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E-mail addresses: hagedorn@math.vt.edu, htoloza@uaeh.reduaeh.mx 
 
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Born-Oppenheimer Approximation, semiclassical approximation, adiabatic approximation
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\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{A Time--Independent Born--Oppenheimer Approximation with
Exponentially Accurate Error Estimates}

\author{George A. Hagedorn\thanks{Partially
supported by National Science Foundation
Grants DMS--0071692 and DMS--0303586.}
\thanks{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.\\[15pt]
and\\[15pt]
Julio H. Toloza\thanks{Partially supported by Secretar\'{\i}a de Educaci\'on 
P\'ublica--PROMEP Grant UAEHGO--PTC--198.}
\thanks{E--mail: htoloza@uaeh.reduaeh.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

%\vskip 1.5cm
\begin{abstract}
We consider a simple molecular--type quantum system in which the
nuclei have one degree of freedom and the electrons have two levels. 
The Hamiltonian has the form
\[
H(\epsilon)\ =\ -\,\frac{\epsilon^4}2\,
\frac{\partial^2\phantom{i}}{\partial y^2}\ +\ h(y),
\]
where $h(y)$ is a $2\times 2$ real symmetric matrix. Near a local
minimum of an electron 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 $E_\epsilon$ and $\Psi_\epsilon$, such that
$\|\Psi_\epsilon\|\,=\,O(1)$ and
\[
\|\,(H(\epsilon)\,-\,E_\epsilon))\,\Psi_\epsilon\,\|\ 
<\ \Lambda\,\exp\,\left(\,-\,{\Gamma}/{\epsilon^2}\,\right),\qquad
\mbox{where}\quad \Gamma>0.
\] \end{abstract}

\newpage

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

The goal of this paper is to construct exponentially accurate quasimodes
for the time--indepen\-dent Schr\"odinger 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 look
at the special case of operators of this type, where $y$ is a single
real variable, and $h(y)$ is a $2\times2$ real symmetric matrix. 
Although this is a very simple model, to the best of our knowledge, 
there are no prior rigorous exponentially
accurate results for the time--independent Born--Oppenheimer
approximation. There are such results for the time--dependent Born--Oppenheimer 
approximation \cite{hagedorn5,martinez3,nenciu}
and for lifetimes of resonances \cite{martinez2,messirdi}.

To make a precise statement of our results, we need some notation and
hypotheses. For small $\epsilon$, we study the eigenvalue problem
%
\begin{equation}
\label{model}
\left[\,-\,\frac{\epsilon^4}{2}\,\frac{d^2}{dy^2}\,+\,h(y)\,\right]\,
\Psi(\epsilon,\,y)
\ =\ E(\epsilon)\ \Psi(\epsilon,\,y),
\end{equation}
%
where $\Psi\in L^2({\mathbb R},{\mathbb C}^2)$.

We rewrite the operator $h(y)$ as
%
\[
h(y)\ =\ R(y)\ \matrix{{\cal E}_1(y)}{0}{0}{{\cal E}_2(y)}\ R^T(y),
\qquad \mbox{where}\quad
R(y)\ =\ \matrix{\cos\Theta(y)}{\sin\Theta(y)}{-\sin\Theta(y)}{\cos\Theta(y)}.
\]
%
Here we assume ${\cal E}_1(y)$, ${\cal E}_2(y)$, and $\Omega(y)=\Theta'(y)$ satisfy
the following requirements:
\begin{enumerate}
\item ${\cal E}_1$, ${\cal E}_2$, and $\Omega$ are $C^{\infty}$
real--valued functions on $\mathbb R$.
\item ${\cal E}_1$ has a non-degenerate minimun. Without loss of generality,
we assume it is at $y=0$ with ${\cal E}_1(0)=0$,
${\cal E}'_1(0)=0$, and ${\cal E}''_1(0)=1$.
\item This minimum does not occur at an electron energy level crossing, {\it i.e.},
${\cal E}_2(0)\,\ne\,0$.
\item ${\cal E}_1$, ${\cal E}_2$, and $\Omega$ have analytic
extensions to a neighborhood of the region
$S_{\delta'}\,=\,\{\,z\in{\mathbb C}:\,|\mbox{Im}\,z|<\delta' +\nu\,\}$
for some $\delta'>0$ and any arbitrarily small $\nu>0$. Without loss
we may assume that $\delta'\leq 1$.
\item We assume ${\cal E}_1$ and ${\cal E}_2$ are bounded below, and that
$|{\cal E}_1(z)|$, $|{\cal E}_2(z)|$, $|\Omega(z)|$, $|\Omega'(z)|$, and
$|\Omega(z)|^2$ are all bounded by $M\,\exp\,(\tau|z|^2)$, for
$z\in S_\delta$, for some positive $\delta\le\delta'$, $M$, and $\tau$.
\end{enumerate}

Under these hypotheses, we have the following result:

\begin{thm}\label{main}
Assume the above hypotheses, and let $\alpha$ be a fixed non-negative integer.
Then, using optimal truncation of a perturbation expansion for
small $\epsilon$, we can construct $E_\epsilon$ and $\Psi_\epsilon$, such that
$\|\Psi_\epsilon\|\,=\,O(1)$,\quad $E_\epsilon\,
=\,(\alpha+1/2)\,\epsilon^2\,+\,O(\epsilon^4)$,\quad and
\[
\|\,(H(\epsilon)-E_\epsilon)\,\Psi_\epsilon\,\|\ <\
\Lambda\,\exp\,\left(\,-\,\Gamma/\epsilon^2\,\right).
\]
This quasimode is associated with the $\alpha^{\mbox{\scriptsize th}}$ vibrational
energy level in the local well of ${\cal E}_1$ near $y=0$.
\end{thm}

\vskip 2mm
\noindent
{\bf Remarks.}\quad {\bf 1.}\quad Our hypotheses allow level crossings of 
${\cal E}_1$ and ${\cal E}_2$, as long as they do not occur at the bottom of the well of
${\cal E}_1$ where $\Psi_\epsilon$ 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 The bottom of the essential spectrum is
greater than or equal to the minimum of the
lim inf of ${\cal E}_1(y)$ and ${\cal E}_2(y)$ as $y$ tends to plus or minus infinity.
For a proof, see \cite{rousse}.\\
{\bf 4.}\quad Suppose the bottom of the essential spectrum is strictly positive.
If ${\cal E}_2$ is bounded below by some positive number and
${\cal E}_1$ 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 5.}\quad The assumption that ${\cal E}_1$ and ${\cal E}_2$ are bounded below
is just used to guarantee self--adjointness of $H(\epsilon)$. It never enters
directly into our calculations, and it could be weakened.

\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}.

Among these papers, our techniques are closest to those of \cite{hagedorn1}.
However, we tried to derive exponentially accurate results by using several
variations of the perturbation expansions of \cite{hagedorn1} without success.
The expansion we use to prove Theorem \ref{main} is different, and
it does not rely on the multiple scales technique that is central to \cite{hagedorn1}.

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 has 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 lead to Bohr--Sommerfeld rules, and he can handle
level crossings and avoided crossings whose gaps have certain $\epsilon$ dependence.
Klein and Martinez and Messirdi
\cite{klein1,martinez1,martinez2,messirdi} have studied resonances whose
lifetimes are finite because of non--adiabatic transitions of the electrons.

\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}. Section \ref{prelim} contains
some technical lemmas that we use throughout the rest of the paper.
In Section \ref{growth}, we study the growth of quantities that occur
in the perturbation expansion. We then use that information in
Section \ref{hotdog} to prove the error bounds that imply Theorem \ref{main}.
A description of the explicit calculation of the error term is presented in
the Appendix.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Perturbation Expansion}\label{pert}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is convenient to restate equation (\ref{model}) in the rescaled variable
$x=y/\epsilon$. The resulting equation is
%
\begin{equation}
\label{modelscaled}
\left[\,-\,\frac{\epsilon^2}{2}\frac{d^2}{dx^2}\
+\ h(\epsilon x)\,\right]\ \Psi(\epsilon,\,x)\
=\ E(\epsilon)\ \Psi(\epsilon,\,x),
\end{equation}
%
We write $\Psi(\epsilon,\,x)$ as a linear combination of the
eigenvectors $\Phi_1(\epsilon x)$ and $\Phi_2(\epsilon x)$
of $h(\epsilon x)$. That is,
%
\[
\Psi(\epsilon,x)\ =\
w(\epsilon,x)\,\Phi_1(\epsilon x)\,+\,v(\epsilon,x)\,\Phi_2(\epsilon x).
\]
%
We substitute this expression into equation (\ref{modelscaled})
and project the resulting equation into
the $\Phi_1(\epsilon x)$ and $\Phi_2(\epsilon x)$ directions.
This transforms equation (\ref{modelscaled}) into the pair
of scalar equations,
%
\begin{equation}
-\,\frac{\epsilon^2}{2}\,\partial_x^2\,w\, -\,\epsilon^3\,\Omega\,\partial_x\,v\,
+\,\frac{\epsilon^4}{2}\,\Omega^2\,w\,
-\,\frac{\epsilon^4}{2}\,\Omega^{\prime}\,v\,
+\,{\cal E}_1\,w
\ =\ E\,w,
\label{eq1}
\end{equation}
and
\begin{equation}
-\,\frac{\epsilon^2}{2}\,\partial_x^2\,v\, +\,\epsilon^3\,\Omega\,\partial_x\,w\,
+\,\frac{\epsilon^4}{2}\,\Omega^2\,v\, +\,\frac{\epsilon^4}{2}\,\Omega^{\prime}\,w\,
+\,{\cal E}_2\,v
\ =\ E\,v,
\label{eq2}
\end{equation}
%
where we have used
%
\[
\frac{d}{dx}\,\Phi_1(\epsilon x)\,=\,-\,\epsilon\,\Omega\,\Phi_2(\epsilon x),
\quad \mbox{and}\quad
\frac{d}{dx}\,\Phi_2(\epsilon x)\,=\,\epsilon\,\Omega\,\Phi_1(\epsilon x).
\]
%

We now replace $E(\epsilon)$, $w(\epsilon,x)$,
and $v(\epsilon,x)$ by their formal Rayleigh--Schr\"odinger series
in powers of $\epsilon$,
%
\begin{eqnarray*}
E(\epsilon) & = & \sum_{n=0}^\infty\,E_n\,\epsilon^n,
\\
w(\epsilon,\,x) & = & \sum_{n=0}^\infty\,w_n(x)\,\epsilon^n,
\qquad\mbox{and}
\\
v(\epsilon,\,x) & = & \sum_{n=0}^\infty\,v_n(x)\,\epsilon^n,
\end{eqnarray*}
%
We also replace the functions $\Omega$, $\Omega^{\prime}$, ${\cal E}_1$,
and ${\cal E}_2$ by their convergent Taylor series in $\epsilon\,x$,
where we employ the following notation:
%
\begin{eqnarray*}
\Omega(\epsilon x) & = &
\sum_{n=0}^\infty\,a_n\,\epsilon^n\,x^n,
\\
\Omega^2(\epsilon x) & = &
\sum_{n=0}^\infty\,b_n\,\epsilon^n\,x^n,
\\
\Omega^{\prime}(\epsilon x) & = &
\sum_{n=0}^\infty\,c_n\,\epsilon^n\,x^n,
\\
{\cal E}_1(\epsilon x) & = &
\frac 12\,\epsilon^2 x^2\,+\,\sum_{n=3}^\infty\,d_n\,\epsilon^n\,x^n,
\qquad\mbox{and}
\\
{\cal E}_2(\epsilon x)& = &
\sum_{n=0}^\infty\,r_n\,\epsilon^n\,x^n.
\\
\end{eqnarray*}
%

We insert all these series into the equations (\ref{eq1}) and (\ref{eq2})
and equate terms of the same orders in $\epsilon$ on the two sides of the
equations. This yields the following:

\noindent
{\bf Order 0\,:}
%
\begin{eqnarray*}
E_0\,w_0&=&0,\qquad\mbox{and}\\[1mm]
r_0\,v_0&=&E_0\,v_0.
\end{eqnarray*}
%
We want $w_0\ne 0$, so the first equation requires $E_0=0$.
Then, since $r_0\ne 0$, the second equation requires $v_0=0$.

\noindent
{\bf Order 1\,:}
%
\begin{eqnarray*}
E_0\,w_1\,+\,E_1\,w_0&=&0,\quad\mbox{and}\\[1mm]
r_0\,v_1\,+\,r_1\,y\,v_0&=&E_0\,v_1\,+\,E_1\,v_0.
\end{eqnarray*}
%
From our earlier conclusions,
the first equation yields $E_1=0$. The second
equation then forces $v_1=0$.

\newpage
\noindent
{\bf Order 2\,:}\\[1.5mm]
%
Using $E_0=E_1=0$, $v_0=v_1=0$ and $d_2=1/2$ we have
\begin{eqnarray*}
-\,\frac{1}{2}\,\partial_x^2\,w_0\,+\,\frac{1}{2}\,x^2\,w_0
&=&E_2\,w_0,\quad\mbox{and}\\[2mm]
r_0\,v_2&=&0.
\end{eqnarray*}
%
From the first equation here, $E_2$ and $w_0$ must be solutions to the eigenvalue
problem for the harmonic oscillator Hamiltonian
%
\[
H_0\ =\ -\,\frac{1}{2}\,\frac{d^2}{dx^2}\,+\,\frac{1}{2}\,x^2,
\]
%
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 clearly requires $v_2=0$.

\noindent
{\bf Order 3\,:}
\begin{eqnarray*}
(H_0-E_2)\,w_1\,-\,a_0\,\partial_x\,v_0\,+\,d_3\,x^3\,w_0
&=&E_3\,w_0,\qquad\mbox{and}\\[2mm]
r_0\,v_3&=&-\,a_0\,\partial_x\,w_0.
\end{eqnarray*}
%
Equating components of the first equation in the direction of
$\phi_\alpha$, we see that $E_3=0$. The components orthogonal to $\phi_\alpha$ 
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 $\phi_\alpha$.
Note that we henceforth arbitrarily assume that $w_n\,\perp\,\phi_\alpha$ for all $n>0$.
We do not worry about normalization during these calculations.

The second equation at Order 3 simply yields
\[
v_3\ =\ -\,\frac{a_0}{r_0}\ \partial_x\,w_0.
\]

\noindent
{\bf Remark}\quad In the Order 3 calculations, $E_3$ vanishes because
$\langle\,\phi_\alpha,\,x^3\,\phi_\alpha\,\rangle\,=\,0$.
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
\noindent
{\bf Order $\mathbf{n\geq 4}$\,:}
%
\begin{eqnarray*}
&&(H_0-E_2)\,w_{n-2}\,
-\ \sum_{m=0}^{n-3}\,a_m\,x^m\,\partial_x\,v_{n-m-3}\
+\ \frac 12\,\sum_{m=0}^{n-4}\,b_m\,x^m\,w_{n-m-4}\
-\ \frac 12\,\sum_{m=0}^{n-4}\,c_m\,x^m\,v_{n-m-4}\
\\[2mm]
&&+\ \sum_{m=0}^n\,d_m\,x^m\,w_{n-m}\
=\ \sum_{m=0}^n\,E_m\,w_{n-m},
\end{eqnarray*}
and
\begin{eqnarray*}
&&-\,\frac 12\,\partial^2_x\,v_{n-2}\
+\ \sum_{m=0}^{n-3}\,a_m\,x^m\,\partial_x\,w_{n-m-3}\
+\ \frac 12\,\sum_{m=0}^{n-4}\,b_m\,x^m\,v_{n-m-4}\
+\ \frac 12\,\sum_{m=0}^{n-4}\,c_m\,x^m\,w_{n-m-4}\
\\[2mm]
&&+\ \sum_{m=0}^n\,r_m\,x^m\,v_{n-m}\
=\ \sum_{m=0}^n\,E_m\,v_{n-m}.
\end{eqnarray*}
We 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 operator
$(H_0-E_2)^{-1}_\perp$.
We then solve the second equation for $v_n$ by dividing
by $r_0$.

Thus, for $n\ge 4$, we have
\begin{eqnarray*}
E_n
&=&
-\ \sum_{m=0}^{n-3}\,
a_m\,\langle\,\phi_\alpha,\,x^m\,\partial_{x}\,v_{n-m-3}\,\rangle\
+\ \frac 12\,\sum_{m=0}^{n-4}\,
b_m\,\langle\,\phi_\alpha,\,x^m\,w_{n-m-4}\,\rangle
\\[2mm]&&
-\ \frac 12\,\sum_{m=0}^{n-4}\,
c_m\,\langle\,\phi_\alpha,\,x^m\,v_{n-m-4}\,\rangle\
+\ \sum_{m=0}^n\,
d_m\,\langle\,\phi_\alpha,\,x^m\,w_{n-m}\rangle,
\\[5mm]
%
w_{n-2}
&=&(H_0-E_2)^{-1}_\perp\ \left[\
\sum_{m=0}^{n-3}\,a_m\,x^m\,\partial_{x}\,v_{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-4}\,c_m\,x^m\,v_{n-m-4}\
-\ \sum_{m=0}^n\,d_m\,x^m\,w_{n-m}\
+\ \sum_{m=0}^n\,E_m\,w_{n-m}\ \right],
\\[2mm]
\mbox{and}
\\[2mm]
v_n
&=&\frac 1{r_0}\ \left[\
\frac 12\,\partial_x^2\,v_{n-2}\
-\ \sum_{m=0}^{n-3}\,a_m\,x^m\,\partial_x\,w_{n-m-3}\
-\ \frac 12\,\sum_{m=0}^{n-4}\,b_m\,x^m\,v_{n-m-4}\,\right.
\\[2mm]
&&\phantom{\frac 1{r_0}\,\ }\left.
-\ \frac 12\,\sum_{m=0}^{n-4}\,c_m\,x^m\,w_{n-m-4}\
-\ \sum_{m=1}^n\,r_m\,x^m\,v_{n-m}\
+\ \sum_{m=0}^n\,E_m\,v_{n-m}\ \right].
\end{eqnarray*}

\vskip 5mm
To facilitate certain estimations, we introduce a bounded operator
$A$ that was originally introduced in \cite{toloza1,toloza2}.
This operator depends on the choice of $\alpha$ and
is diagonal in the basis of Harmonic Oscillator eigenfunctions:
\[
A\,\phi_\beta\ =\ \left\{\,
\begin{array}{cl}
\phi_\beta&\mbox{if}\quad\beta=\alpha,\\[1.5mm]
|\beta-\alpha|^{-1/2}\,\phi_\beta\ &\mbox{if}\quad\beta\ne\alpha.\end{array}
\right.
\]
This operator is useful for the following reasons:
The estimate $\|(H_0-E_2)^{-1}_\perp\,\psi\|\,\le\,\|\psi\|$
is true for all $\psi$, but it is a very poor estimate if 
$\psi\,=\,\sum_{\beta\ge B}\,k_\beta\,\phi_\beta$, where $B$ is large. 
However, the analogous estimate for the inverse of the restriction of
$A\,(H_0-E_2)\,A$ to the orthogonal complement of its kernel is optimal
because $\|\,A\,(H_0-E_2)\,A\,\psi\,\|\,=\,\|\,\psi\,\|$ for any 
$\psi\,\perp\,\phi_\alpha$.

Rather than estimating norms of $w_n$ and $v_n$ directly, we study
$\hat{w}_n$ and $\hat{v}_n$, where $w_n\,=\,A\,\hat{w}_n$ and $v_n\,=\,A\,\hat{v}_n$.
In addition, we multiply equation (\ref{eq1}) by $A$.
Note that $A^{-1}$ is an unbounded, densely defined linear operator,
but that any finite linear combination of $\phi_\beta$'s is in its domain.

After a little bit of rearranging, 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]
\hat{w}_1&=&-\,d_3\,[A(H_0-E_2)A]^{-1}_\perp\,A\,x^3\,A\,\hat{w}_0,
\\[2mm]
\hat{v}_0&=&\hat{v}_1\ =\ \hat{v}_2\ =0,
\\[2mm]
\displaystyle\hat{v}_3&=&-\,\frac{a_0}{r_0}\ A^{-1}\,\partial_x\,A\,\hat{w}_0,
\end{eqnarray*}
and for $n\ge 4$,
\begin{eqnarray*}
E_n
&=&
-\ \sum_{m=0}^{n-3}\,
a_m\,\langle\,A\,\partial_x\,x^m\,A\,\phi_\alpha,\,\hat{v}_{n-m-3}\,\rangle\
+\ \frac 12\,\sum_{m=0}^{n-4}\,
b_m\,\langle\,A\,x^m\,A\,\phi_\alpha,\,\hat{w}_{n-m-4}\,\rangle
\\[2mm]&&
-\ \frac 12\,\sum_{m=0}^{n-4}\,
c_m\,\langle\,A\,x^m\,A\,\phi_\alpha,\,\hat{v}_{n-m-4}\,\rangle\
+\ \sum_{m=0}^n\,
d_m\,\langle\,A\,x^m\,A\,\phi_\alpha,\,\hat{w}_{n-m}\rangle,
\\[4mm]
%
\hat{w}_{n-2}
&=&[A(H_0-E_2)A]^{-1}_\perp\ \left[\
\sum_{m=0}^{n-3}\,a_m\,A\,x^m\,\partial_x\,A\,\hat{v}_{n-m-3}\
-\ \frac 12\,\sum_{m=0}^{n-4}\,b_m\,A\,x^m\,A\,\hat{w}_{n-m-4}\,\right.
\\[2mm]
&&\phantom{[A(H_0-E_2)A]^{-1}_\perp\ \,\ }
+\ \frac 12\,\sum_{m=0}^{n-4}\,c_m\,A\,x^m\,A\,\hat{v}_{n-m-4}\
-\ \sum_{m=0}^n\,d_m\,A\,x^m\,A\,\hat{w}_{n-m}
\\[2mm]
&&\phantom{[A(H_0-E_2)A]^{-1}_\perp\ \,\ }\left.
+\ \sum_{m=0}^n\,E_m\,A^2\,\hat{w}_{n-m}\ \right],
\\[2mm]
\mbox{and}
\\[2mm]
\hat{v}_n
&=&\frac 1{r_0}\ \left[\
\frac 12\,A^{-1}\,\partial^2_x\,A\hat{v}_{n-2}\
-\ \sum_{m=0}^{n-3}\,a_m\,A^{-1}\,x^m\,\partial_x\,A\hat{w}_{n-m-3}\right.
\\[2mm]
&&\phantom{\frac 1{r_0}\,\ \ }
-\ \frac 12\,\sum_{m=0}^{n-4}\,b_m\,A^{-1}\,x^m\,A\,\hat{v}_{n-m-4}\
-\ \frac 12\,\sum_{m=0}^{n-4}\,c_m\,A^{-1}\,x^m\,A\,\hat{w}_{n-m-4}\
\\[2mm]
&&\phantom{\frac 1{r_0}\,\ \ }\left.
-\ \sum_{m=1}^n\,r_m\,A^{-1}\,x^m\,A\,\hat{v}_{n-m}
+\ \sum_{m=0}^n\,E_m\,\hat{v}_{n-m}\ \right].
\end{eqnarray*}

\vskip .5cm

We conclude this section with a crucial observation:\ If we let $P_{i\le n}$ 
denote the projection onto the span of $\phi_i$ with $i\le n$,
then for $n\geq 0$ we have $\wh_{n}\in\mbox{Ran}(P_{i\leq 3n+\alpha})$, and for 
$n\geq 3$ we have $\vh_{n}\in\mbox{Ran}(P_{i\leq 3n+\alpha-8})$.
We prove this by first using our explicit formulas above to see that 
$\wh_0\in\mbox{Ran}(P_{i\le\alpha})$, $\wh_1\in\mbox{Ran}(P_{i\le 3+\alpha})$,
and $\vh_{3}\in\mbox{Ran}(P_{i\leq 1+\alpha})$. We then use an easy induction 
to prove the general results
by using our recursive formulas for $\wh_{n-2}$ and $\vh_n$ for $n\ge 4$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminary Estimates}\label{prelim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we state and prove a number of technical results that we need later.

\begin{lemma}\label{axma} 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 
$\partial_x$. Let $A$ be the operator defined in the previous section. 
For $l\geq 1$ and $n\geq 0$, the following inequalities hold.
\begin{enumerate}
\item[(i)] $\displaystyle{
\|A(\#_{x})^{l}A\,P_{i\leq n}\|\ \leq\ \left\{\,
\begin{array}{ll}
[\,2\,(2+\alpha)\,]^{1/2},&\mbox{when}\quad l=1,\\[1mm]
(2+\alpha)\,2^{l/2}\,\left[\,\displaystyle
\frac{(n+l-1)!}{(n+1)!}\,\right]^{1/2},\ &\mbox{when}\quad l>1.
\end{array}\right.}$
\item[(ii)] $\displaystyle{
\|A^{-1}(\#_{x})^{l}A\,P_{i\leq n}\|\ \leq\
[\,\alpha\,(2+\alpha)\,]^{1/2}\,2^{l/2}\,\left[\frac{(n+l+1)!}{(n+1)!}\right]^{1/2}}$.
\end{enumerate} \end{lemma}

{\bf Proof:}\ Statement (i) has been proved in Lemma 1 of \cite{toloza1}. To prove
(ii), we note that
\begin{equation}
\label{partii}
\|A^{-1}(\#_{x})^{l}A\,P_{i\leq n}\|\
\leq\ \|A^{-1}P_{i\leq n+l}\|\,\|(\#_{x})^{l-1}P_{i\leq n+1}\|\,\|\#_{x}P_{i\leq n}\|
\end{equation}
From the definition of the operator $A$, it is clear that
\[
\|A^{-1}P_{i\leq n+l}\|\ =\ |n+l-\alpha|^{1/2}.
\]
By Lemma 5.1 of \cite{hagedorn4}, the second factor in (\ref{partii}) satisfies the bound
\[
\|(\#_{x})^{l-1}P_{i\leq n+1}\|\
\leq\ 2^{(l-1)/2}\,\left[\frac{(n+l)!}{(n+1)!}\right]^{1/2}.
\]
Finally, the proof of Lemma 1 of \cite{toloza1} shows that
\[
\|\#_{x}P_{i\leq n}\|\
\leq\ [\,2\,(2+\alpha)\,]^{1/2}.
\]
Therefore,
\[
\|A^{-1}(\#_{x})^{l}AP_{i\leq n}\|\
\leq\ (2+\alpha)^{1/2}\,2^{l/2}\
\left[\frac{(n+l+1)!}{(n+1)!}\right]^{1/2}\
\max_{n,l\geq 0}\ \left[\frac{|n+l-\alpha|}{n+l+1}\right]^{1/2}.
\]
This last expression is clearly bounded by the right hand side of (ii).\hfill$\Box$

\begin{lemma}\label{facts} Consider any real number $\kappa\geq 2$.
\begin{enumerate}
\item[(i)] For each $\alpha\geq 0$ there exists a constant $C_{1}=C_{1}(\alpha)$ 
such that, for all $m\geq 0$,
\[
\sum_{l=0}^{m}\,
\left[\,\frac{(m-l+\alpha+1)!\,(l+\alpha+1)!}{(m+\alpha+1)!}\,\right]^{1/2}\ \leq\ C_{1}.
\]
\item[(ii)] For all $\alpha\geq -1$ there exists a constant $C_{2}$ such that, 
for all $m\geq 0$,
\[
\sum_{l=0}^{m}\,\kappa^{-5l/2}\,\left[\frac{(3m-2l+\alpha+1)!\,(m-l+\alpha+1)!}
{(3m-3l+\alpha+1)!\,(m+\alpha+1)!}\,\right]^{1/2}
\ \leq\ C_{2}.
\]
\item[(iii)] For each $\alpha\geq 0$ there exists a constant $C_{3}=C_{3}(\alpha)$ 
such that, for all $m\geq 0$,
\[
\sum_{l=1}^{m}\,\kappa^{-5l/2}\,\left[\,\frac{(m-l+\alpha+1)!\,(l+\alpha+1)!}
{(\alpha+1)!\,(m+\alpha)!}\,\right]^{1/2}
\ \leq\ C_{3}.
\]
\end{enumerate} \end{lemma}

{\bf Proof:} See Lemma 4 of \cite{toloza2}.

\begin{lemma}\label{aboutf}
Let $f$ be a $C^{\infty}$ real-valued function that has an analytic extension to the
region \[
S_\delta\ =\ \{z\in{\mathbb C}:|\mbox{Im}\, z|<\delta +\nu\}
\]
for some $\delta>0$, and any arbitrarily small $\nu>0$. Furthermore, assume that
$f$ is exponentially bounded on $S_{\delta}$, in the sense that there exist certain 
constants $M>0$ and $\tau>0$ for which
\[
|f(z)|\ \leq\ M\exp\left(\tau |z|^2\right)
\] for all $z\in S_{\delta}$. Then,
\begin{enumerate}
\item[(i)] For each $\epsilon\in(0,\,(4\tau)^{-1/2})$,
\[
\|\,f(\epsilon x)\,P_{i\leq n}\,\|\
<\ M\,\left[\,e^{1/4}\,+\,\left(\frac{32}{9}\right)^{1/4}\,\right]\,2^{n}.
\]
\item[(ii)] For each $\epsilon\in(0,\,(8\tau)^{-1/2})$, 
$\eta\in(-\epsilon|x|,\,\epsilon|x|)$, and any pair of integers $j$ and 
$l$ that satisfy $j\geq 1$, $j+l\geq 0$,
\[
\left\|\,\frac{f^{(j)}(\eta)}{j!}\,x^{j+l}\,P_{i\leq n}\,\right\|\ <\
\delta^{-j}\,M\,e^{2\tau\delta^2}\,
\left[\,e^{1/4}+\left(\frac{32}{9}\right)^{1/4}\,\right]
\,2^{\frac{2n+j+l}{2}}\,\left[\frac{(n+j+l)!}{n!}\right]^{1/2}.
\]
\end{enumerate}
\end{lemma}

\vskip 3mm
{\bf Proof:} Set $R=\sqrt{2n+1}$. Let $\chi_{R}$ be the characteristic function of 
the interval $[-R,R]$. We have
\begin{equation}
\label{fp} \|\,f(\epsilon x)\,P_{i\leq n}\,\|\ \leq\
\|\,f(\epsilon x)\,[1-\chi_{R}(x)]\,P_{i\leq n}\,\|
\,+\,\|\,f(\epsilon x)\,\chi_{R}(x)\,P_{i\leq n}\|.
\end{equation}
We begin by estimating the first term. Consider any harmonic oscillator 
eigenfunction $\phi_i$ with $0\leq i\leq n$. We have
\[
\|\,f(\epsilon x)\,[1-\chi_{R}(x)]\,\phi_i(x)\,\|^2\ =\
\int_{R}^{\infty}\,|f(\epsilon x)|^2\,|\phi_i(x)|^2\,dx\,+\,
\int_{-\infty}^{-R}\,|f(\epsilon x)|^2\,|\phi_i(x)|^2\,dx.
\]
By Lemma~3.1 of \cite{hagedorn3}, the Hermite polynomials of order 
$i\le n$ satisfy $|H_i(x)|\leq 2^{i}|x|^{i}$ whenever $|x|\geq R$. Therefore,
\begin{eqnarray}
\|\,f(\epsilon x)\,[1-\chi_{R}(x)]\,\phi_i(x)\,\|^2
&\leq& M^{2}\,\frac{2^{i+1}}{\pi^{1/2}\,i!}\,
\int_{R}^{\infty}\,e^{-(1-2\tau\epsilon^2)x^2}\,x^{2i}\,dx\nonumber
\\[2mm]
& < & M^{2}\,\frac{2^{i+1}}{\pi^{1/2}\,i!}\,\int_{0}^{\infty}\,
e^{-\frac{1}{2}x^2}\,x^{2i}\,dx\label{1/2}
\\[2mm]
& = & \sqrt{\frac{2}{\pi}}\,M^2\,\frac{4^{i}}{i!}\,\Gamma(i+1/2).\nonumber 
\end{eqnarray}
In (\ref{1/2}), we use that $\epsilon^2\leq (4\tau)^{-1}$
implies $1-2\tau\epsilon^2\geq 1/2$. The Schwarz inequality implies \[
\|\,f(\epsilon x)\,[1-\chi_{R}(x)]\,P_{i\leq n}\,\|^{2}\ \leq\
\sqrt{\frac{2}{\pi}}\ M^2\ \sum_{i=0}^{n}\ \frac{4^{i}}{i!}\ \Gamma(i+1/2).
\]
We now use the following crude approximation to estimate the sum in this expression:
\begin{eqnarray}
\sum_{i=0}^{n}\,\frac{4^{i}}{i!}\,\Gamma(i+1/2)
& = & \Gamma(1/2)\ +\ \sum_{i=1}^{n}\,\frac{4^{i}}{i!}\,\Gamma(i+1/2)\nonumber
\\[1mm]
& < & \sqrt{\pi}\ +\ \sum_{i=1}^{n}\,\frac{4^{i}}{i!}\,\Gamma(i+1)\nonumber
\\[1mm]
& < & \sqrt{\pi}\ \sum_{i=0}^{n}\,4^{i}\nonumber
\\[1mm]
& < & \sqrt{\pi}\ \frac{4^{n+1}}{3}.\label{geom1}
\end{eqnarray}
Therefore,
\begin{equation}
\label{fp1}
\|\,f(\epsilon x)\,[1-\chi_{R}(x)]\,P_{i\leq n}\,\|\ <\ M\,
\left(\frac{32}{9}\right)^{1/4}2^{n}.
\end{equation}
To estimate the second term in (\ref{fp}), we notice that the growth condition 
imposed to the function $f$ implies that
\[
|\,f(\epsilon x)\,\chi_{R}(x)\,|\ \leq\ M\,e^{\tau\epsilon^{2}(2n+1)}
\]
Hence,
\begin{equation}
\label{fp2} \|\,f(\epsilon x)\,\chi_{R}(x)\,P_{i\leq n}\,\|\ 
\leq\ M\,e^{\tau\epsilon^{2}(2n+1)}\,\|P_{i\leq n}\|\
\leq\ M\,e^{(2n+1)/4}.
\end{equation}
The inequality (i) then follows from (\ref{fp1}) and (\ref{fp2}).

To prove (ii), we proceed as in the proof of (i), with minor changes. As before, we have
\begin{equation}
\label{djfp} \left\|\frac{f^{(j)}(\eta)}{j!}x^{j+l}P_{i\leq n}\right\|\ \leq\
\left\|\frac{f^{(j)}(\eta)}{j!}x^{j+l}[1-\chi_{R}(x)]P_{i\leq n}\right\|\ +\
\left\|\frac{f^{(j)}(\eta)}{j!}x^{j+l}\chi_{R}(x)P_{i\leq n}\right\|.
\end{equation}
We apply the Cauchy integral formula to $f$ and use the growth condition imposed on 
$f$ to see that
\begin{equation}
\label{cau} \left|\,\frac{f^{(j)}(\eta)}{j!}\,\right|\ \leq\
\delta^{-j}\,M\,e^{\tau(\epsilon|x|+\delta)^{2}}\ \leq\
\delta^{-j}\,M\,e^{2\tau\delta^{2}}\,e^{2\tau\epsilon^2 x^{2}},
\end{equation}
for $\eta\in(-\epsilon |x|,\,\epsilon |x|)$. We now use this inequality to 
estimate the first term in (\ref{djfp}). We have
\begin{eqnarray*}
\left\|\,\frac{f^{(j)}(\eta)}{j!}\,x^{j+l}\,[1-\chi_{R}(x)]\,\phi_i(x)\,\right\|^2
&\leq& \delta^{-2j}\,M^2\,e^{4\tau\delta^2}\,\frac{2^{i+1}}{\pi^{1/2}\,i!}\,
\int_{R}^{\infty}\,e^{-(1-4\tau\epsilon^2)x^2}\,x^{2(i+j+l)}\,dx
\\[2mm]
&\leq& \delta^{-2j}\,M^2\,e^{4\tau\delta^2}\,\frac{2^{i+1}}{\pi^{1/2}\,i!}\,
\int_{0}^{\infty}\,e^{-\frac{1}{2}x^2}\,x^{2(i+j+l)}\,dx
\\[2mm]
& = & \sqrt{\frac{2}{\pi}}\,\delta^{-2j}\,M^2\,e^{4\tau\delta^2}\,
\frac{2^{2i+j+l}}{i!}\,\Gamma(i+j+l+1/2).
\end{eqnarray*}
Consequently,
\[
\left\|\frac{f^{(j)}(\eta)}{j!}x^{j+l}[1-\chi_{R}(x)]P_{i\leq n}\right\|
\ \leq\ \delta^{-j}Me^{2\tau\delta^2}\left(\frac{2}{\pi}\right)^{1/4}
2^{(j+l)/2}\left[\,
\sum_{i=0}^{n}\,4^{i}\,\frac{\Gamma(i+j+l+1/2)}{i!}\,\right]^{1/2}.
\]
We now consider two cases to handle the sum in this expression.
If $j+l=0$, we use the inequality (\ref{geom1}). If $j+l\geq 1$, we have
\begin{eqnarray*}
\sum_{i=0}^{n}\,4^{i}\,\frac{\Gamma(i+j+l+1/2)}{i!}
& < & \sum_{i=0}^{n}\,4^{i}\,\frac{(i+j+l)!}{i!}
\\[1mm]
& < & \frac{(n+j+l)!}{n!}\,\sum_{i=0}^{n}\,4^{i}
\\[1mm]
& < & \frac{4}{3}\,4^{n}\,\frac{(n+j+l)!}{n!}. \end{eqnarray*}
When $j+l=0$, this inequality differs from (\ref{geom1}) by a factor $\sqrt{\pi}$. 
Therefore, in general we have
\[
\left\|\,\frac{f^{(j)}(\eta)}{j!}\,x^{j+l}\,[1-\chi_{R}(x)]\,P_{i\leq n}\,\right\|
\ \leq\ \delta^{-j}Me^{2\tau\delta^2}\left(\frac{32}{9}\right)^{1/4}
2^{(2n+j+l)/2}\left[\frac{(n+j+l)!}{n!}\right]^{1/2}.
\] For the second term in (\ref{djfp}) we have
\begin{eqnarray}
\left\|\,\frac{f^{(j)}(\eta)}{j!}\,x^{j+l}\,\chi_{R}(x)\,P_{i\leq n}\,\right\|
&\leq&\delta^{-j}\,M\,e^{2\tau\delta^2}\,e^{2\tau\epsilon^2R^2}\,
\left\|x^{j+l}P_{i\leq n}\right\|\label{cauchy}
\\[1mm]
&\leq&\delta^{-j}\,M\,e^{2\tau\delta^2}\,e^{(2n+1)/4}\,
2^{(j+l)/2}\,\left[\frac{(n+j+l)!}{n!}\right]^{1/2}\label{hag}
\\[1mm]
& < &\delta^{-j}\,M\,e^{2\tau\delta^2}\,e^{1/4}\,
2^{(2n+j+l)/2}\,\left[\frac{(n+j+l)!}{n!}\right]^{1/2}\label{e}, \end{eqnarray}
where we have used (\ref{cau}) to get (\ref{cauchy}), Lemma~5.1 of \cite{hagedorn4} 
to get (\ref{hag}), and the fact that $e^{1/2}<2$ to finally obtain (\ref{e}).
\hfill$\Box$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Growth of the Perturbation Coefficients}\label{growth}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we prove that the coefficients of the formal Rayleigh--Schr\"odinger 
series for the energy $E(\epsilon)$ and the scalar functions $w(\epsilon x)$ and 
$v(\epsilon x)$ obey the following upper bounds.

\begin{thm}\label{bounds}
Define $\kappa:=2/\delta^{2}$. %
%Let $b_{1}$, $b_{2}$, and $b_{3}$ be certain %constants to be defined later. For %
There exist constants $b_{1}$, $b_{2}$, and $b_{3}$,
such that for $b\geq\{b_{1},b_{2},b_{3},1\}$ and $n\geq 3$ we have
\begin{eqnarray*}
|E_n| & < & b_{1}\,\kappa^{3(n-2)}\,b^{2(n-3)}\,[(n+\alpha-2)!]^{1/2},
\\[3mm]
\|\wh_{n-2}\| & < & b_{2}\,\kappa^{3(n-2)}\,b^{2(n-2)}\,[(n+\alpha-1)!]^{1/2},
\qquad\mbox{and}
\\[3mm]
\|\vh_{n}\| & < & b_{3}\,\kappa^{3n}\,b^{2(n-1)}\,[(n+\alpha-1)!]^{1/2}.
\end{eqnarray*} It follows that
\begin{eqnarray*}
|E_n| & < & \kappa^{3(n-2)}\,b^{2n-5}\,[(n+\alpha-2)!]^{1/2},
\\[3mm]
\|\wh_{n-2}\| & < & \kappa^{3(n-2)}\,b^{2n-3}\,[(n+\alpha-1)!]^{1/2},\qquad\mbox{and}
\\[3mm]
\|\vh_{n}\| & < & \kappa^{3n}\,b^{2n-1}\,[(n+\alpha-1)!]^{1/2}.
\end{eqnarray*} \end{thm}

We prove these estimates by mimicking the techniques used in \cite{toloza2} to study 
the semiclassical limit for the time-independent Schr\"odinger equation.
We begin by using the recursive formulas of Section \ref{pert} to
obtain the following set of inequalities:
\begin{eqnarray*}
|E_{n}|
&\leq& \sum_{m=0}^{n-3}\,|a_{m}|\,\|A\partial_{x}x^{m}A\phi_{\alpha}\|\,\|\vh_{n-m-3}\|
\ +\ \frac{1}{2}\,\sum_{m=0}^{n-4}\,|b_{m}|\,
\|Ax^{m}A\phi_{\alpha}\|\,\|\wh_{n-m-4}\|
\\
& & +\ \frac{1}{2}\,\sum_{m=0}^{n-4}\,|c_{m}|\,\|Ax^{m}A\phi_{\alpha}\|\,
\|\vh_{n-m-4}\|
\ +\ \sum_{m=3}^{n}\,|d_{m}|\,\|Ax^{m}A\phi_{\alpha}\|\,\|\wh_{n-m}\|,
\\[3mm]
\|\wh_{n-2}\|
&\leq& \sum_{m=0}^{n-3}\,|a_{m}|\,\|Ax^{m}\partial_{x}A\vh_{n-m-3}\|\
+\ \frac{1}{2}\,\sum_{m=0}^{n-4}\,|b_{m}|\,\|Ax^{m}A\wh_{n-m-4}\|
\\
& & \ +\ \frac{1}{2}\,\sum_{m=0}^{n-4}\,|c_{m}|\,\|Ax^{m}A\vh_{n-m-4}\|\
+\ \sum_{m=3}^{n}\,|d_{m}|\,\|Ax^{m}A\wh_{n-m}\|\
+\ \sum_{m=3}^{n}\,|E_{m}|\,\|\wh_{n-m}\|,
\\[3mm]
\|\vh_{n}\|
&\leq& |r_{0}|^{-1}\ \left[\ \frac{1}{2}\,\|A^{-1}\partial_{x}^{2}A\vh_{n-2}\|\
+\ \sum_{m=0}^{n-3}\,|a_{m}|\,\|A^{-1}x^{m}\partial_{x}A\wh_{n-m-3}\|\,\right.
\\
& & \qquad\qquad\left. +\
\frac{1}{2}\,\sum_{m=0}^{n-4}\,|b_{m}|\,\|A^{-1}x^{m}A\vh_{n-m-4}\|\
+\ \frac{1}{2}\,\sum_{m=0}^{n-4}\,|c_{m}|\,\|A^{-1}x^{m}A\wh_{n-m-4}\|\right.
\\
& & \qquad\qquad\left. +\
\sum_{m=1}^{n-3}\,|r_{m}|\,\|A^{-1}x^{m}A\vh_{n-m}\|\
+\ \sum_{m=1}^{n-3}\,|E_{m}|\,\|\vh_{n-m}\|\ \right].
\end{eqnarray*}
Because of hypothesis (4), we can use the Cauchy integral formula to
show that there exists a constant $D>0$ such that
\begin{eqnarray*}
|a_{n}| &\leq& D\,\delta^{-n},
\\
|b_{n}| &\leq& (n+1)\,D\,\delta^{-n},
\\
|c_{n}| &\leq& (n+1)\,D\,\delta^{-n-1},
\\
|d_{n}| &\leq& D\,\delta^{-n},\qquad\mbox{and}
\\
|r_{n}| &\leq& D\,\delta^{-n},
\end{eqnarray*} 
for all $n\geq 0$. From the end of Section \ref{pert}, we know that
$\wh_{n}\in\mbox{Ran}(P_{i\leq 3n+\alpha})$ and 
$\vh_{n}\in\mbox{Ran}(P_{i\leq 3n+\alpha-8})$, 
the former for $n\geq 0$ and the latter for $n\geq 3$. Therefore, we have
\begin{eqnarray}
\nonumber
|E_{n}|
&\leq& D\,\sum_{m=0}^{n-6}\,\delta^{-m}\,\|A\partial_{x}x^{m}AP_{i\leq\alpha}\|\,
\|\vh_{n-m-3}\|
\\ \nonumber
& & +\ \frac{D}{2}\,
\sum_{m=0}^{n-4}\,(m+1)\,\delta^{-m}\,\|Ax^{m}AP_{i\leq\alpha}\|\,\|\wh_{n-m-4}\|
\\ \nonumber
& & +\ \frac{D}{2}\,\delta^{-1}\,
\sum_{m=0}^{n-7}\,(m+1)\,\delta^{-m}\,\|Ax^{m}AP_{i\leq\alpha}\|\,\|\vh_{n-m-4}\|
\\ \nonumber
& & +\ D\,\sum_{m=3}^{n}\,\delta^{-m}\,\|Ax^{m}AP_{i\leq\alpha}\|\,\|\wh_{n-m}\|,
\\[2mm] \nonumber
\|\wh_{n-2}\|
&\leq& D\,\sum_{m=0}^{n-6}\,\delta^{-m}\,
\|Ax^{m}\partial_{x}AP_{i\leq 3(n-m-3)+\alpha -8}\|\,\|\vh_{n-m-3}\|
\\ \nonumber
& & +\ \frac{D}{2}\,\sum_{m=0}^{n-4}\,(m+1)\,\delta^{-m}
\,\|Ax^{m}AP_{i\leq 3(n-m-4)+\alpha}\|\,\|\wh_{n-m-4}\|
\\ \nonumber
& & +\ \frac{D}{2}\,\delta^{-1}\,
\sum_{m=0}^{n-7}\,(m+1)\,\delta^{-m}\,\|Ax^{m}AP_{i\leq 3(n-m-4)+\alpha -8}\|
\,\|\vh_{n-m-4}\|
\\ \nonumber
& & +\ D\,\sum_{m=3}^{n}\,\delta^{-m}\,\|Ax^{m}AP_{i\leq 3(n-m)+\alpha}\|\,
\|\wh_{n-m}\|\ +\ \sum_{m=3}^{n}\,|E_{m}|\,\|\wh_{n-m}\|,
\\[2mm]\nonumber
\|\vh_{n}\|
&\leq& \frac{1}{2}\,|r_{0}|^{-1}\,\|A^{-1}\partial_{x}^{2}AP_{i\leq 3(n-2)+\alpha -8}\|
\,\|\vh_{n-2}\|
\\ \nonumber
& & +\ D\,|r_{0}|^{-1}\,\sum_{m=0}^{n-3}\,\delta^{-m}\,
\|A^{-1}x^{m}\partial_{x}AP_{i\leq 3(n-m-3)+\alpha}\|\,\|\wh_{n-m-3}\|
\\ \label{woof}
& & +\ \frac{D}{2}\,|r_{0}|^{-1}\,
\sum_{m=0}^{n-7}\,(m+1)\,\delta^{-m}\,\|A^{-1}x^{m}AP_{i\leq 3(n-m-4)+\alpha-8}\|
\,\|\vh_{n-m-4}\|
\\ \nonumber
& & +\ \frac{D}{2}\,|r_{0}|^{-1}\,\delta^{-1}\,
\sum_{m=0}^{n-4}\,(m+1)\,\delta^{-m}\,\|A^{-1}x^{m}AP_{i\leq 3(n-m-4)+\alpha}\|
\,\|\wh_{n-m-4}\|
\\ \nonumber
& &+\ D\,|r_{0}|^{-1}\,\sum_{m=1}^{n-3}\,\delta^{-m}\,
\|A^{-1}x^{m}AP_{i\leq 3(n-m)+\alpha-8}\|\,\|\vh_{n-m}\|\
+\ |r_{0}|^{-1}\,\sum_{m=1}^{n-3}\,|E_{m}|\,\|\vh_{n-m}\|.
\end{eqnarray}
To obtain these inequalities we have used $E_{0}=E_{1}=0$ and $\vh_{0}=\vh_{1}=\vh_{2}=0$. 
Also, empty sums whose upper limits are less than their lower limits
are regarded as zero. We also regard the first
term on the right hand side of (\ref{woof}) to be equal to zero for $n\leq 4$.

Using Lemma \ref{axma} and recalling that $\kappa=2/\delta^2$, we obtain
\begin{eqnarray}
|E_{n}|
&\leq& (2+\alpha)\,2^{1/2}\,D\,\sum_{m=1}^{n-6}\,\kappa^{m/2}\,
\left[\frac{(m+\alpha)!}{(\alpha+1)!}\right]^{1/2}\,
\|\vh_{n-m-3}\|\nonumber
\\[1mm]
& & +\ (4+2\alpha)^{1/2}\,D\,\|\vh_{n-3}\|\nonumber
\\[1mm]
& & +\ (2+\alpha)\,2^{-1}\,D\,
\sum_{m=2}^{n-4}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(m+\alpha-1)!}{(\alpha+1)!}\right]^{1/2}\,
\|\wh_{n-m-4}\|\nonumber
\\
& & +\ (2+\alpha)^{1/2}\,2^{-1/2}\,D\,\delta^{-1}\,\sum_{m=0}^{1}\,(m+1)\,
\|\wh_{n-m-4}\|\nonumber
\\
& & +\ (2+\alpha)\,2^{-1}\,D\,\delta^{-1}\,
\sum_{m=2}^{n-7}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(m+\alpha-1)!}{(\alpha+1)!}\right]^{1/2}
\,\|\vh_{n-m-4}\|\nonumber
\\
& & +\ (2+\alpha)^{1/2}\,2^{-1/2}\,D\,\delta^{-2}\,\sum_{m=0}^{1}\,(m+1)\,
\|\vh_{n-m-4}\|\nonumber
\\
& & +\ (2+\alpha)\,D\,\sum_{m=3}^{n}\,\kappa^{m/2}\,
\left[\frac{(m+\alpha-1)!}{(\alpha+1)!}\right]^{1/2}\,
\|\wh_{n-m}\|,\label{en}
\\[3mm]
\|\wh_{n-2}\|
&\leq& (2+\alpha)\,2^{1/2}\,D\,\sum_{m=1}^{n-6}\,\kappa^{m/2}\,
\left[\frac{(3(n-3)-2m+\alpha-8)!}{(3(n-3)-3m+\alpha-7)!}\right]^{1/2}\,
\|\vh_{n-m-3}\|\nonumber
\\[1mm]
& & +\ (4+2\alpha)^{1/2}\,D\,\|\vh_{n-3}\|\nonumber
\\[1mm]
& & +\ (2+\alpha)\,2^{-1}\,D\,\sum_{m=2}^{n-4}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(3(n-4)-2m+\alpha-1)!}{(3(n-4)-3m+\alpha+1)!}\right]^{1/2}
\,\|\wh_{n-m-4}\|\nonumber
\\
& & +\ (2+\alpha)^{1/2}\,2^{-1/2}\,D\,\delta^{-1}\,\sum_{m=0}^{1}\,(m+1)\,
\|\wh_{n-m-4}\|\nonumber
\\
& & +\ (2+\alpha)\,2^{-1}\,D\,\delta^{-1}\,\sum_{m=2}^{n-7}\,\kappa^{m/2}\,(m+1)
\left[\frac{(3(n-4)-2m+\alpha-9)!}{(3(n-4)-3m+\alpha-7)!}\right]^{1/2}
\|\vh_{n-m-4}\|\nonumber
\\
& & +\ (2+\alpha)^{1/2}\,2^{-1/2}\,D\,\delta^{-2}\,\sum_{m=0}^{1}\,(m+1)\,
\|\vh_{n-m-4}\|\nonumber
\\ & & +\ (2+\alpha)\,D\,\sum_{m=3}^{n}\,\kappa^{m/2}\,
\left[\frac{(3n-2m+\alpha-1)!}{(3n-3m+\alpha+1)!}\right]^{1/2}\,
\|\wh_{n-m}\|\nonumber
\\
& & +\ \sum_{m=3}^{n}\,|E_{m}|\,\|\wh_{n-m}\|,\label{wn}
\\[3mm]
\|\vh_{n}\|
&\leq& [\alpha(2+\alpha)]^{1/2}\,\frac{1}{|r_{0}|}\,
\left[\frac{(3n+\alpha -11)!}{(3n+\alpha -13)!}\right]^{1/2}
\,\|\vh_{n-2}\| \nonumber
\\[1mm] & & +\ [\alpha\,(4+2\alpha)]^{1/2}\,\frac{D}{|r_{0}|}\,
\sum_{m=0}^{n-3}\,\kappa^{m/2}\,
\left[\frac{(3(n-3)-2m+\alpha+2)!}{(3(n-3)-3m+\alpha+1)!}\right]^{1/2}\,
\|\wh_{n-m-3}\|\nonumber
\\
& & +\ [\alpha\,(2+\alpha)]^{1/2}\,\frac{D}{2|r_{0}|}\,
\sum_{m=0}^{n-7}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(3(n-4)-2m+\alpha-7)!}{(3(n-4)-3m+\alpha-7)!}\right]^{1/2}
\,\|\vh_{n-m-4}\|\nonumber
\\
& & +\ [\alpha\,(2+\alpha)]^{1/2}\,\frac{D}{2\delta|r_{0}|}\,
\sum_{m=0}^{n-4}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(3(n-4)-2m+\alpha+1)!}{(3(n-4)-3m+\alpha+1)!}\right]^{1/2}
\|\wh_{n-m-4}\|\nonumber
\\
& & +\ [\alpha\,(2+\alpha)]^{1/2}\,\frac{D}{|r_{0}|}\,\sum_{m=1}^{n-3}\,\kappa^{m/2}\,
\left[\frac{(3n-2m+\alpha-7)!}{(3n-3m+\alpha-7)!}\right]^{1/2}
\,\|\vh_{n-m}\|\nonumber
\\
& & +\ \frac{1}{|r_{0}|}\,\sum_{m=2}^{n-3}\,|E_{m}|\,\|\vh_{n-m}\|.\label{vn}
\end{eqnarray}

{\bf Proof of Theorem \ref{bounds}:} Since $\alpha$ is fixed, and we can 
increase $b_1$, $b_2$, and $b_3$ if necessary,
we can easily prove Theorem \ref{bounds} for $n=3$. To prove the general result,
we use induction and inequalities (\ref{en}), (\ref{wn}), and (\ref{vn}).

Suppose the estimates are true for $m=3,\,\cdots,\,n-1$, and consider first the bound
(\ref{en}) for $|E_n|$.
Temporarily ignoring trivial factors, we have the following estimate for the
first term in (\ref{en}):
%
%Assume that the estimates are true for
%$m=3,\,\cdots,\,n-1$. We first verify the bound for $|E_{n}|$ by using
%induction on the inequality (\ref{en}). For the first term in (\ref{en}) we have,
\begin{eqnarray*}
\lefteqn{\sum_{m=1}^{n-6}\,\kappa^{m/2}\,
\left[\frac{(m+\alpha)!}{(\alpha+1)!}\right]^{1/2}
\,\|\vh_{n-m-3}\|}
\\
& < & b_{3}\,\kappa^{3(n-3)}\,b^{2(n-5)}\,\sum_{m=1}^{n-6}\,\kappa^{-5m/2}\,
\left[\frac{((n-4)-m+\alpha)!\,(m+\alpha)!}{(\alpha+1)!}\right]^{1/2}
\\
& < & b_{3}\,\kappa^{3(n-3)}\,b^{2(n-5)}\,[(n+\alpha-4)!]^{1/2}\,
\sum_{m=1}^{n-4}\,\kappa^{-5m/2}\,
\left[\frac{((n-4)-m+\alpha+1)!\,(m+\alpha+1)!}{(\alpha+1)!(n+\alpha-4)!}\right]^{1/2}
\\
&\leq& C_{3}\,\kappa^{3(n-3)}\,b^{2n-9}\,[(n+\alpha-4)!]^{1/2},
\end{eqnarray*}
where the last line follows from part (iii) of Lemma \ref{facts} and the inequality
$b_{3}\leq b$.

We treat the last term in (\ref{en}) in a similar fashion:
\begin{eqnarray*}
\lefteqn{\sum_{m=3}^{n}\,\kappa^{m/2}\,
\left[\frac{(m+\alpha-1)!}{(\alpha+1)!}\right]^{1/2}
\,\|\wh_{n-m}\|}
\\
& < & b_{2}\,\kappa^{3n}\,b^{2(n-3)}\,\sum_{m=3}^{n}\,\kappa^{-5m/2}\,
\left[\frac{((n-m+\alpha+1)!\,(m+\alpha-1)!}{(\alpha+1)!}\right]^{1/2}
\\
&\leq& b_{2}\,\kappa^{3n-5}\,b^{2(n-3)}\,[(n+\alpha-2)!]^{1/2}
\,\sum_{l=1}^{n-2}\,\kappa^{-5l/2}\,
\left[\frac{((n-2)-l+\alpha+1)!\,(l+\alpha+1)!}{(\alpha+1)!(n+\alpha-2)!}\right]^{1/2}
\\
&\leq& b_2C_{3}\,\kappa^{3n-5}\,b^{2(n-3)}\,[(n+\alpha-2)!]^{1/2},
\end{eqnarray*}
where we have performed the change of index $l=m-2$ in the third line above.

We now skip to the third term.
\begin{eqnarray*}
\lefteqn{\sum_{m=2}^{n-4}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(m+\alpha-1)!}{(\alpha+1)!}\right]^{1/2}
\,\|\wh_{n-m-4}\|}
\\
& < & b_{2}\,\kappa^{3(n-4)}\,b^{2(n-6)}\,(n-3)\,\sum_{m=2}^{n-4}\,\kappa^{-5m/2}\,
\left[\frac{((n-m+\alpha-3)!\,(m+\alpha-1)!}{(\alpha+1)!}\right]^{1/2}
\\
&\leq& b_{2}\,\kappa^{3n-17}\,b^{2(n-6)}\,(n-3)\,
\left[\frac{(n+\alpha-5)!}{(\alpha+1)!}\right]^{1/2}
\\
& & \times\quad\sum_{l=0}^{n-6}\,\kappa^{-5l/2}\,
\left[\frac{((n-6)-l+\alpha+1)!\,(l+\alpha+1)!}{((n-6)+\alpha+1)!}\right]^{1/2}
\\
& < & C_{1}\,\kappa^{3n-17}\,b^{2n-11}\,
\left[\frac{(n+\alpha-2)!}{(\alpha+1)!}\right]^{1/2}.
\end{eqnarray*}
The last inequality follows from part (i) of Lemma \ref{facts}, along with the inequality
$(n-3)<[(n+\alpha-3)(n+\alpha-2)]^{1/2}$.

Mimicking this calculation with minor changes,
we obtain an estimate for the fifth term. The result is
\[
\sum_{m=2}^{n-7}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(m+\alpha-1)!}{(\alpha+1)!}\right]^{1/2}\|\vh_{n-m-4}\|\
<\ C_{1}\,\kappa^{3n-17}\,b^{2n-13}\,\left[\frac{(n+\alpha-2)!}{(\alpha+1)!}\right]^{1/2}.
\]

The second term satisfies
\[
\|\vh_{n-3}\|\ <\ b_{3}\,\kappa^{3(n-3)}\,b^{2(n-4)}\,[(n+\alpha-4)!]^{1/2}
\ <\ \kappa^{3(n-3)}\,b^{2n-7}\,[(n+\alpha-4)!]^{1/2}.
\]
Finally, because the estimates on $\|\wh_n\|$ and $\|\vh_{n}\|$
are increasing in $n$,
the fourth and sixth terms satisfy
\[
\sum_{m=0}^{1}\,(m+1)\,\|\wh_{n-m-4}\|\
<\ \|\wh_{n-4}\|\,+\,2\,\|\wh_{n-5}\|\
<\ 3\,\kappa^{3(n-4)}\,b^{2n-7}\,[(n+\alpha-3)!]^{1/2},
\]
and \[
\sum_{m=0}^{1}\,(m+1)\,\|\vh_{n-m-4}\|\
<\ \|\vh_{n-4}\|\,+\,2\,\|\vh_{n-5}\|\
<\ 3\,\kappa^{3(n-4)}\,b^{2n-9}\,[(n+\alpha-5)!]^{1/2}.
\]

Putting all these pieces together and including the trivial factors we ignored
in each term,
we get the bound for $|E_{n}|$ in the theorem if
we define $b_1$ to be the maximum of its chosen value for $n=3$ and
\begin{eqnarray*}
\lefteqn{
%b_{1}&:=& &&
(2^{1/2}+b_2\kappa^{4})\,(2+\alpha)\,\kappa^{-3}\,C_{3}\,D
\,+\,(4+2\alpha)^{1/2}\,\kappa^{-3}\,D]}
\\[2mm]
& & 
+\ (1+\delta^{-1})\,(2+\alpha)\,2^{-1}\,\kappa^{-11}\,[(\alpha+1)]^{-1/2}\,C_{1}\,D\,
\,+\,3\,(\delta^{-1}+\delta^{-2})\,(2+\alpha)^{1/2}\,2^{-1/2}\,\kappa^{-6}\,D.
\end{eqnarray*}
Note that our definition of $b_1$ depends on $b_2$. As we shall see below, $b_2$ does
not depend on $b_1$ or $b_3$, and thus, $b_1$ is well defined.

We now turn to the bound on $\|\wh_{n-2}\|$. For the sum in the first term
in (\ref{wn}), we have
%We now use induction on inequality (\ref{wn}). For the sum in the first term we have
\begin{eqnarray*}
\lefteqn{\sum_{m=1}^{n-6}\,\kappa^{m/2}\,
\left[\frac{(3(n-3)-2m+\alpha-8)!}{(3(n-3)-3m+\alpha-7)!}\right]^{1/2}
\,\|\vh_{n-m-3}\|}
\\
& < & b_{3}\,\kappa^{3(n-3)}\,b^{2(n-5)}\,\sum_{m=1}^{n-6}\,\kappa^{-5m/2}\,
\left[\frac{(3(n-3)-2m+\alpha-8)!\,(n-m+\alpha-4)!}{(3(n-3)-3m+\alpha-7)!}\right]^{1/2}
\\
& = & b_{3}\,\kappa^{3(n-3)-5/2}\,b^{2(n-5)}\,\sum_{l=0}^{n-7}\,\kappa^{-5l/2}\,
\left[\frac{(3(n-7)-2l+\alpha+2)!\,((n-7)-l+\alpha+2)!}%
{(3(n-7)-3l+\alpha+2)!}\right]^{1/2}
\\
&\leq& C_{2}\,\kappa^{3(n-3)-5/2}\,b^{2n-9}\,[(n+\alpha-5)!]^{1/2}.
\end{eqnarray*}
We have made a change of index $l=m-1$ and have used part (ii) of Lemma \ref{facts} to 
obtain the last inequality.

We handle the seventh term in (\ref{wn}) in the same way. The result is
\[
\sum_{m=3}^{n}\,\kappa^{m/2}\,
\left[\frac{(3n-2m+\alpha-1)!}{(3n-3m+\alpha+1)!}\right]^{1/2}
\,\|\wh_{n-m}\|\
<\ C_{2}\,\kappa^{3n-5}\,b^{2n-5}\,[(n+\alpha-1)!]^{1/2}.
\]

The sum in the third term is bounded as follows:
\begin{eqnarray*}
\lefteqn{\sum_{m=2}^{n-4}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(3(n-4)-2m+\alpha-1)!}{(3(n-4)-3m+\alpha+1)!}\right]^{1/2}
\,\|\wh_{n-m-4}\|}
\\
& < & b_{2}\,\kappa^{3(n-4)}\,b^{2(n-6)}\,(n-3)\sum_{m=2}^{n-4}\,\kappa^{-5m/2}
\left[\frac{(3(n-4)-2m+\alpha+1)!((n-4)-m+\alpha+1)!}%
{(3(n-4)-3m+\alpha+1)!}\right]^{1/2}
\\[1mm]
&\leq& C_{2}\,b_{2}\,\kappa^{3(n-4)}\,b^{2(n-6)}\,(n-3)\,[(n+\alpha-3)!]^{1/2}
\\[2mm]
& < & C_{2}\,\kappa^{3(n-4)}\,b^{2n-11}\,[(n+\alpha-1)!]^{1/2}.
\end{eqnarray*}
A similar computation allows us to bound the sum in the fifth term.
The result is
\[
\sum_{m=2}^{n-7}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(3(n-4)-2m+\alpha-9)!}{(3(n-4)-3m+\alpha-7)!}\right]^{1/2}
\,\|\vh_{n-m-4}\|
\ <\ C_{2}\,\kappa^{3(n-4)}\,b^{2n-13}\,[(n+\alpha-3)!]^{1/2}. 
\]

The last term in (\ref{wn}) is bounded as follows:
\begin{eqnarray*}
\sum_{m=3}^{n}\,|E_{m}|\,\|\wh_{n-m}\|
& < & b_{1}\,b_{2}\,\kappa^{3(n-2)}\,b^{2(n-3)}\,\sum_{m=3}^{n}\,
[(m+\alpha-2)!\,(n-m+\alpha+1)!]^{1/2}
\\
& < & b_{1}\,b_{2}\,\kappa^{3(n-2)}\,b^{2(n-3)}\,\sum_{l=1}^{n-2}\,
[(l+\alpha+1)!\,((n-2)-l+\alpha+1)!]^{1/2}
\\[1mm]
&\leq& C_{1}\,\kappa^{3(n-2)}\,b^{2(n-2)}\,[(n-\alpha-1)!]^{1/2}. 
\end{eqnarray*}
The last inequality follows after applying part (i) of Lemma~\ref{facts}.

The second, fourth, and sixth terms in (\ref{wn}) are identical to those in (\ref{en}). 
Therefore, the bound for $\|\wh_{n-2}\|$ in the theorem is proved if we define
$b_2$ to be the maximum of its chosen value for $n=3$ and
\begin{eqnarray*}
\lefteqn{
%b_{2}&:=& &&
C_{1}\,+\,(4+2\alpha)^{1/2}\,\kappa^{-6}\,D
\,+\,(2+\alpha)\,(1+2^{1/2}\kappa^{-13/2})\,\kappa\,C_{2}\,D}
\\[2mm]
& & +\ 3\,(\delta^{-1}+\delta^{-2})\,(2+\alpha)^{1/2}\,2^{-1/2}\,\kappa^{-6}\,D\,
+\,2^{-1}\,(2+\alpha)\,(1+\delta^{-1})\,\kappa^{-6}\,C_{2}\,D.
\end{eqnarray*}

Finally, we verify the inequality for $\|\vh_{n}\|$ in the theorem using inequality 
(\ref{vn}). We start with the first term. We have
\begin{eqnarray}
\lefteqn{\left[\frac{(3n+\alpha -11)!}{(3n+\alpha -13)!}\right]^{1/2}
\,\|\vh_{n-2}\|}\nonumber
\\
& < & b_{3}\,\kappa^{3(n-2)}\,b^{2(n-3)}\,[(n+\alpha-1)!]^{1/2}\,
\sup_{n\geq 5}\,\left[\frac{(3n+\alpha-11)\,(3n+\alpha-12)}
{(n+\alpha-1)\,(n+\alpha-2)}\right]^{1/2}\label{g1}
\\
&\leq& g_{1}\,\kappa^{3(n-2)}\,b^{2n-5}\,[(n+\alpha-1)!]^{1/2},\nonumber 
\end{eqnarray}
where $g_{1}$ stands for sup-value in (\ref{g1}).

For the sum in the second term, we have
\begin{eqnarray}
\lefteqn{\sum_{m=0}^{n-3}\,\kappa^{m/2}\,
\left[\frac{(3(n-3)-2m+\alpha+2)!}{(3(n-3)-3m+\alpha+1)!}\right]^{1/2}
\,\|\wh_{n-m-3}\|}\nonumber
\\
& < & b_{2}\,\kappa^{3(n-3)}\,b^{2(n-3)}\,\sum_{m=0}^{n-3}\,\kappa^{-5m/2}\,
\left[\frac{(3(n-3)-2m+\alpha+2)!\,(n-m+\alpha-2)!}
{(3(n-3)-3m+\alpha+1)!}\right]^{1/2}\nonumber
\\[1mm]
& = & b_{2}\,\kappa^{3(n-3)}\,b^{2(n-3)}\,(3(n-3)+\alpha+2)^{1/2}\nonumber
\\[1mm]
& & \times\quad\sum_{m=0}^{n-3}\,\kappa^{-5m/2}\,
\left[\frac{(3(n-3)-2m+\alpha+1)!\,((n-3)-m+\alpha+1)!}
{(3(n-3)-3m+\alpha+1)!}\right]^{1/2}\nonumber
\\
&\leq& C_{2}\,\kappa^{3(n-3)}\,b^{2n-5}\,[(n+\alpha-1)!]^{1/2}\,
\sup_{n\geq 3}\,
\left[\frac{(3(n-3)+\alpha+2)}{((n-3)+\alpha+2)}\right]^{1/2}\label{g2}
\\[1mm]
&\leq& g_{2}\,C_{2}\,\kappa^{3(n-3)}\,b^{2n-5}\,[(n+\alpha-1)!]^{1/2},\nonumber
\end{eqnarray}
where $g_{2}$ is defined by the sup-value in (\ref{g2}).

The fifth term can be treated in the same way. The bound for the sum in it is
\[
\sum_{m=1}^{n-3}\,\kappa^{m/2}\,
\left[\frac{(3n-2m+\alpha-7)!}{(3n-3m+\alpha-7)!}\right]^{1/2}
\,\|\vh_{n-m}\|
\ <\ C_{2}\,\kappa^{3n}\,b^{2n-3}\,[(n+\alpha-1)!]^{1/2}.
\]

The sums in third and fourth terms of inequality (\ref{vn}) are almost identical to
those in the fifth and third terms of inequality (\ref{wn}). 
For that reason we just write down the corresponding bounds:
\[
\sum_{m=0}^{n-7}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(3(n-4)-2m+\alpha-7)!}{(3(n-4)-3m+\alpha-7)!}\right]^{1/2}
\,\|\vh_{n-m-4}\|
\ <\ C_{2}\,\kappa^{3(n-4)}\,b^{2n-9}\,[(n+\alpha-2)!]^{1/2},
\]
and
\[
\sum_{m=0}^{n-4}\,\kappa^{m/2}\,(m+1)\,
\left[\frac{(3(n-4)-2m+\alpha+1)!}{(3(n-4)-3m+\alpha+1)!}\right]^{1/2}
\,\|\wh_{n-m-4}\|
\ <\ C_{2}\,\kappa^{3(n-4)}\,b^{2n-7}\,[(n+\alpha-1)!]^{1/2}.
\]

To deal with the last term in (\ref{vn}), we have,
\begin{eqnarray*}
\sum_{m=2}^{n-3}\,|E_{m}|\,\|\vh_{n-m}\|
& = & |E_{2}|\,\|\vh_{n-2}\|\ +\ \sum_{m=3}^{n-3}\,|E_{m}|\,\|\vh_{n-m}\|
\\[1mm]
& < & (\alpha+1/2)\,b_{3}\,\kappa^{3(n-2)}\,b^{2(n-3)}\,[(n+\alpha-3)!]^{1/2}
\\
& & +\quad b_{1}\,b_{3}\,\kappa^{3(n-2)}\,b^{2(n-4)}\,\sum_{m=3}^{n-3}\,
[(m+\alpha-2)!\,(n-m+\alpha-1)!]^{1/2}
\\[1mm]
&\leq& (\alpha+1/2)\,\kappa^{3(n-2)}\,b^{2n-5}\,[(n+\alpha-3)!]^{1/2}
\\
& & +\quad \kappa^{3(n-2)}\,b^{2(n-2)}\,\sum_{l=0}^{n-6}\,
[(l+\alpha+1)!\,((n-5)-l+\alpha+1)!]^{1/2}
\\[1mm]
& < & (\alpha+1/2+C_{1})\,\kappa^{3(n-2)}\,b^{2(n-2)}\,[(n-\alpha-3)!]^{1/2}. 
\end{eqnarray*}

The estimate for $\|\vh_{n}\|$ is then satisfied if we define
$b_3$ to be the maximum of its chosen value for $n=3$ and
\begin{eqnarray*}
\lefteqn{
%b_{3}&:=&
[\alpha(2+\alpha)]^{1/2}\,|r_{0}|^{-1}\,\kappa^{-6}\,g_{1}\
+\ [\alpha(4+2\alpha)]^{1/2}\,|r_{0}|^{-1}\,\kappa^{-9}\,g_{2}\,C_{2}\,D}
\\[2mm]
&& +\ [\alpha(2+\alpha)]^{1/2}\,|r_{0}|^{-1}\,C_{2}\,D\,
+\ 2^{-1}\,(1+\delta^{-1})\,[\alpha(2+\alpha)]^{1/2}
\,|r_{0}|^{-1}\,\kappa^{-12}\,C_{2}\,D
\\[2mm]
&& +\ (\alpha+1/2+C_{1})\,|r_{0}|^{-1}\,\kappa^{-6}.
\end{eqnarray*}

The theorem's first three inequalities (that we have now proved) immediately imply the
second three inequalities in the theorem.\hfill$\Box$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Estimates for the Norm of the Error}\label{hotdog}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The upper bounds we have proved for $|E_{n}|$, $\|\wh_{n}\|$, and $\|\vh_{n}\|$\
in the previous section allow us to estimate the error made when truncated series 
are inserted into equation (\ref{modelscaled}). To do this, we mimic the
arguments in \cite{toloza1,toloza2}, which are based on a technique developed in 
\cite{hagedorn4}. Concretely, for $N\geq 1$ define
\begin{eqnarray*}
E^{N}&:=&
\sum_{n=0}^{N+2}\,\epsilon^{n}\,E_{n},
\\
w^{N}&:=&
\sum_{n=0}^{N}\,\epsilon^{n}\,w_{n},
\\
v^{N}&:=&
\sum_{n=0}^{N+2}\,\epsilon^{n}\,v_{n},\qquad\mbox{and}
\\[2mm]
\Psi^{N}&:=&
w^{N}\,\Phi_{1}\ +\ v^{N}\,\Phi_{2}.
\end{eqnarray*}
Let $H(\epsilon)$ be the self--adjoint operator in equation (\ref{modelscaled}).
The error commited when truncated Rayleigh--Schr\"odinger series replace the actual 
solution to equation (\ref{modelscaled}) is given by
\begin{equation}
\label{error}
\left(H(\epsilon) - E^{N}\right)\,\Psi^{N}\
=\ S_{N}\,\Phi_{1}\ +\ T_{N}\,\Phi_{2},
\end{equation}
where the residual error terms $S_{N}$ and $T_{N}$ are given explicitly by
\begin{eqnarray*}
S_{N}&:=&
-\,\frac{\epsilon^{2}}{2}\,\partial_{x}^{2}\,w^{N}\,
-\,\epsilon^{3}\,\Omega\,\partial_{x}\,v^{N}\,
+\,\frac{\epsilon^{4}}{2}\,\Omega^{2}\,w^{N}\,
-\,\frac{\epsilon^{4}}{2}\,\Omega^{\prime}\,v^{N}\,
+\,{\cal E}_{1}\,w^{N}\,
-\, E^{N}\,w^{N},\qquad\mbox{and}
\\[3mm]
T_{N}&:=&
-\,\frac{\epsilon^{2}}{2}\,\partial_{x}^{2}\,v^{N}\,
+\,\epsilon^{3}\,\Omega\,\partial_{x}\,w^{N}\,
+\,\frac{\epsilon^{4}}{2}\,\Omega^{2}\,v^{N}\,
+\,\frac{\epsilon^{4}}{2}\,\Omega^{\prime}\,w^{N}\,
+\,{\cal E}_{2}\,v^{N}\,-\,E^{N}\,v^{N}.
\end{eqnarray*}
Since the perturbation coefficients are the solutions to the recursive equations 
computed in Section 2, all terms of order $n\le N+2$
cancel in both $S_{N}$ and $T_{N}$. By a computation that is
described in the Appendix, the residual terms can be written as
\begin{eqnarray}
S_N&=&
-\,\sum_{m=0}^N\,\epsilon^{m+3}\,{\cal A}^{[N-m-1]}\,\partial_x\,v_m\
-\,\epsilon^{N+4}\,{\cal A}^{[-1]}\,\partial_x\,v_{N+1}\,
-\,\epsilon^{N+5}\,{\cal A}^{[-1]}\,\partial_x\,v_{N+2}\nonumber
\\[1mm]
&&+\,\frac 12\,\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\nonumber
\\[1mm]
&&-\,\frac 12\,\sum_{m=0}^{N-1}\,\epsilon^{m+4}\,{\cal C}^{[N-m-2]}\,v_m\
-\,\frac{\epsilon^{N+4}}2\,{\cal C}^{[-1]}\,v_N\,
-\,\frac{\epsilon^{N+5}}2\,{\cal C}^{[-1]}\,v_{N+1}\,
-\,\frac{\epsilon^{N+6}}2\,{\cal C}^{[-1]}\,v_{N+2}\nonumber
\\[1mm]
&&+\,\sum_{m=0}^N\,\epsilon^m\,{\cal D}^{[N-m+2]}\,w_m\nonumber
\\[1mm]
&&-\,\sum_{l=N+3}^{2N+2}\,\epsilon^l\,\sum_{m=l-N-2}^N\,
E_{l-m}\,w_m,\label{SN}
\end{eqnarray}
and
\begin{eqnarray}
T_N&=&
-\,\frac 12\,\epsilon^{N+3}\,\partial_x^2\,v_{N+1}\,
-\,\frac 12\,\epsilon^{N+4}\,\partial_x^2\,v_{N+2}\nonumber
\\[1mm]
&&+\,\sum_{m=0}^N\,\epsilon^{m+3}\,{\cal A}^{[N-m-1]}\,\partial_x\,w_m\nonumber
\\[1mm]
&&+\,\frac 12\,\sum_{m=0}^{N-1}\,\epsilon^{m+4}\,{\cal B}^{[N-m-2]}\,v_m\
+\,\frac{\epsilon^{N+4}}2\,{\cal B}^{[-1]}\,v_N\,
+\,\frac{\epsilon^{N+5}}2\,{\cal B}^{[-1]}\,v_{N+1}\,
+\,\frac{\epsilon^{N+6}}2\,{\cal B}^{[-1]}\,v_{N+2}\nonumber
\\[1mm]
&&+\,\frac 12\,\sum_{m=0}^{N-1}\,\epsilon^{m+4}\,{\cal C}^{[N-m-2]}\,w_m\
+\,\frac{\epsilon^{N+4}}2\,{\cal C}^{[-1]}\,w_N\nonumber
\\[1mm]
&&+\,\sum_{m=0}^{N+2}\,\epsilon^m\,{\cal R}^{[N-m+2]}\,v_m\nonumber
\\[1mm]
&&-\,\sum_{l=N+3}^{2N+4}\,\epsilon^l\,\sum_{m=l-N-2}^{N+2}\,
E_{l-m}\,v_m.\label{TN}
\end{eqnarray}
In these expressions we have used the following conventions.
\begin{eqnarray}
{\cal A}(z)&=&\Omega(z),\label{A}\\
{\cal B}(z)&=&\Omega(z)^2,\label{B}\\
{\cal C}(z)&=&\Omega'(z),\label{C}\\
{\cal D}(z)&=&{\cal E}_1(z),\qquad\mbox{and}\label{D}\\
{\cal R}(z)&=&{\cal E}_2(z).\label{R}
\end{eqnarray}
Also, we have used the following notation for exact Taylor series errors
for any function $f(z)$ expanded around $z=0$:
\[
f^{[n]}(z)\,=\,\left\{\
\begin{array}{cl}\displaystyle
f(z)\ -\ \sum_{m=0}^n\,\frac{f^{(m)}}{m!}\,z^m&\mbox{ if}\quad n\ge 0,
\\[7mm]
f(z)&\mbox{ if}\quad n=-1.
\end{array}\right.
\]
Note that in this notation, the integer $n$ may never be less than $-1$, 
and that $f^{[n]}$ is $O(\epsilon^{n+1})$ since $z=\epsilon x$.

Our main result in this work relies on an upper bound for the $L^{2}$-norm of 
(\ref{error}). Note that, since $\Phi_{1}(x)$ and $\Phi_{2}(x)$ are orthogonal 
for each $x\in{\mathbb R}$, we have
\[
\left\|\,\left(H(\epsilon) - E^{N}\right)\,\Psi^{N}\,\right\|^{2}
\ =\ \left\|S_{N}\right\|^{2}\ +\ \left\|T_{N}\right\|^{2}. \]
We recall that hypothesis (5) ensures that the absolute values of the functions 
(\ref{A})--(\ref{R}) are bounded above by $M\exp(\tau |z|^{2})$.

\begin{thm}\label{errornorm}
There exist constants $G_{1}$, $G_{2}$, $F_{1}$, and $F_{2}$ such that, 
for every $N\geq 3$ and\linebreak $\epsilon\in(0,\,(8\tau)^{-1/2})$,
\begin{eqnarray*}
\left\|S_{N}\right\|
& < & \sum_{m=N+3}^{2N+2}\,G_{1}\,F_{1}^{m}\,\epsilon^{m}\,[(m+\alpha+1)]^{1/2},
\qquad\mbox{and}
\\[2mm]
\left\|T_{N}\right\|
& < & \sum_{m=N+3}^{2N+4}\,G_{2}\,F_{2}^{m}\,\epsilon^{m}\,[(m+\alpha+1)]^{1/2}.
\end{eqnarray*}
As a consequence,
\[
\left\|\,\left(H(\epsilon) - E^{N}\right)\,\Psi^{N}\,\right\|
\ <\ \sum_{m=N+3}^{2N+4}\,G\,F^{m}\,\epsilon^{m}\,[(m+\alpha+1)]^{1/2},
\]
for some constants $G$ and $F$.
\end{thm}

{\bf Proof:}\ Recall that $w_{m}=A\wh_{m},\ \,v_{m}=A\vh_{m}$, and that these
perturbation
coefficients belong to $\mbox{Ran}(P_{i\leq 3m+\alpha})$ and $
\mbox{Ran}(P_{i\leq 3m+\alpha-8})$, respectively. 
Also, $E_{0}=E_{1}=0$ and $\vh_{0}=\vh_{1}=\vh_{2}=0$. 
In what follows, $\tilde{g}_{i}$,
$\tilde{f}_{i}$, $g_{i}$, $f_{i}$, $\tilde{a}_{i}$, $\tilde{b}_{i}$, $a_{i}$, 
and $b_{i}$, with $i=1,2,\cdots,$ denote certain constants that will appear 
along the way. We also regard all the empty sums as zero.

Taking norms in the expression for $S_{N}$, we get
\begin{eqnarray}
\|S_N\|
&\leq&
\sum_{m=3}^{N-1}\,\epsilon^{m+3}\,\|{\cal A}^{[N-m-1]}
P_{i\leq 3m+\alpha-7}\|\,\|\partial_xA\|\,\|\vh_m\|\,+\,
\epsilon^{N+3}\,\|{\cal A}P_{i\leq 3N+\alpha-7}\|\,\|\partial_xA\|\,\|\vh_{N}\|\nonumber
\\[1mm]
&&
+\ \epsilon^{N+4}\,\|{\cal A}P_{i\leq 3(N+1)+\alpha-7}\|\,\|\partial_xA\|\,\|\vh_{N+1}\|
\,+\,\epsilon^{N+5}\,\|{\cal A}P_{i\leq 3(N+2)+\alpha-7}\|\,
\|\partial_xA\|\,\|\vh_{N+2}\|\nonumber
\\[1mm]
&&
+\ \frac{1}{2}\,\sum_{m=0}^{N-2}\,\epsilon^{m+4}\,
\|{\cal B}^{[N-m-2]}AP_{i\leq 3m+\alpha}\|\,\|\wh_m\|
\nonumber
\\[1mm]
&&
+\ \frac{\epsilon^{N+3}}{2}\,\|{\cal B}P_{i\leq 3(N-1)+\alpha}\|\,\|\wh_{N-1}\|
\,+\,\frac{\epsilon^{N+4}}{2}\,\|{\cal B}P_{i\leq 3N+\alpha}\|\,\|\wh_N\|\nonumber
\\[1mm]
&&
+\ \frac{1}{2}\,\sum_{m=3}^{N-2}\,\epsilon^{m+4}\,
\|{\cal C}^{[N-m-2]}AP_{i\leq 3m+\alpha-8}\|\,\|\vh_m\|
\nonumber
\\[1mm]
&&
+\ \frac{\epsilon^{N+3}}{2}\,\|{\cal C}P_{i\leq 3(N-1)+\alpha-8}\|\,\|\vh_{N-1}\|
\,+\,\frac{\epsilon^{N+4}}{2}\,\|{\cal C}P_{i\leq 3N+\alpha-8}\|\,\|\vh_N\|\nonumber
\\[1mm]
&&
+\ \frac{\epsilon^{N+5}}{2}\,\|{\cal C}P_{i\leq 3(N+1)+\alpha-8}\|\,\|\vh_{N+1}\|
\,+\,\frac{\epsilon^{N+6}}{2}\,\|{\cal C}P_{i\leq 3(N+2)+\alpha-8}\|\,\|\vh_{N+2}\|
\nonumber
\\[1mm]
&&
+\ \sum_{m=0}^N\,\epsilon^m\,\|{\cal D}^{[N-m+2]}AP_{i\leq 3m+\alpha}\|\,\|\wh_m\|
\nonumber
\\[1mm]
&&
+\ \sum_{m=N+3}^{2N+2}\,\epsilon^m\,\sum_{l=m-N}^{N-2}\,|E_{l}|\,\|\wh_{m-l}\|.
\label{ugly}
\end{eqnarray}
Note that the operator $A$ has been dropped in several terms.

We apply Taylor's theorem to the function ${\cal A}$, to see that
\[
{\cal A}^{[N-m-1]}(\epsilon x)
\ =\ \frac{{\cal A}^{(N-m)}(\eta)}{(N-n)!}\,\epsilon^{N-n}\,x^{N-n},
\]
for some $\eta\in(-\epsilon|x|,\,\epsilon|x|)$. Also, from the proof of the 
Lemma~\ref{axma}, we know that
\[
\|\partial_xA\|\ \leq\ (4+2\alpha)^{1/2},
\]
and that the same bound holds for $\|xA\|$. Thus, the first term in (\ref{ugly}) 
satisfies
\begin{eqnarray*}
\lefteqn{
\sum_{m=3}^{N-1}\,\epsilon^{m+3}\,\|{\cal A}^{[N-m-1]}P_{i\leq 3m+\alpha-7}\|\,
\|\partial_xA\|\,\|\vh_m\|}
\\
&\leq& (4+2\alpha)^{1/2}\,\epsilon^{N+3}\,\sum_{m=3}^{N-1}\ \left\|\,
\frac{{\cal A}^{(N-m)}(\eta)}{(N-m)!}\,x^{N-m}\,P_{i\leq 3m+\alpha-7}\,\right\|
\ \|\vh_m\|.
\end{eqnarray*}
We now apply part (ii) of Lemma~\ref{aboutf} and the estimate for $\|\vh_{m}\|$. 
This yields
\begin{eqnarray}
\lefteqn{
\sum_{m=3}^{N-1}\,\epsilon^{m+3}\,\|{\cal A}^{[N-m-1]}P_{i\leq 3m+\alpha-7}\|\,
\|\partial_xA\|\,\|\vh_m\|}\nonumber
\\
& < & \epsilon^{N+3}\,\tilde{g}_{1}\,\tilde{f}_{1}^{N}\,\sum_{m=3}^{N-1}\,\kappa^{3m}\,
\left[\frac{(N+2m+\alpha-7)!\,(m+\alpha-1)!}{(3m+\alpha-7)!}\right]^{1/2}\nonumber
\\
& = &
\epsilon^{N+3}\,\tilde{g}_{1}\,\tilde{f}_{1}^{N}\,\kappa^{3N}\,
\sum_{l=1}^{N-3}\,\kappa^{-3l}\,
\left[\frac{(3(N-3)-2l+\alpha+2)!\,((N-3)-l+\alpha+2)!}
{(3(N-3)-3l+\alpha+2)!}\right]^{1/2}.
\label{compare1}
\end{eqnarray}
Now applying Lemma~\ref{facts}, we obtain
\[
\sum_{m=3}^{N-1}\,\epsilon^{m+3}\,\|{\cal A}^{[N-m-1]}P_{i\leq 3m+\alpha-7}\|
\,\|\partial_xA\|\,\|\vh_m\|
\ <\ g_{1}\,f_{1}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha-1)!]^{1/2}.
\]

The second term in inequality (\ref{ugly}) can be estimated using part (i) of
Lemma~\ref{aboutf}. This yields
\begin{eqnarray*}
\lefteqn{\epsilon^{N+3}\,\|{\cal A}P_{i\leq 3N+\alpha-7}\|\,\|\partial_xA\|\,\|\vh_{N}\|}
\\
& < &
\epsilon^{N+3}\,(4+2\alpha)^{1/2}\,M\,\left[e^{1/4}+
\left(\frac{32}{9}\right)^{1/4}\right]\,
2^{3N+\alpha-7}\,\kappa^{3N}\,b^{2N-1}\,[(N+\alpha-1)!]^{1/2}
\\
& = & g_{2}\,f_{2}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha-1)!]^{1/2},
\end{eqnarray*}
The third and fourth terms yield similar bounds. We obtain,
\[
\epsilon^{N+4}\,\|{\cal A}P_{i\leq 3(N+1)+\alpha-7}\|\,\|\partial_xA\|\,\|\vh_{N+1}\|
\ <\ g_{3}\,f_{3}^{N+4}\,\epsilon^{N+4}\,[(N+\alpha)!]^{1/2},
\]
and
\[
\epsilon^{N+5}\,\|{\cal A}P_{i\leq 3(N+2)+\alpha-7}\|\,\|\partial_xA\|\,\|\vh_{N+2}\|
\ <\ g_{4}\,f_{4}^{N+5}\,\epsilon^{N+5}\,[(N+\alpha+1)!]^{1/2}.
\]

We use similar arguments to treat the terms that involve the function $\cal B$. 
Thus, for the fifth term in (\ref{ugly}), we have
\begin{eqnarray}
\lefteqn{ \frac{1}{2}\ \sum_{m=0}^{N-2}\,\epsilon^{m+4}\,
\|{\cal B}^{[N-m-2]}AP_{i\leq 3m+\alpha}\|\,\|\wh_m\|
}\nonumber
\\
&\leq&
\frac{\epsilon^{N+3}}{2}\ \sum_{m=0}^{N-2}\
\left\|\,\frac{{\cal B}^{(N-m-1)}(\eta)}{(N-m-1)!}\,x^{N-m-2}\,
P_{i\leq 3m+\alpha+1}\,\right\|\,
\|xA\|\,\|\wh_m\|\nonumber
\\
& < &
\epsilon^{N+3}\,\tilde{g}_{2}\,\tilde{f}_{2}^{N}\,\kappa^{3N}\
\sum_{l=2}^{N}\,\kappa^{-3l}\,
\left[\frac{(3N-2l+\alpha+1)!\,(N-l+\alpha+1)!}{(3N-3l+\alpha+1)!}\right]^{1/2}\nonumber
\\[1mm]
& < &
g_{5}\,f_{5}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha+1)!]^{1/2}.\label{compare2}
\end{eqnarray} 
Also,
\[
\frac{\epsilon^{N+3}}{2}\,\|{\cal B}P_{i\leq 3(N-1)+\alpha}\|\,\|\wh_{N-1}\|
\ <\ g_{6}\,f_{6}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha)!]^{1/2},
\]
and
\[
\frac{\epsilon^{N+4}}{2}\,\|{\cal B}P_{i\leq 3N+\alpha}\|\,\|\wh_N\|
\ <\ g_{7}\,f_{7}^{N+4}\,\epsilon^{N+4}\,[(N+\alpha+1)!]^{1/2}.
\]

The terms involving the function $\cal C$ in (\ref{ugly}) can be handled in much 
the same way as we did with the previous ones. Thus,
\[
\frac{1}{2}\ \sum_{m=3}^{N-2}\,\epsilon^{m+4}\,
\|{\cal C}^{[N-m-2]}AP_{i\leq 3m+\alpha-8}\|\,\|\vh_m\|
\ <\ g_{8}\,f_{8}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha-1)!]^{1/2},
\]
\vspace{3mm}
\[
\frac{\epsilon^{N+3}}{2}\,\|{\cal C}P_{i\leq 3(N-1)+\alpha-8}\|\,\|\vh_{N-1}\|
\ <\ g_{9}\,f_{9}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha-2)!]^{1/2},
\]
\vspace{3mm}
\[
\frac{\epsilon^{N+4}}{2}\,\|{\cal C}P_{i\leq 3N+\alpha-8}\|\,\|\vh_N\|
\ <\ g_{10}\,f_{10}^{N+4}\,\epsilon^{N+4}\,[(N+\alpha-1)!]^{1/2},
\]
\vspace{3mm}
\[
\frac{\epsilon^{N+5}}{2}\ \|{\cal C}P_{i\leq 3(N+1)+\alpha-8}\|\,\|\vh_{N+1}\|
\ <\ g_{11}\,f_{11}^{N+5}\,\epsilon^{N+5}\,[(N+\alpha)!]^{1/2},
\]
and
\[
\frac{\epsilon^{N+6}}{2}\ \|{\cal C}P_{i\leq 3(N+2)+\alpha-8}\|\,\|\vh_{N+2}\|
\ <\ g_{12}\,f_{12}^{N+6}\,\epsilon^{N+6}\,[(N+\alpha+1)!]^{1/2}.
\]

The next term in (\ref{ugly}) is estimated as follows.
\begin{eqnarray}
\lefteqn{ \sum_{m=0}^N\,\epsilon^m\,\|{\cal D}^{[N-m+2]}AP_{i\leq 3m+\alpha}\|\,
\|\wh_m\|}\nonumber
\\
& < &
\epsilon^{N+3}\,\tilde{g}_{13}\,\tilde{f}_{13}^{N}\,\kappa^{3N}\ \sum_{l=0}^{N}\,
\kappa^{-3l}
\,\left[\frac{(3N-2l+\alpha+3)!\,(N-l+\alpha+1)!}{(3N-3l+\alpha+1)!}\right]^{1/2}
\nonumber
\\
&\leq&
\epsilon^{N+3}\,\tilde{g}_{13}\,\tilde{f}_{13}^{N}\,\kappa^{3N}\ \max_{0\leq l\leq N}\,
\left[\frac{(3N-3l+\alpha+3)(3N-3l+\alpha+2)}{(N-l+\alpha+3)(N-l+\alpha+2)}\right]^{1/2}
\nonumber
\\
& &
\times\quad \sum_{l=0}^{N}\,\kappa^{-3l}\,
\left[\frac{(3N-2l+\alpha+3)!\,(N-l+\alpha+3)!}{(3N-3l+\alpha+3)!}\right]^{1/2}\nonumber
\\[2mm]
&\leq&
\epsilon^{N+3}\,3\,C_{2}\,\tilde{g}_{13}\,\tilde{f}_{13}^{N}\,\kappa^{3N}\,
[(N+\alpha+3)!]^{1/2}\nonumber
\\[3mm]
& = &
g_{13}\,f_{13}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha+3)!]^{1/2}.\label{compare3}
\end{eqnarray}

Finally, for the last term in inequality (\ref{ugly}), we have
\begin{eqnarray}
\lefteqn{
\sum_{m=N+3}^{2N+2}\,\epsilon^m\,\sum_{l=m-N}^{N+2}\,|E_l|\,\|\wh_{m-l}\|}\nonumber
\\[2mm]
& < & \sum_{m=N+3}^{2N+2}\,\epsilon^m\,b^{2(m-2)}\,\kappa^{3(m-2)}\,\sum_{l=m-N}^{N-2}\,
[(l+\alpha-2)!\,(m-l+\alpha+1)!]^{1/2}\nonumber
\\[2mm]
& < &
\sum_{m=N+3}^{2N+2}\,\epsilon^m\,b^{2(m-2)}\,\kappa^{3(m-2)}\,[(m+\alpha+1)!]^{1/2}
\nonumber
\\
& &\quad
\times\quad \sum_{l=0}^{m}\
\left[\frac{(l+\alpha+1)!\,(m-l+\alpha+1)!}{(m+\alpha+1)!}\right]^{1/2}\nonumber
\\
&\leq&
\sum_{m=N+3}^{2N+2}\,g_{14}\,f_{14}^{m}\,\epsilon^m\,[(m+\alpha+1)!]^{1/2}.
\label{compare4}
\end{eqnarray}

The upper bound for $\|S_{N}\|$ in the theorem now follows if we define 
$G_{1}:=\max\,\{\,g_{1},\,\cdots,\,g_{14}\,\}$ and
$F_{1}:=\max\,\{\,f_{1},\,\cdots,\,f_{14}\,\}$.

The proof for the bound for $\|T_{N}\|$ involves the same kind of manipulations as done
with $\|S_{N}\|$. We start with the following inequality:
\begin{eqnarray}
\|T_N\|
&\leq&
\frac{\epsilon^{N+3}}{2}\,\|\partial_{x}P_{i\leq 3(N+1)+\alpha-7}\|\,
\|\partial_{x}A\|\,\|\vh_{N+1}\|\
+\ \frac{\epsilon^{N+4}}{2}\,\|\partial_{x}P_{i\leq 3(N+2)+\alpha-7}\|\,
\|\partial_{x}A\|\,\|\vh_{N+2}\|
\nonumber
\\
&&
+\ \sum_{m=0}^{N-1}\,\epsilon^{m+3}\,\|{\cal A}^{[N-m-1]}P_{i\leq 3m+\alpha+1}\|\,
\|\partial_xA\|\,\|\wh_m\|\ +\ \epsilon^{N+3}\,\|{\cal A}P_{i\leq 3N+\alpha+1}\|\,
\|\partial_xA\|\,\|\wh_{N}\|\nonumber
\\[1mm]
& &
+\ \frac{1}{2}\,\sum_{m=3}^{N-2}\,\epsilon^{m+4}\,
\|{\cal B}^{[N-m-2]}AP_{i\leq 3m+\alpha-8}\|\,\|\vh_m\|
\nonumber
\\[1mm]
& &
+\ \frac{\epsilon^{N+3}}{2}\,\|{\cal B}P_{i\leq 3(N-1)+\alpha-8}\|\,\|\vh_{N-1}\|
\ +\ \frac{\epsilon^{N+4}}{2}\,\|{\cal B}P_{i\leq 3N+\alpha-8}\|\,\|\vh_N\|\nonumber
\\[1mm]
& &
+\ \frac{\epsilon^{N+5}}{2}\,\|{\cal B}P_{i\leq 3(N+1)+\alpha-8}\|\,\|\vh_{N+1}\|
\ +\ \frac{\epsilon^{N+6}}{2}\,\|{\cal B}P_{i\leq 3(N+2)+\alpha-8}\|\,\|\vh_{N+2}\|
\nonumber
\\[1mm]
& &
+\ \frac{1}{2}\,\sum_{m=0}^{N-2}\,\epsilon^{m+4}\,
\|{\cal C}^{[N-m-2]}AP_{i\leq 3m+\alpha}\|\,\|\wh_m\|
\nonumber
\\[1mm]
& &
+\ \frac{\epsilon^{N+3}}{2}\,\|{\cal C}P_{i\leq 3(N-1)+\alpha}\|\,\|\wh_{N-1}\|
\ +\ \frac{\epsilon^{N+4}}{2}\,\|{\cal C}P_{i\leq 3N+\alpha}\|\,\|\wh_N\|\nonumber
\\[1mm]
& &
+\ \sum_{m=3}^{N+2}\,\epsilon^m\,\|{\cal R}^{[N-m+2]}AP_{i\leq 3m+\alpha-8}\|\,
\|\vh_m\|\nonumber
\\[1mm]
& &
+\ \sum_{m=N+3}^{2N+4}\,\epsilon^m\,\sum_{l=m-N-2}^{N+2}\,
|E_{l}|\,\|\vh_{m-l}\|.\label{uglytoo}
\end{eqnarray}

Since $\|\partial_{x}P_{i\leq n}\|\,=\,[(n+1)/2]^{1/2}$, we have
\begin{eqnarray*}
%\lefteqn{ \frac{\epsilon^{N+3}}{2}\,\|\partial_{x}P_{i\leq 3(N+1)+\alpha-7}\|\,
\|\partial_{x}A\|\,\|\vh_{N+1}\|
%}
%\\
& < &
\tilde{a}_{1}\,\tilde{b}_{1}^{N}\,\epsilon^{n+3}\,[(3N+\alpha-3)(N+\alpha)!]^{1/2}
\\[1mm]
&\leq&
\tilde{a}_{1}\,\tilde{b}_{1}^{N}\,\epsilon^{n+3}\,
\max_{N\geq 3}\,\left[\frac{3N+\alpha-3}{N+\alpha+1}\right]^{1/2}[(N+\alpha+1)!]^{1/2}
\\[2mm]
&\leq&
a_{1}\,b_{1}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha+1)!]^{1/2}.
\end{eqnarray*} Similarly,
\[
\frac{\epsilon^{N+4}}{2}\,\|\partial_{x}P_{i\leq 3(N+2)+\alpha-8}\|\,
\|\partial_{x}A\|\,\|\vh_{N+2}\|\
<\ a_{2}\,b_{2}^{N+4}\,\epsilon^{N+4}\,[(N+\alpha+2)!]^{1/2}.
\]

The estimate for the third term in (\ref{uglytoo}) can essentially be read out from 
(\ref{compare1}). The result is \[
\sum_{m=0}^{N-1}\,\epsilon^{m+3}\,
\|{\cal A}^{[N-m-1]}P_{i\leq 3m+\alpha+1}\|\,\|\partial_xA\|\,\|\wh_m\|\
<\ a_{3}\,b_{3}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha+1)!]^{1/2}.
\]
For the next term we have
\[
\epsilon^{N+3}\,\|{\cal A}P_{i\leq 3N+\alpha+1}\|\,\|\partial_xA\|\,\|\wh_{N}\|\
<\ a_{4}\,b_{4}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha+1)!]^{1/2}.
\]

The fifth term in (\ref{uglytoo}) is quite similar to the one in (\ref{ugly}). 
As in (\ref{compare2}), by resorting
to part (ii) of Lemma~\ref{aboutf} and part (ii) of Lemma~\ref{facts}, we get
\[
\frac{1}{2}\,\sum_{m=3}^{N-2}\,\epsilon^{m+4}\,
\|{\cal B}^{[N-m-2]}AP_{i\leq 3m+\alpha-8}\|\,\|\vh_m\|\
<\ a_{5}\,b_{5}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha-1)!]^{1/2}.
\]
The bounds for the other terms that involve the function $\cal B$
do not deserve much comment. We have
\[
\frac{\epsilon^{N+3}}{2}\,\|{\cal B}P_{i\leq 3(N-1)+\alpha-8}\|\,\|\vh_{N-1}\|\
<\ a_{6}\,b_{6}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha-2)!]^{1/2},
\]
\vspace{3mm}
\[
\frac{\epsilon^{N+4}}{2}\,\|{\cal B}P_{i\leq 3N+\alpha-8}\|\,\|\vh_N\|\
<\ a_{7}\,b_{7}^{N+4}\,\epsilon^{N+4}\,[(N+\alpha-1)!]^{1/2},
\]
\vspace{3mm}
\[
\frac{\epsilon^{N+5}}{2}\,\|{\cal B}P_{i\leq 3(N+1)+\alpha-8}\|\,\|\vh_{N+1}\|\
<\ a_{8}\,b_{8}^{N+5}\,\epsilon^{N+5}\,[(N+\alpha)!]^{1/2},
\]
and
\[
\frac{\epsilon^{N+6}}{2}\,\|{\cal B}P_{i\leq 3(N+2)+\alpha-8}\|\,\|\vh_{N+2}\|\
<\ a_{9}\,b_{9}^{N+6}\,\epsilon^{N+6}\,[(N+\alpha+1)!]^{1/2}.
\]

Similarly,
\[
\frac{1}{2}\,\sum_{m=0}^{N-2}\,\epsilon^{m+4}\,
\|{\cal C}^{[N-m-2]}AP_{i\leq 3m+\alpha}\|\,\|\wh_m\|\
<\ a_{10}\,b_{10}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha+1)!]^{1/2},
\]
\vspace{3mm}
\[
\frac{\epsilon^{N+3}}{2}\,\|{\cal C}P_{i\leq 3(N-1)+\alpha}\|\,\|\wh_{N-1}\|\
<\ a_{11}\,b_{11}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha)!]^{1/2},
\]
and
\[
\frac{\epsilon^{N+4}}{2}\,\|{\cal C}P_{i\leq 3N+\alpha}\|\,\|\wh_N\|\
<\ a_{12}\,b_{12}^{N+4}\,\epsilon^{N+4}\,[(N+\alpha+1)!]^{1/2}.
\]

We handle the last two terms in (\ref{uglytoo}) in much the same way that we 
handled (\ref{compare3}) and (\ref{compare4}). Thus,
\[
\sum_{m=3}^{N+2}\,\epsilon^m\,\|{\cal R}^{[N-m+2]}AP_{i\leq 3m+\alpha-8}\|\,\|\vh_m\|\
<\ a_{13}\,b_{13}^{N+3}\,\epsilon^{N+3}\,[(N+\alpha+1)!]^{1/2},
\]
and
\[
\sum_{m=N+3}^{2N+4}\,\epsilon^m\,\sum_{l=m-N-2}^{N+2}\,|E_{l}|\,\|\vh_{m-l}\|\
<\ \sum_{m=N+3}^{2N+4}\,a_{14}\,b_{14}^{m}\,\epsilon^m\,[(m+\alpha+1)!]^{1/2}.
\]

Putting all these estimates together, we obtain the bound on $\|T_N\|$ in the theorem
if we define $G_{2}:=\max\,\{\,a_{1},\,\cdots,\,a_{14}\,\}$
and $F_{2}:=\max\,\{\,b_{1},\,\cdots,\,b_{14}\,\}$.

The last statement of the theorem follows immediately.\hfill $\Box$

\vskip 5mm
By using exactly the same ideas as in the proof of theorem~4 in \cite{toloza2},
Theorem \ref{errornorm} implies the following result, which is a slightly more 
detailed statement of Theorem \ref{main}.

\begin{thm} Let $F$ be the constant defined in Theorem~\ref{errornorm}. 
For each $0< g < F^{-2}$, there is $\epsilon_{g} > 0$ such that
for $0 < \epsilon\leq\epsilon_{g}$, there exists $N(\epsilon)$ that behaves 
like $g/\epsilon^2$, such that 
\[
\left\|\left(H(\epsilon) - E^{N(\epsilon)}\right)\Psi^{N(\epsilon)}\right\|
< \Lambda\exp\left(-\frac{\Gamma}{\epsilon^{2}}\right)
\]
for some constants $\Lambda>0$ and $\Gamma>0$ independent of $\epsilon$.
\end{thm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Appendix}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this appendix we derive formulas (\ref{SN}) and (\ref{TN}) which
are the differences between the two sides
of the equations when the truncated Rayleigh--Schr\"odinger series
are substituted into equations (\ref{eq1}) and (\ref{eq2}).

We truncate the expansions for $w$, $v$, and $E$ at orders
$N$, $N+2$, and $N+2$, respectively. We then insert them into
equation (\ref{eq1}) and expand various functions that appear in their Taylor 
series of appropriate orders.
The order to which we expand depends on the power of $\epsilon$
that it multiplies, as will be seen below.
The results of our perturbation
calculations guarantee that all terms of order $\epsilon^n$
with $n\le N+2$ cancel. The quantity $S_N$ is simply the sum
of the terms that do not have to cancel. Therefore,
we simply need to do the algebra to see that $S_N$ can be put
into the form given in (\ref{SN}).

To simplify some expressions, we define the following expressions, all of which are 
actually equal to $f(z)$.
\[
\tilde{f}^{[n]}(z)\,=\,\left\{\
\begin{array}{cl}\displaystyle
\sum_{m=0}^n\,\frac{f^{(m)}}{m!}\,z^m\ +\ f^{[n]}
&\mbox{ if}\quad n\ge 0,\\[7mm]
f^{[-1]}(z)&\mbox{ if}\quad n\le -1.
\end{array}\right.
\]
We also regard all empty sums, {\it e.g.}, $\displaystyle \sum_{n=4}^2\,\cdots$
as zero.

\vskip 4mm
Before the making use of the cancellations we write $S_N$ as
\begin{eqnarray}\nonumber
&&(H_0-E_2)\,\sum_{m=0}^N\,\epsilon^{n+2}\,w_n\,
-\,\sum_{m=0}^{N+2}\,\epsilon^{m+3}\,
\tilde{{\cal A}}^{[N-m-1]}\,\partial_xv_m
+\,\frac 12\,\sum_{m=0}^N\,\epsilon^{n+4}\,
\tilde{{\cal B}}^{[N-m-2]}\,w_m\\[2mm]
&&-\,\frac 12\,\sum_{m=0}^{N+2}\,\epsilon^{n+4}\,
\tilde{{\cal C}}^{[N-m-2]}\,v_m\,
+\,\sum_{m=0}^N\,\epsilon^m\,
\tilde{{\cal D}}^{[N-m+2]}\,w_m\,
-\,\sum_{k=3}^{N+2}\,\epsilon^k\,E_k\,
\sum_{m=0}^N\,\epsilon^m\,w_m.
\label{grubby1}
\end{eqnarray}
Note that we have used $E_0=E_1=0$, and that we have combined the
$E_2$ term with the first term.
We now rewrite each term in this expression in a way that makes the
$\epsilon$ dependence clearer, and then drop all terms of orders
$\epsilon^n$ with $n\le N+2$.

The first term is simply dropped.

The second term in (\ref{grubby1}) equals
\begin{eqnarray*}
&&-\,\sum_{m=0}^N\,\epsilon^{m+3}\,{\cal A}^{[N-m-1]}\,
\partial_x\,v_m\,
-\,\epsilon^{N+4}\,{\cal A}^{[-1]}\,\partial_x\,v_{N+1}\,
-\,\epsilon^{N+5}\,{\cal A}^{[-1]}\,\partial_x\,v_{N+2}\\[2mm]
&&-\,\sum_{m=0}^N\ \sum_{k=0}^{N-m-1}\,
\epsilon^{m+k+3}\,a_k\,x^k\,\partial_x\,v_m.
\end{eqnarray*}
The first three terms in this expression appear in (\ref{SN}), and the
last term is dropped since all of its contributions are $O(\epsilon^n)$
with $n\le N+2$.

Similarly, the third term in (\ref{grubby1}) can be written as
\[
\frac 12\ \sum_{m=0}^{N-1}\,\epsilon^{m+4}\,{\cal B}^{[N-m-2]}\,w_m\
+\ \frac{\epsilon^{N+4}}2\,{\cal B}^{[-1]}\,w_N\
+\ \frac 12\,\sum_{m=0}^{N-1}\ \sum_{k=0}^{N-m-2}\,
\epsilon^{m+k+4}\,b_k\,x^k\,w_m.
\]
The first two terms here appear in (\ref{SN}), and the
last term is dropped since all of its contributions are $O(\epsilon^n)$
with $n\le N+2$.

The fourth term is handled by the same technique.

The fifth term equals
\[
\sum_{m=0}^N\,\epsilon^m\,{\cal D}^{[N-m+2]}\,w_m\ +\ \sum_{m=0}^{N+2}\ 
\sum_{k=0}^{N-m+2}\,
\epsilon^{m+k}\,d_k\,x^k\,w_m.
\]
The first term here is included in (\ref{SN}), and the second is dropped.

The final term in (\ref{grubby1}) equals
\begin{eqnarray*}
&&\sum_{k=3}^{N+2}\ \sum_{m=0}^N\,\epsilon^{k+m}\,E_k\,w_m\\[2mm]
&=&\sum_{l=3}^{2N+2}\,\epsilon^l\,
\sum_{m=\max\,\{0,\,l-N-2\}}^{\min\,\{N,\,l-3\}}\,E_{l-m}\,w_m\\[2mm]
&=&\sum_{l=3}^{N+2}\,\epsilon^l\,
\sum_{m=0}^{l-3}\,E_{l-m}\,w_m\
+\ \sum_{l=N+3}^{2N+2}\,\epsilon^l\,
\sum_{m=l-N-2}^{N}\,E_{l-m}\,w_m.
\end{eqnarray*}
The second term in the final expression here is contained in (\ref{SN}),
and the first term is dropped.

This finishes the derivation, since the terms we have not dropped are
precisely those given in (\ref{SN}).

The derivation of (\ref{TN}) follows exactly the same procedure, so we
omit the details.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

%\bibitem{bohm} Bohm, A., Kendrick, B., Loewe, M. E., and Boya, L. J., 
%``The Berry connection and Born--Oppenheimer method,'' 
%J. Math. Phys.\ {\bf 33}, 977--989 (1992).

\bibitem{born} Born, M.\ and Oppenheimer, R., `` Zur Quantentheorie der Molekeln,''
Ann. Phys.\ (Leipzig) {\bf 84}, 457--484 (1927).

\bibitem{combes} Combes, J.-M., Duclos, P., and Seiler, R., ``The Born--Oppenheimer
approximation,'' in: {\em Rigorous Atomic and Molecular Physics} 
(eds.\ G. Velo, A. Wightman), New York, Plenum, 185--212 (1981).

\bibitem{hagedorn1} Hagedorn, G. A., ``High order corrections to the time--independent
Born--Oppenheimer aproximation I: smooth potentials,'' Ann. Inst. H. Poincar\'e Sect. A
{\bf 47}, 1--19 (1987).

\bibitem{hagedorn2} Hagedorn, G. A., ``High order corrections to the time--independent 
Born--Oppenheimer approximation. II. Coulomb systems,'' Commun. Math. Phys.\ {\bf 117},
387--403 (1988).

\bibitem{hagedorn3} Hagedorn, G. A.\ and Joye, A., ``Semiclassical dynamics with 
exponentially small error estimates,'' Commun. Math. Phys.\ {\bf 207}, 439--465 (1999).

\bibitem{hagedorn4} Hagedorn, G. A.\ and Joye, A., ``Exponentially accurate 
semiclassical dynamics: prapagation, localization, Ehrenfest times, scattering, 
and more general states,'' Ann. Henri Poincar\'e {\bf 1}, 837--883 (2000).

\bibitem{hagedorn5} Hagedorn, G. A.\ and Joye, A., ``A time--dependent Born--Oppenheimer
approximation with exponentially small error estimates,'' Commun. Math. Phys.\ {\bf 223},
583--626 (2001).

\bibitem{herrin} Herrin, J.\ and Howland, J. S., ``The Born--Oppenheimer approximation:
straight-up and with a twist,'' Rev. Math. Phys.\ {\bf 9}, 467--488 (1997).

\bibitem{klein1} Klein, M., ``On the mathematical theory of
predissociation,'' Ann. Phys. {\bf 178}, 48--73 (1987).

\bibitem{klein2} Klein, M., Martinez, A., Seiler, R., and Wang, X. P., ``On the 
Born--Oppenheimer expansion for polyatomic molecules,'' Commun. Math. Phys.\ {\bf 143},
607--639 (1992).

\bibitem{martinez1} Martinez, A., ``R\'esonances dans l'approximation de 
Born--Oppenheimer I,'' J. Diff. Eq.\ {\bf 91}, 204--234 (1991).

\bibitem{martinez2} Martinez, A., ``R\'esonances dans l'approximation de 
Born--Oppenheimer II. Largeur de r\'esonances," Commun. Math. Phys.\ {\bf 135}, 
517--530 (1991).

\bibitem{martinez3} Martinez, A.\ and Sordoni, V., ``A general reduction scheme for 
the time--dependent Born--Oppenheimer approximation,'' C. R. Math. Acad. Sci. Paris 
{\bf 334}, 185--188 (2002).

%\bibitem{mead} Mead, C. A.\ and Truhlar, D. G.,
%``On the determination of Born--Oppenheimer
%nuclear motion wave functions including complications
%due to conical intersectionsand identical nuclei,''
%J. Chem. Phys.\ {\bf 70}, 2284--2296 (1979).

\bibitem{messirdi} Messirdi, B., ``Asymptotique de Born--Oppenheimer pour la 
pr\'edissociation mol\'eculaire (cas de potentiels r\'eguliers),'' Ann. Inst. H. 
Poincar\'e Sect. A {\bf 61}, 255--292 (1994).

\bibitem{nenciu} Nenciu, G.\ and Sordoni, V., ``Semiclassical limit for multistate
Klein--Gordon systems: almost invariants subspaces and scattering theory,'' Math. Phys. 
Preprint Archive mp\_arc 01-36 (2001).

\bibitem{rousse} Rousse, V., (in preparation).

\bibitem{sordoni} Sordoni, V., ``Born--Oppenheimer expansion for excited states of 
diatomic molecules,'' C. R. Acad. Sci. Paris S\'er. I Math.\ {\bf 320}, 1091--1096 (1995).

\bibitem{toloza1} Toloza, J. H., ``Exponentially accurate error estimates of 
quasiclassical eigenvalues,'' J. Phys. A {\bf 34}, 1203--1218 (2001).

\bibitem{toloza2} Toloza, J. H., ``Exponentially accurate error estimates of 
quasiclassical eigenvalues. II. Several dimensions,'' J. Math. Phys.\ {\bf 44}, 
2806--2838 (2003).

\end{thebibliography}

\end{document}
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