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Proceedings of the Steklov Institute of Mathematics

On a new form of Bethe ansatz equations and separation of variables in the $$\mathfrak{s}\mathfrak{l}_3 $$ Gaudin model

Evgeny Mukhin, Vadim V. Schechtman, V. Tarasov, Alexander N. Varchenko

Published Sep 1, 2007• 3 citations
Publisher

Abstract

A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum $$\mathfrak{s}\mathfrak{l}_3 $$ Gaudin model is presented.

DOI
10.1134/s0081543807030121

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On a new form of Bethe ansatz equations and separation of variables in the sl3\mathfrak{s}\mathfrak{l}_3 sl3​ Gaudin model
A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum sl3\mathfrak{s}\mathfrak{l}_3 sl3​ Gaudin model is presented.
On the new form of Bethe ansatz equations and separation of variables in the sl3sl_3sl3​ Gaudin model
A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum sl3sl_3sl3​ Gaudin model is presented.
We propose new formulas for eigenvectors of the Gaudin model in the sl(3) case. The central point of the construction is the explicit form of some operator P, which is used for derivation of eigenvalues given by the formula ∣w1,w2)=∑n=0∞Pnn!∣w1,w2,0⟩, \left| {w_1 ,w_2 )} \right. = \sum\limits_{n = 0}^\infty {\frac{{P^n }} {{n!}}} \left| {w_1 ,w_2 ,0} \right\rangle , ∣w1​,w2​)=n=0∑∞​n!Pn​∣w1​,w2​,0⟩, where w 1, w 2 fulfil the standard well-know Bethe Ansatz equations.
The Gaudin model has been revisited many times, yet some important issues remained open so far. With this paper we aim to properly address its certain aspects, while clarifying, or at least giving a solid ground to some other. Our main contribution is establishing the relation between the off-shell Bethe vectors with the solutions of the corresponding Knizhnik–Zamolodchikov equations for the non-periodic sℓ(2) Gaudin model, as well as deriving the norm of the eigenvectors of the Gaudin Hamiltonians. Additionally, we provide a closed form expression also for the scalar products of the off-shell Bethe vectors. Finally, we provide explicit closed form of the off-shell Bethe vectors, together with a proof of implementation of the algebraic Bethe ansatz in full generality.