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. It generates phase equivalent potentials. Such equivalence was suggested to be due to the modulo $\pi$ 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, the intervals of which are defined by the zeros of their regular solutions of the Schr\"odinger equation. We give a classification of the ambiguities in terms of indices which include the difference between exact phase modulo $\pi$ and the numbering of the wave function zeros.