Ukrainian Mathematical Journal

, Volume 43, Issue 11, pp 1450–1457 | Cite as

Toward the rank-one singular perturbation theory of self-adjoint operators

  • Y. D. Koshmanenko


The perturbation theory is developed for the case where an arbitrary positive self-adjoint operator is perturbed by the projector on a generalized vector. Similar to the well-known problem −Δ+λδ, we obtain, in the general case, explicit representations for singularly perturbed operators and their resolvents, and we find the point spectrum and an explicit form of the corresponding eigenvectors. Our approach differs from the usual ones and is based on the self-adjoint extension theory of semibounded operators.


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Literature cited

  1. 1.
    S. Albeverio, F. Gesztezy, R. Hoegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer Verlag, Berlin (1988).Google Scholar
  2. 2.
    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
  3. 3.
    Yu. M. Berezanskij, Expansion on the Eigenfunctions of Self-Adjoint Operators, American Mathematical Society, Providence, Rhode Island (1968).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Y. D. Koshmanenko
    • 1
  1. 1.Institute of MathematicsAcademy of Sciences of the UkraineKiev

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