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.
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). - 2.
T. Kato,

*Perturbation Theory for Linear Operators*[Russian translation], Moscow (1972). - 3.
J. F. C. Kingman,

*Poisson Processes*[Russian translation], Moscow (2007). - 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). - 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). - 6.
S. G. Samko, A. A. Kilbas, and O. I. Marichev,

*Fractional Integrals and Derivatives. Theory and Applications*[in Russian], Minsk (1987). - 7.
V. E. Tarasov,

*Models of Theoretical Physics with Integro-differentiation of Fractal Order*[in Russian], Izhevsk (2011). - 8.
V. V. Uchaikin,

*Method of Fractional Derivatives*[in Russian], Ul’yanovsk (2008). - 9.
D. K. Faddeev, B. Z. Vulikh, and N. N. Uraltseva,

*Selected Chapters of Analysis and Higher Algebra*[in Russian], Leningrad Univ., Leningrad (1981).

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Translated from *Zapiski Nauchnykh Seminarov POMI*, Vol. 466, 2017, pp. 257–272.

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Platonova, M.V., Tsykin, S.V. A Probabilistic Approximation of the Cauchy Problem Solution for the Schrödinger Equation with a Fractional Derivative Operator.
*J Math Sci* **244, **874–884 (2020) doi:10.1007/s10958-020-04659-7

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