Recovering asymptotics of potentials from the scattering of relativistic Schr\"odinger operators

TL;DR

This paper demonstrates that for certain decaying potentials in a relativistic Schrödinger operator, the potential's asymptotic behavior can be reconstructed from the scattering matrix at a fixed energy, advancing inverse scattering theory.

Contribution

It establishes a method to recover the asymptotics of poly-homogeneous potentials from fixed-energy scattering data for relativistic Schrödinger operators.

Findings

Potential asymptotics can be uniquely recovered from scattering matrix data.

The method applies to poly-homogeneous potentials decaying at infinity.

Results extend inverse scattering techniques to relativistic operators.

Abstract

We study the stationary scattering for $(-\Delta)^{\frac 12} + V(x)$ on $\mathbb{R}^3$. For poly-homogeneous potentials decaying at infinity, we prove that the asymptotics of the potential can be recovered from the scattering matrix at a fixed energy.