First, we find that spherical harmonics are eigenfunctions of \(\Delta_{\mathbb{S}^d_R}\). To see this, we apply the polar formula of the Laplacian on solid harmonics. Since they are homogeneous, we can write

\[ p(r \theta)=rp(\theta), \]

where \(\theta\in \mathbb{S}^1\).

There are no other eigenfunctions. This follows from the fact that that spherical harmonics are dense in \(L^2(\mathbb{S}^d_R)\). We take an arbitrary eigenfunction \(f\) of the Laplacian. Decomposing \(f\) and \(\Delta f\) into spherical harmonics, and comparing yields that the eigenvalue must coincide with one of the eigenvalues of the spherical harmonics, and this is only possible, if \(f\) is a spherical harmonic [1, Theorem 1.9].

Links Link to heading

• eigenspaces of \(\Delta_{\mathbb{S}^d_R}\)
• solid harmonic
• Laplacian in polar coordinates
• density in \(L^2(\mathbb{S}^d_R\)

References Link to heading

1. J. Gallier. Class Lecture, Topic: Notes on Spherical Harmonics and Linear Representations of Lie Groups.