### (Quasi-) exactly solvable `Discrete' quantum mechanics

**Odake, S ***(Shinshu)*

Monday 23 March 2009, 11:30-12:30

Meeting Room 3, CMS

#### 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.

#### Presentation