Journal of Physics A: Mathematical and General
This is the second in a series of three papers in which it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. In this paper the nonrelativistic Green function is examined in detail; new functional forms are presented and a clear mathematical progression is shown to link these and most other known forms. A linear transformation of the four radial parts of the relativistic Green function is given which allows for the presentation of this function as a simple generalization of the non-relativistic Green function. Thus, many properties of the non-relativistic Green function are shown to have simple relativistic generalizations. In particular, new recursion relations of the radial parts of both the non-relativistic and relativistic Green functions are presented, along with new expressions for the double Laplace transforms and recursion relations between the radial matrix elements. A novel proof of the Sturmian form of the radial Green functions is given in an appendix.
Swainson, R. A. and Drake, Gordon W. F.. (1991). A unified treatment of the non-relativistic and relativistic hydrogen atom II: The Green functions. Journal of Physics A: Mathematical and General, 24 (1), 95-120.