Skip to main content
Log in

Regularized approximations of singular perturbations from the h -2-class

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

For a sequence of singular perturbations belonging to the H -1-class and converging to a given singular perturbation from the H -2-class, we find a method of additive regularization that guarantees the strong resolvent convergence of perturbed operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. V. D. Koshmanenko, Singular Quadratic Forms in Perturbation Theory, Kluwer, Dordrecht 1999.

    MATH  Google Scholar 

  2. S. Albeverio, W. Karwowski, and V. Koshmanenko, “Square power of singularly perturbed operators,” Math. Nachr., 173, 5–24 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Albeverio and V. Koshmanenko, “Singular rank one perturbations of self-adjoint extensions,” Potential Anal., 11, 279–287 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Albeverio and V. Koshmanenko, “On form-sum approximations of singularly perturbed positive self-adjoint operators,” J. Fund. Anal., 169, 32–51 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Albeverio and V. Koshmanenko, “On the problem of the right Hamiltonian under singular form-sum perturbations,” Rev. Math. Phys., 12, No. 1 (2000).

    Google Scholar 

  6. S. Albeverio, V. Koshmanenko, and K. A. Makarov, “Generalized eigenfunctions under singular perturbations,” Meth. Funct. Anal. Topol., 5, No. 1, 13–27 (1999).

    MATH  MathSciNet  Google Scholar 

  7. S. Albeverio, J. F. Brasche, and V. D. Koshmanenko, “Lippmann-Schwinger equation for singularly perturbed operators,” Meth. Funct. Anal. Topol. 3. No. 1, 1–27 (1997).

    MATH  MathSciNet  Google Scholar 

  8. J. F. Brasche, V. D. Koshmanenko, and H. Neidhardt, “New aspects of Krein’s extension theory,” Ukr. Mat. Zh., 46, No. 1, 37–54 (1994).

    MathSciNet  Google Scholar 

  9. F. Gesztesy and B. Simon, “Rank-one perturbations at infinite coupling,” J. Funct. Anal., 128, 245–252 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  10. T. V. Karataeva and V. D. Koshmanenko, “Generalized sum of operators,” Mat. Zametki, 66, No. 5, 671–681 (1999).

    MathSciNet  Google Scholar 

  11. W. Karwowski and V. D. Koshmanenko, “Determination of singular bilinear forms and singular linear operators,” Ukr. Mat. Zh., 45, No. 8, 1084–1089 (1993).

    Article  MathSciNet  Google Scholar 

  12. W. Karwowski, V. Koshmanenko, and S. ôma, “Schrodinger operator perturbed by operators related to null-sets,” Positivity, 2, 77–99 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  13. V. D. Koshmanenko, “Singular bilinear forms and self-adjoint extensions of symmetric operators,” in: Spectral Analysis of Differential Operators [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1980), pp. 37–48.

    Google Scholar 

  14. V. D. Koshmanenko, “Perturbations of self-adjoint operators by singular bilinear forms,” Ukr. Mat. Zh., 41, No. 1, 1–14 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  15. V. D. Koshmanenko, “Singularly perturbed operators,” Operator Theory Adv. Appl., 70, 347–351 (1994).

    MathSciNet  Google Scholar 

  16. V. D. Koshmanenko, “Singular perturbations with infinite coupling constant,” Funkts. Anal. Prilozh., 33, No. 2, 81–84 (1999).

    MathSciNet  Google Scholar 

  17. V. D. Koshmanenko, “Singular operator as a parameter of self-adjoint extensions,” Operator Theory Adv. Appl., 118, 205–225 (2000).

    MathSciNet  Google Scholar 

  18. V. D. Koshmanenko and O. V. Samoilenko, “Singular perturbations of finite rank. Point spectrum,” Ukr. Mat. Zh., 49, No. 9, 1186–1212 (1997).

    Article  MathSciNet  Google Scholar 

  19. V. D. Koshmanenko and S. ôma, “Characteristic properties of singular operators,” Ukr. Mat. Zh., 48, No. 11, 1484–1493 (1996).

    Article  Google Scholar 

  20. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Spaces [in Russian] Nauka, Moscow 1966.

    Google Scholar 

  21. M. G. Krein, “Theory of self-adjoint extensions of semibounded Hermitian operators and its applications. I,” Mat. Sb., 20, 431–495 (1947).

    MathSciNet  Google Scholar 

  22. Yu. M. Berezanskii, Self-Adjoint Operators in Spaces of Functions of Infinitely Many Variables [in Russian] Naukova Dumka, Kiev 1978.

    Google Scholar 

  23. Yu. M. Berezanskii, G. F. Us, and Z. G. Sheftel’, Functional Analysis [in Russian] Vyshcha Shkola, Kiev 1990.

    Google Scholar 

  24. Yu. M. Berezanskii, “Bilinear forms and Hilbert riggings,” in: Spectral Analysis of Differential Operators [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1980), pp. 83–106.

    Google Scholar 

  25. M. G. Krein and V. Ya. Yavryan, “On functions of spectral shift appearing under perturbations of a positive operator,” J. Operator Theory, 6, 155–191 (1981).

    MathSciNet  Google Scholar 

  26. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II. Fourier Analysis, Self-Adjointness, Academic Press, New York 1975.

    Google Scholar 

  27. L. P. Nizhnik, “Point interaction in quantum mechanics,” Ukr. Mat. Zh., 49, No. 11, 1557–1560 (1997).

    Article  MathSciNet  Google Scholar 

  28. S. Albeverio and L. Nizhnik, Approximation of General Zero-Range Potentials, Preprint No. 585, Bonn University, Bonn (1998).

    Google Scholar 

  29. M. Sh. Birman, “On the theory of extensions of positive-definite operators,” Mat. Sb., 38(80), 431–450 (1956).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Koshmanenko, V.D. Regularized approximations of singular perturbations from the h -2-class. Ukr Math J 52, 715–728 (2000). https://doi.org/10.1007/BF02487284

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02487284

Keywords

Navigation