publication . Preprint . 2014

Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrodinger equation

Klibanov, Michael V.; Romanov, Vladimir G.;
Open Access English
  • Published: 28 Dec 2014
Abstract
The inverse scattering problem of the reconstruction of the unknown potential with compact support in the 3-d Schr\"odinger equation is considered. Only the modulus of the scattering complex valued wave field is known, whereas the phase is unknown. It is shown that the unknown potential can be reconstructed via the inverse Radon transform. Therefore, a long standing problem posed in 1977 by K. Chadan and P.C. Sabatier in their book "Inverse Problems in Quantum Scattering Theory" is solved.
Subjects
free text keywords: Mathematical Physics
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27 references, page 1 of 2

[1] T. Aktosun and P.E. Sacks, Inverse problem on the line without phase information, Inverse Problems, 14, 211-224, 1998. [OpenAIRE]

[2] N.F. Berk and C.F. Majkrzak, Statistical analysis of phase-inversion neutron specular reflectivity, Langmuir, 25, 4132-4144, 2009. [OpenAIRE]

[3] K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York, 1977.

[4] D. Dobson, Phase reconstruction via nonlinear least squares, Inverse Problems, 8, 541-557, 1992.

[5] J.R. Fienup, Phase retrieval algorithms: a personal tour [invited], Applied Optics, 52, 45-56, 2013.

[6] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1984.

[7] D. Gerth, B. Hofman, S. Birkholz, S. Koke and G. Steinmeyer, Regularization of an autoconvolution problem in ultrashort laser pulse characterization, Inverse Problems in Science and Engineering, 22, 245-266, 2014.

[8] V. Isakov, Inverse Problems for Partial Differential Equations, Second Edition, Springer, New York, 2006.

[9] O. Ivanyshyn, R. Kress and P. Serranho, Huygens' principle and iterative methods in inverse obstacle scattering, Advances in Computational Mathematics, 33, 413-429, 2010.

[10] O. Ivanyshyn and R. Kress, Inverse scattering for surface impedance from phase-less far field data, J. Computational Physics, 230, 3443-3452, 2011. [OpenAIRE]

[11] R.V. Khachaturov, Direct and inverse problems of determining the parameters of multilayer nanostructures from the angular spectrum of the intensity of reflected X-rays, Computational Mathematics and Mathematical Physics, 49, 1781-1788, 2009. [OpenAIRE]

[12] M.V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74, 392-410, 2014. [OpenAIRE]

[13] M.V. Klibanov, Uniqueness of two phaseless non-overdetermined inverse acoustics problems in 3-d, Applicable Analysis, 93, 1135-1149, 2014.

[14] M.V. Klibanov, On the recovery of a 2-D function from the modulus of its Fourier transform, J. Mathematical Analysis and Applications, 323, 818-843, 2006. [OpenAIRE]

[15] M.V. Klibanov and V.G. Kamburg, Uniqueness of a one dimensional phase retrieval problem, Inverse Problems, 30, 075004, 2014. [OpenAIRE]

27 references, page 1 of 2
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