Global dispersive estimates for defocusing nonlinear Schrodinger equations

Abstract

We consider the defocusing nonlinear Schr{\"o}dinger equation with a gauge invariant power-like nonlinearity. We prove global dispersive estimates in a semi-classical scaling, after rescaling the solution thanks to a suitable distorsion of the natural dispersion rate. We show a uniform bound on the modulus of the solution in some space providing compactness properties. We discuss the consequences of these estimates in the light of both scattering theory and semi-classical analysis.