469 research outputs found
Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties
A classical (or quantum) second order superintegrable system is an integrable
n-dimensional Hamiltonian system with potential that admits 2n-1 functionally
independent second order constants of the motion polynomial in the momenta, the
maximum possible. Such systems have remarkable properties: multi-integrability
and multi-separability, an algebra of higher order symmetries whose
representation theory yields spectral information about the Schroedinger
operator, deep connections with special functions and with QES systems. Here we
announce a complete classification of nondegenerate (i.e., 4-parameter)
potentials for complex Euclidean 3-space. We characterize the possible
superintegrable systems as points on an algebraic variety in 10 variables
subject to six quadratic polynomial constraints. The Euclidean group acts on
the variety such that two points determine the same superintegrable system if
and only if they lie on the same leaf of the foliation. There are exactly 10
nondegenerate potentials.Comment: 35 page
Superintegrability in a two-dimensional space of nonconstant curvature
A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton–Jacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined
Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties
A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n−1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials. ©2007 American Institute of Physic
Complete sets of invariants for dynamical systems that admit a separation of variables
Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton–Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2, ,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj} = {Pi,Pj} = 0, {Qi,Pj} = ij. The 2n–1 functions Q2, ,Qn,P1, ,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the Hamilton–Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion
Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV
This is the second paper on the path integral approach of superintegrable
systems on Darboux spaces, spaces of non-constant curvature. We analyze in the
spaces \DIII and \DIV five respectively four superintegrable potentials,
which were first given by Kalnins et al. We are able to evaluate the path
integral in most of the separating coordinate systems, leading to expressions
for the Green functions, the discrete and continuous wave-functions, and the
discrete energy-spectra. In some cases, however, the discrete spectrum cannot
be stated explicitly, because it is determined by a higher order polynomial
equation.
We show that also the free motion in Darboux space of type III can contain
bound states, provided the boundary conditions are appropriate. We state the
energy spectrum and the wave-functions, respectively
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
Superintegrable systems with spin and second-order integrals of motion
We investigate a quantum nonrelativistic system describing the interaction of
two particles with spin 1/2 and spin 0, respectively. We assume that the
Hamiltonian is rotationally invariant and parity conserving and identify all
such systems which allow additional integrals of motion that are second order
matrix polynomials in the momenta. These integrals are assumed to be scalars,
pseudoscalars, vectors or axial vectors. Among the superintegrable systems
obtained, we mention a generalization of the Coulomb potential with scalar
potential and spin orbital one
.Comment: 32 page
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
Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces
We formulate the necessary conditions for the maximal super-integrability of
a certain family of classical potentials defined in the constant curvature
two-dimensional spaces. We give examples of homogeneous potentials of degree -2
on as well as their equivalents on and for which these
necessary conditions are also sufficient. We show explicit forms of the
additional first integrals which always can be chosen polynomial with respect
to the momenta and which can be of an arbitrary high degree with respect to the
momenta
Infinite families of superintegrable systems separable in subgroup coordinates
A method is presented that makes it possible to embed a subgroup separable
superintegrable system into an infinite family of systems that are integrable
and exactly-solvable. It is shown that in two dimensional Euclidean or
pseudo-Euclidean spaces the method also preserves superintegrability. Two
infinite families of classical and quantum superintegrable systems are obtained
in two-dimensional pseudo-Euclidean space whose classical trajectories and
quantum eigenfunctions are investigated. In particular, the wave-functions are
expressed in terms of Laguerre and generalized Bessel polynomials.Comment: 19 pages, 6 figure
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