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Darren C. Ong

Publications and source records attributed to Darren C. Ong.

At least 19 recordsLinked to original sources
arXiv2026

Bottleneck Effects and Harmonic-Type Velocity Bounds for Periodic Quantum Walks

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 a perturbative regime where one transmission parameter is small, we show that the corresponding almost reflecting coin acts as a bottleneck for transport: the velocity is bounded linearly in this parameter, with an explicit leading order estimate. Second, for arbitrary nonzero transmission parameters, we prove a general a priori bound in terms of their harmonic mean, together with a refined version that detects the spatial variation of neighboring coins. Moreover, we prove a general lower bound on the velocity. These bounds apply directly to the corresponding CMV setting.

math-ph↗
arXiv2025

Average Degree of Graphs Derived From The Ammann A2 Aperiodic Tiling

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 each side of each tile as an edge. We present a closed-form formula for the average degree of the graph corresponding to this Ammann A2 tiling.

math.CO↗
arXiv2023

Almost Everything About the Unitary Almost Mathieu Operator

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 this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. In particular, we show that supercritical, critical, and subcritical behavior all occur in this model. Using Avila's global theory of one-frequency cocycles, we exactly compute the Lyapunov exponent on the spectrum in terms of the given parameters. We also characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. Namely, we show that for almost every frequency and every phase the spectral type is purely absolutely continuous in the subcritical region, pure point in the supercritical region, and purely singular continuous in the critical region. In some parameter regions, we refine the almost-sure results. In the critical case for instance, we show that the spectrum is a Cantor set of zero Lebesgue measure for arbitrary irrational frequency and that the spectrum is purely singular continuous for all but countably many phases.

math.SP↗
arXiv2022

Abstract art generated by Thue-Morse correlation functions

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 autocorrelations of these sequences (roughly, measuring how similar a Thue-Morse sequence is to translates of itself) we can gain understanding of the diffraction patterns of quasicrystals. We generate abstract art images from these Thue-Morse autocorrelation functions, that capture the aperiodic structure of the Thue-Morse sequence in a compelling way.

cond-mat.stat-mech↗
arXiv2022

Subordinacy Theory for Extended CMV Matrices

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 presence or absence of subordinate solutions of the generalized eigenvalue equation. Some corollaries and applications of this result are described as well.

math.SP↗
arXiv2020

On Simon's Hausdorff Dimension Conjecture

Barry Simon conjectured in 2005 that the Szegő matrices, associated with Verblunsky coefficients $\{α_n\}_{n\in\mathbb{Z}_+}$ obeying $\sum_{n = 0}^\infty n^γ|α_n|^2 < \infty$ for some $γ\in (0,1)$, are bounded for values $z \in \partial \mathbb{D}$ outside a set of Hausdorff dimension no more than $1 - γ$. Three of the authors recently proved this conjecture by employing a Prüfer variable approach that is analogous to work Christian Remling did on Schrödinger operators. This paper is a companion piece that presents a simple proof of a weak version of Simon's conjecture that is in the spirit of a proof of a different conjecture of Simon.

math.SP↗
arXiv2020

Simon's OPUC Hausdorff Dimension Conjecture

We show that the Szegő matrices, associated with Verblunsky coefficients $\{α_n\}_{n\in\mathbb{Z}_+}$ obeying $\sum_{n = 0}^\infty n^γ|α_n|^2 < \infty$ for some $γ\in (0,1)$, are bounded for values $z \in \partial \mathbb{D}$ outside a set of Hausdorff dimension no more than $1 - γ$. In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than $1-γ$. This proves the OPUC Hausdorff dimension conjecture of Barry Simon from 2005.

math.SP↗
arXiv2020

Restrictions on the existence of a canonical system flow hierarchy

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 nevertheless share many features with Schrödinger operators. Since this is a very natural generalization, one would expect that it would also be straightforward to build a hierarchy of isospectral evolutions on canonical systems analogous to the KdV hierarchy. Surprisingly, we show that there are many obstructions to constructing a hierarchy of flows on canonical systems that obeys the standard assumptions of the KdV hierarchy. This suggests that we need a more sophisticated approach to develop such a hierarchy, if it is indeed possible to do so.

math.SP↗
arXiv2020

Quasiperiodic Music

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, Raindrops in A minor. We also discuss connections between quasiperiodic music and other works of music theory and composition that make use of aperiodic order or periodic order with large periods, such as Lindenmayer systems, Vuza canons, Messiaen's Quatuor pour la Fin du Temps, and the phase music of Steve Reich.

math.HO↗
arXiv2018

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic Jacobi operators

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 spectrum has been known to be difficult, given that eigenvalues are destroyed easily by small perturbations. However, given a finite or countably infinite set of points within an absolutely continuous band of the original periodic operator (subject only to a very weak non-resonance condition) we are able to construct a diagonal perturbation that preserves the essential spectrum and places eigenvalues in all of those points.

math.SP↗
arXiv2018

Sharp spectral transition for eigenvalues embedded into the spectral bands of perturbed periodic operators

In this paper, we consider the Schrödinger equation, \begin{equation*} Hu=-u^{\prime\prime}+(V(x)+V_0(x))u=Eu, \end{equation*} where $V_0(x)$ is 1-periodic and $V (x)$ is a decaying perturbation. By Floquet theory, the spectrum of $H_0=-\nabla^2+V_0$ is purely absolutely continuous and consists of a union of closed intervals (often referred to as spectral bands). Given any finite set of points $\{ E_j\}_{j=1}^N$ in any spectral band of $H_0$ obeying a mild non-resonance condition, we construct smooth functions $V(x)=\frac{O(1)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$. Given any countable set of points $\{ E_j\}$ in any spectral band of $H_0$ obeying the same non-resonance condition, and any function $h(x)>0$ going to infinity arbitrarily slowly, we construct smooth functions $|V(x)|\leq \frac{h(x)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$. On the other hand, we show that there is no eigenvalue of $H=H_0+V$ embedded in the spectral bands if $V(x)=\frac{o(1)}{1+|x|}$ as $x$ goes to infinity. We prove also an analogous result for Jacobi operators.

math.SP↗
arXiv2018

On a description of the Toda hierarchy using cocycle maps

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 characterization of the Toda hierarchy using cocycle maps is equivalent to the traditional approach using Lax pairs.

math-ph↗
arXiv2018

Generalized Toda flows

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 groups of $C^2$ and entire functions, and in this second case, we also introduce associated cocycles and in fact give center stage to this object.

math.SP↗
arXiv2017

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.

math-ph↗
arXiv2016

A condition for purely absolutely continuous spectrum for CMV operators using the density of states

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 purely absolutely continuous if and only if the density of states is absolutely continuous and the Lyapunov exponent vanishes almost everywhere with respect to the same. Both of these results are CMV operator analogues of theorems obtained by Kotani for Schrödinger operators.

math.SP↗
arXiv2016

Purely Singular Continuous Spectrum for Limit-Periodic CMV Operators with Applications to Quantum Walks

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 measure associated with a stochastic family of CMV matrices and use our construction from the first part to prove that the Craig--Simon result is optimal in general. We discuss applications of these results to a quantum walk model where the coins are arranged according to a limit-periodic sequence. The key ingredient in these results is a new formula which may be viewed as a relationship between the density of states measure of a CMV matrix and its Schur function.

math.SP↗
arXiv2016

Spreading Estimates for Quantum Walks on the Integer Lattice via Power-Law Bounds on Transfer Matrices

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 us to study spreading rates for quantum walks in one spatial dimension. We prove several general results which establish quantitative upper and lower bounds on the spreading of a quantum walk in terms of estimates on a pair of associated matrix cocycles. To demonstrate the power and utility of these methods, we apply them to several concrete cases of interest. In the case where the coins are distributed according to an element of the Fibonacci subshift, we are able to rather completely describe the dynamics in a particular asymptotic regime. As a pleasant consequence, this supplies the first concrete example of a quantum walk with anomalous transport, to the best of our knowldege. We also prove ballistic transport for a quantum walk whose coins are periodically distributed.

math-ph↗