Semiclassical measure of the spherical harmonics by Bourgain on \mathbbS³

2024/11/12 by Han, Xiaolong
#33C55 #35P20 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2411.08146

Abstract

Bourgain used the Rudin-Shapiro sequences to construct a basis of uniformly bounded holomorphic functions on the unit sphere in ℂ². They are also spherical harmonics (i.e., Laplacian eigenfunctions) on \mathbbS³ ⊂ ℝ⁴. In this paper, we prove that these functions tend to be equidistributed on \mathbbS³, 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 \mathbbS³. In particular, this demonstrates a new localization pattern in the study of Laplacian eigenfunctions.

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