Growth of Sobolev Norms of Solutions of Linear Schrödinger Equations on Some Compact Manifolds

We give a new proof of a theorem of Bourgain[4] asserting that solutions of linear Schrödinger equations on the torus, with smooth time-dependent potential, have Sobolev norms growing at most like t^ϵ when $t\rightarrow+\infty$, for any ϵ > 0. Our proof extends to Schrödinger equations on other examples of compact Riemannian manifolds.