  We calculate Bardeen-Cooper-Schrieffer (BCS) state of a unitary Fermi gas of
atoms interacting with the finite-ranged Jost-Kohn potential which has been
recently shown to account for the resonant interactions [2019 {\rm J. Phys. B:
At. Mol. Opt. Phys.} {\bf 52}, 165004]. Using exact scattering solution of the
potential, we derive two-body ${\mathbf T}$-matrix element which is employed to
construct the BCS Hamiltonian in momentum space. We present results on the
energy- and range-dependence of the pairing gap and superfluid density and the
range-dependence of the chemical potential for a wide variation of the
scattering length including the unitary regime. In the zero range limit our
calculated gap at the Fermi energy is found to be nearly equal to that
calculated in mean-field theory with contact potential. The mean gap averaged
over the full width at half maximum of the gap function in the zero range and
unitary limits is found to be $0.42 E_F$ which is quite close to the recent
result of the quantum Monte Carlo simulation [2018 {\rm Phys. Rev.A} {\bf 97},
013601]. The chemical potential in the zero range limit also agrees well with
that for the contact potential.
