Leninskie Gory, Moscow, 119992, Russia
Lomonosov Moscow State University
D.Sci, Professor, Corresponding Member of RAS. Member of Editorial Boards of "Regular and Chaotic Dynamics", "Russian Journal of Mathematical Physics", "Moscow University Mathematics Bulletin".
Tolchennikov A. A., Chernyshev V. L., Shafarevich A. I.
Asymptotic properties and classical dynamical systems in quantum problems on singular spaces
2010, Vol. 6, No. 3, pp. 623-638
In the first part of the article we consider a semiclassical asymptotics for a Cauchy problem for the Schrodinger operator on a metric graph. We discuss the statistical properties of the corresponding classical dynamical system: the behavior of «number of particles» at large times and distribution of «particles» on the graph. We describe the distribution of energy on infinite regular trees. In the second part we describe the asymptotics of the spectrum of the Laplace and Schrodinger operators on a thin torus and on the simplest surfaces with delta-potentials.