Classical and quantum higher order superintegrable systems from coalgebra symmetry

Abstract

The N-dimensional generalization of Bertrand spaces as families of Maximally superintegrable systems on spaces with nonconstant curvature is analyzed. Considering the classification of two dimensional radial systems admitting 3 constants of the motion at most quadratic in the momenta, we will be able to generate a new class of spherically symmetric M.S. systems by using a technique based on coalgebra. The 3-dimensional realization of these systems provides the entire classification of classical spherically symmetric M.S. systems admitting periodic trajectories. We show that in dimension N > 2 these systems (classical and quantum) admit, in general, higher order constants of motion and turn out to be exactly solvable. Furthermore it is possible to obtain non radial M.S. systems by introducing projection of the original radial system to a suitable lower dimensional space.

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• The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
• Coalgebra symmetry for discrete systems
• Superintegrable systems in non-Euclidean plane: hidden symmetries leading to linearity
• Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Discrete-time systems in quasi-standard form and the ${\mathfrak{h}}_6$ coalgebra symmetry
• A maximally superintegrable deformation of the N-dimensional quantum Kepler-Coulomb system
• Exactly solvable deformations of the oscillator and Coulomb systems and their generalization
• Extensions of natural Hamiltonians