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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, for any  ϵ  &gt; 0. Our proof extends to Schrödinger equations on other examples of compact Riemannian manifolds.

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