WHISPERING GALLERY ORBITS IN SINAI OSCILLATOR TRAP

The paper’s Theorem 1 states that, under E > V0(s) and V1(s) > 2k(s)(E − V0(s)) > 0 for all s, any orbit starting sufficiently close to the boundary remains in a uniform O(ε2) slab for all forward time; this is obtained by reducing the dynamics in boundary coordinates, reparametrizing with s as time via K = −(1 + k(s)|r|)√(2E − p_r^2 − 2V), and then using adiabatic/KAM machinery on the reduced non-smooth Hamiltonian H = y^2/(2a(λ)) + b(λ)|x| + εH1, where b(λ)>0 follows exactly from V1 > 2k(E−V0) (equations (3)–(8) and Theorem 1) . The candidate solution proves the same claim by an alternative, elementary route: starting from the same s-time reduction (equivalent to the paper’s K), it establishes uniform bounds c0|p| ≤ |dr/ds| ≤ C0|p| and a negative drift dp/ds ≤ −γ0 + C|r| − c p^2 inside a small slab, where γ0>0 derives directly from V1 − 2k(E−V0) > 0. Iterating across impacts (p→−p at r=0), it constructs a forward-invariant slab |r| ≤ Cε2, |p| ≤ C′ε for all s. These steps are consistent with the paper’s reduction and hypotheses and yield the same near-boundary confinement without invoking KAM, while the paper proceeds further to produce invariant curves via Moser’s twist theorem and to state an adiabatic invariant for the billiard map . Hence both arguments are correct; they rely on the same structural reduction but differ in technique.