Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

Abstract

In this paper we consider linear, time dependent Schrödinger equations of the form i∂tψ=K0ψ+V(t)ψ, where K0 is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and V(t) a smooth in time periodic potential. We give sufficient conditions on V(t) ensuring that K0+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 K0, 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 R and to the half-wave equation on T; in each case, we provide large classes of potentials which are transporters.