Quantum unique ergodicity on negatively curved surfaces
For every compact connected smooth surface without boundary and with strictly negative curvature, does every sequence of normalized Laplace eigenfunctions with eigenvalues tending to infinity have Liouville semiclassical measure, including arbitrary choices wi
Problem statement
For every compact connected smooth surface without boundary and with strictly negative curvature, does every sequence of normalized Laplace eigenfunctions with eigenvalues tending to infinity have Liouville semiclassical measure, including arbitrary choices within degenerate eigenspaces?
Lean 4
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Background
Quantum ergodicity asks how eigenfunctions of a system with chaotic classical limit distribute in phase space at high energy. The Shnirelman–Zelditch–Colin de Verdière theorem says that for an ergodic classical flow, a density-one subsequence of eigenfunctions equidistributes. Quantum Unique Ergodicity (QUE), conjectured by Rudnick–Sarnak, is the stronger statement that the full sequence equidistributes — no exceptional sparse subsequences, and in particular no strong scarring.
What is known: QUE is a theorem in the arithmetic setting. Lindenstrauss proved arithmetic QUE for Hecke–Maass forms on congruence surfaces using measure rigidity for the diagonal action; Soundararajan removed a remaining escape-of-mass issue, and Holowinsky–Soundararajan proved the holomorphic (mass-form) analogue. These rely crucially on the extra Hecke symmetries.
Ergodicity alone is insufficient: Hassell constructed ergodic billiard counterexamples. Such billiards are not compact boundaryless negatively curved surfaces. The restricted geometric question stated here remains open; arithmetic QUE for specially chosen Hecke eigenbases does not settle every eigenfunction sequence on every such surface. Entropy bounds constrain possible exceptional measures but do not force Liouville measure. Related: M9, M10.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Rudnick & Sarnak, 'The behaviour of eigenstates of arithmetic hyperbolic manifolds', Comm. Math. Phys. 161 (1994) link
- [2]Lindenstrauss, 'Invariant measures and arithmetic quantum unique ergodicity', Ann. of Math. 163 (2006) link
- [3]Anantharaman, 'Entropy and the localization of eigenfunctions', Ann. of Math. 168 (2008) link
- [4]Hassell, 'Ergodic billiards that are not quantum unique ergodic' link
- [5]Dyatlov, mathematical account of quantum chaos (2023) link
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