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Resonances; Time-dependent Schroedinger equation
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\def\pcs{P_c^\sharp}
\def\xa{\langle x \rangle}
\def\xam{\langle x\rangle^{-\sigma}} 
\def\xap{\langle x \rangle^{\sigma}}
\def\ta{\langle t \rangle}

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\title{Resonance Theory for Schr\"odinger Operators}
\titlerunning{Resonance Theory}

 



%\subtitle{ aa}
\author{O. Costin\inst{1} \and
  A. Soffer\inst{1}\inst{1}% etc
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\institute{ Department of Mathematics\\ Rutgers University
\\ Piscataway, NJ 08854-8019}
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\date{Received: date / Revised version: }
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\maketitle
%


\begin{abstract}
  
Resonances which result from perturbation of embedded eigenvalues are
studied by time dependent methods.  A general theory is developed
allowing threshold eigenvalues and higher order Fermi Golden rule.
The question of exponential decay rate of resonances is addressed; its
uniqueness in the time dependent picture is shown is certain cases.
The relation to the existence of meromorphic continuation of the
properly weighted Green's function to time dependent resonance is
further elucidated.
\keywords{Resonances; Time-dependent Schr\"odinger equation }%end of abstract
\end{abstract}
%



\section{Introduction and  results}
\label{intro}


Resonances may be defined in different ways, but usually refer to
metastable behavior (in time) of the corresponding system.  The standard
physics definition would be as ``bumps'' in the scattering cross
section, or exponentially decaying states in time, or poles of the
analytically continued $S$ matrix (when such an extension exists).

Mathematically, in the last 25 years one uses a definition close to
the above, by defining $\lambda$ to be a resonance (energy) if it is
the pole of the meromorphic continuation of the weighted Green's
function
$$
\chi(H-Z)^{-1} \chi
$$
with suitable weights $\chi$ (usually, in the Schr\"odinger Theory
context, $\chi$ will be a $C_0^\infty$ function).  $H$ is the
Hamiltonian of the system.  In many cases the equivalence of some of the
above definitions has been shown \cite{G-Sig,H-S,Bal}.  However, the
exponential behavior in time, and the correct estimates on the remainder
are difficult to produce in general \cite{H-Sig}.  It is also not clear how to relate
the time behavior to a resonance, uniquely, and what is the fundamental
role of ``analytic continuation''; see the review \cite{Sim}.  Important
progress on such relations has recently been obtained; Orth \cite{Or}
considered the time dependent behavior of states which can be related to
resonances without the assumption of analytic continuation and
established some preliminary estimates on the remainder terms.  Then,
Hunziker \cite{Hn} was able to develop a quite general relation between
resonances defined via poles of analytic continuations in the context of
Balslev-Combes theory, to exponential decay in time, governed by the
standard Fermi Golden rule.  Here the resonances were small
perturbations of embedded eigenvalues.  In \cite{G-Sig} a definition of
resonance in a time dependent way is given and it is shown to agree with
the one resulting from analytic continuation when it exists, in the
Balslev-Combes theory.  They also get exponential decay and estimates on
the remainder terms.

Exact results, including the case $\varepsilon$ large, for time
dependent potentials have recently been obtained in \cite{CLR}.  Further
notable results on the time dependent behavior of the wave equation were
proved by Tang and Zworski \cite{T-Z}. The construction of states which
resemble resonances, and thus decay approximately exponentially was
accomplished e.g. in \cite{Sk}.

For resonance theory based on Balslev-Combes method the reader is
referred to the book \cite{H-S} and its comprehensive bibliography on
the subject.

Then in a time-dependent approach to perturbation of embedded
eigenvalues developed in \cite{SW} exponential decay and dispersive
estimates on the remainder terms were proved in a general context,
without the assumption of analytic continuation.  The main condition
required that a local decay estimate holds on the continuous spectrum
near the embedded eigenvalue for the {\it unperturbed} Hamiltonian.
The required decay is $O(t^{-2-\eta})$ and the Fermi Golden rule was
required:
$$
\Gamma \geq C\varepsilon^2 > 0.
$$
where  $\varepsilon$ is measure of the size of the perturbation.  The
analysis in this work utilizes in some ways this framework, but
generalizes the results considerably: the required time decay is
$O(t^{-1-\eta})$ and we remove here the assumption of lower bound on
$\Gamma$; it is replaced by
$$
\Gamma \geq C\varepsilon^\frac{2}{1-\eta}
$$
when $\eta < 1,$ and 
$$
\Gamma > 0, \text{ arbitrary}
$$
when $\eta > 1$. This may be important in applications where the lower bound on
$\Gamma$ is hard to prove \cite{Sar}.

  Whenever a meromorphic continuation of the $S$-matrix or
Green's function exists, the poles give an unambiguous definition of
``resonance''.

A time dependent approach or other definitions are less precise, not
necessarily unique, as was observed in \cite{Or}, but usually apply in
more general situations, where analytic continuation is either hard to
prove or not available.

The aim of this work is to generalize the results of \cite{SW} and
to provide some clues about the definition of resonance by time
dependent methods and its relation to the existence of ``analytic
continuation''.

In particular, we will show that in general one can find the
exponential decay rate up to higher order corrections depending on
$\eta$ and $\Gamma$.

In case it is known that analytic continuation exists, our approach
provides a definition of a unique resonance corresponding to the perturbed
eigenvalue. It is given by the solution of some transcendental equation
in the complex plane and it also corresponds to a pole of the weighted
Green's function.  Finally, we analyze the connection between existence
of analytic continuation and the Borel summability of the correction to
the exponential behavior. Typically the remainder term looks like a
stretched exponential times a Borel summable (incompletely) power
series.



Our approach follows the setup of the time dependent theory of
\cite{SW}, combined with Laplace transform techniques.  It is expected
to generalize to the $N$-body case following \cite{M-S}.

We will follow, in part, the notation of \cite{SW}.

  
\centerline{*}

Given $H_0$ a self-adjoint operator on a Hilbert space $\mathcal{H}$, we
assume that $H_0$ has a simple
eigenvalue $\lambda_0$ with normalized eigenvector $\psi_0$:

\begin{equation}
  \label{eq:(1)}
  H_0\psi_0=\lambda_0\psi_0, \|\psi_0\|=1
\end{equation}

\z Our interest is to describe the behavior of solutions of

\begin{equation}
  \label{eq:2}
  i\frac{\partial\phi}{\partial t}=H\phi, \ \ \ H\equiv H_0+\epsilon\, W
\end{equation}

\z with $\phi(0)=E_{\Delta}\phi_0$, where $E_\Delta$ is the spectral
projection of $H$
on the interval $\Delta$ and $\Delta$ is a small interval around
$\lambda_0$. Furthermore, we
will describe, in some cases,  the analytic structure of $(H-z)^{-1}$ in a neighborhood
of $\lambda_0$. $W$ is a symmetric perturbation of $H_0$, such
that $H$ is self-adjoint with same domain as $H_0$.

\subsection{Definitions}

In this section we also introduce certain necessary terminology and
notation. We then state the hypotheses $({\bf H})$ and $({\bf W})$ on
the unperturbed Hamiltonian
$H_0$ and on the perturbation $W$. 

For an operator $L$, $\|L\|$ denotes its norm as an operator from $L^2$
to itself. We interpret functions of a self-adjoint operator as being 
defined by the spectral theorem. In the special case where the operator
is $H_0$, we omit the argument, i.e., $g(H_0)=g$.

For an open interval $\Delta$, we denote an appropriate smoothed
characteristic function of $\Delta$
by $g_\Delta(\lambda)$. In particular, we shall take $g_\Delta(\lambda)$
to be a nonnegative
$C^\infty$ function, which is equal to one on $\Delta$ and zero outside
a neighborhood of $\Delta$. The support of its
derivative
is furthermore chosen to be  small compared to the size of $\Delta$.
We further require that $|g^{(n)}(\lambda)|\le c_n|\Delta|^{-n}, n\ge
1$.


$P_0$ denotes the projection on $\psi_0$, i.e., $P_0
f=(\psi_0,f)\psi_0$.
$P_{1b}$ denotes the spectral projection on $\mathcal{H}_{pp}\cap
\{\psi_0\}^\perp$,
the pure point spectral part of $H_0$ orthogonal to $\psi_0$. That is,
$P_{1b}$ projects onto the subspace of $\mathcal{H}$ spanned by all the
eigenstates other than $\psi_0$.
In our treatment, a central role is played by the subset of the spectrum
of the operator $H_0$, $T^\sharp$ on which a sufficiently rapid local
decay estimate holds. For a decay estimate to hold for $e^{-iH_0 t} $,
one must certainly project out the bound states of $H_0$, but there may
be other obstructions to rapid decay. In scattering theory these are
called threshold energies. Examples of thresholds are: (i)
points of stationary phase of a constant coefficient principal symbol
for two 
body Hamiltonians and (ii) for N--body Hamiltonians, zero and
eigenvalues of subsystems. We will not give a precise definition of
thresholds. For us it is sufficient to say that away from  thresholds
the favorable local decay estimates for $H_0$ hold.

Let $\Delta_*$ be a union of intervals, disjoint from $\Delta$,
containing a neighborhood of infinity and all thresholds of $H_0$ except
possibly those in a small neighborhood of $\lambda_0$. We then let

$$P_1=P_{1b}+g_{\Delta_*}$$

\z where $g_{\Delta_*}=g_{\Delta_*}(H_0)$ is a smoothed characteristic
function of the set 
$\Delta_*$. We also define

\begin{eqnarray}
  \label{eq:2.1}
  \langle x\rangle ^2&=&1+|x|^2\nonumber\\
\overline{Q}&=&I-Q,\ \mbox{and}\nonumber\\
P_c^\sharp&=&I-P_0-P_1
\end{eqnarray}


\z Thus, $\pcs$  is a smoothed out spectral projection of the set
$T^\sharp$
defined as 

\begin{equation}
  \label{eq:2.2}
  T^\sharp=\sigma(H_0)\setminus\{\mbox{eigenvalues, real neighborhoods
    of thresholds and infinity}\}
\end{equation}


\z We expect $e^{-iH_0 t} $ to satisfy good local decay estimates on the
range of $\pcs$; (see ({\bf H4}) below).

\bigskip

Next we state our {\bf hypotheses} on $H_0$.

\bigskip

\z ({\bf H1}) $H_0$ is a self-adjoint operator with dense domain
$\mathcal{D}$, in $L^2(\RR^n)$.

\smallskip

\z ({\bf H2}) $\lambda_0$ is a simple embedded eigenvalue of $H_0$ with
(normalized) eigenfunction $\psi_0$.

\smallskip

\z ({\bf H3}) There is an open interval $\Delta$ containing 
$\lambda_0$ and no other eigenvalue of $H_0$.

\smallskip 



\z ({\bf H4}) {\em Local decay estimate}: Let $r\ge 1+\eta$ and
$\eta>0$. There exists $\sigma>0$ such that if $\xa^\sigma f\in L^2$
then

\begin{equation}
  \label{eq:2.3}
  \| \xa^{-\sigma}e^{-iH_0t} \pcs f\|_2\le C\langle t\rangle ^{-r}\|
  \xa^\sigma f\|_2,
\end{equation}

\z ({\bf H5}) By appropriate choice of a real number $c$, the $L^2$
operator norm of 
$\xa^\sigma(H_0+c)^{-1}\xa^{-\sigma}$ can be made sufficiently small.

\smallskip


\bigskip

\z {\bf Remarks:}

\z (i) We have assumed that $\lambda_0$ is a simple eigenvalue to
simplify the presentation. Our methods
can be easily adapted to the case of multiple eigenvalues.

\smallskip

\z (ii) Note that $\Delta$ does not have to be small and that $\Delta_*$
can be chosen as necessary, depending on $H_0$. 

\smallskip


\z (iii) In certain cases, the above local decay conditions can be
proved even when $\lambda_0$ is a threshold; see \cite{JSS}. 

\smallskip


\z (iv) Regarding the verification of the local decay hypothesis, one
approach is to use techniques based on the Mourre estimate \cite{[26],[45],{Hun}}. If $\Delta$ contains no threshold values, then quite
generally,
the bound (\ref{eq:2.3}) holds with $r$ arbitrary and positive. 

\z We now specify the conditions we require of the perturbation, $W$.

\z {\bf Conditions on $W$}.

\z ({\bf W1}) $W$ is symmetric and $H=H_0+W$ is self-adjoint on
$\mathcal{D}$ and there exists $c\in\RR$ (which can be used in ({\bf
  H5})), such that $c$ lies in the resolvent sets of $H_0$ and $H$.

\z ({\bf W2}) For the same $\sigma$ as in ({\bf H4}) and ({\bf H5}) we
have :

\begin{multline}
  \label{eq:2.5}
  |||W|||\equiv \|\xa^{2\sigma}Wg_\Delta (H_0) \|\nonumber\\+
\|\xa^{\sigma}Wg_\Delta (H_0)\xa^{\sigma} \|+
\|\xa^{\sigma}W (H_0+c)^{-1}\xa^{-\sigma} \|<\infty
\end{multline}

\z and
\begin{equation}
  \label{eq:2.6}
\|\xa^{\sigma}W (H_0+c)^{-1}\xa^{\sigma} \|<\infty
\end{equation}

\z ({\bf W3}) {\em Resonance condition--nonvanishing of the Fermi golden
  rule}:

\z For a suitable choice of $\lambda$ (which will be made precise later)
\begin{equation}
  \label{eq:2.7}
  \Gamma(\lambda)\equiv \pi\, \epsilon^2 (W\psi_0,\delta(H_0-\lambda)(I-P_0)W\psi_0)\ne 0
\end{equation}

In most cases $\Gamma=\Gamma(\lambda_0)$. But in the case 
$\Gamma$ is very small it turns out that the ``correct'' $\Gamma$ will be

$$\Gamma(\lambda_0+\delta)$$

\z with $\delta$ given in the proof of Proposition~\ref{roots}.
See also Section~\ref{S4}. 

\z {\bf Remark:} Let $\mathcal{F}^{H_0}_c $ denote the (generalized)
Fourier transform with respect
to the continuous spectral part of $H_0$. The resonance condition
(\ref{eq:2.7})
can be expressed as


\begin{equation}
  \label{eq:2.8}
  \Gamma\equiv \pi\, \left| \mathcal{F}^{H_0}_c[W\psi_0](\lambda)
\right|^2 > 0
\end{equation}

\section{Main results}

\begin{Theorem}
  \label{2.1}
Let $H_0$ satisfy the conditions {\bf (H1)}...{\bf (H5)} and the perturbation
satisfy the conditions {\bf (W1)...(W3)}. Assume moreover that $\epsilon$ is
sufficiently small and

\z (i) we have good regularity, $\eta>1$ 

\smallskip

\z or

\smallskip

\z (ii) we have lower regularity $0<\eta<1$ supplemented by the conditions
$$\Gamma > C\epsilon^n,\ \ \ n\ge 2$$

\z and $\eta>\frac{n-2}{n}$.

\bigskip
\z 
Under (i) or (ii) we have 


\z a) $H=H_0+\epsilon  W$ has no eigenvalues in $\Delta$.


\smallskip



\z b) The spectrum of $H$ is purely absolutely continuous in $\Delta$,
and 

\begin{equation}
  \label{eq:estn2}
  \|\xa^{-\sigma}e^{-iHt}g_{\Delta}(H)\Phi_0\|_2\le C_\epsilon \langle
  t\rangle ^{-1-\eta} \|\xa ^\sigma \Phi_0\|_2
\end{equation}


\smallskip



\z c) For $\Phi_0$ in the range of $g_\Delta(H)$ we have for $t\ge 0$,


\begin{equation}
  \label{eq:mainEst}
 e^{-iHt}\Phi_0=(I+A_W)
\Big(e^{-i\omega_*t}a(0)\psi_0+ e^{-iH_0t}\phi_d(0)\Big)+R(t)
\end{equation}


\z where $$\|A_W\|\le C\epsilon |||W|||,$$ $a(0)$ and $\phi_d(0)$
are determined by the initial data. The complex frequency $\omega_*$
is given by 

$$-i\omega_*=-is_0-\Gamma$$

\z where $s_0$ solves the equation

\begin{equation}
  \label{eq:w}
 s_0+\omega+\epsilon^2\Im\left\{F(\epsilon,-i\omega-is_0)\right\}=0
\end{equation}

\z (see (\ref{eq:laple}) and (\ref{eq:eql}) below) and 

\begin{equation}
  \label{eq:defGamma}
  \Gamma=\epsilon^2\Re \left\{F(\epsilon,-is_0)\right\}
\end{equation}

\end{Theorem}
\z {\bf Remark:} $\omega_*$ can be found by solving the transcendental
equation (\ref{eq:w}) by either expansion or iteration if sufficient
regularity is present (see also Proposition~\ref{roots} and note
following it and Lemma~\ref{L3}).


\smallskip


In case we have enough regularity ($\eta>1$) then




\begin{Theorem}
  \label{T2.3}

  \begin{equation}
    \label{eq:solw}
    \omega_*=\lambda_0+\epsilon(\psi_0,W\psi_0)+(\Lambda+i\Gamma)(1+o(1))
  \end{equation}

\z where
\begin{equation}
  \label{eq:Lambda}
  \Lambda=\epsilon^2(W\psi_0, P.V. (H_0-\lambda_0)^{-1}W\psi_0)
\end{equation}


\begin{equation}
  \label{eq:Gam}
  \Gamma=\pi\epsilon^2(W\psi_0,\delta(H_0-\lambda_0)(I-P_0)W\psi_0)
\end{equation}
\end{Theorem}

\section{Decomposition and isolation of resonant terms}

We begin with the following decomposition of the solution of
(\ref{eq:2}):

\begin{eqnarray}
  \label{eq:3.1}
  e^{-iHt}\phi_0=\phi(t)=a(t)\psi_0+\tilde{\phi}(t)\\
\Big(\psi_0,\tilde{\phi}(t)\Big)=0,\ \ \ -\infty<t<\infty
\end{eqnarray}

\z Substitution into (\ref{eq:2}) yields

\begin{equation}
  \label{eq:3.3}
  i\partial_t\tilde{\phi}=H_0\phi+\e W\tilde{\phi}-(i\partial_t
  a-\lambda_0 a)\psi_0+a\e W\psi_0
\end{equation}

\z Recall now that $I=P_0+P_1+P_c^\sharp$. Taking the inner product of
(\ref{eq:3.3}) with $\psi_0$ gives the amplitude equation:

\begin{equation}
  \label{eq:3.4}
   i\partial_t a=(\lambda_0+(\psi_0,\e W\psi_0)\,)a+(\psi_0,\e
   WP_1\tilde{\phi})
+(\psi_0,\e W\phi_d),
\end{equation}

\z where

\begin{equation}
  \label{eq:3.5}
  \phi_d\equiv P_c^\sharp\tilde{\phi}
\end{equation}

\z The following equation for $\phi_d$ is obtained by applying
$P_c^\sharp$ to
equation (\ref{eq:3.3}):

\begin{equation}
  \label{eq:3.6}
   i\partial_t\phi_d=H_0\phi_d+P_c^\sharp\e W(P_1\tilde{\phi}+\phi_d)
+aP_c^\sharp\e W\psi_0
\end{equation}


\z To derive a closed system for $\phi_d(t)$ and $a(t)$ we now propose
to obtain an expression for $P_1\tilde{\phi}$, to be used in equations
(\ref{eq:3.4}) and (\ref{eq:3.6}). Since $g_\Delta(H)\phi(\cdot,t)=
\phi(\cdot,t)$ we find
\begin{equation}
  \label{eq:3.7}
  (I-g_\Delta(H))\phi=(I-g_\Delta(H))\Big( a\psi_0+P_1\tilde{\phi}+P_c^\sharp\tilde{\phi}\Big)=0
\end{equation}


\z or


\begin{equation}
  \label{eq:3.8}
 (I-g_\Delta(H)g_I(H_0)) P_1\tilde{\phi}=-\overline{g}_\Delta(H)\Big(a\psi_0+\phi_d\Big)
\end{equation}


\z where $g_I(\lambda) $ is a smooth function which is identically equal
to one on the support of $P_1(\lambda)$, and which has support disjoint
from $\Delta$. Therefore

\begin{equation}
  \label{eq:3.9}
  P_1\tilde{\phi}=-B\overline{g}_\Delta(H)(a\psi_0+\phi_d),
\end{equation}

\z where 

\begin{equation}
  \label{eq:3.9a}
  B=(I-g_\Delta(H)g_I(H_0))^{-1}.
\end{equation}

\z This computation is justified in Appendix B of \cite{SW}.

\begin{Proposition}[\cite{SW}]
  \label{P3.1}

The operator $B$ in (\ref{eq:3.9a}) is a bounded operator on $\mathcal{H}$.
\end{Proposition}

\z From (\ref{eq:3.9}) we get 

\begin{equation}
  \label{eq:3.10}
\phi(t)=a(t)\psi_0+\phi_d+P_1\tilde{\phi}=\tilde{g}_\Delta(H)(a(t)\psi_0+\phi_d(t)),
\end{equation}

\z with

\begin{equation}
  \label{eq:3.101}
  \tilde{g}_\Delta(H)\equiv I-B\overline{g}_\Delta(H)=Bg_\Delta(H)(I-g_I(H_0)).
\end{equation}


\z (see (\ref{eq:2.1}). Although $\tilde{g}_\Delta(H)$ is not really
defined as a function of $H$, we indulge in this mild abuse of notation
to emphasize its dependence on $H$. In fact, in
some sense, $\tilde{g}_\Delta(H)\sim g_\Delta(H) $ to higher order in
$\epsilon$ \cite{SW}.

Substitution of (\ref{eq:3.9}) into (\ref{eq:3.6}) gives:

\begin{equation}
  \label{eq:3.12}
  i\partial_t\phi_d=H_0\phi_d+aP_c^\sharp\e W\tilde{g}_\Delta(H)\psi_0+
P_c^\sharp\e W\tilde{g}_\Delta(H)\phi_d
\end{equation}

\z and

\begin{multline}
  \label{3.13}
i\partial_t a=\Big(\lambda_0+(\psi_0,\e W \tilde{g}_\Delta(H)\psi_0)
\Big)a+(\psi_0,\e W \tilde{g}_\Delta(H)\phi_d)\\=\omega a
+(\omega_1-\omega)a+
(\psi_0,\e W \tilde{g}_\Delta(H)\phi_d)
\end{multline}

\z where

\begin{eqnarray}
  \omega&=&\lambda_0+(\psi_0, \e W\psi_0)\\
\omega_1&=&\lambda_0+(\psi_0, \e W\tilde{g}_\Delta(H)\psi_0)
\end{eqnarray}




\z We write (\ref{eq:3.12}) as an equivalent integral equation.

\begin{multline}
  \label{eq:3.16}
  \phi_d(t)=e^{-iH_0 t}\phi_d(0)-i\int_0^t e^{-iH_0(t-s)}a(s)P_c^\sharp\e
    W\tilde{g}_\Delta(H)\psi_0 ds\\-i\int_0^te^{-iH_0(t-s)}P_c^\sharp\e
    W \tilde{g}_\Delta(H)\phi_d ds
\end{multline}
\begin{Proposition}[\cite{SW}]
  Suppose $|a(t)|\le a_{\infty}\langle t\rangle^{-1-\alpha}$
and assume that $\eta>0$ and $\alpha\ge\eta$. Then for some $C>0$ we have 

$$\|\langle
x\rangle^{-\sigma}\phi_d(t)\|_{L^2}\le C \langle t\rangle^{-1-\eta}\left(
\|\langle
x\rangle^{\sigma}\phi_d(0)\|_{L^2}+a_{\infty}|||W|||\right)$$

\end{Proposition}

We define $K$ as an operator acting on $C(\RR^+,\mathcal{H})$,
the space of continuous functions on $\RR^+$ with values in
$\mathcal{H}$ by

\begin{equation}
  \label{eq:dK}
  \Big(K\, f\Big)(t,x)=\int_0^t e^{-iH_0(t-s)}P_c^\sharp\e
    W\tilde{g}_\Delta(H)f(s,x)ds
\end{equation}

\z We introduce on $C(\RR^+,\mathcal{H})$ the norm

\begin{equation}
  \label{eq:norm}
  \|f\|_\beta =\sup_{t\ge 0}\ta^\beta\|f(\cdot,t)\|_\mathcal{H}
\end{equation}

\z and define the operator norm

\begin{equation}
  \label{eq:normK}
  \|A\|_{\beta;\sigma}=\|\xam A\xap\|_\beta
\end{equation}

\begin{Proposition}
  \label{eK}
If $\epsilon$ is small, $0\le \beta\le r$, $r>1$ and for some
$\beta_1>0$ we have
$\|\xa^{-\sigma}e^{-iH_0 t}P_c^\sharp \xa^{-\sigma} \|\le Ct^{-1-\beta_1}$, then 

\begin{equation}
  \label{eq:nK}
  \|K f\|_{\beta;\sigma}\le C_{\beta;\sigma;r}\e 
\end{equation}
\end{Proposition}

The proof is straightforward.

Using the definition of $K$ given above we see that $K(1-K)^{-1}=
\sum_{n=1}^{\infty}K^n$ is also bounded. We can now rewrite the
equations for 
$\phi_d$ as


\begin{equation}
  \label{eq:phid}
  \phi_d(t)=K\big(a(t)\psi_0\big)+K\phi_d=(I-K)^{-1}\big(a(t)\psi_0\big)
\end{equation}


\z and therefore

\begin{equation}
  \label{eq:eqa}
  i\partial_t a=\omega a +\left(\psi_0,\e W \tilde{g}_\Delta(H)(I-K)^{-1}K\big(a\psi_0\big)\right)
\end{equation}
We now define two operators on $L^\infty$ by

\begin{equation}
  \label{eq:defjt}
  \tilde{j}(a)=\Big(v,\xam K(a\psi_0)\Big);\ \ \ \mbox{where}\ v=\xap \e
  W \tilde{g}_\Delta(H) \psi_0 
\end{equation}

\z and



\begin{equation}
  \label{eq:defj}
  {j}(a)=\Big(v,\xam (I-K)^{-1}K(a\psi_0)\Big)
\end{equation}



\begin{Proposition}
  \label{jj}
The operators $\tilde{j}$ and $j$ are bounded from $L^\infty$ into
itself.
\end{Proposition}

The proof is immediate, using Proposition~\ref{eK} for $\beta=0$.

\z {\bf Remark}. The equation for $a$ can now be written in the
equivalent integral
form

\begin{equation}
  \label{eq:inta}
  a(t)=a(0)e^{-i\omega t}+e^{-i\omega t}\int_0^t e^{i\omega s}j(a)(s)ds
\equiv a(0)e^{-i\omega t}+J(a)
\end{equation}
\begin{Definition}
  Consider the spaces $L^\infty_{T;\nu} $ and $L^\infty_{\nu} $
to be the spaces of functions on $[0,T]$ and $\RR^+$ respectively, in
the norm
\begin{equation}
  \label{eq:nnu}
  \|a\|_\nu =\sup_{s}|e^{-\nu s}a(s)|
\end{equation}

\begin{Remark}
  \label{Remn} 
  
  We note that for $T\in\RR^+$, the norm on $L^\infty_{T;\nu} $ is
  equivalent to the usual norm on $L^\infty[0,T]$.
\end{Remark}

\end{Definition}

\begin{Proposition}
 \label{Pnormj}
 For some constants ${c}$,   $C$ and $\tilde{c}$ {\em independent of
   $T$} we have $\|ja\|_\nu \le c\nu^{-1}\epsilon^2 \|a\|_\nu$, $\|J
 a\|_\nu \le C\nu^{-2}\epsilon^2 \|a\|_\nu$ and $\|\tilde{j}a\|_\nu \le
 \tilde{c}\nu^{-1}\epsilon^2\|a\|_\nu$, and thus $j$, $J$, and
 $\tilde{j}$ are defined on $L^\infty_{T;\nu}$ and $L^\infty_{\nu} $ and
 their norms, in these spaces, are estimated by

\begin{equation}
  \label{eq:njj}
  \|j\|_\nu\le c \nu^{-1}\epsilon^2 ;\ \ \|\tilde{j}\|_\nu \le
\tilde{c}\nu^{-1}\epsilon^2;\ \ \|J\|_\nu\le C\nu^{-2}\epsilon^2
\end{equation}

\end{Proposition}

\z This proof is also immediate.




\begin{Proposition}
 \label{Puniq}
 The equation (\ref{eq:eqa}) has a unique solution in
 $L^1_{loc}(\RR^+)$, and this solution belongs to $L^\infty_{\nu} $ if
 $\nu>\nu_0$ with $\nu_0$ sufficiently large. Thus, in the half-plane
 $\Re(p)>\nu_0$ the Laplace transform of $a$

\begin{equation}
  \label{eq:lapla}
  \hat{a}\equiv \int_0^{\infty} e^{-pt}a(t)dt
\end{equation}

\z exists and is analytic in $p$.
\end{Proposition}

\begin{proof}
  By Proposition~\ref{Pnormj}, and since $\|e^{-i\omega t}\|_\nu=1$, for
  large $\nu$ the equation (\ref{eq:inta}) is contractive in
  $L^\infty_{T;\nu}$ and has a unique solution there. It thus has a
  unique solution in $L^1_{loc}$, by Remark~\ref{Remn}. Since by the
  same argument equation (\ref{eq:inta}) is contractive in
  $L^\infty_{T;\nu}$ and since $L^\infty_{\nu}\subset L^1_{loc}$, the
  unique $L^1_{loc}$ solution of (\ref{eq:inta}) is in $L^\infty_{\nu}$
  as well. The rest is straightforward.

\end{proof}
\begin{Proposition}
 \label{LaplaceTr}
 For $\Re(p)>\nu_0$, the Laplace transform $\hat{a}$ satisfies the
 equation



\begin{multline}
  \label{eq:laple}
  ip\hat{a}=\omega\hat{a}+ia(0)\\
+\left(\psi_0,\e W \tilde{g}_\Delta(H)\left[ \left(
      I+\frac{iI}{p+iH_0}P_c^\sharp\e
      W\tilde{g}_\Delta(H)\right)^{-1}\hat{a}(p)\frac{-iI}{p+iH_0}P_c^\sharp\e W\tilde{g}_\Delta(H)
\right]\psi_0 \right)
\end{multline}


\z or

\begin{equation}
  \label{eq:eql}
  (ip-\omega+i\epsilon^2F(p,\epsilon))\hat{a}(p)=ia(0)
\end{equation}
\end{Proposition}

Our assumptions easily imply that 
 if $\epsilon$ is small enough, then:

(a) $F(p)$ is analytic except for a cut 
along $i\Delta$. $F(p)$ is H\"older continuous of order $\eta>0$ 
($\eta$ depends on $\beta$) at the cut, i.e.

$$\lim_{\gamma\downarrow 0}F(i\tau\pm \gamma)\in H^\eta$$

\z the space of H\"older continuous functions of order $\eta$.


\smallskip


(b) $|F(p)|\le C|p|^{-1}$ for some $C>0$ as $|p|\rightarrow\infty$.  


To
see it we write

  \begin{equation}
    \label{eq:B}
    B=B_1 B_2 \xam;\ \ B_1\equiv \frac{I}{p+iH_0}P_c^\sharp\xam ;\ \
    B_2\equiv
\e \xap W \tilde{g}_\Delta(H)\xap 
  \end{equation}


\z Noting that $P_c^\sharp$ projects on the interval $\Delta$ it is
clear
by the spectral theorem that $\xam B$ is analytic in $p$ on
$\mathcal{D}\equiv\CC\setminus(i\Delta)$. By the assumption on the decay rate
and the Laplace transform of eq. (\ref{eq:2.3}) we have that

\begin{equation}
  \label{eq:HC}
 B_3(p)\equiv  \xam\frac{I}{p+iH_0}P_c^\sharp \xam
\end{equation}

\z is uniformly H\"older continuous, of order $\eta$, as $p\rightarrow
i\Delta$. For $p_0\in i\Delta$, the two sided limits $\lim_{a\downarrow
  0}B_3(p_0\pm a)=B_3^\pm$ will of course differ, in general. A natural
closed domain of definition of $B_3$ is $\mathcal{D}$ together with the
two sides of the cut, $\overline{\mathcal{D}}\equiv
\mathcal{D}\cup\partial \mathcal{D}^+\cup \partial \mathcal{D}^-$. We
then write

\begin{equation}
  \label{eq:unifnorm}
  \left\| B_3 \right\|\le C_1(p)
\end{equation}

\z where we note that $C_1$ can be chosen so that:

\begin{Remark}
  \label{R3}
  $C_1(p)>0$ is uniformly bounded for $p\in\overline{\mathcal{D}}$ and
  $C_1(p)=O(p^{-1})$ for large $p$.
\end{Remark}

\z Hence for some $C_2$ we have
uniformly in $p$ (choosing $\e$ small enough),

\begin{equation}
  \label{eq:normB1B2}
  \|\xam (B_1 B_2)^n\|\le C_2^n\e^n
\end{equation}

\z and therefore the operator 

\begin{equation}
  \label{eq:op3}
  \e W \tilde{g}_\Delta(H)\left[ \left(
      I-\frac{I}{p+iH_0}P_c^\sharp\e
      W\tilde{g}_\Delta(H)\right)^{-1}\frac{I}{p+iH_0}P_c^\sharp\e W\tilde{g}_\Delta(H)
\right]
\end{equation}

\z is analytic in $\mathcal{D}$ and in
in $H^\eta(\overline{\mathcal{D}})$. Finally, part (c) follows from the above 
arguments and Remark~\ref{R3}.

\section{General case}\label{S4}
\subsection{Definition of $\Gamma$}

We have 

\begin{equation}
  \label{eq:ah}
  \hat{a}(p)=\frac{ia(0)}{ip-\omega+i\epsilon^2F(\epsilon,p)}
\end{equation}

\z We are most interested in the behavior of $\hat{a}$ for $p=is$,
$s\in\RR$. $\Gamma$ will be defined in terms of the approximate zeros of
the denominator in (\ref{eq:ah}).  Let $F=:F_1+iF_2$.

\begin{Proposition}
  \label{roots} For $\epsilon$ small enough, the equation
  $s+\omega+\epsilon^2 F_2(\epsilon,is)=0$ has at least one root $s_0$,
  and $s_0=-\omega+O(\epsilon^2)$. If $\eta\ge 1$, then for small enough
  $\epsilon$ the solution is unique. If $\eta<1$ then two solutions
  $s_1$ and $s_2$ differ by at most $O(\epsilon^{\frac{2}{1-\eta}})$.
\end{Proposition}


\begin{proof}
  We write $s=-\omega+\delta$ and get for $\delta$ an equation of the
  form $\delta=\epsilon^2G(\delta)$ where
  $G(x)=-F_2(\epsilon,ix-i\omega)$, and $G(x)\in H^\eta$. The existence
  of a solution for small $\epsilon$ is an immediate consequence of
  continuity and the fact that $\delta-\epsilon^2G(\delta)$ changes sign
  in an interval of size $\epsilon^2\|G\|_\infty$. If $\eta\ge 1$ we
  note that the equation $\delta=\epsilon^2G(\delta)$ is contractive for
  small $\epsilon$ and thus has a unique root. If instead $0<\eta<1$ we
  have, if $\delta_1,\delta_2$ are two roots, then for some $K>0$
  independent of $\epsilon$,
  $|\delta_1-\delta_2|=\epsilon^2|G(\delta_1)-G(\delta_2)|\le \epsilon^2
  K |\delta_1-\delta_2|^\eta$ whence the conclusion.

\end{proof}

\z {\bf Assumption} If $\eta<1$ then we assume that $\epsilon^2
F_1(\epsilon,-i\omega)\gg \epsilon^{\frac{2}{1-\eta}}$ for small
$\epsilon$. When $\eta>1$ this restriction will not be needed,
cf. \S~\ref{Reg1}.


\medskip


\z {\bf Definition} We choose one solution
$s_0=-\omega+\delta$ and let $\Gamma$ be defined by (\ref{eq:defGamma}).



\z {\bf Note.}  
In the case $\eta<1$ the choice of $s_0$ yields, by the
previous assumption a (possible) arbitrariness in the definition of
$\Gamma$ of order $O(\epsilon^{\frac{2}{1-\eta}})=o(\Gamma)$. 


\smallskip



\z {\bf Remarks on the verifiability of condition $\Gamma>0$}. We will
look at various scenarios, which are motivated by concrete examples, in
which the condition of positivity reduces to a condition on
$F(\epsilon,-i\omega)$.

Let 

$$\Gamma_0=\epsilon^2 F_1(\epsilon, -i\omega);\ \ \ \gamma_0=\epsilon^2F_2(\epsilon,-i\omega)$$

\z where we see that $\Gamma_0$ and $\gamma_0$ are $O(\epsilon^2)$.
The equation for $\delta$ reads

$$\delta=-\epsilon^2
[F_2(\epsilon,-i\omega+i\delta)-F_2(\epsilon,-i\omega)]-\gamma_0=\epsilon^2
H(\delta)-\gamma_0$$

\z where $H(0)=0$. We write $\delta=-\gamma_0+\zeta$ and get

$$\zeta=\epsilon^2H(-\gamma_0+\zeta)$$

\z and the definition of $\Gamma$ becomes

$$\Gamma=\epsilon^2 F_1(\epsilon, -i\omega-i\gamma_0+i\zeta)$$



\begin{Proposition}
  (i) Assume that $\eta>1$ and $\gamma_0=o(\epsilon^{-2}\Gamma_0)$. Then
as $\epsilon\rightarrow 0$ we have
 
  \begin{equation}
    \label{eq:condG}
\Gamma=\Gamma_0+o(\Gamma_0)    
  \end{equation}

\z and thus the positivity of $\Gamma$ follows from that of $\Gamma_0$.

(ii) Assume that $\eta<1$, $\gamma_0=o(\epsilon^{-2}\Gamma_0^{1/\eta})$ and
$\Gamma_0\gg
\epsilon^{\frac{2}{1-\eta}}$ as $\epsilon\rightarrow 0$. Then again
(\ref{eq:condG})
holds.

\end{Proposition}
\begin{proof}
(i) Since $\zeta=
O(\epsilon^2\gamma_0)+O(\epsilon^2\zeta)$ we get $\zeta=
O(\epsilon^2\gamma_0)$, implying that

$$\Gamma=\epsilon^2F_1\left[\epsilon, -i\omega-i\gamma_0(1+o(1))\right]
=\Gamma_0+O(\epsilon^2\gamma_0)=\Gamma_0+o(\Gamma_0)   $$


(ii) We have 

\begin{equation}
  \label{eq:111}
\zeta=O(\epsilon^2\gamma_0^\eta)+O(\epsilon^2\zeta^\eta)  
\end{equation}



If $\zeta\le \mbox{const}. \gamma_0$ as $\epsilon\rightarrow 0$, then the proof
is as in part (i). If on the contrary, for some large constant $C$ we
have $\zeta> C\gamma_0$ then by (\ref{eq:111}) we have $\zeta <
\mbox{const}.\epsilon^2\zeta^\eta$ so that $\zeta=O(\epsilon^{2/(1-\eta)})$ and
$\epsilon^2\zeta^\eta=O(\epsilon^{2/(1-\eta)})=o(\Gamma_0)$.  But then

$$\Gamma=\epsilon^2F_1(\epsilon, -i\omega)+O(\epsilon^2\gamma_0^\eta)+O(\epsilon^2\zeta^\eta)=\Gamma_0+o(\Gamma_0)$$

\end{proof}


\subsection{Exponential decay} We let $p=is_0+v$ and get


\begin{Proposition}\label{P0}
Let $r\in\RR^+$. 

(i) As $\epsilon\rightarrow 0$ and $t\Gamma=r$ 
we have

$$ a(t)=e^{-is_0 t}e^{-\Gamma
  t}+O(\epsilon^2\Gamma^{\eta-1})$$

(ii) (For much longer time) as $t\rightarrow\infty$ we have 

$$ a(t)=O(\Gamma^{-1}t^{-\eta-1})$$
\end{Proposition}


\begin{proof}
 (i)  Note first that, taking  $\Re(v)>0$ we get

\begin{multline}
  \label{e}
  F(\epsilon,-is_0+v)=\int_0^\infty
  e^{-is_0t-vt}f(t)dt=\int_0^\infty
e^{-vt}\left(\int_0^t e^{-is_0 u}f(u)du\right)'\\=
v\int_0^\infty e^{-vt}\int_0^t e^{-is_0 u}f(u)du
=
v\int_0^\infty e^{-vt}\left(\int_0^\infty-\int_t^\infty\right)
e^{-is_0 u}f(u)du\\=\int_0^\infty e^{-is_0 u}f(u)du-v
\int_0^\infty e^{-vt}\int_t^\infty e^{-is_0
  u}f(u)du\\=:F(\epsilon,-is_0)
-vL[g](v)
\end{multline}

\z so that with $h(v)=vL[g](v)$ we have

\begin{equation}
  \label{eq:invlap}
  2\pi i a(t)=e^{-is_0 t}\int_{-i\infty}^{i\infty}\frac{e^{vt}}{v+\Gamma+\epsilon^2h(v)}dv
\end{equation}

\z where by construction we have $h\in H^\eta$, $h$ is analytic in
$\CC\setminus i\Delta$ and $h(0)=0$. We write 


\begin{multline}
  \label{eq:expan}
 \int_{-i\infty}^{i\infty}\frac{e^{vt}}{v+\Gamma+\epsilon^2h(v)} dv
=\int_{-i\infty}^{i\infty}\frac{e^{vt}}{(v+\Gamma)\left(1+\epsilon^2
      h(v+\Gamma)^{-1}\right)}dv\\
=\int_{-i\infty}^{i\infty}\frac{e^{vt}dv}{v+\Gamma}-
\epsilon^2\int_{-i\infty}^{i\infty}\frac{1}{v+\Gamma}\frac{h(v+\Gamma)^{-1}}{1+\epsilon^2h(v+\Gamma)^{-1}}e^{vt}dv
\end{multline}



\z We first  need to estimate $L^{-1}\left[{h}(v+\Gamma)^{-1}\right]$
( the transformation is  well defined, since the function is just $(v+\Gamma)^{-1}(F(\epsilon,-is_0+v)-
F(\epsilon,-is_0)))$. We need to write 

\begin{equation}
  \label{eq:tr1}
vL[g](v)=(v+\Gamma)L[g_1](v)  \ \ \mbox{or}\ \ L[g_1]=\left(1-\frac{\Gamma}{v+\Gamma}\right)L[g]
\end{equation}

\z and thus


\begin{equation}
  \label{a1}
  g_1=g-\Gamma e^{-\Gamma t}\int_0^t e^{\Gamma s}g(s)ds
\end{equation}


\z Since $|g(t)|<\mbox{Const.}t^{-\eta}$ we have

\begin{equation}
  \label{a4}
  |g_1(t)|\le \mbox{Const.}t^{-\eta}+e^{-\Gamma t}\int_0^{\Gamma
    t}e^u\left(\frac{u}{\Gamma}\right)^{-\eta}du\le 
\mbox{Const.} t^{-\eta}
\end{equation}

\z  A similar inequality holds for 

\begin{equation}
  \label{eq:Qdef}
  Q=L^{-1}\left[\frac{\frac{
      h}{v+\Gamma}}{1+\frac{\epsilon^2
      h}{v+\Gamma}}\right]
\end{equation}


\z Indeed, we have

\begin{equation}
  \label{eq:Q}
 Q=-L^{-1}\left[\frac{h}{v+\Gamma}\right]+\epsilon^2L^{-1}\left[\frac{h}{v+\Gamma}\right]*Q
\end{equation}

\z It is easy to check that for $t\le r\Gamma^{-1}$ and small
enough $\epsilon$ this equation is
contractive in the norm $\|Q\|=\sup_{s\le t}\langle s\rangle^{\eta}
|Q(s)|$. 

But now, for constants independent of $\epsilon$, 
\begin{multline}
  \epsilon^2 L^{-1}\left[\frac{1}{v+\Gamma}\right]*Q\le \mbox{Const.} e^{-\Gamma
  s}\int_0^te^{\Gamma s}s^{-\eta} ds
\\=\epsilon^2\mbox{Const.} e^{-\Gamma
  s}\Gamma ^{-1}\int_0^{\Gamma t}e^{u}\left(\frac{u}{\Gamma}\right)^{-\eta} du
\le\mbox{Const.}\frac{\epsilon^2}{\Gamma^{1-\eta}}
\end{multline}


(ii)  We now use the fact that 

$$\frac{h}{v+\Gamma}=\frac{F}{v+\Gamma}-\frac{F_0}{v+\Gamma}$$

\z and get for some $C>0$,


$$ H_1=L^{-1}\left[\frac{h}{v+\Gamma}\right]=e^{-\Gamma t}\int_0^t e^{\Gamma
  s}f(s)ds+Ce^{-\Gamma t}$$

\z and thus, proceeding as in the proof of (i) we get for some $C>0$
$H_1\le C \Gamma^{-1}\langle t\rangle^{-\eta-1}$. To evaluate $a(t)$ for
large $t$ we resort again to $Q$ as defined in (\ref{eq:Qdef}) which
satisfies (\ref{eq:Q}). This time we note that the equation is
contractive in the norm $\sup_{s\ge 0}|\langle s\rangle^{1+\eta}\cdot |$
when $\epsilon$ is small enough.
  
\end{proof}

\subsection{Regularity $\eta>1$}\label{Reg1} In this case we obtain  better
estimates.  We write $G(v)=L^{-1}[g](v)$ and (\ref{eq:invlap}) becomes

\begin{equation}
  \label{f}
   a(t)=e^{-is_0 t}\int_{-i\infty}^{i\infty}\frac{e^{vt}}{v+\Gamma+\epsilon^2
     vG(v)}dv
\end{equation}

\z Now

\begin{multline}
  \label{g}
 L^{-1}\left[(v+\Gamma+\epsilon^2vG(v))^{-1}\right]\\
    =L^{-1}\left[\frac{1}{v+\Gamma}\right]-
\epsilon^2 L^{-1}\left[\frac{1}{v+\Gamma}\right]*
L^{-1}\left[\frac{\frac{v}{v+\Gamma}G}{1+\epsilon^2\frac{v}{v+\Gamma}G}\right]
\end{multline}

\begin{Proposition}
  Let 

$$H_2(t)=L^{-1}\left[\frac{\frac{v}{v+\Gamma}G}{1+\epsilon^2\frac{v}{v+\Gamma}G}\right]$$

\z We have 
\begin{equation}
  \label{h}
|H_2|\le \mbox{Const.} \langle 
t\rangle^{-\eta};\ \ \int_0^{\infty}H_2(t)dt=0  
\end{equation}

\end{Proposition}

\begin{proof}
  
 Consider first the function 

$$h_1 =v(v+\Gamma)^{-1}G=G-\Gamma(v+\Gamma)^{-1}G$$

\z we see that (cf. (\ref{e}))

\begin{multline}
H_1=L^{-1}h_1=
\int_t^\infty e^{-is_0
  u}f(u)du-\Gamma\ e^{-\Gamma t}\int_0^t e^{\Gamma s}\int_s^\infty e^{-is_0
  u}f(u)du ds
\end{multline}

\z and thus, for some positive constants $C_i$,
 

\begin{equation}
  \label{eq:g}
|H_1|\le \mbox{Const.}t^{-\eta} +\mbox{Const.}e^{-\Gamma t}\int_0^{\Gamma
  t}e^v\Gamma^{-\eta}\langle v\rangle^{-\eta} dv\
\end{equation}

\z and thus, since $h_1(0)=0$ we have

$$|H_1|\le \mbox{Const.}\langle t\rangle^{-\eta};\ \ \int_0^{\infty}H_1(t)dt=0$$
\z Note now that the function

$$\frac{v}{v+\Gamma}G\left(1+\epsilon^2\frac{v}{v+\Gamma}G\right)^{-1}$$


\z vanishes for $v=0$. Note  furthermore that

$$H_2=H_1-\epsilon^2 H_1*H_2 $$

\z It is easy to check that this integral equation is contractive in the
norm $\|H\|=\sup_{s\le t}|\langle s\rangle^{\eta}H(s)|$ for small enough $\epsilon$;
the proof of the proposition is complete.
\end{proof}
\begin{Proposition}
  

$$L^{-1}\left[(v+\Gamma+\epsilon^2G(v))^{-1}\right]=e^{-\Gamma t}+\Delta(t)$$

\z where for some constant $C$ independent of $\epsilon, t, \Gamma$ we 
have

$$|\Delta|\le C\epsilon^2 \langle t\rangle^{-\eta+1}$$


\end{Proposition}

\begin{proof}
 
We have, by (\ref{g})

 

\begin{multline}
 \Delta(t) =\epsilon^2e^{-\Gamma t}\int_0^t e^{\Gamma
   s}\left(\int_s^{\infty}H_2(u)du\right)'ds\\=\epsilon^2\int_t^{\infty}H_2(s)ds-\Gamma e^{-\Gamma t}\int_0^t e^{\Gamma s}\int_s^{\infty}H_2(u)du
\end{multline}

\z The estimate of the last term is done as in (\ref{eq:g}).
\end{proof}



\section{Analytic case}



Under the assumption



\begin{description}
 \item{({\bf H6})  $\chi (H_0-z)^{-1}P_c^\sharp \chi$ has meromorphic
 continuation from $\Im(z)>0$ to a $\CC\setminus \RR$ neighborhood of
 $\Delta$, and no poles accumulating to $\Delta$, where $\chi$ is an
 appropriate weight.}
\end{description}

\z we can prove
stronger results on the existence of a unique resonance rate
as well as give a more detailed description of the time behavior.

In this case, due to the spectral theorem, the function $F_\epsilon(p)$
will have meromorphic continuation in a neighborhood of $\Delta$, with
branch
points at the end points of the interval $\overline{\Delta}$,
corresponding to the thresholds
of the operator $H_0 g_\Delta(H_0)$ (or $H_0 P_c^\sharp$). We use the
same notation $F_\epsilon$
for this continuation. 


\subsection{Existence and uniqueness of a resonance}

This follows from the following lemma, in which we impose conditions
that were suggested by a number of examples. 

\begin{Lemma}
  \label{L3}
Let $E(p,\epsilon)$ be a function with the following properties:

(i) $E\in H^\eta(\overline{\mathcal{D}}) $ and $E$ is analytic in
$\mathcal{D}$ (this allows for branch-points on the boundary of the
domain, a more general setting that meromorphicity).

(ii) $|E(p,\epsilon)|\le C\epsilon^2$ for some $C$.

(iii) $\lim_{a\downarrow 0} \Re E(-i\omega
-a,\epsilon)=-\Gamma_0<0$.

If (a) $\eta>1$ $E(-i\omega,\epsilon)=o(\Gamma_0/\epsilon^2)$ or (b)
$\eta<1$ and $E(-i\omega,\epsilon)=O(\Gamma_0)$ and $\epsilon$ is small
enough, then the function $G(p,\epsilon)=p+i\omega+E(p,\epsilon)$ has a
unique zero $p=p_z$ in $\overline{\mathcal{D}}$ and furthermore
$\Re(p_z)<0$. In fact, for some $C>0$

\begin{equation}
  \label{eq:impz}
  \Re(p_z)+\Gamma_0=o(\Gamma_0)
\end{equation}


\end{Lemma}

\z {\bf Remark} If the condition For $\eta>1$,
$E(-i\omega,\epsilon)=o(\epsilon^{-2}\Gamma_0)$ is not satisfied, then
we can replace $-i\omega$ by $-i\omega-is_0$ and the uniqueness of the
complex zero will still be true.

 
\begin{proof} The equation $G=0$ can be
  written as

$$0=p+i\omega+E(-i\omega)+[E(p)-E(-i\omega)]$$

\z or, taking $p=-i\omega+\zeta$,
$\phi(\zeta,\epsilon)=E(p)-E(-i\omega)$,


$$\zeta=-E(-i\omega)+\epsilon^2\phi(\zeta,\epsilon)$$



 
  Consider a square centered at $ E(-i\omega)$ with side
  $2|\Re(E(-i\omega))|=2\Gamma_0 $. For both
  parts (a) and (b) of the lemma, note that in our assumptions and by
  the choice of the square we have 

  \begin{equation} \label{eq:sqr}
    \left|\frac{\epsilon^2\phi(\zeta,\epsilon)}{\zeta+E(-i\omega)}\right|\rightarrow 0 \ \ (\mbox{as}\ \e\rightarrow 0)
    \end{equation} \z (on all sides of the square). In
    case (a) on the boundary of the rectangle we have by construction of
    the rectangle, $|\zeta+E(-i\omega)|\ge \Gamma_0$. Also by
    construction, on the sides of the rectangle we have
$|\zeta|\le \Gamma_0$. Still by assumption, $\phi(\epsilon,\zeta)\le
    C\zeta =o(\epsilon^{-2}\Gamma_0)$ and the ratio in (\ref{eq:sqr}) is
$o(1)$. In case (b), we have 

$$\epsilon^2\phi(\epsilon,\zeta) =O(\epsilon^2\zeta^\eta)=O(\epsilon^2\Gamma_0^\eta)=o(\Gamma_0)$$

Thus, on the boundary of the square, the variation of the argument of
    the functions $\zeta+E(-i\omega)-\e ^2\phi(\zeta)$ and that of
    $\zeta+E(-i\omega)$ differ by at most $o(1)$ and thus have to agree
    exactly (being integer multiples of $2\pi i$); thus
    $\zeta+E(-i\omega)-\e ^2\phi(\zeta)$ has exactly one root in the
    square.  The same argument shows that $p+i\omega+E(p,\epsilon)$ has
    no root in any other region in its analyticity domain except in the
    square constructed in the beginning of the proof.

\smallskip

\end{proof}


In the case there is meromorphic continuation of the resolvent and of
the corresponding $S$--matrix in the interval $\Delta$, for a
Schr\"odinger operator of the type $-\Delta+V$, the localized Green's
function is given by an expression of the form

$$\int_{\Delta}g_{\Delta}(k)\frac{e^{i\bf k\cdot
    x}}{k^2-E}f(k)dkd\Omega$$

\z where $f$ is analytic in a neighborhood of $\Delta$. With the choice
that
$g_\Delta$ behaves like $e^{-1/(k-k_i)}$ near the endpoints $k_i$ of the
  interval,
direct calculation shows that the correction to the exponential 
behavior of type $e^{-i\omega_* t}$ of $a(t)$ is given by a remainder of
the form

$$R_1(t)\sim e^{-2\sqrt{t}}\left( e^{i\theta_1 t}\sum_{k=0}^{\infty}\frac{c_{k;1}}{t^{k/4}}+e^{i\theta_2 t}\sum_{k=0}^{\infty}\frac{c_{k;2}}{t^{k/4}}\right)$$

\z where the power series are gotten by Watson's Lemma from Laplace
integrals of the form $\int_0^a e^{-pt}F_{1,2}(p)dp$ (incomplete Borel
summation). For newer results and references in Borel summability see
e.g. \cite{Duke}, \cite{CPAM}.


\begin{Theorem}
  \label{T2}

Under the additional assumptions of this section, the meromorphic
continuation of

$$\chi (H-z)^{-1}\chi,$$

\z exists for $z$ near $\lambda_0$, and has a unique simple pole at
$\omega_*$.



\end{Theorem}

The examples covered by the above approach include those discussed in
\cite{SW} as well as the many cases where analytic continuation has been
established, see e.g. \cite{H-Sig}. Furthermore, following results of
\cite{H-Sig} it follows that under favorable assumptions on $V(x)$,
$-\Delta+V(x)$ has no zero energy bound states in three or more
dimensions extending the results of \cite{SW}, where it was proved for 5
or more dimensions. 


 It is worth mentioning that the possible presence of thresholds inside
$\Delta$ makes it necessary to allow for $\eta<\infty$, and that in the
case where there are finitely many thresholds inside $\Delta$ of known
structure, sharper results may be obtained.



Other applications of our methods involve numerical reconstruction of
resonances from time dependent solutions data, in cases Borel
summability is ensured. This and other implications will be discussed
elsewhere.

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\end{document}
81Q15; 81Q10;35B34; 35B20;  35B40
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