An eigensystem approach to Anderson localization

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• 作者：Alexander Elgart^a ; ¹ ; ^{aelgart@vt.edu" class="auth_mail" title="E-mail the corresponding author} ; Abel Klein^b ; ² ; ^{aklein@uci.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 82B44 ; secondary ; 47B80 ; 60H25 ; 81Q10
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：271
• 期：12
• 页码：3465-3512
• 全文大小：726 K

文摘

We introduce a new approach for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions). Information about eigensystems at a given scale is used to derive information about eigensystems at larger scales. This eigensystem multiscale analysis treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions in a fixed box with high probability. A new feature is the labeling of the eigenvalues and eigenfunctions by the sites of the box.