Linear r-Matrix Algebra for a Hierarchy of One-Dimensional Particle Systems Separable in Parabolic Coordinates

TL;DR

This paper introduces a hierarchy of integrable one-dimensional particle systems with polynomial potentials separable in parabolic coordinates, providing a Lax representation, r-matrix algebra, and discussing classical and quantum aspects.

Contribution

It presents a novel hierarchy of integrable particle systems with explicit Lax pairs and r-matrix algebra, extending known models like Hénon-Heiles.

Findings

Lax representation for the hierarchy of systems

Construction of a dynamical r-matrix algebra

Classical integration and discussion of quantization

Abstract

We consider a hierarchy of many-particle systems on the line with polynomial potentials separable in parabolic coordinates. The first non-trivial member of this hierarchy is a generalization of an integrable case of the H\'enon-Heiles system. We give a Lax representation in terms of $2\times 2$ matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Classical integration in a particular case is carried out and quantization of the system is discussed with the help of separation variables. This paper was published in the rary issues: Sfb 288 Preprint No. 110, Berlin and Nonlinear Mathematical Physics, {\bf 1(3)}, 275-294 (1994)