  The Breith-Wheeler resonant process has been theoretically studied in a
strong X-ray electromagnetic wave field under conditions when the energy of one
of the initial high-energy gamma quanta passes into the energy of a positron or
electron. These resonant conditions have been studied in detail. Analytical
formulas for the resonant differential cross-section of channels A and B of the
reaction are obtained. It is shown that the resonant differential cross-section
significantly depends on the value of the characteristic Breit-Wheeler energy,
which is determined by the parameters of the electromagnetic wave and the
initial gamma quanta. With a decrease in the characteristic Breit-Wheeler
energy, the resonant cross-section increases sharply and may exceed the
corresponding non-resonant cross-section by several orders of magnitude.
