In this paper, we study the Dunkl oscillator model in a generalization of superintegrable Euclidean Hamiltonian systems to the two-dimensional curved ones with a m: n frequency ratio. This defined model of the two-dimensional curved systems depends on a curvature/deformation parameter of the underlying space involving reflection operators. The curved Hamiltonian [Formula: see text] admits the separation of variables in both geodesic parallel and polar coordinates, which generalizes the Cartesian coordinates of the plane. Similar to the behavior of the Euclidean case, which is the κ → 0 limit case of the curved space, the superintegrability of a curved Dunkl oscillator is naturally understood from the factorization approach viewpoint in that setting. Therefore, their associated sets of polynomial constants of motion (symmetries) as well as algebraic relations are obtained for each of them separately. The energy spectrum of the Hamiltonian [Formula: see text] and the separated eigenfunctions are algebraically given in terms of hypergeometric functions and in the special limit case of null curvature occur in the Laguerre and Jacobi polynomials. Finally, the overlap coefficients between the two bases of the geodesic parallel and polar coordinates are given by hypergeometric polynomials.
In the language of the complex formalism, we study the information entropy of a particle on the motion groups from a family of the unitary Cayley-Klein space with constant curvature κ. Hence, in making use of the constant curvature, all the results here presented will be simultaneously valid for the 2D coset space SUκ(2)/U(1) in forms of 2D sphere
, hyperbolic plane
and Euclidean plane
. In addition to physical complex coordinates
, their corresponding components
are expressed in a 2D complex formalism in terms of the constant curvature in Cayley-Klein space. This process enables us to derive information entropies located in the circular well on two spaces, which are the basis for achieving the relationship between Shannon entropy and Fisher information and the inequalities among them. In particular, we notice that the particle has freely behavior in a spherical state
, while it behaves as a constraint particle in the situation of the hyperbolic plane
.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.