The Heun Equation and the Calogero-moser-sutherland System I: the Bethe Ansatz Method

Olshanetsky and Perelomov proposed the family of integrable quantum systems, which is called the Calogero-Moser-Sutherland system or the Olshanetsky-Perelomov system ([5]). In early 90’s, Ochiai, Oshima and Sekiguchi classified the integrable models of quantum mechanics which are invariant under the action of a Weyl group with some assumption ([4]). For the BN (N ≥ 3) case, the generic model coincides with the Inozemtsev’s model. Oshima and Sekiguchi found that the Hamiltonian of the 1-particle Inozemtsev model is transposed to the Heun’s equation with full parameters ([6]). Here, the Heun’s equation is the general second-order differential equation with four regular singular points. In this article we will investigate the solution of the Heun’s equation motivated by the analysis of the Calogero-Moser-Sutherland system. In the articles ([8, 3]), we justified the regular perturbation for the Calogero-Moser-Sutherland model of type AN from the trigonometric one. In ([8]), the author obtained results by applying the Bethe Ansats method for the AN cases. We remark that the Bethe Ansatz for the AN cases was established by Felder and Varchenko ([1]) by investigating