X iv : m at h - ph / 0 10 50 21 v 1 1 5 M ay 2 00 1 Analysis of the Newton - Sabatier scheme for inverting the fixed - energy phase shifts ∗

It is proved that the Newton-Sabatier (NS) procedure 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 q ∈ L 1,1 such that ∞ 0 rq(r)dr = 0, where L 1,1 := {q : q = q, ∞ 0 r|q(r)|dr < ∞, 2) the ansatz (*) K(r, s) = ∞ l=0 c l ϕ l (r)u l (s), ∞ l=0 |c l | < ∞, similar to the one used in the NS procedure is incorrect: the transformation operator corresponding to q ∈ L 1,1 , ∞ 0 rq(r)dr = 0, does not have kernel of the form (*), 3) if one starts with any q ∈ L 1,1 , which is not analytic in a neigborhood of (0, ∞), calculates the corresponding phase shifts, and applies the NS procedure, then, assuming that this procedure is applicable, one does not get the original potential q(r). For example, this is the case if q(r) is compactly supported, and 4) the set of potentials v ∈ L 1,1 , that can possibly be obtained by the NS procedure, is not dense in the set of all L 1,1 potentials in the norm of L 1,1 .