Advertisement

On the inverse problem for Sturm–Liouville-type operators with frozen argument: rational case

  • Sergey ButerinEmail author
  • Maria Kuznetsova
Article
  • 13 Downloads

Abstract

We study the inverse problem of recovering the potential q(x) from the spectrum of the operator \(-y''(x)+q(x)y(a),\) \(y^{(\alpha )}(0)=y^{(\beta )}(1)=0,\) where \(\alpha ,\beta \in \{0,1\}\) and \(a\in [0,1]\) is an arbitrary fixed rational number. We completely describe the cases when the solution of the inverse problem is unique and non-unique. In the last case, we describe sets of iso-spectral potentials and provide various restrictions on the potential under which the uniqueness holds. Moreover, we obtain an algorithm for solving the inverse problem along with necessary and sufficient conditions for its solvability in terms of characterization of the spectrum.

Keywords

Sturm–Liouville-type operator Functional-differential operator Frozen argument Inverse spectral problem 

Mathematics Subject Classification

34A55 34K29 

Notes

Acknowledgements

This research was supported by the Ministry of Education and Science of Russian Federation (Grant 1.1660.2017/4.6).

References

  1. Albeverio S, Hryniv RO, Nizhnik LP (2007) Inverse spectral problems for non-local Sturm–Liouville operators. Inverse Probl 23:523–535MathSciNetzbMATHCrossRefGoogle Scholar
  2. Beals R, Deift P, Tomei C (1988) Direct and inverse scattering on the line, mathematica surveys and monographs, 28. AMS, ProvidencezbMATHCrossRefGoogle Scholar
  3. Bondarenko N, Yurko V (2018) An inverse problem for Sturm–Liouville differential operators with deviating argument. Appl Math Lett 83:140–144MathSciNetzbMATHCrossRefGoogle Scholar
  4. Bondarenko NP, Buterin SA, Vasiliev SV (2019) An inverse spectral problem for Sturm–Liouville operators with frozen argument. J Math Anal Appl 472(1):1028–1041MathSciNetzbMATHCrossRefGoogle Scholar
  5. Borg G (1946) Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Acta Math 78:1–96MathSciNetzbMATHCrossRefGoogle Scholar
  6. Buterin SA (2007) On an inverse spectral problem for a convolution integro-differential operator. Results Math 50(3–4):173–181MathSciNetzbMATHCrossRefGoogle Scholar
  7. Buterin SA, Vasiliev SV (2019) On recovering a Sturm–Liouville-type operator with the frozen argument rationally proportioned to the interval length. J Inverse Ill-Posed Probl 27(3):429–438MathSciNetzbMATHCrossRefGoogle Scholar
  8. Buterin SA, Yurko VA (2019) An inverse spectral problem for Sturm–Liouville operators with a large constant delay. Anal Math Phys 9(1):17–27MathSciNetzbMATHCrossRefGoogle Scholar
  9. Freiling G, Yurko VA (2001) Inverse Sturm-Liouville problems and their applications. NOVA Science Publishers, New YorkzbMATHGoogle Scholar
  10. Freiling G, Yurko VA (2012) Inverse problems for Sturm–Liouville differential operators with a constant delay. Appl Math Lett 25:1999–2004MathSciNetzbMATHCrossRefGoogle Scholar
  11. Levitan BM (1987) Inverse Sturm–Liouville problems, Nauka, Moscow, 1984; English transl. VNU Sci. Press, UtrechtGoogle Scholar
  12. Lomov IS (2014) Loaded differential operators: convergence of spectral expansions. Differ Equ 50(8):1070–1079MathSciNetzbMATHCrossRefGoogle Scholar
  13. Marchenko VA (1986) Sturm–Liouville operators and their applications, Naukova Dumka, Kiev, 1977. English transl, BirkhäuserGoogle Scholar
  14. Nakhushev AM (2012) Loaded equations and their applications. Nauka, MoscowzbMATHGoogle Scholar
  15. Nizhnik LP (2009) Inverse eigenvalue problems for nonlocal Sturm–Liouville operators. Methods Funct Anal Top 15(1):41–47MathSciNetzbMATHGoogle Scholar
  16. Nizhnik LP (2010) Inverse nonlocal Sturm–Liouville problem. Inverse Probl 26:125006MathSciNetzbMATHCrossRefGoogle Scholar
  17. Nizhnik LP (2011) Inverse spectral nonlocal problem for the first order ordinary differential equation. Tamkang J Math 42(3):385–394MathSciNetzbMATHCrossRefGoogle Scholar
  18. Vladičić V, Pikula M (2016) An inverse problem for Sturm–Liouville-type differential equation with a constant delay. Sarajevo J Math 12(24) no.1, 83–88Google Scholar
  19. Yang C-F (2014) Inverse nodal problems for the Sturm–Liouville operator with a constant delay. J Differ Equ 257(4):1288–1306MathSciNetzbMATHCrossRefGoogle Scholar
  20. Yurko VA (2002) Method of spectral mappings in the inverse problem theory. Inverse and Ill-posed Problems Series. VSP, UtrechtzbMATHCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of MathematicsSaratov State UniversitySaratovRussia

Personalised recommendations