An infinite family of superintegrable deformations of the Coulomb potential

TL;DR

This paper introduces a new family of superintegrable Hamiltonians with deformed Coulomb potentials, explores their classical and quantum properties, and reveals their relation to harmonic oscillator deformations through coupling constant metamorphosis.

Contribution

It presents a novel family of superintegrable deformations of the Coulomb potential, extending known systems and establishing their connection to oscillator systems via coupling constant metamorphosis.

Findings

Family is superintegrable for all rational k

Classical trajectories and quantum wave functions are computed

Hamiltonians with oscillator terms relate to Coulomb systems via transformation

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.