  A microscopic model for describing a superconducting mesoscopic weak link is
presented. We consider a model geometry consisting of a narrow channel coupled
to wider superconducting electrodes which act as reservoirs fixing the
asymptotic values of the complex order parameter. For this model, the
Bogoliubov-de Gennes equations are discretized and solved self-consistently
using a non-equilibrium Green functions formalism. The transport properties and
the electronic excitation spectra of this system are studied for the different
regimes that can be reached by varying parameters like coherence length,
constriction length, normal transmission coefficient and temperature. We study
in detail the transition from the point contact limit to the infinite channel
length case, analyzing the maximum Josephson current that can be sustained by
the weak link as a function of its transmission coefficient and length.
