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.
We study probability measures on the unit circle corresponding to orthogonal polynomials whose sequence of Verblunsky coefficients is invariant under the Fibonacci substitution. We focus in particular on the fractal properties of the essential support of these measures.
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.
Avila recently introduced a new method for the study of the discrete Schrödinger Operator with limit periodic potential. I adapt this method to the context of orthogonal polynomials in the unit circle with limit periodic Verblunsky Coefficients. Specifically, I represent these Verblunsky Coefficients as a continuous sampling of the orbits of a Cantor group by a minimal translation. I then investigate the measures that arise on the unit circle as I vary the sampling function.
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.
We provide leading order asymptotics for the size of the gap in the zeros around 1 of paraothogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (-1,0). We also include related results that impose less restrictive conditions on the Verblunsky coefficients.
We show that probability measures on the unit circle associated with Verblunsky coefficients obeying a Coulomb-type decay estimate have no singular continuous component.
We prove that the Szegő function, $D(z)$, of a measure on the unit circle is entire meromorphic if and only if the Verblunsky coefficients have an asymptotic expansion in exponentials. We relate the positions of the poles of $D(z)^{-1}$ to the exponential rates in the asymptotic expansion. Basically, either set is contained in the sets generated from the other by considering products of the form, $z_1 ... z_\ell \bar z_{\ell-1}... \bar z_{2\ell-1}$ with $z_j$ in the set. The proofs use nothing more than iterated Szegő recursion at $z$ and $1/\bar z$.
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…
In the first five sections, we deal with the class of probability measures with asymptotically periodic Verblunsky coefficients of p-type bounded variation. The goal is to investigate the perturbation of the Verblunsky coefficients when we add a pure point to a gap of the essential spectrum. For the asymptotically constant case, we give an asymptotic formula for the orthonormal polynomials in the gap, prove that the perturbation term converges and show the limit explicitly. Furthermore, we prove that the perturbation is of bounded variation. Then we generalize the method to the asymptotically …
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.
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.
We consider Jacobi matrices with eventually increasing sequences of diagonal and off-diagonal Jacobi parameters. We describe the asymptotic behavior of the subordinate solution at the top of the essential spectrum, and the asymptotic behavior of the spectral density at the top of the essential spectrum. In particular, allowing on both diagonal and off-diagonal Jacobi parameters perturbations of the free case of the form $- \sum_{j=1}^J c_j n^{-τ_j} + o(n^{-τ_1-1})$ with $0 < τ_1 < τ_2 < \dots < τ_J$ and $c_1>0$, we find the asymptotic behavior of the $\log$ of spectral density to order $O(\log…
Let $μ$ be a non-trivial probability measure on the unit circle $\partial\bbD$, $w$ the density of its absolutely continuous part, $α_n$ its Verblunsky coefficients, and $Φ_n$ its monic orthogonal polynomials. In this paper we compute the coefficients of $Φ_n$ in terms of the $α_n$. If the function $\log w$ is in $L^1(dθ)$, we do the same for its Fourier coefficients. As an application we prove that if $α_n \in \ell^4$ and $Q(z) = \sum_{m=0}^N q_m z^m$ is a polynomial, then with $\bar Q(z) = \sum_{m=0}^N \bar q_m z^m$ and $S$ the left shift operator on sequences we have $|Q(e^{iθ})|^2 \log w(θ…
The relation between the Toda lattices and similar nonlinear chains and orthogonal polynomials on the real line has been elaborated immensely for the last decades. We examine another system of the differential-difference equations known as the Schur flow within the framework of the theory of orthogonal polynomials on the unit circle. This system can be exhibited in equivalent form as the Lax equation, and the corresponding spectral measure undergoes a simple transformation. The general result is illustrated on the modified Bessel measures on the unit circle and the long time behavior of their …
Let mu be a probability measure on the unit circle and nu be the measure formed by adding a pure point to mu. We give a formula for the Verblunsky coefficients of the perturbed measure, based on a result of Simon.
We consider ergodic families of Verblunsky coefficients generated by minimal aperiodic subshifts. Simon conjectured that the associated probability measures on the unit circle have essential support of zero Lebesgue measure. We prove this for a large class of subshifts, namely those satisfying Boshernitzan's condition. This is accomplished by relating the essential support to uniform convergence properties of the corresponding Szego cocycles.
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…
We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $β^{(l)}$, each of which has rotated bounded variation, i.e., $\sum_{n=0}^\infty | e^{iϕ_l} β_{n+1}^{(l)} - β_n^{(l)} |$ is finite for some $ϕ_l$. This includes discrete Schrödinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann type potentials. For the real line, we prove that in the Leb…
We consider a renormalization transformation $R$ for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of $R$ determine universal constants analogous to the critical exponents in the theory of phase transitions. Restricting to skew-product maps over a circle-rotations by the golden mean, we find several periodic orbits for $R$, and we conjecture that there are infinitely many. Interestingly, all scaling factors that have been determined to high accuracy appear to be algebraically related to the circle-rotation number. We present evid…
