Juan Manuel Reyes, Alberto Ruiz
Published 2009-02-18Version 1
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.
Recovering the singularities of a potential from Backscattering data
On the Formation of Singularities in the Critical O(3) Sigma-Model
On the scattering for the $\bar{\partial}$- equation and reconstruction of convection terms
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