Abstract:
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.
I. A. Ibragimov acknowledges the financial support of the Russian Foundation for Basic Research (project no. 16-01-00258). N. V. Smorodina (who authors the results of Section 3) acknowledges the support of the Russian Science Foundation (project no. 17-11-01136). M. M. Faddeev (who authors the results of Section 4) acknowledges the support of the Russian Science Foundation (project no. 17-11-01136).
Citation:
I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Probabilistic Approximation of the Evolution Operator”, Funktsional. Anal. i Prilozhen., 52:2 (2018), 25–39; Funct. Anal. Appl., 52:2 (2018), 101–112
This publication is cited in the following 10 articles:
M. V. Platonova, “On a Probabilistic Approximation of a Group of Unitary Operators”, J Math Sci, 2024
M. V. Platonova, “Analog formuly Feinmana–Katsa dlya mnogomernogo uravneniya Shrëdingera”, Veroyatnost i statistika. 34, Posvyaschaetsya yubileyu Andreya Nikolaevicha BORODINA, Zap. nauchn. sem. POMI, 525, POMI, SPb., 2023, 96–108
I. A. Alekseev, M. V. Platonova, “Veroyatnostnaya approksimatsiya uravneniya Shrëdingera kompleksnoznachnymi sluchainymi protsessami”, Veroyatnost i statistika. 35, Posvyaschaetsya yubileyu Yany Isaevny BELOPOLSKOI, Zap. nauchn. sem. POMI, 526, POMI, SPb., 2023, 17–28
M. V. Platonova, “An analogue of the Feynman–Kac formula for a high-order operator”, Theory Probab. Appl., 67:1 (2022), 62–76
M. V. Platonova, “O veroyatnostnoi approksimatsii odnoi gruppy unitarnykh operatorov”, Veroyatnost i statistika. 32, Posvyaschaetsya yubileyu Ildara Abdullovicha IBRAGIMOVA, Zap. nauchn. sem. POMI, 510, POMI, SPb., 2022, 211–224
M. V. Platonova, S. V. Tsykin, “On One Limit Theorem Related to the Cauchy Problem Solution for the Schrödinger Equation with a Fractional Derivative Operator of Order $ \upalpha\ \upepsilon \bigcup_{m=3}^{\infty}\left(m-1,m\right) $”, J Math Sci, 258:6 (2021), 912
M. V. Platonova, S. V. Tsykin, “Probabilistic approximation of the solution of the Cauchy problem
for the higher-order Schrödinger equation”, Theory Probab. Appl., 65:4 (2021), 558–569
M. V. Platonova, S. V. Tsykin, “Probabilistic approximation of the evolution operator $e^{itH}$, where $H=\dfrac{(-1)^md^{2m}}{(2m)!dx^{2m}}$”, Dokl. Math., 101:2 (2020), 144–146
I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Approximation of the evolution operator by expectations of
functionals of sums of independent random variables”, Theory Probab. Appl., 64:1 (2019), 12–26
M. V. Platonova, S. V. Tsykin, “Ob odnoi predelnoi teoreme, svyazannoi s resheniem zadachi Koshi dlya uravneniya Shrëdingera s operatorom drobnogo differentsirovaniya poryadka $\alpha\in\bigcup\limits_{m=3}^{\infty}(m-1, m)$”, Veroyatnost i statistika. 28, Zap. nauchn. sem. POMI, 486, POMI, SPb., 2019, 254–264
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