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.
Similar content being viewed by others
References
Ambarzumyan, V.A.: Über eine Frage der Eigenwerttheorie. Z. Phys. 53, 690–695 (1929)
Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math. 78, 1–96 (1946)
Atkinson, F.: Discrete and Continuous Boundary Problems. Academic Press, New York (1964)
Chadan, K., Colton, D., Päivärinta, L., Rundell, W.: An introduction to inverse scattering and inverse spectral problems. Society for Industrial and Applied Mathematics (1997)
Freiling, G., Yurko, V.A.: Inverse Sturm-Liouville Problems and their Applications. Nova Science, New York (2001)
Hochstadt, H., Lieberman, B.: An inverse Sturm-Liouville problem with mixed given data. SIAM J. Appl. Math. 34(4), 676–680 (1978)
Horváth, M.: Inverse spectral problems and closed exponential systems. Ann. Math. 162, 885–918 (2005)
Horváth, M.: On the inverse spectral theory of Schrö dinger and Dirac operators. Trans. Am. Math. Soc. 353(10), 4155–4171 (2001)
Mochizuki, K., Trooshin, I.: Inverse problem for interior spectral data of Sturm-Liouville operator. J. Inverse Ill-posed Probl. 9, 425–433 (2001)
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)
Mochizuki, K., Trooshin, I.: Inverse problem for interior spectral data of the Dirac operator. Com. Kor. Math. Soc. 16(3), 437–444 (2001)
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)
Guo, Y., Wei, G., Yao, R.: Inverse problem for interior spectral data of discontinuous Dirac operator. Appl. Math. Comput. 268, 775–782 (2015)
Yang, C.F., Guo, Y.X.: Determination of a differential pencil from interior spectral data. J. Math. Anal. Appl. 375(1), 284–293 (2011)
Hilger, S.: Analysis on measure chains: a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)
Erbe, L., Hilger, S.: Sturmian theory on measure chains. Differ. Equ. Dyn. Syst. 1, 223–244 (1993)
Erbe, L., Hilger, S.: Eigenvalue conditions and positive solutions. J. Differ. Equ. Appl. 6, 165–191 (2000)
Agarwal, R.P., Bohner, M., Wong, P.J.Y.: Sturm-Liouville eigenvalue problems on time scales. Appl. Math. Comput. 99, 153–166 (1999)
Kong, Q.: Sturm-Liouville problems on time scales with separated boundary conditions. Results Math. 52, 111–121 (2008)
Kuznetsova, M.A.: A Uniqueness Theorem on Inverse Spectral Problems for the Sturm-Liouville Differential Operators on Time Scales. Results Math 75, 44 (2020)
Ozkan, A.S., Adalar, I.: Half-inverse Sturm-Liouville problem on a time scale. Inverse Probl. 36, 025015 (2020)
Ozkan, A.S.: Ambarzumyan-type theorems on a time scale. J. Inverse Ill-posed Probl. 26(5), 633–637 (2018)
Bohner, M., Koyunbakan, H.: Inverse problems for Sturm-Liouville difference equations. Filomat. 30(5), 1297–1304 (2016)
Yurko, V.A.: Inverse problems for sturm-liouville differential operators on closed sets. Tamkang J. Math. 50(3), 199–206 (2019)
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)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhauser, Boston (2001)
Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003)
Ozkan, A.S.: Sturm-Liouville operator with parameter-dependent boundary conditions on time scales. Electron. J. Differ. Equ. 212, 1–10 (2017)
He, X., Volkmer, H.: Riesz bases of solutions of Sturm-Liouville equations. J. Fourier Anal. Appl. 7(3), 297–307 (2001)
Harutyunyan, T., Pahlevanyan, A., Srapıonyan, A.: Riesz bases generated by the spectra of Sturm-Liouville problems. Electron. J. Differ. Equ. 71, 1–8 (2013)
Bartosiewicz, Z., Piotrowska, E.: Lyapunov functions in stability of nonlinear systems on time scales. J. Differ. Equ. Appl. 17(3), 309–325 (2011)
Yang, C.F.: An interior inverse problem for discontinuous boundary-value problems. Integral Equ. Oper. Theory. 65, 593–604 (2009)
Yang, C.F.: Uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions. J. Differ. Equ. 255, 2615–2635 (2013)
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)
Hald, O.H.: Discontiuous inverse eigenvalue problems. Commun. Pure Appl. Math. 37, 539–577 (1984)
Amirov, R.: On Sturm-Liouville operators with discontinuity conditions inside an interval. J. Math. Anal. Appl. 317, 163–176 (2006)
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-020-00402-2