R. G. Novikov, “Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2”, Solitons, geometry, and topology: on the crossroads, Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov, Trudy Mat. Inst. Steklova, 225, Nauka, MAIK «Nauka/Inteperiodika», M., 1999, 301–318; Proc. Steklov Inst. Math., 225 (1999), 285

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 1999, Volume 225, Pages 301–318 (Mi tm728)

This article is cited in 16 scientific papers (total in 16 papers)

Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2

R. G. Novikov

CNRS — Laboratoire de Mathématiques Jean Leray, Département de Mathématiques, Universite de Nantes

Full-text PDF (1544 kB) Citations (16)

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Abstract: For the Schrödinger equation in dimension 2 we reconstruct the potential

$v\in W^{N,1}_{\varepsilon}(\mathbb R^2)$ ,

$\mathbb N\ni N\ge 3$ ,

$\varepsilon>0$ (

$N$ -times smooth potential) from the scattering amplitude

$f$ at fixed energy

$E$ up to

$O(E^{-(N-2)/2})$ in the uniform norm as

$E\to+\infty$ .

Received in December 1998

Bibliographic databases:

UDC: 517.9

Language: Russian

Citation: R. G. Novikov, “Approximate Inverse Quantum Scattering at Fixed Energy in Dimension 2”, Solitons, geometry, and topology: on the crossroads, Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov, Trudy Mat. Inst. Steklova, 225, Nauka, MAIK «Nauka/Inteperiodika», M., 1999, 301–318; Proc. Steklov Inst. Math., 225 (1999), 285–302

Citation in format AMSBIB

\Bibitem{Nov99}

\by R.~G.~Novikov

\paper Approximate Inverse Quantum Scattering at Fixed Energy in Dimension~2

\inbook Solitons, geometry, and topology: on the crossroads

\bookinfo Collection of papers dedicated to the 60th anniversary of academician Sergei Petrovich Novikov

\serial Trudy Mat. Inst. Steklova

\yr 1999

\vol 225

\pages 301--318

\publ Nauka, MAIK «Nauka/Inteperiodika»

\publaddr M.

\mathnet{http://mi.mathnet.ru/tm728}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1725948}

\zmath{https://zbmath.org/?q=an:0980.81058}

\transl

\jour Proc. Steklov Inst. Math.

\yr 1999

\vol 225

\pages 285--302

Linking options:

https://www.mathnet.ru/eng/tm728

https://www.mathnet.ru/eng/tm/v225/p301

This publication is cited in the following 16 articles:

Dmitriev V K., Rumyantseva O.D., “Features of Solving the Direct and Inverse Scattering Problems For Two Sets of Monopole Scatterers”, J. Inverse Ill-Posed Probl., 29:5 (2021), 775–789
Rumyantseva O.D., Shurup A.S., Zotov I D., “Possibilities For Separation of Scalar and Vector Characteristics of Acoustic Scatterer in Tomographic Polychromatic Regime”, J. Inverse Ill-Posed Probl., 29:3 (2021), 407–420
Dmitriev K.V., Rumyantseva O.D., “Features of the Solution of Direct and Inverse Scattering Problems For Inhomogeneities With a Small Wave Size”, Dokl. Phys., 65:9 (2020), 301–307
A. D. Agaltsov, R. G. Novikov, “Examples of solution of the inverse scattering problem and the equations of the Novikov–Veselov hierarchy from the scattering data of point potentials”, Russian Math. Surveys, 74:3 (2019), 373–386
de Hoop M.V., Lassas M., Santacesaria M., Siltanen S., Tamminen J.P., “Positive-energy D-bar method for acoustic tomography: a computational study”, Inverse Probl., 32:2 (2016), 025003
Barcelo J.A., Castro C., Reyes J.M., “Numerical approximation of the potential in the two-dimesional inverse scattering problem”, Inverse Probl., 32:1 (2016), 015006
Santacesaria M., “a Holder-Logarithmic Stability Estimate For An Inverse Problem in Two Dimensions”, J. Inverse Ill-Posed Probl., 23:1 (2015), 51–73
Novikov R.G., “Formulas For Phase Recovering From Phaseless Scattering Data At Fixed Frequency”, Bull. Sci. Math., 139:8 (2015), 923–936
Agaltsov A.D., Novikov R.G., “Riemann–Hilbert Problem Approach For Two-Dimensional Flow Inverse Scattering”, J. Math. Phys., 55:10 (2014), 103502
M. I. Isaev, R. G. Novikov, “Stability estimates for recovering the potential by the impedance boundary map”, St. Petersburg Math. J., 25:1 (2014), 23–41
Grinevich P.G., Novikov R.G., “Faddeev eigenfunctions for point potentials in two dimensions”, Phys Lett A, 376:12–13 (2012), 1102–1106
Beilina L., Klibanov M.V., “The philosophy of the approximate global convergence for multidimensional coefficient inverse problems”, Complex Variables and Elliptic Equations, 57:2–4 (2012), 277–299
Burov V.A., Alekseenko N.V., Rumyantseva O.D., “Multifrequency generalization of the Novikov algorithm for the two–dimensional inverse scattering problem”, Acoustical Physics, 55:6 (2009), 843–856
Novikov R.G., “The partial derivative–approach to monochromatic inverse scattering in three dimensions”, Journal of Geometric Analysis, 18:2 (2008), 612–631
Novikov R.G., “The partial derivative–approachto approximate inverse scattering at fixed energy in three dimensions”, International Mathematics Research Papers, 2005, no. 6, 287–349
Novikov R.G., “Formulae and equations for finding scattering data from the Dirichlet–to–Neumann map with nonzero background potential”, Inverse Problems, 21:1 (2005), 257–270

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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