Exactly and quasi-exactly solvable `discrete' quantum mechanics

Abstract

Brief introduction to the discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators, dynamical symmetry algebras including the $q$-oscillator algebra and the Askey-Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics is presented. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and quasi-exactly solvable ones are constructed. The sinusoidal coordinate plays an essential role.

LaTeX2e with rspublic.cls, amsmath,amssymb, bm, 15 pages. Contribution to the Theme issue of the Philosophical Transactions A, entitled "Nonlinear phenomena, Optical and Quantum solitons," dedicated to the figure of Robin Bullough.

References in corpus (15)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Another set of infinitely many exceptional (X_{\ell}) Laguerre polynomials
• Orthogonal Polynomials from Hermitian Matrices
• Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Properties of the Exceptional ($X_{\ell }$) Laguerre and Jacobi Polynomials
• Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Modification of Crum's Theorem for `Discrete' Quantum Mechanics
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• Crum's Theorem for `Discrete' Quantum Mechanics
• Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations
• q-oscillator from the q-Hermite Polynomial
• Quasi Exactly Solvable Difference Equations
• Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism

Cited by in corpus (3)

• Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials
• Discrete Quantum Mechanics
• Conceptual inconsistencies in finite-dimensional quantum and classical mechanics