Abstract

We consider the multidimensional nonrelativistic Newton equation in a static electromagnetic field

where V ∈ C ²(, ℝ), B(x) is the n × n real antisymmetric matrix with elements B_i,k(x), B_i,k ∈ C ¹(, ℝ) (and B satisfies the closure condition), and ≤ β _|j1|(1 + |x|)^{–(α + |j₁|)} for x ∈ , 1 ≤ |j ₁| ≤ 2, 0 ≤ |j ₂| ≤ 1, |j ₂| = |j ₁| – 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 PB_i,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 Novikov [Ark. Mat. 37: 141–169, 1999] for (∗) with B ≡ 0 and of Jollivet [J. Math. Phys. 47: 062902, 2006] 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.