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

Novikov, Roman G. 

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 3, pp. 279-328

Résumé

We develop the $\overline{\partial }$-approach to inverse scattering at zero energy in dimensions $d\ge 3$ 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 Schrödinger equation from a fixed non-overdetermined (“backscattering” type) restriction $h{|}_{\Gamma }$ 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 $h{|}_{\Gamma }$ 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].

Détail

Classification : 35R30, 81U40, 86A20