Poisson brackets of orthogonal polynomials

Abstract

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.

A note on Poisson brackets for orthogonal polynomials on the unit circle

“…The 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 space of Verblunsky coefficients. In this paper, we compute the complete set of Poisson brackets for the monic orthogonal and the orthonormal polynomials on the unit circle, as well as for the second kind polynomials and the Wall polynomials, This answers a question posed by Cantero and Simon,[3], for the case of measures with finite support. We also show that the results hold for the case of measures with periodic Verblunsky coefficients.…”

“…For example, formula (2.5) becomes, for z = w, [3]. The Poisson bracket that they didn't compute is (2.5) (and its reverse, (2.8)).…”

“…Their purpose was to understand the complete integrability of the defocusing Ablowitz-Ladik (AL) equation with periodic boundary conditions; in the process, they compute certain fairly complicated combinations of the Poisson brackets of the Wall polynomials-see Proposition 2.9. (For the origin of the Ablowitz-Ladik equation, see [1], [2]; for recent results on this integrable system, obtained through its connection to OPUC, see for example [3,4,7,10,11], or the review paper [13].) In [3], Cantero and Simon found all but one of the Poisson brackets for the monic orthogonal and the second kind polynomials.…”

“…(For the origin of the Ablowitz-Ladik equation, see [1], [2]; for recent results on this integrable system, obtained through its connection to OPUC, see for example [3,4,7,10,11], or the review paper [13].) In [3], Cantero and Simon found all but one of the Poisson brackets for the monic orthogonal and the second kind polynomials. In this paper, we give all the Poisson brackets for the OPs (see Theorem 1); they were derived from the combinations of [14], to which they are equivalent (see Proposition 2.9), but we prove them directly by induction.…”

“…Cantero and Simon show in [3] that one can more easily deduce this formula directly from the brackets of the orthogonal polynomials. Indeed, in the finite case they use an analogue for measures with finite support of Theorem 3.2.4 of [15] to connect the Carathéodory function with the orthogonal polynomials of the first and second kind.…”