Phase-shift ambiguities for analytic amplitudes

Ambiguities in reconstructing a unitary elastic amplitude at fixed energy are investigated given the differential cross-section when one assumes analyticity in the cos theta plane. Strong hints are given that not more than a twofold nontrivial ambiguity is present. This is demonstrated for genuine entire functions of finite order. Moreover it is found that within a class of amplitudes which includes polynomials as well as entire functions of order zero, (i) the difference between the two amplitudes with the same cross-section must be a polynomial, (ii) if the cross-section is smaller than 1.38 (4 pi /k/sup 2/) there is no ambiguity. Indications are given on directions for future work. (10 refs).