The spatial $N$-centre problem: scattering at positive energies

For the spatial generalized $N$-centre problem $$\ddot{x}=-\sum _{i=1}^N\frac{m_i(x-c_i)}{\mathrm{\mid }x-c_i{\mathrm{\mid }}^{\alpha +2}},x\in {\mathbb{R}}^3\setminus \{c_1,\dots ,c_N\},$$ where $m_i>0$ and $\alpha \in [1,2)$, we prove the existence of positive energy entire solutions with prescribed scattering angle. The proof relies on variational arguments, within an approximation procedure via (free-time) boundary value problems. A self-contained appendix describing a general strategy to rule out the occurrence of collisions is also included.