The periodic and open Toda lattice

TL;DR

This paper develops an algebro-geometrical framework for solving the open Toda lattice, providing explicit solutions and structures, and extending methods to the 2D case and connecting with the periodic case.

Contribution

It introduces a new algebro-geometrical approach for the open Toda lattice, including explicit inverse spectral solutions and symplectic structures.

Findings

Explicit solutions for finite Jacobi matrices

New symplectic structure and Darboux coordinates for Toda lattice

Extension of methods to 2D open Toda and connection with periodic case

Abstract

We develop algebro-geometrical approach for the open Toda lattice. For a finite Jacobi matrix we introduce a singular reducible Riemann surface and associated Baker-Akhiezer functions. We provide new explicit solution of inverse spectral problem for a finite Jacoby matrix. For the Toda lattice equations we obtain the explicit form of the equations of motion, the symplectic structure and Darboux coordinates. We develop similar approach for 2D open Toda. Explaining some the machinery we also make contact with the periodic case.