Global-in-time smoothing effects for Schr\"odinger equations with inverse-square potentials

TL;DR

This paper establishes that Schrödinger equations with inverse-square potentials exhibit global-in-time smoothing effects, confirming a conjecture and extending understanding of dispersive properties for critical singular potentials.

Contribution

It proves global smoothing effects for Schrödinger equations with inverse-square potentials, answering a previously open question and employing advanced resolvent estimates and perturbation methods.

Findings

Confirmed global-in-time smoothing for inverse-square potentials

Extended dispersive analysis to critical singular potentials

Validated the use of uniform resolvent estimates in this context

Abstract

The purpose of this note is to prove global-in-time smoothing effects for the Schr\"odinger equation with potentials exhibiting critical singularity. A typical example of admissible potentials is the inverse-square potential $a|x|^{-2}$ with $a>-(n-2)^2/4$. This particularly gives an affirmative answer to a question raised by Bui-D'Ancona-Duong-Li-Ly [4]. The proof employs a uniform resolvent estimate proved by Barcel\'o-Vega-Zubeldia [1] and an abstract perturbation method by Bouclet-Mizutani [3].