On some integrable systems in the extended lobachevsky space

Abstract

Some classical and quantum-mechanical problems previously studied in Lobachevsky space are generalized to the extended Lobachevsky space (unification of the real, imaginary Lobachevsky spaces and absolute). Solutions of the Schrodinger equation with Coulomb potential in two coordinate systems of the imaginary Lobachevsky space are considered. The problem of motion of a charged particle in the homogeneous magnetic field in the imaginary Lobachevsky space is treated both classically and quantum mechanically. In the classical case, Hamilton-Jacoby equation is solved by separation of variables, and constraints for integrals of motion are derived. In the quantum case, solutions of Klein-Fock-Gordon equation are found.