Semiclassical resolvent estimates for Holder potentials

Abstract

We first prove semiclassical resolvent estimates for the Schr{\"o}dinger operator in R d , d $\ge$ 3, with real-valued potentials which are H{\"o}lder 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 H{\"o}lder 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 H{\"o}lder, Lipschitz or just L $\infty$ .