Quantum unique ergodicity on negatively curved surfaces
In plain words
On a curved surface where every classical orbit is chaotic, every high-frequency standing wave should spread evenly over the surface. This is proven only for special surfaces with number-theoretic symmetry.
Precise statement
Compact Riemannian surface $M$ of variable negative curvature, eigenfunctions $-\operatorname{Lap} \phi_{j} = \lambda_{j} \phi_{j}$ with $||\phi_{j}||_{2} = 1$. Prove or disprove that $|\phi_{j}|^{2} \,\mathrm{dvol} \to \mathrm{dvol}/\operatorname{vol}(M)$ weakly for the whole sequence (Rudnick-Sarnak conjecture). Answer: yes or no with proof.
What would settle it
A proof for all negatively curved surfaces, or a counterexample sequence of eigenfunctions with a non-uniform limit.
Status in the literature
Proven for Hecke eigenfunctions on arithmetic surfaces (Lindenstrauss, 2006; Soundararajan, 2010); density-one subsequences equidistribute by quantum ergodicity; for all negatively curved surfaces every semiclassical limit has full support (Dyatlov and Jin, 2018; Dyatlov, Jin and Nonnenmacher, 2022); Hassell's 2010 failure of QUE is for stadium billiards, which are not negatively curved surfaces.