Quantum defect theory. XI. Clarification of some aspects of the theory

The theory of Coulomb functions is reviewed for complex values of energy and angular momentum, particular attention being paid to the many-valued nature of some of the functions. The theory for the motion of an electron in the field of a positive ion is simplified by considering a model problem in which a finite system of coupled differential equations contains diagonal Coulomb potentials for an attractive charge Z, together with non-Coulomb potentials U ii' ( rho ) where rho =Zr and r is the radial coordinate for the electron. The energy of the electron in channel i is k i 2 =-Z 2 / kappa i 2 which defines kappa i . In the simple model problem it is assumed that the potentials U ii' ( rho ) are of finite range, U ii' ( rho =0 for rho > rho 1 with rho 1 < infinity . For rho > rho 1 , the solutions of the coupled equations may then be equated to linear combinations of Coulomb functions.