Dmitrií Sadovskií

Quaternionic Dirac oscillator and the universal local form of quaternionic slow-fast systems

We construct a slow-fast dynamical system with co-dimension 1+4 (one general formal parameter and two slow degrees of freedom) degeneracy of its fast semi-quantum eigenvalues on the example of a fast spin-reversal spin-3/2 subsystem coupled to two slow oscillators in spin-oscillator resonance 1:1:2. This is the most elementary dynamical equivalent of the spin-3/2 system with a five-parameter quadrupole Hamiltonian which was introduced by Mead [Phys. Rev. Lett. 59, 161-4 (1987)] and Avron, Sadun, Segert, and Simon, [Commun. Math. Phys. 124(4), 595-627 (1989)] as a time-reversal generalization of the Berry paradigm of systems exhibiting geometric phases.
The spectrum of the quantum universal local form consists of superbands which correspond to the semiquantum eigenvalues, and the sole remaining formal parameter governs the redistribution of quantum states between these superbands. The equivalence to the formal quadrupole system is uncovered through the equality of the numbers of redistributed states and Chern numbers c_2 computed by Avron et al.

Outline:
- Dynamical and formal systems with geometric phases
- Degeneracy of eigenvalues. "Triades" R-C-H of Arnold
- Symmetry Z2 in H-systems: reversal of spin TS and time T
- Geometric phase, Abelian or non-Abelian, and 1st or 2nd Chern classes
- Universal local form and "conical" symmetry of R-C-H systems
- R-system: its elementary form, its Möbius bundle
- Application of the PDE index theories. Topological charge
- Dirac oscillator as an elementary C-system
- Quaternionic Dirac oscillator as an elementary H-system
- "Physical" systems with a compact phase space