Impurity with a resonance in the vicinity of the Fermi energy

We study an impurity with a resonance level whose position coincides with the Fermi energy of the surrounding Fermi gas. Such an impurity causes a rapid variation of the scattering phase shift for fermions at the Fermi surface, introducing a new characteristic length scale into the problem. We investigate manifestations of this length scale in the self-energy of the impurity and in the many-body density function of the bath. Our calculations reveal a model-independent deformation of the density of the Fermi gas, which is determined by the width of the resonance. To provide a broader picture, we investigate time evolution of the density in quench dynamics, and study the behavior of the system at finite temperatures. We also briefly discuss implications of our findings for the Fermi-polaron problem. Overall, our results are relevant for studies that involve quasiparticle interference methods or fermion-mediated RKKY-type interactions in Bose-Fermi mixtures.