  Fermionic superfluidity in atomic Fermi gases across a Feshbach resonance is
normally described by the atom-molecule theory, which treats the closed channel
as a noninteracting point boson. In this work we present a theoretical
description of the resonant superfluidity in analogy to the two-band
superconductors. We employ the underlying two-channel scattering model of
Feshbach resonance where the closed channel is treated as a composite boson
with binding energy $\varepsilon_0$ and the resonance is triggered by the
microscopic interchannel coupling $U_{12}$. The binding energy $\varepsilon_0$
naturally serves as an energy scale of the system, which has been sent to
infinity in the atom-molecule theory. We show that the atom-molecule theory can
be viewed as a leading-order low-energy effective theory of the underlying
fermionic theory in the limit $\varepsilon_0\rightarrow\infty$ and
$U_{12}\rightarrow0$, while keeping the phenomenological atom-molecule coupling
finite. The resulting two-band description of the superfluid state is in
analogy to the BCS theory of two-band superconductors. In the dilute limit
$\varepsilon_0\rightarrow\infty$, the two-band description recovers precisely
the atom-molecule theory. The two-band theory provides a natural approach to
study the corrections because of a finite binding energy $\varepsilon_0$ in
realistic experimental systems. For broad and moderate resonances, the
correction is not important for current experimental densities. However, for
extremely narrow resonance, we find that the correction becomes significant.
The finite binding energy correction could be important for the stability of
homogeneous polarized superfluid against phase separation in imbalanced Fermi
gases across a narrow Feshbach resonance.

Quantum Gases;Superconductivity;Nuclear Theory