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On the distribution of energy of localized solutions of the Schrödinger equation that propagate along symmetric quantum graphs

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Abstract

From the point of view of applications to quantum mechanics, it is natural to pose a question concerning the distribution of energy of localized solutions of a nonstationary Schrödinger equation over the graph (in other words, the probability to find a quantum particle in a given area). This problem is apparently very complicated for general graphs, because the energy distribution is much more sensitive to the form of boundary conditions and to the initial state than the asymptotic behavior of the number of localized functions. Below, we present initial results concerning the distribution of energy in the case of symmetric quantum graphs (this means that the Schrödinger operators on different edges have the same structure). For general local self-adjoint boundary conditions, we describe the process of onestep scattering of the localized solutions and obtain a simple general result of the distribution of energy. Some special cases and specific examples are discussed.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, Russia

    A. I. Shafarevich

  2. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia

    A. I. Shafarevich

  3. Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow Region, Russia

    A. I. Shafarevich

  4. National Research Centre “Kurchatov Institute,”, Moscow, Russia

    A. I. Shafarevich

Authors
  1. A. I. Shafarevich
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Correspondence to A. I. Shafarevich.

Additional information

This work was supported by the Russian Science Foundation (grant 16-11-10069).

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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, 244–250 (2016). https://doi.org/10.1134/S1061920816020096

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  • DOI: https://doi.org/10.1134/S1061920816020096

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