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

Autor:	Han, Xiaolong
Rok vydání:	2024
Předmět:	Mathematics - Classical Analysis and ODEs Mathematics - Analysis of PDEs Mathematics - Spectral Theory 33C55 35P20
Druh dokumentu:	Working Paper
Popis:	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. Comment: 11 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2411.08146 Zobrazit plný text záznamu View this record from Arxiv