Abstract
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 $\ensuremath{\delta}$ is present in the angular-momentum plane. This relation between CDD poles and the Kronecker $\ensuremath{\delta}$ 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 $\ensuremath{\delta}'\mathrm{s}$ in the $l$ plane.
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