Reducibility of 1-d Schroedinger equation with time quasiperiodic unbounded perturbations, I

We study the Schr\"odinger equation on $\R$ with a polynomial potential behaving as $x^{2l}$ at infinity, $1\leq l\in\N$ and with a small time quasiperiodic perturbation. We prove that if the symbol of the perturbation grows at most like $(\xi^2+x^{2l})^{\beta/(2l)}$, with $\beta<l+1$, then the system is reducible. Some extensions including cases with $\beta=2l$ are also proved. The result implies boundedness of Sobolev norms. The proof is based on pseudodifferential calculus and KAM theory.