Spectral Enclosure and Superconvergence for Eigenvalues in Gaps

Abstract

We consider the problem of how to compute eigenvalues of a selfadjoint 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 noá 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, Schrödinger and Dirac operators.

Spectral inclusion and pollution for a class of dissipative perturbations

“…With additional assumptions on the operators s n , for instance that they are projection operators, we prove new types of non-convex enclosures for σ e ((T n )) and conclude for these cases that (T n ) is spectrally exact for T in an open neighbourhood of any eigenvalue of T . The paper [22] gives a similar spectral exactness conclusion for the case that (s n ) are projection operators. However, as well as including different classes of perturbations (s n ), both the statement and the proof of our results in Section 2 are far simpler than those of [22], owing to the use of the limiting essential spectrum.…”

“…The paper [22] gives a similar spectral exactness conclusion for the case that (s n ) are projection operators. However, as well as including different classes of perturbations (s n ), both the statement and the proof of our results in Section 2 are far simpler than those of [22], owing to the use of the limiting essential spectrum.…”

Spectral Inclusion and Pollution for a Class of Dissipative Perturbations

“…Such spectral exactness results are currently limited. We mention, however, the results [28, Theorem 10] and [21,Theorem 4.6] regarding spectral inclusion for isolated eigenvalues and the spectral enclosure result [21,Theorem 3.3].…”