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Semiclassical measure of the spherical harmonics by Bourgain on $\mathbb{S}^3$ https://arxiv.org/abs/2411.08146 #mathCA #mathAP #mathSP

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

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.

arXiv.org
November 15, 2024 at 3:10 AM · · feed2toot · 0 · 0 · 0
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