Interacting families of Calogero-type particles and SU(1,1) algebra

TL;DR

This paper analyzes a one-dimensional multi-family Calogero-type particle model, revealing its universal SU(1,1) algebraic structure, exact eigenstates, and conditions for interaction parameters, establishing equivalences between different systems.

Contribution

It demonstrates the SU(1,1) structure of the multi-family Calogero model and derives exact eigenstates and relations between coupling constants.

Findings

SU(1,1) generators are composed of subsystem generators

Exact eigenenergies are obtained for certain eigenstates

Conditions for absence of three-body interactions are derived

Abstract

We study a one-dimensional model with F interacting families of Calogero-type particles. The model includes harmonic, two-body and three-body interactions. We emphasize the universal SU(1,1) structure of the model. We show how SU(1,1) generators for the whole system are composed of SU(1,1) generators of arbitrary subsystems. We find the exact eigenenergies corresponding to a class of the exact eigenstates of the F-family model. By imposing the conditions for the absence of the three-body interaction, we find certain relations between the coupling constants. Finally, we establish some relations of equivalence between two systems containing F families of Calogero-type particles.