arXiv 0902.16282009-02-10math-phmath.MPmath.SP

矩阵值Anderson模型的局部化

Localization for a matrix-valued Anderson model

Hakim Boumaza

发表信息：: Math. Phys. Anal. Geom. 12(3), 255-286 (2009)
DOI：: 10.1007/s11040-009-9061-3

摘要：

本文研究了一类作用在$L^2(\R)\otimes \C^N$上的一维矩阵值连续随机薛定谔算子的局部化性质，对任意的$N\geq 1$ .我们证明，在适当的假设下，这些运营商的Fürstenberg群，有效的间隔$I\subset \R$，他们表现出本地化性能$I$，在频谱和动力学意义上。在看了李雅普诺夫指数和积分态密度的规律性后，我们证明了一个韦格纳估计，并应用多尺度分析计划来证明这些运营商的本地化。我们还研究了这类算子中的一个例子，我们可以证明关于Fürstenberg群的必要假设。这个群是由转移矩阵生成的，为了证明这些假设，我们可以使用Breuillard和Gelander关于在半单实连通李群中生成稠密李群的代数结果。这里使用的代数方法允许我们处理随机参数的奇异分布。

英文摘要：

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 operators. We also study an example in this class of operators, for which we can prove the required assumptions on the Fürstenberg group. This group being the one generated by the transfer matrices, we can use, to prove these assumptions, an algebraic result on generating dense Lie subgroups in semisimple real connected Lie groups, due to Breuillard and Gelander. The algebraic methods used here allow us to handle with singular distributions of the random parameters.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文