Generalized Periodic Orbits of Time-Periodically Forced Kepler Problem Accumulating the Center and of Circular and Elliptic Restricted Three-Body Problems

The paper proves that a time‑periodically forced Kepler problem admits infinitely many generalized 1‑periodic solutions accumulating the center in all dimensions by: (i) spatial localization; (ii) Kepler rescaling to make the perturbation C1‑small (but not C2‑small) near a Morse–Bott family of periodic manifolds of the regularized Kepler flow; (iii) Moser regularization to a Zoll Reeb flow on S*Sd; and (iv) a localized homotopy‑stretching/Rabinowitz–Floer argument that yields periodic orbits bifurcating from those manifolds for sufficiently large scaling index n, leading to infinitely many distinct generalized periodic orbits with action gaps controlled by Proposition 2.4. The main steps, objects (Λn), and the need for only C1‑smallness are explicitly stated and executed (e.g., non‑degeneracy and action gaps of Λn, the rescaled C1‑estimate O(κn), and the appeal to Theorem A.1 in Appendix A) . The candidate solution tracks the same framework (localization, Kepler scaling, Moser regularization to a neighborhood of the Zoll Reeb flow, and a C1‑persistence/homotopy‑stretching step), so the overall argument aligns with the paper. However, the candidate introduces a few technical inaccuracies: (a) it asserts an O(s^2) smallness and O(s^3) derivative scaling where the paper carefully shows C1‑smallness O(κn) and warns that C2 may blow up due to the fast angle δ; (b) it states that the actions of the resulting orbits “tend to 0,” while the paper computes actions ∼ κn^{-2} → ∞ and uses action gaps to separate orbits; and (c) it informally treats the entire S*Sd × S1 as a single Morse–Bott family of 1‑periodic orbits, whereas the paper works with the discrete Morse–Bott families Λn with prime period 1/n and then rescales to a fixed normalized Λ1 . These do not affect the core conclusion but should be corrected for rigor.