Weak Disorder Localization and Lifshitz Tails

 This paper is devoted to the study of localization of discrete random Schr?dinger Hamiltonians in the weak disorder regime. Consider an i.i.d. Anderson model and assume that its left spectral edge is 0. Let γ be the coupling constant measuring the strength of the disorder. For γ small, we prove a Lifshitz tail type estimate and use it to derive localization in a band starting at 0 going up to a distance of the average of the potential. In this energy region, we show that the localization length at energy E is bounded from above by a constant times the square root of the distance between E and the average of the potential.

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Résumé . Dans cet article, nous étudions la localisation à faible désordre pour des opérateurs de Schr?dinger aléatoires discrets. Considérons un modèle d'Anderson i.i.d. dont le bord spectral gauche vaut 0. Soit γ la constante de couplage mesurant le désordre. Pour γ petit, nous démontrons une estimée de type estimée de Lifshitz pour la densité d'états, et nous utilisons cette estimée pour prouver que le spectre de cet opérateur est localisé dans un intervalle allant de l'énergie 0 jusqu'à une distance de l'ordre de de la moyenne du potentiel. Dans cette région d'énergie, la longueur de localisation à une énergie E est majorée par une constante fois la racine de la distance séparant E de la moyenne du potentiel.

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Received: 7 December 2001 / Accepted: 26 July 2002 Published online: 29 October 2002

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ID="*" It is a pleasure to thank M. Aizenman for his explanations on the paper [4]. The author also gratefully acknowledges support of the FNS 2000 ``Programme Jeunes Chercheurs'     

Lifshitz Tails for the Fractional Anderson Model

Journal of Statistical Physics - We consider the d-dimensional fractional Anderson model $$(-Delta )^alpha + V_omega $$ on $$ell ^2({mathbb {Z}}^d)$$ where $$0<alpha leqslant 1$$....     

Lifshitz Tails for Acoustic Waves in Random Quantum Waveguide

In this study, we consider acoustic operators in a random quantum waveguide. Precisely we deal with an elliptic operator in the divergence form on a random strip. We prove that the integrated density of states of the relevant operator exhibits Lifshitz behavior at the bottom of the spectrum. This result could be used to prove localization of acoustic waves at the bottom of the spectrum. 2000 Mathematics Subject Classification: 81Q10, 35P05, 37A30, 47F05     

Lifshitz Tails on the Bethe Lattice: A Combinatorial Approach

The density of states of disordered hopping models generically exhibits an essential singularity around the edges of its support, known as a Lifshitz tail. We study this phenomenon on the Bethe lattice, i.e. for the large-size limit of random regular graphs, converging locally to the infinite regular tree, for both diagonal and off-diagonal disorder. The exponential growth of the volume and surface of balls on these lattices is an obstacle for the techniques used to characterize the Lifshitz tails in the finite-dimensional case. We circumvent this difficulty by computing bounds on the moments of the density of states, and by deriving their implications on the behavior of the integrated density of states.     

Self-Similarity and Power-Like Tails in Nonconservative Kinetic Models

In this paper, we discuss the large-time behavior of solution of a simple kinetic model of Boltzmann–Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time-monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis.     

Non-Lifshitz Tails at the Spectral Bottom of Some Random Operators

In this paper we continue with the investigation of the behavior of the integrated density of states of random operators of the form H _ω =−∇ ρ _ω ∇. In the present work we are interested in its behavior at 0, the bottom of the spectrum of H _ω . We prove that it converges exponentially fast to the integrated density of states of some periodic operator . Being periodic, cannot exhibit a Lifshitz behaviour. This result relates to the result of S.M. Kozlov (Russ. Math. Surv. 34(4):168–169, 1979) and improves it. Research partially supported by the Research Unity 01/UR/ 15-01 projects.     

Spectral Theory of Massless Pauli-Fierz Models

We study the spectral theory of massless Pauli-Fierz models using an extension of the Mourre method. We prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant. In addition we show that the expectation value of the number operator is finite on all eigenvectors.Supported by Carlsbergfondet     

Drude model and Lifshitz formula

Since nearly 10 years, it has been known that inserting the permittivity of the Drude model into the Lifshitz formula for free energy causes a violation of the third law of thermodynamics. In this paper we show that the standard Matsubara formulation for free energy contains a contribution that is non-perturbative in the relaxation parameter. We argue that the correct formula must have a perturbative expansion and conclude that the standard Matsubara formulation with the permittivity of the Drude model inserted is not correct. We trace the non-perturbative contribution in the complex frequency plane, where it shows up as a self-intersection or a bifurcation of the integration path.     

Lifshitz tails for periodic plus random potentials

We prove that the integrated density of states () for a potentialW =V _per +V has Lifshitz tails where V_per is a periodic potential with reflection symmetry andV is a random potential, e.g., of the formV =q _i ()f(x–i).research partially supported by DFG.research partially supported by USNSF under grant No. MCS-81-20833.     

Al-Li合金光谱标准样品制备和定值

本文叙述了Ａｌ－Ｌｉ合金光谱分析用标准样品的制备过程和定值结果。在制备过程中采用了先进的氩气保护、新型熔剂覆盖盖技术防止锂元素的烧损,首创作了“间歇式半连续铸造法 ”、“分步调整成分法”等先进的工艺,解决了Ａｌ－Ｌｉ标样铸锭不锡成形、成易偏析等技术难关;最后采用多家协作分析定值法和科学的数理统计方法对数据处理给出定值结果和不确定度。     

基于光谱曲线响应特性的油膜厚度估计模型分析

如今,海上溢油事故频发,如何对溢油的油量进行估计,是一个重要课题。如果可以得到溢油量,那么对后续的处理以及损失的评估都会有较大的帮助。高光谱遥感技术的快速发展使对油膜厚度的定量估计成为可能。采用AvaSpec光谱仪测量不同厚度的油膜,然后对得到的光谱曲线的多种曲线特征进行提取,分析其与油膜厚度之间的关系。结果表明,油膜厚度与基于高光谱位置变量的Rg和Ro、三角植被指数的RDVI和TVI以及Haboudane关系式相关性较大。分别采用曲线拟合、BP神经网络和基于SVD的迭代方法建立油膜曲线特征与油膜厚度之间的预测关系,并以此对不同的油膜光谱曲线进行油膜厚度估计,对得到的结果进行精度检测和运行时间分析,最终得出对每个估计模型的分析评价。     

Lifshits Tails for Randomly Twisted Quantum Waveguides

We consider the Dirichlet Laplacian \(H_\gamma \) on a 3D twisted waveguide with random Anderson-type twisting \(\gamma \). We introduce the integrated density of states \(N_\gamma \) for the operator \(H_\gamma \), and investigate the Lifshits tails of \(N_\gamma \), i.e. the asymptotic behavior of \(N_\gamma (E)\) as \(E \downarrow \inf \mathrm{supp}\, dN_\gamma \). In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.     

亚热带土壤铬元素的高光谱响应和反演模型

高光谱遥感技术已成为当前遥感领域的前沿技术,因其高分辨率的特点,可利用地物反射光谱特征定量反演地物的物理化学性质。目前土壤环境质量愈来愈受到关注,土壤重金属含量与土壤环境质量安全密切相关,以往土壤高光谱遥感技术研究多注重于土壤有机成分如土壤碳氮的光谱反演模型,对土壤重金属含量的高光谱反演研究普遍较少。土壤重金属污染已经成为影响土壤质量安全的关键因素,对土壤重金属尤其是污染元素普查是当务之急。传统土壤重金属的测试方法要求条件较高,测试周期较长,试图建立土壤高光谱与土壤铬元素(ICP-MS测定)含量之间的定量预测模型,以实现土壤铬元素的快速准确预测。采集福州市土壤样品135个,对土壤样品在350～2 500 nm的光谱反射率进行倒数、对数、微分等六种变换,筛选出对土壤总铬含量敏感的光谱波段,最后获得福州土壤铬元素高光谱反演优化模型。研究结果表明：亚热带红壤总铬的敏感光谱波段为：可见光520～530 nm和近红外1 440～1 450,2 010～2 020,2 230～2 240 nm;亚热带地区土壤总铬—高光谱反演的优化模型为: y=120.768e-7.037x(相关系数R为0.568,均方根误差为0.619 μg·g-1,检验相关系数R为0.484,均方根误差为1.426 μg·g-1),该模型可以用于福州地区土壤全铬的光谱快速监测。     

Bulk and boundary critical behavior at Lifshitz points

Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard φ⁴ model. Analyzing these models systematically via modern field-theoretic renormalization group methods has been a long-standing challenge ever since their introduction in the middle of 1970s. We survey the recent progress made in this direction, discussing results obtained via dimensionality expansions, how they compare with Monte Carlo results, and open problems. These advances opened the way towards systematic studies of boundary critical behavior atm-axial Lifshitz points. The possible boundary critical behavior depends on whether the surface plane is perpendicular to one of them modulation axes or parallel to all of them. We show that the semi-infinite field theories representing the corresponding surface universality classes in these two cases of perpendicular and parallel surface orientation differ crucially in their Hamiltonian’s boundary terms and the implied boundary conditions, and explain recent results along with our current understanding of this matter.