Laplacian eigenvalues of the round sphere

Example

Let $S^n \subset \R^{n+1}$ be the unit sphere with its round metric, $n \ge 1$, and let $\Delta = -\operatorname{div} \operatorname{grad}$ be its Laplace–Beltrami operator. Its eigenvalues are $$\lambda_k = k(k + n - 1), \quad k = 0, 1, 2, \ldots.$$ The $\lambda_k$-eigenspace consists of the restrictions to $S^n$ of the homogeneous harmonic polynomials of degree $k$ on $\R^{n+1}$, the spherical harmonics of degree $k$. Its dimension is $$\binom{n + k}{n} - \binom{n + k - 2}{n},$$ with the convention that the second term is $0$ when $k < 2$.

Check of the eigenvalue: write a homogeneous harmonic polynomial of degree $k$ as $P = r^k Y$ with $r = \abs{x}$ and $Y$ a function on $S^n$. In polar coordinates the Euclidean Laplacian is $\partial_r^2 + \frac{n}{r} \partial_r - \frac{1}{r^2} \Delta$. Applied to $P$ it gives $r^{k-2} \big( k(k-1) + nk \big) Y - r^{k-2} \Delta Y = 0$, so $\Delta Y = k(k + n - 1) Y$. These eigenfunctions span a dense subspace of $L^2(S^n)$, so there are no other eigenvalues.

For $S^2$ the eigenvalues are $k(k+1)$ with multiplicity $2k + 1$. For $k = 1$ the eigenfunctions are the restrictions of the coordinate functions $x_1, \ldots, x_{n+1}$, with eigenvalue $n$.