Linear Schrödinger equation with an almost periodic potential

We study the reducibility of a linear Schrodinger equation subject to a small unbounded almost periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analyticity, and on the frequency of the almost periodic perturbation, we prove that such an equation is reducible to constant coefficients via an analytic almost periodic change of variables. This implies control of both Sobolev and analytic norms for the solution of the corresponding Schrödinger equation for all times.