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

### Submission summary

 As Contributors: Karol Kozlowski Arxiv Link: https://arxiv.org/abs/1804.01749v2 (pdf) Date submitted: 2018-05-21 02:00 Submitted by: Kozlowski, Karol Submitted to: SciPost Physics Proceedings Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Mathematical Physics Approach: Theoretical

### 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.

### Ontology / Topics

See full Ontology or Topics database.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 1804.01749v3 on 21 June 2018

Submission 1804.01749v2 on 21 May 2018

## Reports on this Submission

### Anonymous Report 1 on 2018-6-12 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1804.01749v2, delivered 2018-06-12, doi: 10.21468/SciPost.Report.497

### Strengths

The paper present a detailed research in one of most complicated areas of the quantum integrable models - solutions of the Baxter type equations.

### Weaknesses

The paper is rather technical so it will be difficult to read this paper for the readers who are not working in this field.

### Report

This paper is devoted to the investigation of the solutions of the Baxter $t-Q$ equations using
auxiliary integral equations for the two types of the generalised Toda chains. This research is a
further generalization of the approach developed by one of the authors together with J. Teschner
in application to the usual Toda chain model. This allows to obtain the quantisation conditions
for the spectra of these models in the form of thermodynamic Bethe ansatz equations.

### Requested changes

1 - Page 6. It is better to use roman "i)" mentioning first item in ii) of the procedure described
at the beginning of the Section 3.
2 - Pages 7 and 8. It seems that the proper citations after formulas (3.7) and (3.11) should be
to the formulas (2.6) and (2.7), not (2.12) and (2.13) where certain parameters of the Baxter
equations for two models are introduced.
3 - Page 10. Before (3.26) - misspelled "thought", last letter 't' is omitted.
4 - Page 10. After (3.260. It seems that better to use notation $(-)$ instead of (-) to signify
the corresponding terms of the Bater equation.
5 - Staring from the page 15 there is a mess with notations of the functions defined by the
equations (3.50) and (3.51). These equations defines the functions $v_\uparrow$ and
$v_\downarrow$, while starting from the beginning of the page 15 authors used notations
$\nu_\uparrow$ and $\nu_\downarrow$ in many places. See, for example, pages 16, 20, 23, 26 and 37.
6 - Page 16. The set of self-dual Baxter equations are introduced in the paper
by the formulas (2.10) and (2.11) while in the text the citation is done only to the formula (2.11).
The same at the beginning of the Appendices C.1 and C.2.
7 - Page 26. Should be ${\rm e}^{-\frac{2\pi}{\omega_2}k\tau_1}$ in the first line, not ${\rm e}^{-\frac{2\pi}{\omega_2}\tau_1}$?
8 - Page 28. Point '.' at end of the page is omitted.

### Attachment

• validity: high
• significance: high
• originality: top
• clarity: good
• formatting: good
• grammar: excellent