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Self-consistent solutions of microscopic Eilenberger theory are presented fora two-dimensional model of a superconducting channel with a geometricconstriction. Magnetic fields, external ones as well as those caused by thesupercurrents, are included and the relevant equations are solved numericallywithout further assumptions. Results concerning the influence of temperature,geometric parameters, of $\kappa=\lambda_L/\xi_0$ and of external magneticfields on the Andreev bound states in the weak link and on the current-phaserelation are presented. We find that the Andreev bound states within thejunction obtain peculiar substructure when a finite supercurrent flows. As longas the London penetration depth is comparable to or bigger than the extensionof the constriction, the Josephson effect is independent of $\kappa$.Furthermore, the weak link is very insensitive to external magnetic fields.Features restricted to a self-consistent calculation are discussed.

Microscopic theory of superconductor-constriction-superconductor Josephson junctions in a magnetic field