Piecewise constant potentials and discrete ambiguities

This work is devoted to the study of discrete ambiguities. For parametrized potentials, they arise when the parameters are fitted to a finite number of phase-shifts, which generate phase-equivalent potentials. Such equivalence was suggested to be due to the modulo π uncertainty inherent to phase determinations. We show that a different class of phase-equivalent potentials exists. To this aim, use is made of piecewise-constant potentials, intervals of which are defined by the zeros of their regular solutions of the Schrödinger equation. We give a classification of the ambiguities in terms of indices which include the difference between exact phase modulo π and the numbering of the wave function zeros.