Failure of $L^p$ Symmetry of Zonal Spherical Harmonics

TL;DR

This paper demonstrates that on the 2-sphere, the symmetry of $L^p$ norms of Laplacian eigenfunctions fails for $p \,\geq\, 6$, revealing asymmetries in eigenfunction behavior at high frequencies.

Contribution

It provides the first explicit construction showing the breakdown of $L^p$ symmetry for spherical eigenfunctions when $p\geq 6$, using properties of Legendre and Bessel functions.

Findings

Existence of eigenfunctions with asymmetric $L^p$ norms for $p\geq 6$

The ratio of positive to negative parts' $L^p$ norms does not tend to 1

Symmetry of $L^p$ norms fails on the 2-sphere for high eigenvalues

Abstract

In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $p\geq 6$. In other words, there exists a sequence of spherical eigenfunctions $\psi_n$, with eigenvalues $\lambda_n\to\infty$ as $n\to\infty$, such that the ratio of the $L^p$ norms of the positive and negative parts of the eigenfunctions does not tend to $1$ as $n\to\infty$ when $p\geq 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.