 Hydrogen radial Hamiltonian factorization

With Bohr radius $a$, define $A_l=a[i{p}_r/\hslash -(l+1)/r+1/((l+1)a)]/\sqrt{2}$. For regular radial bound states the radial inner product and compatible domains give the displayed factorization. Expanding the first-order factors produces the centrifugal term $l(l+1){\hslash }^2/(2m{r}^2)$ and Coulomb term $-{\hslash }^2/(mar)$. factorization converts a second-order spectral problem into a positivity bound and first-order terminating-state equation.