Extended theory of the x-ray-edge singularity

The Nozi\`eres---De Dominicis equations governing the singular behavior of the various x-ray response functions are approximately solved for a Thomas-Fermi model potential of the deep hole which scatters off the Fermi-sea electrons. We are looking for solutions to these equations which are valid not just for t→∞ but over the whole region of time. This enables us to obtain the response function in frequency space away from the edge. Numerical results are given for the L_II,0ex0exIII edge of Na. Our solution approaches the asymptotic Nozi\`eres---De Dominicis solution near the edge within a rather small value of (ω-ω₀)0.01EF, where EF is the Fermi energy, ω₀ is the edge frequency, and ω is the actual x-ray photon frequency.