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We study the reducibility of a Linear Schr\"odinger equation subject to asmall unbounded almost-periodic perturbation which is analytic in time andspace. Under appropriate assumptions on the smallness, analiticity and on thefrequency of the almost-periodic perturbation, we prove that such an equationis reducible to constant coefficients via an analytic almost-periodic change ofvariables. This implies control of both Sobolev and Analytic norms for thesolution of the corresponding Schr\"odinger equation for all times.

Linear Schr\"odinger equation with an almost periodic potential