Commuting families of symmetric differential operators

Ochiai, Hiroyuki; Oshima, Toshio; Sekiguchi, Hideko

doi:10.3792/pjaa.70.62

Cited by 35 publications

(38 citation statements)

References 6 publications

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“…The existence and the explicit expressions are known in ( [8,7,2]) etc. Here, we exhibit the Hasegawa's expression which will be used in the proof of Proposition 4.1.…”

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confidence: 99%

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The Perturbation of the Quantum¶Calogero-Moser-Sutherland System¶and Related Results

Komori

Takemura²

2002

Communications in Mathematical Physics

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Abstract. The Hamiltonian of the trigonometric Calogero-Sutherland model coincides with a certain limit of the Hamiltonian of the elliptic Calogero-Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one.In this article we show the essential self-adjointness of the Hamiltonian of the elliptic Calogero-Moser model and the regularity (convergence) of the perturbation for the arbitrary root system.We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x 1 , . . . , x N ) for the A N−1 -case. As a result, the algebraic calculation of the perturbation is justified.

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“…The existence and the explicit expressions are known in ( [8,7,2]) etc. Here, we exhibit the Hasegawa's expression which will be used in the proof of Proposition 4.1.…”

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confidence: 99%

“…Here, we exhibit the Hasegawa's expression which will be used in the proof of Proposition 4.1. Later we will discuss the relationship between the expression of Ochiai-OshimaSekiguchi ( [7]) and the one of Hasegawa ([2]). …”

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confidence: 99%

The Perturbation of the Quantum¶Calogero-Moser-Sutherland System¶and Related Results

Komori

Takemura²

2002

Communications in Mathematical Physics

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“…the elliptic Calogero-Moser-Sutherland model of type A N ) is a universal completely integrable model of quantum mechanics with the symmetry of the Weyl group of type B N (resp. type A N ), which follows from the classification due to Ochiai, Oshima and Sekiguchi [30,33]. For the case N = 1, the operator (3.2) appears in the elliptic form of Heun's equation (2.4).…”

Section: The Inozemtsev Modelmentioning

confidence: 80%

“…where S N is the symmetric group, [x] is the integral part of x and [30]). The Hamiltonian H is expressed as H = P 2 − P 2 1 /2.…”

Section: The Elliptic Calogero-moser-sutherland Modelmentioning

confidence: 99%

Towards Finite-Gap Integration of the Inozemtsev Model

Takemura¹

2007

SIGMA

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“…After the first draft of this paper was completed, we were informed that Ochiai, Oshima and Sekiguchi [8], [13] have studied all the completely integrable systems with the invariance under the action of the Weyl groups. In their papers, they solved the functional differential equations of the potential function.…”

Section: Substituting These Matrices In the Lax Equation V-1 L = [Mmentioning

confidence: 99%

The Meromorphic Solutions of the Bruschi–Calogero Equation

Kawazumi

Shibukawa

2000

Publ. Res. Inst. Math. Sci.

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We give all the meromorphic functions defined near the origin 0 e C satisfying a functional equation investigated by Bruschi and Calogero [1], [2]. § 0. Introduction It is an important problem to find a Lax pair L and M whose equations of motion are equivalent to the Lax equation [10], [11], [12]. In order to prove their complete integrability it is convenient to use a Lax representation.The systems of Calogero-Sutherland type, which describe one-dimensional n-particle dynamics, are defined by the following Hamiltonian where the potential U has the form Communicated by T. Kawai, July 8, 1999. Revised November 1, 1999. 1991 Here a(jx\ r lf r z ) fs JAe Wefersfross segrafl function, and £(x\ r lf r 2 ) £/z# Weierstrass zeta function. All the solutions except for the case (0-i) extend themselves to meromorphic functions defined on the whole plane C .It should be remarked that a meromorphic function defined near the origin 0 EE C is holomorphic on a sufficiently small punctured disk. Hence our result covers all the meromorphic solutions defined near the origin 0 e C .The methods we use in this paper are quite different from those of Bruschi and Calogero [1], [2].The outline to get all meromorphic solutions is as follows. First, we shall show that the solution 77 is the logarithmic derivative of some meromorphic function

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Commuting families of symmetric differential operators

Cited by 35 publications

References 6 publications

The Perturbation of the Quantum¶Calogero-Moser-Sutherland System¶and Related Results

The Perturbation of the Quantum¶Calogero-Moser-Sutherland System¶and Related Results

Towards Finite-Gap Integration of the Inozemtsev Model

The Meromorphic Solutions of the Bruschi–Calogero Equation

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