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spectral gaps, covering manifolds, type I groups, non-abelian Floquet theory
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%--------------------------------------------------------------------
% YRSgaps.tex       Producing spectral gaps by geometry (YRS proceedings)
%                   Fernando Lled'o, Olaf Post
% 2004-04-20        Institut f"ur Reine und Angewandte Mathematik
%                   RWTH Aachen, Germany
% last change       email: lledo@iram.rwth-aachen.de, post@iram.rwth-aachen.de
% 2005-06-16, 2004-07-09, 2004-05-09, 2004-03-01, 2003-10-24, 2003-04-28
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%------------------------------------------------------------
% begin of the document
%------------------------------------------------------------
\begin{document}

\title{Generating spectral gaps by geometry}

\author{Fernando Lled\'o}
\address{Institut f\"ur Reine und Angewandte Mathematik,
       Rheinisch-Westf\"alische Technische Hochschule Aachen,
       Templergraben 55,
       D-52062 Aachen,
       Germany}
\email{lledo@iram.rwth-aachen.de}

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

%------------------------------------------------------------
% Subject classifications
%------------------------------------------------------------


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

\begin{abstract}
  Motivated by the analysis of Schr\"odinger operators with periodic
  potentials we consider the following abstract situation: Let $\laplacian X$
  be the Laplacian on a non-compact Riemannian covering manifold $X$ with a
  discrete isometric group $\Gamma$ acting on it such that the quotient
  $X/\Gamma$ is a compact manifold.  We prove the existence of a finite number
  of spectral gaps for the operator $\laplacian X$ associated with a suitable
  class of manifolds $X$ with {\em non-abelian} covering transformation groups
  $\Gamma$.  This result is based on the non-abelian Floquet theory as well as
  the Min-Max-principle. Groups of type~I specify a class of examples
  satisfying the assumptions of the main theorem.
  \hfill (\today)
\end{abstract}
\vspace*{-0mm}
\maketitle
%------------------------------------------------------------
% The main part
%------------------------------------------------------------

\section{Introduction}

It is a well known fact that a Schr\"odinger operator $-\Delta + V$ on $\R^d$,
$d\ge 2$, with a suitable periodic potential $V$ has gaps in its spectrum.
This is a quite natural situation in solid state physics, where --- for
example in insulators --- the particles described by the Schr\"odinger
operator have some unreachable energy regions (gaps). This behaviour is
ensured by the following two crucial properties: first, the fact that $V$ is
periodic.  This means that there is a basis $\{\eps_i\}_{i=1}^d$ of $\R^d$
such that the potential satisfies
\begin{displaymath}
 V(x+\eps_i) = V(x)\;,\quad i=1,\ldots,d\,.
\end{displaymath}
In other words, the periodicity of $V$ introduces an action of the discrete
abelian group $\Z^d$ on $\R^d$ and the potential is completely specified on a
fundamental domain $D \subset X$. A typical example for a fundamental domain
is the parallelepiped $D=(0,1)\eps_1 + \dots + (0,1) \eps_d$.  Second, the
potential $V$ has a high barrier near the boundary of $D$. In this way, the
potential $V$ essentially decouples the fundamental domain $D$ from the
neighbouring domains $\eps_i + D$, $i=1,\dots,d$ (see~\cite{hempel-post:03}
for an overview).

A natural question in this context is whether one can replace the effect of
the periodic potentials on the spectrum of the Laplace operator by using
geometry.  Specifically, can we replace $\R^d$ with some Riemannian manifold
$X$ with a suitable discrete group action on it, such that the corresponding
Laplace operator $\laplacian X$ also has gaps in its spectrum (which is purely
essential spectrum)? In other words, has $\spec \laplacian X$ more than one
component as a subset of $[0,\infty)$?  A positive answer to this question was
given in the context of abelian groups in \cite{post:03a} (see also the
references cited therein). The intuitive idea is that the junctions of the
fundamental domains that build up $X$ are small enough (see
Figure~\ref{fig:per-mfd} below). This has a similar effect on the energy of
the particles as the high barriers of the potential in the case of the
Schr\"odinger operator on $\R^d$. Note that the case $d=1$ is uninteresting in
our context since every Laplacian on a one-dimensional (non-compact)
Riemannian manifold is unitary equivalent to the standard one, which has no
gaps (cf.~\cite{davies-harrell:87}).  Moreover, for one-dimensional
Schr\"odinger operators any non-constant potential produces gaps
\cite[Theorem~XIII.91]{reed-simon-4}.

We will show in this paper that this simple idea of scaling down the junctions
also works for many non-abelian discrete groups $\Gamma$. The analysis in the
non-abelian case is more involved because the structure of the dual object
$\widehat \Gamma$ (i.e., the set of equivalence classes of unitary irreducible
representations of $\Gamma$) is less transparent from an algebraic and measure
theoretic point of view.  The purpose of the present paper is to stress the
fundamental ideas that allow to extend the previous result to non-abelian
group actions.  Technical details and further developments will be published
in~\cite{lledo-post:pre05}.  We will not study the nature of the spectrum
outside the gaps. Some papers related to the problem of the band-gap structure
of elliptic operators on covering manifolds are e.g.~\cite{bruening-sunada:92}
or \cite{gruber:01}. In contrast with the Schr\"odinger operator case, the
present geometric setting allows the freedom to choose the dimension $d$ of
the manifold and the number of ``period directions'' $r$ (i.e., number of
generators of $\Gamma$) \emph{independently} from each other. This observation
can probably be useful in further investigations on spectral properties common
for periodic Schr\"odinger operators and Laplacians on manifolds.

The paper is organised as follows: In the following section we set up the
problem, present the geometrical context and state some results that will be
needed later. In Section~\ref{sec:non-abelian} we introduce in detail the
non-abelian Floquet theory, which is at the basis of our analysis. We will
also illustrate the general formulas in the special case $\Gamma=\Z^d$ and
$X=\R^d$. In the next section we prove our main result: the existence of
spectral gaps in the spectrum of $\laplacian X$ for suitable manifolds $X$.
In Section~\ref{sec:examples} we specify a family of discrete groups,
so-called groups of type~I, that satisfy the assumptions of our theorem. For
convenience of the reader we have included in an appendix a short review of
the main results concerning direct integral decompositions of unitary group
representations. This technique is crucial for the Floquet theory.



%------------------------------------------------------------
\section{Notation and background}
\label{sec:prelim}
\subsection{Periodic manifolds and Laplacians}
\label{ssec:per.mfd}
We begin fixing our notation and recalling some results that will 
be useful later on. We denote by $X$ a \emph{non-compact} Riemannian
manifold of dimension $d \ge 2$. We also assume 
the action on $X$ of a finitely generated, discrete group 
$\Gamma$ of isometries of $X$ such that the
quotient $M:= X / \Gamma$ is a \emph{compact} Riemannian manifold 
which also has dimension $d$.
\begin{figure}[h]
  \begin{center}
%------------------------------------------------------------
%    \input{per-mfd.pstex_t}
\begin{picture}(0,0)%
\includegraphics{yrs-v2-fig1}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
\begin{picture}(5244,1381)(259,-575)
\put(900,120){$\eps_2$}
\put(1630,10){$\eps_1$}
\put(3196,-556){$X$}
\put(4950,-100){$D$}
\end{picture}
%------------------------------------------------------------
    \caption{A periodic (or covering) manifold $X$ with group $\Gamma$
      generated by two elements $\eps_1, \eps_2$ and fundamental domain $D$
      (in grey). Here, the group $\Gamma$ is abelian.}
    \label{fig:per-mfd}
  \end{center}
\end{figure}
In other words, $X$ is a \emph{periodic manifold} or \emph{Riemannian covering
  space} of $M$ with covering transformation group $\Gamma$. Moreover, we fix
a \emph{fundamental domain} $D$, i.e., an open set $D \subset X$ such that $\g
D$ and $\g' D$ are disjoint for all $\g \ne \g'$ and $\bigcup_{\g \in \G} \g
\overline D = X$ (cf.\ Figure~\ref{fig:per-mfd}).

As a prototype for an elliptic operator we consider the Laplacian $\laplacian
X$ on $X$ acting on a dense subspace of the Hilbert space $\Lsqr X$
with norm $\norm[X] \cdot$. The positive self-adjoint operator 
$\laplacian X$ can be defined in terms of a suitable 
quadratic form $q_X$ (see e.g.~\cite[Chapter~VI]{kato:95},
\cite{reed-simon-1} or \cite{davies:96}). Concretely we have
\begin{equation}
\label{def:quad.form}
  q_X(u):=\normsqr[X] {d u} = \int_{X} |d u|^2 dX\;,\quad
          u \in \Cci X \;.
\end{equation}
In coordinates we write the pointwise norm of the $1$-form $d u$ as
\begin{displaymath}
  |d u |^2= \sum_{i,j} g^{ij} \partial_i u \, \partial_j\overline u \,,
\end{displaymath}
where $(g^{ij})$ is the inverse of the metric tensor $(g_{ij})$ in a
chart. Taking the closure of the quadratic form we can extend $q_X$ onto
the Sobolev space
\begin{displaymath}
  \Sob X = \set{ u \in \Lsqr X}{ q_X(u) < \infty}\,.
\end{displaymath}
As usual the operator $\laplacian X$ is related with the quadratic form by the
formula $\iprod {\laplacian X u} u = q_X(u)$, $u \in \Cci X$. Since the metric
on $X$ is $\Gamma$-invariant, the Laplacian $\laplacian X$ (i.e., its
resolvent) commutes with the translations on $X$ given by
\begin{equation}
  \label{eq:transl}
   (T_\gamma u )(x) := u(\gamma^{-1}x), \quad u \in \Lsqr X, \gamma \in \Gamma.
\end{equation}
Operators with this property are called \emph{periodic}.

For an open, relatively compact subset $D \subset X$ with sufficiently smooth
boundary $\bd D$ (e.g.~Lipschitz) we define the Dirichlet (respectively,
Neumann) Laplacian $\laplacianD D$ (resp., $\laplacianN D$) via its quadratic
form $q_D^+$ (resp., $q_D^-$) associated to the closure of $q_D$ on $\Cci D$,
the space of smooth functions with compact support, (resp.\, $\Ci {\overline
  D}$, the space of smooth functions with continuous derivatives up to the
boundary).  We also use the notation $\Sobn D = \dom q_D^+$ (resp., $\Sob D =
\dom q_D^-$). Note that the usual boundary condition of the Neumann Laplacian
occurs only in the \emph{operator} domain via the Gau{\ss}-Green formula.
Since $\overline D$ is compact, $\laplacianD D$ has purely discrete spectrum
$\EWD k$, $k \in \N$.  It is written in ascending order and repeated according
to multiplicity.  The same is true for the Neumann Laplacian and we denote the
corresponding purely discrete spectrum by $\EWN k$, $k\in\N$.

One of the advantages of the quadratic form approach is that 
one can easily read off from the inclusion of domains an order relation 
for the eigenvalues. In fact, by the 
the \emph{min-max principle} we have
\begin{equation}
  \label{eq:min.max}
  \EWDN k =
  \inf_{L_k} \sup_{u \in L_k \setminus \{0\} }
      \frac {q_D^\pm(u)}{\normsqr u}\;,
\end{equation}
where the infimum is taken over all $k$-dimensional subspaces $L_k$ of the
corresponding \emph{quadratic} form domain $\dom q_D^\pm$. Then the 
inclusion
\begin{equation}
  \label{eq:dom.mono}
   \dom q_D^+ = \Sobn D  \subset \Sob D = \dom q_D^- 
\end{equation}
implies $ \EWD k \ge \EWN k$, i.e., the Dirichlet eigenvalue is in general
larger than the Neumann eigenvalue and this justifies the choice of the 
labels $+$, respectively, $-$.

%------------------------------------------------------------
\subsection{Spectral gaps in the abelian case}
%------------------------------------------------------------
\label{sec:gaps.abel}
Due to the previous inequality relating the Dirichlet and Neumann 
eigenvalues for any $k\in\N$, we may introduce the following intervals
\begin{equation}
  \label{eq:def.Ik}
  I_k := [\EWN k, \EWD k]\;,\quad k\in\N\,.
\end{equation}
Here, $\EWDN k$ denotes the $k$-th Dirichlet/Neumann eigenvalue on a
fundamental domain $D$.

In this context the existence of spectral gaps of $\laplacian X$ can be
reduced to the question whether $I_k \cap I_{k+1} = \emptyset$ for some $k$.
A class of manifolds with abelian group actions satisfying the previous
intersection condition is specified in \cite{post:03a}:
\begin{theorem}
  \label{thm:ex.gaps.ab}
  Given a finitely generated abelian group $\Gamma$ we can always find a
  covering space $X \to X/\Gamma =: M$ and a fundamental domain $D$ with the
  following property: For every $K \in \N$ there exists a metric $g=g_K$ on
  $M$ such that
  \begin{equation}
    \label{eq:gaps}
    I_k \cap I_{k+1} = \emptyset
  \end{equation}
  for at least $K$ indices $k \in \N$, where $I_k$ is defined as in
  \eqref{eq:def.Ik} for the manifold $D$ with metric $g=g_K$. In particular,
  the Laplacian $\laplacian X$ corresponding to the lifted metric on $X$ has
  at least $K$ gaps.
\end{theorem}
It is important to note that the construction of the covering space $X$ and
the metric $g$ only depends on the quotient $M$, not on the cover $X$. It is
therefore independent whether $\Gamma$ is abelian or not (see the sketch of
the proof of Theorem~\ref{thm:ex.gaps}).  Roughly speaking, we have replaced
the \emph{high} potential barrier in the case of Schr\"odinger operators on
$\R^d$ by \emph{small} junctions between the fundamental domains.  We can say
that now geometry is partly decoupling one fundamental domain from its
neighbours.



%------------------------------------------------------------
\section{Non-abelian Floquet theory}
%------------------------------------------------------------
\label{sec:non-abelian}

In this section we will introduce in several steps the Floquet theory for
non-abelian groups. The main idea is to use the group action on $X$ and a
partial Fourier transformation to decompose the Hilbert space $\Lsqr X$ and
the periodic operators on it into a direct integral of simpler components that
can be analysed more easily. For convenience of the reader we have summarized
in the appendix the main results on the direct integral decompositions of
unitary group representations.

%------------------------------------------------------------
\subsection{Non-abelian Fourier transformation}
%------------------------------------------------------------
\label{ssec:non-ab.fourier}

Consider first the right, respectively, left regular representation $R$,
resp., $L$ on the Hilbert space $\lsqr \G$:
\begin{equation}
  \label{def:reg.rep}
  (R_\g a)_\tg = a_{\tg \g}, \qquad
  (L_\g a)_\tg = a_{\g^{-1}\tg},  \quad\qquad a = (a_\g)_\g \in \lsqr \G, 
      \quad \g,\tg \in \G.
\end{equation}
Let $\mR$ be the von Neumann algebra generated by all unitaries $R_\g$, $\g
\in \G$, i.e.,
\begin{displaymath}
  \mR = \set {R_\g} {\g \in \G}'',
\end{displaymath}
and denote by $\mR'$ the commutant of $\mR$ in $\mB(\lsqr \G)$;
similarly, we define $\mathcal L = \set {L_\g} {\g \in \G}''$.

In this context, we may generalise the Fourier transformation to the
unitary map
\begin{equation}
  \label{eq:fourier}
  \map F {\lsqr \G}{\Zint {H(z)}}
\end{equation}
that transforms the right regular representation $R$ into the following direct
integral representation
\begin{equation}
  \label{eq:reg.rep.trafo}
    \widehat R_\g = F R_\g F^{-1} = \Zint {R_\g(z)}, \qquad \g \in \G.
\end{equation}
By a suitable choice of the measure space $(Z,dz)$ corresponding to an
maximal abelian algebra $\mathcal A$ in $\mathcal R'$ (see the
appendix) we can assume that the unitary representations $R_\g(z)$ are
irreducible on the Hilbert space $H(z)$ a.e. In addition, operators
commuting with all $L_\gamma$ ($\gamma \in \Gamma$), i.e., operators in
$\mathcal L'$, are decomposable, since one can show that $\mathcal
L'=\mathcal R$ and therefore $\mathcal L' \subset \mathcal A'$ (cf.\ 
the appendix).



%------------------------------------------------------------
\subsection{Equivariant Laplacians}
%------------------------------------------------------------
\label{ssec:equiv.lapl}
We will introduce next a new operator that lies ``between''
the Dirichlet and Neumann Laplacians and that will play an important
role in the following section. Consider
on almost each fibre smooth \emph{$R(z)$-equivariant
  functions}, i.e., smooth functions $\map h X {H(z)}$ satisfying
\begin{equation}
  \label{def:equiv.fct}
    h(\g x) = R_\g(z) h(x), \qquad  \g \in \G, x \in X.
\end{equation}  
We denote the corresponding space of smooth $R(z)$-equivariant functions
\emph{restricted to a fundamental domain $D$} by $\CiZ {D,H(z)}$. Note that we
need \emph{vector-valued} functions $\map h X {H(z)}$ since the representation
$R(z)$ acts on the Hilbert space $H(z)$. If $\Gamma$ is non-abelian, then some
of its unitary irreducible representations must be of dimension greater than
one.

We introduce next the so-called \emph{equivariant Laplacian} (w.r.t.\ 
the representation $R(z)$) on $\Lsqr {D,H(z)} \cong \Lsqr D \otimes
H(z)$. Consider the quadratic form given by
\begin{equation}
\label{eq:quad.form}
  \normsqr[D] {dh} := \int_D \normsqr[H(z)] {dh(x)} dX
\end{equation}
for $h \in \CiZ {D,H(z)}$, where the integrand is locally specified by
\begin{displaymath}
    \normsqr[H(z)] {dh(x)} = 
    \sum_{i,j} g^{ij}(x) \, \iprod[H(z)] {\partial_i h(x)} {\partial_j h(x)},
     \qquad x \in D.
\end{displaymath}
This generalises Eq.~\eqref{def:quad.form} to the case of vector-valued
functions. We denote the closure by $q_D^{\mrt eq.}$ and its domain by
$\SobZ{D,H(z)}$. The corresponding non-negative operator on $\Lsqr {D,H(z)}$,
the so-called \emph{$R(z)$-equivariant Laplacian}, will be denoted by
$\laplacianZ D$.


%------------------------------------------------------------
\subsection{Non-abelian Floquet transformation}
%------------------------------------------------------------
\label{ssec:non-ab.floquet}
Next, we analyse the Floquet transformation
\begin{displaymath}
  \map U {\Lsqr X} {\Zint {\Lsqr {D,H(z)}}},
\end{displaymath}
which is the composition of the following three unitary transformations
(denoted with horizontal arrows)
\begin{displaymath}
  \begin{array}{ccr@{\otimes}lcr@{\otimes}lcc}
    \stackrel{T_\g} \curvearrowright & & % first line
    \multicolumn{2}{c}{\stackrel{L_\g \otimes \1} \curvearrowright \quad} && 
    \multicolumn{2}{c}{\stackrel{\widehat L_\g \otimes \1} 
                                                    \curvearrowright \quad} &&
    \\ 
    \Lsqr X & \!\!\!\!\to\!\!\!\! &        % second line
    \lsqr \G & \Lsqr D & \!\!\stackrel {F \otimes \id} \to\!\! & 
    \Zint {H(z)} & \Lsqr D & \!\!\!\!\to\!\!\!\!&
    \Zint {\Lsqr {D,H(z)}}\\[5ex]
    u & \mapsto &                      % third line
    \sum_\g \delta_\g & (T_{\g^{-1}} u \restr D) & &
    b & f & \mapsto & 
    (b(\rho)f)_\rho \;,
  \end{array} % ufff!!!
\end{displaymath}
where $(\delta_\gamma)_\gamma$ is the canonical orthogonal basis of
$\lsqr \G$ and $T_\gamma$ is the translation by $\gamma$ of functions
on $X$ given by~\eqref{eq:transl}.  Each of these transformations
intertwines with the unitary representation of $\G$ which are denoted
with curved arrows in the previous diagram.  The first horizontal
unitary transformation just splits a function on $\Lsqr X$ into a
sequence of $\g$-translates over the fundamental domain $D$. The
second horizontal unitary is essentially the Fourier transformation on
the group part and the last horizontal unitary is clear, since $\Lsqr
D$ is independent of $z\in Z$.

Note that periodic operators on $\Lsqr X$, i.e, operators commuting
with $T_\gamma$, are those commuting with $L_\gamma \otimes \1$ on
$\lsqr \G \otimes \Lsqr D$. Therefore, periodic operators are also
decomposable (recall that $\mathcal L' = \mathcal R \subset \mathcal
A'$).  Note in addition that if $\Gamma$ is not abelian then $\widehat
L_\gamma = F L_\gamma F^{-1}$ does not decompose with respect to the
direct integral specified in Section~\ref{ssec:non-ab.fourier}.

The Floquet transformation $U$ is given explicitely in the following theorem
(see also~\cite{sunada:88} and~\cite[Section~XIII.16]{reed-simon-4}):
\begin{theorem}
  \label{thm:floquet}
  The map
  \begin{equation}
    \label{eq:floquet.short}
    (Uu)(z)(x) = \sum_{\g \in \G} \,u(\gamma x)\;  R_{\g^{-1}}(z) v(z),
    \quad \text{where $v := F \delta_e$,}
  \end{equation}
  is a unitary transformation that intertwines the representations $T$ and
  $\Zint {R(z)}$. In addition, $U$ maps $\Cci X$ into $\Zint {\CiZ {D,H(z)}}$
  and operators on $\Lsqr X$ commuting with all $T_\g$'s are decomposable when
  transformed onto the direct integral.  In particular, the Laplacian
  $\laplacian X$ is unitary equivalent to $\Zint {\laplacianZ D}$ and
  \begin{equation}
  \label{eq:spec.dir.int}
    \spec \laplacian X \subseteq
    \overline {\bigcup_{z \in Z} \spec \laplacianZ D}.
  \end{equation}
\end{theorem}

%Note that $\widehat L_\gamma = F L_\gamma F^{-1}$ does not decompose with
%respect to the direct integral if $\Gamma$ is non-abelian: If
%$L_\gamma$ would be decomposable, then $\mathcal L = \mathcal R'
%\subset \mathcal A'$; since $\mathcal A$ is maximal abelian in
%$\mathcal R'$, this implies $\mathcal A = \mathcal R'$, i.e, $\mathcal
%R'=\mathcal L$ and therefore also $\Gamma$ would be abelian.

%------------------------------------------------------------
\subsection{The special case $\Gamma=\Z^d$ and $X=\R^d$.}
\label{ssec:z.d}
%------------------------------------------------------------
In this case the dual is simply given by the $d$-dimensional torus, i.e., $\hG
= \Torus^d$. Therefore, we can choose $Z = \Torus^d$ with Lebesgue measure
$d\theta$.  The Fourier transformation~\eqref{eq:fourier} reduces to the
standard formula
\begin{displaymath}
    \map F {\lsqr {\Z^d}} {\Lsqr {\Torus^d} = \Aintt {H(\theta)}}\;, \quad
    \mr with.~(Fa)(\theta) = \sum_\g \, \e^{-\im  \theta \cdot \g} a_\g
\end{displaymath}
where $H(\theta)=\C$ for a.e.~$\theta \in \Torus^d = \R^d/\Z^d$ and
$a=(a_\g) \in \lsqr {\Z^d}$.  Here, we only need scalar functions
since every irreducible representation of an \emph{abelian} group is
one-dimensional.  In this case, we can decompose \emph{both} the left
\emph{and} the right regular representation simultaneously; each fibre
of $R$, resp., $L$ is the multiplication with a phase factor
\begin{displaymath}
  \widehat R_\g(\theta) = 
               \e^{\im \theta \cdot \g}, % u(\theta)
     \quad \text{resp.,} \quad
  \widehat L_\g(\theta)  = 
               \e^{-\im \theta \cdot \g} % u(\theta).
\end{displaymath}
 The equivariant condition becomes
\begin{equation}
\label{def:equiv.fct.ab}
   h(x+\gamma) = \e^{-\im \theta \cdot \gamma} h(x)
\end{equation}
for all $x \in \R^d$ and $\gamma \in \Z^d$. A fundamental domain is the cube
$D := (0,1)^d$ and the Floquet transformation is given by
\begin{displaymath}
  (Uu)(\theta)(x) = 
     \sum_{\gamma \in \Z^d} u(x+\gamma) \e^{\im \theta \cdot \gamma}, 
        \qquad x \in D, \theta \in \Torus^d.
\end{displaymath}


%------------------------------------------------------------
\section{Existence of spectral gaps for non-abelian group actions}
%------------------------------------------------------------
\label{sec:main.result}
We will present in this section a method to show the existence of finitely
many spectral gaps for the Laplace operator $\laplacian X$ in the case of
\emph{non-abelian} group actions. Our main assumption on the group is the fact
that the irreducible representation appearing in the
decomposition~\eqref{eq:fourier} are finite-dimensional a.e., in other words
\begin{equation}
  \label{eq:dim.finite}
  \dim H(z) < \infty \qquad \text{for a.e.\ $z \in Z$}.
\end{equation}

The operators $\laplacianD D(z)$, $\laplacianZ D$, resp., $\laplacianN D(z)$
corresponding to the quadratic form~\eqref{eq:quad.form} on $\Sobn {D,H(z)}$,
$\SobZ {D,H(z)}$, resp., $\Sob {D,H(z)}$ have purely discrete spectrum which
we denote by $\EWD m(z)$, $\EWZ m$, resp., $\EWN m (z)$, $m \in \N$.  Recall
that the space $\Sobn {D,H(z)}$ is the $\HS^1$-closure of the space of smooth
functions $\map h D {H(z)}$ with compact support and $\Sob {D,H(z)}$ is the
closure of the space of smooth functions with derivatives continuous up to the
boundary (cf.~Section~\ref{sec:prelim}).

As in \eqref{eq:dom.mono} we obtain from the inclusion of the three
domains
\begin{displaymath}
  \Sob {D,H(z)} \supset \SobZ {D,H(z)} \supset \Sobn {D,H(z)}
\end{displaymath}
that the corresponding eigenvalues satisfy the following reverse inequalities
\begin{displaymath}
 \EWN m (z)  \le \EWZ m  \le \EWD m (z)
\end{displaymath}
for all $m \in \N$ and a.e.\ $z \in Z$.


From the definition of the quadratic form in the Dirichlet, resp., Neumann case
we have that the corresponding vector-valued Laplacians are a direct sum of
the scalar operators since there is no coupling between the components on the
boundary. In particular, if $n = \dim H(z)$, then $\laplacianDN D (z)$ is a
$n$-fold direct sum of the scalar operators $\laplacianDN D$ on $\Lsqr D$.
Therefore the eigenvalues of the corresponding vector-valued Laplace operators
consist of $n$-times repeated eigenvalues of the scalar Laplacians. We can
therefore arrange the former in the following way:
\begin{displaymath}
  \EWDN m (z) = \EWDN k, \qquad
  m=(k-1)n+1, \dots, kn\,,
\end{displaymath}
where $\EWDN k$ denotes the (scalar) $k$-th Dirichlet/Neumann eigenvalue on
$D$.

Recall the definition of the intervals $I_k := [\EWN k, \EWD k]$ in
Eq.~(\ref{eq:def.Ik}).  We may now collect the $n$ eigenvalues
of $\laplacianZ D$ which lie in $I_k$:
\begin{equation}
  \label{eq:band.z}
  B_k(z) := \set {\EWZ m} 
                  {m=(k-1)n+1, \dots, kn}
          \subset I_k,  \qquad n := \dim H(z)\,.
\end{equation}
Moreover we put together all eigenvalues 
corresponding to operators over the base point $z \in Z$ 
that act on Hilbert spaces with the same dimension:
\begin{equation}
  \label{eq:band.r}
    B_k(n) := \bigcup_{z \in Z, \, \dim H(z) = n} B_k(z) \subset I_k.
\end{equation}

\begin{theorem}
  \label{thm:spec.incl}
  Let $\Gamma$ be a finitely generated (in general non-abelian) group  
  satisfying $\dim H(z) < \infty$ for a.e.\ $z \in Z$ in the 
  decomposition~\eqref{eq:fourier}. Then
  \begin{displaymath}
    \spec \laplacian X \subseteq \bigcup_{k \in \N} I_k.
  \end{displaymath}
  In particular, the spectrum of $\laplacian X$ has a gap between $I_k$ and
  $I_{k+1}$ provided 
  \begin{displaymath}
         I_k \cap I_{k+1} = \emptyset\;.
  \end{displaymath}
\end{theorem}
\begin{proof}
  The proof is a consequence of the following chain of inclusions
  \begin{equation}
    \label{eq:spec.incl2}
    \spec \laplacian X \subseteq
    \overline{\bigcup_{z \in Z} \spec \laplacianZ D} =
    \overline{\bigcup_{z \in Z} \bigcup_{k \in \N} B_k(z)} =
    \overline{\bigcup_{n \in \N} \bigcup_{k \in \N} B_k(n)} \subseteq
    \bigcup_{k \in \N} I_k\,.
  \end{equation}
  For the first inclusion, we have applied~\eqref{eq:spec.dir.int}. For the
  second equality we have used $\dim H(z) < \infty$. Note finally that
  $\bigcup_k I_k$ is closed since $\EWDN k \to \infty$ for $k \to \infty$
  implies that $(I_k)_k$ is a locally finite family of closed sets.
\end{proof}

We formulate our main result, i.e., an analogue of
Theorem~\ref{thm:ex.gaps.ab} in the case of non-abelian groups $\Gamma$
satisfying the conditions stated in Section~\ref{sec:prelim}:

\begin{theorem}
  \label{thm:ex.gaps}
  Let $\Gamma$ be a finitely generated (in general non-abelian) group
  satisfying $\dim H(z) < \infty$ for a.e.\ $z \in Z$ in the
  decomposition~\eqref{eq:fourier}. Then we can always find a covering space
  $X \to X/\Gamma$ and a fundamental domain $D$ with the following property:
  For each $K\in\N$ there exist a Riemannian covering space $X \to
  X/\Gamma=:M$ with metric $g=g_K$ such that the Laplacian on $X$ has at least
  $K$ gaps, i.e., $\spec \laplacian X \subset [0, \infty)$ has at least $K$
  components.
\end{theorem}
\begin{proof}[Sketch of the proof:]
  Suppose that $\Gamma$ has $r$ generators $\varepsilon_i$.  To construct $X$
  we begin with a $d$-dimensional compact manifold $M$ with at least $r$
  handles which we may take diffeomorphic to $(0,1)\times\Sphere^{d-1}$. Then
  $D\subset M$ is the open subset obtained from $M$ by removing a section
  $\{1/2\} \times \Sphere^{d-1}$ from each of the $r$ handles. The set $D$ has
  therefore $2r$ cylindrical ends, for each generator a ``left'' and a
  ``right'' one.  Then there exists a covering $X \to X/\Gamma \cong M$ with
  fundamental domain $D$:  Intuitively, one can build up $X$ by glueing
  $\Gamma$ copies $(\gamma D)_\gamma$ of $D$, where one has to identify
  properly the points on the boundary (cf.~Fig.~\ref{fig:per-mfd} in the case
  $r=2$).  Concretely, we identify the ``left'' boundary part of the $i$-th
  cylindrical end of $\gamma_1 D$ with the ``right'' one of $\gamma_2 D$ iff
  $\gamma_2 = \eps_i \gamma_1$.  Finally, we change the metric on the handles
  in order to scale down the junctions between neighbouring copies of the
  fundamental domain (cf.~\cite{post:03a}). Note that the metric depends on
  the minimal number $K$ of gaps.  This implies that Eq.~\eqref{eq:gaps} is
  satisfied for at least $K$ indices $k \in \N$ and the proof is concluded by
  Theorem~\ref{thm:spec.incl}.
  
  Note that we have spectrum in each interval $I_k$, i.e., $\spec
  \laplacian X \cap I_k \ne \emptyset$ for all $k$, since a group
  satisfying~\eqref{eq:dim.finite} is amenable (cf.~the next section);
  therefore, $\spec \laplacian M \subset \spec \laplacian X$ (see
  e.g.~\cite[Prop.~7--8]{sunada:88}). Finally, $\spec \laplacian M
  \cap I_k \ne \emptyset$.
\end{proof}

Note that the previous statement does not give information on the
\emph{maximal} number of spectral gaps.  It still remains an open question if
there are (connected) covering spaces $X$ with an \emph{infinite} number of
spectral gaps.  This problem is related to the so-called
\emph{Bethe-Sommerfeld conjecture} (cf.~\cite{skriganov:87}).

%------------------------------------------------------------
\section{Examples}
%------------------------------------------------------------
\label{sec:examples}

We begin defining a class of a discrete groups that have particularly simple
properties (cf.~\cite{thoma:64}). A discrete group $\Gamma$ is of type~I iff
there is an exact sequence
\begin{displaymath}
  0 \longrightarrow 
  A \longrightarrow
  \G \longrightarrow
  \G_0 \longrightarrow 
  0,
\end{displaymath}
where $A$ is a finitely generated \emph{abelian} normal subgroup of $\Gamma$
and $\G_0 = \G / A$ is a \emph{finite} group (cf.~\cite{thoma:64}). Simple
examples of groups of type~I are abelian groups (in this case $\Gamma_0$ is
trivial) or direct, resp., semi-direct products of an abelian group with a
finite (in general non-abelian) group.

For these type of groups we have that all irreducible representations
are finite dimensional. In addition, such groups are also amenable (as
extensions of amenable groups), cf.~\cite{brooks:81}. Note that the
converse is also true: A discrete group such that all irreducible
representations are finite dimensional is of type~I
(cf.~\cite{moore:72}).

Recall that this was an important assumption in
Section~\ref{sec:main.result}.  Moreover in the direct integral decomposition
of Subsection~\ref{ssec:non-ab.floquet} we may take as measure space $Z$ the
dual $\hG$ of $\Gamma$. For these reasons we have

\begin{proposition}
  \label{thm:groups.type.I}
  Suppose $\G$ is a finitely generated group of type~I.  Then for each $K \in
  \N$ there exist a Riemannian covering space $X \to X/\Gamma$ with metric
  $g=g_K$ such that the Laplacian on $X$ has at least $K$ gaps, i.e., $\spec
  \laplacian X \subset [0, \infty)$ has at least $K$ components.
\end{proposition}
Of course, not all groups are of type~I. For example, free groups with more
than one generator are not of type~I. In~\cite{lledo-post:pre05} we provide
different methods and further classes of groups including the free groups for
which the conclusions of the above proposition remain true.


%------------------------------------------------------------
\section*{Appendix: Direct integral decomposition of unitary 
          group representations}
%------------------------------------------------------------

In the present appendix we will describe in more detail the direct integral
decomposition of the right regular representation given in
Eqs.~\eqref{def:reg.rep} and \eqref{eq:fourier} of
Section~\ref{sec:non-abelian}. It is an application of the direct integral
decomposition of von Neumann algebras. General references are
e.g.~\cite[Chapter~14]{wallach:92} or~\cite[Chapter~2]{mackey:76}.


In this appendix we will consider a more general frame that includes the
particular situation considered in Eqs.~\eqref{eq:fourier} and
\eqref{eq:reg.rep.trafo}, where $\Gamma$ is a discrete group satisfying the
conditions of Section~\ref{sec:prelim} and $R$ is the right regular
representation on $\ell_2(\Gamma)$.

Let $G$ be a separable locally compact group and let $V$ be a continuous
unitary representation of $G$ on a Hilbert space $\HS$. Denote by
\begin{displaymath}
  \al M.:=\{V_g \mid g\in G\}'' 
\end{displaymath}
the von Neumann algebra generated by the representation $V$
and let
\begin{displaymath}
 \al M.'=(V,V):=\{M'\in\al B.(\al H.)\mid M'  V_g=V_g  M'\;,\;g\in G\} 
\end{displaymath}
be the von Neumann algebra of operators commuting with the representation $V$.
If $\al A.$ is an abelian von Neumann subalgebra of $\al M.'$, then there
exists a compact, separable Hausdorff space $Z$, a regular Borel measure $dz$
on $Z$ and a unitary transformation onto a direct integral Hilbert space
\begin{equation}
  \label{eq:dir.int}\tag{A.1}
  \map F {\al H.}{\Zint {\al H.(z)}} ,
\end{equation}
such that
\begin{displaymath}
   F \al A. F^{-1}=\set {M_f} {f\in L_\infty(Z,dz)}
\end{displaymath}
($M_f$ being the multiplication operator with $f$). The von Neumann
algebra $F \mathcal A' F^{-1}$ consists of all decomposable operators
w.r.t.\ the direct integral~\eqref{eq:dir.int}, i.e., if $D \in F
\mathcal A' F^{-1}$ then we can write
\begin{displaymath}
  D = \Zint {D(z)}.
\end{displaymath}
In particular, $V_g \in \mathcal M \subset \mathcal A'$, and therefore
\begin{displaymath}
  F  V_g  F^{-1}= \Zint {V_g(z)},
\end{displaymath}
where $V(z)$ is a unitary representation of $G$ on $H(z)$ a.e. (see
\cite[Section~14.8~ff.]{wallach:92}). There are several natural choices for
the abelian von Neumann algebra $\al A.$:
\begin{itemize}
\item[(i)] If $\al A.=\al M.\cap\al M.'$ is the centre of $\al M.$, then, for
  a.e.~$z\in Z$, the von Neumann algebra generated by the representations
  $V(z)$ are factors, i.e.,
\begin{displaymath}
 \al M.(z)\cap\al M.(z)'
    :=\{V_g(z)\mid g\in G\}''\cap\{V_g(z)\mid g\in G\}'=\C\1_{\al H.(z)}.
\end{displaymath} 
This choice is due to von Neumann.

\item[(ii)] If $\al A.$ is maximal abelian in $\al M.'$, i.e., $\al A.=\al
  A.'\cap\al M.'$, then the components $V(z)$ of the direct integral
  decomposition of $V$ are irreducible a.e.  This choice is due to Mautner and
  was used in Eqs.~\eqref{eq:fourier} and~\eqref{eq:reg.rep.trafo} of
  Section~\ref{sec:non-abelian}.
\end{itemize}

Finally, we mention a class of groups, where the previous decomposition
results become particularly simple.  A group $G$ is of type~I if all its
unitary continuous representations $V$ are of type~I, i.e., each $V$ is
quasi-equivalent to some multiplicity free representation. Compact or abelian
groups are examples of type~I groups. If $G$ is of type~I, then the dual
$\widehat{G}$ (i.e., the set of all equivalence classes of continuous unitary
irreducible representations of $G$) becomes a nice measure space (``smooth''
in the terminology of \cite[Chapter~2]{mackey:76}).  In this case one can take
$\widehat{G}$ as the measure space $Z$ in the Mautner decomposition (ii).  For
discrete groups the previous definition of Type~I is equivalent to the one
given in Section~\ref{sec:examples} (cf.~\cite{thoma:64}).


%------------------------------------------------------------
\section*{Acknowledgements}
%------------------------------------------------------------ 
The second author would like to thank the organisers of the Young Researchers
Symposium at the Instituto Superior T\'ecnico (Lisbon) in July 2003 for the
kind invitation. We would also like to thank Volker En{\ss} for useful remarks
on the manuscript.


%------------------------------------------------------------
% References
%------------------------------------------------------------

% Uncomment the next two lines for bibtex
%\bibliographystyle{amsalpha}% or with numbers: amsplain
%\bibliography{/home/post/Aktuell/BibTeX/literatur}
%\input nc-floquet.bbl

%--------------------------------
% include file "muster.bbl"
%--------------------------------

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\end{thebibliography}


\end{document}

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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
0 slj
 [15 15] 15 sd
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% Polyline
 [15 15] 15 sd
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% Polyline
2 slj
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 cp gs col0 s gr 
% Polyline
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% Polyline
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% Polyline
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% Polyline
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 1269 369 l gs col0 s gr 
% Polyline
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 2674 424 l gs col0 s gr 
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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 4836 424 l gs col0 s gr 
% Polyline
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 4790 308 l 4796 310 l 4802 311 l 4810 312 l 4819 313 l
 4829 314 l 4836 315 l
 4837 315 l gs col0 s gr 
% Polyline
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 4534 364 l 4538 363 l
 4540 362 l gs col0 s gr 
% Polyline
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 4477 355 l 4489 360 l 4498 364 l 4505 367 l 4509 368 l

 4511 369 l gs col0 s gr 
% Polyline
0 slj
 [15 15] 15 sd
n 270 360 m
 405 360 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 4950 360 m
 5085 360 l gs col0 s gr  [] 0 sd
% Polyline
2 slj
 [15 15] 15 sd
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 4168 967 l gs col0 s gr  [] 0 sd
% Polyline
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 1260 720 l gs col0 s gr 
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 [15 15] 15 sd
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% Polyline
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 [15 15] 15 sd
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 cp gs col0 s gr  [] 0 sd
% Polyline
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% Polyline
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 cp gs col7 1.00 shd ef gr 
% Polyline
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 [15 15] 15 sd
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 cp gs col0 s gr  [] 0 sd
% Polyline
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 [15 15] 15 sd
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currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueID 5000768 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 78 /N put
dup 84 /T put
dup 90 /Z put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
/UniqueID 5032014 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
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1B212319637AD43BEA4859A51CBF1D7C68B9B66CE85E68F3C626990890928D20
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cleartomark
%%EndFont 
%%BeginFont: CMTI8
%!PS-AdobeFont-1.1: CMTI8 1.0
%%CreationDate: 1991 Aug 18 21:07:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueID 5000826 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE
3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B
532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470
B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B
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D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5
525003F3DAD7933EB57E7DB1462E9D906F6D8F5BF740206C1EC5F36E00AAFF68
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3A43642CB49A7E81EFF73815E24F4B4FA5EA61086978DCACE8D2CB59B80E4A58
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cleartomark
%%EndFont 
%%BeginFont: MSAM10
%!PS-AdobeFont-1.1: MSAM10 2.1
%%CreationDate: 1993 Sep 17 09:05:00
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSAM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSAM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 3 /square put
dup 22 /harpoonupright put
readonly def
/FontBBox{8 -463 1331 1003}readonly def
/UniqueID 5031981 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
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DB3C
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cleartomark
%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
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cleartomark
%%EndFont 
%%BeginFont: TeX-cmex8
%!PS-AdobeFont-1.0: TeX-cmex8 001.001
% Filtered by type1fix.pl 0.05
%%EndComments
13 dict dup begin
/FontInfo 16 dict dup begin
/Copyright (see\040copyright\040of\040original\040TeX\040font) def
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/Notice (converted\040after\040April\0402001) def
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end readonly def
/FontName /TeX-cmex8 def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 98 /b put
dup 101 /e put
readonly def
/FontBBox {-29 -2957 1554 772} readonly def
/UniqueID 4314416 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueID 5000816 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
/UniqueID 5000829 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 80 /summationtext put
dup 82 /integraltext put
dup 83 /uniontext put
dup 88 /summationdisplay put
dup 90 /integraldisplay put
dup 91 /uniondisplay put
dup 98 /hatwide put
dup 101 /tildewide put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueID 5000792 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
dup 78 /N put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 90 /Z put
dup 121 /archrightdown put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueID 5087386 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueID 5000769 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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 cp gs col7 0.80 shd ef gr 
% Polyline
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 cp gs col7 0.80 shd ef gr 
% Polyline
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 cp gs col7 0.80 shd ef gr 
% Polyline
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 cp gs col7 0.80 shd ef gr 
% Polyline
7.500 slw
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 cp gs col0 s gr 
% Polyline
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% Polyline
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 1988 854 l
 1990 854 l gs col0 s gr 
% Polyline
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 1703 902 l gs col0 s gr 
% Polyline
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 1674 909 l gs col0 s gr 
% Polyline
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 3079 964 l gs col0 s gr 
% Polyline
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 3080 855 l gs col0 s gr 
% Polyline
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 2784 902 l gs col0 s gr 
% Polyline
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% Polyline
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 4160 964 l gs col0 s gr 
% Polyline
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 4161 855 l gs col0 s gr 
% Polyline
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 3864 902 l gs col0 s gr 
% Polyline
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 3835 909 l gs col0 s gr 
% Polyline
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 5241 970 l gs col0 s gr 
% Polyline
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 5241 964 l gs col0 s gr 
% Polyline
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 5242 855 l gs col0 s gr 
% Polyline
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 4945 902 l gs col0 s gr 
% Polyline
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 4916 909 l gs col0 s gr 
% Polyline
0 slj
 [15 15] 15 sd
n 675 900 m
 810 900 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 5355 900 m
 5490 900 l gs col0 s gr  [] 0 sd
% Polyline
2 slj
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 cp gs col0 s gr 
% Polyline
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 1594 424 l gs col0 s gr 
% Polyline
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 1585 314 l gs col0 s gr 
% Polyline
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 1298 362 l gs col0 s gr 
% Polyline
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 1269 369 l gs col0 s gr 
% Polyline
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 2674 424 l gs col0 s gr 
% Polyline
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 1779 226 l 1786 220 l 1793 214 l 1801 209 l 1808 204 l
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 1917 161 l 1929 158 l 1940 156 l 1952 153 l 1964 151 l
 1976 150 l 1987 148 l 1999 146 l 2012 145 l 2025 144 l
 2039 142 l 2054 141 l 2069 140 l 2083 140 l 2097 139 l
 2110 139 l 2123 139 l 2135 139 l 2147 139 l 2160 140 l
 2174 140 l 2187 141 l 2200 142 l 2213 143 l 2225 145 l
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% Polyline
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% Polyline
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 2350 369 l gs col0 s gr 
% Polyline
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 3755 424 l gs col0 s gr 
% Polyline
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 3317 146 l 3328 147 l 3338 149 l 3349 150 l 3360 152 l
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 3628 269 l 3637 274 l 3646 279 l 3655 284 l 3662 288 l
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 3709 308 l 3716 310 l 3722 311 l 3729 312 l 3739 313 l
 3748 314 l 3755 315 l
 3756 315 l gs col0 s gr 
% Polyline
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 3459 362 l gs col0 s gr 
% Polyline
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 3430 369 l gs col0 s gr 
% Polyline
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% Polyline
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 4830 424 l 4835 424 l
 4836 424 l gs col0 s gr 
% Polyline
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 4709 269 l 4718 274 l 4727 279 l 4736 284 l 4743 288 l
 4750 291 l 4758 295 l 4767 299 l 4775 302 l 4783 305 l
 4790 308 l 4796 310 l 4802 311 l 4810 312 l 4819 313 l
 4829 314 l 4836 315 l
 4837 315 l gs col0 s gr 
% Polyline
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 4217 400 l 4248 404 l 4283 406 l 4314 406 l 4343 405 l
 4369 403 l 4394 400 l 4416 396 l 4437 392 l 4456 388 l
 4474 383 l 4491 378 l 4505 374 l 4518 370 l 4528 366 l
 4534 364 l 4538 363 l
 4540 362 l gs col0 s gr 
% Polyline
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 4140 344 l 4158 340 l 4179 336 l 4200 332 l 4224 329 l
 4250 327 l 4278 325 l 4309 325 l 4339 326 l 4366 329 l
 4390 332 l 4411 336 l 4430 341 l 4447 346 l 4463 351 l
 4477 355 l 4489 360 l 4498 364 l 4505 367 l 4509 368 l

 4511 369 l gs col0 s gr 
% Polyline
0 slj
 [15 15] 15 sd
n 270 360 m
 405 360 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 4950 360 m
 5085 360 l gs col0 s gr  [] 0 sd
% Polyline
2 slj
 [15 15] 15 sd
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 4169 966 l
 4168 967 l gs col0 s gr  [] 0 sd
% Polyline
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 1187 630 l 1201 647 l 1216 664 l 1230 682 l 1242 697 l
 1251 709 l 1257 716 l
 1260 720 l gs col0 s gr 
% Polyline
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 1390 626 l 1404 640 l 1417 653 l 1429 664 l 1436 671 l
 1439 674 l
 1440 675 l gs col0 s gr 
% Polyline
0.000 slw
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 1277 530 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
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 2312 630 l 2326 647 l 2341 664 l 2355 682 l 2367 697 l
 2376 709 l 2382 716 l
 2385 720 l gs col0 s gr 
% Polyline
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 2515 626 l 2529 640 l 2542 653 l 2554 664 l 2561 671 l
 2564 674 l
 2565 675 l gs col0 s gr 
% Polyline
0.000 slw
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 2295 582 l 2297 579 l 2303 572 l 2311 565 l 2321 558 l
 2333 551 l 2347 544 l 2361 539 l 2375 534 l 2389 531 l
 2402 530 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
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 3411 709 l 3417 716 l
 3420 720 l gs col0 s gr 
% Polyline
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 3517 515 l 3511 529 l 3507 541 l 3506 552 l 3507 562 l
 3510 571 l 3514 580 l 3520 589 l 3528 600 l 3538 612 l
 3550 626 l 3564 640 l 3577 653 l 3589 664 l 3596 671 l
 3599 674 l
 3600 675 l gs col0 s gr 
% Polyline
0.000 slw
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 3490 555 l 3494 561 l 3498 568 l 3501 574 l 3502 581 l
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 3330 582 l 3332 579 l 3338 572 l 3346 565 l 3356 558 l
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 3437 530 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
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 4500 720 l gs col0 s gr 
% Polyline
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 4630 626 l 4644 640 l 4657 653 l 4669 664 l 4676 671 l
 4679 674 l
 4680 675 l gs col0 s gr 
% Polyline
0.000 slw
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 4410 582 l 4412 579 l 4418 572 l 4426 565 l 4436 558 l
 4448 551 l 4462 544 l 4476 539 l 4490 534 l 4504 531 l
 4517 530 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
 [15 15] 15 sd
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 [15 15] 15 sd
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 cp gs col0 s gr  [] 0 sd
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$F2psEnd
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%%EndDocument
 @endspecial 790 563 a Ft(")836 578 y Fq(2)1173 621 y
Ft(")1219 636 y Fq(1)1995 918 y Ft(X)2916 679 y(D)419
1111 y Fm(Figure)32 b(1.)49 b Fv(A)28 b(p)s(erio)s(dic)h(\(or)e(co)m(v)
m(ering\))j(manifold)e Ft(X)36 b Fv(with)28 b(group)g(\000)g(gen-)419
1227 y(erated)f(b)m(y)h(t)m(w)m(o)g(elemen)m(ts)h Ft(")1459
1242 y Fq(1)1498 1227 y Ft(;)17 b(")1588 1242 y Fq(2)1654
1227 y Fv(and)27 b(fundamen)m(tal)h(domain)g Ft(D)h Fv(\(in)f(grey\).)
419 1343 y(Here,)33 b(the)g(group)g(\000)f(is)h(ab)s(elian.)120
1603 y Ft(D)s Fv(,)f(i.e.,)h(an)f(op)s(en)g(set)h Ft(D)d
Fu(\032)e Ft(X)40 b Fv(suc)m(h)33 b(that)f Ft(\015)5
b(D)35 b Fv(and)d Ft(\015)2153 1567 y Fl(0)2176 1603
y Ft(D)j Fv(are)d(disjoin)m(t)h(for)e(all)h Ft(\015)h
Fu(6)p Fv(=)28 b Ft(\015)3328 1567 y Fl(0)3383 1603 y
Fv(and)120 1645 y Fo(S)203 1749 y Fr(\015)t Fl(2)p Fq(\000)355
1720 y Ft(\015)p 411 1640 84 4 v 5 w(D)j Fv(=)c Ft(X)40
b Fv(\(cf.)33 b(Figure)g(1\).)269 1839 y(As)38 b(a)f(protot)m(yp)s(e)g
(for)g(an)g(elliptic)h(op)s(erator)e(w)m(e)j(consider)f(the)f
(Laplacian)h(\001)3207 1854 y Fr(X)3311 1839 y Fv(on)f
Ft(X)120 1955 y Fv(acting)43 b(on)f(a)g(dense)i(subspace)g(of)e(the)h
(Hilb)s(ert)g(space)h Ft(L)2334 1970 y Fq(2)2374 1955
y Fv(\()p Ft(X)8 b Fv(\))42 b(with)h(norm)f Fu(k\001k)3205
1970 y Fr(X)3272 1955 y Fv(.)73 b(The)120 2072 y(p)s(ositiv)m(e)33
b(self-adjoin)m(t)f(op)s(erator)f(\001)1454 2087 y Fr(X)1554
2072 y Fv(can)h(b)s(e)g(de\014ned)h(in)f(terms)h(of)e(a)g(suitable)i
(quadratic)120 2188 y(form)f Ft(q)393 2203 y Fr(X)493
2188 y Fv(\(see)i(e.g.)f([)p Fw(K95)p Fv(,)f(Chapter)i(VI],)f([)p
Fw(RS80)p Fv(])g(or)f([)p Fw(Da96)p Fv(]\).)44 b(Concretely)34
b(w)m(e)g(ha)m(v)m(e)829 2405 y Ft(q)872 2420 y Fr(X)940
2405 y Fv(\()p Ft(u)p Fv(\))27 b(:=)g Fu(k)p Ft(du)o
Fu(k)1435 2364 y Fq(2)1435 2430 y Fr(X)1530 2405 y Fv(=)1634
2270 y Fo(Z)1689 2495 y Fr(X)1773 2405 y Fu(j)p Ft(du)p
Fu(j)1936 2364 y Fq(2)1974 2405 y Ft(dX)35 b(;)115 b(u)27
b Fu(2)h Ft(C)2537 2364 y Fl(1)2530 2430 y Fq(c)2612
2405 y Fv(\()p Ft(X)8 b Fv(\))27 b Ft(:)508 b Fv(\(2.1\))120
2621 y(In)33 b(co)s(ordinates)g(w)m(e)h(write)f(the)g(p)s(oin)m(t)m
(wise)h(norm)f(of)f(the)h(1-form)f Ft(du)f Fv(as)1359
2802 y Fu(j)p Ft(du)p Fu(j)1522 2761 y Fq(2)1587 2802
y Fv(=)1691 2708 y Fo(X)1725 2918 y Fr(i;j)1851 2802
y Ft(g)1902 2761 y Fr(ij)1962 2802 y Ft(@)2013 2817 y
Fr(i)2042 2802 y Ft(u)17 b(@)2166 2817 y Fr(j)p 2202
2748 56 4 v 2202 2802 a Ft(u)f(;)120 3078 y Fv(where)32
b(\()p Ft(g)489 3042 y Fr(ij)548 3078 y Fv(\))e(is)h(the)g(in)m(v)m
(erse)i(of)c(the)i(metric)g(tensor)g(\()p Ft(g)2150 3093
y Fr(ij)2210 3078 y Fv(\))g(in)f(a)g(c)m(hart.)43 b(T)-8
b(aking)31 b(the)g(closure)120 3194 y(of)h(the)h(quadratic)g(form)g(w)m
(e)g(can)g(extend)h Ft(q)1748 3209 y Fr(X)1848 3194 y
Fv(on)m(to)f(the)g(Sob)s(olev)g(space)1041 3359 y Fu(H)1126
3318 y Fq(1)1165 3359 y Fv(\()p Ft(X)8 b Fv(\))28 b(=)f
Fu(f)17 b Ft(u)27 b Fu(2)h Ft(L)1771 3374 y Fq(2)1811
3359 y Fv(\()p Ft(X)8 b Fv(\))17 b Fu(j)g Ft(q)2081 3374
y Fr(X)2147 3359 y Fv(\()p Ft(u)p Fv(\))27 b Ft(<)g Fu(1)17
b(g)g Ft(:)120 3524 y Fv(As)46 b(usual)h(the)f(op)s(erator)f(\001)1214
3539 y Fr(X)1327 3524 y Fv(is)h(related)g(with)g(the)g(quadratic)h
(form)e(b)m(y)i(the)f(form)m(ula)120 3640 y Fu(h)p Fv(\001)240
3655 y Fr(X)307 3640 y Ft(u;)17 b(u)p Fu(i)58 b Fv(=)d
Ft(q)734 3655 y Fr(X)801 3640 y Fv(\()p Ft(u)p Fv(\),)d
Ft(u)i Fu(2)i Ft(C)1321 3604 y Fl(1)1314 3665 y Fq(c)1395
3640 y Fv(\()p Ft(X)8 b Fv(\).)91 b(Since)50 b(the)f(metric)g(on)g
Ft(X)56 b Fv(is)49 b(\000-in)m(v)-5 b(arian)m(t,)53 b(the)120
3756 y(Laplacian)32 b(\001)646 3771 y Fr(X)746 3756 y
Fv(\(i.e.,)g(its)h(resolv)m(en)m(t\))g(comm)m(utes)h(with)f(the)f
(translations)h(on)e Ft(X)40 b Fv(giv)m(en)33 b(b)m(y)951
3922 y(\()p Ft(T)1046 3937 y Fr(\015)1090 3922 y Ft(u)p
Fv(\)\()p Ft(x)p Fv(\))28 b(:=)f Ft(u)p Fv(\()p Ft(\015)1623
3881 y Fl(\000)p Fq(1)1717 3922 y Ft(x)p Fv(\))p Ft(;)115
b(u)27 b Fu(2)h Ft(L)2195 3937 y Fq(2)2235 3922 y Fv(\()p
Ft(X)8 b Fv(\))p Ft(;)17 b(\015)32 b Fu(2)c Fv(\000)p
Ft(:)630 b Fv(\(2.2\))120 4087 y(Op)s(erators)32 b(with)i(this)f(prop)s
(ert)m(y)g(are)g(called)g Fn(p)-5 b(erio)g(dic)p Fv(.)269
4203 y(F)d(or)50 b(an)g(op)s(en,)55 b(relativ)m(ely)d(compact)e(subset)
i Ft(D)60 b Fu(\032)e Ft(X)g Fv(with)51 b(su\016cien)m(tly)i(smo)s(oth)
120 4320 y(b)s(oundary)44 b Ft(@)5 b(D)48 b Fv(\(e.g.)c(Lipsc)m(hitz\))
i(w)m(e)f(de\014ne)g(the)f(Diric)m(hlet)h(\(resp)s(ectiv)m(ely)-8
b(,)50 b(Neumann\))120 4436 y(Laplacian)45 b(\001)659
4395 y Fq(+)659 4463 y Fr(D)768 4436 y Fv(\(resp.,)k(\001)1163
4395 y Fl(\000)1163 4463 y Fr(D)1227 4436 y Fv(\))c(via)g(its)g
(quadratic)g(form)g Ft(q)2368 4395 y Fq(+)2364 4463 y
Fr(D)2473 4436 y Fv(\(resp.,)k Ft(q)2834 4395 y Fl(\000)2830
4463 y Fr(D)2894 4436 y Fv(\))c(asso)s(ciated)g(to)120
4552 y(the)38 b(closure)i(of)d Ft(q)784 4567 y Fr(D)886
4552 y Fv(on)h Ft(C)1104 4516 y Fl(1)1097 4577 y Fq(c)1179
4552 y Fv(\()p Ft(D)s Fv(\),)h(the)f(space)i(of)d(smo)s(oth)h
(functions)h(with)g(compact)g(sup-)120 4673 y(p)s(ort,)53
b(\(resp.)109 b Ft(C)806 4637 y Fl(1)881 4673 y Fv(\()p
919 4593 84 4 v Ft(D)r Fv(\),)53 b(the)d(space)g(of)e(smo)s(oth)h
(functions)h(with)f(con)m(tin)m(uous)i(deriv)-5 b(a-)120
4798 y(tiv)m(es)42 b(up)g(to)e(the)i(b)s(oundary\).)68
b(W)-8 b(e)42 b(also)f(use)g(the)h(notation)2518 4730
y Fk(\016)2481 4798 y Fu(H)2566 4762 y Fq(1)2606 4798
y Fv(\()p Ft(D)s Fv(\))f(=)h(dom)17 b Ft(q)3173 4757
y Fq(+)3169 4825 y Fr(D)3274 4798 y Fv(\(resp.,)120 4915
y Fu(H)205 4879 y Fq(1)245 4915 y Fv(\()p Ft(D)s Fv(\))46
b(=)h(dom)17 b Ft(q)822 4874 y Fl(\000)818 4942 y Fr(D)882
4915 y Fv(\).)77 b(Note)44 b(that)g(the)g(usual)h(b)s(oundary)f
(condition)g(of)g(the)g(Neumann)120 5031 y(Laplacian)27
b(o)s(ccurs)h(only)g(in)f(the)h Fn(op)-5 b(er)g(ator)37
b Fv(domain)28 b(via)f(the)h(Gau\031-Green)e(form)m(ula.)42
b(Since)p 120 5067 V 120 5147 a Ft(D)47 b Fv(is)f(compact,)i(\001)872
5106 y Fq(+)872 5174 y Fr(D)981 5147 y Fv(has)e(purely)g(discrete)g(sp)
s(ectrum)g Ft(\025)2346 5106 y Fq(+)2346 5175 y Fr(k)2405
5147 y Fv(,)i Ft(k)k Fu(2)d Fs(N)p Fv(.)80 b(It)45 b(is)g(written)h(in)
p eop end
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(al)e(gaps,)k(c)l(overing)e(manifolds)g(and)f(r)l(esidual)t(ly)578
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b(\(2005\).)240 2117 y([M76])132 b(G.)36 b(W.)g(Mac)n(k)n(ey)-7
b(,)37 b Fx(The)i(the)l(ory)f(of)g(unitary)f(gr)l(oup)h(r)l(epr)l
(esentations)p FA(,)h(Univ)n(ersit)n(y)c(of)g(Chicago)578
2217 y(Press,)26 b(Chicago,)g(1976.)240 2316 y([Mo72])90
b(C.)23 b(C.)f(Mo)r(ore,)h Fx(Gr)l(oups)i(with)h(\014nite)f
(dimensional)i(irr)l(e)l(ducible)f(r)l(epr)l(esentations)p
FA(,)e(T)-7 b(rans.)22 b(Amer.)578 2416 y(Math.)28 b(So)r(c.)f
Fb(166)g FA(\(1972\),)f(401{410.)240 2516 y([P03])151
b(O.)27 b(P)n(ost,)g Fx(Perio)l(dic)32 b(manifolds)f(with)g(sp)l(e)l
(ctr)l(al)f(gaps)p FA(,)e(J.)g(Di\013.)g(Equations)f
Fb(187)g FA(\(2003\),)f(23{45.)240 2615 y([RS80])101
b(M.)28 b(Reed)f(and)g(B.)h(Simon,)f Fx(Metho)l(ds)k(of)g(mo)l(dern)f
(mathematic)l(al)h(physics)g(I:)f(F)-6 b(unctional)30
b(anal-)578 2715 y(ysis)p FA(,)e(Academic)g(Press,)e(New)i(Y)-7
b(ork,)27 b(1980.)240 2814 y([RS78])101 b(M.)39 b(Reed)g(and)f(B.)h
(Simon,)j Fx(Metho)l(ds)f(of)g(mo)l(dern)f(mathematic)l(al)i(physics)f
(IV:)g(A)n(nalysis)f(of)578 2914 y(op)l(er)l(ators)p
FA(,)29 b(Academic)e(Press,)f(New)i(Y)-7 b(ork,)27 b(1978.)240
3014 y([Sk87])118 b(M.)35 b(M.)h(Skrigano)n(v,)f Fx(Ge)l(ometric)i(and)
h(arithmetic)g(metho)l(ds)f(in)g(the)g(sp)l(e)l(ctr)l(al)g(the)l(ory)g
(of)h(mul-)578 3113 y(tidimensional)i(p)l(erio)l(dic)g(op)l(er)l(ators)
p FA(,)g(Pro)r(c.)c(Steklo)n(v)g(Inst.)h(Math.)g(\(1987\),)g(no.)g
(171,)g(1{121,)578 3213 y(\(translation\).)240 3313 y([Su88])116
b(T.)39 b(Sunada,)i Fx(F)-6 b(undamental)40 b(gr)l(oups)h(and)f(L)l
(aplacians)p FA(,)k(Geometry)39 b(and)f(analysis)g(on)g(mani-)578
3412 y(folds,)25 b(Lecture)f(Notes)g(Mathematics)g(1339)e(\(Katata/Ky)n
(oto,)g(1987\),)h(Springer,)h(Berlin,)h(1988,)578 3512
y(pp.)j(248{277.)240 3613 y([Th64])102 b(E.)35 b(Thoma,)1026
3592 y Fx(\177)1011 3613 y(Ub)l(er)i(unit\177)-42 b(ar)l(e)36
b(Darstel)t(lungen)h(abz\177)-42 b(ahlb)l(ar)l(er,)41
b(diskr)l(eter)c(Grupp)l(en)p FA(,)h(Math.)d(Ann.)578
3712 y Fb(153)27 b FA(\(1964\),)f(111{138.)240 3812 y([W92])123
b(N.)35 b(R.)g(W)-7 b(allac)n(h,)36 b Fx(R)l(e)l(al)g(r)l(e)l(ductive)h
(gr)l(oups.)g(II)p FA(,)d(Pure)g(and)h(Applied)g(Mathematics,)h(v)n
(ol.)e(132,)578 3912 y(Academic)27 b(Press,)g(Boston,)f(1992.)389
4098 y FB(Institut)c(f)804 4091 y(\177)801 4098 y(ur)g(Reine)h(und)e
(Angew)-7 b(andte)22 b(Ma)-6 b(thema)g(tik,)24 b(Rheinisch-Westf)3170
4091 y(\177)3167 4098 y(alische)f(Tech-)240 4198 y(nische)32
b(Hochschule)e(Aa)n(chen,)h(Templer)n(graben)i(55,)d(D-52062)h(Aa)n
(chen,)g(Germany)389 4298 y Fx(E-mail)g(addr)l(ess)7
b FA(:)38 b Fa(lledo@iram.rwth-)o(aac)o(he)o(n.d)o(e)389
4467 y Fx(E-mail)31 b(addr)l(ess)7 b FA(:)38 b Fa(post@iram.rwth-a)o
(ach)o(en)o(.de)p eop end
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