What domains can we give the Laplacian on the sphere $\mathbb{S}^2$ as an unbounded closed operator? Recall from the theory of spherical harmonics on the sphere $\mathbb{S}^2$ that $L^2(\mathbb{S}^2)$ has an orthonormal basis of smooth eigenfunctions $Y_\ell^m$ for integer $\ell,m$ such that $-\ell \le m\le\ell$, with $$\Delta Y_\ell^m + \ell(\ell+1)Y_\ell^m \;=\; 0$$
Now, if we denote by $D$ the span of the $Y_\ell^m$, then we know that $D$ is dense in $L^2(\mathbb{S}^2)$. But is the operator $\overline{\Delta|_D}$ self-adjoint, where by $\overline{\Delta|_D}$ we mean the closure of the operator $\Delta$ defined on $D$. If not, what precisely is the domain $\mathcal{D}(\overline{\Delta|_D})$?
 A: In fact, yes, that restriction of the invariant Laplacian has a unique self-adjoint extension, which is its graph-closure, as indicated. That is, it is "essentially self-adjoint".
This is not obvious, and the issues are easy to misunderstand, I think, since in general a symmetric, densely-defined operator can have many self-adjoint extensions (or none) (as parametrized by von Neumann).
So the assertion of essential self-adjointness requires proof, since many natural symmetric (densely-defined) operators are not essentially self-adjoint. E.g., in Sturm-Liouville problems, there are typically many different boundary conditions that give a self-adjoint operator.
In a discrete-spectrum situation, such as compact Riemannian manifolds, e.g., spheres, because $L^2$ provably has an orthonormal basis consisting of smooth functions, it is relatively straightforward to prove the essential self-adjointness. This was proven in the early 1950's by M. Gaffney.
EDIT: I should emphasize, without "boundary conditions" (whatever that may mean more precisely), discrete spectrum tends to give essential self-adjointness. In particular, the Sturm-Liouville scenario(s), which do have "boundary conditions" (however one may choose to formalize this), are not essentially self-adjoint.
So, yes, there's no pedestrian way to know, in a given situation. It's not a trivial issue. Often, yes, in "natural" physical situations, there's a unique extension, etc., which is "necessary" to make mathematical sense of various physical phenomena. But, perhaps equally often, there are reasonable boundary-value problems (of physical significance) which have different solution depending on the boundary conditions: Dirichlet? von Neumann? etc.
These details are not pathologies, but are genuine and significant features...
The original refs for von Neumman are something like 
[vonNeumann~1929] J. von Neumann, {\it Allgemeine Eigenwerttheorie
Hermitescher Funktionaloperatoren}, Math. Ann. {\bf 102} (1929),
49-131.
Since then, there are several post-L.Schwartz'-distributions discussions that make things clearer, e.g., G. Grubb's book (with Springer) "Distributions and Operators".
