The quantum ergodicity theorem (Snirelman, Zelditch, Colin de Verdière) says that on a compact Riemannian manifold with ergodic geodesic flow, for any orthonormal basis of eigenfunctions of the Laplacian in L^2, the modulus squared of these eigenfunctions converge weakly as measures to the uniform measure in the limit of large eigenvalues, up to a density 1 subsequence. On the sphere the geodesic flow is not ergodic and it is possible to find sequences of eigenfunctions that don't satisfy the conclusion of the theorem. However, Zelditch has proved that it holds almost surely for random eigenbasis.
We will present a quantum ergodicity theorem on the sphere for joint eigenfunctions of the Laplacian and an averaging operator over a finite set of rotations. The proof also brings a new argument for quantum ergodicity on regular graphs.
Joint work with Shimon Brooks and Elon Lindenstrauss.
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