National Research University 'Higher School of Economics', , Myasnitskaya Street, 20, Moscow, 101978, Russia; and Bauman Moscow State University, 2nd Baumanskaya 5, Moscow, 105005, Russia.
Philos Trans A Math Phys Eng Sci. 2013 Dec 16;372(2007):20130145. doi: 10.1098/rsta.2013.0145. Print 2014 Jan 28.
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.
我们研究了带有局部初始函数的带时变薛定谔方程的度量和装饰图上的柯西问题的半经典渐近性。装饰图是通过用光滑黎曼流形替换顶点从图中获得的拓扑空间。在任意有限时间的渐近解的主要项是高斯包和广义高斯包的和（在某些一维子集中局部化）。我们研究了随着时间的无穷大，包的数量。我们证明，在某些假设下，这个数量随着时间呈多项式增长，并且包在图上均匀填充。我们讨论了一个相反情况的简单例子：在这种情况下，数值实验显示了次指数增长。
