Twenty-four optimal inequalities and several new representations for the Coulomb T matrices in momentum space

Abstract

We derive new series and integral representations for the Coulomb transition matrix in momentum space, 〈p‖Tc‖p′〉, and for its partial-wave projections, 〈p‖Tcl‖p′〉 (l=0,1,...), to be denoted by Tc and Tcl, respectively. We also consider hypergeometric-function representations for Tc and Tcl and discuss their analytic continuation to the whole complex k plane (k2 is the energy). The new integrals are essentially ∫π0 cosh γt (ρ−cos t)−1 dt for Tc and ∫π0 cosh γt ×Ql(uu′+vv′ cos t)dt for Tcl, where γ is Sommerfeld’s parameter and ρ,u,u′,v, and v′ are variables depending on the energy and the momenta; related integrals follow from these. A well-known and convenient series representation for Tc consists essentially of the sum ∑nyn(n2+γ2)−1, where y depends on the energy and the momenta. We derive its analog for Tcl, the corresponding sum being ∑n(n2+γ2)−1Qnl(u) P−nl(u′), 1<u′<u. This sum is a new member of the family of sums of products of Legendre functions that can be evaluated in a relatively simple closed form; other members of this family have been recently obtained by the author. With the new representations for Tc and Tcl we derive a set of twenty-four optimal inequalities (containing two conjectured inequalities) for these Coulomb T matrices, presumably covering all cases relevant for physics. For the proof of these inequalities several different representations, and in particular the newly derived ones, would appear to be indispensable. Many of the inequalities are new. They are valid for fixed real Coulomb strength and fixed real energy≠0. Because of the complexity of exact closed forms for Tc and Tcl, approximations are needed for numerical calculations; the most natural one consists of replacing the Coulomb T matrix by the Coulomb potential. Our inequalities are useful for estimating the accuracy of this approximation.