Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems

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.

25 pages; ref added; to appear in J. Phys. A

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• Superconformal mechanics and nonlinear supersymmetry
• Proof of absence of local conserved quantities in the mixed-field Ising chain
• Onsager algebra and algebraic generalization of Jordan-Wigner transformation
• Nonlinear supersymmetry: from classical to quantum mechanics
• The $q$-tetrahedron algebra and its finite dimensional irreducible modules
• Mock Tridiagonal Systems
• The Tetrahedron algebra, the Onsager algebra, and the ${\mathfrak{sl}}_2$ loop algebra
• Defining relations for quantum symmetric pair coideals of Kac-Moody type
• Matrix equations and trilinear commutation relations