text:
\section{Introduction}

In this paper, we construct exponentially accurate semiclassical
approximations $\psi(x,\,t,\,\hbar)$ to certain normalized
exact solutions $\Psi(x,\,t,\,\hbar)$ of the
$d$--dimensional time--dependent Schr\"od\-ing\-er equation
\begin{equation}\label{sch}
i\,\hbar\,\frac{\partial\Psi}{\partial t}\ =\
-\,\frac{\hbar^2}{2}\,\Delta\,\Psi\,+\,V\,\Psi .
\end{equation}
More precisely, our main result is that for $|t|\le T$ and small values of
$\hbar$, these approximations satisfy error estimates of the form
\begin{equation}\label{expest}
\|\,\psi(x,\,t,\,\hbar)\,-\,\Psi(x,\,t,\,\hbar)\,\|_{L^2(I\kern-.277em R^d)}
\ \le\ \,C\ \exp\left\{\,-\,\gamma/\hbar\,\right\} ,
\end{equation}
where $\gamma >0$.

Our construction of $\psi(x,\,t,\,\hbar)$ is technically complicated, but quite
explicit. It uses a particular collection of semiclassical wave packets
$\{\,\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)\,\}$ that are defined in
\cite{semi4}, \cite{raise}, and the next section. Here $A$ and $B$ are
$d\times d$ complex matrices that satisfy certain conditions. The quantities $a$
and $\eta$ are elements of $\hbox{$\mit I$\kern-.277em$\mit R$}^d$. For fixed $A$, $B$, $\hbar$, $a$, and $\eta$,
$\{\,\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)\,\}$ is an orthonormal basis
of $L^2(\hbox{$\mit I$\kern-.277em$\mit R$}^d)$ as $j$ ranges over all $d$-dimensional multi-indices.

The function $\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)$ is concentrated
near position $a$, and its Fourier transform is concentrated near momentum
$\eta$. Its position and momentum uncertainties are proportional to
$\sqrt{\hbar}$. The position uncertainty is determined by the matrix
$|A|=\sqrt{A\,A^*}$, and the momentum uncertainty is determined by
$|B|=\sqrt{B\,B^*}$.

We construct  $\psi(x,\,t,\,\hbar)$ by applying the idea of ``optimal
truncation'' of an asymptotic expansion.
For initial conditions of the form
\begin{equation}\label{incon}
\Psi(x,\,0,\,\hbar)\,=\,\sum_{|j|\le J}\,c_j\,
\varphi_j(A(0),B(0),\hbar,a(0),\eta(0),x),
\end{equation}
with $\sum_{|j|\le J}\,|c_j|^2\,=\,1$,
there exist (\cite{semi3}, \cite{semi4}) approximate solutions
\begin{equation}\label{app}
\psi_l(x,\,t,\,\hbar)\,=\,e^{iS(t)/\hbar}\
\sum_{|j|\le \widetilde{J}(l)}\,
c_j(l,\,t,\,\hbar)\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x),
\end{equation}
that satisfy
\begin{equation}\label{est}
\sup_{t\in [-T,T]}\ \|\psi_l(x,\,t,\,\hbar)-\Psi(x,\,t,\,\hbar)\|_{L^2(I\kern-.277em R^d)}
\,\leq\,C(l)\,h^{l/2}
\end{equation}
for some constant $C(l)$.
Here $\widetilde{J}(l)=J+3l-3$, and $A(t)$, $B(t)$, $a(t)$, $\eta(t)$, and
$S(t)$ are solutions to the classical equations of motion
\begin{eqnarray}
\dot{a}(t)&=&\eta(t) ,\nonumber \\
\dot{\eta}(t)&=&-\,\nabla V(a(t)),\nonumber \\
\dot{A}(t)&=&i\,B(t),\label{newton} \\
\dot{B}(t)&=&i\,V^{(2)}(a(t))\,A(t),\nonumber \\
\dot{S}(t)&=&\frac{\eta(t)^2}2\,-\,V(a(t)),\nonumber
\end{eqnarray}
where $V^{(2)}$ denotes the Hessian matrix for $V$, and the
initial conditions $A(0)$, $B(0)$, $a(0)$, $\eta(0)$, and
$S(0)=0$ satisfy
\begin{eqnarray}\label{initcond}
A^t(0)\,B(0)\,-\,B^t(0)\,A(0)&=&0,\\
A^*(0)\,B(0)\,+\,B^*(0)\,A(0)&=&2\,I.
\end{eqnarray}
The $c_j(l,\,t,\,\hbar)$ satisfy a linear system of ordinary differential
equations that we describe in the next section.

We carefully estimate the $l$--dependence of $C(l)$ in (\ref{est}).
Then, for each $\hbar$, we choose $l(\hbar)$ to minimize the error
$C(l(\hbar))\,\hbar^{l(\hbar)/2}$ over all choices of $l$. It
turns out that $l(\hbar)$ behaves like a constant times $1/\hbar$. We define
$\psi(x,\,t,\,\hbar)=\psi_{l(\hbar)}(x,\,t,\,\hbar)$ and
prove that (\ref{expest}) is satisfied.

For $t$ in a fixed compact interval, the precise statement of our results is the
following:

\vskip .25cm
\noindent
\begin{thm}\label{mainthm}
Suppose $V$ is a real-valued function on $\hbox{$\mit I$\kern-.277em$\mit R$}^d$ that is bounded below and
has an analytic continuation to the set
\begin{equation}\nonumber
D\,=\,\{\,z\in\hbox{$\mit I$\kern-.6em$\mit C$}^d\,:\ |\mbox{Im }z_j|\,<\,\delta,\ j=1,\,2,\,\cdots,\,d\,\} .
\end{equation}
Suppose further that there exist $M>0$ and $\tau>0$, such that
\begin{equation}\nonumber
|\,V(z)\,|\,\leq\,M\,\exp(\tau |z|^2),\qquad \mbox{for all } z\in D,
\end{equation}
where $|z|^2\,=\,\sum_{j=1}^d\,|z_j|^2$.
Suppose initial conditions $A(0)$, $B(0)$, $a(0)$, $\eta(0)$, $S(0)=0$, and
$c_j$ for $|j|\le J$ are specified that satisfy (\ref{initcond}) and
$\sum_{|j|\le J}\,|c_j|^2\,=\,1$.
Then for any $T>0$, there exist $C$ and $\gamma >0$,
such that the difference between the semiclassical
approximation $\psi(x,\,t,\,\hbar)$ and the exact solution $\Psi(x,\,t,\,\hbar)$
to the Schr\"odinger equation (\ref{sch}) with initial condition
(\ref{incon}) satisfies
$$
\|\,\psi(x,\,t,\,\hbar)\,-\,\Psi(x,\,t,\,\hbar)\,\|_{L^2(I\kern-.277em R^d)}
\ \le\ \,C\ \exp\left\{\,-\,\gamma/\hbar\,\right\} ,
$$
whenever $|t|\,\le\,T$.
\end{thm}

\vskip .25cm
\noindent
{\bf Remarks}\quad 1.\quad Theorem \ref{mainthm} can be generalized to allow
time--dependent potentials. For example, suppose a potential $V(x,\,t)$
depends smoothly on $t$, is bounded below, and satisfies
$$
\frac{1}{m!}\,|D^mV(x,\,t))|\ \leq\
\frac{M\,\exp(\tau |x|^2)}{\delta^{|m|}},
$$
for $|t|\le T$ for all multi-indices $m$.
Suppose further that a classical solution $(a(t),\,\eta(t))$ to
Newton's equations with potential $V(x,\,t)$ is bounded for $|t|\le T$. Then the
conclusion to Theorem \ref{mainthm} holds.\\[7pt]
2.\quad Our results can also be extended to obtain weaker error estimates of the
form\linebreak $C\ \exp\left\{\,-\,\gamma/\hbar^{\sigma}\,\right\}$ for some
$\sigma\in (0,\,1)$, when the potential belongs to a Gevrey class.\\[7pt]
3.\quad Theorem \ref{mainthm} is optimal in the sense that the conclusion fails
if the hypotheses are relaxed slightly.
For example, consider the one-dimensional potential
$V(x)=\exp (-1/x^u)$, for $x>0$, and $V(x)=0$ for $x\leq 0$,
where $u>0$. It is shown in \cite{Jung} that this potential belongs to the
Gevrey class of order $1+1/u$.
For initial conditions $a(0)=0$, $\eta(0)=0$,
$A(0)=B(0)=1$, $S(0)=0$, and $c_j(0)=\delta_{j,0}$, our approximation yields
$\varphi_0(1+it, 1, \hbar, 0, 0)$, for all times. This function
is very simple, and we can write the error term explicitly.
By steepest descent analysis, we can show that there exist $\delta>0$ and
$\Sigma_1>\Sigma_2>0$, such that
$t\in (0,\hbar^{\delta})$ implies
\begin{eqnarray}\nonumber
\exp(-\Sigma_1/\hbar^{u/(1+u)})
&\le&\|\,e^{-itH(\hbar)/\hbar}\,\varphi_0(1,1,\hbar,0,0,\cdot)
\,-\,\varphi_0(1+it,1,\hbar,0,0,\cdot)\,\|\\[6pt] \nonumber
&\le&\exp(-\Sigma_2/\hbar^{u/(1+u)}).
\end{eqnarray}
Note also that if we choose $a(0)=-a<0$ and $ \eta(0)=\eta>0$, it is
easy to check that the error term is $O(\exp(-\gamma /\hbar))$,
for each $t<a/\eta$.\\[7pt]
4.\quad For all practical purposes, we can replace the $c_j$'s by
the corresponding Dyson expansion up to order $l(\hbar)$ without
spoiling our exponential estimate. The normalization of the
approximation, however, will be lost.

\vskip .25cm
Theorem \ref{mainthm} can also be generalized to allow time intervals that grow
like $|\log(\hbar)|$ as $\hbar$ tends to zero. However, we obtain a somewhat
weaker conclusion. Our precise results are summarized by the following theorem.

\vskip .25cm
\noindent
\begin{thm}\label{longtime}
Suppose $V$ is bounded below and analytic in
\begin{equation}\nonumber
D\,=\,\{\,z\in\hbox{$\mit I$\kern-.6em$\mit C$}^d\,:\ |\mbox{Im }z_j|\,<\,\delta,\ j=1,\,2,\,\cdots,\,d\,\} .
\end{equation}
Suppose further that there exist $M>0$ and $\tau>0$, such that
\begin{equation}\nonumber
|\,V(z)\,|\,\leq\,M\,\exp(\tau |z|),\qquad \mbox{for all } z\in D.
\end{equation}
Suppose initial conditions $A(0)$, $B(0)$, $a(0)$, $\eta(0)$, $S(0)=0$, and
$c_j$ for $|j|\le J$ are specified that satisfy (\ref{initcond}) and
$\sum_{|j|\le J}\,|c_j|^2\,=\,1$, and further assume there exist $N>0$
and$\lambda >0$, such that
\begin{equation}\label{expgro}
\|A(t)\|\,\leq\,N\,\exp (\lambda |t|).
\end{equation}
Then for sufficiently small $T'>0$, there exist $C'$, $\gamma'>0$, and
$\sigma>0$, such that the difference between the semiclassical
approximation $\psi(x,\,t,\,\hbar)$ and the exact solution $\Psi(x,\,t,\,\hbar)$
to the Schr\"odinger equation (\ref{sch}) with initial condition
(\ref{incon}) satisfies
$$
\|\,\psi(x,\,t,\,\hbar)\,-\,\Psi(x,\,t,\,\hbar)\,\|_{L^2(I\kern-.277em R^d)}
\ \le\ \,C'\ \exp\left\{\,-\,\gamma'/\hbar^{\sigma}\,\right\} ,
$$
whenever $|t|\,\le\,T'\,|\log(\hbar)|$.
\end{thm}

\vskip .25cm
\noindent
{\bf Remark}\quad
Standard existence and uniqueness theorems for systems of ODE's show that
condition (\ref{expgro}) is satisfied if
the norm of the Hessian $V^{(2)}(a(t))$ is uniformly bounded.
That is the
case if $V$ is the sum of a quadratic polynomial plus a bounded analytic
function. It is also the case if
$E$ denotes the energy of the considered trajectory and
$\|V^{(2)}(x)\|$ is bounded on the connected
component of the classically allowed region $D_E=\{ x\in\hbox{$\mit I$\kern-.277em$\mit R$}^d\,:\,V(x)\leq E\}$
that contains $a(t)$.
This is satisfied for all confining potentials.

\vskip .25cm
The propagation of coherent states is also considered by Combescure
and Robert in \cite{CR}, using an approximation given by a
linear combination of squeezed states. (The squeezed states coincide with our
semiclassical wave packets, although the notation is quite different.)
Their emphasis is on the long time behavior of this approximation. The bound
on the error term is of the form $C_{l}(t)\hbar^{l/2}$, with expicit control
of the time-dependence of $C_{l}(t)$ in terms of classical quantities. The
$l$ behavior is however not investigated.

Results of a flavor similar to ours can be found in the work of
Yajima \cite{Yajima}. They are obtained by means
of the pseudo-differential techniques developed in the analytic
context by Sj\"ostrand in \cite{Sjostrand}. These results concern the
propagation of wave packets of
the form $\varphi(x)=\mbox{e}^{iS(x)/\hbar}f(x)$, where $S$ is analytic and
$f$ belongs to the set
of compactly supported Gevrey functions of order $s>1$.
Assuming the potential $V$ is analytic, Yajima
constructs approximations to the actual evolution of
such wave packets that are valid up to an error, whose $L^2(\hbox{$\mit I$\kern-.277em$\mit R$}^d)$ norm
is of order $\mbox{e}^{-\gamma/\hbar^{1/(2s-1)}}$, with $\gamma >0$ (see
Theorems 1.2, 1.2 and Lemma 2.5 in \cite{Yajima}). However, it should be
possible to make use of the theory \cite{Sjostrand} to recover our results.

Similar issues have been dealt with by Bambusi, Graffi and Paul in
\cite{BGP}. They focus on the validity for large times of the semiclassical
approximation of the Heisenberg evolution of a smooth observable,
under analyticity assumptions on the hamiltonian. They prove that the
semiclassical approximation remains useful for times up to order
$|\ln(\hbar)|$, the Ehrensfest time scale. However, the Hamiltonians
they can accomodate consist more or less of analytic perturbations of
the harmonic oscillator that decay as $x$ and $p$ tend to infinity.

\vskip .25cm
The paper is organized as follows: In the Section 2, we prove Theorem
\ref{mainthm} under the assumption that two types of error terms satisfy
certain bounds. We prove the two required bounds in Sections 3 and 4.
In Section 5 we describe the proof of Theorem \ref{longtime}.

\vskip .5cm
\section{Proof of Theorem 1.1}
\setcounter{equation}{0}

We begin this section by presenting the definition of the semiclassical wave
packets\\
$\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,x)$ that is given in \cite{raise}.
A more explicit, but more complicated definition is given in \cite{semi4}.
Since \cite{raise} provides a detailed discussion of these wave packets, we
do not prove all their properties here.

We adopt the standard multi-index notation. A multi-index
$j=(j_1,\,j_2,\,\dots ,\,j_d)$ is a $d$-tuple of non-negative integers.
We define
$|j|=\sum_{k=1}^d\,j_k,\quad
j!=(j_1!)(j_2!)\cdots(j_d!)$,\\
$x^j=x_1^{j_1}x_2^{j_2}\cdots x_d^{j_d},\quad\mbox{and}\quad
\displaystyle D^j=\frac{\partial^{|j|}}{(\partial x_1)^{j_1}(\partial x_2)^{j_2}\cdots
(\partial x_d)^{j_d}}$.

Throughout the paper we assume $a\in\hbox{$\mit I$\kern-.277em$\mit R$}^d$, $\eta\in\hbox{$\mit I$\kern-.277em$\mit R$}^d$ and $\hbar>0$. We
also assume that $A$ and $B$ are $d\times d$ complex invertible matrices that
satisfy
\begin{eqnarray}\label{cond1}
A^t\,B\,-\,B^t\,A&=&0,\\ \label{cond2}
A^*\,B\,+\,B^*\,A&=&2\,I.
\end{eqnarray}
These conditions guarantee that both the real and imaginary parts of $BA^{-1}$
are symmetric. Furthermore, $\mbox{Re}\,BA^{-1}$ is strictly positive definite,
and $\left(\,\mbox{Re}\,BA^{-1}\,\right)^{-1}\,=\,A\,A^*$. We note that
conditions (\ref{cond1}) and (\ref{cond2}) are preserved under the dynamics
generated by (\ref{newton}).

Our definition of $\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,x)$ is based on the
following raising operators that are defined for $m=1,\,2,\,\dots ,\,d$.
$$
{\cal A}_m(A,B,\hbar,a,\eta)^*
\ =\
\frac{1}{\sqrt{2\hbar}}\,\left[\,\sum_{n=1}^d\,\overline{B}_{n\,m}\,(x_n-a_n)
\ -\ \sum_{n=1}^d\,\overline{A}_{n\,m}\,
(-i\hbar\frac{\partial\phantom{x^n}}{\partial x_n}-\eta_n)\,\right] .
$$

\vskip .25cm
\noindent
{\bf Definition}\quad
For the multi-index $j=0$, we define the normalized complex Gaussian wave packet
(modulo the sign of a square root) by
\begin{eqnarray}\nonumber
&&\varphi_0(A,\,B,\,\hbar,\,a,\,\eta,\,x)\,=\,\pi^{-d/4}\,
\hbar^{-d/4}\,(\det(A))^{-1/2}\\[6pt]
&&\qquad\qquad\times\
\exp\left\{\,-\,\langle\,(x-a),\,B\,A^{-1}\,(x-a)\,\rangle/(2\hbar)\,
+\,i\,\langle\,\eta,\,(x-a)\,\rangle/\hbar\,\right\} .
\end{eqnarray}
Then, for any non-zero multi-index $j$, we define
\begin{eqnarray}\nonumber
\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)&=&
\frac{1}{\sqrt{j!}}\
\left(\,{\cal A}_1(A,B,\hbar,a,\eta)^*\right)^{j_1}\
\left(\,{\cal A}_2(A,B,\hbar,a,\eta)^*\right)^{j_2}\ \cdots \\[5pt]
\nonumber &&\qquad\qquad\times\
\left(\,{\cal A}_d(A,B,\hbar,a,\eta)^*\right)^{j_d}\
\varphi_0(A\,,B,\,\hbar,\,a,\,\eta,\,\cdot\,).
\end{eqnarray}

\vskip .25cm
\noindent
{\bf Remarks}\quad
1.\quad For $A=B=I$, $\hbar=1$, and $a=\eta=0$, the
$\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)$ are just the standard Harmonic
oscillator eigenstates with energies $|j|+d/2$.\\[7pt]
2.\quad For each $A$, $B$, $\hbar$, $a$, and $\eta$,
the set $\{\,\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)\,\}$
is an orthonormal basis for $L^2(\hbox{$\mit I$\kern-.277em$\mit R$}^d)$.\\[7pt]
3.\quad The raising operators can also be given by another formula that was
omitted from \cite{raise} in the multi-dimensional case.
If we set
$$g(A,\,B,\,\hbar,\,a,\,x)\,=\,
\exp\left\{\,-\,\langle\,(x-a),\,
\left(BA^{-1}\right)^*\,(x-a)\,\rangle/(2\hbar)\,-\,i\,
\langle\,\eta,\,(x-a)\,\rangle/\hbar\,\right\} ,$$
then we have
$$\left(\,{\cal A}_m(A,B,\hbar,a,\eta)^*\,\psi\,\right)(x)\,=\,
-\,\sqrt{\frac{\hbar}2}\,\frac 1{g(A,\,B,\,\hbar,\,a,\,x)}\,
\sum_{n=1}^d\,\overline{A}_{n\,m}\,
\frac{\partial\phantom{x^n}}{\partial x_n}\,
\left(\,g(A,\,B,\,\hbar,\,a,\,x)\,\psi(x)\,\right) .$$
4.\quad In \cite{semi4}, the state $\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,x)$ is
defined as a normalization factor times
$$
{\cal H}_j(A;\,\hbar^{-1/2}\,|A|^{-1}\,(x-a))\
\varphi_0(A,\,B,\,\hbar,\,a,\,\eta,\,x).
$$
Here ${\cal H}_j(A;\,y)$ is a
$|j|^{\mbox{\scriptsize th}}$
order polynomial in $y$ that depends on $A$ only through
$U_A$, where $A=|A|\,U_A$ is the polar decomposition of $A$.\\[7pt]
5.\quad By the argument on page 370 of \cite{semi4} or by scaling out the $|A|$
and $\hbar$ dependence and using Remark 3 above, one can show that
${\cal H}_j(A;\,y)\,\mbox{e}^{-y^2/2}$ is an (unnormalized) eigenstate of the usual
Harmonic oscillator with energy $|j|+d/2$.\\[7pt]
6.\quad When the dimension $d$ is $1$, the position and momentum uncertainties
of the\newline
$\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)$ are
$\sqrt{(j+1/2)\hbar}\,|A|$ and $\sqrt{(j+1/2)\hbar}\,|B|$, respectively. In
higher dimensions, they are bounded by
$\sqrt{(|j|+d/2)\hbar}\,\|A\|$ and $\sqrt{(|j|+d/2)\hbar}\,\|B\|$,
respectively.\\[7pt]
7.\quad When we approximately solve the Schr\"odinger equation, the choice of
the sign of the square root in the definition of
$\varphi_0(A,\,B,\,\hbar,\,a,\,\eta,\,\cdot\,)$ is determined by continuity in $t$
after an arbitrary initial choice.

\vskip .25cm
The proof of the theorem depends on the following abstract lemma.

\vskip .25cm
\noindent
\begin{lem}\label{fundcalc}
Suppose $H(\hbar)$ is a family of self-adjoint
operators for $\hbar>0$. Suppose $\psi(t,\,\hbar)$ belongs to the domain
of $H(\hbar)$, is continuously differentiable in $t$, and approximately
solves the Schr\"odinger equation in the sense that
\begin{equation}\label{xidef}
i\,\hbar\,\frac{\partial\psi}{\partial t}(t,\,\hbar)\ =\
H(\hbar)\,\psi(t,\,\hbar)\ +\ \xi(t,\,\hbar),
\end{equation}
where $\xi(t,\,\hbar)$ satisfies
\begin{equation}\label{xiest}
\|\,\xi(t,\,\hbar)\,\|\ \le \,\mu(t,\,\hbar).
\end{equation}
Then, for $t>0$,
\begin{equation}\label{ultimate}
\|\,\mbox{e}^{-itH(\hbar)/\hbar}\,\psi (0,\,\hbar)\ -\ \psi(t,\,\hbar)\,\|\
\le\ \hbar^{-1}\ \int_0^{t}\,\mu(s,\,\hbar)\,ds.
\end{equation}
The analogous statement holds for $t<0$.
\end{lem}

\vskip .25cm
\noindent {\bf Proof}:\quad Assume $t>0$; the proof for $t<0$ is similar.
By the unitarity of the propagator $\mbox{e}^{-itH(\hbar)/\hbar }$
and the fundamental theorem of calculus, the quantity on the
left-hand side of (\ref{ultimate}) can be estimated as follows:
\begin{eqnarray}\nonumber
&&\|\,\mbox{e}^{-itH(\hbar)/\hbar}
\,\psi(0,\,\hbar)\ -\ \psi(t,\,\hbar)\,\|\\[5pt]
\nonumber
&=&\|\,\psi(0,\,\hbar)\ -\ \mbox{e}^{itH(\hbar)/\hbar}
\,\psi(t,\,\hbar)\,\|\\[7pt]
\nonumber
&=&\|\,\int_0^t\ \frac{\partial\phantom{s}}{\partial s}\,\left(
\,\psi(0,\,\hbar)\ -\ \mbox{e}^{isH(\hbar)/\hbar}
\,\psi(s,\,\hbar)\,\right)\ ds\ \| \\[7pt]
\nonumber
&=&\|\,\int_0^t\ \left(\,-i\,\hbar^{-1}\,
\mbox{e}^{isH(\hbar)/\hbar}\,H(\hbar)\,\psi(s,\,\hbar)\ -\
\mbox{e}^{isH(\hbar)/\hbar}
\,\frac{\partial\psi}{\partial s}(s,\,\hbar)\,\right)\ ds\ \| \\[6pt]
\nonumber
&=&\|\,\int_0^t\ i\,\hbar^{-1}\,
\mbox{e}^{isH(\hbar)/\hbar}\,\xi(s,\,\hbar)\ ds\ \|\\[5pt]
\nonumber
&\le&\hbar^{-1}\ \int_0^{t}\,
\mu(s,\hbar)\,ds.
\end{eqnarray}
This proves the lemma. \qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .25cm
Because $V$ is smooth and bounded below, there exist global solutions to
the first two equations of the system (\ref{newton}) for any initial
condition. It then follows immediately that the remaining three
equations of the system (\ref{newton}) have global solutions.
Furthermore, it is not difficult (\cite{semi3}, \cite{semi4}) to prove
that (\ref{cond1}) and (\ref{cond2}) are preserved by the flow.

As mentioned in the introduction, it is proved in \cite{semi3} and \cite{semi4}
that initial conditions of the form (\ref{incon}) give rise to
approximate solutions of the form
$$
\psi_l(x,\,t,\,\hbar)\,=\,e^{iS(t)/\hbar}\
\sum_{|j|\le \widetilde{J}(l)}\,
c_j(l,\,t,\,\hbar)\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x),
$$
with errors whose norms are of order $\hbar^{l/2}$.
Here $\widetilde{J}(l)=J+3l-3$, and $A(t)$, $B(t)$, $a(t)$, $\eta(t)$, and
$S(t)$ satisfy (\ref{newton}). The coefficients $c_j(l,\,t,\,\hbar)$ satisfy the
linear system
\begin{equation}\label{eqcj}
i\,\hbar\,\dot{c}_k(l,\,t,\,\hbar)\,=\,\sum_{|j|\le \widetilde{J}(l)}\,
K_{k\,j}(l,\,t,\,\hbar)\,c_j(l,\,t,\,\hbar),\qquad\qquad
|k|=0,\,1,\,\dots,\,\widetilde{J}(l),
\end{equation}
with initial conditions $c_j(l,\,0,\,\hbar)=c_j$, for $|j|\le J$
in accordance with (\ref{incon}) and $c_j(l,\,0,\,\hbar)=0$ for $|j|>J$.

To specify the $(J+3l-2)\times (J+3l-2)$ matrix $K(l,\,t,\,\hbar)$
that appears in
(\ref{eqcj}), we first decompose the potential as
\begin{equation}\label{defy}
V(x)=W_{a}(x)+Z_a(x)\equiv W_{a}(x)+(V(x)-W_a(x)),
\end{equation}
where $W_a(x)$ denotes the second order Taylor
approximation (with the obvious abuse of notation)
\begin{equation}
W_a(x)\,\equiv\,V(a)\,+\,V^{(1)}(a)\,(x-a)\,+\,V^{(2)}(a)\,(x-a)^2/2,
\end{equation}
Next (reverting to multi-index notation), we approximate $Z_a(x)$ by its Taylor
approximation of order $l+1$,
\begin{equation}\label{ypoly}
Z^{[l]}_a(x)\,=\,\sum_{3\le |m|\le l+1}\,\frac{(D^mV)(a)}{m!}\,(x-a)^m,
\end{equation}
and define the infinite matrix
\begin{eqnarray}\label{defk}
&&\widetilde{K}_{k\,j}(l,\,t,\,\hbar)\ =\nonumber \\[5pt]
&&\langle} \newcommand{\ket}{\rangle\,
\varphi_k(A(t),B(t),\hbar,a(t),\eta(t),x),\
Z^{[l]}_{a(t)}(x)\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x)\,\ket .
\end{eqnarray}
Then, we obtain the matrix $K(l,\,t,\,\hbar)$ from
$\widetilde{K}(l,\,t,\,\hbar)$ by restricting the indices to
$|j|\le\widetilde{J}(l)$ and $|k|\le\widetilde{J}(l)$.

The general strategy \cite{semi3}, \cite{semi4}, \cite{raise} to show that
$\psi_l(x,\,t,\,\hbar)$ is an approximation to the actual solution
$\Psi(x,\,t,\,\hbar)$ of (\ref{sch}) and (\ref{incon}) up to order
$\hbar^{l/2}$ is as follows:
From \cite{raise}, we know that for all multi-indices $j$,
\begin{eqnarray}\label{profi}
&&i\,\hbar\,\frac{\partial }{\partial t}\,
\left[\,\mbox{e}^{iS(t)/\hbar}\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x)\,\right]
\nonumber\\[8pt]
&=&\left(\,-\,\frac{\hbar^2}{2}\,\Delta\,+\,W_{a(t)}(x)\,\right)\,
\left[\,\mbox{e}^{iS(t)/\hbar}\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x)\,\right] .
\end{eqnarray}
Thus, the $\varphi_j$ take into account the kinetic energy and $W_{a(t)}(x)$
parts of the Hamiltonian.
Next, we expand the exact solution as
$$\Psi(x,\,t,\,\hbar)\,=\,\sum_j\,b_j(\hbar,\,t)\,
\mbox{e}^{iS(t)/\hbar}\,\varphi_j(A(t),\,B(t),\,\hbar,\,a(t),\,\eta(t),\,x),$$
where the $b_j(\hbar,\,t)$ satisfy an infinite linear system of ordinary
differential equations whose matrix is obtained from the
$Z_{a(t)}(x)$ term in the Hamiltonian.
In that system,
we make a first approximation by replacing the function $Z_{a(t)}(x)$ by
its Taylor approximation $Z^{[l]}_{a(t)}(x)$. This yields an infinite linear
system whose matrix is
$\widetilde{K}(l,\,t,\,\hbar)$. Its entries are time dependent polynomials
in $\hbar^{1/2}$  of order $l-1$.
We make a second approximation by truncating the infinite
system to obtain (\ref{eqcj}) that is satisfied by the
$c_j(l,\,t,\,\hbar)$. The result (\ref{est}) is proved by using
Lemma \ref{fundcalc} to show that the errors generated
by the Taylor approximation and the truncation approximation are of order
$\hbar^{l/2}$.

As described in the introduction, we construct the exponentially accurate
approximate solution
$\psi(x,\,t,\,\hbar)\,=\,\psi_{l(\hbar)}(x,\,t,\,\hbar)$, by keeping track of
the $l$--dependence of $C(l)$ in (\ref{est}) and then
choosing $l(\hbar)$ in order to minimize the error.

In the remainder of this section, we prove Theorem \ref{mainthm} under the
assumption that the following three technical lemmas are true. We prove the
first lemma in Section 3. It
estimates errors that arise from our replacement of $Z_{a(t)}(x)$ by
$Z^{[l]}_{a(t)}(x)$. The second and third lemmas are proved in Section 4. They
bound certain matrix elements and combinatorial quantities that arise from the
truncation approximation discussed above.

\vskip .25cm
\noindent
\begin{lem}\label{taylorprop}
Suppose $V$ satisfies the hypotheses of Theorem \ref{mainthm}, $|m|=l+2$,
$\widetilde{J}(l)=J+3l-3$, and
$$
\psi_l(x,\,t,\,\hbar)\,=\,\mbox{e}^{iS(t)/\hbar}\
\sum_{|j|\leq\widetilde{J}(l)}\,
c_j(l,\,t,\,\hbar)\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x),
$$
with $\sum_{|j|\leq\widetilde{J}(l)}\,|c_j(l,\,t,\,\hbar)|^2\,=\,1$.\\
Let $\zeta(x,a(t))=(a(t)+\theta_{x,a(t)}(x-a(t)))\in \hbox{$\mit I$\kern-.277em$\mit R$}^d$,
with $\theta_{x,a(t)}\in (0,\,1)$.
There exist constants $g_0$ and $g_1$,
that depend on $d$ and $J$ only, such that for sufficiently small $\hbar$,
\begin{eqnarray}\nonumber
&&\left\|\,\frac{D^mV(\zeta(x,a(t)))}{m!}\ (x -a(t))^m\
\psi_{l}(\cdot,\,t,\,\hbar)\,\right\|\\ \label{goal}
&\le&g_0\,M\,\exp(4\tau(\delta^2 d+a(t)^2))\,
\left(\,g_1\,\sqrt{\hbar(l+2)}\,\| A(t)\|/\delta\,\right)^{l+2}.
\end{eqnarray}
\end{lem}

\vskip .25cm
\noindent
\begin{lem}\label{dlem2}
We define the infinite matrix
\begin{eqnarray}\nonumber
&&\widetilde{X}^m_{j\,k}(t,\,\hbar)\ =
\\[5pt] \label{defx}
&&\langle} \newcommand{\ket}{\rangle\,
\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),\cdot),\
(x-a(t))^m\,\varphi_k(A(t),B(t),\hbar,a(t),\eta(t),\cdot)\,\ket ,
\end{eqnarray}
and then define the finite matrix $X^m(l,\,t,\,\hbar)$ from
$\widetilde{X}^m(t,\,\hbar)$ by restricting its indices to
$|j|\le\widetilde{J}(l)$ and $|k|\le\widetilde{J}(l)$. Then,
$X^m_{j\,k}(l,\,s,\,\hbar)=0$ and
$\widetilde{X}^m_{j\,k}(s,\,\hbar)=0$ if $\bigg| |j|-|k|\bigg| >|m|$ and,
for each $N>0$, there exists $D(N)<\infty$, such that
\begin{eqnarray}\nonumber
&&\sup_{\scriptsize \matrix{|k|\,\leq\widetilde{J}(l)\cr
\widetilde{J}(l)+1\le\,|j|\,\leq \widetilde{J}(l)+l+1}}
\left|\,\left(\widetilde{X}^{m_0}(t_0,\hbar)X^{m_1}(l,t_1,\hbar)
X^{m_2}(l,t_2,\hbar)\cdots X^{m_q}(l,t_q,\hbar)\right)_{j\,k}\,\right| \\
\label{dutil}&\leq&\left(\,
D(N)\,\sqrt{\hbar\,\widetilde{J}(l)}\,
\sup_{t\in\{t_0,t_1,t_2,\cdots t_q\}}\| A(t)\|\,
\right)^{|m_0|+|m_1|+|m_2|\cdots +|m_q|} ,
\end{eqnarray}
for any collection $m_0,\,m_1,\,\dots ,\,m_q$ of multi-indices that satisfy
$|m_j|/\widetilde{J}(l)\le N$.
\end{lem}

\vskip .25cm
\noindent
\begin{lem}\label{truncest} We define
\begin{equation}\label{dcombin}
F_p(n,\,q)\,=\,\sum_{\scriptsize\matrix{1\leq |m_1|,|m_2|,\cdots,|m_q|\leq p \cr
|m_1|+|m_2|+\cdots +|m_q|=n}}1
\end{equation}
to be the number of distinct sets $\{\,m_1,\,m_2,\,\dots,\,m_q\,\}$, where
each $m_j$ is a $d$--dimensional multi-index with $1\le |m_j|\le p$ and
$|m_1|+|m_2|+\cdots +|m_q|=n$.
We note that $F_p(n,\,q)$ is zero unless $q\le n\le qp$.
Suppose that a function $L_q(l)$ satisfies
\begin{equation}\label{Lest}
L_q(l)\,\le\,\frac{C_1^q}{\hbar^q\,q!}\,
\sum_{n=3l-2}^{q(l+1)}\,(C_2\hbar l)^{n/2}\,F_{l+1}(n,\,q),
\end{equation}
where $C_1$ and $C_2$ are constants.
Let $[\kern-.18em [\,\alpha\,]\kern-.18em ]$ denote the greatest integer less than or equal
to $\alpha$.
Then there exists $g^*>0$, such that for any $g\in (0,\,g^*)$, there
exist positive constants $C_3$, $\gamma_1$, and $\hbar^*$, that
depend only on $g$, $C_1$, and $C_2$, such that
\begin{equation}\label{lchoice}
0<\hbar<\hbar^*,\quad
l(\hbar)\,=\,[\kern-.18em [\,g/\hbar\,]\kern-.18em ],\quad {\mbox{and}}\quad
2\le q\le l(\hbar)+2
\end{equation}
imply
\begin{equation}
L_q(l(\hbar))\,\le\,C_3\,\exp\left\{\,-\,\gamma_1/\hbar\,\right\} .
\end{equation}
\end{lem}

\vskip .25cm
\noindent
{\bf Proof of Theorem \ref{mainthm}}:\quad
We define $\psi_l(x,\,t,\,\hbar)$ by (\ref{app}), where the
$c_j(l,\,t,\,\hbar)$ are determined by the system (\ref{eqcj}) and initial
conditions described above.

To apply Lemma \ref{fundcalc} we define
\begin{equation}\label{defxi}
\xi_l(x,\,t,\,\hbar)\,=\,
i\,\hbar\,\frac{\partial\phantom{t}}{\partial t}\psi_l(x,\,t,\,\hbar)\,-\,
\left(\,-\,\frac{\hbar^2}{2}\,\Delta\,+\,
V(x)\,\right)\,\psi_l(x,\,t,\,\hbar).
\end{equation}
By using (\ref{profi}), we see that this can be decomposed as
a sum of two terms,
\begin{equation}\label{defxi1}
\xi^{(1)}_l(x,\,t,\,\hbar)\ =\
\left(\,Z^{[l]}_{a(t)}(x)\,-\,Z_{a(t)}(x)\,\right)\,\psi_l(x,\,t,\,\hbar)
\end{equation}
and
\begin{equation}
\label{defxi2}
\xi^{(2)}_l(x,\,t,\,\hbar)\ =\
-\,P_{\{|j|\ge l+2\}}\,Z^{[l]}_{a(t)}(x)\,\psi_l(x,\,t,\,\hbar),
\end{equation}
where $P_{\{|j|\ge l+2\}}$ is the orthogonal projection onto the span of the
set\\
$\{\,\varphi_j(A(t),\,B(t),\,\hbar,\,a(t),\,\eta(t),\,\cdot\,):\ \,|j|\ge l+2\,\}$.

By the standard Taylor series error formula,
$$Z_{a(t)}(x)\,-\,Z^{[l]}_{a(t)}(x)
\ =\ \sum_{|m|=l+2}\,\frac{D^mV(\zeta(x,a(t)))}{m!}\ (x-a(t))^m,$$
for some $\zeta(x,a)=(a+\theta_{x,a}(x-a))$, with $\theta_{x,a}\in (0,\,1)$.
Thus, by the crude estimate
\begin{eqnarray}\nonumber
\sum_{|m|=n}\,1\ \le\ \sum_{|m|\le n}\,1&\le&
\sum_{j=1}^d\,\sum_{m_j\le n}\,1\\[5pt]
&=&(n+1)^d\ =\ \mbox{e}^{d\,\ln(n+1)}\ \le\
\left(\mbox{e}^d\right)^n,\label{crude}
\end{eqnarray}
and Lemma \ref{taylorprop}, we obtain
\begin{eqnarray}\nonumber
&&\left\|\,\xi^{(1)}_l(\cdot,\,t,\,\hbar)\,\right\|\\[5pt]
\nonumber
&\le&\sum_{|m|=l+2}\,
\left\|\,\frac{D^mV(\zeta(x,a(t)))}{m!}\ (x-a(t))^m\
\psi_{l}(\cdot,\,t,\,\hbar)\,\right\|\\[5pt]
\nonumber
&\le&g_0\,M\,\exp(4\tau(\delta^2 d+a(t)^2))\,
\left(\,g_1\,\sqrt{\hbar(l+2)}\ \| A(t)\|/\delta\,\right)^{l+2}\,
\left(\mbox{e}^d\right)^{l+2}.
\end{eqnarray}
Thus, there exist constants $C_4$ and $C_5$, such that $|t|\,\le\,T$ implies
\begin{equation}\label{taylordone}
\left\|\,\xi^{(1)}_l(\cdot,\,t,\,\hbar)\,\right\|\,\le\,
C_4\,\left(\,C_5\,\sqrt{\hbar(l+2)}\,\right)^{l+2}.
\end{equation}

The quantity $\xi^{(2)}_l(\cdot,\,t,\,\hbar)$ satisfies
$$
\langle} \newcommand{\ket}{\rangle\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x),\
\xi^{(2)}_l(x,\,t,\,\hbar)\,\ket\,=\,0,
\quad\mbox{if}\quad\left\{
\matrix{0\leq|j|\leq\widetilde{J}(l)\quad\mbox{or}\cr
|j|>\widetilde{J}(l)+l+1,}\right.
$$
and
\begin{eqnarray}
&&\langle} \newcommand{\ket}{\rangle\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x),\
\xi^{(2)}_l(x,\,t,\,\hbar)\,\ket\,=
\nonumber\\[8pt]
&&\sum_{|k|\le\widetilde{J}(l)}\,
\langle} \newcommand{\ket}{\rangle\,\varphi_j(A(t),B(t),\hbar,a(t),\eta(t),x),\
Z_{a(t)}(x)\,\varphi_k(A(t),B(t),\hbar,a(t),\eta(t),x)\,\ket\,c_k(t)
\nonumber\\[6pt]
&&=(\widetilde{K}(t)\,{\bf c}(t))_j,\quad
\mbox{ if}\quad\widetilde{J}(l)<|j|\leq\widetilde{J}(l)+l+1,
\end{eqnarray}
where we denote the
$c_j(l,\,t,\,\hbar)$ collectively by the vector ${\bf c}(l,\,t,\,\hbar)$.
We easily verify these facts by using (\ref{eqcj}), (\ref{defk}),
(\ref{profi})
and Lemma \ref{dlem2}.
To estimate the norm of $\xi^{(2)}_l(\cdot,\,t,\,\hbar)$, we use the Dyson
expansion with remainder
to decompose
\begin{equation}\label{split}
{\bf c}(l,\,t,\,\hbar)=\sum_{q=0}^l\,{\bf c}^q(l,\,t,\,\hbar)\,+\,
{\bf r}(l,\,t,\,\hbar),
\end{equation}
where (dropping some arguments)
\begin{equation}\label{cqdef}
{\bf c}^q(t)=\int_0^tds_1\int_0^{s_1}ds_2\cdots\int_0^{s_{q-1}}
ds_q(i\hbar)^{-q} K(s_1)K(s_2)\cdots K(s_q)\,
{\bf c}(0),
\end{equation}
and
\begin{equation}\label{rdef}
{\bf r}(t)=
\int_0^tds_1\int_0^{s_1}ds_2\cdots\int_0^{s_{l}}ds_{l+1}
(i\hbar)^{-(l+1)} K(s_1)K(s_2)\cdots K(s_{l+1})\,
{\bf c}(s_{l+1}).
\end{equation}
Using (\ref{defk}), (\ref{ypoly}), and
(\ref{defx}), we see that each  ${\bf c}^q(l,\,t,\,\hbar)$ is of order
$\hbar^{q/2}$ and that ${\bf r}(l,\,t,\,\hbar)$ is of order $\hbar^{(l+1)/2}$.

To estimate the norm of $\xi^{(2)}_l(\cdot,\,t,\,\hbar)$ we study the
$j^{\mbox{\scriptsize th}}$  component of
$\widetilde{K}(t){\bf c}(l,\,t,\,\hbar)$,
with $|j|>\widetilde{J}(l)$. Because of
(\ref{split}), this coefficient is a sum of two types of terms: those that
arise from ${\bf c}^q(l,\,t,\,\hbar)$ and those that arise from
${\bf r}(l,\,t,\,\hbar)$.
Using (\ref{ypoly}), we expand $K(l,\,t,\,\hbar)$ in (\ref{cqdef}) and
(\ref{rdef}) and $\widetilde{K}(l,\,t,\,\hbar)$ in terms of
$X^m(l,\,t,\,\hbar)$ and $\widetilde{X}^m(t,\,\hbar)$, to obtain
\begin{eqnarray}
&&\widetilde{K}(t){\bf c}^q(t) =
(i\hbar)^{-q}\sum_{n=3(q+1)}^{(l+1)(q+1)}\sum_{\scriptsize\matrix{
m_0,m_1,m_2,\cdots,m_q\cr|m_0|+|m_1|+\cdots+|m_q|=n\cr 3\leq|m_j|\leq l+1}}
\nonumber\\[9pt]
&&\int_0^tds_1\int_0^{s_1}ds_2\cdots\int_0^{s_{q-1}}ds_q
\frac{D^{m_0}V(a(t))D^{m_1}V(a(s_1))\cdots
D^{m_q}V(a(s_q))}{m_0!m_1!m_2!\cdots m_q!}\nonumber\\[4pt]
&&\qquad\qquad\qquad\qquad\qquad\times\ \widetilde{X}^{m_0}(t)
X^{m_1}(s_1)X^{m_2}(s_2)\cdots  X^{m_q}(s_q)\,{\bf c}(0),
\label{expcq}
\end{eqnarray}
and
\begin{eqnarray}
&&\widetilde{K}(t){\bf r}(t) =
(i\hbar)^{-(l+1)}\sum_{n=3(l+2)}^{(l+1)(l+2)}\sum_{\scriptsize\matrix{
m_0,m_1,m_2,\cdots,m_{l+1}\cr|m_0|+|m_1|+\cdots+|m_{l+1}|=n\cr
3\leq|m_j|\leq l+1}}
\nonumber \\[9pt]
&&\int_0^tds_1\int_0^{s_1}ds_2\cdots\int_0^{s_l}ds_{l+1}
\frac{D^{m_0}V(a(t))D^{m_1}V(a(s_1))\cdots
D^{m_{l+1}}V(a(s_{l+1}))}{m_0!m_1!m_2!\cdots m_{l+1}!}\nonumber\\[4pt]
&&\qquad\qquad\qquad\qquad\qquad\times\ \widetilde{X}^{m_0}(t)
X^{m_1}(s_1)X^{m_2}(s_2)\cdots  X^{m_{l+1}}(s_{l+1})\,{\bf c}(s_{l+1}).
\label{exprt}
\end{eqnarray}
The values of $m_j$ that occur in both (\ref{expcq}) and
(\ref{exprt}) satisfy
\begin{equation}\label{Mest}
|m_j|/\widetilde{J}(l)\le (l+1)/(3l-3)\le 1,
\end{equation}
as long as $l\ge 3$. So, we can apply Lemma \ref{dlem2} with $N=1$.

Recall that $X^m_{j\,k}(l,\,s,\,\hbar)=0$ and
$\widetilde{X}^m_{j\,k}(s,\,\hbar)=0$ if $\bigg| |j|-|k|\bigg|>|m|$.
Since $c_k(0)$ is non-zero only for $|k|\le J$, and we need only consider
the $j^{\mbox{\scriptsize th}}$ coefficient of
$\widetilde{K}(t){\bf c}(t)$ for $|j|>\widetilde{J}(l)$, the only relevant
values of $n$ in (\ref{expcq}) must satisfy
\begin{equation}\label{ndl}
n\geq 3l-2.
\end{equation}
This condition is also satisfied for all values of $n$ in (\ref{exprt}), since
the sum begins with $n=3l+6$.

To use the analyticity assumptions to get estimates on
the derivatives of $V$, we define
$$
C_{\delta}(x)=\{z\in\hbox{$\mit I$\kern-.6em$\mit C$}^d\,:\ z_j=x_j+\delta e^{i\theta_j},\,
\theta_j\in [0,2\pi),\,j=1,2,\cdots, d\} .
$$
If $z\in C_{\delta}(a(t))$, then, for all
$j=1,2,\cdots ,d$,
$$
|z_j|\leq\delta +|a_j(t)|.
$$
Hence, writing $\displaystyle \frac{1}{m!}\,D^mV(a(t))$ as a $d$--dimensional Cauchy
integral, we get the bound
\begin{equation}\label{mess88}
\frac{1}{m!}\,|D^mV(a(t))|\ \leq\
 \frac{\sup_{z\in C_{\delta}(a(t))} |V(z)|}{\delta^{|m|}}\ \equiv\
\frac{v(t)}{\delta^{|m|}}
\end{equation}
where
$$
 v(t)\ \leq\ M\,\exp(4\tau(\delta^2 d+a^2(t))).
$$
Furthermore, since $\|A(t)\|$ depends continuously on $t$, there exists
$w(T)$, such that
$$\sup_{t\in [-T,T]}\ \|A(t)\|\ \leq\ w(T).$$

The norm of the vector ${\bf c}(t)$ is 1 since $K(l,\,t,\,\hbar)$ is
self-adjoint. Thus, the non-zero entries of
${\bf c}(0)$ are each bounded by 1, and by a crude estimate, there are
at most $(J+1)^d$ of them. Similarly,
${\bf c}(t)$ has at most $(\widetilde{J}(l)+1)^d$ non-zero entries, each of
which is bounded by 1.
Thus, for $l\ge 3$, (\ref{defk}), (\ref{ypoly}), (\ref{defx}),
(\ref{expcq}), (\ref{exprt}), (\ref{ndl}), (\ref{Mest}), and Lemma
\ref{dlem2} imply the following two estimates when $j$ satisfies
$\widetilde{J}(l)+1\le|j|\le\widetilde{J}(l)+l+1$:
\begin{eqnarray}\nonumber
&&\left|\,(\widetilde{K}(t)\,{\bf c}^q(t))_{j}\,\right| \\[5pt]
&\leq&   \nonumber
v(t)\frac{\left(\int_0^Tv(s)\,ds\right)^{q}}{\hbar^q q!}\,
\sum_{n=3l-2}^{(l+1)(q+1)}\,(D(1)\,w(T)/\delta )^n\,\hbar^{n/2}\,
\widetilde{J}(l)^{n/2}\,F_{l+1}(n,q+1)\,(J+1)^d \\[6pt]\nonumber
&=&\frac{v(t)\hbar(q+1)(J+1)^d}{\int_0^Tv(s)ds}
\frac{\left(\int_0^Tv(s)ds\right)^{q+1}}{\hbar^{q+1}(q+1)!}
\sum_{n=3l-2}^{(l+1)(q+1)}(D(1)w(T)\sqrt{\hbar
\widetilde{J}(l)}/\delta)^nF_{l+1}(n,q+1)\\ \label{cqest1}
\end{eqnarray}
and
\begin{eqnarray}\nonumber
&&\left|\,(\widetilde{K}(t){\bf r}(t))_{j}\,\right|\\[5pt]
\nonumber\!\! \leq&&\!\!\!\!
\frac{v(t)\hbar(l+2)(\widetilde{J}(l)+1)^d}{\int_0^Tv(s)ds}
\frac{\left(\int_0^Tv(s)ds\right)^{l+2}}{\hbar^{l+2}(l+2)!}
\sum_{n=3l-2}^{(l+1)(l+2)}(D(1)w(T)\sqrt{\hbar
\widetilde{J}(l)}/\delta )^nF_{l+1}(n,l+2),\\ \label{rest1}
\end{eqnarray}
where $F_p(n,q)$ is defined by (\ref{dcombin}).

By Lemma \ref{truncest}, (\ref{cqest1}), and (\ref{rest1}), both
$\left|\,(\widetilde{K}(t)\,{\bf c}^q(t))_{j}\,\right|$ and
$\left|\,(\widetilde{K}(t){\bf r}(t))_{j}\,\right|$ are bounded by
$C_3\,\exp\,\left\{\,-\gamma_1/\hbar\,\right\}$ for an appropriate choice of
$l(\hbar)\,=\,[\kern-.18em [\,g/\hbar\,]\kern-.18em ]$ and sufficiently small $\hbar$.
Each of the $l+2$ terms in (\ref{split}) contributes a term of this type to
$\xi^{(2)}_l(x,\,t,\,\hbar)$, so
\begin{eqnarray}\nonumber
\hbar^{-1}\ \left\|\,\xi^{(2)}_l(x,\,t,\,\hbar)\,\right\|
&\le&C_3\,\hbar^{-1}\,(l(\hbar)+2)\,\exp\,\left\{\,-\,\gamma_1/\hbar\,\right\}
\\[5pt] \nonumber
&\le&C_3\,\exp\,\left\{\,-\,\gamma_2/\hbar\,\right\} ,
\end{eqnarray}
for any $\gamma_2<\gamma_1$ when $\hbar$ is sufficiently small.

We shrink $g$ if necessary to make
\begin{equation}\label{george}
C_5^2\,g<1
\end{equation}
and set $l=l(\hbar)$ in
(\ref{taylordone}). This yields a similar estimate
$$\hbar^{-1}\ \left\|\,\xi^{(1)}_l(\cdot,\,t,\,\hbar)\,\right\|\,\le\,
C_6\,\exp\,\left\{\,-\,\gamma_3/\hbar\,\right\} ,$$
for some $\gamma_3>0$.

We combine these two estimates and apply Lemma \ref{fundcalc} to obtain
(\ref{expest}) with\\ $\gamma\,=\,\min\,\{\gamma_2,\,\gamma_3\}$.
This proves the theorem. \qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .5cm
\section{Proof of Lemma 2.2}
\setcounter{equation}{0}

For simplicity, we drop the $t$ dependence in the notation throughout this
section.

To prove Lemma \ref{taylorprop}, we use H\"older's inequality to see that
$\sum_{|j|\leq\widetilde{J}(l)}\,|c_j(l,\,\hbar)|^2\,=\,1$
implies
\begin{eqnarray}\nonumber
\sum_{|j|\leq\widetilde{J}(l)}\,|c_j(l,\,\hbar)|&\le&
\left(\,\sum_{|j|\leq\widetilde{J}(l)}\,|c_j(l,\,\hbar)|^2\,\right)^{1/2}\,
\left(\,\sum_{|j|\leq\widetilde{J}(l)}\,1\,\right)^{1/2}\\ \nonumber
&\le&\left(\,\widetilde{J}(l)+1\,\right)^{d/2}.
\end{eqnarray}
Thus, it is sufficient to prove
\begin{eqnarray}\nonumber
&&\left\|\,\frac{D^mV(\zeta(x,a))}{m!}\ (x-a)^m\
\varphi_{j}(A,\,B,\,\hbar,\,a,\,\eta,\,x)\,\right\|
\\ \label{goal1}
&\le&g_3\,M\,\exp(4\tau(\delta^2 d+a^2))\,
\left(\,g_4\,\sqrt{\hbar(l+2)}\,\| A\|/\delta\,\right)^{l+2},
\end{eqnarray}
for some $g_3$ and $g_4$, whenever $|j|\leq\widetilde{J}(l)$.

We mimic the proof of \ref{mess88} to obtain a bound on
$D^mV(\zeta(x,a))/m!$.
If $z\in C_{\delta}(\zeta(x,a))$, then, for all
$j=1,2,\cdots ,d$,
$$
|z_j|\,\leq\,\delta +|\zeta_j(x,a)|\,\leq\,\delta +|a_j|+|x_j-a_j|.
$$
Using this and applying $(b+c)^2\leq 2(b^2+c^2)$ several times,
we see that
$z\in C_{\delta}(\zeta(x,a))$ implies
$$
|\,V(z)\,|\,\leq\,M\,\exp(2\tau(x-a)^2)\,\exp(4\tau(\delta^2 d+a^2)).
$$
Hence, writing $\displaystyle \frac{1}{m!}\,D^mV(\zeta(x,a))$ as a $d$--dimensional Cauchy
integral, we easily obtain the bound
\begin{equation}\label{mess8}
\frac{1}{m!}\,|D^mV(\zeta(x,a))|\leq
M\ \frac{\exp(4\tau(\delta^2 d+a^2))}{\delta^{|m|}}\
\exp(2\tau(x-a)^2).
\end{equation}
Thus, estimate (\ref{goal1}) follows from the corresponding estimate
on the integral
\begin{equation}\nonumber
I=\int_{\hbox{$\mit I$\kern-.277em$\mit R$}^d}\,(x-a)^{2m}\,\exp(4\tau(x-a)^2)\
|\varphi_{j}(A,\,B,\,\hbar,\,a,\,\eta,\,x)|^2\,dx.
\end{equation}
Performing the change of variables $x\mapsto y=|A|^{-1}(x-a)/\hbar^{1/2}$,
and using the explicit formula for $\varphi_j$, we see that
\begin{equation}\nonumber
I\,=\,\frac{\hbar^{|m|}}{2^{|j|}\,j!\,\pi^{d/2}}\,\int_{\hbox{$\mit I$\kern-.277em$\mit R$}^d}\,
(|A|y)^{2m}\,\exp(-y^2+4\tau\hbar(|A|y)^2)\ |{\cal H}_j(A;\,y)|^2\,dy,
\end{equation}
where ${\cal H}_l(A;\,y)$ is the polynomial described in Remarks 4 and 5
that immediately follow the definition of $\varphi_{j}(A,B,\hbar,a,\eta,x)$ in
Section 2.

We assume henceforth that $\hbar$ is sufficiently small that
\begin{equation}\label{35}
4\,\tau\,\hbar\,\| A\|^2\,\leq\,1/2.
\end{equation}
The estimate $(|A|y)^2\,\leq\,\| A\|^2\,y^2$
implies $(|A|y)_k^2\,\leq\,\| A\|^2\,y^2$,
for $k=1,2,\cdots, d$. From this, we conclude
\begin{equation}\label{Iest}
I\ \leq\ \frac{{(\hbar\,\| A\|^2)}^{|m|}}{2^{|j|}\,j!\,\pi^{d/2}}\ \int\,
y^{2|m|}\,\exp(-y^2/2)\,|{\cal H}_j(A;\,y)|^2\,dy.
\end{equation}
We next need an estimate on  $|{\cal H}_j(A;\,y)|$.
By Remark 5 after the definition in Section 2,
\begin{equation}\label{amazing}
\left(-\Delta+y^2\right)\,{\cal H}_j(A;\,y)\,\mbox{e}^{-y^2/2}
=(2|j|+d)\ {\cal H}_j(A;\,y)\,\mbox{e}^{-y^2/2}.
\end{equation}
This equation states that ${\cal H}_j(A;\,y)\,\mbox{e}^{-y^2/2}$ is a
eigenfunction of $-\Delta+y^2$ corresponding
to the eigenvalue $2|j|+d$. Introducing normalization factors, we conclude
that
\begin{equation}\nonumber
2^{-|j|/2}\,(j!)^{-1/2}\,\pi^{-d/4}\,
{\cal H}_j(A;\,y)\,\mbox{e}^{-y^2/2}\,=\,\sum_{|k|=|j|}\,
b_k\,\varphi_{k_1}(y_1)\varphi_{k_2}(y_2)\cdots \varphi_{k_d}(y_d),
\end{equation}
where $\varphi_{k}(y)\,=\,2^{-k/2}\,(k!)^{-1/2}\,\pi^{-1/4}\,H_k(y)\,\mbox{e}^{-y^2/2}$
is the normalized eigenfunction of
$\displaystyle -\frac{\partial^2\phantom{y}}{\partial y^2}+y^2$
corresponding to the eigenvalue $2k+1$, and the coefficients $b_k$ satisfy
$\sum_{|k|=|j|}\,|b_k|^2\,=\,1$.
We can thus deduce an estimate of ${\cal H}_j(A;\,y)$ from an estimate for
usual Hermite polynomials $ H_k(y)$:
\begin{eqnarray}\nonumber
|{\cal H}_j(A;\,y)|^2&=&|\sum_{|k|=|j|}\,
b_k\,H_{k_1}(y_1)H_{k_2}(y_2)\cdots H_{k_d}(y_d)|^2\\[5pt]
&\leq&\sum_{|k|=|j|}\,|H_{k_1}(y_1)H_{k_2}(y_2) \cdots H_{k_d}(y_d)|^2.
\label{mess18}
\end{eqnarray}
The Hermite polynomials, in turn, satisfy the following bounds.

\vskip .25cm
\noindent
\begin{lem}\label{taylor1}
If $|y|>\sqrt{2k+1}$, then
\begin{equation}\label{joyest}
|H_k(y)|\,\leq\,2^k\,|y|^k,
\end{equation}
and there exists a numerical constant $\kappa >0$, such that for all $y\in\hbox{$\mit I$\kern-.277em$\mit R$}$,
\begin{equation}\nonumber
|H_k(y)|\,\leq\,\kappa\,2^{k/2}\,\sqrt{k!}\,\mbox{e}^{y^2/2}.
\end{equation}
\end{lem}

\vskip .25cm
\noindent
{\bf Proof:}\quad
The second statement is well known. See \cite{AandS}, formula 22.14.17 or
\cite{GrandRy5}, formula 8.954.2.

By symmetry, it is sufficient to prove (\ref{joyest}) for $y>0$.
It is well known that the $k^{\mbox{\scriptsize th}}$ eigenfuction of the
harmonic oscillator is non-zero in the classically forbidden
region $|y|>\sqrt{2k+1}$.
It is also well known that $H_k(y)=2^ky^k+P_k(y)$
where $P_k(y)$ is a polynomial of degree $k-2$. Thus, we have
$$
H_k(y)/(2^ky^k)=1+B_k(y),
$$
where $B_k$ satisfies
$$
B_k(y)\,=\,O(1/y^2)\qquad \mbox{for large } y,
$$
and
$$
B_k(y)>-1\qquad \mbox{if}\quad y>\sqrt{2k+1}.
$$
Using standard properties of the Hermite polynomials, we see that
\begin{eqnarray}\nonumber
B_k'(y)&=&(y\,H_k'(y)\,-\,k\,H_k(y))/(2^k\,y^{k+1})\\[5pt] \nonumber
&=&(2\,k\,y\,H_{k-1}(y)\,-\,k\,H_k(y))/(2^k\,y^{k+1})\\[5pt] \nonumber
&=&k\,(2\,y\,H_{k-1}(y)\,-\,(2\,y\,H_{k-1}(y)\,-\,
2\,(k-1)\,H_{k-2}(y)))/(2^k\,y^{k+1})\\[5pt] \nonumber
&=&k\,(k-1)\,H_{k-2}(y)/(2^{k-1}\,y^{k+1}).
\end{eqnarray}
So, if $y>\sqrt{2k+1}$, we see that $B_k'(y)>0$.
Thus, $y>\sqrt{2k+1}$ implies $-1<B_k(y)<0$,
which implies (\ref{joyest}). \qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .25cm
In (\ref{Iest}), we use the multinomial expansion
$$
(y^2)^{|m|}\,=\,\sum_{|n|=|m|}\,\pmatrix{&|m|&\cr n_1&\cdots&n_d}
\,y_1^{2n_1}\,y_2^{2n_2}\,\cdots\,y_d^{2n_d}
$$
to obtain
$$
I\leq\frac{{(\hbar\,\| A\|^2)}^{|m|}}{2^{|j|}\,j!\,\pi^{d/2}}
\sum_{|n|=|m|}\sum_{|k|=|j|}\pmatrix{&|m|& \cr n_1&\cdots&n_d}\prod_{r=1}^d
\int y_r^{2n_r}|H_{k_r}(y_r)|^2\exp(-y_r^2/2) dy_r.
$$
Using Lemma \ref{taylor1},
the one dimensional integrals in the final factor here can then be estimated as
follows (without indices):
\begin{eqnarray}\nonumber
&&\int\,y^{2n}\,|H_{k}(y)|^2\,\exp(-y^2/2)\,dy\\[7pt]
&\le&\int_{y^2\leq 2k+1}\kappa^2\,2^k\,k!\,y^{2n}\,\exp(y^2/2)\,dy\,+\,
  \int\,4^k\,y^{2(n+k)}\,\exp(-y^2/2)\,dy\nonumber\\[7pt]
&\le&\kappa^2 2^k k! \frac{2}{2n+1} (2k+1)^{n+1/2} \exp(k+1/2)\,+\,
  4^k 2^{(k+n+1/2)} \int z^{2(k+n)}e^{-z^2}dz
\nonumber\\[7pt]
&=&\frac{2^{k+1}}{2n+1} \kappa^2 e^{k+1/2} k! (2k+1)^{n+1/2}\,+\,
  4^k 2^{(k+n+1/2)} \pi^{1/2} \frac{(2(k+n)-1)!}{2^{2(n+k)-1}(k+n-1)!}
\nonumber\\[7pt]
&=&\frac{2^{k+1}}{2n+1} \kappa^2 e^{k+1/2} k! (2k+1)^{n+1/2}\,+\,
  2^{k-n+3/2} \pi^{1/2} \frac{(2(k+n)-1)!}{(k+n-1)!}.\label{mess24}
\end{eqnarray}
Here we have used
$$
\int\,z^{2(k+n)}\,e^{-z^2}\,dz\ =\ \pi^{1/2}\,
\frac{1\cdot 3\cdot 5\cdots (2(k+n)-1)}{2^{k+n}}.
$$
Now, Stirling's formula guarantees the
existence of $a>0$, such that
\begin{equation}\label{stirling}
a^n\,n^n\leq n!\leq n^n,
\end{equation}
for all integers $n\ge 1$.
Using this in (\ref{mess24}), we obtain
$$
\int\,y^{2n}\,|H_{k}(y)|^2\,\exp(-y^2/2)\,dy\le f_0\,f_1^{k+n}\,(k+n)!,
$$
for some constants $f_0$ and $f_1$.
From this we conclude (restoring the indices)
\begin{equation}\label{mess26}
I\ \le\ \frac{{(\hbar\,\| A\|^2)}^{|m|}}{2^{|j|}\,j!\,\pi^{d/2}}\,
\sum_{|n|=|m|}\ \sum_{|k|=|j|}\,f_0^d\,f_1^{|m|+|j|}\
\pmatrix{&|m|& \cr n_1&\cdots&n_d}\,(k+n)!
\end{equation}
In this expression, we have
\begin{eqnarray}\nonumber
(k+n)!&=&(|k|+|n|)!\,
\pmatrix{&|k|+|n|&\cr k_1+n_1&\cdots&k_d+n_d}^{-1}\\[5pt] \nonumber
&=&(|m|+|j|)!\,
\pmatrix{&|m|+|j|&\cr k_1+n_1&\cdots&k_d+n_d}^{-1}\\[7pt]
&\le&(|m|+|j|)!
\end{eqnarray}
We further use
$\displaystyle \sum_{|n|=|m|}\,\pmatrix{&|m|& \cr n_1 & \cdots & n_d}=d^{|m|}$
and (\ref{crude}) in (\ref{mess26}) to obtain
\begin{eqnarray}
&&I\ \leq\ (\hbar\,\| A\|^2)^{|m|}\,
\left(\,\frac{f_0}{\sqrt{\pi}}\,\right)^d\,
(f_1\,d)^{|m|}\,
\left(\,\frac{\mbox{e}^d\,f_1}{2}\right)^{|j|}\,
\frac{(|m|+|j|)!}{j!}\nonumber\\[7pt]
&&=\ (\hbar\,\| A\|^2)^{|m|}\,
\left(\,\frac{f_0}{\sqrt{\pi}}\,\right)^d\,
(f_1\,d)^{|m|}\,
\left(\,\frac{\mbox{e}^d\,f_1}{2}\right)^{|j|}\,
\pmatrix{&|j|&\cr j_1&\cdots &j_d}\,
\pmatrix{|m|+|j|\cr |j|}\,|m|!\nonumber\\[7pt]
&&\le\ (\hbar\,\| A\|^2)^{|m|}\,
\left(\,\frac{f_0}{\sqrt{\pi}}\,\right)^d\,
(2\,f_1\,d)^{|m|}\,
(\mbox{e}^d\,f_1)^{|j|}\,
\pmatrix{&|j|&\cr j_1&\cdots &j_d}\,|m|!\nonumber\\[7pt]
&&\equiv(\hbar\,\| A\|^2 d)^{|m|}\,f_2^d\,f_3^{|m|}\,f_4^{|j|}\,
\pmatrix{&|j|&\cr j_1&\cdots &j_d}\,|m|!,\label{mess30}
\end{eqnarray}
where $f_2$, $f_3$, $f_4$ are numerical constants.

Estimates (\ref{mess8}) and (\ref{mess30}) imply
\begin{eqnarray}\nonumber
&&\left\|\,\frac{D^mV(\zeta(x,a))}{m!}\,(x-a)^m\,
\varphi_{j}(A,\,B,\,\hbar,\,a,\,\eta,\,x)\,\right\|
\\ \nonumber
&&\le\,M\exp(4\tau(\delta^2 d+a^2))
(\hbar\| A\|^2(d/\delta^2)f_3)^{|m|/2}
f_2^{d/2}f_4^{|j|/2}\sqrt{|m|!}
\sqrt{\pmatrix{&|j|&\cr j_1&\cdots &j_d}}.
\end{eqnarray}
So, by the Schwarz inequality, (\ref{app}), (\ref{mess18}), and (\ref{silly}),
\begin{eqnarray}
&&\left\|\,\frac{D^mV(\zeta(x,a))}{m!}\,(x-a)^m\,\psi_{l}(x,t, \hbar)\,\right\|^2
\nonumber\\[5pt]
&&\le\,\sum_{n=0}^{\widetilde{J}(l)}\,\sum_{|j|=n}\,
\left\|\,\frac{D^mV(\zeta(x,a))}{m!}\,(x-a)^m\,
\varphi_{j}(A,\,B,\,\hbar,\,a,\,\eta,\,x)\,\right\|^2\nonumber\\[5pt]
&&\le\,M^2\exp(8\tau(\delta^2 d+a^2))
(\hbar\| A\|^2(d/\delta^2)f_3)^{|m|}f_2^{d}|m|!\,
\sum_{n=0}^{\widetilde{J}(l)}
\sum_{|j|=n}f_4^{|j|}\pmatrix{&|j|&\cr j_1&\cdots &j_d}\nonumber\\[5pt]
&&=\,M^2\,\exp(8\tau(\delta^2 d+a^2))\,
(\hbar\| A\|^2(d/\delta^2)f_3)^{|m|}\,f_2^{d}\,|m|!\,
\sum_{n=0}^{\widetilde{J}(l)}\,f_4^{n}\,d^n\nonumber\\[5pt]
&&=\,M^2\,\exp(8\tau(\delta^2 d+a^2))\,
(\hbar\| A\|^2(d/\delta^2)f_3)^{|m|}\,f_2^{d}\,|m|!\,
\left(\,\frac{(f_4d)^{\widetilde{J}(l)+1}-1}{f_4d-1}\,\right)\nonumber\\[5pt]
&&\le\,M^2\,\exp(8\tau(\delta^2 d+a^2))\,
(\hbar\| A\|^2(d/\delta^2)f_3)^{|m|}\,f_2^{d}\,|m|!\,
2\,(f_4 d)^{\widetilde{J}(l)}.
\end{eqnarray}

The last step depends on the following fact: We can assume without
loss that $q=f_4d\ge 2$, so that for any positive integer $p$, we have
$-1\le q^{p+1}-2q^p$, and hence, $q^{p+1}-1\le 2q^{p+1}-2q^p$.
Thus,
\begin{equation}\label{silly}
(q^{p+1}-1)/(q-1)\le 2q^p.
\end{equation}

We use the hypothesis $|m|=l+2$ and note that $|m|!\,\le\,(l+2)^{l+2}$.
We then conclude that there exist constants $g_0$ and $g_1$,
that depend only on $d$ and $J$, such that (\ref{goal}) holds. This implies the
lemma. \qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .5cm
\section{Proofs of Lemmas 2.3 and 2.4}
\setcounter{equation}{0}

The proof of Lemma 2.3 relies on two preliminary lemmas.

\vskip .25cm
\noindent
\begin{lem}\label{dlem0}
The matrix elements of $(x-a)^m$ satisfy
\begin{eqnarray}\nonumber
&&\left|\,\langle\,\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,x),\,(x-a)^m\,
\varphi_k(A,\,B,\,\hbar,\,a,\,\eta,\,x)\rangle\,\right|\\[7pt]
&\leq&\hbar^{|m|/2}\,(\sqrt{2}d)^{|m|}\,\| A\|^{|m|}\,
\sqrt{(|k|+1)(|k|+2)\cdots (|k|+|m|)},
\end{eqnarray}
and
$$
\langle\,\varphi_j(A,\,B,\,\hbar,\,a,\,\eta,\,x),\,(x-a)^m\,
\varphi_k(A,\,B,\,\hbar,\,a,\,\eta,\,x)\,\rangle\,=\,0,
\quad\mbox{if}\quad\bigg| |j|-|k|\bigg| >|m|.
$$
\end{lem}

\vskip .25cm
\noindent
{\bf Proof:}\quad
For $i=1,2,\cdots, d$, we can use equation (3.28) of \cite{raise} to express
$(x_i-a_i)$ in terms of raising and lowering operators. Doing so, we obtain
\begin{eqnarray}\nonumber
(x_i-a_i)\,\varphi_k(A,\,B,\,\hbar,\,a,\,\eta,\,x)
&=&\sqrt{\hbar/2}\,\sum_{p=1}^d\,A_{ip}\,\sqrt{k_p+1}\,
\varphi_{k'(p)}(A,\,B,\,\hbar,\,a,\,\eta,\,x)\\
\nonumber &&+\,\sqrt{\hbar/2}\,\sum_{p=1}^d\,\bar{A}_{ip}\,\sqrt{k_p}\,
\varphi_{k''(p)}(A,\,B,\,\hbar,\,a,\,\eta,\,x),
\end{eqnarray}
where $k'(p)$ and $k''(p)$ are constructed from $k$ by replacing
the component $k_p$ by $k_p+1$ and $k_p-1$, respectively.
Thus, $(x_i-a_i)\,\varphi_k(A,\,B,\,\hbar,\,a,\,\eta,\,x)$ may be written as a sum
of $2d$ terms, each of which has the form
$b_q\,\varphi_q(A,\,B,\,\hbar,\,a,\,\eta,\,x)$, where $|q|\,\le\,|k|+1$ and
$|b_q|\,\le\,\sqrt{\hbar/2}\,\|A\|\,\sqrt{|k|+1}$.

From this and a simple induction,
$(x-a)^m\,\varphi_k(A,\,B,\,\hbar,\,a,\,\eta,\,x)$
may be written as a sum of $(2d)^{|m|}$ terms, each of which has the form
$b_q\,\varphi_q(A,\,B,\,\hbar,\,a,\,\eta,\,x)$, where
$|k|-|m|\,\le\,|q|\,\le\,|k|+|m|$ and
$|b_q|\,\le\,(\hbar/2)^{|m|}\,\|A\|^{|m|}\,
\sqrt{(|k|+1)(|k|+2)\cdots(|k|+|m|)}.$

This implies the lemma. \qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .25cm
In the next lemma we use the shorthand $X^m(t)$ to denote $X^m(l,\,t,\,\hbar)$ of
Lemma \ref{dlem2}.

\vskip .25cm \noindent
\begin{lem}\label{dlem1}
For each $N>0$, there exists $D(N)<\infty$, such that
\begin{eqnarray}\nonumber
&&\sup_{\{j,k\,:\ |j|,\,|k|\leq \widetilde{J}(l)\}}
\left|\,(X^{m_1}(t_1)\,X^{m_2}(t_2)\,\cdots\,X^{m_q}(t_q))_{j\,k}\,\right|
\\[5pt]\label{muly}&\leq&\left(D(N)\,\sqrt{\hbar}
\sup_{t\in\{t_1,t_2,\cdots t_q\}}\| A(t)\|\right)^{|m_1|+|m_2|\cdots +|m_q|}\,
\widetilde{J}(l)^{(|m_1|+|m_2|+\cdots +|m_q|)/2},
\end{eqnarray}
for any collection $m_1,\,m_2,\,\dots ,\,m_q$ of multi-indices that satisfy
$|m_j|/\widetilde{J}(l)\le N$.
\end{lem}

\vskip .25cm \noindent
{\bf Proof:}\quad
The matrix $X^m$ is obtained from $\widetilde{X}^m$ by restricting its indices
to $|j|,\,|k|\le\widetilde{J}(l)$.
Using Lemma \ref{dlem0}, we see that for such values of the indices,
$$
\left|\,X^m_{j\,k}(t)\,\right|\le
(\sqrt{2\hbar}\,d\,\| A(t)\|)^{|m|}\,\widetilde{J}(l)^{|m|/2}\,
\sqrt{(1+1/\widetilde{J}(l))(1+2/\widetilde{J}(l)))\cdots
(1+|m|/\widetilde{J}(l))}.
$$
Thus, $|m|/\widetilde{J}(l)\le N$ implies
$$
\left|\,X^m_{j\,k}(t)\,\right|\,
\le\,(\sqrt{2\hbar}\,d\,\| A(t)\|)^{|m|}\,(1+N)^{|m|/2}\,
\widetilde{J}(l)^{|m|/2}.
$$

We estimate the absolute value of the $j,\,k$ matrix element of the product
\begin{eqnarray}\nonumber
&&\left|\,(X^{m_1}(t_1)\,X^{m_2}(t_2)\,\cdots\,X^{m_q}(t_q))_{j\,k}\,\right|
\\[10pt]\nonumber
&\le&\sum_{\scriptsize
\matrix{|p_i|\leq \widetilde{J}(l),\; i=1,\cdots , q-1\cr
|p_1-j|\leq |m_1|, |p_2-p_1|\leq |m_2|, \cdots, |k-p_{q-1}|\leq |m_q|}}
\left|\,X^{m_1}_{j\,p_1}(t_1)\,X^{m_2}_{p_1\,p_2}(t_2)\,\cdots\,
X^{m_q}_{p_{q-1}\,k}(t_q)
\,\right|\\ \label{dprod1}
\end{eqnarray}
by bounding the absolute values of each of the matrix elements
$X^{m_i}_{p_{i-1}\,p_i}(t_i)$ by
\begin{equation}\label{dprod2}
\left|\,X^{m_i}_{p_{i-1}\,p_i}(t_i)\,\right|\,\leq
(\sqrt{2\hbar}\,d\,\| A(t_i)\|)^{|m_i|}\,(1+N)^{|m_i|/2}\,
\widetilde{J}(l)^{|m_i|/2},
\end{equation}
and multiplying by a bound on the number of terms.

To estimate the number of terms, we first note that for any multi-index $r$,
the number of multi-indices indices $p$ that satisfy
$\bigg| |p|-|r|\bigg| \leq |m|$ is equal to the number of vectors
$v$ with integer components, such that $|\sum_{i=1}^d\,v_i|\leq |m|$.
This is bounded by the number of vectors with integer components, such
that $|v_i|\leq |m|$ for
$i=1,2,\cdots, d$, which is $(2|m|+1)^d$.

Thus, the number of terms is bounded by
\begin{eqnarray}\nonumber
&&\sum_{\scriptsize
\matrix{|p_i|\leq\widetilde{J}(l),\, i=1,\cdots , q-1\cr
|p_1-j|\leq |m_1|, |p_2-p_1|\leq |m_2|,
\cdots, |k-p_{q-1}|\leq |m_q|}}1
\\[10pt]\nonumber
&\leq&(2|m_1|+1)^d(2|m_2|+1)^d\cdots(2|m_q|+1)^d
\\[10pt]\label{dprod3}
&\leq&(e^2)^{d(|m_1|+|m_2|+\cdots +|m_q|)}.
\end{eqnarray}
The final inequality follows because $1+2|m_j|\le e^{2|m_j|}$.

We obtain the lemma by using (\ref{dprod2}) and (\ref{dprod3}) to bound
(\ref{dprod1}).
\qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .25cm \noindent
{\bf Proof of Lemma \ref{dlem2}}\quad
We mimic the proof of Lemma \ref{dlem1}.
Let $\widetilde{m}=|m_0|+|m_1|+|m_2|+\dots+|m_q|$.
By Lemmas \ref{dlem0} and \ref{dlem1},
\begin{eqnarray}
&&\left(\sqrt{\hbar}
\sup_{t\in\{t_0,t_1,\cdots, t_q\}}\| A(t)\|\right)^{-\widetilde{m}}\
\left|\,
(\widetilde{X}^{m_0}(t_0)\,X^{m_1}(t_1)\,X^{m_2}(t_2)\,
\cdots\,X^{m_q}(t_q))_{j\,k}\,\right|
\nonumber\\[12pt]
&=&\left(\sqrt{\hbar}
\sup_{t\in\{t_0,t_1,\cdots, t_q\}}\| A(t)\|\right)^{-\widetilde{m}}
\sum_{|r|\le\widetilde{J}(l)}
\left|\,
\widetilde{X}^{m_0}_{j\,r}(t_0)\,\right|\,\left|\,
(X^{m_1}(t_1)\,X^{m_2}(t_2)\,\cdots\,X^{m_q}(t_q))_{r\,k}\,\right|
\nonumber\\[12pt]
&\le&\sum_{\scriptsize \matrix{r\le\widetilde{J}(l)\cr |j-r|\le|m_0|}}
2^{|m_0|/2}\,\sqrt{(|j|+1)(|j|+2)\cdots (|j|+|m_0|)}
\nonumber\\&&\qquad\qquad\qquad\qquad\qquad\qquad\times\,
D(N)^{|m_1|+|m_2|+\dots +|m_q|}\,
\widetilde{J}(l)^{(|m_1|+|m_2|+\dots +|m_q|)/2}
\nonumber\\[15pt]
&\le&\sum_{\scriptsize \matrix{|r|\widetilde{J}(l)\cr |j-r|\le|m_0|}}
2^{|m_0|/2}\,\widetilde{J}(l)^{|m_0|/2}\,
\sqrt{
 (1+1/\widetilde{J}(l))(1+2/\widetilde{J}(l))\dots(1+|m_0|/\widetilde{J}(l))}
\nonumber\\&&\qquad\qquad\qquad\qquad\qquad\qquad\times\,
D(N)^{|m_1|+|m_2|+\dots +|m_q|}\,
\widetilde{J}(l)^{(|m_1|+|m_2|+\dots +|m_q|)/2}
\nonumber\\[15pt]
&\le&\sum_{\scriptsize \matrix{|r|\le\widetilde{J}(l)\cr |j-r|\le |m_0|}}
2^{|m_0|/2}\,(1+N)^{|m_0|/2}\,
D(N)^{|m_1|+|m_2|+\dots +|m_q|}\,
\widetilde{J}(l)^{(|m_1|+|m_2|+\dots +|m_q|)/2}
\nonumber\\[12pt]
&\le&2^{|m_0|/2}\,(1+N)^{|m_0|/2}\,
D(N)^{|m_1|+|m_2|+\dots +|m_q|}\,
\widetilde{J}(l)^{(|m_1|+|m_2|+\dots +|m_q|)/2}\,
\sum_{|j-r|\le |m_0|}\, 1.\nonumber
\end{eqnarray}
Since the sum in the last line is $(1+2|m_0|)^d\,\le\,e^{2d|m_0|}$, we
obtain the desired estimate.
\qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .25cm
We now turn to the proof of Lemma \ref{truncest}. Our first step is to study the
combinatorial factor $F_p(n,\,q)$ in the one dimensional case.

\vskip .25cm
\noindent
\begin{lem}\label{lem3}
Let $m_j$ denote positive integers, and let
\begin{equation}\label{combi}
G_p(n,\,q)=\sum_{\scriptsize\matrix{1\leq m_1,m_2,\cdots,m_q\leq p \cr
m_1+m_2+\cdots +m_q=n}}1.
\end{equation}
This quantity is zero if $n<q$ or $n>qp$. Otherwise, it satisfies
\begin{equation}\label{combin}
G_p(n,\,q)\leq \left\{
\matrix{\pmatrix{n-1\cr q-1}&\mbox{if}&q\leq n\leq [q(p+1)/2]\cr
        \pmatrix{q(p+1)-n-1\cr q-1}&\mbox{if}&[q(p+1)/2]+1\leq n\leq qp}\right.
\end{equation}
\end{lem}

\vskip .25cm
\noindent
{\bf Proof:}\quad
If $n<q$ or $n>qp$, the sum in (\ref{combi}) contains no terms, so
$G_p(n,\,q)=0$.

The quantity $G_p(n,\,q)$ is the number of ways that the number $n$ can be
decomposed as $n=m_1+m_2+\dots +m_q$ with each $m_j$ satisfying $1\le m_j\le p$.
We uniquely associate to each such decomposition of $n$, a corresponding
decomposition of $n'=q(p+1)-n=m_1'+m_2'+\dots +m_q'$ by setting
$m_j'=p+1-m_j$. This association is a one-to-one correspondence, so we see that
$G_p(n,\,q)$ satisfies the symmetry relation
\begin{equation}\label{gsym}
G_p(n,\,q)=G_p(q(p+1)-n,\,q).
\end{equation}
As a consequence, the second inequality in (\ref{combin})
(with $[q(p+1)/2]+1\leq n\leq qp$) follows from the first
(with $q\le n\le [q(p+1)/2]$).

To prove the first inequality in (\ref{combin}) we drop the condition
$m_j\le p$ and exactly calculate the resulting function.
Dropping the upper bound on the $m_j$, we clearly have
$G_p(n,\,q)\le G(n,\,q)$, where
$$
G(n,\,q)=\sum_{\scriptsize\matrix{1\leq m_1,m_2,\cdots,m_q\cr
m_1+m_2+\cdots +m_q=n}}1.
$$
So, the lemma will be proved once we establish
\begin{equation}\label{gexact}
G(n,\,q)=\pmatrix{n-1\cr q-1}.
\end{equation}

To prove this, we use induction.
Formula (\ref{gexact}) is trivial to verify for all $n\ge 1$ when $q=1$ and for
all $n\ge 2$ when $q=2$.
Assume (\ref{gexact}) has been verified whenever $n\ge q-1$, and suppose
$n\ge q$.
We have
$$
G(n,\,q)\,
=\,\sum_{m_1=1}^{n-q+1}\,G(n-m_1,\,q-1)
\,=\,\sum_{m_1=1}^{n-q+1}\,\pmatrix{n-m_1-1\cr q-2},
$$
so the induction step will be complete once we show
\begin{equation}\label{mystar}
\sum_{m_1=1}^{n-q+1}\,\pmatrix{n-m_1-1\cr q-2}
\,=\,\pmatrix{n-1\cr q-1}.
\end{equation}

To prove (\ref{mystar}), we do an another induction on $n\ge q$.
For $n=q$ we trivially have
$$
\sum_{m=1}^1\,\pmatrix{q-2\cr q-2}\,=\,1\,=\,\pmatrix{q-1\cr q-1}.
$$
Now assume (\ref{mystar}) is true for some $n\ge q$. For the case of $n+1$, we
have

\begin{eqnarray}
\nonumber
&&\sum_{m=1}^{(n+1)-q+1}\,\pmatrix{(n+1)-m-1\cr q-2}
\\[7pt] \nonumber
&=&\sum_{m=1}^{n-q+1}\,\pmatrix{n-m\cr q-2}
\quad +\quad\pmatrix{(n+1)-\{(n+1)-q+1\}-1\cr q-2}
\\[7pt] \nonumber
&=&\sum_{m=1}^{n-q+1}\,\pmatrix{n-m\cr q-2}
\quad +\quad\pmatrix{q-2\cr q-2}
\\[7pt] \nonumber
&=&\sum_{m=1}^{n-q+1}\,\pmatrix{n-m\cr q-2}
\quad +\quad 1
\\[7pt] \nonumber
&=&\sum_{m'=0}^{n-q}\,\pmatrix{n-m'-1\cr q-2}
\quad +\quad 1
\\[7pt] \nonumber
&=&\pmatrix{n-1\cr q-2}\ +\
\sum_{m'=1}^{n-q+1}\,\pmatrix{n-m'-1\cr q-2}\ -\
\pmatrix{q-2\cr q-2}\ +\ 1
\\[7pt] \nonumber
&=&\pmatrix{n-1\cr q-2}\ +\ \pmatrix{n-1\cr q-1}
\\[7pt] \nonumber
&=&\pmatrix{n\cr q-1}.
\end{eqnarray}
This proves (\ref{mystar}) and completes the proof of the lemma.
\qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .35cm
Our next step is to obtain an estimate for $G_p(n,\,q)$ that depends only on
$n$.

\vskip .25cm
\noindent
\begin{lem}\label{lem4}
Let $G_p(n,\,q)$ be defined by (\ref{combi}).
There exists a constant $C_7$, such that
$p\ge 1$, $q\ge 1$, and $n\ge 1$ imply
\begin{equation}\label{escom}
G_p(n,q))\leq (C_7)^n.
\end{equation}
\end{lem}

\vskip .25cm
\noindent
{\bf Proof:}\quad
We note that $G_p(n,q)$ is zero unless $q\le n\le qp$.
Furthermore, $G_p(q,q)=G_p(qp,q)=G_2(2,1)=1$ and $G_1(n,q)=\delta_{n\,q}$.
So, we need only prove existence of $C_7$,
such that $p\ge 2$, $q\ge 1$, $(p,q)\ne (2,1)$ imply
\begin{equation}\label{escom1}
\sup_{q<n<pq}\ (G_p(n,q))^{1/n}\,\leq\,C_7.
\end{equation}
To prove this, we first study the case where $q<n\leq [q(p+1)/2]$.
Since $q\leq n$, we have
$$
q^{1/n}\,=\,\exp(\ln(q)/n)\,\le\,\exp(\ln(n)/n)\,\le\,\mbox{e}^{1/\mbox{\scriptsize e}}.
$$
Thus, by (\ref{combin}), the condition $q<n\leq [q(p+1)/2]$,
and (\ref{stirling}), we have
\begin{eqnarray}\nonumber
(G_{p}(n,\,q))^{1/n}&\le&\left(\,\frac{(n-1)!}{(q-1)!\,(n-q)!}\,\right)^{1/n}\\
\nonumber&\le& q^{1/n}\ \left(\,\frac{(n-1)!}{q!\,(n-q)!}\,\right)^{1/n}\\
&\le& \mbox{e}^{1/\mbox{\scriptsize e}}\ \frac{n}{aq^{q/n}\,(n-q)^{(n-q)/n}}.\label{above}
\end{eqnarray}
An explicit computation shows that
$$
\frac{\partial\phantom{n}}{\partial n}\ \,\frac{n}{q^{q/n}\,(n-q)^{(n-q)/n}}
\ =\ \frac{n}{q^{q/n}\,(n-q)^{(n-q)/n}}\
     \frac{q}{n^2}\ \left(\,\ln(q)-\ln(n-q)\,\right) .
$$
From this, we deduce that the right hand side of (\ref{above}) attains its
maximum at $n=2q$. Evaluating that maximum, we obtain
\begin{equation}\label{half1}
\sup_{q<n\le [q(p+1)/2]}\ (G_{p}(n,\,q))^{1/n}\ \leq\ \frac{2}{a}\,\mbox{e}^{1/\mbox{\scriptsize e}}.
\end{equation}

For $[q(p+1)/2]<n<pq$, the number $n'=q(p+1)-n$ satisifes
$q<n'\le [q(p+1)/2]$ and $n'/n\le 1$.
Thus, by (\ref{gsym}) and (\ref{half1}), we obtain
\begin{eqnarray}\nonumber
(G_{p}(n,\,q))^{1/n}&=&(G_{p}(n',\,q))^{1/n}\\
\nonumber &\le&\left(\,\frac{2}{a}\,\mbox{e}^{1/\mbox{\scriptsize e}}\,\right)^{n'/n}\\
\nonumber &\le&\frac{2}{a}\,\mbox{e}^{1/\mbox{\scriptsize e}}.
\end{eqnarray}
Thus,
\begin{equation}\label{half2}
\sup_{[q(p+1)/2]<n<pq}\ (G_{p}(n,q))^{1/n}\ \leq\ \frac{2}{a}\,\mbox{e}^{1/\mbox{\scriptsize e}}.
\end{equation}

Inequalities (\ref{half1}) and (\ref{half2}) imply the
existence of $C_7$ for which (\ref{escom1}) holds.
This implies (\ref{escom}),and the lemma is proved. \qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .25cm
We now generalize Lemma \ref{lem4} to the multi-dimensional case.

\vskip .25cm
\noindent
\begin{lem}\label{dlem4}
Let $F_p(n,\,q)$ be defined by (\ref{dcombin}).
For all $p\ge 1$, $q\ge 1$, and $n\ge 1$, we have
\begin{equation}
F_p(n,q)\,\leq\,C_7^{(n+qd)},
\end{equation}
where $C_7$ is the constant of Lemma \ref{lem4}.
\end{lem}

\vskip .25cm
\noindent
{\bf Proof:}\quad
With $m_j$ temporarily denoting numbers instead of multi-indices, we define
$$
\Gamma_p(n,\,q)=\sum_{\scriptsize\matrix{0\leq m_1,m_2,\cdots,m_q\leq p \cr
m_1+m_2+\cdots +m_q=n}}1,
$$
which is zero unless $0\leq n\leq pq$.

By defining $m'_j=m_j+1$, we see that
every decomposition of $n$ as $n=m_1+m_2+\cdots +m_q$, with $0\le m_j\le p$,
corresponds uniquely to a decomposition of $n+q$ as
$n+q=m'_1+m'_2+\cdots +m'_q$, with $1\le m'_j\le p+1$. Therefore,
we have the identity
\begin{equation}\label{shift}
\Gamma_p(n,\,q)\,=\,G_{p+1}(n+q,\,q).
\end{equation}

We now let the $m_j$ denote multi-indices and let $m_j(k)$ denote the
$k^{\mbox{\scriptsize th}}$ component of $m_j$.
Then, using (\ref{shift}) and Lemma \ref{lem4}, we easily obtain
$$
F_p(n,\,q)\ \leq \sum_{\scriptsize\matrix{0\leq m_j(k)\leq p \cr
\sum_{j=1}^q\sum_{k=1}^dm_j(k)=n}}1\ =\ \Gamma_p(n,\,qd)\ \leq\ C_7^{(qd+n)}.
$$
\phantom{}\qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .25in
\noindent
{\bf Proof of Lemma \ref{truncest}:}\quad
Suppose $2\leq q \leq l+2$, and let $L_q(l)$ satisfy (\ref{Lest}), {\it i.e.},
$$
L_q(l)\,\le\,\frac{C_1^q}{\hbar^q\,q!}\,
\sum_{n=3l-2}^{q(l+1)}\,(C_2\,\hbar\,l)^{n/2}\,F_{l+1}(n,\,q).
$$
By Lemma \ref{dlem4} and (\ref{stirling}),
there exist $C_8$ and $C_9$, such that
\begin{eqnarray}
L_q(l)&\leq&\frac{(C_1\,C_7^d)^q}{(a\,\hbar\,q)^q}\,
\sum_{n=3l-2}^{q(l+1)}\,\left(C_2\,\hbar\,l\right)^{n/2}\,C_7^{n}
\nonumber \\ \label{start}
&\leq&\frac{C_9^q}{(\hbar\,q)^q}\,
\sum_{n=3l-2}^{q(l+1)}\,\left(C_8\,\hbar^{1/2}\,l^{1/2}\right)^{n}.
\end{eqnarray}

Note that we can take
\begin{eqnarray}\label{c8}
C_8&=&\max\,\{\,1,\,\sqrt{C_2}\,C_7\,\},\\ \label{c9}
C_9&=&\max\,\{\,1,\,C_1\,C_7^d\,\},
\end{eqnarray}
because we can assume $C_8\ge 1$ and $C_9\ge 1$ without loss of generality.

We arbitrarily choose two positive numbers $\alpha<1$ and $\beta<1$. We then
define
\begin{equation}\label{gchoice}
g\,=\,\frac{1}{C_8^2}\,
\min\,\left\{\,\alpha^2,\,\frac{\beta^2}{C_9^2\,C_8^4\,\mbox{e}^{2/\mbox{\scriptsize e}}}\,\right\} .
\end{equation}
We henceforth assume $l=l(\hbar)$ is chosen to satisfy
(\ref{lchoice}) with this value of $g$. With this choice, we have
\begin{equation}\label{condal}
C_8\,\hbar^{1/2}\,l^{1/2}\,\leq\,\alpha .
\end{equation}
By summing a geometric series in (\ref{start}), we see that
\begin{eqnarray}
L_q(l)&\leq&\frac{C_9^q}{(1-\alpha)\,(\hbar\,q)^q}\
\left(\,C_8\,\hbar^{1/2}\,l^{1/2}\,\right)^{3l-2}\nonumber\\[5pt]
&=&\frac{C_9^q\,C_8^{2q}}{(1-\alpha)\,(\hbar\,q\,C_8^2)^q}\,
\left(\,C_8\,\hbar^{1/2}\,l^{1/2}\,\right)^{3l-2}.
\end{eqnarray}
As a function of $q$,
$$
\left(\frac{l}{q}\right)^q=\exp\left(\,q\,\ln(l/q)\,\right)
$$
is maximized at $q=l/\mbox{e}$, where it has the value $\mbox{e}^{l/\mbox{\scriptsize e}}$.
Thus,
$$
\left(\frac{l}{q}\right)^q\,\le\,\mbox{e}^{l/\mbox{\scriptsize e}}.
$$
This, $q\le l+2$, and $\hbar\,l\,C_8^2\,\leq\,\alpha^2\,<\,1$
imply
$$
\frac{1}{(\hbar\,q\,C_8^2)^q}\ =\ \frac{1}{(\hbar\,l\,C_8^2)^q}\,
\left(\,\frac{l}{q}\,\right)^q
\ \leq\ \frac{\mbox{e}^{l/\mbox{\scriptsize e}}}{(\hbar\,l\,C_8^2)^{l+2}}.
$$
From this, $C_8\geq 1$, and $C_9\geq 1$, we conclude
\begin{eqnarray}\label{etap}
L_q(l)&\leq&\frac{C_9^{l+2}\,C_8^{2(l+2)}\,\mbox{e}^{l/\mbox{\scriptsize e}}}{(1-\alpha)}\
\left(\,C_8\,\hbar^{1/2}\,l^{1/2}\,\right)^{l-6}\nonumber\\[5pt]
&=&\left(\,\mbox{e}^{1/\mbox{\scriptsize e}}\,C_9\,C_8^3\,\hbar^{1/2}\,l^{1/2}\,\right)^l\
\frac{C_9^2}{(1-\alpha)\,(\hbar\,l)^3\,C_8^2}.
\end{eqnarray}
By (\ref{gchoice}), we have
$$
\mbox{e}^{1/e}\,C_9\,C_8^3\,\hbar^{1/2}\,l^{1/2}\ \leq\ \beta\ <\ 1.
$$
Since
$$
\frac g\hbar -1\ \le\ l\ \le\ \frac g\hbar ,
$$
we see from (\ref{etap}) that if $\hbar\leq g/2$, then we have
\begin{eqnarray}
L_q(l)&\leq&\mbox{e}^{-|\ln(\beta)|l}\
\frac{C_9^2}{(1-\alpha)\,(g-\hbar)^3\,C_8^2}\nonumber\\[5pt] \label{425}
&\leq&\frac{2^3\,\mbox{e}\,C_9^2}{(1-\alpha)\,g^3\,C_8^2}\
\mbox{e}^{-|\ln(\beta)\,|g/\hbar}.
\end{eqnarray}
Note that the factor $2^3$ can be replaced
by $(1+\epsilon)$, with $\epsilon$ arbitrarily small, by taking
$\hbar$ sufficiently small.\qquad {\vrule height 10pt width 8pt depth 0pt}

\vskip .25cm \noindent
{\bf Remark}\quad
We can choose $\alpha$ and $\beta$ to satisfy
$\displaystyle \frac{\beta}{\mbox{e}^{1/\mbox{\scriptsize e}}}\,\leq\,\alpha\,<\,\frac 1{\mbox{e}^{1/\mbox{\scriptsize e}}}$.
Then, since $\displaystyle \frac{\beta^2}{C_9^2\,C_8^4\,e^{2/e}}\,\leq\,
\frac{\beta^2}{\mbox{e}^{2/\mbox{\scriptsize e}}}\,\leq\,\alpha^2$,
we obtain the conclusion of Lemma \ref{truncest} with
\begin{eqnarray}\label{gchoice1}
g&=&\frac{\beta^2}{C_9^2\,C_8^6\,\mbox{e}^{2/\mbox{\scriptsize e}}},\\
\label{gamchoice}
\gamma_1&=&\frac{|\ln(\beta)|\,\beta^2}{C_9^2\,C_8^6\,e^{2/e}}\\
\label{Cchoice}
C_3&=&\frac{2^3\,\mbox{e}^{1+6/\mbox{\scriptsize e}}\,C_9^8\,C_8^{10}}{(1-\mbox{e}^{-1/\mbox{\scriptsize e}})\,\beta^6},
\end{eqnarray}
where $C_8$ and $C_9$ are related to $C_1$, $C_2$, and $C_7$
by (\ref{c8}) and (\ref{c9}).
Note that $C_7$ is purely combinatorial and has no time dependence.

\vskip .5cm
\section{Proof of Theorem 1.2}
\setcounter{equation}{0}

To prove Theorem \ref{longtime}, we revisit the proof of Theorem
\ref{mainthm} and make the $T$ dependence of all constants explicit.
We then allow $T$ to grow with $\hbar$ with the restriction that our
approximation remain close to the actual solution as $\hbar\rightarrow 0$.

Lemmas \ref{lem3}, \ref{lem4}, and \ref{dlem4} have no time
dependence, and the
time dependence of Lemmas \ref{fundcalc}, \ref{taylorprop},
\ref{dlem2},
\ref{taylor1}, \ref{dlem0}, and
\ref{dlem1} has been made explicit in their conclusions.

Since $V$ is bounded below and energy is conserved,
$|a(t)|$ grows at most linearly with
time. Thus, there exist $v_1 >0$ and $v_0>0$, such that
$v(t)$ defined by (\ref{mess88}) satisifies
\begin{equation}\label{1}
v(t)\,\leq\,v_0\,\exp(v_1 t).
\end{equation}
From assumption (\ref{expgro}), it follows that
\begin{equation}\label{2}
w(T)\,=\,\sup_{0\leq |t|\leq T}\,\|A(t)\|\,\leq\,N\,\exp (\lambda T).
\end{equation}

In the sequel, we denote all innessential constants that do not depend
on $T$ by the same symbol $c$.

Consider Lemma \ref{taylorprop}.
For $z\in C_{\delta}(\zeta(x,a(t)))$,
we can prove the existence of $\rho>0$ such that
\begin{eqnarray}\nonumber
&&\left\|\,\frac{D^mV(\zeta(x,a(t)))}{m!}\ (x-a(t))^m\
\psi_{l}(x,\,t,\,\hbar)\,\right\|\\[5pt] \label{goal2}
&\le& c\,\exp(\rho |t|)\,
\left(c\,\sqrt{\hbar\,(l+2)}\,\exp (\lambda |t|)\right)^{l+2}.
\end{eqnarray}
Indeed, with our bound on $|V(z)|$ we have
$$\frac{D^mV(\zeta(x,a(t)))}{m!}\,\leq\,c\,\exp(\rho |t|)\,\exp (c|x-a(t)|)
$$ instead of (\ref{mess8}).
With this and the estimate $\exp(cy)\leq\exp(cy^2)+\exp (c)$ for all
$c>0$ and $y>0$, we see that
the estimates in the proof of Lemma \ref{taylorprop} are still valid.
This yields (\ref{goal2}).
Consequently, the
constants $C_4$ and $C_5$ appearing in (\ref{taylordone}) satisfy
\begin{eqnarray}\nonumber
C_4(T)&=&c\,\exp(\rho T)\\ \nonumber
C_5(T)&=&c\,\exp(\lambda T),
\end{eqnarray}
where $\rho>0$.
By using estimates (\ref{1}) and (\ref{2}) in (\ref{cqest1})
and (\ref{rest1}), we see that when we apply Lemma \ref{truncest},
the constants $C_1$ and $C_2$ satisfy
\begin{eqnarray}\nonumber
C_1(T)&=&c\,\exp(\rho T)\\ \nonumber
C_2(T)&=&c\,\exp(\lambda T).
\end{eqnarray}
We still must determine the $T$ dependence of $C$ and $\gamma$ of
Theorem \ref{mainthm}. We do this
by determining the $T$ dependence of $g, \gamma_1$ and
$C_3$ in Lemma \ref{truncest}. Equations
(\ref{c8}) and (\ref{c9}) yield $C_8$ and $C_9$, in terms of which the
above constants are determined.
We find that
\begin{eqnarray}\nonumber
C_8(T)&=&c\,\exp(\lambda T/2)\\ \nonumber
C_9(T)&=&c\,\exp(\rho T).
\end{eqnarray}
Consequently, from (\ref{gchoice1}), (\ref{gamchoice}) ,(\ref{Cchoice}),
we have
\begin{eqnarray}\nonumber
g(T)&=&c\,\exp(-\nu T),\qquad\qquad\nu=2\rho+3\lambda\\ \nonumber
\gamma_1(T)&=&c\,\exp(-\nu T),\\ \nonumber
C_3(T)&=&c\,\exp(\mu T),\qquad\qquad\ \,\mu=8\rho+5\lambda .
\end{eqnarray}

To arrive at these conclusions, we imposed various conditions.
Those conditions were\\[3pt]
\phantom{}\qquad\qquad\qquad
(\ref{35}), that now requires $\hbar\leq c \exp(-2\lambda T)$,\\[3pt]
\phantom{}\qquad\qquad\qquad
(\ref{george}), than now requires $c\exp(2\lambda T)\exp(-\nu T) <1$,
and\\[3pt]
\phantom{}\qquad\qquad\qquad
$ \hbar\leq g(T)/2$ used in (\ref{425}),
that now requires $ \hbar\leq c\exp(-\nu T)$.

\vskip .15cm
These are all satisfied, provided we take
$$
  T(\hbar)\,\leq\,T'\,|\ln(\hbar)|,
$$
with $T'>0$ sufficiently small. Using this in our estimate for the
error term $\xi(t,\hbar)$, we see that for sufficiently small $T'$,
$|t|\,\leq\,T'\,|\ln(\hbar)|$ implies
$$
\| \psi(x,t,\hbar)-\Psi(x,t,\hbar)\|_{L^2(I\kern-.277em R^d)}\,\leq\,
C'\,\exp(-\gamma'/\hbar^{\sigma}),
$$
for some $\gamma'>0$, $\sigma >0$, and $C'>0$.\qquad {\vrule height 10pt width 8pt depth 0pt}

redpajama_set_name: RedPajamaArXiv


text:
Léon Bary (6 June 1880 – 7 January 1954) was a French actor. He appeared in more than 50 films between 1916 and 1955. He was born in Paris, France, and died in Paris, aged 73.

Selected filmography

References

External links

1880 births
1954 deaths
Burials at Père Lachaise Cemetery
French male film actors
French male silent film actors
Male actors from Paris
20th-century French male actors

redpajama_set_name: RedPajamaWikipedia


หุบเขาและทุ่งราบ = Hill farms and paddifields / ริบรินส์ เบอร์ลิง ; แปลและเรียบเรียงโดย ปราณี วงษ์เทศ.
by เบอร์ลิง, ริบรินส์ | ปราณี วงษ์เทศ.
Call number: 306 บ829ห Material type: BookPublisher: กรุงเทพฯ : เจ้าพระยา, 2522Description: 466 หน้า.Availability: Items available for loan: PYULIB-DW [Call number: 306 บ829ห] (1).
พื้นฐานวัฒนธรรมไทย = Foundations of Thai culture / โดย ณรงค์ เส็งประชา.
Call number: 306.09593 ณ215พ Material type: BookPublisher: กรุงเทพฯ : โอเดียนสโตร์, 2531Description: 108 หน้า.Availability: Items available for loan: PYULIB-CL [Call number: 306.09593 ณ215พ] (1).
Call number: 306.4 ม246ส Material type: BookPublisher: กรุงเทพฯ : มหาวิทยาลัยสุโขทัยธรรมาธิราช, 2528Description: 2 เล่ม.Availability: No items available Special Status (1).
Call number: 306.6 จ345ส Material type: BookPublisher: กรุงเทพฯ : แพร่พิทยา, [2525?]Description: 160 หน้า.Availability: Items available for loan: PYULIB-DW [Call number: 306.6 จ345ส] (3).
Call number: 306.6 บ561ส Material type: BookPublisher: กรุงเทพฯ : มหาวิทยาลัยรามคำแหง, 2518Description: 185 หน้า.Availability: Items available for loan: PYULIB-CL [Call number: 306.6 บ561ส] (1), PYULIB-DW [Call number: 306.6 บ561ส] (1).
ศาสนากับสังคมไทย / โดย ส. ศิวรักษ์.
Call number: 306.6 ส113ศ 2525 Material type: BookEdition: พิมพ์ครั้งที่ 2.Publisher: กรุงเทพฯ : เทียนวรรณ, 2525Description: 452 หน้า.Availability: Items available for loan: PYULIB-DW [Call number: 306.6 ส113ศ 2525] (2).
Call number: 306.8 บ561ค Material type: BookPublisher: กรุงเทพฯ : โรงพิมพ์มหาวิทยาลัยรามคำแหง, 2520Description: 221 หน้า : ภาพประกอบ.Availability: Items available for loan: PYULIB-CL [Call number: 306.8 บ561ค] (1).
น้ำ = Water / มูลนิธิโลกสีเขียว.
Call number: 333.91 ม686น Material type: BookPublisher: กรุงเทพฯ : มูลนิธิ, [2536?]Description: 105 หน้า : ภาพประกอบ.Availability: Items available for loan: PYULIB-CL [Call number: 333.91 ม686น] (1).
สังคมวิทยาเบื้องต้น / มงคล หวังสุขใจ, สิริพร ศรีหนองโคตร, ชมพู โกติรัมย์.
by มงคล หวังสุขใจ | สิริพร ศรีหนองโคตร | ชมพู โกติรัมย์.
Call number: 301 ม115ส Material type: BookEdition: พิมพ์ครั้งที่ 1.Publisher: กรุงเทพฯ : ศูนย์เทคโนโลยีทางการศึกษา ฝ่ายเทคโนโลยี มหาวิทยาลัยศรีปทุม, 2538Description: 212 หน้า.Availability: Items available for loan: PYULIB-CL [Call number: 301 ม115ส] (2).
The village in perspective : community and locality in rural Thailand / edited by Philip Hirsch.
Call number: 305.89591 V713 Material type: BookPublisher: Chiang Mai : Social Research Institute, Chiang Mai University, 1993Description: vii, 96 p. : maps.Availability: Items available for loan: PYULIB-CL [Call number: 305.89591 V713] (3).
Chao Rai Thai : 1987 update / Laurence C. Judd.
by Judd, Laurence C | Payap University. Payap Research and Development Center.
Call number: 301 J92C Material type: BookPublisher: Bangkok : Suriyaban, 1987Description: vi, 31 p. : ill.Online access: abstract Availability: Items available for loan: PYULIB-CL [Call number: 301 J92C] (2).
Sociology : the human science, / Elbert W. Stewart.
Call number: 301 S849S Material type: BookPublisher: New York : McGraw-Hill, c1978Description: 607 p. : ill.Availability: Items available for loan: PYULIB-CL [Call number: 301 S849S] (2).
The hill tribes of Northern Thailand : a socio-ethnological report / by Gordon Youg.
Call number: 301.29593 Y72H Material type: BookPublisher: Bangkok : Siam Society, 1962Description: 92 p. : ill.Availability: Items available for loan: PYULIB-CL [Call number: 301.29593 Y72H] (2).
Buddhist monk, Buddhist layman : a study of urdan monastic organization in Central Thailand / Jane Bunnag.
Call number: 301.2 B942B Material type: BookPublisher: London : Cambridge Univerity Press, 1973Description: 219 p.Availability: Items available for loan: PYULIB-CL [Call number: 301.2 B942B] (2).


This week was "inspired by". It could have been taken two different ways and it was great to look through all the contributions to see how people interpreted it. One you could have recreated a piece that you love with your own twist (think lego to create the last supper or something of that nature) or I could have been a photographer or style that you were inspired by and try to create something along that theme.
Me, I'm inspired by all the beautiful lifestyle images up on Clickin Mom's and wanted to try my hand at it. I've always had them turn out too snap shotty so I was reading up on the various tips and pointers that the ladies on the site gave. Luckily I saw this moment with my kiddos – a rare one where they're not trying to hurt one another and ran for my camera.


This a list of all spell effects available in the game, for sale at vendors. For a list of vendors by location, see Spell Merchants.
Spell Damage is Damage * (1 - ( Resistance - Weakness )/100).
Spell Cost (the amount of magicka a spell takes to cast) for custom spells is ( [ Min Magnitude + Max Magnitude ] * [ Duration + 1 ] + Area ) * Base Cost / 40, rounded down. Non-custom spells may have a different cost than their custom equivalent.
If Resistance - Weakness is equal to 100 or more, the effect is resisted.
Resistance/Weakness to Magicka does not affect elemental damage.
"Self" spells have an Area value of 1.
"Target" spells multiply the above spell cost by 1.5.
The lowest value possible on a slider is 1 (one), despite it displaying zero.
Spells without magnitudes (such as Invisibility) behave as though they have magnitudes of 1.
If you are on PC, hold shift and click on the spellname. On Xbox, select the spell and press the White button.
If you want to be able to use an effect for spellmaking/enchanting, keep at least one spell with that effect.
This section lists spells that are cheaper to cast than would normally be expected. It only includes spells, and race and birthsign abilities that operate as spells, not diseases, nor race or birthsign abilities that do not operate as spells (such as attribute enhancement).
Many of these spells are normally unobtainable and are only experienced as effects cast upon the player either by shrines, traps or NPCs, however they can be added to your spellbook by using the AddSpell Console command and used normally thereafter. If no location is specified in the "Obtained" column, it means the spell is only obtainable using the console.
blessed touch Turn Undead 100 pts for 30s on Touch 5 30 Sold by Ygfa. Also available to those born under the sign of the Ritual.
blessed word Turn Undead 100 pts for 30s on Target 10 45 A power obtained from the The Ritual birthsign.
argonian breathing Water Breathing for 120s on Self 5 18 Inbuilt racial spell for Argonian; not to be confused with the spell sold by Marayn Dren.
In contrast to the spells in the previous section, these spells cost more to cast than would normally be expected. Most of them are exclusive to certain NPCs, most especially Ash Vampires.
This page was last modified on 29 September 2018, at 10:05.


The sheep seem to sense your arrival in Bonnie Fisher's Grazing print. 16 x 12 with a lightly distressed frame, black with stained edges, no glass. This item ships from our supplier in 2 to 3 weeks in the continental United States.
Here's a timeless scene captured by Billy Jacobs of a farm far from the hustle & bustle of the city, "Chester County Fieldstone." Pick the size that best fits your home. Frames made of poplar wood painted black & assembled. Simply beautiful work!
The memories of a rocker on the porch on a warm sunny day come from this Summer Afternoon Print. Billy Jacobs captures the countryside of America. Pick the size that best fits your home. Frames are made of poplar wood painted black & assembled.
The air is crisp, the leaves falling and "Harvest Time" is captured in this view of the farmhouse and barn by leading Primitive Artist, Billy Jacobs. Choose your size and frame.


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This entry was posted in collage, digital art, layering, writer, writing and tagged 9 thoughts on writing, adapted, being a writer, collage, digital art, original by ojay ohri, significantly altered by j.kiley, writer, writing. Bookmark the permalink.
Let us write for ourselves. What assurance de we have that these "others" are anything beyond our imaginations?
Far from me may it ever be to seem to disparage another's foundation. Creation is creation however we may depict it.


text:
Department of Information Science and Media Studies
The Cultural Value of Games: Computer Games and Cultural Policy in Europe
Klevjer, Rune
Chapter, Peer reviewed
Accepted version (155.2Kb)
Department of Information Science and Media Studies [681]
In this chapter I want to discuss the concept of 'cultural value' in computer games, as a reflection on why and how games might be a part of a future European cultural policy. To readers who are only vaguely familiar with the various aesthetics and practices of this medium, the general considerations addressed may also serve as a useful first introduction to a relatively new field.
Intellect Ltd
Changing Media, Changing Europe
Copyright Intellect Ltd

redpajama_set_name: RedPajamaCommonCrawl


text:
Мали Буковец је насељено место и седиште општине у Вараждинској жупанији, Хрватска. До територијалне реорганизације у Хрватској налазио се у саставу бивше велике општине Лудбрег.

Становништво

На попису становништва 2011. године, општина Мали Буковец је имала 2.212 становника, од чега у самом Малом Буковцу 729.

Попис 1991.

На попису становништва 1991. године, насељено место Мали Буковец је имало 942 становника, следећег националног састава:

Извори
  Савезни завод за статистику и евиденцију ФНРЈ и СФРЈ, попис становништва 1948, 1953, 1961, 1971, 1981. и 1991. године

Спољашње везе

Насељена места у Хрватској
Насељена места у Вараждинској жупанији
Википројект географија/Насеља у Хрватској
Општине у Хрватској
Општина Мали Буковец


Recently, I had a though—if lemon curd, why not lime or orange curd? They were an easy substitutions and they turned out very nicely.
The next thought, of course, led me to: Can you make fruit curds with something other than citrus fruit? I looked on-line and the short answer is "Yes." I found recipes for apple curd, blueberry curd, melon curd, pomegranate curd, and strawberry curd. Apparently you make a curd out of any fruit you fancy.
Amanda made her curd with fresh pomegranates. I recently made pomegranate jelly and removing all of the pips from a bunch of pomegranates was a bit much. I thought I would take a short cut and use Pom Wonderful.
The finished color of the pomegranate curd is a bit unfortunately purple brown. I even added five drops of red food color, but I could not say it was a vast improvement. I think the citrus juice in some of the recipes was added to improve the color. The taste however, especially on french toast, is wonderful.
1. Set the butter and eggs on the counter and let them come to room temperature, about one hour.
Note: If you are going to put your curd into jars, sterilize them in boiling water and set them on a clean towel, ready to fill.
2. Measure out ¾ of a cup of pomegranate juice and set the measuring cup aside.
3. In a medium bowl, beat the butter and sugar with an electric mixer, about 2 minutes.
Tip: I found that mixer could not break up all of the lumps of butter. Halfway through, I took a fork and mashed the butter bits into the sugar and then continued mixing.
4. Add the eggs to the butter mixture, one egg at a time.
Tip: Mix the eggs in for two minutes on medium high between eggs. You want to break up and separate all of the bits of protein in the whites.
5. Beat in the pomegranate juice and continue beating for 2-4 minutes, until most of the juice in incorporated into the mixture.
Tip: Stop beating and push the mix out of the way, if there is still a lot juice puddled in the bottom of the bowl, keep beating. When the free juice gets down to about a quarter of a cup you are done.
Note: The mixture will look curdled, but it will smooth out as it cooks.
6. Pour the mix into a medium, heavy-based saucepan. Start cooking the mixture over low heat until starts to look smooth.
7. Increase the heat to medium and cook, stirring constantly with a spatula, until the mixture thickens, about 20-30 minutes.
Note: I found that the pomegranate mix did not set as quickly as the citrus curds.
Tip: Constantly scrape the bottom and sides of the pot. You do not want any hot spots or bits sticking, where they may get over cooked. Never let the mix come to a boil! This would cause the eggs and butter to "break" and you would end up with a flavorful mess. Adjust the heat as necessary.
8. Cook the mixture until well thickened and it sticks to the back of the spatula.
Tip: If you find the final color unappealing, stir in a few drops of red food coloring.
Note: Fine Cooking says to cook it to 170° F on a thermometer, but I prefer to go by eye. I cook the curd until it "looks right." Stirring will leave a trail as the curd thickens and you will come to recognize when it is thick enough.
9. Remove the pomegranate curd from the heat and transfer it to a bowl or prepared jars.
Tip: This is more of a spreadable egg custard, not a preserve. Do not leave it out for too long and use it up quickly. Even cooked, egg dishes can develop salmonella.
Note: The curd will thicken further as it cools and will keep in the refrigerator for a week and in the freezer for 2 months.


text:
Letter to Friends and Benefactors
So, Campos is fallen. The two dozens priests with their own bishop who for 20 years from distant Brazil were the re-assuring comrades in arms of the Society of St. Pius X in its lone stand for Catholic Tradition have gone back in with Conciliar Rome. What happenned? What will happen? What does it mean?
What happened can be briefly told in Shakespeare's phrase, the Campos priest had greatness thrust upon them. They laid the burden to one side.
The story of how the diocese of Campos came into the limelight of Catholic Tradition after the Second Vatican Council is known to many readers from the book "The Mouth of the Lion" by Dr. David White. Before the Council, the Brazilian country diocese of the seaside city of Campos, lying three hours by car to the north of Rio de Janeiro, was gifted with a truly Catholic bishop, Antonio de Catro Mayer. As is clear from his marvelous "Catechism of Opportune Truths Opposed to Contemporary Errors", written for his diocese in the 1950's, Bishop de Castro Mayer thoroughly understood the danger of the heresy of modernism. All the pernicious errors then and since devasting the Catholic Church are laid in all their deadly charm. Opposite, the bishop presents the the Catholic truth, less charming but free of poison. Beneath, he explains where the poison lay hidden and why the Church's teaching on the matter is the truth.
This bishop, wether or not he was conscious of the impeding desaster of Vatican II, in any case by his insistence on true doctrine, as Dr. White expresses it, earthquake-proofed his diocese in advance of the earthquake. When the Council happened (1962 to 1965), his priests and people were prepared, so that the bulk of them kept the true Faith, and when the New Mass was introduced (1969), while a few of his priests took it up and left for another diocese, most of his priests made use of his permission to continue saying the old Mass.
Not that the bishop was defiant of Pope Paul VI, or heedless of the Pope's pressure to introduce the New mass. But he had written to the Pope a most respectfull letter, asking for doctrinal problems that he had with the new rite of Mass to be cleared up, and when he received absolutely no answer from the Pope, he acted upon what he knew to be good doctrine, and gave his priests the permission to stay with the rite that was safe and sure.
So his diocese, priests and laity, in large part followed him until his mandatory retirement at age 75 in 1981. The official Church appointed to succeed him a bishop who would by force and violence wrench the diocese away from the old religion and establish it as a "normal" part of the Newchurch. But the priests formed by Bishop de Castro Mayer resisted, almost to a man, and the laity followed their good priests, with the result that when these priests were – of course- thrown out of their parishes, the best of Campos' laity rallied behind them and built –this is not a wealthy diocese!- ten brand-new churches in which to continue the brand-old religion. I can hardly believe my ears when I hear that none of the Campos laity (or priests) are protesting the putting back under Rome of these churches expressly and expensively built to resist that very Rome, but this is what we are told that has happened! Have the priests been up front with their laity?
But back to the story. When Bishop de Catro Mayer had to resign and the Newchurch put in a wrecker to replace him, for ten years the resistance movement flourished in the form of the St. John Mary Vianney Union of Priests. But in 1991, one month to the day after Archbishop died, Bishop de Castro Mayer died. At the Union's request, three SSPX bishops consecrated a successor, Bishop Liciano Rangel, so that the Campos Catholics would continue to receive Confirmations and Ordinations. However, as we know now, replacing Bishop de Castro Mayer as Confirmer and Ordainer was the easy part. The hard part was to replace the anti-liberal leader. And we are back into the mystery of neo-modernism, this incredible mind-rot capable of rottong apparently the most Catholic of minds.
For there can be no doubt of the Catholic Orthodoxy of the Campos priests and laity that Bishop de Castro Mayer left behind him. Nor, at least until the summer of 2000, was there any trace (as far as I know) of deviation from the line of defence of Catholic Tradition laid down by the two great bishops (Lefebvre and de Castro Mayer) in the 1970's and 1980's. But from the time of Tradition's pilgrimage to Rome in August of 2000, maybe discrete contacts between Campos and Rome were re-established (or fortified?). In any case there emerged last year the recent agreement, proclaimed all over the Newchurch media in the middle of last month (January, 2002).
This agreement whereby the Campos Traditionalists are "welcomed back into the Church" looks like a sweetheart deal for the priests and laity of Catholic Tradition. In return for putting an end to their resistance to the Newchurch, they are granted what is called an "apostolic administration", meaning, more or less, a personal diocese coming directly under the Pope, and depending only upon the person of their own bishop, presently Bishop Rangel. They are allowed to keep the Tridentine liturgy, in other words they may continue to say the true Mass. It looks too good to be true!
And it is of course too good to be true. For instance, as everybody knows, Rome better than anybody, Bishop Rangel is stricken with cancer, and his days are numbered. The Campos priests must believe that he will be replaced, but if Rome is in control when he dies, what will stop Rome either from choosing the most liberal priest in the group, or from declaring that the mainstream bishop in Campos is sufficient for all Catholics, mainstream or Traditional, now that they are all in together again? Already, the Pope's theologian in Rome, Fr. Cottier, is reassuring any conciiliarists alarmed by the apparent concessions to Tradition in Campos: "Little by little we must expect other steps, for example that they (the Campos priests) also participate in concelebrations in the reformed rite. However, we must not be in a hurry. What is important is that in their hearts there no longer be rejection. Communion found again in the Church has an internal dynamism of its own that will mature" (Interview, Jan. 20).
Indeed. Indeed. Another classic exemple of "internal dynamism" of "communion maturing" in the Vatican II Church was provided recently by the appearance in Rome last November of a little book entitled "The Jewish people and the Holy Scriptures in the Christian Bible", written by Rome's Biblical Commission and prefaced by Cardinal Ratzinger, no less. This book's thesis is that "The Jewish wait for the Messiah is not in vain", a typically ambiguous statement, capable of meaning that the Messiah will come at the end of the world either a second time (perfectly true), or for the fisrt time (horrendously false).
When questioned about the ambiguity, the Pope's (Opus dei) spokesman, Dr. Navarro Valls, replied: "It means that it would be wrong for a Catholic to wait for the Messiah, but not for a Jew"! In other words Jesus of Nazareth truly was and truly was not the Messiah promised in the Old Terstament! In other words, there is no objective truth. Truth depends on who you are!
So the Campos priest are also losing their minds. They are putting their trust in Romans to protect the absolute truth of Catholic Tradition when these Romans believe in no such thing. What do the Romans believe in? In making us all feel good. And that is what the Campos priests are switching over to. The Campos priests claim they will continue their fight for Tradition from inside the mainstrem Church. But what chance do they have up agiant the insanity of, as continued for instance in another Assisi meeting, clearly condemned by Bishop Fellay on the sheet enclosed?
The poor Campos priests! Having given away the store (the sanity of their mind) in order to come in from the cold (the marginalisation of their hearts), they will amost certainly from now on go along with anything the Romans say, rather than have to go back out into the cold of being "excommunicated", "schismatic", etc.. Like St. Peter's Fraternity, they will have paid so dearly (with their sanity and integrity) for acceptance by Rome, that they will pay anything further in order not to lose acceptance. Which Rome well knows, and will exploit to the full, but "little by little", as the Pope's theologian says.
Incredible. But let us throw no stones. Today's confusion is universal, and is coming from the top – "the shepherd is struck, and the sheep are scattered" (Zach. XIII,7; Mt. XXVI,31). In war, bullets fly, comrades go down. One spends half a minute with one's handkerchief to wipe away a disfigurement, or wipe away a tear, and then the war goes on. Rather than throw stones, let us take thought for ourselves. The Campos priests presently falling in with the insanity and betrayal of Rome have had the true Mass, breviary and traditional prayers for the last 20 years, yet still they have fallen. Who then is safe?
I might say that the Campos priests fell because under Bishop de Castro Mayer they had too easy a passage from a pre- to post-Council, so theirs is just a belated case of Fiftiesism. But as recalled above, they were earthquake-proofed before the Council, and had to rebuild from ground zero afterwards. Was that still not enough to vaccinate them against neo-modernist mind-rot? Apparently not. Truly, if these days are not shortened by God intervening, we all of us risk losing our minds. Kyrie eleison. "But when these things begin to come to pass," says Our Lord, "look up, and lift up your heads, because your redemption is at hand" (Lk. XXI, 28).
Dear readers, God has thrust upon the greatness of not falling in the madness all around us. For love of Our Lord and His Mother of Sorrows, let us not lay the burden to one side: "He that shall persevere to the end, he shall be saved" (Mt. X,22).
With my blessing, in Christ,
+ Richard Williamson


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There's been a fabulous response to the Porch Shed competition which we're helping John Coupe run over at secrets-of-shed-building. While the competition is, unsurprisingly, based around the idea of a shed with a porch there are several excellent entries which include a garden office element and all of them are well worth a browse (pictured above is A Shed by the Stream). Now the judges - Justin Anthony of Materialicious.com, Su Butcher of architects Barefoot and Gilles and garden designer/writer-blogger James Alexander Sinclair (and John and I) must start whittling them down to our top 3, winners to be announced on June 3 when the Shedworking book is officially published.
Is it possible for us to view the entries?


AIG Canada recognizes the value of encouraging groups of employees with common interests to formalize their professional relationships by launching employee resource groups (ERGs). ERG members are dedicated to fostering a diverse and inclusive work environment for all employees. Any employee may become a member of any ERG. Our ERGs often collaborate on initiatives related to the business and to fulfill their individual interests. Initiatives generally fall into one of four areas: career development and networking, cultural awareness and education, community volunteerism and sponsorship, and partnering with leaders to resolve business challenges.
AIG Canada executives actively participate in ERG activities and support ERG goals and objectives.
AIG offers a variety of multicultural heritage ERGs whose mission is to provide support and guidance to our multicultural employees by offering networking, personal and professional development, and mentoring opportunities in an effort to attain greater recruitment and retention, and leadership recognition within AIG.
disAbilities & Allies strive to build a culture of inclusion by leveraging each employee's unique abilities, raising awareness, removing barriers, and encouraging a workplace founded on dignity, trust, and respect, thereby improving productivity and integration of those with disabilities and those with family members with disabilities.
LGBT & Allies strive to foster an inclusive environment where employees feel safe, respected, valued and supported in their workplaces.
Seasoned Professionals aim to provide a platform for sharing experiences, talent and 'know how' within AIG.
Women & Allies strive to create and foster a community of professionals with a shared interest in the professional development, attraction and retention of women within the organization.
Working Families aim to help working families find a healthy balance between personal and professional responsibilities. Working Families represents every family configuration that exists in today's world.
Young Professionals aim to develop, connect, and empower young professionals.


This publication covers the sale of nontimber forest commodities. These include edible wild mushrooms, fresh floral greens, dried flowers, wild berries, seeds, roots, bark, and other plant parts. Information is offered on marketing, harvest sites, harvest feasibility, and pricing.