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A new class of exact solutions of the Klein-Gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium

Exact solutions are presented of the Klein-Gordon equation of a charged particle moving in a classical monochromatic electromagnetic plane wave in a medium of index of refraction n < 1. The solutions are expressed in terms of Ince polynomials, which form a doubly infinite set labeled by two integer quantum numbers. These integer numbers represent quantized spectra of the momentum components of the charged particle along the polarization vector and along the propagation direction of the applied electromagnetic plane wave field. Since this field may represent a laser radiation of arbitrary high intensity propagating in an underdense plasma, the solutions obtained may have relevance, for instance, in describing possible quantum features of laser acceleration of electrons.