# Laplace spectrum of the $2$-Sphere [closed]

The $$2$$-sphere $$S^2$$ endowed with usual round metric, we have a Laplacian operator $$\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$$ acting on functions. The eigenvalues of this operator must be very well known. Where can one see these values written out explicitly?

• Did you try to Google this question? Oct 23, 2019 at 14:41
• The eigenvalues are $\ell(\ell+1)$ of multiplicity $2\ell+1$, where $\ell=0,1,\dots$. See spherical harmonics in Wikipedia. Oct 23, 2019 at 14:41
• Le spectre d’une variété riemannienne (SLN 1971) contains explicit descriptions of the eigenvalues and eigenvectors of the standard basic manifolds including your case. Of historical interest is the treatment in what is arguably the first textbook on physics—-by Tait and Thomson. The latter (a.k.a. Lord Kelvin) used it to estimate the age of the sun. Inaccurately, but not due to errors in the mathematics—-thermonuclear reactions hadn’t yet been discovered. Oct 23, 2019 at 14:45

There is a very simple way to compute the Laplace-Beltrami $$\Delta_S$$ operator of a function on the sphere $$f:\mathbb{S}^{n-1}\to\mathbb{C}$$. You simply extend the function to $$\mathbb{R}^n\setminus\{0\}$$ by $$F(x)=f(x/|x|)$$ and define $$\Delta_Sf(x)=(\Delta F)|_{\mathbb{S}^{n-1}},\quad \text{ where } \quad \Delta F=-\sum_{i=1}^n F_{x_ix_i}.$$ Let $$\mathcal{P}_k$$ be the space of all homogeneous polynomials of degree $$k$$, $$P(x)=\sum_{|\alpha|=k}a_\alpha x^\alpha$$. The elements of the subspace $$\mathcal{H}_k\subset\mathcal{P}_k$$ consisting of polynomials that are harmonic functions are called solid spherical harmonics. Then restrictions of solid spherical harmonics to $$\mathbb{S}^{n-1}$$ are called surface spherical harmonics. The space of surface spherical harmonics is denotes by $$H_k$$.
Theorem. The subspaces $$H_k\subset L^2(\mathbb{S}^{n-1})$$ are mutually orthogonal and $$L^2(\mathbb{S}^{n-1})=H_0\oplus H_1\oplus H_2\oplus\ldots$$ Moreover $$\Delta_S Y(x)=k(k+n-2)Y \quad\text{ for Y\in H_k}$$
In your case $$n=3$$ so the eigenvalues are $$k(k+1)$$.
There are many places where you can find a proof of this result. See for example page 67 in my notes. Note that in my notes I defined the Laplace operator without the minus sign so the eigenvalues are $$-k(k+n-2)$$.