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Session 3
Superintegrable Hamiltonians
Chairs:
Willard Miller Jr. (University of Minnesota, USA)
Pavel Winternitz (Universite de Montreal, Canada)
Kurt Bernardo Wolf (Universidad Nacional Autonoma de Mexico, Mexico)
Titles and Abstracts
Cezary Gonera (University of Lodz, Poland)
Title: Superintegrability of Hamiltonian systems separable in polar coordinates
Abstract: A simple and direct proof of classification of superintegrable systems with
Hamiltonians separable in polar coordinates given by E. Onofri and M. Pauri (Journ.
Math. Phys. 19 (9) (1978), 1850) is presented.
Francisco J. Herranz (University of Burgos, Spain) (Contributed talk)
Title: Quantum superintegrable nonlinear oscillators and curved Kepler-Coulomb
Hamiltonians (joint work with A. Ballesteros, A. Enciso, O. Ragnisco and D. Riglioni)
Abstract: Bertrand's theorem asserts that any spherically symmetric natural
Hamiltonian system
in R3 that has a stable circular trajectory and all whose
bounded trajectories are closed is either a harmonic oscillator or a Kepler system. This
statement finds a natural extension through the definition of Bertrand spacetime
introduced by Perlick [1]; this is a spherically symmetric and static spacetime whose
timelike geodesics satisfy properties analogous to those of the trajectories of the
harmonic oscillator or Kepler systems. Furthermore, if one writes the Lorentzian
metric as where g is a Riemannian metric on a 3-manifold, the timelike geodesics in
spacetime are related to the trajectories of the classical Hamiltonian system. Perlick's
classification of Bertrand spacetimes consists of two multi-parametric families that
correspond to either an oscillator or a Kepler system on a curved manifold [2] and,
moreover, they are superintegrable. The corresponding quantum N-dimensional
Hamiltonians are constructed by applying two different but "gauge-equivalent"
procedures: either through the "direct" Schroedinger quantization or by means of the
Laplace-Beltrami one (with a scalar curvature term!). Several properties of the
resulting quantum superintegrable Bertrand Hamiltonians [3] are also analyzed.
[1] V. Perlick, Class. Quantum Grav. 9 . (1992) 1009.
[2] A. Ballesteros, A. Enciso, F.J. Herranz and O. Ragnisco, Class. Quant. Grav. 25
(2008) 165005.
[3] A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni, in
preparation (2012).
Jonathan Kress (University of New South Wales, Australia)
Title: Invariant classification of superintegrable systems
Abstract: Superintegrable systems with second order constants of the motion have
been extensively studied and all such systems in 3D Euclidean space are known. This
talk will present a classification of second order non-degenerate superintegrable
systems on 3D conformally flat spaces. It will be shown that each such system is
associated with a configuration of 6 points in the extended-complex plane and
invariants of these configurations under the action of the conformal group can be used
to classify the systems. This is a joint work with Joshua Capel.
Willard Miller Jr. (University of Minnesota, USA)
Title: Structure theory for extended Kepler-Coulomb 3D quantum superintegrable
systems
Abstract: A quantum superintegrable system is an integrable n-dimensional
Hamiltonian system with potential: H=Delta_n+V that admits 2n-1 algebraically
independent partial differential operators commuting with the Hamiltonian, the
maximum number possible. The system is of order L if the maximum order of the
symmetry operators is L. Typically, the algebra generated by the symmetry operators
and their commutators has been proven to close polynomally for n<3 and for
systems of order L=2 for general n. However the degenerate 3-parameter potential
for the 3D extended Kepler-Coulomb system (2nd order superintegrable) appeared to
be an exception as Kalnins et al. (2007) showed that it didn't close polynomially.
The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even
2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th
order superintegrable, Tanoudis and Daskaloyannis (2011) showed it closed
polynomially.
We consider an infinite class of quantum extended Kepler-Coulomb systems that we
show to be superintegrable of arbitrarily high order and explain the (apparent)
anomalies of these systems. A main conclusion is that rational, nonpolynomial
closure, is the expected result for higher order superintegrable systems in n> 2
variables; polynomial closure is very special.
Joint work with Ernest Kalnins and Jonathan Kress.
George Pogosyan (Yerevan State University, Armenia)
Title: Superintegrable systems on SO(2,2) hyperboloids
Abstract: The maximal superintegrable systems including the ordinary KeplerCoulomb problem, oscillator and its generalization on SO(2,2) hyperboloid are
discussed.
Ryu Sasaki (Kyoto University, Japan)
Title: Exactly solvable quantum mechanics and Multi-indexed Orthogonal
Polynomials
Abstract: Multi-indexed orthogonal polynomials are discovered as solutions of
exactly solvable 1-d quantum mechanics. They are obtained by multiple Darboux
transformations using virtual state solutions. Multiple (q)-Racah polynomials are also
reported.
Luc Vinet (Université de Montréal, Canada)
Title: A superintegrable finite oscillator in two dimensions with SU(2) symmetry
Abstract: A superintegrable finite model of the quantum oscillator in two dimensions
is introduced. It is defined on a uniform lattice of triangular shape. The constants of
the motion for the model form an SU(2) symmetry algebra. The wavefunctions are
expressed in terms of bivariate Krawtchouk polynomials. These polynomials form a
basis for SU(2) irreducible representations. It is found that the dynamical difference
eigenvalue equation can be written in terms of creation and annihilation operators. In
the continuous limit, when the number of grid points goes to infinity, standard
two-dimensional harmonic oscillators are obtained. The analysis provides the N → ∞
limit of the bivariate Krawtchouk polynomials which is a product of one-variable
Hermite polynomials.
Based on joint work with H. Miki (Kyoto U.), S. Post (U. Hawai), A. Zhedanov
(Donetsk).