Nonrelativistic Particle with Spin in the Lobachevsky Space

Abstract

Nonrelativistic wave equations for particles with spins 1/2 and 1 in the Lobachevsky space are considered. Two formulations of these equations are compared, the first one is based on the limiting procedure in the corresponding relativistic wave equation, and the second one uses the invariant operators of the representations of the symmetry group of the space. Transformations are presented that connect equations for particles with the same spin in different formulations. Separation of variables in the spherical coordinate system is carried out. For a particle with spin 1/2 moving in the Coulomb field the radial equation is reduced to the Heun equation, and expressions for the energy levels of bound states are found in the form of expansions in inverse powers of the radius of curvature of the space. For a particle with spin 1 in the Coulomb field, for some spin states it is possible to find the explicit solutions of the radial equation and an exact expression for the energy levels. For other states, radial functions obey an intricate system of equations, from which asymptotic expressions for the energy levels for the case of small curvature are found. Solutions of the wave equations for free particles are also considered.