  We determine the one-loop and the two-loop back-reaction corrections in the
spectrum of the Hawking radiation for the CGHS model of 2d dilaton gravity by
evaluating the Bogoliubov coefficients for a massless scalar field propagating
on the corresponding backgrounds. Since the back-reaction can induce a small
shift in the position of the classical horizon, we find that a positive shift
leads to a non-Planckian late-time spectrum, while a null or a negative shift
leads to a Planckian late-time spectrum in the leading-order stationary-point
approximation. In the one-loop case there are no corrections to the classical
Hawking temperature, while in the two-loop case the temperature is three times
greater than the classical value. We argue that these results are consistent
with the behaviour of the Hawking flux obtained from the operator quantization
only for the times which are not too late, in accordance with the limits of
validity of the semiclassical approximation.
