Semiclassical Dynamics with Exponentially Small Error Estimates

Abstract

We construct approximate solutions to the time--dependent Schrödinger equation $i\hslash (\partial \psi )/(\partial t)=-({\hslash }^2)/2\Delta \psi +V\psi$ for small values of $\hslash$. 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 $C\exp -\gamma /\hslash$, for some $C$ and $\gamma >0$. Under more restrictive hypotheses, we prove that for sufficiently small $T^{\prime },|t|\le T^{\prime }|\log (\hslash )|$ implies the norms of the errors are bounded by $C^{\prime }\exp -{\gamma }^{\prime }/{\hslash }^{\sigma }$, for some $C^{\prime },{\gamma }^{\prime }>0$, and $\sigma >0$.

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