Statistical analysis of level spacing ratios in pseudo-integrable systems: semi-Poisson insight and beyond

• URL: http://arxiv.org/abs/2505.16656v1
• Date: Thu, 22 May 2025 13:22:37 GMT
• Title: Statistical analysis of level spacing ratios in pseudo-integrable systems: semi-Poisson insight and beyond
• Authors: Afshin Akhshani, Małgorzata Białous, Leszek Sirko,
• Abstract summary: We study the statistical properties of a quantum system in the pseudo-integrable regime.<n>We show that the system exhibits semi-Poisson behavior in the frequency range $8 nu 16 $ GHz.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We studied the statistical properties of a quantum system in the pseudo-integrable regime through the gap ratios between consecutive energy levels of the scattering spectra. A two-dimensional quantum billiard containing a point-like (zero-range) perturbation was experimentally simulated by a flat rectangular resonator with wire antennas. We show that the system exhibits semi-Poisson behavior in the frequency range $8 <\nu < 16 $ GHz. The probability distribution $P(r)$ of the studied system is characterized by the parameter $\xi=0.97 \pm 0.03 $, with the expected value $\xi=1$ for the short-range plasma model. Furthermore, we provide a theoretical expression for the higher-order non-overlapping probability distribution $P_{\mathrm{sP}}^k(r)$, $k \geq 1$, in the semi-Poisson regime, incorporating long-range spectral correlations between levels. The experimental and numerical results confirm the pseudo-integrability of the studied system. The semi-Poisson ensemble, for $k=2$, approaches the GUE distribution. In addition, the uncorrelated Poisson statistics mimic the RMT ensembles at certain $k$ values, $k=4$ for GUE and $k=7$ for GSE. This unexpected scale-dependent convergence shows how spectral statistics can exhibit chaos-like features even in non-chaotic systems, suggesting that scale-dependent analysis bridges integrable and chaotic regimes.

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