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

The develop the ∂̄-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 Schrödinger equation from a fixed non-overdetermined ("backscattering" type) restriction h _Γ of the Faddeev generalized scattering amplitude h in the complex domain at zero energy in dimension d = 3. For sufficiently small potentials ν we formulate also a characterization theorem for the aforementioned restriction h _Γ 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].