Weak Dispersive estimates for Schrödinger equations with long range potentials

Bercelo, J. A.; Ruiz, A.; Vega, L.; Vilela, M. C.

Weak Dispersive estimates for Schrödinger equations with long range potentials

Authors

Description

We prove some local smoothing estimates for the Schrödinger initial value problem with data in $L^2(\mathbb{R}^d)$, $d \geq 2$ and a general class of potentials. In the repulsive setting we have to assume just a power like decay $(1+|x|)^{-γ}$ for some $γ>0$. 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