Nonlinear Schrödinger equation with time dependent potential

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Abstract
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 in this paper is the growth of high Sobolev norms of the solution when the potential is exactly quadratic in space. Such a control is needed to study the large time propagation of nonlinear coherent states.