Topology of the Liouville foliation for the Matveev–Dullin integrable case of the two-dimensional sphere

2009 ◽  
Vol 158 (2) ◽  
pp. 276-287
Author(s):  
A. Yu. Moskvin
Keyword(s):  
Integrable Case ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Liouville Foliation

scholarly journals GEOMETRIC MOMENTUM IN THE MONGE PARAMETRIZATION OF TWO-DIMENSIONAL SPHERE

2013 ◽  
Vol 10 (03) ◽  
pp. 1220031 ◽  
Author(s):  
D. M. XUN ◽  
Q. H. LIU
Keyword(s):  
Quantum Mechanics ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Constrained Motion ◽  
Geometric Invariant ◽  
Gradient Operator ◽  
The Laplace Operator

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

scholarly journals Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

2020 ◽  
pp. 2150006
Author(s):  
Denis Bonheure ◽  
Jean Dolbeault ◽  
Maria J. Esteban ◽  
Ari Laptev ◽  
Michael Loss
Keyword(s):  
Magnetic Field ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Euclidean Spaces ◽  
Hardy Inequalities ◽  
Interpolation Inequalities ◽  
Nonlinear Interpolation ◽  
Magnetic Potentials ◽  
Aharonov Bohm

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

Probabilistic aspects of the SU(2)-harmonic oscillator

2000 ◽  
Vol 37 (1) ◽  
pp. 306-314
Author(s):  
Shunlong Luo
Keyword(s):  
Harmonic Oscillator ◽  
Coherent States ◽  
Probability Distributions ◽  
Quantum Probability ◽  
Dimensional Sphere ◽  
Large Radius ◽  
Two Dimensional ◽  
Bargmann Transform ◽  
Gaussian Distributions ◽  
Binomial Distributions

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

scholarly journals Normal Distribution on the Rotation Group So(3)

1997 ◽  
Vol 29 (3-4) ◽  
pp. 201-233 ◽  
Author(s):  
Dmitry I. Nikolayev ◽  
Tatjana I. Savyolov
Keyword(s):  
Normal Distribution ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Normal Distributions ◽  
The Central Limit Theorem ◽  
Special Cases ◽  
Wrapped Normal ◽  
Circular Normal Distribution

We study the normal distribution on the rotation group SO(3). If we take as the normal distribution on the rotation group the distribution defined by the central limit theorem in Parthasarathy (1964) rather than the distribution with density analogous to the normal distribution in Eucledian space, then its density will be different from the usual (1/2πσ) exp⁡(−(x−m)2/2σ2) one. Nevertheless, many properties of this distribution will be analogous to the normal distribution in the Eucledian space. It is possible to obtain explicit expressions for density of normal distribution only for special cases. One of these cases is the circular normal distribution.The connection of the circular normal distribution SO(3) group with the fundamental solution of the corresponding diffusion equation is shown. It is proved that convolution of two circular normal distributions is again a distribution of the same type. Some projections of the normal distribution are obtained. These projections coincide with a wrapped normal distribution on the unit circle and with the Perrin distribution on the two-dimensional sphere. In the general case, the normal distribution on SO(3) can be found numerically. Some algorithms for numerical computations are given. These investigations were motivated by the orientation distribution function reproduction problem described in the Appendix.

scholarly journals A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

2014 ◽  
Vol 47 (34) ◽  
pp. 345204 ◽  
Author(s):  
Ángel Ballesteros ◽  
Alfonso Blasco ◽  
Francisco J Herranz ◽  
Fabio Musso
Keyword(s):  
Hyperbolic Plane ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Anisotropic Oscillator
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