Piecewise constant potentials and discrete ambiguities

TL;DR

This paper investigates discrete ambiguities in parametrized potentials, revealing a new class of phase-equivalent potentials using piecewise constant potentials and classifying these ambiguities through indices related to phase and wave function zeros.

Contribution

It introduces a novel class of phase-equivalent potentials based on piecewise constant models and provides a classification scheme for these ambiguities.

Findings

Identifies a new class of phase-equivalent potentials.

Classifies ambiguities using indices related to phase and zeros.

Shows differences from the traditional modulo π uncertainty.

Abstract

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.