The R-matrix structure of the Euler-Calogero-Moser model

We construct the $r$-matrix for the generalization of the Calogero-Moser system introduced by Gibbons and Hermsen. By reduction procedures we obtain the $r$-matrix for the $O(N)$ Euler-Calogero-Moser model and for the standard $A_N$ Calogero-Moser model.Comment: 7 page

The quantum angular Calogero-Moser model

The rational Calogero-Moser model of n one-dimensional quantum particles with inverse-square pairwise interactions (in a confining harmonic potential) is reduced along the radial coordinate of R^n to the `angular Calogero-Moser model' on the sphere S^{n-1}. We discuss the energy spectrum of this quantum system, its degeneracies and the eigenstates. The spectral flow with the coupling parameter yields isospectrality for integer increments. Decoupling the center of mass before effecting the spherical reduction produces a `relative angular Calogero-Moser model', which is analyzed in parallel. We generalize our considerations to the Calogero-Moser models associated with Coxeter groups. Finally, we attach spin degrees of freedom to our particles and extend the results to the spin-Calogero system.Comment: 1+19 pages, v2: minor corrections, 4 refs. added, version published in JHE

Algebraic Linearization of Dynamics of Calogero Type for any Coxeter Group

Calogero-Moser systems can be generalized for any root system (including the non-crystallographic cases). The algebraic linearization of the generalized Calogero-Moser systems and of their quadratic (resp. quartic) perturbations are discussed.Comment: LaTeX2e, 13 pages, no figure

Supersymmetric V-systems

We construct ${\mathcal N}=4 \,$ $\, D(2,1;\alpha)$ superconformal quantum mechanical system for any configuration of vectors forming a V-system. In the case of a Coxeter root system the bosonic potential of the supersymmetric Hamiltonian is the corresponding generalised Calogero-Moser potential. We also construct supersymmetric generalised trigonometric Calogero-Moser-Sutherland Hamiltonians for some root systems including $BC_N$.Comment: 31 pages; minor change

Intertwining operator for AG2AG_2 Calogero-Moser-Sutherland system

We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$ associated with a configuration of vectors $AG_2$ on the plane which is a union of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter. We find an intertwining operator between $H$ and the Calogero-Moser-Sutherland Hamiltonian for the root system $G_2$. This gives a quantum integral for $H$ of order 6 in an explicit form thus establishing integrability of $H$.Comment: 24 page

Generalized Calogero-Moser-Sutherland models from geodesic motion on GL(n, R) group manifold

It is shown that geodesic motion on the GL(n, R) group manifold endowed with the bi-invariant metric d s^2 = tr(g^{-1} d g)^2 corresponds to a generalization of the hyperbolic n-particle Calogero-Moser-Sutherland model. In particular, considering the motion on Principal orbit stratum of the SO(n, R) group action, we arrive at dynamics of a generalized n-particle Calogero-Moser-Sutherland system with two types of internal degrees of freedom obeying SO(n, R) \bigoplus SO(n, R) algebra. For the Singular orbit strata of SO(n, R) group action the geodesic motion corresponds to certain deformations of the Calogero-Moser-Sutherland model in a sense of description of particles with different masses. The mass ratios depend on the type of Singular orbit stratum and are determined by its degeneracy. Using reduction due to discrete and continuous symmetries of the system a relation to II A_n Euler-Calogero-Moser-Sutherland model is demonstrated.Comment: 16 pages, LaTeX, no figures. V2: Typos corrected, two references added. V3: Abstract changed, typos corrected, a few formulas and references added. The presentation in the last section has been clarified and it was restricted to the case of GL(3, R) group, the analysis of GL(4, R) will be given elsewhere. V4: Minor corrections in the whole text, more formulas and references added, accepted for publication in PL

The perturbation of the quantum Calogero-Moser-Sutherland system and related results

The Hamiltonian of the trigonometric Calogero-Sutherland model coincides with some limit of the Hamiltonian of the elliptic Calogero-Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In this article we show the essential self-adjointness of the Hamiltonian of the elliptic Calogero-Moser model and the regularity (convergence) of the perturbation for the arbitrary root system. We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x_1, >..., x_N) for the A_{N-1}-case. As a result, the algebraic calculation of the perturbation is justified.Comment: Revised version, 35 page