On inverse scattering at high energies for the multidimensional Newton equation in electromagnetic field

TL;DR

This paper studies inverse scattering for the multidimensional Newton equation in electromagnetic fields at high energies, providing unique determination of the potential and magnetic field from scattering data, extending previous inverse scattering methods.

Contribution

It extends inverse scattering techniques to the nonrelativistic Newton equation with electromagnetic fields, establishing high-energy asymptotics and uniqueness results for reconstructing the potential and magnetic field.

Findings

High-energy scattering data uniquely determine the electromagnetic potential and magnetic field.

Asymptotic formulas for scattering solutions and data are derived for small angle scattering.

The inverse problem is solved for dimensions n ≥ 2, reconstructing (∇V, B) from scattering data.

Abstract

We consider the multidimensional (nonrelativistic) Newton equation in a static electromagnetic field $$\ddot x = F(x,\dot x), F(x,\dot x)=-\nabla V(x)+B(x)\dot x, \dot x={dx\over dt}, x\in C^2(\R,\R^n),\eqno{(*)}$$ where $V \in C^2(\R^n,\R),$ $B(x)$ is the $n\times n$ real antisymmetric matrix with elements $B_{i,k}(x)$, $B_{i,k}\in C^1(\R^n,\R)$ (and $B$ satisfies the closure condition), and $|\pa^{j_1}_xV(x)| +|\pa^{j_2}_xB_{i,k}(x)| \le \beta_{|j_1|} (1+|x|)^{-(\alpha+|j_1|)}$ for $x\in \R^n,$ $1\le|j_1|\le 2,$ $0\le|j_2|\le 1$, $|j_2|=|j_1|-1$, $i,k=1... n$ and some $\alpha > 1$. We give estimates and asymptotics for scattering solutions and scattering data for the equation $(*)$ for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transforms $P\nabla V$ and $PB_{i,k}$ (on sufficiently rich sets of straight lines). Applying results on inversion of the X-ray transform $P$ we obtain that for $n\ge 2$ the velocity valued component of the scattering operator at high energies uniquely determines $(\nabla V,B)$. We also consider the problem of recovering $(\nabla V,B)$ from our high energies asymptotics found for the configuration valued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of [R. Novikov, 1999] for $(*)$ with $B\equiv 0$ and of [Jollivet, 2005] for the relativistic version of $(*)$. We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for $(*)$ on the one hand and for its relativistic version on the other.