A Jost function description of zero-energy resonance and transparency effects in elastic collisions

Abstract

Under certain circumstances, an elastic cross section at very low energies could differ by orders of magnitude above or below any reasonable estimate. The first case occurs near a zero-energy resonance, while the second one is akin to the well-known Ramsauer–Townsend effect. In spite of their intrinsic similarity, both effects are usually explained in very different ways, either in terms of poles of the scattering matrix or by means of a partial-wave analysis, respectively. In this paper we show that a unified description is actually possible. We demonstrate that in a zero-energy resonance or transparency, the standard epsilon2ℓ threshold law of the ℓ-wave cross section is changed into epsilon2(ℓ − 1) (epsilon−1 for ℓ = 0) or epsilon2(ℓ + 1), respectively. Finally, we show that while the zero-energy resonance occurs whenever the scattering length associated to an individual zero of the Jost function diverges, the transparent scattering is a collective effect, where the sum of all the individual scattering lengths plays a relevant role.