A Counterexample to a Uniqueness Result

2008

Mod. Phys. Lett. B

A review of some of the author's results in the area of inverse scattering is given. The following topics are discussed: 1) Property C and applications, 2) Stable inversion of fixed-energy 3D scattering data and its error estimate, 3) Inverse scattering with "incomplete" data, 4) Inverse scattering for inhomogeneous Schrödinger equation, 5) Krein's inverse scattering method, 6) Invertibility of the steps in Gel'fand-Levitan, Marchenko, and Krein inversion methods, 7) The Newton-Sabatier and Cox-Thompson procedures are not inversion methods, 8) Resonances: existence, location, perturbation theory, 9) Born inversion as an ill-posed problem, 10) Inverse obstacle scattering with fixed-frequency data, 11) Inverse scattering with data at a fixed energy and a fixed incident direction, 12) Creating materials with a desired refraction coefficient and wave-focusing properties.

Section: Krein's Inversion Methodsmentioning

confidence: 99%

Some Results on Inverse Scattering

2008

Mod. Phys. Lett. B

“…Recent works [6][7][8][9] and [18][19][20] present new numerical methods for solving this problem. In [21] it is proved that the Newton-Sabatier method for inverting the fixed-energy phase shifts for a potential (see [5,10]) is fundamentally wrong, and in [22] a counterexample is given to a uniqueness theorem claimed in a modification of the Newton scheme.…”

Section: Introductionmentioning

confidence: 98%

Inverse scattering by the stability index method

Gutman

Journal of Inverse and Ill-posed Problems

Scheid

2002

A novel numerical method for solving inverse scattering problem with fixed-energy data is proposed. The method contains a new important concept: the stability index of the inversion problem. This is a number, computed from the data, which shows how stable the inversion is. If this index is small, then the inversion provides a set of potentials which differ so little, that practically one can represent this set by one potential. If this index is larger than some threshold, then practically one concludes that with the given data the inversion is unstable and the potential cannot be identified uniquely from the data. Inversion of the fixed-energy phase shifts for several model potentials is considered. The results show practical efficiency of the proposed method. The method is of general nature and is applicable to a very wide variety of the inverse problems.

“…Starting with [5], [4], and [2] claim N1 was not proved or the proofs given (see [3] were incorrect (see [11]). This equation is uniquely solvable for sufficiently small r > 0, but, in general, it may be not solvable for some r > 0, and if it is solvable for all r > 0, then it yields by formula (1.3) a potential q 1 , which is not equal to the original generic potential q ∈ L 1,1 , as follows from Proposition 1.…”

Section: Let Us Justify These Conclusionmentioning

confidence: 99%

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

2002

Applicable Analysis

Self Cite

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 = q, ∞ 0 r|q(r)|dr < ∞} and recover the original potential: the basic integral equation, introduced by R. 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, ∞) of an analytic function.2) the ansatz ( * ) K(r, s) = ∞ l=0 c l ϕ l (r)u l (s), used by R. Newton, is incorrect: the transformation operator I − 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 Math subject classification: 34R30; PACS: 03.80.+r. 03.65.Nk This paper was written when the author was visiting Institute for Theoretical Physics, University of Giessen. The author thanks DAAD for support and Professor W.Scheid for discussions 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.