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Philos Trans A Math Phys Eng Sci. 2013 Dec 16;372(2007):20130145. doi: 10.1098/rsta.2013.0145. Print 2014 Jan 28.

Statistics of Gaussian packets on metric and decorated graphs.

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National Research University 'Higher School of Economics', , Myasnitskaya Street, 20, Moscow, 101978, Russia; and Bauman Moscow State University, 2nd Baumanskaya 5, Moscow, 105005, Russia.


We study a semiclassical asymptotics of the Cauchy problem for a time-dependent Schrödinger equation on metric and decorated graphs with a localized initial function. A decorated graph is a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds. The main term of an asymptotic solution at an arbitrary finite time is a sum of Gaussian packets and generalized Gaussian packets (localized near a certain set of codimension one). We study the number of packets as time tends to infinity. We prove that under certain assumptions this number grows in time as a polynomial and packets fill the graph uniformly. We discuss a simple example of the opposite situation: in this case, a numerical experiment shows a subexponential growth.


decorated graphs; lattice points; metric graphs; semiclassical approximation

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