DARBOUX-LAMÉ EQUATION AND ISOMONODROMIC DEFORMATION

The Darboux-Lamé equation is defined as the double Darboux transformation of the Lamé equation, and is studied from the viewpoint of the isomonodromic deformation theory. It is shown that the second-order ordinary differential equation of Fuchsian type on P1 corresponding to the second Darboux-Lamé equation is obtained as isomonodromic deformation of some specific Gauss’ hypergeometric differential equation.


Introduction
We consider the nth Lamé equation where n is a natural number and ℘(x, τ) is the Weierstrass elliptic function with the fundamental periods 1 and τ such that τ > 0. If the fundamental period τ and the discrete eigenvalue λ satisfy a kind of degenerate condition obtained in [6], we can construct the nth algebro-geometric elliptic potential u * * n,λ (x,ξ) with the complex parameter ξ by the method of double Darboux transformation. We call the ordinary differential equation

Preliminaries
In this section, the necessary materials are summarized. We refer the reader to [5,6] for more precise information.
We consider the second-order linear ordinary differential operator in the complex domain where u(x) is a meromorphic function. The functions Z n (u), n ∈ N, defined by the recursion relation Z 0 (u) ≡ 1, Z n (u) = Λ(u)Z n−1 (u), n = 1,2,..., (2.2) which are the differential polynomials in u(x), are called the KdV polynomials, where is the Λ-operator associated with the differential operator H(u).
Let V (u) be the linear span of all KdV polynomials over C. If dimV (u) = n + 1, then u(x) is called the nth algebro-geometric potential and we write rankV (u) = n. If u(x) is the nth algebro-geometric potential, then there uniquely exist the polynomials a j (λ), j = 0,1,...,n, in the spectral parameter λ of degree n − j + 1 such that For this fact, see [5,6]. The M-function M(x,λ;u) associated with u(x) is the differential polynomial defined by The spectral discriminant is the polynomial of degree 2n + 1 in λ with constant coefficients. Let which corresponds to the discrete spectrum of the operator H(u). If λ j ∈ SpecH(u), then we have We call M(x,λ j ;u) 1/2 the M-eigenfunction.

Mayumi Ohmiya 513
For f (x) ∈ ker(H(u) − λ) \ {0}, the Darboux transformation is the operator H(u * ) with the potential u * (x) defined by We sometimes call the resulted potential u * (x) itself the Darboux transformation.
This fact is called Darboux's lemma [1]. The Darboux transformation of the algebrogeometric potential u(x) by the corresponding M-eigenfunction is the 1-parameter family of the eigenfunction of H(u * λj ) associated with the eigenvalue λ j , that is, where φ λj (ξ) is an arbitrary function which depends only on ξ. The function φ λj (ξ) will be determined exactly so that the ADDT, which is defined below, of the nth Lamé equation is isomonodromic.

Darboux-Lamé equation and isomonodromic deformation
Let Spec m H(u) = λ j | the multiple roots of ∆(λ;u) = 0 (2.16) which we call the multiple spectrum of H(u). It is shown in [6] that if u(x) is the nth algebro-geometric potential, then u * λj (x) is the (n − 1)th algebro geometric potential if and only if Spec m H(u) = ∅ and λ j ∈ Spec m H(u).
Let M n (x,λ,τ) be the M-function associated with the nth Lamé potential u n (x,τ), that is, and we call it the lattice of degenerate periods associated with the nth Lamé potential u n (x). One can immediately see that the lattice of degenerate periods Θ n is the discrete subset of H + . Now, we enumerate several examples of the degenerate condition for the Lamé potentials. For this purpose, we must carry out elementary but very complicated computation. Hence, here we explain only the simplest case n = 1. See also [3] for another method of computation. KdV polynomials. We have Computation of the M-function M 2 (x,λ,τ). Let ρ 0 and ρ 1 be the constants such that (2.22)
In what follows, we construct exactly the second Darboux-Lamé equation of degenerate type. Suppose that τ * ∈ Θ 2 , then, by the direct calculation parallel to that for M 1 (x, λ,τ), we have Since we have shown g 2 (τ * ) = 0 in Section 2, follows. Hence Spec m H(u 2 (x,τ * )) = {0} and we have Therefore, the ADT u * 2,0 (x) is given by and, by Darboux's lemma, we have On the other hand, we have Hence, by (3.1), follows. Thus we have the following lemma. Moreover, The isospectral property of the potential u * * 2,0 (x,τ * ) will be discussed in the forthcoming paper [7].
Lemma 5.1. Suppose that the second-order ordinary differential equation By the above general criterion, to show that the monodromy matrix associated with the fundamental system f 1 (z,ξ), f 2 (z,ξ) is independent of the parameter ξ, it suffices to show that a(z,ξ) and b(z,ξ), defined by a(z,ξ) = f 1ξ (z,ξ) f 1 (z,ξ) f 2ξ (z,ξ) f 2 (z,ξ)