## Abstract

In classical mechanics (c.m.), and near the semi-classical limit $\hslash\rightarrow 0$ of quantum mechanics (s.c.l.), the enhancement factors $\alpha \equiv \rho_0/\rho_\infty$ are found for scattering by attractive central potentials U(r); here $\rho_{0,\infty}$ (and v$_{0, \infty}$) are the particle densities (and speeds) at the origin and far upstream in the incident beam. For finite potentials (U(0) > -$\infty$), and when there are no turning points, the preceding paper found both in c.m., and near the s.c.l. (which then covers high $\nu_\infty$), $\alpha_1= \nu_\infty/\nu_0, \alpha_2 = 1, \alpha_3 = \nu_0/\nu_\infty$ respectively in one dimension (1D), 2D and 3D. The argument is now extended to potentials (still without turning points), where U(r$\rightarrow$ 0) $\sim$ - C/r$^q$, with 0 < q < 1 in 1D (where r $\equiv$ |x|), and 0 < q < 2 in 2D and 3D, since only for such q can classical trajectories and quantum wavefunctions be defined unambiguously. In c.m., $\alpha_1$ (c.m.) = 0, $\alpha_3$ (c.m.) = $\infty$, and $\alpha_2$ (c.m.) = ($1-\frac{1}{2}$q)N, where N = [integer part of (1-$\frac{1}{2}$q)$^{-1}$] is the number of trajectories through any point (r, $\theta$) in the limit r$\rightarrow$ 0. All features of U(r) other than q are irrelevant. Near the s.c.l. (which now covers low $\nu_\infty$) a somewhat delicate analysis is needed, matching exact zero-energy solutions at small r to the ordinary W.K.B. approximation at large r; for small $\nu_\infty$/u it yields the leading terms $\alpha_1$ (s.c.l.) = $\Lambda_1$(q) $\nu_\infty$/u, $\alpha_2$ (s.c.l.) = (1 - $\frac{1}{2}$q)$^{-1}$, $\alpha_3$ (s.c.l.) = $\Lambda_3$(q)u/$\nu_\infty$, where u $\equiv$ (C/$\hslash^q$ m$^{1-q}$)$^{1/(2-q)}$ is a generalized Bohr velocity. Here $\Lambda_{1,3}$ are functions of q alone, given in the text; as q $\rightarrow$ 0 the $\alpha$ (s.c.l.) agree with the $\alpha$ quoted above for finite potentials. Even in the limit $\hslash$ = 0, $\alpha_2$ (s.c.l.) and $\alpha_2$ (c.m.) differ. This paradox in 2D is interpreted loosely in terms of quantal interference between the amplitudes corresponding to the N classical trajectories. The Coulomb potential -C/r is used as an analytically soluble example in 2D as well as in 3D. Finally, if U(r) away from the origin depends on some intrinsic range parameter a (e.g. U = -C $\exp$(-r/a)/r$^q$), and if, near the s.c.l., $\nu_\infty$/u is regarded as a function not of $\hslash$ but more realistically of $\nu_\infty$, then the expressions $\alpha$ (s.c.l.) above apply only in an intermediate range $1/a \ll mv_\infty/\hslash \ll (mC/\hslash^2)^{1/(2-q)}$ which exists only if $a \gg (\hslash^2/mC)^{1/(2-q)}$.

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