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

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

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

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topNovikov, 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

top- [A] G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal. 27 (1988), 153–172. Zbl0616.35082MR922775
- [BC1] R. Beals and R. R. Coifman, Multidimensional inverse scattering and nonlinear partial differential equations, Proc. Symp. Pure Math. 43 (1985), 45–70. Zbl0575.35011MR812283
- [BC2] R. Beals and R. R. Coifman, The spectral problem for the Davey-Stewartson and Ishimori hierarchies, In: “ Nonlinear evolution equations: integrability and spectral methods”, Proc. Workshop, Como/Italy 1988, Proc. Nonlinear Sci., (1990), 15–23. Zbl0725.35096
- [BLMP] M. Boiti, J. Leon, M. Manna and F. Pempinelli, On a spectral transform of a KDV- like equation related to the Schrödinger operator in the plane, Inverse Problems 3 (1987), 25–36. Zbl0624.35071MR875315
- [BU] R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations 22 (1997), 1009–1027. Zbl0884.35167MR1452176
- [C] A.-P. Calderón, On an inverse boundary value problem, In: “Seminar on Numerical Analysis and its Applications to Continuum Physics” (Rio de Janeiro, 1980), 65–73, Soc. Brasil. Mat. Rio de Janeiro, 1980. Zbl1182.35230MR590275
- [ER] G. Eskin and J. Ralston, The inverse back-scattering problem in three dimensions, Comm. Math. Phys. 124 (1989), 169–215. Zbl0706.35136MR1012864
- [F1] L. D. Faddeev, Growing solutions of the Schrödinger equation, Dokl. Akad. Nauk 165 (1965), 514–517 (in Russian); English Transl.: Sov. Phys. Dokl. 10 (1966), 1033–1035. Zbl0147.09404
- [F2] L. D. Faddeev, Inverse problem of quantum scattering theory II, Itogi Nauki Tekh., Ser. Sovrem. Prob. Math. 3 (1974), 93–180 (in Russian); English Transl.: J. Sov. Math. 5 (1976), 334–396. Zbl0373.35014MR523015
- [G] I. M. Gelfand, Some problems of functional analysis and algebra, Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, 253–276. Zbl0079.32602
- [GN] P. G. Grinevich and S. P. Novikov, Two-dimensional “inverse scattering problem” for negative energies and generalized-analytic functions. I. Energies below the ground state, Funktsional Anal. i Prilozhen 22 (1) (1988), 23–33 (In Russian); English Transl.: Funct. Anal. Appl. 22 (1988), 19–27. Zbl0672.35074MR936696
- [HN] G. M. Henkin and R. G. Novikov, The $\overline{\partial}$- equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk 42 (3) (1987), 93–152 (in Russian); English Transl.: Russian Math. Surveys 42 (3) (1987), 109–180. Zbl0674.35085MR896879
- [KV] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements II, Interior results, Comm. Pure Appl. Math. 38 (1985), 643–667. Zbl0595.35092MR803253
- [Ma] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems 17 (2001), 1435–1444. Zbl0985.35110MR1862200
- [Mos] H. E. Moses, Calculation of a scattering potential from reflection coefficients, Phys. Rev. 102 (1956), 559–567. Zbl0070.21906MR80251
- [Na1] A. I. Nachman, Reconstructions from boundary measurements, Ann. Math. 128 (1988), 531–576. Zbl0675.35084MR970610
- [Na2] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann, Math. 142 (1995), 71–96. Zbl0857.35135MR1370758
- [No1] R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi +\left(v\right(x)-Eu(x\left)\right)\psi =0$, Funktsional Anal. i Prilozhen 22 (4) (1988), 11–22 (in Russian); English Transl.: Funct. Anal. Appl. 22 (1988), 263–272. Zbl0689.35098MR976992
- [No2] R. G. Novikov, The inverse scattering problem at fixed energy level for the two-dimensional Schrödinger operator, J. Funct. Anal. 103 (1992), 409–463. Zbl0762.35077MR1151554
- [No3] R. G. Novikov, Scattering for the Schrödinger equation in multidimensional non-linear $\overline{\partial}$-equation, characterization of scattering data and related results, In: “Scattering”, E. R. Pike and P. Sabatier (eds.), Chapter 6.2.4, Academic, New York, 2002.
- [No4] R. G. Novikov, Formulae and equations for finding scattering data from the Dirichlet-to-Neumann map with nonzero background potential, Inverse Problems 21 (2005), 257–270. Zbl1063.35152MR2146175
- [No5] R. G. Novikov, The $\overline{\partial}$-approach to approximate inverse scattering at fixed energy in three dimensions, International Mathematics Research Papers, 2005:6, (2005), 287–349. Zbl1284.35464MR2202575
- [P] R. T. Prosser, Formal solutions of inverse scattering problem. III, J. Math. Phys. 21 (1980), 2648–2653. Zbl0446.35077MR588937
- [SU] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. Math. 125 (1987), 153–169. Zbl0625.35078MR873380
- [T] T. Y. Tsai, The Schrödinger operator in the plane, Inverse Problems 9 (1993), 763–787. Zbl0797.35140MR1251205

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