  We present an analytical derivation for the quantum decay rate of the
superflow through a weak link in a one-dimensional Bose-Einstein-condensate.
The effective action for the phase difference across the link reduces to that
of a massive particle with damping subject to a periodic potential. We find an
algebraic flow-pressure relation, characteristic for quantum nucleation of
phase slips in the link and show how short-wave length fluctuations
renormalizing the interaction between the Bosons remove the quantum phase
transition expected in this class of systems.
