arXiv subjects

Adrian M. Escobar Ruiz

Publications and source records attributed to Adrian M. Escobar Ruiz.

4 recordsLinked to original sources

arXiv2026

Singular barriers and quartic integrability breaking in the TTW system

We study the competition between resonance-induced integrability breaking and centrifugal confinement in a symmetric quartic deformation of the classical $k=1$ Tremblay--Turbiner--Winternitz (TTW) system, $H=\frac{1}{2}\left(P_X^2+P_Y^2\right)+X^2+Y^2 +\fracγ{X^2}+\fracγ{Y^2}+κX^2Y^2.$ The inverse-square terms are the centrifugal remnants of an $SO(2)\times SO(2)$ reduction of a four-dimensional oscillator, while the $X^2Y^2$ interaction couples the reduced radial modes and connects the Smorodinsky--Winternitz and Contopoulos limits. For $γ>0$, the axes are impenetrable and split configuration space into invariant sectors. For sufficiently small nonzero $κ$, resonant averaging constructs a phase-locked periodic orbit on each energy surface $H=h>4\sqrtγ$ whose fixed-energy reduced Poincaré map has no unit characteristic multiplier. Poincaré's criterion therefore excludes a second independent $C^1$ first integral in any invariant neighbourhood of this orbit. At finite coupling, Poincaré sections and finite-time Lyapunov maps show the breakup of invariant curves and the growth of chaotic layers. Comparison with the barrier-free limit separates two effects: the quartic interaction breaks integrability, whereas the singular barriers reduce phase-space connectivity and chaotic transport without restoring it.

math-ph↗

arXiv2025

Wigner-Husimi phase-space structure of quasi-exactly solvable sextic potential

In this study, we compare the Wigner function $W$, its modulus, and the Husimi distribution $H$ in a one-dimensional quantum system exhibiting a transition from a single-well to a double-well configuration, using the quasi-exactly solvable sextic oscillator as a representative example. High-accuracy variational wavefunctions for the lowest states are used to compute two-dimensional phase-space structures, one-dimensional marginals, and the corresponding Shannon entropies, mutual information, and Cumulative Residual Jeffreys divergences. The analysis shows that the Wigner representation is uniquely responsive to interference effects and displays clear, nonmonotonic entropic behavior as the wells separate, whereas the modulus-Wigner and Husimi distributions account only for geometric splitting or coarse-grained delocalization. These findings establish a quantitative hierarchy in the ability of $W$, $|W|$, and $H$ to resolve structural changes in a quantum state and provide a general framework for assessing the descriptive power of different phase-space representations in systems with emerging bimodality or tunneling.

quant-ph↗

arXiv2025

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 $α\, x^{2}y^{2}$, 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.

nlin.CD↗

arXiv2025

Entropic characterization of Tunneling and State Pairing in a Quasi-Exactly Solvable Sextic Potential

We analyze the (de)localization properties of a quasi-exactly solvable (QES) sextic potential $V_{\text{QES}}(x) = \frac{1}{2}(x^6 + 2x^4 - 2(2λ+ 1)x^2)$ as a function of the tunable parameter $λ\in [-\frac{3}{4}, 6]$. For $λ> -\frac{1}{2}$, the potential exhibits a symmetric double-well structure, with tunneling emerging for the ground state level at $λ\approx 0.732953$. {For the lowest energy states $ n = 0,1,2,3 $, we construct physically meaningful variational wavefunctions that $i)$ respect parity symmetry under the transformation $x \rightarrow -x$, $ii)$ exhibit the correct asymptotic behavior at large distances, and $iii)$ allow for exact analytical Fourier transforms. Variational energies match Lagrange Mesh and available exact analytical QES results with relative errors $\simeq 10^{-8}$ for $n = 0, 1, 2$ and $\simeq 10^{-6}$ for the third excited state $n=3$. We demonstrate that entropic measures (Shannon entropy, Kullback-Leibler, and Cumulative Residual Jeffreys divergences) surpass conventional variance-based methods in revealing tunneling transitions, wavefunction symmetry breaking, and quantum state pairing. Our results confirm that the Beckner-Bialynicki-Birula-Mycielski entropic uncertainty relation holds across all examined values of $n$ and $λ$. The quality of the trial function is also validated by the small $\sim10^{-10}$ Cumulative Residual Jeffreys divergences from the exact QES solutions.

quant-ph↗