Connection between the hydrogen atom and the harmonic oscillator: The zero-energy case

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Abstract
The connection between the three-dimensional hydrogen atom and a four-dimensional harmonic oscillator obtained in previous works, from a hybridization of the infinitesimal Pauli approach to the hydrogen system with the Schwinger approach to spherical and hyperbolical angular momenta, is worked out in the case of the zero-energy point of the hydrogen atom. This leads to the equivalence of the three-dimensional hydrogen problem with a four-dimensional free-particle problem involving a constraint condition. For completeness, the latter result is also derived by using the Kustaanheimo-Stiefel transformation introduced in celestial mechanics. Finally, it is shown how the Lie algebra of SO(4,2) quite naturally arises for the whole spectrum (discrete plus continuum plus zero-energy point) of the three-dimensional hydrogen atom from the introduction of the constraint condition into the Lie algebra of Sp(8,R) associated with the four-dimensional harmonic oscillator.