On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies
Jollivet, Alexandre
arXiv, 0506008 / Harvested from arXiv
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 $d\ge 3.$
Publié le : 2005-06-02
Classification:  Mathematical Physics,  34A55 83A05 78A46 44A12
@article{0506008,
     author = {Jollivet, Alexandre},
     title = {On inverse scattering in electromagnetic field in classical relativistic
  mechanics at high energies},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0506008}
}
Jollivet, Alexandre. On inverse scattering in electromagnetic field in classical relativistic
  mechanics at high energies. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0506008/