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

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.