  The article presents a theoretical study of Oleinik resonances in the process
of scattering a gamma quantum by an ultrarelativistic electron in the field of
a strong electromagnetic wave. It is shown that under resonant conditions, the
scattering channels of the reaction effectively split into two first-order
processes according to a fine structure constant such as the external
field-stimulated Compton effect. And the annihilation channel of the reaction
effectively decays into direct and reverse the external field-stimulated
Breit-Wheeler processes. The significant dependence of the resonant energy of
final particles and resonant cross sections on the outgoing angles of the final
gamma quantum, the number of absorbed and emitted photons of the wave, as well
as the characteristic quantum parameters of the problem is shown. These quantum
parameters are determined by the ratio of the initial particle energies to the
characteristic energies of the Compton effect and the Breit-Wheeler process. An
unambiguous relationship between the outgoing angles of final electrons and
gamma quanta has been obtained, which qualitatively distinguishes the resonant
process from the non-resonant one. The cases when the energy of the initial
electrons significantly exceeds the energy of the initial gamma quanta have
been studied. The conditions under which the energy of high-energy initial
electrons is converted into the energy of final gamma quanta are obtained. At
the same time, the resonant differential cross-section of such a process
significantly (by several orders of magnitude) exceeds the corresponding
non-resonant cross-section. This theoretical study predicts a number of new
physical effects that may explain the high-energy fluxes of gamma quanta born
near neutron stars and magnetars.
