Inelastic Resonances and Castillejo-Dalitz-Dyson Singularities

By considering many coupled two-body-channel scattering amplitudes that possess a unique continuation in the angular-momentum variable $l$, we show: (1) that several methods of obtaining the single-channel $\frac{N}{D}$ solution for the elastic amplitude with an inelasticity factor agree with the corresponding many-channel matrix $\frac{N}{D}$ solution at high $l$; (2) that in order to preserve agreement when continuing to low angular momentum one must generally introduce Castillejo-Dalitz-Dyson (CDD) cuts or poles into the single-channel $D$ function; (3) that the integral equations for $\frac{N}{D}$ may be continued in $l$, automatically yielding the CDD cuts mentioned above. Finally, we introduce a distinction between elastic and inelastic resonance poles based on continuation in $l$ and show that the latter will not appear in calculations that fail to include the requisite CDD singularities.