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

Abstract

We improve the results by Gr\'ebert and Paturel in \cite{GP} and prove that a linear Schr\"odinger equation on $R^d$ with harmonic potential $|x|^2$ and small $t$-quasiperiodic potential as $$ {\rm i}u_t - \Delta u+|x|^2u+\varepsilon V(\omega t,x)u=0, \ (t,x)\in R\times R^d $$ 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.