In our recent paper we proposed new formulas for eigenvectors of the Gaudin model in sl (3) case. Similarly in this paper we used the standard Bethe Ansatz method for finding the eigenvectors and the eigenvalues in the so (5) case in an explicit form.
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∑∞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.
We present a method to construct a basis of singular and nonsingular common eigenvectors for Gaudin Hamiltonians in a tensor product module of the Lie algebra SL(2). The subset of singular vectors is completely described by analogy with covariant differential operators. The relation between singular eigenvectors and the Bethe Ansatz is discussed. In each weight subspace the set of singular eigenvectors is completed to a basis, by a family of nonsingular eigenvectors. We discuss also the generalization of this method to the case of an arbitrary Lie algebra.
On the new form of Bethe ansatz equations and separation of variables in the sl3 Gaudin model
A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum sl3 Gaudin model is presented.
On a new form of Bethe ansatz equations and separation of variables in the sl3 Gaudin model
A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum sl3 Gaudin model is presented.