Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states

Abstract

Various examples of exactly solvable `discrete' quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.

46 pages, 2 figures

References in corpus (19)

• Supersymmetry and Quantum Mechanics
• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Temporally stable coherent states for infinite well and Pöschl-Teller potentials
• Orthogonal Polynomials from Hermitian Matrices
• Unified Theory of Annihilation-Creation Operators for Solvable (`Discrete') Quantum Mechanics
• Quantum vs Classical Integrability in Calogero-Moser Systems
• Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
• Shape Invariant Potentials in "Discrete Quantum Mechanics"
• Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials
• Calogero-Sutherland-Moser Systems, Ruijsenaars-Schneider-van Diejen Systems and Orthogonal Polynomials
• q-oscillator from the q-Hermite Polynomial
• Quasi Exactly Solvable Difference Equations
• On the Equilibrium Configuration of the BC-type Ruijsenaars-Schneider System
• Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials
• Quantum vs Classical Integrability in Ruijsenaars-Schneider Systems
• Quasi-Linear Algebras and Integrability (the Heisenberg Picture)
• New Quasi Exactly Solvable Difference Equation
• Multi-Particle Quasi Exactly Solvable Difference Equations
• The relativistic Calogero model in an external field

Cited by in corpus (43)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Orthogonal Polynomials from Hermitian Matrices
• Nonlinear Supersymmetric Quantum Mechanics: concepts and realizations
• Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
• Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
• Modification of Crum's Theorem for `Discrete' Quantum Mechanics
• Discrete Quantum Mechanics
• Multi-indexed (q-)Racah Polynomials
• The Exceptional (X_{\ell}) (q)-Racah Polynomials
• Exactly Solvable Birth and Death Processes
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• Crum's Theorem for `Discrete' Quantum Mechanics
• Orthogonal Polynomials from Hermitian Matrices II
• Dual Christoffel transformations
• q-oscillator from the q-Hermite Polynomial
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism
• Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials
• A new family of shape invariantly deformed Darboux-Pöschl-Teller potentials with continuous \ell
• New Quasi Exactly Solvable Difference Equation
• Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity
• Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types
• Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q|=1 and Quantum Dilogarithm
• Bound states of moving potential wells in discrete wave mechanics
• Perturbations around the zeros of classical orthogonal polynomials
• Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations
• Exactly and quasi-exactly solvable `discrete' quantum mechanics
• New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics
• Symmetric Morse potential is exactly solvable
• Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
• Reflectionless Potentials for Difference Schrödinger Equations
• Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials
• Non-polynomial extensions of solvable potentials a la Abraham-Moses
• Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts
• Exactly Solvable Quantum Mechanics
• Another Type of Forward and Backward Shift Relations for Orthogonal Polynomials in the Askey Scheme
• Bootstrapping Shape Invariance: Numerical Bootstrap as a Detector of Solvable Systems
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types
• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
• Scalar field stochastic dynamics in de Sitter spacetime from exact solutions of quantum deficient oscillators
• Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme