Abstract
We study an inverse problem with two given spectra for a second-order differential operator with singularity of the type \(\frac{2}{r} + \frac{{\ell (\ell + 1)}}{{r^2 }}\) (here, l is a positive integer or zero) at zero point. It is well known that two spectra {λ n } and {λ n } uniquely determine the potential function q(r) in the singular Sturm-Liouville equation defined on the interval (0, π]. One of the aims of the paper is to prove the generalized degeneracy of the kernel K(r, s). In particular, we obtain a new proof of the Hochstadt theorem concerning the structure of the difference \(\tilde q(r) - q(r)\).
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References
D. I. Blokhintsev, Foundations of Quantum Mechanics [in Russian], Gostekhteorizdat, Moscow (1949); English translation: Reidel, Dordrecht (1964).
V. A. Fock, Fundamentals of Quantum Mechanics [in Russian], Leningrad University, Leningrad (1932).
B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory [in Russian], Nauka, Moscow (1970).
R. Courant and D. Hilbert, Methods of Mathematical Physics, New York (1953).
M. Coz and P. Rochus, “Translation kernels for velocity dependent interaction,” J. Math. Phys., 18, No. 11, 2232–2240 (1977).
V. Y. Volk, “On inverse formulas for a differential equation with a singularity at x = 0,” Usp. Mat. Nauk, 8(56), 141–151 (1953).
R. Kh. Amirov and S. Gulyaz, Proc. Eighth Int. Colloq. Different. Equat. (Plovdiv, Bulgaria, August 18–23, 1998), pp. 17–24.
O. H. Hald, “Discontinuous inverse eigenvalue problems,” Commun. Pure Appl. Math., 37, 539–577 (1984).
B. M. Levitan, “On the determination of the Sturm-Liouville operator from one and two spectra,” Izv. Akad. Nauk SSSR, Ser. Mat., 42, No. 1, 185–199 (1978).
E. S. Panakhov, “The definition of differential operator with peculiarity in zero on two spectrum,” J. Spectral Theory Oper., 8, 177–188 (1987).
R. Carlson, “Inverse spectral theory for some singular Sturm-Liouville problems,” J. Different. Equat., 106, 121–140 (1993).
H. Hochstadt, “The inverse Sturm-Liouville problem,” Commun. Pure Appl. Math., 26, 715–729 (1973).
A. M. Krall, “Boundary values for an eigenvalue problem with a singular potential,” J. Different. Equat., 45, 128–138 (1982).
W. Rundell and P. E. Sacks, “Reconstruction of a radially symmetric potential from two spectral sequences,” J. Math. Anal. Appl., 264, 354–381 (2001).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 132–138, January, 2006.
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Panakhov, E.S., Yilmazer, R. On inverse problem for singular Sturm-Liouville operator from two spectra. Ukr Math J 58, 147–154 (2006). https://doi.org/10.1007/s11253-006-0057-x
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DOI: https://doi.org/10.1007/s11253-006-0057-x