Small angle scattering and X-ray transform in classical mechanics

Abstract

We consider the Newton equation $$\ddot{x} = F(x), \quad F(x) = -∇ \upsilon (x), \quad x \in \mathbf{R}^d,\\ \text{where }\upsilon \in C^2(\mathbf{R}^d, \mathbf{R}), |∂^j_x v(x) |\le c_{|j|}(1+|x|)^{-(\alpha+|j|)}$$ for |j|≤2 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 scattering data at high energies uniquely determine the X-ray transforms PF and Pv. Applying results on inversion of the X-ray transform P we obtain that for d≥2 scattering data at high energies uniquely determine F and v. For the case of potentials with compact support we give a connection between boundary value data and scattering data and for d≥2 we obtain, using known results, a uniqueness theorem in the inverse scattering problem at fixed energy.