A unitarity criterion for the partial $S$-matrix of resonance scattering

We consider the influence of $N$ long-lived states characterized by resonance energies $E_i$ and widths $\Gamma_i(E)$ on the elastic scattering process and obtain an expression for the partial $S$-matrix $S_l(E)$ in the form of a sum over the resonance levels $($poles$)$ at which the residues have the form $\Gamma_i\prod_{\substack{k=1,\\k\ne i}}^N\gamma_{ik}$, where $\gamma_{ik}={(z_i-z_k^*)\imath/2(z_i-z_k)}$ and $z_i=E_i-\imath\Gamma_i/2$. We show that a necessary condition for the unitarity of the partial $S$-matrix in the presence of $N$ resonance levels can be written as $\sum_{i=1}^N \Gamma_i(E)\prod_{\substack{k=1,\\k\ne i}}^N\gamma_{ik}= \sum_{i=1}^N\Gamma_i(E)$.