A limiting absorption principle for the Helmholtz equation with variable coefficients

Abstract

We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions $$ (L+\lambda)v=f, \quad \lambda\in \mathbb{R} $$ under a Sommerfeld radiation condition at infinity. The operator $L$ is a second order elliptic operator with variable coefficients; the principal part is a small, long range perturbation of $-\Delta$, while lower order terms can be singular and large. The main tool is a sharp uniform resolvent estimate, which has independent applications to the problem of embedded eigenvalues and to smoothing estimates for dispersive equations.