Three-body closed chain of interactive (an)harmonic oscillators and the algebra $sl(4)$

TL;DR

This paper investigates the quantum behavior of 2- and 3-body oscillators with pairwise potentials depending on relative distances, exploring integrability, superintegrability, and algebraic structures like $sl(4)$, especially in unequal mass and limiting cases.

Contribution

It introduces a detailed analysis of 3-body oscillators with quadratic and sextic potentials, identifying conditions for superintegrability and connecting the system to the algebra $sl(4)$.

Findings

Two-body harmonic oscillator reduces to a radial Jacobi oscillator.

Certain mass and spring constant relations lead to superintegrability.

The 3-body oscillator exhibits maximal or minimal superintegrability under specific conditions.

Abstract

In this work we study 2- and 3-body oscillators with quadratic and sextic pairwise potentials which depend on relative distances, $|{\bf r}_i - {\bf r}_j |$, between particles. The two-body harmonic oscillator is two-parametric and can be reduced to a one-dimensional radial Jacobi oscillator, while in the 3-body case such a reduction is not possible in general. Our study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only ($S$-states). We pay special attention to the cases where the masses of the particles and spring constants are unequal as well as to the atomic, where one mass is infinite, and molecular, where two masses are infinite, limits. In general, three-body harmonic oscillator is 7-parametric depending on 3 masses and 3 spring constants, and frequency. In particular, the first and second order integrals of the 3-body oscillator for unequal masses are searched: it is shown that for certain relations involving masses and spring constants the system becomes maximally (minimally) superintegrable in the case of two (one) relations.