  Stimulated two-photon emission by Bose-condensed excitons accompanied by a
coherent two-exciton recombination, i.e., by simultaneous recombination of two
excitons with opposite momenta leaving unchanged the occupation numbers of
excitonic states with nonzero momenta, is investigated. Raman light scattering
accompanied by a similar two-exciton recombination (or generation of two
excitons) is also analyzed. The processes under consideration can occur only if
a system contains Bose condensate, therefore, their detection can be used as a
new method to reveal Bose condensation of excitons. The recoil momentum, which
corresponds to a change in the momentum of the electromagnetic field in the
processes, is transferred to phonons or impurities. If the recoil momentum is
transmitted to optical phonons with frequency $\omega_0^s$, the stimulated
two-photon emission with the coherent two-exciton recombination leads to the
appearance of a line at $2\Omega'-\omega$, where $\Omega'=\Omega-\omega_0^s$
and $\Omega$ is the light frequency corresponding to the recombination of an
exciton with zero momentum. Formulas for the cross sections at finite
temperatures are obtained for the processes under consideration. Our estimates
indicate that a spectral line, corresponding to the stimulated two-photon
emission accompanied by the coherent optical phonon-assisted two-exciton
recombination can be experimentally detected in Cu$_2$O.
