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(Quasi-) exactly solvable `Discrete' quantum mechanics

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Discrete Integrable Systems

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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