2009/08/30 by Ian Marquette · 3 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Cartesian coordinate system #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Order (exchange) #Polynomial #Quantum Mechanics and Non-Hermitian Physics #Quintic function #Third order #Unitary state #hep-th #math-ph #math.MP #msc:81R05
paper · pdf · doi:10.1088/1751-8113/43/13/135203
published as J.Phys.A43:135203,2010 · 17 pages
arxiv created 2009/08/30 · openalex publication_date 2010/03/10 · arxiv updated 2010/04/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We present a method to obtain higher order integrals and polynomial algebras for two-dimensional quantum superintegrable systems separable in Cartesian coordinates from ladder operators. All systems with a second- and a third-order integral of motion separable in Cartesian coordinates were studied. The integrals of motion of two of them do not generate a cubic algebra. We construct for these Hamiltonians a higher order polynomial algebra from their ladder operators. We obtain quintic and seventh-order polynomial algebras. We also give for the polynomial algebras of order 7 realizations in terms of deformed oscillator algebras. These realizations and finite-dimensional unitary representations allow us to obtain the energy spectrum. We also apply the construction to the caged anisotropic harmonic oscillator and a system involving the fourth Painlevé transcendent.