2,262 research outputs found
New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces
A generalized version of Bertrand's theorem on spherically symmetric curved
spaces is presented. This result is based on the classification of
(3+1)-dimensional (Lorentzian) Bertrand spacetimes, that gives rise to two
families of Hamiltonian systems defined on certain 3-dimensional (Riemannian)
spaces. These two systems are shown to be either the Kepler or the oscillator
potentials on the corresponding Bertrand spaces, and both of them are maximally
superintegrable. Afterwards, the relationship between such Bertrand
Hamiltonians and position-dependent mass systems is explicitly established.
These results are illustrated through the example of a superintegrable
(nonlinear) oscillator on a Bertrand-Darboux space, whose quantization and
physical features are also briefly addressed.Comment: 13 pages; based in the contribution to the 28th International
Colloquium on Group Theoretical Methods in Physics, Northumbria University
(U.K.), 26-30th July 201
Quantum two-photon algebra from non-standard U_z(sl(2,R)) and a discrete time Schr\"odinger equation
The non-standard quantum deformation of the (trivially) extended sl(2,R)
algebra is used to construct a new quantum deformation of the two-photon
algebra h_6 and its associated quantum universal R-matrix. A deformed one-boson
representation for this algebra is deduced and applied to construct a first
order deformation of the differential equation that generates the two-photon
algebra eigenstates in Quantum Optics. On the other hand, the isomorphism
between h_6 and the (1+1) Schr\"odinger algebra leads to a new quantum
deformation for the latter for which a differential-difference realization is
presented. From it, a time discretization of the heat-Schr\"odinger equation is
obtained and the quantum Schr\"odinger generators are shown to be symmetry
operators.Comment: 12 pages, LaTe
(1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups
All Lie bialgebra structures for the (1+1)-dimensional centrally extended
Schrodinger algebra are explicitly derived and proved to be of the coboundary
type. Therefore, since all of them come from a classical r-matrix, the complete
family of Schrodinger Poisson-Lie groups can be deduced by means of the
Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended
Galilei and gl(2) Lie bialgebras within the Schrodinger classification are
studied. As an application, new quantum (Hopf algebra) deformations of the
Schrodinger algebra, including their corresponding quantum universal
R-matrices, are constructed.Comment: 25 pages, LaTeX. Possible applications in relation with integrable
systems are pointed; new references adde
Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
An infinite family of classical superintegrable Hamiltonians defined on the
N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a
common set of (2N-3) functionally independent constants of the motion. Among
them, two different subsets of N integrals in involution (including the
Hamiltonian) can always be explicitly identified. As particular cases, we
recover in a straightforward way most of the superintegrability properties of
the Smorodinsky-Winternitz and generalized Kepler-Coulomb systems on spaces of
constant curvature and we introduce as well new classes of (quasi-maximally)
superintegrable potentials on these spaces. Results here presented are a
consequence of the sl(2) Poisson coalgebra symmetry of all the Hamiltonians,
together with an appropriate use of the phase spaces associated to Poincare and
Beltrami coordinates.Comment: 12 page
Integrable potentials on spaces with curvature from quantum groups
A family of classical integrable systems defined on a deformation of the
two-dimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed
through Hamiltonians defined on the non-standard quantum deformation of a sl(2)
Poisson coalgebra. All these spaces have a non-constant curvature that depends
on the deformation parameter z. As particular cases, the analogues of the
harmonic oscillator and Kepler--Coulomb potentials on such spaces are proposed.
Another deformed Hamiltonian is also shown to provide superintegrable systems
on the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constant
curvature that exactly coincides with z. According to each specific space, the
resulting potential is interpreted as the superposition of a central harmonic
oscillator with either two more oscillators or centrifugal barriers. The
non-deformed limit z=0 of all these Hamiltonians can then be regarded as the
zero-curvature limit (contraction) which leads to the corresponding
(super)integrable systems on the flat Euclidean and Minkowskian spaces.Comment: 19 pages, 1 figure. Two references adde
Multiparametric quantum gl(2): Lie bialgebras, quantum R-matrices and non-relativistic limits
Multiparametric quantum deformations of are studied through a
complete classification of Lie bialgebra structures. From them, the
non-relativistic limit leading to harmonic oscillator Lie bialgebras is
implemented by means of a contraction procedure. New quantum deformations of
together with their associated quantum -matrices are obtained and
other known quantizations are recovered and classified. Several connections
with integrable models are outlined.Comment: 21 pages, LaTeX. To appear in J. Phys. A. New contents adde
Universal --matrices for non-standard (1+1) quantum groups
A universal quasitriangular --matrix for the non-standard quantum (1+1)
Poincar\'e algebra is deduced by imposing analyticity in the
deformation parameter . A family of ``quantum graded contractions"
of the algebra is obtained; this set of
quantum algebras contains as Hopf subalgebras with two primitive translations
quantum analogues of the two dimensional Euclidean, Poincar\'e and Galilei
algebras enlarged with dilations. Universal --matrices
for these quantum Weyl algebras and their associated quantum groups are
constructed.Comment: 12 pages, LaTeX
Maximal superintegrability on N-dimensional curved spaces
A unified algebraic construction of the classical Smorodinsky-Winternitz
systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie
groups SO(N+1), ISO(N), and SO(N,1) is presented. Firstly, general expressions
for the Hamiltonian and its integrals of motion are given in a linear ambient
space , and secondly they are expressed in terms of two geodesic
coordinate systems on the ND spaces themselves, with an explicit dependence on
the curvature as a parameter. On the sphere, the potential is interpreted as a
superposition of N+1 oscillators. Furthermore each Lie algebra generator
provides an integral of motion and a set of 2N-1 functionally independent ones
are explicitly given. In this way the maximal superintegrability of the ND
Euclidean Smorodinsky-Winternitz system is shown for any value of the
curvature.Comment: 8 pages, LaTe
Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions
The Lie-Poisson algebra so(N+1) and some of its contractions are used to
construct a family of superintegrable Hamiltonians on the ND spherical,
Euclidean, hyperbolic, Minkowskian and (anti-)de Sitter spaces. We firstly
present a Hamiltonian which is a superposition of an arbitrary central
potential with N arbitrary centrifugal terms. Such a system is quasi-maximally
superintegrable since this is endowed with 2N-3 functionally independent
constants of the motion (plus the Hamiltonian). Secondly, we identify two
maximally superintegrable Hamiltonians by choosing a specific central potential
and finding at the same time the remaining integral. The former is the
generalization of the Smorodinsky-Winternitz system to the above six spaces,
while the latter is a generalization of the Kepler-Coulomb potential, for which
the Laplace-Runge-Lenz N-vector is also given. All the systems and constants of
the motion are explicitly expressed in a unified form in terms of ambient and
polar coordinates as they are parametrized by two contraction parameters
(curvature and signature of the metric).Comment: 14 pages. Based on the contribution presented at the "XII
International Conference on Symmetry Methods in Physics", Yerevan (Armenia),
July 2006. To appear in Physics of Atomic Nucle
- …