information  The following paper is LaTeX C version 6.1, LaTeX2e
             (1994/12/10).   
             The paper is 58 pages long. 
BODY
\documentstyle[12pt,doublespace,fullpage]{article}
\setstretch{1.5}
\input mssymb.tex
\newtheorem{defn}{Definition}
\newtheorem{thm}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
\def\RR{{\Bbb R}}
\def\ZZ{{\Bbb Z}}    
\def\EE{{\Bbb E}}
\def\NN{{\Bbb N}}
\def\CC{{\Bbb C}}
\def\ep{\varepsilon}
\def\n{\nabla}
\def\RRn{{\Bbb R}^{N}}
\def\lsh{L^{2}(S^{1}_{\omega},H^{1}(\RRn))}
\def\l2sr{L^{2}(\RRn\times S^{1}_{\omega})}
\def\D{{\cal D}}
\def\bp{\bar{P}}
\def\psre{\psi_{R,\varepsilon}}
\def\psn{\xi_{\alpha}}
\def\psn1{\psi_{\alpha}}
\def\hep{h_{\varepsilon}}
\def\hre{h}
\def\hred{h^{'}}
\def\hredd{h^{''}} 
\def\tn{\psi_{\alpha}}
\def\sss{\scriptscriptstyle}
\begin{document}

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand{\thethm}{\thesection.\arabic{thm}}
\renewcommand{\thelemma}{\thesection.\arabic{lemma}}
\renewcommand{\thecor}{\thesection.\arabic{cor}}
\renewcommand{\theprop}{\thesection.\arabic{prop}}
\setcounter{equation}{0}
\setcounter{thm}{0} 
\setcounter{cor}{0} 

\title{Nonlinear Wave Equations:\ Constraints on Periods and
Exponential Bounds for Periodic Solutions.\thanks{Supported by
     NSERC grant NA7901.}}%\thanks{Submitted to Duke Mathematical
%Journal}}
\author{R.M. Pyke\thanks{e-mail: pyke@math.toronto.edu} 
 and I.M. Sigal 
\\  Department of Mathematics, 
University of Toronto} 
\date{December 1995}
\maketitle
\begin{abstract}
We show there is an upper bound to the allowed frequencies of 
time periodic solutions of a class of nonlinear wave equations:\ if
$\varphi$ is a $2\pi/\omega$ -periodic solution then
\newline $\omega^{2}\leq f'(0)$, where $f$ is the nonlinearity. 
We also prove that 
$$\int_{0}^{2\pi/\omega}\int_{\RRn}
e^{2\alpha|x|}|\varphi(x,t)|^{2}\,d^{{\sss N}}x\,dt\;<\;\infty$$
for all $\alpha^{2}\;<\;f'(0) - \lfloor\sqrt{\frac{f'(0)}
{\omega^{2}} }
\rfloor ^{2}\omega^{2}$\ where $\lfloor a\rfloor$ denotes the 
integer part of $a$. 
\end{abstract}

\section{Introduction}

In this article we study periodic solutions of the nonlinear wave
equation (NLW)
\begin{equation}
\partial_{t}^{2}\varphi - \Delta\varphi + f(\varphi) = 0
\end{equation} 
where $\varphi:\;\RR_{x}^{N}\times\RR_{t}\rightarrow\RR,\;\;f
:\;\RR\rightarrow\RR$
with $f(0) = 0$, and $\partial_{t}^{2} = \partial^{2}/
{\partial t}^{2}$, 
\mbox{$\Delta = \sum_{i=1}^{N}\partial^{2}/{\partial x_{i}}^{2}$.} 
By a periodic solution we understand solutions that are periodic in 
time $t$, and $L^{2}$ in $x$.  
This notion extends, on the one hand, the concept of bound states
of the Schr\"{o}dinger equation and of standing waves for linear 
wave equations and, on the other hand, the concept 
of periodic solutions of 
dynamical systems (equation (1.1) can be viewed as an 
infinite dimensional Hamiltonian system). 
Both concepts are 
among the simplest and most basic in the fields mentioned.

\vspace{.2in} 

To state our results we introduce some notation.
Let $S^{1}_{\omega}$ denote the circle of radius $\omega^{-1}$.  
The class of solutions we consider is the following set:
\begin{eqnarray}
{\cal D}_{\omega}  \equiv \Big\{\varphi\in H^{1}(\RRn\times 
S^{1}_{\omega})&;&  
\mbox{ if }\psi\mbox{ is any of } \varphi,
\partial_{t}\varphi\mbox{ or }
x\cdot\n\varphi,
\mbox{ then }  
\|\psi\|_{L^{\infty}(\RRn\times S^{1}_{\omega} )}<\infty \nonumber \\ 
& & \mbox{ and }\lim_{\mid x\mid\rightarrow 
\infty}\mid\psi(x,t)\mid = 0
\;\mbox{ uniformly in t }\Big\}.     
\end{eqnarray}
(This class of solutions can, probably, be enlarged.) 
Here $H^{1}(\Omega)$ stands for the Sobolev space of order $1$ for 
functions on $\Omega$ and 
$\lsh$ denotes functions $\psi:\;S^{1}_{\omega}
\rightarrow H^{1}(\RRn)$, such 
that $\|\psi\|^{2}_{\l2sr} + \|\n\psi\|^{2}_{\l2sr}\;<\;\infty$.  
\vspace{.2in}

Our main result is a characterization of two fundamental properties
of periodic solutions:\ their frequencies and their spatial
localization. 
More precisely, we prove the following theorems. 

\begin{thm}
 Suppose $f\in C^{3}(\RR,\RR)$. 
 Let $\varphi$ be a nontrivial $2\pi/\omega$-periodic 
solution of NLW on ${\cal D}_{\omega}$. 
Then  $\omega^{2}\leq f'(0)$. 
\end{thm}
\begin{thm}
 Suppose $f\in C^{3}(\RR,\RR)$ and that $f'(0)\neq m^{2}\omega^{2},$
\ $m\in\ZZ$. 
 Let 
$\varphi$  be a $2\pi/\omega$-periodic solution 
of NLW on ${\cal D}_{\omega}$.   
 Then $e^{\alpha |x|}\varphi\in\lsh$ 
 for all 
$\alpha$ satisfying 
$$\alpha^{2} 
< f'(0) - \lfloor\sqrt{\frac{f'(0)}{\omega^{2}}}
\rfloor^{2}\omega^{2}
$$
 where  
$\lfloor a\rfloor$ denotes the integer part of $a$.
\end{thm}

\noindent In Theorem 1.1, by nontrivial periodic solution we 
exclude  
time-independent solutions. 
Theorem 1.2 applies equally well to time-independent solutions.
In that case $\varphi$ solves the nonlinear elliptic equation
\begin{displaymath}
-\Delta\varphi + f(\varphi) = 0. 
\end{displaymath}
Conversely, any solution of this equation is a 
time independent solution of  NLW. 
For time independent 
solutions we can take for the frequency $\omega$ any 
positive number. Taking $\omega$ sufficiently large 
results in $\lfloor\sqrt{\frac{f'(0)}{\omega^{2}}}\rfloor = 0$. 
On the other hand we have that 
if $f'(0)<0$ then $\varphi = 0$. 
This result follows from Theorems 2.2 and 5.1 (below). 
The former theorem states, in the context of Corollary 1.3, that if 
$f'(0)<0$ then $\varphi$ is exponentially bounded with 
arbitrarily large exponent. By the latter theorem $\varphi$  
is then in fact zero.
Thus, we have the following corollary.
\setcounter{cor}{2}
\begin{cor}
Let $f\in C^{3}(\RR,\RR)$ with $f(0)=0$.
Suppose $\varphi\in
 H^{1}(\RRn)$ is a solution of the equation
$$
  \Delta\varphi\;=\;f(\varphi)
$$
such that $\varphi$ and $x\cdot\n\varphi\;\;\in L^{\infty}(\RRn)$ 
and vanish as $|x|\rightarrow\infty$. 
If $f'(0)< 0$, then $\varphi = 0$. 
If $f'(0)>0$, then
$e^{\alpha |x|}\varphi\in H^{1}(\RRn)$ 
for all $\alpha^{2} < f'(0)$.
\end{cor}

We comment on the condition $f(0) = 0$ in the above theorems. 
>From a physical point of view one often looks for 
solutions 
with finite energy $E(\varphi)$, where 
$$
E(\varphi)\;=\;\int_{\RRn}
\frac{1}{2}(\partial_{t}\varphi)^{2} + \frac{1}{2}|\n\varphi|^{2} 
+ F(\varphi),\;\;\;F' = f. 
$$
Note that the potential $F$ is defined only up to a constant.
We see  that for solutions of finite energy,
$$
\partial_{t}\varphi\;\mbox{ and }\;\n\varphi\rightarrow 0
\;\mbox{ as }\;|x|\rightarrow\infty 
$$
(at least along a subsequence).
Therefore, $\varphi\rightarrow\;const.$ as $|x|\rightarrow\infty$.
Since $\varphi$ solves NLW, this implies that
$(\partial_{t}^{2}-\Delta)\varphi\rightarrow 0$ as 
$|x|\rightarrow\infty$ and hence that 
$\varphi\rightarrow c$ as $|x|\rightarrow\infty$ where 
$f(c) = 0$. 
If 
$c\neq 0$, then by defining $\varphi_{c}\equiv\varphi - c$ and 
$f_{c}(z)\equiv f(z+c)$ we have that  
$\partial_{t}^{2}\varphi_{c} - \Delta\varphi_{c} 
+ f_{c}(\varphi_{c}) = 0$ with 
$f_{c}(0) = 0$.  
Thus, in the situation considered the assumption 
$f(0)=0$ is not a restriction. 
\vspace{.2in}

We also make several comments on extensions of these results to 
other equations. Complex valued solutions of NLW for 
nonlinearities of the form $f(\varphi) = g(|\varphi|)\varphi$
can be analysed using our method, the results being the same as above.
In particular, standing wave solutions, that is, solutions of 
the form 
$\varphi(x,t) = e^{i\omega t}\phi(x)$, must satisfy 
$\omega^{2} < g(0)$ while $\phi(x)$ is exponentially bounded 
with any exponent $\alpha$ such that $\alpha^{2} 
< g(0) - \omega^{2}$. These results are consistent with 
previous studies of standing wave solutions ([B], [Str]).

The results of this paper can be extended to a rather wide class of 
hyperbolic equations and some of the results to the nonlinear 
Schr\"{o}dinger equation. This will be done in a separate 
publication. Note however, that frequency bounds of the type 
of Theorem 1.1 are not valid for the nonlinear Schr\"{o}dinger 
equation which conforms with the main premises of our approach 
(see remarks below following the statement of Theorem 2.2).
\vspace{.2in}


To prove Theorems 1.1 and 1.2 we introduce a method into the area of 
nonlinear equations which is related to one that has been developed,
to great success, in  
the scattering theory
of Schr\"{o}dinger operators: 
\ that of positive commutators, and by  
this we mean the following. 
Let $K$ be a self-adjoint operator
on a Hilbert space of functions.  
We say that $K$ satisfies a positive commutator estimate 
on a set $\Omega$ if there is a self-adjoint operator 
$A$ such that 
\ $\langle i[K,\,A]\rangle_{\psi}\geq
\theta\|\psi\|^{2}$ 
for some $\theta>0$ and for all $\psi\in\Omega$. 
This is related to the Mourre estimate [M] of quantum mechanics
(for a discussion of the Mourre estimate and 
its uses see [CFKS] or [HS]). �

Our approach  
is motivated by pioneering
studies of $n$-body Schr\"{o}dinger operators of the form
$H = -\Delta + V$ on $L^{2}(\RRn)$, where  
$V$ is multiplication by a real-valued function, 
([FH], [FHH-OH-O]). In these references positive commutator estimates  
based on the Mourre estimate 
are used to establish 
(among other results) the exponential localization
of eigenfunctions of $H$. 
The situation we are considering here 
has some fundamental differences, though. 
One of them is related to the fact 
that we are dealing with  hyperbolic, not elliptic, operators, 
another, to the fact that the equations in question are nonlinear.  
\vspace{.2in}

We review briefly previous results on the subject.
Exponentially localized periodic solutions
on the half-line were constructed in ([V],[W]) using  
techniques from invariant manifold theory, 
while radially symmetric exponentially decaying periodic solutions
outside of a ball in $\RRn$ (the spatial domain) were found 
in ([S],[Sc]) using 
techniques similar to ([V],[W]).   
Unfortunately, even in theses cases no exponential bounds for a 
reasonable class of periodic solutions was obtained. 
The constraint $\omega^{2}\leq f'(0)$ was obtained for classical 
periodic solutions on $\RR^{1+1}$ in [C] and extended to radially 
symmetric solutions in arbitrary spatial dimensions in [L]. 
In the works above it was 
essential that the spatial variable be effectively one dimensional:
\ the NLW was formulated as a dynamical system in a phase space
of periodic functions with $x$ playing the role of the dynamical 
variable.  
Our results on the frequency restrictions 
extend those of [C] and [L] in that we study NLW
in arbitrary spatial dimensions and we 
assume considerably
less regularity from the solutions ($H^{1}$ in space and time). 
Our results on exponential bounds seem to be new even in the 
one dimensional case. 
\vspace{.2in}

The paper is organized as follows.  
In Section 2 we formulate NLW as a nonlinear eigenvalue 
problem in such a way that if $\varphi$ is a 
$2\pi/\omega$ -periodic solution of NLW, 
then $K_{\varphi}\varphi = 
\lambda\varphi$ where, for a given $2\pi/\omega$ -periodic 
function $\varphi\in{\cal D}_{\omega}$, \ $K_{\varphi}$ 
is a self-adjoint 
operator on a Hilbert space of $2\pi/\omega$ -periodic functions. 
Section 3 collects preliminary results that we use to prove 
exponential bounds for
a class of hyperbolic operators, which is presented in Section 4.
>From this we obtain exponential bounds for 
periodic solutions (Theorem 2) 
by exploiting the relation mentioned above. 
Theorem 1.1 is proven in Section 5.
An appendix describes an operator calculus that we employ.
\vspace{.2in}

\noindent \underline{Notation:}
\ \ For $x\in\RRn$, let $r = 
\mid\!
x\!\mid$ and $\hat{x} = xr^{-1}$. 
$\n$ denotes the gradient operator on $\RRn$:\ $\n\;= 
\;(\partial/\partial x_{1},\ldots,\partial/\partial x_{N})$. 
For an operator $B$ on $\l2sr$,\ $
\;\;\langle B\rangle_{\psi}$ will stand for the expectation value 
\ $\langle B\psi,\;\psi\rangle$.  
\ $\|\psi\|_{\lsh}$ denotes the space-time norm of functions 
$\psi:\;\;S^{1}_{\omega}\rightarrow H^{1}(\RRn)$;
\begin{displaymath}
\|\psi\|_{\lsh}^{2} = \|\psi\|_{\l2sr}^{2} + \|\n\psi\|^{2}_{\l2sr}  
\end{displaymath} 
where $H^{r}(\RRn)$ is the $L^{2}(\RRn)$ Sobolev space of order $r$.
The norm on $\l2sr$ will be denoted by $\|\psi\|$.  
We can decompose $\l2sr$ into a direct sum 
using the eigenspaces of
$i\partial_{t}$;
\begin{displaymath} 
\l2sr\simeq \oplus_{k\in\ZZ} \EE_{k}
\end{displaymath}
where $\EE_{k} = e^{ik\omega t}\RR \otimes L^{2}(\RRn)$. 
For $\psi\in \l2sr$
we can write $\psi = \sum_{k\in\ZZ}e^{ik\omega t}\psi_{k}$ where 
$\psi_{k}(x)= (2\pi/\omega)^{-1}
\int_{S^{1}_{\omega}}\psi(x,t)e^{-ik\omega t}
\,dt\;\in L^{2}(\RRn) $ are the Fourier coefficients 
("modes") of $\psi$. 
$P_{k}$ will denote the projection  
onto $\EE_{k}:\;P_{k}\psi = e^{ik\omega t}\psi_{k}$.  
$\bar{P}_{k} = {\bf 1} - P_{k}$, and
$\Pi_{m} = \sum_{\mid k\mid\leq m}P_{k}$ with  $\bar{\Pi}_{m} = 
{\bf 1} - \Pi_{m}$. 

For an interval $I\subset\RR$  and self-adjoint operator $H$ 
we define a smoothed-out spectral projection $E_{I}(H)$ as 
follows. 
Let $\sigma > 0$ be sufficiently small 
so that $2\sigma < |I|$ and let 
$g\in C^{\infty}_{o}(\RR,\RR)$ be such that $supp\,(g)\subset I$ 
and $g\equiv 1$ on $I_{\sigma}$ where 
$I_{\sigma} = \{\lambda\in I\;;\;dist\,(\lambda,\partial I) 
>\sigma \}$.
We then set $E_{I}(H)\equiv g(H)$. 
The parameter $\sigma$ is not made explicit since it is understood
that it can be taken as small as needed.

 
 
\section{NLW as an eigenvalue problem}
%
%
\renewcommand{\thethm}{\thesection.\arabic{thm}}
\renewcommand{\thelemma}{\thesection.\arabic{lemma}}
\renewcommand{\thecor}{\thesection.\arabic{cor}}
\renewcommand{\theprop}{\thesection.\arabic{prop}}
\setcounter{equation}{0}
\setcounter{thm}{0} 
\setcounter{cor}{0} 
%
%
To apply the technique of positive commutator estimates 
we first formulate NLW  
as a (nonlinear) eigenvalue problem for a self-adjoint operator. 
Let 
\begin{equation}
W(u) = \frac{f(u)}{u} - \kappa,\;\;\;\;\kappa = f'(0).  
\end{equation}
We assume that $f\in C^{3}(\RR,\RR)$. Then,$\;W\in C^{1}(\RR,\RR)$. 
\ For a given function $\varphi\in\l2sr$  define the "potential"
$W_{\varphi}(x,t)\equiv W(\varphi(x,t))$  
acting as an operator of multiplication,  
and the linear operators
\begin{equation}
K_{\varphi} = K_{o} + W_{\varphi},\;\;\;\;K_{o}\equiv 
\partial_{t}^{2}
-\Delta 
\end{equation}
on $\l2sr$. 
Thus, 
\begin{displaymath}
\varphi \mbox { is a } 2\pi/\omega \mbox{-periodic solution to NLW }
\;\;\Longleftrightarrow\;\;   K_{\varphi}\varphi = 
-\kappa\varphi.
\end{displaymath}          �  

\begin{lemma} 
If $\varphi\in{\cal D}_{\omega}$, 
then $W_{\varphi},\;\partial_{t}W_{\varphi}$ and 
$x\cdot\n W_{\varphi}$ are of class 
$L^{q}(\RRn\times S^{1}_{\omega})$ for all $q\in [2,\,\infty]$ 
and vanish as $\mid\! x\!\mid\rightarrow\infty$, uniformly in $t$.
\end{lemma}

\noindent \underline{Proof}:

We denote $W_{\varphi}$ simply by $W$.
By the chain rule, $\partial_{t}W = W'\partial_{t}\varphi$ 
and $x\cdot\n W = W'x\cdot\n\varphi$. Since
$W(u)\in C^{1}(\RR,\RR)$ and $\varphi$ is bounded, 
$W$ and $W'$ are members of $L^{\infty}(\RRn\times S^{1}_{\omega})$.  
Therefore $\partial_{t}W$ and $x\cdot\n W$ are bounded.

There exist constants $c,c^{'}$ and $d$, all strictly positive, 
such that 
$\mid\! W(u)\!\mid < 
c\mid\! u\!\mid$
and $\mid\! W^{'}(u)\! \mid < c^{'}\mid\! u\!\mid$  for
\ $\mid\! u\!\mid < d.$ 
For $q\in[2,\infty)$ we compute; 
\begin{eqnarray}
\|W\|_{L^{q}(\RRn,\times S^{1}_{\omega})}^{q} & = & 
\int_{\mid\varphi\mid\leq d}
\mid W\mid^{q}\;\;+\;\;\int_{\mid\varphi\mid > d}
\mid W\mid^{q} \nonumber \\
&\leq&
c^{q}\|\varphi\|^{q}_{L^{q}(\RRn\times S^{1}_{\omega})}\;\;+\;\;b^{q}
\;meas\,\{\mid\varphi\mid > d\}\;\;=\;\;M,
\end{eqnarray}
where $b = sup\,\{\mid\! W(u)\!\mid\;;\;\mid\! u\!\mid\leq
\|\varphi\|_{L^{\infty}(\RRn\times S^{1}_{\omega})}\}$. Since
$\|\varphi\|_{L^{\infty}(\RRn\times S^{1}_{\omega})}$
is finite, the continuity of $W(u)$ implies that
 $b$ is finite. Hence $M$ is finite.  If $\varphi\in L^{\infty}
(\RRn\times S^{1}_{\omega})$, then so is $W$. Therefore,  
$W\in L^{q}(\RRn\times S^{1}_{\omega})$ 
for all $q\in [2,\,\infty]$. 
A similar argument applies to $W^{'}$, 
and therefore $\partial_{t}W$ and
$x\cdot\n W\in L^{q}(\RRn\times S^{1}_{\omega})$ 
for all $q\in [2,\,\infty]$. 

Since $\varphi$, $\partial_{t}\varphi$ 
and $x\cdot\n \varphi$ vanish as 
$\mid\! x\!\mid\rightarrow\infty$, uniformly in $t$, 
\ $W$, $\partial_{t}W$ and $x\cdot\n W$
vanish in the same manner\ \ \ \ $\Box $
\vspace{.2in}

By  the results of Lemma 2.1, 
for $\varphi\in{\cal D}$, $K_{\varphi}$ as defined by (2.2)
is self-adjoint on $\l2sr$ with domain $D(K_{\varphi}) = 
D(K_{o}) = $ closure  
of $H^{2}(\RRn\times S^{1}_{\omega})$ in the graph norm
$\|\psi\|^{2}_{G} = \|\psi\|^{2}_{\l2sr} + \|K_{o}\psi\|^{2}_{\l2sr}$. 
We saw that if $\varphi\in\l2sr$ is a 
$2\pi/\omega$-periodic solution to NLW, then
$K_{\varphi}\varphi = -\kappa\varphi$. That is, $\varphi$ 
is an eigenfunction
of $K_{\varphi}$ corresponding to the eigenvalue $-\kappa = -f'(0)$. 
For the 
remainder of the paper we will study properties 
of the eigenfunctions of $K_{\varphi}$ 
which we will translate into properties
of periodic solutions of NLW by considering, in particular, the 
eigenfunctions of the operator $K_{\varphi}$  
corresponding to the eigenvalue $-\kappa$. 
\vspace{.3in}


We formulate the theorem concerning 
exponential bounds for eigenfunctions of the operator $K_{\varphi}$,
from which we will derive Theorem~1.2. 


\setcounter{thm}{1}
\begin{thm}
Let $K_{\varphi} = K_{o} + W_{\varphi}$, \ $\varphi
\in{\cal D}_{\omega}
$. 
Suppose $K_{\varphi}\psi = \lambda\psi$ for some 
$\psi\in\lsh$ and for some $\lambda\neq - m^{2}\omega^{2}$ 
$,\;m\in\ZZ$. 
If $\lambda< 0$, then 
\ \ $e^{\alpha r}\psi\in\lsh$ 
for all $\alpha$ satisfying 
$$ \alpha^{2} <  -\lambda - \lfloor\sqrt{\frac{-\lambda}{\omega^{2}}}
\rfloor^{2}\omega^{2} $$ 
where $\lfloor a\rfloor$ denotes the integer part of $a$.
If $\lambda > 0$ then $e^{\alpha r}\psi\in\lsh$ for all $\alpha$.
\end{thm}
\vspace{.2in}
 
\noindent\underline{Remarks}:

If $\varphi$ is a $2\pi/\omega$ -periodic solution of NLW such that
$\varphi\in{\cal D}_{\omega}$, then $\varphi$  
is an eigenfunction of $K_{\varphi} = K_{o} 
+ W_{\varphi}$, corresponding 
to the eigenvalue $-\kappa = -f'(0)$. 
Theorem 1.2  thus follows from Theorem 2.2.
We also have a kind of unique continuation at infinity 
theorem (Theorem 5.1 below) which states that if an eigenfunction 
of $K_{\varphi}$ is in the space $e^{-\alpha r}\lsh$ for all  
$\alpha >0$ then this function is in fact zero. This, coupled with 
the last statement of Theorem 2.2, implies that $K_{\varphi}$ has no 
positive eigenvalues.  
\vspace{.2in}

By separation of variables we see that the spectrum of the 
operator $K_{o}$ is composed of semi-infinite branches of 
essential (continuous) spectrum originating at the points
\mbox{$\{-m^{2}\omega^{2}\;;\;m\in\ZZ\}$.}  
We expect that the essential spectrum of $K_{o}$ will be stable 
under the perturbation $W_{\varphi}$ (this would certainly be true if 
$W_{\varphi}$ was compact relative to $K_{o}$, by Weyl's theorem). 
Thus, due to the nonresonance condition 
$\lambda\neq -m^{2}\omega^{2}$ and the assumption $\lambda <0$,
\ there is an integer $m_{o}\geq 1$ such that
$$ -m_{o}^{2}\omega^{2}\;<\;\lambda\;<\;-(m_{o}-1)^{2}\omega^{2}. 
$$
Then, $\lfloor\sqrt{\frac{-\lambda}{\omega^{2}}}\rfloor^{2} = 
(m_{o}-1)^{2}$.
In keeping with terminology used for Schr\"{o}dinger operators, we
call the set ${\cal E}(K_{\varphi}) = 
\{-m^{2}\omega^{2}\;;\;m\in\ZZ\}$ the 
{\em thresholds} of $K_{\varphi}$.
Therefore, Theorem 2.2 states that eigenfunctions of $K_{\varphi}$ 
are exponentially bounded with any exponent $\alpha$ such that 
$\alpha^{2}$ is less than the distance from $\lambda$ to the 
nearest threshold above (i.e., greater than) $\lambda$ (see 
Figure 1).
\begin{figure}
  \vspace{2in}
  \protect{\caption{(Rotated) Spectrum of $K_{\varphi}$ } }
\end{figure}

The thresholds of $K_{\varphi}$ 
are precisely the spectrum of the operator 
$-\partial_{t}^{2}$ on $L^{2}(S_{\omega}^{1})$,  the operator 
associated to the 
time 
variable, while the branches are due to the continuous 
spectrum of $-\Delta$ on $L^{2}(\RRn)$, the operator associated to  
the spatial variables.
In general, 
the nature of the spectrum of the associated operator 
$K_{\varphi}$  
will depend on the nature of the spectrum of the 
operators associated to the spatial and time variables.
Consequently, the 
statements and 
conclusions of the above theorems will change accordingly.  
For example, in the case of 
the nonlinear Schr\"{o}dinger equation (NLS); 
\ $i\partial_{t}\varphi -\Delta\varphi + g(|\varphi|)\varphi = 0$,
\ the spectrum of 
$i\partial_{t}$ on $L^{2}(S_{\omega}^{1})$ is 
$\{m\omega\;;\;m\in\ZZ\}$. 
Therefore, the spectrum of $i\partial_{t} -\Delta$ on 
$\l2sr$ has branches of continuous spectrum
originating at each integral multiple of $\omega$. 
This conforms with the fact that
periodic solutions of NLS with arbitrarily large (negative) 
frequencies
are possible.
\vspace{.3in}



\noindent\underline{\bf Outline of the proof of Theorems 1.1  
and 2.2:}
\vspace{.2in}

We will omit the subscript $\varphi$ when discussing the operators 
$K_{\varphi}$ and $W_{\varphi}$ 
so that from now on $K = K_{\varphi}$ and $W = W_{\varphi}$.
We begin by presenting an heuristic argument that delineates the 
structure of the proof of Theorem 2.2. 
Let $\psi\in\lsh$ and suppose that $K\psi = \lambda\psi$ for some 
$\lambda < 0$, \ $\lambda \neq -m^{2}\omega^{2},\;m\in\ZZ$. 
For $R>0$ and $\delta \geq 0$ set
\begin{eqnarray*}
\psi_{{\sss R}} & = & 
\chi_{{\sss R}}e^{\delta h(r)}\psi,\;\;\mbox{ and} \\
K^{h} & = & e^{\delta h(r)}Ke^{-\delta h(r)}
\;\;=\;\;K - \delta^{2}|\n h|^{2} + 
 i\delta\gamma_{h}. 
\end{eqnarray*}
Here 
 \ $h(r) = 0,\;r< 2R$,\  $h(r) = r,\;r > 3R$ and  
$\gamma_{h} = \frac{1}{i}(\n h\cdot\n + \n\cdot\n h)$. 
\ $\chi_{\sss R}$ is a smooth cut-off function:
\ $\chi_{{\sss R}}(r) = 0,\;r<R$ and 
$\chi_{{\sss R}}(r) = 1,\;r>2R$. 
The important features of the function $h$ are that $h = 0$ on 
$supp\,(\chi^{'}_{\sss R})$,\ \ $h(r) = r$ near infinity, with 
$|h^{(m)}(r)|\leq c_{m}R^{1-m},\;c_{m}$ independent of $R$.

Our goal is to show that $\psi_{{\sss R}}\in\lsh$ for some 
$\delta > 0$ even though 
for now we are purposely forgetting that 
$\psi_{{\sss R}}$ may not be in $\lsh$. This is just so we 
can elucidate  
the ideas behind the proof, the heuristics presented here 
will illustrate all the essential ideas of the full proof.
 In the rigorous analysis we will 
regularize this function by cutting-off $h$ 
near infinity so that $\psi_{{\sss R}}\in \lsh$, 
regaining $e^{\delta r}\psi$ in a limiting procedure at the end.

The significance of the operator $K^{h}$ is that 
$e^{\delta h(r)}\psi$ is an eigenfunction of $K^{h}$ 
corresponding to the eigenvalue $\lambda$. 
The factor $\chi_{{\sss R}}$ causes $\psi_{{\sss R}}$ not to be 
a bona fide eigenfunction of $K^{h}$, but it is an approximate 
eigenfunction in the sense that 
\begin{equation}      
\|(K^{h} - \lambda)
\psi_{{\sss R}}\|\;\leq\;o_{{\sss R}}(1)\|\psi\|_{\lsh}  
\end{equation} 
where $o_{{\sss R}}(1)$ denotes a quantity that vanishes as 
$R\rightarrow\infty$.
This follows from the formula
\begin{eqnarray} 
(K^{h} - \lambda)\psi_{{\sss R}} &=&  
\;\;e^{\delta h(r)}\chi_{\sss R} 
        (K - \lambda)\psi\;\;+ 
     \;\;e^{\delta h(r)}[-\Delta,\,\chi_{{\sss R}}]\psi
\nonumber \\
& = & 
  (-\Delta\chi_{{\sss R}})\psi\;\;-
\;\;2\n\chi_{{\sss R}}\cdot\n\psi,  
\end{eqnarray} 
where we have used that $h = 0$ on 
$supp\,(\chi^{'}_{\sss R})$, and then the property 
$|\chi_{{\sss R}}^{(m)}|\leq cR^{-m}$ to arrive at (2.4). 

Let $A =  \frac{1}{2i}(x\cdot\n + \n\cdot x)$. 
Since $e^{\delta h(r)}\psi$ is 
an eigenfunction of $K^{h}$, we have that
$$ 0\;\;=\;\;Im\,\langle(K^{h}-\lambda)e^{\delta h(r)}\psi,
\;Ae^{\delta h(r)}\psi\rangle. 
$$
This equation is related to the virial theorem of quantum 
mechanics [CFKS].
Expanding the inner product, we find
\begin{eqnarray*} 
0 &=& Im\,\langle(K^{h}-\lambda)e^{\delta h(r)}\psi,
\;Ae^{\delta h(r)}\psi\rangle  \\  
&=& \frac{1}{2}\langle i[K,\,A]\rangle_{e^{\delta h(r)}\psi}
\;\;+\;\;\delta
Re\,\langle \gamma_{h} A\rangle_{e^{\delta h(r)}\psi}
- \frac{\delta^{2}}{2}\langle i[|\n h|^{2},\,A]\rangle_{e^{\delta
 h(r)}\psi}.
\end{eqnarray*} 
If we substitute $\psi_{{\sss R}}$ for $e^{\delta h(r)}\psi$ 
in this equation the left hand side is no longer 
zero, but since 
$(K^{h} - \lambda)\psi_{{\sss R}}$ is localized to the 
support of $\chi^{'}_{{\sss R}}$ (cf. (2.5))
where $h = 0$, we find that
$$ 
   |Im\,\langle(K^{h} - \lambda)
   \psi_{{\sss R}},\,A\psi_{{\sss R}}
   \rangle|\;\leq\;c\|\psi\|^{2}_{\lsh},   
$$ 
where $c$ is independent of $R$. 
Furthermore, since $\gamma_{h} = \sqrt{h'r^{-1}}A \sqrt{h'r^{-1}}$, 
a simple 
calculation gives
\begin{eqnarray*}
Re\,\langle \gamma_{h} A\rangle_{\psi_{{\sss R}}}
& = & 
Re\,\langle\sqrt{h'r^{-1}}
    A\sqrt{h'r^{-1}}A\rangle_{\psi_{{\sss R}} } \\
& = & 
\langle\sqrt{h'r^{-1}}
   A^{2}\sqrt{h'r^{-1}}\rangle_{\psi_{{\sss R}} } 
+ Re\,\langle\sqrt{h'r^{-1}}A\,[\sqrt{h'r^{-1}},\,A]
   \rangle_{\psi_{{\sss R}} }
\\
& = & 
\langle \sqrt{h'r^{-1}}A^{2}\sqrt{h'r^{-1}}
    \rangle_{\psi_{{\sss R}} } 
+ \frac{1}{2} \langle\sqrt{h'r^{-1}}[A,\,[\sqrt{h'r^{-1}},\,A]\;]
   \rangle
_{\psi_{{\sss R}} } \\
& = &  
\mbox{ positive
term } + o_{{\sss R}}(1)\|\psi_{{\sss R}}\|^{2}, 
\end{eqnarray*}
where we have used that
$\sqrt{h'r^{-1}}[A,\,[\sqrt{h'r^{-1}},\,A]\;]
   \leq cr^{-1}$. We calculate
\begin{eqnarray*}
\Big|[|\n h|^{2},\,A]\Big| &=& \Big|2h'h''r\Big| \\ 
&\leq& c,
\end{eqnarray*}  
independently of $R$ (here we are using the property 
$h''(r) = 0$ for $r>c'R$ for some fixed $c'$, which 
will hold in the rigorous 
analysis), so that
$$
\Big|\frac{\delta^{2}}{2}\langle i[|\n h|^{2},\,A]\rangle
_{\psi_{\sss R} }\Big|\;\leq\;c\delta^{2}\|\psi_{\sss R}\|^{2}.
$$ 
These relations yield
\begin{equation}
 \langle i[K,A]\rangle_{\psi_{{\sss R}}} - o_{{\sss R}}(1)
\|\psi_{{\sss R}}
\|^{2} - c\delta^{2}\|\psi_{\sss R}\|^{2}\;\leq\;c\|\psi\|^{2}_{\lsh}.
\end{equation} 
If we can show that
\begin{equation} 
 \langle i[K,\,A]\rangle_{\psi_{{\sss R}}}\;\geq\;\theta
\|\psi_{{\sss R}}\|^{2}_{\lsh},\;\;\;\mbox{ for some }\theta> 0,
\end{equation}
then from (2.6) and (2.7) it follows that for $R$ sufficiently large
and $\delta$ sufficiently small,
\begin{displaymath}
\|\psi_{{\sss R}}\|_{\lsh}^{2}\;\leq\;c\|\psi\|^{2}_{\lsh} 
\;<\;\infty.
\end{displaymath}
Hence, $e^{\delta r}\psi\in\lsh$. Although  
$\delta$ must be taken sufficiently small,
we will be sure to show that it is still strictly 
positive.  
\vspace{.3in}

Some effort is required to prove the positivity estimate (2.7). 
Evaluating the commutator
\begin{displaymath}
i[K,\,A]\;=\;-2\Delta 
-x\cdot\n W 
\end{displaymath}
and writing $-\Delta = K -\partial_{t}^{2} - W$,
we have 
\begin{displaymath}
\langle i[K,\,A]\rangle_{\psi_{{\sss R}} }\;=\;\langle-\Delta
\rangle_{\psi_{\sss R} } + \langle
 K - \lambda \rangle_{\psi_{{\sss R}} }\;+\;\langle 
\lambda \rangle_{\psi_{{\sss R}} }\;+\;\langle
-\partial_{t}^{2}\rangle_{\psi_{{\sss R}} }\;-\;\langle
W + x\cdot\n W\rangle_{\psi_{{\sss R}} }.
\end{displaymath} 
Now,
\begin{equation} 
K - \lambda \;\;=\;\;K^{h} - \lambda  + 
\delta^{2}|\n h|^{2} - i\delta \gamma_{h},
\end{equation}
and since $K - \lambda $ is self-adjoint and 
$i\gamma_{h}$ is skew-adjoint, 
$$
\langle K - \lambda \rangle_{\psi_{{\sss R}} }\;\;=
\;\;Re\,\langle K^{h} - \lambda\rangle_{\psi_{{\sss R}} }
+ \delta^{2}\langle |\n h|^{2}\rangle_{\psi_{\sss R} }
$$
from which we derive (recalling (2.4)) 
\begin{eqnarray*}
|\langle K - \lambda \rangle_{\psi_{{\sss R}}}|\;
& \leq & 
|\langle K^{h} - \lambda\rangle_{\psi_{{\sss R}} }|  + 
c\delta^{2}\|\psi_{\sss R}\|^{2} \\
& \leq & 
\|(K^{h} - \lambda)\psi_{{\sss R}}\|\,\|\psi_{{\sss R}}\|  + 
c\delta^{2}\|\psi_{\sss R}\|^{2} \\
& \leq &
o_{{\sss R}}(1)\|\psi\|_{\lsh}\,\|\psi_{{\sss R}}\|  + 
c\delta^{2}\|\psi_{\sss R}\|^{2} \\
& \leq &
o_{{\sss R}}(1)\Big[\|\psi\|^{2}_{\lsh}
 + \|\psi_{{\sss R}}\|^{2}_{\lsh}\Big] + c\delta^{2}\|\psi_{\sss R}
\|^{2}.
\end{eqnarray*}
By Lemma 2.1,\ $W$ and $x\cdot\n W$ both vanish as 
$|x|\rightarrow\infty$ uniformly in $t$. We use this property 
and the fact that $\psi_{{\sss R}}$ is supported outside of 
a ball of radius $2R$ to obtain the estimate
$$
 |\langle 2W + 
    x\cdot\n W\rangle_{\psi_{{\sss R}}}|\;\;\leq  
\;\;o_{{\sss R}}(1)
\|\psi_{{\sss R}}\|^{2}. 
$$
Therefore, we have the inequality 
\begin{eqnarray}
\langle i[K,\,A]\rangle_{\psi_{{\sss R}}} &\geq&  
\; \|\n\psi_{\sss R}\|^{2} + \lambda  
\|\psi_{{\sss R}}\|^{2}\; + 
\;\langle -\partial_{t}^{2}\rangle_{\psi_{{\sss R}}} \nonumber \\ 
& &\;\;\;\; - o_{{\sss R}}(1)\Big[\|\psi\|^{2}_{\lsh} 
+ \|\psi_{{\sss R}}\|^{2}_{\lsh}\Big]
- c\delta^{2}\|\psi_{\sss R}\|^{2}. 
\end{eqnarray}
To achieve (2.7) we require that 
\begin{displaymath}
\langle -\partial_{t}^{2}\rangle_{\psi_{{\sss R}}}\;
\;\geq\;(-\lambda + \nu)\|\psi_{{\sss R}}\|^{2}, 
\;\;\;\nu >0
\end{displaymath}
up to a remainder term that we can control.
In fact, due to the discrete nature of the spectrum of 
$-\partial_{t}^{2}$, we  can show that 
\begin{equation}
\langle -\partial_{t}^{2}\rangle_{\psi_{\sss R} }\;\geq
\;m_{o}^{2}\omega^{2}\|\psi_{\sss R}\|^{2},  
\end{equation} 
up to a remainder term,
where $m_{o}\geq 1$ is the integer characterized by the 
relation\newline 
$-m_{o}\omega^{2} < \lambda < -(m_{o}-1)^{2}\omega^{2}$. 
To prove (2.10) we write  
\begin{displaymath}
\psi_{{\sss R}}\;=\;\bar{\Pi}_{m_{o} - 1}\psi_{{\sss R}}\;+
\;\Pi_{m_{o} - 1}\psi_{{\sss R}} 
\end{displaymath}
where $\Pi_{m}$ denotes projection onto the first $m$ modes 
and $\bar{\Pi}_{m} = {\bf 1} - \Pi_{m}$ 
(as defined at the end of Section 1).
Note that $\bar{\Pi}_{m_{o}-1}\psi_{{\sss R}}$ satisfies the 
estimate (2.10). 
Using that $-\partial_{t}^{2}\bar{\Pi}_{m_{o}-1}\geq
m_{o}^{2}\omega^{2}\bar{\Pi}_{m_{o}-1}$
and that
$-\partial_{t}^{2}\Pi_{m_{o}-1}\geq 0$ (these inequalities 
are in the sense of quadratic forms), we obtain 
\begin{eqnarray}
\langle -\partial_{t}^{2}\rangle_{\psi_{{\sss R}}}
& = &
\langle -\partial_{t}^{2}\rangle_{\bar{\Pi}_{m_{o} - 1}
\psi_{{\sss R}} }\;\;+\;\;\langle
-\partial_{t}^{2}\rangle_{\Pi_{m_{o} - 1}\psi_{{\sss R}} }
\nonumber \\
& \geq &
m_{o}^{2}\omega^{2}\langle\bar{\Pi}_{m_{o}-1}\rangle_{
\psi_{{\sss R}} } 
\nonumber \\
& = & 
m_{o}^{2}\omega^{2} \|\psi_{{\sss R}}\|^{2}\;-\;m_{o}^{2}\omega^{2}
\langle\Pi_{m_{o}-1}\rangle_{\psi_{{\sss R}} }
\end{eqnarray}
To estimate the second term on the right hand side 
we proceed as follows. 

If $P_{k}\psi = 0$,\ $|k|\leq m_{o}-1$, then (2.10) is achieved
because $\Pi_{m_{o}-1}\psi_{\sss R} = 0$. This will be the 
situation in Theorem 1.1 (as we will see). 
Otherwise, pick an interval $I\subset\RR$ containing 
$\lambda $ and such that $\sup\,(I)< -(m_{o}-1)^2
\omega^{2}$.  
Let $E_{I}(K)$ be a smoothed-out spectral projection of  
$K$ corresponding to the interval $I$ (as described at the 
end of Section~1) and decompose $\Pi_{m_{o}-1}$ with respect to 
$E_{I}(K)$ and $\bar{E}_{I}(K)$ to obtain
\begin{eqnarray}
\Pi_{m_{o}-1} & = &  
\Big(E_{I}(K) + \bar{E}_{I}(K)\Big) 
\Pi_{m_{o}-1}\Big(E_{I}(K) + \bar{E}_{I}(K)\Big) \nonumber 
\\
& = & 
E_{I}(K)\Pi_{m_{o}-1}E_{I}(K)\;+\;\bar{E}_{I}(K)\Pi_{m_{o}-1}
E_{I}(K)  \nonumber \\
& & \;\;\;\;+\;E_{I}(K)\Pi_{m_{o}-1}\bar{E}_{I}(K)
\;+\;\bar{E}_{I}(K)\Pi_{m_{o}-1}\bar{E}_{I}(K).
\end{eqnarray}
>From this and the Schwarz inequality we then have the bound
\begin{equation}
\Big|\langle\Pi_{m_{o}-1}\rangle_{\psi_{\sss R} }\Big|
\;\leq\;3\|\bar{E}_{I}(K)\psi_{{\sss R}}\|\|\psi_{{\sss R}}\|
\;+\;\langle C\rangle_{\psi_{{\sss R}} }, 
\end{equation}
where $C = E_{I}(K)\Pi_{m_{o}-1}E_{I}(K)$ is a compact operator, 
as we will see shortly. 
Because $\psi_{{\sss R}}$ has support that goes off to 
infinity as $R\rightarrow 
\infty$, this term is of order 
$o_{{\sss R}}(1)\|\psi_{{\sss R}}\|^{2}$ (cf. Lemma 3.3 below). 

To prove that the operator $E_{I}(K)\Pi_{m_{o}-1}E_{I}(K)$ 
is compact, we need some kind of relative 
compactness of $W$. To this end we take advantage of the 
natural decomposition of $K_{o}$ along the eigenspaces 
$\EE_{k}$:
$$
K_{o}\;=\;\oplus_{k\in\ZZ} (-k^{2}\omega^{2} - \Delta),  
$$
from which it follows that 
$$
(K_{o} - z)^{-1}\;=\;\oplus_{k\in\ZZ}
(-k^{2}\omega^{2} - \Delta - z)^{-1},\;\;\;z\in\CC\setminus\RR.
$$
Our assumptions on the solution $\varphi$ guarantees that 
each mode $W_{k}(x)$ of $W$ ($= W_{\varphi}$) is compact 
relative to $-\Delta$ as an operator of multiplication on 
$L^{2}(\RRn)$. 
As a result, $W$ is compact relative to $K_{o}$ when 
restricted to finitely many of the subspaces $\EE_{k}$. 
Introducing the spectral projections $E_{I}(K_{o})$ 
associated to the operator $K_{o}$, we write 
\begin{eqnarray}
E_{I}(K)\Pi_{m_{o}-1}E_{I}(K)
&=&
E_{I}(K_{o})\Pi_{m_{o}-1}
E_{I}(K_{o})
\nonumber \\         
& & \;\;+\;\Big(E_{I}(K) - E_{I}(K_{o})\Big)\Pi_{m_{o}-1}
E_{I}(K)\nonumber \\
& & \;\;+\;E_{I}(K_{o})\Pi_{m_{o}-1}\Big(E_{I}(K) - 
E_{I}(K_{o})\Big).
\end{eqnarray}
The first term on the right hand side is zero by conservation of 
energy. That is, 
\begin{equation} 
   P_{k}E_{I}(K_{o}) = 0\;\;\;\mbox{ for }k<m_{o}. 
\end{equation} 
This relation can be seen as follows. 
On $Ran\,P_{k} = \EE_{k}$,\ \ $K_{o} = -k^{2}\omega^{2} - \Delta$ 
so that
$$
P_{k}E_{I}(K_{o})\;\;=\;\;P_{k}E_{I}(-k^{2}\omega^{2} - \Delta).
$$
Since $\sup\,(I) < -k^{2}\omega^{2}$,\ \ $spec\,(-k^{2}\omega^{2} 
-\Delta) = [-k^{2}\omega^{2},\,\infty)$
is disjoint from $I$. 
Hence,\ $E_{I}(-k^{2}\omega^{2} - \Delta) = 0$.

To treat the other two terms on the right hand side of 
(2.14) it is enough 
to consider the resolvents 
$R(z) = (K - z)^{-1}$ and $R_{o}(z) = (K_{o} - z)^{-1}$ in place  
of the projections $E_{I}(K)$ and $E_{I}(K_{o})$. 
For the second term 
on the right, say, and using the second resolvent equation, we have,
for any $m_{1}\in\NN$, 
\begin{equation}
R(z)WR_{o}(z)\Pi_{m_{o}-1}E_{I}(K)\;=
\;R(z)\Pi_{m_{1}}WR_{o}(z)\Pi_{m_{o}-1}E_{I}(K)\;
+\;R(z)\bar{\Pi}_{m_{1}}WR_{o}(z)\Pi_{m_{o}-1}E_{I}(K).
\end{equation}   
By the relative compactness of $W$,\ \ $WR_{o}(z)$ 
is a compact operator on each $\EE_{k}$, 
and so $\Pi_{m_{1}}WR_{o}(z)\Pi_{m_{o}-1}$ is a compact 
operator for each $m_{1}\in\NN$ since it acts on finitely 
many $\EE_{k}$. Thus the first term on the right hand side of 
equation (2.16) is compact. 
By taking $m_{1}$ sufficiently large we can make 
the second term arbitrarily 
small in norm. This can be seen  by  
noting  that if $W$ is time independent and if $m_{1}>m_{o}-1$
then, because $W$ will 
commute with the projections $P_{k}$,\ \ $\bar{\Pi}_{m_{1}}
WR_{o}(z)\Pi_{m_{o}-1} = 0$.  
The time dependence of $W$ couples the space and time variables 
and can bridge the gap between $\bar{\Pi}_{m_{1}}$ and 
$\Pi_{m_{o}-1}$, but we can estimate this by writing
\begin{eqnarray*}
\bar{\Pi}_{m_{1}}WR_{o}(z)\Pi_{m_{o}-1} & = & 
\partial_{t}^{-1}\partial_{t}\bar{\Pi}_{m_{1}}WR_{o}(z)\Pi_{m_{o}
-1} \\  
& = & 
\partial_{t}^{-1}\bar{\Pi}_{m_{1}}(\partial_{t}W)R_{o}(z)\Pi_{m_{o}
-1}   
\;+\;\partial_{t}^{-1}\bar{\Pi}_{m_{1}}WR_{o}(z)\partial_{t}
\Pi_{m_{o}-1}.
\end{eqnarray*}
If $\partial_{t}W$ is bounded relative to $K_{o}$, as will be 
the case if, in particular, $\partial_{t}W$ is  a bounded function, 
then $(\partial_{t}W)R_{o}(z)$ is a bounded operator. 
Combining this with the 
estimates
$$
\|\partial_{t}^{-1}\bar{\Pi}_{m_{1}}\|\;\leq\;1/m_{1},
\;\;\mbox{ and }\;\;\|\partial_{t}\Pi_{m_{o}-1}\|
\;\leq\;m_{o}-1,
$$ 
we see that $\|\bar{\Pi}_{m_{1}}WR_{o}(z)\Pi_{m_{o}-1}\|$ can be
made arbitrarily small by taking $m_{1}$ sufficiently large.
Therefore, referring to (2.16),
 $R(z)WR_{o}(z)\Pi_{m_{o}-1}E_{I}(K)$, and hence 
$\Big(E_{I}(K) - E_{I}(K_{o})\Big)\Pi_{m_{o}-1}E_{I}(K)$ is 
compact.
\vspace{.2in}

Going back to (2.13), we 
use the fact that $\psi$ is an eigenfunction of $K$ 
corresponding to the eigenvalue $\lambda$  
to show that $\psi_{{\sss R}}$ is essentially localized in $I$ 
with respect to the spectral decomposition of $K$, i.e., that 
$\|\bar{E}_{I}(K)\psi_{{\sss R}}\|$ is small.
More precisely, we will estimate
\begin{eqnarray}
\|\bar{E}_{I}(K)\psi_{{\sss R}}\|\,\|\psi_{{\sss R}}\|
&\leq& 
d^{-1}o_{\sss R}(1)
\Big[\|\psi\|^{2}_{\lsh} + \|\psi_{{\sss R}}\|^{2}_{\lsh}\Big] 
\nonumber \\
& & 
\;\;\;\;\;\;+\;d^{-1}\delta(\delta + 1)\|\psi_{\sss R}\|^{2}_{\lsh}
\end{eqnarray}
where $d = dist\,(\partial I,\,\lambda)$. This follows from 
the formula, derived using the functional calculus,
$$
\|\bar{E}_{I}(K)\psi_{{\sss R}}\|\;\;\leq
\;\;d^{-1}\|(K- \lambda)\psi_{\sss R}\|,
$$ 
and the estimate
$$
\|(K-\lambda)\psi_{\sss R}\|\;\;\leq
\;\;o_{\sss R}(1)\|\psi\|_{\lsh} 
\;+\;\delta(\delta + 1)\|\psi_{\sss R }\|_{\lsh}
$$
which follows from (2.4), (2.8) and the triangle inequality.

We want to emphasize that here we are 
localizing simultaneously in two 
non-commuting operators;   
in $K$ and in $i\partial_{t}$.
That is, we are using the spectral projections associated with 
$K$ and $i\partial_{t}$,\ $E_{I}(K)$ and $P_{k}$ respectively, 
to decompose phase space ($=\lsh$) into regions where 
$-\partial_{t}^{2}$ has certain properties.
In particular, on $Ran\,\bar{\Pi}_{m_{o}-1}$,\ $-\partial_{t}^{2}
\geq m_{o}^{2}\omega^{2}$, i.e., is strictly positive,
 while on $Ran\,\Pi_{m_{o}-1}$,
\ $-\partial_{t}^{2}\leq (m_{o}-1)^{2}\omega^{2}$, i.e., is 
bounded. We 
further decompose $Ran\,\Pi_{m_{o}-1}$ according to the subspaces 
$Ran\,E_{I}(K)$ and $Ran\,\bar{E}_{I}(K)$ where 
$\Pi_{m_{o}-1}$ acts either as a compact operator 
($E_{I}(K)\Pi_{m_{o}-1}E_{I}(K)$), or else is proportional
to $\bar{E}_{I}(K)$ ($\bar{E}_{I}(K)
\Pi_{m_{o}-1}E_{I}(K)$, etc.);\ see Figure 2.
\begin{figure}
  \vspace{3.5in}
  \caption{Phase space decomposition of $\langle -\partial_{t}^{2}
\rangle_{\psi_{\sss R} }$. }
\end{figure}
Either of these latter cases lead to expectation values, evaluated
on $\psi_{\sss R}$, that can be made arbitrarily small by varying 
the parameters $R$ and $\delta$ associated to $\psi_{\sss R}$.
\vspace{.2in}

Combining (2.11), (2.13) and (2.17), we have that 
\begin{eqnarray}  
\langle -\partial_{t}^{2}\rangle_{\psi_{\sss R} } &
\geq&  
\;\;m_{o}^{2}\omega^{2}\|\psi_{\sss }\|^{2}\;-
\;d^{-1}o_{\sss R}(1)\Big[\|\psi\|_{\lsh}^{2} + 
\|\psi_{\sss R}\|^{2}_{\lsh}\Big]
\nonumber \\
& & \;\;\;-\;o_{\sss R}(1)\|\psi_{\sss R}\|^{2}
\;-\;d^{-1}\delta(\delta + 1)\|\psi_{\sss R}\|_{\lsh}^{2}, 
\end{eqnarray}
and so
\begin{eqnarray}
\langle i[K,\,A]\rangle_{\psi_{\sss R} }
&\geq&  
b\|\psi_{\sss R}\|^{2}_{\lsh}\;-
\;(d^{-1} + 1)o_{\sss R}(1)\Big[  
\|\psi\|^{2}_{\lsh} + \|\psi_{\sss R}\|^{2}_{\lsh}\Big]  
\nonumber \\
& & \;\;\;\;\;\;-\;d^{-1}\delta(\delta + 1)\|\psi_{\sss R}\|
_{\lsh}^{2}
\end{eqnarray}
where $b = \min\,(\lambda + m_{o}^{2}\omega^{2},\,1) > 0$.
Thus we have achieved (2.7) for $R$ sufficiently large and 
$\delta$ sufficiently small. 

Since $\delta$ must be taken sufficiently small 
we cannot prove arbitrarily large 
exponential bounds at once. Therefore we will iterate this 
procedure to obtain greater bounds.

\vspace{.3in}


\noindent\underline{Regularization and iterative scheme}

\nopagebreak
\vspace{.2in}
\nopagebreak

To make these heuristics rigorous we cut-off the function 
$h(r) = h_{{\sss R}}(r)$ at infinity so that  
$e^{\delta h(r)}\psi\in\lsh$. 
The cut-off depends on a parameter $\ep$:\ $h(r) = h_{{\sss R},\ep}
(r)$, and is such that   
$\lim_{\ep\rightarrow 0}h_{{\sss R},\ep}(r) = r + const.$ \ for 
$r>R$.  
Our regularized function is then $\chi_{\sss R}e^{\delta h_{{\sss R},
\ep}(r)}\psi$ where $\chi_{\sss R}$ 
is 
a cut-off function with support in a neighborhood of 
infinity.
We then show, following the ideas outlined in the 
heuristics above, that for $R$
sufficiently large and $\delta$ sufficiently small but nonzero, 
\ $\|\chi_{{\sss R}}e^{\delta h_{{\sss R},\ep}(r)}\psi \|_{\lsh} < 
c < \infty$ uniformly in $\ep$.  
By taking the limit $\ep\rightarrow 0$ we conclude that
$e^{\delta r}\psi\in\lsh$. 
\vspace{.2in}

We discuss the regularization procedure. 
For $R>1$ \ let 
$\chi_{{\sss R}}(r)$ be a smooth function such that
$\chi_{{\sss R}} = 0$ for $r\leq R$,\ $\chi_{{\sss R}} = 1$ 
for $r\geq 2R$ 
and $\mid\!\chi_{{\sss R}}^{(m)}\!\mid\leq c_{m}R^{-m}$. 
For $\ep>0$ and the same $R$ 
let $h(r) = h_{{\sss R},\ep}(r)$ be a smooth function such that 
\vspace{.2in}

$\begin{array}{cclrcl} 
  \hre (r) & = & 0,& & r&\leq 2R \\
          & = & r - 3R + 1,& 3R<& r& < 3R + \frac{1}{\ep} \\
          & = & 2 + \frac{1}{\ep},& &r&\geq 2(3R + \frac{1}{\ep}), 
\end{array}$
\vspace{.2in}

\noindent with
$\hre$ defined on the intervals $[2R,\,3R]$ and $[3R+\frac{1}{\ep},\,
\,2(3R+\frac{1}{\ep})]$ in such a way that
\vspace{.2in}

$\begin{array}{cclrcl}
   \mid h^{(m)}(r)\mid & 
\leq & c^{'}_{m}R^{1-m},& 2R< &r& < 3R \\
                          & \leq & c^{''}_{m}(3R + \frac{1}{\ep})^
{1-m},&
                 (3R + \frac{1}{\ep})<&  r& < 2(3R + \frac{1}{\ep}) 
\end{array}$
\vspace{.2in}

\noindent (see Figure 3).
We remark that the constants $c_{m},c^{'}_{m}$ and 
$c^{''}_{m}$ are 
independent of $R$ and $\ep$.
The function $h_{\sss R}$ we define as
$$
h_{\sss R}(r)\;=\;\lim_{\ep\rightarrow 0}h_{{\sss R},\ep}(r)
$$
and is equal to $r + const.$\ in a neighborhood of infinity.
\begin{figure}
    \vspace{3in}
    \caption{The functions $\chi_{{\sss R}}(r)$ and $h_{{\sss R},
\ep}(r)$}
\end{figure}
\vspace{.2in}

The method outlined above secures {\em some} exponential bound 
$\delta$ for 
$\psi$. To achieve a better bound we iterate this method, 
incrementally approaching the optimal bound. 
We begin the iteration by assuming  that $\psi_{\alpha}\equiv  
e^{\alpha r}\psi\in\lsh$ for some 
$\alpha\geq 0$. 
To prove that there exists a $\delta>0$  
such that $e^{\delta r}\psi_{\alpha}\in\lsh$, 
we regularize the function 
$e^{\delta r}\psi_{\alpha}$ as 
$\chi_{{\sss R}}e^{\delta h(r)}\psi_{\alpha}\equiv\xi_{\alpha}$  
which was described above. 
We then show, in the same way as for $\chi_{{\sss R}}e^{\delta h(r)}
\psi$, that for $R$ sufficiently large and $\delta$ sufficiently
small but nonzero, call this $\delta(\alpha)$, 
that $\|\chi_{{\sss R}}e^{\delta(\alpha) h(r)}\psi_{\alpha}\|_{
\lsh}^{2}\;<\;  
\;\infty$  
uniformly in $\ep$ (the $\ep$ appearing in 
$h_{{\sss R},\ep}$). Therefore, after taking the limit 
$\ep\rightarrow 0$, we conclude that   
 \ $e^{\delta(\alpha)r +\alpha r}\psi\in\lsh$ and  
hence our new exponential bound for $\psi$ has exponent 
$\alpha + \delta(\alpha)$.  
Now we set \ $\psi_{\alpha_{1}} = e^{\alpha_{1}r}\psi$, where
$\alpha_{1} = \alpha + \delta(\alpha)$, regularize this function 
as $\chi_{\sss R}e^{\delta h(r)}\psi_{\alpha_{1}}$, and repeat 
the above analysis to determine $\alpha_{2} = \alpha_{1} + 
\delta(\alpha_{1})$. 
Finally, we show that as $n\rightarrow\infty$, 
\ \ $\lambda 
+ \alpha_{n}^{2}\rightarrow -(m_{o}-1)^{2}\omega^{2}$ 
if $\lambda < 0$ (recall that $m_{o}\geq 1$ 
is the largest integer $m$ 
such that $\lambda < -(m-1)^{2}\omega^{2}$),
or else becomes
arbitrarily large if $\lambda > 0$.
\vspace{.3in}

The next main result after Theorem 2.2 
is a sort of unique continuation
theorem at infinity (Theorem 5.1). It states that if eigenfunctions 
of $K$ are exponentially bounded with arbitrarily 
large exponent, then the function is zero. 
This is used in the proof of Theorem 1.1. However, with Theorem 1.1
we work 
with the time dependent part of $\varphi$: 
\ $\varphi - P_{0}\varphi\equiv\bar{\varphi}$ (recall 
that $\Big(P_{o}\varphi\Big)(x) = 
(2\pi/\omega)^{-1}\int_{S_{\omega}^{1}}
\varphi(x,t)\,dt$). 
We first show that $\bar{\varphi}$ is an eigenfunction of an 
operator $\bar{K}$ which is constructed in a similar way as was $K$ 
after first projecting NLW onto $\EE_{0}^{\perp}$: 
\ \ $\bar{K}\bar{\varphi} = -\kappa\bar{\varphi}$, where $\kappa
= f'(0)$. From this we 
prove, along analogous lines as outlined in the heuristics above, 
that 
$e^{\alpha r}\bar{\varphi}\in\lsh$ for all $\alpha$. We are able 
to show this because by using $\bar{K}$ instead of $K$ 
we have removed the only threshold above $-\kappa$,\ the 
threshold $0$ (that this is the only threshold above $-\kappa$ 
follows from the assumption that $\omega^{2}> f'(0)$\ ); see 
Figure 4. 
\begin{figure}
  \vspace{2in}
  \caption{Spectrum of $K_{\varphi}$ when $\omega^{2}>f'(0)$.}
\end{figure}
The unique continuation theorem holds also for 
$\bar{K}$ from which it follows that $\bar{\varphi} = 0$. 
Therefore, $\varphi = P_{0}\varphi$, that is, $\varphi$ is 
independent of time.  
\vspace{.2in}

The proof of Theorem 1.1 is less involved than the 
proof of Theorem 2.2 because we do not 
require a microlocalization of $\langle -\partial_{t}^{2}
\rangle_{\psi}$. 
In Theorem 1.1, we will be dealing with the 
function $\bar{\varphi} = \varphi - P_{0}\varphi$ so that 
$\langle -\partial_{t}^{2}\rangle_{\bar{\varphi} }\geq \omega^{2}
\|\bar{\varphi}\|^{2}$. Therefore,  equation (2.10) is satisfied 
(here $m_{o} = 1$). 
For the proof of Theorem 1.1, then, the reader  
may proceed directly to Section 5.2 for a (mostly) self-contained 
proof.

\section{Preliminary Results}
%
\setcounter{equation}{0}
\setcounter{lemma}{0}
%

In this section we present 
a series of preliminary results that will be used in the proof of 
Theorem 2.2.
We remind the reader
that we will be denoting the operators $K_{\varphi}$ 
and $W_{\varphi}$ simply as $K$ and $W$.
The symbol $o_{\scriptscriptstyle R}(1)$ 
will denote a positive function  that  
depends only on $r$ and vanishes as $R\rightarrow\infty$.
Positive constants will be denoted generically by $c$ and will  
always be independent of the parameters 
$R,\delta,\ep$ and $\alpha$. 
In this way  the explicit dependence on these  
parameters of the estimates 
will be apparent.  

The aim of this section is to prove the estimates 
(2.7) and (2.10)\ -\ this is carried out in Propositions 
3.8 and 3.7 respectively. 
Recall that we are considering the eigenfunction $\psi$ of $K$ 
corresponding to the eigenvalue $\lambda$:\ $K\psi = \lambda\psi$.
We assume that for some $\alpha\geq 0$, \ 
\ $\psi_{\alpha}\equiv e^{\alpha r}\psi\in\lsh$ and for 
$\delta > 0$ define 
$\xi_{\alpha}\equiv \chi_{\sss R}e^{\delta h(r)}\psi_{\alpha}$ 
where the functions $\chi_{\sss R}(r)$ and $h(r)$ have been 
described in the previous section. 

\subsection{Spectral Localization}


Here we determine the localization 
of the function $\xi_{\alpha}$ 
with respect to the spectral 
decomposition of $K$.
Since $K\psi = \lambda\psi$, if we set $E_{I} = 
E_{\{\lambda\}}$, then $\|\bar{E}_{\{\lambda\}}\psi\| = 0$. 
There is no reason to expect that 
 $\xi_{\alpha}$ should also be so well localized near $\lambda$. 
However, it will be enough if we can obtain an upper
bound for this localization. This will follow from Lemma 3.2 below 
via the calculation described presently.

>From spectral theory and noting that for $\beta\in I$, 
\ $(K - \beta)$ is invertible on $\bar{E}_{I}$, we 
calculate $\bar{E}_{I}\psi$ with 
the formula
\begin{equation}
\bar{E}_{I} = \bar{E}_{I}(K - \beta)^{-1}(K - \beta) = 
\int_{\mu\not\in I}(\mu - \beta)^{-1}(\mu - \beta)
\,dE_{\mu}, 
\end{equation}
where $dE_{\mu}$ is the spectral measure on $\RR$ associated 
to $K$. 
This leads to the estimate
\begin{equation}
\|\bar{E}_{I}\psi\| \leq \left(dist\,(\partial I,\;\beta)\right)
^{-1}\|(K - \beta)\psi\|.  
\end{equation}
Lemma 3.2, below, provides 
an estimate for $\|(K -\lambda - \alpha^{2})\xi_{\alpha}\|$, 
but before that we require the following estimate.

\begin{lemma}
Let $K^{H} = e^{H}Ke^{-H}$ where, for $\delta> 0$ and $\alpha\geq 0$,
\ $H(r) = \delta h(r) + \alpha r$ with $h(r)$ as defined in Section 2,
and let $\psi_{\alpha}$ and $\xi_{\alpha}$ be as above. Then 
\begin{equation}
\|(K^{H} - \lambda)\xi_{\alpha} \| \leq (1 + \alpha)
o_{{\sss R}}(1)\|\psi_{\alpha}\|_{\lsh}. 
\end{equation}  
\end{lemma}

\noindent
\underline{Proof}:

\begin{eqnarray}
(K^{H} - \lambda){\xi}_{\alpha} & = & 
e^{H(r)}
\chi_{{\sss R}}(K - \lambda)\psi
\;+ \;e^{H(r)}
[K,\;\chi_{\scriptscriptstyle R}] \psi \nonumber \\
& = &
e^{\alpha r}[-\Delta,\chi_{\scriptscriptstyle R}]\psi
\nonumber \\
& = &
e^{\alpha r}\left( (-\Delta\chi_{{\sss }R})\psi - 
2\n\chi_{{\sss R}}\cdot\n
\psi\right) \nonumber \\
& = & 
(-\Delta\chi_{{\sss R}})\psi_{\alpha} - 2\n\chi_{{\sss R}}\cdot\n
\psi_{\alpha} + 
2\alpha\n\chi_{{\sss R}}\cdot\hat{x}\psi_{\alpha}. 
\end{eqnarray}
Here we have used that $(K - \lambda)\psi = 0$, that $\hre = 0$
on $supp\,(\chi^{'}_{{\sss R}})$, and that 
\begin{equation}
e^{\alpha r}\n\psi =
\n\psi_{\alpha} - \alpha\hat{x}\psi_{\alpha}.
\end{equation}  
Let $c_{1}$ and $c_{2}$  be constants (independent of $R$) 
that satisfy the inequalities
\begin{equation}
\mid\chi^{'}_{\scriptscriptstyle R}\mid\leq c_{1}R^{-1},\;\;\;\;\mid
\chi^{''}_{\scriptscriptstyle R}\mid
\leq c_{2}R^{-2}.
\end{equation}
Combining this with (3.4) and noting that $\n\chi_{{\sss R}} = 
\chi_{{\sss R}}^{'}\hat{x}$, and 
$\Delta\chi_{{\sss R}} = \chi_{{\sss R}} 
^{''} + \frac{N-1}{r}\chi_{{\sss R}}^{'}$, we have, 
\begin{equation}
\|(K^{H}-\lambda)\xi_{\alpha}\| \leq (c_{2} + (N-1)c_{1}/2)R^{-2}
\|\psi_{\alpha}\| + 2c_{1}R^{-1}\|\n\psi_{\alpha}\| + 
2c_{1}\alpha R^{-1}\|\psi_{\alpha}\|
\end{equation}
from which (3.3) follows\ \ \ \ $\Box$

\begin{lemma}
\begin{eqnarray}
\|(K - \lambda - \alpha^{2})\xi_{\alpha}\| &\leq& 
\mu(\delta,\alpha)o_{{\sss R}}(1)\Big[ 
\|\psi_{\alpha}\|_{\lsh} \;\;+
\;\;\|\xi_{\alpha}\|_{\lsh}\Big]  
\nonumber \\ 
& & \;\;\;\;\;\;\;+\;\;2\delta
(\delta + 2\alpha)\|\xi_{\alpha}\|
\end{eqnarray}
where $\mu$ is an  increasing function of both 
variables.
\end{lemma}

\noindent
\underline{Proof}:

Recall from the previous lemma that 
$K^{H} = e^{H}Ke^{-H}$ where $H(r) = \delta h(r) 
+ \alpha r$. Pulling $e^{-H}$ through $K$ we find
\begin{equation}
K^{H}\;=\;K - |\n H|^{2}\;+i\gamma_{\sss H}
\end{equation}
where $\gamma_{\sss H} = \frac{1}{i}(\n H\cdot\n + \n\cdot\n H)$. 
>From the relations 
\begin{eqnarray}
K^{H} - \lambda & = & 
      K - \lambda - |\n H|^{2} + i\gamma_{\sss H} \nonumber \\
       & = & 
 K - \lambda - \alpha^{2} - \delta^{2}\mid\n h\mid^{2} - 
2\delta\alpha\n h\cdot\hat{x} 
+ i\gamma_{\sss H}  
\end{eqnarray}
and  
\begin{equation}
\|(K^{H}- \lambda)\xi_{\alpha}\|^{2} = \langle (K^{H} -\lambda)
\xi_{\alpha},\;
\;(K^{H} -\lambda)\xi_{\alpha}\rangle, 
\end{equation}
we obtain
\begin{eqnarray}
\|(K^{H} - \lambda)\xi_{\alpha}\|^{2} & = &
  \|(K - \lambda - \alpha^{2})\xi_{\alpha}\|^{2} 
+ \|(\delta^{2}(h')^{2} + 2\delta\alpha h' - i\gamma_{\sss H})
    \xi_{\alpha}\|^{2} \nonumber \\
&  &
\;\;-2Re\,\langle(K - \lambda - \alpha^{2})\xi_{\alpha},
\;(\delta^{2}(h')^{2} + 2\delta\alpha h')\xi_{\alpha}\rangle
\nonumber \\ 
& &\;\;+  \langle i[K,\;\gamma_{\sss H}]\rangle_{\xi_{\alpha}}.  
\end{eqnarray}
Here we have written $|\n h|^{2} = (h')^{2}$ and 
$\n h\cdot\hat{x} = h'$. 
Since we are interested in an upper bound to 
$\|(K-\lambda - \alpha^{2})\xi_{\alpha}\|$ 
we need only  estimate the nonpositive
terms on the right. 

Because   
$h'\leq 1$, independently of  
$\ep$ and $R$, from the Schwarz inequality we have that  
\begin{eqnarray}
\Big|2Re\,\langle(K - \lambda - \alpha^{2})\xi_{\alpha},
\;(\delta^{2}(h')^{2} + 2\delta\alpha h')\xi_{\alpha}\rangle
\Big|
\;\; \leq \;\;2\delta(\delta + 2\alpha)
     \|(K-\lambda - \alpha^{2})\xi_{\alpha}\|\;\|\xi_{\alpha}\|.  
\end{eqnarray}

Expanding the commutator in the last line of (3.12) we obtain
\begin{equation}
i[K,\,\gamma_{\sss H}]\;=\;i\delta[K,\,\gamma_{h}]
\;+i\alpha\;[K,\,\gamma]
\end{equation} 
where $\gamma_{h} = \frac{1}{i}(\n h\cdot\n + \n\cdot\n h)$
and $\gamma = \frac{1}{i}(\n r\cdot\n + \n\cdot\n r) \;=\;
\;\frac{1}{i}(\hat{x}\cdot\n + \n\cdot\hat{x})$.
For the first commutator we calculate,
\begin{eqnarray}
i[K,\;\gamma_{\hre}] & = & 
     i[-\Delta + W,\;\gamma_{\hre}] \nonumber \\
 & = & 
   [-\Delta ,\;2h'r^{-1}x\cdot\n]\;\; + 
    \;\;[-\Delta,\,h'' + (N-1)h'r^{-1}] \nonumber \\
& & \;\;\;\;\;\;\;\;\;\;+ \;[W,\;2h'r^{-1}x\cdot\n].
\end{eqnarray}
Because 
\begin{eqnarray}
[-\Delta,\; 2h'r^{-1}x\cdot\n] & = & 
     2h'r^{-1}[-\Delta,\; x\cdot\n] + 
      2[-\Delta,\;h'r^{-1}]x\cdot\n \nonumber \\
& = & 
    -4h'r^{-1}\Delta + 4(h'r^{-3} - 
     h''r^{-2})(x\cdot\n)^{2} \nonumber \\
& & \;\;\;\;\;\;-\left((N+3)h'r^{-3} + 
    (3-N)h''r^{-2} + h''' r^{-1}\right)x\cdot\n\;\;, 
\end{eqnarray}
with $|h^{(m)}(r)|\leq c_{m}R^{1-m}$, we have that
\begin{eqnarray}
\langle [-\Delta,\,2h'r^{-1}x\cdot\n]\rangle_{\xi_{\alpha}} & = & 
\langle 2h'r^{-1}[-\Delta,\,x\cdot\n]\rangle_{\xi_{\alpha}} + 
4\langle \n\cdot x\,(h'r^{-3} - h''r^{-2})x
\cdot\n\rangle_{\xi_{\alpha}} 
\nonumber \\
& & \;\;\;\;\;\;\;\;\;\;\;\;+ cR^{-1}\|\xi_{\alpha}\|^{2}.
\end{eqnarray}
The first term on the right hand side is 
positive while the second term
is bounded above in absolute value by 
$cR^{-1}\|\n\xi_{\alpha}\|^{2}$.  
The absolute value of the real parts of the 
expectation values of the other 
terms in (3.15) 
can be bounded above by 
$o_{{\sss R}}(1)\|\xi_{\alpha}\|^{2}$. 
Thus,
\begin{equation}
\langle i[K,\;\gamma_{\hre}]\rangle_{\xi_{\alpha}}
\;\geq 
\;-o_{{\sss R}}(1)\|\xi_{\alpha}\|^{2}_{\lsh}. 
\end{equation} 
The second commutator, $i[K,\,\gamma]$,  
satisfies the 
same estimate since $i[K,\,\gamma]$ is as (3.15) 
with $h'=1$ and $h''=0$. 
  
Combining the above estimates and dropping the 
positive terms, we have that  
\begin{eqnarray}
\|(K^{H} - \lambda)\xi_{\alpha}\|^{2} & \geq & 
\|(K - \lambda - \alpha^{2})\xi_{\alpha}\|^{2} - 2\delta(\delta + 
  2\alpha)\|(K -\lambda - \alpha^{2})\xi_{\alpha}\|\;\|\xi_{\alpha}\|
\nonumber \\  
& &\;\;\;\;\;\;  
-(\delta + \alpha)o_{{\sss R}}(1)\|\xi_{\alpha}\|^{2}_{\lsh}.
\end{eqnarray}
Rearranging, we write this as
\begin{eqnarray} 
\Big[\; \|(K - \lambda - \alpha^{2})\xi_{\alpha}\| - \delta(\delta  
  + 2\alpha)\|\xi_{\alpha}\|\;\Big]^{2}
&\leq&\|(K^{H}-\lambda)\xi_{\alpha}\|^{2} + 
(\delta + \alpha)o_{{\sss R}}(1)
\|\xi_{\alpha}\|^{2}_{\lsh} \nonumber \\
& & \;\;\;\;+\Big(\delta(\delta + 2\alpha)\|\xi_{\alpha}\|\Big)^{2}.
\end{eqnarray}
Taking the square root of both sides of (3.20), 
using Lemma 3.1 
to estimate $\|(K^{H} - \lambda)\xi_{\alpha}\|$, and
writing $\mu(\delta,\alpha) = 
\max\,\Big(\,(\alpha + \delta)^{1/2},\,(1+\alpha)\,\Big)$, 
we arrive at (3.8)
\ \ \ $\Box$

\subsection{Commutator Estimates}

Our goal in this section is to establish an analogous estimate 
to (2.7) for the commutator $i[K,\,A]$ (cf. Proposition 3.8). 
For this we will require the estimates on the spectral localization 
of $\xi_{\alpha}$ obtained in the previous subsection as well as 
an estimate analogous to (2.10) for the expectation value 
$\langle -\partial_{t}^{2}\rangle_{\xi_{\alpha} }$ (cf. Proposition
3.7). 
In order to establish the latter we 
first show that certain operators constructed 
from the projections $E_{I}(K)$ and $P_{k}$ are compact;\ several
of the following lemmata are for this purpose. 
\vspace{.3in}

The compact operators  that will arise in 
our analysis are not
necessarily small in norm, but we take advantage of the structure of
the  
functions $\xi_{\alpha}$ to control 
these terms as the following lemma 
demonstrates.   


\begin{lemma}
If $C$ is a compact operator, then
\begin{equation}
\mid\langle C\rangle_{\xi_{\alpha}}\mid \;\leq 
\;o_{\scriptscriptstyle R}(1)\|\xi_{\alpha}\|^{2}. 
\end{equation} 
Here, as elsewhere in the paper, $o_{{\sss R}}(1)$ 
vanishes as $R\rightarrow\infty$ uniformly 
in $\ep$ and $\delta$.
\end{lemma}

\noindent 
\underline{Proof}:\nobreak 

Recall that $\xi_{\alpha} = \chi_{\sss R}e^{\delta h(r)}\psi_{\alpha}
$.
Let $F$ be a finite rank operator;
\begin{equation}
F\psi = \sum_{1}^{M}\langle \psi,\;\varphi_{i}\rangle\zeta_{i}, 
\end{equation}
for some $\varphi_{i},\zeta_{i}\in\l2sr$.  
By the Schwarz inequality,
\begin{eqnarray}
\mid\langle \xi_{\alpha},\;\varphi_{i}\rangle\mid  & = & 
\mid\langle\xi_{\alpha},\;\chi_{\scriptscriptstyle R/2}
\varphi_{i}\rangle\mid  \nonumber \\ 
&\leq&
\|\xi_{\alpha}\|\;\|\chi_{\scriptscriptstyle R/2}\varphi_{i}\| 
\end{eqnarray}
where the $R$ appearing in $\chi_{\scriptscriptstyle R/2}$ is the $R$ 
associated to $\xi_{\alpha}$.
For $\eta>0$ choose $R_{i},\;i=1,\ldots,M$, 
sufficiently large such that
for $R^{'}\geq R_{i},\;\;\|\chi_{\scriptscriptstyle R^{'}/2}
\varphi_{i}\|
\leq\eta\|(M\|\zeta_{i}\|)^{-1}$.
Then for $R\geq\;max\,(R_{1},\ldots,R_{M})$,
\begin{eqnarray}
\|F\,\xi_{\alpha}\| &\leq&
\sum_{1}^{M}\mid\langle\xi_{\alpha},\varphi_{i}\rangle\mid\|\zeta_{i}\|
\nonumber \\
&\leq&
\eta\,\|\xi_{\alpha}\|
\end{eqnarray}
independently of $\ep$ and $\delta$ 
(the $\ep$ and $\delta$ appearing in $\xi_{\alpha}$). 
Now, for an arbitrary compact operator $C$ there is a finite rank 
operator $F_{\eta}$ and operator $B_{\eta}$ with $\|B_{\eta}\|
<\eta$, such that $C = F_{\eta}  + B_{\eta}$. By the preceding
analysis, 
\begin{eqnarray}
\|C\,\xi_{\alpha}\| &\leq&
\|F_{\eta}\xi_{\alpha}\| + \|B_{\eta}\,\xi_{\alpha}\| \nonumber \\
&\leq& 2\eta\|\xi_{\alpha}\|
\end{eqnarray}
for $R$ sufficiently large. Applying 
the Schwarz inequality to $\langle C\rangle_{\xi_{\alpha}}$, we 
obtain the desired result
$\ \ \ \Box$ 
 
\begin{lemma}
For any integers $m_{1}\geq 0$, $m_{2}>0$ and any interval 
$I\subset\RR$,
\begin{equation}
\|\Pi_{m_{1}} \,E_{I}\,\bar{\Pi}_{m_{2}}\| \leq const. 
\left(\frac{m_{1} + 1}{m_{2}}\right)\;\;\;\;\mbox{and}\;\;\;\;\| 
\bar{\Pi}_{m_{2}}E_{I}\Pi_{m_{1}}\|\leq const.\left(\frac{m_{1}+1}
{m_{2}}
\right).
\end{equation}
 \end{lemma}
\noindent \underline{Proof}:\nobreak 

Let $g\in C^{\infty}_{o}(\RR,\RR)$ be a smoothed-out characteristic 
function of the interval $I$ such that $E_{I} = g(K)$ as described
at the end of section 1. We are denoting $E_{I}(K)$ simply by 
$E_{I}$.  
Noting that $\partial_{t}\partial_{t}^{-1} = \mbox{ Id }$ on 
$\mbox{ range }\bar{\Pi}_{m_{2}}$, we begin by writing   
\begin{eqnarray}
\Pi_{m_{1}}E_{I}\bar{\Pi}_{m_{2}}   
& = &
\Pi_{m_{1}} E_{I} \partial_{t}\partial_{t}^{-1} \bar{\Pi}_{m_{2}}
\nonumber \\
 & = & 
\partial_{t}\Pi_{m_{1}}E_{I}\partial_{t}^{-1}\bar{\Pi}_{m_{2}}
+ \Pi_{m_{1}}[E_{I},\partial_{t}]\partial_{t}^{-1}\bar{\Pi}_{m_{2}}. 
\end{eqnarray}
We use the operator calculus described in the appendix to  
express the commutator;  
\begin{eqnarray}
[E_{I},\partial_{t}] & = & 
\int_{\RR^{2}}d\tilde{g}(z)\left(K - z\right)^{-1}
[K,\partial_{t}]\left(K - z\right)^{-1} \nonumber \\
& = &
-\int_{\RR^{2}}d\tilde{g}(z)\left(K- z\right)^{-1}\;\partial_{t}W
\;\left(K - z\right)^{-1}.
\end{eqnarray}
Here $z = u + iv$ and $\tilde{g}(z)$ is an almost analytic 
extension of $g$.
As described in the appendix, 
$d\tilde{g}(z) = a(z)\,dudv + b(z)\,dudv,$
with $a(z)$ and $b(z)$ compactly supported, $a(z)$ supported away
from $v = 0$, and $\mid\! b(z) \!\mid < const\;v^{2}$.  
This, combined with the estimate $\|(K - z)^{-1}\| < v^{-1}$ and 
the boundedness of $\partial_{t}W$  
imply that $\|[E_{I},\partial_{t}]\| < \infty$. 
Going back to (3.27), 
$\|\partial_{t}\Pi_{m_{1}}\|\leq m_{1}\omega$ and 
$\|\partial_{t}^{-1}\bar{\Pi}_{m_{2}}\|\leq 1/m_{2}\omega$ 
imply that $\|\Pi_{m_{1}}E_{I}\bar{\Pi}_{m_{2}}\|\leq m_{1}/m_{2}
+ \|[E_{I},\partial_{t}]\|/m_{2}\omega\;\leq const\;(\frac{m_{1}+1}
{m_{2}})$. The same result holds for $\bar{\Pi}_{m_{2}}E_{I}\Pi
_{{m}_{1}}$  since $(\bar{\Pi}_{m_{2}}E_{I}\Pi_{m}) = 
(\Pi_{m_{1}}E_{I}\bar{\Pi}_{m_{2}})^{*}$.  
\ \ \ $\Box$
 
\begin{lemma}
For any $m_{1},m_{2}\in\NN$ and $z\in\CC\setminus\RR ,\;\;\;\Pi
_{m_{1}}\,W\,(K_{o} - z)^{-1}\,\Pi
_{m_{2}}$ is a compact operator on $\l2sr$.
\end{lemma}
\noindent
\underline{Proof}:\nobreak 

First note that $(K_{o} - z)^{-1}$ decomposes along the eigenspaces
of $i\partial_{t}$ as
\begin{equation}
(K_{o} - z)^{-1} P_{k} = (-k^{2}\omega^{2} - \Delta  - z)^{-1},
\end{equation}
and that for $\psi\in\l2sr$,
\begin{equation}
(W\psi)_{l}(x) = \sum_{k}W_{lk}(x)\psi_{k}(x)
\end{equation}
where
\begin{displaymath}
W_{lk}(x) = (2\pi/\omega)^{-1}\int_{S^{1}_{\omega}}
      W(x,t)\,e^{-i(l-k)\omega t}\;dt.
\end{displaymath}
Thus,
\begin{equation}
 \Pi_{m_{1}}\,W\,(K_{o} - z)^{-1}\,\Pi_{m_{2}} = 
\sum_{\stackrel{\mid k\mid\leq m_{2}}{\mid l\mid\leq m_{1}}}
e^{il\omega t}W_{lk}(x)\left(-k^{2}\omega^{2} - \Delta - z\right)
^{-1}\,P_{k}
\end{equation}
The sum in (3.31) is finite so it is enough to show that
\mbox{$e^{il\omega t}\,W_{lk}\left(-k^{2}\omega^{2} - 
\Delta - z\right) 
^{-1}\,P_{k}$} is compact for each $l,k$.
Since $W_{lk}\in L^{\infty}(\RRn)$
and vanishes as $|x|\rightarrow\infty$
(this is a consequence of Lemma 2.1),\  \mbox{ 
$W_{lk}\left(-k^{2}\omega^{2} - \Delta - z\right)^{-1}
$}
is a compact operator on $L^{2}(\RRn)$ (see for example 
[RS-IV]). 
Due to the relation
\begin{equation}
e^{il\omega t}W_{lk}\left(-k^{2}\omega^{2} - \Delta - z\right)^{-1}
\;\;=\;\;e^{i(l-k)\omega t}\otimes W_{lk}(-k^{2}\omega^{2} - \Delta 
- z)^{-1}
\end{equation}
on $\EE_{k}$, each term on the right hand side of (3.31) is compact.
Thus, $\Pi_{m_{1}}W(K_{o} -z)^{-1}\Pi_{m_{2}}$ is compact
\ \ \ $\Box$


\begin{lemma}
Let $E_{I} = E_{I}(K)$ and $E_{I}^{o} = E_{I}(K_{o})$. 
 For any interval $I\subset\RR$ and  any integer $m\in\NN$, 
$(E_{I} - E^{o}_{I})\,\Pi_{m}$ and $\Pi_{m}(E_{I} - E^{o}_{I})$
are compact operators on $\l2sr$.
\end{lemma}
  \noindent
\underline{Proof}: \nobreak 
 
We prove that $(E_{I} - E_{I}^{o})\Pi_{m}$ is compact. Since 
$\Pi_{m}(E_{I} - E_{I}^{o}) = \left( (E_{I} - E_{I}^{o})\Pi_{m}
\right)^{*},$\ this will imply that $\Pi_{m}(E_{I} - E_{I}^{o})$ 
is also compact.    

Let $m_{1} > m$. 
Then
\begin{eqnarray}
(E_{I} - E_{I}^{o})\Pi_{m} & = & 
\Pi_{m_{1}}(E_{I} - E_{I}^{o})\Pi_{m} + 
\bar{\Pi}_{m_{1}}(E_{I} - E_{I}^{o})\Pi_{m} \nonumber \\
& = & 
\Pi_{m_{1}}(E_{I} - E_{I}^{o})\Pi_{m} + 
\bar{\Pi}_{m_{1}}\,E_{I}\,\Pi_{m}.
\end{eqnarray}
We have used the commutativity between $\Pi_{m}$ and $E_{I}^{o}$ in the 
last line.
To compute the first term on the right hand side we use the operator 
calculus described in the appendix  to represent $E_{I}$ as
\begin{equation}
 E_{I}\;\;=\;\;\int_{\RR^{2}}d\tilde{g}(z)\;(K -z)^{-1}
\end{equation}
and similarly for $E_{I}^{o}$
(recall that $E_{I} = g(K)$ and $E_{I}^{o} = g(K_{o})$). 
Next, for $m_{2}>m_{1}$ and using the second resolvent  
equation, we have that
\begin{eqnarray}
\Pi_{m_{1}}(E_{I} - E_{I}^{o})\Pi_{m} & = & 
  \Pi_{m_{1}}\left\{ \int_{\RR^{2}}d\tilde{g}(z)  
   \left( (K - z)^{-1} - (K_{o} - z)^{-1}\right)\right\}
    \Pi_{m} \nonumber \\
& = &
\Pi_{m_{1}}\left\{\int_{\RR^{2}}d\tilde{g}(z)
    \,(K - z)^{-1}\,W\,(K_{o} - z)^{-1}\right\}\Pi_{m}
\nonumber \\ 
& = &
\int_{\RR^{2}}d\tilde{g}(z)\;\Pi_{m_{1}} (K - z)^{-1}
   \,W\,(K_{o} - z)^{-1}\;\Pi_{m} \nonumber \\
& = &
\int_{\RR^{2}}d\tilde{g}(z)
\;\Pi_{m_{1}}\,(K - z)^{-1}\,\Pi_{m_{2}}\,W\,(K_{o} - z)^{-1}
\,\Pi_{m} \nonumber \\
& & \;\;\;\;\;\;\;\;\;\; + \int_{\RR^{2}}
   d\tilde{g}(z)\,\Pi_{m_{1}}\,(K- z)^{-1}\,\bar{\Pi}_{m_{2}}\,W\,(
K_{o} - z)^{-1}\,\Pi_{m}.
\end{eqnarray}
By Lemma 3.5, the first integrand on the right hand side  
in the last line is compact for
each $z\in\CC\setminus\RR$. Since this integral is the norm limit
of Riemann sums, each of which is a compact operator, the 
integral itself is a compact operator 
(which depends on $m_{1}$ and $m_{2}$).
To estimate the second integral we write
\begin{eqnarray}
\lefteqn{
\Pi_{m_{1}}(K -z)^{-1}\bar{\Pi}_{m_{2}}\,W\,(K_{o} - z)
^{-1}\Pi_{m} } 
\nonumber \\
 & &  = \;\;\Pi_{m_{1}}(K -z)^{-1}\partial_{t}^{-1}\partial_{t}  
\bar{\Pi}_{m_{2}}\,W\,(K_{o} - z)^{-1}\Pi_{m}  \nonumber \\
& &  = \;\;\Pi_{m_{1}}(K - z)^{-1}\partial_{t}^{-1}
\bar{\Pi}_{m_{2}}(\partial_{t}W)(K_{o} - z)^{-1}\Pi_{m}
\nonumber \\
& & \;\;\;\;\;\;\;\;\;\; + \;\;\Pi_{m_{1}}
(K -z)^{-1}\partial_{t}^{-1}\bar{\Pi}_{m_{2}}  
\,W\,(K_{o} - z)^{-1}\partial_{t}\Pi_{m}
\end{eqnarray}
where we have commuted $\partial_{t}$ through 
$(K_{o} - z)^{-1}$ in the last line.
Recall from Lemma 3.4 that 
$d\tilde{g}(z) = a(z)\,dudv + b(z)\,dudv$ with $a(z)$ and $b(z)$
as described there. 
This, combined with the estimates
\begin{equation}
\|(K - z)^{-1}\|\leq v^{-1},\;\;\;\;\|(K_{o} - z)^{-1}\|\leq v^{-1},  
\;\;\;\;\|\partial_{t}^{-1}\bar{\Pi}_{m_{2}}\|\leq 1/m_{2}\omega, 
\mbox{ \ and \ }
\|\partial_{t}\Pi_{m}\|\leq m\omega, 
\end{equation}
plus the boundedness of $W$ and $\partial_{t}W$, 
implies that
\begin{equation}
\|\int_{\RR^{2}}d\tilde{g}(z)\,\Pi_{m_{1}}(K -z)^{-1}\,\bar{
\Pi}_{m_{2}}\,W\,(K_{o} -z)^{-1}\,\Pi_{m}\|\leq c/m_{2}  
\end{equation}
(here $c$ depends on $m$ but $m$ is fixed).

Now consider the second term on the right hand side in (3.33).
Applying Lemma 3.4 we find  
\begin{equation}
\|\bar{\Pi}_{m_{1}}\,E_{I}\,\Pi_{m}\| \leq const\, (1/m_{1}).
\end{equation}
Therefore,
\begin{equation}
(E_{I} - E_{I}^{o})\Pi_{m} = C_{m_{1},m_{2}} + B_{m_{1}} + B_{m_{2}}
\end{equation}
where $C_{m_{1},m_{2}}$ is compact for each $m_{1},m_{2}$  
and $\|B_{m_{1}}\|\leq const\, (1/m_{1}),\;\;\;\|B_{m_{2}}\|
\leq const \,(1/m_{2})$. Since
$m_{1}$ and $m_{2}$ are arbitrary, we see that
 $(E_{I}- E_{I}^{o})\Pi_{m}$   
is the norm limit of compact operators, and as such is itself
compact $\;\;\;\;\;\Box$


\setcounter{prop}{6}
\begin{prop}
Let $I\subset\RR$ and
\begin{equation}
\bar{m} = \min\;\{m\in\NN\mid\;- m^{2}\omega^{2} < \sup\;(I)\}.
\end{equation}
Then for any $\psi\in\l2sr$,
\begin{equation}
\langle -\partial_{t}^{2}\rangle_{\psi} \;\;\geq
\;\;\bar{m}^{2}\omega^{2}\|\psi\|^{2} - 3\bar{m}^{2}\omega^{2}
\|\bar{E_{I}}\psi\|\;\|\psi\| + \langle C\rangle_{\psi}
\end{equation}
where $C$ is a compact operator.
\end{prop}  

\noindent
\underline{Remark}:\nobreak 

Of the terms on the right
hand side in (3.42) it is the positive term $\bar{m}^{2}\omega^{2}
\|\psi\|^{2}$ that plays a key role in the proof of Theorem 2.2.  
Notice that this does not necessarily mean that the expectation 
value $\langle -\partial_{t}^{2}\rangle_{\psi}$ is 
positive. 
The usefulness of (3.42) for us is that we will be able to control
the last two terms on the right.  
\vspace{.2in}

\noindent
\underline{Proof of Proposition 3.7}:
 
Since $\Pi_{\bar{m}-1} + \bar{\Pi}_{\bar{m}-1} = {\bf 1}$ we have  
\begin{equation}
-\partial_{t}^{2} = -\partial_{t}^{2}\Pi_{\bar{m}-1} 
 -\partial_{t}^{2}\bar{\Pi}_{\bar{m}-1}.
\end{equation}
Using that the first term on the right hand side is positive and 
that
\begin{eqnarray}
-\partial_{t}^{2}\bar{\Pi}_{\bar{m}-1}
& \geq &
\bar{m}\omega^{2}\bar{\Pi}_{\bar{m}-1} \nonumber \\
& = & 
\bar{m}\omega^{2}\;\;-\;\;\bar{m}\omega^{2}\Pi_{\bar{m}-1}, 
\end{eqnarray}
we obtain
\begin{equation}
-\partial_{t}^{2}\;\geq\;\bar{m}^{2}\omega^{2}\;-\;\bar{m}^{2}
\omega^{2}\Pi_{\bar{m}-1}.
\end{equation} 
For the second term on the right we write 
\begin{eqnarray}
\Pi_{\bar{m}-1} & = &
  E_{I}\Pi_{\bar{m}-1}\,E_{I} 
\nonumber \\
& & \;\;\;\;\;\;\;\;+ \bar{E_{I}}
   \,\Pi_{\bar{m}-1}\,E_{I} + 
   E_{I}\Pi_{\bar{m}-1}\,
   \bar{E_{I}} \nonumber \\
& & \;\;\;\;\;\;\;\;+ \bar{E_{I}}
\Pi_
{\bar{m}-1}\,\bar{E_{I}}.
\end{eqnarray}
The expectation values of the last three terms on the right 
are  estimated as follows. Let $\psi\in\l2sr$. 
Then
\begin{eqnarray}
\mid\langle \bar{E_{I}}\Pi_{\bar{m}-1}
\,E_{I}\rangle_{\psi}\mid & = & 
  \mid\langle\Pi_{\bar{m}-1}\,E_{I}\,\psi,
\;\bar{E}_{I}\,\psi\rangle\mid \nonumber \\
&\leq&
\|E_{I}\,\psi\|\;\|\bar{E}_{I}\,\psi\| 
\nonumber \\    
 &\leq&
\|\bar{E}_{I}\psi\|\;\|\psi\|, 
\end{eqnarray}
using the Schwarz inequality. 
The last two terms on the right in  (3.46) 
satisfy this same estimate.


The first term on the right in (3.46) we write as
\begin{eqnarray}
E_{I}\Pi_{\bar{m}-1}E_{I} & = & 
    E_{I}^{o}\Pi_{\bar{m}-1}E_{I}^{o}
\nonumber \\
& & \;\;\;\;\;\;\;\;+ (E_{I} - E_{I}^{o})
\Pi_{\bar{m}-1}E_{I} \nonumber \\
& & \;\;\;\;\;\;\;\;+ E_{I}^{o}
\Pi_{\bar{m}-1}(E_{I} - E_{I}^{o}).
\end{eqnarray}
The first term on the right in this equation is zero because 
\begin{equation}
P_{k}E_{I}^{o} = 0
\end{equation}
for $|k| < \bar{m}$.  
This was explained in Section 2, but for convenience we repeat the 
argument here.
On $Ran\,P_{k} = \EE_{k}$,\ \ $K_{o} = -k^{2}\omega^{2} - \Delta$ 
so that
$$
P_{k}E_{I}(K_{o})\;\;=\;\;P_{k}E_{I}(-k^{2}\omega^{2} - \Delta).
$$
Since $\sup\,(I) < -k^{2}\omega^{2}$,\ \ $spec\,(-k^{2}\omega^{2} 
-\Delta) = [-k^{2}\omega^{2},\,\infty)$
is disjoint from $I$. 
Hence,\ $E_{I}(-k^{2}\omega^{2} - \Delta) = 0$.

Going back to (3.48), 
 $(E_{I} - E_{I}^{o})\Pi_{\bar{m}-1}$ and
$\Pi_{\bar{m}-1}(E_{I} - E_{I}^{o})$ are compact by Lemma 3.6. 
Thus, $E_{I}\Pi_{\bar{m}-1}E_{I}$ is compact. 
Equations (3.46), (3.47), and (3.48) together show that  
\begin{equation}
\langle \bar{m}^{2}\omega^{2}\Pi_{\bar{m}-1}\rangle_{\psi}\;
\;\leq\;3\bar{m}^{2}\omega^{2} 
\|\bar{E}_{I}\psi\|\,\|\psi\| + \langle C\rangle_{\psi}
\end{equation}
where $C = E_{I}\Pi_{\bar{m}-1}E_{I}$ is compact. 
Combining this with (3.45) yields (3.42)\ \ \ \ $\Box$
\vspace{.3in}

\begin{prop}
Let $I\subset\RR$ be any 
interval containing  
$\lambda + \alpha^{2} $ and let 
\mbox{$A = \frac{1}{2i}(x\cdot\nabla + \nabla\cdot x)$.} Then
\begin{eqnarray}
\lefteqn{ \langle i[K,\;A]\rangle_{\xi_{\alpha}} \geq
   (\lambda + \alpha^{2}   
    + \bar{m}^{2}\omega^{2})\|\xi_{\alpha}\|^{2} 
     + \|\n\xi_{\alpha}\|^{2} } \nonumber \\
& & -\Big(\frac{3\bar{m}^{2}\omega^2}{d_{\alpha}}+ 1\Big)
    \Big\{
(\mu(\delta,\alpha)o_{{\sss R}}(1)
\Big[\|\psi_{\alpha}\|^{2}_{\lsh} + \|\xi_{\alpha}\|^{2}_{\lsh}\Big]  
\nonumber \\
& & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ 2\;\delta
   (\delta + 2\alpha)\|\xi_{\alpha}\|^{2}\Big\} 
 \end{eqnarray}
where $d_{\alpha} = dist\,(\partial I, \lambda + \alpha^{2})$,
\ $\mu$ is as in Lemma 3.2, 
and $\bar{m} = \min\,\{m\in\NN\;|
\;-m^{2}\omega^{2} < \sup\,(I)\}$ (as in Proposition 3.7).
\end{prop}

\noindent\underline{Remark:}

The choice of the interval $I$ is somewhat flexible at this point.
The only requirement now is that it contain $\lambda + \alpha^{2}$. 
However, by making $\alpha$ smaller if necessary we can assume that
$I$ is any interval containing the eigenvalue $\lambda$. 
During the proof of exponential bounds, below, we will 
require that $I$  satisfy additional conditions. 
There it will be important 
that $\sup\,(I)$ be less than the next threshold above $\lambda$
:\ $\sup\,(I) < -(m_{o}-1)^{2}\omega^{2}$ where $m_{o}$ is the 
integer characterized by the relation
$$
-m_{o}^{2}\omega^{2}\;<\;\lambda\;<\;-(m_{o}-1)^{2}\omega^{2}.
$$
At the conclusion of the proof of Theorem 2.2 it will also 
become apparent that $\sup\,(I)$ limits how large of an exponential
bound we can prove for $\psi$. That is, we will be able to 
show that $e^{\alpha r}\psi\in\lsh$ for all $\alpha$ such that 
$\lambda + \alpha^{2}\;<\;\sup\,(I)$. 
But because $\sup\,(I)$ can be arbitrarily close to   
$-(m_{o}-1)^{2}\omega^{2}$, the condition on $\alpha$ is really 
that $\lambda + \alpha^{2}\;<\;-(m_{o}-1)^{2}\omega^{2}$, as 
stated in Theorem 2.2
\vspace{.2in} 
  
\noindent
\underline{Proof of Proposition 3.8}: 

We begin by writing
\begin{equation}
-\Delta = K -\partial_{t}^{2} - W.  
\end{equation}
Then,
\begin{eqnarray}
\langle i[K,\;A]\rangle_{\xi_{\alpha}}& =& 
         \langle -2\Delta \; -  x\cdot\n W\rangle_{\xi_{\alpha}} 
\nonumber \\
 & = &\langle -\Delta\rangle_{\xi_{\alpha}} + 
    \langle K - \lambda - \alpha^{2}\rangle _{\xi_{\alpha}} + 
       \langle \lambda + \alpha^{2}\rangle_{\xi_{\alpha}} + 
       \langle -\partial_{t}^{2}\rangle_{\xi_{\alpha}}
\nonumber \\
& & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;- \langle
      W + x\cdot\n W\rangle_{\xi_{\alpha}} .
\end{eqnarray}
Here we have separated out a $\langle -\Delta\rangle_{\xi_{\alpha}} = 
\|\n\xi_{\alpha}\|^{2}$ 
term because we will need this to compensate for
the negative term $-o_{{\sss R}}(1)\|\n\xi_{\alpha}\|^{2}$  
that will appear below in the estimate for 
$\langle K - \lambda - \alpha^{2}\rangle_{\xi_{\alpha} }$ via 
the use of Lemma 3.2.  


The potential term in (3.53) we estimate as
\begin{equation}
\mid\langle W + x\cdot\n W\rangle_{\xi_{\alpha}}\mid \;=\; o
_{\scriptscriptstyle R}(1)
\|\xi_{\alpha}\|^{2}, 
\end{equation}
since $\mid W + x\cdot\n W\mid\rightarrow 0$ as $\mid\! x\!\mid
\rightarrow\infty$ uniformly in $t$ (cf. Lemma 2.1) and 
$supp\,(\xi_{\alpha})$ is contained outside of a ball of radius 
$R$.  

Letting $I\subset\RR$ be as in the statement of the Proposition,
from Proposition 3.7 and Lemma 3.1 we have 
\begin{equation}
\langle -\partial_{t}^{2}\rangle_{\xi_{\alpha}} \;\;\geq  
\;\;\bar{m}^{2}\omega^{2}\|\xi_{\alpha}\|^{2} - 3\bar{m}^{2}\omega^{2} 
\|\bar{E}_{I}\xi_{\alpha}\|\,\|\xi_{\alpha}\| - 
o_{\scriptscriptstyle R}(1)\|\xi_{\alpha}\|^{2},
\end{equation}  
where, recall,  $\bar{m}= \min\,\{
m\in\NN\;|\;-m^{2}\omega^{2} < \sup\,(I)\}$.  

Applying the usual functional calculus 
we derive the estimate (cf. equations (3.1) and (3.2)) 
\begin{eqnarray}
\|\bar{E}_{I}\xi_{\alpha}\| &=& 
   \|\bar{E}_{I} (K -\lambda - \alpha^{2})^{-1}\,(K - 
    \lambda - \alpha^{2})\,\xi_{\alpha}\|
   \nonumber \\
  &\leq&
  \frac{1}{d_{\alpha}}\|(K - \lambda - \alpha^{2})\xi_{\alpha}\|
\end{eqnarray}
where $d_{\alpha} = dist\,(\partial I,\lambda + \alpha^{2})$.  
>From the 
Schwarz inequality we have that
\begin{equation}
\Big|\langle K - \lambda - \alpha^{2}\rangle_{\xi_{\alpha} }\Big| 
\;\;\leq
\;\;\|(K - \lambda - \alpha^{2})\xi_{\alpha}\|\,\|\xi_{\alpha}\|. 
\end{equation}
Thus,
\begin{eqnarray}
\langle i[K,\,A]\rangle_{\xi_{\alpha}}  
    &\geq&
   (\lambda + \alpha^{2} + \bar{m}^{2}\omega^{2}) 
\|\xi_{\alpha}\|^{2} 
\;\;+\;\;\|\n\xi_{\alpha}\|^{2} \nonumber \\  
& & \;\;\;\;-\Big(\frac{3\bar{m}^{2}
   \omega^{2}}{d_{\alpha}} + 1\Big)\|(K - \lambda - \alpha^{2})  
    \xi_{\alpha}\|\,\|\xi_{\alpha}\| \nonumber \\
& & \;\;\;\;-o_{{\sss R}}(1)\|\xi_{\alpha}\|^{2}.  
\end{eqnarray}
>From Lemma 3.2 we obtain 
\begin{eqnarray}
\|(K - \lambda - \alpha^{2})\xi_{\alpha}\|\,\|\xi_{\alpha}\| &\leq& 
\mu(\delta,\,\alpha)o_{{\sss R}}(1)
\Big[\|\psi_{\alpha}\|^{2}_{\lsh} + \|\xi_{\alpha}\|^{2}_{\lsh}\Big]  
\nonumber \\
& & \;\;\;\;+2\delta(\delta + 2\alpha)\|\xi_{\alpha}\|^{2}.
\end{eqnarray}
This, combined with (3.58) gives (3.51)\ \ \ \ $\Box$ 
\vspace{.3in}

\section{Proof of Theorem 2.2}

\setcounter{equation}{0} 

Our first task
here is to use the results of the previous section to show that
if $\psi_{\alpha} =$ 
               \mbox{$e^{\alpha r}\psi\in\lsh,$}
for some $\alpha\geq 0$ 
where $K\psi = \lambda\psi$,
then there exists a $\delta >0$ such that
$e^{\delta r}\psi_{\alpha}\in\lsh$.   
\vspace{.2in}

\subsection{Exponential Bounds}
\nopagebreak
\vspace{.1in}


Using the relation 
\begin{equation}
K^{H} = K - |\n H|^{2} + i\gamma_{\sss H} 
\end{equation}
we compute
\begin{eqnarray}
Im\;\langle (K^{H} - 
\lambda)\xi_{\alpha},\;A\xi_{\alpha}\rangle  &=& 
 \frac{1}{2}\langle i[K,\;A]\rangle_{\xi_{\alpha}} - 
   \frac{1}{2}\langle i[\mid\n H\mid^{2},\;A]\rangle_ 
    {\xi_{\alpha}} 
\;+\; Re\,\langle\gamma_{\sss H} A\rangle_{\xi_{\alpha}}.  
\end{eqnarray} 
The middle commutator on the right hand side 
is equal to
\begin{eqnarray}
i[|\n H|^{2},\,A] & = & 
i[|\n(\delta h(r) + \alpha r)|^{2},\,A] \nonumber \\
& = & 
i\delta^{2}[|\n h|^{2},\,A]\;+\;2i\delta\alpha
   [\n h\cdot\hat{x},\,A] \nonumber \\
& = & 
-2\delta^{2}h'h''r\;-\;2\delta\alpha h'' r.
\end{eqnarray}
Let $c'_{h}$ and $c''_{h}$ be constants (independent of $R$ and 
$\ep$) such that
\vspace{.2in}

$\begin{array}{lrcl}
\mid \hredd (r)\mid \;\leq\; c'_{h}R^{-1},& 2R<& r& < 3R
\\ 
\mid\hredd (r)\mid \;\leq \;c''_{h}(3R + \frac{1}{\ep})^{-1},& 
3R + \frac{1}{\ep} <& r &< 2(3R + \frac{1}{\ep})
\end{array}$ 
\begin{flushright}
 $(4.4)$
\end{flushright}
\vspace{.2in}

\setcounter{equation}{4}

\noindent (recall the definition of $\hre$) 
and set $c_{h} = \max\,(c'_{h},c''_{h})$. 
Noting that $h'(r)\leq 1$ for all $R ,\ep$  
and that $h''(r)=0$ for $3R<r<3R+\frac{1}{\ep}$ and for 
$r\geq 2(3R+\frac{1}{\ep})$, we have 
\begin{eqnarray}
\mid \hred \hredd r\mid \;\leq\; 3c_{h}, \nonumber \\
\mid\hredd r\mid \;\leq\; 3c_{h}.
\end{eqnarray}
Therefore, 
\begin{equation}
\Big|\langle i[|\n H|^{2},\,A]\rangle_{\xi_{\alpha} }\Big|
\;\leq\;6c_{h}\delta(\delta + \alpha)\|\xi_{\alpha}\|^{2}.      
\end{equation}

Recall that $\langle\gamma_{\sss H}A\rangle_{\xi_{\alpha} }
= \delta\langle\gamma_{h}A\rangle_{\xi_{\alpha} } 
+ \alpha\langle\gamma A\rangle_{\xi_{\alpha} }$ with 
\mbox{$\gamma_{h} = \frac{1}{i}(\n h\cdot\n + \n\cdot\n h)$} 
and \mbox{$\gamma = \frac{1}{i}(\hat{x}\cdot\n + \n\cdot\hat{x})$}.
Since $\gamma = r^{-1/2}A r^{-1/2}$,
\begin{eqnarray}
Re\,\langle \gamma A\rangle_{\xi_{\alpha}}
& = & 
Re\,\langle r^{-1/2}A r^{-1/2}A\rangle_{\xi_{\alpha} } 
\nonumber \\
& = & 
\langle r^{-1/2}A^{2} r^{-1/2} \rangle_{\xi_{\alpha} } 
+ Re\,\langle r^{-1/2}A\,[ 
r^{-1/2},\,A]\rangle_{\xi_{\alpha} }
\nonumber \\
& = & 
\langle r^{-1/2}A^{2} r^{-1/2}\rangle_{\xi_{\alpha} } 
+ \frac{1}{2} \langle r^{-1/2}\,[A,\,[ r^{-1/2},\,A]\,]\rangle
_{\xi_{\alpha} } \nonumber \\
& = &  
\mbox{ positive
term } + o_{{\sss R}}(1)\|\xi_{\alpha}\|^{2}, 
\end{eqnarray}
so that 
\begin{displaymath}
Re\,\langle \gamma A\rangle_{\xi_{\alpha}} \;\;\geq 
\; \;-cR^{-1} 
\|\xi_{\alpha}\|^{2}\;\;\geq \;\;-cR^{-1}\|\xi_{\alpha}\|^{2}_{\lsh}. 
\end{displaymath}
Writing 
$\gamma_{h} = \sqrt{h'r^{-1}}A\sqrt{h'r^{-1}}$
one sees that the same estimate holds for 
$Re\,\langle\gamma_{h} A\rangle
_{\xi_{\alpha} }$. 

Combining these results with Proposition 3.8 we obtain
\begin{eqnarray}
\lefteqn{ Im\,\langle (K^{H} - \lambda)\xi_{\alpha},\;A\xi_{\alpha}
\rangle \;\;\geq  
   \;\;\frac{1}{2}
    (\lambda + \alpha^{2}  
    + \bar{m}^{2}\omega^{2})\|\xi_{\alpha}\|^{2} + 
    \frac{1}{2} \|\n\xi_{\alpha}\|^{2} } \nonumber \\  
& & -\Big(\frac{3\bar{m}^{2}\omega^{2}}{ d_{\alpha}}+ 1\Big)
     \Big\{
    \mu(\delta,\,\alpha)o_{{\sss R}}(1) 
  (\|\xi_{\alpha}\|^{2}_{\lsh} + \|\psi_{\alpha}\|^{2}_{\lsh})
    + 2\delta(\delta + 2\alpha)\|\xi_{\alpha}\|^{2}
   \Big\} \nonumber \\ 
& & - (\delta + \alpha)o_{\scriptscriptstyle R}(1)
                \|\xi_{\alpha}\|^{2}_{\lsh} \nonumber \\
& & -6c_{h}\delta(\delta + \alpha)\|\xi_{\alpha}\|^{2}
\end{eqnarray}
where $d_{\alpha} = dist\,(\partial I,\;\lambda + \alpha^{2})$. 
 Now write this as 
\begin{eqnarray}
\lefteqn{ \frac{1}{2}(\lambda + \alpha^{2} 
     + \bar{m}^{2}\omega^{2})\|\xi_{\alpha}\|^{2} + 
    \frac{1}{2} \|\n\xi_{\alpha}\|^{2} } \nonumber \\
& & \;\;\;\;\;\;\;\;-\Big(\frac{3\bar{m}^{2}\omega^{2}}{d_{\alpha}}
    + 1\Big)\Big\{
\mu_{1}(\delta,\,\alpha)o_{{\sss R}}(1)  
\|\xi_{\alpha}\|^{2}_{\lsh} + 10c_{h}\delta(\delta + \alpha)
\|\xi_{\alpha}\|^{2}\Big\} 
\nonumber \\
& & \leq Im\,\langle 
(K^{H} - \lambda)\xi_{\alpha},\;A\xi_{\alpha}\rangle + 
\left(\frac{3\bar{m}^{2}\omega^{2}}{d_{\alpha}} + 1\right)
\mu_{1}(\delta,\,\alpha)o_{{\sss R}}(1)\|\psi_{\alpha}\|^{2}_{\lsh}
    \nonumber \\ 
& & \leq \;\;\mid\langle(K^{H} - 
\lambda)\xi_{\alpha},\;A\xi_{\alpha}\rangle\mid 
   + \left(\frac{3\bar{m}^{2}\omega^{2}}{d_{\alpha}} + 1\right)
     \mu_{1}(\delta,\,\alpha)o_{{\sss R}}(1)
\|\psi_{\alpha}\|^{2}_{\lsh} 
\end{eqnarray}
where $\mu_{1}(\delta,\alpha) = \mu(\delta,\alpha) 
+ (\delta + \alpha)$.
The term $10c_{h}\delta(\delta + \alpha)\|\xi_{\alpha}\|^{2}$ 
on the left hand side of the inequality comes from combining 
the $6c_{h}\delta(\delta + \alpha)\|\xi_{\alpha}\|^{2}$ and 
$2\delta(\delta + 2\alpha)\|\xi_{\alpha}\|^{2}$ 
terms on the right hand side of (4.8) and estimating their 
sum from above, which preserves the inequality.
To proceed further we require the following estimate.

\setcounter{lemma}{0} 
\begin{lemma}  
\begin{equation}
\mid\langle (K^{H} - \lambda)\xi_{\alpha},\;A\xi_{\alpha}\rangle\mid 
\;\leq 
  \;c(1 + R^{-1})(1+ \alpha)\|\psi_{\alpha}\|^{2}_{\lsh}. 
\end{equation}
\end{lemma}
\noindent \underline{Proof}:

 Recall from Lemma 3.1 that 
\begin{equation}
(K^{H} - \lambda)\xi_{\alpha} = (-\Delta\chi_{\scriptscriptstyle R})\psi_{\alpha} - 
   2\n\chi_{\scriptscriptstyle R}\cdot\n\psi_{\alpha} + 2\alpha\n\chi_{\scriptscriptstyle R}\cdot
    \hat{x}\psi_{\alpha} 
\end{equation}
and that from the definition of $h$ that $h = 0$ on 
$supp\,(\chi_{{\sss R}}^{(m)})$, $m\geq 1$. 
Hence, when restricted to $supp\,(\chi_{{\sss R}}^{(m)})$, 
\ $\xi_{\alpha} = \chi_{\scriptscriptstyle R}\psi_{\alpha}$ 
and  
\begin{eqnarray}
A\xi_{\alpha} & =& -i(x\cdot\n + \frac{N}{2})\chi_{\scriptscriptstyle R}\psi_{\alpha}
\nonumber \\
& = & 
 -i(x\cdot\n\chi_{\scriptscriptstyle R})\psi_{\alpha} - i\chi_{\scriptscriptstyle R} x\cdot\n\psi_{\alpha} 
  - i\frac{N}{2}\chi_{\scriptscriptstyle R}\psi_{\alpha}. 
\end{eqnarray}
Using (4.11) and (4.12) and 
noting 
that $\mid\Delta\chi_{\scriptscriptstyle R}\mid\leq cR^{-2},\;\;\mid 
\n\chi_{\scriptscriptstyle R}\mid\leq cR^{-1}$, 
\ $\mid\! x\!\mid < 2R$ 
on $supp\,(\chi_{\scriptscriptstyle R}^{'})$,     
and so  $\mid x\cdot\n\chi_{\scriptscriptstyle R}\mid \leq c$, 
an application 
of the triangle and Schwarz inequalities 
then leads to (4.10)\ \ \ \ $\Box$
\vspace{.3in}
 

Rewriting (4.9)  and using (4.10) on the right hand side
and using the inequality 
$(1 + \alpha)\leq\mu_{1}(\delta,\,\alpha)$, 
we obtain 
\begin{eqnarray}
\lefteqn{ \frac{1}{2}(\lambda + \alpha^{2} 
     + \bar{m}^{2}\omega^{2})\|\xi_{\alpha}\|^{2} + 
    \frac{1}{2} \|\n\xi_{\alpha}\|^{2} } \nonumber \\
& & \;\;\;\;\;\;\;\;-\Big(\frac{3\bar{m}^{2}\omega^{2}}{d_{\alpha}}
    + 1\Big)\Big\{
\mu_{1}(\delta,\,\alpha)o_{{\sss R}}(1)  
\|\xi_{\alpha}\|^{2}_{\lsh} + 10c_{h}\delta(\delta + \alpha)
\|\xi_{\alpha}\|^{2}\Big\}\nonumber \\ 
& & \leq \mu_{1}(\delta,\,\alpha)\Big[
1 + \left(\frac{3\bar{m}^{2}\omega^{2}}{d_{\alpha}} + 1\right)
o_{{\sss R}}(1)\Big]\|\psi_{\alpha}\|^{2}_{\lsh}.  
\end{eqnarray} 
For sake of clarity we write this here as
\begin{equation}
\frac{1}{2}(\lambda + \alpha^{2} + \bar{m}^{2}
\omega^{2})\|\xi_{\alpha}\|^{2} - \left(\frac{3\bar{m}^{2}\omega^{2}}
{d_{\alpha}} + 1\right)10c_{h}\delta(\delta + \alpha)
\|\xi_{\alpha}\|^{2} \;\;+\;\;e\;\;\leq\;\;g,  
\end{equation}
to distinguish the negative terms containing $\|\xi_{\alpha}\|^{2}$ 
that are controlled by 
$\delta$ alone (rather than by $R$). We will come back to equation 
(4.14) in a moment. 
\vspace{.3in}

We would like to point out that equation (4.13) is rather 
general in that it is valid for any function 
$\xi_{\alpha}$ of the form $\xi_{\alpha} = e^{\delta h}\psi_{
\alpha}$,\ $\psi_{\alpha} = e^{\alpha r}\psi,\;\delta > 0,
\alpha\geq 0$, where $\psi\in\lsh$ is an eigenfunction of 
$K$ corresponding to the eigenvalue $\lambda$. 
The numbers $\bar{m}$ and $d_{\alpha}$ are determined by the 
interval $I$ which, as was pointed out above in the remark 
following the statement of Proposition 3.8, can be any interval 
containing $\lambda + \alpha^{2}$ (or even just containing 
$\lambda$; cf. the same remark). 
This inequality ( equation (4.13) ) is fundamental in 
proving exponential bounds. It is with this purpose in mind that we 
now proceed to determine just what properties we require of $I$. 
Once this is done the numbers $\bar{m}$ and $d_{\alpha}$ will be 
determined and then it remains only to determine $\delta$ and 
$R$ (which up until now have been free parameters).  
\vspace{.3in}

Our objective now is to show 
that for a suitable choice of the parameters
$\bar{m}, \delta$ and $R$ (which we will be denoting by
$m_{o},\delta(\alpha)$ and $R_{o}$) and 
interval $I$,  there exists an $a > 0$ independent of $\ep$   
such that 
\begin{equation} 
\mbox{\ \ \ \ left hand side of (4.13)  }\geq\;a\|\xi_{\alpha}\|^{2}_
{\lsh}.  
\end{equation} 
This will then imply that  
\begin{eqnarray}
\|\xi_{\alpha,{\sss R}_{o},\delta(\alpha),\ep}\|^{2}_{\lsh} &\leq & 
a^{-1}\mu_{1}(\delta(\alpha),\,\alpha)\Big[
1 + \left(\frac{3\bar{m}^{2}\omega^{2}}{d_{\alpha}} + 1\right)
o_{{\sss R}}(1)\Big]\|\psi_{\alpha}\|^{2}_{\lsh} \nonumber \\  
&< & \infty
\end{eqnarray}
for all $\ep> 0$. 
Note that the right hand side of this inequality  
is independent of $\ep$. 
Subsequently (and as will be explained in  detail below), 
from (4.16) and the monotone convergence theorem 
it follows that the function $\chi_{\scriptscriptstyle R_{o}}
e^{\delta(\alpha) 
h_{\scriptscriptstyle R_{o}}(r)} \psi_{\alpha} = 
\lim_{\ep\rightarrow 0}\xi_{\alpha,{\sss R}_{o},
     \delta(\alpha),\ep}$ is in $\lsh$ and
therefore that $e^{(\delta + \alpha)r}\psi\in 
\lsh$.
\vspace{.2in}

We proceed to demonstrate (4.15). 
First we determine what the integer $\bar{m}$ 
should be. This in turn will determine the interval $I$ 
 via Proposition 3.7.  
Then $\delta(\alpha)$
and finally $R_{o}$ will be chosen.

Recall from Proposition 3.8 that the only requirement on $I$ up 
to this point has been that $\lambda + \alpha^{2}\in I$ which 
implies that $\sup\,(I)>\lambda$. 
Now we show that in order to obtain the positivity estimate 
(4.15),\ $\sup\,(I)$ cannot be too large.
More precisely, it must be below the next threshold above 
$\lambda$. This follows from Proposition 3.7, as we will 
presently describe.

To obtain  (4.15) 
we require that the quantity
$\lambda + \bar{m}^{2}\omega^{2}$ 
appearing on the left hand side of  (4.13) be positive.
Let $m_{o}$ to be the smallest integer $m\geq 0$ such that  
$\lambda + m^{2}\omega^{2} > 0$. 
This characterizes the location of $\lambda$ 
with respect to the thresholds ${\cal E}(K)$:        
$$
-m_{o}^{2}\omega^{2}\;<\;\lambda\;<\;-(m_{o}-1)^{2}\omega^{2}.
$$
Note that $m_{o}$ must be the {\em smallest} such integer 
otherwise $\sup\,(I)<\lambda$ and $I$ will not contain 
$\lambda$.
If $m_{o}\geq 1$, ie., if $\lambda < 0$,  
then, referring to Proposition 3.7 
with $m_{o}$ in place of $\bar{m}$ there
(see equations (3.41) and (3.42)), we see that 
the interval $I$ must satisfy  
$-m_{o}^2\omega^2 < \sup(I) < - (m_{o} - 1)^{2}\omega^{2}$.
Unlike $\sup\,(I)$, \ $\inf\,(I)$ can be freely specified. 
We take advantage of this fact to simplify the quantity 
$d_{\alpha} = dist\,(\partial I,\,\lambda + \alpha^{2})$. 
We will assume
that $\inf(I) < \lambda - (\sup(I) - \lambda)$  so that  
$d_{\alpha} = \sup\,(I) - \lambda - \alpha^{2}$.  
An interval $I$ satisfying these conditions is illustrated in 
Figure 5.
\begin{figure}
\vspace{2in}
\caption{ Location of the interval $I$.}
\end{figure}

In the case $m_{o} = 0$, ie., $\lambda > 0$,  we 
do not need to estimate the spectral localization of $\xi_{\alpha}$
with respect to $K$ because then $\lambda + \bar{m}^{2}\omega^{2}$ 
is positive with $\bar{m} = 0$ and the a priori estimate 
$\langle -\partial_{t}^{2}\rangle_{\xi_{\alpha} }\geq 0$ is 
enough. 
Thus, in this case no interval needs to be specified.


For $\lambda < 0$ we have denoted the distance from 
$\lambda + \alpha^{2} $ to $\sup\,(I)$  by $d_{\alpha}$ . 
We will denote the distance from $\lambda$ to 
$\sup\,(I)$ by $d_{0}$:\ $d_{o} = \sup\,(I) - \lambda$.  
Then, $d_{\alpha} =   
d_{o} - \alpha^{2}$.  
Now we solve the equation
\begin{equation}
\left(\frac{3m_{o}^{2}\omega^{2}}{d_{\alpha}}
+ 1\right)10c_{h}\delta(\delta + \alpha) 
\;= \;\frac{1}{8}(\lambda + m_{o}^{2}\omega^{2}+ \alpha^{2})
\end{equation}   
for $\delta$  
:\ see equation (4.14). Here $c_{h}$ is given in (4.5).
In the case $\lambda > 0$ we have $m_{o} = 0$. 
In solving (4.17) we have determined how small $\delta$ must be 
in order  
that the positive 
coefficient dominates the negative coefficient 
of $\|\xi_{\alpha}\|^{2}$ in (4.14). 
Denote by $\delta(\alpha)$ the solution  to (4.17). That is, 
\begin{equation}
\delta(\alpha)\;=\;\frac{-\alpha + \sqrt{\alpha^{2} + 
\frac{1}{20c_{h}a(\alpha)}(\lambda + m_{o}^{2}\omega^{2} 
 + \alpha^{2})}}{2}  
\end{equation}
where $a(\alpha) = (\frac{3m_{o}^{2}\omega^{2}}{d_{\alpha}} + 1)$.
Note that $\delta(\alpha) > 0$.
>From this it follows that for $\delta \leq 
\delta(\alpha)$, 
\begin{equation}
\left(\frac{3m_{o}^{2}\omega^{2}}{d_{\alpha}}
+ 1\right)10c_{h}\delta(\delta + \alpha) 
\|\xi_{\alpha}\|^{2} \leq 
\frac{1}{8}(\lambda + m_{o}^{2}\omega^{2} + \alpha^{2}) 
\|\xi_{\alpha}\|^{2}.
\end{equation}
Having found $\delta(\alpha)$, we next determine how large $R$ 
should be. 
Let
\begin{equation}
 b =  \min\,(\lambda + m_{o}^{2}\omega^{2} + 
\alpha^{2},\;1).  
\end{equation}
Referring to the left hand side of (4.13) and recalling that 
the functions $o_{\sss R}(1)$ vanish as $R\rightarrow\infty$, 
let $R_{o}$ be such that 
\begin{equation} 
\Big(\frac{3m_{o}^{2}\omega^{2}}{d_{\alpha}} + 1\Big) 
  \mu_{1}(\delta(\alpha),\,\alpha)  
o_{{\sss R}}(1)\;\;<\;\;\frac{1}{8}b 
\end{equation}
for all $R \geq R_{o}$. 
\ $R_{o}$ is  
uniform in $\ep > 0$ and in $\delta$ for $\delta\leq
\delta(\alpha)$ 
because the functions  
 $o_{\scriptscriptstyle R}(1)$ are independent of $\ep$ and $\delta$  
and the function $\mu_{1}$ 
is an increasing function of
$\delta$ which is also independent of $\ep$. 

We have then from (4.19) and  (4.21)
that if $R\geq R_{o},\;\delta\leq\delta(\alpha)$ and  
$\bar{m} = m_{o}$, the left side of (4.13) is bounded below by
$\frac{1}{4}b\|\xi_{\alpha}\|^{2}_{\lsh}$ where $b > 0$ is given by 
(4.20),     
 and therefore that
\begin{equation}
\|\xi_{\alpha}\|_{\lsh}^{2} \;\;\leq\;\;4b^{-1}
\mu_{1}(\delta,\,\alpha)\Big[
1 + \left(\frac{3m_{o}^{2}\omega^{2}}{d_{\alpha}}
 + 1 \right)o_{{\sss R}} (1)\Big]\|\psi_{\alpha}\|^{2}_{\lsh}  
\end{equation} 
uniformly in $\ep$ (cf. (4.16)).  In particular,
\begin{equation}
\|\xi_{\alpha,{\sss R_{o}},\delta(\alpha),\ep}\|
_{\lsh}^{2} < c < \infty
\end{equation}
for all $\ep > 0$
where $c$ is the right hand side of (4.22) 
with $\delta(\alpha)$ and $R_{o}$ in place of $\delta$ and $R$.
We point out again that the right hand side of (4.22) is 
independent of $\ep$. 

Now, because $h_{R_{o},\ep}$ and $h_{R_{o},\ep}^{'}$
increase pointwise to $h_{R_{o}}$ and $h_{R_{o}}^{'}$ respectively as 
$\ep\rightarrow 0$,   \ \mbox{$\mid\xi_{\alpha,R_{o},\delta(\alpha),
       \ep}\mid^{2}$} \mbox{$ = 
\mid\chi_{\scriptscriptstyle R_{o}}
e^{\delta(\alpha) h_{R_{o},\ep}}\tn\mid^{2}$} 
\ increases pointwise to 
$\mid\chi_{\scriptscriptstyle R_{o}}e^{\delta(\alpha) 
   h_{R_{o}}}\tn\mid^{2}$ 
 \mbox{$ = 
\mid\xi_{\alpha,{\sss R_{o}},\delta(\alpha),\ep = 0}\mid^{2}$}, and
$\mid\n\xi_{\alpha,{\sss R_{o}},\delta(\alpha),\ep}
\mid^{2}$ \mbox{$ = 
e^{2\delta(\alpha) h_{{\sss R}_{o},\ep}}\mid\delta(\alpha)
 h'_{{\sss R}_{o},\ep}
  \hat{x}\chi_{\scriptscriptstyle R_{o}}\tn + 
\n(\chi_{\scriptscriptstyle R_{o}}\tn)\mid^{2}$} 
increases pointwise to 
\newline $e^{2\delta(\alpha) h_{{\sss R}_{o}}}\mid
\delta(\alpha) h^{'}_{{\sss R}_{o}}\hat{x}
 \chi_{\scriptscriptstyle R_{o}}\psi_{\alpha} 
+ 
\n(\chi_{\scriptscriptstyle R_{o}}\psi_{\alpha})\mid^{2}  =  
\mid\alpha\xi_{n,{\sss R}_{o},\delta(\alpha),\ep = 0}\mid^{2}$   
as $\ep\rightarrow 0$, with both 
$\|\xi_{\alpha,{\sss R}_{o},\delta(\alpha),\ep}\|^{2}$ and
$\|\n\xi_{\alpha,{\sss R}_{o},\delta(\alpha),\ep}\|^{2}$
uniformly bounded in $\ep$ by the right hand side of (4.22).
The monotone convergence theorem implies then that 
$\|\xi_{\alpha,{\sss R}_{o},\delta(\alpha),\ep =0}\|_{\lsh}^{2}$  
is bounded above  
by the right hand side of (4.22). From this it follows that 
$e^{\alpha_{1}r}\psi\in\lsh$ where $\alpha_{1} = \alpha 
+ \delta(\alpha)$.  
\vspace{.2in}

We pause here to comment on the formula (4.22). We can write it 
as
\begin{equation}  
\|\chi_{{\sss R}_{o}}e^{(\alpha + \delta(\alpha))r}
\psi\|^{2}_{\lsh}\;\leq
\;4b^{-1}\mu_{1}(\delta(\alpha),\alpha)\Big[
1 + \Big(\frac{3m_{o}^{2}\omega^{2}}{d_{\alpha}} + 1\Big)
o_{{\sss R}}(1)\Big]
\,\|e^{\alpha r}\psi\|^{2}_{\lsh}
\end{equation}
where $R_{o}$ and $\delta(\alpha)$ are as defined above.
This inequality estimates the norm of the function 
$e^{(\alpha + \delta(\alpha)) r}\psi$ 
restricted to the exterior of a ball of 
radius $R_{o}$ in $\RRn$ by the norm of $e^{\alpha r}\psi$.  
\ $\alpha$ is an exponential bound for $\psi$ and 
$\alpha + \delta(\alpha)$ is the new exponential bound 
for $\psi$.
Recall that 
  $d_{\alpha} = dist\,(\partial I,\,\lambda + \alpha^{2}) = 
\sup\,(I) - (\lambda + \alpha^{2})$ where  
$\sup\,(I)$ 
can be chosen arbitrarily close to, but 
below, $-(m_{o}-1)^{2}\omega^{2}$.   
Since $d_{\alpha}\rightarrow 0$ as $\lambda + \alpha^{2}
\rightarrow \sup\,(I)$,
we see that the right hand side of (4.24) $\nearrow\infty$
as $\lambda + \alpha^{2}\rightarrow 
\sup\,(I)$.
Therefore, it is consistent with our analysis 
that $e^{\alpha r}\psi$ leaves 
$\lsh$ as $\lambda + \alpha^{2}\rightarrow -(m_{o}-1)^{2}
\omega^{2}$, although we are not able to prove this.

\subsection{Iterative Step}

For a given $\alpha\geq 0$ the analysis of the preceding 
subsection defined a function $\delta(\alpha)>0$ which is the 
incremental exponential bound for $\psi_{\alpha} = e^{\alpha r}
\psi$. That is, we showed that $e^{\delta(\alpha)r}\psi_{\alpha}
\in\lsh$. We use this abstract set up to iterate this 
procedure. For example, at the next iteration we start with 
the function $\psi_{\alpha_{1}r}\equiv e^{\alpha_{1}r}\psi$ 
where $\alpha_{1} = \alpha + \delta(\alpha)$ 
and find $\delta(\alpha_{1})>0$ such that 
$e^{\delta(\alpha_{1})r}\psi_{\alpha_{1}}\in\lsh$ by 
considering the regularized function 
$\xi_{\alpha_{1}}\equiv e^{\delta h(r)}\psi_{\alpha_{1}}$. 
In the end 
we will arrive at the inequality (4.13) and subsequently 
will be lead to the equation (4.17) with $\alpha_{1}$ in place of 
$\alpha$  in these two equations.
Keep in mind that the interval $I$ has been fixed at the start, 
that is, it is the same for each iteration so that 
$\sup\,(I)$ and $m_{o}$ are fixed (see the discussion 
preceding equation (4.17)). 
Denote by $\Phi$ the mapping
\begin{equation}
\Phi:\;\alpha\mapsto\;\alpha + \delta(\alpha)
\end{equation}
where $\delta(\alpha)$ is defined by (4.18).
Then $\Phi(0)$ is the exponential bound we find for $\psi$ after the 
first iteration. More generally, $\Phi^{n}(0)$ is the exponential 
bound for $\psi$ after $n$ iterations of the above procedure.
\setcounter{lemma}{1}
\begin{lemma}
If $m_{o}\neq 0$ ($m_{o}$ as in (4.18)), 
then $\lim_{n\rightarrow\infty}
\Big(\Phi^{n}(0)\Big)^{2}= \;d_{o}$, where $d_{o} = 
\sup\,(I)-\lambda$. If $m_{o} = 0$ then 
$\lim_{n\rightarrow\infty}\Phi^{n}(0)  = \infty$. 
\end{lemma}

\noindent
\underline{Proof}:

Recall that the solution $\delta(\alpha)$ to (4.17) is given by
$$
\delta(\alpha)\;=\;\frac{-\alpha + \sqrt{\alpha^{2} + 
\frac{1}{20c_{h}a(\alpha)}(\lambda + m_{o}^{2}\omega^{2} 
 + \alpha^{2})}}{2}
$$
where $a(\alpha) = (\frac{3m_{o}^{2}\omega^{2}}{d_{\alpha}} +  1)$
and $d_{\alpha} = dist\,(\partial I,\,\lambda + \alpha^{2}) =
\sup\,(I) - \lambda - \alpha^{2} = d_{o} - \alpha^{2}$. 
Setting $\alpha_{n} = \Phi^{n}(0)$, 
suppose that $m_{o}\neq 0$ and that $\lim_{n\rightarrow\infty}
\alpha_{n}^{2}< d_{o}$. 
This implies that $a(\alpha_{n})^{-1}$ is bounded away from $0$
for all $n$ (since $d_{\alpha_{n}}$ will be bounded away from 
zero for all $n$). 
It follows then that
\begin{eqnarray}
\alpha_{n} & = & \alpha_{n-1} + \delta(\alpha_{n-1})\nonumber \\
& = &  
\frac{\alpha_{n-1} + \sqrt{
    \alpha_{n-1}^{2} + \frac{1}{20c_{h}a(\alpha_{n-1})}
(\lambda + m_{o}^{2}\omega^{2} + \alpha_{n-1}^{2}) }
}{2} 
\;\geq\; \alpha_{n-1} + c
\end{eqnarray}
where $c>0$ is independent of $n$.  
Hence, $\alpha_{n}\rightarrow\infty$ which contradicts the assumption
that $\lim_{n\rightarrow\infty}\alpha_{n}^{2} < d_{o}$. 
On the other hand, $\lim_{n\rightarrow\infty}\alpha_{n}^{2}$ 
cannot exceed $d_{o}$ because 
$\delta(\alpha_{n})\rightarrow 0$ as $\alpha_{n}^{2}\rightarrow
 d_{o}$.  
Thus, \ $\lim_{n\rightarrow\infty}\alpha_{n}^{2} = d_{o}$.

If $m_{o} = 0$ (that is, if $\lambda > 0$), then
\begin{eqnarray*} 
\delta(\alpha_{n}) & = & 
\frac{-\alpha_{n-1} + \sqrt{\alpha_{n-1}^{2} 
    + \frac{1}{20c_{h}}(\lambda + \alpha_{n-1}^{2})} }
{2} \\
& \geq & 
\frac{-\alpha_{n-1} + \sqrt{\alpha_{n-1}^{2} + \frac{\lambda}{20c_{h} }}}
{2} \\
& > & c\;>\;0
\end{eqnarray*}
for all $n$ and hence $\Phi^{n}(0)\;>\;nc$\ \ \ $\Box$
\vspace{.1in}


If $\lambda < 0$, the results of Lemma 4.2 imply that 
by iterating the    
procedure described in Section 4.1, 
\ $e^{\alpha r}\psi\in\lsh$ for all $\alpha$ such that 
$\alpha^{2} < d_{o} = \sup\,(I) - \lambda$. 
Since $\sup\,(I)$ can be arbitrarily close to, but below,
$- (m_{o}-1)^{2}\omega^{2}$, we have then that 
$e^{\alpha r}\psi\in\lsh$ for all $\alpha$ such that
$\lambda +  \alpha^{2} < - (m_{o}^{2}-1)^{2}\omega^{2}$.
If $\lambda > 0$ then  $m_{o} = 0$ so that
$\Phi^{n}(0)\rightarrow\infty$.
 Thus in this case  we can show that 
$e^{\alpha r}\psi\in\lsh$ for all
$\alpha$. This completes the proof of Theorem 2.2\ \ \ $\Box$


 

\section{Constraints on Periods:\ Proof of Theorem 1.1}
%
\setcounter{equation}{0}
\setcounter{thm}{0}
%
\subsection{Absence of positive eigenvalues} 

We first prove a kind of unique continuation 
theorem at infinity for eigenfunctions of the operator $K$ 
which we will use in the proof of Theorem 1.1.
We show that if an eigenfunction has arbitrarily fast 
exponential decay (in the $L^{2}$ sense) then in fact the 
function is zero. A consequence of this, when
 combined with Theorem 2.2, is that  
$K$ has no positive eigenvalues.

\begin{thm}
Under the definitions and hypothesis of 
Theorem 2.2, if $K\psi = \lambda\psi$  
where $e^{\alpha r}
\psi\in\lsh$ for all $\alpha$, then $\psi = 0$. 
\end{thm}

Our proof follows that of [HS].
We will obtain a contradiction by assuming that $\psi\neq 0$.
We begin by fixing $a$ such that
\begin{equation}
\int_{S^{1}_{\omega}}\int_{r<a}\mid\psi\mid^{2} \leq \int_{S^{1}}
\int_{r>2a}\mid\psi\mid^{2}.
\end{equation}
Let $g(r)\leq r,\;\;g'(r)\geq 0$ with $g(r) = r$ for $r> a$ and 
set $\phi_{\alpha} = e^{\alpha g(r)}\psi\|e^{\alpha g}\psi\|^{-1}$.
Then $\phi_{\alpha}\in\lsh$ for all $\alpha$ and 
\begin{eqnarray}
\int_{S^{1}_{\omega}}\int_{r<a} \mid\phi_{\alpha}\mid^{2}
 &=&
\|e^{\alpha g}\psi\|^{-2}\int_{S^{1}_{\omega}}\int_{r<a}
e^{2\alpha g(r)}\mid\psi\mid^{2} \nonumber \\
 &\leq &
e^{2\alpha a}\|e^{\alpha g}\psi\|^{-2}
\int_{S^{1}_{\omega}}\int_{r<a}\mid\psi\mid^{2} \nonumber \\
 &\leq&
e^{2\alpha a}\|e^{\alpha g}\psi\|^{-2}
\int_{S^{1}_{\omega}}\int_{r>2a}
\mid\psi\mid^{2} \nonumber \\
 & =&
e^{-2\alpha a}\|e^{\alpha g}\psi\|^{-2}
\int_{S^{1}_{\omega}}\int_{r> 2a}
e^{4\alpha a}\mid\psi\mid^{2} \nonumber \\
 &\leq&
e^{-2\alpha a}\|e^{\alpha g}\psi\|^{-2}
\int_{S^{1}_{\omega}}\int_{r>2a}
e^{2\alpha r}  
\mid\psi\mid^{2} \nonumber \\
 &\leq&
e^{-2\alpha a}. 
\end{eqnarray}

We will show that the function $\phi_{\alpha}$ 
is an approximate eigenfunction
of $K$ with eigenvalue $\lambda + \alpha^{2}$ .  
Therefore,  $\langle 
K\rangle_{\phi_{\alpha}}\approx \lambda + \alpha^{2}$. 
On the other hand,
as an approximate eigenfunction the virial theorem implies that 
the expectation value $\langle i[K,\,A]\rangle_{\phi_{\alpha}}$ 
will be small. The fact that $\langle K\rangle_{\phi_{\alpha}}$ 
eventually dominates $\langle i[K,\,A]\rangle_{\phi_{\alpha}}$ 
as $\alpha$ becomes large will lead to a contradiction.   
\vspace{.3in}


\setcounter{lemma}{1}
\begin{lemma}
\begin{equation}
\left\langle i[K,\,A]\right\rangle_{\phi_{\alpha}}
\leq c_{1}\alpha^{2}e^{-2\alpha a} + \alpha c_{1}.
\end{equation}
for some constant $c_{1}$ (not necessarily positive) 
and all $\alpha >0$.
\end{lemma}

\noindent \underline{Proof}:

Set
\begin{eqnarray}
K_{\alpha} &=& 
e^{\alpha g}Ke^{-\alpha g}\;\;\;=
\;\;\;K - \alpha^{2}\mid\n g\mid^{2} + i\alpha\gamma_{g}, \\
\gamma_{g} &=& \frac{1}{i}(\n g\cdot\n + \n\cdot \n g).
\end{eqnarray} 
Then
\begin{equation}
\left\langle i[K,A]\right\rangle_{\phi_{\alpha}} = 
\;2Im\left\langle(K_{\alpha} - \lambda)\phi_{\alpha},\;A\phi_{\alpha}
\right\rangle + \alpha^{2}\left\langle i[\mid\n g\mid^{2},A]
\right\rangle_{\phi_{\alpha}} - 
2\alpha Re\left\langle \gamma_{g} A\right\rangle
_{\phi_{\alpha}}.
\end{equation}
We have
\begin{eqnarray}
(K_{\alpha} - \lambda)\phi_{\alpha} &=& 0, \\
\left\langle i[\mid\n g\mid^{2},A]\right\rangle_{\phi_{\alpha}} &=& 
\int_{S^{1}_{\omega}}\int_{r<a}-(2rg'g'')\mid\phi_{\alpha}\mid^{2}
\;\leq \;ce^{-2\alpha a}, \\ 
Re\left\langle\gamma_{g} A\right\rangle_{\phi_{\alpha}} &=& 
\int_{S^{1}_{\omega}}\int_{\RRn}2r^{-1}g'\mid x\cdot \n\phi_{\alpha}
\mid^{2}\;\;\;\;+\;\;\;\;\mbox{ bounded term},    
\end{eqnarray}
where the 
integral on the right in the last line is positive\ \ \ $\Box$

\begin{lemma}
\begin{equation}
\left\langle K\right\rangle_{\phi_{\alpha}} \geq \lambda + \alpha^{2}
(1 - e^{-2\alpha a}) 
\end{equation}
for all $\alpha >0$.
\end{lemma}           
\noindent \underline{Proof}:

\begin{equation}
\langle K\rangle_{\phi_{\alpha}} = \langle K_{\alpha}\rangle_{
\phi_{\alpha}} 
+ \alpha^{2}\left\langle \mid\n g\mid^{2}\right\rangle_{\phi_{\alpha}}
-\alpha\langle i\gamma_{g}\rangle_{\phi_{\alpha}}.
\end{equation}
We have 
\begin{eqnarray}     
\langle K_{\alpha}\rangle_{\phi_{\alpha}} &=& \lambda, \\
\langle \mid\n g\mid^{2}\rangle_{\phi_{\alpha}} &\geq& 
 \int_{S^{1}_{\omega}}\int_{r>a}\mid\phi_{\alpha}\mid^{2}
\;\;\geq \;\;(1 - e^{-2\alpha a}), \\
\alpha\langle i\gamma_{g}\rangle_{\phi_{\alpha}} &=& 
Im \langle K_{\alpha}\rangle_{\phi_{\alpha}}\;\;=\;\;0
\;\;\;\;\;\;\;\;\;\;\;\;\;\Box   
\end{eqnarray}
\vspace{.3in}

\noindent \underline{Proof of Theorem 5.1}:  

Subtracting (5.10) from (5.3) we obtain 
\begin{eqnarray}
\langle i[K,A]\rangle_{\phi_{\alpha}} - 
\langle K\rangle_{\phi_{\alpha}} &=&
\langle -\partial_{t}^{2} - \Delta - 
x\cdot\n W - W\rangle_{\phi_{\alpha}} \nonumber \\
&\leq&
\alpha^{2}\left(c_{1}e^{-2\alpha a} - (1 - e^{-2\alpha a})\right)
- \lambda + \alpha c_{1}.
\end{eqnarray}
As $\alpha\rightarrow\infty$ the right hand side 
of the inequality tends to $-\infty$
but the left side is bounded from below ($-\partial_{t}^{2}
- \Delta$ is a positive operator while $x\cdot\n W + W$ is bounded). 
This contradiction 
completes the proof of Theorem 5.1\ \ \ $\Box$ 
\vspace{.3in}

During the proof of Theorem 1.1 we will require a slightly modified 
version of Theorem 5.1, 
the difference being with the potential $W$. 
We will require a unique continuation theorem for operators 
$\bar{K}$ of the form $\bar{K} = K_{o} + \bar{\Pi}_{m} V
\bar{\Pi}_{m}$ where $V$ is a real-valued function such that 
$V$ and $x\cdot\n V$ are bounded and vanish as $|x|\rightarrow 
\infty$ uniformly in $t$, just 
like the potentials $W_{\varphi}$, and for 
eigenfunctions of the form $\bar{\psi} = \bar{\Pi}_{m}\psi$. 
That is, if $\bar{K}\bar{\psi}= \lambda\bar{\psi}$ and  
$e^{\alpha r}\bar{\psi}\in\lsh$ for all $\alpha$, then
$\bar{\psi} = 0$.
That this is true follows from the fact that the projections 
$\bar{\Pi}_{m}$ commute with functions 
of $x$ and with derivatives with respect to $x$.  
Therefore, the estimates of Lemmas 5.2 and 5.3 hold for $\bar{K}$ and 
$\bar{\phi}_{\alpha}$, where $\bar{\phi}_{\alpha}$ 
is defined in an analogous 
way as was  $\phi_{\alpha}$. 

\subsection{Proof of Theorem 1.1}
\nopagebreak
\noindent{\bf Preparatory discussion}
\vspace{.1in}

The spectrum of K depends on the 
frequency $\omega$.    
In particular, the thresholds ${\cal E}(K) = 
\{- m^2\omega^2 ;\;m\in\ZZ\}$    
depend on $\omega$. Since we are considering the eigenvalue
$-\kappa = -f'(0)$,  ($K\varphi = 
-\kappa\varphi$), its 
position with respect to the thresholds will change as $\omega$ 
changes. As $\omega$ increases fewer and fewer thresholds will lie 
above the point $-\kappa$. 
For $\omega^{2}>f'(0)$ the only threshold above $-\kappa$ is 
zero (see Figure 4). 
It is the presence of the zero threshold that prevents us 
from proving that the exponential bound $\alpha$ for $\varphi$
is arbitrarily large and hence that $\varphi = 0$ (by Theorem 5.1).
To prove Theorem 1.1, then, we first remove the this threshold
in the following way.

Each threshold corresponds to a point of the spectrum of 
$\partial_{t}^{2}$. Thus, by projecting-out spectral 
subspaces $\EE_{k}$ we can remove points from the spectrum of 
$\partial_{t}^{2}$.
Consequently, we can remove thresholds from $K$. (However, then we 
will be considering $K$ restricted to the compliment of these 
subspaces.) 
Thus, in the case when $\omega^{2} > f'(0)$, by projecting-out  
the subspace $\EE_{0}$ we can remove the one and only threshold above
$-\kappa$.
In removing the zero threshold we also remove the 
zero modes from the set of periodic functions we are considering
so that now any conclusions drawn from this analysis will only be 
about the time-dependent part of a function.
\vspace{.3in}

\noindent{\bf Proof of Theorem 1.1:}
\vspace{.2in}
 

Let $\varphi_{0}\equiv P_{0}\varphi$ and $\bar{\varphi}\equiv
\varphi - \varphi_{0}$. We will show that if $\omega^{2}>
f'(0)$ then $\bar{\varphi}=0$. This implies then that 
$\varphi = \varphi_{0}$, i.e., \ $\varphi$ is independent of time.
\vspace{.2in}

Project\ $\partial_{t}^{2}\varphi -\Delta\varphi + 
f(\varphi) = 0$\ onto  $\EE_{0}^{\perp}$:    
\begin{equation}
0 = \bar{P}_{0}\Big(\partial_{t}^{2}\varphi -\Delta\varphi 
+ f(\varphi)\Big) = K_{o}\bar{\varphi}  + \bar{P}_{0}f(\varphi).
\end{equation}
>From the identity  
\begin{equation}
f(u+v) = f(u) + f(v) + uv\int_{0}^{1}\int_{0}^{1}
f''(au + bv)\,dadb,
\end{equation}
we see that we can write
\begin{equation}
f(\varphi) = f(\varphi_{0} + \bar{\varphi}) = 
f(\varphi_{0}) + f(\bar{\varphi}) + \varphi_{0}\bar{\varphi}\,g
(\varphi_{0},\bar{\varphi})
\end{equation}
where $g(u,v) = \int_{0}^{1}\int_{0}^{1} f''(au+bv)\,dadb$. 
Since $\bar{P}_{0}f(\varphi_{0}) = 0$ we have that  
\begin{equation}
\bar{P}_{0}f(\varphi) = \bar{P}_{0}f(\varphi_{0} + \bar{\varphi}) = 
\bar{P}_{0}\Big(f(\varphi_{0} + \bar{\varphi}) - f(\varphi_{0}) 
\Big).
\end{equation}
We remark here that the property $\bar{P}_{0}f(\varphi_{0}) = 0$ 
means that the branch $[0,\infty)$ of the spectrum of $K$ 
is uncoupled to the other branches (we will discuss this again 
below). 

Combining (5.18) and (5.19) we obtain
\begin{eqnarray}
\bar{P}_{0}f(\varphi)&  =& 
\bar{P}_{0}\Big(f(\bar{\varphi}) + \varphi_{0}\bar{\varphi}
\,g(\varphi_{0},\bar{\varphi})\Big) \nonumber \\
& = & \bar{P}_{0}\Big( \frac{f(\bar{\varphi})}{\bar{\varphi}}  
+ \varphi_{0}\,g(\varphi_{0},\bar{\varphi})\Big)\bar{P}_{0}
\bar{\varphi}\;
\;\;\;\;\;\;(\mbox{note that }\bar{P}_{0}\bar{\varphi} = 
\bar{\varphi}) \nonumber \\
& = & (U + \kappa)\bar{\varphi}, 
\end{eqnarray}
where 
\begin{equation}
U = \bar{P}_{0}V_{\bar{\varphi}}\bar{P}_{0}, 
\;\;\;\;V_{\bar{\varphi}} = \frac{f(\bar{\varphi})}{\bar{\varphi}}  
- \kappa + 
\varphi_{0}\,g(\varphi_{0},\bar{\varphi}),\;\;\;\;\kappa = f'(0). 
\end{equation}
Thus, 
\begin{equation}
\bar{K}\bar{\varphi}\;=\;-\kappa\bar{\varphi}
\;\mbox{ with }\;\bar{K} \;=\;K_{o} + U.
\end{equation}
Because $U$ is bounded and symmetric, $\bar{K}$ is 
self-adjoint.

Our task now is to show that 
$e^{\alpha r}\bar{\varphi}\in\lsh$ for all $\alpha$. 
As in Theorem 2.2, 
set $\bar{\xi}_{\alpha}= \chi_{{\sss R}}e^{\delta h}
\bar{\psi}_{\alpha}$,  
\ $\bar{\psi}_{\alpha}= e^{\alpha r}\bar{\varphi}$,  
where $\delta > 0$ and $\alpha\geq 0$. 
Let $h(r)$ be as described in
Section 2, and
 $H(r) = \delta h(r) + \alpha r$. Then, 
\begin{eqnarray}
\bar{K}^{H} & = & e^{H(r)}
     \bar{K}e^{-H(r)}\;\;=
\;\;e^{H(r)} K_{o}e^{-H(r)}
\;+\;U  \nonumber \\  
& = & \bar{K} - |\n H|^{2}   
   + i\gamma_{\sss H} 
\end{eqnarray}
where $\gamma_{\sss H} = \frac{1}{i}(\n H\cdot\n + \n\cdot\n H)$ as 
before. 
Since $P_{0}\bar{\varphi} = 0$ we have the a priori estimate
\begin{equation}
\langle -\partial_{t}^{2}\rangle_{\bar{\xi}_{\alpha}}   
\;\geq\;\omega^{2}\|\bar{\xi}_{\alpha}\|^{2}.
\end{equation} 
The estimate (5.24) is all we require to prove exponential 
bounds for $\bar{\varphi}$. This is in contrast to the more 
general situation encountered in Theorem 2.2 where a        
microlocalization of $\langle -\partial_{t}^{2}\rangle_{\xi_{
\alpha} }$ was used. 

Proceeding now as in the proof of Theorem 2.2, 
\begin{equation}
Im\;\langle (\bar{K}^{H} + \kappa) 
\bar{\xi}_{\alpha},\;A\bar{\xi}_{\alpha}\rangle  \;= 
\; \frac{1}{2}\langle i[\bar{K},\;A]\rangle_{\bar{\xi}_{\alpha}} - 
   \frac{1}{2}\langle i[\mid\n H\mid^{2},\;A]\rangle_ 
    {\bar{\xi}_{\alpha}}
+ Re\,\langle \gamma_{\sss H}A\rangle_{\bar{\xi}_{\alpha} } 
\end{equation} 
(cf. equation (4.2)).
We compute: 
\begin{eqnarray} 
\langle i[\bar{K},\,A]\rangle_{\bar{\xi}_{\alpha}} & = &  
\;\langle -\Delta\rangle_{\bar{\xi}_{\alpha}} + 
\langle \bar{K} + \kappa - \alpha^{2}\rangle_{\bar{\xi}_{\alpha}} + 
\langle -\kappa +  \alpha^{2}\rangle_{\bar{\xi}_{\alpha}} + 
\langle -\partial_{t}^{2}\rangle_{\bar{\xi}_{\alpha}}
\nonumber \\ 
& & \;\;\;\;\;\;\;\;\;\;\;-\langle
V_{\bar{\varphi}} + x\cdot\n V_{\bar{\varphi}} 
\rangle_{\bar{\xi}_{\alpha}}, 
\end{eqnarray}  
(cf. Proposition 3.8). 
To obtain the expectation value \mbox{$\langle V_{\bar{\varphi}} + 
x\cdot\n V_{\bar{\varphi}}\rangle_{\bar{\xi}_{\alpha}}$} from 
$\langle U + x\cdot\n U\rangle_{\bar{\xi}_{\alpha}}$ we 
have used the fact that $\bar{P}_{0}$ commutes with functions 
of $x$ and derivatives with respect to $x$ .  
This implies that
\begin{equation}
i[U,\,A]\;=\;i\bar{P}_{0}[V_{\bar{\varphi}},\,A]\bar{P}_{0}\;=
\;\bar{P}_{0}(x\cdot\n V_{\bar{\varphi}})\bar{P}_{0}
\end{equation}
and that $\bar{P}_{0}\bar{\xi}_{\alpha} = \bar{\xi}_{\alpha}$. 
>From this we see that
\begin{equation}
\langle i[U,\,A]\rangle_{\bar{\xi}_{\alpha}}\;=\;\langle
\bar{P}_{0}(x\cdot\n V_{\bar{\varphi}})\bar{P}_{0}\rangle_{\bar{\xi}
_{n}}\;=\;\langle  x\cdot\n V_{\bar{\varphi}}\rangle_{\bar{\xi}
_{n}}
\end{equation}
and 
\begin{equation}
\langle U\rangle_{\bar{\xi}_{\alpha}}\;=\;\langle V_{\bar{\varphi}}  
\rangle_{\bar{\xi}_{\alpha}}.
\end{equation}
Using again the commutativity of $\bar{P}_{0}$ with functions of 
$x$, we have 
\begin{eqnarray}  
(\bar{K}^{H} + \kappa)\bar{\xi}_{\alpha} & = & 
e^{H(r)}\chi_{{\sss R}}(\bar{K} + \kappa)
\bar{\varphi}\;+\; e^{H(r)}
[\bar{K},\,\chi_{{\sss R}}]\bar{\varphi} \nonumber \\   
& = & 
e^{\alpha r}[-\Delta,\,\chi_{{\sss R}}]\bar{\varphi} 
\nonumber \\
& = & 
(-\Delta\chi_{{\sss R}})\bar{\psi}_{\alpha} - 
2\n\chi_{{\sss R}}\cdot\n\bar{\psi}_{\alpha} + 
2\alpha\n\chi_{{\sss R}}\cdot\hat{x}\bar{\psi}_{\alpha}.
\end{eqnarray}
Because of this the estimates of Lemmas 3.2 and 4.1 remain valid;
\begin{eqnarray}
\|(\bar{K}+\kappa- \alpha^{2})\bar{\xi}_{\alpha}\| &\leq&  
\mu(\delta,\,\alpha)o_{\sss R}(1)
\Big[\|\bar{\psi}_{\alpha}\|_{\lsh} +  
        \|\bar{\xi}_{\alpha}\|_{\lsh}\Big]  \nonumber \\  
& & \;\;\;\;\;\;\;+\;2\delta(\delta + 2\alpha)\|\bar{\xi}_{\alpha}
\|^{2}  \\ 
|\langle (\bar{K}^{H} +\kappa)\bar{\xi}_{\alpha},
\,A\bar{\xi}_{\alpha}\rangle | &\leq & c
(1 + R^{-1})(1 + \alpha) \|\bar{\psi}_{\alpha}\|
^{2}_{\lsh}.  
\end{eqnarray}
Here $\mu(\delta,\,\alpha)$ is as in Lemma 3.2.
>From the first inequality we estimate the 
expectation value 
$\langle \bar{K} + \kappa - \alpha^{2}\rangle_
{\bar{\xi}_{\alpha}}$;   
\begin{eqnarray}
\mid\langle \bar{K} + \kappa - \alpha^{2}\rangle_{\bar{\xi}_{\alpha}}
\mid
& \leq & 
\|(\bar{K} + \kappa - \alpha^{2})
\bar{\xi}_{\alpha}\|\,\|\bar{\xi}_{\alpha}\| 
\nonumber \\
&\leq &
     \mu(\delta,\alpha) o_{{\sss R}}(1)  
    \Big[\|\bar{\xi}_{\alpha}\|^{2}_{\lsh} + \|\bar{\psi}_{\alpha}\|^
    {2}_{\lsh}\Big] \nonumber \\  
& & \;\;\;\;\;\;\;\;\;\;+ \;2\delta(\delta + 
   2\alpha)\|\bar{\xi}_{\alpha}\|^{2}.
\end{eqnarray}

The potential  $V_{\bar{\varphi}}$ vanishes as $\mid\! x\!\mid
\rightarrow\infty$ uniformly in time. We also require that
$x\cdot\n V_{\bar{\varphi}}$ 
vanish in the same manner. That this is true for 
$\frac{f(\bar{\varphi})}{\bar{\varphi}} 
- \kappa$ follows from the same 
arguments presented in Lemma 2.1 for $W_{\varphi}$.
For the other terms in $V_{\bar{\varphi}}$,\ $\varphi_{o}g(\varphi_
{o},\,\bar{\varphi})$, we note that since
\begin{equation}
 \n g(\varphi_{0},\,\bar{\varphi})\;\;=\;\;\n\varphi_{0}\,g_{u}
(\varphi_{0},\,\bar{\varphi}) + \n\bar{\varphi}\,g_{v}
(\varphi_{0},\,\bar{\varphi}), 
\end{equation}
where $g_{u}$ and $g_{v}$ denote partial derivatives, and 
$x\cdot\n\varphi_{0}$ and $x\cdot\n\bar{\varphi}$ 
vanish uniformly in time as $\mid\!x\!\mid\rightarrow\infty$, it is 
enough that $g_{u}$ and $g_{v}$ be bounded in a 
neighborhood of $(0,0)$.
But this is clear from the formulae
\begin{eqnarray}
g_{u}(u,v)\;\;=\;\;\int_{0}^{1}\int_{0}^{1}af^{(3)}
(au + bv)\,dadb, \\
g_{v}(u,v) \;\;=\;\;\int_{0}^{1}\int_{0}^{1}bf^{(3)}
(au + bv)\,dadb.
\end{eqnarray}
Therefore, 
\begin{equation}
\Big|\langle V_{\bar{\varphi}} + x\cdot\n V_{\bar{\varphi}}\rangle
_{\bar{\xi}_{\alpha}}\Big|\;\leq\;o_{{\sss R}}(1)
\|\bar{\xi}_{\alpha}\|^{2}.
\end{equation}
Combining this with (5.24), (5.26), and (5.33) we obtain
\nobreak
\begin{eqnarray}
\langle i[\bar{K},\,A]\rangle_{\bar{\xi}_{\alpha}}& \geq & 
(-\kappa + \alpha^{2}  
 + \omega^{2})\|\bar{\xi}_{\alpha}\|^{2} + 
\|\n\bar{\xi}_{\alpha}\|^{2}  \nonumber \\
& & \;\;- \mu(\delta,\alpha)
o_{{\sss R}}(1)\Big[\|\bar{\xi}_{\alpha}\|^{2}_{\lsh} + 
\|\bar{\psi}_{\alpha}\|^{2}_{\lsh}\Big] \nonumber \\
& & \;\;-2\delta(\delta + 2\alpha)\|\bar{\xi}_{\alpha}\|^{2}. 
\end{eqnarray}
Using this in (5.25) along with the estimates 
\begin{eqnarray}
\Big|\frac{1}{2}\langle i[|\n H|^{2},\,A]\rangle_{\bar{\xi}_{\alpha}}
\Big| & = &
\Big|\frac{\delta^{2}}{2}\langle i[|\n h|^{2},\,A]\rangle_{\bar
{\xi}_{n}} \;\;+\;\;\delta\alpha
\langle i[\n h\cdot\hat{x},\,A]\rangle_{\bar{\xi}_{\alpha} }
\Big|  \nonumber \\ 
& & \;\;\;\;\leq\;\;3c_{h}\delta(\delta + \alpha)
\|\bar{\xi}_{\alpha}\|^{2}       
\end{eqnarray}
(cf. (4.3) and following) 
and
\begin{equation}
Re\,\langle \gamma_{\sss H}A\rangle_{\bar{\xi}_{\alpha} }\;=
\;\delta\,Re\,\langle\gamma_{h}A\rangle_{\bar{\xi}_{\alpha} } 
\;\;+\;\;\alpha\,Re\langle\gamma A\rangle_{\bar{\xi}_{\alpha} }
\;\;\geq\;\;-(\delta + \alpha)o_{{\sss R}}(1)
\|\bar{\xi}_{\alpha}\|^{2}_{\lsh}  
\end{equation}
(cf. (4.7) and following), we obtain   
\begin{eqnarray}
Im\,\langle(\bar{K} + \kappa)
\bar{\xi}_{\alpha},\;A\bar{\xi}_{\alpha}\rangle
&\geq &
\frac{1}{2}(-\kappa + \alpha^{2}  
+ \omega^{2})\|\bar{\xi}_{\alpha}\|^{2} 
+ \frac{1}{2}\|\n\bar{\xi}_{\alpha}
\|^{2} \nonumber \\
& &   \;\;\;\;\;\;\;\;\;\;
-\mu(\delta,\alpha)o_{{\sss R}}(1)\Big[
\|\bar{\xi}_{\alpha}\|^{2}_{\lsh} + \|\bar{\psi}_{\alpha}\|^{2}_{\lsh}
\Big] 
\nonumber \\
& & \;\;\;\;\;\;\;\;\;\;-(\delta + \alpha)o_{\sss R}(1)\|
\bar{\xi}_{\alpha}\|^{2}_{\lsh} \nonumber \\  
& & \;\;\;\;\;\;\;\;\;\;- 6c_{h}\delta(
             \delta + \alpha)\|\bar{\xi}_{\alpha}\|^{2}.
\end{eqnarray}
This is analogous to equation (4.8) obtained in the proof of Theorem
Proceeding as there and using (5.32) we have,
\begin{eqnarray}
\lefteqn{ \frac{1}{2}(-\kappa + \alpha^{2} 
     + \omega^{2})\|\bar{\xi}_{\alpha}\|^{2} + 
    \frac{1}{2} \|\n\bar{\xi}_{\alpha}\|^{2} } \nonumber \\
& & \;\;\;\;\;\;\;\;-\mu_{1}(\delta,\alpha)
                o_{\scriptscriptstyle R}(1)
                \|\bar{\xi}_{\alpha}\|^{2}_{\lsh}
                         \nonumber \\   
& & \;\;\;\;\;\;\;\;- 10c_{h}\delta(\delta + \alpha)
\|\bar{\xi}_{\alpha}\|^{2} \nonumber \\
& & \leq \mu_{1}(\delta,\,\alpha)
\Big(1 + o_{{\sss R}}(1)\Big)\|\bar{\psi}_{\alpha}\|^{2}_{\lsh}  
\end{eqnarray} 
(cf. (4.13)).
Since $\omega^{2} > f'(0) = \kappa $,\ \ $-\kappa + \omega^{2} > 0$. 
Thus, we solve
\begin{equation}
10c_{h}\delta(\delta + \alpha)\;=\;\frac{1}{8}(
-\kappa + \omega^{2} + \alpha^{2})
\end{equation}
for $\delta$ (see the discussion following (4.17)), the 
solution of which we will denote by 
$\delta(\alpha)$. 
Arguing as in Lemma 4.2, with $m_{o} = 0$,
$\Phi^{n}(0)\rightarrow\infty$ where $\Phi$ is the map 
$\Phi:\;\alpha\mapsto\alpha + \delta(\alpha)$.
By applying 
the same iteration
argument described at the end of  Section 4.2 
we can show that 
$e^{\alpha r}\bar{\varphi}\in\lsh$ for all $\alpha$. The unique
continuation theorem for $\bar{K}$ 
(see the paragraph at the end of section 5.1) then implies that 
$\bar{\varphi} = 0$. This completes the proof of 
Theorem 1.1\ \ \ \ $\Box$.
\vspace{.3in}
 
\noindent\underline{Discussion:}
\vspace{.1in}

One may attempt a similar analysis on $\Pi_{m-1}\varphi$
 for  $m>1$.
That is, perhaps one can prove that if 
$m^{2}\omega^{2}>f'(0)$ then $\bar{\Pi}_{m-1}\varphi=0$.  
However, this would require a very special, and hence nongeneric, 
relationship between the solution and nonlinearity.
For, applying the method used above in the proof of Theorem 1.1, we 
project NLW onto $\EE_{m-1}^{\perp}$ to derive an equation for 
$\bar{\Pi}_{m-1}\varphi$. 
Let us define
$$ \varphi_{{\sss <m}}\equiv\Pi_{m-1}\;,\;\;\;\;\varphi_{{\sss \geq m}}
\equiv\bar{\Pi}_{m-1} $$
and set $\Pi\equiv\Pi_{m-1}$. Then,
\begin{equation}
0 = \bar{\Pi}\Big(\partial_{t}^{2}\varphi
- \Delta\varphi + f(\varphi)\Big) = 
K_{o}\varphi_{{\sss \geq m}} + \bar{\Pi}f(\varphi).   
\end{equation}
We would like to form a potential from the term 
$\bar{\Pi}f(\varphi)$. 
Writing
\begin{equation}
f(\varphi) = f(\varphi_{{\sss <m}} + \varphi_{{\sss \geq m}}) = 
f(\varphi_{{\sss <m}}) + f(\varphi_{{\sss \geq m}}) + 
\varphi_{{\sss < m}} \varphi_{{\sss \geq m}}\,g(\varphi_{{\sss <m}},
\varphi_{{\sss \geq m}}), 
\end{equation}
we have
\begin{eqnarray}
\bar{\Pi}f(\varphi) & =& 
\bar{\Pi}\Big(f(\varphi_{{\sss <m}} + \varphi_{{\sss \geq m}}) - 
f(\varphi_{{\sss <m}} )\Big) + \bar{\Pi}f(\varphi_{{\sss <m}}) 
\nonumber \\
 & = & 
\bar{\Pi}V_{\bar{\varphi}}\bar{\Pi} + \bar{\Pi}
f(\varphi_{{\sss <m}}) \nonumber \\ 
& = & 
U\varphi_{{\sss\geq m}} + \bar{\Pi}f(\varphi_{{\sss <m}}) 
\end{eqnarray}
where here 
\begin{equation}
V_{\bar{\varphi}} = 
\frac{f(\varphi_{{\sss \geq m}})}{\varphi_{{\sss \geq m}}}  
 - \kappa + \varphi_{{\sss <m}} \,g
(\varphi_{{\sss <m}},\varphi_{{\sss \geq m}}),
\;\;\;\kappa = f'(0).
\end{equation}
But now the term $\bar{\Pi}f(\varphi_{{\sss <m}})$ 
may be nonzero because, although 
$\varphi_{{\sss <m}}$ has no modes $\geq m$, 
 \ $f(\varphi_{{\sss <m}})$ 
{\em may} contain
nonzero modes $\geq m$. 
That is, $f$ may generate those modes from the lower 
modes of $\varphi$.
For $\bar{\Pi}f(\varphi_{{\sss <m}})$ to vanish requires a special 
relationship among the modes of $\varphi$, 
a relationship determined by the structure of $f$.
In the case $m=1$ no such special relationship is needed:\ 
\ $\bar{P}_{0}f(P_{0}\varphi)$ is always zero because $f$ 
cannot generate time dependent modes from a time independent 
function.
In terms of the spectrum of $K$, \ $f$ couples the 
branches $[-m^{2}\omega^{2},\infty),\;m> 0$  to one 
another while the zero branch remains uncoupled.

Going back to (5.44) and using (5.46),  
we see that  the equation satisfied by 
$\varphi_{{\sss \geq m}}$ is in fact    
\begin{equation}
K_{o}\varphi_{{\sss \geq m}} + U\varphi_{{\sss \geq m}}\;\;=
\;\;-\bar{\Pi}f(\varphi_{{\sss <m}}). 
\end{equation}
We can think of $-\bar{\Pi}f(\varphi_{{\sss <m}})$ as a 
forcing term:\  the system is not 
closed. The case considered in Theorem 1.1 ($m = 1$)
is then special in that 
projecting NLW  onto $\EE_{0}^{\perp}$
results in a {\em closed} dynamical system 
($\bar{P}_{0}f(\varphi_{0}) = 0$). 

It is worth remarking that in the special case when 
$P_{k}\varphi = 0$ for all
$|k| <m$ then, because $f(0)=0$, \ $\bar{\Pi}f(\varphi_{{\sss <m}})
= 0$ and we {\em do} have a closed system. 
In this case the branches $[-k^{2}\omega^{2},\,\infty),\;|k|
< m$, of the spectrum of $K$ are decoupled from the higher 
branches so that it is possible to project them out. 


\subsection{A nonexistence result}
 
The proof of Theorem 5.1 relied only on the fact that
$e^{\alpha r}\psi\in\lsh$ for all $\alpha\geq 0$.  
By considering the eigenvalue $\lambda = -f'(0)$ and 
corresponding eigenfunction $\varphi$ where $\varphi$ is a 
$2\pi/\omega$ -periodic solution of NLW, that theorem 
implies the following. 
\setcounter{thm}{4}
\begin{thm}
Let $\varphi$ be a $2\pi/\omega$-periodic solution of NLW  
on ${\cal D}_{\omega}$. If $e^{\alpha r}\varphi
\in\lsh$ for all $\alpha$, then $\varphi = 0.$ 
\end{thm}

For example, periodic solutions from ${\cal D}_{\omega}$ 
cannot decay like $e^{-\alpha r^{2}}$ for any $\alpha > 0$, nor
can they have compact spatial support. 

\newpage

\section{Appendix : Operator calculus}
\setcounter{lemma}{0} 
\setcounter{equation}{0} 

We outline here a convenient operator calculus 
first introduced in [Hel,Sj\"{o}] for functions
$g(H)$ of self-adjoint operators based on the representation
$$
 g(H) = \frac{1}{2\pi}\int_{\RR^{2}}\left(H - z\right)^{-1}
\,\partial_{\bar{z}}\tilde{g}(z)\,dudv
 \eqno(A1) $$  
where $z = u + iv$ and  $\partial_{\bar{z}} = \partial_{x} + 
i\partial_{y}$.       
Here $g$ is a complex-valued function in $C^{\infty}_{o}(\RR)$
and $\tilde{g}$ an {\em almost analytic extension} of $g$ 
into the complex plane. By almost analytic we mean that 
$\tilde{g}$ satisfies the Cauchy-Riemann equations on $\RR$:
\begin{displaymath}
\partial_{\bar{z}}\tilde{g}(z) = 0\;\;\;\;\;for\;\;z\in\RR.
\end{displaymath}
We abbreviate (A1) by writing
$$
g(H) = \int\,d\tilde{g}(z)\,\left(H - z\right)^{-1}\;;\;\;\;\;d
\tilde{g}(z) \equiv \frac{1}{2\pi}\partial_{\bar{z}}\tilde{g}
(z)\,dudv.
\eqno(A2) $$ 
For example, we can construct an almost analytic extension of
$g$ by the formula   
$$
\tilde{g}(z) = \left(g(u) + ivg^{'}(u)\right)\chi(z)
\eqno(A3) $$ 
where $\chi\in C^{\infty}_{o}(\CC)$ with $\chi\equiv 1$
on some complex neighborhood of $supp\,(g)$.
\vspace{.2in}

\noindent{\bf Lemma A1}  
\ \ {\em If } $g\in C^{\infty}_{o}(\RR)$ {\em and }
\ $\tilde{g}$ {\em is an almost
analytic extension of}\  $g$ \ {\em as given above, then} 
\ $g(H)$ {\em is given by (A1)}.
\vspace{.2in}  

\noindent \underline{Proof}:

$\partial_{\bar{z}}\tilde{g}$ has compact support and vanishes
on the real axis, so that $\mid\!\partial_{\bar{z}}\tilde{g}
(z)\!\mid < const\mid\!v\!\mid$.
Also, \ $\|\left(H - z\right)^{-1}\|\; < \;\mid\!v\!\mid^{-1}$ so that
the integral (A1) exists as the norm limit of Riemann sums. It is
enough now  to show that   
$$
g_{\ep}(t) \equiv \int_{\mid v\mid >\ep}
  d\tilde{g}(z)\,(t - z)^{-1},\;\;\;\;t\in\RR
\eqno(A4) $$  
converges pointwise to $g(t)$ as $\ep\searrow 0$.
First note that, since $\partial_{\bar{z}}(t - z)^{-1} = 0$
for $z\not\in\RR$, 
\ $\partial_{\bar{z}}\tilde{g}(z)\big(z - t\big)^{-1} = 
\partial_{\bar{z}}\Big(\tilde{g}(z)\big(z - t\big)^{-1}\Big)$.  
Integrating by parts in $u$ and $v$,   
$$
g_{\ep}(t)\;\;=\;\;\frac{i}{2\pi}\int_{\RR}\tilde{g}(u+ iv)
\big(u + iv -t\big)^{-1}\mid^{v = -\ep}_{v = \ep}\;\;du.
\eqno(A5) $$  
Expanding $\tilde{g}(u + iv) = g(u) \pm i\ep g'(u) + {\cal O}(\ep^{2})$ 
we find,     
$$
\tilde{g}(u + iv)\big(u + iv -t\big)^{-1}\mid_{v = \ep}^{v = -\ep}
\;\;=\;\;\frac{1}{\pi}g(u)\frac{\ep}{(u - t)^{2} + \ep^{2}} 
\;\;-\;\;\frac{1}{\pi}g'(u)\frac{\ep (u - t)}{(u - t)^{2} + \ep^{2}}
\;\;+\;\;{\cal O}(\ep).
\eqno(A6) $$  
The last two terms are bounded and vanish pointwise for $u\neq t$ 
as $\ep\rightarrow 0$. Therefore,    
$$
lim\searrow 0\;\;g_{\ep}(t)\;=\;lim\searrow 0\;\;\frac{1}{\pi}  
\int_{\RR}g(u)\frac{\ep}{(u - t)^{2} + \ep^{2}}\;du\;\;=\;\;g(t)
\;\;\;\;\;\;\Box
\eqno(A7) $$ 

\noindent {\bf Commutator Estimates} 
\vspace{.1in}

Suppose $H$ and $A$ are unbounded self-adjoint operators 
on a Hilbert space ${\cal H
}$.  
If $[H,\;A]$ is $H$-bounded, then for $\psi\in{\cal H}$,
\begin{eqnarray*}
\|[H,\;A]\left(H -z\right)^{-1}\psi\| 
 & \leq &
a\|H\left(H - z\right)^{-1}\psi\| + b\|\left(H - z\right)^{-1}\psi
\|,\;\;\;\;a,b > 0,  \\
 & = &
  a\|\psi + z\left(H - z\right)^{-1}\psi\| + b\|\left(
H -z\right)^{-1}\psi\| \\
 & \leq & 
  a\|\psi\| + \left(\mid z\mid + b\right)\|\left(H - z\right)
^{-1}\psi\| \\
 & \leq &
  a\|\psi\| + \left(\mid z\mid + b\right)\mid v\mid^{-1}\|\psi\|
\\   
 &\leq & 
  c\left( 1 + \mid z\mid\right)\mid v\mid^{-1}\|\psi\|. 
\end{eqnarray*}  
That is,  
$$   
\|[H,\;A]\left(H -z\right)^{-1}\|\leq const\,(1 + \mid z\mid)
\mid v\mid^{-1}.
\eqno(A8) $$  
In particular, this holds when $[H,\;A]$ is bounded. 
Thus, if $g\in C_{o}^{\infty}(\RR)$ then the representation 
$$
[g(H),\;A] = \int\,d\tilde{g}(z)\,\left(H -z\right)^{-1}
[H,\;A]\left(H - z\right)^{-1}
\eqno(A9) $$  
requires an almost analytic extension 
$\tilde{g}$ that satisfies $\mid\!\partial_{\bar{z}}\tilde{g}(z)\!\mid\;\leq\;const
\,v^{-2}$. To this end let $\chi\in C^{\infty}_{o}
(\RR)$ with $\chi\equiv 1$ on some open interval containing $0$.
We define an almost analytic extension of $g$ by the formula
$$
\tilde{g}(z) = \chi(v/\langle u\rangle)\sum_{k=0}^{2}
  g^{(k)}(u)\frac{(iv)^{k}}{k!}
\eqno(A10) $$  
where $\langle u\rangle = (1 + u^{2})^{1/2}$. Then, 
\begin{eqnarray*}
\partial_{\bar{z}}\tilde{g}(z) & =&  
\langle u\rangle^{-1}\chi^{'}(v/\langle u\rangle)
\left(i - \frac{uv}{\langle u\rangle^{2}}\right)\sum_{k=0}^{2}
  g^{(k)}(u)\frac{(iv)^{k}}{k!}\;\;+\;\;\chi(v/\langle u\rangle)
g^{(3)}(u)\frac{(iv)^{2}}{2}  \\
 & = &
 a(z)\;\;+\;\;b(z). 
\end{eqnarray*}
 We see that both $a(z)$ and $b(z)$ are compactly supported,  
$a(z)$ away from $v = 0$ (since $\chi^{'}(t)\equiv 0$ in a neighborhood
of $0$). Furthermore, $a(z)$ is bounded and $\mid\! b(z)\!\mid
\leq\;const\,v^{2}$. Thus, $\mid\! 
\partial_{\bar{z}}\tilde{g}(z)\!\mid \leq\;const\,v
^{2}$.  

\newpage

\Large
\noindent{\bf References}
\normalsize
\vspace{.2in}

\noindent
[B]\ \ \ \ Berger, M.S. {\em On the Existence and Structure 
of Stationary States for a Nonlinear Klein-Gordon Equation}.  
J. Fun. Ana.,{\bf 9}, 249-261 (1972). 
\vspace{.1in}

\noindent
[C]\ \ \ \ Coron, J.-M. {\em Minimal Period for a 
Vibrating String of Infinite Length}, C.R. Acad. Sc. Paris, 
{\bf 294}, 127-129 (1982).
\vspace{.1in}

\noindent
[CFKS]\ \ \ Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.
{\em Schr\"{o}dinger Operators}. Springer Verlag,  Berlin (1987).
\vspace{.1in}

\noindent
[FH]\ \ \ \ Froese, R.G., and Herbst, I. {\em Exponential Bounds and 
Absence of Positive Eigenvalues for N-Body Schr\"{o}dinger
Oerators.} Commun. Math. Phys. {\bf 87}, 429-447 (1982). 
\vspace{.1in}

\noindent
[FHH-OH-O]\ \ Froese, R.G., Herbst, I., Hoffman-Ostenhof, M., and
Hoffman-Ostenhof, T. {\em $L^{2}$ Lower Bounds to Solutions of the 
Schr\"{o}dinger Equation}. Commun. Math. Phys. {\bf 87}, 3265-
3286 (1982). 
\vspace{.1in}

\noindent
[HS]\ \ \ \ Hunziker, W., and Sigal, I.M. {\em The General Theory of 
N-Body Quantum Systems}. 
In, {\em Mathematical Quantum Theory II:\ Schr\"{o}dinger
Operators}. J. Feldman, R. Froese, L.M. Rosen eds.
Centre de Recherches Math\'{e}matiques, CRM  
Proceedings and Lecture Notes, vol. 8, American Mathematical 
Society (1995). 
\vspace{.1in}

\noindent
[Hel,Sj\"{o}]\ \ Helffer, B., and Sj\"{o}strand, J.  {\em Equation 
de Schr\"{o}dinger avec champ magnetique et 
equation de Harper}. 
Schr"{o}dinger Operators, Lecture Notes in Mathematics 345,
H.Holden and A. Jensen eds., Springer, Berlin, 118-197 (1989). 

\noindent
[L]\ \ \ \ Levine, H.A. {\em Minimal Periods for Solutions of Semilinear
Wave Equations in Exterior Domains and for Solutions of the 
Equations of Nonlinear Elasticity}. J. Math. Anal. Appl. {\bf 135},
297-308 (1988). 
\vspace{.1in}

\noindent [M]\ \ \ \ Mourre, E. {\em Absence 
of Singular Continuous Spectrum for 
Certain Self-Adjoint Operators}. Commun. Math. Phys. {\bf 78},
391-408 (1981).
\vspace{.1in}

\noindent
[RS-IV]\ \ Reed, M. and Simon, B. {\em Methods of Modern Mathematical
Physics IV. Analysis of Operators}. Academic Press, New York (1978).
\vspace{.1in}

\noindent [S]\ \ \ \ Smiley, M.W. {\em Complex-Valued 
Time-Periodic Solutions and Breathers of Nonlinear Wave Equations}.
Diff. Integral Eq.,{\bf 4}, 851-859 (1991).
\vspace{.1in}

\noindent
[Sc]\ \ \ \ Scarpellini, B. {\em Decay Properties of Periodic, 
Radially Symmetric Solutions of Sine-Gordon-type Equations}.
Math. Meth. Appl. Sci., {\bf 16}, 359-378 (1993).
\vspace{.1in}

\noindent
[Str]\ \ \ Strauss, W.A. {\em Existence of Solitary Waves in 
Higher Dimensions}. Commun. Math. Phys., {\bf 55}, 149-162 (1977).
\vspace{.1in}

\noindent
[V]\ \ \ Vuillermot, P.-A. {\em Periodic Soliton Solutions 
to Nonlinear Klein-Gordon Equations on $\RR^{2}$},
Diff. Integral Eq., {\bf 3}, 541-570 (1990).
\vspace{.1in}

\noindent 
[W]\ \ \ \ Weinstein, A. {\em Periodic Nonlinear Waves 
on a Half Line}.
Comm. Math. Phys. {\bf 99}, 385-388 (1985).
Erratum:\ Comm. Math. Phys. {\bf 107}, 177 (1986). 
\vspace{.1in}

\end{document} 
ENDBODY
