Integrable Many-Body Systems via the Inosemtsev Limit

The Inozemtsev limit (IL), or the scaling limit, is known as a procedure applied to the elliptic Calogero-Moser model. It is a combination of the trigonometric limit, infinite shifts of particle coordinates, and coupling-constant rescalings. This results in an interaction of the exponential type. We show that the IL applied to the $sl(N,\mathbb C)$ elliptic Euler–Calogero–Moser model and to the elliptic Gaudin model produces new Toda-like systems of $N$ interacting particles endowed with additional degrees of freedom corresponding to a coadjoint orbit of $sl(n,\mathbb C)$. The limits corresponding to the complete degeneration of the orbital degrees of freedom lead to recovering only the known periodic and nonperiodic Toda systems. We classify the systems appearing in the IL in the $sl(3,\mathbb C)$ case. This classification is represented on a two-dimensional plane of parameters describing infinite shifts of particle coordinates. This space is subdivided into symmetric domains. In this picture, a mixture of the Toda and trigonometric Calogero-Sutherland potentials emerges on lower-dimensional domain walls. Because of obvious symmetries, this classification can be generalized to an arbitrary number of particles. We also apply the IL to the $sl(2,\mathbb C)$ elliptic Gaudin model on a two-punctured elliptic curve and discuss the main properties of its possible limits. The limits of Lax matrices are also considered.