Abstract
This article focuses on a symmetric block operator spectral problem with two spectral parameters. Under some reasonable restrictions, Levitin and Öztürk 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.
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Acknowledgements
This work was carried out during the author’s doctoral studies at the University of Reading and the author wishes to thank his doctoral supervisor Michael Levitin for his guidance. The author would like to thank anonymous referees for careful reading and helpful suggestions which improved the quality of the paper. The author also gratefully acknowledges the financial support during his education by the Ministry of National Education of the Republic of Türkiye.
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H.M.Ö. wrote the main manuscript text, prepared all the figures, reviewed the document and performed the analysis.
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Communicated by Sanne ter Horst.
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This article is part of the topical collection “Linear Operators and Linear Systems” edited by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz.
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Öztürk, H.M. Asymptotic Behaviour of the Non-real Pair-Eigenvalues of a Two Parameter Eigenvalue Problem. Complex Anal. Oper. Theory 17, 45 (2023). https://doi.org/10.1007/s11785-023-01344-w
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DOI: https://doi.org/10.1007/s11785-023-01344-w
Keywords
- Two-parameter eigenvalue problem
- Complex eigenvalues
- Block-operator matrices
- Multiparameter spectral problems
- Non-self-adjoint problems
- Asymptotic expansion