The core-hole Green's function in the x-ray edge problem is calculated exactly for some model electron-hole interactions: these include contact and separable potentials. Using the theory of determinants, it is shown that the Green's function may be expressed exactly as a series of terms, where the $n$th term has $n$ electron-hole pairs excited. Exact expressions are given for the $n$th term in the series. The series is then resummed using a cumulant expansion. The results show that the electron-hole interaction is renormalized by dispersion integrals.

• Received 9 November 1981

DOI:https://doi.org/10.1103/PhysRevB.25.5021