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The Spectrum of Singular Sturm-Liouville Problems With Eigenparameter Dependent Boundary Conditions and Its Approximation

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We study the spectrum of singular Sturm-Liouville problems with eigenparameter dependent boundary conditions and its approximation with eigenvalues from a sequence of regular problems.

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References

  1. Bailey P.B., Everitt W.N., Weidmann J., Zettl A.: Regular approximations of singular Sturm-Liouville problems. Results Math. 23, 3–22 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bailey P.B., Everitt W.N., Zettl A.: The SLEIGN2 Sturm-Liouville code, ACM TOMS. ACM Trans. Math. Softw. 21, 143–192 (2001)

    Article  Google Scholar 

  3. Binding P.A., Browne P.J., Seddighi K.: Sturm-Liouville problems with eigenparameter dependent boundary conditions. Proc. Edinburgh Math. Soc 37(2), 57–72 (1993)

    MathSciNet  Google Scholar 

  4. Binding P.A., Browne P.J., Watson B.A.: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter, II. J. Comput. Appl. Math. 148, 147–168 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Binding P.A., Browne P.J., Watson B.A.: Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. J. Math. Anal. Appl 291, 246–261 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Calkin J.W.: Abstract symmetric boundary conditions. Trans. Am. Math. Soc 45, 364–442 (1939)

    Article  MathSciNet  Google Scholar 

  7. Edmunds D.E., Evans W.D.: Spectral theory and differential operators. Clarendon Press, Oxford (1987)

    MATH  Google Scholar 

  8. Everitt W.N., Marletta M., Zettl A.: Inequalities and eigenvalues of Sturm-Liouville problems near a singular boundary. J. Inequal. Appl. 6, 405–413 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Everitt W.N., Zettl A.: Generalized symmetric ordinary differential operators I: the basic theory. Nieuw Archief voor Wiskunde XXVII, 363–397 (1979)

    MathSciNet  Google Scholar 

  10. Everitt W.N., Zettl A.: Differential operators generated by a countable number of quasi-differential expressions on the line. Proc. London Math. Soc. 64, 524–544 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fulton C.T.: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 77, 293–308 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fulton C.T., Pruess S.: Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions. J. Math. Anal. Appl 71, 431–462 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fulton C.T.: Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions. Proc. Royal Soc. Edinburgh Sect. A, 87, 1–34 (1980/1981)

  14. Fulton C.T.: Asymptotics of the m-coefficient for eigenvalue problems with eigenparameter in the boundary conditions. Bull. London Math. Soc 13, 547–556 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kato T.: Perturbation Theory for Linear Operators. 2nd edn. Springer- Verlag, Heidelberg (1980)

    MATH  Google Scholar 

  16. Liu J.: Spectral Theory of Ordinary Differential Operators. Science Publishing, Beijing (In Chinese) (2009)

  17. Naimark M.A.: Linear Differential Operators II. Ungar, New York (1968)

    MATH  Google Scholar 

  18. Reed M., Simon B.: Methods of Modern Mathematical Physics, vol. 1. Academic Press, New York (1972)

    Google Scholar 

  19. Walter J.: Regular eigenvalue problems with eigenvalue parameter in the boundary condition. Mathematische Zeitschrift 133, 301–312 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang A., Sun J., Zettl A.: Characterization of domains of self-adjoint ordinary differential operators. J. Differ. Equ. 246, 1600–1622 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Weidmann J.: Linear Operators in Hilbert Spaces, Grad. Texts in Math. Springer-Verlag, Berlin (1980)

    Book  Google Scholar 

  22. Zettl A.: Computing Continuous Spectrum. In: Alavi, Y., Hsieh, P.-F. (eds.) Proceedings of International Symposium, Trends and Developments in Ordinary Differential Equations, World Scientific, pp. 393–406 (1994)

  23. Zettl A.: Sturm-Liouville Theory, Mathematical Surveys and Monographs, vol. 121. American Mathematical Society, Providence (2005)

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Authors and Affiliations

  1. School of Mathematics Sciences, Inner Mongolia University, Hohhot, 010021, China

    Zhang Maozhu & Jiong Sun

  2. Mathematics Department, Northern Illinois University, DeKalb, IL, 60115, USA

    Anton Zettl

  3. College of Mathematics and System Science, Taishan University, Taian, 271021, China

    Zhang Maozhu

Authors
  1. Zhang Maozhu
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  2. Jiong Sun
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  3. Anton Zettl
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Correspondence to Anton Zettl.

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Maozhu, Z., Sun, J. & Zettl, A. The Spectrum of Singular Sturm-Liouville Problems With Eigenparameter Dependent Boundary Conditions and Its Approximation. Results. Math. 63, 1311–1330 (2013). https://doi.org/10.1007/s00025-012-0270-x

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  • DOI: https://doi.org/10.1007/s00025-012-0270-x

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