Weak Dispersive estimates for Schrödinger equations with long range potentials

2008/02/15 by Bercelo, J. A., Ruiz, A., Vega, L. +1
#35P25 #35Q40 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.0802.2161

Abstract

We prove some local smoothing estimates for the Schrödinger initial value problem with data in L²(ℝ^d), d ≥ 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.

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