The semiclassical measure of certain spherical harmonics on S3

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 S3 in R4. In this paper, we identify the semiclassical measure associated to these spherical harmonics by a measure supported on the family of Clifford tori in S3. In particular, these functions equidistribute on S3 in the semiclassical limit.

13 pages; Incorporated referee comments and improved the presentation; To appear in Indiana University Mathematics Journal