  We prove an abstract theorem giving a $\langle t\rangle^\epsilon$ bound
($\forall \epsilon>0$) 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 $t \to \infty$. 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}.
