A fourth-order superintegrable system with a rational potential related to Painleve VI

Abstract

In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order integral of motion and use the classification of such systems to show that the potential gives a rational solution associated with the sixth Painlevé equation. Additionally, we show that the integrals of the motion close to form a cubic algebra and describe briefly deformed oscillator representations of this algebra.

References in corpus (12)

• An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
• An Extension of Bochner's Problem: Exceptional Invariant Subspaces
• Classical and Quantum Superintegrability with Applications
• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Determinant Structure of the Rational Solutions for the Painlevé IV Equation
• Third order superintegrable systems separating in polar coordinates
• New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
• Families of superintegrable Hamiltonians constructed from exceptional polynomials
• Connection between quantum systems involving the fourth Painleve transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite EOP
• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• General $N^{th}$-order superintegrable systems separating in polar coordinates

Cited by in corpus (6)

• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• A family of fourth-order superintegable systems with rational potentials related to Painlevé VI
• Generalized quadratic commutator algebras of PBW-type