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

We develop the ¯-approach to inverse scattering at zero energy in dimensions d3 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].

Classification:  35R30,  81U40,  86A20
@article{ASNSP_2006_5_5_3_279_0,
     author = {Novikov, Roman G.},
     title = {On non-overdetermined inverse scattering at zero energy in three dimensions},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {3},
     year = {2006},
     pages = {279-328},
     zbl = {1121.35143},
     mrnumber = {2274782},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_3_279_0}
}
Novikov, Roman G. On non-overdetermined inverse scattering at zero energy in three dimensions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 3, pp. 279-328. http://www.numdam.org/item/ASNSP_2006_5_5_3_279_0/

[A] G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal. 27 (1988), 153-172. | MR 922775 | Zbl 0616.35082

[BC1] R. Beals and R. R. Coifman, Multidimensional inverse scattering and nonlinear partial differential equations, Proc. Symp. Pure Math. 43 (1985), 45-70. | MR 812283 | Zbl 0575.35011

[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. | Zbl 0725.35096

[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. | MR 875315 | Zbl 0624.35071

[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. | MR 1452176 | Zbl 0884.35167

[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. | MR 590275 | Zbl 1182.35230

[ER] G. Eskin and J. Ralston, The inverse back-scattering problem in three dimensions, Comm. Math. Phys. 124 (1989), 169-215. | MR 1012864 | Zbl 0706.35136

[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. | Zbl 0147.09404

[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. | MR 523015 | Zbl 0373.35014

[G] I. M. Gelfand, Some problems of functional analysis and algebra, Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, 253-276. | Zbl 0079.32602

[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. | MR 936696 | Zbl 0672.35074

[HN] G. M. Henkin and R. G. Novikov, The ¯- 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. | MR 896879 | Zbl 0674.35085

[KV] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements II, Interior results, Comm. Pure Appl. Math. 38 (1985), 643-667. | MR 803253 | Zbl 0595.35092

[Ma] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems 17 (2001), 1435-1444. | MR 1862200 | Zbl 0985.35110

[Mos] H. E. Moses, Calculation of a scattering potential from reflection coefficients, Phys. Rev. 102 (1956), 559-567. | MR 80251 | Zbl 0070.21906

[Na1] A. I. Nachman, Reconstructions from boundary measurements, Ann. Math. 128 (1988), 531-576. | MR 970610 | Zbl 0675.35084

[Na2] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann, Math. 142 (1995), 71-96. | MR 1370758 | Zbl 0857.35135

[No1] R. G. Novikov, A multidimensional inverse spectral problem for the equation -Δψ+(v(x)-Eu(x))ψ=0, Funktsional Anal. i Prilozhen 22 (4) (1988), 11-22 (in Russian); English Transl.: Funct. Anal. Appl. 22 (1988), 263-272. | MR 976992 | Zbl 0689.35098

[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. | MR 1151554 | Zbl 0762.35077

[No3] R. G. Novikov, Scattering for the Schrödinger equation in multidimensional non-linear ¯-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. | MR 2146175 | Zbl 1063.35152

[No5] R. G. Novikov, The ¯-approach to approximate inverse scattering at fixed energy in three dimensions, International Mathematics Research Papers, 2005:6, (2005), 287-349. | MR 2202575 | Zbl pre05034363

[P] R. T. Prosser, Formal solutions of inverse scattering problem. III, J. Math. Phys. 21 (1980), 2648-2653. | MR 588937 | Zbl 0446.35077

[SU] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. Math. 125 (1987), 153-169. | MR 873380 | Zbl 0625.35078

[T] T. Y. Tsai, The Schrödinger operator in the plane, Inverse Problems 9 (1993), 763-787. | MR 1251205 | Zbl 0797.35140