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The problem of determining the scattering amplitude for a given fixed-energy elastic differential cross section is discussed in the spinless case. We show that when the energy is above the inelastic threshold, one may construct an infinte family of unitary scattering amplitudes, by appropriate variation of the elasticity parameters. These amplitudes are analytic in the cosine of the scattering angle throughout the Lehmann ellipse, and all correspond to the same cross section. Hence, even if the cross section is known exactly, there are infinitely many sets of phase shifts. Similar results have been obtained in earlier work, under conditions (on the cross section and elasticities) which seem to be physically unrealistic. In the present paper, the outstanding unrealistic assumptions are avoided. In particular, a finite number of zeros of the dispersive part are now allowed. Each zero reduces the continuum ambiguity by one elasticity parameter, but leaves infinitely many parameters to be varied independently.

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