  In a Feshbach resonance, the effective s-wave scattering length grows when
one moves toward the resonance point, and eventually diverges at this point.
There is one characteristic energy scale, $\delta_c$, defined as, in the
negative side of the resonance point, the detuning energy at which the weight
of the bound state shifts from predominatedly in the open-channel to
predominated in the closed-channel. When the many-body energy scale (e.g. the
Fermi energy, $E_{F}$) is larger than $\delta_c$, the closed-channel weight is
significant and has to be included in the many-body theory. Furthermore, when
two channels share a hyperfine species, the Pauli exclusion between fermions
from two channels also needs to be taken into consideration in the many-body
theory.
  The current thesis addresses the above problem in detail. A set of gap
equations and number equations are derived at the mean-field level. The
fermionic and bosonic excitation spectra are then derived. Assuming that the
uncoupled bound-state of the closed-channel in resonance is much smaller than
the inter-particle distance, as well as the s-wave scattering length, $a_s$, we
find that the basic equations in the single-channel crossover model are still
valid. The correction first comes from the existing of the finite chemical
potential and additional counting complication due to the closed-channel. These
two corrections need to be included into the mean-field equations, i.e. the gap
equations and the number equations, and be solved self-consistently. Then the
correction due to the inter-channel Pauli exclusion is in the order of the
ratio of the Fermi energy and the Zeeman energy difference between two
channels, $E_F/\eta$, which can be analyzed perturbatively over the previous
corrections.
  Fermionic and bosonic excitation modes are studied.
