Counting the Bound States in Short-Range Central Potentials

For the case of a short-range central potential, the quantity ${\alpha }_l$, defined as the zero-energy limit of $k^{2l+1}cot{\delta }_l$, vanishes whenever the range and depth of the potential are such that there is a state of zero binding energy. By solving the zero-energy scattering problem we obtain ${\alpha }_l$ as a function of range and depth and thus determine the number of bound states supportable by a given central potential as a function of the potential parameters without having to solve the associated and more difficult eigenvalue problem. The method is applied to the Debye-Hückel (Yukawa) and Woods-Saxon potentials.