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^{Ch^{-4/3}\log(h^{-1})} , where h is the semiclassical parameter.