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We show that the Foldy-Wouthuysen transformation and its generalizations are simplified if the methods of pseudodifferential operators are used, which also allow estimating the exactness of the transition from the Dirac equation to the reduced equations for electrons and positrons. The methods and techniques used can be useful not only in studying asymptotic solutions of the Dirac equation but also in other problems. Diagonalization of the Dirac equation for a free particleGiven the Dirac equation, the Foldy-Wouthuysen transformation [1] allows obtaining nonrelativistic equations of the type of the Pauli equation for the quantum dynamics of electrons and positrons whose kinetic energy is much smaller than mc 2 (v c). This transformation is applicable under the assumption that the electric and magnetic fields are small and amounts to a procedure of consecutively applying unitary transformations. There are generalizations of the Foldy-Wouthuysen transformation applicable in a wide range of energies up to relativistic ones (see, e.g., [2]).To make the subsequent calculations and transformations more transparent, we recall some well-known facts about the Dirac equation [3]. The Dirac equation describes the quantum relativistic motion of electrons and positrons. We assume that the electric and magnetic fields E and H are described by a four-dimensional vector potential (A, Φ), A = (A 1 , A 2 , A 3 ). In accordance with the Maxwell equations, the electric and magnetic fields are found fromThe Dirac equation in the coordinate-time space R 3 x × R t has the form i ∂Ψ ∂t = HΨ, H = cαπ + βmc 2 + eΦ, (1.1) where α = {α 1 , α 2 , α 3 },

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