Persistence of the Castillejo-Dalitz-Dyson Ambiguity in Relativistic Crossing-Symmetric Amplitudes

The relativistic equations expressing analyticity, crossing, and unitarity are analyzed without approximations. An infinite family of solutions is constructed, corresponding to a Castillejo-Dalitz-Dyson (CDD) ambiguity in the $s$ wave. This ambiguity is in addition to the one resulting from arbitrary inelastic functions. The amplitudes constructed have nonvanishing single spectral functions, and this implies that a Kronecker $\delta$ is present in the angular-momentum plane. This relation between CDD poles and the Kronecker $\delta$ is proved only within a certain limited range of the coupling strength. A computational program for reaching the interesting domain of large couplings is outlined. In the latter domain it is not expected that all CDD poles entail Kronecker $\delta \text{'}\mathrm{s}$ in the $l$ plane.