On the Gaudin model associated to Lie algebras of classical types

TL;DR

This paper derives explicit solutions for the Bethe Ansatz equations of the Gaudin model related to classical Lie algebras, demonstrating completeness and spectrum simplicity in specific tensor products.

Contribution

It provides explicit formulas for solutions and proves the completeness of the Bethe Ansatz for certain tensor products involving classical Lie algebras.

Findings

Explicit formulas for Bethe Ansatz solutions for classical Lie algebras.

Completeness of the Bethe Ansatz in tensor products with mostly vector representations.

Spectrum of Gaudin Hamiltonians is simple except for type D.

Abstract

We derive explicit formulas for solutions of the Bethe Ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.