Relativistic scattering in a multimode external field

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Abstract
A variational principle is applied to the study of the relativistic scattering of a charged particle from a center of force in the presence of a strong, slowly varying external electromagnetic field. The analysis is first given in terms of a spin-zero wave equation and then modified to describe the scattering of a Dirac particle. In contrast to previous treatments of problems of this nature, the field is not assumed to be of the plane-wave type. More realistically, it is modeled as a superposition of modes, each with a different frequency and direction of propagation. Exact solutions of the wave equation describing the asymptotic motion of the particle in the presence of the field are not available for fields of this type. Nevertheless, the variational principle allows for a formulation of the scattering problem and, by a suitable choice of trial functions, generates a gauge-invariant low-frequency approximation for the transition amplitude. The error introduced by the inaccuracy in the asymptotic wave function is compensated for by the inclusion of a variational correction term which properly accounts for the effect of stimulated Compton scattering in initial and final states. It is verified that previously derived low-frequency approximations for one- and two-photon spontaneous bremsstrahlung are reproduced by taking the weak-field limit of the approximate stimulated-bremsstrahlung amplitude.