The quasi-classical asymptotics of the scattering cross-section for asymptotically homogeneous potentials

The total scattering cross-section by a potential $gV(x)$, $x\in\mathbb R^m$, $m\geqslant3$, is considered for'large coupling constants $g$ and .wave numbers $k$. It is supposed that $V(x)\sim\Phi(x/|x|)|x|^{-\alpha}$, $2\alpha>m+1$, as $|x|\to\infty$. It is shown that as $gk^{-1}\to\infty$, $g^{3-\alpha}k^{2(\alpha-2)}\to\infty$ the cross-section asymptotically equals $\theta_\alpha(gk^{-1})^\varkappa$, $\varkappa=(m-1)(\alpha-1)^{-1}$. Here the coefficient $\theta_\alpha$ is determined only by the function $\Phi$ and the number $\alpha$. Under additional assumptions $\Phi>0$, $V>0$ this asymptotics holds in the broader region $gk^{-1}\to\infty$, $gk^{\alpha-2}\geqslant c(gk^{-1})^\delta$, $\delta>0$.