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

We study the Schr"odinger equation on $\mathbb{R}$ with a potential behaving as $x^{2l}$ at infinity, $l\in [1,+\mathrm{\infty })$ 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.