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eigenvalues, spectral gap, perturbation of periodic structures
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%--------------------------------------------------------------------
% eigenvalues.tex   Eigenvalues in Spectral Gaps of a Perturbed
%                   Periodic Manifold
%                   Olaf Post
% 2000-12-06        Institut f"ur Reine und Angewandte Mathematik
%                   RWTH Aachen, Germany
% last change       email: post@iram.rwth-aachen.de 
% 2002-07-12
%--------------------------------------------------------------------

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\begin{document}

\title{Eigenvalues in Spectral Gaps of a Perturbed Periodic Manifold}

\author{Olaf Post}
\address{Institut f\"ur Reine und Angewandte Mathematik,
       Rheinisch-Westf\"alische Technische Hochschule Aachen,
       Templergraben 55,
       52062 Aachen,
       Germany}
\email{post@iram.rwth-aachen.de}
\date{05.12.2001}

%------------------------------------------------------------
% Subject classifications
%------------------------------------------------------------
%Subjclass contains the Classification of the paper following the 1991
%Mathematics Subject Classifications
\subjclass{35P20, 58G18, 47F05}
\keywords{Eigenvalues, spectral gap, perturbation of periodic structures}

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

\begin{abstract}
  We consider a non-compact Riemannian periodic manifold such that the
  corresponding Laplacian has a spectral gap. By continuously perturbing the
  periodic metric locally we can prove the existence of eigenvalues in a gap.
  A lower bound on the number of eigenvalue branches crossing a fixed level is
  established in terms of a discrete eigenvalue problem. Furthermore, we
  discuss examples of perturbations leading to infinitely many eigenvalue
  branches coming from above resp.\ finitely many branches coming from below.
\end{abstract}

\maketitle


%------------------------------------------------------------
% The main part
%------------------------------------------------------------

\section{Introduction}

A \emph{periodic manifold} is a (non-compact) Riemannian manifold $\Mper$ with
metric $g$ such that a finitely generated Abelian group $\Gamma$ acts properly
discontinuously and isometrically on $\Mper$. Furthermore, we assume that the
quotient $\Mper/\Gamma$ is compact. We call the closure of a fundamental
domain a \emph{period cell}. We suppose that the Laplacian $\laplacian \Mper$
on $\Mper$ has a spectral gap $I=(a,b)$ in the (essential) spectrum, i.e.,
\begin{displaymath}
  I \cap \spec \laplacian \Mper = \emptyset
\end{displaymath}
where $a > \inf \essspec \laplacian \Mper$.  We prove that a local
perturbation of the periodic structure leads to eigenvalues in the gap for
large deformation parameters $\tau$.  More precisely, we perturb the periodic
metric $g$ smoothly, starting from $g=\pert g 0$ and we denote the manifold
$\Mper$ with perturbed metric $\pert g \tau$ by $\Mpertau$. Furthermore, we
assume that $\pert g \tau$ and its first derivatives depend continuously on
$\tau$ and that the perturbation is (essentially) contained in the compact set
$\Pert$.  Here, $\Pert$ denotes a finite number of copies of a period cell $M$
of $\Mper$.  Later on we allow perturbations $\Mpertau$ diffeomorphic to
$\Mper$ (see Section~\ref{sec:examples}) or even some cases when the
perturbation changes the topology of $\Mper$ (see
Section~\ref{sec:non.diffeo}).

Donnelly~\cite{donnelly:92} and Li~\cite{li:92} proved the existence of
eigenvalues \emph{below} the essential spectrum of the Laplacian on the simply
connected, complete, hyperbolic plane by a local perturbation of the manifold.
The methods employed there use separation of variables. Therefore the ideas
cannot be generalized to our case.  Furthermore the Min-max principle applies,
since the eigenvalues lie below the essential spectrum. Note also the
introductory article~\cite{hislop:94} concerning results on the spectral
theory of hyperbolic manifolds.

\sloppy In general, it is much more difficult to prove the existence of
eigenvalues \emph{in} a spectral gap. Such problems have been extensively
studied in the case of Schr\"odinger or divergence type operators with a
spectral gap (see e.g.~\cite{hempel-deift:86}, \cite{alama-deift-hempel:92},
\cite{aadh:94} or \cite{hempel-besch:00}). The school of Birman (see e.g.
\cite{birman:97} or \cite{safronov:01} and references given therein) obtained
a vast number of results on the asymptotics of eigenvalue counting functions
using the Birman-Schwinger principle. This principle does not apply to our
case since the perturbation is not additive.  General results on perturbation
of point spectra can be found in~\cite{kato:66} or~\cite{reed-simon-4}.

{}From a physical point of view the Laplacian on a periodic manifold is the
Hamiltonian of an electron confined to a periodically curved material,
cf.~\cite{froese-herbst:00} or \cite{mitchell:01}.  For example, consider a
flat small strip in $\R^2$ (i.e., a \emph{quantum wire} or \emph{quantum wave
  guide}) which is periodically curved. If the strip is thin enough, there
exist gaps in the spectrum of the Dirichlet Laplacian
(cf.~\cite{exner-seba:89}~and \cite{yoshitomi:98}).  The local perturbation of
the periodic structure leads to bound states (i.e.\ square integrable
eigenfunctions) in a spectral gap.  Physically, the perturbation corresponds
to local impurities coming from contamination, e.g., in a semi-conductor. In
particular, we show that local deformations of a periodically curved quantum
wire produce bound states with energy in a spectral gap of the Hamiltonian of
the non-perturbed quantum wire (see Proposition~\ref{prop:qw.blow}).

\subsection*{Assumptions}
The main assumption on the periodic manifold $\Mper$ resp.\ the period cell
$M$ is the following. Let $\EWD k M$ and $\EWN k M$ denote the eigenvalues
(written in increasing order and repeated according to multiplicity) of the
Laplacian on the period cell $M$ with Dirichlet resp.\ Neumann boundary
condition on $\bd M$.  Note that we always have $\EWN k M \le \EWD k M$ in
virtue of the Min-max principle (see the next section for details). We assume
that the following \emph{gap condition} is fulfilled:
\begin{equation}
  \label{eq:gap}
  \exists \, k \in \N \colon \qquad 
  I_k:=\bigl( \EWD k M, \EWN {k+1} M \bigr) \ne \emptyset,
\end{equation}
i.e., $\EWD k M < \EWN {k+1} M$ for some $k$. From Floquet theory and the
Min-max principle it follows that $I_k$ lies \emph{in} a spectral gap for the
Laplacian on $\Mper$. Note that $I_k$ could be \emph{strictly} contained in a
spectral gap. The gap condition~\eqref{eq:gap} is fulfilled if the period cell
decouples from its translates, i.e., if the junctions between different period
cells are small in some sense. Recall that the gap condition is \emph{not}
satisfied for $\Mper=\R^d$ with the standard metric.  In this case, we always
have $\EWD k M \ge \EWN {k+1} M$.  In Section~\ref{sec:mfd.gaps} we give
examples of periodic manifolds with an arbitrary (finite) number of gaps that
were obtained in \cite{post:01b} and \cite{yoshitomi:98}.

If $\Gamma'$ denotes a subset of the Abelian group $\Gamma$ we set
\begin{displaymath}
  \GpM := \bigcup_{\gamma \in \Gamma'} \gamma M,
\end{displaymath}
in particular, $\Gamma M = \Mper$.  Consider a sequence $(\Gamma^n)$ of
subsets of $\Gamma$ such that $\Gamma^n$ has $n$ elements. We assume that
$\Gamma^n \nearrow \Gamma$ and set $\Appr := \GnM$, $\Rest := (\Gamma
\setminus \Gamma^m) M$ and $\RestAppr := (\Gamma^n \setminus \Gamma^m) M$. As
for $\Mpertau$ we denote by $\Apprtau$ resp.\ $\RestApprtau$ the manifold
$\Appr$ resp.\ $\RestAppr$ with perturbed metric $\pert g \tau$.

The assumptions on the family $(\pert g \tau)$ of perturbed metrics on $\Mper$
are the following:
\begin{align}
  \label{eq:unperturbed}
  \pert g 0 = g   & \qquad \text{on $\Mper$} \\
  \label{eq:met.cont}
  \norm[C^1] {\pert g \tau - \pert g {\tau_0}} \to 0
  & \qquad\text{on $\Mper$ as $\tau \to \tau_0$}\\
  \label{eq:pert.compact}
  \pert g \tau = \pert g 0 & \qquad\text{on $\Rest$, $\tau \ge 0$} 
\intertext{for all
    $\tau_0 \ge 0$ and sufficiently large $m \in \N$. The $C^1$-norm of a
    metric is defined in~\eqref{eq:norm.met}. Furthermore, if $\bd M$ is
    smooth we can also allow non-compact perturbations which are small outside
    $\Pert$, i.e.,}
%  \stepcounter{equation}
  \label{eq:met.unif.rest} \tag{1.4'}
  \sup_{\tau \ge 0} \norm[C^1] {\pert g \tau - \pert g 0} \to 0 
  & \qquad\text{on $\Rest$ as $m \to \infty$.}
\end{align}
Note that $\bd M$ is smooth iff $\bd \Rest$ is smooth.

Assumption~\eqref{eq:unperturbed} assures that we start from the periodic
manifold. Assumption~\eqref{eq:met.cont} guarantees the continuous dependence
of the metric and its first derivatives on $\tau$. In particular, eigenvalues
on the approximating problem $\Apprtau$ depend continuously on $\tau$
(cf.~Corollary~\ref{cor:ew.cont}), and the norm resp.\ weak topologies of the
corresponding Hilbert spaces do not depend on $\tau$
(cf.~Corollary~\ref{cor:norm.quad.est}). Note that we only need to control the
metric up to its first derivatives since no higher derivative of the metric
occurs in the coefficients of the Laplacian,
cf.~Equation~\eqref{eq:lapl.local}.

Finally, Assumption~\eqref{eq:pert.compact} resp.~\eqref{eq:met.unif.rest}
says that the support of the perturbation is contained in the compact set
$\Pert$ resp.\ is small outside the compact set $\Pert$. In particular, one
can prove a \emph{decomposition principle} for non-compact perturbations,
i.e., the invariance of the essential spectrum under such perturbations (see
Theorem~\ref{thm:dec.princ}). With other words, the essential spectrum
reflects the geometry at infinity. In our situation, the decomposition
principle assures that the spectral gaps of $\laplacian \Mper$ remain spectral
gaps in the \emph{essential} spectrum of $\laplacian \Mpertau$ for all $\tau
\ge 0$.  Only discrete eigenvalues (possibly accumulating at the band-edges of
$\spec \laplacian \Mper$) can appear in the spectral gaps.


\subsection*{Main result}
To count the number of eigenvalues $\lambda$ (with multiplicity) of the family
$\laplacian {\pert \Mper {\tau'}}$, $0 \le \tau' \le \tau$, we define
\begin{equation}
  \label{eq:def.count.fct}
  \mathcal N(\tau,\lambda) := 
  \sum_{0 \le \tau' \le \tau} 
       \dim \ker (\laplacian {\pert \Mper {\tau'}} - \lambda).
\end{equation}
Note the difference to the ordinary eigenvalue counting function
\begin{displaymath}
  \dimEWD[I] M := 
  \sum_{\lambda \in I} 
       \dim \ker (\laplacianD M - \lambda)
\end{displaymath}
that counts the eigenvalues in the Borel subset $I \subset \R_+$ of the
Dirichlet Laplacian on $M$; similarly, $\dimEWN[I] M$ denotes the
number of eigenvalues in $I$ of the Neumann Laplacian. As an
abbreviation, we set $\dimEWD M := \dimEWD [{[0,\lambda]}] M$.

Our main result is the following:
\begin{theorem}
  \label{thm:eigenvalues}
  Let $\lambda \in I_k$ belong to a spectral gap of $\laplacian \Mper$ then
  \begin{align}
    \label{eq:from.above}
    \mathcal N(\tau,\lambda) &\ge 
    \dimEWD \Perttau - \dimEWD \Pert\\
    \label{eq:from.below}
    \mathcal N(\tau,\lambda) &\ge 
    \dimEWN \Pert - \dimEWN \Perttau
  \end{align}
  for all $\tau \ge 0$ and sufficiently large $m \in \N$.
\end{theorem}
Note that Theorem~\ref{thm:eigenvalues} reduces the eigenvalue problem on the
non-compact manifold $\Mper$ to one on the compact manifold $\Pert$.
Therefore, we can apply the Min-max principle~\eqref{eq:max.min} to assure the
existence of eigenfunctions of the perturbed problem on $\Perttau$.

In a semi-classical picture, the number of bound states for high energy levels
is approximately given by the phase space volume associated with the classical
energy of the quantum system. The classical Hamiltonian on a $d$-dimensional
Riemannian manifold $X$ corresponding to $\laplacian X$ is given by
$h(x,p):=g^*_x(p,p)$ where $g^*$ denotes the metric on the cotangent bundle
$T^*\! X$ and $p \in T^*_x X$. The Hamiltonian $h$ determines a region in the
phase space $T^*\! X$ in which a classical particle with given energy is
allowed to move. The uncertainty principle, however, demands that each bound
state requires a cube of volume $(2\pi)^d$ in phase space, and therefore the
total number of bound states is approximately equal to the phase space volume
divided by $(2\pi)^d$ (see~\cite{reed-simon-4}). Indeed, the Weyl asymptotic
distribution of the (Dirichlet or Neumann) eigenvalues of the Laplacian on $X$
is given by
\begin{equation}
  \label{eq:weyl}
  \dimEW X - 
  \frac {\omega_d}{(2\pi)^d} \lambda^{\frac d 2}  \, \vol(X) =
  O(\lambda^{\frac{d-1}2}), \qquad \lambda \to \infty
\end{equation}
where $\omega_d$ is the volume of the unit ball in $\R^d$
(see~e.g.~\cite{safarov:98}).  Since we expect that the bound states are
localized on the perturbed manifold $\Pert$ we anticipate that
\begin{equation}
\label{eq:phase.diff}
  \frac {\omega_d}{(2\pi)^d} \lambda^{\frac d 2} |\vol(\Pert) - \vol(\Perttau)|
\end{equation}
is a lower bound for the eigenvalue counting
function~\eqref{eq:def.count.fct}.  However, note that
$O(\lambda^{\frac{d-1}2})$ in~\eqref{eq:weyl} depends on the manifold $\Pert$.
Nevertheless in our examples in Sections~\ref{sec:examples}
and~\ref{sec:non.diffeo} we show that an infinite number of eigenvalue
branches crosses the level $\lambda$ from above if the volume of $\Perttau$
increases. In contrast, if we decrease the volume of $\Perttau$, only a finite
number of eigenvalue branches cross the level $\lambda$ from below. Here, by
an \emph{eigenvalue branch} we mean an eigenvalue of (one of) the perturbed
manifold(s) viewed as a function depending on the perturbation parameter
$\tau$.

Under the assumptions given above it is not possible to obtain an upper bound
on the function $\mathcal N(\tau,\lambda)$. Since the dependence on $\tau$ is
not supposed to be analytic in Assumption~\eqref{eq:met.cont}, we could
consider a family of metrics $(\pert g \tau)$ which is constant near $\tau_0$.
If $\lambda$ is an eigenvalue of $\laplacian{\Mpertau[_0]}$, then $\mathcal
N(\tau_0,\lambda)=\infty$.

Nevertheless, the examples in Sections~\ref{sec:examples}
and~\ref{sec:non.diffeo} in which the eigenvalue branches of the problem on
$\Perttau$ are monotonical should be good candidates where an \emph{upper}
bound on the counting function $\mathcal N(\tau,\lambda)$ in terms of
$| \! \dimEW \Perttau - \dimEW \Pert|$ holds (cf.
Remarks~\ref{rem:mono.conf},~\ref{rem:mono.diff} and~\ref{rem:monotonical}).
Under suitable further conditions one should have to assure that the
eigenvalue branches (in the gap) of the \emph{full} problem $\Mpertau$ are
\emph{strictly} monotonical. Note that --- in contrast to the approximating
problem on $\Perttau$ --- the monotonicity of the eigenvalue branches of the
\emph{full} problem $\Mpertau$ cannot be shown by the Min-max principle since
the gap lies above the infimum of the essential spectrum.

Note furthermore that the counting function defined above could be much
greater than the \emph{spectral flow} of the family $(\laplacian \Mtau)$ at
$\lambda$ defined as the difference of the eigenvalue branches crossing the
level $\lambda$ from above and below (cf.~e.g.~\cite{safronov:01}).  Think for
example of an eigenvalue branch which oscillates around $\lambda$.

In~\cite{hempel-deift:86} or~\cite{alama-deift-hempel:92}) a family
$(\laplacian \Mpertau)$ is called \emph{complete} if for each $\lambda \ge 0$
there exists $\tau \ge 0$ such that $\lambda \in \spec \laplacian \Mpertau$.
Note that all our examples given in Sections~\ref{sec:examples}
and~\ref{sec:non.diffeo} are complete in this sense (in
Proposition~\ref{prop:conf.shrink} we have to perturb at least $m=2$ period
cells to ensure that $\mathcal N(\tau, \lambda) \ge 1$ for $\tau$ large
enough).

For the proof of Theorem~\ref{thm:eigenvalues} we cannot directly apply the
Min-max principle since we consider eigenvalues inside a spectral gap.
Therefore, we adopt the following idea given in~\cite{hempel-deift:86},
\cite{alama-deift-hempel:92}, \cite{aadh:94} or~\cite{hempel-besch:00} to our
situation: we show that the eigenfunctions of the full problem on $\Mper$ can
be approximated by eigenfunctions of an approximating problem on $M^n$
(cf.~Theorem~\ref{thm:ef.conv} and Theorem~\ref{thm:count.approx}). Our
situation differs from the case given in the papers mentioned above: in some
sense it is more complicated since we deal with different Hilbert spaces
$\Lsqr \Mpertau$ for each $\tau$. On the other hand, our situation is simpler
since there are no extra eigenvalues in the gap coming from the boundary
conditions on the approximating problem (cf.~Lemma~\ref{lem:common.gap}).

The paper is organized as follows: In Section~\ref{sec:prelim} we provide some
notation and basic results on Laplacians and periodic manifolds. In
Section~\ref{sec:ell.est} we develop elliptic estimates needed for the proof
of Theorem~\ref{thm:eigenvalues} given in Section~\ref{sec:eigenvalues}. In
Section~\ref{sec:mfd.gaps} we recall examples of periodic manifolds with
spectral gaps. Finally, in Sections~\ref{sec:examples}
and~\ref{sec:non.diffeo} we present examples of diffeomorphic and
non-homeomorphic perturbations leading to eigenvalues.

%------------------------------------------------------------
\section{Preliminaries}
\label{sec:prelim}

%------------------------------------------------------------
\subsection*{Laplacian on a manifold}

Throughout this article we study manifolds of dimension $d \ge 2$. For a
Riemannian manifold $M$ (compact or not) without boundary we denote by $\Lsqr
M$ the usual $L_2$-space of square integrable functions on $M$ with respect to
the volume measure on $M$. In a chart, the volume measure has the
density $(\det g)^\frac12$ with respect to the Lebesgue measure, where $\det
g$ is the determinant of the metric tensor $(g_{ij})$ in this chart. The norm
of $\Lsqr M$ will be denoted by $\norm[M] \cdot$. For $u \in \Cci M$, the
space of compactly supported smooth functions, we set
\begin{displaymath}
  \check q_M(u):=\normsqr[M] {\dd u} = \int_{M} |\dd u|^2.
\end{displaymath}
Here the $1$-form $\dd u$ denotes the exterior derivative of $u$ given in
coordinates by $| \dd u |^2= \sum_{i,j} g^{ij} \partial_i u \, \partial_j
\overline u$ where $(g^{ij})$ is the inverse matrix of $(g_{ij})$.

We denote the closure of the non-negative quadratic form $\check q_M$
by $q_M$.  Note that the domain $\dom q_M$ of the closed quadratic
form $q_M$ consists of functions in $L_2(M)$ such that the weak
derivative $\dd u$ is also square integrable (i.e., $q_M(u) <
\infty$).

We define the \emph{Laplacian} $\laplacian M$ (for a manifold without
boundary) as the unique self-adjoint and non-negative operator
associated with the closed quadratic form $q_M$, i.e., operator and
quadratic form are related by
\begin{displaymath}
  q_M(u)=\iprod {\laplacian M u} u
\end{displaymath}
for $u \in \Cci M$ (for details on quadratic forms see e.g.\ 
\cite[Chapter~VI]{kato:66}, \cite{reed-simon-1} or \cite{davies:96}).
Therefore, the Laplacian for smooth functions $u$ is given in a chart by
\begin{equation}
  \label{eq:lapl.local}
  \laplacian M u = 
  -(\det g)^{-\frac12} \sum_{i,j} \partial_i 
  \bigl(
    (\det g)^\frac12 g^{ij} \, \partial_j u
  \bigr).
\end{equation}

If $M$ is a complete manifold with piecewise smooth boundary $\bd M \ne
\emptyset$ we can define the Laplacian with \emph{Dirichlet} resp.\ 
\emph{Neumann boundary condition} in the same way. Here, we start from the
(closure of the) quadratic form $\check q_M$ defined on $\Cci M$, the space of
smooth functions with compact support \emph{away} from the boundary, resp.\ on
the subspace of $\Ci M$ with $\normsqr[M] u$ and $q_M(u)$ finite. Here, $\Ci
M$ denotes the space of smooth functions with derivatives continuous up to the
boundary of $M$. We denote the closure of the quadratic form by $q_M^\Dir$
resp.\ $q_M^\Neu$ and the corresponding self-adjoint operator by $\laplacianD
M$ resp.\ $\laplacianN M$.  If we are only interested in the differential
expression of the Laplacian, we suppress the boundary condition label. If $\bd
M$ can be split into two disjoint closed sets $\bd_1 M$ and $\bd_2 M$ we
define the Laplacian with mixed boundary conditions,. i.e., Dirichlet boundary
condition on $\bd_1 M$ and Neumann boundary condition on $\bd_2 M$ in the
obvious way.

If $M$ is compact the spectrum of $\laplacian M$ (with any boundary condition
if $\bd M \ne \emptyset$) is purely discrete. Note that we always assume that
$\bd M$ is piecewise smooth. We denote the corresponding eigenvalues by $\EW k
M$, $k \in \N$, (resp.\ $\EWD k M$ or $\EWN k M$ in the Dirichlet or Neumann
case) written in increasing order and repeated according to multiplicity. The
corresponding eigenfunctions are $\Ci M$ up to the boundary.  With this
eigenvalue ordering, we can state the \emph{Min-max principle} (for this
version of the Min-max principle see~e.g.~\cite{davies:96}). The $k$-th
eigenvalue of $\laplacianD M$ can be expressed by
\begin{equation}
  \label{eq:max.min}
  \EWD k M = 
  \inf_{L_k} \sup_{u \in L_k, u \ne 0} 
      \frac {\normsqr{\dd u}}{\normsqr u}
\end{equation}
where the infimum is taken over all $k$-dimensional subspaces $L_k$ of $\dom
q_M^\Dir$. Similar results hold for Neumann or other boundary conditions.

Suppose now that $M=M_1 \cup M_2$ such that $M_1$ and $M_2$ have piecewise
smooth boundary and that $M_1 \cap M_2$ has measure $0$. Denote by $\EWN k
{M_1 \dot \cup M_2}$ the eigenvalues of the quadratic form $q^\Neu_{M_1}
\oplus q^\Neu_{M_2}$ and similarly for Dirichlet boundary condition. Then the
Min-max principle implies
\begin{equation}
  \label{eq:dir.neu.brack}
  \EWN k {M_1 \dot \cup M_2} \le 
  \EWN k M \le 
  \EWD k M \le 
  \EWD k {M_1 \dot \cup M_2}
\end{equation}
for all $k \in \N$ since the opposite inclusions hold for the corresponding
quadratic form domains.  This estimate is called \emph{Dirichlet-Neumann
  bracketing}.

%------------------------------------------------------------
\subsection*{Periodic manifolds and Floquet Theory}


Let $\Gamma$ be an Abelian group of infinite order with $r$ generators and
neutral element $1$.  Such groups are isomorphic to $\Z^{r_0} \times
\Z_{p_1}^{r_1} \times \dots \times \Z_{p_a}^{r_a}$ with $r_0>0$ and $r_0 +
\dots + r_a=r$.  Here, $\Z_p$ denotes the Abelian group of order $p$.  A
$d$-dimensional (non-compact) Riemannian manifold $\Mper$ will be called
\emph{periodic} (or \emph{covering manifold}) if $\Gamma$ acts properly
discontinuously, isometrically and cocompactly on $\Mper$. Cocompactness means
that the quotient space $\Mper / \Gamma$ is compact. Note that the quotient
space is a $d$-dimensional compact Riemannian manifold and that the quotient
map $\map \pi \Mper {\Mper/\Gamma}$ is a local isometry. The metric turning
the manifold $\Mper$ into a periodic one will also be called \emph{periodic}.
For details on periodic manifolds see~e.g.~\cite{chavel:93}.

A compact subset $M$ of $\Mper$ is called a \emph{period cell} if $M$ is the
closure of a fundamental domain $D$, i.e., $M=\overline D$, $D$ is open and
connected, $D$ is disjoint from any translate $\gamma D$ for all $\gamma \in
\Gamma$, $\gamma \ne 1$, and the union of all translates $\gamma M$ is equal
to $\Mper$. Furthermore we assume that $M$ has piecewise smooth boundary.

Floquet theory allows us to analyse the spectrum of the Laplacian on $\Mper$
by analysing the spectra of Laplacians with quasi-periodic boundary condition
on a period cell $M$. In order to do this, we define $\theta$-periodic
boundary condition. Let $\theta$ be an element of the dual group $\hat \Gamma
= \Hom(\Gamma,\Torus^1)$ of $\Gamma$, which is isomorphic to a closed subgroup
of the $r$-dimensional torus $\Torus^r = \{ \theta \in \C^r; \, |\theta_i|=1
\text{ for all $i$} \}$. We denote by $q_M^\theta$ the closure of the
quadratic form $\check q_M$ defined on the space of those functions $u \in \Ci
M$ that satisfy
\begin{displaymath}
  u(\gamma x)= \overline{\theta(\gamma)} \, u(x)
\end{displaymath}
for all $x \in \bd M$ and all $\gamma \in \Gamma$ such that $\gamma x \in \bd
M$. The corresponding operator is denoted by $\laplacianT M$. Again,
$\laplacianT M$ has purely discrete spectrum denoted by $\EWT k M$. The
eigenvalues depend continuously on $\theta$
(see~\cite{reed-simon-4} or~\cite{bratteli:99}).  From Floquet theory we
obtain
\begin{displaymath}
  \spec \laplacian \Mper = \bigcup_{\theta \in \hat \Gamma} \spec \laplacianT M
                         = \bigcup_{k \in \N} B_k(\Mper)
\end{displaymath}
where $B_k=B_k(\Mper) = \{ \EWT k M ; \, \theta \in \hat \Gamma\}$ is a
compact interval, called the \emph{$k$-th band} (see e.g. \cite{reed-simon-4},
\cite{donnelly:81}) or \cite{bruening:92}.  Note that we have the following
\emph{Dirichlet-Neumann enclosure}
\begin{equation}
  \label{eq:dir.neu.encl}
  \EWN k M \le \EWT k M \le \EWD k M
\end{equation}
which follows easily from the opposite inclusions of the corresponding
quadratic form domains via the Min-max principle. Therefore, the $k$-th
Dirichlet resp.\ Neumann eigenvalue are an upper resp.\ lower bound for the
$k$-th band. In particular, if the interval $I_k$ defined in~\eqref{eq:gap} is
non-empty, it lies in a spectral gap of $\laplacian \Mper$. In general, we do
not know whether the intervals $I_k$ are empty or not. In many cases, the
Dirichlet-Neumann enclosure is too rough to guarantee the gap
condition~\eqref{eq:gap}.  Nevertheless, in Section~\ref{sec:mfd.gaps} we cite
examples where the gap condition holds.

%------------------------------------------------------------
\section{Elliptic estimates and perturbations of the metric}
\label{sec:ell.est}

First, we construct an atlas of $\Mper$ adapted to the periodic
structure.  This enables us to define a ``natural'' norm on the
Sobolev space on the non-compact periodic manifold $\Mper$. A more
general concept requires only bounds on the Ricci curvature instead of
the periodicity (cf.~e.g.~\cite{hebey:96}). Next, we need some
elliptic estimates. These estimates are used to obtain estimates in which
different Laplacians (defined with respect to different metrics) occur.
 
%------------------------------------------------------------
\subsection*{Periodic atlases and Sobolev spaces} Suppose that $(\widetilde
U_\beta)$, $\beta \in B$, is a finite cover of $\Mper / \Gamma$ and that
$\map{\widetilde \varphi_\beta}{\widetilde U_\beta}{\widetilde V_\beta}$ are
charts with open sets $\widetilde V_\beta \subset \R^d$. We assume that all
transition maps $\widetilde \varphi_{\beta_1} \circ \widetilde
\varphi_{\beta_2}^{-1}$ (when defined) have bounded derivatives up to all
orders.  We can lift this atlas of the quotient to an atlas
$\map{\varphi_\alpha}{U_\alpha}{V_\alpha}$, $\alpha:=(\beta,\gamma) \in A:=B
\times \Gamma$ of the periodic manifold $\Mper$, i.e., $\widetilde
\varphi_\beta \circ \pi = \varphi_\alpha$, $U_{(\beta,\gamma)} = \gamma
U_{(\beta,1)}$ and $\widetilde V_\beta = V_\alpha$. We call such an atlas $A$
\emph{periodic}.

If we consider $\GpM$ instead of the full periodic manifold $\Gamma M=\Mper$
we set $V'_\alpha := \varphi_\alpha(U_\alpha \cap \GpM)$ which is an open set
in $\R_+^d := [0,\infty) \times \R^{d-1}$. Note that if $\Gamma' \ne \Gamma$
then $\GpM$ has non-empty boundary being isometric to (copies of) $\bd M$. We
also call the atlas $A':=B \times \Gamma'$ of $\GpM$ with charts
$\map{\varphi_\alpha}{U_\alpha \cap \GpM}{V'_\alpha}$ \emph{periodic}.

If $\widetilde g_\beta$ denotes the metric of $\Mper / \Gamma$ carried via the
chart $\widetilde \varphi_\beta$ on $\widetilde V_\beta$ and $g_\alpha$ the
periodic metric $g$ of $\Mper$ carried via $\varphi_\alpha$ on
$V_\alpha=\widetilde V_\beta$ then $\widetilde g_\beta=g_\alpha$ for all
$\gamma \in \Gamma$, i.e., the set of different metrics $\set{g_\alpha}{a \in
  A}$ has only $|B| < \infty$ many elements although $A$ is not finite.

In the same way we can lift a partition of unity $(\widetilde \chi_\beta)$
subordinated to the cover $(\widetilde U_\beta)$ to a partition of unity
$(\varphi_\alpha)$ subordinated to $(U_\alpha)$, i.e., we have $\widetilde
\chi_\beta \circ \pi = \chi_\alpha$. We call such a partition of unity
\emph{periodic}. Again, the set of different functions $\set{\chi_\alpha \circ
  \varphi_\alpha^{-1}}{a \in A}$ is finite.

The next definitions are useful for the elliptic estimate in
Theorem~\ref{thm:reg.theory}.
\begin{definition}
  \label{def:unif.ell}
  A metric $h$ on $\GpM$ (not necessarily periodic) will be called
  \emph{uniformly elliptic} (with respect to the atlas $A'$) if there exists a
  constant $c>0$ such that
  \begin{equation}
    \label{eq:unif.ell}
    c^{-1} |v|^2 \le h_\alpha(x)(v,v) \le c |v|^2
  \end{equation}
  for all $v \in \R^d$, all $x \in V'_\alpha$ and all $\alpha \in A'$. We
  will also write $c^{-1} \le h \le c$ for short.
\end{definition}
Next, we define a norm on the space of $C^1$-sections in the bundle of
symmetric bilinear sections on $\GpM$. A Riemannian metric is an element of
this space.
\begin{definition}
  \label{def:norm.met}
  A metric $h$ will be called \emph{$C^1$-bounded} on $\GpM$ (with respect to
  the atlas $A'$) if
  \begin{equation}
    \label{eq:norm.met}
    \norm[C^1] h := 
    \sup_\alpha \norm[C^1] {h_\alpha} := 
    \sup_\alpha \sup_{x \in V'_\alpha} \max_{i,j,k} 
          \{|h_{\alpha,ij}(x)|, |\partial_k h_{\alpha,ij}(x)| \}
  \end{equation}
  is finite.
\end{definition}
We let $\mathcal G(c,c_1,\Gamma')$ be the space of all uniformly elliptic and
$C^1$-bounded metrics $h$ on $\GpM$ such that $c^{-1} \le h \le c$ and
$\norm[C^1] h \le c_1$.

\begin{remark}
  \label{rem:ex.ell.met}
  Note that the atlas can always be chosen in such a way that a periodic
  metric $g$ on $\Mper$ is uniformly elliptic and $C^1$-bounded. If necessary
  we slightly need to make the charts smaller such that $g_\alpha$ has
  derivatives continuous up to the boundary of $V_\alpha$. In the same way,
  any metric $h$ on $\GpM$ is uniformly elliptic if $\Gamma'$ is finite.
\end{remark}
\begin{remark}
  It seems to be unsatisfactory that our definitions depend on the atlas $A'$.
  We also could define the uniform ellipticity in a coordinate free manner by
  comparing a metric $h$ with the periodic metric $g$ globally.  Likewise, we
  could define the $C^1$-boundedness in a global way by defining how to
  derivative a metric. But since we have chosen a periodic atlas one can easily
  see that the local and global definitions are equivalent. For a similar
  concept of $C^1$-convergence of manifolds
  see~e.g.~\cite{hebey:96}.
\end{remark}


For $k \in \N$ and an open set $V' \subset \R^d_+$ let $\Sob[k] {V'}$ be the
usual Sobolev space of order $k$ defined as the closure of $\Ci {V'}$ under
the norm
\begin{displaymath}
  \normsqr[k, V'] u :=
  \sum_{|\kappa| \le k}\normsqr[V']{\partial_\kappa u}.
\end{displaymath}
We define the Sobolev space $\Sob[k] \GpM$ on $\GpM$ as the closure of the
space of all functions $u \in \Ci \GpM$ such that
\begin{equation}
  \label{eq:sob.norm}
  \normsqr[k, \GpM] u := 
  \sum_{\alpha \in A'} \normsqr[k, V'_\alpha] {u_\alpha}
\end{equation}
is finite.  Here, $u_\alpha := (\chi_\alpha u) \circ \varphi_\alpha^{-1}$
denotes the local representation of $\chi_\alpha u$ in the chart
$\varphi_\alpha$.  We denote by $\Sobn[k] \GpM$ the completion of $\Cci \GpM$,
i.e., the space of smooth functions with compact support \emph{away} from $\bd
\GpM$, under the norm defined in \eqref{eq:sob.norm}.

Since the periodic metric $g$ is uniformly elliptic, we conclude
\begin{equation}
  \label{eq:norm.equiv}
  c^{-d/4} \norm[\GpM] u \le
  \norm[0, \GpM] u =
  \sum_\alpha \norm[V_\alpha] {u_\alpha} \le
  c^{+d/4}|B|(2r+1) \, \norm[\GpM] u,
\end{equation}
i.e., $\norm[\GpM] \cdot$ and $\norm[0, \GpM] \cdot$ are equivalent
norms.  Remember that $r$ denotes the number of generators of $\Gamma$. Note
that a chart $U_\alpha$ could intersect with at most $|B|(2r+1)$ other charts.
In the same way we can show that $(\normsqr u + \normsqr{\dd u})^\frac12$ and
$\norm[1, \GpM] u$ are equivalent norms on $\Sob \GpM$. In particular, we
have $\Sobn \GpM = \dom \qD \GpM$ and $\Sob {\GpM} = \dom \qN {\GpM}$.


%------------------------------------------------------------
\subsection*{Elliptic estimates}

The main result of this subsection is the following theorem:
\begin{theorem}
  \label{thm:reg.theory}
  For every $c > 0$, $c_1 > 0$ and $d > 0$ there exists a constant $c_2$ such
  that
  \begin{equation}
    \label{eq:reg.theory}
    \norm[2, U] u \le c_2 \, (\norm[\GpM] u + \norm[\GpM]{\Delta'u})
  \end{equation}
  for all $u \in \Sob[2] U$, all metrics $g' \in \mathcal G(c,c_1,\Gamma')$,
  all open sets $U \subset \GpM$ such that $\dist(U,\bd \GpM) \ge d$ and all
  $\Gamma' \subset \Gamma$. Here, $\Delta'$ denotes the Laplacian on $\GpM$
  with respect to the metric $g'$.
 
  If $\bd M$ is smooth Estimate~\eqref{eq:reg.theory} is valid for $U=\GpM$,
  all $u \in \Sob[2] \GpM \cap \Sobn \GpM$, all metrics $g' \in \mathcal
  G(c,c_1,\Gamma')$ and all $\Gamma' \subset \Gamma$.
\end{theorem}
\begin{proof}
  The proof of the statement in the chart $V'_\alpha$ is standard (see e.g.\ 
  \cite{gilbarg-trudinger:77}). The step from the local to the global estimate
  is possible since the constant in the local estimate has a global bound.
  Here, we need Estimate~\eqref{eq:norm.equiv} and therefore the special
  structure of the atlas and the partition of unity $(\chi_\alpha)$.
  Furthermore, we have to estimate the first order operators
  $[\Delta',\chi_\alpha]$ in terms of the right hand side
  of~\eqref{eq:reg.theory}. This can be done applying the Gau{\ss}-Green
  formula. Note that $u$ vanishes on the boundary.
\end{proof}

The proof of the next lemma is straightforward. Again, we need
\eqref{eq:norm.equiv} for the step from the local to the global estimate.

\begin{lemma}
  \label{lem:lapl.est.sob}
  There exists a constant $c_3>0$ such that
  \begin{align*} 
    \norm[\Pert] {\laplacian \Appr u} & \le c_3 \, \norm[2, \Pert] u
    \intertext{for all $u \in \Sob[2] \Appr$ and all $m, n$ with $n > m$.
      Suppose further that $\pert g \tau$, $\tau \ge 0$, are metrics on
      $\Mper$ such that $\norm[C^1] {\pert g 0 - \pert g \tau} \to 0$ on
      $\RestAppr$ as $m \to \infty$ ($n > m$) uniformly in $\tau \ge 0$.  Then
      there exists a sequence $\delta'_m \to 0$ such that}
    \norm[\RestAppr]{(\laplacian {\pert \Appr 0} - \laplacian \Apprtau)u} &\le
    \delta'_m \, \norm[2, \RestAppr] u
  \end{align*}
  for all $u \in \Sob[2] \RestAppr$, all $\tau \ge 0$ and all $m,n$ with $n >
  m$.
\end{lemma}

Let $m_+$ be the smallest integer such that $\Pert \subset \overcirc
M{}^{m_+}$. Now we prove the estimates needed in the next section:
\begin{lemma}
  \label{lem:lapl.est}
  Let the family $(\pert g \tau)$ satisfy Conditions~\eqref{eq:unperturbed}
  and~\eqref{eq:met.cont}. Suppose further that $\tau_n \to \tau_0 \ge 0$.
  Then there exists a constant $c_4 > 0$ such that
  \begin{align}
    \label{eq:lapl.est}
    \norm[\Pert]{\laplacian \Appr u} & \le c_4 \, (\norm[\Pertplus] u +
    \norm[\Pertplus]{\laplacian {\Apprtau[_n]} u})
    \intertext{for all $u \in
      \Sob[2] \Appr$ and all $m, n$ with $n > m_+$. Furthermore if in addition
      $\bd M$ is smooth and if Condition~\eqref{eq:met.unif.rest} is satisfied
      then there exists a sequence $\delta_m \to 0$ such that}
    \label{eq:lapl.diff}
    \norm[\RestAppr]{(\laplacian \Appr - \laplacian {\Apprtau[_n]})u} & \le
    \delta_m\, (\norm[\RestAppr] u + \norm[\RestAppr]{\laplacian
      {\Apprtau[_n]} u}) 
  \end{align}
  for all $u \in  \Sob[2] \RestAppr \cap \Sobn \RestAppr$ and all
  $m,n$ with $n > m$ and $m$ large enough.
\end{lemma}

\begin{proof}
  For the proof of \eqref{eq:lapl.est} we combine Lemma~\ref{lem:lapl.est.sob}
  with Theorem~\ref{thm:reg.theory}. Here we need a cut-off function $\chi \in
  \Cci \Pertplus$ such that $\chi \restriction \Pert=1$. Therefore
  $\dist(\supp \chi, \bd \Pert) = d >0$, i.e., we do not need the assumption
  that $\bd M$ is smooth in this case. By Remark~\ref{rem:ex.ell.met}, there
  exist $c,c_1 > 0$ such that $\pert g {\tau_0} \in \mathcal
  G(c,c_1,\Gamma^{m_+})$. Furthermore, Condition~\eqref{eq:met.cont} assures
  that there exist constants $c', c_1' > 0$ such that $\pert g {\tau_n} \in
  \mathcal G(c',c_1',\Gamma^{m_+})$ for all $n \in \N$. Note that we still
  have to estimate the first order operator $[\chi, \laplacian
  {\Apprtau[_n]}]$ in term of the right hand side of~\eqref{eq:lapl.est} as in
  the proof of Theorem~\ref{thm:reg.theory}.  Here we also need the fact that
  $\pert g {\tau_n} \in \mathcal G(c',c_1',\Gamma^{m_+})$.
  
  The proof of \eqref{eq:lapl.diff} is similar. Note that $\pert g {\tau_n}$,
  $n \in \N$, are close to the periodic metric $g=g_0$ by
  Condition~\eqref{eq:met.unif.rest} on $\RestAppr$ provided $m$ is large
  enough. Since the periodic metric $g$ is uniformly elliptic, there exist
  constants $c, c_1 > 0$ such that $\pert g {\tau_n} \in \mathcal
  G(c,c_1,\Gamma^n \setminus \Gamma^m)$ for all $n > m$.
\end{proof}

%------------------------------------------------------------
\subsection*{Perturbation of the metric}
Here, we state some results on how to deal with the different Hilbert
spaces $\Lsqr \Mpertau$ resp.\ $\Sob \Mpertau$ depending on $\tau$. In
particular, we show that the norm and weak topology on $\Lsqr
\Mpertau$ resp.\ $\Sob \Mpertau$ are equivalent for all $\tau \ge
0$; the same is true on the submanifolds $\GpM$
(cf.~Corollary~\ref{cor:norm.quad.est}). Furthermore, we need the
continuity of the eigenvalues with respect to $\tau$,
see~Corollary~\ref{cor:ew.cont} and Lemma~\ref{lem:ew.rest}.

\begin{lemma}
  \label{lem:norm.quad.approx}
  Suppose that the family $(g_\tau)$ satisfies Condition~\eqref{eq:met.cont}.
  Suppose furthermore that $\tau_n \to \tau \ge 0$. Then there exists a
  sequence $\eta_n \to 0$ such that
  \begin{align}
    \label{eq:norm.approx}
    \bigl| \normsqr[\Mpertau] u - \normsqr[{\Mpertau[_n]}] u \bigr| & \le
    \eta_n \, \normsqr[\Mpertau] u\\
    \label{eq:quad.approx}
    \bigl| \normsqr[\Mpertau]{\dd u} - \normsqr[{\Mpertau[_n]}] {\dd u} 
    \bigr| & \le
    \eta_n \, \normsqr[\Mpertau]{\dd u}
  \end{align}
  for all $u \in \Lsqr \Mper$ resp.\ $u \in \Sob \Mper$.
\end{lemma}
\begin{proof}
  We only give the idea of how to prove~\eqref{eq:quad.approx}. The other
  inequality can be proven similarly. We have
  \begin{multline*}
    \int_{\Mpertau} |\dd u|^2 - \int_{\Mpertau[_n]} |\dd u|^2 = \\
    = \sum_\alpha \int_{V_\alpha} \chi_\alpha 
      \Bigl( 
        {\Bigl( E -
            \Bigl( \frac {\det G_{\alpha,n}} 
                         {\det G_\alpha} 
            \Bigr)^\frac12
              G_\alpha^{\frac12}  G_{\alpha,n}^{-1} 
              G_\alpha^{\frac12}
         \Bigr) G_\alpha^{-\frac12} \nabla u} 
        \, \cdot \, {G_\alpha^{-\frac12} \nabla u} 
      \Bigr) (\det G_\alpha)^\frac12.
  \end{multline*}
  Here, $G_\alpha$ resp.\ $G_{\alpha,n}$ denotes the matrix corresponding to
  $\pert g \tau$ resp.\ $\pert g {\tau_n}$ in the chart indexed by $\alpha$.
  Furthermore, $E$ denotes the unit $(d \times d)$-matrix. Clearly,
  Estimate~\eqref{eq:quad.approx} follows from Condition~\eqref{eq:met.cont}.
\end{proof}

The next corollary follows easily. 
\begin{corollary}
  \label{cor:norm.quad.est}
  For all $\tau_0 \ge 0$ there exists a constants $c_5=c_5(\tau_0) \ge 1$ such
  that
  \begin{gather}
    \label{eq:norm.est}
    c_5^{-1} \, \normsqr[\pert \Mper 0] u  \le
                \normsqr[\Mpertau] u  \le
    c_5 \,      \normsqr[\pert \Mper 0] u \\
    \label{eq:quad.est}
    c_5^{-1} \, \normsqr[\pert \Mper 0]   {\dd u}  \le
                \normsqr[\Mpertau]{\dd u}  \le
    c_5      \, \normsqr[\pert \Mper 0]   {\dd u}
  \end{gather}
  for all $0 \le \tau \le \tau_0$.
\end{corollary}

{}From Lemma~\ref{lem:norm.quad.approx} and the Min-max principle we conclude
the following corollary.
\begin{corollary}
  \label{cor:ew.cont} The eigenvalue branch $\tau \mapsto \EWD k
  \Apprtau$ is continuous for $\tau \ge 0$. In particular, for every
  $\tau_0 \ge 0$ there exist a modul of continuity for the eigenvalue
  branch on $[0,\tau_0]$, i.e.\ a non-negative monotonical increasing
  function $\eta$ with $\eta(\delta) \to 0$ as $\delta \to 0$ such
  that \begin{displaymath} |\EWD k \Apprtau - \EWD k {\Apprtau[']}|
  \le \eta(|\tau - \tau'|) \end{displaymath} for all $\tau,\tau' \in
  [0,\tau_0]$.
\end{corollary}

In the same way we can show that the boundary conditions on $\RestApprtau$
have nearly no influence:
\begin{lemma}
  \label{lem:ew.rest}
  Suppose that the family $(g_\tau)$ of metrics satisfies
  Condition~\eqref{eq:met.unif.rest}. Then there exists a sequence $\eta_m \to
  0$ as $m \to \infty$ such that
  \begin{displaymath}
    |\EWD k \RestApprtau - \EWD k \RestAppr| \le \eta_m
  \end{displaymath}
  for all $\tau \ge 0$ and $n > m$. The same estimate is true for Neumann or
  mixed boundary conditions on $\RestAppr$.
\end{lemma}


%------------------------------------------------------------
\section{Eigenvalues in Gaps}
\label{sec:eigenvalues}

In this section we prove our main result Theorem~\ref{thm:eigenvalues}. We
approximate eigenfunctions of the perturbed Laplacian on the full manifold
$\Mper$ by eigenfunctions of the perturbed Dirichlet-Laplacian on the
approximating manifold $\Appr$. This idea has already been used in
\cite{hempel-deift:86}, \cite{alama-deift-hempel:92} or \cite{aadh:94}. Our
situation is in some sense simpler: the approximating and the limit problem
have the same spectral gap.  Otherwise our situation is more difficult since
not only the Laplacian but also the Hilbert space depend on the perturbation
parameter. Some of the technical details are already proven in
Section~\ref{sec:ell.est}.

In Theorem~\ref{thm:ef.conv} we show the convergence of the
approximating eigenfunctions. Furthermore, in Theorem~\ref{thm:count.approx}
we give a lower bound on the eigenvalue counting function defined
in~\eqref{eq:def.count.fct} following arguments of~\cite{hempel-besch:00}.

Firstly, we use the decomposition principle
(see~\cite{donnelly:79}) to prove that the essential spectrum remains
invariant under the perturbation.  Therefore, $\laplacian \Mper$ and
$\laplacian {\Mpertau}$ have the same spectral gap. In a spectral
gap of the unperturbed Laplacian, the perturbed Laplacian can only
have discrete eigenvalues (possibly accumulating at the band edges).
It is essential here that the perturbation is (almost) localized on a
compact set.
\begin{theorem}
  \label{thm:dec.princ}
  We have $\essspec \laplacian \Mper = \essspec \laplacian \Mpertau$
  for all $\tau \ge 0$.
\end{theorem}

\begin{proof}
  We only show the inclusion ``$\subset$'' since the opposite inclusion can be
  proven in the same way.
  
  Let $(u_n)$ be a singular sequence for $\lambda \in \essspec \laplacian
  \Mper$, i.e., $u_n \to 0$ weakly in $\Lsqr \Mper$, $\norm[\Mper]{u_n} = 1$
  and $\norm[\Mper]{(\laplacian \Mper - \lambda)u_n} \to 0$ as $n \to \infty$.
  By the decomposition principle (cf.~\cite{donnelly:79} or \cite{post:00}) we
  can assume that $u_n$ has support away from $\Pert$. Furthermore, by
  Corollary~\ref{cor:norm.quad.est} there exists $c_5>0$ such that
  $\norm[\Mpertau]{u_n} \ge c_5^{-1}$ for all $n$.
  
  Now we want to show that $v_n := u_n / \norm[\Mpertau] {u_n}$ is a singular
  sequence for $\lambda$ and $\laplacian \Mpertau$. Clearly, $v_n \to 0$
  weakly in $\Lsqr \Mpertau$. Since $u_n \restriction \Pert = 0$ we have
  \begin{align*}
    \norm[\Mpertau] {(\laplacian \Mpertau - \lambda) v_n} 
              & \le
    c_5 \, \bigl( 
      \norm[\Rest] {(\laplacian \Mpertau - \laplacian \Mper) u_n} +  
      \norm[\Rest] {(\laplacian \Mper    - \lambda) u_n}  
    \bigr) \\ & \le
    \delta_m \, c_5 \, 
      (\norm[\Rest] {u_n} + \norm[\Rest] {\laplacian \Mper u_n}) + 
      c_5 \norm[\Rest] {(\laplacian \Mper    - \lambda) u_n}  
  \end{align*}
  by an estimate similar to~\eqref{eq:lapl.diff}. Here, $\delta_m \to 0$ as $m
  \to \infty$.  Note that we do not need the smoothness of $\bd M$ here even
  if the perturbation lives on the whole manifold since $\supp u_n$ is
  compactly contained in $\Rest$. Therefore the first term converges to $0$.
  The last term converges to $0$ since $(u_n)$ is a singular sequence for
  $\laplacian \Mper$.
\end{proof}


Next, we conclude from the gap condition~\eqref{eq:gap} and the
Dirichlet-Neumann enclosure~\eqref{eq:dir.neu.encl} that no eigenvalue of the
approximating problem lies in the gap. The boundary of $M$ resp.\ $\GpM$ is so
small such that boundary conditions almost have no influence on the
eigenvalues.
\begin{lemma}
  \label{lem:common.gap}
  Suppose that $\Gamma'$ has $n$ elements.  If $\EWD k M < \EWN {k+1} M$ then
  the interval $I_k=(\EWD k M, \EWN {k+1} M)$ is a common spectral gap, i.e.,
  \begin{equation}
    \label{eq:common.gap}
    I_k \cap \spec \laplacian \Mper = \emptyset \quad \text{and} \quad
    I_k \cap \spec \laplacian \GpM   = \emptyset.
  \end{equation}
  Here, $\laplacian \GpM$ denotes the Laplacian on $\GpM$ with Neumann,
  Dirichlet or mixed boundary condition on $\bd \GpM$. Furthermore, if
  $\lambda \in I_k$ then $\dimEW \GpM = k n$ (also for any boundary
  condition).
\end{lemma}

\begin{proof}
  Denote by $\Sigma^n \! M$ the disjoint union of $n$ copies of $M$. Then
  $\EWD j {\Sigma^n \! M} = \EWD k M$ and $\EWN j {\Sigma^n \!  M} = \EWN k M$
  for all $j=(k-1)n+1, \dots, k n$ since every eigenvalue on the disjoint
  union has multiplicity $p n$ if the corresponding eigenvalue on $M$ has
  multiplicity $p$.  By the Dirichlet-Neumann
  bracketing~\eqref{eq:dir.neu.brack} we obtain
  \begin{displaymath}
    \EWN k M =
    \EWN j {\Sigma^n \! M} \le
    \EW  j \GpM \le
    \EWD j {\Sigma^n \! M} =
    \EWD k M.
  \end{displaymath}
  Note that this estimate is true for any boundary condition on $\GpM$.
  Together with the Dirichlet-Neumann enclosure~\eqref{eq:dir.neu.encl} we see
  that $\EWN k M$ resp.\ $\EWD k M$ are lower resp.\ upper bounds for both $\EW
  j \GpM$ and $\EWT k M$. If $\lambda \in I_k$ then $\laplacian \GpM$ has
  exactly $k n$ eigenvalues below $\lambda$.
\end{proof}


Now we prove that approximating eigenfunctions converge to eigenfunctions of
the full problem and that multiplicity is conserved as $n \to \infty$, i.e.,
as the number of copies of period cells increases. Here and later on we
suppress the Dirichlet label at the approximating problem, e.g.\ we write
$\laplacian \Appr$ instead of $\laplacianD \Appr$ until stated otherwise.

The idea of the next theorem is from~\cite{hempel-deift:86},
\cite{alama-deift-hempel:92} or \cite{aadh:94}:
\begin{theorem}
  \label{thm:ef.conv}
  Suppose that the gap condition~\eqref{eq:gap} is satisfied, i.e., $I_k$ is a
  spectral gap of the periodic Laplacian $\laplacian \Mper$ for some $k \in
  \N$. Suppose further that the family $(\pert g \tau)$ satisfies
  Conditions~\eqref{eq:unperturbed},~\eqref{eq:met.cont}
  and~\eqref{eq:pert.compact} or, if $\bd M$ is
  smooth,~\eqref{eq:met.unif.rest}. Let $\tau_n \to \tau$, $\lambda_{n,i} \to
  \lambda \in I_k$ for all $i=1, \dotsc, j$.  Furthermore, suppose that
  $\varphi_{n,1}, \dots, \varphi_{n,j} \in \Lsqr {\Apprtau[_n]}$ are
  orthonormal for all $n \in \N$ and that
  \begin{equation}
    \label{eq:ew.approx}
    \laplacian {\Apprtau[_n]} \varphi_{n,i} = \lambda_{n,i} \, \varphi_{n,i}. 
  \end{equation}
  Then for every $i$ there exists a subsequence of $(\varphi_{n,i})_n$ also
  denoted by $(\varphi_{n,i})_n$ converging weakly to $\varphi_{0,i} \in \dom
  \laplacian \Mpertau$ in $\Sob \Mpertau$ and strongly in $\Lsqrloc \Mpertau$.
  Furthermore,
  \begin{equation}
    \label{eq:ew.limit}
    \laplacian \Mpertau \varphi_{0,i} = \lambda \, \varphi_{0,i} 
  \end{equation}
  and $\varphi_{0,1}, \dots, \chi_{0,j}$ are linearly independent.
\end{theorem}

\begin{proof}
  Let $i \in \{1, \dots j\}$. From Corollary~\ref{cor:norm.quad.est} it
  follows that $(\varphi_{n,i})_n$ is bounded in $\Sob \Mper$.  Therefore we
  can extract a subsequence also denoted by $(\varphi_{n,i})_n$ converging
  weakly in $\Sob \Mper$ to an element $\varphi_{0,i}$.  This subsequence also
  converges strongly in $\Lsqrloc \Mper$ by the Rellich-Kondrachov compactness
  Theorem.  In virtue of Lemma~\ref{lem:norm.quad.approx} and
  Corollary~\ref{cor:norm.quad.est} it is a straightforward calculation to
  show that $\varphi_{0,i}$ is in the domain of $\laplacian \Mpertau$ and that
  the eigenvalue equation \eqref{eq:ew.limit} is satisfied.
  
  The main difficulty is to prove that $\varphi_{0,i}$ are linearly
  independent. Suppose that there exist $\alpha_1, \dots, \alpha_k$, not all
  equal to $0$,
  such that $u_n := \sum_i \alpha_i \, \varphi_{n,i}$ converges to $\sum_i
  \alpha_i \, \varphi_{0,i} =0$ in $\Lsqrloc \Mpertau$. By
  Lemma~\ref{lem:common.gap} the interval $I_k$ is a spectral gap for all
  operators $\laplacian \Appr$, $n \in \N$. In virtue of the spectral theorem
  and Estimate~\eqref{eq:norm.est} we have
  \begin{equation}
    \label{eq:not.zero}
          \normsqr[\Appr]{(\laplacian \Appr - \lambda) u_n} \ge
    c_6^2 \normsqr[\Appr]{u_n} \ge 
    \frac{c_6^2}{c_5} \sum_i |\alpha_i|^2 > 0
  \end{equation}
  for all $n \in \N$ where $c_6 := \dist(\lambda, \R \setminus I_k)$. On the
  other side we have
  \begin{displaymath}
    \norm[\Appr] {(\laplacian \Appr - \lambda) u_n} \le
    \norm[\Appr] 
        {\sum_i (\laplacian \Appr - \lambda_{n,i}) \alpha_i \, \varphi_{n,i}} +
         \sum_i |\alpha_i| \cdot |\lambda_{n,i}-\lambda| \cdot 
         \norm[\Appr] {\varphi_{n,i}}.
  \end{displaymath}
  The last sum converges to $0$ since $\norm[\Appr] {\varphi_{n,i}} \le c_5$.
  The first norm can be split into an integral over $\Pert$ and $\RestAppr$.
  
  The integral over $\Pert$ can be estimated by
  \begin{displaymath}
    \norm[\Pert] {\laplacian \Appr u_n} + 
    \sum_i \lambda_{n,i} \norm[\Pert] {u_n}. 
  \end{displaymath}
  Furthermore, Inequality~\eqref{eq:lapl.est} yields the estimate
  \begin{multline*}
    \norm[\Pert] {\laplacian \Appr u_n} \le 
    c_4 \, c_5 \, \bigl( \, \norm[\Pertplustau] {u_n} + 
                \norm[\Pertplustau] {\laplacian {\Apprtau[_n]} u_n}
      \bigr) \\ \le
    c_4 \, c_5  \, 
    \bigl( (1 + \sum_i \lambda_{n,i}) 
    \bigr)  \norm[\Pertplustau] {u_n}
  \end{multline*}
  for all $n > m_+$. In the last estimate we have used
  Equation~\eqref{eq:ew.approx}. Note that the last term converges to $0$
  since $u_n \to 0$ in $\Lsqrloc \Mper$. 
  
  If the perturbation is contained in $\Pert$ we have a contradiction
  to~\eqref{eq:not.zero}. Otherwise we still have to estimate the integral
  over $\RestAppr$. Here, we need the smoothness assumption on $\bd M$ to be
  able to use~\eqref{eq:lapl.diff}. Together with~\eqref{eq:ew.approx} we
  obtain
  \begin{multline*}
    \norm[\RestAppr]{(\laplacian \Appr - \laplacian{\Apprtau[_n]})u_n} \le
    \delta_m \, 
    \bigr( \norm[\Appr] {u_n} +
           \norm[\Appr] {\laplacian{\Apprtau[_n]} u_n}
    \bigl)
    \\ \le
    \delta_m \, c_5 \,
    \bigr( \sum_i |\alpha_i|^2 \bigl)^\frac12 (1+\max_i \lambda_{n,i})
  \end{multline*}
  tending to $0$ as $m \to \infty$. Again, we have a contradiction
  to~\eqref{eq:not.zero}.
\end{proof}

The next result allows us to estimate the eigenvalue counting functions for
different parameters $\tau$ and fixed $\lambda$ by the eigenvalue counting
function for a fixed $\tau$ and an interval containing $\lambda$ (see
\cite{hempel-besch:00}). We need this lemma since we do not know whether
eigenfunctions corresponding to different parameters $\tau$ are orthogonal. We
do not even know that they are different! But since we want to show that the
multiplicity of eigenvalues is conserved as $n \to \infty$ we need the linear
independence of the approximating eigenfunctions (see the proof of
Theorem~\ref{thm:count.approx}).
\begin{lemma}
  \label{lem:eigen.vert}
  For all $\tau_0 \ge 0$ there exists a monotonical increasing function
  $\eta(\delta) \to 0$ as $\delta \to 0$ such that
  \begin{displaymath}
    |\dimEWD {\Apprtau[+\delta]}  - 
     \dimEWD {\Apprtau[-\delta]}| \le
     \dimEWD[{[\lambda - \eta(\delta),\lambda + \eta(\delta)]}]
             \Apprtau
  \end{displaymath}
  for all $\lambda \ge 0$, $\delta >0$ and all $\tau + \delta \le \tau_0$.
\end{lemma}
\begin{proof}
  Since every eigenvalue branch $\EWD j {\pert \Appr {(\cdot)}}$ is continuous
  by Corollary~\ref{cor:ew.cont}, there exists a parameter $\tau' \in
  [\tau-\delta,\tau+\delta]$ such that $\lambda=\EWD j {\Apprtau[']}$ by the
  intermediate value theorem. Corollary~\ref{cor:ew.cont} yields
  \begin{displaymath}
    \bigl| \lambda               - \EWD j \Apprtau \bigr| = 
    \bigl| \EWD j {\Apprtau[']}  - \EWD j \Apprtau \bigr| \le
    \eta(|\tau-\tau'|) \le
    \eta(\delta),
  \end{displaymath}
    i.e., $\laplacian \Apprtau$ has an eigenvalue in
  $[\lambda - \eta(\delta),\lambda + \eta(\delta)]$.
\end{proof}

Now we give a lower bound on the eigenvalue counting function
following~\cite{hempel-besch:00}:
\begin{theorem}
  \label{thm:count.approx}
  If $\lambda \in I_k$ lies in a spectral gap then
  \begin{equation}
    \label{eq:count.approx}
    \mathcal N(\tau,\lambda) \ge
    \limsup_{n \to \infty} 
       \bigl|\dimEWD \Apprtau - \dimEWD \Appr \bigr|.
  \end{equation}
  for all $\tau \ge 0$.
\end{theorem}

\begin{proof}
  Denote by
  \begin{displaymath}
    T_n := T_n(\lambda) := \set{\tau' \in [0,\tau_0]} 
                           {\lambda \in \spec \laplacian \Apprtau}, 
  \end{displaymath}
  the set of parameters $\tau'$ that produce an eigenvalue $\lambda$. Let
  $T_\infty$ be the set of limit points, i.e., $\hat \tau \in T_\infty$ if and
  only if $\hat \tau \in [0,\tau_0]$ and if there exist sequences $(n_m)_m
  \subset \N$ and $\tau'_m \in T_{n_m}$ such that $\tau'_m \to \hat \tau$.
  
  We have to distinguish two cases. If the cardinality of $T_\infty$ is
  greater or equal to $N_\lambda$, the right hand side of
  \eqref{eq:count.approx}, then we apply Theorem~\ref{thm:ef.conv} with
  fixed eigenvalue $\lambda=\lambda_{n,1}$ and with multiplicity $j=1$
  for each limit point $\hat \tau \in T_\infty$.  As a consequence, there are
  at least $\card T_\infty$ parameters $\hat \tau$ such that $\lambda$ is an
  eigenvalue of $\Delta_{M(\hat\tau)}$. This proves~\eqref{eq:count.approx}.
  
  If $\card T_\infty < N_\lambda$ then $T_\infty$ consists of a finite number
  of points $\hat \tau_1,\dots,\hat \tau_q$, and $T_n \to \{\hat
  \tau_1,\dots,\hat \tau_q\}$.  Furthermore, there exists a sequence $\delta_n
  \to 0$ such that
  \begin{displaymath}
    T_n \subset 
     \bigcup_{p=1}^q \, (\hat \tau_p-\delta_n,\hat \tau_p+\delta_n) \,
       =: \hat T_n
  \end{displaymath}
  for all $n \in \N$. If $n$ is large enough, all these intervals are mutually
  disjoint. As a consequence, $d_{\lambda,n}(\hat \tau):= \dimEWD {\pert \Appr
    {\hat \tau}}$ is constant on each component of $[0,\tau] \setminus \hat
  T_n$. Therefore
  \begin{displaymath}
    \bigl| d_{\lambda,n}(\tau) - d_{\lambda,n}(0) \bigr| \le
    \sum_{p=1}^q 
      \bigl| d_{\lambda,n}(\hat \tau_p-\delta_n) - 
       d_{\lambda,n}(\hat \tau_p+\delta_n) \bigr|. 
  \end{displaymath}
  By Lemma~\ref{lem:eigen.vert} there exist $\eta(\delta) \to 0$ as
  $\delta \to 0$ such that
  \begin{displaymath}
    \bigl| d_{\lambda,n}(\tau) - d_{\lambda,n}(0) \bigr| \le
    \sum_{p=1}^q j_{p,n} 
    \qquad \text{where} \qquad
    j_{p,n} := 
    \dimEWD[{[\lambda - \eta(\delta_n),\lambda + \eta(\delta_n)]}]
          {\pert \Appr {\hat \tau_p}}.
  \end{displaymath}
  By passing to a subsequence we can assume that $|d_{\lambda,n}(\tau) -
  d_{\lambda,n}(0) | \to N_\lambda$. We have even equality if $n$ is large
  enough since the sequence consists of integers. Now we select another
  subsequence $(n_m)_m$ such that $j_p := j_{p,n_m} \in \N$ is independent of
  $m$. This is possible by choosing a convergent subsequence.  Therefore, for
  each $m \in \N$, we have $j_p$ orthonormal eigenfunctions
  $\varphi_{m,1},\dots,\varphi_{m,j_p}$ of the Laplacian on $\pert \Apprm {\hat
    \tau_p}$ with eigenvalues $\lambda_{m,i} \to \lambda$ as $m \to \infty$.
  Hence we can apply Theorem~\ref{thm:ef.conv} with fixed parameter $\hat
  \tau_p$ and conclude that the limit problem $\laplacian \Mpertau$ has
  $\lambda$ as eigenvalue of multiplicity $j_p$. This proves the theorem since
  $\lambda$ hat multiplicity at least $\sum_{p=1}^q j_p \ge N_\lambda$.
\end{proof}

Finally we want a lower bound on the eigenvalue counting function in
terms of $\Pert$ and \emph{not} in terms of $\Appr$
\begin{lemma}
  \label{lem:approx.probl}
  For $m$ sufficiently large we have
  \begin{gather}
    \label{eq:eigen.count1}
    \dimEWD \Apprtau -
    \dimEWD \Appr \ge 
    \dimEWD \Perttau -
    \dimEWD \Pert\\
    \label{eq:eigen.count2}
    \dimEWD \Appr -  
    \dimEWD \Apprtau \ge
    \dimEWN \Pert -
    \dimEWN \Perttau
   \end{gather}
  for all $\tau \ge 0$ and $n > m$.
\end{lemma}

\begin{proof}
  From Lemma~\ref{lem:common.gap} we know that $\dimEW \Appr= k
  n$ and $\dimEW \RestAppr= k (n-m)$ if $\lambda \in I_k$
  independently of the boundary conditions. Furthermore, from
  Lemma~\ref{lem:ew.rest} we obtain 
  \begin{displaymath}
    \dimEWD \RestApprtau = \dimEWD \RestAppr, \qquad
    n > m, \tau \ge 0
  \end{displaymath}
  if $m$ is large enough.  Furthermore, the Dirichlet-Neumann
  bracketing~\eqref{eq:dir.neu.brack} yields
  \begin{multline*}
    \dimEWD \Apprtau - \dimEWD \Appr \\ \ge
    \dimEWD \Perttau + \dimEWD \RestApprtau -
    \bigl(
      \dimEWD \Pert  + \dimEWND \RestAppr
    \bigr) \\ =
    \dimEWD \Perttau - k m.
  \end{multline*}
  where $\dimEWND \RestAppr$ denotes the eigenvalue counting function of the
  Laplacian with Dirichlet boundary condition on $\bd \Appr$ and Neumann
  boundary condition on $\bd \RestAppr \setminus \bd \Appr$.
  Estimate~\eqref{eq:eigen.count2} can be shown similarly.
\end{proof}

To summarize: Theorem~\ref{thm:dec.princ} ensures that a given spectral gap in
the essential spectrum of the periodic manifold remains invariant under local
perturbations. Therefore, $\mathcal N(\tau,\lambda)$ indeed counts the
discrete eigenvalues in the gap.  Theorem~\ref{thm:ef.conv} shows that
eigenfunctions of the approximating problem converge to eigenfunctions of the
full (perturbed) problem and that multiplicity is conserved.
Theorem~\ref{thm:count.approx} follows and with the aid of
Lemma~\ref{lem:approx.probl} we get rid of the approximating index $n$ in
Estimate~\eqref{eq:count.approx}. Thus we have proven our main result
Theorem~\ref{thm:eigenvalues}.


%------------------------------------------------------------
\section{Manifolds with spectral gaps}
\label{sec:mfd.gaps}
For the convenience of the reader, we cite the results on examples of
manifolds with spectral gaps given in \cite{post:01b} and \cite{yoshitomi:98}.
More details and further references can be found therein.


\subsection*{Conformal periodic manifolds}
Let $\Mper$ be a $\Gamma$-periodic manifold of dimension $d$ with periodic
metric $g$.  Furthermore, suppose that $(\rho_\eps)$, $\eps > 0$, is a family
of smooth periodic functions $\map{\rho_\eps} \Mper {(0,\infty)}$, i.e.,
$\rho_\eps(\gamma x)=\rho_\eps(x)$ for all $\gamma \in \Gamma$ and $x \in
\Mper$. We denote by $\Mepsper$ the manifold $\Mper$ with metric $g_\eps :=
\rho_\eps^2 g$.  Similarly we define $\Meps$ for a period cell $M$. We have
the following theorem:
\begin{theorem}
  \label{thm:conf.gaps}
  Suppose $d \ge 3$ and that $\rho_\eps$ converges pointwise on $M$ to the
  indicator function of a closed set $X \subset \overcirc M$ with smooth
  boundary. Then for each $k \in \N$, the $k$-th eigenvalue $\EW k \Meps$ with
  Dirichlet, Neumann or $\theta$-periodic boundary condition converge to $\EWN
  k X$ as $\eps \to 0$ (uniformly in $\theta$). In particular, for all $n \in
  \N$ there exists $\eps>0$ such that $M=\Meps$ satisfies the gap
  condition~\eqref{eq:gap} for $k = 1,\dots,n$ (provided $\EWN k X$ is a
  simple eigenvalue).
\end{theorem}
The precise assumptions on $\rho_\eps$ and $X$ resp.\ $M$ and the proof of
this theorem in the $\theta$-periodic case can be found in \cite{post:00}
resp.\ \cite[Theorem~1.3]{post:01b}. There we also have presented an example
for $d=2$. The proofs for the Dirichlet resp.\ Neumann case are similar.

\subsection*{Attaching small cylindrical ends}
Let $X$ be a compact Riemannian manifold of dimension $d \ge 2$ with metric
$g$. Let $x_1,x_2 \in X$ be two distinct points. On $X \setminus \{x_1,x_2\}$
we change the metric such that a cylinder of radius $\eps$ and length $\eps/2$
is isometrically embedded at each point $x_i$. Here, a cylinder is a product
of the $(d-1)$-dimensional unit sphere $\Sphere^{d-1}$ and an interval. We
denote the (completion of the) resulting manifold by $\Meps$. Note that $\bd
\Meps$ has two additional components $\bd_1 \Meps$ and $\bd_2 \Meps$ each of
them being isometric to $\Sphere^{d-1}$. A typical example of a period cell in
the case when $X$ is a $2$-dimensional torus is drawn in
Figure~\ref{fig:per-cell} on page~\pageref{fig:per-cell}.

For $\gamma \in \Z$ let $\gamma \Meps$ be a copy of $\Meps$. Identifying
$\bd_2 \gamma \Meps$ with $\bd_1 (\gamma+1) \Meps$ pointwise we obtain a
$\Z$-periodic manifold $\Mepsper$ with period cell $\Meps$. Similarly we can
construct $\Gamma$-periodic manifolds by attaching $2r$ cylindrical ends if
$r$ denotes the number of generators of $\Gamma$.  For a detailed construction
we refer to~\cite{post:01b}.

Note that the period cell $\Meps$ always has smooth boundary $\bd M$, i.e., we
can allow non-compact perturbations in the next sections.

Intuitively, the period cell $\Meps$ is close to the original manifold $X$ if
$\eps$ is small. The next theorem shows that the same is true for the
eigenvalues:
\begin{theorem}
  \label{thm:constr.gaps}
  Assume that the periodic manifold $\Mepsper$ and the period cell $\Meps$ are
  constructed as above. Then for each $k \in \N$, the $k$-th eigenvalue
  $\EW k \Meps$ with Dirichlet, Neumann or $\theta$-periodic boundary
  condition converge to $\EW k X$ as $\eps \to 0$ (uniformly in $\theta$). In
  particular, for all $n \in \N$ there exists $\eps>0$ such that $M=\Meps$
  satisfies the gap condition~\eqref{eq:gap} for $k = 1,\dots,n$ (provided
  $\EW k X$ is a simple eigenvalue).
\end{theorem}
Again, the proof of this theorem and related results can be found
in~\cite[Theorem~1.1]{post:01b}. Note that in Theorem~\ref{thm:conf.gaps} as
well as in Theorem~\ref{thm:constr.gaps} the decoupling of the different
period cells $\gamma \Meps$ is responsible for the gaps.

\subsection*{Periodically curved quantum wave guides}
Here, we present an example given by Yoshitomi \cite{yoshitomi:98}. We
consider a $2$-di\-men\-sio\-nal planar strip $\Mepsper$ obtained by
sliding the normal segment of length $\eps$ along a periodically
curved path $\omega$.  Note that the Dirichlet Laplacian on $\Meps$ is
the Hamiltonian for an electron confined in a quantum wire on a planar
substrate, where the vertical dimension is separated
(cf.~\cite{exner-seba:89}).

Suppose $\eps_0 > 0$. Let $\map \kappa \R {(-1/\eps_0,\infty)}$ be a
smooth and $2\pi$-periodic map (i.e., $\kappa(s+2\pi) = \kappa(s)$
for all $s \in \R$). Then
\begin{displaymath}
  \omega(s):= \bigl( x_{\cos}(s),x_{\sin}(s) \bigr) \qquad \text{with} \qquad
  x_f(s):= \int_0^s f \Bigl( -\int_0^{s'} \kappa(s'') \dd s'' \Bigr) \dd s'
\end{displaymath}
for $f=\cos$ or $f=\sin$ is a curve in $\R^2$ with curvature
$\kappa$. We denote by $\dot \omega^\orth(s):=(-\dot x_{\sin}(s), \dot
x_{\cos}(s))$ the normal unit vector with respect to $\dot
\omega(s)$. We set
\begin{equation}
  \label{eq:qwg}
  \Mepsper := 
  \set{\omega(s)+ u \, \dot \omega^\orth(s)}{s \in \R, 0 \le 0 \le \eps}
\end{equation}
and suppose that
\begin{displaymath}
  \map {\Phi_{\eps_0}} {\R \times [0,\eps_0]} \Mepsper, \qquad
  (s,u) \longmapsto \omega(s)+ u \, \dot \omega^\orth(s)
\end{displaymath}
is a diffeomorphism (for the precise assumptions see~\cite{yoshitomi:98}).
This diffeomorphism allows us to calculate the Dirichlet Laplacian on
$\Mepsper$, $0<\eps<\eps_0$, in coordinates $(s,u)$. We furthermore assume
that $\int_0^{2\pi} \kappa(s) \, \dd s =0$. Then $\omega$ is also
$2\pi$-periodic and $\Mepsper$ is indeed a $\Z$-periodic manifold with a
period cell $\Meps$ given by $\Phi_\eps([0,\eps] \times [0, 2\pi])$. We set
\begin{displaymath}
  K := -\frac {\dd ^2}{\dd s^2} - \frac14 \kappa^2.
\end{displaymath}
In $\Lsqr {[0,2\pi]}$ we regard the $\theta$-periodic realisation of $K$ and
denote the corresponding eigenvalues by $\EWT k K$. Yoshitomi proved
that for all $k \in \N$,
\begin{equation}
  \label{eq:qw.asymp}
  \EWT k \Meps = \frac{\pi^2}{\eps^2} + \EWT k K + O(\eps), \qquad \eps \to 0
\end{equation}
uniformly in $\theta$, where $\EWT k \Meps$ denotes the eigenvalues of
the Laplacian on $\Meps$ with Dirichlet boundary condition at $u=0$
and $u=\eps$, and $\theta$-periodic boundary condition at $s=0$ and
$s=2\pi$. A similar identity holds for Dirichlet resp.\ Neumann
boundary condition at $s=0$ and $s=2\pi$ simultaneously for $\laplacian
\Meps$ and $K$.

Furthermore, Yoshitomi proved that if $\kappa \ne 0$ there exists $k
\in \N$ such that the $k$-th band and the $(k+1)$-st band of
$\laplacianD \Mepsper$ are disjoint. He uses classical results about
the inverse problem for Hill's equation
(cf.~\cite{garnett-trubowitz:87}) to show that the periodic
one-dimensional operator $K$ in $\Lsqr \R$ has a gap between the
$k$-th and the $(k+1)$-st band. In our paper, we need the stronger
assumption~\eqref{eq:gap} asserted by the following theorem:
\begin{theorem}
  \label{thm:qw.gaps}
  Assume in addition that the curvature $\kappa$ is even, i.e.,
  $\kappa(-s)=\kappa(s)$.  Then for all $k \in \N$ the $k$-th band $B_k(K)$ of
  the periodic operator $K$ in $\Lsqr \R$ is given by
  \begin{equation}
    \label{eq:qw.gaps}
    B_k(K) = [\EWD k K, \EWN k K].
  \end{equation}
  Here $\EWD k K$ resp.\ $\EWN k K$ denotes the operator $K$ in $\Lsqr
  {[0,2\pi]}$ with Dirichlet resp.\ Neumann boundary conditions.  In particular
  if $\kappa \ne 0$ there exists $k \in \N$ such that the gap
  condition~\eqref{eq:gap} is satisfied for the Dirichlet Laplacian on
  $\Mepsper$ provided $\eps$ is small enough.
\end{theorem}
\begin{proof}
  Since $\kappa$ is even, $K$ has an even potential. In particular,
  \eqref{eq:qw.gaps} holds iff the potential is even
  (cf.~\cite{garnett-trubowitz:87}). The rest follows from
  \cite{yoshitomi:98}.
\end{proof}
Yoshitomi also localized the gaps: he proved that the $k$-th gap of $K$ is
open if the $k$-th Fourier coefficient of $\kappa^2$ is not $0$.
%------------------------------------------------------------
\section{Examples of perturbations}
\label{sec:examples}
In this section we discuss some examples of the great variety of possible
perturbations. The only restriction is the knowledge of eigenvalue estimates
of the perturbed and unperturbed problem to calculate the right hand sides
of~\eqref{eq:from.above} and~\eqref{eq:from.below}. Here, we always assume
that $\Mper$ is a periodic manifold with periodic metric $g$ and period cell
$M$ such that the gap condition~\eqref{eq:gap} is fulfilled.

By the Weyl asymptotic distribution of eigenvalues~\eqref{eq:weyl} and
by~\eqref{eq:phase.diff} we expect that an infinite number of eigenvalue
branches comes from above crossing the level $\lambda$ if we increase the
volume of $\Perttau$ to infinity.  Therefore, we expect an infinite number
of parameters $\tau$, such that $\lambda$ is an eigenvalue of $\laplacian
\Mpertau$ by Theorem~\ref{thm:eigenvalues}.

In contrast, if we shrink the volume of $\Perttau$, we would expect only a
finite number of eigenvalues (depending on the volume of $\Pert$). Here, we
think of finitely many eigenvalue branches crossing the level $\lambda$ from
below.


\subsection*{Conformal perturbations}
First, we give examples of \emph{conformal} perturabtions. Suppose
that $\map {\pert \rho \tau} \Mper {(0,\infty)}$, $\tau \ge 0$, is a
family of smooth functions. The perturbed metric is given by $\pert g
\tau = \pert \rho \tau ^2 g$. Clearly, if the family $(\pert \rho
\tau)$ satisfies
\begin{align}
  \label{eq:conf.unperturbed}
  \pert \rho 0(x) = 1,   & \qquad x \in \Mper \\
  \label{eq:conf.met.cont}
  \norm[C^1] {\pert \rho \tau - \pert \rho {\tau_0}} \to 0
  & \qquad\text{on $\Mper$ as $\tau \to \tau_0$,}\\
  \label{eq:conf.pert.compact}
  \pert \rho \tau(x)= 1, & \qquad x \in \Rest 
\intertext{for all $\tau_0 \ge
    0$ and sufficiently large $m \in \N$ then $(\pert g \tau)$ satisfies
    Conditions~\eqref{eq:unperturbed}, \eqref{eq:met.cont}
    and~\eqref{eq:pert.compact}. Here, the $C^1$-norm of functions is defined
    in the usual way. Furthermore, if $\bd M$ is smooth we can also allow
    non-compact conformal perturbations which are small outside $\Pert$,
    i.e.,}
  \label{eq:conf.met.unif.rest} \tag{6.3'}
  \sup_{\tau \ge 0} \norm[C^1] {\pert \rho \tau - 1} \to 0 
  & \qquad\text{on $\Rest$ as $m \to \infty$.}
\end{align}
If there is a non-compact perturbation we fix $m$ large enough such that
Equation~\eqref{eq:from.above} resp.~\eqref{eq:from.below} holds.

Furthermore, we assume that the conformal factor is constant on some compact
submanifold $K \subset \Pert$ with non-empty interior and piecewise smooth
boundary, i.e.,
\begin{equation}
  \label{eq:conf.const}
  \pert \rho \tau(x)=\pert c \tau, \qquad x \in K
\end{equation}
where $\pert c \tau>0$ is a constant. Again, $\pert K \tau$ denotes the
manifold $K$ with metric $\pert \rho \tau^2 g = \pert c \tau ^2 g$.

\paragraph{Eigenvalues coming from above.}
In our first example we blow up a subset of $\Pert$ by conformal factors:
\begin{proposition}
  \label{prop:conf.blow}
  Suppose that $\lambda \in I_k$ lies between the $k$-th and the $(k+1)$-st
  band and that $\pert c \tau \to \infty$ as $\tau \to \infty$. Then
  \begin{displaymath}
    \mathcal N(\tau,\lambda) \ge
    \frac {\omega_d}{(2\pi)^d} \lambda^{\frac d 2} \vol (\pert K \tau) - km -
      \delta(\tau)
  \end{displaymath}
  where $\delta(\tau) \searrow 0$ as $\tau \to \infty$. In particular,
  $\mathcal N(\tau,\lambda) \to \infty$ as $\tau \to \infty$, i.e., there
  exist an infinite number of parameters $\tau$ such that $\lambda$ is an
  eigenvalue of $\laplacian \Mpertau$.
\end{proposition}

\begin{proof}
  Applying the Min-max principle we conclude
  \begin{displaymath}
    \EWD j \Perttau \le
    \EWD j {\pert K \tau} =
    \pert c \tau ^{-2} \EWD j K \to 0
  \end{displaymath}
  as $\tau \to \infty$ since $\EWD j K > 0$ for all $j \in \N$.  Therefore all
  eigenvalue branches $\tau \mapsto \EWD j \Perttau$, $j \in \N$, cross the
  level $\lambda$ from above (at least once).  Furthermore, there exists a
  positive function $\delta(\tau) \to 0$ as $\tau \to \infty$ such that
  \begin{displaymath}
    \dimEWD \Perttau \ge
    \dimEWD {\pert K \tau}  =
    \dimEWD [\pert c \tau^2 \lambda] K \ge
    \frac {\omega_d}{(2\pi)^d} \lambda^{\frac d 2} \pert c \tau^d \vol (K) -
      \delta(\tau)
  \end{displaymath}
  by the Weyl asymptotic distribution~\eqref{eq:weyl}. The result follows from
  Theorem~\ref{thm:eigenvalues} and Lemma~\ref{lem:common.gap}.
\end{proof}


\paragraph{Eigenvalues coming from below.}
Next we shrink the manifold $\Pert$ on $K$:
\begin{proposition}
  \label{prop:conf.shrink}
  Suppose that $\lambda \in I_k$ and that $\pert c \tau \to 0$ as $\tau \to
  \infty$. Then we have
  \begin{displaymath}
    \mathcal N(\tau,\lambda) \ge k m - 1
  \end{displaymath}
  for $\tau$ sufficiently large. In particular, there exist at least a finite
  number of parameters $\tau$ such that $\lambda$ is an eigenvalue of
  $\laplacian \Mpertau$.
\end{proposition}
\begin{proof}
  The Min-max principle yields
  \begin{displaymath}
    \EWN j \Perttau \ge
    \EWN j {\pert K \tau} =
    \pert c \tau ^{-2} \EWN j K \to \infty
  \end{displaymath}
  as $\tau \to \infty$ for all $j \in \N$, except for the first Neumann
  eigenvalue $\EWN 1 K = 0$.  Therefore the eigenvalue branches $\tau \mapsto
  \EWN j \Perttau$, $j \in \N \setminus \{1\}$ with $\EWN j {\pert \Pert 0} <
  \lambda$, cross the level $\lambda$ from below (at least once).
  Lemma~\ref{lem:common.gap} yields that there are exactly $k m -1$ such
  eigenvalue branches.  Again we apply Theorem~\ref{thm:eigenvalues}.
\end{proof}

\begin{remark}
  \label{rem:mono.conf}
  If $d=2$ one has monotonically decreasing (increasing) eigenvalue branches
  $\tau \mapsto \EWN j \Perttau$ if the conformal factor $\pert \rho \tau$ is
  monotonically increasing (decreasing). Here, the conformal factor only
  occurs in the norm, not in the quadratic form. The monotonicity follows from
  the Min-max principle. This monotonicity can be a first step in the proof
  of an upper bound on the counting function $\mathcal N(\tau,\lambda)$ (see
  the introduction).
\end{remark}

Finally, let us note the following theorem combining results of this section
and of~\cite{post:01b} (cf.~Theorem~\ref{thm:conf.gaps}):
\begin{theorem}
  \label{thm:conformal}
  Let $\Mper$ be a periodic Riemannian manifold of dimension $d \ge 3$ with
  metric $g$. For any $n \in \N$ there exists a periodic metric $g_n$
  conformal to $g$ such that the corresponding Laplacian has at least $n$
  spectral gaps. Furthermore, there exist (non-periodic) metrics on $\Mper$
  conformal to $g$ such that the corresponding Laplacian has an eigenvalue in
  a gap of the essential spectrum.
\end{theorem}

\subsection*{Diffeomorphic perturbations}
\paragraph{Eigenvalues from above.}
Here, we show how perturbations $\tMpertau$ diffeomorphic to $\Mper$ can be
reduced to the case already treated. We think of $\tMpertau$ as being a smooth
deformation of $\Mper$.  For example, consider a quantum wave guide
$\Mper=\Mepsper$ as defined in~\eqref{eq:qwg}. We assume that the gap
condition~\eqref{eq:gap} holds for some fixed $\eps>0$. Let the perturbation
of $\Mper$ be given by cutting the strip at $s=0$ and inserting a ``bubble''
$\pert K \tau$ which blows up as $\tau$ increases.

To be more precise, suppose that $\pert K \tau$ is a simply connected closed
subset of $\R^2$ such that $\pert K \tau$
\begin{itemize}
\item is a rectangle of length $\tau$ and width $\eps$ for small $\tau$, in
  particular, $\pert K 0$ has empty interior;
\item contains a circle of radius $\tau$ or, alternatively, a rectangle of
  length $\tau$ and width $\eps$ for all $\tau$ large enough;
\item has two ends $\pert A {\tau,1}$ and $\pert A {\tau,2}$, each of them
  being isometric to a rectangle of width $\eps$;
\item depends continuously on $\tau$, i.e., the two boundary curves of $\bd
  \pert K \tau \setminus (\pert A {\tau,1} \cup \pert A {\tau,2})$ are smooth
  and depend continuously on $\tau$ and the curve parameter. 
\end{itemize}
\begin{figure}[h]
  \begin{center}
  \begin{picture}(0,0)%
    \includegraphics{figure-1}%
  \end{picture}%
  \setlength{\unitlength}{4144sp}
  \begin{picture}(5563,1112)(310,-413)
    \put(2476,299){$\tau$}
    \put(3208,-295){$\pert A {\tau,2}$}
    \put(1663,-310){$\pert K \tau$}
    \put(1408,425){$\pert A {\tau,1}$}
  \end{picture}
    \caption{A quantum wave guide perturbed by inserting an increasing bubble.}
    \label{fig:qwg-pert}
  \end{center}
\end{figure}
We denote the intersected quantum wave guide together with the
inserted bubble by $\tMpertau$ (see Figure~\ref{fig:qwg-pert}). Note
that there exists a diffeomorphism $\map {\pert \Phi \tau} \Mper
\tMpertau$ such that $\pert \Phi \tau(x)=x$ outside a compact set
(with the obvious identification of the two ends of the unperturbed
and perturbed quantum wave guide).

The Euclidean metric $g$ on $\tMpertau$ is pulled back on $\Mper$ via $\pert
\Phi \tau$, i.e., $\pert g \tau := \pert \Phi \tau^* g$. Note that
$(\tau,x) \mapsto \pert g \tau(x)$ resp.\ $(\tau,x) \mapsto \partial_i \pert g
\tau(x)$ are uniformly continuous maps on $[0,\tau_0] \times \Mper$ for all
$\tau_0>0$ since $\pert g \tau = g$ outside a compact set and since $g$ is
periodic, thus uniformly continuous. Therefore, it is easy to check that the
Assumptions~\eqref{eq:unperturbed} to~\eqref{eq:pert.compact} are satisfied.


\begin{proposition}
\label{prop:qw.blow}
Suppose that $\lambda \in I_k$ and that $\tMpertau$ is the perturbed manifold
$\Mper$ obtained by the above construction. Then we have $\mathcal
N(\tau,\lambda) \to \infty$ as $\tau \to \infty$.
\end{proposition}

\begin{proof}
  The proof is similar to the proof of Proposition~\ref{prop:conf.blow}. Note
  that in the case of an inserted rectangle of length $\tau$, we have $\EWD k
  {\pert K \tau} \le (\pi k)^2/\tau^2$ for all $\tau \ge \tau_0$ where
  $\tau_0$ depends on $k$.
\end{proof}

\begin{remark}
  \label{rem:mono.diff}
  Note that the eigenvalue branches $\tau \mapsto \EWN j {\pert K \tau}$ are
  monotonically decreasing by the Min-max principle at least in the case when
  $\pert K \tau$ is isometric to a rectangle of length $\tau$ and width
  $\eps$. This monotonicity can be a first step in the proof of an upper bound
  on the counting function $\mathcal N(\tau,\lambda)$ (see the introduction).
\end{remark}


\section{Topological perturbations}
\label{sec:non.diffeo}
\paragraph{Eigenvalues from below.}
In this final section we show how to deal with certain non-homeomorphic
perturbations. We allow more general examples of perturbed Riemannian
manifolds $\tMpertau$ diffeomorphic to $\MperP := \Mper \setminus H$ where $H$
consists of discrete points. We think of $\tMpertau$ being a smooth
deformation of $\MperP$. Note that the substraction of a discrete set of
points has no influence on the Laplacian as a self-adjoint operator as we will
see now.

Suppose that $X$ is a complete Riemannian manifold of dimension $d \ge 2$.
Let $H \subset X \setminus \bd X$ be a discrete set of points. On $\pointed X
:= X \setminus H$ we define the Laplacian with Dirichlet resp.\ Neumann
boundary condition on $\bd X$ as in Section~\ref{sec:prelim}. At $H$ we always
assume a Dirichlet boundary condition, i.e., we start from the quadratic form
$\check q_X$ defined for smooth functions with support away from $H$. Note
that the Hilbert spaces $\Lsqr X$ and $\Lsqr {\pointed X}$ agree since $H$ has
measure $0$.  Furthermore, the substraction of a discrete set of points $H$
has no effects on the Laplacian either (with any boundary condition):
\begin{theorem}
  \label{thm:spec.equal}
  Suppose that $\bd X=\emptyset$. For every function $u \in \dom q_X$ there
  exists a sequence $(u_n)$ of smooth functions $u_n \in \dom q_X$ with
  support away from $H$ such that $u_n \to u$ in the form norm $(\normsqr[X] u
  + q_X(u))^\frac12$. In particular, $\dom q_{\pointed X} = \dom q_X$ and
  therefore, the corresponding operators agree as operators in $\Lsqr X$.
  
  If $\bd X \ne \emptyset$ the same is true for the quadratic form with
  Dirichlet resp.\ Neumann (or any other) boundary condition on $\bd X$.
  Again, the corresponding Laplacians on the dotted and undotted manifolds
  agree.
\end{theorem}
\begin{proof}
  For a single point see \cite[Lemma~1]{chavel-feldman:78}. Clearly we can
  generalise this result to a discrete set of points $H$. Since we have
  defined the Laplacian via quadratic forms the equality of the operators
  follows.
\end{proof}

We consider the following example: suppose that we bore a hole around each
point of $H$ in the periodic manifold $\Mper$.  For simplicity, we assume that
$H$ consists of $m$ points $\gamma x_0$ where $\gamma \in \Gamma^m$ (recall
that $\Gamma^m$ is an exhausting sequence of $\Gamma$ as defined in the
beginning) and $x_0 \in M$ with $\dist (x_0,\bd M) < \eps_0/2$. Here, $M$
denotes a fixed period cell of $\Mper$ for which the gap
condition~\eqref{eq:gap} holds and $\eps_0>0$ is the injectivity radius of
$\Mper$, i.e., the exponential map is defined on all balls of radius smaller
than $\eps_0$.

To define properly the smooth dependence of the boundary on the parameter
$\tau$, we need the following construction: let $\map f \Mper {[0,\infty]}$ be
a continuous map.  We think of $f$ quantifying the level of perturbation:
$f(x)=0$ resp.\ $f(x)=\infty$ means that $x \in \Mper$ is always resp.\ never
affected by the perturbation.  In particular, we assume
\begin{itemize}
\item that $f(x)=\dist(x,\gamma x_0)$ for all $x \in \gamma M$ such that
  $\dist(x, \gamma x_0)<\eps_0/2$, in particular $f(\gamma x_0)=0$ for all
  $\gamma \in \Gamma^m$;
\item that $f(x)=\infty$ for all $x \in \Rest=(\Gamma \setminus \Gamma^m) M$
  and all $x \in \Gamma^m \bd M$, i.e., the union of all translates $\gamma
  \bd M$, $\gamma \in \Gamma^m$;
\item that $f$ is smooth on $\set{x \in \Mper}{0 < f(x)< \infty}$ and that all
  $\tau \in (0,\infty)$ are regular values of $f$, i.e., if $f(x)=\tau$ then
  $\dd f_x \ne 0$;
\item that $Z:=\set{x \in \Pert=\Gamma^m M}{f(x)=\infty}$ is the finite union
  of compact smooth submanifolds $Z_i$ of dimension $d_i<d$ with piecewise
  smooth boundary such that $Z_i \cap Z_j$ is either empty
  or of dimension smaller than $d_i$ and $d_j$ (cf.~Figure~\ref{fig:per-cell});
\item that $\set{x \in \Pert}{f(x)>\tau}$ is contained in the
  $1/\tau$-neighbourhood of $Z$, i.e., $\dist(x,Z)<1/\tau$ for all $x \in
  \Pert$ with $f(x)>\tau$.
\end{itemize}
\begin{figure}[h]
  \begin{center}
    \begin{picture}(0,0)%
    \includegraphics{figure-2}%
    \end{picture}%
    \setlength{\unitlength}{4144sp}%
    \begin{picture}(5198,1951)(49,-1198)
      \put(2441,544){$Z_1$}
      \put(2481,-691){$Z_2$}
      \put(2671,-66){$Z_4$}
      \put(5106, 89){$Z_5$}
      \put(2781,-421){$Z_3$}
      \put(5186,-620){$Z_6$}
      \put(5241,-801){$Z_1$}
      \put(5121,364){$Z_2$}
      \put(1498,-971){$x_0$}
    \end{picture}
    \caption{A period cell with the dotted set $Z$ and its smooth components
      $Z_i$. On the right hand side one has to identify opposite sides of the
      square. Note that the complement of $Z$ in the period cell is always
      homeomorphic to an open ball in $\R^d$. The (light and dark) grey set is
      the perturbed manifold for small $\tau>0$, the dark grey set is the
      perturbed manifold for large $\tau$.}
    \label{fig:per-cell}
  \end{center}
\end{figure}
 Let
\begin{align*}
  \widetilde \Mper  & := 
      \set{(\tau,x) \in [0,\infty) \times \Mper} {f(x) > \tau} 
                          \qquad \text{resp.}\\
  Y             & := \set{(\tau,x) \in [0,\infty) \times \Mper} {f(x) = \tau}
\end{align*}
be the fibred manifold with fibres
\begin{align*}
   \tMpertau    & := \set{x \in \Mper} {f(x) > \tau}  \qquad \text{resp.}\\
   \pert Y \tau & := \set{x \in \Mper} {f(x) = \tau}.
\end{align*}
We consider these manifolds as submanifolds of the product of the
Riemannian manifolds $[0,\infty)$ and $\Mper$ (with the induced
metrics).  Note that $\Mper \setminus \tMpertau$ is a manifold
diffemorphic to $m$ copies of a closed ball in $\R^d$ with smooth
boundary $\pert Y \tau$ diffeomorphic to $m$ copies of the sphere
$\Sphere^{d-1}$ provided $0 < \tau < \infty$. Furthermore, note that
$Y$ consists of $m$ cones. In a neighbourhood of (each component of)
$Y$ we introduce \emph{normal (or Fermi) coordinates} given by a chart
$\map \varphi U V$ where
\begin{displaymath}
  U = \set{(\tau,x) \in \widetilde \Mper}{0 < \dist(x,Y) < r_0(\tau)}
\end{displaymath}
(see e.g.~\cite[p. 9-59 -- 9-62]{spivak:70a}). Here, the function $r_0$ is
supposed to be smooth. Then the image of the chart is given by
\begin{displaymath}
  V = \set{(s,\tau,y) \in (0,\infty) \times [0,\infty) \times Y \,}
          {\, s < r_0(\tau)}.
\end{displaymath}
In particular, for fixed $\tau$ we have a chart $\map{\pert \varphi
  \tau}{\pert U \tau}{\pert V \tau}$ for the corresponding fibres.

Note that $\widetilde \Mper$ is diffeomorphic to $[0,\infty) \times \MperP$. We construct the diffeomorphism explicitly. Let
\begin{displaymath}
  \map r {[0,\infty) \times [0,\infty)} {[0,\infty)}
\end{displaymath}
be a continuous map (smooth on the interior) with bounded derivatives such that
\begin{displaymath}
  \map{\pert r \tau := r(\tau,\cdot)}{[0,\infty)}{[\tau,\infty)} 
\end{displaymath}
is bijective and
\begin{displaymath}
  \pert r \tau(0) = \tau \qquad \text{and} \qquad
  \pert r \tau(s) = s, \quad s \ge r_0(\tau)/2.
\end{displaymath}
Then
\begin{displaymath}
  \map \Phi {[0,\infty) \times \MperP} {\widetilde \Mper}, \qquad \qquad
  \Phi(\tau,x):= \varphi^{-1}(\pert r \tau(s),\tau,y)
\end{displaymath}
with $(s,y):=\pert \varphi \tau(x)$ is a diffeomorphism. Clearly, the
corresponding maps on the fibres $\map{\pert \Phi \tau} \MperP \tMpertau$
are also diffeomorphisms.  For a technical reason we do not use $\MperP$ but
\begin{displaymath}
  \hMper := \set{x \in \Mper} {\dist(x,H) > \eps_0/2}
\end{displaymath}
as a reference manifold. Note that there exists a diffeomorphism
\begin{displaymath}
  \map \psi \hMper \MperP.
\end{displaymath}
Therefore, $\hMper$ is also diffeomorphic to each fibre $\tMpertau$ of
$\widetilde \Mper$. The technical reason is that we need an analogue to
Theorem~\ref{thm:reg.theory} where we have used regularity theory.  On
$\MperP$ we would have coefficients which cannot be continuously extended onto
$H$. Blowing up the holes and pulling back the metrics avoids this difficulty.
Therefore, we have a diffeomorphism
\begin{displaymath}
  \map \Psi {[0,\infty) \times \hMper} {\widetilde \Mper}, \qquad \qquad
  \Psi(\tau,x):=(\tau,\pert \Phi \tau(\psi(x))).
\end{displaymath}

There is a natural metric on $\widetilde \Mper$, namely $\widetilde h:=\dd
\tau^2 + g$ where $g$ is the (restriction of the) periodic metric on $\Mper$.
Let $h:= \Psi^* \widetilde h$ be the pull-back of the metric $\widetilde h$.
Note that $h$ (and its derivatives) are uniformly continuous maps on
$[0,\tau_0] \times \hMper$.  Therefore, the restriction $\pert g \tau$ of $h$
onto the fibre $\hMper$ satisfies the continuity property~\eqref{eq:met.cont}.
Furthermore, Condition~\eqref{eq:pert.compact} is fulfilled.

Finally, replacing $\Mper$ by $\hMper$ and $\GpM$ by $\GpM \cap \hMper$ with
metric $\psi^*g$ one can verify that all our results of
Sections~\ref{sec:ell.est} and~\ref{sec:eigenvalues} remain true in virtue of
Theorem~\ref{thm:spec.equal}.

Therefore, we have the following result:
\begin{proposition}
  \label{prop:diff.shrink}
  Suppose that $\lambda \in I_k$. Let $(\tMpertau)$ be the family of manifolds
  constructed above by removing closed sets from $\Pert$. Then we have
  $\widetilde {\mathcal N}(\tau,\lambda) \ge k m$ for $\tau$ sufficiently
  large.  Here, $\widetilde{\mathcal N}(\tau,\lambda)$ denotes the counting
  function for the manifold $\tMpertau$ defined as
  in~\eqref{eq:def.count.fct}. In particular, there exist a finite number of
  parameters $\tau$ such that $\lambda$ is an eigenvalue of $\laplacian
  \tMpertau$.
\end{proposition}
\begin{proof}
  The proof is the same as the proof of Proposition~\ref{prop:conf.shrink}.
  Set $\pert {\widetilde M^m} \tau := \tMpertau \cap \Pert$.  We need the fact
  that $\EWN j {\pert {\widetilde M^m} \tau} \to \infty$ as $\tau \to \infty$
  for all $j \in \N$ when $\vol (\pert {\widetilde M^m} \tau) \to 0$ as $\tau
  \to \infty$. This will be shown in the next lemma.
\end{proof}

\begin{remark}
  Note that we have imposed Neumann boundary condition only at the
  $\tau$-independent component $\bd \Pert = \bd \pert {\widetilde M^m} \tau$ and
  Dirichlet boundary condition at $\pert Y \tau$. If there was only Neumann
  boundary condition the first eigenvalue would be $0$.
\end{remark}

\begin{remark}
  \label{rem:monotonical}
  In this example, the eigenvalue branches $\tau \mapsto \EWN j {\pert
    {\widetilde M^m} \tau}$ are monotonically increasing due to the Min-max
  principle.  This monotonicity can be a first step in the proof of an upper
  bound on the counting function $\mathcal N(\tau,\lambda)$ (see the
  introduction).
\end{remark}

In the next lemma we show that the eigenvalues grow up to infinity
when shrinking the volume of a manifold to $0$.
\begin{lemma}
  \label{lem:ew.mfd.shrink}
  Suppose that $K$ is a $d$-dimensional manifold with piecewise smooth
  boundary $\bd K$. Denote by $(\pert K \tau)$ a family of $d$-dimensional
  closed submanifolds of $K$ such that $\vol \pert K \tau \to 0$ as $\tau \to
  \infty$ and such that $\bd \pert K \tau$ depends smoothly on $\tau$ (defined
  in the same way as before). Furthermore, assume that $\bd \pert K \tau$
  consists of two disjoint components $\pert Y \tau$ and $\bd K$
  where $\pert Y \tau$ is smooth. Besides, suppose that
  \begin{displaymath}
    \pert K \tau \subset 
    \set{x \in K}{\dist(x,Z) < \frac 1 \tau} := 
    \pert N \tau
  \end{displaymath}
  for $\tau$ large. Here, we assume that $Z$ is the finite union of compact
  smooth submanifolds $Z_i$ of dimension $d_i<d$ with piecewise smooth
  boundary such that $Z_i \cap Z_j$ is either empty or of dimension smaller
  than $d_i$ and $d_j$ (cf.~Figure~\ref{fig:per-cell}).  Finally, suppose that
  $\bd K \subset Z$.
  Then $\EWDN k {\pert K \tau} \to \infty$ as $\tau \to \infty$ for all $k \in
  \N$ where we have imposed Dirichlet boundary conditions on $\pert Y \tau$
  and Neumann boundary condition on $\bd K$.
\end{lemma}
\begin{proof}
  We define the following isoperimetric (or Cheeger's) constant
  \begin{displaymath}
    h_{\bd K}(\pert K \tau) := 
    \inf_{\pert \Omega \tau} 
      \frac{\vol_{d-1}(\bd \pert \Omega \tau \setminus \bd K)}
           {\vol_d   (\pert \Omega \tau)}
  \end{displaymath}
  where $\pert \Omega \tau$ ranges over all open subsets of $\pert K \tau$
  such that $\bd \pert \Omega \tau \setminus \bd K$ is smooth and $\bd \pert
  \Omega \tau \cap \pert Y \tau = \emptyset$. The subscript $\bd K$ indicates
  the subset of $\bd \pert K \tau$ where we have imposed Neumann boundary
  conditions. When measuring the surface we exclude the Neumann boundary part
  $\bd K$.  If there were no Dirichlet boundary condition (i.e., $\bd \pert K
  \tau=\bd K$) we could choose $\pert \Omega \tau=\pert K \tau$.  Therefore,
  $\bd \pert \Omega \tau \setminus \bd K = \emptyset$, i.e., $h_{\bd K}(\pert
  K \tau)=0$.
  
  In the same way as in \cite[Chapter~IV]{chavel:84} we can prove the
  following estimate originally due to Cheeger:
  \begin{displaymath}
    \EWDN 1 {\pert K \tau} \ge \frac14 h_{\bd K}(\pert K \tau)^2.
  \end{displaymath}
  \begin{figure}[h]
    \begin{center}
      \begin{picture}(0,0)%
    \includegraphics{figure-3}%
      \end{picture}%
      \setlength{\unitlength}{4144sp}%
      \begin{picture}(2454,1824)(934,-1198)
        \put(3376,-1051){$\bd K$}
        \put(3016, 59){$\pert N {\tau,i}$}
        \put(3139,-521){$\pert Z i$}
      \end{picture}
      \caption{The decomposition of the tubular neighbourhood of $Z$. Again,
        we have to identify opposite sides of the square. Here, $K$ is the
        torus with two open balls cut out, $\pert K \tau$ is marked in dark
        grey and the tubular neighbourhood $\pert N \tau$ of radius $1/\tau$
        is marked in light and dark grey. Note that near $Z_i$,
        we have to decompose the tubular neighbourhood once more into two
        sets.}
      \label{fig:decomp}
    \end{center}
  \end{figure}
  It remains to show that $h_{\bd K}(\pert K \tau) \to \infty$ as $\tau \to
  \infty$.  To this end let $\pert \Omega \tau$ be one of the sets taken in
  the definition of $h_{\bd K}(\pert K \tau)$. Since the set $Z$ is the finite
  union of smooth submanifolds $Z_i$ we decompose the (subset of a) tubular
  neighbourhood $\pert N \tau$ of $Z$ into a finite number of disjoint open
  sets $\pert N {\tau,i}$ such that $Z_i=\pert N {\tau,i} \cap Z$ and such
  that $\pert N \tau \setminus \bigcup_i \pert N {\tau,i}$ has $d$-dimensional
  volume $0$.  If $d_i=d-1$ the set $\pert N {\tau,i}$ could lie on both sides
  of $Z_i$ (cf.\ Figure~\ref{fig:decomp}). In this case we decompose $\pert N
  {\tau,i}$ into its two components of $\pert N {\tau,i} \setminus Z_i$ (again
  denoted by $\pert N {\tau,i}$).
  
  We set $\pert A {\tau,i}:= (\bd \pert \Omega \tau \cap \pert N {\tau,i})$
  and $\pert \Omega {\tau,i}:= \pert \Omega \tau \cap \pert N {\tau,i}$. With
  regard to the next lemma, we have
  \begin{displaymath}
    \vol_{d-1}(\bd \pert \Omega \tau \setminus \bd K) \ge
    \sum_i \vol_{d-1}(\pert A {\tau,i} ) \ge
    c \, \tau \sum_i \vol_d (\pert \Omega {\tau,i}) =
    c \, \tau \vol_d (\pert \Omega \tau)
  \end{displaymath}
  and we are done.
\end{proof}

We still need the following final technical lemma which roughly says that the
``local'' isoperimetric constant in a small strip tends to infinity when the
width of the strip tends to $0$.
\begin{lemma}
  \label{lem:loc.isoper}
  With the notation from above we have
  \begin{displaymath}
     \vol_{d-1}(\pert A {\tau,i} ) \ge
     c \, \tau \vol_d (\pert \Omega {\tau,i})
  \end{displaymath}
  where the constant $c$ only depends on the metric near $Z$.
\end{lemma}
\begin{proof}
  Since $Z_i$ is a smooth submanifold of $K$ we can introduce normal
  coordinates $(y,z)$ on the tubular neighbourhood $\pert N {\tau,i}$ of $Z_i$
  provided $\tau$ is large enough, i.e., $\pert N {\tau,i}$ is diffeomorphic
  to $\pert B \tau \times (Z_i \setminus \bd Z_i)$ where $\pert B \tau:=\set{y
    \in \R^{d-d_i}}{|y| < 1/\tau}$ (resp.\ $\pert B \tau:=(0,1/\tau)$ if
  $d_i=d-1$), see e.g.~\cite[p. 9-59 -- 9-62]{spivak:70a}.  Since $Z_i$ is
  compact and since the metric on $\pert N {\tau,i}$ depends continuously on
  the coordinates and smoothly on $\tau$ it suffices to prove the assertion in
  the case when the metric (in normal coordinates) has product structure $\dd
  y^2 + h$ where $h$ is the metric on $Z_i$.
  
  Denote by $\pert P {\tau,i}$ the image of $\pert A {\tau,i}$ under the
  orthogonal projection onto $Z_i$.  If we parametrize the smooth submanifold
  $\pert A {\tau,i}$ we can show that
  \begin{displaymath}
    \vol_{d-1} (\pert A {\tau,i}) \ge 
    c' \, \tau^{-(d-1-d_i)} \vol_{d_i} (\pert P {\tau,i})
  \end{displaymath}
  where the constant $c'$ depends only on the metric $h$. Finally,
  \begin{displaymath}
     \vol_d (\pert \Omega {\tau,i}) \le
     \vol_d (\pert N {\tau,i}) \le
     c''\, \tau^{-(d-d_i)} \vol_{d_i} (\pert P {\tau,i})
   \end{displaymath}
   where $c''$ depends only on $d-d_i$. Therefore we have finished our proof.
\end{proof}



\section*{Acknowledgements}
  I would like to thank Rainer Hempel for helpful discussions leading to this
  article. Furthermore I would like to thank Wolf Jung for his suggestions
  concerning the proofs of Lemmas~\ref{lem:ew.mfd.shrink}
  and~\ref{lem:loc.isoper}.

%------------------------------------------------------------
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 4228 1250 l 4218 1237 l 4208 1223 l 4200 1210 l 4194 1196 l
 4188 1182 l 4183 1167 l 4179 1152 l 4175 1136 l 4173 1119 l
 4172 1101 l 4171 1084 l 4172 1067 l 4173 1051 l 4175 1036 l
 4177 1022 l 4180 1009 l 4184 995 l 4188 983 l 4193 970 l
 4199 957 l 4205 945 l 4212 932 l 4220 921 l 4228 910 l
 4237 900 l 4246 892 l 4255 884 l 4264 877 l 4274 870 l
 4285 864 l 4296 859 l 4308 854 l 4321 849 l 4335 845 l
 4349 841 l 4362 839 l 4376 836 l 4389 835 l 4401 834 l
 4414 834 l 4426 834 l 4438 834 l 4450 836 l 4463 838 l
 4477 841 l 4490 845 l 4503 850 l 4516 856 l 4528 863 l
 4539 871 l 4550 880 l 4560 890 l 4569 900 l 4578 910 l
 4586 922 l 4596 934 l 4605 947 l 4615 960 l 4625 973 l
 4636 986 l 4646 999 l 4657 1010 l 4667 1020 l 4678 1029 l
 4688 1037 l 4699 1043 l 4711 1047 l 4723 1050 l 4735 1052 l
 4749 1053 l 4763 1052 l 4777 1050 l 4790 1048 l 4804 1044 l
 4817 1039 l 4829 1034 l 4840 1028 l 4850 1022 l 4859 1015 l
 4866 1009 l 4873 1000 l 4879 992 l 4884 982 l 4888 971 l
 4891 959 l 4893 945 l 4895 930 l 4895 914 l 4895 897 l
 4895 879 l 4894 860 l 4894 839 l 4893 825 l 4893 809 l
 4893 792 l 4892 774 l 4892 756 l 4892 736 l 4891 715 l
 4891 694 l 4891 673 l 4891 651 l 4891 630 l 4890 609 l
 4890 589 l 4890 570 l 4889 551 l 4889 534 l 4889 517 l
 4889 502 l 4888 480 l 4887 459 l 4886 439 l 4884 420 l
 4882 402 l 4879 386 l 4875 371 l 4870 357 l 4865 346 l
 4859 336 l 4852 327 l 4844 320 l 4837 316 l 4831 312 l
 4823 308 l 4814 305 l 4803 302 l 4791 299 l 4777 296 l
 4761 294 l 4744 292 l 4724 290 l 4702 288 l 4679 287 l
 4654 286 l 4627 284 l 4598 283 l 4568 283 l 4536 282 l
 4501 281 l 4477 281 l 4452 280 l 4426 280 l 4399 279 l
 4370 279 l 4340 278 l 4309 277 l 4277 277 l 4243 276 l
 4209 276 l 4173 275 l 4138 275 l 4101 274 l 4065 274 l
 4028 273 l 3992 273 l 3956 272 l 3920 272 l 3885 272 l
 3852 271 l 3819 271 l 3787 270 l 3756 270 l 3727 270 l
 3699 269 l 3672 269 l 3646 269 l 3622 269 l 3586 268 l
 3552 268 l 3520 268 l 3489 267 l 3460 267 l 3433 268 l
 3407 268 l 3383 268 l 3361 269 l 3341 270 l 3323 271 l
 3307 272 l 3293 274 l 3280 275 l 3269 277 l 3260 279 l
 3252 282 l 3244 284 l 3236 288 l 3229 293 l 3222 298 l
 3217 305 l 3212 313 l 3208 323 l 3205 334 l 3202 347 l
 3200 361 l 3199 376 l 3198 393 l 3198 411 l 3198 430 l

 cp gs col7 0.80 shd ef gr  [] 0 sd
7.500 slw
 [15 30] 30 sd
1 slc
% Ellipse
n 3690 1080 191 191 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr
 [] 0 sd
 [15 30] 30 sd
% Ellipse
n 4410 1080 191 191 0 360 DrawEllipse gs col7 1.00 shd ef gr gs col0 s gr
 [] 0 sd
0.000 slw
 [30 15 15 15] 0 sd
% Ellipse
n 4050 1620 117 117 0 360 DrawEllipse gs col7 1.00 shd ef gr  [] 0 sd
% Polyline
0 slc
7.500 slw
 [15 30] 30 sd
n 3150 1080 m
 3510 1080 l gs col0 s gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 4590 1080 m
 4950 1080 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2680 1491 m
 3510 2025 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2630 270 m
 3150 765 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2077 455 m
 2430 225 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2423 1491 m
 1380 1051 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2855 856 m
 3525 956 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2730 776 m
 2525 611 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2786 1216 m
 2315 586 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2971 1211 m
 3295 1086 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 4573 964 m
 5101 706 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 4775 1081 m
 5181 1406 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 4944 1556 m
 5234 1586 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 1698 1768 m
 4061 1631 l gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 4734 231 m
 5104 396 l gs col0 s gr  [] 0 sd
% Polyline
n 821 588 m 822 588 l 826 587 l 831 585 l 840 582 l 852 579 l
 867 574 l 886 569 l 906 564 l 929 558 l 953 552 l
 979 546 l 1007 540 l 1035 534 l 1065 529 l 1097 524 l
 1130 520 l 1165 516 l 1203 513 l 1243 511 l 1285 510 l
 1328 510 l 1375 512 l 1419 515 l 1460 519 l 1498 525 l
 1532 531 l 1563 537 l 1592 544 l 1619 552 l 1645 559 l
 1668 567 l 1690 575 l 1710 582 l 1728 589 l 1743 595 l
 1755 600 l 1763 603 l 1769 606 l 1773 607 l
 1774 608 l gs col7 1.00 shd ef gr gs col0 s gr 
% Polyline
n 740 553 m 741 554 l 745 555 l 751 558 l 760 562 l 772 567 l
 788 573 l 807 581 l 828 589 l 852 598 l 878 607 l
 905 617 l 934 626 l 964 635 l 996 644 l 1030 652 l
 1067 660 l 1106 667 l 1148 674 l 1193 679 l 1240 683 l
 1290 686 l 1340 687 l 1387 686 l 1432 684 l 1474 681 l
 1513 676 l 1550 671 l 1584 665 l 1616 659 l 1646 652 l
 1675 645 l 1703 637 l 1728 630 l 1752 623 l 1773 616 l
 1792 610 l 1808 604 l 1820 600 l 1829 597 l 1835 595 l
 1839 594 l
 1840 593 l gs col0 s gr 
% Polyline
0.000 slw
 [30 15 15 15] 0 sd
n 950 1065 m 956 1062 l 962 1060 l 968 1056 l 972 1053 l 977 1049 l
 980 1044 l 983 1039 l 985 1034 l 987 1029 l 988 1023 l
 988 1017 l 988 1011 l 986 1005 l 984 999 l 982 994 l
 979 988 l 975 983 l 971 978 l 966 973 l 961 968 l
 950 961 l 938 954 l 925 947 l 910 941 l 894 936 l
 877 932 l 860 929 l 843 927 l 827 926 l 812 927 l
 798 928 l 786 931 l 780 933 l 775 936 l 769 939 l
 764 942 l 759 945 l 755 949 l 751 953 l 748 958 l
 745 963 l 742 968 l 741 973 l 739 978 l 739 984 l
 739 989 l 739 994 l 741 1000 l 742 1005 l 745 1010 l
 748 1015 l 752 1019 l 756 1024 l 761 1028 l 771 1035 l
 784 1042 l 798 1049 l 815 1055 l 833 1060 l 852 1064 l
 871 1067 l 889 1069 l 907 1070 l 923 1069 l 937 1068 l

 cp gs 1.00 setgray ef gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
n 1396 212 m 1426 213 l 1449 214 l 1465 215 l 1474 216 l 1478 217 l
 1480 217 l 1481 218 l 1485 219 l 1494 221 l 1509 223 l
 1532 226 l 1561 230 l 1587 234 l 1610 237 l 1628 240 l
 1641 242 l 1651 243 l 1657 243 l 1662 244 l 1667 244 l
 1673 245 l 1683 247 l 1696 249 l 1714 253 l 1737 258 l
 1763 264 l 1792 271 l 1815 278 l 1831 282 l 1841 285 l
 1846 287 l 1849 287 l 1851 288 l 1856 290 l 1865 294 l
 1880 300 l 1902 309 l 1928 320 l 1955 332 l 1977 343 l
 1993 350 l 2004 356 l 2012 359 l 2017 361 l 2022 364 l
 2028 367 l 2038 373 l 2053 382 l 2072 395 l 2093 410 l
 2116 429 l 2133 444 l 2143 454 l 2148 461 l 2150 466 l
 2152 470 l 2156 477 l 2162 486 l 2172 498 l 2183 512 l
 2194 526 l 2198 534 l 2197 537 l 2193 537 l 2189 537 l
 2188 538 l 2192 541 l 2202 545 l 2215 548 l 2226 548 l
 2231 548 l 2233 546 l 2236 545 l 2244 543 l 2261 543 l
 2288 542 l 2309 542 l 2328 542 l 2344 543 l 2357 543 l
 2367 545 l 2375 546 l 2384 547 l 2393 548 l 2405 548 l
 2419 547 l 2434 545 l 2450 542 l 2464 536 l 2470 531 l
 2471 525 l 2469 521 l 2464 517 l 2459 512 l 2455 508 l
 2453 503 l 2454 498 l 2459 493 l 2464 491 l 2470 489 l
 2474 488 l 2478 488 l 2481 487 l 2484 486 l 2487 486 l
 2490 487 l 2494 488 l 2499 491 l 2504 496 l 2510 504 l
 2517 518 l 2520 531 l 2521 539 l 2521 545 l 2521 551 l
 2521 561 l 2522 577 l 2523 600 l 2523 623 l 2523 639 l
 2522 649 l 2521 655 l 2520 660 l 2519 669 l 2518 682 l
 2517 699 l 2517 718 l 2518 725 l 2519 725 l 2520 727 l
 2517 733 l 2512 735 l 2508 734 l 2505 731 l 2501 728 l
 2492 724 l 2488 722 l 2484 721 l 2481 721 l 2479 722 l
 2478 723 l 2476 725 l 2475 726 l 2474 727 l 2472 726 l
 2470 724 l 2467 719 l 2465 712 l 2463 698 l 2464 685 l
 2467 675 l 2472 666 l 2476 658 l 2478 647 l 2475 635 l
 2466 622 l 2455 615 l 2444 611 l 2437 610 l 2432 611 l
 2428 613 l 2424 615 l 2417 618 l 2405 619 l 2387 620 l
 2363 620 l 2341 620 l 2321 620 l 2305 621 l 2293 621 l
 2283 621 l 2275 620 l 2267 620 l 2258 620 l 2247 621 l
 2233 621 l 2217 622 l 2202 624 l 2196 625 l 2191 626 l
 2187 626 l 2184 627 l 2182 628 l 2181 628 l 2180 629 l
 2181 629 l 2182 629 l 2184 630 l 2185 630 l 2187 630 l
 2189 630 l 2191 630 l 2192 630 l 2194 630 l 2195 631 l
 2196 631 l 2197 632 l 2197 633 l 2197 634 l 2197 635 l
 2196 637 l 2195 639 l 2193 641 l 2186 650 l 2183 654 l
 2182 654 l 2181 656 l 2174 663 l 2159 679 l 2147 692 l
 2139 702 l 2134 710 l 2133 714 l 2134 718 l 2134 721 l
 2131 726 l 2122 735 l 2104 748 l 2077 765 l 2054 777 l
 2033 788 l 2015 797 l 2001 804 l 1991 808 l 1984 812 l
 1978 814 l 1972 817 l 1963 820 l 1952 824 l 1935 830 l
 1912 838 l 1882 847 l 1848 857 l 1817 865 l 1790 872 l
 1766 877 l 1748 881 l 1734 883 l 1723 885 l 1715 885 l
 1709 886 l 1702 886 l 1695 887 l 1685 887 l 1672 889 l
 1655 891 l 1634 894 l 1609 898 l 1583 902 l 1551 907 l
 1526 910 l 1509 912 l 1499 913 l 1493 914 l 1489 914 l
 1486 914 l 1482 914 l 1475 914 l 1465 915 l 1450 917 l
 1435 919 l 1425 920 l 1417 921 l 1412 922 l 1409 922 l
 1407 922 l 1407 921 l 1408 921 l 1409 920 l 1410 919 l
 1411 919 l 1412 918 l 1412 919 l 1411 921 l 1409 923 l
 1407 926 l 1404 935 l 1404 939 l 1405 939 l 1407 940 l
 1407 950 l 1407 973 l 1406 991 l 1406 1008 l 1406 1021 l
 1405 1032 l 1405 1040 l 1405 1048 l 1404 1058 l 1404 1071 l
 1403 1087 l 1402 1104 l 1401 1112 l 1401 1119 l 1401 1123 l
 1401 1127 l 1402 1129 l 1403 1129 l 1404 1129 l 1405 1129 l
 1406 1128 l 1407 1127 l 1408 1126 l 1409 1125 l 1407 1126 l
 1405 1127 l 1401 1130 l 1396 1132 l 1382 1137 l 1370 1140 l
 1360 1141 l 1352 1142 l 1343 1143 l 1333 1144 l 1319 1145 l
 1304 1147 l 1299 1147 l 1295 1148 l 1292 1148 l 1289 1148 l
 1287 1148 l 1285 1149 l 1284 1149 l 1283 1149 l 1282 1150 l
 1281 1150 l 1281 1151 l 1281 1152 l 1280 1152 l 1280 1151 l
 1280 1150 l 1279 1149 l 1279 1147 l 1279 1145 l 1280 1142 l
 1284 1132 l 1289 1124 l 1294 1119 l 1299 1116 l 1304 1112 l
 1310 1106 l 1316 1094 l 1321 1076 l 1324 1057 l 1324 1043 l
 1324 1035 l 1323 1029 l 1322 1024 l 1320 1016 l 1317 1003 l
 1313 985 l 1306 966 l 1300 958 l 1296 957 l 1293 957 l
 1287 950 l 1280 932 l 1276 916 l 1274 905 l 1272 898 l
 1272 893 l 1272 889 l 1272 882 l 1272 870 l 1273 855 l
 1276 838 l 1280 830 l 1284 830 l 1287 829 l 1291 822 l
 1293 806 l 1293 792 l 1291 782 l 1290 776 l 1288 772 l
 1286 769 l 1284 763 l 1281 753 l 1278 740 l 1276 732 l
 1274 725 l 1271 721 l 1269 718 l 1266 716 l 1263 715 l
 1261 714 l 1259 712 l 1258 710 l 1259 708 l 1261 704 l
 1267 700 l 1280 696 l 1292 694 l 1303 693 l 1313 694 l
 1323 694 l 1334 695 l 1348 696 l 1362 697 l 1366 698 l
 1369 698 l 1371 699 l 1373 699 l 1374 699 l 1375 700 l
 1376 700 l 1376 699 l 1375 699 l 1374 698 l 1373 698 l
 1372 697 l 1371 697 l 1370 696 l 1369 696 l 1368 696 l
 1367 696 l 1367 697 l 1366 698 l 1366 699 l 1366 701 l
 1367 703 l 1368 706 l 1369 710 l 1370 714 l 1374 726 l
 1377 738 l 1379 748 l 1381 756 l 1383 762 l 1385 769 l
 1386 777 l 1389 787 l 1392 799 l 1396 812 l 1398 818 l
 1399 823 l 1400 827 l 1400 830 l 1400 832 l 1399 833 l
 1398 834 l 1397 835 l 1396 835 l 1394 835 l 1393 835 l
 1391 835 l 1390 835 l 1390 836 l 1391 836 l 1393 837 l
 1396 837 l 1400 838 l 1405 838 l 1417 837 l 1428 835 l
 1435 833 l 1439 831 l 1442 829 l 1445 827 l 1451 824 l
 1463 821 l 1482 817 l 1508 812 l 1532 808 l 1553 805 l
 1568 804 l 1578 803 l 1584 804 l 1588 805 l 1592 805 l
 1599 805 l 1611 804 l 1631 801 l 1658 796 l 1692 789 l
 1719 783 l 1744 776 l 1765 771 l 1782 767 l 1796 764 l
 1806 761 l 1814 759 l 1821 758 l 1828 756 l 1835 754 l
 1845 751 l 1858 747 l 1874 741 l 1894 734 l 1916 724 l
 1940 714 l 1969 700 l 1991 688 l 2007 679 l 2016 674 l
 2022 670 l 2025 668 l 2027 665 l 2031 662 l 2037 656 l
 2045 646 l 2056 633 l 2067 617 l 2073 605 l 2078 594 l
 2081 585 l 2084 578 l 2085 573 l 2086 570 l 2087 568 l
 2087 566 l 2087 564 l 2087 562 l 2086 559 l 2084 554 l
 2081 547 l 2076 538 l 2069 527 l 2060 515 l 2045 499 l
 2031 486 l 2019 477 l 2011 471 l 2006 467 l 2001 465 l
 1996 463 l 1989 460 l 1977 454 l 1959 445 l 1934 433 l
 1902 418 l 1876 407 l 1852 397 l 1832 389 l 1816 382 l
 1804 377 l 1795 374 l 1788 371 l 1782 369 l 1776 366 l
 1769 364 l 1760 361 l 1747 357 l 1729 352 l 1707 346 l
 1680 339 l 1651 332 l 1618 325 l 1590 320 l 1569 317 l
 1555 316 l 1546 315 l 1541 316 l 1538 317 l 1535 318 l
 1530 319 l 1521 319 l 1507 319 l 1486 317 l 1458 315 l
 1426 313 l 1393 311 l 1364 309 l 1339 308 l 1320 307 l
 1306 308 l 1295 308 l 1285 309 l 1276 310 l 1266 310 l
 1254 310 l 1239 309 l 1221 306 l 1201 302 l 1182 296 l
 1174 293 l 1167 289 l 1161 286 l 1157 282 l 1153 279 l
 1150 276 l 1148 273 l 1146 270 l 1145 268 l 1144 266 l
 1144 264 l 1144 262 l 1144 260 l 1144 258 l 1144 256 l
 1145 254 l 1146 252 l 1146 250 l 1148 248 l 1149 246 l
 1151 243 l 1153 240 l 1156 237 l 1160 234 l 1164 231 l
 1169 227 l 1175 224 l 1182 221 l 1199 216 l 1216 212 l
 1232 210 l 1246 208 l 1257 208 l 1267 208 l 1275 209 l
 1284 209 l 1295 210 l 1307 211 l 1324 211 l 1345 211 l
 1369 212 l
 cp gs 0.60 setgray ef gr  [] 0 sd
% Polyline
 [30 15 15 15] 0 sd
n 132 549 m 134 557 l 136 565 l 137 575 l 138 586 l 138 596 l
 137 607 l 137 615 l 136 623 l 135 633 l 134 643 l
 132 654 l 131 663 l 129 673 l 127 681 l 124 693 l
 120 704 l 117 715 l 115 724 l 114 732 l 115 737 l
 117 739 l 120 740 l 124 740 l 129 739 l 134 737 l
 139 734 l 145 731 l 149 726 l 155 719 l 160 710 l
 165 700 l 169 690 l 172 680 l 175 671 l 176 665 l
 178 659 l 180 653 l 182 647 l 185 642 l 189 638 l
 193 634 l 199 632 l 206 629 l 215 628 l 225 627 l
 237 626 l 250 625 l 263 625 l 276 625 l 289 625 l
 299 625 l 310 625 l 321 625 l 333 625 l 344 625 l
 355 624 l 366 624 l 375 624 l 384 624 l 392 624 l
 403 625 l 414 626 l 424 628 l 433 631 l 441 635 l
 448 641 l 453 647 l 459 654 l 465 662 l 472 672 l
 480 683 l 488 694 l 497 704 l 506 715 l 513 722 l
 520 730 l 529 737 l 538 746 l 548 754 l 559 763 l
 570 771 l 582 779 l 595 787 l 607 795 l 619 802 l
 632 808 l 643 814 l 655 819 l 668 824 l 681 830 l
 696 835 l 711 841 l 726 846 l 741 851 l 757 856 l
 772 861 l 787 865 l 801 869 l 815 873 l 829 877 l
 843 881 l 857 884 l 871 888 l 886 891 l 901 895 l
 917 898 l 934 902 l 950 905 l 967 909 l 983 912 l
 1000 915 l 1016 918 l 1032 921 l 1048 923 l 1064 926 l
 1081 929 l 1098 931 l 1116 934 l 1135 936 l 1154 938 l
 1172 939 l 1190 941 l 1207 941 l 1223 941 l 1237 940 l
 1250 939 l 1261 937 l 1271 934 l 1277 932 l 1282 929 l
 1286 926 l 1290 923 l 1294 919 l 1297 915 l 1299 910 l
 1301 906 l 1302 901 l 1303 896 l 1303 891 l 1302 886 l
 1301 882 l 1300 877 l 1298 873 l 1295 869 l 1293 865 l
 1290 862 l 1287 859 l 1283 856 l 1275 852 l 1266 848 l
 1256 846 l 1244 845 l 1231 844 l 1218 844 l 1205 845 l
 1192 845 l 1179 846 l 1167 846 l 1156 846 l 1146 846 l
 1135 846 l 1124 845 l 1112 844 l 1099 843 l 1087 841 l
 1074 839 l 1061 837 l 1048 835 l 1036 832 l 1023 830 l
 1011 827 l 999 824 l 986 821 l 973 818 l 959 815 l
 944 811 l 929 807 l 914 803 l 899 799 l 884 795 l
 870 791 l 856 787 l 843 783 l 830 779 l 816 775 l
 801 770 l 786 765 l 771 760 l 756 754 l 740 749 l
 725 742 l 711 736 l 698 730 l 685 723 l 673 717 l
 662 710 l 651 703 l 641 696 l 630 688 l 620 679 l
 609 670 l 599 661 l 590 652 l 581 643 l 573 634 l
 566 626 l 560 619 l 555 612 l 550 604 l 545 596 l
 542 588 l 539 580 l 537 573 l 536 565 l 536 558 l
 537 551 l 539 545 l 541 538 l 545 531 l 549 524 l
 554 517 l 561 509 l 568 501 l 577 492 l 586 484 l
 595 477 l 605 470 l 615 463 l 624 458 l 633 453 l
 644 447 l 656 442 l 668 436 l 681 431 l 694 425 l
 708 420 l 722 415 l 735 410 l 749 405 l 762 401 l
 775 396 l 790 391 l 805 387 l 821 382 l 837 377 l
 854 372 l 871 366 l 887 362 l 903 357 l 919 353 l
 933 349 l 947 345 l 961 341 l 975 338 l 989 334 l
 1004 331 l 1019 328 l 1034 325 l 1050 322 l 1065 319 l
 1079 317 l 1093 315 l 1106 313 l 1119 312 l 1135 311 l
 1151 310 l 1168 309 l 1185 308 l 1201 306 l 1217 304 l
 1231 302 l 1243 299 l 1253 296 l 1261 291 l 1264 289 l
 1267 286 l 1270 283 l 1272 279 l 1273 276 l 1275 272 l
 1275 268 l 1275 264 l 1275 260 l 1274 255 l 1273 251 l
 1270 247 l 1268 243 l 1265 239 l 1261 235 l 1257 232 l
 1253 228 l 1248 226 l 1243 223 l 1238 221 l 1227 217 l
 1215 215 l 1201 213 l 1186 212 l 1169 212 l 1152 212 l
 1133 213 l 1114 214 l 1096 216 l 1077 218 l 1060 220 l
 1043 223 l 1028 225 l 1013 228 l 998 231 l 983 234 l
 967 237 l 950 241 l 934 244 l 918 248 l 902 252 l
 887 256 l 872 259 l 858 263 l 844 267 l 831 270 l
 818 274 l 805 277 l 791 281 l 778 285 l 763 289 l
 749 294 l 735 298 l 720 303 l 706 308 l 693 312 l
 680 317 l 668 322 l 656 327 l 644 332 l 631 338 l
 617 345 l 604 352 l 590 360 l 576 368 l 562 377 l
 549 386 l 536 395 l 525 404 l 514 413 l 505 422 l
 496 431 l 487 441 l 478 453 l 470 464 l 461 476 l
 453 488 l 445 499 l 438 509 l 431 518 l 423 526 l
 416 532 l 408 537 l 400 542 l 390 545 l 379 548 l
 368 550 l 357 551 l 345 552 l 334 553 l 323 553 l
 312 553 l 301 553 l 289 552 l 277 552 l 265 552 l
 253 551 l 241 550 l 230 548 l 220 547 l 211 545 l
 203 542 l 192 537 l 182 531 l 174 524 l 167 516 l
 161 510 l 158 505 l 155 502 l 152 500 l 150 498 l
 148 497 l 145 496 l 143 496 l 141 496 l 138 497 l
 134 498 l 129 499 l 124 500 l 119 501 l 115 502 l
 114 502 l 113 503 l 112 503 l 111 503 l 111 504 l
 110 504 l 110 505 l 110 506 l 111 506 l 111 507 l
 112 508 l 112 509 l 113 510 l 115 514 l 118 519 l
 122 526 l 126 533 l 130 541 l
 cp gs 0.60 setgray ef gr  [] 0 sd
7.500 slw
 [60 30 15 30] 0 sd
% Ellipse
n 4050 1620 30 30 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
 [] 0 sd
 [60 30 15 30] 0 sd
% Ellipse
n 853 993 30 30 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr
 [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 1465 1753 m
 855 998 l gs col0 s gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 135 585 m
 495 585 l gs col0 s gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 2160 585 m
 2520 585 l gs col0 s gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 2522 610 m 2522 628 l 2521 645 l 2519 663 l 2517 680 l 2514 696 l
 2510 710 l 2506 721 l 2502 730 l 2497 735 l 2493 737 l
 2488 736 l 2484 733 l 2480 728 l 2475 720 l 2471 711 l
 2466 700 l 2462 687 l 2459 673 l 2456 659 l 2454 645 l
 2452 630 l 2452 616 l 2452 598 l 2453 580 l 2456 561 l
 2459 543 l 2463 527 l 2468 512 l 2473 499 l 2478 490 l
 2483 485 l 2488 483 l 2493 484 l 2497 489 l 2502 497 l
 2507 509 l 2511 523 l 2514 539 l 2517 557 l 2520 575 l
 2521 593 l
 cp gs col0 s gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 138 615 m 136 639 l 133 659 l 129 675 l 126 688 l 123 698 l
 119 708 l 115 718 l 109 727 l 103 736 l 95 739 l
 88 734 l 82 725 l 77 714 l 73 704 l 71 693 l
 68 683 l 65 669 l 63 653 l 61 632 l 60 608 l
 62 584 l 65 565 l 68 550 l 71 538 l 74 529 l
 77 520 l 81 512 l 86 503 l 92 495 l 100 492 l
 108 497 l 114 505 l 119 515 l 123 524 l 126 534 l
 129 543 l 132 555 l 135 571 l 137 591 l
 cp gs col0 s gr  [] 0 sd
% Polyline
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 1995 836 l 2013 819 l 2026 808 l 2034 803 l 2037 801 l
 2039 801 l 2040 801 l 2045 801 l 2055 798 l 2072 793 l
 2097 786 l 2120 781 l 2140 777 l 2155 774 l 2167 772 l
 2174 771 l 2181 769 l 2187 768 l 2195 768 l 2206 767 l
 2222 767 l 2243 767 l 2267 769 l 2294 773 l 2313 777 l
 2325 781 l 2332 783 l 2335 786 l 2338 788 l 2343 792 l
 2353 797 l 2367 806 l 2385 817 l 2405 832 l 2416 842 l
 2421 849 l 2423 854 l 2423 858 l 2427 864 l 2435 875 l
 2447 889 l 2461 905 l 2467 914 l 2469 918 l 2470 921 l
 2475 926 l 2487 931 l 2490 932 l 2493 932 l 2496 932 l
 2498 933 l 2501 933 l 2502 934 l 2504 935 l 2506 936 l
 2507 937 l 2508 939 l 2509 941 l 2510 942 l 2511 944 l
 2512 946 l 2513 947 l 2514 948 l 2516 948 l 2517 947 l
 2518 945 l 2519 942 l 2521 938 l 2522 933 l 2524 926 l
 2526 917 l 2527 907 l 2529 888 l 2530 868 l 2531 850 l
 2532 833 l 2534 819 l 2535 806 l 2536 795 l 2538 785 l
 2539 774 l 2540 763 l 2540 749 l 2539 734 l 2537 716 l
 2533 695 l 2528 672 l 2520 649 l 2512 630 l 2504 614 l
 2497 601 l 2492 592 l 2489 585 l 2488 581 l 2488 578 l
 2489 576 l 2490 575 l 2491 575 l 2490 574 l 2488 573 l
 2483 571 l 2474 568 l 2461 563 l 2442 558 l 2418 552 l
 2389 547 l 2360 543 l 2331 540 l 2304 539 l 2281 538 l
 2262 538 l 2247 539 l 2235 540 l 2226 542 l 2219 544 l
 2212 546 l 2205 548 l 2197 550 l 2185 552 l 2170 554 l
 2151 555 l 2125 557 l 2092 558 l 2053 559 l 2007 559 l
 1957 559 l 1914 559 l 1872 558 l 1832 558 l 1796 557 l
 1764 557 l 1737 556 l 1713 555 l 1692 554 l 1675 553 l
 1659 552 l 1646 551 l 1632 550 l 1620 549 l 1606 549 l
 1592 548 l 1575 547 l 1557 546 l 1535 545 l 1511 545 l
 1484 544 l 1454 544 l 1422 544 l 1389 544 l 1357 544 l
 1315 545 l 1282 546 l 1260 546 l 1246 546 l 1239 546 l
 1238 545 l 1241 544 l 1246 542 l 1251 541 l 1256 541 l
 1259 541 l 1258 543 l 1254 546 l 1245 551 l 1233 559 l
 1219 570 l 1201 587 l 1186 603 l 1175 616 l 1168 627 l
 1163 635 l 1160 643 l 1156 650 l 1153 659 l 1148 671 l
 1141 687 l 1133 708 l 1125 731 l 1119 757 l 1117 777 l
 1117 791 l 1119 799 l 1123 805 l 1126 810 l 1130 816 l
 1132 826 l 1135 838 l 1136 851 l 1136 854 l 1136 856 l
 1137 858 l 1137 860 l 1137 862 l 1137 863 l 1136 863 l
 1135 862 l 1134 861 l 1134 860 l 1133 858 l 1132 857 l
 1132 856 l 1131 854 l 1130 853 l 1130 851 l 1129 850 l
 1128 848 l 1128 847 l 1127 846 l 1127 844 l 1127 843 l
 1127 842 l 1126 842 l 1127 841 l 1127 840 l 1128 841 l
 1129 841 l 1130 842 l 1131 843 l 1132 844 l 1134 845 l
 1136 847 l 1145 855 l 1153 865 l 1160 875 l 1165 884 l
 1169 893 l 1173 902 l 1179 910 l 1187 917 l 1200 922 l
 1215 922 l 1230 916 l 1241 907 l 1247 898 l 1249 890 l
 1248 881 l 1249 873 l 1251 862 l 1259 850 l 1275 834 l

 cp gs col7 1.00 shd ef gr gs col0 s gr  [] 0 sd
% Polyline
0.000 slw
 [15 30] 30 sd
n 2475 360 m 2610 360 l 2610 945 l 2475 945 l
 cp gs col7 1.00 shd ef gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 1080 360 m 1215 360 l 1215 945 l 1080 945 l
 cp gs col7 1.00 shd ef gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 1080 1215 m 1215 1215 l 1215 1890 l 1080 1890 l
 cp gs col7 1.00 shd ef gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 2475 1215 m 2610 1215 l 2610 1890 l 2475 1890 l
 cp gs col7 1.00 shd ef gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 1170 1395 m 2565 1395 l 2565 1890 l 1170 1890 l
 cp gs col7 1.00 shd ef gr  [] 0 sd
% Polyline
7.500 slw
 [15 30] 30 sd
n 1215 945 m 1080 945 l 1080 364 l 2610 360 l 2610 945 l
 2475 945 l gs col0 s gr  [] 0 sd
% Polyline
 [15 30] 30 sd
n 1215 1215 m 1080 1215 l 1080 1890 l 2610 1890 l 2610 1215 l
 2487 1221 l gs col0 s gr  [] 0 sd
% Polyline
0.000 slw
 [15 15] 15 sd
n 1811 675 m 1890 675 l 1890 1489 l 1811 1489 l
 cp gs col7 1.00 shd ef gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 1170 367 m 2490 367 l 2490 742 l 1170 742 l
 cp gs col7 1.00 shd ef gr  [] 0 sd
% Polyline
7.500 slw
 [60 30 15 30] 0 sd
n 2340 1215 m
 3375 1845 l gs col7 1.00 shd ef gr gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2565 990 m
 2970 765 l gs col7 1.00 shd ef gr gs col0 s gr  [] 0 sd
% Polyline
 [60 30 15 30] 0 sd
n 2608 1075 m
 3113 1310 l gs col7 1.00 shd ef gr gs col0 s gr  [] 0 sd
$F2psEnd
rs

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