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

References

[1] Atkinson, F. and Mingarelli, A., Asymptotics of the number of zeros and of the eigenvalues of general weighted Sturm-Liouville problems. J. Reine Angew. Math. 375/376(1987), 380–393.Google Scholar
[2] Bailey, P., Everitt, W. and Zettl, A., The SLEIGN2 Sturm-Liouville code. ACM Trans. Math. Software, to appear.Google Scholar
[3] Binding, P. and Volkmer, H., Existence and asymptotics of eigenvalues of indefinite systems of Sturm-Liouville and Dirac type. J. Differential Equations 172(2001), 116–133.Google Scholar
[4] Binding, P. and Volkmer, H., Oscillation theory for Sturm-Liouville problems with indefinite coefficients. Proc. Roy. Soc. Edinburgh Sect. A 131(2001), 989–1002.Google Scholar
[5] Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955.Google Scholar
[6] Eastham, M., Kong, Q., Wu, H. and Zettl, A., Inequalities among eigenvalues of Sturm-Liouville problems. J. Inequal. Appl. 3(1999), 25–43.Google Scholar
[7] Everitt, W., Möller, M. and Zettl, A., Discontinuous dependence of the n-th Sturm-Liouville eigenvalue. In: General Inequalities (eds. Bandle, C., Everitt, W., Losonszi, L. and Walter, W.), Birkhäuser, 1997.Google Scholar
[8] Haertzen, K., Kong, Q., Wu, H. and Zettl, A., Geometric aspects of Sturm-Liouville problems, II. Space of boundary conditions for left-definiteness. Trans. Amer.Math Soc., to appear.Google Scholar
[9] Kong, Q., Lin, Q., Wu, H. and Zettl, A., A new proof of the inequalities among Sturm-Liouville eigenvalues. Panamer. Math. J. (2) 10(2000), 1–10.Google Scholar
[10] Kong, Q., Wu, H. and Zettl, A., Dependence of the eigenvalues on the problem. Math. Nachr. 188(1997), 173–201.Google Scholar
[11] Kong, Q., Wu, H. and Zettl, A., Dependence of the n-th Sturm-Liouville eigenvalue on the problem. J. Differential Equations 156(1999), 328–354.Google Scholar
[12] Kong, Q., Wu, H. and Zettl, A., Inequalities among eigenvalues of singular Sturm-Liouville problems. Dynamic Systems Appl. 8(1999), 517–531.Google Scholar
[13] Kong, Q., Wu, H. and Zettl, A., Geometric aspects of Sturm-Liouville problems, I. Structures on spaces of boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 130(2000), 561–589.Google Scholar
[14] Kong, Q., Wu, H. and Zettl, A., Left-definite Sturm-Liouville problems. J. Differential Equations 177(2001), 1–26.Google Scholar
[15] Kong, Q. and Zettl, A., Eigenvalues of regular Sturm-Liouville problems. J. Differential Equations 131(1996), 1–19.Google Scholar
[16]Möller, M., On the unboundedness below of the Sturm-Liouville operator. Proc. Roy. Soc. Edinburgh Sect. A 129(1999), 1011–1015.Google Scholar
[17] Zettl, A., Sturm-Liouville problems. In: Spectral Theory and Computational Methods of Sturm-Liouville Problems (eds. Hinton, D. and Schaefer, P.), Marcel Dekker, 1997.Google Scholar