The Cauchy problem to the wave equation on a homogeneous tree and singular spaces of constant curvature. The distribution of the energy

In the talk we will consider a homogeneous tree, i.e. an infinite tree with a root from which precisely one edge emanates and precisely b>1 edges emanating from each other vertex. We will describe the Laplace operator on this tree and will find the spectrum of the operator. Also we will consider the Cauchy problem to the wave equation in case, when the initial condition is concentrated on the edge emanating from the root. We will present the solution of this problem and will describe the distribution of the energy of the wave, which is the solution, as time tends to infinity.
Also singular spaces, i.e. topological spaces obtained by identifying ends of the graph edges with the points on manifolds, will be discussed. We will study two types of singular spaces: first consists of a three-dimensional Euclidean space with a glued ray, second consists of two three-dimensional Euclidean spaces connected by a segment. The Laplace operator on this objects is defined as the self-adjoint extension of the direct sum of Laplace operators on manifolds and edges, which are restricted on functions vanish in the points of gluing. We will find the solution to the Cauchy problem of the wave equation for every self-adjoint extension. Also we will describe the distribution of the wave energy in case time tends to infinity.