Modeling atom-atom interactions at low energy by Jost-Kohn potentials

More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87, 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s-wave scattering length a(s) being negative. For a(s) > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27, no. 9) using a method based on an adaptation from Gel'fand-Levitan theory (Gel' fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77, 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost-Kohn potentials can account for zero-energy resonances. The s-wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost-Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.