Quantum Dynamics of a One Degree-of-Freedom Hamiltonian Saddle-Node Bifurcation

The paper empirically establishes, and heuristically motivates, four asymptotic behaviors for the saddle-node cubic potential—linear-in-D energy levels, mean position approaching the well minimum, asymptotically constant position uncertainty, and Wigner mass in the classically nonreactive region tending to one—using FD numerics on a finite interval and a harmonic-oscillator Taylor expansion around xe2 (Eqns. (21)–(22)) , with summaries/plots in Fig. 4 and Fig. 6 and a concluding statement that the Wigner mass approaches one as D increases . These results are correct in spirit but lack a rigorous proof and precise limiting hypotheses (e.g., spectral issues on ℝ for an unbounded-below cubic). The model solution supplies a more detailed perturbative argument (HO parity; Hellmann–Feynman; L1-control for Wigner continuity) that, if one assumes discrete eigenpairs (e.g., via box quantization or added confinement) and smooth eigen-branches, does justify the four claims. However, on ℝ the cubic potential admits no true L2 bound states; thus the model’s use of analytic perturbation for simple eigenvalues of H_HO+αy^3 requires an additional confining assumption that is not stated. Consequently, both the paper and the model omit essential spectral hypotheses, so both are incomplete. The conclusions themselves align: Eqn. (22) matches the model’s leading-order spectrum −D+ħ√(2√μ)(n+1/2) , the observed trends in ⟨x⟩ and Δx match the model’s limits (Fig. 4) , and the Wigner mass trend in Fig. 6 matches the model’s limiting claim , but neither side provides a fully rigorous treatment under the true (unboxed) Hamiltonian (1)–(3) .