# Weak Dispersive estimates for Schrödinger equations with long range potentials - Mathematics > Analysis of PDEs

Abstract: We prove some local smoothing estimates for the Schr\-{o}dinger initial valueproblem with data in $L^2\mathbb{R}^d$, $d \geq 2$ and a general class ofpotentials. In the repulsive setting we have to assume just a power like decay$1+|x|^{-\gamma}$ for some $\gamma>0$. Also attractive perturbations areconsidered. The estimates hold for all time and as a consequence a weakdispersion of the solution is obtained. The proofs are based on similarestimates for the corresponding stationary Helmholtz equation and Kato H-smooththeory.

Author: J. A. Bercelo, A. Ruiz, L. Vega, M. C. Vilela

Source: https://arxiv.org/