The Atiyah--Hitchin bracket and the open Toda lattice

TL;DR

This paper explores the Poisson structure of Weyl functions associated with the finite nonperiodic Toda lattice, connecting algebraic-geometrical inverse problems with Atiyah-Hitchin brackets and Hamiltonian systems.

Contribution

It establishes a link between algebraic-geometrical inverse problem methods and the Atiyah-Hitchin Poisson structure for Weyl functions in Toda lattice dynamics.

Findings

Identifies two systems of canonical coordinates from the Atiyah-Hitchin structure.

Connects one system with Jacobi elliptic coordinates.

Shows invariance under linear-fractional transformations leads to commuting Hamiltonians.

Abstract

The dynamics of finite nonperiodic Toda lattice is an isospectral deformation of the finite three--diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one--to--one correspondence with their Weyl functions. These are rational functions mapping the upper half--plane into itself. We consider representations of the Weyl functions as a quotient of two polynomials and exponential representation. We establish a connection between these representations and recently developed algebraic--geometrical approach to the inverse problem for Jacobi matrix. The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We show that an invariance of the AH structure under linear--fractional transformations leads to two systems of canonical coordinates and two families of commuting Hamiltonians. We establish a relation of one of these systems with Jacobi elliptic coordinates.