On non-overdetermined inverse scattering at zero energy in three dimensions

Roman Novikov (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

We develop the d-bar -approach to inverse scattering at zero energy in dimensions d>=3 of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov 2002]. As a result we give, in particular, uniqueness theorem, precise reconstruction procedure, stability estimate and approximate reconstruction for the problem of finding a sufficiently small potential v in the Schrodinger equation from a fixed non-overdetermined ("backscattering type") restriction h on $\Gamma$ of the Faddeev generalized scattering amplitude h in the complex domain at zero energy in dimension d=3. For sufficiently small potentials v we formulate also a characterization theorem for the aforementioned restriction h on $\Gamma$ and a new characterization theorem for the full Faddeev function h in the complex domain at zero energy in dimension d=3. We show that the results of the present work have direct applications to the electrical impedance tomography via a reduction given first in [Novikov, 1987, 1988].

Mots clés: non-overdetermined inverse scattering; d-bar approach; elecrical impedance tomography

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Roman Novikov. On non-overdetermined inverse scattering at zero energy in three dimensions. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2006, 5 (3), pp.279-328. ⟨hal-00077701⟩