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

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.