Resolvent bounds for repulsive potentials

TL;DR

This paper establishes resolvent bounds for semiclassical Schrödinger operators with repulsive potentials, including singularities, and applies these results to demonstrate time decay of wave equation solutions with specific initial conditions.

Contribution

It provides new resolvent bounds for Schrödinger operators with singular repulsive potentials and applies them to analyze wave equation decay.

Findings

Proved resolvent bounds for operators with singular repulsive potentials.

Derived time decay estimates for wave equations with short-range repulsive potentials.

Handled potentials with singularities at the origin in higher dimensions.

Abstract

We prove limiting absorption resolvent bounds for the semiclassical Schr\"odinger operator with a repulsive potential in dimension $n\ge 3$, which may have a singularity at the origin. As an application, we obtain time decay for the weighted energy of the solution to the associated wave equation with a short range repulsive potential and compactly supported initial data.