Rank-one inverse scattering problem: Reformulation and analytic solutions
Kenneth Hartt · Physical Review C · 1984
Using the $K$-matrix formalism, we give a simplified reformulation of the $S$-wave rank-one inverse scattering problem. The resulting Cauchy integral equation, obtained differently by Gourdin and Martin in their first paper, is tailored to rational representations of $F(k)=kcot({\ensuremath{\delta}}_{0})$. Use of such $F(k)$ permits a simple but general solution without integration, giving analytic form factors having a pole structure like the $S$ matrix that are reducible to rational expressions using Pad\'e approximants. Finally, we show a bound state pole condition is necessary, and makes the form factor unique.