Asymptotic Behaviour of the Non-real Pair-Eigenvalues of a Two Parameter Eigenvalue Problem

Abstract

This article focuses on a symmetric block operator spectral problem with two spectral parameters. Under some reasonable restrictions, Levitin and ozturk showed that the real pair-eigenvalues of a two-parameter eigenvalue problem lie in a union of rectangular regions; however, there has been little written about the non-real pair-eigenvalues. This research deals mainly with the non-real pair-eigenvalues. By using formal asymptotic analysis, we prove that as the norm of an off-diagonal operator diverges to infinity there exists a family of non-real pair-eigenvalues, and each component of the pair-eigenvalues lies approximately on a circle in its corresponding complex plane. Afterwards, we establish a Gershgorin-type result for the localisation of the spectrum of a two-parameter eigenvalue problem, which is a more general enclosure result for the pair-eigenvalues, derived from an enclosure result of Feingold and Varga.