This review contains theoretical study of resonant quantum electrodynamics processes in a pulsed laser field. The approximation is examined when the pulse width is considerably greater than the character istic time of wave oscillations. The lepton's interaction with the Coulomb potential of a nucleus and each other is considered in the Born approximation. It is demonstrated that the resonant differential cross section of a process in a pulsed light fields may considerably exceed the corresponding cross section in an absence of a laser field. Results obtained may be experimentally verified by the scientific facilities at the SLAC National Accelerator Laboratory and FAIR (Facility for Antiproton and Ion Research, Darmstadt, Germany) project.

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The gain coefficient of amplified electromagnetic field in the scattering of nonrelativistic electron by a nucleus in the field which is superposition of circularly polarized electromagnetic waves with the different frequencies, propagating in the same direction, is theoretically investigated. The light amplification effect was discovered in the polar-angles range of an initial electron near-by a polarization-plane of the waves. This angular interval is symmetrical concerning a polarization plane and essentially depends from the wave intensities. The greatest gain coefficient takes place in the region of the medium fields. The effect of the field amplification vanishes for moderately strong intensities both waves. It is shown, that in optical frequencies region for the medium fields (F1 ∼ F2 ∼ 10 6 V/cm, the gain coefficient of amplified field maybe a value of order µ ∼ 10 −1 cm −1for the sufficiently intense electron beams.µ/µ