= Near-resonance square-well approximation
{title2=$\sigma_{\rm res}\simeq(4\pi/k^2)E/[E+V_0\cot^2(\kappa a)]$}
The exact <spherical square well> matching equation is $k'\cot(k'a)=k\cot(ka+\delta_0)$. For $E\ll V_0$ near a threshold peak, neglecting $ka$ relative to the chosen <scattering phase shift> and replacing $k'$ by $\kappa$ gives $\sigma_{\rm res}\simeq(4\pi/k^2)E/[E+V_0\cot^2(\kappa a)]$. 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 $\tan(\kappa a)=\kappa a$.
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