Higher Order Quantum Superintegrability: A New “Painlevé Conjecture”

2019/01/01 by Ian Marquette, P. Winternitz, Pavel Winternitz · 12 citations
Mathematics · Physics and Astronomy · #Advanced Fiber Laser Technologies #Cartesian coordinate system #Classical mechanics #Conjecture #Euclidean space #Geometry #Hamiltonian (control theory) #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Motion (physics) #Nonlinear Waves and Solitons #Nonlinear system #Order (exchange) #Physics #Polynomial #Pure mathematics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Type (biology) #math-ph #math.MP

paper · pdf · doi:10.1007/978-3-030-20087-9_4

published as Integrability, Supersymmetry and Coherent States. CRM Series in Mathematical Physics. Springer p.103-131 (2019) · 23 pages, submitted as a contribution to the monographic volume "Integrability, Supersymmetry and Coherent States", a volume in honour of Professor Véronique Hussin. arXiv admin note: text overlap with arXiv:1703.09751

openalex publication_date 2019/01/01 · arxiv created 2019/04/09 · arxiv updated 2020/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order N>2. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For N= 3, 4 and 5 these equations have the Painlevé property. We conjecture that this is true for all N≥3. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any N) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.

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