Eigenfunctions of $L^2$ in Weinberg Towards the end of the p. 37 of his Lectures on Quantum Mechanics, Weinberg claims that the eigenfunctions of $\mathbf{L}^2$ must be of the form
$$\psi(\mathbf{x})=R(r)Y(\theta,\phi)$$
since $\mathbf{L}^2$ acts only on angles.
It is not clear to me how this argument works. I'd greatly appreciate a proof outline or reference.
 A: In spherical coordinates Laplacian can be written as
$${\displaystyle {\begin{aligned}\nabla^2 f&={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}\end{aligned}}} $$
It can also be written as
$${\displaystyle {\begin{aligned}\nabla^2 \psi&={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial \psi}{\partial r}}\right)-\frac{1}{\hbar^2r^2}L^2\psi \end{aligned}}} $$
using the defintion of $L^2$ in spherical coordinates. Clearly the 1st term is only dependent on $r$ and the 2nd term is only dependent on angles.
If the potential is only radial that is $V(r,\theta,\phi)=V(r)$, then substituting $\psi(r,\theta,\phi)=R(r)Y(\theta,\phi)$ in the Schrödinger equation you can find 2 independent equations one for each $R(r)$ and $Y(\theta,\phi)$.
Not every solution $\psi(r,\theta,\phi)$ can be expressed as $R(r)Y(\theta,\phi)$, but every solution $\psi(r,\theta,\phi)$ can be expressed as a linear combination of solutions of the form $R(r)Y(\theta,\phi)$.
