The quantized SU(2) Kepler problem and its symmetry group for negative energies

In a previous paper, the SU(2) Kepler problem was defined and shown to admit the symmetry group SU(4) for negative energies. This paper is a continuation of the previous one, giving the quantization of the SU(2) Kepler problem. In the complex vector bundle associated with an SU(2) bundle R(8) - {0} --> R(5) - {0}, an extension of the Hopf bundle S-7 --> S-4, the quantized SU(2) Kepler problem is defined and analyzed along with its eigenvalues and symmetry. This system, a generalization of the hydrogen atom in five dimensions, describes the motion of a particle with isospin in Yang's monopole field together with the Coulomb potential and a centrifugal potential. It will be shown that the quantized SU(2) Kepler problem of negative energy admits a symmetry group SU(4) congruent to Spin(6), which is indeed represented unitarily in the negative energy eigenspaces. The infinitesimal generators of the symmetry are found in an explicit form for al energies, negative, zero, or positive. Those generators coming from the subgroup Sp(2) congruent to Spin(5) provide the angular momentum operators and the others are viewed as the Runge-Lenz-like operators. According to whether the energy is negative, zero, or positive, the symmetry Lie algebra formed by these generators is shown to be so(6), e(5), or so(1,5), where e(5) is the Lie algebra of the group of motions in R(5).