The d-bar approach to approximate inverse scattering at fixed energy in three dimensions

TL;DR

This paper extends the d-bar approach to inverse scattering in three dimensions, providing a stable method to approximate a potential from scattering data at fixed energy with error decreasing as energy increases.

Contribution

It develops a new stable nonlinear reconstruction method for potentials in 3D inverse scattering, improving error decay rates at high energies.

Findings

Reconstruction error decays as $O(E^{-(n-3-\epsilon)/2})$ as $E o + $

Method applies to n-times smooth potentials with sufficient decay

Provides a stable nonlinear approximation method for inverse scattering

Abstract

We develop the d-bar approach to inverse scattering at fixed energy in dimensions $d\ge 3$ of [Beals, Coifman 1985] and [Henkin, Novikov 1987]. As a result we propose a stable method for nonlinear approximate finding a potential $v$ from its scattering amplitude $f$ at fixed energy $E>0$ in dimension $d=3$. In particular, in three dimensions we stably reconstruct n-times smooth potential $v$ with sufficient decay at infinity, $n>3$, from its scattering amplitude $f$ at fixed energy $E$ up to $O(E^{-(n-3-\epsilon)/2})$ in the uniform norm as $E\to +\infty$ for any fixed arbitrary small $\epsilon >0$ (that is with almost the same decay rate of the error for $E\to +\infty$ as in the linearized case near zero potential).