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Journal of Mathematical Sciences

, Volume 244, Issue 5, pp 874–884 | Cite as

A Probabilistic Approximation of the Cauchy Problem Solution for the Schrödinger Equation with a Fractional Derivative Operator

  • M. V. PlatonovaEmail author
  • S. V. Tsykin
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We construct two types of probabilistic approximations of the Cauchy problem solution for the nonstationary Schrödinger equation with a symmetric fractional derivative of order α ∈ (1, 2) at the right-hand side. In the first case, we approximate the solution by mathematical expectation of point Poisson field functionals, and in the second case, we approximate the solution by mathematical expectation of functionals of sums of independent random variables having a power asymptotics of a tail distribution.

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References

  1. 1.
    L. M. Zelenyi and A. V. Milovanov, “Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics,” Usp. Fiz. Nauk, 174, 809–852 (2004).CrossRefGoogle Scholar
  2. 2.
    T. Kato, Perturbation Theory for Linear Operators [Russian translation], Moscow (1972).Google Scholar
  3. 3.
    J. F. C. Kingman, Poisson Processes [Russian translation], Moscow (2007).Google Scholar
  4. 4.
    I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “A limit theorem on the 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).Google Scholar
  5. 5.
    I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev, “On a limit theorem related to a probabilistic representation of the solution of the Cauchy problem for the Schrödinger equation,” Zap. Nauchn. Semin. POMI, 454 , 158–175 (2016).Google Scholar
  6. 6.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications [in Russian], Minsk (1987).Google Scholar
  7. 7.
    V. E. Tarasov, Models of Theoretical Physics with Integro-differentiation of Fractal Order [in Russian], Izhevsk (2011).Google Scholar
  8. 8.
    V. V. Uchaikin, Method of Fractional Derivatives [in Russian], Ul’yanovsk (2008).Google Scholar
  9. 9.
    D. K. Faddeev, B. Z. Vulikh, and N. N. Uraltseva, Selected Chapters of Analysis and Higher Algebra [in Russian], Leningrad Univ., Leningrad (1981).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical InstituteSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.St. Petersburg State UniversitySt. PetersburgRussia

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