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The extended HeineStieltjes polynomials associated with a special LipkinMeshkovGlick (LMG) model corresponding to the standard two-site BoseHubbard 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. © 2011 IOP Publishing Ltd.

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Journal of Physics A: Mathematical and Theoretical