Quantum Ergodicity and Averaging Operators on the Sphere

Abstract

We prove quantum ergodicity for certain orthonormal bases of $L^2({\mathbb{S}}^2)$, consisting of joint eigenfunctions of the Laplacian on ${\mathbb{S}}^2$ and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs.

arXiv comment27 pages