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

Alexandre Jollivet (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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 $d\ge 3.$

Mots clés: Newton-Einstein equation; scattering data; inverse problems

Domaines: Physique mathématique [math-ph]

Alexandre Jollivet. On inverse scattering in electromagnetic field in classical relativistic mechanics at high energies. Asymptotic Analysis, 2007, 55 (1-2), pp.103-123. ⟨hal-00005114v2⟩