Quantum Ergodicity and Averaging Operators on the Sphere

TL;DR

This paper proves quantum ergodicity for specific eigenfunctions on the sphere, involving averaging over rotations, and extends results to algebraic rotations with quantification, also providing a new proof for regular graphs.

Contribution

It introduces a novel approach to quantum ergodicity using averaging operators on the sphere and extends the results to algebraic rotations with explicit quantification.

Findings

Quantum ergodicity holds for eigenfunctions of the Laplacian and averaging operators on the sphere.

A quantified version of quantum ergodicity is established for algebraic rotations.

A new, simplified proof of quantum ergodicity for large regular graphs is provided.

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.