Supersymmetric many-particle quantum systems with inverse-square interactions

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 vertical bar 2) supersymmetry. Several results related to the 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 the 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 eigenfunctions and eigenvalues. The Calogero-Marchioro model with dynamical SU(1, 1 vertical bar 2) supersymmetry and a quantum system related to the 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.