Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

Abstract

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the $\mathfrak{gl}(n)$ Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

References in corpus (56)

• Supersymmetry and Quantum Mechanics
• An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
• Infinitely many shape invariant potentials and new orthogonal polynomials
• An Extension of Bochner's Problem: Exceptional Invariant Subspaces
• Classical and Quantum Superintegrability with Applications
• Quantum Groups and Their Applications in Nuclear Physics
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Another set of infinitely many exceptional (X_{\ell}) Laguerre polynomials
• Deformed oscillator algebras for two dimensional quantum superintegrable systems
• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• Higher-order supersymmetric quantum mechanics
• Solvable rational extensions of the isotonic oscillator
• Non-linear finite $W$-symmetries and applications in elementary systems
• A quantum exactly solvable non-linear oscillator related with the isotonic oscillator
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials
• New features in supersymmetry breakdown in quantum mechanics
• The Darboux transformation and algebraic deformations of shape-invariant potentials
• Multistep DBT and regular rational extensions of the isotonic oscillator
• Superintegrability and higher order polynomial algebras I
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Solvable rational extensions of the Morse and Kepler-Coulomb potentials
• New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials
• Families of superintegrable Hamiltonians constructed from exceptional polynomials
• New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
• Algebraic Heun operator and band-time limiting
• Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability
• An infinite family of superintegrable systems from higher order ladder operators and supersymmetry
• Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• Supersymmetric Quantum Mechanics and Painlevé IV Equation
• Generalized Deformed su(2) Algebras, Deformed Parafermionic Oscillators and Finite W Algebras
• Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator
• Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials
• Embedding of the Racah Algebra R($\mathbf{n}$) and Superintegrability
• Finite W-algebras
• ABC of ladder operators for rationally extended quantum harmonic oscillator systems
• Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension
• Hidden symmetries of rationally deformed superconformal mechanics
• The Racah algebra: An overview and recent results
• Heun algebras of Lie type
• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
• Two-dimensional superintegrable systems from operator algebras in one dimension
• A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
• A class of quadratic deformations of Lie superalgebras
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• Coherent states for ladder operators of general order related to exceptional orthogonal polynomials
• Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras
• Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra
• A fourth-order superintegrable system with a rational potential related to Painleve VI
• Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials
• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
• General solution of the exceptional Hermite differential equation and its minimal surface representation
• The $SU(3)\supset SO(3)$ missing label problem and the analytical Bethe ansatz

Cited by in corpus (4)

• Recurrence relations and general solution of the exceptional Hermite equation
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Cubic algebras, induced representations and general solution of the exceptional Laguerre equation $X_1$
• Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent