Quantum vs Classical Integrability in Calogero-Moser Systems

Abstract

Calogero-Moser systems are classical and quantum integrable multi-particle dynamics defined for any root system $Δ$. The {\em quantum} Calogero systems having $1/q^2$ potential and a confining $q^2$ potential and the Sutherland systems with $1/\sin^2q$ potentials have "integer" energy spectra characterised by the root system $Δ$. Various quantities of the corresponding {\em classical} systems, {\em 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 "quantised". 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.