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Nelin. Dinam., 2010, Volume 6, Number 3, Pages 623–638 (Mi nd28)  

This article is cited in 9 scientific papers (total in 9 papers)

Asymptotic properties and classical dynamical systems in quantum problems on singular spaces

A. A. Tolchennikova, V. L. Chernyshevb, A. I. Shafarevicha

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b N. E. Bauman Moscow State Technical University

Abstract: In the first part of the article we consider a semiclassical asymptotics for a Cauchy problem for the Schrödinger operator on a metric graph. We discuss the statistical properties of the corresponding classical dynamical system: the behavior of “number of particles” at large times and distribution of “particles” on the graph. We describe the distribution of energy on infinite regular trees. In the second part we describe the asymptotics of the spectrum of the Laplace and Schrödinger operators on a thin torus and on the simplest surfaces with delta-potentials.

Keywords: dynamical systems, quantum, metric graphs, semiclassical theory, spectral properties, Schrödinger operator.

Full text: PDF file (259 kB)
References: PDF file   HTML file

Document Type: Article
UDC: 514.8+517.958+517.984.5
MSC: 34B45, 35R02, 58J50, 81Q10, 81Q20
Received: 29.11.2009

Citation: A. A. Tolchennikov, V. L. Chernyshev, A. I. Shafarevich, “Asymptotic properties and classical dynamical systems in quantum problems on singular spaces”, Nelin. Dinam., 6:3 (2010), 623–638

Citation in format AMSBIB
\Bibitem{TolCheSha10}
\by A.~A.~Tolchennikov, V.~L.~Chernyshev, A.~I.~Shafarevich
\paper Asymptotic properties and classical dynamical systems in quantum problems on singular spaces
\jour Nelin. Dinam.
\yr 2010
\vol 6
\issue 3
\pages 623--638
\mathnet{http://mi.mathnet.ru/nd28}


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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, “Svoistva raspredeleniya gaussovykh paketov na prostranstvennoi seti”, Nauka i obrazovanie, 2011, no. 10, 66–66, MGTU im. N. E. Baumana  elib
    2. 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  elib
    3. M. Kh. Numan Elsheikh, “Operatory Shredingera na razvetvlennykh mnogoobraziyakh”, Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Seriya: Matematika. Fizika, 34:5 (2014), 88–93  elib
    4. M. Kh. Numan Elsheikh, V. Zh. Sakbaev, “Operatory Laplasa dlya uravneniya Shredingera na grafakh”, Trudy Moskovskogo fiziko-tekhnicheskogo instituta, 6:2 (22) (2014), 61–67  mathscinet  elib
    5. 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  crossref  isi
    6. A. V. Tsvetkova, A. I. Shafarevich, “The Cauchy Problem for the Wave Equation on Homogeneous Trees”, Math. Notes, 100:6 (2016), 862–869  mathnet  crossref  crossref  mathscinet  isi  elib
    7. V. I. Bezyaev, N. Kh. Sadekov, “O nekotorykh zadachakh gemodinamiki na grafakh”, Trudy seminara po differentsialnym i funktsionalno-differentsialnym uravneniyam v RUDN pod rukovodstvom A. L. Skubachevskogo, SMFN, 62, RUDN, M., 2016, 5–18  mathnet
    8. 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., 23:2 (2016), 244–250  crossref  isi
    9. 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  crossref  mathscinet  zmath  isi  scopus
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