Skip to main content
SpringerLink
Log in

Recovering singular Sturm-Liouville differential pencils from spectral data

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

Non-self-adjoint Sturm-Liouville differential operators on the half-line with a boundary condition depending polynomially on the spectral parameter are studied. We establish properties of the spectral characteristics and investigate the inverse problem of recovering the operator from the spectral data. For this inverse problem we prove the uniqueness theorem and provide a procedure for constructing the solution by the method of spectral mappings.

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. Collatz L.: Eigenwertaufgaben mit technischen Anwendungen. Akad. Verlagsgesellschaft Geest & Portig, Leipzig (1963)

    Google Scholar 

  2. Mennicken, R., Möller, M.: Non-self-adjoint boundary value problems. In: North-Holland Mathematic Studies, vol. 192. North-Holland, Amsterdam (2003)

  3. Shkalikov, A.A.: Boundary problems for opdinary problems for differential equations with parameter in the boundary conditions. J. Sov. Math. 33, 1311–1342 (1986) (translation from Tr. Semin. im. I.G. Petrovskogo 9, 190–229 (1983))

  4. Tretter Ch.: Boundary eigenvalue problems with differential equations Nη = λ Pη with λ—polynomial boundary conditions. J. Differ. Equ. 170, 408–471 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Marchenko, V.A.: Sturm-Liouville operators and their applications. Naukova Dumka, Kiev (1977) (English transl., Birkhäuser, 1986)

  6. Levitan, B.M.: Inverse Sturm-Liouville Problems. Nauka, Moscow (1984) (Russian) (English transl., VNU Science Press, Utrecht, 1987)

  7. Pöschel J., Trubowitz E.: Inverse Spectral Theory. Academic Press, New York (1987)

    MATH  Google Scholar 

  8. Freiling G., Yurko V.A.: Inverse Sturm-Liouville Problems and their Applications. NOVA Science Publishers, New York (2001)

    MATH  Google Scholar 

  9. Yurko V.A.: Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series. VSP, Utrecht (2002)

    Google Scholar 

  10. Ramm, A.G.: Inverse problems. In: Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York (2005)

  11. Browne P.J., Sleeman B.D.: A uniqueness theorem for inverse eigenparameter dependent Sturm-Liouville problems. Inverse Problems 13(6), 1453–1462 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chugunova M.V.: Inverse spectral problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions. Oper. Theory Adv. Appl. 123, 187–194 (2001)

    MathSciNet  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. Guliyev N.J.: Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary condition. Inverse Problems 21(4), 1315–1330 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Freiling, G., Yurko, V.A.: Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter. Inverse Problems 26(5), p. 17 (2010)

    Google Scholar 

  17. Yurko, V.A.: Recovering non-self-adjoint Sturm-Liouville pencils on the half-line. In: Schriftenreiche des Fachbereichs Mathematik, SM-DU-726 Universitaet Duisburg-Essen, p. 10 (2011)

  18. Coddington E., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Saratov University, Astrakhanskaya 83, Saratov, 410012, Russia

    Vjacheslav A. Yurko

Authors
  1. Vjacheslav A. Yurko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vjacheslav A. Yurko.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yurko, V.A. Recovering singular Sturm-Liouville differential pencils from spectral data. Anal.Math.Phys. 1, 47–67 (2011). https://doi.org/10.1007/s13324-011-0004-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-011-0004-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation