Inverse backscattering for the Schr\"odinger equation in 2D

TL;DR

This paper investigates how to recover singularities of a potential in the 2D Schrödinger equation from backscattering data, showing that the main singularities are captured by the Born approximation.

Contribution

It demonstrates that for non-smooth potentials in 2D, the primary singularities are contained in the diffraction tomography approximation derived from backscattering data.

Findings

Main singularities up to half a derivative are contained in the Born approximation.

Singularities are measured in Sobolev spaces.

Results apply to non-smooth potentials in 2D.

Abstract

We study the inverse backscattering problem for the Schr\"odinger equation in two dimensions. We prove that, for a non-smooth potential in 2D the main singularities up to 1/2 of the derivative of the potential are contained in the Born approximation (Diffraction Tomography approximation) constructed from the backscattering data. We measure singularities in the scale of Hilbertian Sobolev spaces.