= Spherically symmetric bound state of a finite well
{title2=$U>\pi^2\hbar^2/(8ma^2)$}
For well depth $U$ and radius $a$, let $u=r\psi$ for a regular spherically symmetric <wavefunction>. At energy $-U<E<0$, the interior is $u=C\sin(kr)$ and the exterior is proportional to $e^{-\kappa r}$, with $k^2=2m(U+E)/\hbar^2$ and $\kappa^2=-2mE/\hbar^2$. Matching gives $k\cot(ka)=-\kappa$. The first negative-energy solution exists strictly above the displayed threshold. At equality the exterior zero-energy wave behaves as $1/r$ and is not square-integrable, so the threshold is not itself a bound state.
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