  We calculate the spatially resolved optical emission spectrum of a weakly
interacting Bose gas of excitons confined in a three dimensional potential trap
due to interband transitions involving weak direct and phonon mediated
exciton-photon interactions. Applying the local density approximation, we show
that for a non-condensed system the spatio-spectral lineshape of the direct
process reflects directly the shape of the potential. The existence of a
Bose-Einstein condensate changes the spectrum in a characteristic way so that
it directly reflects the constant chemical potential of the excitons and the
renormalization of the quasiparticle excitation spectrum. Typical examples are
given for parameters of the lowest yellow excitons in cuprous oxide.
