Spectral Inclusion and Pollution for a Class of Dissipative Perturbations

TL;DR

This paper investigates spectral inclusion and pollution phenomena for a class of dissipative operator perturbations, providing theoretical results and practical justification for eigenvalue computation methods in spectral gaps.

Contribution

It establishes spectral inclusion and pollution results for operators with dissipative perturbations, both abstractly and in Sturm-Liouville problems, justifying eigenvalue computation techniques.

Findings

Spectral inclusion holds for the studied operator class.

Spectral pollution is characterized and controlled.

Results apply to Sturm-Liouville operators.

Abstract

Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form $T_0 + i \gamma s_n$ on a Hilbert space, where $s_n$ is strongly convergent to the identity operator and $\gamma > 0$. We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.