Constrained Toda hierarchy and turning points of the Ruijsenaars-Schneider model

TL;DR

This paper introduces the constrained Toda hierarchy, a new integrable system related to the 2D Toda lattice, and links its elliptic solutions' zeros to the turning points of the Ruijsenaars-Schneider model.

Contribution

It defines the constrained Toda hierarchy, proves the existence of its tau-function as a square root of the 2D Toda tau-function, and connects elliptic solutions to Ruijsenaars-Schneider model dynamics.

Findings

Constrained Toda hierarchy is a subhierarchy of the 2D Toda lattice.

Tau-function of C-Toda is the square root of the 2D Toda tau-function.

Zeros of elliptic solutions satisfy Ruijsenaars-Schneider equations at turning points.

Abstract

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda hierarchy (C-Toda). It can be regarded as a certain subhierarchy of the 2D Toda lattice obtained by imposing the constraint $\bar {\cal L}={\cal L}^{\dag}$ on the two Lax operators (in the symmetric gauge). We prove the existence of the tau-function of the C-Toda hierarchy and show that it is the square root of the 2D Toda lattice tau-function. In this and some other respects the C-Toda is a Toda analogue of the CKP hierarchy. It is also shown that zeros of the tau-function of elliptic solutions satisfy the dynamical equations of the Ruijsenaars-Schneider model restricted to turning points in the phase space. The spectral curve has holomorphic involution which interchange the marked points in which the Baker-Akhiezer function has essential singularities.