The extended Heine-Stieltjes polynomials associated with a special LMG model

TL;DR

This paper introduces new polynomials linked to a specific LMG model, establishing a correspondence between their zeros and solutions of Bethe ansatz equations, and explores their electrostatic analogues and spectral properties.

Contribution

It derives extended Heine-Stieltjes polynomials for a special LMG model and connects their zeros to Bethe ansatz solutions and electrostatic configurations, providing new analytical tools.

Findings

Zeros of the polynomials correspond to Bethe ansatz solutions.

Electrostatic analogues uniquely match polynomial roots to charge configurations.

Relations between zeros and eigenenergies are established.

Abstract

New polynomials associated with a special Lipkin-Meshkov-Glick (LMG) model corresponding to the standard two-site Bose-Hubbard model are derived based on the Stieltjes correspondence. It is shown that there is a one-to-one correspondence between zeros of this new polynomial and solutions of the Bethe ansatz equations for the LMG model.A one-dimensional classical electrostatic analogue corresponding to the special LMG model is established according to Stieltjes early work. It shows that any possible configuration of equilibrium positions of the charges in the electrostatic problem corresponds uniquely to one set of roots of the Bethe ansatz equations for the LMG model, and the number of possible configurations of equilibrium positions of the charges equals exactly to the number of energy levels in the LMG model. Some relations of sums of powers and inverse powers of zeros of the new polynomials related to the eigenenergies of the LMG model are derived.