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 $i\partial\_t u -- \Delta u + |x|^2 u + \epsilon V (t\omega, x)u = 0, x \in \mathbb{R}^d$ 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. 
Eric Paturel (LMJL), Beno\^it Gr\'ebert (LMJL)
math.AP math.DS