The quantum ergodicity theorem of Shnirelman, Yves Colin de Verdière, and Zelditch 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 → ∞.
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