The partial-wave projected Coulomb T matrix for all l in closed hypergeometric form

Abstract

We obtain relatively simple closed expressions for the partial-wave projections of the off-shell Coulomb T matrix in the momentum representation, 〈 p‖Tcl(k2)‖p′〉, for all l=0,1,2,⋅⋅⋅ . These exact analytic expressions consist of three parts: (i) ℱl, simple combinations of the Jacobi polynomial P(iγ,−iγ)l and the hypergeometric function 2F1(1,iγ;1+iγ;⋅), with different arguments [γ is Sommerfeld’s parameter; the Coulomb potential is Vc(r)=2kγ/r]; (ii) a polynomial ℰl; and (iii) ℒl=a polynomial times ln[( p+p′)2/( p−p′)2]. The polynomials under (ii) and (iii) are given in terms of the Jacobi polynomials P(m,−m)l , m=0,1,...,l. We derive interesting relations, especially valuable for the theory of off-shell Coulomb scattering, and we present simple closed expressions for the special cases l=0, 1, and 2; p′→k, p′→p, p′→∞, and γ→ 0.