Unified theory of exactly and quasiexactly solvable “discrete” quantum mechanics. I. Formalism

Abstract

We present a simple recipe to construct exactly and quasiexactly solvable Hamiltonians in one-dimensional “discrete” quantum mechanics, in which the Schrödinger 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.