The Green formula in the theory of potential scattering and corrections to the eikonal scattering amplitude

A method based on the Green formula is developed for calculating the scattering amplitude of fast charged particles in an external field. The scattering amplitude is representable as an integral over an arbitrary closed surface enveloping the domain of influence of the external field on the particle. Corrections to the eikonal scattering amplitude are simply derived without using the specific form of the potential. The resulting formulas can be used to investigate the interaction between particles and fields of complex configuration.