Quantum unique ergodicity in negative curvature
Let $(M, g)$ be a closed connected smooth Riemannian manifold of dimension at least two with strictly negative sectional curvature, and let $(u_j)$ be $L^2$-normalized eigenfunctions, $-\Delta_g u_j = \lambda_j u_j$, with $\lambda_j \to \infty$. Put $h_j = \lambda_j^{-1/2}$, let $S^*M = \{(x, \xi) \in T^*M : |\xi|_g = 1\}$, let $L$ be the normalized Liouville measure on $S^*M$, and let $\operatorname{Op}_h$ be a standard semiclassical quantization. The geodesic flow of $M$ is ergodic, and by the quantum ergodicity theorem of Shnirelman, Zelditch and Colin de Verdière the convergence below holds along a density-one subsequence of every orthonormal eigenbasis. Anantharaman proved that every semiclassical measure on such a manifold has positive Kolmogorov–Sinai entropy. On surfaces, Dyatlov and Jin (constant curvature) and Dyatlov, Jin and Nonnenmacher (variable curvature) proved that every semiclassical measure has full support in $S^*M$. Lindenstrauss proved the convergence for joint eigenfunctions of the Laplacian and the Hecke operators on compact arithmetic hyperbolic surfaces.
Problem. Is it true that $$\langle \operatorname{Op}_{h_j}(a)\, u_j, u_j \rangle \longrightarrow \int_{S^*M} a \, dL$$ for every such manifold, every such sequence $(u_j)$ and every $a \in C_c^\infty(T^*M)$, without discarding any exceptional eigenfunctions?
Source: Rudnick and Sarnak (1994), the quantum unique ergodicity conjecture; in this phase-space form, see Dyatlov (2022) and Nonnenmacher (2026).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, DS
- Keywords
- AIM Open Applied Problems, Laplace eigenfunctions, quantum ergodicity, semiclassical measures
References
@article{rudnick1994,
author = {Rudnick, Ze{\'e}v and Sarnak, Peter},
title = {The behaviour of eigenstates of arithmetic hyperbolic manifolds},
journal = {Comm. Math. Phys.},
volume = {161},
number = {1},
year = {1994},
pages = {195--213},
doi = {10.1007/BF02099418},
}
@article{dyatlov2022survey,
author = {Dyatlov, Semyon},
title = {Around quantum ergodicity},
journal = {Ann. Math. Qu{\'e}.},
volume = {46},
number = {1},
year = {2022},
pages = {11--26},
doi = {10.1007/s40316-021-00165-7},
eprint = {2103.08093},
archivePrefix = {arXiv},
}
@misc{nonnenmacher2026,
author = {Nonnenmacher, St{\'e}phane},
title = {Quantum ergodicity and semiclassical measures: mathematical results},
year = {2026},
eprint = {2606.12098},
archivePrefix = {arXiv},
}
@article{shnirelman1974,
author = {Shnirelman, A. I.},
title = {Ergodic properties of eigenfunctions},
journal = {Uspekhi Mat. Nauk},
volume = {29},
number = {6(180)},
year = {1974},
pages = {181--182},
}
@article{zelditch1987,
author = {Zelditch, Steven},
title = {Uniform distribution of eigenfunctions on compact hyperbolic surfaces},
journal = {Duke Math. J.},
volume = {55},
number = {4},
year = {1987},
pages = {919--941},
doi = {10.1215/S0012-7094-87-05546-3},
}
@article{colindeverdiere1985,
author = {Colin de Verdi{\`e}re, Yves},
title = {Ergodicit{\'e} et fonctions propres du laplacien},
journal = {Comm. Math. Phys.},
volume = {102},
number = {3},
year = {1985},
pages = {497--502},
doi = {10.1007/BF01209296},
}
@article{anantharaman2008,
author = {Anantharaman, Nalini},
title = {Entropy and the localization of eigenfunctions},
journal = {Ann. of Math. (2)},
volume = {168},
number = {2},
year = {2008},
pages = {435--475},
doi = {10.4007/annals.2008.168.435},
}
@article{dyatlov2018,
author = {Dyatlov, Semyon and Jin, Long},
title = {Semiclassical measures on hyperbolic surfaces have full support},
journal = {Acta Math.},
volume = {220},
number = {2},
year = {2018},
pages = {297--339},
doi = {10.4310/ACTA.2018.v220.n2.a3},
eprint = {1705.05019},
archivePrefix = {arXiv},
}
@article{dyatlov2022,
author = {Dyatlov, Semyon and Jin, Long and Nonnenmacher, St{\'e}phane},
title = {Control of eigenfunctions on surfaces of variable curvature},
journal = {J. Amer. Math. Soc.},
volume = {35},
number = {2},
year = {2022},
pages = {361--465},
doi = {10.1090/jams/979},
eprint = {1906.08923},
archivePrefix = {arXiv},
}
@article{lindenstrauss2006,
author = {Lindenstrauss, Elon},
title = {Invariant measures and arithmetic quantum unique ergodicity},
journal = {Ann. of Math. (2)},
volume = {163},
number = {1},
year = {2006},
pages = {165--219},
doi = {10.4007/annals.2006.163.165},
}
@misc{aim-quantum-unique-ergodicity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Quantum unique ergodicity in negative curvature}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/013-quantum-unique-ergodicity.md},
note = {Entry 013 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00301, "Quantum unique ergodicity in negative curvature", clopen.solutions.