Letters in Mathematical Physics

, Volume 63, Issue 3, pp 219–228 | Cite as

Rank-One Singular Perturbations with a Dual Pair of Eigenvalues

  • Sergio Albeverio
  • Mykola Dudkin
  • Volodymyr Koshmanenko

Abstract

We discuss the eigen-values problem for rank one singular perturbations \(\tilde A = A\tilde + \alpha \langle \cdot ,\omega \rangle \omega \) of a self-adjoint unbounded operator A with a gap in its spectrum. We give a constructive description of operators à which possess at least two new eigenvalues, one in the resolvent set and other in the spectrum of A.

eigen-value problem Krein's formula rank one singular perturbation self-adjoint extension 

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References

  1. 1.
    Albeverio, S., Gesztesy, F., Høegh-Krohn, R. and Holden, H.: Solvable Models in Quantum Mechanics, Springer-Verlag, Berlin, 1988.Google Scholar
  2. 2.
    Albeverio, S. and Koshmanenko, V.: Form-sum approximation of singular perturbation of self-adjoint operators, J. Funct. Anal. 168(1999), 32–51.Google Scholar
  3. 3.
    Albeverio, S. and Koshmanenko, V.: Singular rank one perturbations of self-adjoint operators and Krein theory of self-adjoint extensions, Potential Anal. 11(1999), 279–287.Google Scholar
  4. 4.
    Albeverio, S., Konstantinov, A. and Koshmanenko, V.: The Aronszajn-Donoghue theory for rank one perturbations of the \({\mathcal{H}}\) -2-class, Integral Equations Operator Theory, to appear.Google Scholar
  5. 5.
    Albeverio, S. and Kurasov, P.: Rank one perturbations of not semibounded operators, Integral Equations Operator Theory 27(1997), 379–400.Google Scholar
  6. 6.
    Albeverio, S. and Kurasov, P.: Singular Perturbations of Differential Operators and Solvable Schrodinger Type Operators, Cambridge Univ. Press, 2000.Google Scholar
  7. 7.
    Albeverio, S., Koshmanenko, V., Kurasov, P. and Niznik, L.: On approximations of rank one \({\mathcal{H}}\) -2-perturbations, Proc. Amer. Math. Soc., to appear.Google Scholar
  8. 8.
    Donoghue, W.F.: On the perturbation of spectra, Comm. Pure Appl. Math. 15(1965), 559–579.Google Scholar
  9. 9.
    Dudkin, M. and Koshmanenko, V.: About point spectrum arising under nite rank one perturbations of self-adjoint operators, Ukrainian Math. J., to appear.Google Scholar
  10. 10.
    Gesztesy, F. and Simon, B.: Rank-one perturbations at in nite coupling, J. Funct. Anal. 128(1995), 245–252.Google Scholar
  11. 11.
    Karwowski, W. and Koshmanenko, V.: Generalized Laplace operator, In: F. Gesztesy et al. (eds), L2(R n), in Stochastic Processes, Physics and Geometry: New Interplays. II, Canadian Math. Soc., Conference Proc. 29 (2000), pp. 385–393.Google Scholar
  12. 12.
    Karataeva, T.V. and Koshmanenko, V.D.: Generalized sum of operators, Math. Notes 66(5)(1999), 671–681.Google Scholar
  13. 13.
    Kato, T.: Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980.Google Scholar
  14. 14.
    Kiselev, A. and Simon, B.: Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130(1995), 345–356.Google Scholar
  15. 15.
    Koshmanenko, V.D.: Towards the rank-one singular perturbations of self-adjoint opera-tors, Ukrainian Math. J. 43(11)(1991), 1559–1566.Google Scholar
  16. 16.
    Koshmanenko, V.D.: Singular perturbations at infinite coupling, Funct. Anal. Appl. 33(2)(1999), 81–84.Google Scholar
  17. 17.
    Koshmanenko, V.: Singular Quadratic Forms in Perturbation Theory, Kluwer Acad. Publ., Dordrecht, 1999.Google Scholar
  18. 18.
    Koshmanenko, V.: A variant of the inverse negative eigenvalues problem in singular perturbation theory, Methods Funct. Anal. Topology 8(1)(2002), 49–69.Google Scholar
  19. 19.
    Makarov, K.A. and Pavlov, B.S.: Quantum scattering on a Cantor bar, J. Math. Phys. 35(1994), 188–207.Google Scholar
  20. 20.
    Nizhnik, L.: On rank one singular perturbations of self-adjoint operators, Methods Funct. Anal. Topology 7(3)(2001), 54–66.Google Scholar
  21. 21.
    Simon, B.: Spectral analysis of rank one perturbations and applications In: CRM Proc. Lecture Notes, 8, Amer. Math. Soc., Providence, 1995, pp. 109–149.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
    • 3
    • 4
  • Mykola Dudkin
    • 5
  • Volodymyr Koshmanenko
    • 6
  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany
  2. 2.Bielefeld, Bonn, Germany
  3. 3.IZKSBonnGermany
  4. 4.CERFIM, Locarno and Acc. Arch. (USI)Switzerland
  5. 5.National Technical Uni.KyivUkraine
  6. 6.Institute of MathematicsKyivUkraine

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