Rudnick–Sarnak's quantum unique ergodicity conjecture

Let $N$ be a compact Riemannian manifold with negative sectional curvature, let $(\Psi_n)_{n=0}^{\infty}$ be an orthonormal basis of Laplace eigenfunctions on $N$, and let $\mu_n$ be the Wigner measure associated to $\Psi_n$. Rudnick–Sarnak's quantum unique ergodicity conjecture. The Wigner measures converge weak-* to the Liouville measure:

$$\mu_n\xrightarrow[n\to\infty]{\operatorname{wk-*}}\mu_{L,N}.$$

In other words, the Liouville measure is the unique quantum limit. This strengthens quantum ergodicity, which gives convergence only along a density-one subsequence; the conjecture asserts convergence for the entire eigenbasis on negatively curved manifolds.