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

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. We suggest and numerically study an effective-range expansion for $|a|\ll R$ as well as improved two-parameter effective-range expansions for the cases $|a|>R$ and $|a|<R$. The improved effective-range expansions for $|a|>R$ and $|a|<R$ reduce to the expansions for $|a|/R\gg 1$ and $|a|/R\ll 1$, respectively, in appropriate limits. (paper)