Exactly and quasi-exactly solvable `discrete' quantum mechanics

TL;DR

This paper reviews discrete quantum mechanics, highlighting key solvable systems, algebraic structures, and introduces a method to construct new exactly and quasi-exactly solvable Hamiltonians related to hypergeometric orthogonal polynomials.

Contribution

It presents a simple recipe for constructing new solvable Hamiltonians in discrete quantum mechanics, expanding the class of known exactly and quasi-exactly solvable models.

Findings

Reproduces all known solvable models linked to the Askey scheme.

Constructs several new exactly and quasi-exactly solvable Hamiltonians.

Highlights the role of the sinusoidal coordinate in solvability.

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.