Semiclassical measure of the spherical harmonics by Bourgain on $\mathbb{S}^3$

TL;DR

This paper investigates the equidistribution and semiclassical measures of specific spherical harmonics on , revealing new localization patterns and connecting Rudin-Shapiro sequences with eigenfunction behavior.

Contribution

It proves equidistribution of certain spherical harmonics on  and identifies their semiclassical measures, introducing a novel localization pattern in Laplacian eigenfunction analysis.

Findings

Spherical harmonics constructed from Rudin-Shapiro sequences tend to be equidistributed on .

The semiclassical measure is supported on Clifford tori, indicating a new localization pattern.

Demonstrates a link between sequence auto-correlation and eigenfunction distribution.

Abstract

Bourgain used the Rudin-Shapiro sequences to construct a basis of uniformly bounded holomorphic functions on the unit sphere in $\mathbb{C}^2$. They are also spherical harmonics (i.e., Laplacian eigenfunctions) on $\mathbb{S}^3 \subset \mathbb{R}^4$. In this paper, we prove that these functions tend to be equidistributed on $\mathbb{S}^3$, based on an estimate of the auto-correlation of the Rudin-Shapiro sequences. Moreover, we identify the semiclassical measure associated to these spherical harmonics by the singular measure supported on the family of Clifford tori in $\mathbb{S}^3$. In particular, this demonstrates a new localization pattern in the study of Laplacian eigenfunctions.