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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 492576, 12 pages
The Regularized Trace Formula of the Spectrum of a Dirichlet Boundary Value Problem with Turning Point
1Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21526, Egypt
2Faculty of Industrial Education, Helwan Unversity, Cairo, Egypt
Received 12 May 2012; Revised 17 September 2012; Accepted 17 September 2012
Academic Editor: Toka Diagana
Copyright © 2012 Zaki F. A. El-Raheem and A. H. Nasser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- I. M. Gelfand and B. M. Levitan, “On a simple identity for the characteristic values of a differential operator of the second order,” Doklady Akademii Nauk SSSR, vol. 88, pp. 593–596, 1953.
- L. A. Dikiĭ, “On a formula of Gelfand-Levitan,” Uspekhi Matematicheskikh Nauk, vol. 8, no. 2, pp. 119–123, 1953.
- L. A. Dikiĭ, “The zeta function of an ordinary differential equation on a finite interval,” Izvestiya Akademii Nauk SSSR, vol. 19, pp. 187–200, 1955.
- B. M. Levitan, “Calculation of the regularized trace for the Sturm-Liouville operator,” Uspekhi Matematicheskikh Nauk, vol. 19, no. 1, pp. 161–165, 1964.
- L. D. Faddeev, “On the expression for the trace of the difference of two singular differential operators of the Sturm-Liouville type,” Doklady Akademii Nauk SSSR, vol. 115, no. 5, pp. 878–881, 1957.
- M. G. Gasymov, “On the sum of the differences of the eigenvalues of two self-adjoint operators,” Doklady Akademii Nauk SSSR, vol. 150, no. 6, pp. 1202–1205, 1963.
- V. A. Sadovnichii and V. E. Podolski, “Traces of differential operators,” Differential Equations, vol. 45, no. 4, pp. 477–493, 2009.
- S. Clark and F. Gesztesy, “Weyl-Titchmarsh -function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators,” Transactions of the American Mathematical Society, vol. 354, no. 9, pp. 3475–3534, 2002.
- R. Carlson, “Large eigenvalues and trace formulas for matrix Sturm-Liouville problems,” SIAM Journal on Mathematical Analysis, vol. 30, no. 5, pp. 949–962, 1999.
- Z. F. A. El-Raheem and A. H. Nasser, “On the spectral property of a Dirichlet problem with explosive factor,” Applied Mathematics and Computation, vol. 138, no. 2-3, pp. 355–374, 2003.
- Z. F. A. El-Raheem and A. H. Nasser, “The equiconvergence of the eigenfunction expansion for a singular version of one-dimensional Schrödinger operator with explosive factor,” Boundary Value Problems, vol. 2011, article 45, 2011.
- Z. F. A. El-Raheem and A. H. Nasser, “The eigen function expansion for a Dirichlet problem with explosive factor,” Abstract and Applied Analysis, vol. 2011, Article ID 828176, 16 pages, 2011.
- Z. F. A. El-Reheem, “On some trace formula for the Sturm-Liouville operator,” Pure Mathematics and Applications, vol. 7, no. 1-2, pp. 61–68, 1996.
- V. A. Marchenko, Sturm-Liouville Operators and Applications, Revised Edition, American Mathematical Society, 2011.
- M. G. Gasymov and Z. F. Rekheem, “On the theory of inverse Sturm-Liouville problems with discontinuous sign-alternating weight,” Doklady. Akademiya Nauk Azerbaĭdzhana, vol. 48–50, no. 1–12, pp. 13–16, 1996.