Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 36B / d / Solution

Since $\tan {\delta }_0=\gamma /(K_0-k)$,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 withwhere $\Gamma$ is the full energy width of the Breit-Wigner peak.