5,292 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
Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations
The link between 3D spaces with (in general, non-constant) curvature and
quantum deformations is presented. It is shown how the non-standard deformation
of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that
represent geodesic motions on 3D manifolds with a non-constant curvature that
turns out to be a function of the deformation parameter z. A different
Hamiltonian defined on the same deformed coalgebra is also shown to generate a
maximally superintegrable geodesic motion on 3D Riemannian and (2+1)D
relativistic spaces whose sectional curvatures are all constant and equal to z.
This approach can be generalized to arbitrary dimension.Comment: 7 pages. Communication presented at the 14th Int. Colloquium on
Integrable Systems 14-16 June 2005, Prague, Czech Republi
Superintegrability on sl(2)-coalgebra spaces
We review a recently introduced set of N-dimensional quasi-maximally
superintegrable Hamiltonian systems describing geodesic motions, that can be
used to generate "dynamically" a large family of curved spaces. From an
algebraic viewpoint, such spaces are obtained through kinetic energy
Hamiltonians defined on either the sl(2) Poisson coalgebra or a quantum
deformation of it. Certain potentials on these spaces and endowed with the same
underlying coalgebra symmetry have been also introduced in such a way that the
superintegrability properties of the full system are preserved. Several new N=2
examples of this construction are explicitly given, and specific Hamiltonians
leading to spaces of non-constant curvature are emphasized.Comment: 12 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
(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
Quantum (1+1) extended Galilei algebras: from Lie bialgebras to quantum R-matrices and integrable systems
The Lie bialgebras of the (1+1) extended Galilei algebra are obtained and
classified into four multiparametric families. Their quantum deformations are
obtained, together with the corresponding deformed Casimir operators. For the
coboundary cases quantum universal R-matrices are also given. Applications of
the quantum extended Galilei algebras to classical integrable systems are
explicitly developed.Comment: 16 pages, LaTeX. A detailed description of the construction of
integrable systems is carried ou
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