Solution of Baxter equation for the $q$-Toda and Toda$_2$ chains by NLIE

TL;DR

This paper develops a method to solve the Baxter equation for $q$-Toda and Toda$_2$ chains using auxiliary non-linear integral equations, enabling spectrum quantization via TBA-like equations.

Contribution

It introduces a novel construction of solutions to the Baxter equation for these chains using auxiliary NLIEs, providing new quantization conditions.

Findings

Constructed a basis of solutions for the Baxter equation.

Derived quantization conditions in the form of TBA-like equations.

Facilitated spectral analysis of $q$-Toda and Toda$_2$ models.

Abstract

We construct a basis of solutions of the scalar $\boldsymbol{ \texttt{t} }- \boldsymbol{ \texttt{Q} }$ equation describing the spectrum of the $q$-Toda and Toda$_2$ chains by using auxiliary non-linear integral equations. Our construction allows us to provide quantisation conditions for the spectra of these models in the form of thermodynamic Bethe Ansatz-like equations.