LOCALIZATION AT LARGE DISORDER AND AT EXTREME ENERGIES - AN ELEMENTARY DERIVATION
LOCALIZATION AT LARGE DISORDER AND AT EXTREME ENERGIES - AN ELEMENTARY DERIVATION
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DOI：
10.1007/bf02099760
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发表时间：
1993-10-01
影响因子：
2.4
通讯作者：
MOLCHANOV, S
中科院分区：
文献类型：
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作者：
AIZENMAN, M;MOLCHANOV, S
给出了作用在L2空间中的一类具有随机矩阵元的自伴算子在强无序(lambda>；lambda0)或极能量条件下局部化的简短证明。一个典型的例子是Z(D)上的离散薛定谔算子H=-Delta+U0(X)+lambdaV(X)，d大于或等于1，其中U0(X)是指定的背景势，{V(X)}作为随机变量产生。一般结果适用于用非局部自伴算子T代替-Delta的算子，其矩阵元满足：Sigma(Y)\T(x，y)\S小于或等于常数，在x中一致，对于某些S和lt；1.定域化是指在一定的能量范围内H的谱是纯点型的，或者等价地-波函数在H产生的么正时间演化下不会无限扩散。这种效应是由势或非对角线矩阵元T(x，y)中的强烈无序产生的。在T(x，y)快速衰减的情况下，证明了相应的特征函数也是指数衰减的。该方法是基于预解技术的。其核心技术思想包括使用预解核的低阶矩，即[G(E)(x，y)\S]，其中S足够小(<；1)以避免分布的柯西尾引起的发散，以及有效地利用G(E)(X，y)对H的单个矩阵元素的依赖的简单形式来阐明基本方程(H-E)G(E)(x，X0)=Delta(x，X0)的含义。这种方法简化了以前本地化结果的推导，避免了迄今为止在该主题中遇到的小分母困难。它还得到了一些新的结果，包括在以下几组条件下的局部化：i)具有非齐次非随机部分U0(X)的势，ii)Bethe格子，iii)非对角项(T(x，y)几乎等于1/\x-y\(d+epsilon))中具有非常慢衰减的算子，以及iv)由无序边界条件产生的局部化。
The work presents a short proof of localization under the conditions of either strong disorder (lambda > lambda0) or extreme energies for a wide class of self adjoint operators with random matrix elements, acting in l2 spaces. A prototypical example is the discrete Schrodinger operator H = -DELTA + U0(x) + lambdaV(x) on Z(d), d greater-than-or-equal-to 1, with U0(x) a specified background potential and {V(x)} generated as random variables. The general results apply to operators with - DELTA replaced by a non-local self adjoint operator T whose matrix elements satisfy: SIGMA(y)\T(x,y)\s less-than-or-equal-to Const., uniformly in x, for some s < 1. Localization means here that within a specified energy range the spectrum of H is of the pure-point type, or equivalently - the wave functions do not spread indefinitely under the unitary time evolution generated by H. The effect is produced by strong disorder in either the potential or in the off-diagonal matrix elements T(x,y). Under rapid decay of T(x,y), the corresponding eigenfunctions are also proven to decay exponentially. The method is based on resolvent techniques. The central technical ideas include the use of low moments of the resolvent kernel, i.e., [\G(E)(x,y)\s] with s small enough (< 1) to avoid the divergence caused by the distribution's Cauchy tails, and an effective use of the simple form of the dependence of G(E)(X, y) on the individual matrix elements of H in elucidating the implications of the fundamental equation (H - E)G(E)(x, x0) = delta(x,x0). This approach simplifies previous derivations of localization results, avoiding the small denominator difficulties which have been hitherto encountered in the subject. It also yields some new results which include localization under the following sets of conditions: i) potentials with an inhomogeneous non-random part U0(x), ii) the Bethe lattice, iii) operators with very slow decay in the off-diagonal terms (T(x,y) almost-equal-to 1/\x - y\(d+epsilon)), and iv) localization produced by disordered boundary conditions.