= Hydrogen radial Hamiltonian factorization
{title2=$H_l=\hbar^2[A_l^\dagger A_l-1/(2(l+1)^2)]/(ma^2)$}
With <Bohr radius> $a$, define $A_l=a[i p_r/\hbar-(l+1)/r+1/((l+1)a)]/\sqrt2$. 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 $l(l+1)\hbar^2/(2mr^2)$ and Coulomb term $-\hbar^2/(mar)$. <factorization> converts a second-order spectral problem into a positivity bound and first-order terminating-state equation.
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