Analysis of the Newton-Sabatier Scheme for Inverting Fixed-Energy Phase Shifts

Abstract

Listen

Translate article

Suppose that the inverse scattering problem is understood as follows: given fixed-energy phase shifts, corresponding to an unknown potential q = q ( r ) from a certain class, for example, q ] L 1,1 , recover this potential. Then it is proved that the Newton-Sabatier (NS) procedure does not solve the above problem. It is not a valid inversion method, in the following sense: (1) it is not possible to carry this procedure through for the phase shifts corresponding to a generic potential q ] L 1,1 , where $ L_{1,1} : = \\{ q {:}\\, q = \\overline q, \\int ^\\infty _0 r |q(r)| dr \\lt \\infty \\} $ and recover the original potential: the basic integral equation, introduced by Newton without derivation, in general, may be not solvable for some r > 0, and if it is solvable for all r > 0, then the resulting potential is not equal to the original generic q ] L 1,1 . Here a generic q is any q which is not a restriction to (0, X ) of an analytic function. (2) the ansatz (*) $ K(r,s) = \\sum^\\infty _{l = 0} c_l \\varphi _l (r) u_l (s) $ , used by Newton, is incorrect: the transformation operator I m K , corresponding to a generic q ] L 1,1 , does not have K of the form (*), and (3) the set of potentials q ] 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 above 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 ] 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.