Nonrelativistic Coulomb Green’s function in parabolic coordinates

Abstract

The nonrelativistic Coulomb Green’s function G(+)(r1,r2,k) is evaluated by explicit summation over discrete and continuum eigenstates in parabolic coordinates. This completes the derivation of Meixner, who was able to obtain only the r1=0 and r2→∞ limiting forms of the Green’s function. Further progress is made possible by an integral representation for a product of two Whittaker functions given by Buchholz. We obtain the closed form for the Coulomb Green’s function previously derived by Hostler, via an analogous summation in spherical polar coordinates. The Rutherford scattering limit of the Green’s function is also demonstrated, starting with an integral representation in parabolic coordinates.