14 citations · 77 across the 20 of their papers we have counts for
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Exact mobility edges for almost-periodic CMV matrices via gauge symmetries
Christopher Cedzich, Jake Fillman, Long Li +2
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…
Almost Everything About the Unitary Almost Mathieu Operator
Christopher Cedzich, Jake Fillman, Darren C. Ong
We introduce a unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry--André duality for…
On Simon's Hausdorff Dimension Conjecture
David Damanik, Jake Fillman, Shuzheng Guo +1
Barry Simon conjectured in 2005 that the Szegő matrices, associated with Verblunsky coefficients obeying for…
Simon's OPUC Hausdorff Dimension Conjecture
David Damanik, Shuzheng Guo, Darren C. Ong
We show that the Szegő matrices, associated with Verblunsky coefficients obeying for some , are b…
Subordinacy Theory for Extended CMV Matrices
Shuzheng Guo, David Damanik, Darren C. Ong
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 as…
Restrictions on the existence of a canonical system flow hierarchy
Injo Hur, Darren C. Ong
The KdV hierarchy is a family of evolutions on a Schrödinger operator that preserves its spectrum. Canonical systems are a generalization of Schrödinger operators, that nevertheles…
Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic Jacobi operators
Wencai Liu, Darren C. Ong
We are interested in diagonal perturbations of a periodic Jacobi operator that introduce embedded eigenvalues in its essential spectrum. Embedding multiple points in the essential…
Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators
Wencai Liu, Darren C. Ong
In this paper, we consider the Schrödinger equation, \begin{equation*} Hu=-u^{\prime\prime}+(V(x)+V_0(x))u=Eu, \end{equation*} where is 1-periodic and is a decayin…
Generalized Toda flows
Darren C. Ong, Christian Remling
The classical hierarchy of Toda flows can be thought of as an action of the (abelian) group of polynomials on Jacobi matrices. We present a generalization of this to the larger gro…
A condition for purely absolutely continuous spectrum for CMV operators using the density of states
Jake Fillman, Darren C. Ong
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…
Purely Singular Continuous Spectrum for Limit-Periodic CMV Operators with Applications to Quantum Walks
Jake Fillman, Darren C. Ong
We show that a generic element of a space of limit-periodic CMV operators has zero-measure Cantor spectrum. We also prove a Craig--Simon type theorem for the density of states meas…
Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator
Jake Fillman, Darren C. Ong, Zhenghe Zhang
We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spect…
Purely singular continuous spectrum for CMV operators generated by subshifts
Darren C. Ong
We prove uniform absence of point spectrum for CMV operators corresponding to the period doubling subshift. We also prove almost sure absence of point spectrum for CMV operators co…
Wigner-von Neumann type perturbations of periodic Schrödinger Operators
Milivoje Lukic, Darren C. Ong
We consider decaying oscillatory perturbations of periodic Schrödinger operators on the half line. More precisely, the perturbations we study satisfy a generalized bounded variatio…
Orthogonal Polynomials on the Unit Circle with quasiperiodic Verblunsky Coefficients have generic purely singular continuous spectrum
Darren C. Ong
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-…
The Hölder continuity of spectral measures of an extended CMV matrix
Paul Munger, Darren C. Ong
We prove results about the Hölder continuity of the spectral measures of the extended CMV matrix, given power law bounds of the solution of the eigenvalue equation. We thus arrive…
Limit-Periodic Verblunsky Coefficients for Orthogonal Polynomials on the Unit Circle
Darren C. Ong
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 polynomial…