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On a conjecture of Davies and Levitin

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dc.contributor.author Ozturk, Hasen Mekki
dc.date.accessioned 2024-03-15T11:13:55Z
dc.date.available 2024-03-15T11:13:55Z
dc.date.issued 2023
dc.identifier.citation Öztürk, HM. (2023). On a conjecture of Davies and Levitin. Math. Meth. Appl. Sci., 46(4), 4391-4412. https://doi.org/10.1002/mma.8766 en_US
dc.identifier.issn 0170-4214
dc.identifier.issn 1099-1476
dc.identifier.uri http://dx.doi.org/10.1002/mma.8766
dc.identifier.uri https://www.webofscience.com/wos/woscc/full-record/WOS:000871215300001
dc.identifier.uri http://earsiv.odu.edu.tr:8080/xmlui/handle/11489/4440
dc.description WoS Categories: Mathematics, Applied en_US
dc.description Web of Science Index: Science Citation Index Expanded (SCI-EXPANDED) en_US
dc.description Research Areas: Mathematics en_US
dc.description.abstract Let Hc$$ {H}_c $$ be a (2n)x(2n)$$ (2n)\times (2n) $$ symmetric tridiagonal matrix with diagonal elements c is an element of Double-struck capital R$$ c\in \mathbb{R} $$ and off-diagonal elements one, and S$$ S $$ be a (2n)x(2n)$$ (2n)\times (2n) $$ diagonal matrix with the first n$$ n $$ diagonal elements being plus ones and the last n$$ n $$ being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil Ac=Hc-lambda S$$ {\mathcal{A}}_c={H}_c-\lambda S $$ as 2n$$ 2n $$ approaches to infinity. It was conjectured by DL that for any n is an element of N$$ n\in \mathbb{N} $$ the non-real eigenvalues lambda$$ \lambda $$ of Ac$$ {\mathcal{A}}_c $$ satisfy both |lambda+c|<2$$ \mid \lambda +c\mid and |lambda-c|<2$$ \mid \lambda -c\mid . The conjecture has been verified numerically for a wide range of n$$ n $$ and c$$ c $$, but so far the full proof is missing. The purpose of the paper is to support this conjecture with a partial proof and several numerical experiments which allow to get some insight in the behaviour of the non-real eigenvalues of Ac$$ {\mathcal{A}}_c $$. We provide a proof of the conjecture for n <= 3$$ n\le 3 $$, and also in the case where |lambda+c|=|lambda-c|$$ \mid \lambda +c\mid =\mid \lambda -c\mid $$. In addition, numerics indicate that some phenomena may occur for more general linear pencils. en_US
dc.description.sponsorship Ministry of National Education of the Republic of Turkiye en_US
dc.language.iso eng en_US
dc.publisher WILEY-HOBOKEN en_US
dc.relation.isversionof 10.1002/mma.8766 en_US
dc.rights info:eu-repo/semantics/openAccess en_US
dc.subject chebyshev polinomials of the second kind, eigenvalues, linear operator pencils, non-self-adjoint matrices, spectral theory en_US
dc.title On a conjecture of Davies and Levitin en_US
dc.type article en_US
dc.relation.journal MATHEMATICAL METHODS IN THE APPLIED SCIENCES en_US
dc.contributor.department Ordu Üniversitesi en_US
dc.contributor.authorID 0000-0002-4524-651X en_US
dc.identifier.volume 46 en_US
dc.identifier.issue 4 en_US
dc.identifier.startpage 4391 en_US
dc.identifier.endpage 4412 en_US


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