Smoothing effect and quantum-classical correspondence for the Schr{\"o}dinger equation with confining potential

Antoine Prouff

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arXiv · 2412.01209

Smoothing effect and quantum-classical correspondence for the Schr{\"o}dinger equation with confining potential

Antoine Prouff

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Abstract

The smoothing effect states that solutions to the Schr{\"o}dinger equation in the Euclidean space have, for almost-every time, a local-in-space improved regularity (gain of half a derivative in Sobolev spaces). In this note, we show that, for the Schr{\"o}dinger equation with a sub-quadratic confining potential, the smoothing effect is equivalent to an escape rate estimate on the associated classical flow. The proof relies on an Egorov theorem proved in~\cite{P:24Egorovinprep}.

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Antoine Prouff. 2024-12-02. Smoothing effect and quantum-classical correspondence for the Schr{\"o}dinger equation with confining potential. https://arxiv.org/abs/2412.01209

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