Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension

TL;DR

This paper introduces a new superintegrable Kepler-Coulomb system in N-dimensional space, analyzes its algebraic structure, and derives its energy spectrum using quadratic algebra and special functions.

Contribution

It presents a novel superintegrable system with a quadratic algebra structure and provides an algebraic method to derive its spectrum, extending understanding of higher-dimensional integrable models.

Findings

System is multiseparable in hyperspherical and hyperparabolic coordinates.

Quadratic algebra with three generators deforms the $so(N+1)$ symmetry.

Energy spectrum derived from algebraic structure using deformed oscillators.

Abstract

We introduce a new superintegrable Kepler-Coulomb system with non-central terms in $N$-dimensional Euclidean space. We show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates. We present the wave function in terms of special functions. We give a algebraic derivation of spectrum of the superintegrable system. We show how the $so(N+1)$ symmetry algebra of the $N$-dimensional Kepler-Coulomb system is deformed to a quadratic algebra with only 3 generators and structure constants involving a Casimir operator of $so(N-1)$ Lie algebra. We construct the quadratic algebra and the Casimir operator. We show this algebra can be realized in terms of deformed oscillator and obtain the structure function which yields the energy spectrum.