Spectral enclosure and superconvergence for eigenvalues in gaps

TL;DR

This paper introduces a simplified perturbation method for computing eigenvalues of self-adjoint operators, outperforming quadratic methods, and provides a new spectral enclosure for operators of the form A+iB, with applications to various physical operators.

Contribution

The paper proposes a new perturbation approach that requires no prior information and offers rigorous convergence analysis, improving eigenvalue computation in challenging regions.

Findings

The new method outperforms quadratic methods in convergence.

A spectral enclosure for A+iB operators is established.

Demonstrated effectiveness on magnetohydrodynamics, Schrödinger, and Dirac operators.

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.