BEC-BCS crossover with Feshbach resonance for a three-hyperfine-species model

We consider the behavior of an ultracold Fermi gas across a narrow Feshbach resonance, where the occupation of the closed channel may not be negligible. While the corrections to the single-channel formulas associated with the nonzero chemical potential and with particle conservation have been considered in the existing literature, there is a further effect, namely, the "interchannel Pauli exclusion principle," associated with the fact that a single hyperfine species may be common to the two channels. We focus on this effect and show that, as intuitively expected, the resulting corrections are of the order of E_F/η, where E_F is the Fermi energy of the gas in the absence of interactions and η is the Zeeman energy difference between the two channels. We also consider the related corrections to the fermionic excitation spectrum, and briefly discuss the collective modes of the system.