  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.
