Algebraic properties of three-four dimensional anisotropic oscillator potentials through Hamiltonian chains

TL;DR

This paper explores the algebraic properties of three- and four-dimensional anisotropic oscillator potentials using Hamiltonian chains, revealing their superintegrability, Poisson algebra structures, and quantum eigenvalues, with potential extension to higher dimensions.

Contribution

It introduces the concept of Hamiltonian chains for anisotropic oscillators in 3D and 4D, linking superintegrability with algebraic structures and quantum spectra, extending to n dimensions.

Findings

Hamiltonian chains describe superintegrable anisotropic oscillators.

Poisson and quantum algebra structures are characterized.

Eigenvalues of the Hamiltonian chain members are determined.

Abstract

In this work the notion of Hamiltonian chain is presented as applied to anisotropic oscillator potentials especially defined on three and four dimensional Euclidean spaces. A Hamiltonian chain is a sequence of superintegrable Hamiltonians which, in addition, constitute integrals of motion of a new superintegrable system. Along with the Poisson algebras the quantum counterparts of the chain is given as well as the eigenvalues of each member of the chain. The method can be extended to cover n dimensional cases.