A class of Calogero type reductions of free motion on a simple Lie group

The reductions of the free geodesic motion on a non-compact simple Lie group G based on the $G_+ \times G_+$ symmetry given by left- and right multiplications for a maximal compact subgroup $G_+ \subset G$ are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some special values the `spin' degrees of freedom are absent and we obtain the standard $BC_n$ Sutherland model with three independent coupling constants from SU(n+1,n) and from SU(n,n). This generalization of the Olshanetsky-Perelomov derivation of the $BC_n$ model with two independent coupling constants from the geodesics on $G/G_+$ with G=SU(n+1,n) relies on fixing the right-handed momentum to a non-zero character of $G_+$. The reductions considered permit further generalizations and work at the quantized level, too, for non-compact as well as for compact G.Comment: shortened to 13 pages in v2 on request of Lett. Math. Phys. and corrected some spelling error

Supersymmetry, Shape Invariance and Solvability of AN−1A_{N-1} and BCNBC_{N} Calogero-Sutherland Model

Using the ideas of supersymmetry and shape invariance we re-derive the spectrum of the $A_{N-1}$ and $BC_N$ Calogero-Sutherland model. We briefly discuss as to how to obtain the corresponding eigenfunctions. We also discuss the difficulties involved in extending this approach to the trigonometric models.Comment: 15 pages, REVTeX,No figure

Exchange Operator Formalism for Integrable Systems of Particles

We formulate one dimensional many-body integrable systems in terms of a new set of phase space variables involving exchange operators. The hamiltonian in these variables assumes a decoupled form. This greatly simplifies the derivation of the conserved charges and the proof of their commutativity at the quantum level.Comment: 8 page