On higher-dimensional superintegrable systems: A new family of classical and quantum Hamiltonian models

TL;DR

This paper introduces a new family of higher-dimensional Hamiltonian systems that unify several known superintegrable models and conjectures the existence of parameter values where these systems are superintegrable.

Contribution

The paper presents a novel family of $n$-dimensional Hamiltonian models that generalize and encompass several classical superintegrable systems, proposing new superintegrability cases.

Findings

Includes special reductions to known superintegrable systems.

Conjectures existence of parameter values for superintegrability.

Unifies multiple classical models within a single framework.

Abstract

We introduce a family of $n$-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay-Turbiner-Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable.