Compton scattering of twisted photons is investigated within a non-relativistic framework using first-order perturbation theory. We formulate the problem in the density matrix theory, which enables one to gain new insights into scattering processes o
f twisted particles by exploiting the symmetries of the system. In particular, we analyze how the angular distribution and polarization of the scattered photons are affected by the parameters of the initial beam such as the opening angle and the projection of orbital angular momentum. We present analytical and numerical results for the angular distribution and the polarization of Compton scattered photons for initially twisted light and compare them with the standard case of plane-wave light.
Two--photon decay of hydrogen--like ions is studied within the framework of second--order perturbation theory, based on relativistic Diracs equation. Special attention is paid to the effects arising from the summation over the negative--energy (inter
mediate virtual) states that occurs in such a framework. In order to investigate the role of these states, detailed calculations have been carried out for the $2s_{1/2} - 1s_{1/2}$ and $2p_{1/2} - 1s_{1/2}$ transitions in neutral hydrogen H as well as for hydrogen--like xenon Xe$^{53+}$ and uranium U$^{91+}$ ions. We found that for a correct evaluation of the total and energy--differential decay rates, summation over the negative--energy part of Diracs spectrum should be properly taken into account both for high--$Z$ and low--$Z$ atomic systems.
