CMV matrices in random matrix theory and integrable systems: a survey

Abstract

Abstract. We present a survey of recent results concerning a remarkable class of unitary matrices, the CMV matrices. We are particularly interested in the role they play in the theory of random matrices and integrable systems. Throughout the paper we also emphasize the analogies and connections to Jacobi matrices.

OPUC, CMV matrices and perturbations of measures supported on the unit circle

“…These matrices play a similar role among unitary matrices as Jacobi matrices among all Hermitian matrices, for instance, see [45,61]. This analogy is illustrated in many fields of application such as random matrix theory and integrable systems [52], Dirichlet data of a circular periodic problem [53] and scattering problem [58]. The spectral analysis of CMV matrices has also attracted much attention in the last years [2,12,13,31,60].…”

Algebro-Geometric Finite-Band Solutions of the Ablowitz-Ladik Hierarchy

“…On the other hand, by (3.44)-(3.47) one obtains 50) and hence the part of (3.48) concerning ω In the following it will be convenient to use the abbreviation…”

“…In this sense the discussion in [35] is a purely stationary one and connections with a zero-curvature formalism, theta function representations, and integrable hierarchies are not made in [35] (but in this context we refer to the discussion concerning [33] in the next paragraph). More recently, the defocusing case with periodic and quasi-periodic coefficients was studied in great detail by Deift [25], Golinskii and Nevai [43], Killip and Nenciu [44], Li [45], Nenciu [49], [50], and Simon [52,Ch. 11], [53], [54].…”

“…The Ablowitz-Ladik Lax pair in the special defocusing case, where β = α, in the finite-dimensional context, was recently derived by Nenciu [50].…”

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

“…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.…”