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Journal of Spectral Theory

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Volume 8, Issue 4, 2018, pp. 1349–1392
DOI: 10.4171/JST/229

Published online: 2018-10-09

A limiting absorption principle for the Helmholtz equation with variable coefficients

Federico Cacciafesta[1], Piero D'Ancona[2] and Renato Lucà[3]

(1) Università degli Studi di Padova, Italy
(2) Università di Roma La Sapienza, Italy
(3) Universität Basel, Switzerland

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.

Keywords: Smoothing estimates, Helmholtz equation, variable coefficients, limiting absorption principle

Cacciafesta Federico, D'Ancona Piero, Lucà Renato: A limiting absorption principle for the Helmholtz equation with variable coefficients. J. Spectr. Theory 8 (2018), 1349-1392. doi: 10.4171/JST/229