We explain the finite as well as infinite degeneracy in the spectrum of a particular system of spin-$\frac{1}{2}$ fermions with spin-orbit coupling in three spatial dimensions. Starting from a generalized Runge-Lenz vector, we explicitly construct a complete set of symmetry operators, which span a noncompact $\text{SO}(3,2)$ algebra. The degeneracy of the physical spectrum only involves an infinite, so-called singleton representation. In the branch where orbital and spin angular momentum are aligned, the full representation appears, constituting a three-dimensional analog of Landau levels. Antialigning the spin leads to a finite degeneracy due to a truncation of the singleton representation. We conclude the paper by constructing the spectrum generating algebra of the problem.

• Received 25 September 2013
• Revised 24 December 2013

DOI:https://doi.org/10.1103/PhysRevA.89.032105