Singular and non-singular eigenvectors for the Gaudin model

TL;DR

This paper introduces a method to construct and analyze singular and non-singular eigenvectors for Gaudin Hamiltonians in SL(2) modules, linking them to the Bethe Ansatz and extending to general Lie algebras.

Contribution

It provides a new systematic approach to construct eigenvectors for Gaudin models, including a complete basis of singular and non-singular vectors, and discusses generalizations to arbitrary Lie algebras.

Findings

Complete basis of singular eigenvectors described

Relation between singular eigenvectors and Bethe Ansatz established

Method generalized to arbitrary Lie algebras

Abstract

We present a method to construct a basis of singular and non-singular 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 non-singular eigenvectors. We discuss also the generalization of this method to the case of an arbitrary Lie algebra.