On inverse scattering for the multidimensional relativistic Newton equation at high energies

Abstract

Consider the Newton equation in the relativistic case (that is the Newton-Einstein equation) $$\eqalign{\dot p = F(x),& F(x)=-\nabla V(x),\cr p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}},& \dot p={dp\over dt}, \dot x={dx\over dt}, x\in C^1(\R,\R^d),}\eqno{(*)}$$ $${\rm where\}V \in C^2(\R^d,\R), |\pa^j\_x V(x)| \le \beta\_{|j|}(1+|x|)^{-(\alpha+|j|)}$$ for $|j| \le 2$ 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 transform $PF.$ Applying results on inversion of the X-ray transform $P$ we obtain that for $d\ge 2$ the velocity valued component of the scattering operator at high energies uniquely determines $F$. In addition we show that our high energy asymptotics found for the configuration valued component of the scattering operator doesn't determine uniquely $F$. The results of the present work were obtained in the process of generalizing some results of Novikov [R.G. Novikov, Small angle scattering and X-ray transform in classical mechanics, Ark. Mat. 37, pp. 141-169 (1999)] to the relativistic case.