On inverse scattering at high energies for the multidimensional relativistic Newton equation in a long range electromagnetic field

TL;DR

This paper investigates the inverse scattering problem for the relativistic Newton equation in multidimensional long-range electromagnetic fields, establishing uniqueness of potential recovery at high energies and analyzing scattering data behavior in various regimes.

Contribution

It generalizes previous work by considering long-range electromagnetic fields and provides estimates and uniqueness results for scattering data at high energies.

Findings

Scattering data at high energies uniquely determine the short-range part of the electromagnetic field.

Provides estimates on scattering solutions and data in the presence of long-range fields.

Analyzes the behavior of scattering data in different asymptotic regimes.

Abstract

We define scattering data for the relativistic Newton equation in an electric field $-\nabla V\in C^1(\R^n,\R^n)$, $n\ge 2$, and in a magnetic field $B\in C^1(\R^n,A_n(\R))$ that decay at infinity like $r^{-\alpha-1}$ for some $\alpha\in (0,1]$, where $A_n(\R)$ is the space of $n\times n$ antisymmetric matrices. We provide estimates on the scattering solutions and on the scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of $(\nabla V,B)$ up to the knowledge of the long range tail of $(\nabla V,B)$. The Born approximation at fixed energy of the scattering data is also considered. We then change the definition of the scattering data to study their behavior in other asymptotic regimes. This work generalizes [Jollivet, 2007] where a short range electromagnetic field was considered.