On the squared eigenfunction symmetry of the Toda lattice hierarchy

TL;DR

This paper explicitly constructs the squared eigenfunction symmetry for the Toda lattice hierarchy, linking it to additional symmetries and deriving its action on the tau function, thus extending the understanding of integrable systems.

Contribution

It provides an explicit construction of the squared eigenfunction symmetry using Kronecker products and connects it to the ASvM formulas for the Toda hierarchy.

Findings

Explicit form of squared eigenfunction symmetry derived.

Connection established between symmetry and tau function actions.

Fay-like identities and wave function relations analyzed.

Abstract

The squared eigenfunction symmetry for the Toda lattice hierarchy is explicitly constructed in the form of the Kronecker product of the vector eigenfunction and the vector adjoint eigenfunction, which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively. Then after the Fay-like identities and some important relations about the wave functions are investigated, the action of the squared eigenfunction related to the additional symmetry on the tau function is derived, which is equivalent to the Adler-Shiota-van Moerbeke (ASvM) formulas.