# Completeness of the Bethe states for the rational, spin-1/2 Richardson--Gaudin system

### - Published as SciPost Phys. 3, 007 (2017)

• Other versions of this Submission (with Reports) exist:

### Submission summary

 As Contributors: Jon Links Arxiv Link: http://arxiv.org/abs/1603.03542v5 Date accepted: 2017-07-17 Date submitted: 2017-07-11 Submitted by: Links, Jon Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Mathematical Physics

### Abstract

Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson--Gaudin system it will be argued that, for generic values of the system's coupling parameters, the Bethe states are complete. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothesis, and is generalisable to a wider class of systems.

#### Current status:

I thank the Editor-in-charge, and the referees, for their helpful comments on improvements.

### List of changes

The following changes have been made in response to the Editor-in-charge:

[1.] The explanation that has been requested has been provided in the final two sentences appearing on page 3 of the revised version.
[2.] The Editor is correct, the change has been made to the equation which is now numbered (4).
[3.] Regarding equation (9) (previously (8)), explanatory text has been added before its appearance on page 5, and also preceding Proposition 2.
[4.] $t(u)$ in the equation now numbered (20) was a typographical error. It has now been corrected to read $T(u)$.
[5.] Equation numbering has been added to those unnumbered equations which were referred to in the Editorial Recommendation.

The following changes have been made in response to the Anonymous Report 169:

[1.] It was requested that the publication titled {\it On the separation of variables and completeness of the Bethe Ansatz for quantum ${\mathfrak gl}_{N}$ Gaudin model} by Mukhin, Tarasov, and Varchenko is cited and discussed. The paper by Mukhn et al. deals with a Gaudin model, as opposed to a Richardson-Gaudin model. That is, it is dealing with the $\alpha=0$ case which can be seen from the Bethe Ansatz equations which appear on page 141 of the published version, or page 5 of the arXiv version. This publication has been cited as [23] and referred in the text, within the second last paragraph preceding Section 2, on page 4.
[2.] Reference to the identity of [28] (previously [30]) has been added at the end of the proof for Proposition 1.
[3.] The typographical error referred to on page 11 has been corrected.

There were no changes requested in the Anonymous Report 165.