A uniqueness theorem for eigenvalue problem having special potential type


Erdal Bas - Department of Mathematics, Faculty of Science, Firat University, Elazig, 23119, Turkey Etibar S. Panakhov - Department of Mathematics, Faculty of Science, Firat University, Elazig, 23119, Turkey Resat Yilmazer - Department of Mathematics, Faculty of Science, Firat University, Elazig, 23119, Turkey


In this study, a uniqueness theorem is given for Sturm-Liouville problem with special singular potential. We prove that singular potential function can be uniquely determined by the spectral set \( \left\{ \lambda _{n}\left( q_{0},h_{m}\right) \right\} _{m=1}^{+\infty }.\)



[1] V. A. Ambarzumjan, Über eine frage der eigenwerttheorie, Z. Phys., 53 (1929), 690–695.
[2] Y. Aygar, E. Bairamov, Jost solution and the spectral properties of the matrix-valued difference operators, Appl. Math. Comput., 218 (2012), 9676–9681.
[3] G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Bestimmung der Differentialgleichung durch die Eigenwerte, (German) Acta Math., 78 (1946), 1–96.
[4] N. L. Carothers, Real analysis, Cambridge University Press, Cambridge, (2000).
[5] Z. M. Gasımov, On the determination of the Singular Sturm Liouville differential equation, Academic Congress of Kravchcuk Kiev, (1992).
[6] Z. M. Gasımov, Solved inverse problems for Singular Sturm Liouville differential equation from two spectra, Ph.D. thesis, Baku State University, (1992).
[7] Z. M. Gasımov, Inverse problem with two spectra for a singular Sturm-Liouville equation, Dokl. RAN., 365 (1999), 304–305.
[8] I. M. Gel’fand, B. M. Levitan, On the determination of a differential equation by its spectral function, (Russian) Doklady Akad. Nauk SSSR (N.S.), 77 (1951), 557–560.
[9] F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential, II, The case of discrete spectrum, Trans. Amer. Math. Soc., 352 (2000), 2765–2787.
[10] O. H. Hald, Inverse eigenvalue problems for the mantle, Geophys. J. Int., 62 (1980), 41–48.
[11] H. Hochstadt, B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34 (1978), 676–680.
[12] O. R. Hryniv, Y. V. Mykytyuk, Half-inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 20 (2004), 1423–1444.
[13] B. M. Levitan, On the determination of the SturmLiouville operator from one and two spectra, Math. USSR Izv., 12 (1978), 179–193.
[14] M. M. Malamud, Questions of uniqueness in inverse problems for systems of differential equations on a finite interval, (Russian); translated from Tr. Mosk. Mat. Obs., 60 (1999), 199–258, Trans. Moscow Math. Soc., 1999 (1999), 173– 224.
[15] V. A. Marčenko, Some questions of the theory of one-dimensional linear differential operators of the second order, I, (Russian) Trudy Moskov. Mat. Obsč., 1 (1952), 327–420.
[16] J. R. McLaughlin, W. Rundell, A uniqueness theorem for an inverse Sturm-Liouville problem, J. Math. Phys., 28 (1987), 1471–1472.
[17] E. S. Panakhov, E. Bas¸, On inverse problem for singular Sturm-Liouville operator from two spectra, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 28 (2008), 85–92.
[18] E. S. Panakhov, M. Sat, Reconstruction of potential function for Sturm-Liouville operator with Coulomb potential, Bound. Value Probl., 2013 (2013), 9 pages.
[19] A. N. Tihonov, On the uniqueness of the solution of the problem of electric prospecting, (Russian) Doklady Akad. Nauk SSSR (N.S.), 69 (1949), 797–800.
[20] V. Y. Volk, On inversion formulas for a differential equation with a singularity at \(x = 0\), Uspehi Matem. Nauk (N.S.), 8 (1953), 141–151.
[21] Y.-P. Wang, A uniqueness theorem for indefinite Sturm-Liouville operators, Appl. Math. J. Chinese Univ. Ser. B, 27 (2012), 345–352.