New exact solutions of the Dirac equation of a charged particle interacting with an electromagnetic plane wave in a medium

Exact solutions are presented of the Dirac equation of a charged particle moving in a classical monochromatic electromagnetic plane wave in a medium of index of refraction n < 1. The found solutions are expressed in terms of new complex trigonometric polynomials, which form a doubly infinite set labeled by two integer quantum numbers. These quantum numbers represent quantized spectra of the energy-momentum components of the charged particle along the polarization vector and along the propagation direction of the applied electromagnetic plane wave field (which is considered as a laser field of arbitrary high intensity, propagating in an underdense plasma). These new solutions may serve as a basis for the description of possible quantum features of mechanisms of acceleration of electrons by high-intensity laser fields.