467 research outputs found

    Locus configurations and ∨\vee-systems

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    We present a new family of the locus configurations which is not related to ∨\vee-systems thus giving the answer to one of the questions raised in \cite{V1} about the relation between the generalised quantum Calogero-Moser systems and special solutions of the generalised WDVV equations. As a by-product we have new examples of the hyperbolic equations satisfying the Huygens' principle in the narrow Hadamard's sense. Another result is new multiparameter families of ∨\vee-systems which gives new solutions of the generalised WDVV equation.Comment: 12 page

    Bethe ansatz for the Ruijsenaars model of BC1- type

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    We consider one-dimensional elliptic Ruijsenaars model of type BC1. We show that when all coupling constants are integers, it has meromorphic eigenfunctions expressed by a variant of Bethe ansatz. This result generalizes the Bethe ansatz formulas known in the A1-case

    A class of Baker-Akhiezer arrangements

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    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 quasi-invariants 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

    On algebraic integrability of the deformed elliptic Calogero--Moser problem

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    Algebraic integrability of the elliptic Calogero--Moser quantum problem related to the deformed root systems \pbf{A_{2}(2)} is proved. Explicit formulae for integrals are found

    A remark on rational isochronous potentials

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    We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form U(x)=1/2ω2x2U(x) = 1/2 \omega^2 x^2 or U(x)=1/8ω2x2+c2x−2.U(x) = 1/8 \omega^2 x ^2 + c^2 x^{-2}.Comment: 5 pages, contribution to a special issue of JNMP dedicated to F. Calogero, slightly revised versio

    Generalized Lame operators

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    We introduce a class of multidimensional Schr\"odinger operators with elliptic potential which generalize the classical Lam\'e 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 Lam\'e 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.Comment: 38 pages, latex; in the new version a reference was adde

    Quantum Lax Pairs via Dunkl and Cherednik Operators

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    We establish a direct link between Dunkl operators and quantum Lax matrices L for the Calogero–Moser systems associated to an arbitrary Weyl group W (or an arbitrary finite reflection group in the rational case). This interpretation also provides a companion matrix A so that L,A form a quantum Lax pair. Moreover, such an A can be associated to any of the higher commuting quantum Hamiltonians of the system, so we obtain a family of quantum Lax pairs. These Lax pairs can be of various sizes, matching the sizes of orbits in the reflection representation of W, and in the elliptic case they contain a spectral parameter. This way we reproduce universal classical Lax pairs by D’Hoker–Phong and Bordner–Corrigan–Sasaki, and complement them with quantum Lax pairs in all cases (including the elliptic case, where they were not previously known). The same method, with the Dunkl operators replaced by the Cherednik operators, produces quantum Lax pairs for the generalised Ruijsenaars systems for arbitrary root systems. As one of the main applications, we calculate a Lax matrix for the elliptic BCn case with nine coupling constants (van Diejen system), thus providing an answer to a long-standing open problem
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