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

TL;DR

This paper proves that a quantum harmonic oscillator with a small quasiperiodic potential can be simplified to an autonomous system, ensuring solutions are almost periodic and bounded in Sobolev norms for most frequencies.

Contribution

It demonstrates the reducibility of the Schrödinger equation with quasiperiodic potential on ^d, extending the understanding of quantum harmonic oscillators with time-dependent perturbations.

Findings

The equation reduces to an autonomous system for most frequency vectors.

Solutions are almost periodic in time.

Solutions remain bounded in all Sobolev norms.

Abstract

We prove that a linear d-dimensional Schr{\"o}dinger equation on $\mathbb{R}^d$ with harmonic potential $|x|^2$ and small t-quasiperiodic potential $i\partial\_t u -- \Delta u + |x|^2 u + \epsilon V (t\omega, x)u = 0, x \in \mathbb{R}^d$ 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.