Geodesic
Probabilistic Approximation of the Evolution Operator
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 2, pp. 25-39 Cet article a éte moissonné depuis la source Math-Net.Ru

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A method for approximation of the operator $e^{-itH}$, where $H=-\frac{1}{2}\frac{d^2}{dx^2}+V(x)$, in the strong operator topology is proposed. The approximating operators have the form of expectations of functionals of a certain random point field.
Mots-clés : evolution equation, Feynman–Kac formula.
Keywords: limit theorem 
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I. A. Ibragimov; N. V. Smorodina; M. M. Faddeev. Probabilistic Approximation of the Evolution Operator. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 2, pp. 25-39. http://geodesic.mathdoc.fr/item/FAA_2018_52_2_a2/

[1] A. D. Venttsel, Kurs teorii sluchainykh protsessov, 2-e izd., dop., Fizmatlit, M., 1996 | MR

[2] A. M. Vershik, N. V. Smorodina, “Nesingulyarnye preobrazovaniya simmetrichnykh ustoichivykh protsessov Levi”, Zap. nauchn. semin. POMI, 408 (2012), 102–114

[3] N. Danford, Dzh. Shvarts, Lineinye operatory. Obschaya teoriya, IL, M., 1962 | MR

[4] H. Doss, “Sur une résolution stochastique de l'equation de Schrödinger à coefficients analytiques”, Commun. Math. Phys., 73:3 (1980), 247–264 | DOI | MR | Zbl

[5] J. Glimm, A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer-Verlag, New York–Heidelberg–Berlin, 1987 | MR

[6] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Ob odnoi predelnoi teoreme, svyazannoi s veroyatnostnym predstavleniem resheniya zadachi Koshi dlya uravneniya Shrëdingera”, Zap. nauchn. semin. POMI, 454 (2016), 158–176

[7] M. M. Faddeev, I. A. Ibragimov, N. V. Smorodina, “A stochastic interpretation of the Cauchy problem solution for the equation $\frac{\partial u}{\partial t}=\frac{\sigma^2}{2}\frac{\partial^2 u}{\partial x^2}+V(x)u$ with complex $\sigma$”, Markov Processes Related Fields, 22:2 (2016), 203–226 | MR | Zbl

[8] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR

[9] Dzh. Kingman, Puassonovskie protsessy, Mir, MTsNMO, M., 2007

[10] D. Reitzner, D. Nagaj, V. Bužek, “Quantum walks”, Acta Physica Slovaca, 61:6 (2011), 603–725

[11] M. V. Platonova, “Veroyatnostnoe predstavlenie resheniya zadachi Koshi dlya evolyutsionnogo uravneniya s operatorom differentsirovaniya vysokogo poryadka”, Zap. nauchn. sem. POMI, 454 (2016), 220–237

[12] M. V. Platonova, “Veroyatnostnoe predstavlenie resheniya zadachi Koshi dlya evolyutsionnogo uravneniya s operatorom Rimana–Liuvillya”, TVP, 61:3 (2016), 417–438 | DOI | MR

[13] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 2, Mir, M., 1978 | MR

[14] A. V. Skorokhod, Sluchainye protsessy s nezavisimymi prirascheniyami, Nauka, M., 1986 | MR

[15] O. G. Smolyanov, E. T. Shavgulidze, Kontinualnye integraly, Izd. 2-e pererab. i susch. dop., LENAND, M., 2015 | MR