  We propose a semiclassical method to calculate S-matrix elements for
two-stage gravitational transitions involving matter collapse into a black hole
and evaporation of the latter. The method consistently incorporates
back-reaction of the collapsing and emitted quanta on the metric. We illustrate
the method in several toy models describing spherical self-gravitating shells
in asymptotically flat and AdS space-times. We find that electrically neutral
shells reflect via the above collapse-evaporation process with probability
exp(-B), where B is the Bekenstein-Hawking entropy of the intermediate black
hole. This is consistent with interpretation of exp(B) as the number of black
hole states. The same expression for the probability is obtained in the case of
charged shells if one takes into account instability of the Cauchy horizon of
the intermediate Reissner-Nordstrom black hole. Our semiclassical method opens
a new systematic approach to the gravitational S-matrix in the non-perturbative
regime.
