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

Research paper by **Alexandre Jollivet**

Indexed on: **12 Jul '06**Published on: **12 Jul '06**Published in: **Mathematical Physics**

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#### Abstract

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 $d\ge 3.$