  We calculate the finite temperature X-ray photoemission spectroscopy for the
Falicov-Kimball model using a Weiner-Hopf sum equation approach. In the
metallic state, the core-hole spectral function shows two side peaks
corresponding to the creation of a core-hole on an empty site (or a doubly
occupied site) and also has two nearly degenerate central peaks (because of our
choice of the model parameters) corresponding to the creation of a core-hole on
a singly occupied site. The nearly doubly degenerate central peaks merge into a
single peak at higher temperatures. In the insulating state, we obtain two
peaks and a strongly temperature dependent low-energy peak corresponding to the
creation of a core-hole on a thermally excited empty site. These results for
the insulating state should be similar to those of the more general Hubbard
model. Also, the strong correlations suggest that even without any additional
broadening due to Auger like processes, the core-hole lifetime will be short.
