14 citations · 77 across the 20 of their papers we have counts for
26 papers
Bottleneck Effects and Harmonic-Type Velocity Bounds for Periodic Quantum Walks
Houssam Abdul-Rahman, Thomas A. Jackson, Darren C. Ong
We prove explicit upper bounds on the propagation velocity of one-dimensional quantum walks with periodic coins of arbitrary period. We treat two complementary settings. First, in…
Average Degree of Graphs Derived From The Ammann A2 Aperiodic Tiling
Xinyan Xu, Darren C. Ong
The Ammann A2 tiling is a simple aperiodically ordered tiling of the plane. We consider the graph derived from this tiling, by treating each corner of each tile as a vertex and eac…
Fractals corresponding to the metallic means sequences
Y. S. J. Liang, Darren C. Ong
In this work, we introduce a class of fractal subsets of corresponding to the aperiodically ordered metallic means sequences. We find simple formulas for the fractal dimens…
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…
Abstract art generated by Thue-Morse correlation functions
Darren C. Ong
The Thue-Morse sequence is an aperiodically ordered infinite binary sequence. It is used as a one-dimensional way to model the structure of a quasicrystal. For example, taking auto…
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…
Quasiperiodic Music
Darren C. Ong
Using the definition of quasiperiodic function as a motivation, we introduce the idea of quasiperiodic music and detail the composition process of a quasiperiodic music piece, Rain…
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…
On a description of the Toda hierarchy using cocycle maps
Darren C. Ong
The Toda hierarchy refers to a family of integrable flows on Jacobi operators that have many applications in mathematics and physics. We demonstrate carefully that an alternative c…
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…
Spectral Approximation for Ergodic CMV Operators with an Application to Quantum Walks
Jake Fillman, Darren C. Ong, Tom Vandenboom
We establish concrete criteria for fully supported absolutely continuous spectrum for ergodic CMV matrices and purely absolutely continuous spectrum for limit-periodic CMV matrices…
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…
Spreading Estimates for Quantum Walks on the Integer Lattice via Power-Law Bounds on Transfer Matrices
David Damanik, Jake Fillman, Darren C. Ong
We discuss spreading estimates for dynamical systems given by the iteration of an extended CMV matrix. Using a connection due to Cantero--Grünbaum--Moral--Velázquez, this enables u…
Generalized Prüfer variables for perturbations of Jacobi and CMV matrices
Milivoje Lukic, Darren C. Ong
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 Sim…
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-…