98-760 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 time--dependent 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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