Asymptotic properties and classical dynamical systems in quantum problems on singular spaces

In the first part of the article we consider a semiclassical asymptotics for a Cauchy problem for the Schrödinger 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 Schrödinger operators on a thin torus and on the simplest surfaces with delta-potentials.