Limiting absorption principle for the dissipative Helmholtz equation

TL;DR

This paper extends Mourre's commutator method to dissipative operators, establishing a limiting absorption principle for the Helmholtz equation with complex potentials, and provides resolvent estimates in various functional spaces.

Contribution

It introduces a novel adaptation of Mourre's method for dissipative operators and derives resolvent estimates for high frequency Helmholtz equations with trapping and complex potentials.

Findings

Proves a limiting absorption principle for dissipative operators.

Provides resolvent estimates for high frequency Helmholtz equations.

Establishes resolvent bounds in Besov spaces.

Abstract

Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the resolvent estimates for the high frequency Helmholtz equation when trapped trajectories meet the set where the imaginary part of the potential is non-zero. We also give the resolvent estimates in Besov spaces.