Quantum ergodicity of Wigner induced random spherical harmonics

Abstract

We introduce a new notion of a ‘random orthonormal basis of spherical harmonics’ of $L^(S^2)$ using generalized Wigner ensembles and show that such a random basis is almost surely quantum ergodic. On one hand, because of how our random basis is constructed, this generalizes previous results for random eigenfunctions that are defined using Haar measures on unitary groups. On the other hand, this provides a semi-classical realization of the probabilistic ‘local quantum unique ergodicity’ of Wigner eigenvectors.