= Scattering length of a delta-shell potential
{title2=$a_0=R\frac{g}{1+g},\quad g=2mV_0R/\hbar^2$}
The zero-energy limit of the <phase shift of a delta-shell potential> gives the displayed <scattering length>. It vanishes at zero coupling and has a pole at $g=-1$. For repulsion $g>0$, $0<a_0<R$; for weak attraction $-1<g<0$, $a_0<0$; for $g<-1$, $a_0>R$. Both infinite-coupling limits tend to $R$. The pole is the threshold for an S-wave <bound state>: writing its energy as $-\hbar^2\kappa^2/(2m)$ gives $g=-2\kappa R/[1-e^{-2\kappa R}]$, with threshold $g\to-1$ as $\kappa\downarrow0$. The <delta-shell Born amplitude> agrees with the exact low-energy amplitude only in their common weak-coupling and low-energy regime, $|g|\ll1$ and $kR\ll1$, where both tend to $-gR$.
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