Systems of Calogero-Moser Type

Abstract

We survey results on Galilei-and Poincare-invariant CalogeroMoser and Toda N-particle systems, both in the context of classical mc hanics and of quantum mechanics. Special attention is given to integrability issues and interconnections between the various models. Action-angle and joint eigenfunction transforms are also considered, and some novel results on N = 2 eigenfunctions of hyperbolic Askey-Wilson type and of relativistic elliptic type are sketched.

The Ruijsenaars Self-Duality Map as a Mapping Class Symplectomorphism

“…For example, the dual of the hyperbolic Sutherland system is the rational RuijsenaarsSchneider system, and the rational Calogero system is self-dual. See the review [14] for the other cases. Incidentally, at the quantum mechanical level, all these systems are known to enjoy the related bispectral property [2], too.…”

Degenerately Integrable Systems

“…These integrable systems were studied quite a lot. For the duality in the non-spin case see [27] [21][4] [9]. Duality in the context of superintegrability in the non-spin case was futher explored in [2], where it was shown that for spinless scattering systems the duality implies the superintegrability.…”

A Recursive Construction of Joint Eigenfunctions for the Hyperbolic Nonrelativistic Calogero-Moser Hamiltonians

“…In Section 4.3 of [17], it was shown that the ordering ensuring commutativity is normal ordering: the procedure of putting x-dependent coefficients to the left of monomials in the momentum operators −ih∂ x j , j = 1, . .…”

“…Recursive Construction of Joint Eigenfunctions for Calogero-Moser Hamiltonians 5 follows from Section 4.3 of [17]. More specifically, this reference dealt with the elliptic versions of the commuting PDOs.…”