DIS

Abstract

This talk is based on the collaboration with Ryu Sasaki. `Discrete' quantum mechanics is a quantum mechanical system whose Schr\"{o}dinger equation is a difference equation instead of differential in ordinary quantum mechanics. We present a simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. An essential role is played by the sinusoidal coordinate, which generates the closure relation and the Askey-Wilson algebra together with the Hamiltonian. We also present the Crum's Theorem for `discrete' quantum mechanics.