Effective interaction in an unbalanced Fermion mixture

A one dimensional Fermi mixture with delta--interaction is investigated in the limit of extreme imbalance. In particular we consider the cases of only one or two minority Fermions which interact with the Fermi-sea of the majority Fermions. We calculate dispersion relation and polaron mass for the minority Fermions as well as equal time density-density correlators. Within a cluster expansion we derive an expression for the effective interaction potential between minority Fermions. For our calculations we use a reformulation of the exact wave functions, originally obtained by Yang and Gaudin by a nested Bethe ansatz, in terms of determinants.