CMV: the unitary analogue of Jacobi matrices

TL;DR

This paper explores the properties of CMV matrices, a class of unitary matrices analogous to Jacobi matrices, highlighting their structural features, integrable systems connections, and implications for orthogonal polynomials.

Contribution

It establishes the analogy between CMV and Jacobi matrices, detailing their geometric, algebraic, and dynamical properties, and constructs action/angle variables for related integrable systems.

Findings

CMV matrices share properties with Jacobi matrices, including foliation by co-adjoint orbits.

The paper describes a symplectic structure and reduction algorithms for CMV matrices.

It constructs action/angle variables for the Ablowitz-Ladik hierarchy and analyzes its long-term behavior.

Abstract

We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behaviour of this system.