What is the right domain for this Hamiltonian I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy that the function itself and its derivative is continuous at the boundary of the intervals of $\theta$ and $\phi$.)
$$   \left\{ -\frac{1}{\sin^2\theta} \left[
\sin\theta\frac{\partial}{\partial\theta} \Big(\sin\theta\frac{\partial}{\partial\theta}\Big)
+\frac{\partial^2}{\partial \phi^2}\right]\right\} \psi(\theta,\phi)+V(\theta)\psi(\theta,\phi) = E \psi(\theta,\phi),$$
where $V \in C^{\infty}([0,2\pi]) \rightarrow \mathbb{R}$.
One normally wants that this operator shall be self-adjoint (therefore it should be a dense domain). Could anybody explain to me how to find a proper domain for this operator?

You have to better specify the boundary conditions, because you may have different self-adjoint extensions, depending on them (and you have to be sure your operator maps you in the same space with same boundary conditions).
I suppose that, once you have fixed the suitable conditions, $C^\infty([0,2\pi]\times [0,2\pi])$ should be a domain of essential self-adjointness. Furthermore, your operator is of the form $H=H_0+V$ with $V$ very regular, so you may apply Kato-Rellich theorem to have self-adjointness on the domain of $H_0$ (angular momentum operator).
A discussion of boundary conditions for a general class of operators that include yours can be found in this paper of Calkin.
