In this talk some recent results on the long-time behavior of
dispersive equations with potentials will be presented. More
precisely, we will focus on the NLS equation in 3D with quadratic
nonlinearity and two types of potentials (electric time dependent on
one hand, and electromagnetic time independent on the other). These
equations are of interest since they constitute good models for the
linearization of dispersive equations around special solutions.
The methods described consist in bringing together the space-time
resonance theory of Germain, Masmoudi and Shatah with tools used in
the study of the linear Schroedinger equation (Strichartz and
smoothing estimates, boundedness and representation of wave
operators).