It ha~ been shown previo~sly by Newton that his solution (and its extension by Sabatier) of the problem of find~ng a cent:al potentIal from a knowledge of all phase shifts at fixed energy yields a series whose expansl~n coefficlen~s converge slowly unless the first moment of the potential vanishes. In particular, any truncatIOn of the serl~s after a finite number of terms necessarily results in potentials which have vanishing firs.t moments. In thIS ~aper 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 ~ew exact. solut~ons Of. the SchrOding~r equation at fixed energy. A closed-form expression for the scattermg amplItude IS obtamed for a specIfic example. The problem of constructing the "new series from the phase shifts is not discussed.

It is proved that the solutions recently obtained by the authors [J. Math. Phys. 11, 805 (1970)1 of the Regge-Newton integral equation (of interest in connection with the inverse scattering problem at fixed energy) are, for a given kernel, inhomogeneity, and boundary condition, uniquely determined.

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