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

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

2009-02-18

arxiv.org/abs/0902.3096v1

Mathematics

Analysis of PDEs

33 pages

Scientific paper

We prove that the singularities of a potential in the two and three dimensional Schr\"odinger equation are the same as the singularities 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. The key point is 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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