  In this paper we investigate in more detail our previous formulation of the
dilaton-gravity theory by Bilal--Callan--de~Alwis as a $SL_2$-conformal affine
Toda (CAT) theory. Our main results are: i) a field redefinition of the
CAT-basis in terms of which it is possible to get the black hole solutions
already known in the literature; ii) an investigation the scattering matrix
problem for the quantum black hole states. It turns out that there is a range
of values of the $N$ free-falling shock matter fields forming the black hole
solution, in which the end-point state of the black hole evaporation is a zero
temperature regular remnant geometry. It seems that the quantum evolution to
this final state is non-unitary, in agreement with Hawking's scenario for the
black hole evaporation.
