Skip to main content
SpringerLink
Log in

Pauli–Villars Regularization for Some Models with Singular Perturbations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We show how Pauli–Villars regularization works in the construction of renormalized Hamiltonian for two exemplars of quantum systems with singular perturbations. The systems are the scalar 3-dimensional particle interacting with δ-potential and the infrared extensions of the quadratic forms of the Gaussian functional of the ground state in the quantum field theory.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. G. Krein, “The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications,” Rec. Math. (Mat. Sbornik) N.S., 20, no. 62, 431–495 (1947).

  2. S. Albeverio, F. Gesztesy, R. Hegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer, 1988.

  3. A. Kiselev and B. Simon, “Rank one perturbations with infinitesimal coupling,” J. Funct. Anal. 130, 345–356 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Albeverio and P. Kurasov, “Rank one perturbations, approximations and selfadjoint extensions,” J. Funct. Anal. 148, 152–169 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  5. F. A. Berezin and L. D. Faddeev, “A Remark on Schrödinger’s equation with a singular potential,” Sov. Math. Dokl. 2, 372 (1961). [Dokl. Akad. Nauk Ser. Fiz. 137, 1011 (1961)].

  6. S. Albeverio and P. Kurasov, Singular Perturbation of Differential Operators. Solvable Schr¨odinger Type Operators, Cambridge University Press, 2000.

  7. A. Alonso and B. Simon, “The Birman–Krein–Vishik theory of selfadjoint extensions of semibounded operators,” J. Operator Theory 4, 251–270 (1980).

    MathSciNet  MATH  Google Scholar 

  8. W. Pauli and F. Villars, “On the invariant regularization in relativistic quantum theory,” Rev. Mod. Phys. 21, 434–444 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Gesztesy and E. Tsekanovskii, “On Matrix-Valued Herglotz Functions,” arXiv:funct-an/9712004.

  10. L. D. Faddeev, “Notes on divergences and dimensional transmutation in Yang–Mills theory,” Theor. Math. Phys. 148, 986 (2006). [Teor. Mat. Fiz. 148, 133 (2006)].

  11. T. A. Bolokhov, “Infrared extensions of the quadratic form of the ground state of scalar field theory,” J. Math. Sci. 264, 244–251 (2022).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg Department of Steklov Mathematical Institute RAS, St. Petersburg, Russia

    T. A. Bolokhov

Authors
  1. T. A. Bolokhov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. A. Bolokhov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 509, 2021, pp. 54–70.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bolokhov, T.A. Pauli–Villars Regularization for Some Models with Singular Perturbations. J Math Sci 275, 259–270 (2023). https://doi.org/10.1007/s10958-023-06678-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06678-6

Search

Navigation