On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies

Details On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies

2005-06-02

arxiv.org/abs/math-ph/0506008v2

Asympotic Analysis 55, 1-2 (2007) 103-123

Physics

Mathematical Physics

Scientific paper

We consider the multidimensional Newton-Einstein equation in static electromagnetic field $$\eqalign{\dot p = F(x,\dot x), F(x,\dot x)=-\nabla V(x)+{1\over c}B(x)\dot x,\cr p={\dot x \over \sqrt{1-{|\dot x|^2 \over c^2}}}, \dot p={dp\over dt}, \dot x={dx\over dt}, x\in C^1(\R,\R^d),}\eqno{(*)}$$ where $V \in C^2(\R^d,\R),$ $B(x)$ is the $d\times d$ real antisymmetric matrix with elements $B\_{i,k}(x)={\pa\over \pa x\_i}\A\_k(x)-{\pa\over \pa x\_k}\A\_i(x)$, and $|\pa^j\_x\A\_i(x)|+|\pa^j\_x V(x)| \le \beta\_{|j|}(1+|x|)^{-(\alpha+|j|)}$ for $x\in \R^d,$ $|j| \le 2,$ $i=1..d$ 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 transforms $P\nabla V$ and $PB\_{i,k}$ for $i,k=1..d,$ $i\neq k.$ 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 $(V,B)$. In addition we show that our high energy asymptotics found for the configuration valued component of the scattering operator doesn't determine uniquely $V$ when $d\ge 2$ and $B$ when $d=2$ but that it uniquely determines $B$ when $d\ge 3.$

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