 Spherically symmetric bound state of a finite well

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 $k^2=2m(U+E)/{\hslash }^2$ and ${\kappa }^2=-2mE/{\hslash }^2$. 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.