Some features of the direct and inverse double Compton effect as applied
  to astrophysics
  In the present paper, we consider the process of inverse double Compton (IDC)
scattering in the context of astrophysical applications. It is assumed that the
two hard X-ray photons emitted from an astrophysical source are scattered on a
free electron and converted into a single soft photon of optical range. Using
QED S-matrix formalism for the derivation of a cross-section of direct double
Compton (DDC) and assuming detailed balance conditions we give an analytical
expression for the cross-section of the IDC process. It is shown that at fixed
energies of incident photons the inverse cross-section has no infra-red
divergences and its behavior is completely defined by the spectral
characteristics of the photon source itself, in particular, by the finite
interaction time of radiation with an electron. Thus, even for the direct
process, the problem of resolving infrared divergence actually refers to a real
physical source of radiation in which photons are never actually plane waves.
As a result the physical frequency profile of the scattered radiation for
direct as well as inverse double Compton processes is a function of both the
intensity and line shape of the incident photon field.
