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Sturm-Liouville Problems Whose Leading Coefficient Function Changes Sign

Published online by Cambridge University Press:  20 November 2018

Xifang Cao ,
Qingkai Kong ,
Hongyou Wu  and
Anton Zettl
Xifang Cao
Affiliation:
Department of Mathematics, Yangzhou University, Yangzhou, Jiangsu 225002, China Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA
Qingkai Kong
Affiliation:
Department of Mathematics, Yangzhou University, Yangzhou, Jiangsu 225002, China Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA
Hongyou Wu
Affiliation:
Department of Mathematics, Yangzhou University, Yangzhou, Jiangsu 225002, China Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA
Anton Zettl
Affiliation:
Department of Mathematics, Yangzhou University, Yangzhou, Jiangsu 225002, China Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA
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Abstract

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For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Prüfer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.

Keywords

Primary: 34B2434C10secondary: 34L0534L1534L20

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 4 , 01 August 2003 , pp. 724 - 749
Copyright
Copyright © Canadian Mathematical Society 2003

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