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 $\Theta(\theta)e^{in\phi}$.

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 ?