Growth of Sobolev Norms in Linear Schrödinger Equations with Quasi-Periodic Potential

Abstract

In this paper, we consider the following problem. Let iu t +Δu+V(x,t)u= 0 be a linear Schrodinger equation ( periodic boundary conditions) where V is a real, bounded, real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector λ. Denote S(t) the corresponding flow map. Thus S(t) preserves the L 2-norm and our aim is to study its behaviour on H s (T D ), s> 0. Our main result is the growth in time is at most logarithmic; thus if φ∈H s , then More precisely, (*) is proven in 1D and 2D when V is small. We also exhibit examples showing that a growth of higher Sobolev norms may occur in this context and (*) is thus essentially best possible.