Semiclassical resolvent bound for compactly supporte $L^{\infty }$ potentials

Abstract

We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger 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 distributional derivatives in $L^{\infty }({\mathbb{R}}^n)$. The weighted resolvent norm is bounded by $e^{C{h}^{-4/3}\log (h^{-1})}$, where $h$ is the semiclassical parameter.