Perturbed quantum billiards on the hyperbolic plane

• URL: http://arxiv.org/abs/2607.03786v1
• Date: Sat, 04 Jul 2026 09:19:57 GMT
• Title: Perturbed quantum billiards on the hyperbolic plane
• Abstract summary: We study the emergence of generic quantum chaos from an arithmetic billiard in the hyperbolic plane.<n>We show a perturbation-dependent crossover from Poisson-like arithmetic behaviour to the Gaussian ensemble (GOE)<n>We quantify phase-space localization using an entropy-based measure.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the emergence of generic quantum chaos from an arithmetic billiard in the hyperbolic plane. Starting from an arithmetic triangle, we introduce small area-preserving geometric perturbations and compute long consecutive sequences of eigenvalues and eigenstates. The spectral statistics show a perturbation-dependent crossover from Poisson-like arithmetic behaviour to the Gaussian orthogonal ensemble (GOE), with the crossover scale moving to higher wavenumbers as the perturbation decreases. To characterize the nearest-neighbour spacing distribution, we compare the data with Brody and generalized-gamma fits. The generalized-gamma form is used only as a two-parameter diagnostic, allowing the small- and intermediate-spacing structure to vary independently from the large-spacing tail, rather than as a universal crossover law. For the weakest perturbation, we also compare the data with an effective block-GOE model, which diagnoses residual block-like structure deep into the computed spectral range. We complement the spectral analysis with Poincaré-Husimi representations of the eigenstates and quantify phase-space localization using an entropy-based measure. Its distribution is well described by Beta distributions whose width decays as a power law in the wavenumber. Before the spectral crossover, the localization statistics retain memory of the near-arithmetic regime. After the GOE regime is reached, the localization-fluctuation widths collapse onto a common decay law, sigma(Beta) ~ k^{-eta}, with eta approximately 0.37--0.38. These results provide a combined spectral and eigenfunction-level picture of the arithmetic-to-GOE crossover in hyperbolic quantum billiards.

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