Core-hole propagator in Wiener-Hopf sum equation approach

We apply Wiener-Hopf sum equation approach to calculate the propagator for a core-hole, added as an additional localized level to the Falicov-Kimball model. The Wiener-Hopf sum equation approach was resently applied by Shvaika {\it et. al.} [1] to calculate the finite temperature real time $f$-electron propagator. We present further iterative improvement of the method and its application to the core-hole propagator. Finally, time permitting, we will show some results on the application of the core-hole propagator on Resonant Inelastic X-ray Scattering (RIXS) spectra.\\[4pt] [1] A.M.Shvaika and J.K.Freericks, Condensed Matter Physics, {\bf 11}, 425-442 (2008).