On non-overdetermined inverse scattering at zero energy in three dimensions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 3, pp. 279-328.

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].

Classification : 35R30,  81U40,  86A20
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Novikov, Roman G. On non-overdetermined inverse scattering at zero energy in three dimensions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 3, pp. 279-328. http://www.numdam.org/item/ASNSP_2006_5_5_3_279_0/

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