Weyl-Titchmarsh Theory for CMV Operators Associated with Orthogonal Polynomials on the Unit Circle

Analogs of the M-Function in the Theory of Orthogonal Polynomials on the Unit Circle

We show that the multitude of applications of the Weyl-Titchmarsh m-function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function.

Uniform Hyperbolicity for Szegő Cocycles and Applications to Random CMV Matrices and the Ising Model

We consider products of the matrices associated with the Szegő recursion from the theory of orthogonal polynomials on the unit circle and show that under suitable assumptions, their norms grow exponentially in the number of factors. In the language of dynamical systems, this result expresses a uniform hyperbolicity statement. We present two applications of this result. On the one hand, we identify explicitly the almost sure spectrum of extended CMV matrices with non-negative random Verblunsky coefficients. On the other hand, we show that no Ising model in one dimension exhibits a phase transit…

Orthogonal Polynomials on the Unit Circle with quasiperiodic Verblunsky Coefficients have generic purely singular continuous spectrum

As an application of the Gordon lemma for orthogonal polynomials on the unit circle, we prove that for a generic set of quasiperiodic Verblunsky coefficients the corresponding two-sided CMV operator has purely singular continuous spectrum. We use a similar argument to that of the Boshernitzan-Damanik result that establishes the corresponding theorem for the discrete Schrödinger operator.

Finite-gap CMV matrices: Periodic coordinates and a Magic Formula

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 latter class arises as spectrally-dependent operator Möbius transforms of certain generating CMV matrices which are periodic up to a rotational phase; for this reason we call them "MCMV". Such matrices are related to a choice of orthogonal rational functions on the unit circle, and their correspondence to the isospectral torus follows from a functional mode…

带有衰减扰动的周期CMV矩阵的定量谱稳定性与精确傅里叶估计

Quantitative spectral stability and sharp Fourier estimates for periodic CMV matrices with decaying perturbations

该研究针对Verblunsky系数为周期序列衰减扰动的半直线CMV矩阵，建立了扰动谱测度的Hölder连续性与精确傅里叶估计，为相关单位圆正交多项式（OPUC）研究提供了新结论。

Orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift

I give an example of a family of orthogonal polynomials on the unit circle with Verblunsky coefficients given by the skew-shift for which the associated measures are supported on the entire unit circle and almost-every Aleksandrov measure is pure point. Furthermore, I show in the case of the two dimensional skew-shift the zeros of para-orthogonal polynomials obey the same statistics as an appropriate irrational rotation. The proof is based on an analysis of the associated CMV matrices.

Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle

For suitable classes of random Verblunsky coefficients, including independent, identically distributed, rotationally invariant ones, we prove that if \[ \mathbb{E} \biggl(\int\frac{dθ}{2π} \biggl|\biggl(\frac{\mathcal{C} + e^{iθ}}{\mathcal{C} -e^{iθ}} \biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-κ_1 |k-\ell|} \] for some $κ_1 >0$ and $p<1$, then for suitable $C_2$ and $κ_2 >0$, \[ \mathbb{E} \bigl(\sup_n |(\mathcal{C}^n)_{k\ell}|\bigr) \leq C_2 e^{-κ_2 |k-\ell|} \] Here $\mathcal{C}$ is the CMV matrix.

Sharp asymptotics of the quasimomentum

We consider the Schrödinger operator with a periodic potential $p$ on the real line. We assume that $p$ belongs to the Sobolev space $\mH_m$ on the circle for some $m\ge -1$, and we determine the asymptotics of the quasimomentum and the Titchmarsh-Weyl functions, the Bloch functions at high energy.

A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure

We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-κ^2) =-κ- \int_0^a A(α) e^{-2ακ} dα+O(e^{-(2a -ε)κ}) for all ε> 0. We discuss five issues here. First, we extend the theory to general q in L^1 ((0,a)) for all a, including q's which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure ρ: A(α) = -2\int_{-\infty}^\infty λ^{-\frac12} \sin (2α\sqrtλ)\, dρ(λ) (since the integral is divergent, this f…