  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.
