A particle scatters quantum mechanically off a spherically symmetric potential $V(r)$. In the $l=0$ sector, and assuming $\hbar^{2} / 2 m=1$, the radial wavefunction $u(r)$ satisfies

$$-\frac{d^{2} u}{d r^{2}}+V(r) u=k^{2} u$$

and $u(0)=0$. The asymptotic behaviour of $u$, for large $r$, is

$$u(r) \sim C\left(S(k) e^{i k r}-e^{-i k r}\right)$$

where $C$ is a constant. Show that if $S(k)$ is analytically continued to complex $k$, then

$$S(k) S(-k)=1 \quad \text { and } \quad S(k)^{*} S\left(k^{*}\right)=1$$

Deduce that for real $k, S(k)=e^{2 i \delta_{0}(k)}$ for some real function $\delta_{0}(k)$, and that $\delta_{0}(k)=-\delta_{0}(-k) .$

For a certain potential,

$$S(k)=\frac{(k+i \lambda)(k+3 i \lambda)}{(k-i \lambda)(k-3 i \lambda)}$$

where $\lambda$ is a real, positive constant. Evaluate the scattering length $a$ and the total cross section $4 \pi a^{2}$.

Briefly explain the significance of the zeros of $S(k)$.