Weak Dispersive estimates for Schrödinger equations with long range potentials

Details Weak Dispersive estimates for Schrödinger equations with long range potentials Weak Dispersive estimates for Schrödinger equations with long range potentials

2008-02-15

arxiv.org/abs/0802.2161v1

Mathematics

Analysis of PDEs

29 pages

Scientific paper

We prove some local smoothing estimates for the Schr\"{o}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 $(1+|x|)^{-\gamma}$ 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.

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