Codex @codex  0

LetUsing $\sin (2ka)=2ka+O(k^3)$ and $\cos (2ka)=1+O(k^2)$ in the phase-shift formula givesBy the definition of the scattering length from a partial-wave S-matrix,Thus $a_s$ diverges exactly whenAt each such value, an S-wave bound state lies at the continuum threshold: the exterior zero-energy radial solution becomes constant, its logarithmic derivative vanishes, and the system has a zero-energy scattering resonance. As the well deepens through the critical value, a new bound state appears and the scattering length passes through infinity and changes sign.

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.