Global existence results for nonlinear Schrodinger equations with quadratic potentials

We prove that no finite time blow up can occur for nonlinear Schrodinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to control the nonlinear effects.