Skip to main content
Log in

An interior inverse Sturm–Liouville problem on a time scale

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

Abstract

In this paper, a regular Sturm–Liouville operator is considered on time scale \(\mathbb {T}=[0,a_{1}]\cup [a_{2},l].\) We study an interior inverse problem for this kind operator and give a Mochizuki–Trooshin type theorem.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ambarzumyan, V.A.: Über eine Frage der Eigenwerttheorie. Z. Phys. 53, 690–695 (1929)

    Article  Google Scholar 

  2. Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math. 78, 1–96 (1946)

    Article  MathSciNet  Google Scholar 

  3. Atkinson, F.: Discrete and Continuous Boundary Problems. Academic Press, New York (1964)

    MATH  Google Scholar 

  4. Chadan, K., Colton, D., Päivärinta, L., Rundell, W.: An introduction to inverse scattering and inverse spectral problems. Society for Industrial and Applied Mathematics (1997)

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

    MATH  Google Scholar 

  6. Hochstadt, H., Lieberman, B.: An inverse Sturm-Liouville problem with mixed given data. SIAM J. Appl. Math. 34(4), 676–680 (1978)

    Article  MathSciNet  Google Scholar 

  7. Horváth, M.: Inverse spectral problems and closed exponential systems. Ann. Math. 162, 885–918 (2005)

    Article  MathSciNet  Google Scholar 

  8. Horváth, M.: On the inverse spectral theory of Schrö dinger and Dirac operators. Trans. Am. Math. Soc. 353(10), 4155–4171 (2001)

    Article  Google Scholar 

  9. Mochizuki, K., Trooshin, I.: Inverse problem for interior spectral data of Sturm-Liouville operator. J. Inverse Ill-posed Probl. 9, 425–433 (2001)

    Article  MathSciNet  Google Scholar 

  10. Mochizuki, K., Trooshin, I.: Inverse problem for interior spectral data of the Dirac operator on a finite interval. Publ. RIMS, Kyoto Univ. 38, 387–395 (2002)

    Article  MathSciNet  Google Scholar 

  11. Mochizuki, K., Trooshin, I.: Inverse problem for interior spectral data of the Dirac operator. Com. Kor. Math. Soc. 16(3), 437–444 (2001)

    MathSciNet  MATH  Google Scholar 

  12. Gesztesy, F., Simon, B.: Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum. Trans. Am. Math. Soc. 352(6), 2765–2787 (2000)

    Article  MathSciNet  Google Scholar 

  13. Guo, Y., Wei, G., Yao, R.: Inverse problem for interior spectral data of discontinuous Dirac operator. Appl. Math. Comput. 268, 775–782 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Yang, C.F., Guo, Y.X.: Determination of a differential pencil from interior spectral data. J. Math. Anal. Appl. 375(1), 284–293 (2011)

    Article  MathSciNet  Google Scholar 

  15. Hilger, S.: Analysis on measure chains: a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

    Article  MathSciNet  Google Scholar 

  16. Erbe, L., Hilger, S.: Sturmian theory on measure chains. Differ. Equ. Dyn. Syst. 1, 223–244 (1993)

    MathSciNet  MATH  Google Scholar 

  17. Erbe, L., Hilger, S.: Eigenvalue conditions and positive solutions. J. Differ. Equ. Appl. 6, 165–191 (2000)

    Article  MathSciNet  Google Scholar 

  18. Agarwal, R.P., Bohner, M., Wong, P.J.Y.: Sturm-Liouville eigenvalue problems on time scales. Appl. Math. Comput. 99, 153–166 (1999)

    MathSciNet  MATH  Google Scholar 

  19. Kong, Q.: Sturm-Liouville problems on time scales with separated boundary conditions. Results Math. 52, 111–121 (2008)

    Article  MathSciNet  Google Scholar 

  20. Kuznetsova, M.A.: A Uniqueness Theorem on Inverse Spectral Problems for the Sturm-Liouville Differential Operators on Time Scales. Results Math 75, 44 (2020)

    Article  MathSciNet  Google Scholar 

  21. Ozkan, A.S., Adalar, I.: Half-inverse Sturm-Liouville problem on a time scale. Inverse Probl. 36, 025015 (2020)

    Article  MathSciNet  Google Scholar 

  22. Ozkan, A.S.: Ambarzumyan-type theorems on a time scale. J. Inverse Ill-posed Probl. 26(5), 633–637 (2018)

    Article  MathSciNet  Google Scholar 

  23. Bohner, M., Koyunbakan, H.: Inverse problems for Sturm-Liouville difference equations. Filomat. 30(5), 1297–1304 (2016)

    Article  MathSciNet  Google Scholar 

  24. Yurko, V.A.: Inverse problems for sturm-liouville differential operators on closed sets. Tamkang J. Math. 50(3), 199–206 (2019)

    Article  MathSciNet  Google Scholar 

  25. Atıcı, F.M., Guseinov, G.: On Green’s functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math. 141(1–2), 75–99 (2002)

    Article  MathSciNet  Google Scholar 

  26. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhauser, Boston (2001)

    Book  Google Scholar 

  27. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003)

    Book  Google Scholar 

  28. Ozkan, A.S.: Sturm-Liouville operator with parameter-dependent boundary conditions on time scales. Electron. J. Differ. Equ. 212, 1–10 (2017)

    MathSciNet  Google Scholar 

  29. He, X., Volkmer, H.: Riesz bases of solutions of Sturm-Liouville equations. J. Fourier Anal. Appl. 7(3), 297–307 (2001)

    Article  MathSciNet  Google Scholar 

  30. Harutyunyan, T., Pahlevanyan, A., Srapıonyan, A.: Riesz bases generated by the spectra of Sturm-Liouville problems. Electron. J. Differ. Equ. 71, 1–8 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Bartosiewicz, Z., Piotrowska, E.: Lyapunov functions in stability of nonlinear systems on time scales. J. Differ. Equ. Appl. 17(3), 309–325 (2011)

    Article  MathSciNet  Google Scholar 

  32. Yang, C.F.: An interior inverse problem for discontinuous boundary-value problems. Integral Equ. Oper. Theory. 65, 593–604 (2009)

    Article  MathSciNet  Google Scholar 

  33. Yang, C.F.: Uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions. J. Differ. Equ. 255, 2615–2635 (2013)

    Article  MathSciNet  Google Scholar 

  34. Yang, C.F., Yang, X.P.: An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions. Appl. Math. Lett. 22(9), 1315–1319 (2009)

    Article  MathSciNet  Google Scholar 

  35. Hald, O.H.: Discontiuous inverse eigenvalue problems. Commun. Pure Appl. Math. 37, 539–577 (1984)

    Article  Google Scholar 

  36. Amirov, R.: On Sturm-Liouville operators with discontinuity conditions inside an interval. J. Math. Anal. Appl. 317, 163–176 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to İbrahim Adalar.

Ethics declarations

Conflict of interest

This work does not have any conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Adalar, İ., Ozkan, A.S. An interior inverse Sturm–Liouville problem on a time scale. Anal.Math.Phys. 10, 58 (2020). https://doi.org/10.1007/s13324-020-00402-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-020-00402-2

Keywords

Mathematics Subject Classification

Navigation