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 vertical bar a vertical bar is much larger than the range R of the scattering potential: vertical bar a vertical bar >> 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 vertical bar a vertical bar << R as well as improved two-parameter effective-range expansions for the cases vertical bar a vertical bar > R and vertical bar a vertical bar < R. The improved effective-range expansions for vertical bar a vertical bar > R and vertical bar a vertical bar < R reduce to the expansions for vertical bar a vertical bar/R >> 1 and vertical bar a vertical bar/R << 1, respectively, in appropriate limits.