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Adiabatic Evolution and Shape Resonances
About this Title
Michael Hitrik, Andrea Mantile and Johannes Sjoestrand
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 280, Number 1380
ISBNs: 978-1-4704-5421-0 (print); 978-1-4704-7280-1 (online)
DOI: https://doi.org/10.1090/memo/1380
Published electronically: October 7, 2022
Table of Contents
Chapters
- 1. Introduction and main results
- 2. Formal adiabatic solutions for an isolated eigenvalue
- 3. Some further adiabatic results
- 4. General facts about operators and escape functions
- 5. Microlocal approach to resonances [16]
- 6. Semi-boundedness in $H(\Lambda _{\upsilon G})$ spaces
- 7. Far away improvement
- 8. Resolvent estimates
- 9. Back to adiabatics
Abstract
Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schrödinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter $\varepsilon$ with $\ln \varepsilon \asymp -1/h$, where $h$ denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length $\varepsilon ^{-N}$ with an error ${\mathcal O}(\varepsilon ^N)$. Here $N>0$ is arbitrary.
\center Résumé \endcenter
Motivés par un problème d’approximation à un mode pour une évolution avec accumulation de charge dans des puits de potentiel, nous considérons un problème d’évolution linéaire pour un opérateur de Schrödinger avec un potentiel dépendant du temps avec un puits dans une île. En particular, nous montrons que nous pouvons choisir le paramètre adiabatique $\varepsilon$ avec $\ln \varepsilon \asymp -1/h$, où $h$ désigne le paramètre semi-classique, et obtenir des approximations adiabatiques de solutions exactes sur des intervalles de temps de longueur $\varepsilon ^{-N}$ avec une erreur ${\mathcal O}(\varepsilon ^N)$. Ici $N>0$ est arbitraire.
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