  We present the microscopic theory for the BEC-BCS crossover of an atomic
Fermi gas in an optical lattice, showing that the Feshbach resonance underlying
the crossover in principle induces strong multiband effects. Nevertheless, the
BEC-BCS crossover itself can be described by a single-band model since it
occurs at magnetic fields that are relatively far away from the Feshbach
resonance. A criterion is proposed for the latter, which is obeyed by most
known Feshbach resonances in ultracold atomic gases.
