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. It is also shown that for a constriction size ranging from zero to several times the coherence length, most of the current is carried, inside the constriction region, by bound states within the superconducting energy gap. These states correspond to Cooper pairs with binding energies smaller than the superconducting gap and which are spatially extended along the channel region, decaying exponentially inside the reservoirs. The importance of the self-consistent determination of the order parameter along the weak link isillustrated by analyzing different profiles obtained for channel lengths of the order of the coherence length. For temperatures not very close to Tc, our microscopic calculation predicts the appearance of features which cannot be obtained from Ginzburg-Landau theory. PACS numbers: 74.50.+r, 85.25.Cp, 73.20.Dx I
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