Exactly Solvable 'Discrete' Quantum Mechanics; Shape Invariance, Heisenberg Solutions, Annihilation-Creation Operators and Coherent States

2008/02/07 by Satoru Odake, Ryu Sasaki · 1 citation
Mathematics · Physics and Astronomy · #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #hep-th #math-ph #math.CA #math.MP #nlin.SI #quant-ph

paper · pdf · doi:10.1143/ptp.119.663

published as Prog. Theor. Phys. 119 (2009), 663-700 · 46 pages, 2 figures

arxiv created 2008/02/07 · openalex publication_date 2008/04/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials
• ASSOCIATED STURM-LIOUVILLE SYSTEMS
• Quantum Inozemtsev model, quasi-exact solvability and 𝒩-fold supersymmetry
• Supersymmetry and Quantum Mechanics
• The relativistic Calogero model in an external field
• Quasi-Linear Algebras and Integrability (the Heisenberg Picture)
• Orthogonal polynomials from Hermitian matrices

Cited by

• Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials