A new family of $N$ dimensional superintegrable double singular oscillators and quadratic algebra $Q(3)\oplus so(n) \oplus so(N-n)$

TL;DR

This paper introduces a new class of N-dimensional superintegrable quantum models with double singular oscillators, constructing their algebraic structure and deriving their energy spectra using polynomial algebra methods.

Contribution

The paper presents a novel family of superintegrable models with explicit algebraic structures and solutions, extending previous dualities and providing a framework for analyzing similar systems.

Findings

Constructed the quadratic algebra $Q(3)$ and its direct sum with $so(n)$ and $so(N-n)$.

Derived finite-dimensional unitary representations of the algebra.

Provided an algebraic derivation of the energy spectrum.

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 $(n,N-n)$ double-hyperspherical 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. $Q(3)\oplus so(n) \oplus so(N-n)$ ). 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.