On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2. II.

Abstract

The properties of several noncentral n=2 harmonic oscillators are examined on spaces of constant curvature. All the mathematical expressions are presented using the curvature κ as a parameter, in such a way that particularizing for κ>0, κ=0, or κ<0, the corresponding properties are obtained for the system on the sphere S2, the Euclidean plane E2, or the hyperbolic plane H2, respectively. First, the separability in several κ-dependent systems of coordinates, as well as the existence of four families of κ-dependent superintegrable potentials related with the harmonic oscillator, are studied. Then three harmonic oscillators (1:1, 2:1 and 12:1) are studied by using two different methods: superseparability and complex factorization. The second part deals with the problem of the existence of superintegrable but not superseparable systems. Several κ-dependent superintegrable harmonic oscillators with higher-order constants of motion are studied. The constants of motion are obtained by making use of the method of the complex factorization.