Spherical square well ($V(r)=-V_0{1}_{\{r<a\}}$)

A spherical square well is a central potential equal to a negative constant inside a sphere and zero outside it. For an S-wave scattering state with exterior wave number $k$, the interior wave number obeys ${\kappa }^2=k^2+2m{V}_0/{\hslash }^2$. Continuity of the reduced radial wave and its derivative gives $\kappa \cot (\kappa a)=k\cot (ka+{\delta }_0)$, relating the depth to the scattering phase shift.

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

For an attractive spherical square well, let $\kappa =\sqrt{2m{V}_0}/\hslash$. At zero energy, the outer reduced radial wave is proportional to $r-a_s$ and the regular inner wave to $\sin (\kappa r)$. Matching their logarithmic derivatives gives $a_s=a-\tan (\kappa a)/\kappa$. Away from a threshold and at sufficiently low energy, the total S-wave scattering scattering cross-section tends to $4\pi a_s^2$. The length can be negative, zero or divergent.

For a spherical square well of radius $a$, depth $V_0$ and reduced mass $\mu$, the regular zero-orbital angular momentum solution has reduced radial wavefunction $U=A\sin kr$ inside and $U=B{e}^{-qr}$ outside. Wavefunction matching at a finite potential step gives $k\cot (ka)=-q$, where $k^2=2\mu (V_0+E)/{\hslash }^2$ and $q^2=-2\mu E/{\hslash }^2$. The $n$th radial branch first reaches $E=0$ at $ka=(n+1/2)\pi$, for $n=0,1,\dots$. A normalizable negative-energy state on that branch requires $V_0>V_{s,n}$. Thus exactly one such state exists for $V_{s,0}<V_0\le V_{s,1}$; the new zero-energy $s$-wave solution at the upper equality is not normalizable.

For the first $\ell =1$ branch, the zero-energy exterior solution has $U\propto r^{-1}$. The regular interior solution is $U\propto r{j}_1(kr)$, where $j_1$ is a Spherical Bessel function. Matching logarithmic derivatives gives $1+z{j}_1^{\prime }(z)/j_1(z)=-1$, or $j_0(z)=0$, with $z=a\sqrt{2\mu V_0}/\hslash$. Its first threshold is $z=\pi$, so $V_{p,0}={\pi }^2{\hslash }^2/(2\mu a^2)$. Higher values of orbital angular momentum increase the positive centrifugal potential, so none can bind below this first $p$-wave threshold. Spin-dependent nuclear forces are absent from this central-well model.

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.