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

TL;DR

This paper explores exactly solvable discrete quantum systems focusing on shape invariance, operator solutions, symmetry algebras, and coherent states, with eigenfunctions linked to hypergeometric orthogonal polynomials.

Contribution

It provides explicit examples and detailed analysis of shape invariance, symmetry reductions, and algebraic structures in discrete quantum mechanics, connecting to orthogonal polynomial systems.

Findings

Explicit solutions for discrete quantum systems using hypergeometric polynomials

Detailed analysis of symmetry algebra reductions in Askey-Wilson systems

Construction of coherent states and operator solutions in discrete quantum mechanics

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 Schr\"odinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.