相关论文: Absolutely Continuous Spectrum for CMV Matrices Wi…

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…

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…

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…

We prove an averaging formula for the derivative of the absolutely continuous part of the density of states measure for an ergodic family of CMV matrices. As a consequence, we show that the spectral type of such a family is almost surely…

We investigate the spectral structure of multi-frequency quasi-periodic CMV matrices with Verblunsky coefficients defined by shifts on the $d$-dimensional torus. Under the positive Lyapunov exponent regime and standard Diophantine frequency…

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…

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…

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.…

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…

Recent results of Denisov and Kaluzhny-Shamis describe the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an l^2 bounded variation condition with step p and are asymptotically periodic. We extend these results…

We review recent developments in the spectral theory of continuum one-dimensional quasicystals, yielding purely singular continuous spectrum for these Schr\"odinger operators. Allowing measures as potentials we can generalize some results…

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these…

Avila and Jitomirskaya prove that the quasi-periodic Schr\"{o}dinger operator $H_{\lambda v,\alpha,\theta}$ has purely absolutely continuous spectrum for $\alpha $ in sub-exponential regime (i.e., $\beta(\alpha)=0$) with small $\lambda$, if…

We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schr\"odinger operator problems for the twenty-first century.

We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures. As an elementary application…

We develop subordinacy theory for extended CMV matrices. That is, we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the…

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…

In this note, we show that the limiting spectral distribution of symmetric random matrices with stationary entries is absolutely continuous under some sufficient conditions. This result is applied to obtain sufficient conditions on a…

I prove that quasi-periodic Schr\"odinger operators in arbitrary dimension have some absolutely continuous spectrum.

We investigate the symmetries of so-called generalized extended CMV matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant…