
     \delta \norm[0] f \norm[0] u\\
  \label{eq:j.comm.}
  \tag{\ref{eq:j.comm}'}
  |\wt{\qf h} (J_1 f, u) - \qf h (f, J_1' u)| & \le
     \delta \norm[k] f \norm[1] u\\
  \label{eq:j.inv.}
  \tag{\ref{eq:j.inv}'}
  \norm[0] {f - J' J f} & \le \delta \norm[1] f, & 
  \norm[0] {u - J J' u} & \le \delta\ \norm[1] u\\ 
  \label{eq:j.bdd.}
  \tag{\ref{eq:j.bdd}'}
  \norm[0] {J f}   & \le 2 \norm[0] f, & 
  \norm[0] {J' u} & \le 2 \norm[0] u
\end{align}
for all $f, u$ in the appropriate spaces. Here, $\qf h$ and
$\wt{\qf h}$ denote the sesquilinear forms associated to $H$ and $\wt
H$, i.e., $\qf h(f,g)=\iprod {H^{1/2}f} {H^{1/2}g}$ for $f,g \in
\HS_1$ and similarly for $\wt{\qf h}$.

Let us illustrate the above abstract setting in the following example
of norm resolvent convergence in a fixed Hilbert space:
\begin{example}
  \label{ex:nrconv}
  Suppose that $\wt \HS = \HS$, $J=J' = \1$, $J_1 = J_1' = \1$, $k=1$
  and $\delta = \delta_n \to 0$ as $n \to \infty$. Assume in addition
  that the quadratic form domains of $H$ and $\wt H = H_n$ agree. Now
  the only non-trivial assumption in \Def{closeness} is \Eq{j.comm},
  which is equivalent to
  \begin{displaymath}
    \norm[1 \to -1]{H_n - H} = 
    \norm[0 \to 0] {R_n^{1/2}(H_n - H) R^{1/2}} \to 0
  \end{displaymath}
  whereas $H_n \to H$ in norm resolvent convergence means
  \begin{displaymath}
    \norm[0 \to 0] {R_n - R} = 
    \norm[0 \to 0] {R_n(H_n - H) R} =
    \norm[2 \to -2] {H_n - H} \to 0
  \end{displaymath}
  as $n \to \infty$.  Therefore, we see that our
  assumption~\eqref{eq:j.comm} implies the norm resolvent convergence
  but not vice versa.
\end{example}

\begin{remark}
  \label{rem:closeness}
  We have expressed the closeness of certain quantities in dependence
  on the initial closeness data $\delta>0$.  Although, in our
  applications, $(\wt H,\wt \HS)$ will depend on some parameter $\eps>0$
  with $\delta=\delta(\eps) \to 0$ as $\eps \to 0$ we prefer to
  express the dependence only in terms of $\delta$. In
  particular, an assertion like $\norm{JR - \wt R J} \le 4 \delta$
  means that it is true for all $(H,\HS)$ and $(\wt H,\wt \HS)$ being
  $\delta$-close with respect to $(J,J_1)$ and $(J',J_1')$. In this
  sense, $(H,\HS)$ and $(\wt H,\wt \HS)$ should be considered as
  ``variables'' being close to each other.
\end{remark}


We deduce the following simple estimates:
\begin{lemma}
  \label{lem:j.iso}
  Suppose that Assumption~\eqref{eq:j.adj}, \eqref{eq:j.inv}
  and~\eqref{eq:j.bdd} are fulfilled, then
  \begin{equation}
    \label{eq:j.iso}
    \norm[0] f - \delta' \norm[1] f \le 
    \norm[0]{Jf} \le 
    \norm[0] f + \delta' \norm[1] f \qquad \text{with} \qquad
    \delta' := \sqrt{3\delta}
  \end{equation}
  and similarly for $J'$.
\end{lemma}

\begin{proof}
  We calculate
  \begin{multline*}
    \big| \normsqr{Jf} - \normsqr f \big| =
    \big| \iprod {(J^*J - \1)f} f \big| \le
    \big| \iprod {(J^* - J')Jf} f \big| +
       \big| \iprod {(J'J - \1) f} f \big| \\ \le
    \norm[0 \to 0] {J^* - J'} \norm[0]{Jf} \norm[0] f +
       \norm[1 \to 0]{J' J - \1} \norm[1] f \norm[0] f \le
    3\delta \normsqr[1] f
  \end{multline*}
  and the result follows.
\end{proof}


%------------------------------------------------------------
\subsection{Resolvent convergence and functional calculus}
\label{sec:res.conv}
%------------------------------------------------------------

In this section we prove our result on resolvent convergence.  More
precisely, we estimate the errors in terms of $\delta$.  All the
results below are valid for pairs of non-negative operators and
Hilbert spaces $(H,\HS)$ and $(\wt H,\wt\HS)$ which are $\delta$-close
of order $k$. We set
\begin{equation}
  \label{eq:m}
   m := \max \{ 0, k-2\} 
\end{equation}
as regularity order for the resolvent difference. Note that $m=0$ if
$k=1$ (as in our application) or $k=2$.
\begin{theorem}
\label{thm:res}
  Suppose~\eqref{eq:j.scale}, \eqref{eq:j.adj} and~\eqref{eq:j.comm}, then
  \begin{gather}
  \label{eq:res}
    \norm[m \to 0] {\wt R J - JR}  = 
    \norm[2+m \to -2]{JH - \wt H J} \le 4\delta,\\
  \label{eq:res.j}
    \norm[m \to 0] {\wt R^j J - JR^j}  \le 4j \delta
  \end{gather}
  for all $j \in \N$.
\end{theorem}

\begin{proof}
  We start with the equation
  \begin{equation*}
%    \label{eq:op.diff}
    JH - \wt H J = 
    (J - J'^*)H + (J'-J_1')^*H + (J_1'^* H - \wt H J_1) + \wt H (J_1 - J)
  \end{equation*}
  considered as bounded operator from $\HS_{2+m}$ to $\wt \HS_{-2}$.
  Using~\eqref{eq:norm.adj} and~\eqref{eq:norm.op.ineq} yields
  \begin{multline*}
    \norm[m \to 0]{\wt R J - JR} =
    \norm[m \to 0]{\wt R (JH - \wt H J) R} =
    \norm[2+m \to -2] {JH - \wt H J} \\\le
    \norm[m \to -2] {J-{J'}^*} + 
        \norm[2 \to -m] {J' - J_1'} + 
        \norm[2+m \to -2] {J_1'^* H - \wt H J_1} +
        \norm[2+m \to 0] {J_1 - J} \\ \le
    \norm[0 \to 0] {J-{J'}^*} + 
        \norm[1 \to 0] {J' - J_1'} + 
        \norm[k \to -1] {J_1'^* H - \wt H J_1} +
        \norm[1 \to 0] {J_1 - J} \le
    4\delta,
  \end{multline*}
  i.e., the assertion~\eqref{eq:res}. For the second estimate we use
  the resolvent identity
  \begin{displaymath}
    \wt R^j J - J R^j = 
    \sum_{i=0}^{j-1} \wt R^{j-1-i} (\wt R J - JR) R^i,
  \end{displaymath}
  and conclude
  \begin{equation*}
    \norm[m \to 0] {\wt R^j J - JR^j} \le
    \sum_{i=0}^{j-1}
         \norm[0 \to 0] {\wt R^{j-1-i}}
         \norm[m \to 0] {\wt R J - JR} 
         \norm[m \to m] {R^i} \le
     4 j \delta
  \end{equation*}
  using the estimate for $j=1$. Note that $\norm[m \to m] R \le 1$ for
  any $m$ and similarly for $\wt R$.
\end{proof}

\begin{remark}
  Observe that we cannot obtain a better result using the
  quasi-unitary operator $J$ although we loose regularity order at
  some stages. The best what we can expect (in the case $k=1$)
  is the estimate
  \begin{equation}
    \label{eq:res.better}
     \norm[-1 \to 1]{\wt R J_1'^* - J_1R} \le 4 \delta    
  \end{equation}
  which follows from
  \begin{multline*}
    (\wt H + 1)^{1/2}(\wt R J_1'^* - J_1 R)(H + 1)^{1/2} \\=
    \wt R^{1/2} \bigl[
       (J_1'^*H - \wt H J_1) + 
          (J_1'^* - J'^*) + (J'^* - J) + (J - J_1)
    \bigr] R^{1/2}
  \end{multline*}
  and the assumptions.
  
  On the other hand, if we assume that $\norm[2+m \to -2] {J H - \wt H
    J} \le \wt \delta$, i.e.,
  \begin{equation}
    \label{eq:j.op}
    | \iprod {JHf} u - \iprod {Jf} {\wt H u}| \le 
    \wt \delta \norm[2+m] f \norm[2] u
  \end{equation}
  for all $f \in \HS_{2+m}$, $u \in \wt \HS_2$ then we directly obtain
  the resolvent estimate~\eqref{eq:res} with $\wt \delta = 4\delta$.
  Although Assumption~\eqref{eq:j.op} is weaker
  than~\eqref{eq:j.scale}--\eqref{eq:j.comm} (cf.\ \Ex{nrconv}) it is
  often easier in our applications to deal with the quadratic form
  domains, even if one needs the additional operators $J'$, $J_1$ and
  $J_1'$ and the stronger
  estimates~\eqref{eq:j.scale}--\eqref{eq:j.comm}.
\end{remark}


We want to extend our results to more general functions $\phi(H)$ of
the operator $H$ and similarly for $\wt H$. We start with continuous
functions on $\R_+ := [0, \infty)$ such that $\lim_{\lambda \to
  \infty} \phi(\lambda)$ exist, i.e., with functions continuous on
$\overline \R_+ := [0, \infty]$. We denote this space by $\Cont
{\overline \R_+}$.

\begin{theorem}
\label{thm:cont}
Suppose that~\eqref{eq:j.scale}, \eqref{eq:j.adj}, \eqref{eq:j.comm}
and~\eqref{eq:j.bdd} are fulfilled, then
  \begin{equation}
    \label{eq:cont}
    \norm[m \to 0] {\phi(\wt H) J - J \phi(H)} \le \eta_\phi(\delta)
  \end{equation}
  for all $\phi \in \Cont {\overline \R_+}$ where $\eta_\phi(\delta)
  \to 0$ as $\delta \to 0$.
\end{theorem}

\begin{proof}
  Let $p(\lambda) := \sum_{j=0}^n a_j (\lambda+1)^{-j}$ be a
  polynomial in $(\lambda+1)^{-1}$. Then
  \begin{multline*}
    \norm[m \to 0]{\phi(\wt H) J - J\phi(H)} \\ \le
    \norm[0 \to 0]{(\phi - p)(\wt H)} \norm[m \to 0] J +
       \norm[0 \to 0] J \norm[m \to 0] {(\phi - p)(H)} \qquad\qquad \\ {}+
       \sum_{j=0}^n |a_j| \norm[m \to 0] {\wt R^j J - JR^j} \le
    4 \norm[\infty] {\phi - p} + \sum_{j=0}^n |a_j| 4 j \delta =:
   \eta_\phi(\delta,p)
  \end{multline*}
  using~\eqref{eq:norm.op.ineq}, the spectral calculus,
  \eqref{eq:j.bdd} and~\eqref{eq:res.j}. Here, $\norm[\infty] \phi$
  denotes the supremum norm of $\phi$.
  
  Suppose $\eta > 0$. By the Stone-Weierstrass theorem there exists a
  polynomial $p$ such that $\norm[\infty]{p - \phi} \le \eta/8$. If
  \begin{displaymath}
    0 < \delta \le \frac \eta {8 \sum_{j=0}^n |a_j| j}    
  \end{displaymath}
  then $\eta_\phi(\delta) := \eta_\phi(\delta,p) \le \eta/2 + \eta/2
  = \eta$ and therefore $\eta_\phi(\delta) \to 0$ as $\delta \to 0$.
\end{proof}


In a second step we extend the previous result to certain bounded
measurable functions $\map \psi {\overline \R_+} \C$.
\begin{theorem}
  \label{thm:meas}  
  Suppose that $U \subset \overline \R_+$ and that $\map \psi
  {\overline \R_+} \C$ is a measurable, bounded function, continuous
  on $U$ such that $\lim_{\lambda \to \infty} \psi(\lambda)$ exist.
  Then
  \begin{equation}
    \label{eq:meas}
    \norm[m \to 0] {\psi(\wt H) J - J \psi(H)}  \le \eta_\psi(\delta)
  \end{equation}
  for all pairs of non-negative operators and Hilbert spaces $(H,\HS)$
  and $(\wt H, \wt \HS)$ which are $\delta$-close provided
  \begin{equation*}
    \spec H \subset U \qquad \text{or} \qquad \spec {\wt H} \subset U.
  \end{equation*}
  Furthermore, $\eta_\phi(\delta) \to 0$ as $\delta \to 0$.
\end{theorem}

\begin{proof}
  Let $\chi_1$ be a continuous function on $\overline \R_+$ satisfying
  $0 \le \chi_1 \le 1$, $\chi_1 = 1$ on $\spec H \cup \{\infty\}$
  (resp.\ $\chi_1 = 1$ on $\spec {\wt H} \cup \{\infty\}$ if $U$ is a
  neighbourhood of $\spec {\wt H}$) and $\supp \chi_1 \subset U$. Then
  $\chi_1 \psi$ and $\chi_2 = 1 - \chi_1$ are continuous functions on
  $\overline \R_+$ and
  \begin{multline*}
    \norm[m \to 0]{\psi(\wt H) J - J\psi(H)} \\ \le
    \norm[m \to 0]{(\chi_1\psi)(\wt H)J - J (\chi_1\psi)(H)} +
       \norm[m \to 0]{(\chi_2\psi)(\wt H)J - J (\chi_2\psi)(H)}.
  \end{multline*}
  In the case that $U$ is a neighbourhood of $\spec H$ we can estimate
  the norm with $\chi_2$ by
  \begin{displaymath}
   \norm[\infty]{\psi} 
      \norm[m \to 0]{\chi_2(\wt H)J - J \chi_2(H)}
  \end{displaymath}
  using the fact that $(\chi_2 \psi)(H) = \chi_2(H) = 0$ since
  $\chi_2=0$ on $\spec H$.
  
  In the case that $U$ is a neighbourhood of $\spec {\wt H}$ and if $m \ge
  1$ then we can estimate the norm with $\chi_2$ by
  \begin{displaymath}
      \norm[m \to 0]{J(\chi_2 \psi)(H)} \le
         \norm[0 \to 0] J \norm[m \to 0]{(\chi_2 \psi)(H)} \le
         2 \norm[\infty] \psi \norm[m \to 0]{\chi_2(H)}
  \end{displaymath}
  again using the fact that $(\chi_2 \psi)(\wt H) = \chi_2(\wt H) = 0$
  since $\chi_2=0$ on $\spec {\wt H}$. Now
  \begin{equation*}
    \norm[m \to 0]{\chi_2(H)} \le
    \norm[1 \to 0] {\1 - J' J} \norm[m \to 1] {\chi_2(H)} + {}%\\ +
      \norm[0 \to 0] {J'} \norm[m \to 0] {J \chi_2(H) - \chi_2(\wt H) J}.
  \end{equation*}
  Note that $\norm[m \to 1] {\chi_2(H)} \le 1$ since $m \ge 1$. If
  $m=0$ then use the fact that
  \begin{displaymath}
    \norm[0 \to 0]{\psi(\wt H) J - J\psi(H)} =
    \norm[0 \to 0]{\psi(H) J^* - J^* \psi(\wt H)}
  \end{displaymath}
  and argue as in the case where $\spec H \subset U$ with the roles of
  $H$ and $\wt H$ interchanged.

  Applying the preceding %sic! Nur ein e!
  theorem twice (in each of the above cases), we
  have the error estimate
  \begin{displaymath}
    \eta_\psi(\delta):=
      \eta_{\chi_1 \psi}(\delta) + 
        2 \norm[\infty]{\psi} (2 \eta_{\chi_2}(\delta) + \delta).
  \end{displaymath}
\end{proof}

\begin{example}
  Consider $\psi=\1_I$ with an interval $I$ such that $\bd I \cap
  \spec H = \emptyset$ or $\bd I \cap \spec {\wt H} = \emptyset$ then
  the spectral projections satisfy
  \begin{equation}
    \label{eq:spec.proj}
    \norm[m \to 0]{\1_I(\wt H) J - J \1_I(H)} \le \eta_{\1_I}(\delta).
  \end{equation}
\end{example}


Finally we show the following estimates from the ones already
considered:
\begin{theorem}
  \label{thm:other.est}
  Suppose that \eqref{eq:j.adj}, \eqref{eq:j.inv}, \eqref{eq:j.bdd}
  and
  \begin{displaymath}
    \norm[m \to 0] {\phi(\wt H) J - J \phi(H)} \le \eta
  \end{displaymath}
  for some function $\phi$ and some constant $\eta>0$. Then we have
  \begin{align}
    \label{eq:dual}
    \norm[0 \to -m] {\phi(H) J' - J' \phi(\wt H)} & \le 
        2\norm[\infty] \phi \delta + \eta\\
    \label{eq:sandwich}
    \norm[m \to 0] {\phi(H) - J' \phi(\wt H) J} & \le 
        C \delta + 2 \eta\\
    \label{eq:sandwich.}
    \norm[0 \to 0] {\phi(\wt H) - J \phi(H) J'} & \le 
        5C \delta + 2 \eta
  \end{align}
  provided $m=0$ for the last estimate. Here, $C:= \norm[\infty]
  \phi$ if $m\ge 1$ and $C>0$ is a constant satisfying
  $|\phi(\lambda)| \le C (\lambda+1)^{-1/2}$ for all $\lambda$ if
  $m=0$.
\end{theorem}

\begin{proof}
  The first estimate follows from
  \begin{displaymath}
    \norm[0 \to -m] {\phi(H) J' - J' \phi(\wt H)} \le
    2 \norm[\infty] \phi \norm[0 \to 0] {J' - J^*} +
      \norm[0 \to -m] {\phi(H) J^* - J^* \phi(\wt H)}
  \end{displaymath}
  and~\eqref{eq:norm.adj}; the second from
  \begin{multline*}
    \norm[m \to 0] {\phi(H) - J' \phi(\wt H) J} \\ \le
    \norm[1 \to 0] {\1 - J' J} \norm[m \to 1] {\phi(H)}
      + \norm[0 \to 0] {J'} \norm[m \to 0] {J \phi(H) - \phi(\wt H) J}
  \end{multline*}
  and the third from
  \begin{multline*}
    \norm[0 \to 0] {\phi(\wt H) - J \phi(H) J'} \\ \le
    \norm[1 \to 0] {\1 - J J'} \norm[0 \to 1] {\phi(\wt H)}
      + \norm[0 \to 0] J \norm[0 \to 0] {J' \phi(\wt H) - \phi(H) J'}
  \end{multline*}
  together with~\eqref{eq:dual}.
\end{proof}



%------------------------------------------------------------
\subsection{Spectral convergence}
\label{sec:spec}
%------------------------------------------------------------

We now prove some convergence results for spectral projections and
(parts) of the spectrum.
\begin{theorem}
  \label{thm:proj}
  Let $I$ be a measurable and bounded subset of $\R$.  Then there
  exists $\delta_0=\delta_0(I,k)>0$ such that for all $\delta>0$ we
  have
  \begin{equation*}
    \dim P = \dim \wt P
  \end{equation*}
  for all pairs of non-negative operators and Hilbert spaces $(H,\HS)$
  and $(\wt H,\wt \HS)$ which are $\delta$-close of order $k$ provided
  \begin{equation*}
    \bd I \cap \spec H = \emptyset \qquad \text{or} \qquad 
    \bd I \cap \spec {\wt H}  = \emptyset.
  \end{equation*}
  Here, $P := \1_I(H)$ and $\dim P := \dim P(\HS)$, similarly for
  $\wt H$.
\end{theorem}
\begin{proof}
  Let us first show the inequality $\dim P \le \dim \wt P$:
  Suppose $f \in P(\HS)$. Then $\norm[m] f \le C_{I,m} \norm[0] f$ with
  \begin{displaymath}
    C_{I,m} := \sup_{\lambda \in I} (1+\lambda)^{m/2} < \infty
  \end{displaymath}
  since $I$ is bounded. Furthermore,
  \begin{multline*}
    \norm[0] {\wt P J f} \ge 
    \norm[0] {J P f} - \norm[0] {(\wt P J - JP)f} \\ \ge
    \norm[0] {J f} - \norm[m \to 0] {\wt P J - J P} \norm[0] f \ge
    (1 - \delta' C_{I,1} - \eta_{\1_I}(\delta)) \norm[0] f
  \end{multline*}
  using \Lem{j.iso} and \Thm{meas}. Since $\delta' \to
  0$ and $\eta_{\1_I}(\delta) \to 0$ as $\delta \to 0$ there exists
  $\delta_0 > 0$ such that
  \begin{equation}
    \label{eq:pj.lower}
    \norm[0] {\wt P J f} \ge \frac 12 \norm[0] f
  \end{equation}
  provided $0 < \delta \le \delta_0$. Therefore, $\wt P J \restr
  {P(\HS)}$ is injective. If $f_1, \dots, f_d$ are linear independent in
  $P(\HS)$, the same is true for $\wt P J f_1, \dots, \wt P J f_d$ in
  $\wt P(\wt \HS)$. If $P(\HS)$ is infinite dimensional so is
  $\wt P(\wt \HS)$. Thus we have shown $\dim P \le \dim \wt P$.
  
  The other inequality is more difficult due to the asymmetry in the
  norm convergence $\norm[m \to 0] \cdot$ if $m>0$. Suppose that $u
  \in \wt P(\wt \HS)$ and that $\chi_i \in \Cont {\overline \R_+}$ with
  $\chi_1 + \chi_2 + \chi_3 = 1$. Suppose in addition that $\supp
  \chi_1$ and $\supp \chi_2$ are compact, that $\supp \chi_1$ and
  $\supp \chi_3$ are disjoint and that $\supp \chi_2 \cap I =
  \emptyset$. Then
  \begin{multline*}
    \norm[-m] {P J^* u} \ge 
    \norm[-m] {J^* \wt P u} - \norm[-m] {(\wt P J^* - J^* \wt P)u} \\ \ge
    \norm[-m] {\chi_1(H) J^* u} - 
        \norm[-m] {\chi_2(H) J^* u} -
        \norm[-m] {\chi_3(H) J^* u} -
        \norm[m \to 0] {\wt P J - J P} \norm[0] u \\ \ge
     C'_{I,m} \norm[0] {J^* \wt P u} -
        \norm[-m] {(\chi_2(H) J^* - J^* \chi_2(\wt H))\wt P u} -
        \eta_{\1_I}(\delta) \norm[0] u
  \end{multline*}
  by \Thm{meas} and the fact that $\chi_2(\wt H) \wt P = (\chi_2
  \1_I)(\wt H) = 0$ since the support of $\chi_2$ and $I$ are
  disjoint. Here,
  \begin{displaymath}
    C'_{I,m} := \inf_{\set{\lambda}{\chi_1(\lambda)=1}} 
                            (1+\lambda)^{-m/2} - 
                \sup_{\lambda \in \supp \chi_3} (1+\lambda)^{m/2}
  \end{displaymath}
  by the spectral calculus. Since $\chi_1$ and $\chi_3$ have disjoint
  support, $C'_{I,m}>0$. Next, the norm involving $\chi_2$ can be
  estimated from above by
  \begin{displaymath}
    \eta_{\chi_2}(\delta) \norm[0] u
  \end{displaymath}
  using \Thm{cont}. Furthermore,
  \begin{displaymath}
    \norm[0] {J^* u} \ge 
    \norm[0] {J' u} - \norm[0] {(J^* - J') u} \ge
    (1 - C_{I,1} \delta' - \delta) \norm[0] u
  \end{displaymath}
  by \Lem{j.iso} and \eqref{eq:j.adj}. Finally, we have shown that
  \begin{displaymath}
    \norm[-m] {P J^* u} \ge 
    \big(
       C'_{I,m}(1 - C_{I,1} \delta' - \delta) - \eta_{\chi_2}(\delta) -
          \eta_{\1_I} (\delta)
    \big)
    \norm[0] u.
  \end{displaymath}
  The inequality $\dim P \ge \dim \wt P$ follows as before.
\end{proof}

In the case of $1$-dimensional projections we can even show the
convergence of the corresponding eigenvectors. Note that generically,
the eigenvalues are simple (cf.~\cite{uhlenbeck:76}):
\begin{theorem}
  \label{thm:eigenvectors}
  Suppose that $\phi$ is a normalised eigenvector of $H$ with
  eigenvalue $\lambda$ and that $\dim \1_I(H)=1$ for some open,
  bounded interval $I \subset [0,\infty)$ containing $\lambda$. Then
  there exists $\delta_0=\delta(I,k)>0$ such that $\wt H$ has only one
  eigenvalue $\wt \lambda$ of multiplicity $1$ in $I$ for all $(\wt H,
  \wt \HS)$ being $\delta$-close of order $k$ to $(H,\HS)$ and all
  $0<\delta<\delta_0$.
  
  In addition, there exist a unique eigenvector $\wt \phi$ (up to a
  unitary scalar factor close to $1$) and functions
  $\eta_{1,2}(\delta) \to 0$ as $\delta \to 0$ depending only on
  $\lambda$ and $k$ such that
  \begin{align*}
    \norm{J \phi - \wt \phi} &\le \eta_1(\delta), &
    \norm{J' \wt \phi - \phi} &\le \eta_2(\delta).
  \end{align*}
\end{theorem}
\begin{proof}
  Denote the corresponding eigenprojections by $P$ resp.\ $\wt P$. The
  first assertion follows from \Thm{proj}. For the second, note that
  \begin{equation*}
    \wt \phi = \frac 1 {\iprod {\wt P J \phi} {J\phi}} \, \wt P J \phi
  \end{equation*}
  since $\wt P$ is a $1$-dimensional projection. Note in addition that
  \begin{equation*}
    \iprod {\wt P J \phi} {J \phi} =
    \normsqr {\wt P J \phi} \ge
    \frac 1 4 \normsqr \phi =
    \frac 1 4, \qquad 0 < \delta < \delta_0
  \end{equation*}
  for some $\delta_0=\delta_0(I,k)$ due to~\eqref{eq:pj.lower}. Now,
  \begin{multline*}
    \norm{J \phi - \wt \phi} =
    \Bignorm{J P \phi - 
          \frac 1 {\iprod {\wt P J \phi} {J \phi}} \, \wt P J \phi} \\ \le
    \norm{(JP - \wt PJ)\phi} + 
       \Bigl| 1 - \frac 1  {\iprod {\wt P J \phi} {J \phi}} \Bigr|
       \norm{\wt P J \phi} \\ \le
    \eta_{\1_I}(\delta) +
        8 \bigr| \iprod {(\wt P J - J P)\phi} {J\phi} + 
           \normsqr{J \phi} - \normsqr \phi \bigr| \le
    17 \eta_{\1_I}(\delta) + 3 \delta =: \eta_1(\delta)
  \end{multline*}
  since $\phi=P\phi$ and $\norm \phi=1$ using~\eqref{eq:j.bdd}
  and~\eqref{eq:j.iso}. The second estimate follows immediately from
  \begin{equation*}
    \norm{J' \wt \phi - \phi} \le
    \norm{J'(\wt \phi - J \phi)} + \norm{(J'J-\1)\phi} \le
    2 \eta_1(\delta) + \delta(1+\lambda) =: \eta_2(\delta).
  \end{equation*}
  All estimates are valid for $0<\delta<\delta_0$. Note that
  $\delta_0$ and $\eta_i(\delta)$ depend also on $I$ and therefore on
  $\lambda$.
\end{proof}


We now show that the spectrum of the resolvents $R=(H+1)^{-1}$ and
$\wt R =(\wt H + 1)^{-1}$ are close in the Hausdorff distance defined
by
\begin{equation}
  \label{eq:hausdorff}
  d(A,B) := \max \bigl\{ \sup_{a \in A} d(a,B), \sup_{b \in B} d(b,A) \bigr\}
\end{equation}
for subsets $A,B$ of $\R$ where $d(a,B) := \inf_{b \in B} |a-b|$.
Furthermore, we set\footnote{Strictly speaking, $(\spec H + 1)^{-1} =
  \spec R \setminus \{0\}$, but the point $0$ plays no special role
  since $d(A,B)=d(\clo A,\clo B)$.}
\begin{equation}
  \label{eq:hausdorff.res}
  \overline d(A,B) := d((A+1)^{-1},(B+1)^{-1})
\end{equation}
for closed subsets of $[0, \infty)$ (cf.
also~\cite[Appendix~A]{herbst-nakamura:99}, where an equivalent
characterisation of the convergence $\overline d(A_n, A) \to 0$ as $n
\to \infty$ is given).
\begin{theorem}
  \label{thm:spectrum}
  There exists $\eta(\delta)>0$ with $\eta(\delta) \to 0$ as $\delta
  \to 0$ such that
  \begin{displaymath}
    \overline d\bigl(\spec[\bullet] H, \spec[\bullet] {\wt H}\bigr) 
           \le \eta(\delta)
  \end{displaymath}
  for all pairs of non-negative operators and Hilbert spaces $(H,\HS)$
  and $(\wt H,\wt \HS)$ which are $\delta$-close. Here, $\spec[\bullet]
  H$ denotes either the entire spectrum, the essential or the discrete
  spectrum of $H$.

  Furthermore, the multiplicity of the discrete spectrum is preserved,
  i.e., if $\lambda \in \disspec H$ has multiplicity $m>0$ then $\dim
  \1_I(\wt \HS) = m$ for $I:=(\lambda - \eta(\delta), \lambda +
  \eta(\delta))$ provided $\delta$ is small enough.
\end{theorem}
\begin{proof}
  We start with the discrete spectrum. Let $\eta>0$ and $z=(\lambda +
  1)^{-1}>0$, $\lambda \in \disspec H$. By the definition of the
  discrete spectrum, there exists an open interval $I$ containing
  $\lambda$ such that $I \cap \spec H = \{\lambda\}$ and $0 < \dim
  \1_I(H) < \infty$. Without loss of generality, we assume that $I
  \subset (\lambda - \eta, \lambda + \eta)$. From \Thm{proj} it
  follows that $\dim \1_I(H) = \dim \1_I(\wt H)$ provided $0 < \delta <
  \delta_z$ for some $\delta_z >0$. In particular, the multiplicity is
  preserved and there exists $\wt \lambda \in I \cap \disspec {\wt H}$,
  i.e.,
  \begin{equation}
    \label{eq:sp.est}
    d(z, \wt S) \le |z - \wt z| \le |\lambda - \wt \lambda| < \eta
  \end{equation}
  where $\wt S = (\disspec {\wt H} + 1)^{-1}$ and $\wt z = (\wt
  \lambda + 1)^{-1}$.  Now let $\delta(\eta)$ be the minimum of all
  $\delta_z$ where $z$ runs through the finite set $S \cap [\eta, 1]$
  with $S:=(\disspec H + 1)^{-1}$.  Then~\eqref{eq:sp.est} holds for
  all $z \in S \cap [\eta, 1]$ and $0 < \delta < \delta(\eta)$. If
  $\disspec H$ is finite, we just have to assure that $\eta < \inf S$.
  If $\disspec H$ is infinite, so is $\disspec {\wt H}$ and in
  particular, $\wt S \cap (0,\eta) \ne \emptyset$ for all $\eta>0$.
  Therefore, if $z \in (0,\eta) \cap S$ then $d(z, \wt S) \le \eta$.
  Finally,~\eqref{eq:sp.est} holds for all $z \in S$ and $0<\delta <
  \delta(\eta)$.
  
  Interchanging the roles of $H$ and $\wt H$ leads to the inequality
  $d(\wt z, S) \le \eta$ for all $\wt z \in \wt S$ and therefore
  $\overline d(\disspec H, \disspec {\wt H}) \le \eta(\delta)$ where
  $\eta(\delta)$ is the smallest constant satisfying the previous
  estimate for all $(H,\HS)$, $(\wt H, \wt \HS)$ being $\delta$-close.

  
  For the essential spectrum we argue similarly: Let $\eta>0$ and $z
  =(\lambda + 1)^{-1}>0$ with $\lambda \in \essspec H$. Let $I$ be an
  open interval with $\lambda \in I$ and $\bd I \cap \spec H =
  \emptyset$. If $I$ can be chosen in such a way that $I \subset
  (\lambda - \eta, \lambda + \eta)$ then $\infty = \dim \1_I(H) = \dim
  \1_I(\wt H)$ for all pairs $(\wt H, \wt \HS)$ being $\delta$-close,
  $0 < \delta < \delta_z$ for some fixed $\delta_z>0$ due to
  \Thm{proj}. In particular, $I \cap \essspec {\wt H} \ne \emptyset$
  and therefore $d(z, \wt S) < \eta$ as in~\eqref{eq:sp.est} where now
  $\wt S=(\essspec {\wt H} + 1)^{-1}$.
  
  If no such interval $I$ exist, then there is $0<\eta_0<\eta$ such
  that $I_0:=(\lambda-\eta_0, \lambda+\eta_0) \subset \essspec H$. We
  want to show that in this case, $I_0 \subset \essspec {\wt H}$ and
  in particular, $d(z, \wt S) \le \eta_0 < \eta$ provided
  $0<\delta<\delta_z$ for some fixed $\delta_z$: Suppose that this is
  not true.  Then there were $\wt \lambda \in I_0$ and an open
  interval $J$ containing $\wt \lambda$ which is disjoint from the
  closed set $\essspec {\wt H}$ for all $\delta>0$ and all $(\wt H,\wt \HS)$
  being $\delta$-close. But then, \Thm{proj} implies $0 = \dim \1_J
  (\wt H) = \dim \1_J (H)$ contradicting the fact that $J \subset
  \essspec H$.
  
  A compactness argument shows that there exists $\delta(\eta)>0$ such
  that~\eqref{eq:sp.est} is true for all $z$ in the compact set $S
  \cap [\eta, 1]$ and all $(\wt H,\wt \HS)$ being $\delta$-close,
  $\delta < \delta(\eta)$ where $S=(\essspec H + 1)^{-1}$.  If
  $\essspec H$ is bounded (from above) then $S \cap [\eta, 1] = S$
  provided $\eta < \inf S$ and we are done. If $\essspec H$ is
  unbounded, a similar reasoning as before shows that the same is true
  for $\essspec {\wt H}$. In particular, $(0, \eta) \cap \wt S \ne
  \emptyset$ and $d(z, \wt S) < \eta$ for $z \in (0,\eta) \cap S$,
  i.e.,~\eqref{eq:sp.est} holds for all $z \in S$. The assertion
  follows as in the discrete case by symmetry.
  
  The case of the entire spectrum can be shown similarly.
\end{proof}
We have the following immediate consequences when $\disspec H =
\emptyset$ resp.\ $\essspec H= \emptyset$:
\begin{corollary}
  \label{cor:ess.sp}
  Suppose that $H$ has purely essential spectrum. Then for each
  $\lambda \in \essspec H$ there is essential spectrum close to
  $\lambda$ for $\wt H$ being $\delta$-close to $H$. Either $\wt H$
  has no discrete spectrum or the discrete spectrum merges into the
  essential spectrum as $\delta \to 0$.
\end{corollary}
\begin{corollary}
\label{cor:ev.conv}
Suppose that $H$ has purely discrete spectrum denoted by $\lambda_k$
(repeated according to multiplicity). Then the infimum of the
essential spectrum of $\wt H$ tends to infinity (if there where any)
and there exists $\eta_k(\delta)>0$ with $\eta_k(\delta) \to 0$ as
$\delta \to 0$ such that
  \begin{equation}
  \label{eq:ev.est.k}
    |\lambda_k - \wt \lambda_k| \le \eta_k(\delta)
  \end{equation}
  for all $(\wt H, \wt \HS)$ being $\delta$-close. Here, $\wt
  \lambda_k$ denotes the discrete spectrum of $\wt H$ (below the
  essential spectrum) repeated according to multiplicity.
\end{corollary}
Note that the convergence $\eta_k(\delta) \to 0$ is \emph{not} uniform
in $k$. The convergence of the eigenvalues can also be seen by a
direct argument using the min-max principle:

\begin{remark}
  \label{rem:ew.better}
  If we assume that
  \begin{align}
    \label{eq:j1.quad}
    \qf h(f)       &\ge \wt {\qf h} (J_1 f) - \delta \normsqr[1] f, &
    \wt {\qf h}(u) &\ge \qf h (J_1'u) - \delta \normsqr[1] u,\\
    \label{eq:j1.norm}
    \normsqr f &\ge \normsqr {J_1 f} + \delta \normsqr[1] f, &
    \normsqr u &\ge \normsqr {J_1' u} + \delta \normsqr[1] u
  \end{align}
  we obtain the more concrete eigenvalue estimate
  \begin{equation*}
    | \lambda_k - \wt \lambda_k| \le
       \frac {\bigl(\lambda_k + 2 + 
         \frac{(\lambda_k + 2)^2}{1 - \delta(\lambda_k+1)}\delta\bigr)^2}
              {1 - \delta\bigl(\lambda_k + 1 + 
        \frac{(\lambda_k+2)^2}{1 - \delta(\lambda_k+1)}\delta\bigr)}
                \cdot \delta
                = O(\delta)
  \end{equation*}
  using the min-max principle where $O(\delta)$ depends on $\lambda_k$
  (cf.~\cite[Lemma~2.1]{exner-post:05}). Note that the
  assumptions~\eqref{eq:j1.quad} and~\eqref{eq:j1.norm} are equivalent
  to the estimates
  \begin{align}
    \label{eq:j1.quad.}
    \tag{\ref{eq:j1.quad}'}
    H - J_1^* \wt H J_1 + \delta (H+1) &\ge 0, &
    \wt H - J_1'^* H J_1' + \delta (\wt H+1) &\ge 0,\\
    \label{eq:j1.norm.}
    \tag{\ref{eq:j1.norm}'}
    J_1^* J_1 - \1 + \delta(H+1) &\ge 0, &
    J_1'^* J_1' - \1 + \delta(\wt H+1) &\ge 0
  \end{align}
  in the sense that $\map A{\HS_1}{\HS_{-1}} \ge 0$ iff $\iprod {Af} f
  \ge 0$ for all $f \in \HS_1$ and similarly on $\wt \HS$. Note
  that~\eqref{eq:j1.quad} and~\eqref{eq:j1.norm} do not follow from
  the closeness assumptions~\eqref{eq:j.scale}--\eqref{eq:j.bdd};
  e.g.\ for~\eqref{eq:j1.norm} one needs in addition that $\norm[1 \to
  1]{J_1} \le C$ for some constant $C>0$ and similarly for $J_1'$. The
  estimates~\eqref{eq:j1.quad}--\eqref{eq:j1.norm} have been used
  e.g.\ in~\cite{exner-post:05, kuchment-zeng:01,
    rubinstein-schatzman:01} in the graph model and the verification
  of~\eqref{eq:j1.quad}--\eqref{eq:j1.norm} is quite similar to the
  proof of the closeness
  assumptions~\eqref{eq:j.scale}--\eqref{eq:j.bdd} as we have seen
  in~\Sec{graph}.
\end{remark}


%------------------------------------------------------------
\subsection*{Acknowledgments}
%------------------------------------------------------------
It is a pleasure to thank Pavel Exner, Peter Hislop, Fernando Lled\'o,
Gianfausto Dell'Antonio and Luca Tenuta for fruitful discussions and
helpful comments on this manuscript. Part of this work has been done
while the author was funded by the DFG under the grant Po-1034/1-1.

%\enlargethispage*{2cm}

%\bibliographystyle{amsalpha}
%\bibliography{/home/post/Aktuell/BibTeX/literatur}
%\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
%\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
%  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{AGHKH05}

\bibitem[AADH94]{aadh:94}
S.~Alama, M.~Avellaneda, P.A. Deift, and R.~Hempel, \emph{{On the existence of
  eigenvalues of a divergence-form operator $A+\lambda B$ in a gap of
  $\sigma(A)$}}, Asymptotic Anal. \textbf{8} (1994), 311--344.

\bibitem[AEL94]{ael:94}
J.~E. Avron, P.~Exner, and Y.~Last, \emph{{Periodic Schroedinger operators with
  large gaps and Wannier-Stark ladders }}, Phys. Rev. Lett. \textbf{72} (1994),
  no.~6, 896--899.

\bibitem[AGHH05]{aghh:05}
S.~Albeverio, F.~Gesztesy, R.~H{\o}egh-Krohn, and H.~Holden, \emph{Solvable
  models in quantum mechanics}, second ed., AMS Chelsea Publishing, Providence,
  RI, 2005. %, With an appendix by Pavel Exner.

\bibitem[AS00]{aizenman-schenker:00}
M.~Aizenman and J.~H. Schenker, \emph{The creation of spectral gaps by graph
  decoration}, Lett. Math. Phys. \textbf{53} (2000), 253--262.

\bibitem[BCG01]{bcg:01}
Martin Barlow, Thierry Coulhon, and Alexander Grigor'yan, \emph{Manifolds and
  graphs with slow heat kernel decay}, Invent. Math. \textbf{144} (2001),
  no.~3, 609--649.

\bibitem[BEG03]{beg:03}
J.~Br{\"u}ning, P.~Exner, and V.~A. Geyler, \emph{Large gaps in point-coupled
  periodic systems of manifolds}, J. Phys. A \textbf{36} (2003), no.~17,
  4875--4890. \MR{MR1984016}

\bibitem[BGL05]{bgl:05}
J.~Br\"uning, V.~Geyler, and I.~Lobanov, \emph{Spectral properties of
  {S}chr\"odinger operators on decorated graphs}, Mat. Zametki \textbf{77}
  (2005), no.~1, 152--156.

\bibitem[Br81a]{brooks:81}
R.~Brooks, \emph{The fundamental group and the spectrum of the {L}aplacian},
  Comment. Math. Helv. \textbf{56} (1981), 581--598.

\bibitem[Br81b]{brooks:81b}
\bysame, \emph{A relation between growth and the spectrum of the {L}aplacian},
  Math. Z. \textbf{178} (1981), no.~4, 501--508.

\bibitem[Ca97]{cattaneo:97}
Carla Cattaneo, \emph{The spectrum of the continuous {L}aplacian on a graph},
  Monatsh. Math. \textbf{124} (1997), no.~3, 215--235. 
 
\bibitem[DaS92]{davies-simon:92}
E.~B. Davies and B.~Simon, \emph{Spectral properties of {N}eumann {L}aplacian
  of horns}, Geom. Funct. Anal. \textbf{2} (1992), no.~1, 105--117.

\bibitem[EvHa89]{evans-harris:89}
W.~D. Evans and D.~J. Harris, \emph{On the approximation numbers of {S}obolev
  embeddings for irregular domains}, Quart. J. Math. Oxford Ser. (2)
  \textbf{40} (1989), no.~157, 13--42. 

\bibitem[EvS00]{evans-saito:00}
W.~D. Evans and Y.~Sait{\=o}, \emph{Neumann {L}aplacians on domains and
  operators on associated trees}, Q. J. Math. \textbf{51} (2000), no.~3,
  313--342.

\bibitem[E95]{exner:95}
P.~Exner, \emph{Lattice {K}ronig-{P}enney models}, Phys. Rev. Lett. \textbf{74}
  (1995), no.~18, 3503--3506.

\bibitem[E97]{exner:97b}
\bysame, \emph{A duality between {S}chr\"odinger operators on graphs and
  certain {J}acobi matrices}, Ann. Inst. H. Poincar\'e Phys. Th\'eor.
  \textbf{66} (1997), no.~4, 359--371.

\bibitem[EP05]{exner-post:05}
P.~Exner and O.~Post, \emph{Convergence of spectra of graph-like thin
  manifolds}, Journal of Geometry and Physics \textbf{54} (2005), 77--115.

\bibitem[EP06]{exner-post:pre06}
\bysame, \emph{{Convergence of resonances on thin branched quantum wave
  guides}}, (in preparation) (2006).

\bibitem[HeN99]{herbst-nakamura:99}
I.~Herbst and S.~Nakamura, \emph{{Schr{\"o}dinger operators with strong
  magnetic fields: Quasi-periodicity of spectral orbits and topology}},
  American Mathematical Society. Transl., Ser. 2, Am. Math. Soc.
  \textbf{189(41)} (1999), 105--123.

\bibitem[KaP88]{karp-pinsky:88}
L.~Karp and M.~Pinsky, \emph{First-order asymptotics of the principal
  eigenvalue of tubular neighborhoods}, Geometry of random motion (Ithaca,
  N.Y., 1987), Contemp. Math., vol.~73, Amer. Math. Soc., Providence, RI, 1988,
  pp.~105--119.

\bibitem[KoS99]{kostrykin-schrader:99}
V.~Kostrykin and R.~Schrader, \emph{Kirchhoff's rule for quantum wires}, J.
  Phys. A \textbf{32} (1999), no.~4, 595--630.

\bibitem[Ku02]{kuchment:02}
P.~Kuchment, \emph{Graph models for waves in thin structures}, Waves Random
  Media \textbf{12} (2002), no.~4, R1--R24.

\bibitem[Ku04]{kuchment:04}
\bysame, \emph{Quantum graphs: {I}. {S}ome basic structures}, Waves Random
  Media \textbf{14} (2004), S107--S128.

\bibitem[Ku05]{kuchment:05}
\bysame, \emph{Quantum graphs. {II}. {S}ome spectral properties of quantum and
  combinatorial graphs}, J. Phys. A \textbf{38} (2005), no.~22, 4887--4900.

\bibitem[KuZ01]{kuchment-zeng:01}
P.~Kuchment and H.~Zeng, \emph{Convergence of spectra of mesoscopic systems
  collapsing onto a graph}, J. Math. Anal. Appl. \textbf{258} (2001), no.~2,
  671--700.

\bibitem[KuZ03]{kuchment-zeng:03}
\bysame, \emph{Asymptotics of spectra of {N}eumann {L}aplacians in thin
  domains}, Advances in differential equations and mathematical physics
  (Birmingham, AL, 2002), Contemp. Math., vol. 327, Amer. Math. Soc.,
  Providence, RI, 2003, pp.~199--213.

\bibitem[Lo01]{lott:01}
J.~Lott, \emph{On the spectrum of a finite-volume negatively-curved manifold},
  Amer. J. Math. \textbf{123} (2001), no.~2, 185--205.

\bibitem[LP04]{lledo-post:pre04}
F.~Lled\'o and O.~Post, \emph{Generating spectral gaps by geometry},
  arXiv:math-ph/0406032 (2004).

\bibitem[LP05]{lledo-post:pre05}
\bysame, \emph{Existence of spectral gaps, covering manifolds and residually
  finite groups}, arXiv:math-ph/0503005 (2005).

\bibitem[M68]{milnor:68}
J.~Milnor, \emph{A note on curvature and fundamental group}, J. Differential
  Geometry \textbf{2} (1968), 1--7. 

\bibitem[P03a]{post:03b}
O.~Post, \emph{Eigenvalues in spectral gaps of a perturbed periodic manifold},
  Mathematische Nachrichten \textbf{261--262} (2003), 141--162.

\bibitem[P03b]{post:03a}
\bysame, \emph{Periodic manifolds with spectral gaps}, J. Diff. Equations
  \textbf{187} (2003), 23--45.

\bibitem[P05]{post:05a}
\bysame, \emph{Branched quantum wave guides with {D}irichlet boundary
  conditions: the decoupling case}, Journal of Physics A: Mathematical and
  General \textbf{38} (2005), no.~22, 4917--4931.

\bibitem[RuS53]{ruedenberg-scherr:53}
K.~Ruedenberg and C.~W. Scherr, \emph{{Free--electron network model for
  conjugated systems, I.~Theory}}, J. Chem. Phys. \textbf{21} (1953),
  1565--1581.

\bibitem[RS80]{reed-simon-1}
M.~Reed and B.~Simon, \emph{{Methods of modern mathematical physics I:
  Functional analysis}}, Academic Press, New York, 1980.

\bibitem[RS01]{rubinstein-schatzman:01}
J.~Rubinstein and M.~Schatzman, \emph{Variational problems on multiply
  connected thin strips. {I}. {B}asic estimates and convergence of the
  {L}aplacian spectrum}, Arch. Ration. Mech. Anal. \textbf{160} (2001), no.~4,
  271--308.

\bibitem[Sa00]{saito:00}
Y~Saito, \emph{{The limiting equation for Neumann Laplacians on shrinking
  domains.}}, Electron. J. Differ. Equ. \textbf{31} (2000), 25 p.

\bibitem[T98]{teplyaev:98}
Alexander Teplyaev, \emph{Spectral analysis on infinite {S}ierpi\'nski
  gaskets}, J. Funct. Anal. \textbf{159} (1998), no.~2, 537--567.

\bibitem[U76]{uhlenbeck:76}
K.~Uhlenbeck, \emph{Generic properties of eigenfunctions}, Amer. J. Math.
  \textbf{98} (1976), no.~4, 1059--1078.

\end{thebibliography}


\end{document}

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 

---------------0601111605739
Content-Type: application/postscript; name="graph.emb.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="graph.emb.eps"

%!PS-Adobe-2.0 EPSF-2.0
%%Title: graph.emb.fig
%%Creator: fig2dev Version 3.2 Patchlevel 4
%%CreationDate: Thu Oct 20 11:30:54 2005
%%For: post@lingen (Olaf Post)
%%BoundingBox: 0 0 448 200
%%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} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
newpath 0 200 moveto 0 0 lineto 448 0 lineto 448 200 lineto closepath clip newpath
-6.1 203.4 translate
1 -1 scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
 /DrawEllipse {
	/endangle exch def
	/startangle exch def
	/yrad exch def
	/xrad exch def
	/y exch def
	/x exch def
	/savematrix mtrx currentmatrix def
	x y tr xrad yrad sc 0 0 1 startangle endangle arc
	closepath
	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
10 setmiterlimit
0 slj 0 slc
 0.06299 0.06299 sc
%
% Fig objects follow
%
% 
% here starts figure with depth 60
% Polyline
 [15 7 15 7] 0 sd
n 2347 825 m 2680 860 l 2657 1046 l 2629 1191 l 2909 1150 l 2856 1724 l
 2403 1688 l 2259 1087 l 2304 973 l
 2328 886 l gs col7 0.85 shd ef gr  [] 0 sd
% Polyline
 [15 7 15 7] 0 sd
n 1614 1180 m 1902 2083 l 2223 2037 l 2497 2014 l 2205 884 l 1855 1055 l

 cp gs col7 0.85 shd ef gr  [] 0 sd
% Polyline
7.500 slw
n 4770 765 m 4770 1845 l 5400 1845 l 5400 1530 l 6030 1530 l 6030 1080 l
 5748 1080 l 5908 817 l 5615 635 l 5400 990 l 5400 765 l

 cp gs col7 0.85 shd ef gr gs col0 s gr 
% here ends figure;
% 
% here starts figure with depth 55
% Polyline
7.500 slw
 [30 15 15 15] 0 sd
n 5805 1080 m 6030 1080 l 6030 1530 l 5805 1530 l
 cp gs col7 0.70 shd ef gr gs col0 s gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
n 4770 765 m 4995 765 l 4995 1845 l 4770 1845 l
 cp gs col7 0.70 shd ef gr gs col0 s gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
n 5901 814 m 5789 1008 l 5506 826 l 5614 637 l
 cp gs col7 0.70 shd ef gr gs col0 s gr  [] 0 sd
% here ends figure;
% 
% here starts figure with depth 50
% Arc
7.500 slw
 [30 15 15 15] 0 sd
n 5519.3 1121.5 601.2 -59.8 16.8 arc
gs col0 s gr
 [] 0 sd
% Arc
 [30 15 15 15] 0 sd
n 2368.6 1441.4 185.3 -73.3 -5.7 arc
gs col0 s gr
 [] 0 sd
% Ellipse
n 2345 1447 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 5409 1300 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2205 881 m
 2255 1097 l gs col0 s gr 
% Polyline
n 2400 1683 m
 2495 2026 l gs col0 s gr 
% Polyline
 [30 15 15 15] 0 sd
n 1616 1172 m
 1903 2086 l gs col0 s gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
n 2910 1166 m
 2856 1726 l gs col0 s gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
n 2363 832 m
 2683 859 l gs col0 s gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
n 6120 1305 m
 4635 1305 l gs col7 0.85 shd ef gr gs col0 s gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
n 5400 1305 m
 5850 585 l gs col7 0.85 shd ef gr gs col0 s gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
gs  clippath
6360 1545 m 6420 1545 l 6420 1394 l 6390 1514 l 6360 1394 l cp
6420 1065 m 6360 1065 l 6360 1216 l 6390 1096 l 6420 1216 l cp
eoclip
n 6390 1080 m
 6390 1530 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 6420 1216 m 6390 1096 l 6360 1216 l 6420 1216 l  cp gs col7 1.00 shd ef gr  col0 s
% arrowhead
n 6360 1394 m 6390 1514 l 6420 1394 l 6360 1394 l  cp gs col7 1.00 shd ef gr  col0 s
% Polyline
 [30 15 15 15] 0 sd
gs  clippath
6045 1740 m 6045 1680 l 5894 1680 l 6014 1710 l 5894 1740 l cp
5385 1680 m 5385 1740 l 5536 1740 l 5416 1710 l 5536 1680 l cp
eoclip
n 5400 1710 m
 6030 1710 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 5536 1680 m 5416 1710 l 5536 1740 l 5536 1680 l  cp gs col7 1.00 shd ef gr  col0 s
% arrowhead
n 5894 1740 m 6014 1710 l 5894 1680 l 5894 1740 l  cp gs col7 1.00 shd ef gr  col0 s
% Polyline
 [30 15 15 15] 0 sd
gs  clippath
5415 615 m 5415 555 l 5264 555 l 5384 585 l 5264 615 l cp
4755 555 m 4755 615 l 4906 615 l 4786 585 l 4906 555 l cp
eoclip
n 4770 585 m
 5400 585 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 4906 555 m 4786 585 l 4906 615 l 4906 555 l  cp gs col7 1.00 shd ef gr  col0 s
% arrowhead
n 5264 615 m 5384 585 l 5264 555 l 5264 615 l  cp gs col7 1.00 shd ef gr  col0 s
% Polyline
 [30 15 15 15] 0 sd
gs  clippath
2893 1742 m 2953 1747 l 2968 1597 l 2927 1714 l 2908 1591 l cp
3009 1158 m 2949 1153 l 2934 1303 l 2976 1187 l 2994 1309 l cp
eoclip
n 2978 1171 m
 2925 1730 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 2994 1309 m 2976 1187 l 2934 1303 l 2994 1309 l  cp gs col7 1.00 shd ef gr  col0 s
% arrowhead
n 2908 1591 m 2927 1714 l 2968 1597 l 2908 1591 l  cp gs col7 1.00 shd ef gr  col0 s
% Polyline
 [30 15 15 15] 0 sd
n 2628 1180 m 2326 1207 l 2357 1690 l
 2401 1681 l gs col0 s gr  [] 0 sd
% Polyline
2 slj
n 455 3217 m 457 3216 l 461 3213 l 470 3207 l 482 3199 l 500 3188 l
 522 3173 l 548 3155 l 579 3135 l 612 3112 l 647 3088 l
 684 3063 l 721 3038 l 757 3013 l 793 2988 l 827 2963 l
 859 2940 l 889 2917 l 918 2896 l 944 2875 l 969 2855 l
 993 2835 l 1015 2816 l 1035 2798 l 1055 2779 l 1073 2761 l
 1091 2742 l 1108 2724 l 1126 2703 l 1144 2683 l 1162 2661 l
 1179 2640 l 1196 2617 l 1214 2594 l 1232 2571 l 1249 2547 l
 1267 2523 l 1285 2499 l 1304 2475 l 1322 2450 l 1341 2426 l
 1360 2402 l 1380 2379 l 1399 2356 l 1419 2334 l 1439 2313 l
 1459 2292 l 1479 2273 l 1499 2254 l 1520 2237 l 1541 2220 l
 1563 2205 l 1585 2190 l 1607 2176 l 1629 2164 l 1652 2153 l
 1676 2143 l 1702 2133 l 1729 2124 l 1757 2115 l 1788 2107 l
 1822 2099 l 1858 2091 l 1896 2084 l 1937 2077 l 1981 2070 l
 2027 2063 l 2075 2057 l 2125 2051 l 2175 2045 l 2225 2039 l
 2274 2034 l 2319 2029 l 2361 2025 l 2397 2022 l 2428 2019 l
 2452 2017 l 2469 2016 l 2480 2015 l 2486 2014 l
 2489 2014 l gs col0 s gr 
% Polyline
n 109 3037 m 110 3034 l 111 3029 l 114 3018 l 119 3002 l 124 2981 l
 132 2954 l 140 2923 l 149 2889 l 159 2853 l 169 2817 l
 179 2780 l 188 2744 l 197 2710 l 206 2677 l 214 2647 l
 221 2618 l 228 2591 l 234 2566 l 240 2541 l 246 2518 l
 251 2496 l 256 2473 l 261 2452 l 266 2428 l 271 2404 l
 276 2380 l 281 2355 l 286 2331 l 292 2305 l 298 2280 l
 304 2254 l 310 2228 l 317 2202 l 325 2176 l 333 2150 l
 341 2124 l 350 2099 l 359 2075 l 369 2052 l 379 2029 l
 390 2007 l 401 1986 l 413 1965 l 426 1945 l 440 1926 l
 452 1910 l 464 1894 l 478 1878 l 493 1862 l 509 1845 l
 526 1829 l 544 1812 l 563 1796 l 583 1778 l 605 1761 l
 627 1744 l 651 1726 l 676 1708 l 702 1690 l 728 1672 l
 755 1655 l 783 1637 l 812 1619 l 841 1601 l 870 1584 l
 900 1566 l 930 1549 l 960 1531 l 991 1514 l 1023 1496 l
 1055 1478 l 1079 1465 l 1104 1452 l 1129 1438 l 1155 1424 l
 1182 1410 l 1210 1395 l 1239 1379 l 1270 1363 l 1302 1346 l
 1336 1328 l 1372 1309 l 1410 1290 l 1449 1269 l 1491 1247 l
 1535 1224 l 1580 1201 l 1628 1176 l 1676 1151 l 1726 1125 l
 1777 1099 l 1827 1073 l 1877 1048 l 1926 1023 l 1972 999 l
 2016 976 l 2056 956 l 2091 937 l 2122 921 l 2148 908 l
 2169 897 l 2185 889 l 2196 883 l 2203 880 l 2207 878 l

 2209 877 l gs col0 s gr 
% Polyline
n 2262 137 m 2263 140 l 2264 146 l 2266 157 l 2269 174 l 2274 197 l
 2279 225 l 2285 258 l 2292 294 l 2299 332 l 2305 371 l
 2312 409 l 2319 447 l 2324 482 l 2330 515 l 2334 546 l
 2338 575 l 2342 601 l 2344 626 l 2347 649 l 2348 670 l
 2349 690 l 2350 710 l 2350 728 l 2350 752 l 2348 776 l
 2346 799 l 2343 822 l 2338 845 l 2333 870 l 2326 896 l
 2319 923 l 2310 951 l 2301 980 l 2291 1008 l 2282 1034 l
 2273 1056 l 2267 1074 l 2262 1087 l 2259 1094 l
 2258 1097 l gs col0 s gr 
% Polyline
n 2406 1689 m 2409 1689 l 2414 1689 l 2424 1690 l 2440 1691 l 2462 1692 l
 2489 1694 l 2521 1696 l 2557 1699 l 2596 1701 l 2637 1704 l
 2678 1707 l 2719 1710 l 2760 1714 l 2798 1717 l 2835 1720 l
 2870 1723 l 2903 1727 l 2934 1730 l 2964 1734 l 2992 1738 l
 3018 1741 l 3044 1746 l 3069 1750 l 3094 1754 l 3119 1759 l
 3143 1764 l 3167 1770 l 3192 1776 l 3217 1782 l 3243 1789 l
 3270 1796 l 3298 1804 l 3327 1813 l 3359 1823 l 3391 1833 l
 3426 1844 l 3462 1856 l 3500 1869 l 3538 1882 l 3576 1895 l
 3614 1908 l 3650 1921 l 3683 1932 l 3713 1943 l 3738 1952 l
 3758 1959 l 3772 1964 l 3781 1967 l 3787 1969 l
 3789 1970 l gs col0 s gr 
% Polyline
n 2622 1186 m 2625 1186 l 2631 1185 l 2641 1184 l 2657 1182 l 2678 1179 l
 2704 1176 l 2734 1172 l 2766 1168 l 2799 1164 l 2832 1160 l
 2865 1156 l 2897 1152 l 2928 1149 l 2957 1145 l 2985 1142 l
 3011 1138 l 3037 1135 l 3063 1132 l 3088 1129 l 3114 1126 l
 3140 1123 l 3162 1120 l 3186 1117 l 3210 1114 l 3235 1111 l
 3262 1108 l 3290 1104 l 3320 1100 l 3352 1096 l 3387 1092 l
 3423 1088 l 3463 1083 l 3504 1078 l 3547 1072 l 3592 1067 l
 3637 1061 l 3681 1056 l 3724 1050 l 3763 1046 l 3798 1041 l
 3828 1038 l 3851 1035 l 3869 1033 l 3880 1031 l 3886 1030 l

 3889 1030 l gs col0 s gr 
% Polyline
n 2615 70 m 2615 73 l 2616 78 l 2617 89 l 2619 105 l 2622 126 l
 2626 154 l 2630 186 l 2634 222 l 2639 262 l 2644 302 l
 2648 344 l 2653 385 l 2658 424 l 2662 462 l 2666 498 l
 2669 532 l 2672 563 l 2675 593 l 2677 620 l 2679 645 l
 2681 669 l 2682 692 l 2683 713 l 2683 734 l 2684 754 l
 2684 779 l 2684 804 l 2683 828 l 2681 852 l 2679 877 l
 2677 902 l 2674 928 l 2670 956 l 2665 985 l 2661 1014 l
 2655 1045 l 2650 1074 l 2645 1103 l 2640 1128 l 2635 1149 l
 2632 1166 l 2630 1177 l 2629 1183 l
 2628 1186 l gs col0 s gr 
% Polyline
 [15 30] 30 sd
gs  clippath
3836 1561 m 3843 1502 l 3693 1483 l 3809 1528 l 3686 1542 l cp
eoclip
n 270 3150 m 271 3148 l 273 3145 l 277 3138 l 283 3127 l 291 3112 l
 302 3092 l 317 3067 l 333 3037 l 353 3003 l 375 2964 l
 399 2922 l 425 2877 l 453 2831 l 481 2783 l 510 2734 l
 540 2685 l 569 2637 l 598 2590 l 627 2545 l 656 2501 l
 684 2458 l 711 2418 l 738 2379 l 764 2343 l 790 2308 l
 815 2275 l 840 2243 l 865 2213 l 889 2185 l 914 2157 l
 939 2131 l 964 2105 l 990 2080 l 1016 2056 l 1043 2033 l
 1068 2010 l 1095 1989 l 1122 1967 l 1150 1946 l 1178 1925 l
 1207 1904 l 1238 1883 l 1269 1862 l 1301 1842 l 1334 1822 l
 1367 1802 l 1402 1782 l 1437 1763 l 1473 1743 l 1509 1725 l
 1547 1706 l 1584 1688 l 1623 1670 l 1661 1653 l 1700 1637 l
 1739 1621 l 1778 1605 l 1818 1590 l 1857 1576 l 1896 1563 l
 1935 1550 l 1974 1538 l 2012 1526 l 2050 1515 l 2088 1505 l
 2125 1496 l 2162 1487 l 2199 1479 l 2235 1471 l 2271 1465 l
 2307 1458 l 2342 1453 l 2378 1448 l 2413 1443 l 2448 1439 l
 2484 1435 l 2519 1432 l 2556 1430 l 2593 1428 l 2631 1427 l
 2669 1426 l 2709 1426 l 2751 1427 l 2793 1428 l 2838 1430 l
 2884 1432 l 2932 1435 l 2981 1438 l 3033 1442 l 3086 1446 l
 3140 1451 l 3196 1456 l 3253 1462 l 3310 1468 l 3367 1474 l
 3423 1480 l 3478 1487 l 3531 1493 l 3581 1499 l 3627 1505 l
 3669 1510 l 3706 1514 l 3738 1518 l 3764 1522 l 3785 1525 l
 3801 1527 l
 3825 1530 l gs col0 s gr gr
 [] 0 sd
% arrowhead
0 slj
n 3686 1542 m 3809 1528 l 3693 1483 l 3686 1542 l  cp gs col7 1.00 shd ef gr  col0 s
% Polyline
2 slj
 [15 30] 30 sd
n 2430 90 m 2430 93 l 2431 99 l 2433 109 l 2436 126 l 2440 148 l
 2444 176 l 2449 209 l 2455 247 l 2462 287 l 2468 329 l
 2475 372 l 2481 414 l 2487 455 l 2492 494 l 2497 531 l
 2502 566 l 2506 599 l 2509 630 l 2512 658 l 2514 685 l
 2516 710 l 2517 734 l 2518 757 l 2518 779 l 2518 800 l
 2518 827 l 2517 854 l 2515 881 l 2513 907 l 2510 933 l
 2506 959 l 2502 985 l 2498 1010 l 2493 1035 l 2488 1060 l
 2482 1083 l 2476 1106 l 2470 1128 l 2464 1149 l 2459 1168 l
 2453 1187 l 2447 1204 l 2441 1220 l 2436 1236 l 2431 1251 l
 2424 1271 l 2417 1290 l 2409 1309 l 2402 1329 l 2394 1349 l
 2386 1371 l 2377 1394 l 2368 1417 l 2359 1439 l 2351 1458 l
 2345 1472 l 2342 1481 l 2340 1484 l
 2340 1485 l gs col0 s gr  [] 0 sd
% here ends figure;
$F2psEnd
rs
showpage

---------------0601111605739
Content-Type: application/postscript; name="graph1.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="graph1.eps"

%!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} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
newpath 0 158 moveto 0 0 lineto 334 0 lineto 334 158 lineto closepath clip newpath
-23.0 177.0 translate
1 -1 scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
 /DrawEllipse {
	/endangle exch def
	/startangle exch def
	/yrad exch def
	/xrad exch def
	/y exch def
	/x exch def
	/savematrix mtrx currentmatrix def
	x y tr xrad yrad sc 0 0 1 startangle endangle arc
	closepath
	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
%%Page: 1 1
10 setmiterlimit
 0.06299 0.06299 sc
%
% Fig objects follow
%
% Polyline
7.500 slw
n 426 984 m
 3036 1389 l gs col0 s gr 
% Polyline
n 476 716 m
 3086 1121 l gs col0 s gr 
% Polyline
2 slj
 [15 45] 45 sd
n 419 813 m 415 826 l 412 839 l 410 853 l 408 867 l 408 881 l
 409 894 l 411 906 l 414 917 l 418 926 l 422 934 l
 427 939 l 432 942 l 438 943 l 444 942 l 451 938 l
 458 933 l 466 926 l 474 918 l 482 908 l 490 896 l
 497 885 l 504 873 l 509 860 l 515 848 l 519 835 l
 523 822 l 527 808 l 529 794 l 531 781 l 532 768 l
 532 756 l 531 745 l 529 737 l 526 730 l 522 725 l
 517 722 l 511 721 l 504 722 l 496 725 l 487 729 l
 477 736 l 467 744 l 457 754 l 448 765 l 439 776 l
 431 788 l 425 801 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 3000 1225 m 2996 1238 l 2993 1251 l 2991 1265 l 2989 1279 l 2989 1293 l
 2990 1306 l 2992 1318 l 2995 1329 l 2999 1338 l 3003 1346 l
 3008 1351 l 3013 1354 l 3019 1355 l 3025 1354 l 3032 1350 l
 3039 1345 l 3047 1338 l 3055 1330 l 3063 1320 l 3071 1308 l
 3078 1297 l 3085 1285 l 3090 1272 l 3096 1260 l 3100 1247 l
 3104 1234 l 3108 1220 l 3110 1206 l 3112 1193 l 3113 1180 l
 3113 1168 l 3112 1157 l 3110 1149 l 3107 1142 l 3103 1137 l
 3098 1134 l 3092 1133 l 3085 1134 l 3077 1137 l 3068 1141 l
 3058 1148 l 3048 1156 l 3038 1166 l 3029 1177 l 3020 1188 l
 3012 1200 l 3006 1213 l
 cp gs col0 s gr  [] 0 sd
% Ellipse
n 426 2085 45 45 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Ellipse
n 2991 2490 45 45 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Polyline
0 slj
n 426 2085 m
 3036 2490 l gs col0 s gr 
 [15 30] 30 sd
1 slc
% Ellipse
n 5280 1080 45 45 0 360 DrawEllipse gs col7 0.45 shd ef gr gs col0 s gr
 [] 0 sd
% Polyline
0 slc
 [15 45] 45 sd
n 5277 1095 m
 4685 1380 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5285 1080 m
 4445 937 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5292 1080 m
 4655 510 l gs col0 s gr  [] 0 sd
% Polyline
2 slj
 [15 45] 45 sd
n 5019 1411 m 5013 1417 l 5004 1421 l 4994 1425 l 4982 1427 l 4968 1427 l
 4953 1427 l 4937 1425 l 4920 1421 l 4904 1416 l 4887 1411 l
 4872 1404 l 4857 1396 l 4844 1388 l 4832 1379 l 4822 1370 l
 4813 1361 l 4805 1350 l 4797 1340 l 4791 1328 l 4785 1316 l
 4780 1304 l 4777 1292 l 4774 1280 l 4773 1268 l 4772 1258 l
 4772 1247 l 4774 1238 l 4776 1230 l 4779 1222 l 4784 1216 l
 4788 1211 l 4794 1206 l 4801 1203 l 4809 1200 l 4817 1199 l
 4827 1198 l 4837 1198 l 4848 1200 l 4859 1202 l 4871 1206 l
 4882 1210 l 4893 1215 l 4905 1221 l 4915 1228 l 4926 1235 l
 4936 1243 l 4948 1253 l 4959 1265 l 4970 1277 l 4981 1290 l
 4991 1304 l 5000 1318 l 5008 1332 l 5015 1347 l 5020 1360 l
 5024 1373 l 5025 1385 l 5025 1395 l 5023 1403 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 4966 818 m 4980 812 l 4993 805 l 5005 798 l 5018 789 l 5030 779 l
 5041 768 l 5052 756 l 5062 744 l 5071 731 l 5079 718 l
 5086 705 l 5091 692 l 5095 679 l 5097 667 l 5098 656 l
 5098 645 l 5096 635 l 5093 625 l 5087 615 l 5079 606 l
 5070 597 l 5058 590 l 5045 583 l 5030 577 l 5014 572 l
 4997 568 l 4978 565 l 4960 563 l 4941 563 l 4923 564 l
 4905 566 l 4888 568 l 4872 572 l 4856 578 l 4843 583 l
 4830 589 l 4817 596 l 4805 605 l 4794 613 l 4783 623 l
 4772 634 l 4763 645 l 4755 656 l 4747 668 l 4741 680 l
 4737 692 l 4733 703 l 4731 715 l 4731 725 l 4732 736 l
 4734 746 l 4737 755 l 4743 765 l 4750 775 l 4759 784 l
 4770 793 l 4783 801 l 4797 808 l 4813 814 l 4830 820 l
 4847 824 l 4865 827 l 4883 829 l 4900 829 l 4918 828 l
 4934 826 l 4951 822 l
 cp gs col0 s gr  [] 0 sd
% Polyline
n 5022 562 m 5024 564 l 5030 568 l 5039 575 l 5053 586 l 5071 600 l
 5092 616 l 5115 635 l 5139 653 l 5163 672 l 5186 691 l
 5208 708 l 5228 724 l 5247 739 l 5264 753 l 5280 766 l
 5294 778 l 5308 790 l 5322 801 l 5335 812 l 5350 826 l
 5366 840 l 5381 854 l 5397 868 l 5412 882 l 5427 897 l
 5442 912 l 5456 927 l 5469 941 l 5482 955 l 5493 968 l
 5504 981 l 5513 993 l 5521 1005 l 5527 1016 l 5533 1026 l
 5538 1036 l 5542 1046 l 5545 1055 l 5548 1065 l 5549 1075 l
 5549 1085 l 5548 1095 l 5546 1105 l 5543 1115 l 5539 1124 l
 5534 1134 l 5528 1143 l 5521 1152 l 5513 1161 l 5504 1170 l
 5495 1179 l 5484 1188 l 5473 1197 l 5460 1206 l 5447 1216 l
 5432 1227 l 5416 1237 l 5399 1248 l 5382 1259 l 5364 1269 l
 5346 1280 l 5328 1290 l 5311 1299 l 5294 1309 l 5277 1317 l
 5261 1325 l 5246 1333 l 5231 1340 l 5216 1346 l 5200 1353 l
 5184 1360 l 5167 1367 l 5148 1374 l 5129 1381 l 5107 1389 l
 5086 1397 l 5064 1405 l 5045 1412 l 5029 1417 l 5017 1421 l
 5010 1424 l
 5007 1425 l gs col0 s gr 
% Polyline
n 4632 1095 m 4633 1095 l 4638 1095 l 4651 1096 l 4670 1097 l 4694 1098 l
 4718 1100 l 4740 1102 l 4759 1104 l 4775 1106 l 4788 1108 l
 4799 1110 l 4808 1113 l 4815 1116 l 4821 1119 l 4826 1123 l
 4831 1127 l 4834 1131 l 4837 1135 l 4840 1140 l 4841 1145 l
 4842 1149 l 4842 1154 l 4842 1158 l 4841 1162 l 4840 1166 l
 4839 1170 l 4834 1177 l 4826 1184 l 4814 1193 l 4801 1201 l
 4792 1206 l
 4790 1207 l gs col0 s gr 
% Polyline
 [15 45] 45 sd
n 4586 955 m 4582 968 l 4579 981 l 4577 995 l 4575 1009 l 4575 1023 l
 4576 1036 l 4578 1048 l 4581 1059 l 4585 1068 l 4589 1076 l
 4594 1081 l 4599 1084 l 4605 1085 l 4611 1084 l 4618 1080 l
 4625 1075 l 4633 1068 l 4641 1060 l 4649 1050 l 4657 1038 l
 4664 1027 l 4671 1015 l 4676 1002 l 4682 990 l 4686 977 l
 4690 964 l 4694 950 l 4696 936 l 4698 923 l 4699 910 l
 4699 898 l 4698 887 l 4696 879 l 4693 872 l 4689 867 l
 4684 864 l 4678 863 l 4671 864 l 4663 867 l 4654 871 l
 4644 878 l 4634 886 l 4624 896 l 4615 907 l 4606 918 l
 4598 930 l 4592 943 l
 cp gs col0 s gr  [] 0 sd
% Polyline
n 4707 750 m 4708 751 l 4713 754 l 4723 763 l 4739 775 l 4758 791 l
 4777 806 l 4793 821 l 4807 833 l 4816 842 l 4822 850 l
 4825 856 l 4825 860 l 4822 862 l 4815 864 l 4806 864 l
 4792 863 l 4773 861 l 4751 857 l 4727 853 l 4703 848 l
 4682 844 l 4669 841 l 4663 840 l
 4662 840 l gs col0 s gr 
 [15 30] 30 sd
1 slc
% Ellipse
n 5280 2497 45 45 0 360 DrawEllipse gs col7 0.45 shd ef gr gs col0 s gr
 [] 0 sd
% Polyline
0 slj
0 slc
 [15 45] 45 sd
n 5277 2512 m
 4685 2797 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5285 2497 m
 4445 2354 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5292 2497 m
 4655 1927 l gs col0 s gr  [] 0 sd
% Polyline
 [45] 0 sd
gs  clippath
5130 1605 m 5070 1605 l 5070 1790 l 5100 1640 l 5130 1790 l cp
eoclip
n 5100 2115 m
 5100 1620 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 5130 1790 m 5100 1640 l 5070 1790 l 5100 1760 l 5130 1790 l 
 cp gs col7 1.00 shd ef gr  col0 s
% Polyline
 [45] 0 sd
gs  clippath
5106 564 m 5132 618 l 5269 554 l 5148 578 l 5243 499 l cp
eoclip
n 5643 345 m
 5133 585 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 5243 499 m 5148 578 l 5269 554 l 5243 499 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
 [45] 0 sd
gs  clippath
4679 770 m 4653 716 l 4516 780 l 4638 757 l 4542 835 l cp
eoclip
n 4173 975 m
 4653 750 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 4542 835 m 4638 757 l 4516 780 l 4542 835 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
 [45] 0 sd
gs  clippath
1650 1605 m 1590 1605 l 1590 1790 l 1620 1640 l 1650 1790 l cp
eoclip
n 1620 2115 m
 1620 1620 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 1650 1790 m 1620 1640 l 1590 1790 l 1620 1760 l 1650 1790 l 
 cp gs col7 1.00 shd ef gr  col0 s
% Polyline
 [45] 0 sd
gs  clippath
2924 1024 m 2981 1044 l 3031 901 l 2963 1005 l 2974 881 l cp
eoclip
n 3123 555 m
 2958 1020 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 2974 881 m 2963 1005 l 3031 901 l 2974 881 l  cp gs 0.00 setgray ef gr  col0 s
% Polyline
 [45] 0 sd
gs  clippath
2871 1420 m 2814 1400 l 2764 1543 l 2833 1440 l 2821 1563 l cp
eoclip
n 2673 1890 m
 2838 1425 l gs col0 s gr gr
 [] 0 sd
% arrowhead
n 2821 1563 m 2833 1440 l 2764 1543 l 2821 1563 l  cp gs 0.00 setgray ef gr  col0 s
$F2psEnd
rs

---------------0601111605739
Content-Type: application/postscript; name="graph2.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="graph2.eps"

%!PS-Adobe-2.0 EPSF-2.0
%%Title: graph2.fig
%%Creator: fig2dev Version 3.2 Patchlevel 4
%%CreationDate: Tue Sep 27 22:09:07 2005
%%For: post@lingen (Olaf Post)
%%BoundingBox: 0 0 395 202
%%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} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
newpath 0 202 moveto 0 0 lineto 395 0 lineto 395 202 lineto closepath clip newpath
-19.3 236.3 translate
1 -1 scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
 /DrawEllipse {
	/endangle exch def
	/startangle exch def
	/yrad exch def
	/xrad exch def
	/y exch def
	/x exch def
	/savematrix mtrx currentmatrix def
	x y tr xrad yrad sc 0 0 1 startangle endangle arc
	closepath
	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
10 setmiterlimit
0 slj 0 slc
 0.06299 0.06299 sc
%
% Fig objects follow
%
% 
% here starts figure with depth 50
% Ellipse
7.500 slw
 [15 30] 30 sd
1 slc
n 3645 1890 45 45 0 360 DrawEllipse gs 0.50 setgray ef gr gs col0 s gr
 [] 0 sd
% Ellipse
 [15 30] 30 sd
n 4815 2970 45 45 0 360 DrawEllipse gs col7 0.45 shd ef gr gs col0 s gr
 [] 0 sd
% Ellipse
 [15 30] 30 sd
n 6210 2295 45 45 0 360 DrawEllipse gs col7 0.45 shd ef gr gs col0 s gr
 [] 0 sd
% Ellipse
 [15 30] 30 sd
n 5085 1305 45 45 0 360 DrawEllipse gs col7 0.45 shd ef gr gs col0 s gr
 [] 0 sd
% Polyline
0 slc
 [15 45] 45 sd
n 5085 1305 m 3645 1890 l 4815 2970 l 6255 2295 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 3645 1890 m
 6255 2295 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 4809 2973 m
 4676 3740 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5096 1293 m
 5236 560 l gs col0 s gr  [] 0 sd
% Polyline
2 slj
n 4561 1661 m 4536 1670 l 4510 1679 l 4484 1687 l 4458 1696 l 4432 1705 l
 4405 1714 l 4380 1723 l 4354 1732 l 4330 1741 l 4307 1749 l
 4286 1758 l 4267 1767 l 4250 1775 l 4236 1783 l 4225 1791 l
 4216 1799 l 4211 1806 l 4209 1813 l 4210 1820 l 4215 1827 l
 4223 1833 l 4235 1840 l 4246 1844 l 4259 1849 l 4274 1853 l
 4291 1858 l 4310 1863 l 4332 1867 l 4355 1872 l 4380 1877 l
 4407 1881 l 4436 1886 l 4466 1891 l 4498 1896 l 4532 1901 l
 4566 1906 l 4602 1911 l 4638 1916 l 4675 1921 l 4713 1926 l
 4750 1931 l 4788 1936 l 4825 1941 l 4862 1946 l 4899 1951 l
 4935 1956 l 4970 1960 l 5005 1965 l 5039 1970 l 5072 1974 l
 5104 1979 l 5135 1984 l 5177 1990 l 5218 1997 l 5258 2003 l
 5297 2010 l 5335 2016 l 5372 2023 l 5409 2029 l 5443 2035 l
 5477 2040 l 5508 2045 l 5538 2050 l 5565 2053 l 5589 2056 l
 5611 2058 l 5630 2059 l 5647 2059 l 5660 2058 l 5671 2056 l
 5678 2053 l 5683 2048 l 5686 2043 l 5685 2036 l 5682 2028 l
 5677 2019 l 5668 2008 l 5658 1996 l 5645 1983 l 5630 1968 l
 5613 1952 l 5594 1935 l 5573 1918 l 5551 1899 l 5528 1880 l
 5504 1860 l 5479 1840 l 5454 1820 l 5430 1800 l 5405 1781 l
 5382 1762 l 5359 1744 l 5338 1726 l 5317 1709 l 5298 1693 l
 5280 1677 l 5257 1657 l 5236 1637 l 5217 1618 l 5198 1600 l
 5181 1583 l 5165 1567 l 5149 1552 l 5135 1539 l 5120 1528 l
 5106 1518 l 5093 1510 l 5079 1503 l 5065 1499 l 5051 1496 l
 5037 1495 l 5021 1496 l 5009 1498 l 4996 1500 l 4981 1503 l
 4966 1507 l 4950 1512 l 4932 1518 l 4914 1524 l 4894 1531 l
 4873 1539 l 4851 1547 l 4829 1556 l 4806 1565 l 4782 1575 l
 4758 1585 l 4733 1595 l 4709 1605 l 4684 1614 l 4660 1624 l
 4635 1634 l 4610 1643 l 4586 1652 l
 cp gs col0 s gr 
% Polyline
n 5751 2359 m 5761 2364 l 5768 2370 l 5770 2376 l 5770 2383 l 5766 2391 l
 5759 2399 l 5748 2407 l 5735 2417 l 5719 2427 l 5700 2437 l
 5679 2448 l 5656 2459 l 5632 2471 l 5606 2483 l 5579 2495 l
 5552 2507 l 5524 2520 l 5497 2533 l 5469 2546 l 5441 2560 l
 5418 2571 l 5394 2583 l 5370 2596 l 5346 2608 l 5321 2621 l
 5296 2635 l 5271 2648 l 5245 2661 l 5218 2675 l 5192 2688 l
 5165 2701 l 5139 2713 l 5113 2724 l 5087 2735 l 5062 2745 l
 5038 2754 l 5014 2762 l 4991 2769 l 4969 2775 l 4948 2779 l
 4928 2783 l 4908 2784 l 4890 2785 l 4872 2784 l 4852 2782 l
 4834 2778 l 4815 2772 l 4797 2764 l 4779 2755 l 4760 2744 l
 4742 2732 l 4723 2718 l 4704 2703 l 4686 2686 l 4667 2668 l
 4648 2650 l 4628 2630 l 4609 2610 l 4590 2590 l 4571 2569 l
 4552 2548 l 4533 2527 l 4514 2507 l 4494 2486 l 4475 2466 l
 4454 2446 l 4431 2423 l 4407 2401 l 4382 2379 l 4357 2356 l
 4331 2333 l 4304 2311 l 4278 2288 l 4253 2266 l 4229 2245 l
 4206 2224 l 4184 2204 l 4165 2186 l 4149 2169 l 4136 2154 l
 4125 2140 l 4118 2129 l 4115 2119 l 4115 2111 l 4119 2105 l
 4127 2100 l 4134 2098 l 4144 2097 l 4156 2097 l 4171 2097 l
 4187 2098 l 4205 2100 l 4226 2103 l 4249 2107 l 4273 2111 l
 4300 2116 l 4329 2121 l 4359 2127 l 4391 2134 l 4424 2141 l
 4458 2148 l 4493 2155 l 4530 2163 l 4566 2171 l 4604 2179 l
 4641 2187 l 4679 2195 l 4716 2202 l 4753 2210 l 4790 2217 l
 4827 2224 l 4864 2231 l 4900 2237 l 4936 2244 l 4972 2250 l
 5008 2255 l 5045 2261 l 5081 2266 l 5118 2271 l 5155 2275 l
 5193 2280 l 5230 2285 l 5267 2289 l 5304 2293 l 5340 2297 l
 5376 2302 l 5412 2306 l 5446 2309 l 5479 2313 l 5511 2317 l
 5541 2320 l 5570 2324 l 5597 2327 l 5622 2331 l 5645 2334 l
 5666 2338 l 5685 2341 l 5702 2344 l 5718 2348 l 5731 2351 l
 5742 2355 l
 cp gs col0 s gr 
% Polyline
 [15 45] 45 sd
n 4629 2933 m 4643 2927 l 4656 2920 l 4668 2913 l 4681 2904 l 4693 2894 l
 4704 2883 l 4715 2871 l 4725 2859 l 4734 2846 l 4742 2833 l
 4749 2820 l 4754 2807 l 4758 2794 l 4760 2782 l 4761 2771 l
 4761 2760 l 4759 2750 l 4756 2740 l 4750 2730 l 4742 2721 l
 4733 2712 l 4721 2705 l 4708 2698 l 4693 2692 l 4677 2687 l
 4660 2683 l 4641 2680 l 4623 2678 l 4604 2678 l 4586 2679 l
 4568 2681 l 4551 2683 l 4535 2687 l 4519 2693 l 4506 2698 l
 4493 2704 l 4480 2711 l 4468 2720 l 4457 2728 l 4446 2738 l
 4435 2749 l 4426 2760 l 4418 2771 l 4410 2783 l 4404 2795 l
 4400 2807 l 4396 2818 l 4394 2830 l 4394 2840 l 4395 2851 l
 4397 2861 l 4400 2870 l 4406 2880 l 4413 2890 l 4422 2899 l
 4433 2908 l 4446 2916 l 4460 2923 l 4476 2929 l 4493 2935 l
 4510 2939 l 4528 2942 l 4546 2944 l 4563 2944 l 4581 2943 l
 4597 2941 l 4614 2937 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5244 2986 m 5238 2992 l 5229 2996 l 5219 3000 l 5207 3002 l 5193 3002 l
 5178 3002 l 5162 3000 l 5145 2996 l 5129 2991 l 5112 2986 l
 5097 2979 l 5082 2971 l 5069 2963 l 5057 2954 l 5047 2945 l
 5038 2936 l 5030 2925 l 5022 2915 l 5016 2903 l 5010 2891 l
 5005 2879 l 5002 2867 l 4999 2855 l 4998 2843 l 4997 2833 l
 4997 2822 l 4999 2813 l 5001 2805 l 5004 2797 l 5009 2791 l
 5013 2786 l 5019 2781 l 5026 2778 l 5034 2775 l 5042 2774 l
 5052 2773 l 5062 2773 l 5073 2775 l 5084 2777 l 5096 2781 l
 5107 2785 l 5118 2790 l 5130 2796 l 5140 2803 l 5151 2810 l
 5161 2818 l 5173 2828 l 5184 2840 l 5195 2852 l 5206 2865 l
 5216 2879 l 5225 2893 l 5233 2907 l 5240 2922 l 5245 2935 l
 5249 2948 l 5250 2960 l 5250 2970 l 5248 2978 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 4209 1779 m 4203 1785 l 4194 1789 l 4184 1793 l 4172 1795 l 4158 1795 l
 4143 1795 l 4127 1793 l 4110 1789 l 4094 1784 l 4077 1779 l
 4062 1772 l 4047 1764 l 4034 1756 l 4022 1747 l 4012 1738 l
 4003 1729 l 3995 1718 l 3987 1708 l 3981 1696 l 3975 1684 l
 3970 1672 l 3967 1660 l 3964 1648 l 3963 1636 l 3962 1626 l
 3962 1615 l 3964 1606 l 3966 1598 l 3969 1590 l 3974 1584 l
 3978 1579 l 3984 1574 l 3991 1571 l 3999 1568 l 4007 1567 l
 4017 1566 l 4027 1566 l 4038 1568 l 4049 1570 l 4061 1574 l
 4072 1578 l 4083 1583 l 4095 1589 l 4105 1596 l 4116 1603 l
 4126 1611 l 4138 1621 l 4149 1633 l 4160 1645 l 4171 1658 l
 4181 1672 l 4190 1686 l 4198 1700 l 4205 1715 l 4210 1728 l
 4214 1741 l 4215 1753 l 4215 1763 l 4213 1771 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5949 2626 m 5943 2632 l 5934 2636 l 5924 2640 l 5912 2642 l 5898 2642 l
 5883 2642 l 5867 2640 l 5850 2636 l 5834 2631 l 5817 2626 l
 5802 2619 l 5787 2611 l 5774 2603 l 5762 2594 l 5752 2585 l
 5743 2576 l 5735 2565 l 5727 2555 l 5721 2543 l 5715 2531 l
 5710 2519 l 5707 2507 l 5704 2495 l 5703 2483 l 5702 2473 l
 5702 2462 l 5704 2453 l 5706 2445 l 5709 2437 l 5714 2431 l
 5718 2426 l 5724 2421 l 5731 2418 l 5739 2415 l 5747 2414 l
 5757 2413 l 5767 2413 l 5778 2415 l 5789 2417 l 5801 2421 l
 5812 2425 l 5823 2430 l 5835 2436 l 5845 2443 l 5856 2450 l
 5866 2458 l 5878 2468 l 5889 2480 l 5900 2492 l 5911 2505 l
 5921 2519 l 5930 2533 l 5938 2547 l 5945 2562 l 5950 2575 l
 5954 2588 l 5955 2600 l 5955 2610 l 5953 2618 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 4967 1486 m 4961 1492 l 4952 1496 l 4942 1500 l 4930 1502 l 4916 1502 l
 4901 1502 l 4885 1500 l 4868 1496 l 4852 1491 l 4835 1486 l
 4820 1479 l 4805 1471 l 4792 1463 l 4780 1454 l 4770 1445 l
 4761 1436 l 4753 1425 l 4745 1415 l 4739 1403 l 4733 1391 l
 4728 1379 l 4725 1367 l 4722 1355 l 4721 1343 l 4720 1333 l
 4720 1322 l 4722 1313 l 4724 1305 l 4727 1297 l 4732 1291 l
 4736 1286 l 4742 1281 l 4749 1278 l 4757 1275 l 4765 1274 l
 4775 1273 l 4785 1273 l 4796 1275 l 4807 1277 l 4819 1281 l
 4830 1285 l 4841 1290 l 4853 1296 l 4863 1303 l 4874 1310 l
 4884 1318 l 4896 1328 l 4907 1340 l 4918 1352 l 4929 1365 l
 4939 1379 l 4948 1393 l 4956 1407 l 4963 1422 l 4968 1435 l
 4972 1448 l 4973 1460 l 4973 1470 l 4971 1478 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 4007 2340 m 4021 2334 l 4034 2327 l 4046 2320 l 4059 2311 l 4071 2301 l
 4082 2290 l 4093 2278 l 4103 2266 l 4112 2253 l 4120 2240 l
 4127 2227 l 4132 2214 l 4136 2201 l 4138 2189 l 4139 2178 l
 4139 2167 l 4137 2157 l 4134 2147 l 4128 2137 l 4120 2128 l
 4111 2119 l 4099 2112 l 4086 2105 l 4071 2099 l 4055 2094 l
 4038 2090 l 4019 2087 l 4001 2085 l 3982 2085 l 3964 2086 l
 3946 2088 l 3929 2090 l 3913 2094 l 3897 2100 l 3884 2105 l
 3871 2111 l 3858 2118 l 3846 2127 l 3835 2135 l 3824 2145 l
 3813 2156 l 3804 2167 l 3796 2178 l 3788 2190 l 3782 2202 l
 3778 2214 l 3774 2225 l 3772 2237 l 3772 2247 l 3773 2258 l
 3775 2268 l 3778 2277 l 3784 2287 l 3791 2297 l 3800 2306 l
 3811 2315 l 3824 2323 l 3838 2330 l 3854 2336 l 3871 2342 l
 3888 2346 l 3906 2349 l 3924 2351 l 3941 2351 l 3959 2350 l
 3975 2348 l 3992 2344 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5364 1575 m 5378 1569 l 5391 1562 l 5403 1555 l 5416 1546 l 5428 1536 l
 5439 1525 l 5450 1513 l 5460 1501 l 5469 1488 l 5477 1475 l
 5484 1462 l 5489 1449 l 5493 1436 l 5495 1424 l 5496 1413 l
 5496 1402 l 5494 1392 l 5491 1382 l 5485 1372 l 5477 1363 l
 5468 1354 l 5456 1347 l 5443 1340 l 5428 1334 l 5412 1329 l
 5395 1325 l 5376 1322 l 5358 1320 l 5339 1320 l 5321 1321 l
 5303 1323 l 5286 1325 l 5270 1329 l 5254 1335 l 5241 1340 l
 5228 1346 l 5215 1353 l 5203 1362 l 5192 1370 l 5181 1380 l
 5170 1391 l 5161 1402 l 5153 1413 l 5145 1425 l 5139 1437 l
 5135 1449 l 5131 1460 l 5129 1472 l 5129 1482 l 5130 1493 l
 5132 1503 l 5135 1512 l 5141 1522 l 5148 1532 l 5157 1541 l
 5168 1550 l 5181 1558 l 5195 1565 l 5211 1571 l 5228 1577 l
 5245 1581 l 5263 1584 l 5281 1586 l 5298 1586 l 5316 1585 l
 5332 1583 l 5349 1579 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5896 2033 m 5910 2027 l 5923 2020 l 5935 2013 l 5948 2004 l 5960 1994 l
 5971 1983 l 5982 1971 l 5992 1959 l 6001 1946 l 6009 1933 l
 6016 1920 l 6021 1907 l 6025 1894 l 6027 1882 l 6028 1871 l
 6028 1860 l 6026 1850 l 6023 1840 l 6017 1830 l 6009 1821 l
 6000 1812 l 5988 1805 l 5975 1798 l 5960 1792 l 5944 1787 l
 5927 1783 l 5908 1780 l 5890 1778 l 5871 1778 l 5853 1779 l
 5835 1781 l 5818 1783 l 5802 1787 l 5786 1793 l 5773 1798 l
 5760 1804 l 5747 1811 l 5735 1820 l 5724 1828 l 5713 1838 l
 5702 1849 l 5693 1860 l 5685 1871 l 5677 1883 l 5671 1895 l
 5667 1907 l 5663 1918 l 5661 1930 l 5661 1940 l 5662 1951 l
 5664 1961 l 5667 1970 l 5673 1980 l 5680 1990 l 5689 1999 l
 5700 2008 l 5713 2016 l 5727 2023 l 5743 2029 l 5760 2035 l
 5777 2039 l 5795 2042 l 5813 2044 l 5830 2044 l 5848 2043 l
 5864 2041 l 5881 2037 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 4286 1975 m 4282 1988 l 4279 2001 l 4277 2015 l 4275 2029 l 4275 2043 l
 4276 2056 l 4278 2068 l 4281 2079 l 4285 2088 l 4289 2096 l
 4294 2101 l 4299 2104 l 4305 2105 l 4311 2104 l 4318 2100 l
 4325 2095 l 4333 2088 l 4341 2080 l 4349 2070 l 4357 2058 l
 4364 2047 l 4371 2035 l 4376 2022 l 4382 2010 l 4386 1997 l
 4390 1984 l 4394 1970 l 4396 1956 l 4398 1943 l 4399 1930 l
 4399 1918 l 4398 1907 l 4396 1899 l 4393 1892 l 4389 1887 l
 4384 1884 l 4378 1883 l 4371 1884 l 4363 1887 l 4354 1891 l
 4344 1898 l 4334 1906 l 4324 1916 l 4315 1927 l 4306 1938 l
 4298 1950 l 4292 1963 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5516 2170 m 5512 2183 l 5509 2196 l 5507 2210 l 5505 2224 l 5505 2238 l
 5506 2251 l 5508 2263 l 5511 2274 l 5515 2283 l 5519 2291 l
 5524 2296 l 5529 2299 l 5535 2300 l 5541 2299 l 5548 2295 l
 5555 2290 l 5563 2283 l 5571 2275 l 5579 2265 l 5587 2253 l
 5594 2242 l 5601 2230 l 5606 2217 l 5612 2205 l 5616 2192 l
 5620 2179 l 5624 2165 l 5626 2151 l 5628 2138 l 5629 2125 l
 5629 2113 l 5628 2102 l 5626 2094 l 5623 2087 l 5619 2082 l
 5614 2079 l 5608 2078 l 5601 2079 l 5593 2082 l 5584 2086 l
 5574 2093 l 5564 2101 l 5554 2111 l 5545 2122 l 5536 2133 l
 5528 2145 l 5522 2158 l
 cp gs col0 s gr  [] 0 sd
% Polyline
n 4930 3389 m 4931 3386 l 4933 3379 l 4938 3367 l 4944 3350 l 4952 3328 l
 4961 3303 l 4972 3277 l 4984 3250 l 4995 3225 l 5007 3201 l
 5020 3178 l 5032 3157 l 5045 3138 l 5059 3120 l 5074 3103 l
 5090 3086 l 5108 3070 l 5122 3057 l 5137 3045 l 5153 3032 l
 5171 3019 l 5191 3005 l 5212 2991 l 5236 2976 l 5262 2960 l
 5290 2943 l 5320 2925 l 5353 2905 l 5388 2885 l 5424 2865 l
 5461 2844 l 5497 2824 l 5531 2805 l 5562 2788 l 5589 2773 l
 5611 2761 l 5627 2753 l 5638 2747 l 5644 2743 l 5647 2742 l
 5650 2741 l 5656 2739 l 5667 2735 l 5684 2728 l 5707 2720 l
 5736 2709 l 5770 2696 l 5808 2681 l 5849 2665 l 5891 2649 l
 5935 2632 l 5977 2616 l 6018 2600 l 6057 2584 l 6093 2570 l
 6127 2556 l 6159 2543 l 6188 2531 l 6214 2519 l 6239 2509 l
 6261 2498 l 6282 2489 l 6301 2479 l 6319 2470 l 6335 2461 l
 6350 2453 l 6364 2445 l 6377 2437 l 6390 2428 l 6402 2420 l
 6413 2411 l 6424 2402 l 6433 2393 l 6442 2384 l 6450 2375 l
 6457 2365 l 6463 2355 l 6468 2345 l 6472 2335 l 6475 2324 l
 6477 2314 l 6477 2303 l 6477 2292 l 6475 2281 l 6473 2269 l
 6469 2258 l 6464 2246 l 6459 2234 l 6452 2222 l 6444 2210 l
 6435 2198 l 6426 2185 l 6415 2172 l 6403 2157 l 6389 2142 l
 6373 2127 l 6357 2110 l 6339 2093 l 6320 2075 l 6299 2056 l
 6277 2037 l 6254 2017 l 6230 1996 l 6205 1975 l 6180 1954 l
 6153 1932 l 6126 1910 l 6099 1888 l 6072 1866 l 6044 1845 l
 6017 1823 l 5990 1802 l 5964 1782 l 5938 1761 l 5912 1741 l
 5887 1722 l 5863 1703 l 5839 1685 l 5816 1667 l 5791 1647 l
 5767 1628 l 5743 1609 l 5719 1589 l 5695 1570 l 5671 1551 l
 5647 1531 l 5624 1512 l 5601 1492 l 5578 1473 l 5557 1453 l
 5535 1434 l 5515 1415 l 5495 1397 l 5477 1379 l 5460 1362 l
 5444 1345 l 5429 1329 l 5415 1313 l 5402 1298 l 5391 1284 l
 5380 1270 l 5371 1257 l 5363 1244 l 5352 1225 l 5344 1207 l
 5338 1188 l 5334 1169 l 5331 1149 l 5331 1127 l 5332 1104 l
 5335 1080 l 5339 1055 l 5344 1031 l 5349 1009 l 5353 991 l
 5357 978 l 5359 971 l
 5360 967 l gs col0 s gr 
% Polyline
 [15 45] 45 sd
n 4924 3416 m 4917 3424 l 4908 3432 l 4896 3440 l 4882 3448 l 4866 3456 l
 4848 3463 l 4829 3469 l 4809 3475 l 4789 3479 l 4770 3482 l
 4751 3484 l 4734 3484 l 4718 3483 l 4703 3481 l 4693 3478 l
 4684 3474 l 4675 3469 l 4667 3463 l 4660 3456 l 4653 3449 l
 4648 3441 l 4643 3432 l 4639 3423 l 4636 3414 l 4634 3404 l
 4633 3395 l 4633 3386 l 4634 3377 l 4635 3369 l 4638 3362 l
 4641 3355 l 4645 3348 l 4651 3341 l 4658 3334 l 4667 3328 l
 4677 3323 l 4688 3319 l 4700 3316 l 4714 3313 l 4728 3312 l
 4742 3311 l 4757 3312 l 4771 3313 l 4786 3315 l 4800 3317 l
 4813 3320 l 4824 3323 l 4835 3326 l 4845 3330 l 4856 3334 l
 4866 3338 l 4876 3343 l 4886 3348 l 4895 3354 l 4904 3360 l
 4911 3366 l 4917 3372 l 4922 3379 l 4926 3385 l 4928 3391 l
 4929 3398 l 4929 3404 l 4927 3410 l
 cp gs col0 s gr  [] 0 sd
% Polyline
n 5047 962 m 5045 965 l 5041 972 l 5035 983 l 5025 999 l 5013 1018 l
 4999 1038 l 4984 1060 l 4969 1080 l 4953 1099 l 4937 1117 l
 4920 1134 l 4903 1150 l 4884 1165 l 4864 1180 l 4842 1195 l
 4827 1204 l 4811 1213 l 4794 1223 l 4777 1233 l 4758 1243 l
 4738 1254 l 4717 1264 l 4695 1275 l 4672 1287 l 4647 1298 l
 4622 1310 l 4597 1322 l 4570 1334 l 4543 1346 l 4516 1357 l
 4488 1369 l 4460 1381 l 4432 1393 l 4404 1404 l 4376 1415 l
 4349 1426 l 4321 1437 l 4294 1448 l 4267 1458 l 4240 1468 l
 4214 1478 l 4187 1488 l 4160 1498 l 4132 1509 l 4104 1519 l
 4076 1529 l 4047 1539 l 4018 1550 l 3988 1561 l 3959 1572 l
 3929 1582 l 3900 1593 l 3870 1604 l 3841 1615 l 3813 1626 l
 3785 1637 l 3759 1647 l 3733 1657 l 3708 1668 l 3685 1677 l
 3663 1687 l 3642 1696 l 3622 1705 l 3604 1714 l 3587 1723 l
 3571 1731 l 3557 1740 l 3543 1748 l 3531 1756 l 3519 1764 l
 3509 1773 l 3499 1781 l 3490 1790 l 3483 1799 l 3476 1808 l
 3471 1817 l 3466 1827 l 3463 1837 l 3460 1847 l 3459 1858 l
 3459 1868 l 3460 1879 l 3462 1890 l 3465 1902 l 3469 1913 l
 3473 1925 l 3479 1936 l 3486 1948 l 3493 1960 l 3501 1972 l
 3510 1985 l 3520 1998 l 3530 2011 l 3541 2024 l 3553 2039 l
 3566 2053 l 3580 2069 l 3594 2085 l 3610 2102 l 3627 2120 l
 3645 2138 l 3663 2157 l 3683 2177 l 3703 2197 l 3725 2218 l
 3747 2240 l 3769 2261 l 3792 2283 l 3815 2305 l 3838 2327 l
 3862 2349 l 3886 2371 l 3909 2393 l 3933 2415 l 3956 2436 l
 3980 2457 l 4003 2478 l 4026 2499 l 4049 2520 l 4071 2540 l
 4093 2560 l 4115 2580 l 4138 2600 l 4161 2620 l 4184 2641 l
 4208 2663 l 4231 2684 l 4255 2706 l 4279 2728 l 4304 2751 l
 4327 2773 l 4351 2795 l 4374 2817 l 4397 2839 l 4419 2861 l
 4441 2882 l 4461 2903 l 4481 2923 l 4500 2943 l 4517 2962 l
 4534 2980 l 4549 2998 l 4564 3015 l 4577 3031 l 4589 3047 l
 4600 3062 l 4610 3076 l 4624 3098 l 4635 3119 l 4643 3139 l
 4649 3159 l 4653 3179 l 4655 3200 l 4654 3221 l 4651 3243 l
 4647 3266 l 4641 3289 l 4635 3311 l 4627 3332 l 4621 3350 l
 4615 3364 l 4611 3374 l 4608 3379 l
 4607 3382 l gs col0 s gr 
% Polyline
 [15 45] 45 sd
n 5336 1009 m 5329 1017 l 5320 1025 l 5308 1033 l 5294 1041 l 5278 1049 l
 5260 1056 l 5241 1062 l 5221 1068 l 5201 1072 l 5182 1075 l
 5163 1077 l 5146 1077 l 5130 1076 l 5115 1074 l 5105 1071 l
 5096 1067 l 5087 1062 l 5079 1056 l 5072 1049 l 5065 1042 l
 5060 1034 l 5055 1025 l 5051 1016 l 5048 1007 l 5046 997 l
 5045 988 l 5045 979 l 5046 970 l 5047 962 l 5050 955 l
 5053 948 l 5057 941 l 5063 934 l 5070 927 l 5079 921 l
 5089 916 l 5100 912 l 5112 909 l 5126 906 l 5140 905 l
 5154 904 l 5169 905 l 5183 906 l 5198 908 l 5212 910 l
 5225 913 l 5236 916 l 5247 919 l 5257 923 l 5268 927 l
 5278 931 l 5288 936 l 5298 941 l 5307 947 l 5316 953 l
 5323 959 l 5329 965 l 5334 972 l 5338 978 l 5340 984 l
 5341 991 l 5341 997 l 5339 1003 l
 cp gs col0 s gr  [] 0 sd
% Ellipse
n 360 1890 45 45 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Ellipse
n 1800 1305 45 45 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Ellipse
n 2925 2295 45 45 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Ellipse
n 1530 2970 45 45 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Polyline
0 slj
n 1800 1305 m 360 1890 l 1530 2970 l 2970 2295 l
 cp gs col0 s gr 
% Polyline
n 360 1890 m
 2970 2295 l gs col0 s gr 
% Polyline
n 1817 1292 m
 1907 812 l gs col0 s gr 
% Polyline
n 1537 2992 m
 1457 3392 l gs col0 s gr 
% Polyline
 [15 45] 45 sd
n 1457 3382 m
 1427 3592 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 1917 822 m
 1957 562 l gs col0 s gr  [] 0 sd
% here ends figure;
$F2psEnd
rs
showpage

---------------0601111605739
Content-Type: application/postscript; name="sierpinski.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="sierpinski.eps"

%!PS-Adobe-2.0 EPSF-2.0
%%Title: sierpinski.fig
%%Creator: fig2dev Version 3.2 Patchlevel 4
%%CreationDate: Thu Nov  3 23:42:47 2005
%%For: post@lingen (Olaf Post)
%%BoundingBox: 0 0 369 264
%%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} bind def
/col9 {0.000 0.000 0.690 srgb} bind def
/col10 {0.000 0.000 0.820 srgb} bind def
/col11 {0.530 0.810 1.000 srgb} bind def
/col12 {0.000 0.560 0.000 srgb} bind def
/col13 {0.000 0.690 0.000 srgb} bind def
/col14 {0.000 0.820 0.000 srgb} bind def
/col15 {0.000 0.560 0.560 srgb} bind def
/col16 {0.000 0.690 0.690 srgb} bind def
/col17 {0.000 0.820 0.820 srgb} bind def
/col18 {0.560 0.000 0.000 srgb} bind def
/col19 {0.690 0.000 0.000 srgb} bind def
/col20 {0.820 0.000 0.000 srgb} bind def
/col21 {0.560 0.000 0.560 srgb} bind def
/col22 {0.690 0.000 0.690 srgb} bind def
/col23 {0.820 0.000 0.820 srgb} bind def
/col24 {0.500 0.190 0.000 srgb} bind def
/col25 {0.630 0.250 0.000 srgb} bind def
/col26 {0.750 0.380 0.000 srgb} bind def
/col27 {1.000 0.500 0.500 srgb} bind def
/col28 {1.000 0.630 0.630 srgb} bind def
/col29 {1.000 0.750 0.750 srgb} bind def
/col30 {1.000 0.880 0.880 srgb} bind def
/col31 {1.000 0.840 0.000 srgb} bind def

end
save
newpath 0 264 moveto 0 0 lineto 369 0 lineto 369 264 lineto closepath clip newpath
-31.2 272.1 translate
1 -1 scale

/cp {closepath} bind def
/ef {eofill} bind def
/gr {grestore} bind def
/gs {gsave} bind def
/sa {save} bind def
/rs {restore} bind def
/l {lineto} bind def
/m {moveto} bind def
/rm {rmoveto} bind def
/n {newpath} bind def
/s {stroke} bind def
/sh {show} bind def
/slc {setlinecap} bind def
/slj {setlinejoin} bind def
/slw {setlinewidth} bind def
/srgb {setrgbcolor} bind def
/rot {rotate} bind def
/sc {scale} bind def
/sd {setdash} bind def
/ff {findfont} bind def
/sf {setfont} bind def
/scf {scalefont} bind def
/sw {stringwidth} bind def
/tr {translate} bind def
/tnt {dup dup currentrgbcolor
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add
  4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb}
  bind def
/shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul
  4 -2 roll mul srgb} bind def
 /DrawEllipse {
	/endangle exch def
	/startangle exch def
	/yrad exch def
	/xrad exch def
	/y exch def
	/x exch def
	/savematrix mtrx currentmatrix def
	x y tr xrad yrad sc 0 0 1 startangle endangle arc
	closepath
	savematrix setmatrix
	} def

/$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def
/$F2psEnd {$F2psEnteredState restore end} def

$F2psBegin
10 setmiterlimit
0 slj 0 slc
 0.06299 0.06299 sc
%
% Fig objects follow
%
% 
% here starts figure with depth 50
% Ellipse
7.500 slw
n 3259 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3741 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3500 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3500 3603 m 3250 4036 l 3751 4036 l
 cp gs col0 s gr 
% Ellipse
n 3741 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4223 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3982 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3982 3603 m 3732 4036 l 4233 4036 l
 cp gs col0 s gr 
% Ellipse
n 3500 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3982 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3741 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3741 3170 m 3491 3603 l 3991 3603 l
 cp gs col0 s gr 
% Ellipse
n 4223 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4705 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4464 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 4464 3603 m 4214 4036 l 4715 4036 l
 cp gs col0 s gr 
% Ellipse
n 4705 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4946 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 5186 4036 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 4946 3603 m 4695 4036 l 5197 4036 l
 cp gs col0 s gr 
% Ellipse
n 4464 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4946 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4705 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 4705 3170 m 4455 3603 l 4956 3603 l
 cp gs col0 s gr 
% Ellipse
n 3741 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4223 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3982 2736 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3982 2736 m 3732 3170 l 4233 3170 l
 cp gs col0 s gr 
% Ellipse
n 4223 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4705 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4464 2736 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 4464 2736 m 4214 3170 l 4715 3170 l
 cp gs col0 s gr 
% Ellipse
n 3982 2736 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4464 2736 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4223 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 4223 2302 m 3973 2736 l 4474 2736 l
 cp gs col0 s gr 
% Ellipse
n 2296 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2778 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2537 1869 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2537 1869 m 2286 2302 l 2787 2302 l
 cp gs col0 s gr 
% Ellipse
n 2778 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3259 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3019 1869 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3019 1869 m 2768 2302 l 3269 2302 l
 cp gs col0 s gr 
% Ellipse
n 2537 1869 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3019 1869 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2778 1436 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2778 1436 m 2528 1869 l 3028 1869 l
 cp gs col0 s gr 
% Ellipse
n 3259 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3741 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3500 1869 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3500 1869 m 3250 2302 l 3751 2302 l
 cp gs col0 s gr 
% Ellipse
n 3741 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 4223 2302 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3982 1869 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3982 1869 m 3732 2302 l 4233 2302 l
 cp gs col0 s gr 
% Ellipse
n 3500 1869 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3982 1869 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3741 1436 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3741 1436 m 3491 1869 l 3991 1869 l
 cp gs col0 s gr 
% Ellipse
n 2778 1436 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3259 1436 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3019 1002 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3019 1002 m 2768 1436 l 3269 1436 l
 cp gs col0 s gr 
% Ellipse
n 3259 1436 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3741 1436 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3500 1002 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3500 1002 m 3250 1436 l 3751 1436 l
 cp gs col0 s gr 
% Ellipse
n 3019 1002 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3500 1002 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3259 569 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3259 569 m 3009 1002 l 3510 1002 l
 cp gs col0 s gr 
% Ellipse
15.000 slw
n 1332 4036 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 1814 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 1573 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 1573 3603 m 1322 4036 l 1824 4036 l
 cp gs col0 s gr 
% Ellipse
n 1814 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2296 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2055 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2055 3603 m 1804 4036 l 2305 4036 l
 cp gs col0 s gr 
% Ellipse
n 1573 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2055 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 1814 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 1814 3170 m 1563 3603 l 2064 3603 l
 cp gs col0 s gr 
% Ellipse
n 2296 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2778 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2537 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2537 3603 m 2286 4036 l 2787 4036 l
 cp gs col0 s gr 
% Ellipse
n 2778 4036 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3259 4036 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3019 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 3019 3603 m 2768 4036 l 3269 4036 l
 cp gs col0 s gr 
% Ellipse
n 2537 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 3019 3603 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2778 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2778 3170 m 2528 3603 l 3028 3603 l
 cp gs col0 s gr 
% Ellipse
n 1814 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2296 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2055 2736 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2055 2736 m 1804 3170 l 2305 3170 l
 cp gs col0 s gr 
% Ellipse
n 2296 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2778 3170 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2537 2736 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2537 2736 m 2286 3170 l 2787 3170 l
 cp gs col0 s gr 
% Ellipse
n 2055 2736 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2537 2736 24 24 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Ellipse
n 2296 2302 45 45 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr

% Polyline
n 2296 2302 m 2045 2736 l 2546 2736 l
 cp gs col0 s gr 
% Polyline
7.500 slw
 [15 45] 45 sd
n 3249 574 m
 3494 148 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 3274 569 m
 3765 569 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5189 4028 m
 5434 3602 l gs col0 s gr  [] 0 sd
% Polyline
 [15 45] 45 sd
n 5201 4029 m
 5692 4029 l gs col0 s gr  [] 0 sd
% here ends figure;
$F2psEnd
rs
showpage

---------------0601111605739--
