Reducibility of 1-d Schr\"odinger equation with time quasiperiodic unbounded perturbations, II

We study the Schr\"odinger equation on $\R$ with a potential behaving as $x^{2l}$ at infinity, $l\in[1,+\infty)$ and with a small time quasiperiodic perturbation. We prove that, if the perturbation belongs to a class of unbounded symbols including smooth potentials and magnetic type terms with controlled growth at infinity, then the system is reducible.