Smoothing effect and quantum-classical correspondence for the Schr{ö}dinger equation with confining potential
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915044178001920 |
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| author | Prouff, Antoine |
| author_facet | Prouff, Antoine |
| contents | The smoothing effect states that solutions to the Schr{ö}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{ö}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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Smoothing effect and quantum-classical correspondence for the Schr{ö}dinger equation with confining potential Prouff, Antoine Analysis of PDEs The smoothing effect states that solutions to the Schr{ö}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{ö}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}. |
| title | Smoothing effect and quantum-classical correspondence for the Schr{ö}dinger equation with confining potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.01209 |