Skip to main content
SpringerLink
Log in

On a Limit Theorem Related to Probabilistic Representation of Solution to the Cauchy Problem for the Schrödinger Equation

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

A new method of probabilistic approximation of solution to the Cauchy problem for the unperturbed Schrödinger equation by expectations of functionals of a random walk is suggested. In contrast to earlier papers of the authors, the existence of exponential moment for each step of the random walk is not assumed.

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. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, [in Russian], 2th ed., Lan’ (2010).

  2. I. M. Gelfand and G. E. Shilov, Generalized Functions: Properties and Operations [in Russian], Fizmatgiz, Moscow (1958).

    Google Scholar 

  3. N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, Izd. Inostr. Lit., Moscow (1962).

  4. I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “A limit theorem on convergence of random walk functionals to a solution of the Cauchy problem for the equation \( \frac{\partial u}{\partial t}=\frac{\sigma^2}{2}\Delta u \) with complex σ,” Zap. Nauchn. Semin. POMI, 420, 88–102 (2013).

  5. I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “A complex analog of the central limit theorem and a probabilistic approximation of the Feynman integral,” Dokl. Ross. Akad. Nauk, 459, No. 3, 400–402 (2014).

    MathSciNet  MATH  Google Scholar 

  6. T. Kato, Perturbation Theory of Linear Operators [in Russian], Mir, Moscow (1972).

    Google Scholar 

  7. M. Reed and B. Simon, Methods of Modern Mathematical Physics. Vol. II [in Russian], Mir, Moscow (1978).

  8. D. K. Faddeev, B. Z. Vulikh, N. N. Uraltseva, et al., Selected Chapters of Analysis and Higher Algebra [in Russian], Izd. Leningr. Univ., Leningrad (1981).

  9. M. V. Fedoryuk, Asymptotics: Integrals and Series [in Russian], Nauka, Moscow (1987).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St.Petersburg Department of the Steklov Mathematical Institute, St. Petersburg State University, St. Petersburg, Russia

    I. A. Ibragimov & N. V. Smorodina

  2. St. Petersburg State University, St. Petersburg, Russia

    M. M. Faddeev

Authors
  1. I. A. Ibragimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. V. Smorodina
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. M. Faddeev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. A. Ibragimov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 454, 2016, pp. 158–175.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ibragimov, I.A., Smorodina, N.V. & Faddeev, M.M. On a Limit Theorem Related to Probabilistic Representation of Solution to the Cauchy Problem for the Schrödinger Equation. J Math Sci 229, 702–713 (2018). https://doi.org/10.1007/s10958-018-3709-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-018-3709-0

Search

Navigation