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On reducibility of quantum harmonic oscillator on d with quasiperiodic in time potential
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 5, pp. 977-1014.

We prove that a linear d-dimensional Schrödinger equation on d with harmonic potential |x| 2 and small t-quasiperiodic potential

itu-Δu+|x|2u+εV(tω,x)u=0,xd

reduces to an autonomous system for most values of the frequency vector ω n . As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms.

On montre que l’équation de Schrödinger d-dimensionnelle avec potentiel harmonique |x| 2 , perturbée par un petit potentiel quasipériodique en temps

itu-Δu+|x|2u+εV(tω,x)u=0,xd

est réductible à un système autonome pour la plupart des valeurs du vecteur de fréquences ω n . En conséquence, toute solution d’une telle EDP linéaire est presque-périodique en temps et toutes ses normes de Sobolev restent bornées.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1619
Keywords: Reducibility, Quantum harmonic oscillator, KAM Theory
Benoît Grébert 1; Eric Paturel 1

1 Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2 rue de la Houssinière, 44322 Nantes Cedex 03, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {On reducibility of quantum harmonic oscillator on $\protect \mathbb{R}^d$ with quasiperiodic in time potential},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {977--1014},
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Benoît Grébert; Eric Paturel. On reducibility of quantum harmonic oscillator on $\protect \mathbb{R}^d$ with quasiperiodic in time potential. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 5, pp. 977-1014. doi : 10.5802/afst.1619. https://afst.centre-mersenne.org/articles/10.5802/afst.1619/

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