Bose-Fermi Pairs in a Mixture and the Luttinger Theorem within a Nozieres-Schmitt-Rink like Approach

Bose-Fermion pair correlations in a mixture are considered at zero temperature in the T-matrix approximation. Special attention is paid to the Luttinger theorem. In a strict RPA 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.