Noncompact dynamical symmetry of a spin-orbit coupled oscillator

We explain the finite as well as infinite degeneracy in the spectrum of a particular system of spin-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 SO(3,2) algebra. The degeneracy of the physical spectrum only involves a particular, infinite, so called singleton representation. In the branch where orbital and spin angular momentum are aligned the full representation appears, constituting a 3D analogue of Landau levels. Anti-aligning 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.