Perturbations of the Laplacian with variable coefficients in exterior domains and differentiability of the resolvent

Abstract

We consider self‐adjoint, strictly elliptic, short‐range perturbations of the Laplacian with variable coefficients, that operate on square integrable functions defined in an exterior domain. According to the limiting absorption principle, the resolvent is extended to the positive real line, which is the continuous spectrum of these operators. In suitable weighted spaces, where the weight also depends on the grade of differentiation, we show that the extended resolvent is strongly differentiable w.r.t. the spectral parameter. Moreover, we obtain estimates for the derivatives of the extended resolvent in these weighted spaces, which are locally uniform in the spectral parameter.