On the Inverse Scattering Problem at Fixed Energy for Potentials Having Nonvanishing First Moments

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Abstract
It has been shown previously by Newton that his solution (and its extension by Sabatier) of the problem of finding a central potential from a knowledge of all phase shifts at fixed energy yields a series whose expansion coefficients converge slowly unless the first moment of the potential vanishes. In particular, any truncation of the series after a finite number of terms necessarily results in potentials which have vanishing first moments. In this paper we propose a new, but formally somewhat similar, series for the potential which, for such truncations, does not suffer from this physically rather severe restriction. The series also furnishes new exact solutions of the Schrödinger equation at fixed energy. A closed‐form expression for the scattering amplitude is obtained for a specific example. The problem of constructing the new series from the phase shifts is not discussed.