Symmetry and degeneracy of the curved Coulomb potential on the S3 ball

Abstract

The ‘curved’ Coulomb potential on the S3 ball, whose isometry group is SO(4), takes the form of a cotangent function, and when added to the four-dimensional squared angular momentum operator, one of the so(4) Casimir invariants, a Hamiltonian is obtained which describes a perturbance of the free geodesic motion that results in several peculiar aspects. The spectrum of such a motion has been studied on various occasions and is known to unexpectedly carry so(4) degeneracy patterns despite the non-commutativity of the perturbance with the Casimir operator. We suggest here an explanation for this behavior in designing a set of operators which close the so(4) algebra and whose Casimir invariant coincides with the Hamiltonian of the perturbed motion at the level of the eigenvalue problem. The above operators are related to the canonical geometric SO(4) generators on S3 by a non-unitary similarity transformation of the scaling type. In this fashion, we identify a complementary option to the deformed dynamical so(4) Higgs algebra constructed in terms of the components of the ordinary angular momentum and a related Runge–Lenz vector.