Do attractive scattering potentials concentrate particles at the origin in one, two and three dimensions? III. High energies in quantum mechanics

Abstract

The enhancement factor α D ≡ 1+ δα 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 δα 1 = ─δα 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 δα 3 ~ ─δα 1 ~ ─ mU (0)/ p 2 , and δα 2 = 0 to the same order h °. It is shown that if U ( r →0) ~ U (0) + r U' (0) + ..., then δα 2 ~ (2 m U' (0) h / p 3 )(π/16), the value of U (0) being irrelevant. If U ( r →0) ~ ─ C / r q , with 0 < q < 2, then δα 3 ~ ─δα 1 ~ (2 mC / h q p 2- q ) π ½ Γ(1-½ q )/2Γ(½+½ q ); and, with -1 < q < 2, δα 2 ~ (2 m C / h q p 2- q )π ½ Γ 2 (1-½ q )/2Γ(½ q )Γ(3/2-½ 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of Fourier integrals; but in 2D one needs to develop corresponding methods for integrals where the sine or cosine has been replaced by a product J 0 Y 0 of two Bessel functions.