Family of N-dimensional superintegrable systems and quadratic algebra structures

23rd International Conference on Integrable Systems and Quantum Symmetries

Hoque, Md Fazlul^1 ; Marquette, Ian^1 ; Zhang, Yao-Zhong^1

School of Mathematics and Physics, University of Queensland, Brisbane

QLD

4072, Australia^1

关键词: Algebraic derivation;    Applied mathematics;    Finite dimensional;    Integrals of motion;    Polynomial algebra;    Quadratic algebras;    Separation of variables;    Unitary representations;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/670/1/012024/pdf
DOI  :  10.1088/1742-6596/670/1/012024

来源: IOP

【 摘 要 】

Classical and quantum superintegrable systems have a long history and they possess more integrals of motion than degrees of freedom. They have many attractive properties, wide applications in modern physics and connection to many domains in pure and applied mathematics. We overview two new families of superintegrable Kepler-Coulomb systems with non-central terms and superintegrable Hamiltonians with double singular oscillators of type (n, N - n) in N-dimensional Euclidean space. We present their quadratic and polynomial algebras involving Casimir operators of so(N + 1) Lie algebras that exhibit very interesting decompositions Q(3) ⊕ so(N - 1), Q(3) ⊕ so(n) ⊕ so(N - n) and the cubic Casimir operators. The realization of these algebras in terms of deformed oscillator enables the determination of a finite dimensional unitary representation. We present algebraic derivations of the degenerate energy spectra of these systems and relate them with the physical spectra obtained from the separation of variables.