Can I see separation variable as tensor product? For example in a radial potential the separation of variable brings to the solution $R(r)\Theta(\theta) \Phi (\phi)$. This sounds like an element of the space spanned by $|r\rangle\otimes|\theta\rangle\otimes |\phi\rangle$ where $R(r)$ lives in the space spanned by $|R\rangle$ and so on