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

The extended Heine-Stieltjes polynomials associated with a special LipkinMeshkov-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 analog 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 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.