  In this paper we consider linear, time dependent Schr\"odinger equations of
the form ${\rm i} \partial_t \psi = K_0 \psi + V(t) \psi$, where $K_0$ is a
strictly positive selfadjoint operator with discrete spectrum and constant
spectral gaps, and $V(t)$ a time periodic potential. We give sufficient
conditions on $V(t)$ ensuring that $K_0+V (t)$ generates unbounded orbits. The
main condition is that the resonant average of $V(t)$, namely the average with
respect to the flow of $K_0$, has a nonempty absolutely continuous spectrum and
fulfills a Mourre estimate. These conditions are stable under perturbations.
The proof combines pseudodifferential normal form with dispersive estimates in
the form of local energy decay. We apply our abstract construction to the
Harmonic oscillator on $\mathbb R$ and to the half-wave equation on $\mathbb
T$; in each case, we provide large classes of potentials which are
transporters.
