Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems

TL;DR

This paper explores noncommutative quantum systems with gauge invariance, revealing how Dolan-Grady relations influence their algebraic structure and spectrum, especially on fuzzy geometries like the sphere and hyperbolic plane.

Contribution

It introduces a covariant method using intrinsic algebra to analyze spectral problems of noncommutative systems governed by Dolan-Grady relations.

Findings

Partial algebraization of spectra on fuzzy geometries.

Restriction of structure functions to quadratic polynomials.

Development of a covariant spectral analysis method.

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.