From hardcore Bosons to free Fermions with Painlev{\'e} V

We calculate zero temperature Green's function, the density--density correlation and expectation values of a one--dimensional quantum particle which interacts with a Fermi--sea via a $\delta$--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 analysed further using Painlev{\'e} 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.