Superintegrable quantum u(3) systems and higher rank factorizations

Abstract

A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant intertwining operators we arrive at a so(6) dynamical algebra and its Hamiltonian hierarchies. We pay attention to those associated to certain unitary irreducible representations that can be displayed by means of three-dimensional polyhedral lattices. We also discuss the role of superpotentials in this new context.

Intertwining Symmetry Algebras of Quantum Superintegrable Systems on Constant Curvature Spaces

“…As far as this ground state is square integrable (with the invariant measure mentioned above) we will get an irreducible representation (for special values of the parameters it is also unitary [2]). This type of representation has a symmetric character due to the invariance under the reflections I j (see also the figures corresponding to the particular representations displayed in [2,3]).…”

“…Observe that taking the adjoint of relation (8) There are two properties worth to mention here. First, it is easy to check that the operator L satisfying (8) will transform eigenfunctions of H into eigenfunctions of H having the same eigenvalue [2,3] (although it can change the square integrability character of the eigenfunctions involved). Second, concerning the structure of L displayed in (7), it can be shown [11,20,21] that the differential term D(s) must be an element of the symmetry algebra of the kinetic part T (s) of H ; thus, in our case it will be an element of so(p, q).…”

“…, n. In order to see how the fundamental states are described by this formula, we will consider two particular cases. For the so(3) ⊂ su(3) ⊂ so (6) Hamiltonian hierarchy [2] we have…”

“…Our main concern here is to set up the adequate frame to extend the properties derived in the particular cases [2,3] and to present them in the simplest and most systematic form. We would like this paper to constitute a motivation in using the intertwining approach to integrable systems, leaving aside many details that should deserve a larger extension.…”

The Harmonic Oscillator on Three-Dimensional Spherical and Hyperbolic Spaces: Curvature Dependent Formalism and Quantization

“…[8][9][10][11] and references therein) studied the quantum oscillator on curved spaces. We must also mention that Higgs studied in 1979 the existence of dynamical symmetries in a spherical geometry [12] and that since then a certain number of authors have considered [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] the problem of the symmetries or some other properties characterizing the Hamiltonian systems on curved spaces (the studies of Schrödinger and Higgs were concerned with a spherical geometry but other authors applied their ideas to the hyperbolic space).…”