Abstract and Applied Analysis

The Regularized Trace Formula of the Spectrum of a Dirichlet Boundary Value Problem with Turning Point

Zaki F. A. El-Raheem and A. H. Nasser

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Abstract

We calculate the regularized trace formula of the infinite sequence of eigenvalues for some version of a Dirichlet boundary value problem with turning points.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 492576, 12 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475869

Digital Object Identifier
doi:10.1155/2012/492576

Mathematical Reviews number (MathSciNet)
MR2984035

Zentralblatt MATH identifier
06116411

Citation

El-Raheem, Zaki F. A.; Nasser, A. H. The Regularized Trace Formula of the Spectrum of a Dirichlet Boundary Value Problem with Turning Point. Abstr. Appl. Anal. 2012 (2012), Article ID 492576, 12 pages. doi:10.1155/2012/492576. https://projecteuclid.org/euclid.aaa/1364475869


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References

  • 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 $M$-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.