Quantum ergodicity of Wigner induced spherical harmonics

TL;DR

This paper proves that Wigner induced random bases of spherical harmonics are almost surely quantum ergodic, extending previous results and providing a semi-classical probabilistic perspective on quantum unique ergodicity.

Contribution

It introduces a new probabilistic framework for quantum ergodicity using Wigner ensembles, generalizing prior Haar measure-based results.

Findings

Wigner induced bases are almost surely quantum ergodic.

The results provide a semi-classical realization of quantum unique ergodicity.

Extension of Zelditch's results to Wigner ensemble-based random bases.

Abstract

We show that a Wigner induced random orthonormal basis of spherical harmonics is almost surely quantum ergodic. Here, a random basis is identified with an element of the product probability space of unitary groups, each endowed with the measure induced by the generalized Wigner ensemble. This yields a semi-classical realization of the probabilistic quantum unique ergodicity result for Wigner eigenvectors of Bourgade-Yau. At the same time, this generalizes a similar result due to Zelditch, who uses Haar measure on the unitary groups in defining the notion of a random basis.