多维高能相对论性牛顿方程的逆散射
On inverse scattering for the multidimensional relativistic Newton equation at high energies
摘要：
考虑相对论情况下的牛顿方程（即牛顿-爱因斯坦方程）$$\eqalign{\dot p = F(x),& F(x)=-\nabla V(x),\cr p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}},& \dot p={dp\over dt}, \dot x={dx\over dt}, x\in C^1(\R,\R^d),}\eqno{(*)}$$ $${\rm where\}V \in C^2(\R^d,\R), |\pa^j\_x V(x)| \le β\_{|j|}(1+|x|)^{-(α+|j|)}$$对于$|j| \le 2$和某些$α> 1$。对于小角散射的情形，我们给出了方程$(*)$的散射解和散射数据的估计和渐近性。我们表明，在高能量的散射算子的速度值的分量唯一地确定X射线变换$PF.$-应用结果反演的X射线变换$P$，我们得到的$d\ge 2$的速度值的分量的散射算子在高能量唯一地确定$F$。此外，我们发现我们的高能量渐近散射算子的配置值组件不确定唯一的$F$。本文的结果是在推广Novikov [R.G.陈晓，小角散射和X射线变换在经典力学中的应用。Mat. 37，pp. 141-169（1999）]相对论的情况。
英文摘要：
Consider the Newton equation in the relativistic case (that is the Newton-Einstein equation) $$\eqalign{\dot p = F(x),& F(x)=-\nabla V(x),\cr p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}},& \dot p={dp\over dt}, \dot x={dx\over dt}, x\in C^1(\R,\R^d),}\eqno{(*)}$$ $${\rm where\}V \in C^2(\R^d,\R), |\pa^j\_x V(x)| \le β\_{|j|}(1+|x|)^{-(α+|j|)}$$ for $|j| \le 2$ and some $α> 1$. 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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