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\begin{document}
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\theoremstyle{remark}
\newtheorem*{prf}{Proof of Lemma~\ref{lem1}}
\newcommand{\ess}{\operatorname{ess}}
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\title[Spectral Mapping Theorems and
Invariant Manifolds]{A Spectral Mapping Theorem and
Invariant Manifolds for Nonlinear Schr\"odinger Equations}
\author[Gesztesy, Jones, Latushkin, and Stanislavova]
{F.~Gesztesy, C.~K.~R.~T.~Jones, Y.~Latushkin, and
M.~Stanislavova}
\address{Department of Mathematics,
University of
Missouri, Columbia, MO
65211, USA}
\email{fritz@math.missouri.edu\newline
\indent{\it URL:}
http://www.math.missouri.edu/people/fgesztesy.html}
\address{Division of Applied Mathematics, Brown University,
Providence, RI 02912, USA}
\email{ckrtj@cfm.brown.edu}
\address{Department of Mathematics, University of
Missouri, Columbia, MO 65211, USA}
\email{yuri@math.missouri.edu}
\address{Department of Mathematics, University of
Missouri, Columbia, MO 65211, USA}
\email{mstanis@pascal.math.missouri.edu}
\begin{abstract} A spectral mapping theorem is proved
that
resolves a key problem in applying invariant manifold
theorems to
nonlinear Schr\" odinger type equations. The theorem
is applied to
the operator that arises as the linearization of the
equation around
a standing wave solution. We cast the problem in the
context of
space-dependent nonlinearities that arise in optical
waveguide problems.
The result is, however, more generally applicable
including to
equations in higher dimensions and even systems. The
consequence
is that stable, unstable, and center manifolds exist in
the neighborhood
of a (stable or unstable) standing wave, such as a
waveguide mode,
under simple and commonly verifiable spectral
conditions.
\end{abstract}
%%\subjclass{Primary: 35Q55, 47D03; Secondary: 47D06}
%%\keywords{Nonlinear Schr\"odinger equation, center manifolds,
%%semigroups, spectral mapping theorem}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main Results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The local behavior near some distinguished solution,
such as a steady state, of an evolution equation, can be
determined through a decomposition into invariant
manifolds,
that is, stable, unstable and center manifolds. These
(locally invariant)
manifolds are characterized by decay estimates. While
the flows on the
stable and unstable manifolds are determined by exponential
decay in
forward and backward time respectively, that on the
center manifold is
ambiguous. Nevertheless, a determination of the flow on
the center
manifold can lead to a complete characterization of the
local flow and
thus this decomposition,
when possible, leads to a reduction of this problem to
one of identifying
the flow on the center manifold.
This strategy has a long history for studying the local
behavior near
a critical point of an ordinary differential equation,
or a fixed point
of a map, and it has gained momentum in the last few
decades in the
context of nonlinear wave solutions of evolutionary
partial differential
equations. Extending the ideas to partial differential
equations has,
however, introduced a number of new issues. In infinite
dimensions,
the relation between the linearization and the full
nonlinear equations
is more delicate. This issue, however, turns out to be
not so difficult
for the invariant manifold decomposition and has largely
been resolved,
see, for instance, \cite{Ball}, \cite{BJ}.
A more subtle issue arises at the linear level. All of
the known proofs
for the existence of invariant manifolds are based upon
the use of the
group (or semigroup) generated by the linearization.
The hypotheses of
the relevant theorems are then formulated in terms of
estimates on the
appropriate projections of these groups onto stable,
unstable and center
subspaces. These amount to spectral estimates that come
directly from a
determination of the spectrum of the group. However, in
any actual
problem, the information available will, at best, be of
the spectrum of
the infinitesimal generator, that is, the linearized
equation and not its
solution operator. Relating the spectrum of the
infinitesimal generator
to that of the group is a spectral mapping problem
that is often
non-trivial.
In this paper, we resolve this issue for
nonlinear Schr\"odinger equations. We formulate the
results for the
case of space-dependent nonlinearities in arbitrary
dimensions.
This class of equations is motivated by the one space
dimension
case that appears in
the study of optical waveguides, see \cite{WGpap}, and has
attracted the
attention of many authors. In particular, there is
extensive literature on
the existence and instability of standing waves, see, for
instance, \cite{Gril88,Gril90,GSS,JonesED,JonesJDE} and
the references therein. In many
instances the questions of the existence
of standing waves and the structure of the spectrum of the
linearization
of the nonlinear equation around the standing wave are
well-understood, see \cite{WGpap}.
In the case considered in this paper, the interesting
examples are known to have the
spectrum of their
linearization ${\mathcal A}$ enjoying a disjoint
decomposition:
the
essential spectrum is positioned on the imaginary axis, and
there are several isolated
eigenvalues off the imaginary axis, see
\cite{Gril88,JonesED,JonesJDE}.
However, as mentioned above, this spectral information
about the linearization ${\mathcal A}$ is not sufficient
to guarantee the
existence of invariant manifolds. The general theory gives
the existence of these manifolds for
a semilinear equation with linear part ${\mathcal A}$
only when the spectrum of the operator $e^{t{\mathcal
A}}$, $t>0$, rather than that of ${\mathcal A}$, admits a
decomposition into disjoint components.
It is not {\em a priori} clear that the spectrum of the
operator
$e^{t{\mathcal A}}$ is obtained from the spectrum of
${\mathcal A}$ by
exponentiation. Indeed, in the present case,
the operator ${\mathcal A}$ does not generate a semigroup
for which this property is known (such as for analytic
semigroups).
Thus, we prove such a spectral mapping theorem
(cf.~Theorem~\ref{SMTh}) in
this paper. This spectral mapping theorem is derived
from a known
abstract result in the theory of strongly continuous
semigroups of
linear operators (see Theorem~\ref{GGHP}). To apply this
abstract result
one needs to prove that the norm of the resolvent of
${\mathcal A}$ is
bounded along vertical lines in the complex plane. The
corresponding proof
is based on Lemmas~\ref{lem1} and \ref{lem1prime}. The
main technical step
in the proof of these lemmas concerns a result about the
high-energy decay
of the norm of a Birman-Schwinger-type operator
(cf.~Proposition~\ref{maintech}), a well-known device
borrowed from
quantum mechanics.
We consider the following Schr\"odinger equation with
space-dependent nonlinearity,
\begin{equation}\label{NLS}
iu_t=\Delta u+f(x,|u|^2)u+\beta u,
\quad u=u(x,t)\in {\Bbb C}, \quad x\in {\Bbb R}^n, \quad
t\geq 0,\quad
\beta\in {\Bbb R},
\end{equation}
where
$\Delta = \sum\limits_{j=1}^n \frac{\partial^2}
{\partial x_j} $
denotes the Laplacian, $n \in {\bf N},$ and $f$ is
real-valued.
Rewriting \eqref{NLS} in terms of its real and
imaginary parts, $u=v+iw,$
one obtains
\begin{equation}\begin{split} v_t&=
\Delta w+f(x,v^2+w^2)w+\beta w,\\
w_t&=-\Delta v-f(x,v^2+w^2)v-
\beta v.\label{RINLS}\end{split}\end{equation}
A standing wave of frequency $\beta $ for \eqref{NLS}
is
a time-independent real-valued solution
$\hat{u}=\hat{u}(x)$ of
\eqref{RINLS}. Suppose the standing wave $\hat{u}$
is given
{\em a priori}.
Consider the linearization of \eqref{RINLS}
around
$\hat{u}$ (recalling $\hat{w}=0$),
\begin{align*}
p_t&=\Delta q+f(x,\hat{u}^2)q+\beta q,\\
q_t&=-\Delta p-f(x,\hat{u}^2)p-2\partial _2
f (x,\hat{u}^2)\hat{u}^2p-\beta p,
\end{align*}
where $\partial _2 f(x,y)=f_y(x,y)$. Thus, the linearized
stability of the standing wave is determined by
the operator
\begin{equation}\label{defA}
{\mathcal A}=
\left[\begin{array}{cc}0 & -L_R\\ L_I & 0\end{array}\right],
\text{ where } L_R=-\Delta-\beta
+Q_1,\quad L_I=-\Delta-\beta +Q_2,
\end{equation}
and the potentials $Q_1$ and $Q_2$ are explicitly given
by the formulas
$$ Q_1(x)=-f(x,\hat{u}^2(x)),\quad
Q_2(x)=-f(x,\hat{u}^2(x))-2\partial_2
f (x,\hat{u}^2 (x))\hat{u}^2(x).$$
We impose the following conditions on $f,\beta$ and the
standing wave $\hat{u}$ (see
\cite{Gril88,JonesED,JonesJDE}):
\begin{enumerate}\item[(H1)] $f:{\Bbb R}^{n+1}\to {\Bbb R}$
is $C^3$ and all derivatives are bounded on a set of the
form ${\Bbb R}\times U$, where $U$ is a neighborhood of
$0\in {\Bbb R}^n$;
\item[(H2)] $f(x,0)\to 0$ exponentially as $|x|\to \infty$;
\item[(H3)] $\beta <0$;
\item[(H4)] $|\hat{u}(x)|\to 0$ exponentially as
$|x|\to \infty$.\end{enumerate}
As a result, the potentials $Q_1$ and $Q_2$
exponentially decay at infinity. The operator
${\mathcal A}$ is
considered on $L^2({\Bbb R^n})\oplus L^2({\Bbb R^n})$;
the domain ${\mathcal D}\left(-\Delta \right)$
is chosen to be the standard
Sobolev space $H^2({\Bbb R}^n)$, and the domain of
${\mathcal A}$ is then
$H^2({\Bbb R}^n)\oplus H^2({\Bbb R}^n)$. Note, that
$\left[\begin{array}{cc} 0 & \Delta\\
-\Delta & 0\end{array}\right]$ generates a strongly
continuous group on $L^2({\Bbb R}^n)\oplus L^2({\Bbb R}^n)$.
Thus, its bounded perturbation ${\mathcal A}$ generates
a strongly continuous group
$\{e^{t{\mathcal A}}\}_{t \in{\mathbb R}}$ as well.
It was proved in \cite[Thm.~3.1]{Gril88} that
$\sigma_{\ess}({\mathcal A} )=\{ i\xi
:\xi \in {\Bbb R},|\xi |\geq -\beta \}$.
In addition, it was proved in
\cite{Gril88,JonesED,JonesJDE} that, under the above hypotheses,
$\sigma ({\mathcal A})\backslash \sigma_{\ess}({\mathcal A})$
consists of finitely many eigenvalues, symmetric with
respect to both coordinate axes.
We prove the following result that
relates the spectrum of the semigroup
$\{e^{t{\mathcal A}}\}_{t \geq 0}$
and the spectrum of its generator.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm} [Spectral Mapping Theorem] \label{SMTh} For
each $n \in {\bf N}$ one has
$$
\sigma (e^{t{\mathcal A}})\backslash \{0\}=
e^{t\sigma ({\mathcal A})} \text{ for all } t>0.
$$
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
See \cite{CL,Nagel,Renardy, vanN} for a discussion of the
spectral mapping theorems for strongly continuous
semigroups and examples where the spectral mapping property
as in Theorem~\ref{SMTh} fails. Since the
spectral mapping theorem always holds for the point spectrum,
Theorem~\ref{SMTh} implies, in particular, that
$\sigma_{\ess} (e^{t{\mathcal A}})\subseteq {\Bbb T}
=\{|z|=1\}$. A spectral mapping theorem was proved in
\cite{kapsan}
for a nonlinear Schr\" odinger equation with a specific
potential
and in the case $n=1$. Their proof also uses
Theorem~\ref{GGHP} that
is the key to our result. To the best of our knowledge
the work of
Kapitula and Sandstede \cite{kapsan} was the first to
use the
Gearhart-Greiner-Herbst-Pr\" uss Theorem in this context.
It follows that there will be only finitely many
eigenvalues of
$e^{t{\mathcal A}}$ off the unit circle and therefore
general
results on the existence of invariant
manifolds for semilinear equations can be invoked
(see, e.g., \cite{BJ}
and compare also with \cite{JBsmall} and the literature
cited in \cite[p.~4]{LW97}).
The (local) stable manifold is defined as the set of
initial data
whose solutions stay in the prescribed neighborhood and
tend to
$\hat{u}$ exponentially as $t \rightarrow +\infty$.
The unstable
manifold is defined analogously but in backward time.
The center
manifold is complementary to these two and contains
solutions with
neutral decay behavior (although they can decay, they
will not do
so exponentially). In particular, the center manifold
contains all
solutions that stay in the neighborhood in both forward
and backward
time. For details see, for instance, \cite{BJ}.
Concerning the equations under consideration here, we
have the
following main theorem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}\label{ECM} Assuming (H1)-(H4), in a
neighborhood of
the standing wave solution $\hat {u}$ of \eqref{NLS} there are
locally invariant stable, unstable and center manifolds.
Moreover the stable and unstable manifolds are of (equal)
finite
dimension and the center manifold is infinite-dimensional.
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
All of the examples of standing waves considered in \cite{kup}
and \cite{WGpap} satisfy the hypotheses given here and thus
enjoy a local decomposition of the flow by invariant
manifolds. Some of
these waveguide modes are stable, while others are unstable. In
the unstable cases, the above results show that the
instabilities
are controlled by finite-dimensional (mostly, just
one-dimensional)
unstable manifolds. A natural question is whether the
waveguide modes
are stable relative to the flow on the center manifold.
Such a result
was shown for the case of nonlinear Klein-Gordon
equations in
\cite{JBsmall} using an energy argument. Whether such
an argument
will work for nonlinear Schr\" odinger equations is open.
It is more
than of academic interest, as stability on the center
manifold has the
consequence that the center manifold is unique,
see \cite{BJ}, and armed
with such a result, a complete description of the local
flow can be
legitimately claimed. Cases of standing waves in higher
dimensions are
given in \cite{JonesJDE}. Some of these are unstable
and the above
considerations again apply.
We also wish to stress that the spectral mapping theorem
developed here
is not restricted to a single equation. Indeed, the
results formulated
here are easily adaptable to systems of nonlinear
Schr\" odinger
equations. This is particularly important as such
systems arise in,
among other problems, second harmonic generation in
waveguides and
wave-division multiplexing in optical fibers.
%%%%%!!!!!!!!!!!!!!
The case of systems is considered in Section~\ref{Sec4}.
In the next section, we give the basic set-up that
will be used and
formulate the necessary lemmas for proving
Theorem~\ref{SMTh}. The
proofs are given in Section~\ref{Sec3}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic lemmas}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Fix an $a\in{\Bbb R}\backslash\{0\}$ such that the line
$\{\xi =a+i\tau :\tau
\in {\Bbb R}\}$ does not intersect $\sigma ({\mathcal
A})$. To prove Theorem~\ref{SMTh}, one needs to show that
$\sigma (e^{t{\mathcal A}})$ does not intersect the circle
with radius $e^{ta}$ centered at the origin. We will use
the following
abstract result known as the Gearhart-Greiner-Herbst-Pr\"uss
theorem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm} {\rm (}see, e.g., \cite[p.~95]{Nagel}.{\rm )}
\label{GGHP} Let ${\mathcal A}$ be a generator of a strongly
continuous semigroup on a complex Hilbert space.
Assume
$$
\sigma ({\mathcal A})\cap \{\xi =a+i\tau :\tau \in
{\Bbb R}\}=\emptyset, \quad a\in{\Bbb R}\backslash\{0\}.
$$
Then $\sigma (e^{t{\mathcal A}})\cap \{|z|=e^{ta}\}=
\emptyset$ for all $t>0$, if and only if the norm of the
resolvent of
${\mathcal A}$ is bounded along the line $\{\xi =
a+i\tau :\tau \in {\Bbb R}\}$, that is,
\begin{equation}
\sup_{\tau \in {\Bbb R}}\|(a+i\tau
-{\mathcal A})^{-1}\|<\infty.\label{BDRes}\end{equation}
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Therefore, to prove Theorem~\ref{SMTh}, one needs to
show that $\|(a+i\tau -{\mathcal A})^{-1}\|$ remains
bounded as $|\tau|\to\infty$ for the operator ${\mathcal A}$
defined in \eqref{defA}.
We denote $D=-\Delta -\beta$ and recall that
$\beta <0$ by (H3). Moreover, we have
$$\sigma (D)=\sigma
\left(-\Delta\right)-\beta = [-\beta,\infty).$$
We note, that $D^2$ with domain $H^4({\Bbb R}^n)$ is
a self-adjoint operator. Thus, for $\xi =a+i\tau$ with
$\tau \neq 0$ one has $-\xi ^2\notin \sigma (D^2)$.
Moreover, we write
\begin{equation}\begin{split}\label{f4.1} \xi -
{\mathcal A} & =\left(\begin{array}{cc} \xi & L_R\\ -L_I & \xi
\end{array}\right)
= \left(\begin{array}{cc} \xi & D\\ -D & \xi
\end{array}\right) +\left( \begin{array}{cc} 0 & Q_1\\
-Q_2 & 0
\end{array}\right)\\ & =\left(\begin{array}{cc} \xi
& D\\ -D & \xi \end{array}\right) \left[I +\left(
\begin{array}{cc} \xi & D\\ -D & \xi \end{array}\right)^{-1}
\left(\begin{array}{cc} 0 & Q_1\\ -Q_2 &
0\end{array} \right) \right],\end{split}\end{equation}
where, by a direct computation with operator-valued matrices,
\begin{equation}\label{f4.2} \left(\begin{array}{cc} \xi
& D\\ -D & \xi \end{array}\right)^{-1} =\left(
\begin{array}{cc}\xi [\xi^2+D^2]^{-1} & -[\xi^2+D^2]^{-1}D\\
\left[\xi ^2+D^2\right]^{-1}D & \xi
[\xi^2+D^2]^{-1}\end{array}\right) .\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lem} \label{lem2} For $\xi =a+i\tau,$
$a\in{\Bbb R}\backslash\{0\},$ $\tau\in\Bbb R,$ the norm of
the operator \eqref{f4.2} remains bounded as $|\tau |\to
\infty$.
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The elementary proof of this lemma is given in the next
section.
Next, we denote
\begin{equation}\label{f5.1} T(\xi)=\left(\begin{array}{cc}
\xi & D\\ -D & \xi \end{array}\right) ^{-1}
\left(\begin{array}{cc} 0 & Q_1 \\-Q_2 & 0\end{array}\right)
=\left[\begin{array}{cc} [\xi^2+D^2]^{-1}DQ_2 & \xi
[\xi^2+D^2]^{-1}Q_1\\
-\xi [\xi^2+D^2]^{-1}Q_2 & [\xi^2+D^2]^{-1}DQ_1\end{array}
\right].
\end{equation}
The main step in the proof of Theorem~\ref{SMTh} is
contained in the
next two lemmas. They imply that the
norm of the operator $(I+T(\xi))^{-1}$ is bounded as
$|\tau|\to\infty$.
Assume $Q$ is a real-valued potential
exponentially decaying at infinity and let $\xi =a+i\tau $.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lem}\label{lem1}
\begin{enumerate}
\item[(a)] For $n=1,$ $\|Q[\xi^2+D^2]^{-1}D\| \to 0
$ as $|\tau | \to \infty ;$
\item[(b)] For $n \geq 1,$ $\|\xi [\xi^2+D^2]^{-1}
Q\| \to 0 $ as $|\tau | \to \infty $.
\end{enumerate}
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lem}\label{lem1prime}
For each $n\ge2$ there exists a $\tau_0>0$ such that
for $|\tau |\geq \tau_0$ the operator
$ I + [\xi^2+D^2]^{-1}DQ $ has a bounded inverse. Moreover,
\begin{equation}\label{BDIn}
\sup_{|\tau|\geq \tau_0} \|[I+[\xi^2+D^2]^{-1}DQ ]^{-1}\|
< \infty.
\end{equation}
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The proofs of Lemmas~\ref{lem1} and \ref{lem1prime} are
given in the next
section. We proceed finishing the proof of Theorem~\ref{SMTh}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent {\it Proof of Theorem~\ref{SMTh}.}
In the case $n=1,$
by passing to the adjoint operator of
$\xi[\xi^2+D^2]^{-1}Q_1$ and $[\xi^2+D^2]^{-1}DQ_2$,
parts (a) and (b) of Lemma~\ref{lem1} imply that the norm
of each of the
four block-operators in the right-hand side of \eqref{f5.1}
is
strictly less than 1 for $|\tau|$ sufficiently large.
Thus, $\|T(\xi)\|<1$. By
\eqref{f4.1}, one infers
$$\|(a+i\tau -{\mathcal
A})^{-1}\|=\left\|(I+T(\xi))^{-1}\left(\begin{array}{cc}
\xi & D\\ -D & \xi \end{array}\right)^{-1}\right\| \leq
\frac{1}{1-\|T(\xi)\|}\left\| \left(\begin{array}{cc}
\xi &D\\
-D & \xi \end{array}\right)^{-1}\right\|.$$
Using Lemma~\ref{lem2}, one obtains \eqref{BDRes} and hence
Theorem~\ref{GGHP} implies the result.
For $n \geq 2$, we write $I+T(\xi)=I+M(\xi)+N(\xi)$, where
\begin{equation*}\begin{split}
M(\xi)&= \left[\begin{array}{cc}[\xi^2+D^2]^{-1}DQ_2 & 0\\
0 & [\xi^2+D^2]^{-1}DQ_1 \end{array} \right] , \\
N(\xi)&= \left[\begin{array}{cc}0
& \xi [\xi^2+D^2]^{-1}Q_1 \\
\xi [\xi^2+D^2]^{-1}Q_2 & 0 \end{array} \right].
\end{split}\end{equation*}
By part (b) of Lemma~\ref{lem1}, one has
$\|N(\xi)\| \to 0 $ as $|\tau|
\to \infty$.
Lemma~\ref{lem1prime} yields that $I+M(\xi)$ is invertible
for $|\tau| \geq \tau_0,$
with $K:=\sup_{|\tau| \geq \tau_0} \|(I+M(\xi))^{-1}\|
\leq \infty$.
Choosing $\tau'_0 \geq \tau_0$ such that $K\|N(\xi)\| \leq 1$
for $|\tau| \geq \tau'_0,$ the norm of the operator
$ (I+T(\xi))^{-1}=
[I+(I+M(\xi))^{-1}N(\xi)]^{-1}(I+M(\xi))^{-1} $ is
bounded for $|\tau| \geq \tau'_0$ and, as before,
Theorem~\ref{GGHP}
implies the result. \hfill $\square$\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proofs}\label{Sec3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we give the proofs of Lemmas~\ref{lem2} --
\ref{lem1prime}. The proof of Lemma~\ref{lem1} is
based on a direct estimate of the trace norms.
We will give two proofs of Lemma~\ref{lem1prime}.
The first proof is applicable to all $n \geq 2$
and uses estimates for the norm of the resolvent on
weighted spaces of
$L^2$-functions. The second proof
works for the cases $n=2$ and $n=3$ and is based on
explicit estimates for the integral kernel of the
resolvent of the Laplacian.
The main tool in the proof of parts (a) and (b) of
Lemma~\ref{lem1} is the following well-known
result.~Denote by
${\mathcal J}_q$ the set of operators
$A\in {\mathcal L}(L^2({\Bbb R}^n))$, $n\ge 1$
such that $\| A\|_{{\mathcal
J}_q}=(\tr(|A|^q))^{1/q}<\infty$, $q\geq 1,$
where $\tr(\cdot)$ denotes the trace of operators in
$L^2({\Bbb R}^n).$ We recall that $\|A\|\leq
\| A\|_{{\mathcal J}_q}$ for all $q\geq 1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm} {\rm (}see, e.g., \cite[Theorem XI.20]{RS}.{\rm )}
\label{ReedSim}
Suppose $2\leq q<\infty$ and let
$f,g\in L^q({\Bbb R}^n).$ Then $f(x)g(-i\nabla )\in {\mathcal
J}_q$ and
$$\| f(x)g(-i\nabla )\|_{{\mathcal J}_q}\leq
(2\pi )^{-n/2} \| f\|_{L^q}\| g\|_{L^q}.$$
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We will apply Theorem~\ref{ReedSim}
to the exponentially decaying $Q=f\in L^q$, $q>1$ and an
appropriate choice of $g$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\noindent{\it Proof of Lemma~\ref{lem1}.}
First, let $g(x)=(|x|^2-\beta )(\xi^2+(|x|^2-\beta )^2)^{-1}$.
Then, for $n\ge 1,$ one has $g(-i\nabla )=[\xi^2+
(-\Delta-\beta)^2]^{-1}(-\Delta-\beta)
=g(-i d/dx)=[\xi^2+D^2]^{-1}D$. For
$r=|x|$ one infers,
\begin{equation*}\begin{split} \|g\|^q_{L^q}&=
\int^\infty_0
r^{n-1}(r^2-\beta )^q [(a^2-\tau ^2+(r^2-\beta )^2)^2
+4a^2\tau^2]^{-q/2}dr\\
&=(2|a\tau )|^{-q}\int^\infty_0r^{n-1}(r^2-\beta )^q
\left[\left(\frac{(r^2-\beta )^2-(\tau^2-a^2)}{2|a\tau
|}\right)^2+1\right]^{-q/2}dr.\end{split}\end{equation*}
The change of variable $y=((r^2-\beta )^2-(\tau^2-a^2))
/(2|a\tau |)$ shows that
$$\|g\|^q_{L^q}=k(\tau )
\int^\infty_{\frac{\beta^2}{2|a\tau |}
-\tau '}(y^2+1)^{-q/2} \left( \frac{1}{\tau
'}y+1\right)^{\frac{q-1}{2}}\left[\left(\frac{1}{\tau '}y+
1\right)^{\frac{1}{2}}+\frac{\beta } {(\tau
^2-a^2)^{\frac{1}{2}}}\right]^{\frac{n-2}{2}}dy,$$
where
$$k(\tau )=\frac{1}{4} (2|a\tau |)^{1-q} (\tau ^2-
a^2)^{\frac{q-1}{2}+\frac{n-2}{4}}= O(|\tau
|^{\frac{n-2}{2}}) \text{ as } |\tau |\to \infty$$
and
$$\tau'=
(\tau^2-a^2)/(2|a\tau |)=O(|\tau|) \text{ as } |\tau
|\to \infty.$$
We recall, that $\beta <0$ by (H3). Thus,
\begin{equation}\label{esti}
\|g\|^q_{L^q}\leq c|\tau |^{\frac{n-2}{2}}I(-\tau ',\infty),
\end{equation}
denoting
$$I(a,b)=\int^b_a(y^2+1)^{-q/2}\left(\frac{1}{\tau '}y+
1\right)^{\frac{q-1}{2}+\frac{n-2}{4}}dy.$$
Next, one observes that $|\tau |^{\frac{n-2}{2}}I(\tau ',
\infty)\to 0$ as $|\tau |\to \infty$ for
$q>\max\left\{1,\frac{n}{2}\right\}$. Indeed, for
$y\geq \tau'$ one has
$$
|\tau |^{\frac{n-2}{2}}I
(\tau ',\infty )\leq |\tau |^{\frac{n-2}{2}}\int_{\tau '}
^{\infty }(y^2)^{-q/2}\left(2\frac{y}{\tau '}
\right)^{\frac{2q+n-4}{4}}dy
=c|\tau |^{\frac{n-2}{2}}{\tau'}^{1-q}\to 0
\text{ as } |\tau |\to \infty.
$$
To prove part (a) of Lemma~\ref{lem1} for $n=1$ we
remark that
$|\tau |^{\frac{n-2}{2}}I(-\tau ',\tau ')\to 0$ as
$|\tau |\to \infty$.
We stress, that this assertion does not hold for
$n \geq 2$; that is why one needs Lemma~\ref{lem1prime}.
Indeed, assuming $n=1$,
for $y\leq \tau '$ one concludes
$$|\tau |^{-\frac{1}{2}}I(-\tau ',\tau )\leq
2^{(2q-3)/4}|\tau
|^{-\frac{1}{2}}\int^\infty_{-\infty}
(y^2+1)^{-q/2}dy\to 0 \text{ as } |\tau |\to \infty.$$
Therefore, for $n=1$ one infers $\|g\|_{L^q}\to 0$ as
$|\tau |\to \infty$ and, using Theorem~\ref{ReedSim},
$$\|Q[\xi^2+D^2]^{-1}D\|\leq \|Q(x)g(-i\nabla )
\|_{{\mathcal J}_q}\leq c\|Q\|_{L^q}\|g\|_{L^q}
\to 0 \text{ as } |\tau|\to \infty.$$
Thus, part (a) of Lemma~\ref{lem1} is proved.
To prove part (b) of Lemma~\ref{lem1}, let
$g(x)=\xi (\xi^2+(|x|^2-\beta )^2)^{-1}.$
Then, for $n\ge 1$, one obtains $g(-i\nabla )=\xi
[\xi^2+(-\Delta-\beta)^2]^{-1}$.
For $r=|x|$ one concludes as above,
$$\|g\|^q_{L^q}=
l(\tau )\int^\infty_{\frac{\beta^2}{2|a\tau |}
-\tau '} (y^2+1)^{-q/2}\left(\frac{1}{\tau
'}y+1\right) ^{-\frac{1}{2}} \left[ \left(\frac{1}{\tau
'}y+1\right)^{\frac{1}{2}}+\frac{\beta}{(\tau^2-
q^2)^{\frac{1}{2}}}\right]^{\frac{n-2}{2}}dy,$$
where
$$l(\tau )=\frac{1}{4}(2|a\tau |)^{1-q}(\tau ^2-
a^2)^{\frac{n-2}{4}}=O(|\tau|^{\frac{n-2q}{2}})
\text{ as } |\tau |\to \infty.$$
Since $\beta <0$ one has
$$\|g\|^q_{L^q}\leq c|\tau |^{\frac{n-2q}{2}}J(-\tau ',
\infty ),$$
denoting
$$J(a,b)=\int^b_a(y^2+1)^{-q/2}\left(\frac{1}{\tau '}y+
1\right)^{\frac{n-4}{4}}dy.$$
Next, fix $q>\max \{1,(n+2)/2\}$. We claim that
$|\tau |^{\frac{n-
2q}{2}}J(\tau ', \infty )\to 0$ as $|\tau |\to
\infty$. Indeed, for $y\geq \tau '$ one concludes,
$$
|\tau |^{\frac{n-2q}{2}}J(\tau ',\infty )
\leq |\tau
|^{\frac{n-2q}{2}}\int^\infty_{\tau '} (y^2)^{-q/2}
\left(2\frac{y}{\tau '}\right)^{\frac{n-4}{4}}dy
=c|\tau |^{\frac{2+n-4q}{2}}\to 0 \text{ as }
|\tau |\to \infty.
$$
On the other hand, for $y\leq \tau '$ one has
$$|\tau |^{\frac{n-2q}{2}}J(-\tau ',\tau ')
\leq 2^{\frac{n-4}{4}}
|\tau |^{\frac{n-2q}{2}}\int^\infty_{-\infty}
(y^2+1)^{-q/2} dy\to 0 \text{ as } |\tau |\to \infty.$$
Therefore, $\| g\|_{L^q}\to 0$ as $|\tau |\to \infty$ and,
using Theorem~\ref{ReedSim}, one obtains for $n\ge 1,$
$$\|Q\xi [\xi^2+(-\Delta-\beta)^2]^{-1}\|\leq \|Q (x)
g(-i\nabla )\|_{{\mathcal J}_q}\leq c\|Q\|_{L^q}\|g\|_{L^q}
\to 0 \text{ as } |\tau
|\to \infty.$$
Thus, part (b) of Lemma~\ref{lem1} is proved. \hfill
$\square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The proof of Lemma~\ref{lem1prime} is based on the
following proposition. Denote
\begin{equation}\label{DefQ12}
|Q|^{1/2}(x)=|Q(x)|^{1/2},\quad
Q^{1/2}(x)= |Q|^{1/2}(x)\sgn [Q(x)],
\end{equation}
so that
$Q=Q^{1/2} |Q^{1/2}|$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}\label{maintech}
Assume $Q$ is a real-valued potential
exponentially decaying at infinity. Then for $n\ge2$
and $\im (\omega) > 0$ one infers
\begin{equation}\label{T9.3}
\||Q|^{1/2}(-\Delta-\omega^2)^{-1}Q^{1/2}\|_{{\mathcal L}
(L^2({\Bbb R}^n))}
\rightarrow 0 \
\text{ as } \ |\re (\omega^2)| \rightarrow \infty.
\end{equation}
\end{prop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We postpone the proof of the proposition and proceed
with the proof of Lemma~\ref{lem1prime}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\noindent {\em Proof of Lemma~\ref{lem1prime}.} Recall
the following elementary fact:
If $A$ and $B$ are bounded operators, then $I+AB$ is
invertible provided $I+BA$ is invertible; moreover,
$$ (I+AB)^{-1}= I - A (I+BA)^{-1} B. $$
Thus, using $Q=Q^{1/2}|Q|^{1/2}$ and letting
$ A= [\xi^2+D^2]^{-1} D Q^{1/2}$ and $B=|Q|^{1/2},$
the desired inequality ~\eqref{BDIn} is implied by
the following claim
\begin{equation}
\label{CLA}
\| |Q|^{1/2}[\xi^2+D^2]^{-1} D Q^{1/2} \| \to 0
\text{ as } |\tau| \to \infty .
\end{equation}
Indeed, if ~\eqref{CLA} holds, then, for some
$\tau_0 > 0$, one has $\|BA\| < 1$ and
$\sup_{|\tau| \geq \tau_0} \| (I+BA)^{-1} \| < \infty$.
To prove $\sup_{|\tau| \geq \tau_0} \| A \| < \infty$
we use the identity
\begin{equation}
\label{Decom}
[\xi^2+D^2]^{-1} D=\frac{1}{2} [D+i \xi]^{-1}
+ \frac{1}{2} [D-i \xi]^{-1}.
\end{equation}
Since $D$ is self-adjoint,
$$\|[D \pm i \xi]^{-1}\| = 1/|\im (\pm i \xi)| =
1/|a| < \infty.$$
This and $\|B\| = \||Q|^{1/2}\|_\infty < \infty$
implies $\sup_{|\tau| \geq \tau_0} \| (I+AB)^{-1} \|
< \infty$, which is the desired relation~\eqref{BDIn}.
To prove the claim \eqref{CLA} one recalls that
$D=-\Delta-\beta $ and
similarly to \eqref{Decom} one obtains
\begin{equation}\label{16.1}
|Q|^{1/2}[\xi^2+D^2]^{-1} D Q^{1/2}=
\frac{1}{2}|Q|^{1/2}[-\Delta-\omega_1^2]^{-1}Q^{1/2}+
\frac{1}{2}|Q|^{1/2}[-\Delta-\omega_2^2]^{-1}Q^{1/2}.
\end{equation}
Here $\omega_1$ and $\omega_2$ are chosen such that
$\im (\omega_1)>0$,
$\im (\omega_2)>0$ and
$\omega_1^2=\beta -i \xi$, $\omega_2^2=\beta +i \xi$.
Recall that $\xi=a+i\tau$. Thus, if $|\tau|\to\infty$ then
$|\re (\omega_j^2)|\to\infty$ for $j=1,2$.
An application of Proposition~\ref{maintech} to each summand
in the right-hand side of
\eqref{16.1} then proves the claim \eqref{CLA}.
\hfill $\square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We will give two proofs of Proposition~\ref{maintech}.
The first proof is based on an estimate
for the norm of the resolvent for the Laplacian acting
between weighted $L^2$-spaces.
This estimate is given in Lemma~\ref{Thm9} below. In fact,
Lemma~\ref{Thm9} is just
a minor refinement of Lemma~XIII.8.5 in \cite{RS4}. Results
of this type go back to Agmon~\cite{Ag75}, Ikebe and
Saito~\cite{IS72}, and others (cf. the discussion in
\cite[p.~345--347]{RS4}). Much more
sophisticated results of this type can be found in the
work by
Jensen~\cite{AJ} and the bibliography therein.
For $n \geq 2$ and $s>1/2$ let $\rho_s(x)=(1+|x|^2)^{s/2}$,
for $x \in {\Bbb R}^n$, and
consider the weighted $L^2$-spaces
$$
L^2_s({\Bbb R}^n)=\{ f: \|f\|_s:=
\|\rho_s f\|_{L^2({\Bbb R}^n)} < \infty \}
$$
and
$$
L^2_{-s}({\Bbb R}^n)=\{ f: \|f\|_{-s}:=
\|\rho_s^{-1} f\|_{L^2({\Bbb R}^n)} < \infty \}.
$$
Note that $L^2_s({\Bbb R}^n)
\hookrightarrow L^2({\Bbb R}^n) \hookrightarrow
L^2_{-s}({\Bbb R}^n)$. Also, let
${\mathcal S}({\Bbb R}^n)$ denote the Schwartz class of
rapidly decaying functions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lem}\label{Thm9} There exists a constant $d=d(n,s)$
depending only on $n$ and $s$,
such that for all $\lambda \in {\Bbb C}$ with
$\im (\lambda) \neq 0$ and
$| \re (\lambda) | \geq 1, $
and all $\varphi \in {\mathcal S}({\Bbb R}^n)$ the following
estimate holds
\begin{equation}\label{T9.1}
\|\varphi\|_{-s} \leq d |\re (\lambda)|^{-1/2}
\|(-\Delta-\lambda) \varphi\|_s .
\end{equation}
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof} For completeness, we briefly sketch a
modification of the proof of Lemma XIII.8.5
in \cite{RS4} to prove inequality \eqref{T9.1}.
An elementary calculation shows that for each
$ \lambda \in {\Bbb C}$ with $\im (\lambda) \neq 0$ the
expression
$$ \gamma ^{-2} := \inf_{y \in {\Bbb R}} \left[ |y^2-
\lambda|^2 + |y|^2 \right]=
\inf_{y \geq 0} \left[ y^2+(1-2 \re (\lambda))y +
(\re (\lambda))^2+(\im (\lambda))^2 \right]$$
is equal to $\re (\lambda)+(\im (\lambda))^2-1/4$ for
$\re (\lambda)>1/2$ and to $|\lambda|^2$
for $\re (\lambda) \leq 1/2$. Therefore, $\gamma
\leq 2 |\re (\lambda)|^{-1/2}$ for
$|\re (\lambda)| \geq 1$.
For a positive $\alpha$ to be selected below, let
$\rho_{1,\alpha}(x)=(1+\alpha |x|^2)^{1/2}$. Arguing
as in the proof of
Lemma XIII.8.5 in \cite{RS4} (see the corresponding
equations (61)--(62) in \cite{RS4}), one
infers $$ \|\rho_{1,\alpha}^{-s} \varphi \|_{L^2} \leq
\gamma \|\rho_{1,\alpha}^{-s} (-\Delta - \lambda)
\varphi \|_{L^2}+
\gamma (2s \alpha^{1/2}+1) \sum_{j=1}^n
\|\rho_{1,\alpha}^{-s} \partial_j \varphi \|_{L^2}
+ \gamma (d \alpha
+ n \alpha^{1/2} s) \|\rho_{1,\alpha}^{-s}
\varphi \|_{L^2}, $$
where $d$ depends only on $n$ and $s$. Next, pick
$\alpha < 1$
such that
$ d \alpha +n \alpha^{1/2} s < 1/4$. Note that $\alpha$
depends only on $n$ and $s$.
Moreover, $\gamma (d \alpha +n \alpha^{1/2} s) < 1/2$
uniformly for
$|\re (\lambda) | \geq 1,$ since $\gamma \leq 2
|\re (\lambda) |^{-1/2}$. Since $s>1/2$,
one obtains
$$ \frac{1}{2} \| \rho_{1,\alpha}^{-s} \varphi
\|_{L^2} \leq
\gamma (2s+1) (\|\rho_{1,\alpha}^{-s} (-\Delta - \lambda)
\varphi \|_{L^2}+
\sum_{j=1}^n \|\rho_{1,\alpha}^{-s} \partial_j
\varphi \|_{L^2}).$$
This and the inequality $\rho_s^{-s} \leq \rho_{1,\alpha}^{-s}
\leq \alpha^{-s/2} \rho_s^{-s}$
show that
$$
\| \varphi \|_{{-s}}=\| \rho_s^{-s} \varphi \|_{L^2} \leq
\| \rho_{1,\alpha}^{-s} \varphi \|_{L^2} \leq
\gamma (2s+1) \alpha^{-s/2} (\| (-\Delta - \lambda)
\varphi \|_{{-s}}+
\sum_{j=1}^n \| \partial_j \varphi \|_{{-s}}).
$$
Now \eqref{T9.1} follows from the inequalities
$\|\cdot\|_{{-s}} \leq \|\cdot\|_{{s}}$ and
$\|\partial_j \varphi \|_{{-s}} \leq C \|(-\Delta - \lambda)
\varphi \|_{{s}},$
where $C$ is an absolute constant (see Lemma XIII.8.4
in \cite{RS4}).
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If $\im (\lambda) \neq 0$, then the resolvent
$ (-\Delta-\lambda)^{-1}$ is a bounded operator on
$L^2({\Bbb R}^n)$. Consider its restriction
$R_{s,-s}(\lambda):=(-\Delta-\lambda)^{-1}
\big|_{L^2_s({\Bbb R}^n)}$
as an operator from $ L^2_s({\Bbb R}^n)$ to
$ L^2_{-s}({\Bbb R}^n)$. Inequality \eqref{T9.1}
shows that $R_{s,-s}(\lambda)$ is a bounded operator from
$ L^2_s({\Bbb R}^n)$ to $ L^2_{-s}({\Bbb R}^n)$ and that
\begin{equation}\label{T9.2}
\|R_{s,-s}(\lambda)\|_{{\mathcal L }(L^2_s,L^2_{-s})}
\leq d \ | \re (\lambda) |^{-1/2} \rightarrow 0
\text{ as } |\re (\lambda)| \rightarrow \infty.
\end{equation}
This relation with $\lambda=\omega^2$ will be used in
the first
proof of Proposition~\ref{maintech}.\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent {\it First proof of Proposition~\ref{maintech}.}
Define the multiplication operators $(M_s f ) (x)=
\rho_s (x) f(x)$ and $(M_{-s} f ) (x)= \rho_s^{-1} (x) f(x)$.
If $Q$ is a real-valued potential exponentially decaying at
infinity, then using notation \eqref{DefQ12},
one observes that the operators $M_s Q^{1/2}$ and $|Q|^{1/2}
M_{-s}^{-1} = |Q|^{1/2} M_s$
are bounded operators on $L^2({\Bbb R}^n)$. On the
other hand,
$M_s : L^2_s({\Bbb R}^n) \rightarrow L^2({\Bbb R}^n) $ and
$M_{-s} : L^2_{-s}({\Bbb R}^n) \rightarrow L^2({\Bbb R}^n) $
are isometric isomorphisms.
This fact and the identity
$$|Q|^{1/2}(-\Delta-\omega^2)^{-1}Q^{1/2} f =
\left[|Q|^{1/2} M_{-s}^{-1} \right]
\left[ M_{-s}R_{s,-s}(\omega^2)M_{s}^{-1} \right]
\left[M_s Q^{1/2} \right] f $$
for $f \in L^2({\Bbb R}^n)$, yields the following estimate
\begin{eqnarray*}
&& \||Q|^{1/2}(-\Delta-\omega^2)^{-1}Q^{1/2}\|_{{\mathcal L}
(L^2)} \\
&& \quad \leq \||Q|^{1/2} M_{-s}^{-1} \|_{{\mathcal L} (L^2)}
\| M_{-s}R_{s,-s}(\omega^2)M_{s}^{-1} \|_{{\mathcal L} (L^2)}
\|M_s Q^{1/2} \|_{{\mathcal L} (L^2)} \\
&& \quad \leq \||Q|^{1/2} M_{-s}^{-1} \|_{{\mathcal L} (L^2)}
\| R_{s,-s} (\omega^2) \|_{{\mathcal L} (L^2_s,L^2_{-s})}
\|M_s Q^{1/2} \|_{{\mathcal L} (L^2)}.
\end{eqnarray*}
Using relation \eqref{T9.2} for $\lambda=\omega^2$ one concludes
that \eqref{T9.3} holds.\hfill
$\square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The main tool in the second proof of
Proposition~\ref{maintech} in the case $n=3$ is
Theorem~I.23 in \cite{Simon}, inspired by previous results of
Zemach and Klein \cite{ZK58}. Assume $n=3$ and
suppose that
the potential $Q$ satisfies the following (Rollnik) condition
(see \cite[p.~3] {Simon})
\begin{equation}\label{RolCon}
\int\int\limits_{{\Bbb R}^6} \frac{|Q(x)|
|Q(y)|}{|x-y|^2} dx dy
< \infty.\end{equation}
Note that ~\eqref{RolCon} trivially holds for exponentially
decaying continuous $Q$.
Consider on
$L^2({\Bbb R}^3)$ the operator
$ {\mathcal R}_{\omega} $ with integral kernel
\begin{equation}\label{defR}
R_\omega (x,y)=|Q|^{1/2}(x) \frac{e^{i\omega |x-y|}}
{4\pi |x-y|} Q^{1/2}(y),
\quad x,y \in {\Bbb R}^3, \, x \neq y, \, \im (\omega) \geq 0.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm}\label{Th7} {\rm (}\cite[Theorem I.23]{Simon}.{\rm )}
Assume~\eqref{RolCon}. Then
${\mathcal R}_k,$ $ k\in {\Bbb R}$, is a Hilbert-Schmidt
operator in
$L^2({\Bbb R}^3)$ with $\lim_{|k| \to \infty}
\|{\mathcal R}_k \| = 0$,
$k \in {\Bbb R}$.
Moreover,
\begin{equation}
\label{trform}
\tr ({\mathcal R}^*_k{\mathcal R}_k{\mathcal R}^*_k
{\mathcal R}_k)\to 0 \text{ as }
|k| \to \infty, \quad k \in {\Bbb R}.
\end{equation}
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Relation~\eqref{trform} follows from a direct
calculation of
$\tr ({\mathcal R}^*_k{\mathcal R}_k{\mathcal R}^*_k
{\mathcal R}_k)$,
resulting in
\[\int\int\int\int_{{\Bbb R}^{12}} e^{-i k (|x-y|-|y-z|
+|z-w|-|w-x|)}
\frac{ |Q(x)| |Q(y)| |Q(w)|
|Q(z)|}{(4\pi)^4|x-y||y-z||z-w||w-x|}
dx dy dz dw,\]
see \cite[p.~24]{Simon} (and compare also with \eqref{TRS}
below),
and from the Riemann-Lebesgue lemma. The first assertion
in Theorem~\ref{Th7} follows
from \eqref{trform} and the estimate $\|{\mathcal R}_k
\|^4 = \|{\mathcal R}^*_k{\mathcal R}_k \|^2 \le
\|{\mathcal R}^*_k{\mathcal R}_k \|^2_{{\mathcal J}_2}
= \tr ({\mathcal R}^*_k{\mathcal R}_k{\mathcal R}^*_k
{\mathcal R}_k)$.
In the case $n=2$ one
replaces the integral kernel
${e^{i\omega |x-y|}}/{(4\pi |x-y|)}$
in \eqref{defR} by the Hankel function
$(i/4)H^{(1)}_0(\omega |x-y|)$
of order zero and first kind (cf.~\cite{FG}). We use the
following result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thm} {\rm (}\cite{BDO}, \cite[p.~1449]{Cheney}.{\rm )}
\label{Th8} Assume $Q \in L^2({\Bbb R}^2)
\cap L^{4/3}({\Bbb R}^2)$.
Consider on
$L^2({\Bbb R}^2)$ the operator
$ {\mathcal K}_\omega $ with integral kernel
\begin{equation}\label{defK}
K_{\omega}(x,y)=|Q|^{1/2}(x)
\frac{i}{4}H^{(1)}_0(\omega|x-y|) Q^{1/2}(y),
\quad x,y \in {\Bbb R}^2, \, x \neq y,
\, \im (\omega) \geq 0, \, \omega\neq 0.
\end{equation}
Then
\begin{equation}
\label{2dim}
\|{\mathcal K}_\omega \|_{{\mathcal J}_2}
\leq c\ |\omega|^{-1/2}
\|Q\|_{L^{4/3}({\Bbb R}^2)},
\quad \im (\omega) \geq 0, \, \omega \neq 0.
\end{equation}
\end{thm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Note that $Q \in L^2({\Bbb R}^2) \cap L^{4/3}({\Bbb R}^2)$
for exponentially decaying continuous potentials $Q$.
To establish the connection with the discussion in the
current paper, we recall that
$(-\Delta - \omega^2)^{-1} $ for $\im (\omega) > 0$ is
an integral
operator with kernel ${e^{i \omega |x-y|}}/{(4\pi |x-y|)}$
in the case $n=3$ and with kernel
$(i/4)H^{(1)}_0(\omega |x-y|)$ in the case $n=2$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\noindent {\it Second proof of Proposition~\ref{maintech}.}
In the case $n=2$ we use Theorem~\ref{Th8} for the operator
${\mathcal K}_{\omega}:= |Q|^{1/2}
(-\Delta-\omega^2)^{-1}Q^{1/2}$ in $L^2({\Bbb R}^2)$
to conclude that
$\||Q|^{1/2}(-\Delta-\omega^2)^{-1}Q^{1/2}
\| \to 0$ as $|\re (\omega^2)| \to \infty.$ This proves
Proposition~\ref{maintech} for $n=2.$
In the case $n=3$ one needs one more calculation.
If $\omega=k + i m$ with $m>0$, $k \in {\Bbb R}$,
then the integral kernel $S_{\omega} (x,y) $
of the Hilbert-Schmidt operator
${\mathcal S}_{\omega}:= |Q|^{1/2}
(-\Delta-\omega^2)^{-1}Q^{1/2}$ in $L^2({\Bbb R}^3)$
is given by the formula
$$S_{\omega} (x,y) = R_{k} (x,y) e^{-m |x-y|},
\quad x,y\in{\mathbb R}^3,\, x\neq y,$$
where $R_{k} (x,y)$ is defined in \eqref{defR} (with
$\omega =k,$ $\im(\omega)=0$ in \eqref{defR}).
A direct calculation using $m >0$ then shows
\begin{align}
\tr({\mathcal S}_{\omega}^*{\mathcal S}_{\omega}
{\mathcal S}_{\omega}^*{\mathcal S}_{\omega}) &=
\int\int\int\int_{{\Bbb R}^{12}}
e^{-i k (|x-y|-|y-z|+|z-w|-|w-x|)}
\frac{ |Q(x)| |Q(y)| |Q(w)| |Q(z)|}{|x-y||y-z||z-w||w-x|}
\times \nonumber\\
& \quad \times (4\pi)^{-4}
e^{- m (|x-y|+|y-z|+|z-w|+|w-x|)} dx dy dz dw
\leq \tr ({\mathcal R}^*_{k}{\mathcal R}_{k}
{\mathcal R}^*_{k}{\mathcal R}_{k}).
\label{TRS}\end{align}
But then relation \eqref{trform} yields
$\|{\mathcal S}_{\omega}\|^4 \leq
\tr ({\mathcal R}^*_{k}{\mathcal R}_{k}
{\mathcal R}^*_{k}{\mathcal R}_{k})
\to 0$ as $|k|\to\infty.$ Similarly,
$\|{\mathcal S}_{\omega}\|^4 \leq
\tr({\mathcal S}_{\omega}^*{\mathcal S}_{\omega}
{\mathcal S}_{\omega}^*{\mathcal S}_{\omega})
\to 0$ as $m\to +\infty$ by the dominated convergence
theorem and hence $\|{\mathcal S}_{\omega}\|\to 0$
as $|\re (\omega^2)|=|k^2-m^2| \to \infty.$
Thus, $\||Q|^{1/2}[-\Delta-\omega^2]^{-1}Q^{1/2}
\| \to 0$ as $|\re (\omega^2)| \to \infty$, and hence
Proposition~\ref{maintech} is proved. \hfill $\square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\noindent {\it Proof of Lemma~\ref{lem2}.} We recall that
$\xi=a+i \tau,$ $a\in{\Bbb R}\backslash\{0\},$ $\tau\in\Bbb R$.
Since $D^2$ is self-adjoint,
$$\| \xi (\xi^2+D^2)^{-1}\| \leq \frac{|\xi |}
{|\im (\xi ^2)|}=\frac{(a^2+\tau ^2)^{1/2}}{2|a\tau |}
\leq c
\text{ uniformly with respect to } |\tau|.$$
On the other hand, $[\xi^2+D^2]^{-1}D=h(D)$ with $h$
defined as $h(p)=p(\xi^2+p^2)^{-1}$. Therefore,
\begin{equation}\label{f4.3} \| (\xi^2+D^2)^{-1}D\|=
\max_{p\in \sigma (D)}|h(p)|=\max_{p\geq
-\beta}|p|[(a^2-\tau^2+p^2)^2+4a^2
\tau^2]^{-\frac{1}{2}}.\end{equation}
To verify that the right-hand side of \eqref{f4.3} is
bounded as
$|\tau |\to \infty$, one considers two cases. First, let
$p^2\geq 2\tau^2$. Since $p^2-\tau^2\geq \frac{1}{2}p^2$,
$$\max_{|p|\geq \sqrt{2}|\tau |}|p|[(a^2-
\tau^2+p^2)^2+4a^2\tau ^2]^{-\frac{1}{2}}\leq
\max_{p\in{\Bbb R}}|p|\left[a^2+
\frac{p^2}{2}\right]^{-\frac{1}{2}}
\leq \sqrt{2}.$$
Second, for $p^2\leq 2\tau^2,$ one has
$$\max_{|p|\leq \sqrt{2}|\tau |}
|p|\left[(a^2-\tau^2+p^2)^2+4a^2\tau^2\right]^{-\frac{1}{2}}
\leq\max_{|p|\leq \sqrt{2}|\tau
|}\frac{1}{2|a|}\frac{|p|}{|\tau |} \leq
\frac{1}{\sqrt{2}|a|}.$$
This completes the proof of Lemma~\ref{lem2}.\hfill $\square$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Systems of nonlinear Schr\"{o}dinger
equations}\label{Sec4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Systems of nonlinear Schr\" odinger equations arise in many
applications of nonlinear optics. From the perspective of
the current
paper, the fundamental issue is the same as for a single
equation, namely
the existence and stability of standing wave solutions. A
particularly
interesting example of multiple pulses was studied recently
by Yew
\cite{yew1, yew2, yew3}. The problem of second harmonic
generation occurs
in a slab waveguide with a quadratically nonlinear response.
Yew showed
that multiple pulses could be generated from the base
pulse through a
resonant homoclinic bifurcation. The details of this
process are not
pertinent for the current work, only the outcome, which
is the presence of
multiple pulses that Yew has shown to be unstable due to
real, positive
eigenvalues. It can be shown easily that the essential
spectrum for these
pulses remains on the imaginary axis and hence the spectral
configuration
for the linearization at the multiple pulses is analogous
to the cases
considered above for scalar equations. Since the
technology of this
paper can be applied to this system,
Theorem~\ref{ECM} holds in
this case. In the following we show how the above
considerations can be
adapted to this case of systems.
The equations governing the second harmonic generation
problem are
a system of coupled nonlinear Schr\"{o}dinger equations
of the form
\begin{align}\label{sysEq}
i\frac{\partial w}{\partial t}&+\Delta w-
\theta w+\overline{w}v = 0, \\
i\sigma\frac{\partial v}{\partial t}&+\Delta v-\alpha v
+\frac12 w^2=0,
\notag\end{align}
where $w=w(t,x)$ and $v=v(t,x)$, $x\in{\Bbb R}^n$,
$n\ge 1$,
are complex valued functions, and
$\alpha$, $\sigma$, and $\theta$ are positive parameters.
For the
one-dimensional case
$n=1$ the questions of the existence of standing
waves for \eqref{sysEq} and, as mentioned above, the
structure of the
spectrum of the linearization around the
standing waves are well-understood, see \cite{yew2, yew3}.
The linearization at a standing wave
$\hat{u}=(\varphi,\psi)$ is given by
the operator $\displaystyle{
{\mathcal A}=\left[\begin{array}{cc}0 & -L_R\\ L_I &
0\end{array}\right]}$ that has the same structure as
in \eqref{defA},
but $L_R$ and $L_I$ are now $2\times 2$ operator
matrices defined as
follows,
\begin{equation}\label{sysA}
L_R:=\left[\begin{array}{cc}-\Delta+\theta-\psi
& -\varphi\\
-{\varphi}/{\sigma} & (-\Delta+\alpha)/\sigma
\end{array}\right],
\quad
L_I:=\left[\begin{array}{cc}-\Delta+\theta+\psi
& -\varphi\\
-{\varphi}/{\sigma} & (-\Delta
+\alpha)/\sigma\end{array}\right],
\end{equation}
where the functions $\varphi$ and $\psi$ are assumed
to be
continuous and exponentially decaying at infinity.
\begin{thm}\label{th12} The Spectral Mapping Theorem
holds
for the group generated on the space
$[L^2({\Bbb R}^n)]^4$ by the operator
${\mathcal A}$ with $L_R$ and $L_I$ defined in
\eqref{sysA}.
\end{thm}
\begin{proof} As above, one needs to show that
$\|(a+i\tau-{\mathcal
A})^{-1}\|$ remains bounded as $|\tau|\to\infty$.
Denote
\[D=\left[\begin{array}{cc}D_1&0\\0&D_2\end{array}
\right],\quad
\text{where}\quad
D_1=-\Delta+\theta,\,D_2=
-\frac{1}{\sigma}\Delta+\frac{\alpha}{\sigma},\]
and consider the following matrix potentials
exponentially decaying at
infinity,
\[Q_1=\left[\begin{array}{cc}-\psi
&-\varphi\\-{\varphi}/{\sigma}&0
\end{array}\right]\quad\text{and}\quad Q_2=
\left[\begin{array}{cc}\psi&-\varphi\\
-{\varphi}/{\sigma}&0
\end{array}\right].\]
With this new notation and $\xi=a+i\tau$ formula
\eqref{f4.1} still holds.
By transposing the second and third rows and columns
in the $4\times 4$
matrix
$\displaystyle{\left[\begin{array}{cc}\xi&D\\
-D&\xi\end{array}\right
]}$, we remark that this matrix is similar to the
block-diagonal matrix
with the blocks
$$\left[\begin{array}{cc}\xi&D_1\\
-D_1&\xi\end{array}\right
]\quad\text{and}\quad
\left[\begin{array}{cc}\xi&D_2\\-D_2&\xi\end{array}
\right]$$
on the main diagonal and zero remaining entries.
Lemma~\ref{lem2},
applied to each of the diagonal blocks, shows that
the norm of the
operator \eqref{f4.2} in the matrix case
remains bounded as $|\tau|\to\infty$.
Formula \eqref{f5.1} is also valid in the new notation.
In particular,
one infers,
\[\xi[\xi^2+D^2]^{-1}Q_1=\left[\begin{array}{cc}
-\xi[\xi^2+D_1^2]^{-1}
\psi&-\xi[\xi^2+D_1^2]^{-1}\varphi\\-\xi[\xi^2
+D_2]^{-1}\varphi/\sigma&0
\end{array}\right].\]
By part (b) of Lemma~\ref{lem1}, the norm of each
entry in the last
matrix decays to zero as $|\tau|\to\infty$. Similarly
for
$\xi[\xi^2+D^2]^{-1}Q_2$, and, as a result, the norms
of the off-diagonal
entries of $T(\xi)$, defined as in \eqref{f5.1}, decay
to zero as
$|\tau|\to\infty$.
To handle the diagonal entries of $T(\xi)$, let
$Q:=[q_{j,k}]_{j,k=1}^2$
be a matrix-valued function with exponentially decaying
$q_{ij}$. Note
that
$$ [\xi^2+D^2]^{-1}DQ=
\left[
\begin{matrix}
\,[\xi^2+D_1^2]^{-1}D_1q_{11} &
[\xi^2+D_1^2]^{-1}D_1q_{12}\, \\
\,{[\xi^2+D_2^2]^{-1}D_2q_{21}} &
[\xi^2+D_2^2]^{-1}D_2q_{22}\,
\end{matrix}
\right]. $$
In the case $n=1$, one can apply part (a) of
Lemma~\ref{lem1} to each
entry of this matrix. This shows that
$\|T(\xi)\|\to 0$ as
$|\tau|\to\infty$ and the proof of Theorem~\ref{th12}
for $n=1$ is finished.
In the case $n\ge 2$ we claim that the conclusion of
Lemma~\ref{lem1prime}
holds for the matrix-valued case (and, hence,
Theorem~\ref{th12}
holds). Indeed, to verify \eqref{BDIn}, one considers
the polar
decomposition $Q(x)=|Q(x)| U(x)$, where
$|Q(x)|:=[Q^*(x)Q(x)]^{1/2}$
and $U(x)$ is a partial isometry. To be consistent
with our previous
notations in \eqref{DefQ12}, we denote
$|Q|^{1/2}(x)=|Q(x)|^{1/2}$
and $Q^{1/2}(x)=|Q(x)|^{1/2}U(x)$, so that
$Q=|Q|^{1/2}Q^{1/2}$.
Similarly to the proof of Lemma~\ref{lem1prime} in
the scalar case,
we put $A=[\xi^2+D^2]^{-1}DQ^{1/2}$ and $B=|Q|^{1/2}$
and observe that
$$ [I+[\xi^2+D^2]^{-1}DQ]^{-1}=I-A(I+BA)^{-1}B.$$
As above, the estimate \eqref{BDIn} in the matrix case
follows
from the relation
\[ \||Q|^{1/2}[\xi^2+D^2]^{-1}D|Q|^{1/2}\|\to 0
\quad\text{as}\quad
|\tau|\to\infty,\]
(cf. claim \eqref{CLA}). To verify this relation,
we denote
$|Q|^{1/2}(x):=[a_{j,k}(x)]_{j,k=1}^2$ and observe
that the entries
of the matrix operator $|Q|^{1/2}[\xi^2+
D^2]^{-1}D|Q|^{1/2}$ are
scalar operators of the type
$$a_{j,1}[\xi^2+D_1^2]D_1a_{1,k}+
a_{j,2}[\xi^2+D_2^2]^{-1}D_2a_{2,k}, \quad j,k=1,2.$$
But the norm of each summand in these entries tends
to zero as
$|\tau|\to\infty$. To verify the latter fact one can
apply identity
\eqref{16.1}, Proposition~\ref{maintech}, and
Lemma~\ref{Thm9}
for the scalar case, and the result follows.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{3mm}
\noindent {\bf Acknowledgments.}
We are indebted to Arne Jensen for valuable correspondence
in connection with clarifying the origin of results of
the type of Lemma~\ref{Thm9}.
C. Jones was supported by the
National Science Foundation under grant number DMS-9704906.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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