Intertwining Symmetry Algebras of Quantum Superintegrable Systems on Constant Curvature Spaces

Abstract

A class of quantum superintegrable Hamiltonians defined on a hypersurface in a n+1 dimensional ambient space with signature (p, q) is considered and a set of intertwining operators connecting them are determined. It is shown that the intertwining operators can be chosen such that they generate the su(p, q) and so(2p, 2q) Lie algebras and lead to the Hamiltonians through Casimir operators. The physical states corresponding to the discrete spectrum of bound states as well as the degeneration are characterized in terms of some particular unitary representations.

The general Racah algebra as the symmetry algebra of generic systems on pseudo-spheres

“…In the case of generic metric g µν (see Ref. [11] were the signature was considered) the second order symmetries of the above Hamiltonian (2.2)have the form…”

“…The symmetries for the generic Euclidean case have been known from some time ago [15]. In the case of generic metric g μν (see reference [11] were the signature was considered) the second order symmetries of the above Hamiltonian (2.2) have the form…”

“…The main problem is that the method of Daskaloyannis needs to be extended to higher rank algebras, for instance along the lines shown in [16,17]. On the other hand, the spectrum, degeneracy and eigenfunctions can be obtained by separation of variables in different ways [18,14,10,11].…”

“…This means that their symmetry algebras can be identified to the same Racah algebra R(N + 1). This algebra is the commutant of ⊕ N +1 o(2) in the Lie algebra o(2p, 2q + 2) [10,11], which should be the isomorphic to the commutant in so(2N + 2), i.e., independent of the signature. We will analyse the possible discrete spectra for the particular case R(3), by following the method of Daskaloyannis [12].…”