Applications of Polynomial Algebras to 2-Dimensional Deformed Oscillators

The idea of using physical systems symmetries to study degenerate energy levels has been adopted since the early days of quantum mechanics. So ladder operators which connect all the eigen-states with a given energy lead a good method to solve this problem. For linear systems, such as Hydrogen atom and isotropic harmonic oscillator, Lie algebra can work out these problems well. Generally, the N -dimensional hydrogen atom has the so(N + 1) and the oscillator has the su(N) symmetry. Afterwards, Higgs [1] and Leemon [2] introduced a generalization of the hydrogen atom and isotropic harmonic oscillator in a space with constant curvature. In Higgs’ literature [1], he constructed a new algebra isomoriphic to so(3) and su(2) to describe the symmetry of hydrogen atom and isotropic harmonic oscillator on 2-dimensional sphere and this new algebra is called Higgs algebra which is also used in two-body Calogero-Sutherland model [3] and Karassiov-Klimov model [4]. Then, additional examples, like the Fokas-Lagerstrom potential [5], the Smorodinsky-Winternitz potential [6], and the Holt potential [7], were finally solved by Dennis Bonatsos et al [8] in the method of ladder operators. The polynomial algebra [9] is a deformation of normal angular algebra su(2), which owns three generators J0, J+ and J−. However, the commutative relation of J+ and J− appears the polynomial of J0. su(2) and Higgs algebra are both special cases of polynomial algebra. It can be represented as J(Ω), where Ω is a positive integer which expresses the highest power of the polynomial. The generators J0, J+ and J− of J (Ω) satisfy