RationalSmatrices and NN interactions

The S matrix for central NN interactions is represented in the s wave as a rational function of k. Single and double Yukawa potential models of ${}_{}{}^1{}_{}{}^{}\mathrm{S}_0^{}{}_{}{}^{}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{S}_1^{}{}_{}{}^{}$ interactions are the starting point. Twelve terms of the effective range expansion are found, Padé approximants are calculated, and poles and zeros of the associated rational S matrix are located. For all the potentials, rational S matrices are shown to give excellent agreement with data through medium energies while satisfying Levinson’s theorem. Inverse scattering theory is easily applied to recover phase shift equivalent potentials, either local or nonlocal. Bound state and antibound state poles are precisely determined, suggesting this approach is a viable alternative for finding bound state eigenvalues. We truncate our potentials beyond a range R which we vary from 8 to 21 fm. Rigorously, the S matrix of such truncated potentials has no cuts, and the Jost functions are entire. Our analysis introduces distribution of poles and zeros that, as characteristic of Padé approximants, is seen to bear a relation to the Yukawa cuts of the full potentials. Statistical determinations of rational S matrices from experimental phase shifts, already found to be useful, are further supported by the present results.