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

We define scattering data for the relativistic Newton equation in an electric field $-\mathrm{\nabla }V\in C^1({\mathbb{R}}^n,{\mathbb{R}}^n)$, $n\ge 2$, and in a magnetic field $B\in C^1(\R^n,A_n(\R))$thatdecayatinfinitylike$r^{-\alpha-1}$forsome$\alpha\in (0,1]$,where$A_n(\R)$isthespaceof$n\times n$ 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 $(\mathrm{\nabla }V,B)$ up to the knowledge of the long range tail of $(\mathrm{\nabla }V,B)$. 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.