Semiclassical resolvent bound for compactly supported $L^\infty$ potentials

TL;DR

This paper provides an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators with compactly supported, bounded potentials, establishing bounds without requiring potential derivatives.

Contribution

It introduces a new, simplified proof technique for resolvent bounds applicable to potentials in L-infinity with compact support, without derivative assumptions.

Findings

Weighted resolvent norm bounded by exponential of h^{-4/3} log(h^{-1})

Applicable to potentials in L-infinity with compact support

No derivative conditions required for the potential

Abstract

We give an elementary proof of a weighted resolvent estimate for semiclassical Schr\"odinger operators in dimension $n \ge 1$. We require the potential belong to $L^\infty(\mathbb{R}^n)$ and have compact support, but do not require that it have derivatives in $L^\infty(\mathbb{R}^n)$. The weighted resolvent norm is bounded by $e^{Ch^{-4/3}\log(h^{-1})}$, where $h$ is the semiclassical parameter.