Description of a free quantum-mechanical particle in the Lobachevsky space based on the integral equation

The quantum mechanical problem of the motion of a free particle in the three-dimensional Lobachevsky space is interpreted as space scattering. The quantum case is considered on the basis of the integral equation derived from the Schrödinger equation. The work continues the problem considered in [1] studied within the framework of classical mechanics and on the basis of solving the Schrödinger equation in quasi-Cartesian coordinates. The proposed article also uses a quasi-Cartesian coordinate system; however after the separation of variables, the integral equation is derived for the motion along the axis of symmetry horosphere axis coinciding with the z axis. The relationship between the scattering amplitude and the analytical functions is established. The iteration method and finite differences for solution of the integral equation are proposed.

On the classification of stably reflective hyperbolic Z[√2]-lattices of rank 4

In this paper we prove that the fundamental polyhedron of a ℤ2-arithmetic reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces is small enough. Using this fact we obtain a classification of stably reflective hyperbolic ℤ2-lattices of rank 4.

Billiards on constant curvature spaces and generating functions for systems with constraints

In this note we consider a method of generating functions for systems with constraints and, as an example, we prove that the billiard mappings for billiards on the Euclidean space, sphere, and the Lobachevsky space are sympletic. Further, by taking a quadratic generating function we get the skew-hodograph mapping introduced by Moser and Veselov, which relates the ellipsoidal billiards in the Euclidean space with the Heisenberg magnetic spin chain model on a sphere. We define analogous mapping for the ellipsoidal billiard on the Lobachevsky space. It relates the billiard with the Heisenberg spin model on the light-like cone in the Lorentz-Poincare-Minkowski space.