On reducibility of quantum harmonic oscillator on $\protect \mathbb{R}^d$ with quasiperiodic in time potential

Benoît Grébert; Eric Paturel

doi:10.5802/afst.1619

On reducibility of quantum harmonic oscillator on

ℝ^{d}

with quasiperiodic in time potential

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

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 5, pp. 977-1014.

Abstract
Abstract

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

i \partial_{t} u - Δ u + {| x |}^{2} u + ε V (t ω, x) u = 0, x \in ℝ^{d}

reduces to an autonomous system for most values of the frequency vector $ω \in ℝ^{n}$ . 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

i \partial_{t} u - Δ u + {| x |}^{2} u + ε V (t ω, x) u = 0, x \in ℝ^{d}

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

Received: 2016-11-17
Accepted: 2017-09-21
Published online: 2020-04-23

Zbl

DOI: 10.5802/afst.1619

Keywords: Reducibility, Quantum harmonic oscillator, KAM Theory

Author's affiliations:

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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     author = {Beno{\^\i}t Gr\'ebert and Eric Paturel},
     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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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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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