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

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 Schrödinger 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.

References in corpus (3)

• On the structure of critical energy levels for the cubic focusing NLS on star graphs
• Degeneracies in the length spectra of metric graphs
• Clustering of periodic orbits and ensembles of truncated unitary matrices

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• The semiclassical limit on a star-graph with Kirchhoff conditions
• A Metric Graph for Which the Number of Possible Endpoints of a Random Walk Grows Minimally