## Abstract

The enhancement factor $\alpha_D \equiv 1+\delta\alpha_D$ is defined (in D dimensions) as the ratio of the particle density at the origin to the density far upstream in the incident beam. At high incident momentum p (and for regular potentials) the first Born approximation is known to be adequate in 3D and 1D, and is assumed to be adequate also in 2D; it entails $\delta\alpha_1$ = -$\delta\alpha_3$ if U(r) in 3D equals U(x) in 1D when r = x. For central potentials U(r) with U(0) finite, previous work implies $\delta\alpha_3 \sim - \delta\alpha_1 \sim$ -mU(0)/p$^2$, and $\delta\alpha_2$ = 0 to the same order $\hslash^0$. It is shown that if U(r$\rightarrow$ 0) $\sim$ U(0) + rU$'$(0) + $\ldots$, then $\delta\alpha_2 \sim$ (2mU$'$(0)$\hslash$/p$^3$)($\pi$/16), the value of U(0) being irrelevant. If U(r$\rightarrow$ 0) $\sim$ -C/r$^q$, with 0 < q < 2, then $\delta\alpha_3 \sim -\delta\alpha_1 \sim (2mC/\hslash^qp^{2-q})\pi^\frac{1}{2}\Gamma (1-\frac{1}{2}q)/2\Gamma(\frac{1}{2}+\frac{1}{2}q);$ and, with -1 < q < 2, $\delta\alpha_2 \sim (2mC/\hslash^qp^{2-q})\pi^\frac{1}{2}\Gamma^2(1- \frac{1}{2}q)/2\Gamma(\frac{1}{2}q)\Gamma(\frac{3}{2}-\frac{1}{2}q).$ The only mathematics needed in 3D and 1D is the standard asymptotic estimation of Fourier integrals; but in 2D one needs to develop corresponding methods for integrals where the sine or cosine has been replaced by a product J$_0$Y$_0$ of two Bessel functions.

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