Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions

TL;DR

This paper reviews supersymmetric many-particle quantum systems with inverse-square interactions, focusing on exactly solvable models like the Calogero model, and discusses their properties, formulations, and related unsolved systems.

Contribution

It provides a comprehensive review of supersymmetric inverse-square interaction models, including exact solutions, shape invariance, and pseudo-hermitian formulations, highlighting recent developments.

Findings

Exact solutions for supersymmetric Calogero model

Formulation of pseudo-hermitian supersymmetric systems

Discussion of unsolved inverse-square interaction models

Abstract

The development in the study of supersymmetric many-particle quantum systems with inverse-square interactions is reviewed. The main emphasis is on quantum systems with dynamical OSp(2|2) supersymmetry. Several results related to exactly solved supersymmetric rational Calogero model, including shape invariance, equivalence to a system of free superoscillators and non-uniqueness in the construction of the Hamiltonian, are presented in some detail. This review also includes a formulation of pseudo-hermitian supersymmetric quantum systems with a special emphasis on rational Calogero model. There are quite a few number of many-particle quantum systems with inverse-square interactions which are not exactly solved for a complete set of states in spite of the construction of infinitely many exact eigen functions and eigenvalues. The Calogero-Marchioro model with dynamical SU(1,1|2) supersymmetry and a quantum system related to short-range Dyson model belong to this class and certain aspects of these models are reviewed. Several other related and important developments are briefly summarized.