A limiting absorption principle for the Helmholtz equation with variable coefficients

We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions: egin{equation*} (L+lambda)v=f, qquad lambdain mathbb{R} end{equation*} 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.