S-wave bound-state threshold in a spherical square well ($\kappa a=(n+1/2)\pi$)

For binding energy $\epsilon >0$, match the inner reduced radial wave $\sin (Kr)$ to an outer decaying exponential. The resulting equation is $K\cot (Ka)=-\chi$, with $K^2=2m(V_0-\epsilon )/{\hslash }^2$ and ${\chi }^2=2m\epsilon /{\hslash }^2$. A bound level reaches zero energy when $\kappa a=(n+1/2)\pi$. These are divergences of the spherical square-well scattering length. The exact zero-energy threshold wave is not square-integrable; a normalizable bound state appears on the more attractive side.

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 $\tan (\kappa a)=\kappa a$.