Unified theory of exactly and quasi-exactly solvable ‘Discrete’ quantum mechanics: I. Formalism

We present a simple recipe to construct exactly and quasi-exactly 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.