  We analyze a system of two-component fermions which interact via a Feshbach
resonance in the presence of a three-dimensional lattice potential. By
expressing a two-channel model of the resonance in the basis of Bloch states
appropriate for the lattice, we derive an eigenvalue equation for the
two-particle bound states which is nonlinear in the energy eigenvalue. Compact
expressions for the interchannel matrix elements, numerical methods for the
solution of the nonlinear eigenvalue problem, and a renormalization procedure
to remove ultraviolet divergences are presented. From the structure of the
two-body solutions we identify the relevant degrees of freedom which describe
the resonance behavior in the lowest Bloch band. These degrees of freedom,
which we call dressed molecules, form an effective closed channel in a
many-body model of the resonance, the Fermi resonance Hamiltonian (FRH). It is
shown how the properties of the FRH can be determined numerically by solving a
projected lattice two-channel model at the two-particle level. As opposed to
single-channel lattice models such as the Hubbard model, the FRH is valid for
general s-wave scattering length and resonance width. Hence, the FRH provides
an accurate description of the BEC-BCS crossover for ultracold fermions on an
optical lattice.
