Inverse Scattering at a Fixed Energy for Long-Range Potentials

TL;DR

This paper demonstrates that the long-range component of a potential in the Schrödinger equation can be uniquely identified from the scattering amplitude's singularity at a fixed energy, advancing inverse scattering theory.

Contribution

It introduces a method to uniquely reconstruct long-range potentials from fixed-energy scattering data, focusing on the leading forward singularity.

Findings

Long-range potentials can be uniquely reconstructed from scattering data.

The leading forward singularity encodes essential information about the potential.

The approach applies to Schrödinger equations in dimensions three and higher.

Abstract

In this paper we consider the inverse scattering problem at a fixed energy for the Schr\"odinger equation with a long-range potential in $\ere^d, d\geq 3$. We prove that the long-range part can be uniquely reconstructed from the leading forward singularity of the scattering amplitude at some positive energy.