Family of $N$-dimensional superintegrable systems and quadratic algebra structures

TL;DR

This paper introduces two new families of superintegrable systems in N-dimensional space, explores their quadratic and polynomial algebra structures, and derives their energy spectra algebraically and via separation of variables.

Contribution

It presents novel superintegrable systems with complex algebraic structures and provides algebraic methods to determine their spectra, advancing understanding of high-dimensional integrable models.

Findings

New superintegrable Kepler-Coulomb systems with non-central terms

Superintegrable Hamiltonians with double singular oscillators in N dimensions

Algebraic derivation of energy spectra matching separation of variables results

Abstract

Classical and quantum superintegrable systems have a long history and they possess more integrals of motion than degrees of freedom. They have many attractive properties, wide applications in modern physics and connection to many domains in pure and applied mathematics. We overview two new families of superintegrable Kepler-Coulomb systems with non-central terms and superintegrable Hamiltonians with double singular oscillators of type $(n,N-n)$ in $N$-dimensional Euclidean space. We present their quadratic and polynomial algebras involving Casimir operators of $so(N+1)$ Lie algebras that exhibit very interesting decompositions $Q(3)\oplus so(N-1)$, $Q(3)\oplus so(n)\oplus so(N-n)$ and the cubic Casimir operators. The realization of these algebras in terms of deformed oscillator enables the determination of a finite dimensional unitary representation. We present algebraic derivations of the degenerate energy spectra of these systems and relate them with the physical spectra obtained from the separation of variables.