Bound-state ladder termination by norm positivity ($\Vert A_l{\psi }_E{\Vert }^2=m{a}^{2}E/{\hslash }^2+1/[2(l+1{)}^2]$)

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.