Geodesic
A probabilistic approximation of the Cauchy problem solution for the Schrödinger equation with a fractional derivative operator
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 26, Tome 466 (2017), pp. 257-272 
M. V. Platonova; S. V. Tsykin. A probabilistic approximation of the Cauchy problem solution for the Schrödinger equation with a fractional derivative operator. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 26, Tome 466 (2017), pp. 257-272. http://geodesic.mathdoc.fr/item/ZNSL_2017_466_a16/
@article{ZNSL_2017_466_a16,
     author = {M. V. Platonova and S. V. Tsykin},
     title = {A probabilistic approximation of the {Cauchy} problem solution for the {Schr\"odinger} equation with a~fractional derivative operator},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {257--272},
     year = {2017},
     volume = {466},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_466_a16/}
}
TY  - JOUR
AU  - M. V. Platonova
AU  - S. V. Tsykin
TI  - A probabilistic approximation of the Cauchy problem solution for the Schrödinger equation with a fractional derivative operator
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2017
SP  - 257
EP  - 272
VL  - 466
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2017_466_a16/
LA  - ru
ID  - ZNSL_2017_466_a16
ER  - 
%0 Journal Article
%A M. V. Platonova
%A S. V. Tsykin
%T A probabilistic approximation of the Cauchy problem solution for the Schrödinger equation with a fractional derivative operator
%J Zapiski Nauchnykh Seminarov POMI
%D 2017
%P 257-272
%V 466
%U http://geodesic.mathdoc.fr/item/ZNSL_2017_466_a16/
%G ru
%F ZNSL_2017_466_a16

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We construct two types of probabilistic approximations of the Cauchy problem solution for the nonstationary Schrödinger equation with a symmetric fractional derivative of order $\alpha\in(1,2)$ on the right hand side. In the first case we approximate the solution by a mathematical expectation of point Poisson field functionals and in the second case we approximate the solution by a mathematical expectation of functionals of sums of independent random variables with a power asymptotics of a tail distribution.

[1] L. M. Zelenyi, A. V. Milovanov, “Fraktalnaya topologiya i strannaya kinetika: ot teorii perkolyatsii k problemam kosmicheskoi elektrodinamiki”, Uspekhi fizicheskikh nauk, 174:8 (2004), 809–852 | DOI

[2] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, Moskva, 1972 | MR

[3] Dzh. Kingman, Puassonovskie protsessy, MTsNMO, Moskva, 2007

[4] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Predelnaya teorema o skhodimosti funktsionalov ot sluchainogo bluzhdaniya k resheniyu zadachi Koshi dlya uravneniya $\frac{\partial u}{\partial t}=\frac{\sigma^2}2\Delta u$ c kompleksnym parametrom $\sigma$”, Zap. nauchn. semin. POMI, 420, 2013, 88–102

[5] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Ob odnoi predelnoi teoreme, svyazannoi s veroyatnostnym predstavleniem resheniya zadachi Koshi s operatorom Shredingera”, Zap. nauchn. semin. POMI, 454, 2016, 158–175 | MR

[6] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, “Nauka i tekhnika”, Minsk, 1987 | MR

[7] V. E. Tarasov, Modeli teoreticheskoi fiziki s integro-differentsirovaniem drobnogo poryadka, Izhevskii institut kompyuternykh issledovanii, 2011

[8] V. V. Uchaikin, Metod drobnyx proizvodnykh, “Artishok”, Ulyanovsk, 2008

[9] D. K. Faddeev, B. Z. Vulikh, N. N. Uraltseva, Izbrannye glavy analiza i vysshei algebry, Izd-vo Leningradskogo universiteta, 1981