Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 2133–2150 | Cite as

On Borg’s method for non-selfadjoint Sturm–Liouville operators

  • Sergey ButerinEmail author
  • Maria Kuznetsova


We prove local solvability and stability for the inverse problem of recovering a complex-valued square integrable potential in the Sturm–Liouville equation on a finite interval from spectra of two boundary value problems with one common boundary condition. For this purpose we generalize classical Borg’s method to the case of multiple spectra.


Inverse spectral problem Non-selfadjoint Sturm–Liouville operator Borg’s method Nonlinear integral equation Stability 

Mathematics Subject Classification

34A55 34B24 47E05 



This work was supported in part by the Ministry of Education and Science of Russian Federation (Grant 1.1660.2017/4.6) and by the Russian Foundation for Basic Research (Grant 19-01-00102).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


  1. 1.
    Marchenko, V.A.: Sturm–Liouville Operators and Their Applications, Naukova Dumka, Kiev (1977) [English transl., Birkhäuser, 1986]Google Scholar
  2. 2.
    Levitan, B.M.: Inverse Sturm–Liouville Problems. Nauka, Moscow (1984) [English transl., VNU Sci. Press, Utrecht, 1987]Google Scholar
  3. 3.
    Freiling, G., Yurko, V.A.: Inverse Sturm–Liouville Problems and Their Applications. NOVA Science Publishers, New York (2001)Google Scholar
  4. 4.
    Borg, G.: Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Acta Math. 78, 1–96 (1946)Google Scholar
  5. 5.
    Karaseva, T.M.: On the inverse Sturm–Liouville problem for a non-Hermitian operator. Mat. Sb. 32(74), 477–484 (1953). (Russian)Google Scholar
  6. 6.
    Marchenko, V.A., Ostrovskii, I.V.: A characterization of the spectrum of the Hill operator. Mat. Sb. 97, 540–606 (1975) [English transl. in Math. USSR-Sb. 26 (1975), no. 4, 493–554]Google Scholar
  7. 7.
    Savchuk A.M., Shkalikov, A.A.: Inverse problems for Sturm–Liouville operators with potentials in Sobolev spaces: uniform stability. Funk. Anal. i ego Pril. 44(4), 34–53 (2010) [English transl. in Funk. Anal. Appl. 44 (2010) no. 4, 270–285]Google Scholar
  8. 8.
    Hryniv, R.O., Mykytyuk, Y.V.: Inverse spectral problems for Sturm–Liouville operators with singular potentials. Inverse Probl. 19, 665–684 (2003)Google Scholar
  9. 9.
    Yurko, V.A.: Method of Spectral Mappings in the Inverse Problem Theory. Inverse and Ill-Posed Problems Series. Amsterdam, Utrecht-VSP (2002)Google Scholar
  10. 10.
    Yurko, V.A.: An inverse problem for integro-differential operators. Mat. Zametki 50(5), 134–146 (1991) (Russian) [English transl. in Math. Notes 50 (1991), no. 5–6, 1188–1197]Google Scholar
  11. 11.
    Buterin, S.A.: On the reconstruction of a convolution perturbation of the Sturm–Liouville operator from the spectrum. Diff. Uravn. 46, 146–149 (2010) [English transl. in Diff. Eqns. 46 (2010), 150–154]Google Scholar
  12. 12.
    Bondarenko, N.P., Buterin, S.A.: On a local solvability and stability of the inverse transmission eigenvalue problem. Inverse Probl. 33(11), 115010 (2017)Google Scholar
  13. 13.
    Buterin, S.A.: On the reconstruction of a non-selfadjoint Sturm–Liouville operator. In: Matematika. Mekhanika, vol. 2, pp 10–13. Saratov Univ., Saratov (2000) (Russian)Google Scholar
  14. 14.
    Tkachenko, V.: Non-selfadjoint Sturm–Liouville operators with multiple spectra. In: Interpolation Theory, Systems Theory and Related Topics, Oper. Theory Adv. Appl., vol. 134, pp. 403–414. Birkhäuser, Basel (2002)Google Scholar
  15. 15.
    Brown, B.M., Peacock, R., Weikard, R.: A local Borg–Marchenko theorem for complex potentials. J. Comput. Appl. Math. 148, 115–131 (2002)Google Scholar
  16. 16.
    Marletta, M., Weikard, R.: Weak stability for an inverse Sturm–Liouville problem with finite spectral data and complex potential. Inverse Probl. 21, 1275–1290 (2005)Google Scholar
  17. 17.
    Buterin, S.A.: On inverse spectral problem for non-selfadjoint Sturm–Liouville operator on a finite interval. J. Math. Anal. Appl. 335(1), 739–749 (2007)Google Scholar
  18. 18.
    Albeverio, S., Hryniv, R., Mykytyuk, Y.: On spectra of non-self-adjoint Sturm–Liouville operators. Sel. Math. N. Ser. 13, 571–599 (2008)Google Scholar
  19. 19.
    Horváth, M., Kiss, M.: Stability of direct and inverse eigenvalue problems: the case of complex potential. Inverse Probl. 27, 095007 (2011)Google Scholar
  20. 20.
    Buterin, S.A., Shieh, C.-T., Yurko, V.A.: Inverse spectral problems for non-selfadjoint second-order differential operators with Dirichlet boundary conditions. Bound. Value Probl. 2013(1), 1–24 (2013). Google Scholar
  21. 21.
    Buterin, S.A.: On an inverse spectral problem for a convolution integro-differential operator. Results Math. 50(3–4), 173–181 (2007)Google Scholar

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© Springer Nature Switzerland AG 2019
corrected publication 2019

Authors and Affiliations

  1. 1.Department of MathematicsSaratov State UniversitySaratovRussia

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