Linear Schr\"odinger equation with an almost periodic potential

Abstract

We study the reducibility of a Linear Schr\"odinger equation subject to a small unbounded almost-periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analiticity 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 Schr\"odinger equation for all times.