Document Type: Research Paper
Authors
- Mohammad Shahriari ^{} ^{1}
- Reza Akbari ^{2}
- Mostafa Fallahi ^{1}
^{1} Department of Mathematics, Faculty of Science, University of Maragheh, P.O. Box 55136-553, Maragheh, Iran.
^{2} Department of Mathematical Sciences, Payame Noor University, Iran.
Abstract
In this paper, we study the inverse problem for Dirac differential operators with discontinuity conditions in a compact interval. It is shown that the potential functions can be uniquely determined by the value of the potential on some interval and parts of two sets of eigenvalues. Also, it is shown that the potential function can be uniquely determined by a part of a set of values of eigenfunctions at an interior point and parts of one or two sets of eigenvalues.
Keywords
Main Subjects
[1] R.Kh. Amirov, On Sturm--Liouville operators with discontinuity conditions inside an interval, J. Math. Anal. Appl., 317 (2006), pp. 163-176.
[2] R.Kh. Amirov, On system of Dirac differential equations with discontinuity conditions inside an interval, Ukrainian Math. J., 57 (2005), pp. 712-727.
[3] G. Freiling and V.A. Yurko, Inverse Sturm--Liouville problems and their applications, NOVA Science Publishers, New Yurk, 2001.
[4] M.G. Gasymov and B.M. Levitan, The inverse problem for a Dirac system, Dokl. Akad. Nauk SSSR, 167 (1966), pp. 967-970.
[5] Y. Guo, G. Wei, and R. Yao, Inverse problem for interior spectral data of discontinuous Dirac operator, Appl. Math. Comput., 268 (2015), pp. 775-782.
[6] O. Hald, Discontinuous inverse eigenvalue problem, Commun. Pure. Appl. Math., 37 (1984), pp. 539-577.
[7] H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math., 34 (1978), pp. 676-680.
[8] M. Kobayashi, A uniqueness proof for discontinuous inverse Sturm-Liouville problems with symmetric potentials, Inverse Probl., 5 (1989), pp. 767-781.
[9] B.J. Levin, Distribution of zeros of entire functions, AMS. Transl. Vol. 5, Providence, 1964.
[10] B.Ya. Levin, Entire functions, MGU, Moscow, 1971.
[11] B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac operators, Kluwer Academic Publishers, Dodrecht, Boston, London, 1991.
[12] K. Mochizuki and I. Trooshin, Inverse problem for interior spectral data of the Dirac operator, Comm. Korean Math. Soc., 16 (2001), pp. 437-442.
[13] K. Mochizuki and I. Trooshin, Inverse problem for interior spectral data of Sturm-Liouville operator, J. Inverse Ill-posed Probl., 9 (2001), pp. 425-433.
[14] K. Mochizuki and I. Trooshin, Inverse problem for interior spectral data of the Dirac operator on a finite interval, Publ. RIMS, KyotoUniv., 38 (2002), pp. 387-395.
[15] M. Shahriari, A.J. Akbarfam, and G. Teschl, Uniqueness for inverse Sturm-Liouville problems with a finite number of transmission conditions, J. Math. Anal. Appl., 395 (2012), pp. 19-29.
[16] Z. Wei, Y. Guo, and G. Wei, Incomplete inverse spectral and nodal problems for Dirac operator, Adv. Difference Equ., 2015 (2015), 188.
[17] C. Willis, Inverse Sturm-Liouville problems with two discontinuities, Inverse Probl., 1 (1985), pp. 263-289.
[18] C.F. Yang, Hochstadt-Lieberman theorem for Dirac operator with eigenparameter dependent boundary conditions, Nonlinear Anal., 74 (2011), pp. 2475-2484.
[19] V. Yurko, Integral transforms connected with discontinuous boundary value problems, Int. Trans. Spec. Functions, 10 (2000), pp. 141-164.