Rudnick–Sarnak Quantum Unique Ergodicity conjecture

Let $(M,g)$ be a compact manifold of negative curvature without boundary, and let $(u_n)_{n\geq 1}$ be any orthogonal eigenbasis of the Laplacian. A semiclassical measure is a weak-* limit of the microlocal probability measures associated with a sequence of high-frequency eigenmodes; the Liouville measure is the normalized invariant measure on $S^*M$. Quantum Unique Ergodicity conjecture. There are no exceptional eigenmodes: the full sequence of eigenmodes is quantum ergodic, and there is a unique semiclassical measure, namely the Liouville measure. Thus all high-frequency eigenmodes equidistribute on $S^*M$, and

$$|u_n(x)|^2\,dg(x)\rightharpoonup \frac{dg(x)}{\operatorname{vol}(M)}$$

as $n\to\infty$. This strengthens the Quantum Ergodicity theorem by excluding exceptional density-zero subsequences and is expected to describe eigenmodes on negatively curved manifolds despite the many invariant measures of their geodesic flows. The conjecture remains open, even for manifolds of constant negative curvature.