整数集上具有长程跳跃的安德森 - 伯努利模型的定位与唯一延拓

The Integrated Density of States of the 1D Discrete Anderson-Bernoulli Model at Rational Energies

We show there is a countable dense set of energies at which the integrated density of states of the 1D discrete Anderson-Bernoulli model can be given explicitly and does not depend on the disorder parameter, provided the latter is above an energy-dependent threshold.

关于具有长程跳跃的合金型Anderson-Bernoulli模型的局域化

On localization for the alloy-type Anderson-Bernoulli model with long-range hopping

本文针对具有指数长程跳跃的合金型Anderson-Bernoulli模型，结合Bourgain的多尺度分析与Klopp基于Floquet-Bloch理论的初始尺度估计方法，证明了谱边缘附近的Anderson局域化，并将结果推广到$\mathbb{R}^d$上的类似模型。

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.

强无序一维Anderson-Bernoulli模型积分态密度的精确界

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。

Localization for an Anderson-Bernoulli model with generic interaction potential

We present a result of localization for a matrix-valued Anderson-Bernoulli operator, acting on $L^2(\R)\otimes \R^N$, for an arbitrary $N\geq 1$, whose interaction potential is generic in the real symmetric matrices. For such a generic real symmetric matrix, we construct an explicit interval of energies on which we prove localization, in both spectral and dynamical senses, away from a finite set of critical energies. This construction is based upon the formalism of the Fürstenberg group to which we apply a general criterion of density in semisimple Lie groups. The algebraic nature of the objec…

关于周期Anderson-Bernoulli模型的谱

On the Spectrum of the Periodic Anderson-Bernoulli Model

本文研究周期Anderson-Bernoulli模型下离散薛定谔算子的谱，通过分析SL(2,R)转移矩阵生成的一致双曲性，证明交替随机势能的几乎必然谱至多含4个区间。

Dynamical Localization for the Singular Anderson Model in $\mathbb{Z}^d$

We prove that once one has the ingredients of a ``single-energy multiscale analysis (MSA) result'' on the $\mathbb{Z}^d$ lattice, several spectral and dynamical localization results can be derived, the most prominent being strong dynamical localization (SDL). In particular, given the recent progress at the bottom of the spectrum for the $\mathbb{Z}^2$ and $\mathbb{Z}^3$ cases with Bernoulli single site probability distribution, our results imply SDL in these regimes.

Localization for Anderson Models on Metric and Discrete Tree Graphs

We establish spectral and dynamical localization for several Anderson models on metric and discrete radial trees. The localization results are obtained on compact intervals contained in the complement of discrete sets of exceptional energies. All results are proved under the minimal hypothesis on the type of disorder: the random variables generating the trees assume at least two distinct values. This level of generality, in particular, allows us to treat radial trees with disordered geometry as well as Schrödinger operators with Bernoulli-type singular potentials. Our methods are based on an i…

在高对比度介质中随机球形包体的安德森局域化

Anderson localization in high-contrast media with random spherical inclusions

本文首次证明在高对比度介质中随机球形包体的偏微分算子在带边缘附近发生安德森局域化，揭示了极限谱行为。

Parametric Furstenberg Theorem on Random Products of $SL(2, \mathbb{R})$ matrices

We consider random products of $SL(2, \mathbb{R})$ matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense $G_δ$ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for disc…

Scale-free unique continuation estimates and applications to random Schrödinger operators

We prove a unique continuation principle or uncertainty relation valid for Schrödinger operator eigenfunctions, or more generally solutions of a Schrödinger inequality, on cubes of side $L\in 2\NN+1$. It establishes an equi-distribution property of the eigenfunction over the box: the total $L^2$-mass in the box of side $L$ is estimated from above by a constant times the sum of the $L^2$-masses on small balls of a fixed radius $δ>0$ evenly distributed throughout the box. The dependence of the constant on the various parameters entering the problem is given explicitly. Most importantly, there is…