Quantum unique ergodicity on negatively curved manifolds

Prove the quantum unique ergodicity conjecture for compact negatively curved Riemannian manifolds by showing that all Laplace–Beltrami eigenfunctions equidistribute in the high-frequency limit.

Background

In the historical context presented by the authors, quantum ergodicity was first established for compact manifolds under ergodic geodesic flow, showing equidistribution for a density-one subsequence of eigenfunctions. Rudnick and Sarnak formulated the stronger quantum unique ergodicity (QUE) conjecture asserting equidistribution for all eigenfunctions.

Despite significant progress over the past decades, the full QUE conjecture for compact negatively curved manifolds remains unresolved, and the authors explicitly note its open status in their introduction.

References

Later on, Rudnick and Sarnak made a stronger, so-called quantum unique ergodicity conjecture: for compact negatively curved manifolds, all Laplace eigenfunctions should become equidistributed in the high frequency limit. The conjecture is still widely open, but significant progress has been made in the last twenty years.

— Quantum ergodicity for Dirichlet-truncated operators on $\mathbb{Z}^d$  (2505.02339 - Cao et al., 5 May 2025) in Section 1 (Introduction)

Whether any such sequence survives on a negatively curved manifold is the subject of the quantum unique ergodicity conjecture of Rudnick and Sarnak, which says the exceptional set is empty there.

— Classical and Quantum Chaos from Foundations to Holography  (2608.19131 - Shukla, 19 Aug 2026) in Section 3, subsection “Where quantum signatures of chaos appear” (label sec:L3_signatures)