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arXiv · 1004.4712

Exactly and quasi-exactly solvable `discrete' quantum mechanics

Ryu Sasaki
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Abstract

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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Ryu Sasaki. 2010-04-27. Exactly and quasi-exactly solvable `discrete' quantum mechanics. https://doi.org/10.1098/rsta.2010.0262

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