Ambiguities in Scattering Amplitudes Resulting from Insufficient Experimental Data

We present a systematic study of the problem of reconstructing the scattering matrix from experimental data. Insufficient data lead to ambiguities; a general technique for finding ambiguities is given. The cases of isospinless spin-0-spin-$\frac{1}{2}$ scattering (two amplitudes) and pion-nucleon scattering (four amplitudes) are treated in detail. The ambiguities present in both cases when the $R$ and $A$ parameters are unknown have a profound effect on partial-wave analysis, leading to confusion and misidentification of resonances. It is shown that only eight measurements are needed to reconstruct the pion-nucleon amplitudes, rather than nine as averred by Bilenkii and Ryndin.