In a certain spherically symmetric potential, the radial wavefunction for particle scattering in the $l=0$ sector ( $S$-wave), for wavenumber $k$ and $r \gg 0$, is

$$R(r, k)=\frac{A}{k r}\left(g(-k) e^{-i k r}-g(k) e^{i k r}\right)$$

where

$$g(k)=\frac{k+i \kappa}{k-i \alpha}$$

with $\kappa$ and $\alpha$ real, positive constants. Scattering in sectors with $l \neq 0$ can be neglected. Deduce the formula for the $S$-matrix in this case and show that it satisfies the expected symmetry and reality properties. Show that the phase shift is

$$\delta(k)=\tan ^{-1} \frac{k(\kappa+\alpha)}{k^{2}-\kappa \alpha}$$

What is the scattering length for this potential?

From the form of the radial wavefunction, deduce the energies of the bound states, if any, in this system. If you were given only the $S$-matrix as a function of $k$, and no other information, would you reach the same conclusion? Are there any resonances here?

[Hint: Recall that $S(k)=e^{2 i \delta(k)}$ for real $k$, where $\delta(k)$ is the phase shift.]