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generalized eigenfunctions, generalized eigenfunction expansion, 
classical waves in inhomogeneous media, 
classical wave operators
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\begin{document}

\title{\textbf{Generalized eigenfunctions for waves in 
inhomogeneous media}}
\author{\textbf{Abel Klein}
\thanks{This author was supported in part by the NSF Grant
 DMS-9800883.}\\
University of California, Irvine\\
Department of Mathematics\\
Irvine, CA 92697-3875, USA\\
aklein@math.uci.edu
 \and \textbf{Andrew Koines}\\
Texas A \& M University\\
Department of Mathematics\\
College Station, TX 77843-3368, USA\\
Andrew.Koines@math.tamu.edu\\
 \and \textbf{Maximilian Seifert}\\
University of California, Irvine\\
Department of Mathematics\\
Irvine, CA 92697-3875, USA\\
mseifert@math.uci.edu}
\date{}
\maketitle

\newpage

\abstract{Many wave propagation phenomena in classical physics are governed
 by equations that can be recast in Schr\"odinger form.
 In this approach the classical wave equation (e.g.,
Maxwell's equations, acoustic equation, elastic equation) is rewritten in
Schr\"odinger form, leading to the study of the spectral theory of its
\emph{classical wave operator},
 a  self-adjoint, partial differential operator on a  
 Hilbert space of vector-valued, square integrable functions. 
  Physically interesting 
inhomogeneous media give rise to nonsmooth
coefficients. 
We construct a generalized eigenfunction expansion for 
classical wave operators with nonsmooth coefficients.  Our construction
yields  polynomially bounded generalized 
eigenfunctions, 
the  set of generalized eigenvalues forming a 
subset of the operator's spectrum with full spectral measure.}



\section{Introduction}

A self-adjoint operator in a finite dimensional Hilbert spaces can 
always be diagonalized in an orthonormal basis of eigenvectors
 (i.e., it has a complete set of eigenvectors).  In infinite dimensional 
Hilbert spaces only self-adjoint operators with pure point spectrum (e.g.,
compact operators) have this property, but  this has not stopped physicists
from writing generalized eigenfunction expansions for arbitrary
self-adjoint operators (e.g., \cite{Di,Scf}).

Generalized eigenfunction expansions have been developed and applied to
elliptic partial differential operators with smooth coefficients by
Gel'fand and Kostyucenko \cite{Gel},
 Berezanskii \cite{B1}, Browder \cite{Bro},
 Kac \cite{Kac}, and others; we refer to
Berezanskii's  book \cite{B} (see also \cite{B2}).  These expansions were 
extended to Schr\"odinger operators with singular potentials (Simon
 \cite{Si} and references therein; see also \cite{PGW,PS}).

 These  expansions construct polynomially bounded generalized 
eigenfunctions for
a set of generalized eigenvalues  with full spectral measure.  
These generalized eigenfunctions were first exploited
 by Pastur \cite{Pa}, followed by Martinelli and Scoppola \cite{MS}, to prove that 
certain Schr\"odinger operators with
random potentials have no absolutely continuous spectrum.  They played 
a crucial role in the work by Fr\"ohlich, Martinelli, Spencer and Scoppola
\cite{FMSS} and by von Dreifus and Klein \cite{DK} on Anderson localization
(pure point spectrum) of random Schr\"odinger operators;  the 
exponential decay of finite volume Green's functions 
(obtained by a multiscale analysis) forces polynomially bounded 
generalized eigenfunctions to be bona fide eigenfunctions.

Many wave propagation phenomena in classical physics are governed
 by equations that can be recast in Schr\"odinger form \cite{W,FK4,KK}.
These Schr\"odinger-like equations lead to the study of 
\emph{classical wave operators} (as in \cite{KK}), 
a class of self-adjoint, partial differential operators on  
 Hilbert spaces of vector-valued, square integrable functions on 
$\mathbb{R}^d$. 
In this approach the classical wave equation (e.g.,
Maxwell's equations, acoustic equation, elastic equation) is rewritten in
Schr\"odinger form, leading to the study of the spectral theory of its
 classical wave operator.  The method is particularly suitable for the
 study of  phenomena normally associated with quantum
mechanical electron waves, especially Anderson localization in
random media \cite{FK3,FK4,KK}. 




In this article we study generalized eigenfunctions of
classical wave operators with nonsmooth coefficients.  
Physically interesting inhomogeneous media give
rise to nonsmooth coefficients in the classical wave equations, and
 hence in their  classical wave operators. (E.g., a medium 
composed of two different homogeneous materials will be represented
 by piecewise constant coefficients.) 
 We must also take into account that many classical
wave operators are not elliptic operators (e.g., the Maxwell operator -- see 
\cite{FK4,KK}).  We construct
a generalized eigenfunction expansion for classical wave operators which
yields  polynomially bounded generalized 
eigenfunctions, 
the  set of generalized eigenvalues forming a 
subset of the operator's spectrum with full spectral measure.  These
generalized eigenfunctions are used
in the proofs
of Anderson localization of classical waves in random media
\cite{FK3,FK4,KK}.  The  generalized eigenfunction expansion itself
plays a key role in recent work by Germinet and Klein \cite{GK}, who  show 
 that strong dynamical localization follows from a multiscale analysis,
both for classical waves in random media and random 
 Schr\"odinger operators.  

This article is organized as follows: Classical wave operators are introduced in 
Section \ref{secCWO}.  A trace estimate for partially elliptic
classical wave operators, crucial for the 
construction of generalized eigenfunction expansions, is given in
 Theorem \ref{tHS}, and proven in Section \ref{sHS}.  We  define generalized eigenfunctions and eigenvalues
 of classical wave operators in Section \ref{secgeneigen}; our main result
on the generalized point spectrum of a classical wave operator is Theorem
\ref{thm1}, with  Corollary \ref{corused} being the result   used in the
 proofs of Anderson localization
for classical waves in random media.  In Section \ref{berez}
we rewrite the standard results on  generalized eigenfunction
expansions (as in  Berezanskii \cite{B})  in the framework of this article,
making additions and modifications needed for classical wave operators;
see Theorems \ref{athm} and  \ref{athm22}. 
In Section \ref{spaces}  we introduce some mathematical machinery: we
 construct a chain of functional spaces 
 and ``extensions" and ``restrictions" of classical wave operators.
 The key technical result is a generalization of interior
estimates (Theorem \ref{tinterior}). The main results about
 ``extensions" and ``restrictions" of  second order classical wave operators
  are summarized in Theorem \ref{realeig}.
  We also prove that second order classical wave operators have an operator
 core consisting of  functions  with compact support (Corollary \ref{core2}).
In  Section \ref{sec3}, the results of Sections  \ref{berez} and \ref{spaces}
are combined to construct a  generalized eigenfunction expansion for
partially elliptic classical wave
operators (Theorem \ref{Bhyp}), and prove Theorem \ref{thm1}.  Section
\ref{eigenexpansions} analyzes the case when a second order classical wave
operator is partially elliptic, but the corresponding first order 
classical wave operator is not.  We show that we still
have a generalized eigenfunction expansion for the corresponding  
classical waves;
 the main result is Theorem \ref{tQ}.  Section \ref{sHS} contains the proof
 of Theorem \ref{tHS}. A proof that $0$ is at the
 bottom of the spectrum of proper classical wave operators is given in
Appendix A. 
 Appendix B contains a review  of densely defined operators
in locally convex spaces, which are  used in Section \ref{spaces}.
In Appendix C we summarize some results of the theory of 
Bochner integrals and  vector valued
Borel measures, including  the Radon-Nikodym Theorem 
and  Lebesgue's Differentiation Theorem.



\section{Classical wave operators} \label{secCWO}
 

 Given an operator $T$ on a Hilbert space 
$\mathcal{H}$, we denote its kernel by $\mbox{ker}\,T$ and its range
by $\mbox{ran}\,T$; note
 $\mbox{ker}\,T^*T= \mbox{ker}\,T$.  If $T$ is 
self-adjoint, it leaves invariant 
the orthogonal complement of its kernel; \emph{ the restriction of $T$  to 
 $\left(\mathrm{ker}\,T\right)^\perp$ will be denoted by  $T_\perp$}.  Note that
 $T_\perp$ is a self-adjoint operator on the Hilbert space 
$\left(\mbox{ker}\,T\right)^\perp= P^\perp_T \mathcal{H}$, where 
$ P^\perp_T$ denotes the orthogonal projecction onto $\left(\mbox{ker}\,T\right)^\perp$.




Let us set
\begin{equation} \label{H}
\mathcal{H}^{(n)}=L^2(\mathbb{R}^d, dx;\mathbb{C}^n) \ .
\end{equation}

\begin{definition} \label{CPDO} 
A constant coefficient, first order, partial differential operator 
 $D$ from 
$\mathcal{H}^{(n)}$ to 
$\mathcal{H}^{(m)}$ ($CPDO^{(1)}_{n,m}$)   is of the form $D~=~D(-i\nabla)$, 
where 
\begin{equation}  \label{Dij}
D(k)=\left[D(k)_{r,s}\right]_{\stackrel{r=1,\ldots,m}{s=1,\ldots,n}} \ ,\ \
\mbox{with}\ \ D(k)_{r,s}= a_{r,s}\cdot k\  ,\ \ 
a_{r,s}\in\mathbb{C}^d \  . 
\end{equation}
\end{definition}


Defined on
\begin{equation} \label{DD}
\mathcal{D}(D)=
\{\psi\in\mathcal{H}^{(n)} :\ D\psi\in\mathcal{H}^{(m)} \, 
\mbox{ in distributional sense}
\}\  ,
\end{equation}
a $CPDO^{(1)}_{n,m}$ $D$ is a closed, densely defined operator, and 
$C^{\infty}_{0}(\mathbb{R}^d;\mathbb{C}^n)$ (the space of infinitely
differentiable functions with compact support) is an operator core for D. 
We will denote by $D^*$ the $CPDO^{(1)}_{m,n}$ given by the formal 
adjoint of the matrix in (\ref{Dij}). 

 For later use, we set $D_+$ to be the norm of the
matrix $\left[|a_{r,s}|\right]_{\stackrel{r=1,\ldots,m}{s=1,\ldots,n}}$,
so $\|D(k)\| \le D_+ |k|\,$.

\begin{definition} \label{essell} 
A $CPDO^{(1)}_{n,m}$ $D$ is said to be
 partially elliptic if there exists a
$CPDO^{(1)}_{n,q} \,$  
 $D^{\perp}$ (for  some $q$), satisfying the following two properties:
\begin{eqnarray}
D^{\perp}D^* & = & 0\ ,\label{3}\\
D^*D+(D^{\perp})^*D^{\perp} & \geq & \Upsilon\ [(-\Delta)\otimes I_n] \ ,
\label{4}
\end{eqnarray}
 $\Upsilon$ being a positive constant . ($\Delta=\nabla\cdot\nabla$ is the Laplacian
 on
$L^2(\mathbb{R}^d, dx)$;  $I_n$  denotes the $n \times n$
identity matrix.) 
\end{definition}


If $D$ is partially elliptic, we have
\begin{equation}\label{5}
\mathcal{H}^{(n)}=\mbox{ker}\,D^{\perp}\oplus\mbox{ker}\,D \ ,
\end{equation}
and
\begin{equation}\label{05}
D^*D+(D^{\perp})^*D^{\perp}=(D^*D)_\perp \oplus ((D^{\perp})^*D^{\perp})_\perp \ .
\end{equation}
Note that $D$ is elliptic if and only it is  partially elliptic with
$D^{\perp}=0$. 

\begin{remark}  \label{remexample}
A $CPDO^{(1)}_{n,m}$ $D$ may be
partially elliptic with $D^*$ not being  partially elliptic; e.g.,  the
 two dimensional  $CPDO^{(1)}_{2,3}$   given by 
\begin{equation} \label{DDD2}
D=\left[\begin{array}{cc}
i\frac{\partial}{\partial x_1}&0\\
0 & i\frac{\partial}{\partial x_2} \\
i\frac{\partial}{\partial x_2}&i\frac{\partial}{\partial x_1}
\end{array}\right]  \ .
\end{equation}
\end{remark}



\begin{definition} \label{CO} 
A   coefficient operator 
$S$ on  $\mathcal{H}^{(n)}$ ($CO_n$)  is a bounded, invertible operator
 given by multiplication by an  $n\times n$ 
matrix -valued measurable function
 $S(x)$, with
\begin{equation}\label{7}
 0<S_-I_n\leq S(x)\leq S_+I_n \ ,
\end{equation}
for some positive, finite constants  $S_{\pm}$. 
\end{definition}

\begin{definition} \label{MPDO} 
A  multiplicative coefficient, first order, partial differential operator  
 from $\mathcal{H}^{(n)}$ to $\mathcal{H}^{(m)}$ ($MPDO^{(1)}_{n,m}$) is 
   of the form
\begin{equation}\label{6}
A={R}DS \ \ \mbox{on} \ \ \mathcal{D}(A) = S^{-1}\mathcal{D}(D) \ ,
\end{equation}
where $D$ is a $CPDO^{(1)}_{n,m}\,$, $S$ is a  $CO_n$ and   $R$ is a $CO_m$.  
\end{definition}

An $MPDO^{(1)}_{n,m}$ $A$ is a closed, densely defined operator with
$A^*=SD^*R$  an $MPDO^{(1)}_{m,n}$. Note that 
$S^{-1}C^{\infty}_{0}(\mathbb{R}^d;\mathbb{C}^n)$ is an operator core for $A$.

\begin{definition} \label{CWO1} 
A   first order classical operator on   $\mathcal{H}^{(n)}$ 
 ($CWO^{(1)}_{n}$) is a  $MPDO^{(1)}_{n,n}$ of the form
$SDS$, with $D$ a self-adjoint  $CPDO^{(1)}_{n,n}$ and  $S$  a  $CO_n$.
If $D$ is  partially elliptic, the $CWO^{(1)}_{n}$ will also be said to be
 partially elliptic.
\end{definition}


To every  $MPDO^{(1)}_{n,m}$ $A$ we associate the $CWO^{(1)}_{n+m}$  
$\mathbb{W}_A$ given by:
\begin{equation} \label{AA*}
\mathbb{W}_A=\left[\begin{array}{cc}
0 &A^*\\
A& 0
\end{array}\right]  \ \ \mbox{on} \ \ 
\mathcal{H}^{(n+m)} \cong \mathcal{H}^{(n)} \oplus \mathcal{H}^{(m)} \ .
\end{equation}
To see that $\mathbb{W}_A$ is indeed a $CWO^{(1)}_{n+m}$,  we
  write $\mathbb{W}_A= K\mathbb{W}_D K$, where $K= S \oplus R$ is a
$CO_{n+m}$ and $\mathbb{W}_D$ is a
 self-adjoint   $CPDO^{(1)}_{n+m,n+m}\,$. 
Note that
\begin{equation} \label{cwo2}
(\mathbb{W}_A)^2=\left[\begin{array}{cc}
A^* A & 0\\
0& AA^*
\end{array}\right]  \ .
\end{equation}



 
\begin{remark} \label{remexample9}  $\mathbb{W}_{A}$ is 
partially elliptic  if, and only if, both $A^* A$ and $AA^*$ are
  partially elliptic; i.e., 
$\mathbb{W}_{D}$ is 
partially elliptic  if, and only if, both $D$ and $D^*$ are  partially elliptic.
To see this, first note that if $D$ and $D^*$ are  partially elliptic, it is
 immediate that $\mathbb{W}_{D}$ is 
partially elliptic with
\begin{equation} \label{DDD}
(\mathbb{W}_D)^\perp=\left[\begin{array}{cc}
0 &\left(D^*\right)^\perp\\
D^\perp & 0
\end{array}\right]  \ .
\end{equation}
To prove the converse,  suppose $\mathbb{W}_{D}$ is 
partially elliptic satisfying (\ref{3})  
 and (\ref{4}) for a $CPDO^{(1)}_{m+m,q} \ (\mathbb{W}_D)^\perp$.  Let us write
$ (\mathbb{W}_D)^\perp = [ D_1 \ D_2   ]$, where $D_1$ is a 
$CPDO^{(1)}_{n,q}$ and $D_2$ a $CPDO^{(1)}_{m,q}$.  It follows that
$D$ and $D^*$ satisfy  (\ref{3})   and (\ref{4}) with
$D^\perp=D_1$ and $\left(D^*\right)^\perp=D_2$.
\end{remark}


The  Schr\"odinger-like equations for classical waves are of the form 
 \cite{W,FK4,KK}:
\begin{equation} \label{schr}
-i \frac{\partial}{\partial t} \mathbf{\Psi}_t=   
(\mathbb{W}_A)_\perp \mathbf{\Psi}_t \ ,
 \ \  \mathbf{\Psi}_t \in
 (\mbox{ker}\,\mathbb{W}_A)^\perp =  (\mbox{ker}\,{A})^\perp \oplus
 (\mbox{ker}\,{A^*})^\perp , \quad
 \end{equation}
with  $\mathbb{W}_A$ a  $CWO^{(1)}_{n+m}$ as in (\ref{AA*}).  Examples include 
Maxwell's equations and  the equations of  acoustics and elasticity.


If $\mathbf{\Psi}_t =
(\psi_t, \phi_t) \in \mathcal{H}^{(n)} \oplus \mathcal{H}^{(m)}$ is a solution
of  (\ref{schr}), it follows from (\ref{cwo2}) that
 its components satisfy the second order wave equations
\begin{eqnarray}
\frac{\partial^2}{\partial t^2}  \psi_t &=& - (A^*A)_\perp  \psi_t \ , 
 \ \ \mbox{with} \ \  \psi_t \in  (\mbox{ker}\,{A})^\perp \ , \\
\frac{\partial^2}{\partial t^2}  \phi_t &=& - (AA^*)_\perp  \phi_t \ , 
 \ \ \mbox{with} \ \  \phi_t \in  (\mbox{ker}\,{A^*})^\perp \ .
\end{eqnarray}

It turns out that the operators  $(A^*A)_\perp$ and  $(AA^*)_\perp$ are unitarily 
equivalent.  The operator $U$ defined by 
\begin{equation} \label{unitary0}
U\psi= A(A^*A)_\perp^{-\frac12} \psi \ \  \mbox{for} \ \ \psi \in \
\mathrm{ran} \, (A^*A)_\perp^{\frac12} \, ,
\end{equation}
extends to a unitary operator from $(\mbox{ker}\,{A})^\perp$ to 
$(\mbox{ker}\,{A^*})^\perp$, and 
\begin{equation} \label{unitary}
 (AA^*)_\perp = U (A^*A)_\perp U^* \ .
\end{equation}

In addition, if 
\begin{equation} \label{unit22}
\mathbb{U}=\frac1{\sqrt{2}}\left[\begin{array}{cc}
I_A&I_A\\
U &-U 
\end{array}\right]  \ , \ \
\mbox{with $I_A$ the identity on $(\mbox{ker}\,{A})^\perp$} \ , 
\end{equation}
$\mathbb{U}$ is a unitary operator from  
$(\mbox{ker}\,{A})^\perp \oplus (\mbox{ker}\,{A})^\perp $ to 
$(\mbox{ker}\,{A})^\perp \oplus(\mbox{ker}\,{A^*})^\perp$, 
and we have the unitary equivalence: 
\begin{equation} \label{unitequiv}
\mathbb{U}^* (\mathbb{W}_A)_\perp\mathbb{U} = 
 (A^*A)_\perp^{\frac12}  \oplus
\left[-  (A^*A)_\perp^{\frac12}\right] \ .
\end{equation}
Thus the operator
$(A^*A)_\perp$ contains full information about the spectral theory
of the operator 
$(\mathbb{W}_A)_\perp $  (e.g., \cite{FK4,KK}).
In particular
\begin{equation} \label{=sp}
\sigma((\mathbb{W}_A)_\perp)= 
\sigma\left( (A^*A)_\perp^{\frac12}\right) \cup
 \left(-\sigma\left( (A^*A)_\perp^{\frac12}\right)\right) \ .
\end{equation}



\begin{definition} \label{CWO2} 
A second order classical wave operator on $\mathcal{H}^{(n)}$ 
($CWO^{(2)}_{n}$) is 
 of the form $A^*A$, with $A$ an $MPDO^{(1)}_{n,m}$
for some $m$.  If $D$ in (\ref{6}) is  partially elliptic, the 
$CWO^{(2)}_{n}$
will also be called  partially elliptic.
\end{definition}

\begin{remark} \label{notpe}
 If $A$ is an $MPDO^{(1)}_{n,m}\,$, both $A^*A$ and $AA^*$ are
second order classical wave operators.  But  $A^*A$ may be 
partially elliptic
 with $AA^*$  not
  partially elliptic, since a $CPDO^{(1)}_{n,m}$ $D$ may be 
 partially elliptic with $D^*$ not being  partially elliptic.
 (See Remark \ref{remexample}.)  
\end{remark}

\begin{remark} \label{cwo12}
If $W$ is a $CWO^{(1)}_{n}$, then $W^2$ is a $CWO^{(2)}_{n}$. If
 $W$ is  partially elliptic so is $W^2$.
\end{remark}

\begin{definition} \label{proper}
A classical wave operator ($CWO$) is either a 
$CWO^{(1)}_{n}$ or a $CWO^{(2)}_{n}$. 
If the operator  $W$ is  a $CWO$, we call
$W_\perp$ a proper $CWO$.
\end{definition}



\begin{remark} A proper $CWO$ $W$  has a trivial kernel by 
construction, so $0$ is not an eigenvalue.  However, using a dilation 
argument, one can show that 0 is in the spectrum of  $W_\perp$
 (Theorem \ref{thmC1}),   so
 $W_\perp$ and $W$ have the same spectrum.
\end{remark}

Partially elliptic CWOs satisfy a trace estimate crucial for the 
construction of
generalized eigenfunction expansions.  
In view of Remark \ref{cwo12} it suffices to consider second order 
classical wave operators. 





\begin{theorem} \label{tHS}
Let $W$ be a partially elliptic $CWO^{(2)}_{n}$, and let $ P_W^\perp$ denote
the orthogonal projection onto $ (\mbox{ker}\,{W})^\perp$.   Then 
\begin{equation}\label{tr}
\mathrm{tr}\, (V^*P_W^\perp (W+I)^{-2r}V)\leq 
C_{d,n,S_{\pm},R_{\pm},D_+,D_+^{\perp},\Upsilon}\,\|V\|^2_{\infty,2}<\infty\ \ ,
\end{equation}
for $r\geq\nu$, where $\nu$ is the smallest integer satisfying  $\nu>\frac d
4$.
 $V$ is the bounded operator on $\mathcal{H}^{(n)}$
 given by multiplication by an  $n\times n$ 
matrix-valued measurable function $V(x)$, with 
\begin{equation}
\|V\|_{\infty,2}^2=
\sum_{y \in {\mathbb{Z}}^d}\Vert \chi _{y,1}(x){ V}^*(x) { V}(x) \Vert
_{\infty } < \infty \ . \label{vv}
\end{equation}
($ \chi _{y,L}$ denotes the characteristic function of a cube of
 side length $L$ centered at $y$.)
The  constant $C_{d,n,S_{\pm},R_{\pm},D_+,D_+^{\perp},\Upsilon}$ depends only on
the fixed parameters $d,n,S_{\pm},R_{\pm},D_+,D_+^{\perp},\Upsilon$.
 \end{theorem}
Theorem \ref{tHS} is proved in Section \ref{sHS}. It generalizes 
\cite[Theorem 18]{FK4}.

\section{Generalized eigenfunctions and eigenvalues} \label{secgeneigen}
 


We now define generalized eigenfunctions and eigenvalues
 of classical wave operators.

A   vector valued measurable function $\psi$ 
 is said to be \emph{polynomially $L^2$-bounded}, if 
\begin{equation}
\|(1+|x|^2)^{-p}\psi\|_2<\infty  \  \ \mbox{for some}\ p>0\ .  \label{polb}
\end{equation}

\begin{definition}\label{def01}
A non-zero, $\mathbb{C}^n$-valued  polynomially $L^2$-bounded function 
$\psi$ is said to be 
a generalized eigenfunction of a $CWO^{(1)}_{n}$ $W=SDS$, 
with generalized eigenvalue $\lambda\in\mathbb{C}$, if 
\begin{eqnarray} \label{09}
\langle SDS \phi, \psi \rangle = \lambda \langle  \phi, \psi \rangle \ \ 
\mbox{for all} \ \ \phi \in S^{-1}C^{\infty}_{0}(\mathbb{R}^d;\mathbb{C}^n)\, .
\end{eqnarray}
\end{definition}


\begin{definition}\label{def1}
A non-zero, $\mathbb{C}^n$-valued  polynomially $L^2$-bounded function 
$\psi$ is said to be 
a generalized eigenfunction of the $CWO^{(2)}_{n}$
$W=A^*A$,  $A=RDS$,
 with generalized eigenvalue $\lambda\in\mathbb{C}$, if 
\begin{eqnarray}
DS\psi \;  \mbox{\ is a polynomially $L^2$-bounded function} \ , 
\label{9a}
\end{eqnarray}
in the sense of distributions, and
\begin{eqnarray}
\langle DS \phi, R^2 DS\psi \rangle = \lambda \langle  \phi, \psi \rangle \ \ 
\mbox{for all} \ \ \phi \in S^{-1}C^{\infty}_{0}(\mathbb{R}^d;\mathbb{C}^n) 
 \, .\label{9b}
\end{eqnarray}
\end{definition}

Property (\ref{9a}) is a regularity property needed for the formulation of
 the eigenvalue equation (\ref{9b}).   It  
is explicitly used in  \cite{FK3,FK4,KK}.  

Definition \ref{def1} may be recast in an analogous form to Definition
\ref{def01}, once we have an appropriate operator core for $A^*A$, as 
 provided by Corollary \ref{core2}. (See Proposition \ref{eigeq}.)

 

Any eigenfunction of a $CWO$ is easily seen to be 
also a generalized eigenfunction. 

A  $MPDO^{(1)}_{m,n}$ $A$ gives raise to a $CWO^{(1)}_{n+m}$ 
$\mathbb{W}_A$ (as in (\ref{AA*})), and also to a  $CWO^{(2)}_{n}$ $A^*A$ and a   $CWO^{(2)}_{m}$
$AA^*$, the diagonal entries in   $(\mathbb{W}_A)^2$.  It turns out that if
$\mathbf{\Psi} =(\psi, \phi)$ is a generalized eigenfunction of 
$\mathbb{W}_A$ with generalized eigenvalue $\lambda$, then $\psi$ is a generalized
eigenfunction of $A^*A$ and $\phi$ is a generalized
eigenfunction of $AA^*$, both with generalized eigenvalue $\lambda^2$. 
 Conversely, if $\psi$ is a generalized
eigenfunction of $A^*A$ with generalized eigenvalue $\lambda^2 \not=0$, 
we can find    generalized
eigenfunctions $\phi_\pm$ of $AA^*$ also with generalized eigenvalue 
$\lambda^2$, such that  $(\psi, \phi_\pm)$ is a generalized eigenfunction of 
$\mathbb{W}_A$ with generalized eigenvalue $\pm \lambda\,$. 
 (See Proposition \ref{eigeq22}.)



\begin{definition}
The generalized point spectrum of a classical wave operator $W$  is the set of
 all its generalized eigenvalues; it wil be denoted by $\sigma_{gp}(W)$.  
\end{definition}

 


\begin{theorem}\label{thm1}
The generalized point spectrum of  a 
classical wave operator $W$ is a subset of 
its spectrum. 
For a generalized eigenfunction $\psi$ of   $W$ 
with generalized eigenvalue $\lambda$, we have:
\begin{description}
\item[(i)] If $\psi$ 
 is an $L^2$-function, then
$\psi$ is an eigenfunction of  $W$ with eigenvalue
$\lambda$ (i.e., $\psi \in \mathcal{D}(W)$ and 
$W \psi = \lambda \psi$). 
\item[(ii)] If $\psi$  is  not an $L^2$-function, then $\lambda$ 
is in the essential spectrum of $W$.
\end{description}
Moreover, if $W$ is  partially elliptic, or $W=\mathbb{W}_A$  as 
in (\ref{AA*}) with $A^*A$  partially elliptic, then  
the difference between its spectrum and generalized point spectrum has
 zero spectral measure.
\end{theorem}

We have two immediate corollaries.

\begin{corollary} Let $W$ be  a partially elliptic classical wave operator,
 or let $W=\mathbb{W}_A$  as 
in (\ref{AA*}) with $A^*A$  partially elliptic.
Then  its spectrum is the closure of its generalized point 
spectrum.
\end{corollary}
\begin{corollary} \label{corused}
 Let $W$ be  a partially elliptic classical wave operator,
 or let $W=\mathbb{W}_A$  as 
in (\ref{AA*}) with $A^*A$  partially elliptic.
Let $\Omega$ be a Borel subset of its spectrum. If  every  generalized
 eigenfunction with generalized 
eigenvalue $\lambda \in \Omega $  is an $L^2$-function,
 then $W$ has
pure point spectrum in $\Omega$.     
\end{corollary}

Note that Corollary \ref{corused} is the result  
 used in the proofs of Anderson localization
for classical waves in random media \cite{FK3,FK4,KK}, which also required
the regularity property (\ref{9a}). (The analogous of Corollary \ref{corused} 
 for Schr\"odinger operators  is used in \cite{FMSS,DK}.) In 
\cite{FK3,FK4}, such a result was erroneously taken  as following directly 
from the results in \cite{B}.

Simon \cite{Si} obtained the equivalent of 
Theorem \ref{thm1} and its corollaries for
Schr\"odinger operators with singular potentials. Simon's approach is based
 on the study of the Schr\"odinger semigroup $ \textrm{e}^{-tH}$,  $H$
being the Schr\"odinger operator, via the Feynman-Kac formula. His methods 
allow the treatment of Schr\"odinger operators for which 
$C_0^\infty(\mathbb{R}^d)$ is only a form core; thus he also has a regularity
 condition  in the definition of generalized eigenfunctions 
\cite[Theorem C.1.1]{Si}.  Simon's methods are not applicable
 to classical wave operators.



\section{Generalized eigenfunction expansions: generalities}\label{berez}

In this section  we derive a  generalized eigenfunction
expansion in the framework of this article, in a form suitable
 for classical wave operators.  These results strengthen and extend the
standard generalized eigenfunction
expansion, as in the books \cite{B,B2}.



\bigskip
\noindent{\textbf{Hypothesis I}} \  \emph{$T$ is a self-adjoint operator
 on a Hilbert space $\mathcal{H}$ with a bounded
 inverse $T^{-1}$}.
\smallskip 

The  domain of $T$,  $\mathcal{D}(T)$, equipped with the norm
$\|\phi\|_+ = \|T\phi\|$, is a Hilbert space,  denoted by
$\mathcal{H}_+\,$.  The Hilbert space $\mathcal{H}_-$ is defined as
 the completion of  $\mathcal{H}$ in the norm $\|\psi\|_- = \|T^{-1}\psi\|$.
The sesquilinear form $\langle \ , \ \rangle$ on 
$\mathcal{H}_+\times \mathcal{H}_-\,$, defined as the inner product in
 $\mathcal{H}$ 
 for $\phi\in\mathcal{H}_+\,,\psi\in\mathcal{H}$, 
 turns the spaces $\mathcal{H}_+$ and 
$\mathcal{H}_-$ into conjugate duals of each other. The adjoint of an
 operator $O$ with respect to this duality will be denoted by $O^{\dagger}$, whereas the
 adjoint with respect to the usual Hilbert space self-duality is given  by
$O^*$. 

 By construction, 
$\mathcal{H}_+\subset\mathcal{H}\subset\mathcal{H}_-\,$, and the
natural injections  
 $\imath_+: \mathcal{H}_+\rightarrow\mathcal{H}$ and
$\imath_-:\mathcal{H}\rightarrow\mathcal{H}_-$  are
continuous with dense range, with $\imath_+^\dagger =\imath_-\,$. 
 The operators $T_+: \mathcal{H}_+ \to \mathcal{H}$
and $T_-: \mathcal{H} \to \mathcal{H}_-$, defined by
$T_+= T \imath_+\,$, 
 $T_- = \imath_- T $ on $ \mathcal{D}(T)$, are unitary
with $T_-=T_+^\dagger$.


\bigskip
\noindent{\textbf{Hypothesis II}} \  \emph{$H$ is a self-adjoint operator
 on  $\mathcal{H}$, such that
\begin{equation} \label{D+}
\mathcal{D}_+ =\{\phi \in\mathcal{D}(H)\cap\mathcal{H}_+: \ 
H\phi  \in \mathcal{H}_+\}
 \end{equation}
is dense in 
$\mathcal{H}_+\,$.} 

\smallskip 
 $H_+$, defined by $H_+\phi=H\phi$ for
$\phi \in \mathcal{D}_+$, is a closed densely defined operator on 
$\mathcal{H}_+\,$.  The operator $H$ can be considered as an operator in
$ \mathcal{H}_-$.  We have 

\begin{lemma} \label{h-h} The operator $H$ is closable, densely defined as
 an operator on $ \mathcal{H}_-$.  If $H_-$ denotes its closure in 
 $ \mathcal{H}_-$, we have
\begin{equation} \label{ap1}
H_-= H_+^\dagger \ .
\end{equation}
\end{lemma}

\begin{proof}
$H$ is closable in   $ \mathcal{H}_-$ since $H \subset   H_+^\dagger$.  
To prove (\ref{ap1}),  let 
$\psi \in  \mathcal{D}( H_+^\dagger)$ be such that, for all 
$\phi \in \mathcal{D}(H)$, 
\begin{eqnarray}
0&=&\langle \phi, \psi\rangle_- +
\langle H\phi,  H_+^\dagger\psi\rangle_- \\
&=&  \langle\phi, T^{-2}\psi\rangle +
\langle H\phi,T^{-2} H_+^\dagger\psi\rangle \\
&=&  \langle\phi, ,T^{-2}\psi\rangle +
\langle H\phi, H_{+}^* T^{-2} \psi\rangle \ ,
\end{eqnarray}
where we used $H_+^\dagger= T^2 H_+^*T^{-2}$.
Thus $H_{+}^* \,  T^{-2} \psi \in 
\mathcal{D}(H)$, and 
 we have
$H H_{+}^* \,  T^{-2} \psi  = - T^{-2} \psi$.  We conclude that
$ H H_{+}^* \,  T^{-2} \psi \in  \mathcal{H}_+$, and hence that
$  H_{+}^* \,  T^{-2} \psi \in   \mathcal{D}_+\,$.  Thus
$H_+ H_{+}^* \,  T^{-2} \psi = - T^{-2} \psi$.  Since
 $H_{+} H_{+}^*$ is a positive operator on 
$ \mathcal{H}_{+} \,$, we must have $\psi=0$.
\end{proof}
 



\begin{definition} \label{defgeneig} A generalized eigenfunction of $H$ is defined
as an  eigenfunction of the operator $H_-$ on $\mathcal{H}_-\,$: 
$\psi \in \mathcal{H}_-$ is a generalized eigenfunction of $H$ with generalized 
eigenvalue $\lambda$, if and only if $\psi \in \mathcal{D}(H_-)$ and
$H_- \psi=\lambda \psi$, i.e.,
\begin{equation} \label{eigb}
\langle H_+ \phi,\psi \rangle = \lambda \langle \phi,\psi \rangle
\ \ \mbox{for all} \ \ \phi \in \mathcal{D}_+\,.
\end{equation}
\end{definition}

An eigenfunction of $H$ is always a generalized eigenfunction.

\bigskip 
\noindent{\textbf{Hypothesis III}}  \ \emph{$\mathcal{D}_+$ is an operator core for $H$}.

\smallskip 

Let  $(H_-)_\mathcal{H}$  denote the
restriction of $H_-$ to $\mathcal{H}$, i.e., 
$(H_-)_\mathcal{H} \,\phi = H_- \phi$  on
\begin{equation} \label{HCHK}
\mathcal{D}\left((H_-)_\mathcal{H}\right)= 
\{\phi \in \mathcal{D}(H_-)\cap\mathcal{H}: \ 
(H_-)_\mathcal{H}\,\phi \in \mathcal{H}\} \, .
\end{equation}
(Note that $H_+ = H_{\mathcal{H}_+}$.)

\begin{lemma} \label{lemmaapp2} We have
$H=(H_-)_\mathcal{H}$ if and only if 
$\mathcal{D}_+$ is an operator core for $H$.
 \end{lemma}

\begin{proof}
It follows from Lemma \ref{h-h} that for  
$\psi \in  \mathcal{D}((H_-)_\mathcal{H})$,   $\phi \in \mathcal{D}_+\,$, 
\begin{equation}
\langle \phi, H_-\psi\rangle = 
\langle H_+\phi, \psi\rangle =\langle H\phi, \psi\rangle \  .
\end{equation}
If $\mathcal{D}_+$ is an operator core for $H$,  we can conclude that
$\psi \in  \mathcal{D}(H)$ and $H\psi= H_-\psi$.

To prove the converse, let $H_{+,0}$ denote $H_+$ as an operator on 
$\mathcal{H}$. Let 
$\psi \in  \mathcal{D}(H_{+,0}^*)$, we have
\begin{equation}
\langle H_+\phi, \psi\rangle = 
\langle \phi, H_{+,0}^* \,\psi\rangle 
\end{equation}
 for all $\phi \in \mathcal{D}_+\,$.  Again using Lemma \ref{h-h}, we obtain
  $\psi \in  \mathcal{D}((H_-)_\mathcal{H})$, 
$(H_-)_\mathcal{H}\,\psi =  H_{+,0}^* \,\psi\,$.  Since $H=(H_-)_\mathcal{H}$,
we conclude that $\mathcal{D}_+$ is an operator core for $H$.
 \end{proof}

This lemma has an important immediate consequence.

\begin{proposition}  \label{L2}  Assume Hypotheses I-III.
 Whenever
 a generalized eigenfunction $\psi$ of $H$ is  in  $\mathcal{H}$,  it is also an
 eigenfunction of $H$: $\psi \in \mathcal{D}(H)$ and $H\psi=\lambda \psi$.
\end{proposition}


 By $E(\Omega)$ we denote the
 spectral projections
 of the self-adjoint operator $H$ for Borel sets $\Omega$. We use
$\textrm{tr}_{\mathcal{H}}$ to denote the usual trace in $\mathcal{H}$.
If $1\le q <\infty$,
 $\mathcal{ T}_{q}(\mathcal{H})$ denotes the Banach space
 of bounded operators $Y$ on $\mathcal{H}$ with
 $\|Y\|_q= \left| \textrm{tr}_{\mathcal{H}} |Y|^q\right|^{\frac1q} <\infty$; with
 $\mathcal{ T}_{q,+}(\mathcal{H})$ being  the subset of positive operators.



\bigskip
\noindent{\textbf{Hypothesis IV}} \ \emph{There exist a 
Borel set $\Delta$ and a bounded, continuous  function $f$, 
strictly positive on the spectrum of $H$, such that 
\begin{equation} \label{KK1}
\textrm{tr}_{\mathcal{H}}\,T^{-1}E(\Delta) f(H)T^{-1}<\infty \ .
\end{equation}
}

\smallskip
This implies 
\begin{equation}\label{012}
\mu(\Omega)=
\textrm{tr}_{\mathcal{H}}\,T^{-1}{E(\Omega\cap \Delta)}T^{-1}<\infty \ ,
\end{equation}
for all  bounded Borel sets $\Omega$.   Thus
$\Omega \mapsto T^{-1}{E(\Omega\cap \Delta)}T^{-1}$ is a
 $\mathcal{ T}_{1,+}(\mathcal{H})$-valued Borel
measure, and
$\mu$ is a spectral measure for the restriction of $H$
to the Hilbert space $E(\Delta) \mathcal{H}$.  
(See Appendix C.) 



The map $\tau: \mathcal{B}(\mathcal{H})  \to 
\mathcal{ B}(\mathcal{H}_+,\mathcal{H}_-)$, with 
$\tau(C)=T_-CT_+ \,$, is a Banach space isomorphism, as $T_\pm$
are unitary operators.  If $1\le q<\infty$, we define
\begin{equation} \label{trace}
\mathcal{ T}_{q}(\mathcal{H}_+,\mathcal{H}_-)=
\tau\left(\mathcal{ T}_{q}(\mathcal{H})\right) \ ; \ \
\mathcal{ T}_{q,+}(\mathcal{H}_+,\mathcal{H}_-)=
\tau\left(\mathcal{ T}_{q,+}(\mathcal{H})\right) \ .
\end{equation}
By construction,  $\mathcal{ T}_{q}(\mathcal{H}_+,\mathcal{H}_-)$, equipped with 
the norm  $\| B\ \|_q=\|  \tau^{-1}(B)\|_q $, is a Banach space isomorphic to 
$\mathcal{ T}_{q}(\mathcal{H})$, a separable dual  Banach space.   
$\mathcal{ T}_{q,+}(\mathcal{H}_+,\mathcal{H}_-)$ is the subset 
of positive  operators with respect to the pairing between $\mathcal{H}_+$
and $\mathcal{H}_-\,$:  $\langle \phi, B \phi \rangle \ge 0$ for 
$\phi \in \mathcal{H}_+$. 

For $B \in \mathcal{ T}_{1}(\mathcal{H}_+,\mathcal{H}_-)$, we set
\begin{eqnarray}
\textrm{tr} \, B &=&\textrm{tr}_{\mathcal{H}} \  \tau^{-1}(B) = 
\textrm{tr}_{\mathcal{H}} \ T^{-1}_-BT^{-1}_+ \ .
\end{eqnarray}
If  $\{\phi_n\}$ is an orthonormal basis  of $\mathcal{H}_+$, we have
\begin{equation}
\textrm{tr} \, B = \sum_n \langle \phi_n, B \phi_n \rangle \ .
\end{equation}
 Moreover,
for  $B \in \mathcal{ T}_{2}(\mathcal{H}_+,\mathcal{H}_-)$, $\|B\|_2$ 
is the usual Hilbert-Schmidt norm:
\begin{equation}
\|B\|_2^2= \sum_n \| B \phi_n \|_-^2  \ .
\end{equation}










It follows that
$\Omega \mapsto \imath_-{E(\Omega\cap \Delta)}\imath_+$ is a 
$\mathcal{ T}_{1,+}(\mathcal{ H}_+,\mathcal{H}_-)$-valued
measure, with
$\mu(\Omega)=\textrm{tr} \, \imath_-{E(\Omega\cap \Delta)}\imath_+\,$,
as 
\begin{equation}
\textrm{tr}  \   \imath_-{C}\imath_+ =
\textrm{tr}_{\mathcal{H}} \ T^{-1}CT^{-1} \, , \ \ \textrm{if}
\ \  C \in \mathcal{ B}(\mathcal{H}) \,  , \   
 T^{-1}CT^{-1}\in \mathcal{ T}_{1}(\mathcal{H}) \ .
\end{equation}
 


\begin{theorem}\label{athm}  Assume Hypotheses I-IV.
 There exits a $\mu$-locally integrable function $P(\lambda)$
from the real line  into 
$\mathcal{ T}_{1,+}(\mathcal{H}_+,\mathcal{H}_-)$, 
such that
\begin{equation}\label{a16}
\imath_-{E(\Omega\cap \Delta)}\imath_+=\int_{\Omega}P(\lambda)\,d\mu(\lambda)\  \ \
\mbox{for bounded Borel sets}\ \Omega\ ,
\end{equation} 
where the integral is the Bochner integral of 
$\mathcal{ T}_{1}(\mathcal{H}_+,\mathcal{H}_-)$-valued 
functions.  The
function $P(\lambda)$ satisfies
\begin{equation}\label{a16a} \textrm{tr}\,P(\lambda)=1 \ 
 \ \ \mathrm{for} \ \mu-\mathrm{a.e.} \ \lambda\ ,
\end{equation}
\begin{equation}\label{a16b}
 P(\lambda)=\lim_{\delta\downarrow 0}\
\frac{\imath_-{E(I_{\lambda,\delta}\cap \Delta)}\imath_+}
{\mu(I_{\lambda,\delta})}\ \
 \textrm{in}\ \mathcal{ T}_{1}(\mathcal{ H}_+,\mathcal{H}_-)\,,\ \textrm{for} \ 
\mu-\textrm{a.e.} \ \lambda\ ,
\end{equation} where $I_{\lambda,\delta}$ is the interval
$[\lambda-\delta,\lambda+\delta]$.  In addition, we have
\begin{equation}\label{017}
\imath_-{E(\Delta)g(H)}\,\imath_+
=\int g(\lambda)P(\lambda)\,d\mu(\lambda) \  ,
\end{equation}
 for all $\mu$-integrable, bounded, scalar Borel functions $g$. 
\end{theorem}

\begin{proof}In view of  (\ref{012}), it
is a consequence of the Radon-Nikodym Theorem (Theorem
\ref{thmA2}), that there exits a locally integrable function $P(\lambda)$
from the real line  into 
$\mathcal{ T}_{1,+}(\mathcal{H}_+,\mathcal{H}_-)$, 
such that (\ref{a16}) holds.
 According to (\ref{A4}) and (\ref{A5}), the
function $P(\lambda)$ satisfies (\ref{a16a}) and (\ref{a16b}).  Using
 (\ref{a16}), and taking limits of
Borel step functions, we obtain (\ref{017}).  \end{proof}

We will also use the following variant of Hypothesis IV.

\bigskip
\noindent{\textbf{Hypothesis IV$_\mathbf{2}$}} \ \emph{There exist a 
Borel set $\Delta$ and a spectral measure $\mu$  for the restriction of $H$
to the Hilbert space $E(\Delta) \mathcal{H}$,
such that
$\Omega \mapsto \imath_-{E(\Omega\cap \Delta)}\imath_+$ is a
 $\mathcal{ T}_{2,+}(\mathcal{H}_+,\mathcal{H}_-)$-valued Borel
measure, with the property that for each compact interval $I$ 
we have
\begin{equation} \label{nu-bound}
\|\imath_-{E(\Omega\cap \Delta)}\imath_+ \|_2 \le C_I \, 
\mu(\Omega)  \ \ \mbox{for all Borel sets} \ \ \Omega \subset I \ , 
\end{equation}
with  $C_I < \infty$.}

\begin{theorem}\label{athm22}  Assume Hypotheses I-III and IV$_2$.
There exists a $\mu$-locally integrable function $P(\lambda)$
from the real line  into 
$\mathcal{ T}_{2,+}(\mathcal{H}_+,\mathcal{H}_-)$, 
such that
\begin{equation}\label{a1622}
\imath_-{E(\Omega\cap \Delta)}\imath_+=\int_{\Omega}P(\lambda)\,d\mu(\lambda)\  \ \
\mbox{for bounded Borel sets}\ \Omega\ ,
\end{equation} 
where the integral is the Bochner integral of 
$\mathcal{ T}_{2}(\mathcal{H}_+,\mathcal{H}_-)$-valued 
functions.  The
function $P(\lambda)$ is  is $\mu$-locally bounded and  given by
\begin{equation}\label{a16b22}
 P(\lambda)=\lim_{\delta\downarrow 0}\
\frac{\imath_-{E(I_{\lambda,\delta}\cap \Delta)}\imath_+}
{\mu(I_{\lambda,\delta})}\ \
 \textrm{in}\ \mathcal{ T}_{2}(\mathcal{ H}_+,\mathcal{H}_-)\,,\ \textrm{for} \ 
\mu-\textrm{a.e.} \ \lambda\ ,
\end{equation} where $I_{\lambda,\delta}$ is the interval
$[\lambda-\delta,\lambda+\delta]$.  In addition, we have
\begin{equation}\label{01722}
\imath_-{E(\Delta)g(H)}\,\imath_+
=\int g(\lambda)P(\lambda)\,d\mu(\lambda) \  ,
\end{equation}
 for all bounded, scalar Borel functions $g$ with 
$g(\lambda)\|P(\lambda)\|_2$ $\mu$-integrable. 
\end{theorem}

\begin{proof} 
 The proof  proceeds as in the proof of Theorem 
\ref{athm}, using (\ref{nu-bound}). \end{proof}

It follows from either Theorem \ref{athm} or Theorem \ref{athm22} that 
the values of the
operator-valued  Radon-Nikodym derivative $P(\lambda)$ are generalized
eigenprojectors, in a sense made precise in  the following corollary.

\begin{corollary}\label{athm2}  Assume either Hypotheses I-IV or
Hypotheses I-III and IV$_2$.
 Then, for almost every $\lambda \in \Delta$  with respect to the spectral measure for
$H$, we have  $H_-P(\lambda)=\lambda P(\lambda)$,
and
$P(\lambda)\phi  \in \mathcal{H}_-$ is a
generalized eigenfunction of $H$
with generalized eigenvalue $\lambda$ for any $\phi\in\mathcal{H}_+$.
\end{corollary}

\begin{proof}
Using (\ref{a16b}) or (\ref{a16b22}),  Lemma \ref{lemmaapp2}, (\ref{017})
or  (\ref{01722}),  and  Lebesgue's Differentiation Theorem (Theorem
 \ref{thmA1}), we obtain
\begin{equation}\label{a23}
\lim_{\delta\downarrow 0}\, H_-
 \frac{\imath_-{E(I_{\lambda,\delta}\cap \Delta)}
\imath_+}{\mu(I_{\lambda,\delta})}=
\lim_{\delta\downarrow 0}\,  \imath_- H
 \frac{{E}(I_{\lambda,\delta}\cap \Delta)}
{\mu(I_{\lambda,\delta})}\imath_+=
\lambda
P(\lambda)
\end{equation}
in either $\mathcal{ T}_{1}(\mathcal{H}_+,\mathcal{H}_-)$ or
$\mathcal{ T}_{2}(\mathcal{H}_+,\mathcal{H}_-)$, for 
almost every $\lambda \in \Delta$  with respect to the spectral measure for
$H$.
Since $H_-$ is closed by Lemma \ref{h-h}, 
it follows from either (\ref{a16b})  or 
(\ref{a16b22})  and (\ref{a23}), that 
$\mathrm{ran} \, P(\lambda) \subset \mathcal{D}(H_-)$
and $H_-P(\lambda)=\lambda P(\lambda)$  for 
almost every $\lambda \in \Delta$  with respect to the spectral measure for
$H$. \end{proof}





\begin{corollary}\label{acor2}  Assume either Hypotheses I-IV or
Hypotheses I-III and IV$_2$.
Then,
almost every $\lambda \in \Delta$  with respect to the spectral measure for
$H$  is a
generalized eigenvalue of $H$.
\end{corollary} 

Corollary \ref{acor2} is an immediate consequence of Corollary \ref{athm2} 
and either (\ref{a16}) or  (\ref{a1622}) . 
\bigskip



\section{Spaces and operators} \label{spaces}

In this section 
we  construct ``extensions" and ``restrictions" of classical wave operators.
   Along the 
way we  prove that a second order classical wave operator has an 
operator core consisting of functions with compact support.  
 

We start by introducing
 the following functional spaces:

$\bullet$ $\mathcal{H}^{(n)}_\tau =
 L^2(\mathbb{R}^d,\tau(x)^2 dx;\mathbb{C}^n)$,
with  $\tau(x) \ge 0$ a locally bounded measurable function; a Hilbert space
whose norm and inner product will be denoted by $\| \ \|_\tau$ and 
$\langle \ , \ \rangle_\tau$.
 
$\bullet$ $\mathcal{H}^{(n)}_{\mathrm{max}}=
L^2_{loc}(\mathbb{R}^d, dx;\mathbb{C}^n)$, the space of 
$\mathbb{C}^n$-valued, 
locally $L^2$-functions with the locally convex topology given by $L^2$
convergence on bounded sets; a Fr\'echet space  (e.g., \cite[Section V.2]{RS1}).

$\bullet$ $\mathcal{H}^{(n)}_{\mathrm{min}}=
L^2_{0}(\mathbb{R}^d, dx;\mathbb{C}^n) $,
 the space of $\mathbb{C}^n$-valued, $L^2$-functions with compact support; 
a complete locally convex space as the strict inductive 
limit of the Hilbert spaces 
$L^2(B_r, dx;\mathbb{C}^n)$, $B_r\subset \mathbb{R}^d$ being the ball 
of radius $r$ centered at the origin (e.g., \cite[Section V.4]{RS1}).  

The spaces   $\mathcal{H}^{(n)}_{\mathrm{max}}$ and
 $\mathcal{H}^{(n)}_{\mathrm{min}}$ are used to prove the existence
of an operator core of compactly supported functions for a  $CWO^{(2)}_{n}$ . 
To study generalized eigenfunctions we use  spaces $\mathcal{H}^{(n)}_\alpha$ and 
$\mathcal{H}^{(n)}_{1/\alpha}$, where 
 \begin{equation} \label{alpha}
\alpha \in C^1( \mathbb{R}^d), \ \ \mbox{with} \ \ \alpha(x) \ge 1  \ \
\mbox{and} \ \
\frac{\nabla\alpha}{\alpha} \ \ \mbox{bounded .}
\end{equation}
  (In Section \ref{sec3} we will take
$\alpha(x) = (1 + |x|^2)^{p}$, $p>0$.) Note that 
$\frac{\nabla(1/\alpha)}{1/\alpha}=-\frac{\nabla\alpha}{\alpha}$,
so it is also bounded.
 The sesquilinear form
\begin{equation} \label{14}
\langle\phi,\psi\rangle
=\int\overline{\phi(x)}\cdot\psi(x)\,dx \, ,
\end{equation}
defined for either 
$\phi\in\mathcal{H}^{(n)}_{\mathrm{min}}\,,
\psi\in\mathcal{H}^{(n)}_{\mathrm{max}}$ or  
$\phi\in\mathcal{H}^{(n)}_{\alpha}\,,
\psi\in\mathcal{H}^{(n)}_{1/\alpha} \, $,
turns  $\mathcal{H}^{(n)}_{\mathrm{\mathrm{min}}}$ and 
$\mathcal{H}^{(n)}_{\mathrm{max}}$, and $\mathcal{H}^{(n)}_{\alpha}$ and
$\mathcal{H}^{(n)}_{1/\alpha} \, $,  into conjugate duals of each other. We denote the adjoint of an
 operator $O$ with respect to this duality by $O^{\dagger}$, whereas the
 adjoint with respect to the usual Hilbert space self-duality in 
$\mathcal{H}^{(n)}$ is denoted by
$O^*$. (Unbounded operators on locally convex spaces are discussed in
Appendix B.)  It is easy to see that
$$\mathcal{H}^{(n)}_{\mathrm{\mathrm{min}}}\subset  
 \mathcal{H}^{(n)}_{\alpha}  \subset
\mathcal{H}^{(n)}\subset\mathcal{H}^{(n)}_{1/\alpha} 
\subset\mathcal{H}^{(n)}_{\mathrm{max}} \ ,$$ the
natural injections   being
continuous with dense range.  Note 
 $\mathcal{H}^{(n)} = \mathcal{H}^{(n)}_\alpha$ for  $\alpha(x) \equiv 1$.




The following notation will be useful:  
Let 
$\mathcal{H}_i,\, \mathcal{K}_i$, $i=1,2$,  be locally convex spaces,  with 
$\mathcal{H}_2\subset \mathcal{H}_1$ and 
$\mathcal{K}_2\subset \mathcal{K}_1$, the natural injections being 
continuous.  Given an operator $C$ from  $\mathcal{H}_1$ to 
 $\mathcal{K}_1$, we will denote by $C_{\mathcal{H}_2,\mathcal{K}_2}$
its restriction to an operator from  $\mathcal{H}_2$ to 
 $\mathcal{K}_2$, i.e.,
\begin{equation} \label{CHK}
C_{\mathcal{H}_{2},\mathcal{K}_{2}}\phi = C\phi \ \, \mbox{on} \ \,
\mathcal{D}\left(C_{\mathcal{H}_{2},\mathcal{K}_{2}}\right)= 
\{\phi \in \mathcal{D}(C)\cap\mathcal{H}_2: \ C\phi \in \mathcal{K}_2\} \, .
\end{equation}
Note that if $C$ is a closed operator so is
 $C_{\mathcal{H}_{2},\mathcal{K}_{2}}$.  If 
$\mathcal{H}_i,= \mathcal{K}_i$, $i=1,2$, we write 
$C_{\mathcal{H}_{2}}$ for $C_{\mathcal{H}_{2},\mathcal{H}_{2}}$.


Given a $CPDO^{(1)}_{n,m}$ $D$, we
define $D_{\mathrm{max}}$ as
the closure of $D|_{C^{\infty}_{0}({{\mathbb{R}}^d},\mathbb{C}^n)}$ as an operator
 from $\mathcal{H}^{(n)}_{\mathrm{max}}$ into
$\mathcal{H}^{(m)}_{\mathrm{max}}$; its domain is given by
\begin{equation}\label{D+-} 
\mathcal{D}(D_{\mathrm{max}})=
\{\psi\in\mathcal{H}^{(n)}_{\mathrm{max}} :\ D\psi\in
\mathcal{H}^{(m)}_{\mathrm{max}}\ \mbox{in distributional sense}\} \ .
\end{equation}
From (\ref{DD}) and  (\ref{D+-}) we immediately see that 
$D = (D_{\mathrm{max}})_{\mathcal{H}^{(n)},\mathcal{H}^{(m)}}\,$.  We define
$D_{\sigma} = 
 (D_{\mathrm{max}})_{\mathcal{H}^{(n)}_\sigma,\mathcal{H}^{(m)}_\sigma}$, for
$\sigma = \mathrm{min}, \alpha, 1/\alpha\,$; they all have
$C^{\infty}_{0}(\mathbb{R}^d,\mathbb{C}^n)$ as an operator core. 
We similarly define $D^*_{\sigma}$,  for
$\sigma = \mathrm{min}, \alpha, 1/\alpha, \mathrm{max}$.


 A $CO_{n}$ $S$, defined  by multiplication
by the  matrix-valued measurable function
 $S(x)$ as in (\ref{7}), is easily seen to be
a bounded operator with a 
bounded inverse in $\mathcal{H}^{(n)}_{\sigma}$, for all
$\sigma = \mathrm{min}, \alpha, 1/\alpha, \mathrm{max}$.  


We now consider an $MPDO^{(1)}_{n,m}$ $A=R D S $, and 
 introduce  closed densely defined operators $A_\sigma$
from  $\mathcal{H}^{(n)}_\sigma$ into
$\mathcal{H}^{(m)}_\sigma$, 
$\sigma = \mathrm{min}, \alpha, 1/\alpha, \mathrm{max}$, by 
\begin{equation}\label{20} 
A_\sigma=
R D_\sigma S  \ \ 
\mbox{on} \ \ \mathcal{D}(A_\sigma) =
 S^{-1}\mathcal{D}(D_\sigma)  \ ;
\end{equation}
note that $S^{-1}C^{\infty}_{0}(\mathbb{R}^d,\mathbb{C}^n)$ is an 
operator core for $A_\sigma$. The same considerations apply to
 $A^*_\sigma= S D^*_\sigma{R}$. 

 We  use the notation 
$1/\mathrm{min} =\mathrm{max}, 1/\mathrm{max} =\mathrm{min}$, and the ordering \  
$ \mathrm{min}< \alpha< 1 < 1/\alpha< \mathrm{max}$.  We have, for 
$\sigma, \sigma' \in \{\mathrm{min}, \alpha, 1,  1/\alpha, \mathrm{max}\}$,
with $\sigma <\sigma'$,
\begin{eqnarray}
A_\sigma &=& (A_{\sigma'})_{\mathcal{H}^{(n)}_{\sigma},\mathcal{H}^{(m)}_{\sigma}} 
 \  , \label{e1}\\
A_{1/\sigma}^\dagger &=& A^*_\sigma=
 (A^*_{\sigma'})_{\mathcal{H}^{(m)}_\sigma,\mathcal{H}^{(n)}_\sigma}= 
 (A^\dagger_{1/\sigma'})_{\mathcal{H}^{(m)}_\sigma,
\mathcal{H}^{(n)}_\sigma}\  .  \label{e3} 
\end{eqnarray}
We also have, for  
$\sigma \in \{\mathrm{min}, \alpha, 1,  1/\alpha, \mathrm{max}\}$,
\begin{eqnarray}  \label{min1}
\mathcal{D}(A_\mathrm{min})&=& 
\mathcal{D}(A_\sigma) \cap \mathcal{H}^{(n)}_\mathrm{min} \ , \\
\mathcal{D}(A^\dagger_\mathrm{max})&=& 
\mathcal{D}(A^\dagger_{1/\sigma}) \cap \mathcal{H}^{(m)}_\mathrm{min} \ .  \label{min2}
\end{eqnarray}


 The following lemma is easily 
verifiable.

\begin{lemma} \label{pho} Let 
$\sigma \in \{\mathrm{min}, \alpha, 1, 1/\alpha, \mathrm{max}\}$, then
if  $\rho\in C^{1}(\mathbb{R}^d)$,  
 with   
$\rho$ and $\nabla\rho$ bounded, we have
$\rho\psi \in \mathcal{D}(A_\sigma)$ for any 
$\psi \in \mathcal{D}(A_\sigma)$, and
\begin{equation}
A_\sigma\rho\psi =\rho A_\sigma\psi + {R}\,D[\rho]S \psi \ , \label{ar}
\end{equation}
where $D[\rho]$, given by   multiplication
by the matrix valued function $D(-i\nabla\rho(x))$,   is a bounded operator from
$\mathcal{H}^{(n)}_\sigma$ to $\mathcal{H}^{(m)}_\sigma$, with norm bounded
by $D_+ \|\nabla\rho\|_\infty\,$. 
\end{lemma} 


We  now turn our attention to  the operators 
$A_{1/\sigma}^\dagger A_\sigma= A_{\sigma}^* A_\sigma$ on 
$\mathcal{H}^{(n)}_{\sigma}$, defined on
their 
natural domains: 
\begin{equation} 
{D}(A_{1/\sigma}^\dagger A_\sigma) = 
\{\phi\in\mathcal{H}^{(n)}_{\sigma} : 
\ \phi\in\mathcal{D}( A_\sigma) \ \mathrm{and} 
\  A_\sigma\phi\in\mathcal{D}(A_{1/\sigma}^\dagger)\} \ .
\end{equation}

 The following result is a generalization of interior estimates
(e.g., \cite{FK3,FK4}). 

\begin{theorem} \label{tinterior}
Let $\rho \in C^{1}(\mathbb{R}^d)$ and
$\tau \in L^\infty_{loc}(\mathbb{R}^d, dx)$, with $0 \le \rho(x) \le \tau(x)$
and $|\nabla \rho(x)| \le c \tau(x) $ a.e., where $c$ is a  finite constant. 
Then
for any $a>0$ we can find $b < \infty$, depending only on
 $a$ and $c$,  such
that
\begin{equation}  \label{int00}
\| A_{\mathrm{max}} \psi\|_{\rho} \le 
a\|A_{\mathrm{min}}^\dagger A_{\mathrm{max}} \psi\|_{\tau}
+b\|  \psi\|_{\tau}  \ ,
\end{equation}
for any $\psi \in 
\mathcal{D}(A_{\mathrm{min}}^\dagger A_{\mathrm{max}})$. 
\end{theorem}

\begin{proof}  Let $\chi_L$ denote the characteristic function of the cube of side 
$L$ centered at the origin.   We can find a sequence 
$\rho_L \in  C^{1}(\mathbb{R}^d)$, with 
$\chi_L(x)\rho(x) \le \rho_L(x)\le \chi_{L+ 4d }(x) \rho(x)$ and
$|\nabla \rho_L(x)| \le (c + 1) \tau(x) $.

Now let  
$\psi \in \mathcal{D}(A_{\mathrm{min}}^\dagger A_{\mathrm{max}})$;
 using Lemma \ref{pho} we have  
$\rho_L \psi  \in \mathcal{D}( A_{\mathrm{max}})$
and, since 
$\rho_L A_{\mathrm{max}}\psi \in \mathcal{H}^{(n)}_{\mathrm{min}}\,$,
\begin{eqnarray}
\lefteqn{\| A_{\mathrm{max}}\psi\|_{\rho_L}^2= 
\langle\rho^2_L A_{\mathrm{max}}\psi, A_{\mathrm{max}}\psi\rangle} \\
&& =\langle A_{\mathrm{min}}\rho^2_L \psi, A_{\mathrm{max}}\psi\rangle
- 2\langle\rho_LR\,D[\rho_L]S \psi , A_{\mathrm{max}}\psi\rangle \\
&& = \langle\rho^2_L \psi, 
A_{\mathrm{min}}^\dagger  A_{\mathrm{max}}\psi\rangle 
- 2\langle R\,D[\rho_L]S \psi , \rho_LA_{\mathrm{max}}\psi\rangle \\
&&\le \| \psi\|_{\tau}
\| A_{\mathrm{min}}^\dagger  A_{\mathrm{max}}\psi\|_{\tau} +
c_1\| \psi\|_{\tau}\| A_{\mathrm{max}}\psi\|_{\rho_L} \ ,
\end{eqnarray}
 where $c_1$ is a constant depending only on $c$.  We may now let 
$L \to \infty$; (\ref{int00}) follows from the resulting inequality plus 
the fact that $2rs \le tr^2 +\frac1{t} {s^2}$ for  $r,s,t > 0$.
 \end{proof}

Let $\alpha$ be  as in (\ref{alpha}); applying  Theorem \ref{tinterior} with
 $\rho = \tau =\sigma$, where $\sigma =\alpha, 1/\alpha$, we can find
a constant $c<\infty$, such that
\begin{equation}  \label{int0011}
\| A_{\mathrm{max}} \psi\|_{\sigma } \le 
\|A_{\mathrm{min}}^\dagger A_{\mathrm{max}} \psi\|_{\sigma }
+c\|  \psi\|_{\sigma }  \ ,
\end{equation}
for any $\psi \in 
\mathcal{D}(A_{\mathrm{min}}^\dagger A_{\mathrm{max}})$.  Thus we
obtain the
second inclusion in\begin{equation}
\mathcal{D}(A_{1/\sigma}^\dagger A_\sigma) \subset
\mathcal{D}((A_{\mathrm{min}}^\dagger
 A_{\mathrm{max}})_{\mathcal{H}^{(n)}_\sigma}) \subset
\mathcal{D}(A_\sigma) \ ,
\end{equation}
the first inclusion being evident.  Hence we must have equality, so 
$A_{1/\sigma}^\dagger A_\sigma = 
(A_{\mathrm{min}}^\dagger A_{\mathrm{max}})_{\mathcal{H}^{(n)}_\sigma}\;$.
Thus we have
\begin{equation} \label{equal}
A_{1/\sigma}^\dagger A_\sigma =
 (A_{1/\sigma'}^\dagger A_{\sigma'})_{\mathcal{H}^{(n)}_\sigma} \ \ \mbox{for}
\ \  \sigma < \sigma' \in 
\{\mathrm{min}, \alpha, 1,  1/\alpha, \mathrm{max}\} \ ,
\end{equation}
since the case $\sigma=\mathrm{min}$ follows immediately
 from (\ref{min1}) and (\ref{min2}).

Applying again Theorem \ref{tinterior}, this time with
 $\rho =\rho_L\sigma$ and $\tau= \chi_{L+1}\sigma$,
where $\rho_L \in C^1_0(\mathbb{R}^d)$, with 
$\chi_L(x) \le \rho_L(x) \le \chi_{L+1}(x)$ and $\nabla \rho_L$ uniformly
bounded in $L$, we get that for any $a>0$ we can find $b < \infty$,
independent of $L$, such that
\begin{equation}  \label{int01}
\|\chi_L A_{\mathrm{max}} \psi\|_{\sigma} \le 
a\|\chi_{L+1}A_{\mathrm{min}}^\dagger A_{\mathrm{max}} \psi\|_{\sigma}
+b\| \chi_{L+1} \psi\|_{\sigma}  \ ,
\end{equation}
for any $\psi \in 
\mathcal{D}(A_{\mathrm{min}}^\dagger A_{\mathrm{max}})$.  From
(\ref{int01}) with $\sigma=1$ and the fact that $ A_{\mathrm{max}}$
and $A_{\mathrm{min}}^\dagger$ are closed operators, we can conclude that
$A_{\mathrm{min}}^\dagger A_{\mathrm{max}}$ is a closed operator. Using
 (\ref{equal}), we have that $A_{1/\sigma}^\dagger A_\sigma$
is a closed operator for $\sigma = 
\mathrm{min}, \alpha, 1,  1/\alpha, \mathrm{max}$.

We  now show that 
\begin{equation} \label{equal1}
(A_{1/\sigma}^\dagger A_\sigma)^\dagger =
 A_{\sigma}^\dagger A_{1/\sigma} \ \ \mbox{for}
\ \  \sigma  \in 
\{\mathrm{min}, \alpha, 1,  1/\alpha, \mathrm{max}\} \ .
\end{equation}
It is immediate that $ A_{\sigma}^\dagger A_{1/\sigma}  \subset
(A_{1/\sigma}^\dagger A_\sigma)^\dagger $.  (For operators
$C_1, C_2$, $C_1\subset C_2$ means that
$\mathcal{D}(C_1) \subset \mathcal{D}(C_2)$ and $C_1= C_2$ on
$\mathcal{D}(C_1)$.)  To prove the converse, note that
since $ A^*A = 
(A_{1/\sigma}^\dagger A_{\sigma})_{\mathcal{H}^{(n)}}$ for 
$\sigma = 1/\alpha, \mathrm{max}$
by (\ref{equal}), we have 
 \begin{equation} \label{equal2}
(A_{1/\sigma}^\dagger A_{\sigma})^\dagger \subset
((A^*A)^\dagger)_{\mathcal{H}^{(n)}_{1/\sigma}}=
 (A^*A)_{\mathcal{H}^{(n)}_{1/\sigma}}=
 A_{\sigma}^\dagger A_{1/\sigma} \ ,
\end{equation}
so $(A_{1/\sigma}^\dagger A_{\sigma})^\dagger=
 A_{\sigma}^\dagger A_{1/\sigma} $ for 
$\sigma = 1/\alpha, \mathrm{max}$.  Since 
$((A_{1/\sigma}^\dagger A_{\sigma})^{\dagger} )^{\dagger}=
A_{1/\sigma}^\dagger A_{\sigma}\,$, we also have  (\ref{equal1}) for
$\sigma =\mathrm{min}, \alpha $.

Since $A^*A$ is densely defined, it follows from (\ref{equal})
 and (\ref{equal1}) that
$A_{1/\sigma}^\dagger A_{\sigma}$ is densely defined for 
$\sigma= 
\mathrm{min}, \alpha, 1,  1/\alpha, \mathrm{max}\,$.  In fact,
we have more. 

\begin{lemma} \label{core}
The domain of 
$A_{\mathrm{max}}^\dagger  A_{\mathrm{min}}$ is an operator
core for $A^\dagger_{1/\sigma} A_\sigma$, for $\sigma= \alpha, 1, 1/\alpha$.
\end{lemma}

\begin{proof} 
Let $H_\sigma = A^\dagger_{1/\sigma} A_\sigma$.  If  
$\sigma= \alpha, 1, 1/\alpha$, then $\mathcal{H}^{(n)}_\sigma$
is a Hilbert space, and 
$H_\sigma = H_{1/\sigma}^\dagger =
 \frac{1}{\sigma^2} H_{1/\sigma}^* \,  \sigma^2  $,
where $ H_{1/\sigma}^*$ is the adjoint of the closed, densely defined
operator $ H_{1/\sigma}$ in the Hilbert space $\mathcal{H}^{(n)}_{1/\sigma} \,$.

To show that $\mathcal{D}(H_{\mathrm{min}})$ is  an operator core for 
$H_\sigma$, we need to show that  $\Gamma(H_\sigma)$ is the closure of 
$\Gamma(H_{\mathrm{min}})$ in 
$\mathcal{H}^{(n)}_{\sigma} \times \mathcal{H}^{(n)}_{\sigma}\,$. So let 
$\psi \in  \mathcal{D}(H_\sigma)$ be such that, for all 
$\phi \in \mathcal{D}(H_{\mathrm{min}})$, 
\begin{eqnarray}
0&=&\langle \phi, \psi\rangle_\sigma +
\langle H_{\mathrm{min}}\phi, H_\sigma\psi\rangle_\sigma \\
&=&  \langle\phi, \sigma^2\psi\rangle +
\langle H_{\mathrm{min}}\phi,\sigma^2 H_\sigma\psi\rangle \\
&=&  \langle\phi, \sigma^2\psi\rangle +
\langle H_{\mathrm{min}}\phi, H_{1/\sigma}^* \,  \sigma^2 \psi\rangle \ .
\end{eqnarray}
Thus $H_{1/\sigma}^* \,  \sigma^2 \psi \in 
\mathcal{D}(H_\mathrm{min}^\dagger)$, and, recalling 
$H_\mathrm{max} =H_\mathrm{min}^\dagger\,$, we have
$H_\mathrm{max}H_{1/\sigma}^* \,  \sigma^2 \psi = - \sigma^2 \psi$.  But, 
since $\sigma^2 \psi, \ H_{1/\sigma}^* \,  \sigma^2 \psi 
\in  \mathcal{H}^{(n)}_{1/\sigma} \,$, we have 
$H_{1/\sigma}^* \,  \sigma^2 \psi \in \mathcal{D}(H_{1/\sigma})$ and 
$H_{1/\sigma} H_{1/\sigma}^* \,  \sigma^2 \psi=  - \sigma^2 \psi$.  Since
$H_{1/\sigma} H_{1/\sigma}^*$ is a positive operator on 
$ \mathcal{H}^{(n)}_{1/\sigma} \,$, we must have $\psi=0$.
\end{proof}

We summarize our results in the following theorem:

 \begin{theorem} \label{realeig}
 $A^\dagger_{1/\sigma} A_\sigma$ is a closed, densely defined  operator
on  $ \mathcal{H}^{(n)}_{\sigma} \,$, for each 
$\sigma= \mathrm{min},\alpha, 1, 1/\alpha,\mathrm{max}$. 
 We have $A_{1/\sigma}^\dagger A_\sigma =
 (A_{1/\sigma'}^\dagger A_{\sigma'})_{\mathcal{H}_\sigma}$  for
 $\sigma < \sigma'$, and $(A_{1/\sigma}^\dagger A_\sigma)^\dagger =
 A_{\sigma}^\dagger A_{1/\sigma}$. In addition, the domain of 
$A_{\mathrm{max}}^\dagger  A_{\mathrm{min}}$ is an operator
core for $A^\dagger_\sigma A_\sigma$, for $\sigma= \alpha, 1, 1/\alpha$.
\end{theorem}

The following corollary is a nontrivial statement when $A^*A$ 
has nonsmooth coefficients.

 \begin{corollary} \label{core2}
The  functions  in the domain of  $A^*A$ with compact
 support form an operator core.
\end{corollary}

Note that $A^*A$ is a local operator, in the sense that if 
$\psi \in \mathcal{D}(A^*A)$ and $\psi=0$ in some open set $\Omega$,
then we must also have $A^*A\psi=0$ in $\Omega$.  This follows from the
same property for $A$ and $A^*$.




\section{Generalized eigenfunction expansions for classical 
wave operators}\label{sec3}

In this section we prove Theorem \ref{thm1}. 
   
Let us set the weight $\alpha(x)$ in (\ref{alpha}) to be
\begin{equation} \label{alphap}
\alpha_p (x) = (1 + |x|^2)^{p} \ , \ \ \mbox{with} \ \  p>0 \ ,
\end{equation}
 and  simplify the notation by replacing $\alpha_p$ and ${1/\alpha_p}$
in subscripts by $+$ and $-$, respectively.  Thus  
\begin{eqnarray}
\mathcal{H}^{(n)}_{\pm}=L^2(\mathbb{R}^d,(1+|x|^2)^{\pm 2p}dx; 
\, \mathbb{C}^n) \ . \label{13} 
\end{eqnarray}
(These spaces depend on our choice of $p >0$, but we omit $p$ from
the notation.)
Note that $\mathcal{H}^{(n)}_-$ 
 is the space of $\mathbb{C}^n$-valued polynomially $L^2$-bounded functions that 
satisfy  (\ref{polb}) with the same  $p$ as in (\ref{13}). 
We use 
$\| \ \|_\pm \ , \langle  \ ,\ \rangle_\pm$
for the corresponding norms and inner products. 

  





\begin{theorem} \label{Bhyp} Hypotheses I-III 
 (Section \ref{berez}) are satisfied for classical wave operators W, with
 $T$ being the operator given
by multiplication by $(1 + |x|^2)^{p}$, $p>0$.  
If $W$ is a first order classical wave operator, $W_\pm$ are as in 
 (\ref{20}). If  $W=A^*A$ is a second order classical wave operator,
we have
$W_+ =A_{-}^\dagger A_+$ and  $W_- = A_+^\dagger A_-$.

 If the classical wave operator $W$ is
partially elliptic  and $p > \frac d4\,$, Hypothesis IV  is also satisfied with
 $\Delta = \mathbb{R} \backslash\{0\}$, so $W$ has the generalized eigenfunction 
expansion of Theorem \ref{athm}.
\end{theorem}

\begin{proof}   If $W=SDS$ is a first order classical wave operator, Hypotheses I-III
 are satisfied, since 
$S^{-1}C^{\infty}_{0}(\mathbb{R}^d;\mathbb{C}^n)\subset \mathcal{D}_+$
and  is 
an operator core for $W$, and   
  $W_\pm$ are as in (\ref{20}).  For a second order classical wave operator
$W=A^*A$, it follows from (\ref{equal}) and Theorem \ref{realeig}
  that $W_+ =A_{-}^\dagger A_+$, 
a densely defined operator , whose domain is an operator
core for $W$.
Thus Hypotheses I-III of
 Section \ref{berez} are also satisfied in this case, and
  $W_- = A_+^\dagger A_-$ by (\ref{equal1}) and Lemma \ref{h-h}. 

If the classical wave operator $W$ is partially elliptic and
 $p > \frac d4\,$, Hypothesis IV
is a direct consequence of Theorem \ref{tHS}
 \end{proof}

It turns out that Definitions \ref{def01} and \ref{def1} are equivalent to
 Definition \ref{defgeneig}, for an appropriate choice
of   $p>0$. 

\begin{proposition}  \label{eigeq}  
A non-zero, $\mathbb{C}^n$-valued  polynomially $L^2$-bounded function  
$\psi$ is a generalized eigenfunction
of a classical wave operator  $W$ with generalized eigenvalue $\lambda$
 if, and only if,
 $\psi$ is an eigenfunction of  $W_-$ with eigenvalue 
$\lambda$ for some $p>0$,
 i.e., $\psi \in \mathcal{D}(W_-)$ and 
$W_- \psi= \lambda \psi$.    
\end{proposition}

\begin{proof}
If $W=SDS$ is first order,  (\ref{09}) is equivalent to  (\ref{eigb})
 for an appropriate choice
of   $p>0$, since  
$S^{-1}C^{\infty}_{0}(\mathbb{R}^d;\mathbb{C}^n)\subset \mathcal{D}_+$
and  is 
an operator core for $W$. 

If $W=A^*A$ is second order, it is easy to see that,  for an appropriate choice
of   $p>0$, (\ref{9a}) is equivalent to $\psi \in \mathcal{D}(A_-)$,
and  (\ref{9b}) is equivalent to  $ A_-\psi\in\mathcal{D}(A_+^\dagger)$
with $A_+^\dagger A_- \psi= \lambda \psi$.  \end{proof}

\begin{proposition}  \label{eigeq22}  
Let  $A$ be a $MPDO^{(1)}_{m,n}$; fix $p>0$.  
If
$\mathbf{\Psi} =(\psi, \phi)\in   
\mathcal{H}^{(n+m)}_- \cong \mathcal{H}^{(n)}_-\oplus  \mathcal{H}^{(m)}_- $
 is a generalized eigenfunction of 
$\mathbb{W}_A$ with generalized eigenvalue $\lambda$, then $\psi$ is a generalized
eigenfunction of $A^*A$ and $\phi$ is a generalized
eigenfunction of $AA^*$, both with generalized eigenvalue $\lambda^2$. 
 Conversely, if $\lambda \not=0$ and $\psi \in \mathcal{H}_-^{(n)}$ is a generalized
eigenfunction of $A^*A$ with generalized eigenvalue $\lambda^2$, 
we can find    generalized
eigenfunctions $\phi_\pm \in \mathcal{H}_-^{(m)}$ of $AA^*$ also
 with generalized eigenvalue 
$\lambda^2$, such that  $(\psi, \phi_\pm)$ is a generalized eigenfunction of 
$\mathbb{W}_A$ with generalized eigenvalue $\pm \lambda\,$.  Moreover,
if $\psi \in \mathcal{H}^{(n)}$, we also have  
$\phi_\pm \in \mathcal{H}^{(m)}$, so
 $(\psi, \phi_\pm)\in \mathcal{H}^{(n+m)}$.

We have the following relation for the generalized point spectra:
\begin{equation}  \label{gps}
\sigma_{gp}(\mathbb{W}_A) = \left\{ \lambda \in \mathbb{C}: \ \lambda^2 \in
\sigma_{gp}({A^*A})  \right\} \ .
\end{equation}

\end{proposition}

\begin{proof}
We use Proposition \ref{eigeq}.  It is easy to see that
\begin{equation} \label{AA*-}
(\mathbb{W}_A)_-=\left[\begin{array}{cc}
0 &A^*_-\\
A_-& 0
\end{array}\right]  \ ,
\ \ (\mathbb{W}_A)_-^2=\left[\begin{array}{cc}
A^*_- A_- & 0\\
0& A_-A^*_-
\end{array}\right]  \ .
\end{equation}
If $\mathbf{\Psi} =(\psi, \phi)\in \mathcal{D}( (\mathbb{W}_A)_-)$, with
$(\mathbb{W}_A)_-\Psi=\lambda \Psi$, we have 
$\Psi \in \mathcal{D}( (\mathbb{W}_A)_-^2)$, 
$(\mathbb{W}_A)_-^2 \Psi=\lambda^2 \Psi$, i.e.,   
$\psi \in  \mathcal{D}( A^*_- A_-)$, $A^*_- A_-\psi=\lambda^2 \psi$, and
$\phi \in  \mathcal{D}( A_-A^*_-)$, $ A_-A^*_-\phi=\lambda^2 \phi$.

Conversely,   let $\lambda \not=0$ and
$\psi \in  \mathcal{D}( A^*_- A_-)$, with $A^*_- A_-\psi=\lambda^2 \psi$.  
Set $\phi_\pm = \pm \frac 1 \lambda A_- \psi$.  Then 
$\phi_\pm \in  \mathcal{D}( A^*_- )$, so 
$\Psi_\pm =(\psi, \phi_\pm) \in  \mathcal{D}((\mathbb{W}_A)_-)$ 
and we can
check that
$(\mathbb{W}_A)_-\Psi_\pm= \pm \lambda \Psi_\pm$.  Moreover, if
$\psi \in \mathcal{H}^{(n)}$, we have
 $\psi \in \mathcal{D}{( A^* A)}$
by (\ref{equal}), so $\psi \in \mathcal{D}{(  A)}$.  Thus  
$\phi_\pm \in \mathcal{H}^{(m)}$ by (\ref{e1}). 

The relation (\ref{gps}) follows.
\end{proof}

We are now ready to prove  Theorem \ref{thm1}.
In view of Theorem \ref{Bhyp} and  Proposition \ref{eigeq},
Theorem \ref{thm1}(i) follows from Proposition \ref{L2}. 


To prove Theorem \ref{thm1}(ii):  it follows from Proposition 
\ref{eigeq} and Theorem \ref{thmC1}
that it suffices to 
show that, if $\lambda \not=0$
is an eigenvalue of  $W_-$ with a corresponding eigenfunction
$\psi \notin  \mathcal{H}^{(n)}$, then $\lambda$ is in the essential spectrum of 
$W$.  In view of Proposition  \ref{eigeq22}, it is enough to consider the case
 when $W$ is a first order classical wave operator, say $W=SDS$.

So let $\psi  \in \mathcal{D}(W_-)$, $W_- \psi =\lambda \psi$. 
We pick a sequence
$\rho_L \in C_0^\infty(\mathbb{R}^d)$, such that
$ \chi_L(x)\le \rho_L(x) \le \chi_{L+1}(x)$ ($\chi_L$ denotes the 
characteristic function of the cube of side 
$L$ centered at the origin) and
$\sup_L \|\nabla\rho_L\|_\infty < \infty$,
and set
 $\psi_L= \rho_L \psi$; it follows from Lemma \ref{pho} and (\ref{e1})
 that $\psi_L \in \mathcal{D}(W)$, and
\begin{eqnarray} \label{cphi2}
W\psi_L = \rho_L W_-\psi + S D[\rho_L]S \psi =
 \lambda \psi_L +S D[\rho_L]S \psi \ .
\end{eqnarray}


Since
 $\|{\psi}_L\| \ge  \|\chi_L{\psi}\|$, it follows from (\ref{cphi2}) that
\begin{eqnarray}
\frac{\left\| (W - \lambda){\psi}_L \right\|}{\|{\psi}_L\|} \le 
C\frac{ \left\| (\chi_{L+1} - \chi_L){\psi} \right\| }{ \|\chi_L{\psi}\|}=
C \left(\frac{ \left\| \chi_{L+1}{\psi} \right\|^2 }
{ \|\chi_L{\psi}\|^2} -1\right)^{\frac12} \, ,
 \end{eqnarray}
where $C$ is a finite constant.

Using an argument of Sch'nol (see \cite[Theorem C.4.1]{Si}), we can show 
that the
 polynomial boundedness of $\psi$ implies
\begin{equation}
\liminf_{L \to \infty} \frac{\|\chi_{L+1} \psi\| }{ \|\chi_{L}{\psi}\| }=1 \ .
\end{equation}
Theorem \ref{thm1}(ii) now follows from  Weyl's criterion.

It follows from  (i)  and (ii)  that the generalized point 
spectrum of a classical wave operator is a subset of its spectrum:
$\sigma_{gp}(W)\subset\sigma(W)$.

If the classical wave operator $W$ is partially elliptic, we can use
 Corollary \ref{acor2} to conclude that
 $\sigma(W) \backslash \sigma_{gp}(W)$
has zero spectral measure.

If  $W=\mathbb{W}_A$  as 
in (\ref{AA*}) with $A^*A$  partially elliptic, we  also conclude that
 $\sigma(W) \backslash \sigma_{gp}(W)$
has zero spectral measure from (\ref{gps}), (\ref{=sp}), (\ref{unitequiv}),
 and the  result for
 $A^*A$.

Theorem \ref{thm1} is proved.


\section{Generalized eigenfunction expansions of classical waves}
\label{eigenexpansions}

Finite energy classical waves are the solutions of the 
classical wave equation (\ref{schr}) in the Hilbert space 
$(\mbox{ker}\,\mathbb{W}_A)^\perp = 
P^\perp_{\mathbb{W}_A}\mathcal{H}^{(n+m)} $; they are of the form 
\begin{equation} \label{solution}
 \mathbf{\Psi}_t=   \mathrm{e}^{it{\mathbb{W}_A}}
 P^\perp_{\mathbb{W}_A}\mathbf{\Phi}_0 \ , 
\ \ \mbox{} \ \  \mathbf{\Phi}_0 \in \mathcal{H}^{(n+m)} \ .
  \end{equation}

If the first order classical wave operator $\mathbb{W}_A$
 is partially elliptic, the second order  classical wave operators 
$(\mathbb{W}_A)^2$,
$A^*A$ and $AA^*$ are also partially elliptic, so they all have generalized 
eigenfunction expansions given  by Theorems \ref{Bhyp} and \ref{athm}.
 In particular, if   $p > \frac d4\,$,  
$\mu_{\mathbb{W}_A}$ given by (\ref{012}) is a spectral measure for
$(\mathbb{W}_A)_\perp$, we have generalized eigenprojectors 
${P}_{\mathbb{W}_A} (\lambda)$ as in Theorem \ref{athm} and
 Corollary \ref{athm2}, and
 it  follows from (\ref{017}) that  a finite energy classical wave
 $\Psi_t$, as in (\ref{solution}),
has an expansion in generalized eigenfunctions:
\begin{equation} \label{finiteenergyexp}
\Psi_t = \lim_{j \to \infty}
\int_{(-j,j)} \mathrm{e}^{it\lambda} {P}_{\mathbb{W}_A} (\lambda)\Phi_0\,
d\mu_{\mathbb{W}_A} (\lambda)  \ \ 
\mathrm{in} \ \ \mathcal{H}^{(n+m)}_- \ .
\end{equation}

It may happen that $A^*A$ is partially elliptic with
$\mathbb{W}_A$ and $AA^*$ not being partially elliptic, as discussed in
 Remark \ref{notpe}.  It turns out that we still have generalized
 eigenfunction expansions for $\mathbb{W}_A$ and $AA^*$, and
an expansion similar to (\ref{finiteenergyexp}).

\begin{theorem} \label{tQ}
 Let $A$ be a $MPDO^{(1)}_{n,m}$  with  $A^*A$  partially elliptic,
and let $\mu_{A^*A}$ be the spectral measure for $(A^*A)_\perp$ given by
(\ref{012}) with $\Delta= \mathbb{R} \backslash \{0\}$. 
Then  Hypothesis IV$_2$  (Section \ref{berez})  is  satisfied 
by  $AA^*$ and $\mathbb{W}_A$ with
 $\Delta = \mathbb{R} \backslash(-\varepsilon,\varepsilon)$, for any
$\varepsilon >0$, and  spectral measure $\mu=\mu_{A^*A}$ for
$(AA^*)_\perp$ and  $\mu(\Omega)= \mu_{A^*A}(\Omega^2)$ for  
$(\mathbb{W}_A)_\perp$,
 with $T$ being the operator given
by multiplication by $(1 + |x|^2)^{p}$, $p>\frac d 4$.  Thus
$AA^*$ and $\mathbb{W}_A$  have the generalized
 eigenfunction expansion of Theorem \ref{athm22}.  In particular,
if $P_{A^*A}(\lambda)$ is as in Theorem \ref{athm}, we 
have
\begin{equation}\label{a16223}
\imath_-{E_{AA^*}(\Omega)}\imath_+=
\int_{\Omega}\textstyle{\frac1\lambda}A_-P_{A^*A}(\lambda)A^*_+\,
d\mu_{A^*A}(\lambda) \ ,
\end{equation}
and
 \begin{eqnarray}\label{a162234}
\lefteqn{\imath_-{E_{\mathbb{W}_A}(\Omega)}\imath_+=}\\
&&{\textstyle{\frac12} }  \int_{\Omega}
\left[\begin{array}{cc} 
 P_{A^*A}(\lambda^2) &
{\textstyle{\frac1\lambda} } P_{A^*A}(\lambda^2)A_+^*
 \\[.1in]
{\textstyle{\frac1\lambda} }A_- P_{A^*A}(\lambda^2) &
 {\textstyle{\frac1{\lambda^2}} }
A_- P_{A^*A}(\lambda^2)A_+^*  
\end{array}\right] 
d\mu_{A^*A}(\lambda^2)  \ ,\nonumber
\end{eqnarray}
for all bounded Borel sets
 $\Omega$ with $0\notin \bar{\Omega}$, as Bochner
 integrals in  $\mathcal{ T}_{2}(\mathcal{H}_+^{(m)},\mathcal{H}_-^{(m)})$
and  $\mathcal{ T}_{2}(\mathcal{H}_+^{(n+m)},\mathcal{H}_-^{(n+m)})$.
\end{theorem}

\begin{proof}
If $H$ is a self-adjoint
operator, we denote by $E_H^\perp(\Omega)$
 the spectral projections of $H_\perp$ on $(\mathrm{ker} \, H)^\perp$.
It follows from  (\ref{unitary})  that
\begin{equation} \label{555}
E_{AA^*}^\perp(\Omega) =
A(A^*A)_\perp^{-\frac12} 
E_{A^*A}^\perp(\Omega)(A^*A)_\perp^{-\frac12}A^* \ ,
\end{equation}
on an the domain of $A^*$.
Similarly, it follows from  (\ref{unitequiv})  that
\begin{eqnarray} \label{bigmatrix}
\lefteqn{E_{\mathbb{W}_A}^\perp(\Omega) =
 \mathbb{U} \left\{E_{\sqrt{A^*A}}^\perp(\Omega_+ ) \oplus
E_{\sqrt{A^*A}}^\perp(\Omega_- )\right\}
 \mathbb{U}^*
 =}\\
&& \hspace{-.3in}\frac1{{2}}\left[\begin{array}{cc}
\left\{E_{{\sqrt{A^*A}}}^\perp(\Omega_+ ) +
E_{{\sqrt{A^*A}}}^\perp(\Omega_- )\right\}&
\left\{E_{{\sqrt{A^*A}}}^\perp(\Omega_+ )-
E_{{\sqrt{A^*A}}}^\perp(\Omega_- )\right\}U^* \\
U\left\{E_{{\sqrt{A^*A}}}^\perp(\Omega_+ ) -
E_{{\sqrt{A^*A}}}^\perp(\Omega_- )\right\} &
U\left\{E_{{\sqrt{A^*A}}}^\perp(\Omega_+ ) +
E_{{\sqrt{A^*A}}}^\perp(\Omega_- )\right\}U^*   \nonumber
\end{array}\right] \, ,
\end{eqnarray}
where $\Omega_\pm= (\pm\Omega)\cap (0,\infty)$.
Note that 
\begin{eqnarray} \label{bigmatrix2}
E_{{\sqrt{A^*A}}}^\perp(\Omega_\pm )  =
 E_{{{A^*A}}}^\perp(\Omega_\pm^2 ) 
\end{eqnarray}
In addition,
we have
\begin{eqnarray} \label{bigmatrix3}
U E_{{\sqrt{A^*A}}}^\perp(\Omega_\pm ) =
A(A^*A)_\perp^{-\frac12}
E_{{{A^*A}}}^\perp(\Omega_\pm^2 ) \ .
\end{eqnarray}


Since $A^*A$  is partially elliptic, we have 
$\imath_-E_{A^*A}(\Omega\backslash\{0\})\imath_+ \in
  \mathcal{ T}_{1}(\mathcal{H}_+,\mathcal{H}_-)$ for any bounded
 Borel set $\Omega$.  Let $\Omega \subset  [\theta, \Theta]$ with
$0< \theta \le \Theta <\infty$. We have, using  the  estimate (\ref{int0011})
and the definition of the Hilbert-Schmidt norm, that
\begin{eqnarray} 
 \|\imath_-A(A^*A)^{-\frac12} E_{A^*A}(\Omega ) \imath_+\|_2
&=&\|A_-\imath_-(A^*A)^{-\frac12} E_{A^*A}(\Omega ) \imath_+\|_2
\nonumber\\
&\le & ({\Theta}^{\frac12} + c {\theta}^{-\frac12})
 \|\imath_- E_{A^*A}(\Omega ) \imath_+\|_2 \nonumber\\
&\le & 
  ({\Theta}^{\frac12} + c {\theta}^{-\frac12})
\|\imath_- E_{A^*A}(\Omega ) \imath_+\|_1 \nonumber\\
&= & ({\Theta}^{\frac12} + c {\theta}^{-\frac12})\mu_{A^*A}(\Omega) \ ,
\label{bigmatrix4}
\end{eqnarray}
so we can conclude that 
$\imath_-A(A^*A)^{-\frac12} E_{A^*A}(\Omega ) \imath_+ \in
 \mathcal{ T}_{2}(\mathcal{H}_+^{(n)},\mathcal{H}_-^{(m)})$.  Moreover, the same
 argument shows that
\begin{eqnarray}  
\lefteqn{ \|\imath_-A(A^*A)^{-\frac12} E_{A^*A}(\Omega ) 
(A^*A)^{-\frac12}A^*\imath_+\|_2
}\nonumber \\ 
&&\le  ({\Theta}^{\frac12} + c {\theta}^{-\frac12})
 \|\imath_-E_{A^*A}(\Omega ) 
(A^*A)^{-\frac12}A^*\imath_+\|_2
\nonumber \\
&&= ({\Theta}^{\frac12} + c {\theta}^{-\frac12})
\|\imath_-A(A^*A)^{-\frac12} E_{A^*A}(\Omega ) \imath_+\|_2\nonumber \\
&&\le  ({\Theta}^{\frac12} + c {\theta}^{-\frac12})^2
 \|\imath_- E_{A^*A}(\Omega ) \imath_+\|_1 \nonumber\\
&&=  ({\Theta}^{\frac12} + c {\theta}^{-\frac12})^2
\mu_{A^*A}(\Omega) \ , \label{hs2}
\end{eqnarray}
so we also conclude that 
$$\imath_-A(A^*A)^{-\frac12} E_{A^*A}(\Omega ) 
(A^*A)^{-\frac12}A^*\imath_+  \in
  \mathcal{ T}_{2}(\mathcal{H}_+^{(m)},\mathcal{H}_-^{(m)}) \ .$$

 Since   any spectral
 measure for $(A^*A)_\perp$ is also a spectral measure for $(AA^*)_\perp$
by  (\ref{unitary}),
$\mu_{A^*A}$  is  a spectral measure for
$(AA^*)_\perp$.  It now follows from (\ref{555}) and (\ref{hs2}) that 
Hypothesis IV$_2$   is  satisfied 
by  $AA^*$ with
 $\Delta = \mathbb{R} \backslash(-\varepsilon,\varepsilon)$, for any
$\varepsilon >0$, and spectral measure $\mu_{A^*A}$.  We can thus apply Theorem \ref{athm22} to obtain 
(\ref{a16223}).


We now turn to the first order wave operator $\mathbb{W}_A$.  Using 
(\ref{bigmatrix})-(\ref{hs2}), we get that
$\imath_-E_{\mathbb{W}_A}(\Omega)\imath_+  \in
   \mathcal{ T}_{2}(\mathcal{H}_+^{(n+m)},\mathcal{H}_-^{(n+m)})$
for all bounded Borel sets $\Omega$ with $0\notin \bar{\Omega}$.
   Moreover, the measure $\nu_{\sqrt{A^*A}}$,
given by 
\begin{eqnarray} \label{square22}
d\nu_{\sqrt{A^*A}} (\lambda)=\left\{ 
\begin{array}{cc} 
d\mu_{A^*A} (\lambda^2)  & if \ \lambda >0 \\
0 & if  \ \lambda \le 0 
\end{array}\right. \ ,
\end{eqnarray}
is a spectral measure for $(\sqrt{A^*A})_\perp\,$.  It then follows from
(\ref{unitequiv}) that the measure $\nu_{\mathbb{W}_A}$, defined by
$d\nu_{\mathbb{W}_A} (\lambda)= d\mu_{A^*A} (\lambda^2) $, is a 
spectral measure for $(\mathbb{W}_A)_\perp$.  
Thus Hypothesis IV$_2$   is  satisfied 
by  $\mathbb{W}_A$ with
 $\Delta = \mathbb{R} \backslash(-\varepsilon,\varepsilon)$, for any
$\varepsilon >0$, and spectral measure $\nu_{\mathbb{W}_A}$.  
Applying Theorem 
\ref{athm22}, we get (\ref{a162234}). \end{proof}

\begin{remark}  We have explicit bounds on the  
$\mathcal{T}_2$- valued Radon-Nikodym derivatives 
in (\ref{a16223}) and (\ref{a162234}).   
Proceeding as in
 (\ref{bigmatrix4}), and using   Corollary \ref{athm2}, we get 
\begin{eqnarray} \label{bigmatrix45}
 \|A_- P_{A^*A}(\lambda ) \|_2
\le  ({|\lambda|}+ c)
 \| P_{A^*A}(\lambda)\|_2\le 
   ({|\lambda|}+ c) \ .
\end{eqnarray}
Similarly, as in   (\ref{hs2})., we obtain
\begin{eqnarray}  \label{hs211}
\lefteqn{ \|A_- P_{A^*A}(\lambda )A_+^*\|_2
\le ({|\lambda|}+ c)
 \| P_{A^*A}(\lambda)A_+^*\|_2}\\ 
&&= ({|\lambda|}+ c)
 \|A_- P_{A^*A}(\lambda)\|_2 \le
({|\lambda|}+ c)^2
 \| P_{A^*A}(\lambda)\|_2  \le ({|\lambda|}+ c)^2 \nonumber \ .
\end{eqnarray}
\end{remark}

\begin{remark} If $A^*A$ is partially elliptic,  a finite energy classical wave
 $\Psi_t$, as in (\ref{solution}),
has an expansion in generalized eigenfunctions:
\begin{eqnarray} \label{finiteenergyexp2}
\lefteqn{\Psi_t =}\\
&& \lim_{j \to \infty}
\int_{(-j,j)}  {\textstyle{\frac12} }  \mathrm{e}^{it\lambda}
\left[\begin{array}{cc} 
 P_{A^*A}(\lambda^2) &
{\textstyle{\frac1\lambda} } P_{A^*A}(\lambda^2)A_+^*
 \\[.1in]
{\textstyle{\frac1\lambda} }A_- P_{A^*A}(\lambda^2) &
 {\textstyle{\frac1{\lambda^2}} }
A_- P_{A^*A}(\lambda^2)A_+^*  
\end{array}\right] 
 \Phi_0\,
d\mu_{A^*A} (\lambda^2)  \nonumber
\end{eqnarray}
in $\mathcal{H}^{(n+m)}_-$.
\end{remark}

\begin{remark} If both $A^*A$ and $AA^*$ are  partially elliptic, 
or equivalently,
if the first order classical wave operator $\mathbb{W}_A$
 is partially elliptic (see Reamark \ref{remexample9}), the operators $AA^*$ and $\mathbb{W}_A$
 have generalized eigenfunction expansions given  by 
Theorem  \ref{athm}.  Since $A^*A$ is partially elliptic,
they also have  the expansions given in Theorem \ref{tQ}.  Comparing
 these expansions, we conclude that
\begin{eqnarray}
  P_{AA^*}(\lambda)
= \textstyle{\frac1\lambda}
\left(\frac{d\mu_{AA^*}}{d\mu_{A^*A}}(\lambda)\right)^{-1}
A_-P_{A^*A}(\lambda)A^*_+ 
\end{eqnarray} 
for $\mu_{A^*A}$-a.e. $\lambda$, and 
\begin{eqnarray}
\lefteqn{P_{\mathbb{W}_A}(\lambda) =}\\ && 
\left(1 +\frac{d\mu_{AA^*}}{d\mu_{A^*A}}(\lambda^2)\right)^{-1}
\left[\begin{array}{cc} 
 P_{A^*A}(\lambda^2) &
{\textstyle{\frac1\lambda} } P_{A^*A}(\lambda^2)A_+^*
 \\[.1in]
{\textstyle{\frac1\lambda} }A_- P_{A^*A}(\lambda^2) &
 {\textstyle{\frac1{\lambda^2}} }
A_- P_{A^*A}(\lambda^2)A_+^*  
\end{array}\right] \ \nonumber
\end{eqnarray}
for  $\mu^{(2)}_{A^*A}$-a.e. $\lambda$, with 
$d\mu^{(2)}_{A^*A}(\lambda)=d\mu_{A^*A}(\lambda^2)$.
\end{remark}


\section{Proof of the trace estimate}  \label{sHS}

In this section we prove Theorem \ref{tHS}.
Note that it suffices to prove (\ref{tr}) for $r=\nu$ and, since
\begin{equation}
\label{tr2}
 \mathrm{tr}\,  (V^*P_W^\perp (W+I)^{-2r}V)= 
\mathrm{tr}\,  (P_W^\perp (W+I)^{-r}VV^*P_W^\perp (W+I)^{-r}) \ ,
\end{equation}
and the constant on the right-hand-side of (\ref{tr}) is tranlation
invariant, we need only to consider the case when  $V$ is multiplication by the 
scalar function $\chi_{\ell}\equiv\chi_{0,\ell}$ with $\ell =1$.


 Tha partially elliptic $CWO_n^{(2)}$ 
$W$ is of the form $A^*A$, where
$A=RDS$ is an $MPDO^{(1)}_{n,m}$ with $D$ partially elliptic. Thus
 there exists a $CPDO^{(1)}_{n,q}$ 
$D^{\perp}$, as in Definition \ref{essell}. Let us define the operators
\begin{equation}B=D^{\perp}S^{-1}\ \ ,\ \ H=A^*A+B^*B\ \ ,\ \ R=(H+I)^{-1}
\ \ .\end{equation} 
Note that, because of (\ref{5}) and (\ref{05}),
\begin{equation}
\mathcal{H}^{(n)}=\mbox{ker}\,A\oplus\mbox{ker}\,B\ \ ,\ \ 
H=(A^*A)_{\perp}\oplus(B^*B)_{\perp}\ \ ,
\end{equation} 
so $H$ is a self-adjoint operator.  In addition, 
\begin{equation}
P_W^\perp(W+I)^{-2r}=P_W^\perp(H+I)^{-2r}\ \ .
\end{equation}
Thus, if we establish an estimate on the Hilbert-Schmidt norm of the
form
\begin{equation}\label{HS}
\|R^{\nu}\chi_{1}\|_2\leq
C_{d,n,S_{\pm},R_{\pm},D_+,D_+^{\perp},\Upsilon}<\infty \ ,
\end{equation}
 the theorem follows.  Here and on what  follows
 $C_{\alpha, \beta,\dots,}$ denotes a constant depending only on 
  $\alpha, \beta, \ldots \ $.

We first prove the analogous result on the  torus
 $\Lambda_\ell=\left[-\frac \ell 2,\frac\ell 2\right]^d$ and then carry it over to
$\mathbb{R}^d$.  We set 
$\mathcal{H}_\ell^{(n)}=L^2(\Lambda_\ell, dx;\mathbb{C}^n)$.
A $CPDO^{(1)}_{n,m} \ D$ defines a closed densely defined 
 operator $D_\ell$ from $\mathcal{H}_\ell^{(n)}$ to
$\mathcal{H}_\ell^{(m)}$ with periodic boundary condition; 
an operator core is given by 
$C^\infty_{\mathrm{per}}(\Lambda_\ell, \mathbb{C}^n)$, the
infinitely differentiable, periodic $\mathbb{C}^n$-valued
 functions on $\Lambda_\ell$. The restriction of a $CO_n$ $S$ to  
$\Lambda_\ell$ gives the bounded, invertible operator
  $S_\ell$ on $\mathcal{H}_\ell^{(n)}$.
 
Thus we define  the restrictions $ A_\ell =R_\ell D_\ell S_\ell$,
$ B_\ell=D^{\perp}_\ell S_\ell^{-1}$ of the operators $A$, $B$ to the torus,
and set
\begin{equation}
H_\ell=A_\ell^*A_\ell+B_\ell^*B_\ell\ \ ,\ \
R_\ell=(H_\ell+I)^{-1} \ .
\end{equation}
As before, $H_\ell$ is a self-adjoint operator on 
$L^2( \Lambda_\ell, dx;\mathbb{C}^n)$, since the kernels of
 $A_\ell$ and $B_\ell$ are mutually orthogonal. 

\begin{lemma}\label{torus} If  $p>\frac d 2$, we have 
\begin{equation} \label{trtorus}
{\rm tr}\,R_\ell^p
\leq C_{d,n,R_{-},S_{\pm},\Upsilon,\ell,p}<\infty \ .
\end{equation}
\end{lemma}

\begin{proof} First we notice that the lemma is valid for $-\Delta_\ell$, the
Laplacian on the torus with periodic boundary conditions. Namely, using
the Fourier series representation for $L^2(\Lambda_\ell)$,  we find
\begin{equation}
{\rm
tr}((-\Delta_\ell+I)^{-p})=
\sum_{{k}\in\frac{2\pi}{\ell}\mathbb{Z}^d}(1+|{k}|^2)^{-p}
<\infty\ \ . \label{Delta}
\end{equation}

In what follows,we write $F\simeq G$ if the positive self-adjoint operators 
$F$ and $G$ are unitarily equivalent, and we write 
$F\preceq G$ if $F\simeq J$ for some positive self-adjoint operator
$J \le G$.

Recall that if $C$ is a closed, densely defined
operator
 on a Hilbert space, we always have
$(C^*C)_{\perp}\simeq (CC^*)_{\perp}$ (as in (\ref{unitary})). Thus,
\begin{eqnarray}
\lefteqn{(H_\ell)_{\perp}=(S_\ell D_\ell^*R_\ell^2 D_\ell S_\ell)_{\perp}\oplus
(S_\ell^{-1} D^{\perp *}_\ell D^{\perp}_\ell S_\ell^{-1})_{\perp}}\nonumber\\
&&\geq (R_-^2S_\ell D_\ell^*D_\ell S_\ell)_{\perp}\oplus
(S_\ell^{-1}D^{\perp *}_\ell D^{\perp}_\ell S_\ell^{-1})_{\perp}\nonumber\\
&&\simeq(R_-^2D_\ell S_\ell^2D_\ell^*)_{\perp}\oplus
(D^{\perp}_\ell S_\ell^{-2}D^{\perp *}_\ell)_{\perp}\nonumber\\
&&\geq(S_-^2R_-^2D_\ell D_\ell^*)_{\perp}\oplus
(S_+^{-2}D^{\perp}_\ell D^{\perp *}_\ell)_{\perp}\nonumber\\ 
&&\simeq (S_-^2R_-^2D_\ell^*D_\ell)_{\perp}\oplus
(S_+^{-2}D^{\perp *}_\ell D^{\perp}_\ell)_{\perp}\nonumber\\ 
&& \ge \min(S_-^2R_-^2,S_+^{-2}) \, \left[ 
(D_\ell^*D_\ell)_{\perp}\oplus
(D^{\perp *}_\ell D^{\perp}_\ell)_{\perp}\right]\nonumber \\
&&\geq \Upsilon  \min(S_-^2R_-^2,S_+^{-2}) \ (-\Delta_\ell\otimes I_n)_{\perp}\ \ ,
\label{ranH}
\end{eqnarray}
where we used  (\ref{4}) to obtain the last inequality. 


If $\mathcal{E}$ and $\mathcal{F}$ are (not necessarily closed) subspaces
of some Hilbert space with $\mathcal{E}\subset \mathcal{F}$, by 
$\mathrm{codim}_{\mathcal{F}}(\mathcal{E})$ we denote the dimension of 
the orthogonal complement of $\mathcal{E}$ in $\overline{\mathcal{F}}$.
Again using  (\ref{4}), we obtain
\begin{eqnarray}
\lefteqn{{\rm dim}\,({\rm ker}\,H_l) ={\rm codim}\,_{{\rm
ker}A_l^*A_l}\,({\rm ran}\,B_l^*B_l)}\nonumber\\
&& ={\rm codim}\,_{S_\ell^{-1}{\rm
ker}D_\ell}(S_\ell^{-1} ({\rm ran}\,D_\ell^{\perp *}))
={\rm codim}\,_{{\rm ker}D_\ell}({\rm ran}\,D_\ell^{\perp *})\nonumber\\
&& ={\rm codim}\,_{{\rm ker}D_\ell^*D_\ell}({\rm ran}\,
D_\ell^{\perp *}D_\ell^{\perp})
={\rm dim}\,({\rm ker}\,(D_\ell^*D_\ell +D^{\perp *}_\ell D^{\perp}_\ell))\nonumber\\
&& ={\rm dim}\,({\rm ker}\,(-\Delta_\ell\otimes I_n))=n\ \ .\label{kerH}
\end{eqnarray}

Combining (\ref{ranH}) and (\ref{kerH}), we obtain
\begin{equation}
H_l\succeq C_{R_-,S_{\pm},\Upsilon}\ (-\Delta_\ell \otimes I_n)\ \ ,\ \ 
 C_{R_-,S_{\pm},\Upsilon} >0\ \ .\label{HDelta}
\end{equation}

If $0\leq F\preceq G$, then ${\rm tr}f(G)\leq {\rm tr}f(F)$, for
any positive, decreasing function $f$ on $[0,\infty)$. Thus, the lemma
follows from (\ref{Delta}) and (\ref{HDelta}). \end{proof}

To make the connection between the torus $\Lambda_\ell$ and $\mathbb{R}^d$,  
for any positive integer
$q$, let
 $\imath_\ell
: L^2(\Lambda_\ell,dx; \mathbb{C}^q )\rightarrow 
L^2(\mathbb{R}^d,dx; \mathbb{C}^q)$ be the isometry that
extends functions from $\Lambda_\ell$ to $\mathbb{R}^d$ by setting them
equal to zero outside $\Lambda_\ell$. (We omit $q$ from the notation.)
 Its adjoint, $\imath_\ell^*$, is the partial
isometry that restricts square integrable functions on $\mathbb{R}^d$ to
$\Lambda_\ell$, i.e., 
\begin{equation}\label{injection}
\imath_\ell^*\imath_\ell=I\ \ ,\ \ \imath_\ell \imath_\ell^*=\chi_\ell\ \ .
\end{equation}

Given a real-valued function 
$\phi\in C_0^\infty(\mathbb{R}^d)$, wth
$\mathrm{supp}\,\phi\subset\Lambda_{\ell^\prime}$,  ${\ell^\prime}<\ell$,
we do not distinguish in the notation between $\phi$ as a multiplication operator on
the torus $\Lambda_\ell$ and on the whole space. 
If $D$ is a $CPDO^{(1)}_{n,m}\, $,
we can verify that 
\begin{equation}
{D} \imath_\ell \phi = \imath_\ell {D}_\ell \phi \ \ \mbox{on}
\ \ \mathcal{D}({D}_\ell) \ .
\end{equation}
It follows that, if $Y$ is a $MPDO^{(1)}_{n,m}\, $, we have
\begin{equation} \label{restr}
Y \imath_\ell \phi = \imath_\ell Y_\ell\phi \ \ \mbox{on}
\ \ \mathcal{D}(Y_\ell) \ .
\end{equation}
 We let
\begin{equation}
A(\phi)=RD[\phi]S\ \ ,\ \ B(\phi)=D^{\perp}[\phi]S^{-1}\ \ ,
\end{equation}
where $D[\phi]$, $D^{\perp}[\phi]$ are as in Lemma \ref{pho}. 
Thus  $A(\phi)$ and $B(\phi)$ are bounded operators given by
multiplication by matrix-valued, measurable functions, with
\begin{equation} \label{abp}
\|A(\phi)\|\leq D_+ R_+S_+\|\nabla \phi\|_{\infty}\ \ ,\ \ 
\|B(\phi)\|\leq{D^{\perp}_+} S_-^{-1}\|\nabla \phi\|_{\infty}\ .
\end{equation}
We denote by
$A_\ell(\phi)$ and $B_\ell(\phi)$ the corresponding restrictions to the torus
$\Lambda_\ell$;  they also satisfy the bounds (\ref{abp}).
Furthermore, we set, with $X$ standing for
either $A$
or $B$, 
\begin{eqnarray*}
F^X_\ell(\phi) & = &
\sqrt{R}X(\phi)^*\imath_\ell( X_\ell\sqrt{R}_\ell)-(X\sqrt{R})^*
\imath_\ell X_\ell(\phi)\sqrt{R}_\ell\ \ ,\nonumber\\
\widehat{F}^X_\ell(\phi) & = &
\sqrt{R}_\ell X_\ell(\phi)^*(X_\ell \sqrt{R}_\ell)-(X_\ell \sqrt{R}_\ell)^* 
X_\ell(\phi)\sqrt{R}_\ell\ \ ,
\end{eqnarray*}
\begin{equation}
F_\ell(\phi)=F^A_\ell(\phi)+F^B_\ell(\phi)\ \ ,\ \
\widehat{F}_\ell(\phi)=\widehat{F}^A_\ell(\phi)+\widehat{F}^B_\ell(\phi)\
\ .\label{Fl}
\end{equation}
It is easy to see that
\begin{equation}
\|X\sqrt{R}\|\leq 1\ \ ,\ \ \|X_\ell\sqrt{R}_\ell\|\leq 1\ ,
\label{Restimate}
\end{equation}
so
\begin{equation} \label{Festimate2}
\|F^X_\ell(\phi)\|\leq C_{R_+,S_{\pm},D_+,D_+^{\perp}}\|\nabla\phi\|_{\infty}\ .  
\end{equation}
In addition, it follows from Lemma \ref{torus} that if $p > d $
 we have
\begin{equation}
\|\widehat{F}^X_\ell(\phi)\|_p\leq
C_{d,n,R_{\pm},S_{\pm},\Upsilon,\ell,p}\,\|\nabla\phi\|_{\infty}\ .
\label{Festimate}
\end{equation}

\begin{lemma} \label{recursion1lemma} Let
 $\phi\in C^{\infty}_0(\mathbb{R}^d)$ be
real-valued with 
$\mathrm{supp}\,\phi\subset\Lambda_{\ell^\prime}$,  ${\ell^\prime}<\ell$.
 Then
\begin{equation}\label{recursion1}
R\phi=\imath_\ell\phi
R_\ell\imath_\ell^*+\sqrt{R}F_\ell(\phi)\sqrt{R}_\ell\imath_\ell^*\ .
\end{equation}
\end{lemma}
\begin{proof} Because of (\ref{injection}) and $\phi\chi_\ell=\phi$, equation
(\ref{recursion1}) is the same as
\begin{equation}\label{recursion1b}
R\phi\imath_\ell=\phi\imath_\ell
R_\ell+\sqrt{R}F_\ell(\phi)\sqrt{R}_\ell\ \ .
\end{equation}
To prove (\ref{recursion1b}), let $u\in L^2(\mathbb{R}^d,dx; \mathbb{C}^n)$ and
$v\in L^2(\Lambda_\ell,dx; \mathbb{C}^n)$. We compute, as in
 \cite[Lemma 24]{FK3},
\begin{eqnarray}
\langle u,(R\phi\imath_\ell-\phi\imath_\ell R_\ell)v\rangle
&=&\langle \phi Ru,\imath_\ell(H_\ell+I)R_\ell v\rangle-\langle
(H+I)Ru,\phi\imath_\ell R_\ell v\rangle\nonumber\\
&=& \langle \phi Ru,\imath_\ell H_\ell R_\ell v\rangle-\langle
HRu,\phi\imath_\ell R_\ell v\rangle\label{commutator1} \ .
\end{eqnarray}
For $X=A$ or $B$, we may use Lemma \ref{pho} (which  is 
valid also on the torus)  and (\ref{restr}) twice to obtain
\begin{eqnarray}
\lefteqn{\langle \phi Ru,\imath_\ell X^*_\ell X_\ell R_\ell v\rangle
=\langle Ru,\imath_\ell X^*_\ell \phi X_\ell R_\ell v\rangle +\langle Ru,\imath_\ell X_\ell(\phi)^*X_\ell R_\ell v\rangle
}\nonumber\\
&&=\langle XRu,\imath_\ell \phi X_\ell R_\ell v\rangle+
\langle Ru,X(\phi)^*\imath_\ell X_\ell R_\ell v\rangle
\nonumber\\
&&=\langle XRu,\imath_\ell X_\ell\phi R_\ell v\rangle
-\langle XRu,\imath_\ell X(\phi)R_\ell v\rangle
+\langle Ru,X(\phi)^*\imath_\ell X_\ell R_\ell u\rangle \nonumber\\
&&=\langle X^*XRu,\phi\imath_\ell R_\ell v\rangle +
\langle u,\sqrt{R}F^X_\ell(\phi)\sqrt{R}_\ell v\rangle 
 \ .\label{commutator2}
\end{eqnarray}
The identity (\ref{recursion1b}) now follows from (\ref{commutator1})
and (\ref{commutator2}). \end{proof}


\begin{lemma}\label{recursion2lemma}
Let $\phi,\psi\in C^{\infty}_0(\mathbb{R}^d)$ be real-valued     with 
${\rm supp}\,\phi,\, {\rm supp}\,\psi\subset\Lambda_{\ell^\prime}$,  
${\ell^\prime}<\ell$, with $\psi=1$ on ${{\rm supp}\,\phi}$. Then
\begin{equation}\label{recursion2}
\sqrt{R}F_\ell(\phi)=\imath_\ell\psi\sqrt{R}_\ell\widehat{F}_\ell(\phi)
+\sqrt{R}F_\ell(\psi)\widehat{F}_\ell(\phi) \ .
\end{equation}
\end{lemma}

\begin{proof} We observe that 
\[D(R^{-\frac 1 2}_\ell)=D(A_\ell)\cap D(B_\ell)\supset
\imath_\ell^*\psi\,(D(A)\cap D(B))=\imath_\ell^*\psi\,{\rm ran}\,R^{\frac 1 2}\ \
.\] Thus, $R_\ell^{-1/2}\imath_\ell^*\psi R^{1/2}$ is a closed, everywhere defined
operator,  hence bounded, and so is its adjoint $R^{1/2}\psi\imath_\ell
R^{-1/2}_\ell$. Furthermore, we note that 
${\rm supp}\,\phi \cap {\rm supp}\,\nabla\psi = \emptyset$ and
${\rm supp}\,X(\phi)\subset{\rm supp}\,\nabla\phi\subset{\rm
supp}\,\phi$, for $X=A$ or $B$. From this we conclude
\begin{equation}
\psi X(\phi)=X(\phi)\  ,\ \ X(\phi)^*X\psi=X(\phi)^*X\ .
\end{equation}
Using the definitions (\ref{Fl}), we get
\begin{eqnarray}
F^X_\ell(\phi)&=&R^{\frac 1 2}\psi X(\phi)^*\imath_\ell 
(X_\ell R_\ell^{\frac 1 2})
-(X(\phi)^*X\psi R^{\frac 1 2})^*\imath_\ell R^{\frac 1 2}_\ell \nonumber \\
&=& (R^{\frac 1 2}\psi\imath_\ell R^{-\frac 1 2}_\ell)\widehat{F}_\ell(\phi)\  \ .
\end{eqnarray}
We now apply Lemma \ref{recursion1lemma} in the form
 (\ref{recursion1b}), 
obtaining
\begin{equation}
R^{\frac 1 2}F_\ell(\phi)=\left(R\psi\imath_\ell R^{-\frac 1
2}_\ell\right)\widehat{F}_\ell(\phi)=\left(\psi\imath_\ell R_\ell^{\frac 1 2}+R^{\frac 1
2}F_\ell (\psi)\right)\widehat{F}_\ell (\phi)\ \ ,
\end{equation}
which proves proves the lemma. \end{proof}

We may now finish the proof of  Theorem \ref{tHS}.
We pick $\rho\in C_0^{\infty}(\mathbb{R}^d)$,  positive with ${\rm
supp}\,\rho\subset\Lambda_{\frac12}$ and $\int
\rho(x)\,dx=1$, and set $\phi_j=\rho\ast \chi_{j+\frac 1 2}$ for $j >0$; note
that $\phi_j \in
 C_0^{\infty}(\mathbb{R}^d)$ and $\chi_j\leq\phi_j\leq\chi_{j+1}$.


Let  $0< \ell + \tau  < L$, where $\tau$ is  a positive integer.  Applying
 (\ref{recursion1}) and then  iterating (\ref{recursion2})
$\tau-1$ times, we get
\begin{eqnarray}
\lefteqn{R \chi_\ell =[ R \phi_\ell] \chi_\ell = 
\imath_L \phi_\ell R_L  \imath_L^*\chi_\ell + 
\left[\sqrt{R} F_L(\phi_\ell)\right]\sqrt{R_L}\imath_L^*\chi_\ell} \\
&&= \imath_L \phi_\ell R_L  \imath_L^*\chi_\ell +  
\imath_L \phi_{\ell+1} \sqrt{R_L} \widehat{F}_L(\phi_\ell)\sqrt{R_L}
 \imath_L^*\chi_\ell  \nonumber\\
 && \hspace{1.2in} + 
\left[\sqrt{R} F_L(\phi_{\ell+1})\right]\widehat{F}_L(\phi_\ell)\sqrt{R_L} 
\imath_L^*\chi_\ell \\
&& \vdots  \hspace{1.2in} \vdots \nonumber \\
&& =\left\{ \imath_L \phi_\ell R_L  \imath_L^*\chi_\ell +
 \sum_{k=1}^{\tau - 1} \imath_L  \phi_{\ell+k} \sqrt{R_L} 
\prod_{j=k-1}^{0} \widehat{F}_L(\phi_{\ell+j}) 
\sqrt{R_L}
 \imath_L^*\chi_\ell\right\}  \nonumber\\
&&  \hspace{.5in} + \sqrt{R} F_L(\phi_{\ell+\tau -1})
\widehat{F}_L(\phi_{\ell+\tau -2})
 \ldots \widehat{F}_L(\phi_\ell)\sqrt{R_L}
 \imath_L^*\chi_\ell \\
&&\equiv \Gamma_\ell^{(\tau,L)} + \Xi_\ell^{(\tau,L)} \ . \label{iterate}
\end{eqnarray}
The last line defines the operators $\Gamma_\ell^{(\tau,L)}$ and 
$ \Xi_\ell^{(\tau,L)}$
on $L^2(\mathbb{R}^d,dx; \mathbb{C}^n)$.   Note that
\begin{equation} \label{ggg}
\Gamma_\ell^{(\tau,L)}= \chi_{\ell + \tau}\Gamma_\ell^{(\tau,L)} \ .
\end{equation}
 Moreover, it follows from (\ref{trtorus}),   (\ref{Festimate2}), 
and (\ref{Festimate}) that
\begin{equation}\label{GXestimate1}
\| \Gamma_\ell^{(\tau,L)}\|_p\leq 
 C_{d,n,R_{\pm},S_{\pm},D_+,D_+^\perp, \Upsilon,L,\ell, \tau,p}<\infty \ ,\ \
\mbox{if} \ \ p>\frac d 2\ ,
\end{equation}
\begin{equation}\label{GXestimate}
\|\ \Xi_\ell^{(\tau,L)}\|_q \leq 
C_{d,n,R_{\pm},S_{\pm},D_+,D_+^\perp, \Upsilon,L,\ell, \tau,q}<\infty \ ,\ \
\mbox{if} \ \ q>\frac d {\tau} \ .
\end{equation}
These constants also depend on our choice of the function $\rho$, but we fixed 
$\rho$ once for all, so we omit the dependence.

We now let  $0< \ell + \theta\tau  < L$, where $\tau$ and $\theta$
are  positive integers.  Applying  (\ref{iterate}) and   (\ref{ggg}) 
successively $\theta$ times, 
 we obtain
\begin{eqnarray}\label{recurrencerelation}
\lefteqn{R^{\theta}\chi_\ell=R^{\theta-1}\left(\Xi_\ell^{(\tau,L)} +
\Gamma_\ell^{(\tau,L)}\right)= R^{\theta-1}\Xi_\ell^{(\tau,L)} +
R^{\theta-2}\left[R\chi_{\ell + \tau}\right]\Gamma_\ell^{(\tau,L)}}
 \nonumber \\
&& = R^{\theta-1}\Xi_\ell^{(\tau,L)} +  
R^{\theta-2}\Xi_{\ell+\tau}^{(\tau,L)}\Gamma_\ell^{(\tau,L)} +
R^{\theta-2}\Gamma_{\ell+\tau}^{(\tau,L)}\Gamma_\ell^{(\tau,L)} 
\hspace{.5in} \nonumber \\
&& \vdots   \hspace{1in} \vdots  \label{vdot}\\
&& = R^{\theta-1}\Xi_\ell^{(\tau,L)} +
 \sum_{j=2}^{\theta}  R^{\theta-j} \,\Xi_{\ell+(j-1)\tau}^{(\tau,L)}
\prod_{r=j-2}^{0}\Gamma_{\ell+r\tau}^{(\tau,L)}  \nonumber
  \\
&& \hspace{1.3in}   +
\Gamma_{\ell+(\theta-1)\tau}^{(\tau,L)}
\Gamma_{\ell+(\theta-2)\tau}^{(\tau,L)} \ldots \Gamma_\ell^{(\tau,L)} \ .
\nonumber
\end{eqnarray}
 
Let us choose $\tau =[ [\frac d 2]] $, where $[[x]]$ denotes the
the smallest integer bigger than $x$, and 
$\theta=\nu \equiv [[\frac d 4]]$.
 It then follows from
  (\ref{GXestimate1}), (\ref{GXestimate})  and (\ref{vdot}) that 
\begin{equation} \label{estimate8}
\|R^{\nu}\chi_\ell \|_2\leq
 C_{d,n,R_{\pm},S_{\pm},D_+,D_+^\perp, \Upsilon,L,\ell}<\infty \ \ .
\end{equation}

If we now take $\ell=1$, $L = [ [\frac d 2]]\,[ [\frac d 4]]+ 2 $, the 
desired estimate (\ref{HS}) follows from (\ref{estimate8}).
This concludes the proof of  Theorem \ref{tHS}.




\appendix{The bottom of the spectrum} \label{appC}


\begin{theorem}\label{thmC1}  
Let $W$ be a classical wave operator.
The bottom of the spectrum of 
$W_\perp$ is at $0$, so  $W_\perp$ and $W$ have the
 same spectrum. 
\end{theorem}

\begin{lemma}\label{lemC1} 
Let $H_0$ be a self-adjoint operator with a gap $(0,\gamma)$ in its 
spectrum, and let $H= Q H_0Q$, where $Q$ is a strictly positive, 
bounded operator, say $0< Q_- \le Q \le Q_+$ for some constants $Q_\pm\,$.
 Then $(0,\gamma Q_-^2)$ is 
a gap in the spectrum of $H$. 
\end{lemma}

\begin{proof} 
Let $\lambda >0$, $ \mu \in (0,\gamma)$.  Then
\begin{eqnarray}
\lefteqn{H - \lambda = Q(H_0 - \mu)Q + (\mu Q^2 - \lambda)}\\
&&= Q \left( I + \left(\mu - \lambda  Q^{-2}\right)(H_0-\mu)^{-1}\right)
(H_0-\mu)Q \ . \nonumber
\end{eqnarray}
We  fix $ \mu= \frac  \gamma 2$, so 
$\|(H_0-\frac  \gamma 2)^{-1}\| \le \frac 2 \gamma$.  Since $Q$ is invertible with
a bounded inversee, 
it suffices to show that 
\begin{equation} \label{gam}
\left\| \left(\textstyle{\frac  \gamma 2 }- \lambda  Q^{-2}\right)
(H_0-\textstyle{\frac  \gamma 2 })^{-1} \right\| <1
\end{equation}
for $\lambda \in (0, \gamma Q_-^2)\,$.  But for such $\lambda$
we have
\begin{equation}
-  \textstyle{\frac  \gamma 2 } <  \textstyle{\frac  \gamma 2 }- \lambda  Q_-^{-2} \le 
\textstyle{\frac  \gamma 2 }- \lambda  Q^{-2} \le
 \textstyle{\frac  \gamma 2 }- \lambda  Q_+^{-2}<  
\textstyle{\frac  \gamma 2 }\ ,
\end{equation}
so (\ref{gam}) follows.  \end{proof}

We may now prove Theorem \ref{thmC1}.  In view of
(\ref{cwo2}) and (\ref{unitary}), 
it suffices to consider the case of a first order classical wave operator
$W=SDS$. 
In this case $0$ is in the spectrum of  $W_\perp$ if and only if
 $W$ has no spectral gap of the form
$(-\gamma,\gamma)$, and  it follows from  Lemma \ref{lemC1} that
 we can take  $W =  D$.

Thus it suffices to show that $0$ is in the spectrum of $D_\perp$. 
   For $r>0$, let us define 
\begin{equation}\label{C3}
\varphi_r(x)=r^{d/2}\varphi(rx) 
\end{equation} 
for any vector-valued measurable function $\varphi(x)$ on $\mathbb{R}^d$. 
Note that $\|\varphi_r\|_2= \|\varphi\|_2$.  Let us pick 
 $\psi\in \mathcal{D}(D) \cap (\textrm{ker}\,D)^\perp$, with $\|\psi\|=1$.
It is easy to see that 
$\psi_r\in \mathcal{D}(D) \cap (\textrm{ker}\,D)^\perp$, $\|\psi_r\|=1$,
 and
$D\psi_r= r (D\psi)_r\,$.
Thus
$
\lim_{r\rightarrow 0} \,  D\psi_r=0$ in $\mathcal{H}^{(m)}$, so
 we can use  Weyl's criterion to conclude that that $0$ is in the
spectrum of $D_\perp$. 

Theorem \ref{thmC1} is proved.

\appendix{Operators on locally convex spaces} \label{lcs}

In this appendix we review some basic facts about unbounded operators
on locally convex spaces.  

Let $\mathcal{E}$ and $\mathcal{F}$ be locally convex spaces.
 An operator $C$ from  $\mathcal{E}$ to $\mathcal{F}$ is a linear mapping
from  $\mathcal{D}(C)$ to $\mathcal{F}$,  the domain $\mathcal{D}(C)$
being a linear subspace of  $\mathcal{E}$.  $C$ is densely defined if 
$\mathcal{D}(C)$ is dense in  $\mathcal{E}$. $C$ is closed  if its graph,
$\Gamma(C)=\{ (\phi,C\phi): \ \phi \in  \mathcal{D}(C)  \} $, is closed in 
 $\mathcal{E} \times \mathcal{F}$. $C$ is closable if 
$\overline{\Gamma(C)}= \Gamma(\overline{C})$ for a (necessarily unique)
 operator $\overline{C}$ from  $\mathcal{E}$ to $\mathcal{F}$, 
the closure of $C$, i.e.,    if 
$\overline{\Gamma(C)}\cap \{0\}\times\mathcal{F}= \{(0,0)\}$. 

We set
$J_{\mathcal{E},\mathcal{F}}:\mathcal{E}\times\mathcal{F}\to  
\mathcal{F}\times\mathcal{E}$  to be the isomorphism given by 
$J_{\mathcal{E},\mathcal{F}} (\phi,\psi) =  (-\psi,\phi)$; note
$J_{\mathcal{E},\mathcal{F}}J_{\mathcal{F},\mathcal{E}}=
 -I_{\mathcal{E} \times \mathcal{F}}$.    


We say that two  locally convex spaces $\mathcal{E}_\pm$  form a dual pair,
if there exists   a sesquilinear form $\langle \ , \ \rangle$ on 
$\mathcal{E}_+ \times \mathcal{E}_-$, which turns  $\mathcal{E}_\pm$ into
 the duals of each other. (Example:  $\mathcal{E}_+=L^2_{0}(\mathbb{R}^d, dx;\mathbb{C}^q) $ and
$\mathcal{E}_- = L^2_{loc}(\mathbb{R}^d, dx;\mathbb{C}^q)$ as in 
Section \ref{spaces}, with the sesquilinear form given in (\ref{14}).)

Let   $\mathcal{E}_\pm$ be a dual pair.  Given $S\subset \mathcal{E}_+$ we set
\begin{equation}
S^\perp = \{\psi \in \mathcal{E}_-: \ \langle \phi , \psi \rangle=0 
\ \ \mbox{for all} \ \ \phi \in S\} \ .
\end{equation}
It is easy to see  that $S^\perp$ is closed and  $S^\perp = \bar{S}^\perp$. 

\begin{proposition} \label{perp}
Let $V$ be a linear subspace of $\mathcal{E}_+$, then 
$\bar{V} =(V^\perp)^\perp$.
\end{proposition}

\begin{proof}
We can assume that $V$ is closed.  Obviously, 
${V} \subset (V^\perp)^\perp$.  To see the converse, let 
$\phi \in \mathcal{E}_+$, $\phi \notin V$.
According to the Hahn-Banach Theorem, there exists 
$\psi \in \mathcal{E}_-$, such that  $\langle \phi , \psi \rangle\not=0 $ but
$\langle \phi' , \psi \rangle=0 $ for all $\phi' \in V$.  Thus $\psi \in V^\perp$,
hence $\phi \notin (V^\perp)^\perp$. \end{proof}


Let us consider dual pairs $\mathcal{E}_\pm$ and $\mathcal{F}_\pm$;
note that   $\mathcal{E}_\pm \times\mathcal{F}_\pm$ also forms a
dual pair.  Let   
$C$ be a densely defined operator from  $\mathcal{E}_-$ to $\mathcal{F}_-$.Its 
adjoint,  $C^\dagger$, is defined as the operator from 
$\mathcal{F}_+$ to $\mathcal{E}_+$ with domain
\begin{equation}
\mathcal{D}(C^\dagger)= \{ \psi \in \mathcal{F}_+: 
\ \phi \in \mathcal{D}(C) \mapsto  \langle \psi , C\phi \rangle \ \ 
\mbox{is continuous}\} \ ,
\end{equation}
with $C^\dagger\psi$,  $ \phi \in \mathcal{D}(C^\dagger)$,
 being the unique element of  $\mathcal{E}_+$ such
 that $ \langle C^\dagger\psi , \phi \rangle =  \langle \psi , C\phi \rangle$
for all  $\phi \in \mathcal{D}(C)$. It follows that 
\begin{equation} \label{graphad}
\Gamma( C^\dagger)= (J_{\mathcal{E}_-, \mathcal{F}_-}\Gamma( C))^\perp
=J_{\mathcal{E}_+, \mathcal{F}_+}\Gamma( C)^\perp
\end{equation}.

The following theorem is identical to the familiar result for 
operators in Hilbert spaces (e.g., \cite[Theorem VIII.1]{RS1}). 

\begin{theorem}
 Let $\mathcal{E}_\pm$ and $\mathcal{F}_\pm$ be dual pairs 
of locally convex spaces, and let $C$ be a densely defined operator
 from  $\mathcal{E}_-$ to $\mathcal{F}_-$.  Then:
\begin{description}
\item[(i)]  $C^\dagger$ is closed.
\item[(ii)]  $C$ is closable if and only if $C^\dagger$  is densely defined, in 
which case $\bar{C}= (C^\dagger)^\dagger$.
\item[(iii)] If $C$ is closable, then  $\bar{C}^\dagger = {C}^\dagger$.
\end{description}
\end{theorem} 

\begin{proof}
(i) follows immediately from (\ref{graphad}).

To prove (ii), note that if $C$ is not closable, we can find 
$\phi \in  \mathcal{F}_-\,$, $\phi \not=0$, such that 
$(0,\phi) \in \overline{\Gamma(C)}$, so 
$(\phi,0) \in \Gamma( C^\dagger )^\perp$ by  (\ref{graphad}) and  Proposition
\ref{perp}. Thus $\langle \eta,\phi \rangle=0$ for all 
$\eta \in \mathcal{D}(C^\dagger)$, hence $C^\dagger$ is not densely defined.

On the other hand, if  $C^\dagger$ is not densely defined, it follows from
the Hahn-Banach Theorem that there is $\phi \in  \mathcal{F}_-\,$,
$\phi \not=0$, such that $\langle \eta,\phi \rangle=0$ for all 
$\eta \in \mathcal{D}(C^\dagger)$.  Thus 
$(\phi,0) \in \Gamma( C^\dagger )^\perp$, so 
$(0,\phi) \in  \overline{\Gamma(C)}$ by  (\ref{graphad}) and  Proposition
\ref{perp}. 

Thus, if $C$ is closable, we get from  (\ref{graphad}) that
\begin{eqnarray} \label{graphad2}
\Gamma( (C^\dagger)^\dagger)&= &
(J_{\mathcal{F}_+, \mathcal{E}_+}\Gamma( C^\dagger))^\perp =
(J_{\mathcal{F}_+, \mathcal{E}_+}
J_{\mathcal{E}_+, \mathcal{F}_+}\Gamma( C)^\perp)^\perp\\
&= &(\Gamma( C)^\perp)^\perp = \overline{\Gamma(C)}
 ={\Gamma(\bar{C})} \ .
\end{eqnarray}

To finish the proof, it suffices to note that (iii) follows from (ii).  
\end{proof}



\appendix{Bochner integrals and vector valued measures}\label{appA}

In this appendix, we summarize some results of the theory of Bochner
 integrals and  vector valued
Borel measures \cite{DU}. We briefly discuss the Radon-Nikodym Theorem 
and  Lebesgue's Differentiation Theorem. All results are true for Borel measures
in $\mathbb{R}^d$. For the sake of simplicity, everything is stated for the
real line, $\mathbb{R}$. 

Let $\mathcal{B}$ be a separable Banach space. A function
$f(\lambda):\mathbb{R}\rightarrow\mathcal{B}$ is said to be weakly Borel
measurable if $l(f(\lambda))$ is Borel measurable, for any $l\in\mathcal{B}^*$.
$f$ is said to be strongly Borel measurable if it is the pointwise limit of a
sequence of finitely valued Borel step functions. Since $\mathcal{B}$ is
separable, strong measurability and weak measurability are equivalent.
Hence, such a function is simply called measurable. If $f(\lambda)$ is
measurable, then so is $\|f(\lambda)\|$. 

Let $\mu$ be a positive Borel measure on $\mathbb{R}$. A Borel
measurable, $\mathcal{B}$-valued function $f(\lambda)$ is called integrable if
$\|f(\lambda)\|$ is integrable. Then, there exists a sequence $f_n(\lambda)$
of finitely valued Borel step functions such that 
\[\lim_{n\rightarrow\infty}\int
\|f(\lambda)-f_n(\lambda)\|\,d\mu(\lambda)\ =\ 0\  .\] 
 The norm limit
\begin{equation}\label{A1}
\int f(\lambda)\,d\mu(\lambda)=\lim_{n\rightarrow\infty}\int
f_n(\lambda)\,d\mu(\lambda)
\end{equation} 
exists and is independent of the sequence $f_n$; 
it is  called the Bochner integral. A Borel measurable, $\mathcal{B}$-valued function $f(\lambda)$ is said to be locally integrable if
$\|f(\lambda)\|$ is locally integrable.

Let $I_{\lambda,\delta}$ denote the closed interval
$[\lambda-\delta,\lambda+\delta]$. The separability of $\cal B$ allows to
prove Lebesgue's Differentiation Theorem for vector valued functions
using the same result for scalar valued functions
 (e.g., \cite[Theorem 9 on page 49]{DU}): 
\begin{theorem}[Lebesgue's Differentiation Theorem] \label{thmA1} If
$f(\lambda)$ is a $\mathcal{B}$-valued, $\mu$-locally integrable function, then
\begin{equation}\label{A2}
\lim_{\delta\downarrow 0}\
\frac{1}{\mu(I_{\lambda,\delta})}
\int_{I_{\lambda,\delta}}\|f(\lambda')-f(\lambda)\|\,d\mu(\lambda')\
=\ 0   \  \ \ \mbox{for} \ \mu-\mbox{a.e.}\ \lambda\ .
\end{equation}
\end{theorem}

A $\mathcal{B}$-valued Borel measure on $\mathbb{R}$ is a 
countably additive
set function $E(\Omega)$, defined on the family of bounded Borel subsets 
of  $\mathbb{R}$ and
taking its values in $\mathcal{B}$.  $E(\Omega)$ is said to be
 of local bounded variation
if the positive set function $\|E(\Omega)\|$ is of bounded variation in any
bounded Borel subset of $\mathbb{R}$.
 $E(\Omega)$ is said to be absolutely continuous with
respect to a Borel measure $\mu$ if $E(\Omega)=0$ whenever
$\mu(\Omega)=0$.  If  $\mathcal{B}$ is a separable dual space, one has the
  Radon-Nikodym Theorem \cite[Theorem 1 on page 79]{DU}.

If $\mathcal{H}$ is a separable Hilbert space, then 
 $\mathcal{T}_p(\mathcal{H})$,  the Banach
space of bounded operators $S$ on $\mathcal{H}$ with 
$\|S\|_p= (\mathrm{tr} \,|S|^p)^{1 \over p}<\infty$,
 is a separable dual space for $1 \le p<\infty$.
 Let
$\mathcal{T}_{p,+}(\mathcal{H})$ denote the positive  operators in 
 $\mathcal{T}_p(\mathcal{H})$. 
If $p=1$ and $E(\Omega)$ is a
$\mathcal{T}_{1,+}(\mathcal{H})$-valued Borel measure on $\mathbb{R}$,
then $E(\Omega)$ is of local bounded variation,
$\mu(\Omega)={\rm tr}\,E(\Omega)$ is a positive Borel measure, and $E$
is absolutely continuous with respect to $\mu$. For 
$\mathcal{T}_{p,+}(\mathcal{H})$-valued Borel measures we can state
 the Radon-Nikodym theorem as follows (see also 
\cite[Theorem 1.1 on page 321]{B}, \cite[Theorem 1.1 on page 187]{B2}):

\begin{theorem}\label{thmA2} Let  $\mathcal{H}$ be a separable Hilbert 
space and $1 \le p<\infty$.  If $E(\Omega)$ is a 
$\mathcal{T}_{p,+}(\mathcal{H})$-valued Borel measure on $\mathbb{R}$,
of local bounded variation, and
 absolutely
continuous with respect to a  positive Borel measure $\mu$
 on $\mathbb{R}$,
 there exists a $\mu$-locally integrable,
$\mathcal{T}_{p,+}(\mathcal{H})$-valued function $\frac{dE}{d\mu}$
on $\mathbb{R}$, the Radon-Nikodym
derivative, such that
\begin{equation}\label{A3}
E(\Omega)=\int_{\Omega}\frac{dE}{d\mu}(\lambda)\,d\mu(\lambda)\ \
\mbox{for any bounded Borel set}\ \Omega\ .
\end{equation} 

If $p=1$, $E(\Omega)$ is always of local bounded varation and, if 
$\mu(\Omega)~=~{\rm tr}\,E(\Omega)$, the Radon-Nikodym derivative satisfies
\begin{equation}\label{A4} {\rm tr}\,\frac{dE}{d\mu}(\lambda)=1\ \
\mbox{for} \ \mu-{\rm a.e.}\ \lambda\ .
\end{equation}
\end{theorem} 

Using  Theorems \ref{thmA1} and \ref{thmA2}, we obtain

\begin{corollary}\label{corA1} Under the hypotheses of 
Theorem \ref{thmA2},
the Radon-Nikodym derivative is given by
\begin{equation}\label{A5}
\frac{dE}{d\mu}(\lambda)=\lim_{\delta\downarrow 0}\
\frac{E(I_{\lambda,\delta})}{\mu(I_{\lambda,\delta})}\ \ \mbox{for} \ 
 \mu-{\rm a.e.}\ \lambda\ ,
\end{equation} 
where the limit is in $\mathcal{T}_{p}(\mathcal{H})$.
\end{corollary}




\begin{thebibliography}{99}

\bibitem{B1} Ju.M. Berezanskii, On expansion according to eigenfunctions
of general self-adjoint differential operators, Dokl. Akad. Nauk SSSR 
\textbf{108}, 379-382 (1956).
\bibitem{B}  Ju.M. Berezanskii, { ``Expansions in Eigenfuncions of
Selfadjoint Operators"}, American Mathematical Society, 
Providence, RI, 1968.
\bibitem{B2} Y.M. Berezansky, Z.G. Sheftel and G.F. Us, ``Functional Analysis",
vol. II, Birkhauser, Basel, Switzerland, 1996.

\bibitem{Bro} F. Browder, Eigenfunction expansions for formally
 self-adjoint partial differential operators. I, II, Proc. Nat. Acad. Sci. U.S.A.
\textbf{42}, 769,771, 870-872 (1956).

\bibitem{DU}  J. Diestel and J.J. Uhl, jr., ``Vector Measures", 
Mathematical Surveys \textbf{15}, American Mathematical Society, 
Providence, RI, 1977.

\bibitem{Di} P.A.M. Dirac, ``The Principles of Quantum Mechanics", Oxford University
Press, London, UK, 1958.

\bibitem{DK}  H. von Dreifus and A. Klein, {A new proof of localization
in the Anderson tight binding model}, Commun. Math. Phys. \textbf{124},
285-299 (1989).



\bibitem{FK3}  A. Figotin and A. Klein, {Localization of Classical Waves I: Acoustic 
Waves}, Commun. Math. Phys. \textbf{180}, 439-482 (1996).
\bibitem{FK4}  A. Figotin and A. Klein, {Localization of Classical
Waves II: Electromagnetic Waves}, Commun. Math. Phys.  \textbf{184}, 411-441
(1997).

\bibitem{FMSS}  J. Fr\"ohlich, F.  Martinelli, E. Scoppola and 
T. Spencer,
{ Constructive proof of localization in the Anderson tight binding
model},  Commun. Math. Phys. \textbf{101}, 21-46 (1985).



\bibitem{Gel} I.M. Gel'fand and A.G. Kostyucenko, Expansion in 
eigenfunctions of differential and other operators, Dokl. Akad. Nauk SSSR
\textbf{103}, 349-352 (1955).

\bibitem{GK} F. Germinet and A. Klein, in preparation.

\bibitem{Kac} G.I. Kac,   Expansion in characteristic functions of self-adjoint
 operators, Dokl. Akad. Nauk SSSR \textbf{119}, 19-22 (1958). 



\bibitem{KK}  A. Klein and A. Koines, {A General Framework for  
Localization of Classical Waves}, in preparation.

\bibitem{MS} F.  Martinelli and E. Scoppola, Remark on the absence 
of absolutely continuous spectrum for $d$-dimensional Schr\"odinger
 operators with random potential for large disorder or low energy,
  Commun. Math. Phys. \textbf{97}, 465-471 (1985). 

\bibitem{Pa} L. Pastur, Spectral properties of disordered systems in 
one-body approximation,  Commun. Math. Phys. \textbf{75}, 179-196
 (1980). 
\bibitem{PS} T. Poerschke and G. Stolz, On eigenfunction  expansions
 and scattering theory, Math. Z. \textbf{212}, 337-357 (1993).
\bibitem{PGW} T. Poerschke, G. Stolz and J. Weidman, Expansions in 
generalized eigenfunctions of self-adjoint operators, Math. Z. 
\textbf{202}, 397-408 (1989).

\bibitem{RS1}  M. Reed and B. Simon,  ``Methods of Modern
Mathematical Physics, Vol.I, Analysis of Operators", Academic Press, San Diego,
 1972.

\bibitem{Scf} L.I. Schiff, ``Quantum Mechanics", McGraw-Hill, New York, 
1955.

\bibitem{SW} J. Schulenberger and C. Wilcox,  Coerciveness inequalities
for nonelliptic systems of partial differential equations, Arch. Rational Mech. Anal. 
\textbf{88}, 229-305 (1971).

\bibitem{Si}  B. Simon, Schr\"odinger semigroups, Bull. Am. Math. Soc. 
\textbf{7}, 447-526 (1982).

\bibitem{W} C. Wilcox, Wave operators and asymptotic solutions of wave
 propagation problems of classical physics, Arch. Rational Mech. Anal. 
\textbf{22}, 37-78 (1966).


\end{thebibliography}
\end{document}


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