An infinite family of superintegrable deformations of the Coulomb potential

Abstract

We introduce a new family of Hamiltonians with a deformed Kepler- Coulomb potential dependent on an indexing parameter k. We show that this family is superintegrable for all rational k and compute the classical trajectories and quantum wave functions. We show that this system is related, via coupling constant metamorphosis, to a family of superintegrable deformations of the harmonic oscillator given by Tremblay, Turbiner and Winternitz. In doing so, we prove that all Hamiltonians with an oscillator term are related by coupling constant metamorphosis to systems with a Kepler-Coulomb term, both on Euclidean space. We also look at the effect of the transformation on the integrals of the motion, the classical trajectories and the wave functions and give the transformed integrals explicitly for the classical system.

11 pages, typos corrected and references updated.

References in corpus (10)

• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Reduction of superintegrable systems: the anisotropic harmonic oscillator
• Superintegrability of the Caged Anisotropic Oscillator
• A New Superintegrable Hamiltonian
• Third order superintegrable systems separating in polar coordinates
• Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd $k$
• Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion
• Coupling constant metamorphosis and Nth order symmetries in classical and quantum mechanics
• Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds
• Families of classical subgroup separable superintegrable systems

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