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

p^·=F(x, x^·), F(x, x^·)=−∇V(x)+(1/c)B(x)x^·,

p=x^·/\sqrt(1−|x^·|²/c²)), p^·=dp/dt, x^·=dx/dt, x∈C¹(R, R^d),  (*)

where V∈C²(R^d, R), B(x) is the d×d real antisymmetric matrix with elements B_i,k(x)=∂/∂x_i A_k(x)−∂/∂x_k A_i(x), and |∂_x^jA_i(x)|+|∂_x^jV(x)|≤β_|j|(1+|x|)^−(α+|j|) for x∈R^d, |j|≤2, i=1, …, d, and some α>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∇V and PB_i,k for i, k=1, …, d, i≠k. Applying results on inversion of the X-ray transform P we obtain that for d≥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≥2 and B when d=2 but that it uniquely determines B when d≥3.