Spectral stability, spectral flow and circular relative equilibria for the Newtonian n-body problem

The paper’s Theorem 1.1 states the exact parity criterion on E3 (using 8n−16 or 8n−20 depending on the planar/non-collinear case) that implies spectral instability of the circular relative equilibrium, after reducing by symmetries and detailing the dimensions of E3 . In Section 4 the authors prove a mod 2 spectral-flow identity: if σ(JA) ⊂ iR then n−(A) − n−(A|ker(JA)2p) ≡ 0 (mod 2) (Theorem 4.1), which is exactly the parity obstruction used to derive Theorem 5.1 and hence Theorem 1.1 . The candidate solution reaches the same conclusion by invoking the finite-dimensional Hamiltonian–Krein index formula to deduce that an odd parity forces k_r ≥ 1 and hence a non-purely-imaginary eigenvalue, i.e., spectral instability. Thus, both are correct; the paper uses spectral flow (with a new treatment for degenerate starting points), while the model uses the Hamiltonian–Krein index. The model’s argument is slightly stronger in that it identifies a real positive eigenvalue when the parity is odd, consistent with the paper’s weaker instability conclusion.