On the inverse problem at fixed energy in the "Iterative" range

Abstract

The scattering amplitude by a spherically symmetric potential at fixed energy is given in the Born approximation by a filtered Fourier transform, whose inverse is not unique. It is well known that matrix methods enable one to study exactly the problem at fixed energy in classes of potentials parametrised by sequences of numbers. In the range of potentials (or of phase shifts) where these methods can be managed by iteration, Born case is a limit. This article is a brief survey of the inverse problem (scattering amplitude → potential?) recalling how the nonuniqueness predicted in the Born approximation appears in these exact methods, showing henceforth that the inverse problem ill-posedness corresponds to physical features of the potential on which experiments at finite energy are unable to give information.

Keywords:

• Nonlinear ill-posed problems
• Scattering theory
• Mathematics Subject Classification: 47J06
• 47A40

Notes

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Additional information

Notes on contributors

P.C. Sabatier *

E-mail: [email protected]