Hard-core spherical square-well potential

For radii $0<a<b$, a hard-core spherical square-well potential has $V(r)=\infty$ for $r\le a$, $V(r)=-V_0$ for $a<r\le b$, and $V(r)=0$ for $r>b$. Its radial wavefunction vanishes at the hard-core radius and is matched continuously with its radial derivative at $r=b$.

For outer radius $b=2a$ and zero-energy well wavenumber ${\kappa }_0=\sqrt{2m{V}_0}/\hslash$, matching the S-wave scattering logarithmic derivative givesIt diverges when ${\kappa }_{0}a=(n+\frac{1}{2})\pi$, where the well has a zero-energy scattering resonance.