Kustaanheimo–Stiefel transformation and sub-Riemannian geometry for systems of quantum particles

We recast the Kustaanheimo–Stiefel transformation to be applicable for systems of two interacting point particles moving in two-dimensional space around a heavy source of a central force. This arrangement of particles in the presence of Coulomb attractive forces is mapped onto a family of harmonic oscillators with an additional interaction term given by a potential with no radial dependence. We analyze a variety of bundle sections to reduce redundant symmetries. In particular, we indicate affine sections and a connection of the mathematical formalism developed with sub-Riemannian geometry. The method can be successfully generalized to a bigger number of particles moving in three-dimensional space.