  Within the scope of the relativistic quantum theory for electron-laser
interaction in a medium and using the resonant approximation for the two
degenerated states of an electron in a monochromatic radiation field [1] a
nonperturbative solution of the Dirac equation (nonlinear over field solution
of the Hill type equation) are obtained. The multiphoton cross sections of
electrons coherent scattering on the plane monochromatic wave at the Cherenkov
resonance are obtained taking into account the specificity of induced Cherenkov
process [1, 2] and spin-laser interaction as well. In the result of this
resonant scattering the electron beam quantum modulation at high frequencies
occurs that corresponds to a quantity of an electron energy exchange at the
coherent reflection from the ''phase lattice'' of slowed plane wave in a
medium. So, we can expect to have a coherent X-ray source in induced Cherenkov
process, since such beam is a potential source of coherent radiation itself.
