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

Roman G. Novikov

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 3, page 279-328
  • ISSN: 0391-173X

Abstract

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We 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 v 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].

How to cite

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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 5.3 (2006): 279-328. <http://eudml.org/doc/241734>.

@article{Novikov2006,
abstract = {We develop the $\bar\{\partial \}$-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 $h\big |_\{\Gamma \}$ 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 $h\big |_\{\Gamma \}$ 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].},
author = {Novikov, Roman G.},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {279-328},
publisher = {Scuola Normale Superiore, Pisa},
title = {On non-overdetermined inverse scattering at zero energy in three dimensions},
url = {http://eudml.org/doc/241734},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Novikov, Roman G.
TI - On non-overdetermined inverse scattering at zero energy in three dimensions
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 3
SP - 279
EP - 328
AB - We develop the $\bar{\partial }$-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 $h\big |_{\Gamma }$ 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 $h\big |_{\Gamma }$ 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].
LA - eng
UR - http://eudml.org/doc/241734
ER -

References

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