A representation is obtained for the quasiclassical Green functions of the
Dirac and Klein-Gordon equations allowing for the first nonvanishing correc
tion in an arbitrary localized potential which generally possesses no spher
ical symmetry. This is used to obtain a solution of these equations in an a
pproximation similar to the Furry-Sommerfeld-Maue approximation. It is show
n that the quasiclassical Green function does not reduce to the Green funct
ion obtained in the eikonal approximation and has a wider range of validity
. This is illustrated by calculating the amplitude of small-angle scatterin
g of a charged particle and the amplitude of Delbruck forward scattering. A
correction proportional to the scattering angle was obtained for the ampli
tude of charged particle scattering in a potential possessing no spherical
symmetry. The real part of the Delbruck forward scattering amplitude was ca
lculated in a screened Coulomb potential. (C) 2000 MAIK "Nauka/Interperiodi
ca".