Span of eigenstates of angular momentum

Question

Do the eigenstates of the angular momentum operator span the space of all possible wave functions even if the potential is not central?

Answer 1

The spherical harmonics span the space of square integrable functions of the angular variables $\theta,\varphi$. This is a statement about math which is independent of quantum mechanics, let alone the particular Hamiltonian you choose.

However, this does not mean they are always useful. An important point in the study of quantum mechanics is symmetry...very much like it is in classical Lagrangian/Hamiltonian mechanics. An operator generates a symmetry if it commutes with the Hamiltonian. It is an independent math fact that commuting operators can be simultaneously diagonalized. So while we can always represent any function of the angular coordinates as a sum over spherical harmonics, we can only represent a wavefunction in terms of both the spherical harmonics and the energy eigenfunction (in the hydrogen atom, this is what the "radial" functions are) when the angular momentum commutes with the Hamiltonian and hence is a symmetry. The most prototypical example of this occurrence is a central potential.

So the answer in the end is sort of, yes, if by your statement we understand that the spherical harmonics only span the angular coordinates (even in the hydrogen atom we need the radial functions, so the spherical harmonics don't actually span, but rather the tensor product of spherical harmonics with radial functions spans). So, this fact is really only useful when the potential is central.