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

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

2005-02-16

arxiv.org/abs/math-ph/0502040v2

Journal of Mathematical Physics 47, 6 (2006) 062902

Physics

Mathematical Physics

Scientific paper

10.1063/1.2206690

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.

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