 98760 George A. Hagedorn, Alain Joye
 Semiclassical Dynamics with Exponentially Small Error Estimates
(69K, Latex)
Dec 21, 98

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Abstract. We construct approximate solutions to the timedependent Schr\"odinger equation
$$
i\,\hbar\,\frac{\partial \psi}{\partial t}\ =\
\,\frac{\hbar^2}{2}\,\Delta\,\psi\,+\,V\,\psi
$$
for small values of $\hbar$.
If $V$ satisfies appropriate analyticity and growth hypotheses and
$t\le T$, these solutions agree with exact solutions up to errors
whose norms are bounded by
$$
C\ \exp\left\{\,\,\gamma/\hbar\,\right\} ,
$$
for some $C$ and $\gamma>0$.
Under more restrictive hypotheses, we prove that for sufficiently
small $T'$, $t\le T'\,\log(\hbar)$ implies the norms of the
errors are bounded by
$$
C'\ \exp\left\{\,\,\gamma'/\hbar^{\sigma}\,\right\} ,
$$
for some $C'$, $\gamma'>0$, and $\sigma>0$.
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