Near-resonance square-well approximation (${\sigma }_{\mathrm{res}}\simeq (4\pi /k^2)E/[E+V_0{\cot }^2(\kappa a)]$)

The exact spherical square well matching equation is $k^{\prime }\cot (k^{\prime }a)=k\cot (ka+{\delta }_0)$. For $E\ll V_0$ near a threshold peak, neglecting $ka$ relative to the chosen scattering phase shift and replacing $k^{\prime }$ by $\kappa$ gives ${\sigma }_{\mathrm{res}}\simeq (4\pi /k^2)E/[E+V_0{\cot }^2(\kappa a)]$. Its maxima approach the S-wave bound-state threshold in a spherical square well. Its formal zeros at $\kappa a=n\pi$ are outside that approximation's validity: the exact zero-energy scattering length is then $a$, whereas genuine leading-order zeros satisfy $\tan (\kappa a)=\kappa a$.