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Weak Dispersive Estimates for Schrödinger Equations with Long Range Potentials

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 05 Mar 2009Embargo end date: 01 Jan 2008Publisher:Informa UK LimitedJournal:Communications in Partial Differential Equations, volume 34, pages 74-105 (issn: 0360-5302, eissn: 1532-4133,

Authors: Luis Vega; Alberto Ruiz; Juan Antonio Barceló; M. C. Vilela;

doi: 10.1080/03605300802683596 , 10.48550/arxiv.0802.2161

arXiv: http://arxiv.org/abs/0802.2161

Weak Dispersive Estimates for Schrödinger Equations with Long Range Potentials

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Abstract

We prove some local smoothing estimates for the Schr��dinger 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 $(1+|x|)^{-��}$ for some $��>0$. 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

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Keywords

Mathematics - Analysis of PDEs, 35Q40, FOS: Mathematics, 35Q40; 35P25, FOS: Physical sciences, Mathematical Physics (math-ph), Mathematical Physics, 35P25, Analysis of PDEs (math.AP)

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