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(Quasi-) exactly solvable `Discrete' quantum mechanics

Monday 23rd March 2009 - 11:30 to 12:30
Center for Mathematical Sciences
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.
Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons