Eikonal approximation for fast-decreasing potentials. I

The Schrodinger equation with a potential $gq(x)$, decreasing quicker than any power of $|x|^{-1}$ at infinity, is considered at an energy $k^2$. The full asymptotic expansion of its wave function is constructed for $k\to\infty$, $g\leq Ck^{2-\gamma}$, $\gamma>0$. This expansion is used to derive the asymptotics of the forward scattering amplitude and of the total scattering cross-section.