  Renormalization group methods are used to study the low-energy behavior of
the unscreened Coulomb interaction in a one-dimensional electron system. By
applying a GW approximation, a strong wavefunction renormalization is found in
the model, as well as renormalization of the Fermi velocity. This may have
significant effects depending on the filling level of the system. The effective
couplings remain bounded at arbitrarily low energies so that, in spite of the
long-range character of the interaction, the system still falls into the
Luttinger liquid universality class.
