Non-localization of eigenfunctions on large regular graphs

Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg's theorem. That is, a Schrödinger operator in $\ell^2(\mathbb{Z})$ whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typical…

一维非稳态安德森模型的定位

Localization of the 1D Non-Stationary Anderson Model

研究非稳态安德森模型的谱定位，利用非稳态矩阵乘积工具证明谱和动力学定位，考虑非恒定分布的势能。

非平稳 Anderson 局域化

Non-stationary Anderson Localization

本文研究具有非同分布有界随机势的离散 Schrödinger 算子，利用非平稳参数 Furstenberg 定理证明了其谱局域化与动力学局域化。