Classical n-body scattering with long-range potentials

The paper rigorously establishes: (i) the explicit finally free region F^+_loc (Def. 1.9) is forward invariant and every free orbit eventually enters it, with linear growth 1/2 v_ij(0) t ≤ q_ij(t)−q_ij(0) ≤ 3/2 v_ij(0) t and ∂F^+_loc a global C^0 surface of section (Theorem 2.2 and (2.5)) ; (ii) the asymptotic velocity map v^+ is C^{k−1} on F^+ (Theorem 3.3) ; (iii) for short range (α>1) the Møller map Ω: F_0→F^+ is a C^{k−1} symplectomorphism with quantitative near-identity bounds (4.3)–(4.4) (Theorem 4.2) ; and (iv) for long range α∈(1/2,1] the inverse Dollard–Møller maps Ω^{-1,±} exist, are C^{k−2} symplectomorphisms that conjugate the flow and yield the affine fiber structure over fixed asymptotic velocity (Theorem 5.3) , with the Dollard correction W(t;p) explicitly defined in (5.3) . The candidate solution largely mirrors these arguments but makes a critical error in Section (2): it asserts that the ‘impact parameter’ b^+(x)=lim_{t→∞}(q−t v^+(x)) exists for long-range potentials because v(t)−v^+ is L^1. This is false for α≤1, where q(t) has logarithmic or t^{1−α} corrections (Chazy-type asymptotics) unless one subtracts the Dollard term W(t;Mv) . The paper explicitly avoids this pitfall by using Dollard’s modification and proves the affine fiber statement via Theorem 5.3(4) rather than a naive b^+ limit .