Shape invariance in the Calogero and Calogero-Sutherland models

Abstract

We show that the Calogero and Calogero-Sutherland models possess an N -body generalization of shape invariance. We obtain the operator representation that gives rise to this result, and discuss the implications of this result, including the possibility of solving these models using algebraic methods based on this shape invariance. Our representation gives us a natural way to construct supersymmetric generalizations of these models, which are interesting both in their own right and for the insights they offer in connection with the exact solubility of these models.

Supersymmetry, shape invariance, and solvability ofAN−1andBCNCalogero-S

“…It is interesting to observe that this system also shares the property of SI as found in [7] in the case of the A N −1 model. Superpotential corresponding to this model is…”

“…In this paper we demonstrate that using SUSY QM, SI and exchange operator formalism [8], the spectrum of the rational A N −1 CSM, and also of all its generalizations like B N , D N and BC N can be obtained algebraically. It is worth mentioning that the SI in our case is somewhat different from that of Efthimiou and Spector [7]. So far as we are aware off, this is the first instance when an N-particle quantum system has been solved using the techniques of SUSY QM and SI.…”

“…It is also shown in this section that the BC N model posses SI even if the exchange operator formalism is not employed. This is a generalization of Efthimiou et al's work [7] on A N −1 model to the BC N case. Finally, in Sec.…”

“…Following [7], define A i (A † i ) = ±∂ i + W i ,from which Hamiltonian (29) with l 2 = 0 can be expressed as…”

“…In particular, whether the spectrum of the celebrated Calogero and other models could be obtained algebraically by using the ideas of SI and SUSY QM. The first step in that direction was taken recently by Efthimiou and Spector [7] who showed that the well known Calogero model (also termed as A N −1 CSM) exhibits SI. However, they were unable to obtain the spectrum algebraically.…”