Semiclassical measure of the spherical harmonics by Bourgain on $\mathbb{S}^3$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916478574395392 |
|---|---|
| author | Han, Xiaolong |
| author_facet | Han, Xiaolong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semiclassical measure of the spherical harmonics by Bourgain on $\mathbb{S}^3$ Han, Xiaolong Classical Analysis and ODEs Analysis of PDEs Spectral Theory 33C55, 35P20 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. |
| title | Semiclassical measure of the spherical harmonics by Bourgain on $\mathbb{S}^3$ |
| topic | Classical Analysis and ODEs Analysis of PDEs Spectral Theory 33C55, 35P20 |
| url | https://arxiv.org/abs/2411.08146 |