Physics – Mathematical Physics
Scientific paper
2008-07-17
J.Math.Phys.50:012101,2009
Physics
Mathematical Physics
33 pages, references added, misprints corrected
Scientific paper
10.1063/1.3013804
We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general cubic algebra and we present specific realizations. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in Cartesian coordinates with a third order integral are known. The general formalism is applied to quantum reducible and irreducible rational potentials separable in Cartesian coordinates in E2. We also discuss these potentials from the point of view of supersymmetric and PT-symmetric quantum mechanics.
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