Invariant treatment of the two-body problem with central interaction on simply connected spaces of constant sectional curvature

The classical and quantum problems of two particles with central interaction on simply connected spaces of a constant curvature with an arbitrary dimension are considered from the coordinate-free point of view. We give a representation of Hamiltonians via generators of the symmetry group. Some ordinary differential equations for energy levels of this system are obtained. For some potentials these levels are calculated explicitly.