  Quantum Accelerator Modes were discovered in experiments with Kicked Cold
Atoms in the presence of gravity. They were shown to be tightly related to
resonances of the Quantum Kicked Rotor. In this paper a spinor formalism is
developed for the analysis of Modes associatedwith resonances of arbitrary
order q>1. Decoupling of spin variables from orbital ones is achieved by means
of an ansatz of the Born-Oppenheimer type, that generates q independent band
dynamics. Each of these is described, in classical terms, by a map, and the
stable periodic orbits of this map give rise to quantum accelerator modes,
which are potentially observable in experiments. The arithmetic organization of
such periodic orbits is briefly discussed.
