Some Analytic Properties of Scattering Amplitudes for Long-Range Forces

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Abstract
We examine the properties of the partial-wave amplitude a(l, k) and the full amplitude A(k, cos θ) for scattering by a long-range potential made up of a Coulomb part 2α/r and a short-range part V(r). The properties of a(l, k) as an analytic function of l and k are shown to be quite similar to those of the usual short-range amplitude, except in the neighborhood of the threshold k = 0, which point we examine in detail. The full amplitude is treated as a function of cos θ for fixed physical momentum k; using the Sommerfeld-Watson transformation, we show that A(k, cos θ) is analytic in the cut plane of cos θ.