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?



Laplace spectrum of the $2$-Sphere
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 $-k(k+n-2)$.


**P. Hajlasz,** *Functional Analysis.* Available at:


<http://www.google.com/url?q=http%3A%2F%2Fwww.pitt.edu%2F~hajlasz%2FNotatki%2FFunctional%2520Analysis2.pdf&sa=D&sntz=1&usg=AFQjCNEgjTnVgLE_daoXzgSa6ggIkXuiXA>


