  We investigate the evolution of the Josephson and related effects throughout
the BCS-BEC crossover, by performing a systematic numerical solution of the
(time-independent) Bogoliubov-de Gennes equations at zero temperature in a
fully self-consistent fashion. The numerical strategies and algorithms we have
adopted are described in detail, with the aim of easing further independent
studies. Several results are obtained by the present calculation. The profiles
of the magnitude and phase of the gap parameter across the Josephson barrier
are determined under a variety of conditions. We find that the Josephson
current is considerably enhanced at about unitarity for all barriers we have
considered. A related enhancement is also found in the contribution to the
total current from the Andreev bound states, which stem from the depression of
the gap profile about the barrier.For vanishing barrier height, we find that
the critical Josephson current approaches the limiting value predicted by the
Landau criterion, which is determined by either pair-breaking or sound-mode
excitations depending on the coupling value. In the BCS limit, we reveal the
presence of Friedel oscillations in the oscillatory modulations of the gap and
density profiles. In this limit, we also emphasize the special role played by
the Andreev bound state in determining the critical Josephson current in the
presence of a barrier. Finally, the stability of the two branches, out of which
the Josephson characteristics are composed, is analyzed by calculating the
energy required to produce a given spatial profile of the gap parameter.
