A solution of the $S$-wave Schrödinger equation at large distances for a particle of mass $m$ with momentum $\hbar k$ and energy $E=\hbar^{2} k^{2} / 2 m$, has the form

$$\psi_{0}(\boldsymbol{r}) \sim \frac{A}{r}[\sin k r+g(k) \cos k r] .$$

Define the phase shift $\delta_{0}$ and verify that $\tan \delta_{0}(k)=g(k)$.

Write down a formula for the cross-section $\sigma$, for a particle of momentum $\hbar k$ scattering on a radially symmetric potential of finite range, as a function of the phase shifts $\delta_{l}$ for the partial waves with quantum number $l$.

(i) Suppose that $g(k)=-k / K$ for $K>0$. Show that there is a bound state of energy $E_{B}=-\hbar^{2} K^{2} / 2 m$. Neglecting the contribution from partial waves with $l>0$ show that the cross section is

$$\sigma=\frac{4 \pi}{K^{2}+k^{2}} .$$

(ii) Suppose now that $g(k)=\gamma /\left(K_{0}-k\right)$ with $K_{0}>0, \gamma>0$ and $\gamma \ll K_{0}$. Neglecting the contribution from partial waves with $l>0$, derive an expression for the cross section $\sigma$, and show that it has a local maximum when $E \approx \hbar^{2} K_{0}^{2} / 2 \mathrm{~m}$. Discuss the interpretation of this phenomenon in terms of resonant behaviour and derive an expression for the decay width of the resonant state.