Spherical square-well scattering length ($a_s=a-\tan (\kappa a)/\kappa$)

For an attractive spherical square well, let $\kappa =\sqrt{2m{V}_0}/\hslash$. At zero energy, the outer reduced radial wave is proportional to $r-a_s$ and the regular inner wave to $\sin (\kappa r)$. Matching their logarithmic derivatives gives $a_s=a-\tan (\kappa a)/\kappa$. Away from a threshold and at sufficiently low energy, the total S-wave scattering scattering cross-section tends to $4\pi a_s^2$. The length can be negative, zero or divergent.