Ukrainian Mathematical Journal

, Volume 58, Issue 12, pp 1876–1890 | Cite as

Jacobi matrices associated with the inverse eigenvalue problem in the theory of singular perturbations of self-adjoint operators

  • V. D. Koshmanenko
  • H. V. Tuhai
Article

Abstract

We establish the relationship between the inverse eigenvalue problem and Jacobi matrices within the framework of the theory of singular perturbations of unbounded self-adjoint operators.

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References

  1. 1.
    S. Albeverio and V. Koshmanenko, “Singular rank one perturbations of self-adjoint operators and Krein theory of self-adjoint extensions,” Potent. Anal., 11, 279–287 (1999).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators and Solvable Schrödinger Type Operators, Cambridge University Press, Cambridge (2000).Google Scholar
  3. 3.
    V. D. Koshmanenko, “Towards the rank-one singular perturbations of self-adjoint extensions,” Ukr. Mat. Zh., 43, No. 11, 1559–1566 (1991).MathSciNetGoogle Scholar
  4. 4.
    V. D. Koshmanenko, Singular Bilinear Forms in Perturbation Theory of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1993).Google Scholar
  5. 5.
    V. Koshmanenko, “A variant of the inverse negative eigenvalue problem in singular perturbation theory,” Meth. Funct. Anal. Topol., 8, No. 1, 49–69 (2002).MATHMathSciNetGoogle Scholar
  6. 6.
    M. E. Dudkin and V. D. Koshmanenko, “On the point spectrum of self-adjoint operators that appears under singular perturbations of finite rank,” Ukr. Mat. Zh., 55, No. 9, 1269–1276 (2003).MATHMathSciNetGoogle Scholar
  7. 7.
    V. D. Koshmanenko and H. V. Tuhai, “On the structure of the resolvent of a singularly perturbed operator that solves an eigenvalue problem,” Ukr. Mat. Zh., 56, No. 9, 1292–1297 (2004).MATHCrossRefGoogle Scholar
  8. 8.
    S. Albeverio, A. Konstantinov, and V. Koshmanenko, “On inverse spectral theory for singularly perturbed operator: point spectrum,” Inverse Probl., 21, 1871–1878 (2005).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    S. Albeverio, M. Dudkin, A. Konstantinov, and V. Koshmanenko, “On the point spectrum of H −2 singular perturbations,” Math. Nachr., 280, No. 1–2, 1–8 (2005).MathSciNetGoogle Scholar
  10. 10.
    L. Nizhnik, “The singular rank-one perturbations of self-adjoint operators” Meth. Funct. Anal. Topol., 7, No. 3, 54–66 (2001).MATHMathSciNetGoogle Scholar
  11. 11.
    Yu. M. Berezanskii, Expansion of Self-Adjoint Operators in Eigenfunctions [in Russian], Naukova Dumka, Kiev (1965).Google Scholar
  12. 12.
    V. Koshmanenko, “Singular operator as a parameter of self-adjoint extensions,” in: Proceedings of the Mark Krein International Conference on Operator Theory and Applications (Odessa, August 18–22, 1997), Birkhäuser, Basel (2000), pp. 205–223.Google Scholar
  13. 13.
    A. Posilicano, “A Krein-like formula for singular perturbations of self-adjoint operators and applications,” J. Funct. Anal., 183, 109–147 (2001).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    T. V. Karataeva and V. D. Koshmanenko, “A generalized sum of operators,” Mat. Zametki, 66, No. 5, 671–681 (1999).MathSciNetGoogle Scholar
  15. 15.
    T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1966).MATHGoogle Scholar
  16. 16.
    M. G. Krein, “Theory of self-adjoint extensions of semibounded Hermitian operators and its applications. I,” Mat. Sb., 20(62), No. 3, 431–495 (1947).MathSciNetGoogle Scholar
  17. 17.
    F. Gesztesy and B. Simon, “Rank-one perturbations at infinite coupling,” J. Funct. Anal., 128, 245–252 (1995).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    V. D. Koshmanenko, “Singular perturbations with infinite coupling constant,” Funkts. Anal. Prilozhen., 33, No. 2, 81–84 (1999).MathSciNetGoogle Scholar
  19. 19.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1. Functional Analysis, Academic Press, New York (1972).Google Scholar
  20. 20.
    B. Simon, “The classical moment problem as a self-adjoint finite difference operator,” Adv. Math., 137, 82–203 (1998).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Yu. M. Berezansky, “Some generalizations of the classical moment problem,” Integral Equat. Oper. Theory, 44, 255–289 (2002).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. D. Koshmanenko
    • 1
  • H. V. Tuhai
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKyiv

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