Finite-range model potentials for resonant interactions

We show that it is possible to model two-body resonant interactions at low energy with a class of finite-range potentials based on the methods of Jost and Kohn. These potentials are expressed in terms of the effective range r(0) and the s-wave scattering length a(s). We derive continuum solutions of these potentials. By writing V-+/-(r) = V-0(r) + V-+/-(epsilon)(r), where the sign +(-) refers to positive(negative) scattering length, V-0(r) is of the form of Poschl-Teller potential and V-+/-(epsilon) is expressed as a power series of the small parameter epsilon = (root 1 - 2r(0)/a(s))(-1) - 1 when a(s) is large, we derive Green's function of V-0(r). Using the Green's function, solutions of V-+/-(r) for |a(s)| >> r(0) can be obtained numerically by treating V-+/-(epsilon)(r) as a perturbation. We describe the threshold behavior of scattering phase shift for V-0(r). This study may be important for developing a better understanding of physics of strongly interacting ultracold atomic gases with tunable interactions.