Exactly and quasi-exactly solvable 'discrete' quantum mechanics.

  • Ryu Sasaki
  • 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.

DOI: 10.1098/rsta.2010.0262

Cite this paper

@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} }