General structures of Hamiltonian hierarchies

Three dimensional exactly solvable quantum potentials for which an extra term of form 1/r² has been added are shown to maintain their functional form which allows the construction of the Hamiltonian hierarchy and the determination of the spectra of eigenvalues and eigenfunctions within the Supersymmetric Quantum Mechanics formalism. For the specific cases of the Harmonic Oscillator and the Coulomb potentials, known as Pseudo-Harmonic Oscillator and Pseudo-Coulomb potentials, it is shown here that the inclusion of the new term corresponds to rescaling the angular momentum and it is responsible for maintaining their exact solvability.