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

Novikov, Roman G.

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

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

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

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

[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 | Zbl

[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. | MR | Zbl

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

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

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

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

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

[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. | MR | Zbl

[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 | Zbl

[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. | MR | Zbl

[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. | MR

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

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

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