16 research outputs found
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
Hopf maps and Wigner's little groups
We present the explicit formulae relating Hopf maps with Wigner's little
groups. They, particularly, explain simple action of group on a fiber for the
first and second Hopf fibrations, and present most simplified form for the
third one. Corresponding invariant Lagrangians are presented, and their
possible reductions are discussed.Comment: 9pp, published versio
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
Action-angle variables for the particle near extreme Kerr throat
We construct the action-angle variables for the spherical part of conformal
mechanics describing the motion of a particle near extreme Kerr throat. We
indicate the existence of the critical point (with
being the mass of the particle, denoting the speed of light, being the Schwarzschild radius of a black hole with mass
, and denoting the gravitational constant), where these variables
are expressed in terms of elementary functions. Away from this point the
action-angle variables are defined by elliptic integrals.
The proposed formulation allows one to easily reconstruct the whole dynamics
of the particle both in initial coordinates, as well as in the so-called
conformal basis, where the Hamiltonian takes the form of conventional
non-relativistic conformal mechanics.
The related issues, such as semiclassical quantization and
supersymmetrization are also discussed.Comment: 8 pages, PACS numbers: 04.70.Bw, 45.10.Na; we corrected a mistak
Second Hopf map and Yang-Coulomb system on 5d (pseudo)sphere
Using the second Hopf map, we perform the reduction of the eight-dimensional
(pseudo)spherical (Higgs)oscillator to a five-dimensional system interacting
with a Yang monopole. Then, using a standard trick, we obtain, from the latter
system, the pseudospherical and spherical generalizations of the Yang-Coulomb
system (the five dimensional analog of MICZ-Kepler system). We present the
whole set of its constants of motions, including the hidden symmetry generators
given by the analog of Runge-Lenz vector. In the same way, starting from the
eight-dimensional anisotropic inharmonic Higgs oscillator, we construct the
integrable (pseudo)spherical generalization of the Yang-Coulomb system with the
Stark term.Comment: 10 pages, PACS: 03.65.-w, 02.30.Ik, 14.80.H
Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems
We propose the integrable (pseudo)spherical generalization of the
four-dimensional anisotropic oscillator with additional nonlinear potential.
Performing its Kustaanheimo-Stiefel transformation we then obtain the
pseudospherical generalization of the MICZ-Kepler system with linear and
potential terms. We also present the generalization of the
parabolic coordinates, in which this system admits the separation of variables.
Finally, we get the spherical analog of the presented MICZ-Kepler-like system.Comment: 7 page