Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
arXiv:2309.12622 · doi:10.1088/1751-8121/ad2e3f
Abstract
We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins, Kress and Miller (2010). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (2011). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay, Turbiner and Winternitz (2009) as well as a deformation discovered by Post, Vinet and Zhedanov (2011) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realisations with finite-dimensional irreps which fill a gap in the literature.
28 pages
References in corpus (15)
- Superintegrability of the Caged Anisotropic Oscillator
- A maximally superintegrable system on an n-dimensional space of nonconstant curvature
- Superintegrable Systems with a Third Order Integrals of Motion
- A Recurrence Relation Approach to Higher Order Quantum Superintegrability
- Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties
- Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
- Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension
- Invariant Classification and Limits of Maximally Superintegrable Systems in 3D
- The Tremblay-Turbiner-Winternitz system as extended Hamiltonian
- Toward a classification of semidegenerate 3D superintegrable systems
- On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields
- A fourth-order superintegrable system with a rational potential related to Painleve VI
- On higher-dimensional superintegrable systems: A new family of classical and quantum Hamiltonian models
- A new way to classify 2D higher order quantum superintegrable systems
- A family of fourth-order superintegable systems with rational potentials related to Painlevé VI