The Journal of Analysis

, Volume 27, Issue 4, pp 1065–1079 | Cite as

The inverse spectral problem of some singular Sturm–Liouville problems with sign-valued weights

  • Shimaa A. M. HagagEmail author
  • Zaki F. A. El-Raheem
Original Research Paper


In this paper we study the inverse spectral problem for singular Sturm–Liouville operator with sign valued weight. We define the spectral data of the problem, we construct the main integral equation. Solve the inverse spectral problem and use the inverse problem by two spectrum.


Singular Sturm–Liouville problem Spectral data The main integral equation Uniqueness theorem 

Mathematics Subject Classification

34B05 58C40 34L25 


Compliance with ethical standards

Conflict of interest

The author(s) declare(s) that there is no conflict of interest regarding the publication of this paper.


  1. 1.
    Ambarzumian, V. 1929. \(\ddot{U}\)ber eine Frage der Eigenwertheorie, Z.physik. 53:690–695.Google Scholar
  2. 2.
    Borg, G. 1946. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertfrage. Acta Mathematica 78: 1–96.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Gasymov, M.G., 1965. Determination of a Sturm-Liouville equation with asingularity by two spectra, Doki. Akad. Nauk SSSR 161:274–276. (English transl. in Sov. Math. Doki., 6, (1965), 396–399)Google Scholar
  4. 4.
    Gasymov, M.G. 1977. Direct and inverse problems of spectral analysis for a class of equations with discontinuous coefficients. In Proceedings of the international conference on non-classical methods in geophysics, 37–44. Novosibirsk (Russian)Google Scholar
  5. 5.
    Gasymov, M.G., A.S. Kakhramanov, and S.K. Petrosyan. 1987. On the spectral theory of linear differential operators with discontinuous coefficients. Doklady Akad. Nauk Azerbaidzhan. SSR 43: 13–16.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gasymov, M.G., and Z.F.A. El-Reheem. 1993/1994. On the theory of inverse SturmLiouville problems with discontinuous sign-alternating weight. Dokl. Akad. Nauk Azerbaidzhana 48/50:13-16.Google Scholar
  7. 7.
    El-Raheem, Z.F.A. 1995. On the scattering problem for the Sturm–Liouville equation on the half line with sign valued weight coefficient. Applicable Analysis 57: 333–339.MathSciNetCrossRefGoogle Scholar
  8. 8.
    El-Raheem, Z.F.A. 2015. The inverse spectral problem of some singular version of one-dimensional Schr\(\ddot{o}\)dinger operator with explosive factor in finite interval. Journal of the Eqyptian Mathematical Society 23: 271–277.Google Scholar
  9. 9.
    levitan, B.M., and G.M. Gasymov. 1964. Determination of a differential equation by two of its spectra (English translation). Russian Math. Surveys 2:1–63.Google Scholar
  10. 10.
    Amrein, O.W., M.A. Hinz, and P.D. Pearson. 2005. Sturm–Liouville theory: past and present. New York: Springer.CrossRefGoogle Scholar
  11. 11.
    Aliev, A.R., and E.H. Eyvazov. 2012. The resolvent equation of the one dimensional Schr\(\ddot{o}\)dinger operator on the whole axis. Sibirsk. Mat. Zh. 6:1201–1208. (English transl. Sib. Math. J. 6: 957–964.)Google Scholar
  12. 12.
    Efendiev, R.F., and H.D. Orudzhev. 2010. Inverse wave spectral problem with discontinuous wave speed. Mathematical Physics Analysis and Geometry 3: 255–265.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Efendiev, R.F. 2011. Spectral analysis for one class of second-order indefinite non-self-adjoint differential operator pencil. Applicable Analysis 12: 1837–1849.MathSciNetCrossRefGoogle Scholar
  14. 14.
    El-Raheem, Z.F.A., and S.A.M. Hagag. 2017. On the spectral study of singular Sturm-Liouville problem with sign valued weight. Electronic Journal of Mathematical Analysis and Applications 2: 98–115.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hagag, S.A.M., and Z.F.A. El-Raheem. 2017. The Eigenfunction Expansion of singular Sturm-Liouville problem with sign-valued weight. Journal of Contemporary Applied Mathematices 7(1).Google Scholar
  16. 16.
    El-Raheem, Z.F.A. 2015. The inverse spectral problem of some singular version of one-dimensional Schrodinger operator with explosive factor in finite interval. Journal of the Egyptian Mathematical Socitey 23: 271–277.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Marchenko, V.A. 1986. Sturm–Liouville operators and their applications. Naukova Dumka Kieb, (1977), English translate Birkhauser.Google Scholar
  18. 18.
    Bainov. D. 1999. Inverse spectral problem for differential equation of the second order with singularity. 9th Int. Colloquium on Differential Equations 7–14.Google Scholar

Copyright information

© Forum D'Analystes, Chennai 2018

Authors and Affiliations

  1. 1.Faculty of EducationAlexandria UniversityAlexandriaEgypt

Personalised recommendations