Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$

Abstract

In a recent FTC by Tremblay {\sl et al} (2009 {\sl J. Phys. A: Math. Theor.} {\bf 42} 205206), it has been conjectured that for any integer value of $k$, some novel exactly solvable and integrable quantum Hamiltonian $H_k$ on a plane is superintegrable and that the additional integral of motion is a $2k$th-order differential operator $Y_{2k}$. Here we demonstrate the conjecture for the infinite family of Hamiltonians $H_k$ with odd $k\ge 3$, whose first member corresponds to the three-body Calogero-Marchioro-Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some $D_{2k}$-extended and invariant Hamiltonian $\chh_k$, which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a $D_{2k}$-invariant integral of motion $\cyy_{2k}$, from which $Y_{2k}$ can be obtained by projection in the $D_{2k}$ identity representation space.

14 pages, no figure; change of title + important addition to sect. 4 + 2 more references + minor modifications; accepted by JPA as an FTC

References in corpus (5)

• An infinite family of solvable and integrable quantum systems on a plane
• Superintegrability and higher order constants for classical and quantum systems
• Periodic orbits for an infinite family of classical superintegrable systems
• Superintegrability and higher order constants for quantum systems
• Exchange operator formalism for an infinite family of solvable and integrable quantum systems on a plane

Cited by in corpus (39)

• Superintegrability and higher order constants for classical and quantum systems
• Third order superintegrable systems separating in polar coordinates
• An infinite family of superintegrable deformations of the Coulomb potential
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Superintegrability and higher order constants for quantum systems
• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces
• Note on Superintegrability of TTW model
• Supersymmetric Quantum Mechanics with Reflections
• Integrable generalizations of oscillator and Coulomb systems via action-angle variables
• An infinite family of superintegrable systems from higher order ladder operators and supersymmetry
• Tools for Verifying Classical and Quantum Superintegrability
• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces : Superintegrability, curvature-dependent formalism and complex factorization
• Rationally-extended Dunkl oscillator on the line
• Third-order superintegrable systems separable in parabolic coordinates
• Superintegrable systems on spaces of constant curvature
• A nonseparable quantum superintegrable system in 2D real Euclidean space
• A new approach to the higher-order superintegrability of the Tremblay-Turbiner-Winternitz system
• Infinite families of superintegrable systems separable in subgroup coordinates
• Higher-order superintegrability of a Holt related potential
• N=2 supersymmetric extension of the Tremblay-Turbiner-Winternitz Hamiltonians on a plane
• Quantum Integrals from Coalgebra Structure
• Structure results for higher order symmetry algebras of 2D classical superintegrable systems
• Analytical Solution of Two-Dimensional Scarf II Model by Means of SUSY Methods
• Superintegrable deformations of superintegrable systems : Quadratic superintegrability and higher-order superintegrability
• An infinite family of superintegrable Hamiltonians with reflection in the plane
• Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion
• On higher-dimensional superintegrable systems: A new family of classical and quantum Hamiltonian models
• Comments on dihedral and supersymmetric extensions of a family of Hamiltonians on a plane
• Chiral super-Tremblay-Turbiner-Winternitz Hamiltonians and their dynamical superalgebra
• From two-dimensional (super-integrable) quantum dynamics to (super-integrable) three-body dynamics
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions
• Searching for integrable Hamiltonian systems with Platonic symmetries
• Action-Angle Variables In Conformal Mechanics
• Superintegrable systems with position dependent mass: master symmetry and action-angle methods
• Superintegrability of the Post-Winternitz system on spherical and hyperbolic spaces
• Coherent States for Tremblay-Turbiner-Winternitz Potential