Math Conjectures

Quantum unique ergodicity

OPENLandmarkConjectureProposed 1994 · Standard version

Canonical statement

Let (M,g)(M,g) be a closed connected Riemannian manifold with strictly negative sectional curvature, and let −Δgφj=λj2φj-\Delta_g\varphi_j=\lambda_j^2\varphi_j, ∥φj∥L2=1\|\varphi_j\|_{L^2}=1, with λj→∞\lambda_j\to\infty. For every classical order-zero pseudodifferential operator A∈Ψ0(M)A\in\Psi^0(M),
⟨Aφj,φj⟩⟶∫S∗Mσ0(A) dL. \langle A\varphi_j,\varphi_j\rangle\longrightarrow\int_{S^*M}\sigma_0(A)\,dL.
Here S∗M={(x,ξ)∈T∗M:∣ξ∣g=1}S^*M=\{(x,\xi)\in T^*M:|\xi|_g=1\}, σ0(A)\sigma_0(A) is the degree-zero principal symbol restricted to S∗MS^*M, and LL is normalized Liouville probability measure.
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Let \((M,g)\) be a closed connected Riemannian manifold with strictly negative sectional curvature, and let \(-\Delta_g\varphi_j=\lambda_j^2\varphi_j\), \(\|\varphi_j\|_{L^2}=1\), with \(\lambda_j\to\infty\). For every classical order-zero pseudodifferential operator \(A\in\Psi^0(M)\), \[ \langle A\varphi_j,\varphi_j\rangle\longrightarrow\int_{S^*M}\sigma_0(A)\,dL. \] Here \(S^*M=\{(x,\xi)\in T^*M:|\xi|_g=1\}\), \(\sigma_0(A)\) is the degree-zero principal symbol restricted to \(S^*M\), and \(L\) is normalized Liouville probability measure.
MATH-PHYS-008:[Shnirelman1974ErgodicEigenfunctions][RudnickSarnak1994BehaviourEigenstates][Anantharaman2008EntropyEigenfunctions][LeMasson2026QuantumErgodicitySurvey]
Mathematical physicsGeometry and topologyDynamical systems

Notes

The quantum unique ergodicity conjecture concerns a closed Riemannian manifold of strictly negative sectional curvature, whose geodesic flow is a model of classical chaos. It asserts that the whole sequence of Laplace eigenfunctions equidistributes in phase space: for every order-zero pseudodifferential operator AA, the matrix elements ⟨Aφj,φj⟩\langle A\varphi_j,\varphi_j\rangle converge to the Liouville average of the principal symbol over the unit cosphere bundle. The conjecture was posed by Rudnick and Sarnak in 1994 in their study of eigenstates of arithmetic hyperbolic manifolds [RudnickSarnak1994BehaviourEigenstates].

Quantum ergodicity, going back to Shnirelman [Shnirelman1974ErgodicEigenfunctions], already yields the conclusion along a density-one subsequence whenever the geodesic flow is ergodic; the conjecture asks that no exceptional subsequence exist at all. Entropy and observability estimates significantly restrict the possible quantum limits — in negative curvature, limit measures must carry substantial entropy, which excludes concentration on a single closed geodesic [Anantharaman2008EntropyEigenfunctions] — and in the arithmetic setting, for joint eigenfunctions of the Hecke operators, the full equidistribution statement is known.

For a general negatively curved manifold, however, scarring along some exceptional sequence of eigenfunctions has not been ruled out, and the conjecture remains open; see [LeMasson2026QuantumErgodicitySurvey] for a recent survey of the semiclassical-measure picture.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.