Wencai Liu, W. -M. Wang
It is classical, following Furstenberg's theorem on positive Lyapunov exponent for products of random SL$(2, \mathbb R)$ matrices, that the one dimensional random Schrödinger operator has Anderson localization at arbitrary disorder. This paper proves a nonlinear analogue, thereby establishing a KAM-type persistence result for a non-integrable system.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Anderson localized states for the quasi-periodic nonlinear wave equation on $\mathbb Z^d$
研究在$\mathbb Z^d$上的准周期非线性波方程中建立大量安德森局域态，将非线性安德森局域化从随机环境扩展到确定性设置。
We consider the Anderson model at large disorder on $\mathbb{Z}^2$ where the potential has a symmetric Bernoulli distribution. We prove that Anderson localization happens outside a small neighborhood of finitely many energies. These finitely many energies are Dirichlet eigenvalues of the minus Laplacian restricted on some finite subsets of $\mathbb{Z}^{2}$.
Our recently established criterion for the formation of extended states on tree graphs in the presence of disorder is shown to have the surprising implication that for bounded random potentials, as in the Anderson model, there is no transition to a spectral regime of Anderson localization, in the form usually envisioned, unless the disorder is strong enough.
We explore the properties of discrete random Schroedinger operators in which the random part of the potential is supported on a sub-lattice. In particular, we provide new conditions on the sub-lattice under which Anderson localisation happens at strong disorder, and provide examples in which it can be ruled out.
The purpose of the present work is to establish decorrelation estimates for the locally renormalized eigenvalues of the discrete Anderson model near two distinct energies inside the localization region. In dimension one, we prove these estimates at all energies. In higher dimensions, the energies are required to be sufficiently far apart from each other.
The regular tree Anderson model at low disorder
该研究证明了无限正则树安德森模型在低无序度下的离域性，填补谱缺失部分以扩展早期成果，论证可推广至特定无序情况，并证明无序度消失时李雅普诺夫指数的连续性。
Complete localization is shown to hold for the $d$-dimensional Anderson model with uniformly distributed random potentials provided the disorder strength $λ>λ_{And}$ where $λ_{\text{And}}$ satisfies $λ_{\text{And}}=μ_d e \ln λ_{\text{And}}$ with $μ_d$ the self-avoiding walk connective constant for the lattice $\mathbb{Z}^d$. Notably $λ_{\text{And}}$ is precisely the large disorder threshold proposed by Anderson in 1958.
We report on recent results on the spectral statistics of the discrete Anderson model in the localized phase. Our results show, in particular, that, for the discrete Anderson Hamiltonian with smoothly distributed random potential at sufficiently large coupling, the limit of the level spacing distribution is that of i.i.d. random variables distributed according to the density of states of the random Hamiltonian. This text is a contribution to the proceedings of the conference "Spectra of Random Operators and Related Topics" held at Kyoto University, 02-04/12/09 organized by N. Minami and N. Uek…
We prove localization and probabilistic bounds on the minimum level spacing for the Anderson tight-binding model on the lattice in any dimension, with single-site potential having a discrete distribution taking N values, with N large.
Sharp Bounds for the Integrated Density of States of a Strongly Disordered 1D Anderson-Bernoulli Model
针对强无序下一维离散Anderson-Bernoulli模型的积分态密度，本文给出了全谱范围内与无序强度一致的上下界，证明存在不依赖无序参数的显式IDS取值的能量序列。
We study the higher-order correlation functions of covariant families of observables associated with random Schrödinger operators on the lattice in the strong disorder regime. We prove that if the distribution of the random variables has a density analytic in a strip about the real axis, then these correlation functions are analytic functions of the energy outside of the planes corresponding to coincident energies. In particular, this implies the analyticity of the density of states, and of the current-current correlation function outside of the diagonal. Consequently, this proves that the cur…
We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Schlag, Shubin and Wolff for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by $O(\frac{λ^{-2}}{\log\frac1λ})$, where $λ$ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erdös and Yau to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lat…
Anderson localization transition in disordered hyperbolic lattices
本研究针对双曲晶格的无序紧束缚模型，借助计算群论方法构建大系统并通过周期边界逼近热力学极限，数值证明{8,3}和{8,8}晶格存在安德森局域化转变，测得大临界无序强度、临界指数及强有限尺寸效应。
Disorder on the hyperbolic square lattice I: Anderson delocalization and absolutely continuous spectrum
研究双曲格上的安德森模型，证明弱无序下正测度集合上绝对连续谱存在且无奇异谱，并在特定条件下获得纯绝对连续谱。
This paper concerns spectral properties of linear Schrödinger operators under oscillatory high-amplitude potentials on bounded domains. Depending on the degree of disorder, we prove the existence of spectral gaps amongst the lowermost eigenvalues and the emergence of exponentially localized states. We quantify the rate of decay in terms of geometric parameters that characterize the potential. The proofs are based on the convergence theory of iterative solvers for eigenvalue problems and their optimal local preconditioning by domain decomposition.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。