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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.

The first author would also like to thank Professors Maxim Zinchenko and Alexander Katsevich for their valuable comments and suggestions on the presentation of the ideas. In addition, he thanks his friends Jie Liang, Byron Conley, and Brent Perreault for their feedback after proofreading portions of this manuscript.

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