Theoretical and Mathematical Physics

, Volume 102, Issue 2, pp 133–143 | Cite as

Commutative properties of singularly perturbed operators

  • N. E. Dudkin
  • V. D. Koshmanenko
Article

Abstract

Suppose the self-adjoint operatorA in the Hilbert spaceH commutes with the bounded operatorS. Suppose another self-adjoint operatorĀ is singularly perturbed with respect toA, i.e., it is identical toA on a certain dense set inH. We study the following question: Under what conditions doesĀ also commute withS? In addition, we consider the case whenS is unbounded and also the case whenS is replaced by a singularly perturbed operator S. As application, we consider the Laplacian inL2(Rq) that is singularly perturbed by a set of δ functions and commutes with the symmetrization operator inRq,q=2, 3, or with regular representations of arbitrary isometric transformations inRq,q≤3.

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References

  1. 1.
    S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden,Solvable Models in Quantum Mechanics, Springer-Verlag, New York (1988).Google Scholar
  2. 2.
    S. Albeverio, W. Karwowski, and V. D. Koshmanenko “Square power of singularly perturbed operators,” SFT 237 Preprint No. 176, Ruhr-Univ.-Bochum (1992) (to appear inMath. Nachr.).Google Scholar
  3. 3.
    A. Alonso and B. Simon,J. Oper. Theory, No. 4, 251 (1980).Google Scholar
  4. 4.
    N. I. Akhiezer and I. M. Glazman,The Theory of Linear Operators in Hilbert Space, Pitman, Boston, Mass. (1981).Google Scholar
  5. 5.
    Yu. M. Berezanskii,Self-Adjoint Operators in Spaces of Functions of Infinitely Many Variables [in Russian], Naukova Dumka, Kiev (1978).Google Scholar
  6. 6.
    Yu. E. Bokhonov, Ukr. Mat. Zh.,42, 695 (1990).Google Scholar
  7. 7.
    P. E. T. Jørgensen,Am. J. Math.,103, 273 (1980).Google Scholar
  8. 8.
    V. Koshmanenko,Singular Bilinear Forms in the Theory of Perturbations of Self-Adjoint Operators [in Russian], Naukova Dumka. Kiev (1993).Google Scholar
  9. 9.
    V. Koshmanenko, “Dense subspace inA-scale of Hilbert spaces,” ITP UWr Preprint No. 835 (1993).Google Scholar
  10. 10.
    A. N. KochubeiFunktsional. Analiz i Ego Prilozhen,13, 77 (1979).Google Scholar
  11. 11.
    S. Ôta,Proc. Am. Math. Soc.,117, No. 4, 8 (1993).Google Scholar
  12. 12.
    S. Ôta,Proc. Am. Math. Soc.,118, No. 2, 5 (1993).Google Scholar
  13. 13.
    K. Schmüdgen,Unbounded Operator Algebras and Representations Theory, Akademie-Verlag, Berlin (1988).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • N. E. Dudkin
  • V. D. Koshmanenko

There are no affiliations available

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