Ukrainian Mathematical Journal

, Volume 55, Issue 9, pp 1532–1541 | Cite as

On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

  • M. E. Dudkin
  • V. D. Koshmanenko


We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \), Im z ≠ 0, where Bz are finite-rank operators such that dom Bz ∩ dom A = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ i } ∩ dom A = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\), we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\). We prove the uniqueness of \(\tilde A\) under the condition that rank Bz = n.


Hilbert Space Real Number Eigenvalue Problem Singular Perturbation Point Spectrum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Albeverio, W. Karwowski, and V. Koshmanenko, “Square power of singularly perturbed operators,” Math. Nachr., 173, 5-24 (1995).Google Scholar
  2. 2.
    S. Albeverio and V. Koshmanenko, “Singular rank one perturbations of self-adjoint operators and Krein theory of self-adjoint extensions,” Potential Anal., 11, 279-287 (1999).Google Scholar
  3. 3.
    S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators and Solvable Schrödinger Type Equations, Cambridge Univ. Press, Cambridge (2000).Google Scholar
  4. 4.
    W. Karwowski, V. Koshmanenko, and S. Ota, “Schrödinger operator perturbed by operators related to null-sets,” Positivity, 77, No. 2, 18-34 (1998).Google Scholar
  5. 5.
    L. Nizhnik, “The singular rank-one perturbations of self-adjoint operators,” Meth. Funct. Anal. Topology, 7, No. 3, 54-66 (2001).Google Scholar
  6. 6.
    V. D. Koshmanenko, “On rank-one singular perturbations of self-adjoint operators,” Ukr. Mat. Zh., 43, No. 11, 1559-1566 (1991).Google Scholar
  7. 7.
    V. D. Koshmanenko, Singular Bilinear Forms in the Theory of Perturbations of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1993).Google Scholar
  8. 8.
    V. D. Koshmanenko, Singular Quadratic Forms in Perturbation Theory, Kluwer, Dordrecht (1999).Google Scholar
  9. 9.
    S. Albeverio and P. Kurasov, “Rank one perturbations, approximations and self-adjoint extensions,” J. Funct. Anal., 148, 152-169 (1997).Google Scholar
  10. 10.
    F. Gesztesy and B. Simon, “Rank-one perturbations at infinite coupling,” J. Funct. Anal., 128, 245-252 (1995).Google Scholar
  11. 11.
    M. G. Krein, “Theory of self-adjoint extensions of semibounded Hermite operators and its applications. I,” Mat. Sb., 20 (62), No. 3, 431-495 (1947).Google Scholar
  12. 12.
    N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1966).Google Scholar
  13. 13.
    S. Albeverio, W. Karwowski, and V. Koshmanenko, “On negative eigenvalues of generalized Laplace operator,” Repts Math. Phys., 45, No. 2, 307-325 (2000).Google Scholar
  14. 14.
    A. Alonso and B. Simon, “The Birman - Krein - Vishik theory of self-adjoint extensions of semibounded operators,” J. Operator Theory, 4, 251-270 (1980).Google Scholar
  15. 15.
    V. D. Koshmanenko and O. V. Samoilenko, “Singular perturbations of finite rank. Point spectrum,” Ukr. Mat. Zh., 49, No. 11, 1186-1212 (1997).Google Scholar
  16. 16.
    A. A. Posilicano, “Krein-like formula for singular perturbations of self-adjoint operators and applications,” J. Funct. Anal., 183, 109-147 (2001).Google Scholar
  17. 17.
    V. A. Derkach and M. M. Malamud, “General resolvents and the boundary value problem for Hermitian operators with gaps,” J. Funct. Anal., 95, 1-95 (1991).Google Scholar
  18. 18.
    J. F. Brasche, M. Malamud, and H. Neidhardt, “Weyl function and spectral properties of self-adjoint extensions,” Integral Equat. Operator Theory, 43, 264-289 (2002).Google Scholar
  19. 19.
    V. D. Koshmanenko, “A variant of the inverse negative eigenvalues problem in singular perturbation theory,” Meth. Funct. Anal. Topology, 8, No. 1, 49-69 (2002).Google Scholar
  20. 20.
    T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).Google Scholar
  21. 21.
    A. I. Plesner, Spectral Theory of Linear Operators [in Russian], Nauka, Moscow (1965).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • M. E. Dudkin
    • 1
  • V. D. Koshmanenko
    • 2
  1. 1.Kiev Polytechnic InstituteKiev
  2. 2.Institute of MathematicsUkrainian Academy of SciencesKiev

Personalised recommendations