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

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

2009-03-15

arxiv.org/abs/0903.2604v1

Physics

Mathematical Physics

32 pages

Scientific paper

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.

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