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

Abstract

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.

Effective Rb–Rb inter-atomic potential from ultracold Bose-gas collisions

“…Inverse scattering formulas We employ the fixed energy inverse scattering method of Cox and Thompson (CT) [17] in order to derive model independent potentials from the given phase shifts [15,16] corresponding to the measurements. The CT inversion method has a number of useful properties [18,19]: we may work with a finite set of N experimental phase shifts and obtain inversion potentials of non-zero first momentum, rV (r)dr = 0, and of finite value at the origin.…”

A Counterexample to a Uniqueness Result

“…where T is a finite set of distinct real numbers from the interval (−0.5, ∞), and the sets T and S are disjoint. This uniqueness result is crucial for the arguments in [4].…”

“…In this note a counterexample to the uniqueness claim from [3] is constructed. This counterexample invalidates the arguments in [3] and [4]. The uniqueness of the solution of similar equations in [2] (see equations (12.1.2) and (12.2.1) on pp.195-196 in [2]) and [5] does not hold for some r > 0, in general, also.…”

Development of a Cox–Thompson inverse scattering method to charged particles

“…In the recent decade another procedure, the Cox-Thompson (CT) inverse quantum scattering method [12] has been investigated [13,14,15]. This method has the advantage that it requires a finite set of phase shifts and possesses a non-zero first momentum of the potential generated.…”

“…In nuclear physics the modified Newton-Sabatier (mNS) method [5,6] has been used to determine, in general, complex valued optical potentials describing various systems, e.g., 12 C -12 C elastic scattering [7,8] or to guess the spin-orbit potentials arising in the p -α and n -α collisions [9]. The Newton-Sabatier (NS) method [10,11] has the property that it requires an infinite set of phase shifts otherwise the first moment of the potential generated vanishes.…”