On the Furstenberg measure and density of states for the Anderson-Bernoulli model at small disorder

An application of group expansion to the Anderson-Bernoulli model

We establish smoothness of the density of states for 1D lattice Schrodinger operators with potential taking values $\pmλ$, for $λ$ in a class of small algebraic numbers and energy $E\in)-2, 2($ suitably restricted away from $\pm2$.

On the local eigenvalue spacings for certain Anderson-Bernoulli Hamiltonians

The aim of this work is to extend the results from [B2] on local eigenvalue spacings to certain 1D lattice Schrodinger with a Bernoulli potential. We assume the disorder satisfies a certain algebraic condition that enables one to invoke the recent results from [B1] on the regularity of the density of states. In particular we establish Poisson local eigenvalue statistics in those models

安德森局部化在$\mathbb{Z}^d$上的分层安德森-伯努利模型中

Anderson Localization for the hierarchical Anderson-Bernoulli model on $\mathbb{Z}^d$

本文证明了在任意维数的晶格上，具有几何分层结构和独立同分布伯努利随机变量诱导波动的分层安德森-伯努利模型的安德森局部化，同时方法也适用于证明$\mathbb{Z}^d$上的概率唯一持续性结果。

Anderson-Bernoulli localization at large disorder on the 2D lattice

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 $\mathbb{Z}^{2}$.

Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice

We consider a Hamiltonian given by the Laplacian plus a Bernoulli potential on the two dimensional lattice. We prove that, for energies sufficiently close to the edge of the spectrum, the resolvent on a large square is likely to decay exponentially. This implies almost sure Anderson localization for energies sufficiently close to the edge of the spectrum. Our proof follows the program of Bourgain--Kenig, using a new unique continuation result inspired by a Liouville theorem of Buhovsky--Logunov--Malinnikova--Sodin.

Nonlinear Anderson localized states at arbitrary disorder

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.