Nonlinear Schrodinger equation with time dependent potential

We prove a global well-posedness result for defocusing nonlinear Schrodinger equations with time dependent potential. We then focus on time dependent harmonic potentials. This aspect is motivated by Physics (Bose–Einstein condensation), and appears also as a preparation for the analysis of the propagation of wave packets in a nonlinear context. The main aspect considered here is the growth of high Sobolev norms of the solution.