On the Integrability of Classical Ruijsenaars-Schneider Model of BC2BC_{2} Type

The problem of finding most general form of the classical integrable relativistic models of many-body interaction of the $BC_{n}$ type is considered. In the simplest nontrivial case of $n=2$,the extra integral of motion is presented in explicit form within the ansatz similar to the nonrelativistic Calogero-Moser models. The resulting Hamiltonian has been found by solving the set of two functional equations.Comment: 10 pages, LaTeX2e, no figure

Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models

For any root system $\Delta$ and an irreducible representation ${\cal R}$ of the reflection (Weyl) group $G_\Delta$ generated by $\Delta$, a {\em spin Calogero-Moser model} can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member $\mu$ of ${\cal R}$, to be called a "site", we associate a vector space ${\bf V}_{\mu}$ whose element is called a "spin". Its dynamical variables are the canonical coordinates $\{q_j,p_j\}$ of a particle in ${\bf R}^r$, ($r=$ rank of $\Delta$), and spin exchange operators $\{\hat{\cal P}_\rho\}$ ($\rho\in\Delta$) which exchange the spins at the sites $\mu$ and $s_{\rho}(\mu)$. Here $s_\rho$ is the reflection generated by $\rho$. For each $\Delta$ and ${\cal R}$ a {\em spin exchange model} can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For $\Delta=A_r$ and ${\cal R}=$ vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for {\em degenerate} potentials.Comment: 18 pages, LaTeX2e, no figure

Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry

Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser models. Because of the additional q^6 (rational models) or sin^2(2q) (trigonometric models) potentials, their quantum versions are not exactly solvable in contrast with Calogero-Moser models. We show that quantum Inozemtsev models can be deformed to be a widest class of partly solvable (or quasi-exactly solvable) multi-particle dynamical systems. They posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A new method for identifying and solving quasi-exactly solvable systems, the method of pre-superpotential, is presented.Comment: LaTeX2e 28 pages, no figure

Equivalence of the super Lax and local Dunkl operators for Calogero-like models

Following Shastry and Sutherland I construct the super Lax operators for the Calogero model in the oscillator potential. These operators can be used for the derivation of the eigenfunctions and integrals of motion of the Calogero model and its supersymmetric version. They allow to infer several relations involving the Lax matrices for this model in a fast way. It is shown that the super Lax operators for the Calogero and Sutherland models can be expressed in terms of the supercharges and so called local Dunkl operators constructed in our recent paper with M. Ioffe. Several important relations involving Lax matrices and Hamiltonians of the Calogero and Sutherland models are easily derived from the properties of Dunkl operators.Comment: 25 pages, Latex, no figures. Accepted for publication in: Jounal of Physics A: Mathematical and Genera

Quantum versus classical integrability in Calogero-Moser systems

Calogero–Moser systems are classical and quantum integrable multiparticle dynamics defined for any root system Δ. The quantum Calogero systems having 1/q2 potential and a confining q2 potential and the Sutherland systems with 1/sin2q potentials have 'integer' energy spectra characterized by the root system Δ. Various quantities of the corresponding classical systems, e.g. minimum energy, frequencies of small oscillations, the eigenvalues of the classical Lax pair matrices etc, at the equilibrium point of the potential are investigated analytically as well as numerically for all root systems. To our surprise, most of these classical data are also 'integers', or they appear to be 'quantized'. To be more precise, these quantities are polynomials of the coupling constant(s) with integer coefficients. The close relationship between quantum and classical integrability in Calogero–Moser systems deserves fuller analytical treatment, which would lead to better understanding of these systems and of integrable systems in general