Weak Dispersive estimates for Schrödinger equations with long range potentials

Abstract

We prove some local smoothing estimates for the Schrödinger initial value problem with data in $L^2({\mathbb{R}}^d)$, $d\ge 2$ and a general class of potentials. In the repulsive setting we have to assume just a power like decay $(1+|x|{)}^{-\gamma }$ for some $\gamma >0$. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.

29 pages