Growth of Sobolev norms for abstract linear Schrodinger equations

We prove an abstract theorem giving a < t >(epsilon) bound (for all epsilon > 0) on the growth of the Sobolev norms in linear Schrodinger equations of the form i(psi)over dot = H-0 psi V(t)psi as t -> infinity. The abstract theorem is applied to several cases, including the cases where (i) H-0 is the Laplace operator on a Zoll manifold and V(t) a pseudodifferential operator of order smaller than 2; (ii) H-0 is the (resonant or nonresonant) harmonic oscillator in R-d and V(t) a pseudodifferential operator of order smaller than that of H-0 depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the results of [MR17].