3,919 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
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
On the gravitational content of molecular clouds and their cores
(Abridged) The gravitational term for clouds and cores entering in the virial
theorem is usually assumed to be equal to the gravitational energy, since the
contribution to the gravitational force from the mass distribution outside the
volume of integration is assumed to be negligible. Such approximation may not
be valid in the presence of an important external net potential. In the present
work we analyze the effect of an external gravitational field on the
gravitational budget of a density structure. Our cases under analysis are (a) a
giant molecular cloud (GMC) with different aspect ratios embedded within a
galactic net potential, and (b) a molecular cloud core embedded within the
gravitational potential of its parent molecular cloud. We find that for
roundish GMCs, the tidal tearing due to the shear in the plane of the galaxy is
compensated by the tidal compression in the z direction. The influence of the
external effective potential on the total gravitational budget of these clouds
is relatively small, although not necessarily negligible. However, for more
filamentary GMCs, the external effective potential can be dominant and can even
overwhelm self-gravity, regardless of whether its main effect on the cloud is
to disrupt it or compress it. This may explain the presence of some GMCs with
few or no signs of massive star formation, such as the Taurus or the
Maddalena's clouds. In the case of dense cores embedded in their parent
molecular cloud, we found that the gravitational content due to the external
field may be more important than the gravitational energy of the cores
themselves. This effect works in the same direction as the gravitational
energy, i.e., favoring the collapse of cores. We speculate on the implications
of these results for star formation models.Comment: Accepted for publication in MNRA
Integrable deformations of oscillator chains from quantum algebras
A family of completely integrable nonlinear deformations of systems of N
harmonic oscillators are constructed from the non-standard quantum deformation
of the sl(2,R) algebra. Explicit expressions for all the associated integrals
of motion are given, and the long-range nature of the interactions introduced
by the deformation is shown to be linked to the underlying coalgebra structure.
Separability and superintegrability properties of such systems are analysed,
and their connection with classical angular momentum chains is used to
construct a non-standard integrable deformation of the XXX hyperbolic Gaudin
system.Comment: 15 pages, LaTe
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