Superintegrability and higher-order constants for classical and quantum systems

@article{Kalnins2011SuperintegrabilityAH,
  title={Superintegrability and higher-order constants for classical and quantum systems},
  author={Ernie G. Kalnins and Willard Miller and George S. Pogosyan},
  journal={Physics of Atomic Nuclei},
  year={2011},
  volume={74},
  pages={914-918}
}
We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate that there are closed orbits in the corresponding classical system if k is rational, and for a number of examples there are generating quantum symmetries that are higher order differential operators than two. Indeed they conjectured that for a general class… 
Tools for verifying classical and quantum superintegrability.
Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n 1 symmetries polynomial in
Structure results for higher order symmetry algebras of 2 D classical superintegrable systems
Recently the authors and J.M. Kress presented a special function recurrence relation method to prove quantum superintegrability of an integrable 2D system that included explicit constructions of
A Recurrence Relation Approach to Higher Order Quantum Superintegrability
We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates.
Superintegrability and higher order integrals for quantum systems
We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrodinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex,
Classical and quantum superintegrability with applications
A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum
Chiral super-Tremblay–Turbiner–Winternitz Hamiltonians and their dynamical superalgebra
The family of Tremblay–Turbiner–Winternitz (TTW) Hamiltonians Hk on a plane, corresponding to any positive real value of k, is shown to admit another supersymmetric extension than that previously
N = 2 supersymmetric extension of the Tremblay-Turbiner-Winternitz Hamiltonians on a plane
The family of Tremblay–Turbiner–Winternitz Hamiltonians Hk on a plane, corresponding to any positive real value of k, is shown to admit an supersymmetric extension of the same kind as that introduced
Quantum Integrals from Coalgebra Structure
Quantum versions of the hydrogen atom and the harmonic oscillator are studied on non Euclidean spaces of dimension N. integrals, of arbitrary order, are constructed via a multi-dimensional version of
An infinite family of superintegrable deformations of the Coulomb potential
We introduce a new family of Hamiltonians with a deformed Kepler–Coulomb potential dependent on an indexing parameter k. We show that this family is superintegrable for all rational k and compute the
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 26 REFERENCES
Second-order superintegrable quantum systems
A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the
Models for quadratic algebras associated with second order superintegrable systems in 2D
There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study
Coupling constant metamorphosis and Nth-order symmetries in classical and quantum mechanics
We review the fundamentals of coupling constant metamorphosis (CCM) and the Stackel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on
Superintegrability and higher order integrals for quantum systems
We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrodinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex,
Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties
A classical (or quantum) superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n − 1 functionally independent constants of the motion polynomial in the
Infinite-order symmetries for quantum separable systems
We develop a calculus to describe the (in general) infinite-order differential operator symmetries of a nonrelativistic Schrödinger eigenvalue equation that admits an orthogonal separation of
FAST TRACK COMMUNICATION: An infinite family of solvable and integrable quantum systems on a plane
An infinite family of exactly solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of
Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform
This paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. Here
A new superintegrable Hamiltonian
We identify a new superintegrable Hamiltonian in three degrees of freedom, obtained as a reduction of pure Keplerian motion in six dimensions. The new Hamiltonian is a generalization of the Keplerian
Families of classical subgroup separable superintegrable systems
TLDR
A method is described for determining complete sets of integrals for a classical Hamiltonian that separates in orthogonal subgroup coordinates, polynomial in the momenta, for some families of generalized oscillator and Kepler-Coulomb systems, hence demonstrating their superintegrability.
...
1
2
3
...