Results in Mathematics

, Volume 63, Issue 3–4, pp 1057–1070 | Cite as

The Finite Spectrum of Sturm–Liouville Problems with Transmission Conditions and Eigenparameter-Dependent Boundary Conditions

  • Ji-jun Ao
  • Jiong Sun
  • Mao-zhu Zhang


We study the finite spectrum of Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions. For any positive integers m and n, we construct a class of regular Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions, which have at most m + n + 4 eigenvalues.

Mathematics Subject Classification

Primary 34B24 34L15 Secondary 34L05 


Sturm–Liouville problems Eigenvalue Finite spectrum Transmission conditions Eigenparameter-dependent boundary conditions 


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  1. 1.
    Chanane B.: Sturm-Liouville problems with impulse effects. Appl. Math. Comput. 190, 610–626 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Gesztesy F., Macedo C., Streit L.: An exactly solvable periodic Schröedinger operator. J. Phys. A: Math. Gen. 18, 503–507 (1985)CrossRefGoogle Scholar
  3. 3.
    Mukhtarov O.Sh., Kandemir M.: Asymptotic behavior of eigenvalues for the discontinuous boundary-value problem with functional-transmission conditions. Acta Math. Sci. 22 B(3), 35–345 (2002)MathSciNetGoogle Scholar
  4. 4.
    Mukhtarov O.Sh., Kadakal M., Muhtarov F.S.: Eigenvalues and normalized eigenfunctions of discontinuous Sturm-Liouville problem with transmission conditions. Reports Math. Phys. 54(1), 41–56 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Mukhtarov O.Sh., Yakubov S.: Problems for differential equations with transmission conditions. Appl. Anal. 81, 1033–1064 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Sun, J., Wang, A.: Sturm-Liouville operators with interface conditions. In: The Progress of Research for Math., Mech., Phy. and High New Tech., vol. 12, pp. 513–516. Science Press, Beijing (2008)Google Scholar
  7. 7.
    Titeux I., Yakubove Ya.: Completeness of root functions for thermal condition in a strip with piecewise continuous coefficients. Math. Models Methods Appl. Sc. 7, 1035–1050 (1997)zbMATHCrossRefGoogle Scholar
  8. 8.
    Fulton C., Pruess S.: Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions. J. Math. Anal. Appl. 71, 431–462 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Walter J.: Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133, 301–312 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fulton C.: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb. Sect. A. 77, 293–308 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Akdoğan Z., Demirci M., Mukhtarov O.Sh.: Green function of discontinuous boundary-value problem with transmission conditions. Math. Meth. Appl. Sci. 30, 1719–1738 (2007)zbMATHCrossRefGoogle Scholar
  12. 12.
    Binding P.A., Browne P.J., Watson B.A.: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, II. J. Comp. Appl. Math. 148, 147–168 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Binding P.A., Browne P.J., Watson B.A.: Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. J. Lond. Math. Soc. 62(1), 161–182 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Atkinson F.V.: Discrete and Continuous Boundary Value Problems. Academic Press, New York (1964)Google Scholar
  15. 15.
    Kong Q., Wu H., Zettl A.: Sturm-Liouville problems with finite spectrum. J. Math. Anal. Appl. 263, 748–762 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kong Q., Volkmer H., Zettl A.: Matrix representations of Sturm-Liouville problems with finite spectrum. Result Math. 54, 103–116 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Ao J.J., Sun J., Zhang M.Z.: The finite spectrum of Sturm-Liouville problems with transmission conditions. Appl. Math. Comput. 218, 1166–1173 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Everitt W.N., Race D.: On necessary and sufficient conditions for the existence of Caratheodory solutions of ordinary differential equations. Quaest. Math. 3, 507–512 (1976)MathSciNetGoogle Scholar
  19. 19.
    Zettl A.: Sturm-Liouville Theory, Amer. Math. Soc., Mathematical Surveys and Monographs, vol. 121 (2005)Google Scholar
  20. 20.
    Volkmer H.: Eigenvalue problems of Atkinson, Feller and Krein and their mutual relationship. Electron. J. Diff. Equ. 2005(48), 1–15 (2005)MathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesInner Mongolia UniversityHohhotChina
  2. 2.College of SciencesInner Mongolia University of TechnologyHohhotChina
  3. 3.School of Mathematics and System SciencesTaishan UniversityTaianChina

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