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E-mail addresses: hagedorn@math.vt.edu, jtoloza@leibniz.iimas.unam.mx  
  
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Born-Oppenheimer approximation, semiclassical approximation, adiabatic approximation
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\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{Exponentially Accurate Quasimodes for the Time--Independent 
Born--Oppenheimer Approximation on a One--Dimensional Molecular System
}

\author{George A. Hagedorn\thanks{Partially
supported by National Science Foundation
Grant DMS--0303586. E--mail: hagedorn@math.vt.edu.}\\
Department of Mathematics and\\
Center for Statistical Mechanics and Mathematical Physics\\
Virginia Polytechnic Institute and State University\\
Blacksburg, Virginia\ \, 24061--0123,\quad U.S.A.\\[10pt]
and\\[10pt]
Julio H. Toloza\thanks{Partially supported by Secretar\'{\i}a de Educaci\'on 
P\'ublica--PROMEP Grant UAEHGO--PTC--198. Current address: Instituto de Investigaciones
en Matem\'aticas Aplicadas y en Sistemas, Universidad Nacional Aut\'onoma de M\'exico,
Apartado Postal 20-726, D.F. 01000, M\'exico. E--mail: jtoloza@leibniz.iimas.unam.mx.}
\\
Centro de Investigaci\'on en Matem\'aticas\\ Universidad Aut\'onoma del 
Estado de Hidalgo\\
Carretera Pachuca--Tulancingo Km.\ 4.5\\
Pachuca de Soto, Hidalgo, CP 42090,\quad M\'exico}

\date{}

\maketitle

\vspace*{-7mm}

\begin{abstract}
We consider the eigenvalue problem for a one-dimensional  
molecular--type quantum Hamiltonian that has the form
\[
H(\epsilon)\ =\ -\,\frac{\epsilon^4}2\,
\frac{\partial^2\phantom{i}}{\partial y^2}\ +\ h(y),
\]
where $h(y)$ is an analytic
family of self-adjoint operators that has an discrete, 
nondegenerate electronic level ${\cal E}(y)$
for $y$ in some open subset of ${\mathbb R}$. 
Near a local minimum of the electronic level ${\cal E}(y)$ that is not at a 
level crossing, we construct quasimodes
that are exponentially accurate in the square of the Born--Oppenheimer
parameter $\epsilon$ by optimal truncation of the Rayleigh--Schr\"odinger
series. That is, we construct an energy $E_\epsilon$ and
a wave function $\Xi_\epsilon$, such that
the $L^2$-norm of $\Xi_\epsilon$ is ${\cal O}(1)$ and the $L^2$-norm of
$(H(\epsilon)\,-\,E_\epsilon)\,\Xi_\epsilon$ is bounded by\ \,
$\Lambda\,\exp\,\left(\,-\,{\Gamma}/{\epsilon^2}\,\right)\ $ with
$\Gamma>0$.
\end{abstract}

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{equation}{0}

In this paper we construct exponentially accurate quasimodes
for the time--indepen\-dent Schr\"o\-ding\-er equation for a simple molecular
system. The small parameter that governs the approximation is the usual
Born--Oppenheimer parameter $\epsilon$, where $\epsilon^4$ is
the electron mass divided by the mean nuclear mass. Under appropriate
circumstances, the quasimodes we produce correspond exactly to the
low--lying energy levels of the system. In that case, the exact eigenvalues
and our quasimode energies differ by at most\ \,
$\Lambda\,\exp\,\left(\,-\,\Gamma/\epsilon^2\,\right)$. A bound of
the same form holds
for the norm of the difference between the quasimodes and the
exact eigenvectors.

Hamiltonians for molecular systems can generally be put in the form
\[
H(\epsilon)\ =\ -\ \frac{\epsilon^4}2\ \Delta_y\ +\ h(y),
\]
where the variable $y$ describes the nuclear configuration vector and
the operator $h(y)$ is the electron Hamiltonian. In this paper we examine
the special case of operators of this type, where $y$ is a single
real variable, and $h(y)$ is a family of (possibly unbounded) self--adjoint
operators. 
There are three motivations for studying this somewhat unphysical model. First, 
few rigorous exponentially accurate results for the time--independent 
Born--Oppenheimer approximation have been published,
and they do not cover this model.
Indeed, the only previous results of this kind, to the best of our 
knowledge, is our study of the special case where $h(y)$ is a $2\times 2$ real 
symmetric matrix \cite{hagtol1}. Second, we consider the treatment of this model as
an intermediate stage toward the study of more realistic molecular systems. We hope 
to be able to construct exponentially accurate quasimodes for diatomic molecules, 
using the techniques developed in \cite{hagtol1} and in this paper, 
although several technical difficulties are yet to be overcome. Finally,
we believe that the results we present are nevertheless interesting in themselves,
because they generalize those of \cite{hagtol1} to a much larger class of situations
with a relatively small amount of work. Most of the heavy computations have been
done in \cite{hagtol1}. Although the technical details are somewhat different,
we essentially reduce the present
problem to the one studied in \cite{hagtol1}.

To state our results precisely, we need some notation and
hypotheses. For small $\epsilon$, we study the eigenvalue problem 
\[
\left[\,-\,\frac{\epsilon^4}{2}\,\frac{d^2}{dy^2}\,+\,h(y)\,\right]\,
\Psi(\epsilon,\,y)\ =\ E(\epsilon)\ \Psi(\epsilon,\,y),
\]
where $h(y)$ denotes a family of operators that satisfies the following 
conditions:
\begin{enumerate}
\item $h(y):{\cal H}_e\rightarrow{\cal H}_e$ is a self--adjoint operator for 
      every $y\in{\mathbb R}$, where the electronic Hilbert space 
      ${\cal H}_e$ is separable. We furthermore assume 
      that $\ \inf\,\sigma(h(y))\ $ is uniformly bounded from below.
%
\item $h(y)$ is an analytic family of type (A) in a neighborhood 
      $S\subset{\mathbb C}$ of the origin. 
       From hypothesis 1, it follows that $h(y)^\star=h(\overline{y})$\
       for $y\in S$.
      % The last condition 
%      is equivalent to saying that $h(y)$ is self-adjoint on ${\mathbb R}\cap S$.
%
\item We assume $h(0)$ has an isolated, non-degenerate eigenvalue ${\cal E}(0)$.
\end{enumerate}
By standard results, these hypotheses imply that $h(y)$ has 
an isolated, non-degenerate eigenvalue ${\cal E}(y)$ 
that depends analytically on $y$ in a
neighborhood $S'\subset S$ of the origin.
The associated projection $P(y)$ is analytic
for $y\in S'$, and it is orthogonal for real $y$.
The $y$-independent domain $D$ of
$h(y)$ is a dense subspace of the electronic Hilbert space ${\cal H}_e$. 
Finally we assume: 
\begin{enumerate}
\item[4.] ${\cal E}(y)$ has a non-degenerate local minimum at $y=0$,
      {\it i.e.}, ${\cal E}'(0)= 0$ and  ${\cal E}''(0)> 0$.
      Without loss we assume that ${\cal E}(0)=0$ and ${\cal E}''(0)=1$.
\end{enumerate}
%
An eigenfunction $\Psi(\epsilon,y)$ of $H(\epsilon)$
belongs to $L^2({\mathbb R},{\cal H}_e)$,
whose norm we denote by $\norm{\cdot}$.

Under these hypotheses, we have the following result:

\begin{thm}\label{main}
Assume hypotheses 1--4, and let $\alpha$ be a fixed non-negative integer.
Then, using optimal truncation of a perturbation expansion in powers of
$\epsilon$, we can construct $E_\epsilon$ and $\Xi_\epsilon$, such that
$\norm{\,\Xi_\epsilon\,}\,=\,{\cal{O}}(1)$,\quad $E_\epsilon\,
=\,(\alpha+1/2)\,\epsilon^2\,+\,{\cal O}(\epsilon^4)$,\quad and
\[
\norm{\,(H(\epsilon)-E_\epsilon)\,\Xi_\epsilon\,}\ <\
\Lambda\,\exp\,\left(\,-\,\Gamma/\epsilon^2\,\right).
\]
This quasimode is associated with the $\alpha^{\mbox{\scriptsize th}}$ vibrational
energy level in the local well of ${\cal E}$ near $y=0$.
\end{thm}

\vskip 2mm
\noindent
{\bf Remarks.}\quad {\bf 1.}\quad Our hypotheses allow level crossings as well as
merging of ${\cal E}$ into the continuous spectrum, as long as these phenomena do not 
occur at the bottom of the local well of
${\cal E}$ where $\Xi_\epsilon$ is concentrated.\\
{\bf 2.}\quad Our quasienergy $E_{\epsilon}$
may lie within the essential spectrum. In that case, one would expect a resonance 
near our quasienergy associated with the system being temporarily trapped in the 
well.\\ 
{\bf 3.}\quad
%Suppose the bottom of the essential spectrum is strictly positive.
If $\,\inf\,(\sigma(h(y))\,\setminus\,{\cal E}(y))\,>\,0$,
$\ \liminf_{|y|\to\infty}\,{\cal E}(y)\,>\,0$, and
${\cal E}$ has a unique global minimum at zero, then for small $\epsilon$,
the quasimode of the theorem is an exponentially accurate approximation
to the $\alpha^{\mbox{\scriptsize th}}$ eigenvalue of $H(\epsilon)$. This is
proved by combining our results with those of \cite{combes}.\\
{\bf 4.}\quad The assumption that $\inf\sigma(h(y))$ is bounded below
is just used to guarantee self--adjointness of $H(\epsilon)$. It never enters
directly into our calculations, and it could be weakened.\\
{\bf 5.}\quad Our estimates depend in complicated ways on various parameters of the
problem. We have not kept track of the dependence of $\Lambda$ and $\Gamma$ on 
these parameters. We anticipate that doing so would be very tedious.

\vskip 5mm
The time--independent Born--Oppenheimer approximation has a long history that dates
back to \cite{born}. The first mathematically rigorous result about its validity
for low--lying states
is \cite{combes}, in which the expansion for the energy
is proved to be accurate through fourth order
in $\epsilon$. Rigorous expansions to all orders are developed for systems with 
smooth potentials \cite{hagedorn1}, diatomic Coulomb systems \cite{hagedorn2},
and general Coulomb systems \cite{klein2}. As we have already mentioned, the only
previous construction of approximate solutions up to exponentially small errors
is developed by the authors in \cite{hagtol1}. 

Various other authors have studied time--independent Born--Oppenheimer limits for
other problems. Sordoni \cite{sordoni} has extended the results mentioned above
to include high angular momentum states of diatomic molecules.
Herrin and Howland \cite{herrin} have studied a situation in which the bundle of
eigenvectors for the electron Hamiltonian had a non--trivial Berry phase.
Rousse \cite{rousse} has constructed quasimodes at fixed energies above the
bottoms of wells when the nuclei have one degree of freedom.
His results involved Bohr--Sommerfeld rules, and he handled
level crossings and avoided crossings whose gaps had certain $\epsilon$ dependence.
Klein, Martinez, and Messirdi
\cite{klein1,martinez1,martinez2,messirdi} have studied resonances whose
lifetimes are finite because non--adiabatic transitions of the electrons
coupled them to the continuous spectrum.
In closely related subjects, we note rigorous exponentially 
accurate results for the time--dependent Born--Oppenheimer 
approximation \cite{hagedorn5,martinez3,nenciu}
and for lifetimes of resonances \cite{martinez2,messirdi}.

%\vskip 5mm

The paper is organized as follows:
In Section~\ref{pert}, we derive the formal perturbation expansion
that we use to prove Theorem~\ref{main}. In Section~\ref{estim}, we study 
the growth of quantities that occur
in the perturbation expansion. Then in Section~\ref{hotdog}, we
prove the error bounds that imply 
Theorem~\ref{main}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Perturbation Expansion}\label{pert}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Our construction of the quasimodes involves the computation of formal
Rayleigh--Schr\"odinger series in powers of $\epsilon$, whose coefficients
belong to the following Hilbert spaces:
\begin{enumerate}
\item The nuclear Hilbert space $L^2({\mathbb R})$ (with Lebesgue measure),
      whose inner product and norm we denote respectively by 
      $(\,\cdot,\cdot\,)$ and $\|\cdot\|$.
\item The electronic Hilbert space ${\cal H}_e$.
      The inner product and the norm we denote by $\inner{\,\cdot}{\cdot\,}$ and 
      $\|\cdot\|_e$.
\item The molecular Hilbert space $L^2({\mathbb R,{\cal H}_e})$. The inner
      product is defined by
      \[
      \int_{\mathbb R} \inner{\Phi(x)}{\Psi(x)} dx.
      \]
      As already mentioned, we denote its norm by $\norm{\cdot}$.
\end{enumerate}

The operator $h(y)$ can be decomposed as
\[
h(y)={\cal E}(y)P(y) + h_\perp(y),
\]
where $h_\perp(y):= h(y)(1-P(y))$ is an analytic family of type (A) in 
some neighborhood $S''\subset S'\subset S$ around the origin.
By shrinking $S''$ if necessary, we may assume that
0 belongs to the resolvent set of the restriction of $h_\perp(y)$
to the range of $(1-P(y))$ for $y\in S''$. 
We let $h_\perp(0)^{-1}$ denote the reduced resolvent that is 0 in the range 
on $P(0)$.
Henceforth, we assume that all the analytic properties used in the 
treatment of this problem are valid in the region $S''$.

After making the convenient scaling $\ y=\epsilon\,x$,\ \,the eigenvalue equation becomes
\begin{equation}\label{model}
\left[\,-\,\frac{\epsilon^2}{2}\ \frac{d^2}{dx^2}\ +\ h(\epsilon x)\,\right]\
\Psi(\epsilon,\,x)\ =\ E(\epsilon)\ \Psi(\epsilon,\,x),
\end{equation}
where $\Psi(\epsilon,\,x)\,\in\,L^2({\mathbb R},{\cal H}_e)$. 
With a slight abuse of notation, we henceforth denote
the Hamiltonian operator in (\ref{model}) by $H(\epsilon)$.

The function $\Psi$ can be decomposed as
\[
\Psi(\epsilon,\,x)\
=\ w(\epsilon,\,x)\ \Phi(\epsilon x)\ +\ 
                                 \Psi_\perp(\epsilon,\,x),
\]
where $\Phi(y)$ is the eigenvector associated to ${\cal E}(y)$
and $\langle\Phi(\epsilon x),\,\Psi_\perp(\epsilon, x)\rangle\,=\,0$
for each $x$. We choose $\Phi(y)$ to be
analytic for $y\in S''$ and normalized when $y\in S''$ is
real. We choose its phase so that $\inner{\Phi(y)}{\Phi'(y)}=0$.
Since $\Psi_\perp(\epsilon,\,x)\,=\,\left(1-P(\epsilon x)\right)\
\Psi_\perp(\epsilon,\,x)$,
the first and second derivatives of $\Psi(\epsilon,\,x)$
are given by
\begin{eqnarray*}
\partial_x\Psi
&=& (\partial_x w)\,\Phi + \epsilon\,w\,\Phi' 
    - \epsilon\,P'\,\Psi_\perp + \left(1-P\right)\,\partial_x\Psi_\perp,
    \qquad\mbox{and}
    \\[2mm]
\partial^2_x\Psi
&=& (\partial^2_x w)\,\Phi 
    + 2\,\epsilon\,(\partial_x w)\,\Phi' 
    + \epsilon^2\,w\,\Phi'' - \epsilon^2\,P''\,\Psi_\perp 
    - 2\,\epsilon\,P'\,\partial_x\Psi_\perp + \left(1-P\right)\,
    \partial^2_x\Psi_\perp
    \\[2mm]
&=& \left(\partial^2_x w 
    + 2\,\epsilon\,\langle\Phi,\Phi'\rangle\,\partial_x w
    - 2\,\epsilon\,\langle\Phi,P'\partial_x\Psi_\perp\rangle 
    + \epsilon^2\,w\,\langle\Phi,\Phi''\rangle
    - \epsilon^2\,\langle\Phi,P''\Psi_\perp\rangle\right)\Phi      
     \\[2mm]
& & +\ (1-P)\left(\partial^2_x\Psi_\perp 
    + 2\,\epsilon\,(\partial_x w)\,\Phi' 
    - 2\,\epsilon\,P'\,\partial_x\Psi_\perp 
    + \epsilon^2\,w\,\Phi'' 
    - \epsilon^2\,P''\,\Psi_\perp\right).
\end{eqnarray*}
Thus, in a neighborhood of 0, equation (\ref{model}) is equivalent to the 
following pair of equations:
\begin{equation}\label{one}
- \frac{\epsilon^2}{2}\,\partial^2_x w 
- \epsilon^3\,\langle\Phi,\Phi'\rangle\,\partial_x w
+ \epsilon^3\,\langle\Phi,P'\partial_x\Psi_\perp\rangle 
- \frac{\epsilon^4}{2}\,w\,\langle\Phi,\Phi''\rangle
+ \frac{\epsilon^4}{2}\,\langle\Phi,P''\Psi_\perp\rangle
+ {\cal E} w\ =\ E w,
\end{equation}
and
\begin{eqnarray}
\lefteqn{-\frac{\epsilon^2}{2}(1-P)\partial^2_x\Psi_\perp 
-\epsilon^3(\partial_x w)(1-P)\Phi'
+\,\epsilon^3(1-P)P'\partial_x\Psi_\perp}\nonumber\\
& &
\qquad\qquad-\,\frac{\epsilon^4}{2}w(1-P)\Phi''
+\frac{\epsilon^4}{2}(1-P)P''\Psi_\perp
+h_\perp(1-P)\Psi_\perp\ =\ E(1-P)\Psi_\perp.\quad\label{two}
\end{eqnarray}
In order to take full advantage of the work done in 
\cite{hagtol1}, it is convenient to cast these equations in forms
that resemble equations (3) and (4) of that paper. We first note that
\begin{eqnarray*}
P     
&=& \inner{\Phi}{\cdot\,}\Phi,
       \\
P'    
&=& \inner{\Phi}{\cdot\,}\Phi'\ +\ \inner{\Phi'}{\cdot\,}\Phi,
       \\
P''
&=& \inner{\Phi}{\cdot\,}\Phi'' +\ 2\inner{\Phi'}{\cdot\,}\Phi' \ 
    +\ \inner{\Phi''}{\cdot\,}\Phi,
       \\
PP'
&=& \inner{\Phi'}{\cdot\,}\Phi,\quad \mbox{and}
       \\
PP''
&=& \inner{\Phi}{\Phi''}\inner{\Phi}{\cdot\,}\Phi\ +\ \inner{\Phi''}{\cdot\,}\Phi.
\end{eqnarray*}
The last two identities follow from our phase condition
$\inner{\Phi}{\Phi'}=0$ on $\Phi$.
It thus follows that 
$\inner{\Phi'}{\Phi'}=-\inner{\Phi}{\Phi''}=-\inner{\Phi''}{\Phi}$. 
Also, from $\inner{\Phi}{\Psi_\perp}=0$, we easily obtain
\begin{eqnarray*}
\inner{\Phi}{\partial_x\Psi_\perp}
&=& -\,\epsilon\,\inner{\Phi'}{\Psi_\perp},\qquad\mbox{and}
\\[2mm]
\inner{\Phi}{\partial_x^2\Psi_\perp}
&=& -\,2\,\epsilon\,\inner{\Phi'}{\partial_x\Psi_\perp}\,+\,
     \epsilon^2\,\inner{\Phi''}{\Psi_\perp}.
\end{eqnarray*}
With the use of these identities, equations (\ref{one}) and (\ref{two}) become
\begin{equation}\label{lessnice}
-\,\frac{\epsilon^2}{2}\partial^2_x w\,+\, 
\epsilon^3\,\inner{\Phi'}{\partial_x\Psi_\perp}\,-\, 
\frac{\epsilon^4}{2}\,w\,\inner{\Phi''}{\Phi}\,+\,
\frac{\epsilon^4}{2}\,\inner{\Phi''}{\Psi_\perp}\,+\,
{\cal E}\,w\ =\ E\,w,
\end{equation}
and
\begin{eqnarray}
\lefteqn{
-\,\frac{\epsilon^2}{2}\,\partial^2_x\Psi_\perp\,
-\,\epsilon^3\,\left(\,\partial_xw\,\Phi'\,+\, 
\inner{\Phi'}{\partial_x\Psi_\perp}\,\Phi\,\right)}\nonumber\\
& &\qquad\qquad\qquad
-\,\frac{\epsilon^4}{2}\,w\,\left(\Phi''-\inner{\Phi}{\Phi''}\Phi\right) \,
-\,\frac{\epsilon^4}{2}\,\inner{\Phi''}{\Psi_\perp}\,\Phi\,
+\,h_\perp\,\Psi_\perp\ 
=\ E\,\Psi_\perp.
\label{lessugly}
\end{eqnarray}

%\begin{eqnarray}
%\lefteqn{
%-\,\frac{\epsilon^2}{2}\,\left(\,\partial^2_x\Psi_\perp\,
%-\,\inner{\Phi}{\partial^2_x\Psi_\perp}\,\Phi\,\right)\,
%-\,\epsilon^3\,\left(\,\partial_x w\,-\,
%\inner{\Phi}{\partial_x\Psi_\perp}\,\right)\,\Phi'}\nonumber\\
%& &\qquad\qquad\qquad
%-\,\frac{\epsilon^4}{2}\,w\,\left(\Phi''\,-\,\inner{\Phi}{\Phi''}\,\Phi\,\right)
%\,+\,\epsilon^4\,\inner{\Phi'}{\Psi_\perp}\,\Phi'\,
%+\,h_\perp\,\Psi_\perp\ =\ E\,\Psi_\perp.
%\label{uglysimo}
%\end{eqnarray}
%From $\inner{\Phi}{\Psi_\perp}=0$, we easily obtain
%\begin{eqnarray*}
%\inner{\Phi}{\partial_x\Psi_\perp}
%&=& -\,\epsilon\,\inner{\Phi'}{\Psi_\perp},\qquad\mbox{and}
%\\[2mm]
%\inner{\Phi}{\partial_x^2\Psi_\perp}
%&=& -\,2\,\epsilon\,\inner{\Phi'}{\partial_x\Psi_\perp}\,+\,
%     \epsilon^2\,\inner{\Phi''}{\Psi_\perp}.
%\end{eqnarray*}
%With these identities, equation (\ref{uglysimo}) can be written as
%\begin{eqnarray}
%\lefteqn{
%-\,\frac{\epsilon^2}{2}\,\partial^2_x\Psi_\perp\,
%-\,\epsilon^3\,\left(\,\partial_xw\,\Phi'\,+\, 
%\inner{\Phi'}{\partial_x\Psi_\perp}\,\Phi\,\right)}\nonumber\\
%& &\qquad\qquad\qquad
%-\,\frac{\epsilon^4}{2}\,w\,\left(\Phi''-\inner{\Phi}{\Phi''}\Phi\right) \,
%-\,\frac{\epsilon^4}{2}\,\inner{\Phi''}{\Psi_\perp}\,\Phi\,
%+\,h_\perp\,\Psi_\perp\ 
%=\ E\,\Psi_\perp.
%\label{lessugly}
%\end{eqnarray}

In order to simplify some expressions, we introduce the following notation:
\begin{eqnarray*}
{\cal A}(z)&=&\Phi'(z),\\[1mm]
{\cal B}(z)&=&\inner{\Phi''(z)}{\Phi(z)},\\[1mm]
{\cal C}(z)&=&\Phi''(z),\\
{\cal F}(z)&=&\Phi''(z)\,-\,\inner{\Phi(z)}{\Phi''(z)}\,\Phi(z),\\[1mm]
{\cal G}(z)&=&\inner{\Phi'(z)}{\cdot\,}\,\Phi(z),
              \qquad\mbox{and}\\[1mm]
{\cal K}(z)&=&\inner{\Phi''(z)}{\cdot\,}\,\Phi(z).            
\end{eqnarray*}
Note that $\cal A$, $\cal C$ and $\cal F$ are vector--valued functions, whereas
$\cal G$ and $\cal K$ are operator--valued functions. All these functions are
analytic in $S''$. Equations (\ref{lessnice}) and 
(\ref{lessugly}) can now be written as
\begin{equation}\label{nice}
-\,\frac{\epsilon^2}{2}\,\partial^2_x w\, 
+\,\epsilon^3\,\inner{{\cal A}}{\partial_x\,\Psi_\perp}\, 
-\,\frac{\epsilon^4}{2}\,{\cal B}\,w\,
+\,\frac{\epsilon^4}{2}\,\inner{{\cal C}}{\Psi_\perp}\,
+\,{\cal E}\,w\,=\,E\,w,
\end{equation}
and
\begin{equation}\label{ugly}
-\,\frac{\epsilon^2}{2}\,\partial^2_x\Psi_\perp\,
-\,\epsilon^3\,\left(\,\partial_x w \,{\cal A}\,
+\,{\cal G}\,\partial_x\Psi_\perp\,\right)\,
-\,\frac{\epsilon^4}{2}\,w\,{\cal F}\, 
-\,\frac{\epsilon^4}{2}\,{\cal K}\,\Psi_\perp\,
+\,h_\perp\,\Psi_\perp\,=\,E\,\Psi_\perp.
\end{equation}

The functions defined above, along with ${\cal E}(z)$ and $\Phi(z)$, 
have convergent Taylor series in a neighborhood of 0.
We denote these as
\begin{eqnarray*}
{\cal E}(\epsilon x)
&=&
\frac{1}{2}\,\epsilon^2\,x^2\,+\ \sum_{n=3}^\infty\,d_n\,\epsilon^n\,x^n,
\\
\Phi(\epsilon x)
&=&
\sum_{n=0}^\infty\,\Phi_n\,\epsilon^n\,x^n,
\\
{\cal A}(\epsilon x)
&=&
\sum_{n=0}^\infty\,a_n\,\epsilon^n\,x^n,
\\
{\cal B}(\epsilon x)
&=&
\sum_{n=0}^\infty\,b_n\,\epsilon^n\,x^n,
\\
{\cal C}(\epsilon x)
&=&
\sum_{n=0}^\infty\,c_n\,\epsilon^n\,x^n,
\\
{\cal F}(\epsilon x)
&=&
\sum_{n=0}^\infty\,f_n\,\epsilon^n\,x^n,
\\
{\cal G}(\epsilon x)
&=&
\sum_{n=0}^\infty g_n\,\epsilon^n\,x^n,\qquad\mbox{and}
\\
{\cal K}(\epsilon x)
&=&
\sum_{n=0}^\infty k_n\,\epsilon^n\,x^n.
\end{eqnarray*}
The coefficients $\Phi_n$, $a_n$, $c_n$, and $f_n$ are vectors in ${\cal H}_e$. 
The coefficients $g_n$ and $k_n$ are bounded linear operators on ${\cal H}_e$ that
satisfy the identities
\begin{eqnarray*}
g_n&=&\sum_{p=0}^n\,\inner{a_p}{\cdot\,}\,\Phi_{n-p},\qquad\mbox{and}\\
k_n&=&\sum_{p=0}^n\,\inner{c_p}{\cdot\,}\,\Phi_{n-p}.
\end{eqnarray*}
A similar expansion can be performed on the (unbounded) operator--valued
function $h_\perp(\epsilon x)$. Since $h_\perp$ is an analytic family in 
the sense of Kato, there exists a collection of operators 
$h_{\perp,n}:D\rightarrow{\cal H}_e$, all relatively bounded with respect to
$h_{\perp,0}:=h_\perp(0)$, such that
\[
h_\perp(\epsilon x)\,\Upsilon\ =\
\sum_{n=0}^\infty\,\epsilon^n\,x^n\,h_{\perp,n}\,\Upsilon
\]
in the strong sense, for every $\epsilon x$ close
to 0 and every $\Upsilon\in D$.

We now replace $E(\epsilon)$, 
$w(\epsilon,x)$ and $\Psi_\perp(\epsilon,x)$ by their formal 
Rayleigh--Schr\"odinger series,
\begin{eqnarray*}
E(\epsilon)
&=&
\sum_{n=0}^\infty\,E_n\,\epsilon^n,
\\
w(\epsilon,\,x)
&=&
\sum_{n=0}^\infty\,w_n(x)\,\epsilon^n,
\\
\Psi_\perp(\epsilon,x)
&=&
\sum_{n=0}^\infty\,\Psi_{\perp,n}(x)\,\epsilon^n.
\end{eqnarray*}

We insert all these series into the equations (\ref{nice}) and (\ref{ugly})
and equate terms of the same orders in $\epsilon$ on the two sides of the
equations. This yields the following:

{\bf Order 0\,:}
\begin{eqnarray*}
E_0\ w_0&=&0,\qquad \mbox{and}
\\[2mm]
h_{\perp,0}\ \Psi_{\perp,0}&=&E_0\ \Psi_{\perp,0}.
\end{eqnarray*}
Since we want $w_0\ne 0$, the first equation requires $E_0\,=\,0$. Since
$0\not\in\sigma\left(h_{\perp, 0}\right)$, the second equation forces 
$\Psi_{\perp,0}\,=\,0$.

{\bf Order 1\,:}
\begin{eqnarray*}
E_0\ w_1\ +\ E_1\ w_0&=&0,\qquad \mbox{and}
\\[2mm]
h_{\perp,0}\ \Psi_{\perp,1}\ +\ x\ h_{\perp,1}\ \Psi_{\perp,0}&
=&E_0\ \Psi_{\perp,1}\ +\ E_1\ \Psi_{\perp,0}.
\end{eqnarray*}
The first equation requires $E_1\,=\,0$. The second equation then forces 
$\Psi_{\perp,1}\,=\,0$.

{\bf Order 2\,:}
\begin{eqnarray*}
-\ \frac{1}{2}\ \partial_x^2\ w_0\ +\ \frac{1}{2}\,x^2\ w_0
&=&E_2\ w_0,\qquad \mbox{and}
\\[2mm]
h_{\perp,0}\ \Psi_{\perp,2}&=&0.
\end{eqnarray*}
Clearly $E_2$ and $w_0$ must be solutions to the eigenvalue
problem for the harmonic oscillator Hamiltonian
\[
H_0\ =\ -\ \frac{1}{2}\ \frac{d^2\phantom{t}}{dx^2}\ +\ \frac{1}{2}\ x^2,
\]
whose eigenvalues and normalized eigenfunctions are
$(\alpha+1/2)$ and $\phi_\alpha(x)$, where $\alpha=0,1,2,3,\ldots$
We fix a choice of $\alpha$ and then have $E_2\,=\,(\alpha+1/2)$, 
and $w_0\,=\,\phi_\alpha$. The second equation requires $\Psi_{\perp,2}\,=\,0$.

{\bf Order 3\,:}
\begin{eqnarray*}
(H_0-E_2)\ w_1\ +\ \inner{a_0}{\partial_x\Psi_{\perp,0}}\ +\ d_3\ x^3\ w_0
&=&E_3\ w_0,\qquad \mbox{and}
\\[2mm]
-\ \left(\partial_xw_0\right)\ a_0\ +\ h_{\perp,0}\ \Psi_{\perp,3}
&=&0.
\end{eqnarray*}
Equating components in the direction
of $\phi_\alpha$ in the first equation
leads to $E_3\ =\ 0$. The orthogonal components then require
\[
w_1\ =\ -\ d_3\ (H_0-E_2)^{-1}_\perp\ x^3\ w_0,
\]
%
where $(H_0-E_2)^{-1}_\perp$ denotes the inverse of the restriction
of $(H_0-E_2)$ to the subspace orthogonal to $\phi_\alpha$.
We henceforth arbitrarily assume that $w_n\,\perp\,\phi_\alpha$ 
for all $n>0$. The second equation yields
\[
\Psi_{\perp,3}\ =\ \left(\partial_xw_0\right)\ (h_{\perp,0})^{-1}\ a_0.
\]

\vskip 2mm
{\bf Remark}\quad In the Order 3 calculations, $E_3$ vanishes because
$(\,\phi_\alpha,\,x^3\,\phi_\alpha\,)\,=\,0$.
The symmetry that forces this inner product to vanish also causes $E_n=0$ for all 
odd values of $n$ in the higher order calculations.
We do not make use of this fact in our estimates.
\vskip 2mm

{\bf Order $\mathbf{n\geq 4}$\,:}
\begin{eqnarray*}
%\lefteqn
{
(H_0-E_2)\ w_{n-2}\ 
+\ \sum_{m=0}^{n-3}\ x^m\ \inner{a_m}{\partial_x\,\Psi_{\perp,n-m-3}}
-\ \frac 12\ \sum_{m=0}^{n-4}\ b_m\ x^m\ w_{n-m-4}}&&
\\[1mm]
+\ \frac 12\ \sum_{m=0}^{n-4}\ x^m\ \inner{c_m}{\Psi_{\perp,n-m-4}}\
+\ \sum_{m=3}^n\ d_m\ x^m\ w_{n-m}\
&=&\ \sum_{m=2}^n\ E_m\ w_{n-m},
\end{eqnarray*}
and
\begin{eqnarray*}
&&\hspace{-18pt}
-\ \frac 12\ \partial^2_x\ \Psi_{\perp,n-2}\
-\ \sum_{m=0}^{n-3}\,a_m\,x^m\ \partial_xw_{n-m-3}\
-\ \sum_{m=0}^{n-3}\,g_m\,x^m\ \partial_x\Psi_{\perp,n-3-m}\
-\ \frac 12\ \sum_{m=0}^{n-4}\,f_m\ x^m\ w_{n-m-4}
\\[2mm]
&& \hspace{74pt} 
-\ \frac 12\ \sum_{m=0}^{n-4}\ x^m\ k_m\ \Psi_{\perp,n-4-m}
+\ \sum_{m=0}^n\ x^m\ h_{\perp,m}\ \Psi_{\perp,n-m}\quad
=\quad \sum_{m=2}^n\ E_m\ \Psi_{\perp,n-m}.
\end{eqnarray*}

We now project the two sides of the first equation into the
$\phi_\alpha$ direction to obtain a formula for $E_n$. Then
projecting onto the subspace orthogonal to $\phi_\alpha$, 
we obtain an equation for $(H_0-E_2)\,w_{n-2}$. 
We determine $w_{n-2}$ by applying the reduced resolvent
$(H_0-E_2)^{-1}_\perp$.
We then solve the second equation for $\Psi_{\perp,n}$ by applying the reduced
resolvent $(h_{\perp,0})^{-1}$ to both sides of that equation. 
We therefore have, for $n\geq 4$,
\begin{eqnarray*}
E_n&=&
\sum_{m=0}^{n-6}\ (\phi_\alpha,\,x^m\,\inner{a_m}{\partial_x\,\Psi_{\perp,n-m-3}})\
-\ \frac 12\ \sum_{m=0}^{n-4}\ b_m\ (\phi_\alpha,\,x^m\,w_{n-m-4})
\\[2mm]
&&
+\ \frac 12\ \sum_{m=0}^{n-7}\ (\phi_\alpha,\,x^m\,
 \inner{c_m}{\Psi_{\perp,n-m-4}})\
+\ \sum_{m=3}^n\ d_m\ (\phi_\alpha,\,x^m\,w_{n-m})\ ,
\\[5mm]
w_{n-2}&=&
(H_0-E_2)^{-1}_\perp\ \left[\
-\ \sum_{m=0}^{n-6}\ x^m\ \inner{a_m}{\partial_{x}\,\Psi_{\perp,n-m-3}}\
+\ \frac 12\ \sum_{m=0}^{n-4}\ b_m\ x^m\ w_{n-m-4}\ \right.
\\[2mm]
&&
\phantom{(H_0-E_2)^{-1}_\perp\ \,\ }\left.
-\ \frac 12\,\sum_{m=0}^{n-7}\,x^m\,\inner{c_m}{\Psi_{\perp,n-m-4}}\,
-\,\sum_{m=0}^n\,d_m\,x^m\,w_{n-m}\,
+\,\sum_{m=2}^n\,E_m\,w_{n-m}\ \right],
\\[2mm]
\mbox{and}
\\[2mm]
\Psi_{\perp,n}&=&
(h_{\perp,0})^{-1}\ \left[\
\frac 12\ \partial_x^2\ \Psi_{\perp,n-2}\
+\ \sum_{m=0}^{n-3}\ a_m\ x^m\ \partial_x\ w_{n-m-3}\
+\ \sum_{m=0}^{n-6}\ x^m\ g_m\ \partial_x\ \Psi_{\perp,n-3-m}\,\right.
\\[2mm]
&&
\phantom{(h_{\perp,0})^{-1}\,\ }\left.
-\ \frac 12\,\sum_{m=0}^{n-4}\ f_m\ x^m\ w_{n-m-4}\
-\ \frac 12\ \sum_{m=0}^{n-7}\ x^m\ k_m\ \Psi_{\perp,n-4-m}\,\right.
\\[2mm]
&&
\phantom{(h_{\perp,0})^{-1}\,\ }\left.
+\ \sum_{m=1}^{n-3}\ x^m\ h_{\perp,m}\Psi_{\perp,n-m}\
+\ \sum_{m=2}^{n-3}\ E_m\ \Psi_{\perp,n-m}\ \right].
\end{eqnarray*}

In the sequel, we use the fact that  
$L^2({\mathbb R},{\cal H}_e)$ is isomorphic to $L^2({\mathbb R})\otimes{\cal H}_e$.
Let\newline
$U:L^2({\mathbb R})\otimes{\cal H}_e\rightarrow L^2({\mathbb R},{\cal H}_e)$ 
denote the natural isomorphism defined by
\[
U(\phi(x)\otimes\Upsilon)=\phi(x)\Upsilon.
\]

Although the set of recursive equations given above involves certain unbounded 
operators, they can be reduced to bounded operators because of the 
following result.

\begin{lemma} \label{p}
Let $P_{i\le n}$ denote the projection onto the 
span of $\phi_i$'s with $0\leq i\le n$. Then
\[
\begin{array}{ll}
w_{n}\in\mbox{Ran}\,(P_{i\leq 3n+\alpha})\qquad 
& \mbox{for all}\quad n\geq 0,\\[3mm]
\Psi_{\perp,n}\in\mbox{Ran}\,(U\,(P_{i\leq 3n+\alpha-8}\otimes I)\,U^\star)\qquad 
& \mbox{for all}\quad n\geq 3.
\end{array}
\]
\end{lemma}

{\em Proof:\ } We prove this by an easy induction
using the equations that define 
$w_{n-2}$ and $\Psi_{n}$.\hfill $\Box$
\vskip 5mm

In order to obtain estimates that are sharp enough for our
purposes, we transform the problem, using a method originally
developed in \cite{toloza1,toloza2}. We introduce a bounded operator $A$,
which depends on the choice of $\alpha$, by defining its action on
the basis of the harmonic oscillator Hamiltonian:
\[
A\,\phi_\beta\ =\ \left\{\,
\begin{array}{cl}
\phi_\beta&\mbox{if}\quad\beta=\alpha,\\[1.5mm]
|\beta-\alpha|^{-1/2}\,\phi_\beta\ &\mbox{if}\quad\beta\ne\alpha.\end{array}
\right.
\]
We then define  $\tilde{A}:L^2({\mathbb R},{\cal H}_e)
\rightarrow L^2({\mathbb R},{\cal H}_e)$ as
$\tilde{A}=U(A\otimes I)U^\star$. These operators satisfy the following 
identity.

\begin{lemma}\label{product} 
Consider any $\Psi(x)\in L^2({\mathbb R},{\cal H}_e)$ and $\Theta\in{\cal H}_e$.
Then,
\[
\inner{\Theta}{U(A\otimes I)U^\star\Psi(x)}\,=\,A\inner{\Theta}{\Psi(x)}.
\]
As a consequence, the operators $g_m$ and $k_m$ all commute with $\tilde{A}$.
\end{lemma}

{\em Proof:\ } Let $\{\Upsilon_m\}$ be a basis of ${\cal H}_e$ and
$\{\phi_\alpha(x)\}$ a basis of $L^2({\mathbb R})$. 
Then $\{\phi_\alpha(x)\Upsilon_m\}$ is a basis of $L^2({\mathbb R},{\cal H}_e)$
and
\begin{eqnarray*}
\inner{\Theta}{U(A\otimes I)U^\star(\phi_\alpha(x)\Upsilon_m)}
&=&
\inner{\Theta}{U(A\otimes I)(\phi_\alpha(x)\otimes\Upsilon_m)}
\\[2mm]
&=&
\inner{\Theta}{U((A\phi_\alpha)(x)\otimes\Upsilon_m)}
\\[2mm]
&=&
\inner{\Theta}{(A\phi_\alpha)(x)\Upsilon_m}
\\[2mm]
&=&
(A\phi_\alpha)(x)\inner{\Theta}{\Upsilon_m}
\\[2mm]
&=&
A\inner{\Theta}{\phi_\alpha(x)\Upsilon_m}.
\end{eqnarray*}
Now extend by linearity.\hfill $\Box$
\vskip 5mm

Rather than estimating the norms of $w_n$ and $\Psi_{\perp,n}$ directly, we study
$\wh_n$ and $\Psih_{\perp,n}$, where 
$w_n=A\,\wh_n$ and $\Psi_{\perp,n}=h_{\perp,0}^{-1}\,\tilde{A}\,\Psih_{\perp,n}
=\tilde{A}\,h_{\perp,0}^{-1}\,\Psih_{\perp,n}$. 
These definitions make sense because all finite linear 
combinations of $\phi_i$'s are in the domain of the unbounded operator
$A^{-1}$. We define $\tilde{A}^{-1}=U(A^{-1}\otimes I)U^\star$. 
In addition, we multiply (\ref{nice}) and (\ref{ugly}) by
$A$ and $\tilde{A}$ respectively. Afterward we use 
Lemma~\ref{product}. In this manner we obtain the following:
\begin{eqnarray*}
E_0&=&E_1\ =\ E_3\ =\ 0,
\\[2mm]
E_2&=&\alpha+1/2,
\\[2mm]
\hat{w}_0&=&\phi_\alpha,
\\[2mm]
\wh_1&=&-\,d_3\,[A(H_0-E_2)A]^{-1}_\perp\,A\,x^3\,A\,\wh_0,
\\[2mm]
\Psih_{\perp,0}&=&\Psih_{\perp,1}\ =\ \Psih_{\perp,2}\ =0,
\\[2mm]
\Psih_{\perp,3}&=&\tilde{A}^{-1}\left(\partial_xw_0\right) a_0,
\end{eqnarray*}
and for $n\ge 4$,
\begin{eqnarray*}
E_n&=&
\sum_{m=0}^{n-6}
 \ \left(A\,\partial_x\,x^m\,A\,\phi_\alpha,
 \,\inner{a_m}{\,h_{\perp,0}^{-1}\,\Psih_{\perp,n-m-3}}\right)\
-\ \frac 12\ \sum_{m=0}^{n-4}\ b_m\,(A\,x^m\,A\,\phi_\alpha,\,\wh_{n-m-4})
\\[2mm]
&&
+\ \frac 12\ \sum_{m=0}^{n-7}\ \left(A\,x^m\,A\,\phi_\alpha,\,
 \inner{c_m}{\,h_{\perp,0}^{-1}\,\Psih_{\perp,n-m-4}}\right)\
+\ \sum_{m=3}^n\ d_m\ (A\,x^m\,A\,\phi_\alpha,\,\wh_{n-m})\ ,
\\[5mm]
%
%
\wh_{n-2}\hspace{-1pt}&=&\hspace{-1pt}
[A(H_0-E_2)A]^{-1}_\perp \left[
-\sum_{m=0}^{n-6} A\,x^m\,\partial_{x}\,A\inner{a_m}{h_{\perp,0}^{-1}\,\Psih_{\perp,n-m-3}}
 +\frac 12\,\sum_{m=0}^{n-4} b_m\,A\,x^m A\,\wh_{n-m-4}\,\right.
\\[2mm]
&&
\phantom{[A(H_0-E_2)A]^{-1}_\perp\ \,}\left.
-\ \frac 12\,\sum_{m=0}^{n-7}\,A\,x^m\,A\inner{c_m}{h_{\perp,0}^{-1}\,\Psih_{\perp,n-m-4}}\ 
-\ \sum_{m=3}^n\,d_m\,A\,x^m A\,\wh_{n-m}\,\right.
\\[2mm]
&&
\phantom{[A(H_0-E_2)A]^{-1}_\perp\ \,}\left.
+\ \sum_{m=2}^nE_m\,A^2\,\wh_{n-m}\ \right],
\\[2mm]
%
%
\mbox{and}
\\[2mm]
\Psih_{\perp,n}&=&
\tilde{A}^{-1}\ \left[\
\frac 12\,\partial_x^2\,\tilde{A}\,h_{\perp,0}^{-1}\,\Psih_{\perp,n-2}\
+\ \sum_{m=0}^{n-3}\,a_m\,x^m\,\partial_x\,A\,\wh_{n-m-3}\,\right.
\\[2mm]
&&
\phantom{\tilde{A}^{-1}\ \,\ }\left.
+\ \sum_{m=0}^{n-6}\,x^m\,\partial_x\,\tilde{A}\,g_m\,h_{\perp,0}^{-1}\,\Psih_{\perp,n-3-m}\
-\ \frac 12\,\sum_{m=0}^{n-4}\,f_m\,x^m\,A\,\wh_{n-m-4}\,\right.
\\[2mm]
&&
\phantom{\tilde{A}^{-1}\ \,\ }\left.
-\ \frac 12\,\sum_{m=0}^{n-7}\,x^m\,\tilde{A}\,k_m\,h_{\perp,0}^{-1}\,\Psih_{\perp,n-4-m}\
+\ \sum_{m=1}^{n-3}\,x^m\,h_{\perp,m}\,\tilde{A}\,h_{\perp,0}^{-1}\,\Psih_{\perp,n-m}\,\right.
\\[2mm]
&&
\phantom{\tilde{A}^{-1}\ \,\ }\left.
+\ \sum_{m=2}^{n-3}\,E_m\,\tilde{A}\,h_{\perp,0}^{-1}\,\Psih_{\perp,n-m}\ \right].
\end{eqnarray*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Growth of the Perturbation Coefficients}\label{estim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we obtain nice upper bounds for the growth of the norms of
$|E_n|$, $\|\wh_{n-2}\|$ and $|\!|\!|\Psih_n|\!|\!|$. We do this by first
getting recursive inequalities for $|E_n|$, $\|\wh_{n-2}\|$
and $|\!|\!|\Psih_n|\!|\!|$ from the equations described in the previous section. 
Then an induction argument yields the desired estimates.

Let us first note that the region $S''$ contains the complex disc of radius $\delta$ 
centered at the origin. Since certain functions involved in this problem are 
analytic in $S''$,
we can use the Cauchy integral formula to conclude that 
$\nre{\Phi_m}$, $\nre{a_m}$, $|b_m|$, 
$\nre{c_m}$, $\nre{f_m}$, and $|d_m|$ are all bounded by 
$Z\,\delta^{-m}$, for some constant $Z>0$. Also, $\nre{g_m}$ and $\nre{k_m}$
are bounded by $(m+1)\,Z^2\,\delta^{-m}$.
Furthermore, various operators that
show up in the last three equations of Section \ref{pert}
satisfy some of the following statements.

\begin{lemma}\label{bits}
Let $\#_{x}$ denote either $x$ or $\partial_x$, and let $(\#_x)^l$ 
      denote any product of $l$ factors, each of which is either $x$ or 
      $\partial_x$. Then
\begin{enumerate}
\item The operators $h_{\perp,m}$ and  $h_{\perp,0}^{\ -1}$ conmute with 
      the operators $\tilde{A}$, $\tilde{A}^{-1}$ and $(\#_{x})^l$.
\item \(
      \norm{\tilde{A}^{-1}\,
      (\#_{x})^l\,\tilde{A}\,U\,(P_{i\leq n}\otimes I)\,U^\star} = 
      \left\| A^{-1}\,(\#_{x})^l\,A\,P_{i\leq n}\right\| .
      \)
\end{enumerate}
Also,
\begin{enumerate}
\item[3.] The operators 
      $h_{\perp,m}\,h_{\perp,0}^{-1}:{\cal H}_e\rightarrow {\cal H}_e$ satisfy\ 
      \(
      \nre{h_{\perp,m}\,h_{\perp,0}^{-1}}\leq Z\,\delta^{-m},
      \)
      where $Z$ is the constant introduced above, properly redefined if
      necessary.      
\end{enumerate}
\end{lemma}

{\em Proof:\ }\quad % {\bf To be written.}\hfill $\Box$
1.\ \,This result follows since $h_{\perp,m}$ and  $h_{\perp,0}^{\ -1}$ act in the
electron variables, while $x$ and $\partial_x$ act in the nuclear variables.\\
2.\ \,This result is a consequence of the electronic variables playing no 
significant role in the operators involved.\\
3.\ \,We have assumed that $h(\cdot)$ is an analytic family of type (A) with domain $D$.
By results of Section 2 of Chapter 7 of \cite{Kato}, it follows that
$h(y)\,h_{\perp,0}^{\ -1}$ is a bounded analytic family on ${\cal H}_e$. The
Taylor series coefficients of this operator valued function are
$h_{\perp,m}\,h_{\perp,0}^{-1}$, and the radius of convergence is at least $\delta$.
The result follows from the standard Cauchy estimates.\hfill $\Box$

\vskip .5cm
Using Lemmas \ref{p} and \ref{bits} we obtain the following set 
of inequalities:
\begin{eqnarray*}
|E_n|&\leq& 
Z\,\sum_{m=0}^{n-6}
 \delta^{-m}\|A\,\partial_x\,x^m A\,P_{i\leq\alpha}\|
 \norm{\Psih_{\perp,n-m-3}}\
+\ 
 \frac Z2\,\sum_{m=0}^{n-4}\delta^{-m}\|A\,x^m A\,P_{i\leq\alpha}\|
 \|\wh_{n-m-4}\|
\\[2mm]
&&
+\ \frac Z2\,\sum_{m=0}^{n-7}\delta^{-m}\|\,A\,x^mA\,P_{i\leq\alpha}\|
 \norm{\Psih_{\perp,n-m-4}}\
+\ Z\,\sum_{m=3}^n\delta^{-m}\|A\,x^m A\,P_{i\leq\alpha}\|\|\wh_{n-m}\|\, ,
\\[5mm]
\|\wh_{n-2}\|&\leq&
Z\,\sum_{m=0}^{n-6}
 \delta^{-m}\|A\,\partial_x\,x^m A\,P_{i\leq 3(n-m-3)+\alpha-8}\|
 \norm{\Psih_{\perp,n-m-3}}\
\\[2mm]
&&
+\ 
 \frac Z2\,\sum_{m=0}^{n-4}\delta^{-m}\|A\,x^m A\,P_{i\leq 3(n-m-4)+\alpha}\|
 \|\wh_{n-m-4}\|
\\[2mm]
&&
+\ \frac Z2\,
 \sum_{m=0}^{n-7}\delta^{-m}\|\,A\,x^mA\,P_{i\leq 3(n-m-4)+\alpha-8}\|
 \norm{\Psih_{\perp,n-m-4}}\
\\[2mm]
&&
+\ Z\,\sum_{m=3}^n\delta^{-m}\|A\,x^m A\,P_{i\leq 3(n-m)+\alpha}\|\|\wh_{n-m}\|
\ +\ \sum_{m=2}^n |E_m|\|\wh_{n-m}\|\, ,
\\[2mm]
\mbox{and}
\\[2mm]
\norm{\Psih_{\perp,n}}&\leq&
\frac{\nre{h_{\perp,0}^{-1}}}{2}\,
\left\|A^{-1}\,\partial_x^2\,\tilde{A}\,P_{i\leq 3(n-2)+\alpha-8}\right\|
\norm{\Psih_{\perp,n-2}}
\\[2mm]
&&
+\ Z\,
\sum_{m=0}^{n-3}\,\delta^{-m}\,
\left\|A^{-1}\,x^m\,\partial_x\,A\,P_{i\leq 3(n-m-3)+\alpha}\right\|
\left\|\wh_{n-m-3}\right\|
\\[2mm]
&&
+\ Z^2\nre{h_{\perp,0}^{-1}}\,
\sum_{m=0}^{n-6}\,\delta^{-m}\,(m+1)
\left\|A^{-1}\,x^m\,\partial_x\,A\,P_{i\leq 3(n-m-3)+\alpha-8}\right\|
\norm{\Psih_{\perp,n-m-3}}
\\[2mm]
&&
+\ \frac{Z}{2}\,
\sum_{m=0}^{n-4}\,\delta^{-m}\,
\left\|A^{-1}\,x^m\,A\,P_{i\leq 3(n-m-4)+\alpha}\right\|
\left\|\wh_{n-m-4}\right\|
\\[2mm]
&&
+\ \frac{Z^2\nre{h_{\perp,0}^{-1}}}{2}\,
\sum_{m=0}^{n-7}\,\delta^{-m}\,(m+1)
\left\|A^{-1}\,x^m\,A\,P_{i\leq 3(n-m-4)+\alpha-8}\right\|
\norm{\Psih_{\perp,n-m-4}}
\\[2mm]
&&
+\ Z\, 
\sum_{m=1}^{n}\,\delta^{-m}\,
\left\|A^{-1}\,x^m\,A\,P_{i\leq 3(n-m)+\alpha-8}\right\|
\norm{\Psih_{\perp,n-m}}
\\[2mm]
&&
+\ \nre{h_{\perp,0}^{-1}}
\sum_{m=0}^n\,|E_m|\norm{\Psih_{\perp,n-m}}\ .
\end{eqnarray*}

These inequalities differ only in unimportant ways from those
preceding (16) in \cite{hagtol1}. This leads us to the following
result.

\begin{thm}\label{growth}
Define $\kappa:=2/\delta^{2}$. There exists $b>1$ such that for every 
$n\geq 3$ we have
\begin{eqnarray*}
\left|E_n\right| & < & \kappa^{3(n-2)}\,b^{2n-5}\,[(n+\alpha-2)!]^{1/2},
\\[3mm]
\left\|\wh_{n-2}\right\| & < & \kappa^{3(n-2)}\,b^{2n-3}\,[(n+\alpha-1)!]^{1/2},
\qquad\mbox{and}
\\[3mm]
\norm{\Psih_{ \perp,n}} & < & \kappa^{3n}\,b^{2n-1}\,[(n+\alpha-1)!]^{1/2}.
\end{eqnarray*}
\end{thm}

The rather tedious proof of this theorem is based on a simple induction argument.
We refer to the proof of Theorem~2 of \cite{hagtol1}
for the details.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Estimates for the Norm of the Error}\label{hotdog}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The construction of the exponentially accurate quasimodes is based on 
truncation of the formal Rayleigh--Schr\"odinger series whose coefficients
obey the growth estimates stated in Theorem~\ref{growth}. Here we mimic 
to a certain extent the
arguments in \cite{toloza1,toloza2}, which are based on a technique developed in 
\cite{hagedorn4}. The main change in our present case is that
the recursion relations that generate these coefficients make sense only in a 
neighborhood of the bottom of the energy surface ${\cal E}(y)$. This issue
will be dealt with by using a suitable cut--off function.
In order to prove certain estimates, we 
henceforth assume that the region $S''\subset{\mathbb C}$ contains the
open set
\[
\{z\in{\mathbb C}: |z+s|<\delta+\mu\}\cup
        \{z\in{\mathbb C}: |z-s|<\delta+\mu\}\cup
        \{z\in{\mathbb C}: |\mbox{Re}\,z|<s\ ,\ |\mbox{Im}\,z|<\delta+\mu\}, 
\]
for some fixed $s>0$, $1\geq\delta>0$ and $\mu>0$ arbitrarily small. 

For $N\geq 1$ define
\begin{eqnarray*}
E^{N}&:=&
\sum_{n=0}^{N+2}\,\epsilon^{n}\,E_{n},
\\
w^{N}(\epsilon,x)&:=&
\sum_{n=0}^{N}\,\epsilon^{n}\,w_{n}(\epsilon,x),
\\
\Psi_\perp^{N}(\epsilon,x)&:=&
\sum_{n=0}^{N+2}\,\epsilon^{n}\,\Psi_{\perp,n}(\epsilon,x),\qquad\mbox{and}
\\[3mm]
\Psi^{N}(\epsilon,x)&:=&
w^{N}(\epsilon,x)\,\Phi(\epsilon x)\ +\ \Psi_\perp^{N}(\epsilon,x).
\end{eqnarray*}

Let $f$ be a real $C^2$ function, such that $f(y)=1$ for 
$y\in[-r/\epsilon,r/\epsilon]$ and $f(y)=0$ for $y\not\in[-s/\epsilon,s/\epsilon]$,
where $s>r>0$. The support of $f'$ is then contained in 
$[-s/\epsilon,-r/\epsilon]\cup[r/\epsilon,s/\epsilon]$. For later use,
let $\chi_{f'}$ denote the characteristic function of the support of $f'$.
 
We now define
\[
\Xi^N(\epsilon,x)\ :=\ f(x)\,\Psi^{N}(\epsilon,x).
\]
Let $H(\epsilon)$ denote the self--adjoint operator defined in equation 
(\ref{model}). The error commited when truncated Rayleigh--Schr\"odinger series 
replace the actual solution to equation (\ref{model}) is given by
\begin{equation}\label{cuco}
\left(H(\epsilon) - E^N\right)\Xi^N\ =\ f\left(H(\epsilon) - E^N\right)\Psi^N
-\frac{\epsilon^2}{2}\left[\partial_x^2,f\right]\Psi^N.
\end{equation}

The first term can be written as (temporarily ignoring the cut--off) 
\[
\left(H(\epsilon) - E^N\right)\Psi^N\ =\ S_N\,\Phi\ +\ T_N,
\]
where the residual terms $S_N$ and $T_N$ are given explicitly by
\begin{eqnarray*}
S_N &:=&
-\,\frac{\epsilon^2}{2}\,\partial^2_x\,w^N + 
\epsilon^3\inner{{\cal A}}{\partial_x\,\Psi_\perp^N} - 
\frac{\epsilon^4}{2}\,{\cal B}\,w^N +
\frac{\epsilon^4}{2}\,\inner{{\cal C}}{\Psi_\perp^N} +
{\cal E}\,w^N\,-\,E^N\,w^N,\quad\mbox{and}
\\[3mm]
T_N &:=&
-\,\frac{\epsilon^2}{2}\,\partial^2_x\,\Psi_\perp^N\,
-\,\epsilon^3\left(\partial_x\,w^N\,{\cal A}\, 
+\,{\cal G}\,\partial_x\,\Psi_\perp^N\right)\,
-\,\frac{\epsilon^4}{2}\,w^N\,{\cal F}\,
-\,\frac{\epsilon^4}{2}\,{\cal K}\,\Psi_\perp^N
\\[3mm]
&& 
+\,h_\perp\Psi_\perp^N\, 
-\,E^N\,\Psi_\perp^N.
\end{eqnarray*}
Since the perturbation coefficients are the solutions to the recursive equations
obtained in Section~2, all terms of order $n\leq N+2$ cancel in both $S_N$ and 
$T_N$. Thus, the residual terms can be written as
\begin{eqnarray*}
S_N &=&
\sum_{m=0}^N\,\epsilon^{m+3}\inner{{\cal A}^{[N-m-1]}}{\partial_x\Psi_{\perp,m}}\,
+\,\epsilon^{N+4}\inner{{\cal A}^{[-1]}}{\partial_x\Psi_{\perp,N+1}}\,
+\,\epsilon^{N+5}\inner{{\cal A}^{[-1]}}{\partial_x\Psi_{\perp,N+2}}
\\[1mm]
&&
-\,\frac{1}{2}\,\sum_{m=0}^{N-1}\,\epsilon^{m+4}\,{\cal B}^{[N-m-2]}\,w_m\,
-\,\frac{\epsilon^{N+4}}{2}\,{\cal B}^{[-1]}\,w_N
\\[1mm]
&&
+\,\frac{1}{2}\,
   \sum_{m=0}^{N-1}\,\epsilon^{m+4}\inner{{\cal C}^{[N-m-2]}}{\Psi_{\perp,m}}
\\[1mm]
&&
+\,\frac{\epsilon^{N+4}}{2}\inner{{\cal C}^{[-1]}}{\Psi_{\perp,N}}\,
+\,\frac{\epsilon^{N+5}}{2}\inner{{\cal C}^{[-1]}}{\Psi_{\perp,N+1}}\,
+\,\frac{\epsilon^{N+6}}{2}\inner{{\cal C}^{[-1]}}{\Psi_{\perp,N+2}}
\\[1mm]
&&
+\,\sum_{m=0}^N\,\epsilon^{m}\,{\cal E}^{[N-m+2]}\,w_m
\\[1mm]
&&
-\,\sum_{l=N+3}^{2N+2}\epsilon^l\sum_{m=l-N-2}^{N}E_{l-m}\,w_m\, ,
\\
\mbox{and}
\\
T_N &=&
-\,\frac 12\,\epsilon^{N+3}\,\partial_x^2\,\Psi_{\perp,N+1}\,
-\,\frac 12\,\epsilon^{N+4}\,\partial_x^2\,\Psi_{\perp,N+2}
\\[1mm]
&&
-\,\sum_{m=0}^N\,\epsilon^{m+3}\,(\partial_x\,w_m)\,{\cal A}^{[N-m-1]}
\\[1mm]
&&
-\,\sum_{m=0}^N\,\epsilon^{m+3}\,{\cal G}^{[N-m-1]}\,\partial_x\,\Psi_{\perp,m}\,
-\,\epsilon^{N+4}\,{\cal G}^{[-1]}\,\partial_x\,\Psi_{\perp,N+1}\,
-\,\epsilon^{N+5}\,{\cal G}^{[-1]}\,\partial_x\,\Psi_{\perp,N+2}
\\[1mm]
&&
-\,\frac 12\,\sum_{m=0}^{N-1}\,\epsilon^{m+4}\,{\cal F}^{[N-m-2]}\,w_m\,
-\,\frac{\epsilon^{N+4}}2\,{\cal F}^{[-1]}\,w_N
\\[1mm]
&&
-\,\frac 12\,\sum_{m=0}^{N-1}\,\epsilon^{m+4}\,{\cal K}^{[N-m-2]}\,\Psi_{\perp,m}\,
\\[1mm]
&&
-\,\frac{\epsilon^{N+4}}2\,{\cal K}^{[-1]}\,\Psi_{\perp,N}\,
-\,\frac{\epsilon^{N+5}}2\,{\cal K}^{[-1]}\,\Psi_{\perp,N+1}\,
-\,\frac{\epsilon^{N+6}}2\,{\cal K}^{[-1]}\,\Psi_{\perp,N+2}
\\[1mm]
&&
+\,\sum_{m=0}^{N+2}\,\epsilon^m\,h_{\perp}^{[N-m+2]}\,\Psi_{\perp,m}
\\[1mm]
&&
-\,\sum_{l=N+3}^{2N+4}\,\epsilon^l\,\sum_{m=l-N-2}^{N+2}\,
E_{l-m}\,\Psi_{\perp,m}\, . 
\end{eqnarray*}
We have used above the following notation for exact Taylor series errors
for any function $g(z)$ expanded around $z=0$:
\[
g^{[n]}(z)\ :=\ \,\left\{\
\begin{array}{cl}\displaystyle
g(z)\ -\ \sum_{m=0}^n\ \frac{g^{(m)}}{m!}\,z^m\ &\mbox{ if}\quad n\ge 0,
\\[7mm]
g(z)&\mbox{ if}\quad n=-1.
\end{array}\right.
\]
Note that in this notation, the integer $n$ may never be less than $-1$, 
and that $g^{[n]}$ is $O(\epsilon^{n+1})$ since $z=\epsilon\,x$. Similarly,
\[
h_\perp^{[n]}(z)\ \Upsilon\ \,:=\ \left\{\
\begin{array}{cl}\displaystyle
h_{\perp}(z)\ \Upsilon\ -\ \sum_{m=0}^n\ z^m\,h_{\perp,m}\ \Upsilon\
&\mbox{ if}\quad n\ge 0,
\\[7mm]
h_\perp(z)\ \Upsilon&\mbox{ if}\quad n=-1,
\end{array}\right.
\]
for every $\Upsilon\in D$.

Our main result in this work relies on an upper bound for the 
$L^2({\mathbb R},{\cal H}_e)$-norm of (\ref{cuco}). We obviously have
\begin{eqnarray}
\norm{\left(H(\epsilon) - E^N\right)\Xi^N}
&\leq&
\norm{f\left(H(\epsilon) - E^N\right)\Psi^N}\ +\ \frac{\epsilon^2}{2}
\norm{\left[\partial_x^2,f\right]\Psi^N}\nonumber
\\[2mm]
&\leq&
\norm{f\,S_N\,\Phi}\ +\ \norm{f\,T_N}\ +\ \frac{\epsilon^2}{2}
\norm{\left[\partial_x^2,f\right]\Psi^N}\nonumber
\\[2mm]
&\leq&
\|f\,S_N\|\ +\ \norm{f\,T_N}\ +\ \frac{\epsilon^2}{2}
\norm{\left[\partial_x^2,f\right]\Psi^N}\label{badday}
\end{eqnarray}

We use the following three propositions to estimate the last line
in this expression.

\vskip 3mm

\begin{prop}\label{goodguy1}
There exist constants $G_1$ and $F_1$, such that
\[
\nr{f\,S_N}\ <\ \sum_{m=N+3}^{2N+2}\ G_1\,F_1^m\,\epsilon^m\,[(m+\alpha+1)!]^{1/2},
\]
for every $\epsilon>0$.
\end{prop}

\vskip 2mm

\begin{prop}\label{goodguy2}
There exist constants $G_2$ and $F_2$, such that
\[
\norm{f\,T_N}\ <\ 
\sum_{m=N+3}^{2N+2}\ G_2\,F_2^m\,\epsilon^m\,[(m+\alpha+1)!]^{1/2},
\]
for every $\epsilon>0$.
\end{prop}

\vskip 2mm
For the third proposition, we introduce 
$\Psih^N$, which is defined as
\[
\Psih^N(\epsilon,x)\ :=\
\wh^N(\epsilon,x)\,\Phi(\epsilon x)\ +\ \Psih_\perp^N(\epsilon,x). 
\]
Here $\wh^N$ and $\Psih_\perp^N$ are given by $w^N=A\,\wh^N$ and
$\Psi_\perp^N=h_{\perp,0}^{-1}\,\tilde{A}\,\Psih_\perp^N$.

\begin{prop}\label{goodguy3}
Let $r$ and $s$ be the numbers that define the support of $f'$. Then, for 
every $\epsilon>0$ that satisfies $r^2/\epsilon^2\,\geq\,6N+2\alpha+3$, we have
\[
\norm{\left[\partial_x^2,f\right]\Psi^N}
\ <\
G_0(\epsilon)\ 
2^{3N+\alpha}\ \frac{\exp\left(-\frac{r^2}{2\epsilon^2}\right)}
{\left(\pi\frac{r}{\epsilon}\right)^\frac 12}
\ \norm{\Psih^N},
\]
where
\begin{eqnarray*}
G_0(\epsilon) 
&:=& 
\|f''\|_\infty\left(1+\nre{h_{\perp,0}^{-1}}^2\right)^{1/2}
   \ +\  8\,\|f'\|_\infty\,\|\partial_x\,A\|\,\nre{h_{\perp,0}^{-1}}
\\
&&
   \mbox{\hspace{44.3mm}}  +\  8\,\|f'\|_\infty\left(\,\|\partial_x\,A\|^2
   \ +\ \epsilon^2\sup_{y\in[-s,s]}\,\|\Phi'(y)\|^2\,\right)^{1/2}.
\end{eqnarray*}
\end{prop}

The proof of Proposition~\ref{goodguy1} is based on norm estimates for
operators that are rank one in ${\cal H}_e$ for each $x$. In particular, let 
${\cal R}:{\mathbb R}\rightarrow{\cal H}_e$ be continuous (or
just locally $L^2({\mathbb R},{\cal H}_e)$). Then clearly
$f(x){\cal R}(x)\in L^2({\mathbb R},{\cal H}_e)$, where $f$ is a cut--off
function. Furthermore, 
$f(x)\inner{{\cal R}(x)}{\ \cdot\ }:
L^2({\mathbb R},{\cal H}_e)\rightarrow L^2({\mathbb R})$
is a bounded linear operator. Since
\begin{eqnarray*}
\nr{f(x)\inner{{\cal R}(x)}{\Psi(x)}}^2
&=&
\int_{{\mathbb R}}\ f^2(x)\ \left|\inner{{\cal R}(x)}{\Psi(x)}\right|^2\ dx
\\[1mm]
&\leq&
\int_{{\mathbb R}}\ f^2(x)\ \nre{{\cal R}(x)}^2\ \nre{\Psi(x)}^2\ dx
\\[1mm]
&\leq&
\sup_{y\in\mbox{\scriptsize supp}\,(f)}\,
\nre{{\cal R}(y)}^2\ \int_{{\mathbb R}}\ \nre{\Psi(x)}^2\ dx. ,
\end{eqnarray*}
we have, 
\[
\nr{f(x)\inner{{\cal R}(x)}{\ \cdot \ }}\
\leq\ \sup_{y\in\mbox{\scriptsize supp}\,(f)}\,\nre{{\cal R}(y)}.
\]
Further consequences of this result are summarized in the following lemma:

\begin{lemma}\label{tool1}
Let ${\cal R}$ be an analytic ${\cal H}_e$--valued function, defined on the 
region $S''$ (which depends on $s>0$, $1\geq\delta>0$ and $\mu>0$). Let
$f$ be a cut--off function with support on 
$[-s/\epsilon,s/\epsilon]$. Then there exists a constant $M_{\cal R}>0$,
independent of $\epsilon>0$, such that:
\begin{enumerate}
\item[(i)]  $\displaystyle{\nr{\,f(x)\,\inner{{\cal R}(\epsilon x)}{\ \cdot \ }\,}
            \ \leq\  M_{\cal R}}$.
\item[(ii)] For each $\eta\in(-|x|,|x|)$ and any pair of integers $j$ and $l$ 
            that satisfy $j\leq i$ and $j+l\leq 0$,
            \[     
            \nr{\,f(x)\,\frac{x^{j+l}}{j!}\,\inner{{\cal R}^{(j)}(\epsilon\eta)}
            {U(P_{i\leq n}\otimes I)U^\star\ \cdot\ }\,}\ \leq\
            M_{\cal R}\ \delta^{-j}\ 2^{\frac{j+l}{2}}
            \left[\frac{(n+j+l)!}{n!}\right]^{1/2}.
            \]
\end{enumerate}
\end{lemma}

{\em Proof:} Statement (i) is obvious. To prove (ii), first note that
\[
\nr{\,f(x)\,\frac{x^{j+l}}{j!}\,\inner{{\cal R}^{(j)}(\epsilon\eta)}
{U(P_{i\leq n}\otimes I)U^\star\ \cdot\ }\,}
\ \leq\ 
\nr{\,f(x)\,\frac{1}{j!}\,\inner{{\cal R}^{(j)}(\epsilon\eta)}{\ \cdot\ }\,}
\ \nr{x^{j+l}P_{i\leq n}}.
\]
Now use (i), the Cauchy Integral Formula to estimate 
$\nre{{\cal R}^{(j)}(\epsilon\eta)}/j!$, and Lemma~5.1 of \cite{hagedorn4} to 
estimate $\nr{x^{j+l}P_{i\leq n}}$.\hfill$\Box$
\vskip 5mm

A similar result holds for analytic complex--valued functions, which we state
without proof.

\begin{lemma}\label{tool2}
Let ${\cal D}$ be an analytic complex--valued function, defined on the 
region $S''$. Under the same conditions as stated in Lemma~\ref{tool1}, there
exists a constant $M_{\cal D}>0$, independent of $\epsilon>0$, such that:
\begin{enumerate}
\item[(i)]  $\displaystyle{\nr{\,f(x)\,{\cal D}(\epsilon x)\,}
            \ \leq\  M_{\cal D}}$.
\item[(ii)] For each $\eta\in(-|x|,\,|x|)$ and any pair of integers $j$ and $l$ 
            that satisfy $j\leq i$ and $j+l\leq 0$,
            \[     
            \nr{\,f(x)\ \frac{x^{j+l}}{j!}\ {\cal D}^{(j)}(\epsilon\eta)\
            P_{i\leq n}\,}\ \leq\
            M_{\cal D}\ \delta^{-j}\ 2^{\frac{j+l}{2}}\
            \left[\frac{(n+j+l)!}{n!}\right]^{1/2}.
            \]
\end{enumerate}
\end{lemma}

{\em Sketch of the proof of Proposition~\ref{goodguy1}:\ } We have
\begin{eqnarray}\nonumber
\nr{f\,S_N} &\leq&
\sum_{m=0}^N\ \epsilon^{m+3}\
        \nr{\,f\,\inner{{\cal A}^{[N-m-1]}}{\partial_x\Psi_{\perp,m}}\,}
\\[1mm]\nonumber
&&      
+\ \epsilon^{N+4}\ \nr{\,f\,\inner{{\cal A}^{[-1]}}{\partial_x\Psi_{\perp,N+1}}\,}\
+\ \epsilon^{N+5}\ \nr{\,f\inner{{\cal A}^{[-1]}}{\partial_x\Psi_{\perp,N+2}}\,}
\\[1mm]\nonumber
&&
+\ \frac{1}{2}\ \sum_{m=0}^{N-1}\ \epsilon^{m+4}\ \nr{\,f\,{\cal B}^{[N-m-2]}\,w_m\,}\ 
+\ \frac{\epsilon^{N+4}}{2}\ \nr{\,f\,{\cal B}^{[-1]}\,w_N\,}
\\[1mm]\nonumber
&&
+\ \frac{1}{2}\ 
   \sum_{m=0}^{N-1}\ \epsilon^{m+4}\
   \nr{\,f\inner{{\cal C}^{[N-m-2]}}{\Psi_{\perp,m}}\,}
+\ \frac{\epsilon^{N+4}}{2}\ \nr{\,f\inner{{\cal C}^{[-1]}}{\Psi_{\perp,N}}\,}
\\[1mm]\nonumber
&&
+\ \frac{\epsilon^{N+5}}{2}\ \nr{\,f\inner{{\cal C}^{[-1]}}{\Psi_{\perp,N+1}}\,}\
+\ \frac{\epsilon^{N+6}}{2}\ \nr{\,f\inner{{\cal C}^{[-1]}}{\Psi_{\perp,N+2}}\,}
\\[1mm]\label{snowball}
&&
+\ \sum_{m=0}^N\ \epsilon^{m}\ \nr{\,f\,{\cal E}^{[N-m+2]}\,w_m\,}\
+\ \sum_{l=N+3}^{2N+2}\ \epsilon^l\ \sum_{m=l-N-2}^{N}\ |E_{l-m}|\,\nr{w_m}\,.
\end{eqnarray}
We now apply Lemma~\ref{tool1}, Lemma~\ref{tool2}, and Theorem~\ref{growth}.
This leads to an upper bound for $\nr{f\,S_N}$ that looks nearly
identical to the right-hand side of inequality (30) of \cite{hagtol1}. 
The statement of the proposition then follows from the proof of Theorem~3 of 
that paper.\hfill$\Box$  

\vskip 5mm

{\em Sketch of the proof of Proposition~\ref{goodguy2}:\ }
The strategy is the same as for Proposition~\ref{goodguy1}. However, there is
a complication because our expression for $T_N$ contains more terms than in the
corresponding expression in \cite{hagtol1}.

By the triangle inequality and our expression for $T_N$, we have
\begin{eqnarray}\nonumber
\norm{f\,T_N} &=&
\frac 12\ \epsilon^{N+3}\ \norm{\,f\ \partial_x^2\,\Psi_{\perp,N+1}\,}\
+\ \frac 12\ \epsilon^{N+4}\ \norm{\,f\ \partial_x^2\,\Psi_{\perp,N+2}\,}
\\[1mm]\nonumber
&&
+\ \sum_{m=0}^N\ \epsilon^{m+3}\ \norm{\,f\ (\partial_x\,w_m)\,{\cal A}^{[N-m-1]}\,}\
+\ \sum_{m=0}^N\ \epsilon^{m+3}\
\norm{\,f\ {\cal G}^{[N-m-1]}\ \partial_x\,\Psi_{\perp,m}\,}
\\[1mm]\nonumber
&&
+\ \epsilon^{N+4}\ \norm{\,f\ {\cal G}^{[-1]}\ \partial_x\,\Psi_{\perp,N+1}\,}\
+\ \epsilon^{N+5}\ \norm{\,f\ {\cal G}^{[-1]}\ \partial_x\,\Psi_{\perp,N+2}\,}
\\[1mm]\nonumber
&&
+\ \frac 12\ \sum_{m=0}^{N-1}\ \epsilon^{m+4}\ \norm{\,f\ {\cal F}^{[N-m-2]}\ w_m\,}\
+\ \frac{\epsilon^{N+4}}2\ \norm{\,f\ {\cal F}^{[-1]}\ w_N\,}
\\[1mm]\nonumber
&&
+\ \frac 12\ \sum_{m=0}^{N-1}\ \epsilon^{m+4}\ 
\norm{\,f\ {\cal K}^{[N-m-2]}\ \Psi_{\perp,m}\,}\
+\ \frac{\epsilon^{N+4}}2\ \norm{\,f\ {\cal K}^{[-1]}\ \Psi_{\perp,N}\,}
\\[1mm]\nonumber
&&
+\ \frac{\epsilon^{N+5}}2\ \norm{\,f\ {\cal K}^{[-1]}\ \Psi_{\perp,N+1}\,}\
+\ \frac{\epsilon^{N+6}}2\ \norm{\,f\ {\cal K}^{[-1]}\ \Psi_{\perp,N+2}\,}
\\[1mm]\label{exhausted}
&&
+\ \sum_{m=0}^{N+2}\ \epsilon^m\ \norm{\,f\ h_{\perp}^{[N-m+2]}\ \Psi_{\perp,m}\,}\
+\ \sum_{l=N+3}^{2N+4}\ \epsilon^l\ \sum_{m=l-N-2}^{N+2}\ 
|E_{l-m}|\ \norm{\Psi_{\perp,m}}\, . 
\end{eqnarray}
We refer to the terms in this expression (in order) as 
terms 1, 2, \dots, 14.
Terms 1--3, 7--14 are completely analogous to
terms 1--11 of equation (35) of \cite{hagtol1} (in the same order),
and they respectively satisfy the corresponding bounds
as in \cite{hagtol1}.
As with Proposition~\ref{goodguy1} the proof of this requires
the use of Lemmas~\ref{tool1}, \ref{tool2} (with slight modifications), and 
Theorem~\ref{growth}.
Terms 4--6 in (\ref{exhausted}) do not occur in equation (35) of \cite{hagtol1},
but they arise in the same way as the first three terms in (\ref{snowball})
(or equivalently as the first four terms in equation (35) of \cite{hagtol1}).
Since the operator norms of ${\cal G}^{[j]}$ and
vector norms of ${\cal A}^{[j]}$ satisfy the same types of bounds,
the norms of terms 4--6 of (\ref{exhausted}) satisfy the same
bounds as the norms of the first three terms of (\ref{snowball}).

The Proposition follows since the norm of every term 
in (\ref{exhausted}) is bounded by some
constant times $F_2^m\,\epsilon^m\,[(m+\alpha+1)!]^{1/2}$, for some $F_2$,
where $N+2\le m\le 2N+2$.\hfill$\Box$ 
%
%XXXXXXX  Last Quack of the Lame Duck!
%

\vskip 5mm
Proposition~\ref{goodguy3} is essentially a consequence of  
$\chi_{f'}$ picking up only the exponentially decaying portion 
of the lower--order eigenfunctions of the harmonic oscillator Hamiltonian $H_0$. 
More precisely, we have the following lemma.

\begin{lemma}\label{tango}
For any non-negative integer $n$ and 
every $\epsilon>0$ that satisfy $r^2/\epsilon^2\geq 2n+1$, we have
\[
\left\|\,\chi_{f'}\ P_{i\leq n}\,\right\|
\ \,<\,\
\frac{2^n\ \exp\left(-\,\frac{r^2}{2\epsilon^2}\right)}
{\left(\pi\,\frac{r}{\epsilon}\right)^\frac 12}.
\]
\end{lemma}

{\em Proof:} Consider any $\varphi\,=\,\sum_{i=0}^n\,d_i\,\phi_i$, where the 
$\phi_i$'s are the normalized eigenfunctions of $H_0$. Then,
\[
\left\|\,\chi_{f'}\,P_{i\leq n}\,\varphi\,\right\|
\ \leq\ \sum_{i=0}^n\, |d_i|\,\left\|\,\chi_{f'}\phi_i\,\right\|
\ \leq\ \left(\,\sum_{i=0}^n\,|d_i|^2\,\right)^\frac 12
     \left(\,\sum_{i=0}^n\,\left\|\,\chi_{f'}\,\phi_i\,\right\|^2\right)^\frac 12.
\]
Therefore,
\[
\left\|\,\chi_{f'}\,P_{i\leq n}\,\right\|\
\leq\ 
\left(\,\sum_{i=0}^n\,\left\|\,\chi_{f'}\,\phi_i\,\right\|^2\right)^\frac 12.
\]
Recall that
$\phi_i(x)=\sqrt{\pi^{1/2}\,2^i\,i!}\ \exp(-x^2/2)\ H_i(x)$, where
the Hermite polynomials
$H_i(x)$ satisfy the inequality $|H_i(x)|\leq 2^i |x|^i$ for $x\geq\sqrt{2i+1}$, 
as proved in Lemma~3.1 of \cite{hagedorn3}. Thus,
\begin{eqnarray*}
\sum_{i=0}^n\ \left\|\,\chi_{f'}\,\phi_i\,\right\|^2
&  = &
\sum_{i=0}^n\ \int_{\mathbb R}\ \chi_{f'}(x)\ |\phi_i(x)|^2\ dx
\\[1mm]
&\leq&
2\ \sum_{i=0}^n\ \int_{r/\epsilon}^\infty\ |\phi_i(x)|^2\ dx
\\[1mm]
&\leq&
\frac{2}{\pi^{1/2}}\
\sum_{i=0}^n\ \frac{2^i}{i!}\ \int_{r/\epsilon}^\infty\ e^{-x^2}\ x^{2i}\ dx
\\[1mm]
&  < &
\frac{2}{\pi^{1/2}}\ 4^n\
\int_{r/\epsilon}^\infty\ e^{-x^2}\
\sum_{i=0}^n\ \frac{1}{i!}\ \left(\frac{x^2}{2}\right)^i\ dx
\\[1mm]
&  < &
\frac{2}{\pi^{1/2}}\ 4^n\
\int_{r/\epsilon}^\infty\ e^{-x^2/2}\ dx.
\end{eqnarray*}
Finally, use inequality 7.1.13 of \cite{abramowitz} to 
complete the proof.\hfill $\Box$

\vskip 5mm

{\em Proof of Proposition~\ref{goodguy3}:\ } We have
\begin{eqnarray*}
\norm{\left[\partial_x^2,f\right]\Psi^N}
&\leq&
\norm{f''\,\Psi^N}\ +\ 
2\,\norm{f'\,\partial_x\,\Psi^N}\
\\[2mm]
&\leq&
\|f''\|_\infty
\norm{\chi_{f'}\,\Psi^N}\ +\
2\,\|f'\|_\infty
\norm{\chi_{f'}\,\partial_x\,\Psi^N}.
\end{eqnarray*}

The first term is estimated as follows:
\begin{eqnarray*}
\norm{\chi_{f'}\,\Psi^N}^2
& =  &
\nr{\chi_{f'}\,w^N}^2\ +\ \norm{\chi_{f'}\,\Psi_{\perp}^N}^2
\\[2mm]
&\leq&
\nr{\chi_{f'}\,A\,P_{i\leq 3N+\alpha}}^2\nr{\wh^N}^2\ +\ 
\norm{\chi_{f'}\,h_{\perp,0}^{-1}\,U(A\,P_{i\leq 3N+\alpha-2}\otimes I)U^\star}^2
\norm{\Psih_{\perp}^N}^2
\\[2mm]
&\leq&
\nr{\chi_{f'}\,P_{i\leq 3N+\alpha}}^2\nr{\wh^N}^2\ +\
\nre{h_{\perp,0}^{-1}}^2\nr{\chi_{f'}\,P_{i\leq 3N+\alpha-2}}^2
\norm{\Psih_{\perp}^N}^2
\\[2mm]
&  < &
\left(1\ +\ \nre{h_{\perp,0}^{-1}}^2\right)\nr{\chi_{f'}\,P_{i\leq 3N+\alpha}}^2
\norm{\Psih^N}^2.
\end{eqnarray*}

The second term requires a little more work. We have,
\[
\norm{\chi_{f'}\,\partial_x\,\Psi^N}\ \leq \ 
\norm{\chi_{f'}\,\partial_x\left(w^N\,\Phi\right)}\ +\
\norm{\chi_{f'}\,\partial_x\,\Psi_{\perp}^N}.
\]
Since $\inner{\Phi}{\Phi'}=0$, it follows that
\[
\inner{\partial_x\left(w^N\,\Phi\right)}{\partial_x\left(w^N\,\Phi\right)}
\ =\  \left|\partial_x\,w^N\right|^2\ 
   +\ \epsilon^2\,\left|w^N\right|^2\nre{\Phi'}^2.
\]
Therefore,
\begin{eqnarray*}
\norm{\chi_{f'}\,\partial_x\left(w^N\,\Phi\right)}^2
&  = &
\nr{\chi_{f'}\,\partial_x\,w^N}^2 \ +\ 
\epsilon^2\int_{\mathbb R}\chi_{f'}\,\left|w^N\right|^2\nre{\Phi'}^2\,dx
\\[2mm]
&\leq&
\nr{\chi_{f'}\,\partial_x\,w^N}^2 \ +\ 
\epsilon^2\sup_{x\in[-s/\epsilon,s/\epsilon]}\nre{\Phi'(\epsilon x)}^2
\nr{\chi_{f'}\,w^N}^2
\\[2mm]
&  < &
\nr{\chi_{f'}\,P_{i\leq 3N+\alpha+1}}^2\left(\nr{\partial_x\,A}^2\ +\ 
\epsilon^2\sup_{y\in[-s,s]}\nre{\Phi'(y)}^2\right)\nr{\wh^N}^2
\\[2mm]
&\leq&
\nr{\chi_{f'}\,P_{i\leq 3N+\alpha+1}}^2\left(\nr{\partial_x\,A}^2\ +\ 
\epsilon^2\sup_{y\in[-s,s]}\nre{\Phi'(y)}^2\right)\norm{\Psih^N}^2.
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
\norm{\chi_{f'}\,\partial_x\,\Psi_{\perp}^N}
&  = &
\norm{\chi_{f'}\,\partial_x\,h_{\perp,0}^{-1}\,\tilde{A}\,\Psih_{\perp}^N}
\\[2mm]
&\leq&
\nre{h_{\perp,0}^{-1}}\norm{\chi_{f'}\,
\partial_x\,U(A\,P_{i\leq 3N+\alpha-2}\otimes I)U^\star\,\Psih_{\perp}^N}
\\[2mm]
&\leq&
\nre{h_{\perp,0}^{-1}}\nr{\chi_{f'}\,\partial_x\,A\,P_{i\leq 3N+\alpha-2}}
\norm{\Psih_{\perp}^N}
\\[2mm]
&\leq&
\nre{h_{\perp,0}^{-1}}\nr{\chi_{f'}\,P_{i\leq 3N+\alpha-1}}\nr{\partial_x\,A}
\norm{\Psih^N}.
\end{eqnarray*}

We now put all the pieces together and use Lemma~\ref{tango}.\hfill$\Box$

\vskip 5mm

An immediate consequence of Propositions~\ref{goodguy1}--\ref{goodguy3} is:
\begin{thm}\label{errornorm}
There exist constants $G$, $F$, and $Q$ such that 
\begin{equation}
\label{dope}
\norm{\left(H(\epsilon) - E^N\right)\Xi^N}
\ <\ \sum_{m=N+3}^{2N+4}\,G\,F^{m}\,\epsilon^{m}\,[(m+\alpha+1)!]^{1/2}\,
+\, Q\,2^{3N}\,\exp\left(-\frac{r^2}{2\epsilon^2}\right)\,
\norm{\Psih^N},
\end{equation}
for every $N\geq 3$ and $1\geq\epsilon>0$ that satisfy the inequality 
$r^2/\epsilon^2\geq 6N+2\alpha+3$.
\end{thm}

\vskip 5mm

An easy estimate for $\norm{\Psih^N}$, which follows from  
Theorem~\ref{growth}, together with a redefinition of the constants $G$ and $F$, yield the following inequality:
\[
\norm{\left(H(\epsilon) - E^N\right)\Xi^N}
<\exp\left(-\frac{r^2}{2\epsilon^2}\right)
\sum_{m=0}^{N+2}GF^{m}\epsilon^{m}\,[(m+\alpha+1)!]^{1/2}
+
\sum_{m=N+3}^{2N+4}GF^{m}\epsilon^{m}\,[(m+\alpha+1)!]^{1/2},
\]
also valid under the conditions given in Theorem~\ref{errornorm}. This inequality
is the starting point for the proof of our main result:

\vskip 5mm

\begin{thm} Let $r$ and $F$ be the constants defined above. 
Assume $\epsilon_0 > 0$ is small enough to satisfy the inequality 
$r^2/\epsilon_0^2\geq 6N+2\alpha+3$ with 
$N\geq 3$. Define $X=\min\left\{F^{-2},r^2/3\right\}$.
Then, for every $g$ that
satisfies $0<g<X$ and for every $\epsilon_0\geq\epsilon>0$, there exists 
$N(\epsilon)$ that behaves like $g/\epsilon^2$, such that
\[
\norm{\,\left(H(\epsilon)-E^{N(\epsilon)}\right)\,\Xi^{N(\epsilon)}\,}\ <\
\Lambda\,\exp\,\left(\,-\,\Gamma/\epsilon^2\,\right).
\]
for some $\Lambda>0$ and $\Gamma>0$ independent of $\epsilon$.
\end{thm}

{\em Proof:\ } Choose $0<g<X$. Then $gF^2=e^{-\zeta}$ for some $\zeta>0$. 
Set $N(\epsilon)=\llbracket g/2\epsilon^2-\alpha/2-5/2\rrbracket$. Then
$N(\epsilon)$ satisfies both\, $r^2/\epsilon^2\geq 6N(\epsilon)+2\alpha+3$
\ and\,\, $g/\epsilon^2-\alpha-5\geq 2N(\epsilon)\geq g/\epsilon^2-\alpha-7$.

Stirling's formula implies
\[
\sum_{m=M_1}^{M_2}\,G\,F^{m}\,\epsilon^{m}\,[(m+\alpha+1)!]^{1/2}
\ <\ \Lambda_0\,\sum_{m=M_1}^{M_2}\,m^{1/4}\,e^{-m}\,
\left[F^2\,\epsilon^2\,(m+\alpha+1)\right]^{(m+\alpha+1)/2},
\]
where $\Lambda_0$ is some appropiate constant, and the sum range
$[M_1,\,M_2]$ is 
either $[1,\,N(\epsilon)+2]$ or $[N(\epsilon)+3,\,2N(\epsilon)+4]$ 
(the term with $m=0$ is ignored without consequence).    
Now, since $m\leq 2N(\epsilon)+4\leq g/\epsilon^2 -\alpha-1$,
\begin{eqnarray*}
\sum_{m=M_1}^{M_2}m^{1/4}e^{-m}
\left[F^2\,\epsilon^2\,(m+\alpha+1)\right]^{(m+\alpha+1)/2}
&\leq &
\sum_{m=M_1}^{M_2}m^{1/4}e^{-m}(gF^2)^{(m+\alpha+1)/2}
\\
& = &
\sum_{m=M_1}^{M_2}m^{1/4}e^{-m}e^{-\zeta(m+\alpha+1)/2}
\\
&< &
e^{-\zeta(M_1+\alpha+1)/2}\sum_{m=1}^{\infty}m^{1/4}e^{-m},
\end{eqnarray*}
where the series in the last inequality is clearly convergent. It thus follows
that
\[
\norm{\,\left(H(\epsilon)-E^{N(\epsilon)}\right)\,\Xi^{N(\epsilon)}\,}\ <\
\Lambda_1 \exp\left(-\frac{r^2}{2\epsilon^2}\right) \, +\,
\Lambda_2 \exp\left(-\frac{\zeta g}{2\epsilon^2}\right),
\]
for some $\Lambda_1$ and $\Lambda_2$. \hfill$\Box$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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