Quantum ring models and action-angle variables

We suggest to use the action-angle variables for the study of properties of (quasi)particles in quantum rings. For this purpose we present the action-angle variables for three two-dimensional singular oscillator systems. The first one is the usual (Euclidean) singular oscillator, which plays the role of the confinement potential for the quantum ring. We also propose two singular spherical oscillator models for the role of the confinement system for the spherical ring. The first one is based on the standard Higgs oscillator potential. We show that, in spite of the presence of a hidden symmetry, it is not convenient for the study of the system's behaviour in a magnetic field. The second model is based on the so-called CP(1) oscillator potential and respects the inclusion of a constant magnetic field.Comment: 9 pages, nofigure

Noncommutative integrability and action-angle variables in contact geometry

We introduce a notion of the noncommutative integrability within a framework of contact geometry.Comment: 22 pages,1 figure, Theorem 5 slightly modifie

Global Action-Angle Variables for Non-Commutative Integrable Systems

In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which exist as soon as the fibers of the momentum map of such an integrable system are compact, global action-angle variables rarely exist. This fact was first observed and analyzed by Duistermaat in the case of Liouville integrable systems on symplectic manifolds and later by Dazord-Delzant in the case of non-commutative integrable systems on symplectic manifolds. In our more general case where phase space is an arbitrary Poisson manifold, there are more obstructions, as we will show both abstractly and on concrete examples. Our approach makes use of a few new features which we introduce: the action bundle and the action lattice bundle of the NCI system (these bundles are canonically defined) and three foliations (the action, angle and transverse foliation), whose existence is also subject to obstructions, often of a cohomological nature

Action-angle variables for dihedral systems on the circle

A nonrelativistic particle on a circle and subject to a cos^{-2}(k phi) potential is related to the two-dimensional (dihedral) Coxeter system I_2(k), for k in N. For such `dihedral systems' we construct the action-angle variables and establish a local equivalence with a free particle on the circle. We perform the quantization of these systems in the action-angle variables and discuss the supersymmetric extension of this procedure. By allowing radial motion one obtains related two-dimensional systems, including A_2, BC_2 and G_2 three-particle rational Calogero models on R, which we also analyze.Comment: 8 pages; v2: references added, typos fixed, version for PL

"A Solvable Hamiltonian System" Integrability and Action-Angle Variables

We prove that the dynamical system charaterized by the Hamiltonian $H = \lambda N \sum_{j}^{N} p_j + \mu \sum_{j,k}^{N} {{(p_j p_k)}^{1\over 2}} \{ cos [ \nu ( q_j - q_k)] \}$ proposed and studied by Calogero [1,2] is equivalent to a system of {\it non-interacting} harmonic oscillators. We find the explicit form of the conserved currents which are in involution. We also find the action-angle variables and solve the initial value problem in simple form.Comment: 12 pages, Latex, No Figure