Growth of Sobolev norms for abstract linear Schr\"odinger Equations

We prove an abstract theorem giving a $⟨t{⟩}^{ϵ}$ bound ($\mathrm{\forall }ϵ>0$) on the growth of the Sobolev norms in linear Schr"odinger equations of the form $i\dot{\psi }=H_0\psi +V(t)\psi$ when the time $t\to \mathrm{\infty }$. 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 then 2; (ii) $H_0$ is the (resonant or nonresonant) Harmonic oscillator in $R^d$ and $V(t)$ a pseudodifferential operator of order smaller then $H_0$ 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 \cite{MaRo}.