Quantum unique ergodicity

On a compact Riemannian manifold of negative curvature, every orthonormal basis of Laplace eigenfunctions becomes equidistributed in phase space in the high-energy limit — no subsequence can scar on a closed geodesic.

Shnirelman (1974), Zelditch and Colin de Verdière proved quantum ergodicity: a density-one subsequence of eigenfunctions equidistributes. Rudnick–Sarnak (1994) conjectured the full sequence does on negatively curved manifolds. Lindenstrauss (2006) proved it for arithmetic surfaces using ergodic theory on homogeneous spaces — work cited in his 2010 Fields Medal. Anantharaman–Nonnenmacher showed macroscopic mass cannot concentrate too far (entropy bounds), but scars on unstable orbits are still not ruled out in general: the conjecture is open beyond the arithmetic case.