A new family of $N$ dimensional superintegrable double singular oscillators and quadratic algebra $Q(3)\oplus so(n)\oplus so(N-n)$

Abstract

We introduce a new family of $N$-dimensional quantum superintegrable model consisting of double singular oscillators of type $(n,N-n)$. The special cases $(2,2)$ and $(4,4)$ were previously identified as the duals of 3- and 5-dimensional deformed Kepler-Coulomb systems with $u(1)$ and $su(2)$ monopoles respectively. The models are multiseparable and their wave functions are obtained in $(n,N-n)$ double-hyperspherical coordinates. We obtain the integrals of motion and construct the finitely generated polynomial algebra that is the direct sum of a quadratic algebra $Q(3)$ involving three generators, $so(n)$, $so(N-n)$ (i.e. $Q(3)\oplus so(n)\oplus so(N-n)$ ). The structure constants of the quadratic algebra themselves involve the Casimir operators of the two Lie algebras $so(n)$ and $so(N-n)$. Moreover, we obtain the finite dimensional unitary representations (unirreps) of the quadratic algebra and present an algebraic derivation of the degenerate energy spectrum of the superintegrable model.

References in corpus (5)

• Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras
• 4D singular oscillator and generalized MIC-Kepler system
• The su(1,1) Dynamical Algebra for the Generalized MICZ-Kepler Problem from the Schrödinger Factorization
• Supersymmetric quantum mechanical generalized MIC-Kepler system
• Four-dimensional double singular oscillator

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• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials
• Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• Generalized MICZ-Kepler systems on three-dimensional sphere and hyperboloid