The hierarchy of Poisson brackets for the open Toda lattice and its' spectral curves

TL;DR

This paper introduces a novel spectral curve-based representation of the infinite Poisson bracket hierarchy for the open Toda lattice, linking classical and higher brackets through Abelian differentials and spectral parameters.

Contribution

It provides a new spectral curve framework for understanding the entire Poisson hierarchy in the open Toda lattice, unifying classical and higher brackets.

Findings

Classical Poisson bracket expressed as a contour integral of Abelian differential.

Higher brackets generated by multiplying the differential by powers of the spectral parameter.

Establishes a spectral curve approach to the Toda lattice hierarchy.

Abstract

We establish a new representation of the infinite hierarchy of Pois- son brackets (PB) for the open Toda lattice in terms of its spectral curve. For the classical Poisson bracket (PB) we give a representation in the form of a contour integral of some special Abelian differential (meromorphic one-form) on the spectral curve. All higher brackets of the infinite hierarchy are obtained by multiplication of the one-form by a power of the spectral parameter.