Generalised Weyl theorems and spectral pollution in the Galerkin method

TL;DR

This paper develops a general framework to understand spectral pollution in the Galerkin method, characterizing it through Weyl sequences and analyzing how perturbations affect the limiting set of spectral pollution.

Contribution

It introduces a novel characterization of spectral pollution via Weyl sequences and identifies conditions under which perturbations preserve the limiting set of spectral pollution.

Findings

Spectral pollution is characterized by particular Weyl sequences.

Perturbations satisfying certain conditions keep the limiting set of spectral pollution unchanged.

The behavior of spectral pollution under perturbations resembles that of the essential spectrum.

Abstract

We consider a general framework for investigating spectral pollution in the Galerkin method. We show how this phenomenon is characterised via the existence of particular Weyl sequences which are singular in a suitable sense. For a semi-bounded selfadjoint operator A we identify relative compactness conditions on a selfadjoint perturbation B ensuring that the limiting set of spectral pollution of A and B coincide. Our results show that, under perturbation, this limiting set behaves in a similar fashion as the essential spectrum.