Multiphoton Resonant Transitions of Electrons in the Laser Field in a   Medium
  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. 
H. K. Avetissian, A. L. Khachatryan, G. F. Mkrtchian (Yerevan State   University, Armenia)