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

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.

On polynomial symmetry algebras underlying superintegrable systems in Darboux spaces

“…Superintegrable systems in 2D Darboux spaces were classified [3,4] and it was found that there exist 12 distinct classes of second order superintegrable systems in the Darboux spaces. In [1] we presented exact solutions via purely algebraic means for the energies of all the 12 classes of superintegrable systems in four different 2D Darboux spaces. This was 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.…”

“…Applying the deformed oscillator technique, in [1] we gave algebraic derivations of the spectra for the 12 superintegrable systems in the 2D Darboux spaces. As an example, we here review some of the results for the superintegrable system in the Darboux space II with the Hamiltonian Ĥ =…”

“…Solution of the Schrödinger equation has the separable form Ψ(x, y) = X(x)Y (y), where X(x) and Y (y) satisfy the second order ODE X ′′ + λφ(x)X = 0 and Y ′′ − λY = 0 with separation constant λ. As shown in [1], the integrals form cubic algebra Q(3) with the following commutation relations…”

“…Finite and Infinite-dimensional representations of symmetry algebras play a significant role in determining the spectral properties of physical Hamiltonians. In [1,2] we focused on the representations of polynomial symmetry algebras underlying superintegrable systems in 2D Darboux spaces. As a result, we are able to construct a large number of states in terms of the Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways.…”