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

Abstract

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.

32 pages

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• Bound states of moving potential wells in discrete wave mechanics
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• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials V : Racah and $q$-Racah types
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• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
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