  We study the stationary Josephson effect for neutral fermions across the
BCS-BEC crossover, by solving numerically the Bogoliubov-de Gennes equations at
zero temperature. The Josephson current is found to be considerably enhanced
for all barriers at about unitarity. For vanishing barrier, the Josephson
critical current approaches the Landau limiting value which, depending on the
coupling, is determined by either pair-breaking or sound-mode excitations. In
the coupling range from the BCS limit to unitarity, a procedure is proposed to
extract the pairing gap from the Landau limiting current.
