if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta\_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on the azimuthal coordinate explicitely, then the substitution $\psi(\theta,\phi) = \Theta(\theta)e^{i n \phi}$ leaves you with the differential equation


$$- \Theta''(\theta)- \cot(\theta) \Theta'(\theta) + \frac{n^2}{\sin^2(\theta)} \Theta(\theta)+ V(\theta) \Theta(\theta) = E \Theta(\theta).$$


For $n=0$ this simplifies to 
$$- \Theta''(\theta)- \cot(\theta) \Theta'(\theta) + V(\theta) \Theta(\theta) = E \Theta(\theta).$$


Has this equation ever been studied? I mean if $V \in C^{\infty}$, then it should have a discrete spectrum, as the initial PDE has a discrete spectrum. But is there a further literature on this kind of problem?


I mean, by just googling Schrödinger operator on sphere, I got this result for example [on arxiv](http://arxiv.org/pdf/1301.1210.pdf), but if you are aware of better overview articles, please do not hesitate to give them to me. But apparently, the special case, where you can separate that easily is not (as far as I see) that well-studied or am I wrong?



Schrödinger operators on a sphere
I presume that $V$ is real valued. Then the equation on the sphere is $L\psi=E\psi$, an eigenvalue equation for the self-adjoint operator $L$, with compact resolvant. Therefore the eigenvalues are real and the normalized eigen-functions form an orthonormal basis of $L^2(S)$. If $H\subset L^2$ is an invariant subspace, then you can choose the basis in such a way that some of the eigen-functions form a basis of $H$. This applies to the subspace of functions of the form $\Theta(\theta)e^{in\phi}$.


**Edit** OK, the reduced equation is $\ell^n\Theta=E\Theta$, an eigenvalue problem for an ordinary differential linear second-order operator. Thus $\ell^n$ is Liouville type. It can be written as a self-adjoint operator for some suitable weight. Liouville's theory tells you that at fixed $n$, the eigenvalues 
$$E^n\_0<E^n\_1<\cdots$$
are simple, and the eigenfunction $\Theta^n\_k$ vanishes precisely at $k$ points. Is it what you wished ?


