Quantum Unique Ergodicity: equidistribution of all high-energy Laplace eigenfunctions on negatively curved manifolds, and the removal of the Hecke eigenfunction assumption on arithmetic hyperbolic surfaces

The Quantum Unique Ergodicity (QUE) conjecture of Rudnick and Sarnak (1994), stated as Conjectures 1.2 and 1.3 of the source paper (Lindenstrauss, Annals of Mathematics 163 (2006) 165-219), asserts that on a compact Riemannian manifold of strictly negative sectional curvature every sequence of Laplace eigenfunctions equidistributes: the only quantum limit is the uniform (Liouville) measure on the unit tangent bundle. The conjecture is open even for compact hyperbolic surfaces, and the only setting in which QUE-type results are proved is the arithmetic one, where Lindenstrauss's measure-rigidity theorem, combined with Soundararajan's elimination of escape of mass (2010) and with Holowinsky-Soundararajan (2010) for holomorphic forms, settles the conjecture for joint eigenfunctions of the Laplacian and the Hecke operators — Hecke-Maass forms on compact and non-compact congruence hyperbolic surfaces alike, and holomorphic Hecke eigenforms on non-compact ones (QUE for holomorphic forms on compact arithmetic quotients remaining open by those methods).

Background

Let $M$ be a compact Riemannian manifold and $\Delta$ its Laplace-Beltrami operator, a self-adjoint elliptic operator with discrete spectrum $0=\lambda_0<\lambda_1\le\lambda_2\le\cdots\to\infty$. A normalized eigenfunction $\phi_j\in L^2(M)$ with $\Delta\phi_j=\lambda_j\phi_j$ represents a stationary quantum state, and $\lambda_j\to\infty$ is the high-energy (semi-classical) limit. A central question of quantum chaos is the macroscopic distribution of $\phi_j$: do the probability measures $|\phi_j(x)|^2\,d\mathrm{vol}(x)$, which give the position distribution of a quantum particle in state $\phi_j$, spread uniformly over $M$, or can they concentrate ("scar") on proper subsets?

The relevant classical dynamics is the geodesic flow on the unit tangent bundle $SM$, which for strictly negatively curved $M$ is Anosov (uniformly hyperbolic) and in particular ergodic. The quantum ergodicity theorem (announced by Šnirel'man (1974) and proved by Colin de Verdière (1985) and Zelditch (1987)) asserts that if the geodesic flow is ergodic, then for any orthonormal basis of eigenfunctions, along a subsequence of density one the measures $|\phi_j|^2d\mathrm{vol}$ converge weak-* to the normalized volume; nothing is said about the exceptional density-zero subsequence. The strongest formulation of the problem passes to the microlocal lift: every weak-* limit of the lifted measures on $SM$ (called a quantum limit, or semiclassical measure) is automatically invariant under the geodesic flow, and conversely any such limit projects to a limit of $|\phi_j|^2d\mathrm{vol}$. The Quantum Unique Ergodicity (QUE) conjecture of Rudnick and Sarnak (1994) — Conjectures 1.2 and 1.3 of the source paper — asserts that when $M$ is compact with negative sectional curvature, there are no exceptions at all: the only quantum limit is the normalized Liouville (volume) measure $d\mathrm{vol}_{SM}$, which in particular implies that for every orthonormal sequence of eigenfunctions with $\lambda_j\to\infty$, $|\phi_j|^2d\mathrm{vol}\to d\mathrm{vol}/\mathrm{vol}(M)$. The conjecture is open even for compact surfaces of constant curvature $-1$. It is known to fail in neighbouring settings where the hypotheses are weakened: certain chaotic Euclidean billiards with marginally stable orbits, and quantized torus maps with large spectral multiplicities, admit non-equidistributing eigenstates, so any proof must use negative curvature in an essential way.

The only general framework in which QUE-type results are proved is arithmetic. On an arithmetic (congruence) hyperbolic surface $M=\Gamma\backslash\mathbb{H}$, where $\Gamma$ is e.g. $\mathrm{SL}(2,\mathbb{Z})$ or a congruence subgroup (a subgroup containing all matrices congruent to the identity modulo some integer $N$), or the unit group of an order (a lattice subring) in a quaternion algebra over the rationals, the Hecke operators $T_n$ form a commuting family of self-adjoint averaging operators on $L^2(M)$ induced by the commensurator of $\Gamma$ (the group of rational matrices that conjugate $\Gamma$ to a group commensurable with it, i.e. intersecting it in finite index) and coming from the adelic (number-theoretic) structure; they commute with the Laplacian, and joint eigenfunctions of the Laplacian and all $T_n$ are called Hecke eigenfunctions. Restricting attention to such joint eigenfunctions ("arithmetic quantum limits"), Lindenstrauss (2006) proved arithmetic QUE: if $M$ is compact, the only arithmetic quantum limit is the volume. The proof combines (a) a measure classification theorem (Theorem 1.1 of that paper) for probability measures on quotients $\Gamma\backslash\mathrm{SL}(2,\mathbb{R})\times L$ ($L$ an $S$-algebraic group, i.e. a product of real and $p$-adic algebraic groups) that are invariant under the diagonal subgroup $A$ of $\mathrm{SL}(2,\mathbb{R})$ — whose flow on $SM$ is conjugate to the geodesic flow — recurrent under $L$, and of positive Kolmogorov-Sinai entropy (the measure-theoretic entropy of the invariant measure under the flow) on every $A$-ergodic component; and (b) the input, due to Bourgain and Lindenstrauss (2003), that every arithmetic quantum limit has all $A$-ergodic components of entropy at least $2/9$ (the volume has entropy $2$). For non-compact $M$ of finite volume, e.g. the modular surface, the classification only yields limits of the form $c\,d\mathrm{vol}_{SM}$ with $0\le c\le 1$: a positive fraction of the $L^2$ mass could "escape" into the cusps (the non-compact ends of the surface). The source paper records that the absence of escape of mass was known conditionally on the generalized Riemann hypothesis (GRH) from T. C. Watson's Princeton thesis (cited there as 2001; the thesis was conferred in 2002). Unconditionally, Soundararajan (2010) eliminated escape of mass for Hecke-Maass forms (non-holomorphic Laplace eigenfunctions that are Hecke eigenfunctions) on the modular surface (with the stated argument extending to all congruence subgroups), completing arithmetic QUE for Hecke-Maass forms on non-compact arithmetic surfaces; the holomorphic analog on non-compact quotients was proved by Holowinsky and Soundararajan (2010). Brooks and Lindenstrauss (2014) subsequently weakened the Hecke input to joint eigenfunctions of the Laplacian and a single Hecke operator (and to suitable joint quasimodes), and Shem-Tov and Silberman (published 2026) extended arithmetic QUE to congruence quotients of products of hyperbolic $2$- and $3$-spaces — all still within the Hecke framework.

Thus every known proof of a QUE-type statement uses the arithmetic structure essentially, and two further facts delimit the remaining gap. First, if the Laplacian spectrum on a congruence surface were simple, i.e. if every eigenspace were one-dimensional (the spectral simplicity conjecture), every eigenfunction would automatically be a Hecke eigenfunction and arithmetic QUE would apply — but spectral simplicity is believed to be even less accessible than QUE itself. Second, on general negatively curved manifolds the best unconditional information about arbitrary eigenfunctions is far from equidistribution: Anantharaman (2008) proved that every quantum limit has positive Kolmogorov-Sinai entropy, and Dyatlov and Jin (2018) proved that on compact hyperbolic surfaces every semiclassical measure has full support; neither result determines the measure. The open target is therefore QUE without the arithmetic crutch: the general conjecture on compact negatively curved manifolds, and its derived arithmetic variant — QUE for every Laplace eigenfunction on a congruence hyperbolic surface, without assuming joint Hecke eigenfunctions, including absence of escape of mass in the non-compact case.

Problem Statement

Prove, or refute, the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak, in the formulation of Conjectures 1.2 and 1.3 of the source paper: (A) show that for every compact Riemannian manifold $M$ of strictly negative sectional curvature and every orthonormal sequence $\{\phi_j\}$ of Laplace eigenfunctions with $\lambda_j\to\infty$, the probability measures $|\phi_j(x)|^2\,d\mathrm{vol}(x)$ converge in the weak-* topology to the normalized volume measure (Conjecture 1.2 of the source paper) — or, in the slightly stronger form also due to Rudnick and Sarnak (Conjecture 1.3 of the source paper), that the only quantum limit (microlocal lift to the unit tangent bundle $SM$ of a weak-* limit of $|\phi_j|^2d\mathrm{vol}$) on any such $M$ is the normalized Liouville measure $d\mathrm{vol}_{SM}$, a statement which implies the projected form; or (B), for the derived arithmetic variant that is the source paper's own remaining gap, show the same equidistribution for every sequence of $L^2$-normalized Laplace eigenfunctions on an arithmetic hyperbolic surface $\Gamma\backslash\mathbb{H}$ with $\Gamma$ a congruence lattice (compact or of finite volume) — without assuming that the eigenfunctions are joint eigenfunctions of the Hecke operators — including, in the non-compact case, that no $L^2$ mass escapes into the cusps. The compact case of (B) follows from (A); the finite-volume case of (B) additionally requires the absence of escape of mass, and (B) is the full QUE statement for arithmetic surfaces. The problem is also answerable by refutation: exhibit a compact negatively curved manifold, or an arithmetic hyperbolic surface, together with a sequence of $L^2$-normalized Laplace eigenfunctions whose associated measures have a weak-* limit (possibly after passing to a subsequence) that is not the uniform distribution — such a 'scarring' construction would disprove the conjecture.

References

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14. Semyon Dyatlov and Long Jin, 'Semiclassical measures on hyperbolic surfaces have full support', Acta Mathematica 220 (2018), no. 2, 297-339. DOI: 10.4310/acta.2018.v220.n2.a3, https://doi.org/10.4310/acta.2018.v220.n2.a3
15. Manfred Einsiedler and Elon Lindenstrauss, 'Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations', preprint arXiv:2307.04163 (2023). arXiv ID: 2307.04163, https://arxiv.org/abs/2307.04163
16. Zvi Shem-Tov and Lior Silberman, 'Arithmetic quantum unique ergodicity for products of hyperbolic 2- and 3-spaces', Journal d'Analyse Mathématique 158 (2026), no. 2, 413-448 (preprint posted 2022, revised 2025). DOI: 10.1007/s11854-025-0430-z, https://doi.org/10.1007/s11854-025-0430-z; arXiv ID: 2206.05955, https://arxiv.org/abs/2206.05955
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