Symmetric Fock space and orthogonal symmetric polynomials associated with the Calogero model

Using a similarity transformation that maps the Calogero model into N decou pled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions.The simultaneous eigenfunction is a deformation of the symmetrized number stat e (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This orthogonal basis is different from the known one that i s a variant of the Jack polynomial, i.e., the Hi-Jack polynomial. This fact shows that the conserved operators derived by the similarity transformatio n and those derived by the Dunkl operator formulation do not commute. Thus we conclude that the Calogero model has two, algebraically inequivalent set s of mutually commuting conserved operators, as is the case with the hydrog en atom. We also confirm the same story for the B-N-Calogero model. (C) 200 0 Elsevier Science Ltd. All rights reserved.