A uniqueness theorem for Bessel operator from interior spectral data. (English) Zbl 1275.47097

Sat, Murat; Panakhov, Etibar S.

Abstr. Appl. Anal. 2013, Article ID 713654, 6 p. (2013).

The singular Sturm-Liouville operator \(L\) is considered: \[ Ly = - y^{\prime \prime} + \left [ \frac{l(l+1)}{x^2} +q(x) \right ]y = \lambda y,\quad 0<x<1, \]
\[ y(0)=0, \quad y^{\prime}(1, \lambda) +Hy(1, \lambda), \] where the real valued function \(q(x) \in L_1(0,1)\), \(l \in \mathbb{N}_0\), \(H\in\mathbb{R}\). The operator \(L\) is self adjoint on \(L_2(0,1)\) and has a discrete spectrum \(\{\lambda\}\). The objective of this paper is to study the inverse problem of reconstructing the singular Sturm-Liouville operator on the basis of spectral data including the information on eigenfunctions at the internal point.

Reviewer: Denis Sidorov (Irkutsk)

MSC:

47E05	General theory of ordinary differential operators
34B24	Sturm-Liouville theory
34A55	Inverse problems involving ordinary differential equations
34B30	Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)

Keywords:

Bessel operator; singular Sturm-Liouville operator; spectral data; singular point

References:

[1]	Borg, G., Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Mathematica, 78, 1-96 (1946) · Zbl 0063.00523
[2]	Ambarzumyan, V. A., Über eine frage der eigenwerttheorie, Zeitschrift Für Physik, 53, 690-695 (1929) · JFM 55.0868.01
[3]	Levinson, N., The inverse Sturm-Liouville problem, Mathematica Tidsskrift B, 1949, 25-30 (1949) · Zbl 0045.36402
[4]	Levitan, B. M., On the determination of the Sturm-Liouville operator from one and two spectra, Mathematics of the USSR-Izvestiya, 12, 179-193 (1978) · Zbl 0401.34022
[5]	Hochstadt, H., The inverse Sturm-Liouville problem, Communications on Pure and Applied Mathematics, 26, 715-729 (1973) · Zbl 0281.34015
[6]	Marchenko, V. A., Certain problems of the theory of one dimensional linear di \(¤\) erential operators of the second order, Trudy Moskovskogo Matematicheskogo Obshchestva, 1, 327-340 (1952)
[7]	Carlson, R., A Borg-Levinson theorem for Bessel operators, Pacific Journal of Mathematics, 177, 1, 1-26 (1997) · Zbl 0868.34061
[8]	Stashevskaya, V. V., On inverse problems of spectral analysis for a certain class of differential equations, Doklady Akademii Nauk SSSR, 93, 409-412 (1953)
[9]	Albeverio, S.; Hryniv, R.; Mykytyuk, Ya., Inverse spectral problems for coupled oscillating systems: reconstruction from three spectra, Methods of Functional Analysis and Topology, 13, 2, 110-123 (2007) · Zbl 1150.34314
[10]	Freiling, G.; Yurko, V., Inverse problems for differential operators with singular boundary conditions, Mathematische Nachrichten, 278, 12-13, 1561-1578 (2005) · Zbl 1090.34007
[11]	Yurko, V. A., On the reconstruction of Sturm-Liouville differential operators with singularities inside the interval, Matematicheskie Zametki, 64, 1, 143-156 (1998) · Zbl 0922.34022
[12]	Yurko, V. A., An inverse problem for differential equations with a singularity, Differentsial’nye Uravneniya, 28, 8, 1355-1362 (1992) · Zbl 0799.34015
[13]	Yurko, V. A., On higher-order differential operators with a singular point, Inverse Problems, 9, 4, 495-502 (1993) · Zbl 0786.34027
[14]	Gesztesy, F.; Simon, B., Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum, Transactions of the American Mathematical Society, 352, 6, 2765-2787 (2000) · Zbl 0948.34060
[15]	Panakhov, E. S.; Sat, M., On the determination of the singular Sturm-Liouville operator from two spectra, CMES. Computer Modeling in Engineering & Sciences, 84, 1, 1-11 (2012) · Zbl 1356.34027
[16]	Mochizuki, K.; Trooshin, I., Inverse problem for interior spectral data of the Sturm-Liouville operator, Journal of Inverse and Ill-Posed Problems, 9, 4, 425-433 (2001) · Zbl 1035.34008
[17]	Yang, C.-F.; Yang, X.-P., An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions, Applied Mathematics Letters, 22, 9, 1315-1319 (2009) · Zbl 1173.34306
[18]	Watson, G. A., Treatise on the Theory of the Bessel Functions (1962), Cambridge, UK: Cambridge University Press, Cambridge, UK
[19]	Koyunbakan, H., Inverse spectral problem for some singular differential operators, Tamsui Oxford Journal of Mathematical Sciences, 25, 3, 277-283 (2009) · Zbl 1202.34033
[20]	Rundell, W.; Sacks, P. E., Reconstruction of a radially symmetric potential from two spectral sequences, Journal of Mathematical Analysis and Applications, 264, 2, 354-381 (2001) · Zbl 1011.35129
[21]	Volk, V. Ya., On inversion formulas for a differential equation with a singularity at \(x = 0\), Uspekhi Matematicheskikh Nauk, 8, 4, 141-151 (1953) · Zbl 0053.24501
[22]	Levin, B. Ja., Distribution of Zeros of Entire Functions, 5, viii+493 (1964), Providence, RI, USA: American Mathematical Society, Providence, RI, USA