Toda-Darboux maps and vertex operators

TL;DR

This paper investigates Toda-Darboux transforms for Toda lattice operators, showing how they induce transformations on tau-vectors via vertex operators and on eigenfunctions through Wronskians, enriching the understanding of integrable systems.

Contribution

It introduces a novel framework connecting Toda-Darboux transforms with vertex operators and Wronskians, providing new insights into the structure of integrable systems.

Findings

Toda-Darboux transforms induce tau-vector transformations via vertex operators

Eigenfunctions are transformed through Wronskian relations

The framework links Darboux transforms with integrable system symmetries

Abstract

The purpose of this paper is to study Toda-Darboux transforms, i.e., Darboux transforms for operators L(t) flowing according to the Toda lattice. Each element of the null-space $L(t)-z$ specifies a factorization for all t and thus a Toda-Darboux transform on $L(t)$. The Toda-Darboux map induces a transformation on the tau-vectors, given by a certain vertex operator, and on eigenfunctions, given by a Wronskian. .