Exact description of a Dirac particle in the presence of wave fields (generalization of the Volkov problem)

Exact solutions of the Dirac equation in the presence of external wave fields are studied. In contrast to the classical Volkov problem (an electron in the field of a plane electromagnetic wave), consideration is given to wave fields of different tensor structure [scalar, vector (electromagnetic), tensor, pseudovector, and pseudoscalar] and all possible superpositions of them. Situations are identified in which there can be exact solutions without separation of the variables, and with separation of the variables, in the language of differential equations of the first or second order. An essential feature of the present approach is the effective use of a specific representation of singular matrices, which makes it possible to reduce the problem of exact solution of the Dirac equation to the study of a comparatively simple system of equations, including two ordinary differential equations and two functional algebraic equations. Planck's constant, the speed of light, and the electron charge are taken to be equal to unity