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arXiv · hep-th/0212117

Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems

Sergey M. Klishevich·Mikhail S. Plyushchay
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Abstract

We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives obeying the nonlinear Dolan-Grady relations. This restricts the structure function of the deformed oscillator algebra to a quadratic polynomial. The cases when the coordinates form the su(2) and sl(2,R) algebras are investigated in detail. Reducing the Hamiltonian to 1D finite-difference quasi-exactly solvable operators, we demonstrate partial algebraization of the spectrum of the corresponding systems on the fuzzy sphere and noncommutative hyperbolic plane. A completely covariant method based on the notion of intrinsic algebra is proposed to deal with the spectral problem of such systems.

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Sergey M. Klishevich, Mikhail S. Plyushchay. 2003-09-11. Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems. https://doi.org/10.1088/0305-4470%2F36%2F44%2F009

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