Some dispersive estimates for Schrödinger equations with repulsive potentials

We prove the local smoothing effect for Schrödinger equations with repulsive potentials for n?3. The estimates are global in time and are proved using a variation of Morawetz multipliers. As a consequence we give sharp constants to measure the attractive part of the potential and its rate of decay, which turns out to be different whether dimension 3 or higher are considered. Also a notion of zero resonance arises in a natural way. Our smoothing estimate allows us to use Sobolev inequalities and treat nonradial perturbations.