|SLMath: Eisenbud Auditorium
The quantum ergodicity theorem says that on a compact Riemannian manifold with ergodic geodesic flow, for any orthonormal basis of eigenfunctions of the Laplacian in L2, the modulus squared of these eigenfunctions converge weakly as probability measures to the uniform measure, in the limit of large eigenvalues and up to a subsequence of density 0.
On the sphere the geodesic flow is not ergodic and it is possible to find subsequences of eigenfunctions with positive density that do not satisfy the conclusion of the theorem. However, 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 is based on a new argument for quantum ergodicity on regular graphs.