Extended theory of the x-ray-edge singularity

The Nozières—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 $t\to \infty$ 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 $L_{\mathrm{I}\mathrm{I},\mathrm{I}\mathrm{I}\mathrm{I}}$ edge of Na. Our solution approaches the asymptotic Nozières—De Dominicis solution near the edge within a rather small value of $(\omega -{\omega }_0)\lesssim 0.01E_F$, where $E_F$ is the Fermi energy, ${\omega }_0$ is the edge frequency, and $\omega$ is the actual x-ray photon frequency.