The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations

Abstract

We obtain isomonodromic transformations for Heun's equation by generalizing the Darboux transformation, and we find pairs and triplets of Heun's equation which have the same monodromy structure. By composing generalized Darboux transformations, we establish a new construction of the commuting operator which ensures that the finite-gap property is satisfied. As an application, we prove some previous conjectures in part III.

The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

“…Then the mean field equations Proof. It was proved by Takemura [26,Section 4] that H((n, 0, 0, 0), B, τ) and H((n 0 , n 1 , n 2 , n 3 ), B, τ) are isomonodromic (i.e. their monodromy representations are the same) for any (B, τ).…”

Integral representation of solutions to Fuchsian system and Heun's equation

“…Finite-gap integration is applicable for the case γ, δ, , α − β ∈ Z + 1/2, t ∈ C \ {0, 1} and all q, and results on the integral representation of solutions [13], the Bethe Ansatz [13], the Hermite-Krichever Ansatz [1,16], the monodromy formulae by hyperelliptic integrals [15], the hyperelliptic-to-elliptic reduction formulae [16] and relationships with the Darboux transformation [17] have been obtained. In this paper, we obtain integral formulae of solutions for the case γ, δ, , α + 1/2, β + 1/2 ∈ Z, t ∈ C \ {0, 1} and all q, which then facilitates a calculation of the monodromy.…”

On the Heun equation

“…Other examples of the finite-gap potential were found by Treibich andVerdier. Around 1987-1992, they found that the potential P 3 iZ0 l i ðl i C 1Þ §ðx C u i Þ (u 0 Z0, u 1 Z1/2, u 2 ZK(tC1)/2 and u 3 Zt/2), which is called the Treibich & Verdier (1992) potential, is an algebro-geometric finite-gap potential iff l i 2Z for all i2{0, 1, 2, 3}. They proved it by using a theory of elliptic soliton, which is closely related to the geometry of Riemann surfaces.…”

“…(ii) (The case l 0 Z2,l 1 Z1,l 2 Z0,l 3 Z1) The functionsL 1 ðw; qÞ,L 2 ðw; qÞ,QðqÞ, Q 1 ðqÞ andQ t ðqÞ are calculated as follows: (Takemura 2005) and Darboux transformation (Takemura 2006b). Consequently, conjectures 3.1 and 3.2 were proved.…”