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

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 $ipartial_t u -- Delta u + |x|^2 u + epsilon V (tomega, x)u = 0, x in mathbb{R}^d$ 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.