On inverse scattering for the multidimensional relativistic Newton equation at high energies
Résumé
The purpose of this article is the generalization of part of the work 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. Consider the Newton equation in the relativistic case (that is the Newton-Einstein equation) $$\dot p =F(x), \F(x)=-\nabla v(x),\ p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}},\ x\in \R ^d, \leqno (*)$$ $${\rm where\ }v \in C^2(\R^d,\R),\ |\pa^j_x v(x)| \le \beta_{|j|}(1+|x|)^{-(\alpha+|j|)}$$ 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 first 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 first component of the scattering operator at high energies uniquely determines $F$. In addition we show that our high energy asymptotics found for the second component of the scattering operator doesn't determine uniquely $F$.