New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces

TL;DR

This paper extends Bertrand's theorem to curved spaces, classifies superintegrable Hamiltonian systems with position-dependent mass, and explores their quantum properties, revealing new models with potential applications in physics.

Contribution

It introduces a generalized Bertrand's theorem on curved spaces and links superintegrable systems to position-dependent mass models, expanding the understanding of integrable systems.

Findings

Two families of superintegrable Hamiltonians are identified as Kepler or oscillator types.

Explicit relationship established between Bertrand Hamiltonians and position-dependent mass systems.

Example of a superintegrable nonlinear oscillator on a Bertrand-Darboux space is analyzed, including quantization.

Abstract

A generalized version of Bertrand's theorem on spherically symmetric curved spaces is presented. This result is based on the classification of (3+1)-dimensional (Lorentzian) Bertrand spacetimes, that gives rise to two families of Hamiltonian systems defined on certain 3-dimensional (Riemannian) spaces. These two systems are shown to be either the Kepler or the oscillator potentials on the corresponding Bertrand spaces, and both of them are maximally superintegrable. Afterwards, the relationship between such Bertrand Hamiltonians and position-dependent mass systems is explicitly established. These results are illustrated through the example of a superintegrable (nonlinear) oscillator on a Bertrand-Darboux space, whose quantization and physical features are also briefly addressed.