title: Exactly and quasi-exactly solvable 'discrete' quantum mechanics.
abstract: A brief introduction to discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators and dynamical symmetry algebras, including the q-oscillator algebra and the Askey-Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable (QES) Hamiltonians in one-dimensional 'discrete' quantum mechanics is presented. It reproduces all the known Hamiltonians whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and QES Hamiltonians are constructed. The sinusoidal coordinate plays an essential role.
authors.affiliation: Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan. ryu.yukawa.kyoto-u.ac.jp
authors.forename: Ryu
authors.initials: R
authors.lastname: Sasaki