  The dc Josephson current across a multi-channel SNS junction is computed by
summing contributions from sub-gap Andreev bound states, as well as from
continuum states propagating within the superconducting leads. We show that, in
a long multi-channel SNS-junction, at low temperatures, all these contributions
add up, so that the current can be entirely expressed in terms of
single-particle normal- and Andreev reflection amplitudes at the Fermi level at
both SN interfaces. Our derivation applies to a generic number of channels in
the normal region and/or in the superconducting leads, without assumptions
about scattering processes at the SN interfaces: if the channels within the
central region have the same dispersion relation, it leads to simple analytical
formulas for the current at low temperatures; if the channels within the
central region have different dispersion relations, it allows for expressing
the current in terms of a simple integral involving only scattering amplitudes
at the Fermi level. Our result motivates using a low energy effective boundary
Hamiltonian formalism for computing the current, which is crucial for treating
Luttinger liquid interaction effects.
