Essential numerical ranges for linear operator pencils

TL;DR

This paper introduces new concepts of essential numerical ranges for linear operator pencils, enabling better understanding and control of spectral pollution in operator approximation methods, with applications to block operator matrices and quantum operators.

Contribution

The paper develops novel essential numerical range concepts for operator pencils, improving spectral pollution analysis and providing tighter spectral enclosures for various operators.

Findings

Describes the set of spectral pollution in operator pencil approximations.

Provides non-convex spectral enclosures for operator eigenvalues.

Improves results for Dirac and Schrödinger operators.

Abstract

We introduce concepts of essential numerical range for the linear operator pencil $\lambda\mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=\lambda x$ into the pencil problem $BTx=\lambda Bx$ for suitable choices of $B$, we can obtain non-convex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of non-selfadjoint Schr\"{o}dinger operators which it has not been possible to treat with existing methods.