Unified theory of exactly and quasi-exactly solvable `Discrete' quantum mechanics: I. Formalism

TL;DR

This paper introduces a unified formalism for constructing exactly and quasi-exactly solvable discrete quantum mechanical Hamiltonians, connecting the Askey scheme of orthogonal polynomials with algebraic structures like the Askey-Wilson algebra.

Contribution

It provides a simple recipe to generate solvable Hamiltonians in discrete quantum mechanics, unifying known cases and predicting new ones using the sinusoidal coordinate and algebraic relations.

Findings

Reproduces all known solvable models in the Askey scheme

Predicts new solvable Hamiltonians in discrete quantum mechanics

Clarifies the relationship between the closure relation and Askey-Wilson algebra

Abstract

We present a simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics, in which the Schr\"{o}dinger equation is a difference equation. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. The recipe also predicts several new ones. An essential role is played by the sinusoidal coordinate, which generates the closure relation and the Askey-Wilson algebra together with the Hamiltonian. The relationship between the closure relation and the Askey-Wilson algebra is clarified.