= S-wave bound-state threshold in a spherical square well
{c}
{title2=$\kappa a=(n+1/2)\pi$}
For <binding energy> $\varepsilon>0$, match the inner reduced radial wave $\sin(Kr)$ to an outer decaying exponential. The resulting equation is $K\cot(Ka)=-\chi$, with $K^2=2m(V_0-\varepsilon)/\hbar^2$ and $\chi^2=2m\varepsilon/\hbar^2$. 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.
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