Algebraic integrability for the Schr�dinger equation and finite reflection groups

Веселов, А. П.; Styrkas, Konstantin; Chalykh, Oleg

doi:10.1007/bf01019330

Cited by 61 publications

(67 citation statements)

References 11 publications

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“…However, it is not isomorphic to the spectral surface as an affine variety (though they are birationally equivalent). This can be seen already in the trigonometric limit τ → +i∞, by using the information about the spectral variety from [10]. Namely, the results of [10] imply that these four operators L, .…”

mentioning

confidence: 94%

“…On the other hand, in [8] it was suggested to consider the quantum elliptic Calogero-Moser problem and its versions related to the root systems [9] as natural multidimensional analogues of the Lamé operator. More specifically, a conjecture from [8] says that for integer values of the coupling parameters the corresponding Schrödinger operators are algebraically integrable (this is a proper generalization of the properties of the algebro-geometric operators to higher dimensions, see [10,11] and Section 3 below). For the rational and trigonometric versions of the Calogero-Moser problem this was proved in [10].…”

Section: Introductionmentioning

confidence: 99%

“…More specifically, a conjecture from [8] says that for integer values of the coupling parameters the corresponding Schrödinger operators are algebraically integrable (this is a proper generalization of the properties of the algebro-geometric operators to higher dimensions, see [10,11] and Section 3 below). For the rational and trigonometric versions of the Calogero-Moser problem this was proved in [10]. Elliptic version, however, turned out to be more difficult: until now it was known for A n case only, due to [11].…”

Section: Introductionmentioning

confidence: 99%

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Generalized Lam� Operators

Chalykh

Etingof

Oblomkov

2003

Communications in Mathematical Physics

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Abstract. We introduce a class of multidimensional Schrödinger operators with elliptic potential which generalize the classical Lamé operator to higher dimensions. One natural example is the Calogero-Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finite-gap property of the Lamé operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh-Veselov conjecture for the elliptic Calogero-Moser problem for all root systems. We also establish algebraic integrability in all known two-dimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to B 2 case, another one is a certain deformation of the A 2 case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald-Ruijsenaars type.

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mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Lam� Operators

Chalykh

Etingof

Oblomkov

2003

Communications in Mathematical Physics

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“…where One can interpret these formulas either as a new way of representing of BA function or as an explicit evaluation of the Selberg-type integral (36) in terms of the BA function, which can be computed by other methods as well (see [4,6,30]). The same of course is true for general n and in the deformed case.…”

Section: Examplesmentioning

confidence: 99%

“…The notion of rational Baker-Akhiezer (BA) function related to a configuration of hyperplanes with multiplicities was introduced in [4,6,30] as a multi-dimensional version of Krichever's axiomatic approach [15]. Such a function exists only for special configurations, in particular for all Coxeter configurations.…”

Section: Introductionmentioning

confidence: 99%

Baker–akhiezer Function as Iterated Residue and Selberg-Type Integral

Felder¹,

Веселов²

2009

Glasgow Math. J.

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Abstract.A simple integral formula as an iterated residue is presented for the Baker-Akhiezer function related to A n -type root system in both the rational and trigonometric cases. We present also a formula for the Baker-Akhiezer function as a Selberg-type integral and generalise it to the deformed A n,1 -case. These formulas can be interpreted as new cases of explicit evaluation of Selberg-type integrals.2002 Mathematics Subject Classification. 33E30, 81R12.

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Solution of a restricted Hadamard problem on Minkowski spaces

Berest¹

1997

Comm. Pure Appl. Math.

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Algebraic integrability for the Schr�dinger equation and finite reflection groups

Cited by 61 publications

References 11 publications

Generalized Lam� Operators

Generalized Lam� Operators

Baker–akhiezer Function as Iterated Residue and Selberg-Type Integral

Solution of a restricted Hadamard problem on Minkowski spaces

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