Phase shift of a delta-shell potential ($\tan {\delta }_0=-\frac{g{\sin }^2(kR)}{kR+g\sin (kR)\cos (kR)}$)

For $V(r)=V_0\delta (r-R)$, put $g=2m{V}_{0}R/{\hslash }^2$. 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 $g\ne 0$, an equivalent formula is $k\cot {\delta }_0=-k\cot (kR)-k(kR)/[g{\sin }^2(kR)]$. The exact S-wave scattering amplitude is $[k\cot {\delta }_0-ik{]}^{-1}$.