Higher Order Quantum Superintegrability: a new "Painlevé conjecture"

Abstract

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order $N>2$. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For $N=3$, 4 and 5 these equations have the Painlevé property. We conjecture that this is true for all $N\ge 3$. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any $N$) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.

arXiv comment23 pages, submitted as a contribution to the monographic volume "Integrability, Supersymmetry and Coherent States", a volume in honour of Professor Véronique Hussin. arXiv admin note: text overlap with arXiv:1703.09751

References in corpus (49)

• A "missing" family of classical orthogonal polynomials
• Higher-Derivative Supersymmetry and the Witten Index
• Classical and Quantum Superintegrability with Applications
• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• An infinite family of solvable and integrable quantum systems on a plane
• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Superintegrability with third order invariants in quantum and classical mechanics
• Exact Solvability of Superintegrable Systems
• Exceptional orthogonal polynomials and the Darboux transformation
• A systematic construction of completely integrable Hamiltonians from coalgebras
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics II:Painleve transcendent potentials
• Systems with Higher-Order Shape Invariance: Spectral and Algebraic Properties
• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Superintegrability in two dimensions and the Racah-Wilson algebra
• Reduction of superintegrable systems: the anisotropic harmonic oscillator
• Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials
• Periodic orbits for an infinite family of classical superintegrable systems
• Third order superintegrable systems separating in polar coordinates
• Superintegrable Systems with a Third Order Integrals of Motion
• An infinite family of superintegrable deformations of the Coulomb potential
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials
• Families of superintegrable Hamiltonians constructed from exceptional polynomials
• The anisotropic oscillator on curved spaces: A new exactly solvable model
• (Super)integrability from coalgebra symmetry: formalism and applications
• Fourth order Superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials
• 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
• Three-dimensional superintegrable systems in a static electromagnetic field
• 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
• Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
• New exactly solvable systems with Fock symmetry
• Higher-order superintegrability of separable potentials with a new approach to the Post-Winternitz system
• General $N^{th}$-order superintegrable systems separating in polar coordinates
• Third-order superintegrable systems separable in parabolic coordinates
• A nonseparable quantum superintegrable system in 2D real Euclidean space
• Quantum superintegrable system with a novel chain structure of quadratic algebras
• Superintegrable systems with spin induced by coalgebra symmetry
• Laplace-Runge-Lenz vector for arbitrary spin
• Symmetry algebra for the generic superintegrable system on the sphere
• Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials
• A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra
• Two-dimensional superintegrable systems from operator algebras in one dimension
• Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials
• Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
• Superintegrable systems, polynomial algebra structures and exact derivations of spectra
• The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion

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• Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$