J. Bourgain, S. Jitomirskaya
We study regularity properties of the Lyapunov exponent L of quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove that L is jointly continuous, in frequency and energy, at every irrational frequency.
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It is known that the Lyapunov exponent of analytic 1-frequency quasiperiodic cocycles is continuous in cocycle and, when the frequency is irrational, jointly in cocycle and frequency. In this paper, we extend a result of Bourgain to show the same continuity result for multifrequency quasiperiodic $M(2,\mathbb{C})$ cocycles. Our corollaries include applications to multifrequency Jacobi cocycles with periodic background potentials.
We construct discontinuous point of the Lyapunov exponent of quasiperiodic Schrödinger cocycles in the Gevrey space $G^{s}$ with $s>2$. In contrast, the Lyapunov exponent has been proved to be continuous in the Gevrey space $G^{s}$ with $s<2$ \cite{klein,cgyz}. This shows that $G^2$ is the transition space for the continuity of the Lyapunov exponent.
We consider an m-dimensional analytic cocycle with underlying dynamics given by an irrational translation on the circle. Assuming that the d-dimensional upper left corner of the cocycle is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets-Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of ba…
Pointwise modulus of continuity of the Lyapunov exponent and integrated density of states for analytic multi-frequency quasiperiodic $M(2, \mathbb{C})$ cocycles
本文针对解析多频准周期余环，建立了 Lyapunov 指数在余环和频率上的逐点模量连续性，并由此得到态密度积分在能量、势和频率上的正则性。
Furstenberg Theory of Mixed Random-Quasiperiodic Cocycles
本文推导了一般混合随机-拟周期线性上链最大Lyapunov指数的正性准则，适用于随机扰动拟周期势的Schrödinger算子，并建立了Oseledets定理中的平均一致收敛。
We study lower bounds on the Lyapunov exponent associated with one-frequency quasiperiodic Schrödinger operators with an added finite valued background potential. We prove that, for sufficiently large coupling constant, the Lyapunov exponent is positive with a uniform (in energy and background) minoration.
Anderson localization and Hölder continuity of the integrated density of states for analytic quasiperiodic Schrödinger operators
本文研究了准周期薛定谔算子在扰动范围内，对任意非常数解析势和任意迪ophantine频率的安德森局域化及积分态密度的霍尔德连续性，提出了一种新的格林函数控制方法以消除双重共振。
We study discrete quasiperiodic Schrödinger operators on $\ell^2(\zee)$ with potentials defined by $γ$-Hölder functions. We prove a general statement that for $γ>1/2$ and under the condition of positive Lyapunov exponents, measure of the spectrum at irrational frequencies is the limit of measures of spectra of periodic approximants. An important ingredient in our analysis is a general result on uniformity of the upper Lyapunov exponent of strictly ergodic cocycles.
We prove estimates on the Hölder exponent of the density of states measure for discrete Schrödinger operators with potential of the form $V(n) = λ(\lfloor(n+1)β\rfloor - \lfloor nβ\rfloor)$, with $λ$ large enough, and conclude that for almost all values of $β$, the density of states measure is not Hölder continuous.
Sharp arithmetic delocalization for quasiperiodic operators with potentials of semi-bounded variation
本文针对一类半有界变差势的一维拟周期Schrödinger算子，证明了尖锐算术Gordon定理，在Lyapunov指数受频率有理逼近速率限制的能量集上排除特征值，核心工具是有界变差上闭迭代的新均匀上界。
We study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show the continuity of the entropy spectrum at the boundary of Lyapunov spectrum in the sense that $h_{top}(E(α_{t}))\ \rightarrow h_{top}(E(β(\mathcal{A}))$ for generic cocycles, where $E(α)=\{x\in X: \lim_{n\rightarrow \infty}\frac{1}{n}\log \|\mathcal{A}^{n}(x)\|=α\}$. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles over mixing subshifts of finite type. Moreover, we prove the restricted variational principle…
We establish localization type dynamical bounds as a corollary of positive Lyapunov exponents for general operators with quasiperiodic potentials defined by piecewise Holder functions.
We prove that the Lyapunov exponent of quasi-periodic cocyles with singularities behaves continuously over the analytic category. We thereby generalize earlier results, where singularities were either excluded completely or constrained by additional hypotheses. Applications are one-parameter families of analytic Jacobi operators, such as extended Harper's model describing crystals subject to external magnetic fields.
We study discrete Schroedinger operators $(H_{α,θ}ψ)(n)= ψ(n-1)+ψ(n+1)+f(αn+θ)ψ(n)$ on $l^2(Z)$, where $f(x)$ is a real analytic periodic function of period 1. We prove a general theorem relating the measure of the spectrum of $H_{α,θ}$ to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of $H_{α,θ}$ are positive. For the almost Mathieu operator ($f(x)=2λ\cos 2πx$) it follows that the measure of the spectrum is equal to $4|1-|λ||$ for all real $θ$, $λ\ne\pm 1$, and all irrational $α$.
An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We prove large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we establish continuity properties of the Lyapu…
Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a centra…
We consider continuous one-dimensional multifrequency Schrödinger operators, with analytic potential, and prove Anderson localization in the regime of positive Lyapunov exponent for almost all phases and almost all Diophantine frequencies.
We consider an open quantum system in $M_{d}(\mathbb{C})$ governed by quasiperiodic Hamiltonian with rationally independent frequencies and under assumption of Lyapunov-Perron reducibility of associated Schroedinger equation. We construct the Markovian Master Equation and resulting CP-divisible evolution in weak coupling limit regime, generalizing our previous results from periodic case. The analysis is conducted with application of projection operator techniques and concluded with some results regarding stability of solutions and existence of quasiperiodic global steady state.
For analytic quasi-periodic Schrödinger cocycles, You and Zhang [9] proved that the Lyapunov exponent is Hölder continuous for weak Liouville frequency. In this paper, we prove that the Hölder continuity also holds if the analytic potential is weakened to Gevrey potential.
We consider quasiperiodic Jacobi matrices of size N with analytic coefficients. We show that, in the positive Lyapunov exponent regime, after removing some small sets of energies and frequencies, any eigenvalue is separated from the rest of the spectrum by $N^{-1}(\log N)^{-p}$, with p>15.
Obstructions to the regularity of the Lyapunov exponents for non-compact random Schrödinger cocycles
该研究针对非紧支撑的随机薛定谔上循环，揭示其李雅普诺夫指数正则性的关键障碍，明确赫尔德连续性需满足指数矩条件。
We study multi-frequency quasiperiodic Schrödinger operators on $\mathbb{Z} $. We prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval. The result is a consequence of a criterion for the spectrum to contain an interval at a given location that we establish non-perturbatively in the regime of positive Lyapunov exponent.
For quasiperiodic Schrödinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schrödinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy. From spectral theory side, the "Schrödinger conjecture" and the "Last's intersection spectrum conjecture" have been verified. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, see open pr…
We establish Anderson localization for quasiperiodic operator families of the form $$ (H(x)ψ)(m)=ψ(m+1)+ψ(m-1)+λv(x+mα)ψ(m) $$ for all $λ>0$ and all Diophantine $α$, provided that $v$ is a $1$-periodic function satisfying a Lipschitz monotonicity condition on $[0,1)$. The localization is uniform on any energy interval on which Lyapunov exponent is bounded from below.
The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schrödinger operators
本文在低正则性 C^2 余弦型准周期薛定谔算子中解决了 Dry Ten Martini 问题，证明解析性假设可由几何条件替代，并进一步得到谱齐次性与积分态密度的绝对连续性。
We provide an example of a Schrödinger cocycle over a mixing Markov shift for which the integrated density of states has a very weak modulus of continuity, close to the log-Hölder lower bound established by W. Craig and B. Simon. This model is based upon a classical example due to Y. Kifer of a random Bernoulli cocycle with zero Lyapunov exponents which is not strongly irreducible. It follows that the Lyapunov exponent of a Bernoulli cocycle near this Kifer example cannot be Hölder or weak-Hölder continuous, thus providing a limitation on the modulus of continuity of the Lyapunov exponent of r…
Let us concern the quasi-periodic Schrödinger operator in the continuous case, \begin{equation*} (Hy)(x)=-y^{\prime\prime}(x)+V(x,ωx)y(x), \end{equation*} where $V:(\R/\Z)^2\to \R$ is piecewisely $γ$-Hölder continuous with respect to the second variable. Let $L(E)$ be the Lyapunov exponent of $Hy=Ey$. Define $β(ω)$ as \begin{equation*} β(ω)= \limsup_{k\to \infty}\frac{-\ln ||kω||}{k}. \end{equation*} We prove that $H$ admits no eigenvalue in regime $\{E\in\R:L(E)<γβ(ω)\}$.
We devise an abstract, modular scheme to prove continuity of the Lyapunov exponents for a general class of linear cocycles. The main assumption is the availability of appropriate large deviation type (LDT) estimates which are uniform in the data. We provide a modulus of continuity that depends explicitly on the sharpness of the LDT estimate. Our method uses an inductive procedure based on the deterministic, general Avalanche Principle in [4]. The main advantage of this approach, besides the fact that it provides quantitative estimates, is its versatility, as it applies to quasi-periodic cocycl…
Sharp Logarithmic Quantum Dynamics for Quasiperiodic Schrödinger Operators
针对带偶势的单频拟周期薛定谔算子，证明其量子动力学的对数尺度及矩阶数依赖均为尖锐值，通过适配双局域中心的反射型半一致局域本征函数得到时间序列的对数下界。
We consider an integer lattice quasiperiodic Schrodinger operator. The underlying dynamics is either the skew-shift or the multi-frequency shift by a Diophantine frequency. We assume that the potential function belongs to a Gevrey class on the multi-dimensional torus. Moreover, we assume that the potential function satisfies a generic transversality condition, which we show to imply a Lojasiewicz type inequality for smooth functions of several variables. Under these assumptions and for large coupling constant, we prove that the associated Lyapunov exponent is positive for all energies, and con…
We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates which lead to optimal Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.
The proof of Anderson localization for the 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona-Klein-Martinelli in 1987, is based in part on the multi-scale analysis. Later, in the 90s, it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. In this paper we present a short proof along these lines, for the Anderson mode…
Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices
本文证明了具Diophantine频率的离散拟周期d×d Jacobi块矩阵态密度积分的Hölder连续性，给出指数范围并推广且加强薛定谔算子情形结果，其证明基于新方案可从全局零计数得有限体积特征多项式局部零计数。
Hölder continuity of the integrated density of states for Liouville frequencies
本文针对一般解析势能，在 Lyapunov 指数满足特定不等式条件下，证明了 Lyapunov 指数与态密度积分的 Hölder 连续性。
We show Hölder continuity for the integrated density of states of a quasi-periodic Jacobi operator with analytic coefficients, in the regime of positive Lyapunov exponent and with a strong Diophantine condition on the frequency. In particular, when the coefficients are trigonometric polynomials we express the Hölder exponent in terms of the degrees of the coefficients.
We study a matrix-valued Schrödinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that away from a discrete set its Lyapunov exponents do not vanish. For this we use a criterion by Gol'dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the Hölder continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prov…
Quantitative Hölder Regularity, Concentration, and Spectral Applications for Lyapunov Exponents of Random $\operatorname{GL}(2,\mathbb{R})$ Cocycles, with Extensions to $\operatorname{GL}(d,\mathbb{R})$
本文研究了随机矩阵乘积的Lyapunov指数的定量正则性，扩展到GL(d,R)。通过分析测度的Hölder指数和Lyapunov间隙，推导出Hölder指数和对数Hölder指数，并提出谱间隙方法的应用。
We consider dissipative dynamical systems represented by a smooth compressible flow in a finite domain. The density evolves according to the continuity (Liouville) equation. For a general, non-degenerate flow the result of the infinite time evolution of an initially smooth density is a singular measure. We give a condition for the non-degeneracy which allows to decide for a given flow whether the infinite time limit is singular. The condition uses a Green-Kubo type formula for the space-averaged sum of forward and backward-in-time Lyapunov exponents. We discuss how the sums determine the fluct…
A doubled Gordon threshold for palindromic quasiperiodic Schrödinger operators
研究单频拟周期薛定谔算子，开发新戈登型方法，利用近似重复和回文对称性。证明对于完全共振相位，若李雅普诺夫指数\(L(E)<2\beta(\alpha)\)，\(E\)不是特征值。应用于几乎马修算子，解决相关猜想的无特征值部分。
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