The discrete spectrum of a q-analogue of the hydrogen atom is obtained from a deformation of the Pauli equations. As an alternative, the spectrum is derived from a deformation of the four-dimensional oscillator arising in the application of the Kustaanheimo-Stiefel transformation to the hydrogen atom. A model of the 2s−2p Dirac shift is proposed in the context of q-deformations.

Preliminary results concerning non-quadratic (and non-bijective) transformations that exibit a degree of parentage with the well known Levi-Civita, Kustaanheimo-Stiefel, and Fock transformations are reported in this article. Some of the new transformations are applied to non-relativistic quantum dynamical systems in two dimensions.

We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovász. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds provide examples of highly connected, highly symmetric manifolds.

In this paper we show, in a systematic way, how to relate the Kepler problem to the isotropic harmonic oscillator. Unlike previous approaches, our constructions are carried over in the Lagrangian formalism dealing, with second order vector fields. We therefore provide a tangent bundle version of the Kustaanheimo-Stiefel map.

The exact Green function of a relativistic Coulomb system is given by the transformation method. The earlier treatments are based on the multiple-valued transformation of Kustaanheimo and Stiefel as well as the perturbation expansions. The method presented in this paper relates the relativistic Coulomb path integral to the simple ones and may apply to a large class of problems.