Mourre's method for a dissipative form perturbation

TL;DR

This paper extends Mourre's commutator method to establish uniform resolvent estimates and the limiting absorption principle for dissipative perturbations of self-adjoint operators, with applications to PDEs with boundary dissipation.

Contribution

It adapts Mourre's method to dissipative operators in an abstract setting, providing new resolvent estimates and principles applicable to PDEs with boundary dissipation.

Findings

Established uniform resolvent estimates for dissipative perturbations

Proved the limiting absorption principle in this setting

Derived uniform estimates for derivatives of the resolvent

Abstract

We prove uniform resolvent estimates for an abstract operator given by a dissipative perturbation of a self-adjoint operator in the sense of forms. For this we adapt the commutators method of Mourre. We also obtain the limiting absorption principle and uniform estimates for the derivatives of the resolvent. This abstract work is motivated by the Schr{\"o}dinger and wave equations on a wave guide with dissipation at the boundary.