Abstract
In this paper, a class of inverse Sturm–Liouville problems of Atkinson type with transmission conditions is investigated. Based on the method of well-known theory for inverse matrix eigenvalue problems and the spectral properties of Sturm–Liouville problems with transmission conditions of Atkinson type, the corresponding Sturm–Liouville problems can be constructed using two sets of interlacing real numbers and some other conditions. We solve the problems not only under the separated boundary conditions, but also under the real coupled boundary conditions and the general real self-adjoint transmission conditions.
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References
Akdoğan, Z., Demirci, M., Mukhtarov, O.S.: Green function of discontinuous boundary-value problem with transmission conditions. Math. Methods Appl. Sci. 30, 1719–1738 (2007)
Ambarzumyan, V.A.: Über eine frage der eigenwerttheorie. Z. Phys. 53, 690–695 (1929)
Ao, J.J., Sun, J., Zhang, M.Z.: The finite specturm of Sturm-Liouville problems with transmission conditions. Appl. Math. Comput. 218, 1166–1173 (2011)
Ao, J.J., Sun, J., Zhang, M.Z.: The matrix representations of Sturm–Liouville problems with transmission conditions. Comput. Math. Appl. 63, 1335–1348 (2012)
Atkinson, F.V.: Discrete and Continuous Boundary Problems. Academic Press, New York (1964)
Aydemir, K., Mukhtarov, O.S.: Completeness of one two-interval boundary value problem with transmission conditions. Miskolc Math. Notes 15(2), 293–303 (2014)
Bebinao, N., Providência, J.: Inverse problems for pseudo-Jacobi matrices: existence and uniqueness results. Inverse Probl. 27, 025005 (2011)
Borg, G.: Eine umkehrung der Sturm–Liouvilleschen eigenwertaufgabe. Acta Math. 78, 1–96 (1946)
Cai, J., Zheng, Z.: Inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type. Appl. Math. Comput. 327, 22–34 (2018)
Feller, W.: The general diffusion operator and positivity preserving semi-groups in one dimension. Ann. Math. 60, 417–436 (1954)
Ferguson, W.E.: The construction of Jacobi and periodic Jacobi matrices with prescribed spectrum. Math. Comp. 35, 1203–1220 (1980)
Freiling, G., Yurko, V.A.: Inverse Sturm–Liouville Problems and Their Applications. NOVA Science Publishers, New York (2001)
Fu, S.Z., Xu, Z.B., Wei, G.S.: The interlacing of spectra between continuous and discontinuous Sturm-Liouville problems and its application to inverse problems. Taiwan. J. Math. 16, 651–663 (2012)
Hochstadt, H.: On some inverse problems in matrix theory. Arch. Math. 18, 201–207 (1967)
Kadakal, M., Mukhtarov, O.S.: Sturm–Liouville problems with discontinuities at two points. Comput. Math. Appl. 54, 1367–1379 (2007)
Kong, Q., Volkmer, H., Zettl, A.: Matrix representations of Sturm–Liouville problems with finite spectrum. Results Math. 54, 103–116 (2009)
Kong, Q., Wu, H., Zettl, A.: Sturm–Liouville problems with finite spectrum. J. Math. Anal. Appl. 263, 748–762 (2001)
Kong, Q., Zettl, A.: Inverse Sturm–Liouville problems with finite spectrum. J. Math. Anal. Appl. 386, 1–9 (2012)
Krein, M.G.: Determination of the density of a symmetrical inhomogeneous string from its spectrum of frequencies. Dokl. Akad. Nauk SSSR 76, 345–348 (1951)
Krein, M.G.: On inverse problems for an inhomogeneous string. Dokl. Akad. Nauk SSSR 82, 669–672 (1952)
Levinson, N.: The inverse Sturm–Liouville problem. Mat. Tidskr. B, 25–30 (1949)
Titeux, I., Yakubove, Y.: Completeness of root functions for thermal condition in a strip with piecewise continuous coefficients. Math. Model. Methods Appl. Sci. 7, 1035–1050 (1997)
Volkmer, H.: Eigenvalue problems of Atkinson, Feller and Krein and their mutual relationship. Electron. J. Differ. Equ. 48, 15–24 (2005)
Volkmer, H., Zettl, A.: Inverse spectral theory for Sturm–Liouville problems with finite spectrum. Proc. Am. Math. Soc. 135(4), 1129–1132 (2007)
Xu, S.F.: An Introduction to Inverse Algebraic Eigenvalue Problems. Peking University Press, Beijing (1998)
Xu, Y.H., Jiang, E.X.: An inverse eigenvalue problem for periodic Jacobi matrices. Inverse Probl. 23, 165–181 (2007)
Yang, C.F., Yang, X.P.: An interior inverse problem for the Sturm–Liouville operator with discontinuous conditions. Appl. Math. Lett. 22, 1315–1319 (2009)
Yurko, V.A.: Inverse problems for the matrix Sturm–Liouville equation on a finite interval. Inverse Probl. 22, 1139–1149 (2006)
Yurko, V.A.: An inverse problem for higher order differential operators on star-type graphs. Inverse Probl. 23, 893–903 (2007)
Yurko, V.A.: Inverse problems for second order integro-differential operators. Appl. Math. Lett. 74, 1–6 (2017)
Zhang, L., Ao, J.J.: Inverse spectral problems for Sturm-Liouville operator with coupled eigenparameter-dependent boundary conditions of Atkinson type. Inverse Probl. Sci. Eng. 27(12), 1689–1702 (2019)
Zhang, X.Y., Sun, J.: The determinants of fourth order dissipative operators with transmission conditions. J. Math. Anal. Appl. 410, 55–69 (2014)
Acknowledgements
The authors sincerely thank the referees for their helpful comments and detailed suggestions. These have significantly improved the presentation of this paper. This work was supported by National Natural Science Foundation of China (Grant Nos. 11661059, 11301259) and Natural Science Foundation of Inner Mongolia (Grant No. 2017JQ07).
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Ao, Jj., Zhang, L. An Inverse Spectral Problem of Sturm–Liouville Problems with Transmission Conditions. Mediterr. J. Math. 17, 160 (2020). https://doi.org/10.1007/s00009-020-01598-0
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DOI: https://doi.org/10.1007/s00009-020-01598-0
Keywords
- Inverse Sturm–Liouville problems
- transmission conditions
- inverse matrix eigenvalue problems
- Atkinson type
- finite spectrum