The Galerkin Method for Perturbed Self-Adjoint Operators and Applications

TL;DR

This paper analyzes the Galerkin method for approximating spectra of perturbed self-adjoint operators, establishing reliability and introducing a new technique for eigenvalue identification and spectral pollution detection.

Contribution

It provides a new approach for spectral approximation of perturbed operators, especially in identifying eigenvalues and spectral pollution using the form domain.

Findings

Galerkin method reliably approximates spectra of perturbed operators

New technique identifies eigenvalues on the form domain

Method detects spectral pollution effectively

Abstract

We consider the Galerkin method for approximating the spectrum of an operator $T+A$ where $T$ is semi-bounded self-adjoint and $A$ satisfies a relative compactness condition. We show that the method is reliable in all regions where it is reliable for the unperturbed problem - which always contains $\mathbb{C}\backslash\mathbb{R}$. The results lead to a new technique for identifying eigenvalues of $T$, and for identifying spectral pollution which arises from applying the Galerkin method directly to $T$. The new technique benefits from being applicable on the form domain.