# Quantum ergodicity

${\displaystyle U_{t}=\exp(it{\sqrt {\Delta }})}$
where ${\displaystyle {\sqrt {\Delta }}}$ is the square root of the Laplace–Beltrami operator. The quantum ergodicity theorem of Shnirelman 1974, Zelditch, and Yves Colin de Verdière states that a compact Riemannian manifold whose unit tangent bundle is ergodic under the geodesic flow is also ergodic in the sense that the probability density associated to the nth eigenfunction of the Laplacian tends weakly to the uniform distribution on the unit cotangent bundle as n → ∞ in a subset of the natural numbers of natural density equal to one. Quantum ergodicity can be formulated as a non-commutative analogue of the classical ergodicity (T. Sunada).