Abstract
We study the asymptotics of the distribution function and compute the regularized trace of a boundary value problem for the operator-differential equation with the boundary value depending on a spectral parameter.
Similar content being viewed by others
References
Gorbachuk V. I. and Rybak M. A., “On boundary value problems for the operator Sturm-Liouville equation with some spectral parameter in the equation and the boundary condition,” in: Direct and Inverse Scattering Problems [in Russian], Kiev, 1981, pp. 3–13.
Rybak M. A., “Asymptotic distribution of the eigenvalues of some boundary-value problems for operator Sturm-Liouville equations,” Ukrainian Math. J., 32, No. 2, 159–162 (1980).
Aliev B. A., “Asymptotic behaviour of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation,” Ukrainian Math. J., 58, No. 8, 1298–1306 (2006).
Gel’fand I. M. and Levitan B. M., “One simple identity for the eigenvalues of a second-order differential operator,” Dokl. Akad. Nauk SSSR, 88, No. 4, 593–596 (1953).
Dikii L. A., “On a formula of Gel’fand-Levitan,” Uspekhi Mat. Nauk, 8, No. 2, 119–123 (1953).
Gasymov M. G., “The sum of differences of eigenvalues of two selfadjoint operators,” Soviet Math. Dokl., 4, 838–842 (1963).
Lidskii V. B. and Sadovnichii V. A., “Regularized sums of zeros of a class of entire functions,” Soviet Math. Dokl., 8, 1082–1085 (1967).
Maksudov F. G., Bayramoglu M., and Adiguzelov E., “On the regularized trace of the Sturm-Liouville operator with an unbounded operator coefficient on a finite interval,” Soviet Math. Dokl., 30, 169–173 (1984).
Savchuk A. M. and Shkalikov A. A., “A trace formula for Sturm-Liouville operators with singular potentials,” Math. Notes, 69, No. 3, 387–400 (2001).
Dubrovskii V. V., “Abstract formulas for regularized traces of elliptic smooth differential operators defined on compact manifolds,” Differentsial’nye Uravneniya, 27, No. 12, 2164–2166 (1991).
Sadovnichii V. A. and Podol’skii V. E., “Traces of operators with relatively compact perturbations,” Sb.: Math., 193, No. 2, 279–302 (2002).
Sadovnichii V. A. and Podolskii V. E., “Traces of operators,” Russian Math. Surveys, 61, No. 5, 885–953 (2006).
Albayrak I., Bayramoglu M., and Adiguzelov E., “The second regularized trace formula for the Sturm-Liouville problem with spectral parameter in a boundary condition,” Methods Funct. Anal. Topology, 6, No. 3, 1–8 (2000).
Lions J.-L. and Magenes E., Inhomogeneous Boundary Value Problems and Their Applications [Russian translation], Mir, Moscow (1971).
Yakubov S. and Yakubov Y., Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton (2000).
Naimark M. A., Linear Differential Operators. Pts. 1 and 2 [in Russian], Nauka, Moscow (1969).
Smirnov V. I., A Course in Higher Mathematics. Vol. 5 [in Russian], Nauka, Moscow (1959).
Watson G. N., A Treatise on the Property of Bessel Functions. Vol. 1 [Russian translation], Izdat. Inostr. Lit., Moscow (1949).
Polya G. and Szego G., Problems and Theorems in Analysis. Vol. 2 [Russian translation], Nauka, Moscow (1978).
Gradshtein I. S. and Ryzhik I. M., Tables of Integrals, Sums, Series, and Products [in Russian], Nauka, Moscow (1971).
Gorbachuk V. I. and Gorbachuk M. L., “On some classes of boundary value problems for the Sturm-Liouville equation with operator potential,” Ukrain. Mat. Zh., 24, No. 3, 291–305 (1972).
Gorbachuk V. I. and Gorbachuk M. L., Boundary-Value Problems for Operator-Differential Equations [in Russian], Naukova Dumka, Kiev (1984).
Gashimov I. F., “Calculation of the regularized trace of the operator Sturm-Liouville equation with singularity on a finite interval,” submitted to VINITI 12.12.89, No. 7340-B89.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2010 Aslanova N. M.
Baku. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 4, pp. 721–737, July–August, 2010.
Rights and permissions
About this article
Cite this article
Aslanova, N. About the spectrum and the trace formula for the operator Bessel equation. Sib Math J 51, 569–583 (2010). https://doi.org/10.1007/s11202-010-0059-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11202-010-0059-7