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Zapiski Nauchnykh Seminarov LOMI, 1985, Volume 147, Pages 155–178
(Mi znsl5346)
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This article is cited in 1 scientific paper (total in 1 paper)
Phase analysis in the problem of scattering by a radial potential
A. V. Sobolev, D. R. Yafaev
Abstract:
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.
Citation:
A. V. Sobolev, D. R. Yafaev, “Phase analysis in the problem of scattering by a radial potential”, Boundary-value problems of mathematical physics and related problems of function theory. Part 17, Zap. Nauchn. Sem. LOMI, 147, "Nauka", Leningrad. Otdel., Leningrad, 1985, 155–178
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https://www.mathnet.ru/eng/znsl5346 https://www.mathnet.ru/eng/znsl/v147/p155
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