On inverse scattering at high energies for the multidimensional Newton equation in electromagnetic field

where V ∈ C(R,R), B(x) is the n × n real antisymmetric matrix with elements Bi,k(x), Bi,k ∈ C(R,R) (and B satisfies the closure condition), and |∂j1 x V (x)| + |∂j2 x Bi,k(x)| ≤ β|j1|(1 + |x|)−(α+|j1|) for x ∈ R, 1 ≤ |j1| ≤ 2, 0 ≤ |j2| ≤ 1, |j2| = |j1| − 1, i, k = 1 . . . n 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 PBi,k (on sufficiently rich sets of straight lines). Applying results on inversion of the X-ray transform P we obtain that for n ≥ 2 the velocity valued component of the scattering operator at high energies uniquely determines (∇V,B). We also consider the problem of recovering (∇V,B) from our high energies asymptotics found for the configuration valued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of [R. Novikov, 1999] for (∗) with B ≡ 0 and of [Jollivet, 2005] for the relativistic version of (∗). We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for (∗) on the one hand and for its relativistic version on the other.