Spherical harmonics and point configurations on the sphere

TL;DR

This paper introduces a framework for constructing spherical harmonics on the sphere using Gaussian beams based on point configurations, linking their properties to the geometry of these configurations.

Contribution

It presents a systematic method to build spherical harmonics from Gaussian beams with poles arranged in specific configurations, connecting geometry to harmonic properties.

Findings

Harmonics exhibit quantum ergodicity with equidistributed poles.

L^ norms are controlled by clustering of poles.

Framework relates pole geometry to harmonic behavior.

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.