Improved effective-range expansions for small and large values of scattering length

TL;DR

This paper proposes improved effective-range expansions for scattering systems with either small or large scattering lengths relative to the potential range, enhancing analysis accuracy across different regimes.

Contribution

It introduces and numerically studies new two-parameter effective-range expansions tailored for cases where the scattering length is either much larger or smaller than the potential range.

Findings

Improved expansions reduce to textbook form for large scattering length.

Numerical validation shows better accuracy across regimes.

Applicable to low-energy scattering phenomenology.

Abstract

The textbook effective-range expansion of scattering theory is useful in the analysis of low-energy scattering phenomenology when the scattering length $|a|$ is much larger than the range $R$ of the scattering potential: $|a|\gg R$. Nevertheless, the same has been used for systems where the scattering length is much smaller than the range of the potential, which could be the case in many scattering problems. We suggest and numerically study improved two-parameter effective-range expansions for the cases $|a| > R$ and $|a| < R$. The improved effective-range expansion for $|a| > R$ reduces to the textbook expansion for $|a|/R \gg 1$.