Analysis of the Newton-Sabatier scheme for inverting fixed-energy phase shifts

TL;DR

This paper demonstrates that the Newton-Sabatier method for inverse scattering at fixed energy is fundamentally flawed, as its core equations are often unsolvable and it cannot reliably recover potentials from phase shift data.

Contribution

The paper provides a rigorous critique showing the Newton-Sabatier scheme is invalid as an inversion method for fixed-energy phase shifts, highlighting its mathematical and practical limitations.

Findings

The basic integral equation may be unsolvable for some r>0.

The ansatz used by Newton is incorrect for generic potentials.

The set of potentials obtainable by NS is not dense in all L_{1,1} potentials.

Abstract

It is proved that the Newton-Sabatier (NS) procedure does not solve the inverse scattering problem with fixed-energy data and is not a valid inversion method, in the following sense: 1) the basic integral equation, introduced by R. Newton without derivation, in general, may be not solvable for some $r>0$, and in this case NS procedure breaks down: it produces a potential which is not locally integrable. 2) the ansatz $(\ast)$ $K(r,s) = \sum^\infty_{l=0} c_l \phi_l (r) u_l (s)$, used by R. Newton, is incorrect: the transformation operator $I-K$, corresponding to a generic does not have $K$ of the form $(\ast),$ and 3) the set of potentials $q \in L_{1,1},$ that can possibly be obtained by NS procedure, is not dense in the set of all $L_{1,1}$ potentials in the norm of $L_{1,1}$. Therefore one cannot justify NS procedure even for approximate solution of the inverse scattering problem with fixed-energy phase shifts as data. Thus, the NS procedure, if considered as a method for solving the inverse scattering problem, is based on an incorrect ansatz, the basic integral equation of NS procedure is, in general, not solvable for some $r>0$, and in this case this procedure breaks down, and NS procedure is not an inversion theory: it cannot recover generic potentials $q \in L_{1,1}$ from their fixed-energy phase shifts. Suppose now that one considers another problem: given fixed-energy phase shifts, corresponding to some potential, find a potential which generates the same phase shifts. Then NS procedure does not solve this problem either: the basic integral equation, in general, may be not solvable for some $r>0$, and then NS procedure breaks down.