On the limit behaviour of second order relative spectra of self-adjoint operators

TL;DR

This paper investigates the approximation properties of second order relative spectra for self-adjoint operators, revealing they do not generally approximate the entire essential spectrum despite avoiding spectral pollution.

Contribution

It proves that second order relative spectra fail to approximate the full essential spectrum of bounded self-adjoint operators.

Findings

Second order relative spectra do not generally approximate the entire essential spectrum.

They are effective in approximating isolated eigenvalues of finite multiplicity.

Standard projection methods can cause spectral pollution in spectrum recovery.

Abstract

It is well known that the standard projection methods allow one to recover the whole spectrum of a bounded self-adjoint operator but they often lead to spectral pollution, i.e. to spurious eigenvalues lying in the gaps of the essential spectrum. Methods using second order relative spectra are free from this problem, but they have not been proven to approximate the whole spectrum. L. Boulton (2006, 2007) has shown that second order relative spectra approximate all isolated eigenvalues of finite multiplicity. The main result of the present paper is that second order relative spectra do not in general approximate the whole of the essential spectrum of a bounded self-adjoint operator.