# Poisson brackets of orthogonal polynomials

@article{Cantero2009PoissonBO,
title={Poisson brackets of orthogonal polynomials},
author={M. J. Cantero and B. Simon},
journal={J. Approx. Theory},
year={2009},
volume={158},
pages={3-48}
}
• Published 2009
• Computer Science, Mathematics
• J. Approx. Theory
For the standard symplectic forms on Jacobi and CMV matrices, we compute Poisson brackets of OPRL and OPUC, and relate these to other basic Poisson brackets and to Jacobians of basic changes of variable.

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#### References

SHOWING 1-10 OF 102 REFERENCES
Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices
• Physics, Mathematics
• 2000
We introduce a family of compatible Poisson brackets on the space of rational functions with denominator of a fixed degree and use it to derive a multi-Hamiltonian structure for a family ofExpand
Poisson brackets for orthogonal polynomials on the unit circle
The recently discovered connection of orthogonal polynomials on the unit circle (OPUC) to the defocusing Ablowitz-Ladik integrable system involves the definition of a Poisson structure on the spaceExpand
Some remarks on CMV matrices and dressing orbits
The CMV matrices are the unitary analogs of Jacobi matrices. In the finite case, it is well-known that the set of Jacobi matrices with a fixed trace is nothing but a coadjoint orbit of the lowerExpand
A family of the Poisson brackets compatible with the Sklyanin bracket
We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the Sklyanin bracket, and use it to derive a multi-Hamiltonian structure for a setExpand
Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle
We prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators. In particular, we obtain new details on the interlacing of zerosExpand
Orthogonal Polynomials
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
Algebro-Geometric Solutions of a Discrete System Related to the Trigonometric Moment Problem
• Mathematics, Physics
• 2004
We derive theta function representations of algebro-geometric solutions of a discrete system governed by a transfer matrix associated with (an extension of) the trigonometric moment problem studiedExpand
Toda flows with infinitely many variables
• Mathematics
• 1985
Abstract The Toda flow and related flows extend naturally to operators in Hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties.
On Schur flows
• Mathematics
• 1999
For the finite Schur (dmKdV) flows, a non-local Poisson structure is introduced and shown to be linked via Backlund-Darboux transformations to linear and quadratic Poisson structures for the TodaExpand
Algebro-Geometric Solutions of the Baxter–Szegő Difference Equation
• Mathematics
• 2005
We derive theta function representations of algebro-geometric solutions of a discrete system governed by a transfer matrix associated with (an extension of) the trigonometric moment problem studiedExpand