= Phase shift of a delta-shell potential
{title2=$\tan\delta_0=-\frac{g\sin^2(kR)}{kR+g\sin(kR)\cos(kR)}$}
For $V(r)=V_0\delta(r-R)$, put $g=2mV_0R/\hbar^2$. The regular reduced radial <wave function> is proportional to $\sin kr$ inside the shell and to $\sin(kr+\delta_0)$ outside. It is continuous at $R$, while its radial derivative jumps by $(g/R)u(R)$. Dividing the jump condition by the common value gives $k\cot(kR+\delta_0)-k\cot(kR)=g/R$, which implies the displayed <phase shift> modulo $\pi$. At nodes, the undivided matching equations give the continuous limiting interpretation. For $g\ne0$, an equivalent formula is $k\cot\delta_0=-k\cot(kR)-k(kR)/[g\sin^2(kR)]$. The exact <S-wave scattering> amplitude is $[k\cot\delta_0-ik]^{-1}$.
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