Weak Dispersive estimates for Schr\"odinger equations with long range potentials

TL;DR

This paper establishes local smoothing and weak dispersive estimates for Schrödinger equations with long-range potentials, including both repulsive and attractive cases, using stationary Helmholtz and Kato H-smooth theory.

Contribution

It extends dispersive estimates to Schrödinger equations with general long-range potentials, covering both repulsive and attractive scenarios.

Findings

Local smoothing estimates hold for potentials with decay $(1+|x|)^{-eta}$, $eta>0$.

Weak dispersion of solutions is demonstrated for all time.

Results are derived via stationary Helmholtz estimates and Kato H-smooth theory.

Abstract

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.