From Hardcore Bosons to Free Fermions with Painlevé V

We calculate zero temperature Green’s function, the density–density correlations and expectation values of a one-dimensional quantum particle which interacts with a Fermi-sea via a δ-potential. The eigenfunctions of the Bethe-Ansatz solvable model can be expressed as a determinant. This allows us to obtain a compact expression for the Green’s function of the extra particle. In the hardcore limit the resulting expression can be analyzed further using Painleve V transcendents. It is found that depending on the extra particles momentum its Green’s function undergoes a transition of that for hardcore Bosons to that of free Fermions.