Sum Rule for Resonance Reactions

It is shown that in the absence of direct reactions, the averages of the diagonal elements of the pole-residue matrices $-i{\mathrm{G}}_{\mu }$ of the $S$ matrix (sometimes called partial widths) satisfy $\frac{2\pi {〈G_{\mu \mathrm{cc}}〉}_{\mu }}{D}=\frac{\exp \left(2i{\xi }_c\right)T_c}{{(1-T_c)}^{\frac{1}{2}}},$ where $D$ is the average pole spacing, $T_c$ is the optical-model transmission coefficient, and ${\xi }_c$ is the real part of the optical-model phase shift.