213 research outputs found
Inverse problem for sl(2) lattices
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
B\"acklund Transformation for the BC-Type Toda Lattice
We study an integrable case of n-particle Toda lattice: open chain with
boundary terms containing 4 parameters. For this model we construct a
B\"acklund transformation and prove its basic properties: canonicity,
commutativity and spectrality. The B\"acklund transformation can be also viewed
as a discretized time dynamics. Two Lax matrices are used: of order 2 and of
order 2n+2, which are mutually dual, sharing the same spectral curve.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Factorisation of Macdonald polynomials
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 -matrix algebras
We consider representations of quadratic -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
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
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 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
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