title: Statistics of Gaussian packets on metric and decorated graphs.
abstract: 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.
authors.affiliation: National Research University 'Higher School of Economics', , Myasnitskaya Street, 20, Moscow, 101978, Russia; and Bauman Moscow State University, 2nd Baumanskaya 5, Moscow, 105005, Russia.
authors.forename: V L
authors.initials: VL
authors.lastname: Chernyshev
authors.forename: A I
authors.initials: AI
authors.lastname: Shafarevich
keywords: decorated graphs; lattice points; metric graphs; semiclassical approximation