Related papers: A note on circular trace formulae

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 lower triangular group. In this note, we will give…

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…

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…

For Schrodinger operators, there is a well known and widely used formula connecting the transfer matrices and Dirichlet determinants. No analog of this formula was previously known for CMV matrices. In this paper we fill this gap and…

We show that the Jacobi polynomials that are orthogonal on the unit circle (the Jacobi OPUC) are CMV bispectral. This means that the corresponding Laurent polynomials in the CMV basis satisfy two dual ordinary eigenvalue problems: a…

We prove a bijective unitary correspondence between 1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula. This…

Recently, a trace formula for non-self-adjoint periodic Schr\"odinger operators in $L^2(\mathbb{R})$ associated with Dirichlet eigenvalues was proved in [9]. Here we prove a corresponding trace formula associated with Neumann eigenvalues.…

We study CMV matrices (a discrete one-dimensional Dirac-type operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients,…

We consider the inverse spectral problem for periodic Jacobi matrices in terms of the vertical slits on the quasi-momentum domain plus the Dirichlet eigenvalues, i.e., the Marchenko-Ostrovsky mapping. Moreover, we show that the gradients of…

CMV matrices are the unitary analog of Jacobi matrices; we review their general theory.

We recall criteria on the spectrum of Jacobi matrices such that the corresponding isospectral torus consists of periodic operators. Motivated by those known results for Jacobi matrices, we define a new class of operators called GMP…

M. Derevyagin, L. Vinet and A. Zhedanov introduced in Constr. Approx. 36 (2012) 513-535 a new connection between orthogonal polynomials on the unit circle and the real line. It maps any real CMV matrix into a Jacobi one depending on a real…

We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence…

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the…

The Wigner-von Neumann method, which was previously used for perturbing continuous Schr\"{o}dinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary $T$-periodic Jacobi…

We consider dynamic inverse problems for a dynamical system associated with a finite Jacobi matrix and for a system describing propagation of waves in a finite Krein-Stieltjes string. We offer three methods of recovering unknown parameters:…

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related with the theory of CMV matrices. It contains an arbitrary parameter which leads to a linear…

In this note we investigate the discrete spectrum of Jacobi matrix corresponding to polynomials defined by recurrence relations with periodic coefficients. As examples we consider a)the case when period $N$ of coefficients of recurrence…

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a Jacobi or CMV operator from a finite gap isospectral torus. The special case of eventually periodic operators…

It is shown that traces of mapping classes of finite order may be expressed by Verlinde-like formulae. The 3D topological argument is explained, and the resulting trace identities for modular matrix elements are presented.