56 research outputs found

    Inverse problem for sl(2) lattices

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    We consider the inverse problem for periodic sl(2) lattices as a canonical transformation from the separation to local variables. A new concept of a factorized separation chain is introduced allowing to solve the inverse problem explicitly. The method is applied to an arbitrary representation of the corresponding Sklyanin algebra.Comment: 16 pages, LaTeX, talk at SPT 2002, 19-26/5, Cala Gonone, Sardinia; corrected voffset-optio

    Factorisation of Macdonald polynomials

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    We discuss the problem of factorisation of the symmetric Macdonald polynomials and present the obtained results for the cases of 2 and 3 variables.Comment: 13 pages, LaTex, no figure

    Gauss hypergeometric function and quadratic RR-matrix algebras

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    We consider representations of quadratic RR-matrix algebras by means of certain first order ordinary differential operators. These operators turn out to act as parameter shifting operators on the Gauss hypergeometric function and its limit cases and on classical orthogonal polynomials. The relationship with W. Miller's treatment of Lie algebras of first order differential operators will be discussed.Comment: 26 page

    Eigenproblem for Jacobi matrices: hypergeometric series solution

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    We study the perturbative power-series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The(small) expansion parameters are being the entries of the two diagonals of length d-1 sandwiching the principal diagonal, which gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series of 3d-5 variables in the generic case, or 2d-3 variables for the eigenvalue growing from a corner matrix element. To derive the result, we first rewrite the spectral problem for a Jacobi matrix as an equivalent system of cubic equations, which are then resolved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.Comment: Latex, 20 pages; v2: corrected typos, added section with example

    Separation of variables and B\"acklund transformations for the symmetric Lagrange top

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    We construct the 1- and 2-point integrable maps (B\"acklund transformations) for the symmetric Lagrange top. We show that the Lagrange top has the same algebraic Poisson structure that belongs to the sl(2)sl(2) Gaudin magnet. The 2-point map leads to a real time-discretization of the continuous flow. Therefore, it provides an integrable numerical scheme for integrating the physical flow. We illustrate the construction by few pictures of the discrete flow calculated in MATLAB.Comment: 19 pages, 2 figures, Matlab progra

    Backlund transformations for the sl(2) Gaudin magnet

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    Elementary, one- and two-point, Backlund transformations are constructed for the generic case of the sl(2) Gaudin magnet. The spectrality property is used to construct these explicitly given, Poisson integrable maps which are time-discretizations of the continuous flows with any Hamiltonian from the spectral curve of the 2x2 Lax matrix
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