Let \(d\ge 2\). A spherical Harmonic \(Y_{l,m}\) is an eigenfunction of the Laplace-Beltrami operator on the \(d\)-sphere \(\mathbb{S}^{d}_R\) with radius \(R>0\) with eigenvalues \(-\frac{l(l+d-1)}{R^2}\), \(l\in \mathbb{N}_0\) and \(m=1,\ldots, n_l\), where \(n_l\) denotes the dimension of the corresponding eigenspace.

The dimension of the \(l\)-th eigenspace is given by

\[ n_l = \Bigl(1+\frac{2l}{d-1}\Bigr)\binom{l+d-2}{l}. \]

Examples Link to heading

Eigenvalues Link to heading

The first eigenvalues \(\lambda_l\) are:

• \(\lambda_0 = 0\)
• \(\lambda_1 = \frac{d}{R^2}\)
• \(\lambda_2 = \frac{2d+2}{R^2}\)
• \(\lambda_3 = \frac{3d+6}{R^2}\)