Anne Boutet de Monvel, Daniel Lenz, Peter Stollmann
We prove a simple uncertainty principle and show that it can be applied to prove Wegner estimates near fluctuation boundaries. This gives new classes of models for which localization at low energies can be proven.
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We consider a two dimensional magnetic Schroedinger operator with a spatially stationary random magnetic field. We assume that the magnetic field has a positive lower bound and that it has Fourier modes on arbitrarily short scales. We prove the Wegner estimate at arbitrary energy, i.e. we show that the averaged density of states is finite throughout the whole spectrum. We also prove Anderson localization at the bottom of the spectrum.
We prove a Wegner estimate for a large class of multiparticle Anderson Hamiltonians on the lattice. These estimates will allow us to prove Anderson localization for such systems. A detailed proof of localization will be given in a subsequent paper.
We present a new approach to the eigensystem multiscale analysis (EMSA) for random Schrödinger operators that relies on the Wegner estimate. The EMSA treats all energies of the finite volume operator in an energy interval at the same time, simultaneously establishing localization of all eigenfunctions with eigenvalues in the energy interval with high probability. It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization). The new method removes the restrictive level spacing hypothesis used in the previous ver…
We study Schrödinger operators on $L^2 (\RR^d)$ and $\ell^2(\ZZ^d)$ with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation a Wegner estimate which is polynomial in the volume of the box and linear in the size of the energy interval holds. We apply the established Wegner est…
We prove localization at the bottom of the spectrum for a random Schrödinger operator in the continuum with a single-site potential probability distribution supported by a Cantor set of zero Lebesgue measure. This distribution is too singular to be treated by the usual methods. In particular, an "a priori" Wegner estimate is not available. To prove the result we perform a multiscale analysis following the work of Bourgain and Kenig for the Bernoulli-Anderson Hamiltonian, and obtain the required Wegner estimate scale by scale. To do so, we generalize their argument based on Sperner's Lemma by r…
We prove a unique continuation principle for spectral projections of Schr\" odinger operators. We consider a Schr\" odinger operator $H= -Δ+ V$ on $\mathrm{L}^2(\mathbb{R}^d)$, and let $H_Λ$ denote its restriction to a finite box $Λ$ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type $χ_I (H_Λ) W χ_I (H_Λ) \ge κ\, χ_I (H_Λ) $ with $κ>0$ for appropriate potentials $W\ge 0$ and intervals $I$. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schr\" odinger operators with alloy{-type random…
We prove the Wegner bounds for the one-dimensional interacting multi-particle Anderson models in the continuum. The results apply to singular probability distribution functions such as the Bernoulli's measures. The proofs need the amplitude of the inter-particle interaction potential to be sufficiently weak. As a consequence, the results imply the Anderson localization via the multi-scale analysis.
Localization and unique continuation for non-stationary Schrödinger operators on the 2D lattice
本文基于二维晶格上非稳态薛定谔算子的随机特性，通过统一界和正下界条件，利用伯努利分解方法实现了谱底的定位与唯一延续。
This paper is devoted to the study of the random displacement model on $\R^d$. We prove that, in the weak displacement regime, Anderson and dynamical localization holds near the bottom of the spectrum under a generic assumption on the single site potential and a fairly general assumption on the support of the possible displacements. This result follows from the proof of the existence of Lifshitz tail and of a Wegner estimate for the model under scrutiny.
We prove Anderson localization at the internal band-edges for periodic magnetic Schr{ö}dinger operators perturbed by random vector potentials of Anderson-type. This is achieved by combining new results on the Lifshitz tails behavior of the integrated density of states for random magnetic Schr{ö}dinger operators, thereby providing the initial length-scale estimate, and a Wegner estimate, for such models.
In the present note we show dynamical localization for an Anderson model with missing sites in a discrete setting at the bottom of the spectrum in arbitrary dimension $d$. In this model, the random potential is defined on a relatively dense subset of $\mathbb Z^d$, not necessarily periodic, i.e., a Delone set in $\mathbb Z^d$. To work in the lower band edge we need no further assumption on the geometric complexity of the Delone set. We use a spatial averaging argument by Bourgain-Kenig to obtain a uniform Wegner estimate and an initial length scale estimate, which yields localization through t…
We study spectral properties of ergodic random Schrödinger operators on $L^2 (\RR^d)$. The density of states is shown to exist for a certain class of alloy type potentials with single site potentials of changing sign. The Wegner estimate we prove implies Anderson localization under certain additional assumptions. For some examples we discuss briefly some properties of the common and conditional densities of the random coupling constants used in the proof of the Wegner estimate.
We prove spectral and dynamical localization for the multi-dimensional random displacement model near the bottom of its spectrum by showing that the approach through multiscale analysis is applicable. In particular, we show that a previously known Lifshitz tail bound can be extended to our setting and prove a new Wegner estimate. A key tool is given by a quantitative form of a property of a related single-site Neumann problem which can be described as "bubbles tend to the corners".
We study localization and derive stochastic estimates (in particular, Wegner and Minami estimates) for the eigenvalues of weakly correlated random discrete Schrödinger operators in the localized phase. We apply these results to obtain spectral statistics for general discrete alloy type models where the single site perturbation is neither of finite rank nor of fixed sign. In particular, for the models under study, the random potential exhibits correlations at any range.
Localization for non-stationary Anderson models in three dimensions
本文利用确定性定量唯一延拓定理和组合分解方法，证明了三维非平稳安德森模型在谱底附近的局域化性质。
The purpose of the present work is to establish decorrelation estimates for some random discrete Schrodinger operator in dimension one. We prove that the Minami estimates are consequences of the Wegner estimates and Localization. We also prove decorrelation estimates at distinct energies for the random hopping model and Schrodinger operators with alloy-type potentials. These results are used to give a description of the spectral statistics.
Localization for one-dimensional Anderson-Dirac models
本文针对一维 Anderson-Dirac 模型，利用散射性质与 Fürstenberg 定理证明 Lyapunov 指数为正，并通过新 Thouless 公式和 Wegner 估计建立谱与动力学局域化。
A "lifting" method for exponential large deviation estimates and an application to certain non-stationary 1D lattice Anderson models
本文提出一种“提升”方法，将平稳模型的指数大偏差估计推广到非平稳模型，并据此推导一维晶格上特定非平稳势的Anderson局域化。
We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finite-area Green's functions. The proof of…
We study localization properties for a class of one-dimensional, matrix-valued, continuous, random Schrödinger operators, acting on $L^2(\R)\otimes \C^N$, for arbitrary $N\geq 1$. We prove that, under suitable assumptions on the Fürstenberg group of these operators, valid on an interval $I\subset \R$, they exhibit localization properties on $I$, both in the spectral and dynamical sense. After looking at the regularity properties of the Lyapunov exponents and of the integrated density of states, we prove a Wegner estimate and apply a multiscale analysis scheme to prove localization for these op…
We present a new, short, self-contained proof of localization properties of multi-dimensional continuum random Schrödinger operators in the fluctuation boundary regime. Our method is based on the recent extension of the fractional moment method to continuum models in by Aizenman et al, but does not require the random potential to satisfy a covering condition. Applications to random surface potentials and potentials with random displacements are included.
We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes $Λ\subset R^d$. We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes $Λ\subset R^d$, for $d \geq 1$. The result hol…
It has been empirically observed that eigenfunctions of Laplace's equation $-Δϕ= λϕ$ with Neumann boundary conditions sometimes localize near the boundary of the domain if that boundary is rough (say, fractal). This has some nontrivial implications in acoustics that has been put to real-life use (sound attenuation by noise-protective walls); this short paper describes the mathematical mechanism responsible for this and describes the quantitative strength of the phenomenon for some examples.
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