Solution
= Solution
Since $\tan\delta_0=\gamma/(K_0-k)$,
$$
\boxed{\sigma_0(k)=\frac{4\pi}{k^2}
\frac{\gamma^2}{(K_0-k)^2+\gamma^2}}.
$$
The exact nearby stationary point satisfies $K_0-k=\gamma^2/K_0+O(\gamma^4/K_0^3)$, so to leading order in $\gamma/K_0$ the local maximum is at $k=K_0$. This is a <scattering resonance> with
$$
E_R=\frac{\hbar^2K_0^2}{2m},
\qquad
\boxed{\Gamma=\frac{2\hbar^2K_0\gamma}{m}},
$$
where $\Gamma$ is the full energy width of the Breit-Wigner peak.