A limiting absorption principle for the Helmholtz equation with variable coefficients

TL;DR

This paper establishes a limiting absorption principle for a variable coefficient Helmholtz equation on exterior domains, providing key resolvent estimates with applications to spectral theory and dispersive PDEs.

Contribution

It proves a new limiting absorption principle for elliptic operators with variable coefficients, including singular and large lower order terms, extending previous results for constant coefficient cases.

Findings

Sharp uniform resolvent estimate established

Applications to embedded eigenvalues problem

Implications for smoothing estimates in dispersive equations

Abstract

We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions \begin{equation*} (L+\lambda)v=f, \qquad \lambda\in \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.