GOE level spacings in a desymmetrized Sinai billiard
Let $$\Omega = \{(x, y) \in \mathbb{R}^2 : 0 < y < x < 1,\ x^2 + y^2 > 1/4\},$$ one eighth of the Sinai billiard obtained by removing the disk of radius $1/2$ about the origin from the square $(-1, 1)^2$; Dirichlet conditions on the two symmetry cuts keep a single symmetry class of that billiard. Let $0 < \lambda_1 \le \lambda_2 \le \cdots$ be the Dirichlet eigenvalues of $-\Delta$ on $\Omega$, counted with multiplicity, and set $\mu_j = |\Omega| \lambda_j / (4\pi)$. Let $F_{\mathrm{GOE}}$ be the bulk nearest-neighbour spacing distribution of the Gaussian orthogonal ensemble: if $\nu_1 \le \cdots \le \nu_n$ are the eigenvalues of an $n \times n$ real symmetric random matrix whose entries above the diagonal are independent centred Gaussians of variance $1/n$ and whose diagonal entries are independent centred Gaussians of variance $2/n$, and $k = \lfloor n/2 \rfloor$, then $$F_{\mathrm{GOE}}(s) = \lim_{n \to \infty} \mathbb{P}\Bigl\{ \frac{n}{\pi} (\nu_{k+1} - \nu_k) \le s \Bigr\}.$$ Bohigas, Giannoni and Schmit found the level fluctuations of the quantum Sinai billiard consistent with the Gaussian orthogonal ensemble, and conjectured that this agreement holds in general for time-reversal-invariant quantum systems whose classical analogues are strongly chaotic.
Problem. Is it true that $$\lim_{N \to \infty} \frac{1}{N} \#\{1 \le j \le N : \mu_{j+1} - \mu_j \le s\} = F_{\mathrm{GOE}}(s) \quad \text{for every } s \ge 0\,?$$
Source: Bohigas, Giannoni and Schmit (1984), as a conjecture for time-reversal-invariant quantum systems whose classical analogues are strongly chaotic, with Sinai's billiard as their example. Stated by AIM — Open Applied Problems for this domain and the nearest-neighbour spacing distribution.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, level spacings, quantum chaos, random matrices
References
@article{bohigas1984,
author = {Bohigas, O. and Giannoni, M. J. and Schmit, C.},
title = {Characterization of chaotic quantum spectra and universality of level fluctuation laws},
journal = {Phys. Rev. Lett.},
volume = {52},
number = {1},
year = {1984},
pages = {1--4},
doi = {10.1103/PhysRevLett.52.1},
}
@article{bohigas1984jpl,
author = {Bohigas, O. and Giannoni, M. J. and Schmit, C.},
title = {Spectral properties of the {L}aplacian and random matrix theories},
journal = {J. Physique Lett.},
volume = {45},
number = {21},
year = {1984},
pages = {L1015--L1022},
doi = {10.1051/jphyslet:0198400450210101500},
}
@misc{aim-sinai-goe-spacing,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{GOE level spacings in a desymmetrized Sinai billiard}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/211-sinai-goe-spacing.md},
note = {Entry 211 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00472, "GOE level spacings in a desymmetrized Sinai billiard", clopen.solutions.