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 Tr. Mat. Inst. Steklova, 2010, Volume 270, Pages 249–265 (Mi tm3027)

Time-dependent Schrödinger equation: Statistics of the distribution of Gaussian packets on a metric graph

V. L. Chernyshevab

a Moscow State University, Moscow, Russia
b Bauman Moscow State Technical University, Moscow, Russia

Abstract: We consider a time-dependent Schrödinger equation in which the spatial variable runs over a metric graph. The boundary conditions at the vertices of the graph imply the continuity of the function and the zero sum of the one-sided derivatives taken with some weights. In the semiclassical approximation, we describe a propagation of Gaussian packets on the graph that are localized at a point at the initial instant of time. The main focus is placed on the statistics of the behavior of asymptotic solutions as time increases. We show that the calculation of the number of quantum packets on a graph is related to the well-known number-theoretic problem of finding the number of integer points in an expanding simplex. We prove that the number of Gaussian packets on a finite compact graph grows polynomially. Several examples are considered. In a particular case, Gaussian packets are shown to be distributed on a graph uniformly with respect to the edge travel times.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 270, 246–262

Bibliographic databases:

UDC: 517.958+517.938

Citation: V. L. Chernyshev, “Time-dependent Schrödinger equation: Statistics of the distribution of Gaussian packets on a metric graph”, Differential equations and dynamical systems, Collected papers, Tr. Mat. Inst. Steklova, 270, MAIK Nauka/Interperiodica, Moscow, 2010, 249–265; Proc. Steklov Inst. Math., 270 (2010), 246–262

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Tolchennikov, V. L. Chernyshev, A. I. Shafarevich, “Asimptoticheskie svoistva i klassicheskie dinamicheskie sistemy v kvantovykh zadachakh na singulyarnykh prostranstvakh”, Nelineinaya dinam., 6:3 (2010), 623–638
2. A. A. Tolchennikov, V. L. Chernyshev, “Svoistva raspredeleniya gaussovykh paketov na prostranstvennoi seti”, Nauka i obrazovanie, 2011, no. 10, 1–10, MGTU im. N. E. Baumana
3. A. A. Tolchennikov, V. L. Chernyshev, “Kolichestvo tochek, dvizhuschikhsya po metricheskomu grafu: zavisimost ot perestanovki reber”, Nauka i obrazovanie, 2012, no. 12, 143–152, MGTU im. N. E. Baumana
4. Chernyshev V.L., Shafarevich A.I., “Statistics of Gaussian Packets on Metric and Decorated Graphs”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 372:2007 (2014), 20130145
5. Vsevolod L. Chernyshev, Anton A. Tolchennikov, Andrei I. Shafarevich, “Behavior of Quasi-particles on Hybrid Spaces. Relations to the Geometry of Geodesics and to the Problems of Analytic Number Theory”, Regul. Chaotic Dyn., 21:5 (2016), 531–537
6. Shafarevich A.I., “On the distribution of energy of localized solutions of the Schrödinger equation that propagate along symmetric quantum graphs”, Russ. J. Math. Phys., 23:2 (2016), 244–250
7. Chernyshev V.L., Tolchennikov A.A., “Correction to the Leading Term of Asymptotics in the Problem of Counting the Number of Points Moving on a Metric Tree”, Russ. J. Math. Phys., 24:3 (2017), 290–298
8. Allilueva A.I., Shafarevich A.I., “On the Distribution of Energy of Localized Solutions of the Schrodinger Equation That Propagate Along Symmetric Quantum Graphs”, Russ. J. Math. Phys., 24:2 (2017), 139–147
9. Vsevolod L. Chernyshev, Anton A. Tolchennikov, “The Second Term in the Asymptotics for the Number of Points Moving Along a Metric Graph”, Regul. Chaotic Dyn., 22:8 (2017), 937–948
10. Chernyshev V.L., Tolchennikov A.A., “Asymptotic Estimate For the Counting Problems Corresponding to the Dynamical System on Some Decorated Graphs”, Ergod. Theory Dyn. Syst., 38:5 (2018), 1697–1708
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