Bose-Fermi pairs in a mixture and the Luttinger theorem within a Nozieres-Schmitt-Rink-like approach

Boson-fermion pair correlations in amixture are considered at zero temperature in the T-matrix approximation. Special attention is paid to the Luttinger theorem. In a strict random-phase approximation variant of the Nozieres-Schmitt-Rink approach, it is shown that this theorem is respected also in the homogeneous infinite matter case. We calculate the corresponding occupation numbers of fermions and bosons and the condensate depletion. We also show that in the limit of very small boson density, our results are in good agreement with the results found in the literature for the Fermi polaron in strongly imbalanced Fermi-Fermi mixtures.