Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

Abstract

In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defined as Abelian, Heisenberg, Conformal and Ladder case of operator algebras, we re-derived these models. These new techniques also provided further examples of superintegrable Hamiltonian with integrals of arbitrary order. It has been conjectured that all quantum superintegrable potentials that do not satisfy any linear equation satisfy nonlinear equations having the Painlevé property. In addition, it has been discovered that their integrals naturally generate finitely generated polynomial algebras and the representations can be exploited to calculate the energy spectrum. For certain very interesting cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. It has been demonstrated that alternative sets of integrals which can be build and used to provide a complete solution. This this allow to make another conjecture i.e. that higher order superintegrable systems can be solved algebraically, they require alternative set of integrals than the one provided by a direct approach.

References in corpus (41)

• An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
• An Extension of Bochner's Problem: Exceptional Invariant Subspaces
• Quantum Groups and Their Applications in Nuclear Physics
• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• 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
• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• 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 in two dimensions and the Racah-Wilson algebra
• 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
• Krein-Adler transformations for shape-invariant potentials and pseudo virtual states
• Extended Krein-Adler theorem for the translationally shape invariant potentials
• Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators
• Third order superintegrable systems separating in polar coordinates
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion
• New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
• Pure quantum integrability
• 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
• Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
• Connection between quantum systems involving the fourth Painleve transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite EOP
• Construction of classical superintegrable systems with higher order integrals of motion from ladder operators
• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• On realizations of polynomial algebras with three generators via deformed oscillator algebras
• Third-order superintegrable systems separable in parabolic coordinates
• Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension
• Quantum superintegrable system with a novel chain structure of quadratic algebras
• A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra
• Structure results for higher order symmetry algebras of 2D classical superintegrable systems
• Two-dimensional superintegrable systems from operator algebras in one dimension
• Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems
• The stationary KdV hierarchy and so(2,1) as a spectrum generating algebra
• Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
• New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies
• Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras

Cited by in corpus (5)

• Symmetries of Schroedinger equation with scalar and vector potentials
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Symmetries of the Schroedinger-Pauli equations for charged particles and quasirelativistic Schroedinger equations
• Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras