Inverse scattering at high energies for the multidimensional Newton equation in a long range potential

TL;DR

This paper develops an inverse scattering method for the multidimensional Newton equation with long-range potentials, showing high-energy scattering data uniquely determine the short-range potential part, extending previous approaches.

Contribution

It introduces a new inverse scattering framework for the Newton equation with long-range potentials, including high-energy and other asymptotic regimes, building on Novikov's approach.

Findings

Scattering data at high energies uniquely determine the short-range potential.

Estimates on scattering solutions and data for long-range potentials.

Extension of inverse scattering methods to various asymptotic regimes.

Abstract

We define scattering data for the Newton equation in a potential $V\in C^2(\R^n,\R)$, $n\ge2$, that decays at infinity like $r^{-\alpha}$ for some $\alpha\in (0,1]$. We provide estimates on the scattering solutions and scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of the potential up to the knowledge of the long range tail of the potential. The Born approximation at fixed energy of the scattering data is also considered. We then change the definition of the scattering data to study inverse scattering in other asymptotic regimes. These results were obtained by developing the inverse scattering approach of [Novikov, 1999].