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Article

Свойства распределения гауссовых пакетов на пространственной сети

Чернышев В.Л., Толченников А.

The description of the statistical behavior of Gaussian packets on a metric graph is considered. Semiclassical asymptotics of solutions of the Cauchy problem for the Schroeodinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situa- tion where all times for packets to pass over edges 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 final compact graph packets almost always are distributed evenly. The formula for the leading coefficient of the asymp- totic behavior of the number of packets with an increasing time is obtained. The situation is also discussed where the edge travel times are not linearly independent over the rationals.