Nuclear Physics B 674 [PM] (2003) 615–633 www.elsevier.com/locate/npe
Generalized Hermite polynomials in superspace as eigenfunctions of the supersymmetric rational CMS model Patrick Desrosiers a , Luc Lapointe b , Pierre Mathieu a a Département de Physique, de Génie Physique et d’Optique, Université Laval, Québec, G1K 7P4, Canada b Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile
Received 7 May 2003; received in revised form 14 July 2003; accepted 1 August 2003
Abstract We present an algebraic construction of the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero–Moser–Sutherland model with harmonic confinement. These eigenfunctions are the superspace extension of the generalized Hermite (or Hi-Jack) polynomials. The conserved quantities of the rational supersymmetric model are related to their trigonometric relatives through a similarity transformation. This leads to a simple expression between the corresponding eigenfunctions: the generalized Hermite superpolynomials are written as a differential operator acting on the corresponding Jack superpolynomials. As an aside, the maximal superintegrability of the supersymmetric rational Calogero–Moser–Sutherland model is demonstrated. 2003 Elsevier B.V. All rights reserved.
1. Introduction Operator constructions of the eigenfunctions of the rational Calogero–Moser–Sutherland1 model with confinement (rCMS) [1], E-mail addresses:
[email protected] (P. Desrosiers),
[email protected] (L. Lapointe),
[email protected] (P. Mathieu). 1 We use the qualitative ‘Calogero–Moser–Sutherland’ to describe the generic class of models that includes the models studied by Calogero and Sutherland in the quantum case and by Moser in the classical case. In this work, we only treat the quantum case, and in this context the name of Moser is often omitted. The rationale for its inclusion is due to the fundamental importance of the Lax formulation in the quantum case, which is a direct extension of the classical one that he introduced. Since the quantum rational model with confinement was studied by Calogero, what we call the rCMS model is often referred to as the Calogero model. 0550-3213/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2003.08.003
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P. Desrosiers et al. / Nuclear Physics B 674 [PM] (2003) 615–633
H=
N β(β − 1) ∂2 1 − 2 + ω2 xi2 + , 2 ∂xi xij2 i=1 1i
