Padé approximants,NNscattering, and hard core repulsions

Padé approximants to the scattering function $F=kcot\left({\delta }_0\right)$ are studied in terms of the variable $x=k^2$, using four examples of potential models which possess features of the $\mathrm{np}$ ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ state. Strategies are thereby developed for analytically continuing $F$ when only approximate partial knowledge of $F$ is available. Results are characterized by high accuracy of interpolation. It is suggested that a physically realistic inverse scattering problem begins with such an analytically continued $F$. When it exists, the solution of this problem in terms of the Marchenko equation is a local potential of the Bargmann type. Some strategies for carrying out this program lead to a stably defined potential, while others do not. With hard core repulsions present, low order Padé approximants accurately describe $F$ for $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}\le 300$ MeV. However, since the condition $\delta \left(\infty \right)-\delta \mathrm{}\left(0\right)=0\mathrm{}$ is not satisfied in any of our examples containing hard core repulsions, the Marchenko method does not have a solution for them. A possible physical consequence of this result is discussed. Another inverse scattering method is proposed for application to hard core problems.