Low-energy expansion of the Jost function for long-range potentials

Abstract

The low-energy behavior of the partial-wave Jost function for scattering by a long-range $1∕{r}^{s}$ $(s>2)$ central potential is investigated analytically using the linear variant of the variable-phase equation. An exact expansion of the Jost function in powers of the wave number $k$ is derived iteratively and shown to be simpler compared to the modified effective-range expansion of the phase shift. Improved expansions are determined explicitly for $s=3$ and $s=4$. It is suggested that the Jost function offers a practical alternative for interpolating low-energy cross sections and extracting scattering lengths; this is illustrated by fitting the Jost function, up to a normalizing constant, to the integral cross section for elastic collisions of slow electrons with ${\mathrm{N}}_{2}$ molecules.