A new family ofNdimensional superintegrable double singular oscillators and quadratic algebraQ(3) ⨁so(n)⨁so(N-n)

Abstract

We introduce a new family of N -dimensional quantum superintegrable model consisting of double singular oscillators of type (n, N −n). The special cases (2, 2) and (4, 4) were previously identified as the duals of 3-and 5-dimensional deformed Kepler-Coulomb systems with u(1) and su(2) monopoles respectively. The models are multiseparable and their wave functions are obtained in (n, N − n) doublehyperspherical coordinates. We obtain the integrals of motion and construct the finitely generated polynomial algebra that is the direct sum of a quadratic algebra Q(3) involving three generators, so(n), so(N − n) (i.e. Q(3) ⊕ so(n) ⊕ so (N − n) ). The structure constants of the quadratic algebra themselves involve the Casimir operators of the two Lie algebras so(n) and so(N − n). Moreover, we obtain the finite dimensional unitary representations (unirreps) of the quadratic algebra and present an algebraic derivation of the degenerate energy spectrum of the superintegrable model.

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

“…Another main result of this paper is the construction of universal quadratic algebras for these models with arbitrary functions. We have presented two new families of superintegrable systems with partition of coordinates, generalizing the ones studied in [45] and [44]. By using the separation of variables realized by blocked spherical coordinates, both systems can be transformed to familiar forms but parametrized by operators which are central elements.…”

Superintegrable systems, polynomial algebra structures and exact derivations of spectra

“…After the introductory and preliminary chapters 1 and 2, we present the thesis by the style of "thesis by publication". The thesis incorporates our peer-reviewed and published papers [89][90][91][92][93][94][95][96] in such a way that each paper can be considered as a stand alone chapter.…”

“…This thesis has thus been divided into eleven chapters according to the key mathematical themes of the publications [89][90][91][92][93][94][95][96].…”

“…[90]. I have incorporated text of that paper [90]. In this chapter, we introduce a new family of N -dimensional quantum superintegrable model consisting of double singular oscillators of type (n, N − n).…”

“…We review the results of our recent two papers [89,90] and present the algebraic derivation of the complete energy spectrum of the N -dimensional superintegrable Kepler-Coulomb system with non-central terms and superintegrable double singular oscillators.…”