Multidimensional Baker–Akhiezer Functions and Huygens' Principle

Chalykh, Oleg; Feigin, Misha; Веселов, А. П.

doi:10.1007/pl00005521

Cited by 76 publications

(191 citation statements)

References 36 publications

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“…A further remark is on the class of locus configurations BA w that admit the weaker version of the Baker-Akhiezer function [3]. In the two-dimensional case these configurations are described in [5,8] (see also [9]).…”

Section: Discussionmentioning

confidence: 99%

“…In that context, the BA function is a special common eigenfunction of the Calogero-Moser operator and its quantum integrals. Chalykh, Styrkas, Veselov, and one of the authors studied the BA functions associated with finite sets of vectors in C N taken with integer multiplicities [2,3]. Besides the relevance to quantum integrable systems of CalogeroMoser type, it was established in [3] that the BA functions are closely related to the Huygens' Principle in the Hadamard sense (see also [4,5]).…”

Section: Introductionmentioning

confidence: 99%

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A Class of Baker–Akhiezer Arrangements

Feigin

Johnston

2014

Commun. Math. Phys.

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Abstract:We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh-Veselov Baker-Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker-Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasiinvariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero-Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Class of Baker–Akhiezer Arrangements

Feigin

Johnston

2014

Commun. Math. Phys.

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“…The latter fact is due to the quasi-invariance of u, see [27] for the one-dimensional case and [13], section 2 for a discussion in the multivariable setting.…”

Section: Definition Let Us Say That a Schrödinger Operatormentioning

confidence: 99%

“…Other known examples of the generalized Lamé operators in dimension > 1 are related to deformed root systems, which appeared in [13]. Below we describe the set of linear functionals A = {α} and the corresponding multiplicities m α .…”

Section: Deformed Root Systemsmentioning

confidence: 99%

“…Such operator was considered by Hietarinta [19] who showed that it admits a commuting operator of order 3. Its rational version ℘(z) = z −2 corresponds to a specific choice of the parameters in the family of two-dimensional Schrödinger operators introduced by Berest and Lutsenko [20] in connection with Huygens' Principle (see section 4 of [13] for details). Note that the potential in (1.3) is real-valued (for real x, y) if α is real and the period τ is pure imaginary.…”

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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Macdonald Polynomials and Algebraic Integrability

Chalykh

2002

Advances in Mathematics

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We construct explicitly (nonpolynomial) eigenfunctions of the difference operators by Macdonald in the case t=q k , k ¥ Z. This leads to a new, more elementary proof of several Macdonald conjectures, proved first by Cherednik. We also establish the algebraic integrability of Macdonald operators at t=q k (k ¥ Z), generalizing the result of Etingof and Styrkas. Our approach works uniformly for all root systems including the BC n case and related Koornwinder polynomials. Moreover, we apply it for a certain deformation of the A n root system where the previously known methods do not work.

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Multidimensional Baker–Akhiezer Functions and Huygens' Principle

Cited by 76 publications

References 36 publications

A Class of Baker–Akhiezer Arrangements

A Class of Baker–Akhiezer Arrangements

Generalized Lam� Operators

Macdonald Polynomials and Algebraic Integrability

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