Bohigas–Giannoni–Schmit (BGS) conjecture for Laplacian spectra on negatively curved surfaces

Establish that for any closed, connected Riemannian surface with negative curvature, in the high-energy regime and for energy windows [λ, λ+W] satisfying 1 ≪ W ≪ √λ, the number variance of the eigenvalues of the positive Laplacian Δ_g equals that of the Gaussian Orthogonal Ensemble in the time-reversal invariant case (and the Gaussian Unitary Ensemble when time-reversal symmetry is broken), as predicted by the Bohigas–Giannoni–Schmit conjecture.

Background

The paper reviews the Bohigas–Giannoni–Schmit (BGS) conjecture, which predicts that quantum systems with chaotic classical dynamics exhibit universal spectral statistics matching random matrix ensembles. In the context of closed negatively curved surfaces, the geodesic flow is Anosov (chaotic), and the conjecture predicts GOE/GUE statistics for Laplacian eigenvalues in the high-energy limit.

Despite extensive numerical evidence supporting BGS for negatively curved surfaces, there is currently no rigorously proved instance for any surface. Moreover, certain arithmetic hyperbolic surfaces are known to deviate from GOE statistics, underlining the challenge of proving the conjecture in general.