Quantum spin chains and integrable many-body systems of classical mechanics

Abstract

This note is a review of the recently revealed intriguing connection between integrable quantum spin chains and integrable many-body systems of classical mechanics. The essence of this connection lies in the fact that the spectral problem for quantum Hamiltonians of the former models is closely related to a sort of inverse spectral problem for Lax matrices of the latter ones. For simplicity, we focus on the most transparent and familiar case of spin chains on N sites constructed by means of the GL(2)-invariant R-matrix. They are related to the classical Ruijsenaars-Schneider system of N particles, which is known to be an integrable deformation of the Calogero-Moser system. As an explicit example the case N=2 is considered in detail.