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 $V_{\pm}(r) = V_{0}(r) + V_{\pm}^{\epsilon}(r)$, where the sign +(-) refers to positive(negative) scattering length, $V_{0}(r)$ is of the form of P\"{o}schl-Teller potential and $V_{\pm}^{\epsilon}$ is expressed as a power series of the small parameter $\epsilon = (\sqrt{1 - 2 r_0/a_s})^{-1} - 1$ when $a_s$ is large, we derive Green function of $V_{0}(r)$. Using the Green function, solutions of $V_{\pm}(r)$ for $|a_s| >\!> r_0$ can be obtained numerically by treating $V_{\pm}^{\epsilon}(r)$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.