CMV matrices: Five years after

多频准周期CMV矩阵的有限和全尺度局域化

Finite and full scale localization for the multi-frequency quasi-periodic CMV matrices

研究多频准周期CMV矩阵的有限和全尺度局域化方法，提出与Goldstein等人关于多频准周期Schrödinger算子局域化结果相对应的CMV矩阵结果。

由双曲环面自同构定义的Verblunsky系数的CMV矩阵的Anderson局域化

Anderson localization for CMV matrices with Verblunsky coefficients defined by the hyperbolic toral automorphism

本文针对由双曲环面自同构动力学定义Verblunsky系数的CMV矩阵，证明了大偏差估计与Anderson局域化，并将Chulaevsky-Spencer和Bourgain-Schlag的相关结果从Schrödinger算子推广到CMV矩阵。

多频拟周期CMV矩阵的谱包含区间

The spectrum of the multi-frequency quasi-periodic CMV matrices contains intervals

本文研究多频拟周期CMV矩阵的谱结构，在正Lyapunov指数和Diophantine条件下证明其谱在单位圆上包含区间。

A formula related to CMV matrices and Szego cocycles

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 provide the CMV analog of this formula.

Localization for random CMV matrices

We prove Anderson localization (AL) and dynamical localization in expectation (EDL, also known as strong dynamical localization) for random CMV matrices for arbitrary distribution of i.i.d. Verblunsky coefficients.

Anderson localization for the quasi-periodic CMV matrices with Verblunsky coefficients defined by the skew-shift

In this paper, we study quasi-periodic CMV matrices with Verblunsky coefficients given by the skew-shift. We prove the positivity of Lyapunov exponents and Anderson localization for most frequencies, which establish the analogous results of one-dimensional Schrödinger operators proved by Bourgain, Goldstein and Schlag.

Cantor Spectrum for CMV and Jacobi Matrices with Coefficients arising from Generalized Skew-Shifts

We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.

Purely Singular Continuous Spectrum for Sturmian CMV Matrices via Strengthened Gordon Lemmas

The Gordon Lemma refers to a class of results in spectral theory which prove that strong local repetitions in the structure of an operator preclude the existence of eigenvalues for said operator. We expand on recent work of Ong and prove versions of the Gordon Lemma which are valid for CMV matrices and which do not restrict the parity of scales upon which repetitions occur. The key ingredient in our approach is a formula of Damanik-Fillman-Lukic-Yessen which relates two classes of transfer matrices for a given CMV operator. There are many examples to which our result can be applied. We apply o…

Reflectionless CMV matrices and scattering theory

Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associated to the coupled and decoupled operators are derived. In particular, it is shown that a CMV matrix is reflectionless iff the scattering matrix is off-diagonal which in turn provides a short proof of an important result of [Breuer-Ryckman-Simon]. These developments parallel those recently obtained for Jacobi matrices.

电量子行走的区间谱及相关的斜移CMV矩阵

Interval spectrum for electric quantum walk and related skew-shift CMV matrices

本文研究含电场量子行走模型的谱性质，证明对任何无理场其谱均为单位圆，该结果对斜移定义的相关CMV矩阵同样成立，并推广至高维环面上的斜移CMV矩阵。

Anderson Localization for Quasi-Periodic CMV Matrices and Quantum Walks

We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.

多频拟周期CMV矩阵与量子游走的Anderson局域化

Anderson localization for the multi-frequency quasi-periodic CMV matrices and quantum walks

本文通过构造半代数集并结合雪崩原理与大偏差定理，证明了多频拟周期CMV矩阵的Anderson局域化，并经由酉等价性推广至量子游走。

Generalized Prüfer variables for perturbations of Jacobi and CMV matrices

Prüfer variables are a standard tool in spectral theory, developed originally for perturbations of the free Schrödinger operator. They were generalized by Kiselev, Remling, and Simon to perturbations of an arbitrary Schrödinger operator. We adapt these generalized Prufer variables to the setting of Jacobi and Szegő recursions. We present an application to random $L^2$ perturbations of Jacobi and CMV matrices, and an application to decaying oscillatory perturbations of periodic Jacobi and CMV matrices.

On Scattering for CMV Matrices

Adamjan-Arov (Lax--Phillips) model space is considered as a scattering representation space for a CMV matrix in context of an extended Marchenko--Faddeev scattering theory. That is, there exists a basis in which the multiplication by independent variable is a CMV matrix. This basis as well as Verblunski coefficients are computed explicitly in terms of Nehari interpolation. Asymptotically the Verblynski coefficients go to zero. Moreover, relations between the basis and wandering subspaces are established. Transformation from scattering representation to spectral representation is given.

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。

Right limits and reflectionless measures for CMV matrices

We study CMV matrices by focusing on their right-limit sets. We prove a CMV version of a recent result of Remling dealing with the implications of the existence of absolutely continuous spectrum, and we study some of its consequences. We further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of Khrushchev. To demonstrate the analogy with the Jacobi case, we recover corresponding previous results of Simon using the same approach.

在严格遍历的 Jacobi 与 CMV 矩阵谱中打开谱隙

Opening Gaps in the Spectrum of Strictly Ergodic Jacobi and CMV Matrices

本文证明一般连续采样函数对应的动力系统 Jacobi 与 CMV 矩阵可实现 Gap Labelling Theorem 预测的全部谱隙，并基于共振舌边界的解析性给出准周期解析情形的亚临界谱隙打开机制。

Spectral Approximation for Ergodic CMV Operators with an Application to Quantum Walks

We establish concrete criteria for fully supported absolutely continuous spectrum for ergodic CMV matrices and purely absolutely continuous spectrum for limit-periodic CMV matrices. We proceed by proving several variational estimates on the measure of the spectrum and the vanishing set of the Lyapunov exponent for CMV matrices, which represent CMV analogues of results obtained for Schrödinger operators due to Y.\ Last in the early 1990s. Having done so, we combine those estimates with results from inverse spectral theory to obtain purely absolutely continuous spectrum.

通过规范对称性得到近周期CMV矩阵的精确迁移率边

Exact mobility edges for almost-periodic CMV matrices via gauge symmetries

该研究针对标准扩展CMV矩阵反射对称性问题的处理难点，通过显式对角酉矩阵转入广义扩展CMV矩阵类别开展对称性研究，构造了mosaic unitary almost-Mathieu算子族并证明其存在可精确计算的、分隔绝对连续谱与纯点谱的精确迁移率边。

Generic Spectral Results for CMV Matrices with Dynamically Defined Verblunsky Coefficients

We consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic in the sense that for a fixed base transformation, the set of continuous sampling functions for which the spectral phenomenon occurs is residual. Among the phenomena we discuss are the absence of absolutely continuous spectrum and the vanishing of the Lebesgue measure of the spectrum.

Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem

B. Simon proved the existence of the wave operators for the CMV matrices with Szego class Verblunsky coefficients, and therefore the existence of the scattering function. Generally, there is no hope to restore a CMV matrix when we start from the scattering function, in particular, because it does not contain any information about the (possible) singular measure. Our main point of interest is the solution of the inverse scattering problem (the heart of the Faddeev--Marchenko theory), that is, to give necessary and sufficient conditions on a certain class of CMV matrices such that the restrictio…

Cantor Spectrum for CMV Matrices With Almost Periodic Verblunsky Coefficients

We consider extended CMV matrices with analytic quasi-periodic Verblunsky coefficients with Diophantine frequency vector in the perturbatively small coupling constant regime and prove the analyticity of the tongue boundaries. As a consequence we establish that, generically, all gaps of the spectrum that are allowed by the Gap Labelling Theorem are open and hence the spectrum is a Cantor set. We also prove these results for a related class of almost periodic Verblunsky coefficients and present an application to suitable quantum walk models on the integer lattice.

通过反射证明量子行走与CMV矩阵无束缚态

Absence of Bound States for Quantum Walks and CMV Matrices via Reflections

本文利用反射对称性建立判据，证明分裂步量子行走与CMV矩阵无点谱，并应用于酉几乎Mathieu算子的算术离域化及Thue–Morse子移位生成CMV矩阵的奇异连续谱。

Characterizations of Uniform Hyperbolicity and Spectra of CMV Matrices

We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of $\mathrm{GL}(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.

Contractions with rank one defect operators and truncated CMV matrices

The main issue we address in the present paper are the new models for completely non-unitary contractions with rank one defect operators acting on some Hilbert space of dimension $N\leq\infty$. This model complements nicely the well-known models of Liv$\rm{\check{s}}$ic and Sz.-Nagy--Foias. We show that each such an operator is unitarily equivalent to some truncated CMV matrix obtained from the ``full'' CMV matrix by deleting the first row and the first column, and acting in $\ell^2$ ($\dC^N$). This result can be viewed as a nonunitary version of the famous characterization of unitary operator…

Anderson localization for electric quantum walks and skew-shift CMV matrices

We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the L…

CMV matrices with asymptotically constant coefficients. Szegö over Blaschke class, Scattering Theory

We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We demonstrate that an orthonormal system in a certain "weighted'' Hilbert space, which we call the Fadeev-Marchenko (FM) space, behaves asymptotically as the system in the standard (free) case. The duality between the two types of Hardy subspaces in it plays the key role in the proof of all asymptotics involved. We show that the traditional (Faddeev-Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representa…

Quantum Intermittency for Sparse CMV Matrices with an Application to Quantum Walks on the Half-Line

We study the dynamics given by the iteration of a (half-line) CMV matrix with sparse, high barriers. Using an approach of Tcheremchantsev, we are able to explicitly compute the transport exponents for this model in terms of the given parameters. In light of the connection between CMV matrices and quantum walks on the half-line due to Cantero-Grünbaum-Moral-Velázquez, our result also allows us to compute transport exponents corresponding to a quantum walk which is sparsely populated with strong reflectors. To the best of our knowledge, this provides the first rigorous example of a quantum walk …

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 Fibonacci Verblunsky Coefficients, II. Applications

We consider CMV matrices with Verblunsky coefficients determined in an appropriate way by the Fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics. In our first application we estimate the spreading rates of quantum walks on the line with time-independent coins following the Fibonacci sequence. The estimates we obtain are explicit in terms of the parameters of the system. In our second application, we establish a connection between the classical nearest neighbor Ising model on the one-dimensional lattice in the complex magn…

单位元算子中无球形运动而近球形运动的存在性

Absence of ballistic motion and presence of almost-ballistic motion for unitary operators with pure point spectrum

研究揭示了单位元算子中纯点谱与近球形运动之间的关系，展示了扩展CMV矩阵在纯点谱下接近球形运动的实例。

Poisson Brackets of Orthogonal Polynomials

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.

Equality of the Spectral and Dynamical Definitions of Reflection

For full-line Jacobi matrices, Schrödinger operators, and CMV matrices, we show that being reflectionless, in the sense of the well-known property of $m$-functions, is equivalent to a lack of reflection in the dynamics in the sense that any state that goes entirely to $x=-\infty$ as $t\to -\infty $ goes entirely to $x=\infty$ as $t\to\infty$. This allows us to settle a conjecture of Deift and Simon from 1983 regarding ergodic Jacobi matrices.

非酉带状矩阵的谱性质

Spectral Properties of Non-Unitary Band Matrices

本文研究由 CMV 矩阵形变得到的随机非酉压缩算子，证明其谱随参数变化的确定性与几乎必然性质，并用于分析树或格上随机量子游走的动力学。

Killip-Simon-classes of Jacobi matrices with essential spectrum on two symmetric and of SMP matrices on two arbitrary intervals

Jacobi matrices probably are the most classical object in spectral theory, while CMV matrices are a comparably fresh one, although they are related to a very classical topic, namely to orhtogonal polynomials on the unit circle (in the same way as Jacobi matrices are related to orhogonal polynomials on the real axis). We will discuss the third member of this family. Our matrices are generated by the orthonormal systems of functions related to the so-called Strong Moment Problem. For this reason we call them SMP matrices. For instance, one can describe the spectral sets of periodic SMP matrices.…

Killip-Simon problem and Jacobi flow on GMP matrices

One of the first theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class, which destroys completely the absolutely continuous spectrum of A (von Neumann). However, if A is the discrete free 1-D Schrödinger operator and B is a Jacobi matrix the a.c. spectrum remains perfectly the same. Moreover, Killip and Simon described explicitly the spectral properties for such A+B. Jointly with Damanik they generalized this result to the case of perturbations of periodic Jacobi matrices. Recall that the spectrum of a period…

Killip-Simon problem and Jacobi flow on GSMP matrices

One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class with arbitrary small operator norm, which destroys completely the absolutely continuos (a.c.) spectrum of the initial operator A (von Neumann). However, if A is the discrete free 1-D Schrödinger operator and B is an arbitrary Jacobi matrix (of Hilbert-Schmidt class) the a.c. spectrum remains perfectly the same, that is, the interval [-2,2]. Moreover, Killip and Simon described explicitly the spectral properties…