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\begin{document}

\title[Geometric operators]{On spectra of geometric operators on open
manifolds and differentiable groupoids}

\author[R. Lauter]{Robert Lauter}
\address{Universit\"at
Mainz, Fachbereich 17-Mathematik, D-55099 Mainz, Germany}
\email{lauter@mathematik.uni-mainz.de}
\thanks{Lauter was partly supported by a scholarship of the German
Academic Exchange Service (DAAD) within the {\em
Hochschulsonderprogramm III von Bund und L\"andern}, and the
Sonderforschungsbereich 478 {\em Geometrische Strukturen in der
Mathematik} at the University of M\"unster. Nistor was partially
supported by NSF Young Investigator Award DMS-9457859 and NSF Grant
DMS-9971951.}


\author[V. Nistor]{Victor Nistor}
\address{Pennsylvania State
University, Department of Mathematics, University Park, PA 16802}
\email{nistor@math.psu.edu}
\urladdr{http://www.math.psu.edu/nistor/}


\commby{Michael Taylor}
\date{May 30, 2000}
\revdate{February 14, 2001}
\subjclass[2000]{Primary 58J50; Secondary 58H05, 47G30, 58J40}
\keywords{Laplace operator, pseudodifferential operator,
$C^*$-algebra, groupoid, essential spectrum}

\begin{abstract}
We use a pseudodifferential calculus on differentiable groupoids to
obtain new analytical results on geometric operators on certain
noncompact Riemannian manifolds. The first step is to establish that
the geometric operators belong to a pseudodifferential calculus on an
associated differentiable groupoid. This then leads to Fredholmness
criteria for geometric operators on suitable noncompact manifolds, as
well as to an inductive procedure to compute their essential spectra.
As an application, we answer a question of Melrose on the essential
spectrum of the Laplace operator on manifolds with multicylindrical
ends.
\end{abstract}

\maketitle


\section*{Introduction}

On a compact manifold, the spectrum of an elliptic selfadjoint
operator of positive order consists of isolated eigenvalues of finite
multiplicities. In particular, the essential spectrum of such an
operator is empty. These facts are part of the ``elliptic package'',
which also includes boundedness, Fredholmness, and compactness
criteria for (pseudo)differential operators acting between Sobolev
spaces. It is well-known that the assumption of compactness of the
underlying manifold is essential
%N
for some
of these results.

For example, on a noncompact manifold with cylindrical ends the
essential spectrum of the Laplace operator is nonempty.  A manifold
with multicylindrical ends is (locally at infinity) a product of
manifolds with cylindrical ends.  In \cite{MelroseScattering}
(Conjecture 7.1), Melrose conjectured that the spectrum of the Laplace
operator on a manifold with multicylindrical ends has a precise form
determined by Laplace operators on certain canonical lower-dimensional
manifolds. In Theorem \ref{mainth} we answer this question of Melrose
in the affirmative. In a certain sense, one can view this theorem as
part of an ``elliptic package'' for manifolds with multicylindrical
ends.
%%%>>> MOVED BELOW
%However, it should be pointed out that, at this point,
%understanding the fine-structure of the spectrum requires some
%additional analysis that depends on more specific properties of the
%geometry of the noncompact manifold studied.

Indeed, we show that there exists an elliptic package for a larger
class of noncompact manifolds. The first part of this note is devoted
to reviewing some results in this direction. We then use some of these
results to answer Melrose's question.
%%%>>>
However, it should be pointed out that, at this point, understanding
the fine structure of spectra requires some additional analysis that
depends on more specific properties of the geometry of the noncompact
manifold studied.

Our results apply to manifolds modeled by groupoids and use the
pseudodifferential calculus on groupoids developed in
\cite{ConnesF,LMN,LN,MelroseFB,Monthubert} and \cite{NWX}.  We begin
by quickly and informally reviewing some of the necessary definitions,
%%%>>>
%first,
including the definition of a differential groupoid $\GR$ and that of
the algebra $\tPS{\infty}$ of pseudodifferential operators on $\GR$.
The precise relation between the open manifold $M_0$ we study and
groupoids is that $M_0$ has a compactification to a manifold with
corners $M$, on which we are given a vector bundle $A$ such that
$A\vert_{M_0} \cong TM_0$, and $\Gamma(A)$ is naturally a Lie algebra
with respect to the bracket induced by the Lie bracket of vector
fields. (Thus, the vector bundle $A$ is a Lie algebroid of a
particular kind.)  This is in the spirit of Melrose's approach to a
pseudodifferential analysis on manifolds with corners and geometric
scattering theory \cite{MelroseScattering}.


A metric on $A$ as above restricts to a metric on $TM_0$, so that
$M_0$ becomes, naturally, a Riemannian manifold. We are interested in
studying the geometric operators associated to this metric (Dirac,
Laplace, and so on). Our approach is along the lines of
\cite{MelroseScattering}. Thus, we first integrate $A$ to a
differential groupoid $\GR$. This step has to be carried out in
detail, as it is not true that every Lie algebroid is integrable.
Then we check that the geometric operators on $M_0$ belong to our
calculus $\tPS{\infty}$ (or $\PS{\infty}$, if they act on a vector
bundle $E \to M$).

The restriction (an operation that has to be properly defined) of an
operator in $\tPS{\infty}$ to a hyperface $H$ of $M$ belongs to the
pseudodifferential calculus on the restricted groupoid $\GR_H$.  The
restriction of a geometric operator to a hyperface is in fact also a
geometric operator of the same kind.  As expected from the work of
Melrose on the $b$-calculus and related algebras, the results in the
elliptic package for the action on $M_0$ of an operator $P \in
\PS{\infty}$ are formulated not only in terms of the principal symbol
of $P$, but also in terms of the restriction of $P$ to the hyperfaces
of $M$.  For suitable $\GR$, these results include:
\begin{itemize}
\item {\em Order zero operators are bounded on Sobolev spaces.}
\item {\em An operator $P \in \tPS{\infty}$ is Fredholm between
appropriate Sobolev spaces if, and only if, it is elliptic} (\ie its
principal symbol is invertible) {\em and its restrictions to all
hyperfaces are invertible as operators between Hilbert spaces.}
\item {\em An order zero operator is compact if, and only if, its
principal symbol and all its restrictions to hyperfaces vanish.}
\end{itemize}
Some of these results were proved before for certain classes of
manifolds, most notably, for manifolds with multicylindrical ends by
Melrose and Piazza \cite{mepi92}.  An extensive list of references can
be found in \cite{LMN,LN,sur}.

It is worth pointing out at this point that the well-known
difficulties in connection with a parametric construction in the
$b$-calculus were circumvented by studying the structure of the
%%%>>>
%$C^{*}$-closures
$C^{*}$-algebra closures of the operators of order $0$, respectively,
$-\infty$. In fact, the $C^{*}$-algebra setting is sufficient for a
reasonable Fredholm theory for pseudodifferential operators of order
$0$. Operators of positive order, {\em e.g.}\ differential operators
then can be treated using appropriate families of order reducing
operators; for the spectral properties of selfadjoint elliptic
operators of positive order, we use the Cayley transform to reduce to
the case of norm limits of operators of order $0$.

\begin{acknowledgments}
We would like to thank Richard Melrose and Bertrand Monthubert for
useful discussions.
\end{acknowledgments}



\section{Pseudodifferential operators on groupoids}
\label{Sec.prelim}



We shall need groupoids endowed with various additional structures
(see \cite{connes} and the references therein).  Recall that a {\em
groupoid} is a small category $\GR$ all of whose morphisms are
invertible.  More precisely, a groupoid $\GR$ is a pair $(\Gr0,\Gr1)$
of sets together with structural morphisms $d,r,\mu,u$, and $\iota$.
Here the first set, $\Gr0$, represents the objects (or units) of the
groupoid and the second set, $\Gr1$, represents the set of morphisms
of $\GR$. Usually, we shall denote the space of units of $\GR$ by $M$,
and we shall identify $\GR$ with $\Gr1$. Each object of $\GR$ can be
identified with a morphism of $\GR$, the identity morphism of that
object, which leads to an injective map $u : M := \Gr0 \to \GR$ used
to identify $M$ with a subset of $\GR$.  Each morphism $g \in \GR$ has
a ``domain'' and a ``range'', which are denoted by $d(g)$, the {\em
domain}, and $r(g)$, the {\em range} of $g$, respectively.  The
multiplication (or composition) $\mu(g,h)=gh$ of two morphisms $g, h
\in \GR$ is only defined when $d(g) = r(h)$.  The map $\iota : \GR \to
\GR: g\mapsto g^{-1}$ is the ``inverse'' map.  The structural maps are
required to satisfy the usual compatibility relations satisfied by the
morphisms of a category.

A {\em differentiable groupoid} is a groupoid $
\GR=(\Gr0,\Gr1,d,r,\mu,u,\iota) $ such that $\Gr 0$ and $\Gr1$ are
manifolds with corners, the structural maps $d,r,\mu,u,$ and $\iota$
are differentiable, the domain map $d$ is a submersion, and all the
spaces $\Gr0$ and $\GR_x := d^{-1}(x)$, $x\in \GR_0$, are Hausdorff.

The Lie algebroid $A(\GR)$ associated to a differentiable groupoid
$\GR$ is a bundle on $M=\Gr0$ such that its smooth sections identify
with the sections of the $d$-vertical tangent bundle to $\GR$ that are
invariant with respect to right translations on $\GR$.  Consequently,
$\Gamma(A(\GR))$ has a natural Lie algebra structure.


\begin{definition}[\cite{NWX}] \label{Def.Differentiable}\
A {\em pseudodifferential operator on $\GR$} is a uniformly supported
family $P = (P_x)$ of pseudodifferential operators on the fibers
$\GR_x:=d^{-1}(x)$, which form a smooth family invariant with respect
to right translations by elements in $\GR$.
\end{definition}

This definition was first presented in July 1996 at the joint
SIAM-AMS-MAA Meeting on Quantization in Mount Holyoke. A similar
definition was independently announced in
\cite{MonPie}.

The support condition means that the support of $k_{P}(g) =
K_{d(g)}(g,d(g))$ is compact in $\GR$; here $K_{x}(g,g')$ denotes the
Schwartz kernel of the operator $P_{x}$. The set of pseudodifferential
operators of order $m$ on $\GR$ will be denoted by $\tPS{m}$. If $E
\to M$ is a vector bundle, we can consider operators on $r^*(E)$,
which leads to the algebra $\PS{m}$. Both $\tPS{\infty} := \bigcup_m
\tPS{m}$ and $\PS{\infty}:= \bigcup_m \PS{m}$ are $*$-algebras that
satisfy the usual symbolic properties of pseudodifferential operators,
provided that one uses $A^*(\GR)$, the dual of the Lie algebroid of
$\GR$, instead of the cotangent bundle of $M$.  This is the same idea
as the one behind the introduction of various compressed cotangent
bundles by Melrose.  It is interesting to mention here that for the
particular choice of the product groupoid $\GR=M\times M$, we recover
the usual pseudodifferential calculus on a smooth manifold $M$; for
this choice of $\GR$ we have $A(\GR) = TM$.
%%%>>>
The principal symbol map $\sigma_m : \PS{m} \to \CI(A^*(\GR)
\smallsetminus 0)$ is defined in analogy with the classical case, by
defining it first for the operators on each of the spaces $\GR_x$.

%%%>>>
%Moreover,
The algebra
$\tPS{\infty}$ acts on $\CIc(M)$ and on each of the spaces
$\CIc(\GR_x)$. We denote these actions, or representations, by $\pi$
and by $\pi_x$, respectively. The action on $\GR_x$ is simply by
restriction, and the action on $M$ is determined by the relation
$(\pi(P)f) \circ r = (Pf) \circ r$. If $P \in \tPS{0}$, then both
$\pi(P)$ and $\pi_x(P)$ act as bounded operators.  The representation
$\pi$ is called the {\em vector representation}.



\section{Geometric operators}
\label{Sec.GO}



For two vector bundles $E_{0}, E_{1}$ on $M$, we shall denote by
$\Diff{\GR;E_0,E_1}$ the space of differential operators $D :
\Gamma(\GR;r^{*}E_0) \to \Gamma(\GR;r^{*}E_1)$ with smooth
coefficients that differentiate only along the fibers of $d: \GR \to
M$ and that are right invariant.  Then, $\Diff{\GR;E_0,E_1}$ is
exactly the space of differential operators in
$\Psi^m(\GR;E_0,E_1)$. The elements of $\Diff{\GR;E_0,E_1}$ will be
called {\em differential operators on $\GR$}. For example, the choice
of a metric on $A(\GR)$ gives rise to a metric on each of the fibers
$\GR_x$ and hence to Hodge-Laplace operators acting on $p$-forms on
these fibers; these operators form then a family $\Delta_p^\GR \in
\Psi^{2}(\GR;\Lambda^{p})$.

Suppose $Y \subset M$ is a submanifold such that $r^{-1}(Y) =
d^{-1}(Y)$.  Then $Y$ is called an {\em invariant} subset, and
$\GR_Y:=d^{-1}(Y)$ is also a differentiable groupoid, so we can
consider the restriction of a family $P = (P_x) \in \tPS{m}$ to a
family in $\Psi^m(\GR_Y)$. The induced map $\tPS{m} \to \Psi^m(\GR_Y)$
will be denoted by $\inn_Y$ and called the {\em restriction} (or {\em
indicial}) morphism. For example, $\inn_Y(\Delta_p^\GR) =
\Delta_p^{\GR_Y} $.

Consider a bounded, nondegenerate representation
$\varrho:\tPS{-\infty}\longrightarrow\End(\cH)$ on a Hilbert space
$\cH$. Then $\varrho$ extends to a representation of $\tPS{\infty}$ by
densely defined operators \cite{LMN,LN}.


\begin{proposition} \label{ess}
Let $m>0$, and let $A=A^{*}\in\tPS{m}$ be elliptic. Then the unbounded
operator $\varrho(A): \varrho(\tPS{-\infty})\cH\to\cH$ is densely
defined and essentially selfadjoint.
\end{proposition}


Let us denote by $\alg\GR$ the closure of $\tPS0$ in the maximal
(enveloping) $C^*$-norm $\|\:\cdot \, \|$ and by $\ideal\GR$ the closure
of $\tPS{-\infty}$ in the same norm. The norm $\| \: \cdot \, \|_r := \sup
\| \pi_x(\:\cdot \, ) \|$ is called the reduced norm.  The following
theorem allows us to define Sobolev spaces $H^{s}(\cH,\varrho)$
associated to a representation $\varrho$ of $\tPS{\infty}$ as the
domains (or duals of the domains) of a fixed positive element.
%%%>>>
As a rule, when we consider algebras of operators acting on sections
of a vector bundle $E$, we replace ``$\GR$'' by ``$\GR;E$'' in the
notation.


\begin{theorem}\label{theorem.red}\
Fix a differentiable groupoid $\GR$ whose space of units, $M$, is
compact. Let $D \in \tPS{m}$, $m > 0$, be such that $D \ge 1$ and
$\sigma_m(D) > 0$.  Then $D^{-s} \in \ideal\GR$, for all $s >
0$. Moreover, if $P$ has order $\le k$, then $PD^{-k/m} \in \alg\GR$.
\end{theorem}


The following corollary on the Cayley transform is especially useful.


\begin{corollary}\label{cor.Cayley}\ If $A=A^{*} \in \tPS{m}$, $m >0$,
is elliptic, then the Cayley transform $(A + i)(A - i)^{-1}$ of $A$
belongs to $ \alg\GR$.
\end{corollary}

The proofs of the above results depend essentially on the results of
Landsman and Ramazan \cite{LandsmanRamazan}.

The above corollary is an important technical tool to handle
parametrices (or more precisely the lack of parametrices) in certain
pseudodifferential calculi, like the $b$-calculus.  Also, it shows
that while the inverse of an elliptic $b$-pseudodifferential operator
may not be a $b$-pseudodifferential operator (in the so-called small
calculus) anymore, it is nevertheless a uniform limit of such
operators.


\begin{assumption}
From now on we shall assume that $M_0$ is a smooth manifold
without corners which is diffeomorphic to (and will be identified
with) an open dense subset of a compact manifold with corners $M$, and
$\GR$ will be a differentiable groupoid with units $M$, such that
$M_0$ is an invariant subset and $ \GR_{M_0}\cong M_0 \times M_0$.
\end{assumption}

Note that the last condition ensures that on the open part $M_0$
elements in $\tPS{\infty}$ coincide with usual pseudodifferential
operators. The special structure of the groupoid, however, takes care
of the behavior at infinity.

The choice of a metric on $A(\GR)$ leads to geometric operators on
$\GR$ and $M$, which correspond
%%%>>>
to each other
under the representation $\pi$ of
$\tPS{\infty}$ on $\CIc(M)$.
%%%>>>
For instance, $\pi$ maps the Laplace operator on $\GR$ to the Laplace
operator on $M$.

The next example fits into this setting if $M_0$ is taken to be the
interior of the manifold with corners $M$.  The proof of Melrose's
conjecture will be obtained by applying the general results we explain
below to the following example.  (Actually, for $c_H=1$
[respectively,\ $c_H =2$] we essentially recover Melrose's $b$-
[respectively,~$c$]-calculus).


\begin{example}\
\label{excn} {\em The ``very small'' $c_n$-calculus.}
Let $M$ be a compact manifold with corners, and associate to each
hypersurface $H \subset M$ an integer $c_H \ge 1$.  Choose also on $M$
a metric $h$ which can be written in the neighborhood of each point
$p$ in the interior of a face $F\subseteq M$ of codimension $k$ as $h
= h_F + (dx_1)^2 + \dots +(dx_k)^2$, with $x_1, \dots, x_k$ being the
defining functions of $F$ and $h_F$ being a metric on $F$.

On $M$ we consider the vector fields $X$ locally of the form $X = X_F
+ \sum_{j=1}^k x_j^{c_j} \pa_{x_j},$ with $c_j$ being the integer
associated to the hyperface $\{x_j = 0\}$ and $X_F$ being the lift of
a vector field on $F$.  The set of all vector fields with these
properties forms a Lie subalgebra of the algebra of all vector fields
on $M$, which can be identified with the smooth sections of a vector
bundle $A(M,c)$.

For the interior $S$ of a boundary face $F$ of codimension $k$, let
$\GR_S = S \times S \times \RR^k$, and let $\GR := \bigcup_F S \times
S \times \RR^k $ be the groupoid with the obvious induced structural
maps.  As a set, $\GR$ does not depend on $c$.
\end{example}

\begin{proposition}\
There exists on $\GR$ a smooth structure which makes it a
differentiable groupoid with $A(\GR) \cong A(M,c)$. This smooth
structure is uniquely determined.
\end{proposition}



\section{Spectral properties}\label{Sec.SP}


Our analysis of geometric operators on $M_0$ depends on the structure
of the algebras $\alg\GR$ and $\ideal\GR$.  Let $\mI =
\ideal{\GR_{M_0}}$; then $\mI$ is isomorphic to
$\mathcal{K}(L^2(M_0))$, the algebra of compact operators on $L^2(M_0)
= L^2(M)$, the isomorphism being induced by the vector representation
$\pi$, or by any of the representations $\pi_x$, $x \in M_0$, and the
isometry
%%%>>>
%$\GR_x \simeq M$.
$\GR_x \simeq M_0$.
For $x \notin M_0$, $\pi_x$ descends to a
representation of $Q(\GR) : = \alg\GR/\mI$.  Some of the results of
this section were also obtained by Monthubert in his thesis; see
\cite{LMN} and the definitions therein.

We denote by $\sigma(P)$ the spectrum of an element $P \in \alg\GR$
and by $\sigma_{Q(\GR)}(P)$ the spectrum of the image of $P$ in
$Q(\GR) = \alg\GR/\ideal{\GR_{M_0}}$.  These definitions extend to
elliptic, selfadjoint elements $P \in \tPS{m}$, $m >0$, using the
Cayley transform, as follows. Let $f (t) = (t + i)/(t -i)$ and
$f(P):=(P + i)(P - i)^{-1} \in \alg{\GR}$ be its Cayley transform,
which is defined by Corollary \ref{cor.Cayley}.  We define then
\begin{equation*}
\sigma(P) := f^{-1}(\sigma(f(P)))\,, \quad
\mbox{ and } \quad
        \sigma_{Q(\GR)}(P) := f^{-1}(\sigma_{Q(\GR)}(f(P))).
\end{equation*}

The spectrum and essential spectrum of an element $T$ acting as an
unbounded operator on a Hilbert space will be denoted by $\sigma(T)$
and, respectively, by $\sigma_{ess}(T)$.

We shall formulate all results below for operators acting on vector
bundles. Fix an elliptic operator $A \in \PSF{m}$, $m >0$; then for
any $P \in \PSF{k}$, we have $P_1 := P (1 + A^*A)^{-k/2m} \in
\alg{\GR;E}$, by Theorem \ref{theorem.red}. If $\pi$ is the vector
representation of $\PSF{0}$ on $L^2(M;E) = L^2(M_0;E)$, then the
spaces $H^s(M;E) = H^s(L^2(M),\pi)$ are the usual Sobolev spaces
associated to the manifold of bounded geometry $M_0$.


\begin{theorem}\label{theorem.I}\
For $P \in \PSF{m}$, let $P_1:=P(1 + A^*A)^{-m/2k} \in \alg{\GR;E}$.

(i)\ If $P \in \PSF{m}$ is such that the image of $P_1$ in $Q(\GR;E)$
   is invertible, then $\pi(P)$ extends to a bounded Fredholm operator
   $H^{m}(M;E) \to L^2(M;E)$.

(ii)\ If $P_{1}$ maps to zero in $Q(\GR;E)$, then $\pi(P):H^{m}(M;E)
\to L^2(M;E)$ is a compact operator.

In particular, if $P \in \PSF{0}$ or $P \in \PSF{m}$, $m > 0$, is
selfadjoint and elliptic, then $\sigma(\pi(P)) \subseteq \sigma(P)$ and
$\sigma_{ess}(\pi(P)) \subseteq \sigma_{Q(\GR;E)}(P)$.
\end{theorem}

\begin{proof}
The inverse of $P_{1}$ up to $\mI$ leads directly to an inverse up to
compact operators of $\pi(P)$.
As for (ii), the operator $\pi(P) : H^{m}(M;E) \to L^2(M;E)$ is the product
of the bounded operator $\pi(1 + A^*A)^{m/2k} : H^{m}(M;E) \to
L^2(M;E)$ and of the compact operator $\pi(P_1)$.

For $m=0$, the result on the spectrum is a general property of
$C^{*}$-algebras. In general, the case $m>0$ can be reduced to $m=0$
using the Cayley transform and Corollary~\ref{cor.Cayley}.
\end{proof}

This result can be sharpened to a necessary and sufficient condition
for Fredholmness, respectively for compactness.


\begin{theorem}\label{theorem.II}\
Assume that the vector representation $\pi$ is injective on
$\ideal\GR$, and let $P\in\PSF{m}$ be arbitrary.
If $P \in \PSF{0}$ or $P \in \PSF{m}$, $m > 0$, is
selfadjoint and elliptic, then we have $\sigma(\pi(P)) =\sigma(P)$ and
$\sigma_{ess}(\pi(P)) = \sigma_{Q(\GR;E)}(P)$.
In particular:

(i)\ If $\pi(P)$ defines a Fredholm operator $H^{m}(M;E) \to L^2(M;E)$,
then the image of $P(1 + A^*A)^{-m/2k} \in \alg{\GR;E}$ in $Q(\GR;E)$
is invertible.

(ii)\ If $\pi(P):H^{m}(M;E) \to L^2(M;E)$ is compact, then $P(1 +
A^*A)^{-m/2k}$ vanishes in $Q(\GR;E)$.
\end{theorem}

\begin{proof}
Let $P_{0}=P\in\alg{\GR;E}$ if $m=0$, or $P_0 = f(P)\in\alg{\GR;E}$ if
$m>0$.  By the very definition, it then suffices to consider $P_{0}$.
An injective morphism of $C^{*}$-algebras preserves the spectrum;
thus, $\sigma(P_0) = \sigma(\pi(P_0))$.  Let $\mathcal B$
[respectively,\ $\mathcal{K}$] be the algebra of bounded
[respectively,\ compact] operators on $L^2(M;E)$.  Since the map
$\pi':Q(\GR;E) \to \mathcal{B/K}$ induced by the vector representation
$\pi$ is injective as well, we obtain $\sigma_{Q(\GR;E)}(P_0) =
\sigma_{ess}(\pi(P_0))$.
\end{proof}

%%%>>> (ADDED THE NOTATIONS FOR THE NORMS)
Recall that a groupoid $\GR$ is called {\em amenable} if, and only if,
the enveloping $C^*$-norm $\|\:\cdot \, \|$ and the reduced norm
$\|\:\cdot \, \|_r$ coincide.


\begin{theorem}\label{theorem.III}\
Suppose the restriction of $\GR$ to $M \smallsetminus M_0$ is
amenable, and the vector representation $\pi$ is injective. Then,

(i)\ $P : H^s(M;E) \to L^2(M;E)$ is Fredholm if, and only if, $P$ is
elliptic and $\pi_x(P): H^s(\GR_x,r^*E) \to L^2(\GR_x,r^*E)$ is
invertible, for any $x \not \in M_0$.

(ii)\ $P : H^s(M;E) \to L^2(M;E)$ is compact if, and only if, its
principal symbol vanishes, and $\pi_x(P) = 0$, for all $x \not \in
M_0$.

(iii)\ For $P \in \PSF{0}$, we have
\[\sigma_{ess}(\pi(P)) = \bigcup_{x
     \not \in M_0}\sigma(\pi_x(P)) \cup
      \bigcup_{\xi\in S^{*}\GR}{\rm spec}(\sigma_{0}(P)(\xi))\,,\]
      where ${\rm spec} (\sigma_{0}(P)(\xi))$ denotes the spectrum
      of the linear map $\sigma_{0}(P)(\xi):E_x\rightarrow E_x$.

(iv)\ If $P \in \PSF{m}$, $m >0$, is formally selfadjoint and
elliptic, then we have $\sigma_{ess}(\pi(P)) = \bigcup_{x \not \in
M_0}\sigma(\pi_x(P))$.
\end{theorem}


\begin{proof}
We can certainly assume $E=\CC$.  The assumption $\alg\GR = \ralg\GR$
implies $\alg\GR/\mI = \ralg\GR/\mI$. Since the groupoid obtained by
reducing $\GR$ to $M \smallsetminus M_0$ is amenable, the
representation $\varrho := \prod \pi_x$, $x \not \in M_0$ is injective
on $Q(\GR)$. This gives $\sigma_{Q(\GR)}(T) = \bigcup_x
\sigma(\pi_x(T))$, $x \not \in M_0$, for all $T \in \alg\GR$.
Together with Theorem \ref{theorem.II} this gives (iii).  Part (iv) is
proved similarly using the Cayley transform, and (i) [respectively,\
(ii)] follow from (iii) [respectively,\ (iv)].
\end{proof}

Another explicit criterion is contained in the theorem below.


\begin{theorem}\label{theorem.IV}\
Suppose the vector representation $\pi$ is injective and $M \setminus
M_0$ can be written as a union $\bigcup_{j=1}^rZ_j$ of closed,
invariant manifolds with corners $Z_j \subset M$.

(i)\
%%%>>> ASSUMPTION ADDED, $\GR_{Z_j}$ INCLUDED
Assume also that each $\GR\vert_{Z_j}$ is amenable.
Let $P\in
\PSF{m}$, then $P : H^s(M;E) \to L^2(M;E)$ is Fredholm if, and only
if, it is elliptic and $\inn_{Z_j}(P):
H^s(\GR\vert_{Z_j};E\vert_{Z_{j}}) \to
L^2(\GR\vert_{Z_j};E\vert_{Z_{j}})$ is invertible, for all $j$.

(ii)\ Let $P\in \PSF{m}$; then $P : H^s(M;E) \to L^2(M;E)$ is compact
if, and only if, its principal symbol vanishes and $\inn_{Z_j}(P) =
0$, for all $j$.

(iii)\ For $P \in \PSF{0}$, we have
\[\sigma_{ess}(\pi(P)) =
       \bigcup_{j=1}^r\sigma(\inn_{Z_j}(P)) \cup \bigcup_{\xi\in
       S^{*}\GR}{\rm spec}(\sigma_{0}(P)(\xi))\,.\]

(iv)\ If $P \in \PSF{m}$, $m >0$, is formally selfadjoint and
elliptic, then we have $\sigma_{ess}(\pi(P)) =
\bigcup_{j=1}^r\sigma(\inn_{Z_j}(P))$.
\end{theorem}


\begin{proof}\ We assume that $E = \CC$. The morphism
\[\alg\GR/\mI \rightarrow \bigoplus_j
\alg{\GR_{Z_j}}\oplus\mathcal{C}(S^{*}\GR;\End(E))\]
given by the
restrictions $\inn_{Z_{j}}$ and the homogeneous principal symbol is
injective. This gives (iii) and (iv).  For $m>0$ note that we have
$\sigma_{0}(f(P))=1$ for the Cayley transform
$f(P)=(P+i)(P-i)^{-1}\in\alg{\GR}$ of $P$, and $f^{-1}(1)=\{\infty\}$.

%%%>>> SEVERAL CHANGES IN THE PARAGRAPH BELOW
To obtain (i) and (ii) from (iii) as above, it is enough to observe
that the operator $P_1 = P(1 + A^*A)^{-m/2k}$ (with $A$ elliptic of
order $k$, fixed) belongs to $\mI = \ideal{\GR_{M_0}}$ if, and only
if, $\sigma_0(P_1) = 0$ and $\inn_{Z_j}(P_1) = 0$ for all $j$. Moreover,
$\inn_{Z_j}(P_1) = 0$ if, and only if, $\inn_{Z_j}(P) = 0$.
\end{proof}

We are now going to apply the results of this section to the groupoid
$\GR(M,c)$ of Example $\ref{excn}$.  The main result is an inductive
method for the determination of the essential spectrum of
Hodge-Laplace operators. Because $b$-differential operators on $M$
correspond to the differential operators in $\Psi^{\infty}(\GR(M,c))$
if $c_{H}=1$ for all boundary hyperfaces $H$ of $M$, we in particular
answer a question of Melrose on the essential spectrum of the
$b$-Laplacian on the interior of a compact manifold $M$ with corners
\cite[Conjecture 7.1]{MelroseScattering} and natural complete metric
(the interior of $M_0$ with this metric is sometimes also called a
{\em manifold with multicylindrical ends}).


\begin{lemma}
\label{lemma7}
The groupoid $\GR(M,c)$ is amenable, and the vector representation of
$\alg{\GR(M,c)}$ is injective.
\end{lemma}


Fix now a metric $h$ on $A=A(\GR(M,c))$, and let
$\Delta_p^{c}:=\Delta_{p}^{\GR(M,c)}$ be the corresponding
Hodge-Laplacian acting on $p$-forms.

Let us introduce some more notation.  For each hyperface $H$ of $M$,
we consider the system $c^{(H)}$ determined by $c^{(H)}_{F}=c_{F'}$
for all boundary hyperfaces $F'$ of $M$ with $F$ an open component of
$H\cap F'\neq \emptyset$, as in Example \ref{excn}. By the
construction of $\GR(M,c)$,
\[
        \GR(M,c)_H \cong \GR(H,c^{(H)}) \times \RR.
\]
It is convenient to use the Fourier transform to switch to the dual
representation in the $\RR$ variable, so that the action of the group
by translation becomes an action by multiplication with the dual
variable $\lambda \in \RR^* \cong \RR$.  This reasoning then gives
\begin{equation*}\label{ide}
        \inn_{H}(\Delta_{p}^{c})=\Delta_{p}^{\GR(M,c)_{H}} =
        \begin{cases} \lambda^{2} + \Delta_{0}^{ c^{(H)} } & \text{
        if } p = 0, \\ \big (\lambda^{2} + \Delta_{p}^{ c^{(H)} } \big
        )\oplus \big ( \lambda^{2} + \Delta_{p-1}^{ c^{(H)} } \big )
        & \text{ if } p > 0. \end{cases}
\end{equation*}

Denote $m_H^{(p)} = \min \sigma(\Delta_{p}^{c^{(H)}})$ and 
$m^{(p)} = \min_H m_H^{(p)}$.  Then $m^{(p)}\geq0$ because the
Hodge-Laplace operators $\Delta_{p}^{c^{(H)}}$ are positive operators.

On the other hand, note that $\pi(\Delta_p^c)$ is equal to
$\Delta_p$, the Hodge-Laplace operator acting on $p$-forms on the
complete manifold $M_0 := M \smallsetminus \pa M$, with the induced
metric from $A(M,c)$.


\begin{theorem}\label{mainth}\
Consider the open manifold $M_0$ which is the interior of a compact
manifold with corners $M$, with the metric induced from $A(M,c)$. Then
the essential spectrum of the Hodge-Laplacian $\Delta_{p}$ acting on
$p$-forms on $M_0$ is $[m,\infty)$, with $m = m^{(0)}$, if $p = 0$, or
$m = \min \{ m^{(p)} , m^{(p-1)}\}$, if $p>0$.
\end{theorem}


\begin{proof}\
The proof of the above theorem is obtained by applying Theorem
\ref{theorem.IV} (iv), with $Z_j$ ranging through the set of
hyperfaces of $M$, using also Lemma \ref{lemma7}.  Then we notice that
the restriction (or indicial) operators associated to these faces are
the (family of) operators $\Delta_{p}(Z_j) + t^2$, where
$\Delta_{p}(Z_j)$ are the Laplace operators on the interior of the
hyperfaces $Z_j$ and $t$ is the variable dual to the variable normal
to the hyperface~$Z_j$.
\end{proof}


In particular, the spectrum of $\Delta_{p}$ itself is the union of
$[m,\infty)$ and a discrete set consisting of eigenvalues of finite
multiplicity.  As pointed out to us by Melrose, $m^{(0)} = 0$
%%%>>>
because $M$ is compact, and hence every minimal face of $M$ is also
compact. Also, it is necessary to mention that our results say nothing
about the more precise structure of the spectral decomposition of
operators on manifolds with corners. It goes without saying, though,
that these results are very interesting to pursue.
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\end{document}