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

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.