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

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 $V\in C^2({\mathbb{R}}^d,\mathbb{R}),$ $B(x)$ 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 $|\pa^j\_x\A\_i(x)|+|\pa^j\_x V(x)| \le β\_{|j|}(1+|x|)^{-(α+|j|)}$ for $x\in {\mathbb{R}}^d,$ $|j|\le 2,$ $i=\mathrm{1..}d$ and some $\alpha >1$. We give estimates and asymptotics for scattering solutions and scattering data for the equation $(\ast )$ 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 $i,k=\mathrm{1..}d,$ $i\ne k.$ 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 $d\ge 3.$

References in corpus (1)

• The d-bar approach to approximate inverse scattering at fixed energy in three dimensions

Cited by in corpus (4)

• Aharonov-Bohm Effect and High-Momenta Inverse Scattering for the Klein-Gordon Equation
• On inverse scattering at high energies for the multidimensional Newton equation in electromagnetic field
• On inverse scattering at high energies for the multidimensional relativistic Newton equation in a long range electromagnetic field
• Inverse scattering at fixed energy for the multidimensional Newton equation in short range radial potentials