Nondegenerate 3D complex Euclidean superintegrable systems and algebraic varieties

Abstract

A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n-1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multi-separability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schroedinger operator, deep connections with special functions and with QES systems. Here we announce a complete classification of nondegenerate (i.e., 4-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in 10 variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly 10 nondegenerate potentials.

35 pages

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Cited by in corpus (25)

• Classical and Quantum Superintegrability with Applications
• Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials
• Superintegrability and higher order constants for classical and quantum systems
• Superintegrability and higher order constants for quantum systems
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• Invariant classification of second-order conformally flat superintegrable systems
• Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions
• Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
• Tools for Verifying Classical and Quantum Superintegrability
• Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
• Leonard Euler: addition theorems and superintegrable systems
• Toward a classification of semidegenerate 3D superintegrable systems
• Structure results for higher order symmetry algebras of 2D classical superintegrable systems
• Bôcher Contractions of Conformally Superintegrable Laplace Equations
• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension
• Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems
• Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
• Stäckel Equivalence of Non-Degenerate Superintegrable Systems, and Invariant Quadrics
• Projectively equivalent 2-dimensional superintegrable systems with projective symmetries
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• Torsion-free connections of second-order maximally superintegrable systems
• Laplace-type equations as conformal superintegrable systems