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- W3106424119 abstract "Consider the Newton equation in the relativistic case (that is the Newton-Einstein equation) $$eqalign{dot p = F(x),& F(x)=-nabla V(x),cr p={dot x over sqrt{1-{|dot x|^2 over c^2}}},& dot p={dpover dt}, dot x={dxover dt}, xin C^1(R,R^d),}eqno{(*)}$$ $${rm where}V in C^2(R^d,R), |pa^j_x V(x)| le beta_{|j|}(1+|x|)^{-(alpha+|j|)}$$ 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 $dge 2$ 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." @default.
- W3106424119 created "2020-11-23" @default.
- W3106424119 creator A5034590869 @default.
- W3106424119 date "2006-06-01" @default.
- W3106424119 modified "2023-10-07" @default.
- W3106424119 title "On inverse scattering for the multidimensional relativistic Newton equation at high energies" @default.
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- W3106424119 doi "https://doi.org/10.1063/1.2206690" @default.
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