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

Grébert, Benoît; Paturel, Eric

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

On montre que l’équation de Schrödinger $d$-dimensionnelle avec potentiel harmonique ${\left|x\right|}^2$, perturbée par un petit potentiel quasipériodique en temps

$$i{\partial }_{t}u-\Delta u+{\left|x\right|}^{2}u+\epsilon V(t\omega ,x)u=0,x\in {ℝ}^d$$

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

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

$$i{\partial }_{t}u-\Delta u+{\left|x\right|}^{2}u+\epsilon V(t\omega ,x)u=0,x\in {ℝ}^d$$

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

Mots clés : Reducibility, Quantum harmonic oscillator, KAM Theory

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