Treibich-Verdier potentials and the stationary (m)KDV hierarchy

Gesztesy, Fritz; Weikard, Rudi

doi:10.1007/bf02572375

Cited by 52 publications

(79 citation statements)

References 34 publications

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“…(4.5) (l 0 + l 1 + l 2 + l 3 : even) or Eq. (4.10) (l 0 + l 1 + l 2 + l 3 : odd), in agreement with the results in [3,13].…”

supporting

confidence: 91%

“…Following the results of Treibich and Verdier [14], who showed that if l i ∈ Z ≥0 for all i ∈ {0, 1, 2, 3}, the potential of operator (1.5) is an algebro-geometric finite-gap potential, this potential is now referred to as the Treibich-Verdier potential. The interested reader may refer to [3,6,7,9,13] for further results in this area. The algebro-geometric finite-gap property in turn provides a possible means of determining the eigenfunctions and monodromy of the operator H (l 0 ,l 1 ,l 2 ,l 3 ) .…”

Section: )mentioning

confidence: 99%

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

Takemura¹

2006

JNMP

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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.

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“…(4.5) (l 0 + l 1 + l 2 + l 3 : even) or Eq. (4.10) (l 0 + l 1 + l 2 + l 3 : odd), in agreement with the results in [3,13].…”

supporting

confidence: 91%

Section: )mentioning

confidence: 99%

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

Takemura¹

2006

JNMP

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“…Note that Gesztesy and Weikard [17] obtained similar results. The monodromy of the functionΛ(x) in equation (2.12) is written as…”

Section: Proposition 9 ([42] Theorem 312) (I)supporting

confidence: 59%

“…Subsequently several others [17,38,42,44,45,46] have produced more precise statements and concerned results on this subject. Namely, integral representations of solutions [38,42], the Bethe Ansatz [17,42], the global monodromy in terms of the hyperelliptic integrals [44], the Hermite-Krichever Ansatz [45] and a relationship with the Darboux transformation [46] were studied. In this paper, we discuss some approaches to finite-gap integration for multivariable cases.…”

Section: Introductionmentioning

confidence: 99%

Towards Finite-Gap Integration of the Inozemtsev Model

Takemura¹

2007

SIGMA

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“…7) v(x) = for the Schrödinger operator −d 2 /dx 2 + v(x), where l i (i = 0, 1, 2, 3) are integers and ω 1 , ω 3 , ω 0 (= 0), ω 2 (= ω 1 − ω 3 ) are half-periods. Subsequently several studies [6,30,19,23,24,25,26] have further added to understanding of this subject. Note that the function in Eq.…”

Section: Introductionmentioning

confidence: 99%

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

Takemura

2008

Math. Z.

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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.

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Treibich-Verdier potentials and the stationary (m)KDV hierarchy

Cited by 52 publications

References 34 publications

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

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

Towards Finite-Gap Integration of the Inozemtsev Model

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

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