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arXiv · 1111.3945

The properties of the distribution of Gaussian packets on a spatial network

V. L. Chernyshev·A. A. Tolchennikov
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Abstract

The article deals with the description of the statistical behavior of Gaussian packets on a metric graph. Semiclassical asymptotics of solutions of the Cauchy problem for the Schr\"{o}dinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situation where all times for packets to pass over edges ("edge travel times") are linearly independent over the rational numbers, a description of the behavior of such systems is related to the number-theoretic problem of counting the number of lattice points in an expanding polyhedron. In this paper we show that for a finite compact graph packets almost always are distributed evenly. A formula for the leading coefficient of the asymptotic behavior of the number of packets with an increasing time is obtained. The article also discusses a situation where the times of passage over the edges are not linearly independent over the rationals.

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V. L. Chernyshev, A. A. Tolchennikov. 2011-11-16. The properties of the distribution of Gaussian packets on a spatial network. https://doi.org/10.1098/rsta.2013.0145

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