Classical-quantum study of confinement in the chaotic $x^{2}y^{2}$ Yang-Mills Hamiltonian

TL;DR

This work resolves the paradox of classical chaos and quantum confinement by showing that quantum fluctuations restore a discrete spectrum in two planar $x^{2}y^{2}$ Hamiltonians: the Contopoulos model with quadratic confinement and the pure Yang–Mills potential. Classical analysis via Poincaré sections, Lyapunov heat maps, and symmetry-line periodic orbits reveals distinct escape channels (diagonals for $\alpha<0$ and axes for $\alpha>0$) and chaotic regimes. Quantum mechanically, variational and Lagrange-m mesh methods yield high-precision spectra, with YM ($\alpha>0$) showing discrete, degenerate levels organized by $D_4$ symmetry, while Contopoulos experiences degeneracy lifting but remains confined. The semiclassical mechanism—transverse zero-point motion generating a linear barrier along open directions—provides a unified explanation for confinement across regimes, corroborated by 1D and 2D numerics, WKB analyses, and analog electronic simulations that validate the theoretical framework. These results illuminate the quantum-classical correspondence in chaotic confinement and point to experimental platforms for studying chaos-driven confinement and mobility-edge phenomena.

Abstract

We analyze how quantum mechanics reinstates confinement in Hamiltonian systems that are classically unstable and exhibit chaotic dynamics. Specifically, we consider two paradigmatic models: the Contopoulos Hamiltonian, an isotropic oscillator perturbed by the quartic coupling $α\, x^{2}y^{2}$, and the purely quartic Yang--Mills Hamiltonian $H=\tfrac{1}{2}(p_{x}^{2}+p_{y}^{2})+α\, x^{2}y^{2}$. Classical dynamics, characterized through Poincaré sections, Lyapunov exponents, and periodic orbits, reveals distinct escape mechanisms: in the Contopoulos system, trajectories destabilize along the diagonal valleys $x=\pm y$ for $α<0$, whereas in the Yang--Mills case with $α>0$, escape occurs along the coordinate axes $x=0$ or $y=0$. In sharp contrast, the quantum Yang--Mills Hamiltonian with $α>0$ admits only discrete, normalizable eigenstates. Semiclassical WKB and full two--dimensional analyses further show that these quantum states are localized along the classical escape channels, illustrating how transverse zero--point motion generates an effective confining barrier. Our study combines global Lyapunov--exponent heat maps with high--precision quantum spectra obtained via variational and Lagrange--mesh methods, providing quantitatively controlled results across regimes. In addition, we corroborate the classical predictions through analog electronic simulations based on operational--amplifier circuit models, offering an experimentally inspired validation of the theoretical framework.

Figures (25)

• Figure 1: Potential landscapes of Hamiltonian \ref{['hamiltoniano']}.
• Figure 2: Poincaré sections (right) and largest Lyapunov exponents (left) on the $(y,p_y)$ plane for the Contopoulos Hamiltonian with $\alpha>0$ at different energies $E$. Symmetry lines are indicated in each plot. Subfigures (a)–(b), (c)–(d), (e)–(f), and (g)–(h) correspond to regions (I), (II), (III), and (IV) in \ref{['positivo']}. As the energy increases, motion remains bounded but chaotic regions grow, illustrating the progressive destruction of invariant tori.
• Figure 3: Poincaré sections (right) and Lyapunov exponents (left) on the $(y,p_y)$ plane for the Contopoulos Hamiltonian with $\alpha<0$ at different energies $E$. Symmetry lines are displayed. Subfigures (a)–(b) and (c)–(d) correspond to regions (A) and (B) in \ref{['negativo']}. For $\alpha<0$, real saddles open escape channels along $x=\pm y$: at low energy trajectories remain confined, while above the threshold chaotic escape dominates.
• Figure 4: Global Lyapunov heat map of chaos. The average LLE in the $(\alpha,E)$ plane. The green curve marks the classical escape threshold. For $\alpha<0$, trajectories above this line escape along $x=\pm y$, while for $\alpha>0$ motion remains bounded but increasingly chaotic. Lines in blue and violet denote the ground- and first excited energies of the quantum Contopoulos system, respectively.
• Figure 5: Representative periodic orbit $\gamma_1$ at $(\alpha=-1/2,E=3/5)$ obtained from symmetry–line intersections. Panels show (a) a phase–space magnification of \ref{['mapa_neg_rojo']} highlighting the orbit, and (b) the time series $y(t)$. The corresponding initial conditions are given in \ref{['B']}.