Solutions to Approximate Integral Equations for Regge Pole Parameters

Abstract

A method described in a previous paper is tested in the light of potential theory. The method is based on dispersion relations for Regge pole parameters. The approximations consist in coupling the $\beta \text{'}\mathrm{s}$ to the $\alpha \text{'}\mathrm{s}$ by applying unitarity at $l=\alpha$ and considering only a few poles. When the generalized potential is replaced by a nonrelativistic potential, the coupling equations and solutions to the integral equations can be compared to exact results. Various representations are tested, and it is found that the "modified Khuri" representation for $A(l, s)$ gives good results for ${\alpha }_1\mathrm{}\left(s\right)\mathrm{}$ in the one-trajectory approximation for Yukawa potentials strong enough to cause bound $S$ states. The results for ${\beta }_1\mathrm{}\left(s\right)\mathrm{}$ are less satisfactory. The effect of coupling in the second trajectory is considered.