Spherical harmonics and point configurations on the sphere

Abstract

We develop a systematic framework for constructing spherical harmonics on the two-dimensional unit sphere as superpositions of Gaussian beams whose poles form well-separated point configurations. The distributional and analytic properties of the resulting spherical harmonics are determined by the geometry of these poles: when the configuration is equidistributed, the sequence of harmonics exhibits quantum ergodicity, while their $L^{\infty }$ norms are quantitatively controlled by the maximal clustering of poles within small neighborhoods of great circles.

21 pages, 5 figures