  In this paper we consider time dependent Schr{\"o}dinger linear PDEs of the
form i$\partial$t$\psi$ = L(t)$\psi$, where L(t) is a continuous family of
self-adjoint operators. We give conditions for well-posedness and polynomial
growth for the evolution in abstract Sobolev spaces. If L(t) = H + V (t) where
V (t) is a perturbation smooth in time and H is a self-adjoint positive
operator whose spectrum can be enclosed in spectral clusters whose distance is
increasing, we prove that the Sobolev norms of the solution grow at most as t
$\epsilon$ when t $\rightarrow$ $\infty$, for any $\epsilon$ \textgreater{} 0.
If V (t) is analytic in time we improve the bound to (log t) $\gamma$ , for
some $\gamma$ \textgreater{} 0. The proof follows the strategy, due to Howland,
Joye and Nenciu, of the adiabatic approximation of the flow. We recover most of
known results and obtain new estimates for several models including 1-degree of
freedom Schr{\"o}dinger operators on R and Schr{\"o}dinger operators on Zoll
manifolds.
