Intertwining Symmetry Algebras of Quantum Superintegrable Systems

Abstract

Abstract. We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like (su(n), so(2n)) or (su(p, q), so(2p, 2q)). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness.

Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions

“…Up to now, only the first one of these Lie algebraic approaches, namely that of potential algebras, has been applied to some D-dimensional superintegrable systems [76,77,78,79,80,81,82,83].…”

Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras

“…Among them are the co-algebra and tensor product approaches, which allowed one to obtain models on curved spaces [25,26]. Other approaches based on factorizations or intertwining relations [27,28] have also been used to generate Ddimensional superintegrable Hamiltonians [29,30]. In the context of classical mechanics, families of subgroup separable superintegrable systems were introduced and studied in [31].…”

Integrable and Superintegrable Systems with Higher Order Integrals of Motion: Master Function Formalism

“…In Refs. [9,10], the spectrum of these systems has been calculated by an algebraic method using the realization of some Lie groups.…”