  Using solutions of the discrete Bethe ansatz equations, we study in detail
the quantum impurity problem of a spin-down fermion immersed into a fully
ploarized spin-up Fermi sea with weak attraction. We prove that this impurity
fermion in the one-dimensional (1D) fermionic medium behaves like a polaron for
weak attraction. However, as the attraction grows, the spin-down fermion binds
with one spin-up fermion from the fully-polarized medium to form a tightly bond
molecule. Thus it is seen that the system undergos a cross-over from a mean
field polaron-like nature into a mixture of excess fermions and a bosonic
molecule as the attraction changes from weak attraction into strong attraction.
This polaron-molecule cross-over is universal in 1D many-body systems of
interacting fermions. In thermodynamic limit, we further study the relationship
between the Fredholm equations for the 1D spin-1/2 Fermi gas with weakly
repulsive and attractive delta-function interactions.
