Parabolic solutions for the planar $N$-centre problem: multiplicity and scattering

For the planar $N$-centre problem $\ddot x = - \sum_{i=1}^N \frac{m_i (x-c_i)}{| x - c_i|^{α+2}}, \qquad x \in \mathbb{R}^2 \setminus \{ c_1,\ldots,c_N \},$ where $m_i > 0$ for $i=1,\ldots,N$ and $α\in [1,2)$, we prove the existence of entire parabolic trajectories, having prescribed asymptotic directions for $t \to \pm\infty$ and prescribed topological characterization with respect to the set of the centres.

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