Phase analysis in the problem of scattering by a radial potential

Let $\sigma(k,g)$ be the total cross-section for scattering of a three-dimensional quantum particle of energy $k^2$ by a radial potential $gV(r)$, $r=|x|$. Under the assumption $V(r)\sim v_0|r|^{-\alpha}$, $\alpha>2$, $r\to\infty$ it is shown that in the region $gk^{-1}\to\infty$, $g^{3-\alpha}k^{2(\alpha-2)}\to\infty$ the asymptotics $\sigma(k,g)\sim\varkappa_\alpha(|v_0|gk^{-1})^{2\lambda_\alpha}$, $\lambda_\alpha=(\alpha-1)^{-1}$ is valid; the coefficient $\varkappa_\alpha$ is expressed explicitly in terms of the $\Gamma$-function. For nonnegative potentials this asymptotics holds even in the broader region. For potentials with a strong positive singularity $V(r)\sim v_0r^{-\beta}$, $v_0>0$, $\beta>2$, $r\to0$ the asymptotics $\sigma(r,g)\sim\varkappa_\beta(v_0gk^{-1})^{2\lambda_\beta}$ as $gk^{-1}\to0$, $gk^{\beta-2}\to\infty$ is established. Similar results are obtained for the forward scattering amplitude.