MATHPH
In the literature: open
Random-matrix pair correlation for a specific chaotic billiard
In plain words
Prove for one concrete chaotic shape, such as a stadium-shaped drum, that its high vibration frequencies have random-matrix spacing statistics. No single example is proven.
Precise statement
Dirichlet Laplacian on the Bunimovich stadium or the Sinai billiard, eigenvalues $\lambda_n \to \infty$ unfolded by Weyl's law. Prove that the pair correlation converges to the GOE two-point function, or at least that the spectral form factor satisfies $K(\tau) = 2 \tau - \tau \log(1 + 2 \tau)$ for small $\tau > 0$. Answer: a proof.
What would settle it
A proof for one billiard or one negatively curved surface without arithmetic symmetry.
Related problems
- Special case of Why chaotic quantum spectra follow random-matrix statistics