On reducibility of Quantum Harmonic Oscillator on ${\mathbb{R}}^d$ with quasiperiodic in time potential

Abstract

We prove that a linear d-dimensional Schr{ö}dinger equation on ${\mathbb{R}}^d$ with harmonic potential $|x{|}^2$ and small t-quasiperiodic potential $i\partial \_tu--\Delta u+|x{|}^{2}u+\epsilon V(t\omega ,x)u=0,x\in {\mathbb{R}}^d$ reduces to an autonomous system for most values of the frequency vector $\omega \in {\mathbb{R}}^n$. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms.