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

TL;DR

This paper demonstrates that in 2D and 3D Schrödinger equations, the singularities of a potential can be accurately reconstructed from backscattering data, matching the singularities of the Born approximation with high precision.

Contribution

It establishes the equivalence of potential singularities and Born approximation singularities in 2D and 3D, analyzing smoothing properties of the scattering amplitude's quartic term.

Findings

Singularities of potential and Born approximation coincide within 1/2^- derivative accuracy.

Analysis of smoothing properties of the quartic term in the Neumann-Born expansion.

Development of a Leibniz formula for multiple scattering applicable in any dimension.

Abstract

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.