Integrable Many-Body Systems via Inozemtsev Limit

TL;DR

This paper explores the Inozemtsev limit applied to elliptic integrable systems, classifying resulting Toda-like models and analyzing their properties, including limits of Lax matrices, with a focus on $sl(3,\mathbb{C})$ and $sl(2,\mathbb{C})$ cases.

Contribution

It introduces a classification scheme for integrable systems obtained via the Inozemtsev limit, especially for the $sl(3,\mathbb{C})$ case, and analyzes their structures and limits.

Findings

Classification of $sl(3,\mathbb{C})$ systems via IL on a parameter space.

Emergence of Toda and Calogero-Sutherland potentials on domain walls.

Analysis of Lax matrix limits for different models.

Abstract

The Inozemtsev limit (IL) or the scaling limit is known to be a procedure applied to the elliptic Calogero Model. It is a combination of the trigonometric limit, infinite shifts of particles coordinates and rescalings of the coupling constants. As a result, one obtains an exponential type of interaction. In the recent paper it is shown that the IL applied to the $sl(N,\bf C)$ elliptic Euler-Calogero Model and the elliptic Gaudin Model produces new Toda-like systems of $N$ interacting particles endowed with additional degrees of freedom corresponding to a coadjoint orbit in $sl(n,\bf C)$. The limits corresponding to the complete degeneration of the orbital degrees provide only ordinary periodic and non periodic Toda systems. We introduce a classification of the systems appearing in the $sl(3,\bf C)$ case via IL. The classification is represented on two-dimensional space of parameters describing the infinite shifts of the coordinates. This space is subdivided into symmetric domains. The mixture of the Toda and the trigonometric Calogero-Sutherland potentials emerges on the low dimensional domain walls of this picture. Due to obvious symmetries this classification can be generalized to the arbitrary number of particles. We also apply IL to $sl(2,\bf C)$ elliptic Gaudin Model with two marked points on the elliptic curve and discuss main features of its possible limits. The limits of Lax matrices are also considered.