The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials

Abstract

Abstract. Several results including integral representation of solutions and HermiteKrichever Ansatz on Heun's equation are generalized to a certain class of Fuchsian differential equations, and they are applied to equations which are related with physics. We investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin's solution. The relationship with finite-gap potential is also discussed. We find new finite-gap potentials. Namely, we show that the potential which is written as the sum of the Treibich-Verdier potential and additional apparent singularities of exponents −1 and 2 is finite-gap, which extends the result obtained previously by Treibich. We also investigate the eigenfunctions and their monodromy of the Schrödinger operator on our potential. IntroductionIt is well known that a Fuchsian differential equation with three singularities is transformed to a Gauss hypergeometric equation, and plays important roles in substantial fields in mathematics and physics. Several properties of solutions to the hypergeometric equation have been explained in various textbooks.A canonical form of a Fuchsian equation with four singularities is written aswith the conditionand is called Heun's equation. Heun's equation frequently appears in physics, i.e., general relativity [21], fluid mechanics [3] and so on. Despite that Heun's equation was resolved in the 19th century; several results of solutions have only been recently revealed. Namely, integral representations of solutions, global monodromy in terms of hyperelliptic integrals, relationships with the theory of finite-gap potential and the Hermite-Krichever Ansatz for the case γ, δ, ǫ, α − β ∈ Z + 1/2 are contemporary (see [1,6,19,23,24,25,26,28] etc.), though they are not written in a textbook on Heun's equation [17].2000 Mathematics Subject Classification. 82B23,34M55,33E10,33E15.

Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I

“…The formula (1.8) was first obtained by Takemura [26] and also obtained in [6] by a different argument. We will see that (1.8) plays a fundamental role for studying the poles of λ(t).…”

The geometry of generalized Lamé equation, I

“…Then Φ(z) := y 1 (z)y 2 (z) is a solution of (2.14), and also an even elliptic function due to (2.7)-(2.9). The following result was proved by Takemura [18], but we give a proof here for the convenience of readers because it plays a fundamental role in our theory. Proof.…”

“…To normalize the even elliptic solution Φ(z), we apply the following result due to Takemura [18]. [2].…”

“…Then we could obtain that all the coefficients are polynomials in A after normalization. For details, we refer the readers to [18,19].…”

“…all solutions of (1.11) are free of logarithmic singularity at any singular point. See [4,5,7,18] for recent developments of (1.11). Like (1.2), we could associate a hyperelliptic curve Γ n,p (τ ) := {(A, W )|W 2 = Q n,p (A; τ )} and an addition map σ n,p with (1.11).…”