Quasi-Exactly Solvable Deformations of Gaudin Models and ``Quasi-Gaudin Algebras''

TL;DR

This paper introduces a new class of integrable models that are deformations of Gaudin models, exhibiting quasi-exact solvability, and develops the underlying algebraic structure called quasi-Gaudin algebra.

Contribution

It constructs quasi-exactly solvable deformations of Gaudin models and defines the quasi-Gaudin algebra as their algebraic foundation.

Findings

Models are quasi-exactly solvable with partial spectrum solutions.

The quasi-Gaudin algebra is a non-Lie algebraic deformation of the Gaudin algebra.

The algebraic Bethe ansatz applies only to parts of the spectrum.

Abstract

A new class of completely integrable models is constructed. These models are deformations of the famous integrable and exactly solvable Gaudin models. In contrast with the latter, they are quasi-exactly solvable, i.e. admit the algebraic Bethe ansatz solution only for some limited parts of the spectrum. An underlying algebra responsible for both the phenomena of complete integrability and quasi-exact solvability is constructed. We call it "quasi-Gaudin algebra" and demonstrate that it is a special non-Lie-algebraic deformation of the ordinary Gaudin algebra.