## Exceptional Askey–Wilson type polynomials through Darboux

- S. Odake, R. Sasaki
- 2010

- Published 2011 in Philosophical transactions. Series A…

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.

@article{Sasaki2011ExactlyAQ,
title={Exactly and quasi-exactly solvable 'discrete' quantum mechanics.},
author={Ryu Sasaki},
journal={Philosophical transactions. Series A, Mathematical, physical, and engineering sciences},
year={2011},
volume={369 1939},
pages={1301-18}
}