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

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 $\alpha x^2{y}^2$, and the purely quartic Yang--Mills Hamiltonian $H=\frac{1}{2}(p_x^2+p_y^2)+\alpha x^2{y}^2$. 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 $\alpha <0$, whereas in the Yang--Mills case with $\alpha >0$, escape occurs along the coordinate axes $x=0$ or $y=0$. In sharp contrast, the quantum Yang--Mills Hamiltonian with $\alpha >0$ 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.

arXiv comment25 pages, 25 figures