Is it possible to factorize $$(-\partial^2+\phi^2(r))^2-\left(\frac{\partial\phi(r)}{\partial r}\right)^2,$$ where $\phi(r)$ is a function of $r\equiv\sqrt{x^2+y^2+z^2+\xi^2}$ and $\partial^2$ is the 4-dimensional Laplacian? Actually since we have the $SO(4)$ symmetry, we can separate the angular part and focus on the radial part. That is, we can take $$\partial^2\rightarrow \frac{1}{r^3}\frac{\partial}{\partial r}r^3\frac{\partial }{\partial r}\equiv \hat{L}(r)$$. Note that we can not simply factorize it as $$\left(-\hat{L}(r)+\phi^2(r)+\left(\frac{\partial\phi(r)}{\partial r}\right)\right)\cdot\left(-\hat{L}(r)+\phi^2(r)-\left(\frac{\partial\phi(r)}{\partial r}\right)\right)$$ because of the non-commutativity of derivative operators. Here I just wonder whether there is a subject that show us how to factorize operator polynomials systematically?
If it can not be factorized, can anyone give a proof?