Statistical analysis of level-spacing ratios in pseudointegrable systems: Semi-Poisson insight and beyond

We studied the statistical properties of a quantum system in the pseudointegrable 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 using a flat rectangular resonator with wire antennas. We show that the system exhibits semi-Poisson behavior in the frequency range 8<ν<16GHz. The probability distribution P(r) of the studied system is characterized by the parameter ξ=0.97±0.03, with the expected value ξ=1 for the short-range plasma model. Furthermore, we provide a theoretical expression for the higher-order nonoverlapping probability distribution P_{sP}^{k}(r), k≥1 in the semi-Poisson regime, incorporating long-range spectral correlations between levels. The experimental and numerical results confirm the pseudointegrability of the studied system. The semi-Poisson ensemble for k=2 approaches the Gaussian orthogonal ensemble distribution. In addition, the uncorrelated Poisson statistics mimic Random Matrix Theory ensembles at certain k values, k=4 for the Gaussian unitary ensemble and k=7 for the Gaussian symplectic ensemble. This unexpected scale-dependent convergence shows how spectral statistics can exhibit chaos-like features even in nonchaotic systems, suggesting that scale-dependent analysis bridges integrable and chaotic regimes.