  More than 65 years ago, Jost and Kohn [R. Jost and W. Kohn, {Phys. Rev.} {\bf
87}, 977 (1952)] 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 [R. Jost and W. Kohn, { Dan. Mat. Fys.
Medd} {\bf 27}, 1 (1953)] using a method based on an adaptation from
Gelfand-Levitan theory [I. M. Gel'fand and B. M. Levitan, { Dokl. Akad. Nauk.
USSR} {\bf 77}, 557-560 (1951)] 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; and 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.
