On the convergence of second order spectra and multiplicity

TL;DR

This paper investigates how the second order spectrum of a self-adjoint operator reveals the multiplicity of its isolated eigenvalues, supported by theoretical analysis and numerical experiments.

Contribution

It provides a general framework linking second order spectra to eigenvalue multiplicities, enhancing understanding of spectral pollution in numerical methods.

Findings

Second order spectrum encodes eigenvalue multiplicity information.

Theoretical framework for analyzing spectral pollution.

Numerical experiments validate the framework on differential operators.

Abstract

Let A be a self-adjoint operator acting on a Hilbert space. The notion of second order spectrum of A relative to a given finite-dimensional subspace L has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. We establish in this paper a general framework allowing us to determine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of A. Our theoretical findings are supported by various numerical experiments on the computation of inclusions for eigenvalues of benchmark differential operators via finite element bases.