Reconstruction of the singularities of a potential frombackscattering data in 2D and 3D

Juan Manuel Reyes; Alberto Ruiz

doi:10.3934/ipi.2012.6.321

Article Contents

2012, Volume 6, Issue 2: 321-355. Doi: 10.3934/ipi.2012.6.321

This issue Previous Article Photo-acoustic inversion in convex domains Next Article Strongly convex programming for exact matrix completion and robust principal component analysis

Reconstruction of the singularities of a potential from backscattering data in 2D and 3D

Juan Manuel Reyes^1, and
Alberto Ruiz^2,

1.
Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049-Madrid

Received: May 31, 2011

Revised: November 30, 2011

Published: April 2012

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Abstract

We prove that the singularities of a potential in two and three dimensional Schrödinger equation are the same as those of the Born approximation (Diffraction Tomography), obtained from backscattering inverse data, with an accuracy of $1/2^-$ derivative in the scale of $L^2$-based Sobolev spaces. This improves previous results, see [30] and [20], removing several constrains on the a priori regularity of the potential. The improvement is based on the study of the smoothing properties of the quartic term in the Neumann-Born expansion of the scattering amplitude in 3D, together with a Leibniz formula for multiple scattering valid in any dimension.

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Mathematics Subject Classification: Primary: 34A55, 35P25, 81U40.

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References

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