Skip to main content
SpringerLink
Log in

Asymptotic behavior of the solutions and nodal points of Sturm-Liouville differential expressions

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We obtain asymptotic formulas for the values of second-order differential operators with the coefficient of bounded variation depending on the spectral parameter. We give a counterexample showing that the requirement of boundedness for the variation is essential for the preservation of the error of the so-obtained asymptotic formulas.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Suetin P. K., Classical Orthogonal Polynomials [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  2. Szegö G., Orthogonal Polynomials, Amer. Math. Soc., Providence (1975).

    MATH  Google Scholar 

  3. Badkov V. M., “The asymptotic behavior of orthogonal polynomials,” Math. USSR-Sb., 37, No. 1, 39–51 (1980).

    Article  MATH  Google Scholar 

  4. Badkov V. M., “Asymptotic properties of orthogonal polynomials,” in: Constructive Function Theory. Vol. 81, Sofia, 1983, pp. 21–27.

    MathSciNet  Google Scholar 

  5. Naĭmark M. A., Linear Differential Operators. Pts. 1 and 2, Frederick Ungar Publishing Co., New York (1968, 1969).

    Google Scholar 

  6. Sansone G., Ordinary Differential Equations. Vol. 1 [Russian translation], Izdat. Inostr. Lit., Moscow (1953).

    Google Scholar 

  7. Levitan B. M. and Sargsyan I. S., Sturm-Liouville and Dirac Operators, Kluwer Acad. Publ., Dordrecht etc. (1990).

    MATH  Google Scholar 

  8. Vinokurov V. A. and Sadovnichiĭ V. A., “Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm-Liouville boundary value problem on a segment with a summable potential,” Izv. Math., 64, No. 4, 695–754 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  9. Yurko V. A., Introduction to the Theory of Inverse Spectral Problems [in Russian], Fizmatlit, Moscow (2007).

    Google Scholar 

  10. Savchuk A. M. and Shkalikov A. A., “Sturm-Liouville operators with singular potentials,” Math. Notes, 66, No. 6, 741–753 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  11. Savchuk A. M., “On the eigenvalues and eigenfunctions of the Sturm-Liouville operator with a singular potential,” Math. Notes, 69, No. 2, 245–252 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  12. Savchuk A. M. and Shkalikov A. A., “On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces,” Math. Notes, 80, No. 6, 814–832 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  13. Pokornyi Yu. V. and Pryadiev V. L., “Some problems of the qualitative Sturm-Liouville theory on a spatial network,” Russian Math. Surveys, 59, No. 3, 515–552 (2004).

    Article  MathSciNet  Google Scholar 

  14. Pokornyi Yu. V., Zvereva M. B., and Shabrov S. A., “Sturm-Liouville oscillation theory for impulsive problems,” Russian Math. Surveys, 63, No. 1, 109–153 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  15. Pokornyi Yu. V., Zvereva M. B., Ishchenko A. S., and Shabrov S. A., “An irregular extension of the oscillation theory of the Sturm-Liouville spectral problem,” Math. Notes, 82, No. 4, 518–521 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  16. Mukhtarov O. Sh. and Kadakal M., “Some spectral properties of one Sturm-Liouville type problem with discontinuous weight,” Siberian Math. J., 46, No. 4, 681–694 (2005).

    Article  MathSciNet  Google Scholar 

  17. Aigunov G. A., “The boundedness of the orthonormal eigenfunctions of a certain class of non-linear Sturm-Liouville type operators with a weight function of unbounded variation on a finite interval,” Russian Math. Surveys, 55, No. 4, 815–816 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  18. Buchaev Ya. G., “Estimates of norms of eigenfunctions of the Strum-Liouville problem in Sobolev spaces,” Math. Notes, 73, No. 4, 582–584 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  19. Guseĭnov I.M., Nabiev A. A., and Pashaev R. T., “Transformation operators and asymptotic formulas for the eigenvalues of a polynomial pencil of Sturm-Liouville operators,” Siberian Math. J., 41, No. 3, 453–464 (2000).

    Article  MathSciNet  Google Scholar 

  20. Nabiev I. M., “Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators,” Math. Notes, 67, No. 3, 309–319 (2000).

    Article  MathSciNet  Google Scholar 

  21. Zhidkov P. E., “Completeness of systems of eigenfunctions for the Sturm-Liouville operator with potential depending on the spectral parameter and for one non-linear problem,” Sb.: Math., 188, No. 7, 1071–1084 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  22. McLaughlin J. R., “Inverse spectral theory using nodal points as data-a uniqueness result,” J. Differential Equations, 73, No. 2, 354–362 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  23. Whittaker E. T. and Watson G. N., A Course of Modern Analysis, Cambridge Univ. Press, Cambridge (1996).

    MATH  Google Scholar 

  24. Shilov G. E. and Gurevich B. L., Integral, Measure, and Derivative: A Unified Approach, Dover Publications, New York (1977).

    MATH  Google Scholar 

  25. Hartman Ph., Ordinary Differential Equations, Wiley, New York (1964).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Saratov State University, Saratov, Russia

    A. Yu. Trynin

Authors
  1. A. Yu. Trynin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Yu. Trynin.

Additional information

Original Russian Text Copyright © 2010 Trynin A. Yu.

The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00060).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 3, pp. 662–675, May–June, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Trynin, A.Y. Asymptotic behavior of the solutions and nodal points of Sturm-Liouville differential expressions. Sib Math J 51, 525–536 (2010). https://doi.org/10.1007/s11202-010-0055-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-010-0055-y

Keywords

Search

Navigation