Superintegrable systems in Darboux spaces

2003/07/18 by E. G. Kalnins, J. M. Kress, W. Miller +2 · 1 citation
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #math-ph #math.MP #msc:37K05 #msc:70H20

paper · pdf · doi:10.1063/1.1619580

published as J. Math. Phys. 44 (2003) 5811-5848

arxiv created 2003/07/18 · openalex publication_date 2003/11/22 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via “coupling constant metamorphosis” (or equivalently, via Stäckel multiplier transformations). We present a table of the results.

Citations

• Completeness of superintegrability in two-dimensional constant-curvature spaces
• Superintegrability in a two-dimensional space of nonconstant curvature

Cited by

• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models

Related

• Superintegrability in a two-dimensional space of nonconstant curvature
• Completeness of superintegrability in two-dimensional constant-curvature spaces
• Suprintegrable models on riemannian surfaces of revolution with\n integrals of any integer degree (II)
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Hamiltonians separable in Cartesian coordinates and third-order integrals of motion