521 research outputs found
Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras
We present the quadratic algebra of the generalized MICZ-Kepler system in
three-dimensional Euclidean space and its dual the four dimensional
singular oscillator in four-dimensional Euclidean space . We present
their realization in terms of a deformed oscillator algebra using the
Daskaloyannis construction. The structure constants are in these cases function
not only of the Hamiltonian but also of other integrals commuting with all
generators of the quadratic algebra. We also present a new algebraic derivation
of the energy spectrum of the MICZ-Kepler system on the three sphere
using a quadratic algebra. These results point out also that results and
explicit formula for structure functions obtained for quadratic, cubic and
higher order polynomial algebras in context of two-dimensional superintegrable
systems may be applied to superintegrable systems in higher dimensions with and
without monopoles.Comment: 15 page
Integrable and superintegrable systems with spin
A system of two particles with spin s=0 and s=1/2 respectively, moving in a
plane is considered. It is shown that such a system with a nontrivial
spin-orbit interaction can allow an 8 dimensional Lie algebra of first-order
integrals of motion. The Pauli equation is solved in this superintegrable case
and reduced to a system of ordinary differential equations when only one
first-order integral exists.Comment: 12 page
Exact Solvability of Superintegrable Systems
It is shown that all four superintegrable quantum systems on the Euclidean
plane possess the same underlying hidden algebra . The gauge-rotated
Hamiltonians, as well as their integrals of motion, once rewritten in
appropriate coordinates, preserve a flag of polynomials. This flag corresponds
to highest-weight finite-dimensional representations of the -algebra,
realized by first order differential operators.Comment: 14 pages, AMS LaTe
Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion
We consider a superintegrable Hamiltonian system in a two-dimensional space
with a scalar potential that allows one quadratic and one cubic integral of
motion. We construct the most general associative cubic algebra and we present
specific realizations. We use them to calculate the energy spectrum. All
classical and quantum superintegrable potentials separable in cartesian
coordinates with a third order integral are known. The general formalism is
applied to one of the quantum potentials
Third-order superintegrable systems separable in parabolic coordinates
In this paper, we investigate superintegrable systems which separate in
parabolic coordinates and admit a third-order integral of motion. We give the
corresponding determining equations and show that all such systems are
multi-separable and so admit two second-order integrals. The third-order
integral is their Lie or Poisson commutator. We discuss how this situation is
different from the Cartesian and polar cases where new potentials were
discovered which are not multi-separable and which are expressed in terms of
Painlev\'e transcendents or elliptic functions
Lie Point Symmetries and Commuting Flows for Equations on Lattices
Different symmetry formalisms for difference equations on lattices are
reviewed and applied to perform symmetry reduction for both linear and
nonlinear partial difference equations. Both Lie point symmetries and
generalized symmetries are considered and applied to the discrete heat equation
and to the integrable discrete time Toda lattice
Generalized Kadomtsev-Petviashvili equation with an infinite dimensional symmetry algebra
A generalized Kadomtsev-Petviashvili equation, describing water waves in
oceans of varying depth, density and vorticity is discussed. A priori, it
involves 9 arbitrary functions of one, or two variables. The conditions are
determined under which the equation allows an infinite dimensional symmetry
algebra. This algebra can involve up to three arbitrary functions of time. It
depends on precisely three such functions if and only if it is completely
integrable.Comment: AMSLaTeX, 16 pages, no figures, corrected some typos and added two
new section
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