Jonathan Breuer
We introduce a family of trees that interpolate between the Bethe lattice and $\bbZ$. We prove complete localization for the Anderson model on any member of that family.
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We prove spectral and dynamical localization for Anderson models on locally finite graphs using the fractional moment method. Our theorems extend earlier results on localization for the Anderson model on $\ZZ^d$. We establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder.
We establish Anderson localization for general analytic $k$-frequency quasi-periodic operators on $\mathbb{Z}^d$ for \textit{arbitrary} $k,d$.
Localization for heavy-tailed Anderson models
本文利用随机矩阵乘积的均匀大偏差估计结果，证明了一维重尾安德森模型的局域化特性。
We consider a one-dimensional continuum Anderson model where the potential decays in average like $|x|^{-α}$, $α>0$. We show dynamical localization for $0<α<\frac12$ and provide control on the decay of the eigenfunctions.
In this note we review some results on localization and related properties for random Schrödinger operators arising in aperiodic media. These include the Anderson model associated to disordered quasycrystals and also the so-called Delone operators, operators associated to deterministic aperiodic structures.
We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.
We consider a one-frequency, quasi-periodic, block Jacobi operator, whose blocks are generic matrix-valued analytic functions. We establish Anderson localization for this type of operator under the assumption that the coupling constant is large enough but independent of the frequency. This generalizes a result of J. Bourgain and S. Jitomirskaya on localization for band lattice, quasi-periodic Schroedinger operators.
We consider the multi-particle Anderson model on the lattice with infinite range but sub-exponentially decaying interaction and show the Anderson localization consisting of the spectral exponential and the strong dynamical localization. In particular, the dynamical localization is proved int he Hilbert-Schmidt norm. The results concern very singular probability distributions such as the Bernoulli's measures.
We consider the Anderson model on a strip. Assuming that potentials have bounded density with considerable tails we get a lower bound for the fluctuations of the logarithm of the Green's function in a finite box. This implies an effective estimate by $ \exp(CW^2) $ for the localization length of the Anderson model on the strip of width $ W $. The results are obtained, actually, for a more general model with a non-local operator in the vertical direction.
Anderson localization of long-range quasi-periodic operators via Dynamical Rigidity
本文通过动力学刚性论证，证明了长程准周期算子在大三角函数势和迪奥菲antine频率下的安德森局域化。
We propose a new method to prove Anderson localization for quasiperiodic Schrödinger operators and apply it to the quasiperiodic model considered by Sinai and Fröhlich-Spencer-Wittwer. More concretely, we prove Anderson localization for even $C^2$ cosine type quasiperiodic Schrödinger operators with large coupling constants, Diophantine frequencies and Diophantine phases.
We give a widely self-contained introduction to the mathematical theory of the Anderson model. After defining the Anderson model and determining its almost sure spectrum, we prove localization properties of the model. Here we discuss spectral as well as dynamical localization and provide proofs based on the fractional moments (or Aizenman-Molchanov) method. We also discuss, in less self-contained form, the extension of the fractional moment method to the continuum Anderson model. Finally, we mention major open problems. These notes are based on several lecture series which the author gave at t…
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.
We prove Anderson localization for the discrete Laplace operator on radial tree graphs with random branching numbers. Our method relies on the representation of the Laplace operator as the direct sum of half-line Jacobi matrices whose entries are non-degenerate, independent, identically distributed random variables with singular distributions.
This work establishes the Anderson localization in both the spectral exponential and the strong dynamical localization for the multi-particle Anderson tight-binding model with correlated but strongly mixing random external potential. The results are obtained near the lower edge of the spectrum of the multi-particle Hamiltonian. In particular, the exponential decay of the eigenfunctions is proved in the max-norm and the dynamical localization in the Hilbert-Schmidt norm. The proofs need the conditional probability distribution function of the random external stochastic processes to obey the uni…
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.
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 $\mathbb{Z}^{2}$.
We establish the complete spectral exponential, and the strong Hilbert-Schmidt dynamical localization for the one-dimensional multi-particle Anderson tight-binding model and for weakly interacting particles system. In other words, we show stability of the one-dimensional localization from the single-particle to multi-particle systems with an arbitrary large but finite number of particles and for sufficient weakly interacting models. The proof uses the multi-scale analysis estimates for multi-particle systems. The common probability distribution function of the random external potential in the …
The arithmetic version of Anderson localization (AL), i.e., AL with explicit arithmetic description on both the localization frequency and the localization phase, was first given by Jitomirskaya \cite{J} for the almost Mathieu operators (AMO). Later, the result was generalized by Bourgain and Jitomirskaya \cite{bj02} to a class of {\it one dimensional} quasi-periodic long-range operators. In this paper, we propose a novel approach based on an arithmetic version of Aubry duality and quantitative reducibility. Our method enables us to prove the same result for the class of quasi-periodic long-ra…
We consider the random Schrödinger operator on $\mathbb{R}$ obtained by perturbing the Laplacian with a white noise. We prove that Anderson localization holds for this operator: almost surely the spectral measure is pure point and the eigenfunctions are exponentially localized. We give two separate proofs of this result. We also present a detailed construction of the operator and relate it to the parabolic Anderson model. Finally, we discuss the case where the noise is smoothed out.
Localization and unique continuation for the Anderson-Bernoulli model with long-range hopping on $\mathbb{Z}$
研究整数集上带长程跳跃的安德森 - 伯努利模型谱边缘附近的安德森定位，对有理洛朗符号跳跃证明定量唯一延拓原理及定位，给出一般情况反例并证明弱结果。
In this paper, we study block Jacobi operators on $\mathbb{Z}$ with quasi-periodic meromorphic potential. We prove the non-perturbative Anderson localization for such operators in the large coupling regime.
We prove that at large disorder, Anderson localization in $\Z^d$ is stable under localized time-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schrödinger equations.
Anderson localization for CMV matrices with Verblunsky coefficients defined by the hyperbolic toral automorphism
本文针对由双曲环面自同构动力学定义Verblunsky系数的CMV矩阵，证明了大偏差估计与Anderson局域化，并将Chulaevsky-Spencer和Bourgain-Schlag的相关结果从Schrödinger算子推广到CMV矩阵。
In this paper, we study quasi-periodic CMV matrices with Verblunsky coefficients given by the skew-shift. We prove the positivity of Lyapunov exponents and Anderson localization for most frequencies, which establish the analogous results of one-dimensional Schrödinger operators proved by Bourgain, Goldstein and Schlag.
For the almost Mathieu operator $ (H_{λ,α,θ}u) (n)=u(n+1)+u(n-1)+ λv(θ+nα)u(n)$, Avila and Jitomirskaya guess that for every phase $ θ\in \mathscr{R} \triangleq\{θ\in \mathbb{R}\;| \; 2θ+ α\mathbb{Z} \in \mathbb{Z}\}$, $H_{λ,α,θ}$ satisfies Anderson localization if $ |λ| > e^{ 2 β}$. In the present paper, we show that for every phase $ θ\in \mathscr{R} $, $H_{λ,α,θ}$ satisfies Anderson localization if $ |λ| > e^{ 7 β}$.
Anderson localization for the multi-frequency quasi-periodic CMV matrices and quantum walks
本文通过构造半代数集并结合雪崩原理与大偏差定理，证明了多频拟周期CMV矩阵的Anderson局域化，并经由酉等价性推广至量子游走。
We prove that at large disorder, with large probability and for a set of Diophantine frequencies of large measure, Anderson localization in $\Bbb Z^d$ is {\it stable} under localized time-quasi-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The main tools are the Fröhlich-Spencer mechanism for the random component and the Bourgain-Goldstein-Schlag mechanism for the quasi-periodic component. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schrödinger equations.
For the almost Mathieu operator $(H_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+2λ\cos2π(θ+nα)u_n$, Avila and Jitomirskaya guess that for a.e. $θ$, $H_{λ,α,θ}$ satisfies Anderson localization if $ |λ| > e^{ β} $, and they establish this for $ |λ| > e^{\frac{16}{9} β}$. In the present paper, we extend their result to regime $ |λ| > e^{\frac{3}{2} β}$.
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