Time-dependent Schr\"odinger equation: Statistics of the distribution of Gaussian packets on a~metric graph

V. L. Chernyshev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 249-265

Résumé

We consider a time-dependent Schrödinger equation in which the spatial variable runs over a metric graph. The boundary conditions at the vertices of the graph imply the continuity of the function and the zero sum of the one-sided derivatives taken with some weights. In the semiclassical approximation, we describe a propagation of Gaussian packets on the graph that are localized at a point at the initial instant of time. The main focus is placed on the statistics of the behavior of asymptotic solutions as time increases. We show that the calculation of the number of quantum packets on a graph is related to the well-known number-theoretic problem of finding the number of integer points in an expanding simplex. We prove that the number of Gaussian packets on a finite compact graph grows polynomially. Several examples are considered. In a particular case, Gaussian packets are shown to be distributed on a graph uniformly with respect to the edge travel times.