Spectral enclosure and superconvergence for eigenvalues in gaps

Hinchcliffe, James and Strauss, Michael (2016) Spectral enclosure and superconvergence for eigenvalues in gaps. Integral Equations and Operator Theory, 84 (1). pp. 1-32. ISSN 0378-620X

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Abstract

We consider the problem of how to compute eigenvalues of a self-adjoint operator when a direct application of the Galerkin (finite-section) method is unreliable. The last two decades have seen the development of the so-called quadratic methods for addressing this problem. Recently a new perturbation approach has emerged, the idea being to perturb eigenvalues off the real line and, consequently, away from regions where the Galerkin method fails. We propose a simplified perturbation method which requires no a priori information and for which we provide a rigorous convergence analysis. The latter shows that, in general, our approach will significantly outperform the quadratic methods. We also present a new spectral enclosure for operators of the form $A+iB$ where $A$ is self-adjoint, $B$ is self-adjoint and bounded. This enables us to control, very precisely, how eigenvalues are perturbed from the real line. The main results are demonstrated with examples including magnetohydrodynamics, Schr\"odinger and Dirac operators.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Michael Strauss
Date Deposited: 08 Feb 2016 10:35
Last Modified: 08 Mar 2017 05:51
URI: http://sro.sussex.ac.uk/id/eprint/58996

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