Superintegrable systems, polynomial algebra structures and exact derivations of spectra
arXiv:1802.08410 · doi:10.14264/uql.2018.140
Abstract
Superintegrable systems are a class of physical systems which possess more conserved quantities than their degrees of freedom. The study of these systems has a long history and continues to attract significant international attention. This thesis investigates finite dimensional quantum superintegrable systems with scalar potentials as well as vector potentials with monopole type interactions. We introduce new families of -dimensional superintegrable Kepler-Coulomb systems with non-central terms and double singular harmonic oscillators in the Euclidean space, and new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We show their multiseparability and obtain their Schrödinger wave functions in different coordinate systems. We show that the wave functions are given by (exceptional) orthogonal polynomials and Painlevé transcendents (of hypergeometric type). We construct higher-order algebraically independent integrals of motion of the systems via the direct and constructive approaches. These integrals form (higher-rank) polynomial algebras with structure constants involving Casimir operators of certain Lie algebras. We obtain finite dimensional unitary representations of the polynomial algebras and present the algebraic derivations for degenerate energy spectra of these systems. Finally, we present a generalized superintegrable Kepler-Coulomb model from exceptional orthogonal polynomials and obtain its energy spectrum using both the separation of variable and the algebraic methods.
arXiv commentPhD Thesis, School of Mathematics and Physics, The University of Queensland, Australia, January 2018, 175 pages
References in corpus (60)
- 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
- Physics and Mathematics of Calogero particles
- Quantum Groups and Their Applications in Nuclear Physics
- Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
- Two-step Darboux transformations and exceptional Laguerre polynomials
- Exact Solvability of the Calogero and Sutherland Models
- Relation of the oscillator and Coulomb systems on spheres and pseudospheres
- Two families of superintegrable and isospectral potentials in two dimensions
- Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials
- 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
- Superintegrability of the Caged Anisotropic Oscillator
- Integrable and superintegrable quantum systems in a magnetic field
- Non-Abelian magnetic monopole in a Bose-Einstein condensate
- The MICZ-Kepler Problems in All Dimensions
- Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators
- Coulomb-oscillator duality in spaces of constant curvature
- A Recurrence Relation Approach to Higher Order Quantum Superintegrability
- Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion
- Hidden symmetry of hyperbolic monopole motion
- Coupling constant metamorphosis and Nth order symmetries in classical and quantum mechanics
- New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems
- The equitable Racah algebra from three su(1,1) algebras
- Superintegrability of (generalized) Calogero models with oscillator or Coulomb potential
- Hidden Symmetry of the Yang--Coulomb System
- 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
- Multi-center MICZ-Kepler system, supersymmetry and integrability
- Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
- Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
- Classical ladder operators, polynomial Poisson algebras and classification of superintegrable systems
- 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
- Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
- On realizations of polynomial algebras with three generators via deformed oscillator algebras
- An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric
- The Scattering amplitude for Rationally extended shape invariant Eckart potentials
- Second Hopf map and Yang-Coulomb system on 5d (pseudo)sphere
- Scattering Amplitudes for Multi-indexed Extensions of Solvable Potentials
- Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension
- On the use of the group SO(4,2) in atomic and molecular physics
- A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant Curvature with Accidental Degeneracy in the Spectrum
- Superintegrable systems with spin induced by coalgebra symmetry
- Quantum Integrals from Coalgebra Structure
- The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion
- Generalized Kaluza-Klein monopole, quadratic algebras and ladder operators
- Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
- Oscillator as a hidden non-Abelian monopole
- Recurrence approach and higher rank cubic algebras for the -dimensional superintegrable systems
- Four-dimensional double singular oscillator
- 8D oscillator as a hidden SU(2) - monopole
- The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
- Noncommutative Coulombic Monopole
- Polynomial symmetries of spherical Lissajous systems
- Fifth-order superintergrable quantum system separating in Cartesian coordinates. Doubly exotic potentials
- On the solution of the Calogero model and its generalization to the case of distinguishable particles
- Superintegrability and higher order polynomial algebras II
- Contractions of Degenerate Quadratic Algebras, Abstract and Geometric