Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2010, Article ID 982749, 10 pages
http://dx.doi.org/10.1155/2010/982749
Research Article

Non-Self-Adjoint Singular Sturm-Liouville Problems with Boundary Conditions Dependent on the Eigenparameter

1Department of Mathematics, Science Faculty, Ankara University, 06100 Ankara, Turkey
2Department of Mathematics, Science and Art Faculty, Usak University, 64200 Campus-Uşak, Turkey

Received 6 December 2009; Accepted 16 February 2010

Academic Editor: Ağacik Zafer

Copyright © 2010 Elgiz Bairamov and M. Seyyit Seyyidoglu. 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.

Linked References

  1. M. A. Naĭmark, “Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis,” American Mathematical Society Translations, vol. 16, pp. 103–193, 1960. View at Google Scholar · View at MathSciNet
  2. R. R. D. Kemp, “A singular boundary value problem for a non-self-adjoint differential operator,” Canadian Journal of Mathematics, vol. 10, pp. 447–462, 1958. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. G. Gasymov, “On the decomposition in a series of eigenfunctions for a non-selfadjoint boundary value problem of the solution of a differential equation with a singularity at zero point,” Soviet Mathematics. Doklady, vol. 6, pp. 1426–1429, 1965. View at Google Scholar
  4. B. S. Pavlov, “On seperation conditions for spectral components of a dissipative operator,” Mathematics of the USSR-Izvestiya, vol. 9, pp. 113–137, 1975. View at Google Scholar
  5. V. E. Lyance, “A differential operator with spectral singularities—I,” American Mathematical Society Translations, vol. 2, no. 60, pp. 185–225, 1967. View at Google Scholar
  6. V. E. Lyance, “A differential operator with spectral singularities—II,” American Mathematical Society Translations, vol. 2, no. 60, pp. 227–283, 1967. View at Google Scholar
  7. A. M. Krall, “The adjoint of a differential operator with integral boundary conditions,” Proceedings of the American Mathematical Society, vol. 16, pp. 738–742, 1965. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. M. Krall, “A nonhomogeneous eigenfunction expansion,” Transactions of the American Mathematical Society, vol. 117, pp. 352–361, 1965. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. M. Krall, “Second order ordinary differential operators with general boundary conditions,” Duke Mathematical Journal, vol. 32, pp. 617–625, 1965. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. E. Bairamov, Ö. Çakar, and A. O. Çelebi, “Quadratic pencil of Schrödinger operators with spectral singularities: discrete spectrum and principal functions,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 303–320, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. E. Bairamov, Ö. Çakar, and A. M. Krall, “An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities,” Journal of Differential Equations, vol. 151, no. 2, pp. 268–289, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. E. Bairamov, Ö. Çakar, and A. M. Krall, “Non-selfadjoint difference operators and Jacobi matrices with spectral singularities,” Mathematische Nachrichten, vol. 229, pp. 5–14, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. Bairamov and A. O. Çelebi, “Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators,” The Quarterly Journal of Mathematics, vol. 50, no. 200, pp. 371–384, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. E. Kir, “Spectrum and principal functions of the non-self-adjoint Sturm-Liouville operators with a singular potential,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1247–1255, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. A. M. Krall, E. Bairamov, and Ö. Çakar, “Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition,” Journal of Differential Equations, vol. 151, no. 2, pp. 252–267, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. M. Krall, E. Bairamov, and Ö. Çakar, “Spectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities,” Mathematische Nachrichten, vol. 231, pp. 89–104, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. M. A. Naimark, Linear Differential Operators II, Ungar, New York, NY, USA, 1968.
  18. P. A. Binding, P. J. Browne, W. J. Code, and B. A. Watson, “Transformation of Sturm-Liouville problems with decreasing affine boundary conditions,” Proceedings of the Edinburgh Mathematical Society, vol. 47, no. 3, pp. 533–552, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. P. A. Binding, P. J. Browne, and K. Seddighi, “Sturm-Liouville problems with eigenparameter dependent boundary conditions,” Proceedings of the Edinburgh Mathematical Society, vol. 37, no. 1, pp. 57–72, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. I,” Proceedings of the Edinburgh Mathematical Society, vol. 45, no. 3, pp. 631–645, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. P. A. Binding, P. J. Browne, and B. A. Watson, “Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. II,” Journal of Computational and Applied Mathematics, vol. 148, no. 1, pp. 147–168, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. P. A. Binding, P. J. Browne, and B. A. Watson, “Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter,” Journal of Mathematical Analysis and Applications, vol. 291, no. 1, pp. 246–261, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. V. A. Marchenko, Sturm-Liouville Operators and Applications, vol. 22 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1986. View at MathSciNet
  24. E. P. Dolzhenko, “Boundary-value uniqueness theorems for analytic functions,” Mathematical Notes of the Academy of Sciences of the USSR, vol. 25, no. 6, pp. 437–442, 1979. View at Publisher · View at Google Scholar · View at Scopus
  25. L. Carleson, “Sets of uniqueness for functions regular in the unit circle,” Acta Mathematica, vol. 87, pp. 325–345, 1952. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. N. F. G. Martin, “A note on metric density of sets of real numbers,” Proceedings of the American Mathematical Society, vol. 11, no. 3, pp. 344–347, 1960. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C. Goffman, “On Lebesgue's density theorem,” Proceedings of the American Mathematical Society, vol. 1, no. 3, pp. 384–388, 1950. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet