  Using exact Bethe ansatz (BA) solutions, we show that a spin-down fermion
immersed into a fully polarized spin-up Fermi sea with a weak attraction is
dressed by the surrounding spin-up fermions to form the one-dimensional analog
of a polaron. As the attraction becomes strong, the spin-down fermion binds
with one spin-up fermion to form a tightly bound molecule. Throughout the whole
interaction regime, a crossover from the polaron to a molecule state is fully
demonstrated through exact results of the excitation spectrum, the effective
mass, binding energy and kinetic energy. Furthermore, a clear distinction
between the polaron and molecule is conceived by the probability distribution,
single particle reduced density matrix and density-density correlations, which
are calculated directly from the Bethe ansatz wave function. Such a
polaron-molecule crossover presents a universal nature of an impurity immersed
into a fermionic medium with an attraction in one dimension.
