  We investigate a recently proposed model for a full quantum description of
two-dimensional black hole evaporation, in which a reflecting boundary
condition is imposed in the strong coupling region. It is shown that in this
model each initial state is mapped to a well-defined asymptotic out-state,
provided one performs a certain projection in the gravitational zero mode
sector. We find that for an incoming localized energy pulse, the corresponding
out-going state contains approximately thermal radiation, in accordance with
semi-classical predictions. In addition, our model allows for certain acausal
strong coupling effects near the singularity, that give rise to corrections to
the Hawking spectrum and restore the coherence of the out-state. To an
asymptotic observer these corrections appear to originate from behind the
receding apparent horizon and start to influence the out-going state long
before the black hole has emitted most of its mass. Finally, by putting the
system in a finite box, we are able to derive some algebraic properties of the
scattering matrix and prove that the final state contains all initial
information.
