Subgroup Type Coordinates and the Separation of Variables in Hamilton-Jacobi and Schrödinger Equations

Separable coordinate systems are introduced in complex and real four-dimensional flat spaces. We use maximal Abelian subgroups to generate coordinate systems with a maximal number of ignorable variables. The results are presented (also graphically) in terms of subgroup chains. Finally the explicit solutions of the Schrödinger equation in the separable coordinate systems are computed.