Scattering length from a partial-wave S-matrix

The $s$-wave scattering length isand the low-energy total cross-section is $4\pi a_s^2$.

For a barrier of radius $a$ and potential ${\hslash }^2{\gamma }^2/(2m)$, matching the regular inner hyperbolic-sine wave to the outer sine wave gives $a_s=a-\tanh (\gamma a)/\gamma$. The low-energy $s$-wave scattering amplitude is $-a_s$.

A zero-energy scattering resonance occurs when an S-wave bound state reaches the continuum threshold. The zero-energy radial solution then approaches a nonzero constant rather than a generic linear function, and the scattering length diverges and changes sign.