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Daxiong Piao

Publications and source records attributed to Daxiong Piao.

At least 19 recordsLinked to original sources
arXiv2026

Positivity and log-Hölder Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators

We prove the positivity and continuity of the Lyapunov exponent for one-dimensional discrete Schrödinger operators with multi-frequency skew-shift potentials. For the operator $H_{λ,ω} = Δ+ λv(T_ω^n(x,y))$ on $\ell^2(\mathbb{Z})$, where $T_ω$ is a skew-shift on $\mathbb{T}^{d}\times\mathbb{T}^{d}$ $(d\geq1)$ and $v$ is a non-constant real-analytic function on $\mathbb{T}^{2d}$, we establish that for Diophantine frequency vectors $ω$ and large coupling $λ\gg 1$, the Lyapunov exponent satisfies $L(λ,E) \geq c\logλ> 0$ uniformly in $E$ (with $c>0$), and is log-Hölder continuous in $E$. This work extends the known results of Lyapunov exponents--previously developed for one-frequency or simpler quasi-periodic models--to the genuinely multi-frequency skew-shift setting.

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arXiv2026

Construction and Characterization of Oscillatory Chain Sequences

This paper initiates a theoretical investigation of $\frac{1}{4}$-oscillatory chain sequences $\{a_n\}$, generalizing Szwarc's classical framework for non-oscillatory chains \cite{Sz94, Sz98, Sz02, Sz03} to sequences fluctuating around $\frac{1}{4}$. We prove the existence of a fixed point for the critical map $f(x)=1-\frac{1}{4x}$ and establish convergence properties linking oscillatory behavior to parameter sequences $\{g_n\}$. A complete characterization is provided via a necessary and sufficient condition, exemplified by explicit solutions $a_n=\frac{1}{4}\left(1+(-1)^{n}\varepsilon_{n}\right)$. Crucially, we construct oscillatory chain sequences for which the series $\sum_{n=1}^{\infty} \left(a_n - \frac{1}{4}\right)$ diverges, demonstrating fundamentally different behavior outside the hypothesis $a_n \ge \frac{1}{4}$ required by Chihara's bound.

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arXiv2026

Necessary Conditions for Single-Critical-Point Higher-Order Szegő Sum Rules in OPUC

We prove the necessity part of the higher-order Szegő theorem on the unit circle for the single-critical-point weights $H_m(e^{iθ})=(1-\cosθ)^m$, $m\ge1$. If $\{α_n\}_{n\ge0}$ are the Verblunsky coefficients of a nontrivial probability measure $dμ=w(θ)dθ/(2π)+dμ_{\mathrm s}$, then the weighted Szegő condition $\int_0^{2π} (1-\cosθ)^m\log w(θ)\frac{dθ}{2π}>-\infty$ implies $Δ^mα\in\ell^2, \,\, α\in\ell^{2m+2}.$ The proof uses a finite-volume version of Yan's higher-order sum rule. The quadratic part yields the $m$-th difference energy, and the logarithmic tail yields the $\ell^{2m+2}$-control. The non-sign-definite critical terms are treated in two steps. First, the quartic principal critical block is isolated using the Yan quotient-algebra normal representative and shown to have a positive semidefinite Gram representation. Second, the remaining non-principal critical terms are controlled by the diagonal-vanishing property $\mathcal Y_{k,\mathrm{crit}}^{(m)} \in \mathfrak I_k^{\,m+1-k}, \,\, 2\le k\le m,$ together with the Breuer--Simon--Zeitouni normal form, discrete interpolation, and Young's inequality. These estimates yield a uniform finite-volume coercive bound, from which the necessity theorem follows for all $m\ge1$.

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arXiv2026

On the Genericity of the Spectrum Intervalization for Multi-Frequency Quasiperiodic Schrödinger Operators

This paper proves a genericity conjecture by Goldstein, Schlag, and Voda[Invent. Math.\textbf{217}(2019)] for multi-frequency quasiperiodic Schrödinger operators. Specifically, we show that for almost all coefficients of real trigonometric polynomial potentials, the spectrum forms a single interval under strong coupling conditions. This confirms a long-standing intuition by Chulaevky and Sinai[Comm.Math.Phys.\textbf{125}(1989)] that the spectrum typically intervals for generic potentials, and extends the existence results of Goldstein et al. to a full measure setting. Our proof relies on tools from differential topology, measure theory, and analytic function theory.

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arXiv2026

Single-Point Higher-Order Szegő Sum Rules in OPUC: Necessity for $m=1,2,3$

We give a direct algebraic proof of the necessity direction in the single-point higher-order Szegő sum rules on the unit circle for $m=1,2,3$. More precisely, for $H_m(e^{iθ})=(1-\cosθ)^m$, we show that $\int_0^{2π}H_m(e^{iθ})\log w(θ)\frac{dθ}{2π}>-\infty$ implies $(S-1)^mα\in\ell^2,\qquad α\in\ell^{2m+2}.$ The proof is carried out within Yan's algebraic model for higher-order sum rules. The main point is to obtain coercive lower bounds for the nonlogarithmic part of the truncated sum rule: the quadratic component yields the principal finite-difference energy, while the higher-order correction terms are controlled by telescoping cancellations and relative bounds. The logarithmic remainder then gives the required $\ell^{2m+2}$-summability. The purpose is to isolate explicit low-order necessity arguments within the algebraic framework.

math.CA↗
arXiv2026

Anderson Localization for Schrödinger Operators with Monotone Potentials Generated by the Doubling Map

In this paper, we consider the Schrödinger operators on $ \ell^{2}(\N) $, defined for all $ x\in\mathbb{T} $ by \begin{equation} (H(x)u)_n = u_{n+1} + u_{n-1} + λf(2^{n} x) u_n, \quad \text{for } n \geq 0,\notag \end{equation} with the Dirichlet boundary condition $ u_{-1}=0 $. Building on Zhang's recent breakthrough work [Comm.Math.Phys.405:231(2024)] that resolved Damanik's open problem [Proc.Sympos. Pure Math.76,Amer.Math.Soc.(2007)] on the uniform positivity of the Lyapunov exponent, for the potential $ f \in C^{1}(0,1)$ with $ \|f\|_{C^{1}(0,1)} < C $ and $ \inf_{x \in (0,1)} |f^{\prime}(x)| > c>0 $, we obtain the large deviation estimate and prove that for a.e. $ x \in \mathbb{T} $ and sufficiently large $ λ> λ_{0} $, the operators $ H(x) $ display Anderson localization. Furthermore, if the potentials also have zero mean, our analysis reveals that the doubling map models can exhibit localization behavior for both small and large coupling constants $ λ$.

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arXiv2026

A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szegő Theorems

This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $dμ= w(θ) \frac{dθ}{2π} + dμ_s$ with Verblunsky coefficients $α=\{α_n\}_{n=0}^\infty$, distinct singular points $(θ_k)_{k=1}^{\ell}$, and multiplicities $(m_k)_{k=1}^{\ell}$, we establish the equivalence between the entropy condition \[ \int_0^{2π} \prod_{k=1}^{\ell} [1 - \cos(θ- θ_k)]^{m_k} \log w(θ) \frac{dθ}{2π} > -\infty \] and the decomposition condition \[ \exists β^{(1)}, \ldots, β^{(\ell)} : α= \sum_{k=1}^\ell β^{(k)} \,\, \text{with} \,\, (S - e^{-iθ_k})^{m_k} β^{(k)} \in \ell^2, \,\, β^{(k)} \in \ell^{2m_k + 2}. \] The proof synthesizes unitary transformations, discrete Sobolev-type inequalities, higher-order Szegő expansions, and a novel algebraic decomposition technique. Our resolution affirms that spectral theory is fundamentally local-global behavior emerges from the superposition of local resonances, each governed by its intrinsic scale.

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arXiv2025

Anderson localization for the multi-frequency quasi-periodic CMV matrices and quantum walks

In this paper we prove Anderson localization for multi-frequency quasi-periodic extended CMV matrices with analytic Verblunsky coefficients in the regime of positive Lyapunov exponents. By constructing a suitable semialgebraic set and combining the Avalanche Principle with a Large Deviation Theorem, we overcome the key obstruction of eliminating double resonances along the orbit, where multi-frequency potentials introduce significant challenges compared to the single-frequency case. As a direct application, we establish Anderson localization for corresponding analytic multi-frequency quasi-periodic quantum walks via unitary equivalence.

math.SP↗
arXiv2025

The spectrum of the multi-frequency quasi-periodic CMV matrices contains intervals

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 conditions, we establish that the spectrum of these operators contains intervals on the unit circle.

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arXiv2024

Anderson localization for CMV matrices with Verblunsky coefficients defined by the hyperbolic toral automorphism

In this paper, we prove the large deviation estimates and Anderson localization for CMV matrices on $\ell^2(\mathbb{Z}_+)$ with Verblunsky coefficients defined dynamically by the hyperbolic toral automorphism. Part of positivity results on the Lyapunov exponents of Chulaevsky-Spencer and Anderson localization results of Bourgain-Schlag on Schrödinger operators with strongly mixing potentials are extended to CMV matrices.

math.SP↗
arXiv2023

Invariant curves of low smooth quasi-periodic reversible mappings

In this paper, we obtain the invariant curves of quasi-periodic reversible mappings with finite smoothness. Since the reversible property is difficult to maintain in the process of approximating smooth functions by analytical ones, Rüssmann's method in \cite{HR} is invalid. Inspired by the recent work of Li, Qi and Yuan in \cite{LJ}, we turn to regard the reversible mapping as the Poincaré map of a reversible differential equation. By constructing a KAM theorem for a reversible differential equation which is quasi-periodic in time, we obtain the invariant curves of the reversible mapping. Beyond that, we establish some variants of invariant curve theorems for quasi-periodic reversible mappings.

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arXiv2023

Invariant curves of quasi-periodic reversible mappings and its application

We consider the existence of invariant curves of real analytic reversible mappings which are quasi-periodic in the angle variables. By the normal form theorem, we prove that under some assumptions, the original mapping is changed into its linear part via an analytic convergent transformation, so that invariants curves are obtained. In the iterative process, by solving the modified homological equations, we ensure that the transformed mapping is still reversible. As an application, we investigate the invariant curves of a class of nonlinear resonant oscillators, with the Birkhoff constants of the corresponding Poincar$\acute{e}$ mapping all zeros or not.

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arXiv2018

Invariant Curves of Almost Periodic Reversible Mappings

In this paper, we prove some invariant curve theorems for the planar almost periodic reversible mappings. As an application, we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the nonlinear oscillator $x"+g(x)x'+\varpi^{2}x+φ(x)=f(t)$ with $f(t)$ almost periodic.

math.CA↗
arXiv2013

Boundedness of semilinear Duffing equations at resonance with oscillating nonlinearities

In this paper, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+ψ'(x)=p(t)$ with the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $ψ'(x)$ are periodic and $g(x)$ is bounded. For the critical situation that $\big |\int_0^{2π}p(t)e^{int}dt \big|=2\big|g(+\infty)-g(-\infty)\big|$, we also prove a sufficient and necessary condition for the boundedness if $ψ'(x)\equiv0$.

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