Hydrogen radial Hamiltonian factorization ($H_l={\hslash }^2[A_l^{†}{A}_l-1/(2(l+1{)}^2)]/(m{a}^2)$)

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.

The Coulomb radial factors satisfy $[A_l,A_l^{†}]=a^2(l+1)/r^2=m{a}^2(H_{l+1}-H_l)/{\hslash }^2$. Commuting $A_l$ through its factorized Hamiltonian gives the displayed intertwining relation. A nonzero transformed radial eigenfunction has the same energy in the next angular-momentum sector; this changes the radial sector, rather than being the usual magnetic-quantum-number raising operator within a fixed $l$ multiplet.

For a normalized bound eigenstate of a factorized Coulomb radial Hamiltonian, the displayed squared norm must be nonnegative. At fixed negative $E$, increasing $l$ indefinitely would make it negative. The intertwining chain must therefore terminate with zero norm, giving $E=-{\hslash }^2/[2m{a}^2(L+1{)}^2]$. The positivity contradiction does not apply to the positive-energy continuum; domain preservation and normalizability are necessary assumptions.