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 $8<\nu <16\mathrm{GHz}$. The probability distribution $P\left(r\right)$ 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 nonoverlapping probability distribution $P_{\mathrm{sP}}^k\left(r\right)$, $k\ge 1$ 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.

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• Quantum chaos
• Wave chaos