Reducibility of quantum harmonic oscillator on $R^d$ with differential and quasi-periodic in time potential

Abstract

We improve the results by Grébert and Paturel in \cite{GP} and prove that a linear Schrödinger equation on $R^d$ with harmonic potential $|x{|}^2$ and small $t$-quasiperiodic potential as $$\mathrm{i}{u}_t-\Delta u+|x{|}^{2}u+\epsilon V(\omega t,x)u=0,(t,x)\in R\times R^d$$ reduces to an autonomous system for most values of the frequency vector $\omega \in R^n$. The new point is that the potential $V(\theta ,\cdot )$ is only in ${\mathcal{C}}^{\beta }(T^n,{\mathcal{H}}^s(R^d))$ with $\beta$ large enough. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in some suitable Sobolev norms.