The representation of scattering data by phase shifts is by now a deeply-rooted tradition accepted universally by each new generation of physicists. After measuring a differential cross-section,
(1) dσ / dΩ = Ιf(θ)Ι2
at a certain energy (here it is assumed that there is no spin dependence), one seeks to represent the data by means of a set of real angles such that
where k is the momentum in the center-of-mass system(l). The advantage of such a representation is in situations in which one expects and finds that only a small number of non-vanishing phase shifts will suffice. It is an implicit assumption, never proved, that there will be a unique set of such phase shifts, aside from well-known and obvious ambiguities. (We exclude here a consideration of experimental uncertainties.) We show that this assumption is false by exhibibiting a counter-example.
The problem here is that specification of the phase shifts also specifies the phase of the scattering amplitude f, whereas only its absolute value is determined experimentally. Thus we inquire if it is possible for two or more different sets of phase shifts to satisfy the experimental data. The ambiguities which we discuss here for spin-independent scattering are somewhat analogous to the famous Fermi-Yang ambiguities encountered in the early history of π p scattering.
Crichton, James H. (1966) "Phase Shift Ambiguities for Spin-independent Scattering," Journal of the Institute for Research: Ser. A: No. 4, Article 2. Available at: http://digitalcommons.spu.edu/jir/vol5/iss4/2