Maximal estimates for Schrödinger equations with inverse-square potential

Changxing Miao, Junyong Zhang and Jiqiang Zheng

Vol. 273 (2015), No. 1, 1–19

DOI: 10.2140/pjm.2015.273.1

Abstract

We consider maximal estimates for the solution to an initial value problem of the linear Schrödinger equation with a singular potential. We show a result about the pointwise convergence of solutions to this special variable coefficient Schrödinger equation with initial data $u_0\in H^s\left({ℝ}^n\right)$ for $s>\frac{1}{2}$ or radial initial data $u_0\in H^s\left({ℝ}^n\right)$ for $s\ge \frac{1}{4}$ and that the solution does not converge when $s<\frac{1}{4}$.

Keywords

inverse square potential, maximal estimate, spherical harmonics

Mathematical Subject Classification 2010

Primary: 35B65, 35Q55, 47J35

Milestones

Received: 23 May 2013

Revised: 1 January 2014

Accepted: 25 February 2014

Published: 6 December 2014