Discrete Quantum Mechanics

TL;DR

This paper provides a comprehensive review of discrete quantum mechanics, covering theoretical foundations, solution structures, and introducing new orthogonal polynomial families, advancing the understanding of exactly solvable models.

Contribution

It introduces two new infinite families of orthogonal polynomials, X_ll Meixner-Pollaczek and X_ll Meixner, and offers a unified theory of solvability in discrete quantum mechanics.

Findings

Introduction of X_ll Meixner-Pollaczek polynomials

Introduction of X_ll Meixner polynomials

Detailed analysis of solution spaces and algebraic structures

Abstract

A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics. The main subjects to be covered are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modification), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, the unified theory of exact and quasi-exact solvability based on the sinusoidal coordinates, the infinite families of new orthogonal (the exceptional) polynomials. Two new infinite families of orthogonal polynomials, the X_\ell Meixner-Pollaczek and the X_\ell Meixner polynomials are reported.