Exactly and quasi-exactly solvable `discrete' quantum mechanics

Details Exactly and quasi-exactly solvable `discrete' quantum mechanics Exactly and quasi-exactly solvable `discrete' quantum mechanics

2010-04-27

arxiv.org/abs/1004.4712v1

Physics

Mathematical Physics

LaTeX2e with rspublic.cls, amsmath,amssymb, bm, 15 pages. Contribution to the Theme issue of the Philosophical Transactions A,

Scientific paper

Brief introduction to the 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, dynamical symmetry algebras including the $q$-oscillator algebra and the Askey-Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics is presented. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and quasi-exactly solvable ones are constructed. The sinusoidal coordinate plays an essential role.

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