Semiclassical resolvent estimates for Hölder potentials

Abstract

We first prove semiclassical resolvent estimates for the Schrödinger operator in ${ℝ}^d$, $d\ge 3$, with real-valued potentials which are Hölder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in ${ℝ}^d$, $d\ge 2$, and real-valued potentials which are Hölder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be Hölder, Lipschitz or just $L^{\infty }$.