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A Uniqueness Theorem on Inverse Spectral Problems for the Sturm–Liouville Differential Operators on Time Scales

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Abstract

In the paper, Sturm–Liouville differential operators on time scales consisting of a finite number of isolated points and segments are considered. Such operators unify differential and difference operators. We obtain properties of their spectral characteristics including asymptotic formulae for eigenvalues and weight numbers. Uniqueness theorem is proved for recovering the operators from the spectral characteristics.

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References

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

    Book  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Atici, F.M., Biles, D.C., Lebedinsky, A.: An application of time scales to economics. Math. Comput. Model. 43, 718–726 (2006)

    Article  MathSciNet  Google Scholar 

  5. Prasad, K., Md, K.: Stability of positive almost periodic solutions for a fishing model with multiple time varying variable delays on time scales. Bull. Int. Math. Virtual Inst. 9, 521–533 (2019)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Rynne, B.P.: L2 spaces and boundary value problems on time-scales. J. Math. Anal. Appl. 328, 1217–1236 (2007)

    Article  MathSciNet  Google Scholar 

  8. Agarwal, R.P., Bohner, M., O’Regan, D.: Time scale boundary value problems on infinite intervals. J. Comput. Appl. Math. 141, 27–34 (2002)

    Article  MathSciNet  Google Scholar 

  9. Eckhardt, J., Teschl, G.: Sturm–Liouville operators on time scales. J. Differ. Equ. Appl. 18(11), 1875–1887 (2012)

    Article  MathSciNet  Google Scholar 

  10. Amster, P., De Nápoli, P., Pinasco, J.P.: Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals. J. Math. Anal. Appl. 343, 573–584 (2008)

    Article  MathSciNet  Google Scholar 

  11. Zhang, Y., Ma, L.: Solvability of Sturm–Liouville problems on time scales at resonance. J. Comput. Appl. Math. 233, 1785–1797 (2010)

    Article  MathSciNet  Google Scholar 

  12. Ozkan, A.S., Adalar, I.: Half-inverse Sturm–Liouville problem on a time scale. Inverse Probl. (2019). https://doi.org/10.1088/1361-6420/ab2a21

    Article  MATH  Google Scholar 

  13. Yurko, V.: Inverse problems for Sturm–Liouville differential operators on closed sets. Tamkang J. Math. 50(3), 199–206 (2019)

    Article  MathSciNet  Google Scholar 

  14. Marchenko, V.A.: Sturm–Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977. English transl, Birkhäuser (1986)

  15. Levitan, B.M.: Inverse Sturm–Liouville Problems, Nauka, Moscow, 1984; Engl. transl., VNU Sci. Press, Utrecht (1987)

  16. Freiling, G., Yurko, V.A.: Inverse Sturm–Liouville Problems and Their Applications. NOVA Science Publishers, New York (2001)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  18. Guseinov, G.S.: Determination of an infinite non-self-adjoint Jacobi matrix from its generalized spectral function. Mat. Zametki 23(2), 237–248 (1978)

    MathSciNet  Google Scholar 

  19. Guseinov, G.S., Tuncay, H.: On the inverse scattering problem for a discrete one-dimensional Schrödinger equation. Commun. Fac. Sci. Univ. Ank. Ser. A1(44), 95–102 (1995)

    MATH  Google Scholar 

  20. Yurko, V.A.: An inverse problem for operators of a triangular structure. Results Math. 30, 346–373 (1996)

    Article  MathSciNet  Google Scholar 

  21. Bohner, M., Kemaloğlu, H.: Inverse problems for Sturm–Liouville difference equations. Filomat 30, 1297–1304 (2016)

    Article  MathSciNet  Google Scholar 

  22. Ignatiev, M., Yurko, V.: Numerical methods for solving Inverse Sturm–Liouville problems. Results Math. 52, 63–74 (2008)

    Article  MathSciNet  Google Scholar 

  23. Bondarenko, N.P.: An inverse problem for Sturm–Liouville operators on trees with partial information given on the potentials. Math. Methods Appl. Sci. 42, 1512–1528 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by Grant 19-71-00009 of the Russian Science Foundation.

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

    M. A. Kuznetsova

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  1. M. A. Kuznetsova
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Kuznetsova, M.A. A Uniqueness Theorem on Inverse Spectral Problems for the Sturm–Liouville Differential Operators on Time Scales. Results Math 75, 44 (2020). https://doi.org/10.1007/s00025-020-1171-z

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