Recently, Nozières et al. have shown that Mahan's model x-ray problem (coupling between the deep hole and electron-hole excitations) could be solved exactly for the singular exponents. Here, it is shown that Lundqvist's model x-ray probelm (coupling between the deep hole and plasmons) may also be solved exactly. Furthermore, an exact solution for the singular exponents may be obtained for the more general problem where both effects exist simultaneously. Finally, a perturbation method is developed to treat both effects on an equal footing for a real interacting electron gas.

• Received 4 August 1969

DOI:https://doi.org/10.1103/PhysRevB.1.471