Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces

TL;DR

This paper provides algebraic solutions for the energy spectra of 12 classes of superintegrable systems in 2D Darboux spaces, introducing new algebraic structures and explicit representations.

Contribution

It introduces algebraic methods to exactly solve superintegrable systems in Darboux spaces, including new cubic and quintic symmetry algebras and their representations.

Findings

Exact algebraic solutions for all 12 classes of systems

Construction of deformed oscillator realizations for symmetry algebras

Presentation of three new superintegrable systems with cubic symmetry

Abstract

Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.