Title: Quantum ergodicity of Wigner induced random orthonormal basis of spherical harmonics on S^2
Speaker: Robert Chang
Speaker Info: Northwestern University
It is well-known that on a compact Riemannian manifold, classical ergodicity of the geodesic flow is reflected in the quantum ergodicity of Laplacian eigenfunctions. Roughly speaking, quantum ergodicity refers to the diffuseness of eigenbases on phase in the high frequency limit. This is not the case for spherical harmonics on the standard 2-sphere, but Zelditch proves that a random basis, obtained by applying Haar-distributed unitary matrices to the spherical harmonics, is almost surely quantum ergodic.Date: Tuesday, October 27, 2015
In this talk we show that quantum ergodicity holds for a more general class of random bases, which are constructed according to measures on the unitary group induced by the generalized Wigner ensemble. The key tool we use is asymptotic Gaussianity of Wigner eigenvectors. We explain a recent proof of this result using Dyson Brownian motion due to Bourgade-Yau, and contrast their notion of probabilistic QE of eigenvectors viewed as functions on an index set with semi-classical QE of Laplacian eigenfunctions on a manifold.