Heun operator of Lie type and the modified algebraic Bethe ansatz

TL;DR

This paper connects the Heun operator of Lie type to Gaudin magnet models, using a modified algebraic Bethe ansatz to find its spectrum and relate Bethe roots to polynomial solutions of the Heun equation.

Contribution

It introduces a novel approach to diagonalize the Heun operator of Lie type via the modified algebraic Bethe ansatz, linking it to Gaudin models and polynomial solutions.

Findings

Spectrum of the Heun operator expressed in Bethe roots.

Bethe roots related to polynomial solutions of the Heun equation.

Applications demonstrated in representation theory and entanglement entropy.

Abstract

The generic Heun operator of Lie type is identified as a certain $BC$-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We show also that these Bethe roots are intimately associated to the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of $O(3)$ and the computation of the entanglement entropy for free Fermions on the Krawtchouk chain.