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.