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

Growth of Sobolev norms in linear Schr\"odinger equations as a dispersive phenomenon